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PhysRevB.78.035135.pdf | Collective properties of magnetobiexcitons in quantum wells and graphene superlattices
Oleg L. Berman,1Roman Ya. Kezerashvili,1and Yurii E. Lozovik2
1Physics Department, New York City College of Technology, The City University of New York, Brooklyn, New York 11201, USA
2Institute of Spectroscopy, Russian Academy of Sciences, 142190 Troitsk, Moscow Region, Russia
/H20849Received 28 May 2008; revised manuscript received 12 June 2008; published 30 July 2008 /H20850
The Bose-Einstein condensation and superfluidity of quasi-two-dimensional spatially indirect magnetobiex-
citons in a slab of superlattice with alternating electron and hole layers consisting from the semiconductingquantum wells /H20849QWs /H20850and graphene superlattice in high magnetic field are considered. The two different
Hamiltonians of a dilute gas of magnetoexcitons with a dipole-dipole repulsion in superlattices, consisting ofboth QWs and graphene layers /H20849GLs /H20850in the limit of high magnetic field, have been reduced to one effective
Hamiltonian—a dilute gas of two-dimensional excitons with the renormalized effective mass of the magne-toexciton, which depends on the magnetic field. This Hamiltonian does not include the vector potential.Moreover, for Nexcitons we have reduced the problem of 2 N/H110032-dimensional space onto the problem of N
/H110032-dimensional space by integrating over the coordinates of the relative motion of an electron and a hole. The
instability of the ground state of the system of interacting two-dimensional indirect magnetoexcitons in a slabof superlattice with alternating electron and hole layers in high magnetic field is found. The stable system ofindirect quasi-two-dimensional magnetobiexcitons, consisting of a pair of indirect excitons with oppositedipole moments, is considered. The density of the superfluid component n
s/H20849T/H20850and the temperature of the
Kosterlitz-Thouless phase transition to the superfluid state in the system of two-dimensional indirect magne-tobiexcitons, interacting as electrical quadrupoles, are obtained for both the QW and graphene realizations.
DOI: 10.1103/PhysRevB.78.035135 PACS number /H20849s/H20850: 71.35.Ji, 68.65.Cd, 71.35.Lk
I. INTRODUCTION
The many-particle systems of the spatially indirect exci-
tons in coupled quantum wells /H20849CQWs /H20850in the magnetic field
B, as well as without magnetic field, have been the subject of
recent experimental investigations.1–4These systems are of
interest, in particular, in connection with the possibility ofthe Bose-Einstein condensation /H20849BEC /H20850and superfluidity of
indirect excitons or electron-hole pairs, which would mani-fest itself in the CQW as persistent electrical currents in eachwell, and also through coherent optical properties and Jo-sephson phenomena.
5–9In strong magnetic fields /H20849B/H110227T /H20850
two-dimensional /H208492D/H20850excitons survive in a substantially
wider temperature region, as the exciton binding energiesincrease with magnetic field.
10–16The problem of the essen-
tial interest is also collective properties of magnetoexcitonsin high magnetic fields in superlattices and layered system.
17
In this paper we propose a new physical realization of the
magnetoexcitonic BEC and superfluidity in superlatticeswith alternating electronic and hole layers, that is in asense representing array of the CQWs or GLs with spatiallyseparated electrons /H20849e/H20850and holes /H20849h/H20850in high magnetic field.
Recent technological advances have allowed the productionof graphene, which is a 2D honeycomb lattice of carbonatoms that form the basic planar structure in graphite.
18,19
Graphene has been attracting a great deal of experimental
and theoretical attention because of unusual properties in itsbandstructure.
20–23It is a gapless semiconductor with mass-
less electrons and holes, which have been described asDirac-fermions.
24Since there is no gap between the conduc-
tion and valence bands in graphene without magnetic field,the screening effects result in the absence of excitons ingraphene without magnetic field. A strong magnetic fieldproduces a gap since the energy spectrum formed by Landaulevels becomes discrete. The gap reduces screening and leads
to the formation of the magnetoexcitons. We also considermagnetoexcitons in the superlattices with alternating elec-tronic and hole GLs. We suppose that recombination timesmay be much greater than relaxation times
/H9270rdue to small
overlapping of the spatially separation of e- and h-wave
functions in the CQW or GLs. In this case electrons andholes are characterized by different quasiequilibrium chemi-cal potentials. Therefore, in the system of indirect excitons insuperlattices, as well as in CQW,
5,9the quasiequilibrium
phases appear.25While coupled-well structures with spatially
separated electrons and holes are typically considered to beunder applied electric field, which separates electrons andholes in different quantum wells,
2,3we assume that there are
no external electric fields applied to a slab of superlattice. If“electron” and “hole” quantum wells alternate, there are ex-citons with parallel dipole moments in one pair of wells, butdipole moments of excitons in another neighboring pairs ofneighboring wells have an opposite direction, which causesthe attraction between the excitons with opposite dipole mo-ments. This fact leads to the essential distinction of proper-ties of e−hsystem in superlattices and coupled quantum
wells with spatially separated electrons and holes, where in-
direct exciton system is stable due to the dipole-dipole repel-ling of all excitons. This difference manifests itself alreadybeginning from three-layer e−h−eorh−e−hsystem.
In this paper we reduce the problem of magnetoexcitons
in the QWs and GLs superlattices in the strong magneticfield to the problem of excitons by deriving an effectiveHamiltonian with renormalized mass. It is shown that theinstability of the ground state of the system of interactingindirect excitons in the slab of superlattice with alternating e
andhlayers is established in the strong magnetic field. Two-
dimensional indirect magnetobiexcitons, consisting of the in-direct magnetoexcitons with opposite dipole moments, arePHYSICAL REVIEW B 78, 035135 /H208492008 /H20850
1098-0121/2008/78 /H208493/H20850/035135 /H208499/H20850 ©2008 The American Physical Society 035135-1considered in the strong magnetic field. The radius and the
binding energy of the indirect magnetobiexciton are calcu-lated. These magnetobiexcitons repel as electrical quadru-poles at long distances. As a result, the system of the indirectmagnetobiexcitons is stable. The collective spectrum of theweakly interacting by the quadrupole law two-dimensionalindirect magnetobiexcitons is considered in the ladder ap-proximation. The superfluid density n
s/H20849T/H20850of interacting two-
dimensional indirect magnetobiexcitons in superlattices iscalculated at low temperatures T. We analyze the dependence
of the Kosterlitz-Thouless transition temperature,
26as well as
superfluid density on magnetic field.
This paper is organized as follows. In Sec. II, we derive
an effective Hamiltonian for magnetoexcitons in both CQWsand two graphene layers in strong magnetic field. In Sec. III,we prove the instability of dipole magnetoexcitons in theQWs and graphene superlattices due to the attraction of op-positely directed dipoles. The superfluidity of quadrupolemagnetobiexcitons in the QWs and graphene superlattices isanalyzed and discussed in Sec. IV. The results of calculationsand discussion are presented in Sec. V. Conclusions follow inSec. VI.
II. EFFECTIVE HAMILTONIAN FOR
MAGNETOEXCITONS IN STRONG MAGNETIC FIELD
A. Hamiltonian for magnetoexcitons in the CQWs
Let’s start with two interacting excitons in the CQW in
the presence of the external magnetic field B, and for the
description of the motion of the exciton center of mass andrelative motion of the electron and hole with the masses m
e
andmhand coordinates reandrhalong the QW, introduce
the standard set of Jacobi coordinates
R=mere+mhrh
me+mh,
r=re−rh. /H208491/H20850
Without loss of a generality we can take the magnetic field to
be in the zdirection and it is convenient to work with the
symmetric gauge for electrons and holes vector potentialA
e/H20849h/H20850=1 /2/H20851B/H11003re/H20849h/H20850/H20852. Using these notations and coordinates
/H20851Eq. /H208491/H20850/H20852the Hamiltonian Hfor 2D spatially separated exci-
tons in magnetic field can be written as
Hˆ=/H20885dR/H20885dr/H20851/H9274ˆ†/H20849R,r/H20850Hˆ/H9274ˆ/H20849R,r/H20850/H20852
+1
2/H20885dR1/H20885dr1/H20885dR2/H20885dr2/H9274ˆ†/H20849R1,r1/H20850/H9274ˆ†/H20849R2,r2/H20850
/H11003/H20858
i,j=e,hUij/H20849ri1−rj2/H20850/H9274ˆ/H20849R2,r2/H20850/H9274ˆ/H20849R1,r1/H20850, /H208492/H20850
where Hˆis the operator of an isolated electron-hole pair
given byHˆ=Hˆ0−e2
/H9280/H20881/H20849re−rh/H208502+D2, /H208493/H20850
with the operator Hˆ0for the noninteracting electron-hole pair
in magnetic field presented by
Hˆ0=/H20858
i=e,h/H208751
2mi/H20873pi+e
cAi/H208742/H20876, /H208494/H20850
In Eqs. /H208492/H20850–/H208494/H20850/H9274ˆ†/H20849R,r/H20850and/H9274ˆ/H20849R,r/H20850are the creation and
annihilation operators for magnetoexcitons, reandrhare
two-dimensional vectors of coordinates of an electron andhole, respectively, Dis the distance between electron and
hole quantum wells, eis the charge of an electron, cis the
speed of light, and
/H9280is a dielectric constant. In Eq. /H208492/H20850
for electrons and holes from two different excitons,we use the two-particle potentials U
ijfor the electron-
electron, hole-hole, electron-hole, and hole-electroninteraction: U
ee/H20849re1−re2/H20850=e2//H20849/H9280/H20841re1−re2/H20841/H20850,Uhh/H20849rh1−rh2/H20850
=e2//H20849/H9280/H20841rh1−rh2/H20841/H20850, Ueh/H20849re1−rh2/H20850=−e2//H20849/H9280/H20881/H20841re1−rh2/H208412+D2/H20850,
Uhe/H20849rh1−re2/H20850=−e2//H20849/H9280/H20881/H20841rh1−re2/H208412+D2/H20850.
A conserved quantity for an isolated electron-hole pair in
magnetic field Bis the exciton generalized momentum Pˆ
defined as
Pˆ=−i/H6036/H11612e−i/H6036/H11612h+e
c/H20849Ae−Ah/H20850−e
c/H20851B/H11003/H20849re−rh/H20850/H20852 /H20849 5/H20850
for the Dirac equation in the GLs,27as well as for the
Schrödinger equation in the CQWs.10,12,28
The Hamiltonian /H20851Eq. /H208494/H20850/H20852of a single isolated magnetoex-
citon is commutated with Pˆ, and hence, they have the same
eigenfunctions, which have the following form /H20849see Refs. 10
and28/H20850:
/H9023kP/H20849R,r/H20850= exp/H20877iR
/H6036/H20873P+e
cB/H11003r/H20874+i/H9253Pr
2/H6036/H20878/H9021˜k/H20849P,r/H20850,
/H208496/H20850
where /H9021˜k/H20849P,r/H20850is a function of the internal coordinates rand
the eigenvalue Pof the generalized momentum, and krepre-
sents the quantum numbers of the exciton internal motion.
The wave function /H9021˜/H20849P,r/H20850is provided in Refs. 10,14, and
28:
/H9021˜k/H20849P,r/H20850=/H9021n1,n2/H20849P,r/H20850
=/H208492/H9266/H20850−1 /22−/H20841m/H20841/2n˜!
/H20881n1!n2!1
rBsgn/H20849m/H20850mr/H20841m/H20841
rB/H20841m/H20841
/H11003exp/H20875−im/H9278−r2
4rB2/H20876Ln˜/H20841m/H20841/H20873r2
2rB2/H20874, /H208497/H20850
where Ln˜/H20841m/H20841denotes Laguerre polynomials, m=n1−n2,n˜
=min /H20849n1,n2/H20850, and sgn /H20849m/H20850m→1 for m=0. In high magnetic
fields the magnetoexcitonic quantum numbers k=/H20853n+,n−/H20854for
an electron in the Landau level n+and a hole in level n−and
/H9253=/H20849mh−me/H20850//H20849mh+me/H20850.BERMAN, KEZERASHVILI, AND LOZOVIK PHYSICAL REVIEW B 78, 035135 /H208492008 /H20850
035135-2B. Hamiltonian for magnetoexcitons in the bilayer graphene
If the Coulomb interaction between the electron and hole
is neglected, the four-component Hamiltonian of electron-hole pairs in the bilayer graphene with spatially separatedelectrons and holes in one valley in magnetic field Bcan be
obtained by substituting in Eq. /H208492/H20850the following four-
component Hamiltonian Hˆ
0for an isolated electron-hole
pair:27
Hˆ0=vF/H208980 pex+ipey 0 0
pex−ipey 0 00
000 phx−iphy
0 0 phx+iphy 0/H20899,
/H208498/H20850
where
pe=−i/H6036/H11612e+e
cAe,ph=−i/H6036/H11612h−e
cAh. /H208499/H20850
In Eq. /H208498/H20850vF=/H208813at//H208492/H6036/H20850is the Fermi velocity of electrons in
graphene, where a=2.566 Å is a lattice constant, and t
/H110152.71 eV is the overlap integral between the nearest carbon
atoms.29It should be mentioned that for the bilayer graphene
the terms corresponding to the Coulomb attraction betweenelectron and hole in an isolated pair and dipole-dipole inter-action between two different electron-hole pairs are the sameas for the CQWs and are given by Eqs. /H208493/H20850and /H208492/H20850, respec-
tively.
The wave function /H9021
˜/H20849P,r/H20850of the relative coordinate for e
and hspatially separated in the different GLs can be ex-
pressed in terms of the two-dimensional harmonic oscillatoreigenfunctions /H9021
n1,n2/H20849r/H20850given by Eq. /H208497/H20850/H20849/H9253=0/H20850. For an elec-
tron in Landau level n+and a hole in level n−, the four-
component spinor wave function of the relative coordinateis
27
/H9021˜n+,n−/H20849P,r/H20850=/H20849/H208812/H20850/H9254n+,0+/H9254n−,0−2/H20898s+s−/H9021/H20841n+/H20841−1,/H20841n−/H20841−1/H20849P,r/H20850
s+/H9021/H20841n+/H20841−1,/H20841n−/H20841/H20849P,r/H20850
s−/H9021/H20841n+/H20841,/H20841n−/H20841−1/H20849P,r/H20850
/H9021/H20841n+/H20841,/H20841n−/H20841/H20849P,r/H20850/H20899.
/H2084910/H20850
The corresponding energy of the electron-hole pair En+,n−/H208490/H20850,
which is the eigenvalue of the Hamiltonian /H208498/H20850is given by27
En+,n−/H208490/H20850=/H6036vF
rB/H208812/H20851sgn/H20849n+/H20850/H20881/H20841n+/H20841− sgn /H20849n−/H20850/H20881/H20841n−/H20841/H20852, /H2084911/H20850
where s/H11006=sgn /H20849n/H11006/H20850andrB=/H20881c/H6036//H20849eB/H20850is a magnetic length.
C. Effective Hamiltonian for magnetoexcitons
In a strong magnetic field at low densities, n/H11270rB−2, indi-
rect magnetoexcitons repel as parallel dipoles, and we havefor the pair interaction potentialU/H20849/H20841R
1−R2/H20841/H20850 /H11013Uee+Uhh+Ueh+Uhe/H11229e2D2
/H9280/H20841R1−R2/H208413.
/H2084912/H20850
Let’s expand the magnetoexciton field operators in terms of a
single magnetoexciton basis set /H9023kP/H20849R,r/H20850:
/H9274ˆ†/H20849R,r/H20850=/H20858
kP/H9023kP/H11569/H20849R,r/H20850aˆkP†,
/H9274ˆ/H20849R,r/H20850=/H20858
kP/H9023kP/H20849R,r/H20850aˆkP, /H2084913/H20850
where aˆkP†andaˆkPare the corresponding creation and anni-
hilation operators of a magnetoexciton in /H20849k,P/H20850space and
substitute the expansions for the field creation and annihila-tion operators into Eq. /H208492/H20850. Due to the orthonormality of the
wave functions /H9021
n+,n−/H208490,r/H20850the projection of the Hamiltonian
/H20851Eq. /H208492/H20850/H20852onto the lowest Landau level results in the effective
Hamiltonian, which does not reflect the spinor nature of thefour-component magnetoexcitonic wave functions ingraphene. Since typically, the value of risr
B, and P/H11270/H6036 /rB
in this approximation, the effective Hamiltonian Hˆeffin the
magnetic momentum representation Pin the subspace the
lowest Landau level, which are n+=n−=0 for QWs, and n+
=n−=1 for GLs, has the same form /H20849compare with Ref. 9/H20850as
for the two-dimensional boson system without a magneticfield, but with the magnetoexciton magnetic mass m
B/H20849which
depends on BandD; see below /H20850instead of the exciton mass
/H20849M=me+mh/H20850, magnetic momenta instead of ordinary mo-
menta:
Hˆeff=/H20858
P/H92550/H20849P/H20850aˆP†aˆP
+1
2/H20858
P1,P2,P3,P4/H20855P1,P2/H20841Uˆ/H20841P3,P4/H20856aˆP1†aˆP2†aˆP3aˆP4,/H2084914/H20850
where the matrix element /H20855P1,P2/H20841Uˆ/H20841P3,P4/H20856is the Fourier
transform of the pair interaction potential U/H20849R/H20850=e2D2//H9280R3,
and for the lowest Landau level we denote the spectrum ofthe single exciton /H9255
0/H20849P/H20850/H11013/H925500/H20849P/H20850. For an isolated magnetoex-
citon on the lowest Landau level at the small magnetic mo-menta under consideration, /H9255
0/H20849P/H20850/H11015P2//H208492mB/H20850, where mBis
the effective magnetic mass of a magnetoexciton in the low-
est Landau level and is a function of the distance Dbetween
eandhlayers and magnetic field B/H20849see Ref. 14/H20850. In strong
magnetic fields at D/H11271rBthe exciton magnetic mass is
mB/H20849D/H20850=/H9280D3//H20849e2rB4/H20850for the QWs /H20849Ref. 14/H20850and mB/H20849D/H20850
=/H9280D3//H208494e2rB4/H20850for the graphene layers.30The projection of
the electron-hole Hamiltonian in magnetic field for theCQWs /H20851Eq. /H208492/H20850/H20852and GLs /H20851Eq. /H208498/H20850/H20852onto the lowest Landau
level results in the effective Hamiltonian /H20851Eq. /H2084914/H20850/H20852with
renormalized mass and where term related to the vector po-tential is missing. The magnetic field in the effective Hamil-tonian /H20851Eq. /H2084914/H20850/H20852is present in the renormalized mass of the
magnetoexciton m
B. Therefore, Hamiltonian for the spatially
separated electrons and holes in two-layer system /H20851Eq. /H208492/H20850/H20852
with the operator of the kinetic energy for an isolatedCOLLECTIVE PROPERTIES OF MAGNETOBIEXCITONS IN … PHYSICAL REVIEW B 78, 035135 /H208492008 /H20850
035135-3electron-hole pair Hˆ0defined by Eq. /H208494/H20850for the CQWs and
Eq. /H208498/H20850for the bilayer graphene can be reduced in high mag-
netic field to the effective Hamiltonian /H20851Eq. /H2084914/H20850/H20852. Magnetic
field Bis reflected by the effective Hamiltonian /H20851Eq. /H2084914/H20850/H20852
only through the effective magnetic mass of a magnetoexci-
tonmBin the expression for /H92550/H20849P/H20850in the first term of Hˆeff.
The only difference in the effective Hamiltonian /H20851Eq. /H2084914/H20850/H20852
for the CQWs and bilayer graphene realizations of two-layersystems is that m
Bfor the bilayer graphene is four times less
than for the CQWs due to the four-component spinor struc-ture of the wave function of the relative motion for the iso-lated noninteracting electron-hole pair in magnetic field /H20851Eq.
/H2084910/H20850/H20852.
Transitions between Landau levels due to the Coulomb
electron-hole attraction for the large electron-hole separationD/H11271r
Bcan be neglected if the following condition is valid:
Eb=e2//H20849/H9280bD/H20850/H11270/H6036/H9275c=/H6036eB/H20849me+mh/H20850//H208492memhc/H20850for the QWs
andEb=4e2//H20849/H9280D/H20850/H11270/H6036vF/rBfor the GLs, where Eband/H9275care
the magnetoexcitonic binding energy and the cyclotron fre-quency, respectively. This corresponds to the high magneticfield B, the large interlayer separation D, and large dielectric
constant of the insulator layer between the graphene layers.
As it was defined previously, here
vF=/H208813at//H208492/H6036/H20850is the
Fermi velocity of electrons.
It was shown in Refs. 31and32that taking into account
the spin degree of freedom can qualitatively modify the re-sults for exciton-polariton condensation at magnetic fields
lower than critical magnetic field. We assume that magneticfield Bconsidered in this paper is above the critical one, and,
therefore, Zeeman splitting does not effect on the spectrumof collective excitations according to Fig. 1 in Ref. 31.
III. INSTABILITY OF DIPOLE MAGNETOEXCITONS IN
QW AND GRAPHENE SUPERLATTICES
Let us show that the low-density system of weakly inter-
acting two-dimensional indirect magnetoexcitons in superlat-tices is instable, contrary to the two-layer system in the
CQW. At the small densities of nr
B2/H112701 the system of indirect
excitons at low temperatures is the two-dimensional weaklynonideal Bose gas with normal to wells dipole moments din
the ground state /H20849d=eD,Dis the interwell separation /H20850, in-
creasing with the distance Dbetween wells. In contrast to
ordinary excitons, for the low-density spatially indirect mag-netoexciton system the main contribution to the energy isoriginated from the dipole-dipole interactions U
−andU+of
magnetoexcitons with opposite /H20849see Fig. 1/H20850and parallel di-
poles, respectively. The potential energy of interaction be-tween two indirect magnetoexcitons with parallel U
+/H20849R/H20850and
opposite U−/H20849R/H20850dipoles is a function of the distance Rbe-
tween indirect magnetoexcitons along the quantum wells orgraphene layers:
U
+/H20849R/H20850=2e2
/H9280R−2e2
/H9280/H20881R2+D2,
U−/H20849R/H20850=e2
/H9280R−2e2
/H9280/H20881R2+D2+e2
/H9280/H20881R2+4D2. /H2084915/H20850
The behavior of the potential energies U+/H20849R/H20850andU−/H20849R/H20850as
the functions of the distance between two excitons Risshown in Fig. 2. We suppose that D/R/H112701 and L/R/H112701,
where Lis the mean distance between dipoles normal to the
wells. We consider the case, when the number of quantum
wells kin superlattice is restricted k/H112701//H20849D/H20881/H9266n/H20850. This is
valid for small kor for sufficiently low exciton density.
The distinction between magnetoexcitons and bosons
manifests itself in exchange effects.9,33The exchange inter-
action in the spatially separated system is suppressed in con-trast to the e−hsystem in one well due to smallness of the
tunnel exponent Tconnected with the penetration through
barrier of the dipole-dipole interaction. Hence, at D/H11271r
Bex-
change phenomena, connected with the distinction betweenexcitons and bosons, can be neglected for both the QWs andgraphene layers.
30Two indirect exciton in a dilute system
interact according to Eq. /H2084915/H20850, where Ris the distance be-
tween exciton dipoles along the QWs and GLs. Small tun-neling parameter connected with this barrier is
30B
FIG. 1. Two-dimensional indirect magnetobiexcitons consisting
of indirect magnetoexcitons with opposite dipole moments, locatedin neighboring pairs of QWs or graphene layers.
Distance between magnetoexcitons, R/DPotential energy o findirect magnetoexcitons, U(R)/Eex
00.5 11.5 22.5 33.5 44.5 5-0.050.050.150.250.350.450.550.650.75
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.050.050.150.250.350.450.550.650.75
U-(R)
U+(R)
FIG. 2. The potential energy of the interaction of indirect mag-
netoexcitons with parallel U+/H20849R/H20850and opposite U−/H20849R/H20850dipoles, lo-
cated in neighboring pairs of QWs or GLs /H20849in units of the binding
energy of the indirect magnetoexciton Eex=e2//H9280D/H20850, as a function of
the distances Rbetween magnetoexcitons along the QWs or GLs /H20849in
units of D/H20850.BERMAN, KEZERASHVILI, AND LOZOVIK PHYSICAL REVIEW B 78, 035135 /H208492008 /H20850
035135-4exp/H20877−1
/H6036/H20885
r0R0/H208812mB/H20875U+/H20849−/H20850−/H92602
2mB/H20876dR/H20878,
where the characteristic momentum of the system is given by
/H9260=/H208812mB/H9262d/H20851/H9262dis the chemical potential of the very dilute
system of magnetoexcitons, which is in a dilute 2D weaklynonideal Bose gas very small and proportional to n/logn
/H20849Ref. 34/H20850/H20852,r
0is the 2D radius of magnetoexciton, R0is the
classical turning point for the dipole-dipole interaction deter-mined from the condition U
+/H20849−/H20850/H20849R0/H20850=/H9262d. Hence, for the par-
allel dipoles R0is given by R0=/H208492mBe2D2//H92602/H208501/3. Since /H9262dis
very small, for the opposite dipole the classical turning pointcan be approximated by the condition U
−/H20849R0/H20850=0 and R0
/H11015D/H20849see Fig. 2/H20850. In high magnetic fields the small parameter
mentioned above has the form exp /H20851−2/H6036−1/H20849mB/H208501/2eDr0−1 /2/H20852.S o
at zero temperature T=0 the dilute gas of magnetoexcitons,
which is a boson system, form the Bose-Einsteincondensate.
35,37Therefore, the system of indirect magnetoex-
citons can be treated by the formalism applicable for a bosonsystem.
For the analysis of a stability of the ground state of the
weakly nonideal Bose gas of indirect excitons in superlat-tices, let us apply the Bogoliubov approximation. The total
Hamiltonian Hˆ
totof the low-density system of the indirect
magnetoexcitons in the superlattice, which is the generaliza-tion of the effective Hamiltonian of magnetoexcitons in atwo-layer system /H20851Eq. /H2084914/H20850/H20852, is given by
Hˆ
tot=Hˆ0+Hˆint. /H2084916/H20850
Here Hˆ0is the effective Hamiltonian of the system of non-
interacting magnetoexcitons:
Hˆ0=/H20858
P/H92550/H20849p/H20850/H20849aP+aP+bP+bP+a−P+a−P+b−P+b−P/H20850. /H2084917/H20850
In Eq. /H2084917/H20850/H92550/H20849p/H20850=p2//H208492mB/H20850is the spectrum of isolated two-
dimensional indirect magnetoexciton, Prepresents the exci-
tonic magnetic momentum, aP+,aP,bP+,bPare creation and
annihilation operators of the magnetoexcitons with up anddown dipoles. The Hamiltonian which describes the interac-tion between magnetoexcitons is
Hˆ
int=/H208492S/H20850−1/H20858P1+P2=P3+P4/H20851U+/H20849aP4+aP3+aP2aP1+bP4+bP3+bP2bP1/H20850
−U−aP4+bP3+aP2bP1/H20852, /H2084918/H20850
where U+andU−are the 2D Fourier images of U+/H20849R/H20850and
U−/H20849R/H20850atP=0, respectively, and Sis the surface of the sys-
tem. Let us mention that the appropriate cutoff parameter forthis Fourier transform is the classical turning point of thedipole-dipole interaction. Note that the cutoff parameter R
0
for the potential U+/H20849R/H20850is much greater than for U−/H20849R/H20850/H20851the
cutoff parameters R0for the both potentials can be repre-
sented in Fig. 2by the points where the curves corresponding
toU+/H20849R/H20850andU−/H20849R/H20850are crossed by the chemical potential /H9262d
represented by the horizontal straight line placed right above
but close to U+/H20849−/H20850/H20849R/H20850=0/H20852. Therefore, we claim that U+/H110220,
U−/H110210, and /H20841U−/H20841/H11022/H20841U+/H20841.The physical meaning of the terms of the type
aP4+bP3+aP2bP1is just the scattering between two excitons with
the opposite orientation of dipole /H20849the momenta change /H20850. Af-
ter scattering both magnetoexcitons stay in their initial lay-ers.
Let us consider the temperature T=0. Assuming the ma-
jority of particles are in the condensate /H20851/H20849N−N
0/H20850/N0/H112701,
where Nand N0are the total number of particles and the
number of the particles in condensate, respectively /H20852,w ea c -
count as in Bogoliubov approximation only the interactionbetween condensate particles and excited particles with con-densate particles, and neglect the interaction between non-
condensate particles. Then the total Hamiltonian Hˆ
tottrans-
forms to
Hˆtot=1
2/H20858
P/HS110050/H20853/H20851/H92550/H20849p/H20850+/H20849U++U−/H20850n/H20852/H20849aP+aP+bP+bP+a−P+a−P
+b−P+b−P/H20850+2U+n/H20849aP+a−P++aPa−P+b−P+bP++b−PbP/H20850
+U−n/H20849aP+b−P++aPb−P+a−P+bP++a−PbP+aP+bP+a−P+b−P
+aPbP++a−Pb−P+/H20850/H20854. /H2084919/H20850
Let us diagonalize Hamiltonian Hˆtotby using the
Bogoliubov-type unitary transformation35
aP=1
/H208811−AP2−BP2−CP2/H20849/H9251P+AP/H9251−P++BP/H9252−P++CP/H9252P/H20850,
bP=1
/H208811−AP2−BP2−CP2/H20849/H9252P+AP/H9252−P++BP/H9251−P++CP/H9251P/H20850,
/H2084920/H20850
where the coefficients AP,BP, and CPare found from the
condition of vanishing of the coefficients at nondiagonalterms in the Hamiltonian. As a result we obtain
Hˆ
tot=/H20858
P/HS110050/H9255/H20849p/H20850/H20849/H9251P+/H9251P+/H9252P+/H9252P/H20850/H20849 21/H20850
with the spectrum of quasiparticles /H9255/H20849P/H20850:
/H925512/H20849P/H20850=/H925502/H20849P/H20850+2nU+/H92550/H20849P/H20850,
/H925522/H20849P/H20850=/H925502/H20849P/H20850+2n/H20849U++U−/H20850/H92550/H20849P/H20850. /H2084922/H20850
Since U+/H110220 and U−/H110210, we have /H925512/H20849P/H20850/H11022/H925522/H20849P/H20850at
P/H110220. Therefore, at low temperatures the quasiparticles only
with the spectrum /H925522/H20849P/H20850will be excited, since the excitation
of these quasiparticles requires less energy than for the qua-
siparticles with the spectrum /H925512/H20849P/H20850. Since U++U−/H110210, it is
easy to see from Eq. /H2084922/H20850that for the small momenta
P/H11021/H208814mBn/H20841U++U−/H20841the spectrum of excitations becomes
imaginary. Hence, the system of weakly interacting indirectmagnetoexcitons in the slab of the superlattice is unstable. Itcan be seen that the condition of the instability of magne-toexcitons becomes stronger as magnetic field becomeshigher, because m
Bincreases with the increment of the mag-
netic field, and, therefore, the region of Presulting in the
imaginary collective spectrum increases as Bincreases.COLLECTIVE PROPERTIES OF MAGNETOBIEXCITONS IN … PHYSICAL REVIEW B 78, 035135 /H208492008 /H20850
035135-5The mathematically similar system was considered in Ref.
36, where 2D spinor polaritons were studied, with attractive
interactions between the polaritons of the opposite circularpolarization and repulsive interaction for polaritons with thesame circular polarization. Note that our results for magne-toexcitonic spectrum given by Eq. /H2084922/H20850are analogous to that
in Ref. 36. While in Ref. 36the repulsion exceeds the attrac-
tion between two different polaritons /H20849/H20841U
+/H20841/H11022/H20841U−/H20841/H20850,i n
multilayer superlattice with parallel and opposite dipolesthere is an opposite case /H20849/H20841U
+/H20841/H11021/H20841U−/H20841/H20850. Therefore, there is an
instability in multilayer magnetoexciton system while thereis no instability in polariton system considered in Ref. 36.
The analogous instability was studied theoretically in 2D
atomic gas with the attractive and repulsive interaction be-tween bosons.
38
IV. SUPERFLUIDITY OF QUADRUPOLE
MAGNETOBIEXCITONS IN QW AND GRAPHENE
SUPERLATTICES
Let us consider the low-density weakly nonideal gas of
two-dimensional indirect magnetobiexcitons, created by theindirect magnetoexcitons with opposite dipoles in the neigh-boring pairs of wells as the ground state of the system /H20849Fig.
1/H20850. The ratio of the magnetobiexciton and magnetoexciton
energies, as well as the ratio of the magnetoexciton and mag-netobiexciton radii along the QWs or GLs, is the parameterfor the adiabatic approximation used in Ref. 39. These pa-
rameters are small and they are even smaller than analogousparameters for atoms and molecules. The smallness of theseparameters will be verified below by the calculations of thecorresponding energy and radius of the indirect magneto-biexciton. Here it was assumed that the distance Dbetween
the QWs or GLs is greater than the radius of indirect mag-netobiexciton r
0,D/H11022r0.A tr/H110221.11Dindirect magnetoexci-
tons attract and at r/H110211.11Dthey repel. The minimum of the
potential energy U/H20849r/H20850locates at r=r0/H110151.67Dbetween the
indirect excitons and this r0is the mean radius of the mag-
netobiexciton along wells or graphene layers. At large Done
can expand the potential energy U/H20849r/H20850in terms of the param-
eter /H20849r−r0/H20850/D/H112701:
U/H20849r/H20850= − 0.04e2
/H9280D+ 0.44e2
/H9280D3/H20849r−r0/H208502+ ... . /H2084923/H20850
If we restrict ourselves by the first two terms of this expan-
sion it easy to see that at large Dmagnetobiexciton levels
correspond to the two-dimensional harmonic oscillator withthe frequency
/H9275=0.88 e2//H20849mB/H9280D3/H20850:
En= − 0.04e2
/H9280D+2/H208812E0/H20873r/H11569
D/H208743/2
/H20849n+1/H20850, /H2084924/H20850
where E0=mBe˜4//H20849/H60362/H9280/H20850,r/H11569=/H60362/H9280//H208492mBe˜2/H20850, and e˜2=0.88 e2.I n
the ground state the characteristic spread of the magnetobiex-citon a
balong the QWs or GLs near the mean radius r0is
ab=/H208812/H6036
mB/H9275=/H208498r/H11569/H208501/4D3/4= 1.03 aex, /H2084925/H20850
where rex=/H208498rex/H208501/4D3/4andaex=/H60362/H9280//H208492mBe2/H20850are the radius
magnetoexciton and the two-dimensional effective Bohr ra-dius with the effective magnetic mass mB, respectively.
Hence, the ratio of the binding energies of the magnetobiex-citon E
bexand magnetoexciton EexisEbex /Eex=0.04 /H112701a t
D/H11271rex, and the ratio of radii of the magnetoexciton and
magnetobiexciton is r0/rex=0.67 /H208498aex/H208501/4D−1 /4/H112701. Thus, the
adiabatic condition is valid.
The mean dipole moment of the indirect magnetobiexci-
ton equals to zero. However, the quadrupole moment is non-zero and equal to Q=3eD
2, and the large axis of the quad-
rupole is normal to the quantum wells or graphene layers.Hence, the indirect magnetobiexcitons interact at long dis-tances R/H11271Das parallel quadrupoles: U/H20849R/H20850=9e
2D4//H20849/H9280R5/H20850.
We account the scattering of the magnetobiexciton on
magnetobiexciton by using the results of the theory of two-dimensional Bose gas.
5The chemical potential /H9262for two-
dimensional biexcitons, repelled by the quadrupole law, inthe ladder approximation, has the form /H20849compare to Refs. 5
and9/H20850
/H9262=2/H9266/H60362nb
mBlog/H20853/H60364/3/H92802/3//H208518/H9266/H2084918mBe2D4/H208502/3nb/H20852/H20854. /H2084926/H20850
where nb=n//H208492s/H20850is the density of magnetobiexcitons in the
QWs and nb=n//H208498s/H20850in the GLs. The chemical potential /H20851Eq.
/H2084926/H20850/H20852is different than the chemical potential for two-
dimensional excitons with the dipole-dipole repulsion givenin Refs. 5and9/H20850.
At small momenta the collective spectrum of the magne-
tobiexciton system is the soundlike /H9255/H20849p/H20850=c
sp, where cs
=/H20881/H9262//H208492mB/H20850is the sound velocity, and satisfied by the Lan-
dau criterion for superfluidity. The density of the superfluidcomponent n
S/H20849T/H20850for the two-dimensional system with this
sound spectrum can be estimated as37
nS/H20849T/H20850=nb−3/H9256/H208493/H20850
4/H9266/H60362kB3T3
mBcs4, /H2084927/H20850
where /H9256/H20849z/H20850is the Riemann zeta function, and /H9256/H208493/H20850/H112291.202.
The second term in Eq. /H2084927/H20850is the temperature-dependent
normal density taking into account the gas of phonons
/H20849“bogolons” /H20850with the dispersion law /H9255/H20849p/H20850=/H20881/H9262//H208492mB/H20850p,
where /H9262is given by Eq. /H2084926/H20850.
In a 2D system, superfluidity of magnetobiexcitons ap-
pears below the Kosterlitz-Thouless transition temperatureT
c=/H9266nS/H20849T/H20850//H208494mB/H20850, where only coupled vortices are
presented.26Employing nS/H20849T/H20850for the superfluid component,
we obtain the cubic equation for the Kosterlitz-Thoulesstransition temperature T
c, which has the following solution:
Tc=/H20875/H208731+/H2088132
27/H208738mBkBTc0
/H9266/H60362nb/H208743
+1/H208741/3
−/H20873/H2088132
27/H208738mBkBTc0
/H9266/H60362nb/H208743
+1−1/H208741/3/H20876Tc0
21/3. /H2084928/H20850
Here, Tc0is an auxiliary quantity, equal to the temperature at
which the superfluid density vanishes in the mean-field ap-
proximation, i.e., nS/H20849Tc0/H20850=0,Tc0=kB−1/H208534/H9266/H60362ncs4mB//H208513/H9256/H208493/H20850/H20852/H208541/3.
The temperature Tc0=Tc0/H20849B,D/H20850may be used to estimate the
crossover region where the local superfluid density appearsBERMAN, KEZERASHVILI, AND LOZOVIK PHYSICAL REVIEW B 78, 035135 /H208492008 /H20850
035135-6for magnetoexcitons on a scale smaller or of the order of the
mean intervortex separation in the system. The local super-
fluid density can manifest itself in the local optical or trans-port properties.V. RESULTS AND DISCUSSION
We calculated the dependence of the Kosterlitz-Thouless
transition temperature Tcon the density of the magnetoexci-
tons for the superlattice consisting of the quantum wells andgraphene layers. The results of the calculations for the tran-sition temperature T
cas a function of the density of the mag-
netoexcitons for the different values of the magnetic field B
for the QWs and GLs are presented in Fig. 3. Analysis of
these results shows that, firstly, for the given thickness of theQWs or GLs and for the fixed magnetic field the Kosterlitz-Thouless transition temperature T
cin good approximation
linearly increases with the increase in the magnetoexcitonsdensity for the QWs, as well as for GLs. This is due to thefact that the denominator in Eq. /H2084926/H20850for the chemical poten-
tial and, therefore, the sound velocity, weakly depends on n.
Our calculations show that the slope of the function T
c/H20849n/H20850
decreases with nvery weakly and it is almost constant. Sec-
ondly, for the same magnetoexcitons density the transitiontemperature T
cfor the GLs is always higher than for the
QWs, because the magnetic mass of magnetoexciton in theGLs is four times smaller than in the CQWs due to the four-component spinor structure of the magnetoexciton wavefunction in the GLs. Moreover, the transition temperaturestrongly depends on the magnetic field: with the increase inthe magnetic field the transition temperature T
cdramatically
increases for the GLs, as well as for the QWs. This is due to
FIG. 3. /H20849Color online /H20850Dependence of the Kosterlitz-Thouless
transition temperature Tc=Tc/H20849B/H20850for the superlattice consisting of
the QWs for GaAs/AlGaAs, /H9280=13; and for GLs separated by the
layer of SiO 2with/H9280=4.5 on the magnetoexciton density natD
=10 nm at different magnetic fields. The solid, dashed, and thinsolid curves for the QWs, dotted, dashed-dotted, and thin dottedcurves for the GL at B:B=20 T, B=15 T, and B=10 T,
respectively.
THgr
THrTca)
Tc(K)
n(cm-2) B(T)Graphene
D = 10nmTc(K)b)
n(cm-2)B(T)QWs
D = 10nm
×1010×1010
D(nm)n(cm-2)QWs
B = 10TTc(K)c)
Tc(K)
D(nm)
n(cm-2)Graphene
B = 10Td)
×1010×1010
FIG. 4. Dependence of the Kosterlitz-Thouless transition temperature Tc=Tc/H20849B/H20850for the superlattice consisting of the QWs /H20849for GaAs/
AlGaAs /H9280=13 /H20850and of the GLs /H20849for GLs separated by the layer of SiO 2/H9280=4.5 /H20850on the magnetoexciton density nand magnetic fields B
presented by /H20849a/H20850and /H20849b/H20850and on the magnetoexciton density nand the interlayer separation Dpresented by /H20849c/H20850and /H20849d/H20850, respectively.COLLECTIVE PROPERTIES OF MAGNETOBIEXCITONS IN … PHYSICAL REVIEW B 78, 035135 /H208492008 /H20850
035135-7the increment of mBas a function of BandD. The Tcde-
creases as B−1 /2atD/H11270rBand as B−2when D/H11271rB. The three-
dimensional plots of the dependence of the Kosterlitz-Thouless transition temperature T
con the magnetic field and
magnetoexcitons density for the superlattice consisting fromthe QWs and GLs are shown in Fig. 4.
The advantage of the observation of the magnetoexciton
superfluidity and the BEC in graphene in comparison withthis in the CQWs consists of essentially weak influence ofthe random field on T
cdue to the fact that the density of
defects in graphene is sufficiently lower than in the CQWs/H20849due to the absence of the roughness of the QWs bound-
aries /H20850. As known, the influence of impurities is suppressed
due to the existence of the Berry phase.
40Particularly, Berry
phase makes impossible backscattering of the excitons on theimpurities. Due to this fact, the influence of impurities on theBEC occurs to be smaller than for the semiconductor QWs.The disorder more likely suppresses the superfluidity and theKosterlitz-Thouless temperature for magnetobiexcitons inthe QW superlattices analogously to the case of magnetoex-citons in the CQWs.
41
VI. CONCLUSIONS
It is shown that the low-density system of the indirect
magnetoexcitons in a slab of the superlattice of the secondtype or consisting of the QWs or GLs in high magnetic fieldoccur to be instable due to the attraction of magnetoexcitons
with opposite dipoles at large distances. The instability of theground state of the system of interacting two-dimensionalindirect magnetoexcitons in a slab of superlattice with thealternating electron and hole layers for both QWs and GLs inhigh magnetic field is claimed. This instability is due to theattraction between the indirect excitons with the oppositedirected dipole moments. The stable system of indirect
quasi-two-dimensional magnetobiexcitons, consisting fromthe indirect excitons with opposite directed dipole momentsin the superlattices, is stable due to the quadrupole-
quadrupole repulsion. Therefore, at the pumping increase atlow temperatures the excitonic line must vanish and only themagnetobiexcitonic line survives. Note that in spite of boththe QW and graphene realizations represented by completelydifferent Hamiltonians, the effective Hamiltonian in a strongmagnetic field was obtained to be the same. The projectionof the electron-hole Hamiltonian for the CQWs /H20851Eq. /H208492/H20850/H20852and
GLs /H20851Eq. /H208498/H20850/H20852in magnetic field onto the lowest Landau level
leads to the effective Hamiltonian /H20851Eq. /H2084914/H20850/H20852without the vec-
tor potential and renormalized effective mass of the magne-toexciton, which depends on the magnetic field. The mag-netic field in the effective Hamiltonian /H20851Eq. /H2084914/H20850/H20852is
presented into the renormalized mass m
Bof the magnetoex-
citon. Moreover, for Nexcitons we have reduced the number
of the degrees of freedom from 2 N/H110032t o N/H110032 by integrat-
ing over the coordinates of the relative motion of the elec-trons and holes. The Kosterlitz-Thouless transition to the su-perfluid state is calculated for the system of indirectmagnetobiexcitons. The temperature T
cfor the onset of su-
perfluidity due to the Kosterlitz-Thouless transition at a fixed
magnetobiexciton density decreases as a function of mag-
netic field Band interlayer separation D, and almost linearly
increases when the magnetoexcitons density of the QWs orGLs increases. At the same exciton density nthe Kosterlitz-
Thouless temperature T
cis higher for the superlattice con-
sisting of the GLs than for the superlattice consisting of thequantum wells.
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PhysRevB.83.205415.pdf | PHYSICAL REVIEW B 83, 205415 (2011)
Fano resonance in electron transport through single dopant atoms
L. E. Calvet,1,*J. P. Snyder,2and W. Wernsdorfer3
1Institut d’Electronique Fondamentale-CNRS UMR 8622, Universit ´e Paris-Sud, FR-91405 Orsay, France
2Volovox, LLC, Minnesota 55425, Minneapolis, USA
3Institut N ´eel, CNRS and Universit ´e J. Fourier, BP 166, FR-38042 Grenoble Cedex 9, France
(Received 14 September 2010; revised manuscript received 28 January 2011; published 19 May 2011)
Antiresonances are observed in electron transport through a resonant dopant atom situated near a metal-
semiconductor interface in a Schottky barrier metal-oxide-semiconductor field-effect transistor. The lineshapesdo not significantly change in magnetic fields up to 5 T, but are modified by small dc bias voltages. We arguethat these effects are the result of quantum interference between two tunneling paths and can be explained in thecontext of a Fano lineshape.
DOI: 10.1103/PhysRevB.83.205415 PACS number(s): 73 .63.−b, 85.35.−p
Quantum interference has been a longstanding topic of
research in mesoscopic physics because it probes the wavenature of electrons. While signatures of interference areobserved in the magnetoresistance of confined nanostructures,giving rise to weak localization and universal conductancefluctuations,
1even more intriguing are experiments in which
interference can be directly measured. The most typical ex-ample is the Aharonov-Bohm ring, where electrons traversingtwo different paths interfere as a perpendicular magnetic fieldis varied.
2In this paper, we show that a localized dopant in a
semiconductor can act as an interferometer when two tunnelingpaths result in quantum interference, also known as the Fanoeffect.
3
Unlike a discrete resonance in which the resulting char-
acteristics exhibit Lorentzian peaks, the Fano effect resultsin a family of lineshapes depending on the phase differencebetween the two paths. They have been experimentallypredicted and observed in a large variety of semiconductorheterostructures
4and are of interest because the interfer-
ence can provide valuable information about the coherenceof the transport.
5,6The Fano effect has been widely in-
vestigated in microfabricated quantum dots in which thedesign of the structure can serve as a playground forexperimentalists. Researchers have investigated geometriesin which the dot-lead coupling can be tuned,
7,8in which
an additional path has been included into the geometry9
and embedded a quantum dot into an Aharonov-Bohmring.
10
Understanding the nature of quantum interference on
the scale of a single nano-object, however, can be quitechallenging. This topic is of great interest both theoretically
11
and experimentally12–14because observations can provide
additional information about the measured object and serveas the basis of novel device concepts.
15,16In this paper, we
demonstrate quantum interference in the electrical transportthrough dopant atoms in silicon. There has been a definitiveobservation of Fano resonances between paths through twoAs atoms in a nanosized Si field-effect transistor (FinFET).
17
In the research considered here, we use micron size devicesto demonstrate a quantum interference effect on a muchsmaller length scale, i.e., that approaching the Bohr radiusof the atoms and thus similar to the length scale in singlemolecules.I. EXPERIMENTAL DETAILS
We explore transport through dopant atoms located near
the metallic contact in a p-type Schottky barrier metal-oxide-
semiconductor field-effect transistor (SBMOSFET).18These
devices have been widely researched in the past 15 years as analternative to traditional MOSFETs for nanoscaled devices,
19
and a complementary metal-oxide-semiconductor (CMOS)
technology has recently been demonstrated.20In the device,
shown schematically in Fig. 1, metallic silicide source and
drain electrodes are formed instead of the p-njunctions found
in conventional MOSFETs. The variable temperature mea-surements indicate that, at low drain-source bias voltages V
ds,
the transport is dominated by tunneling through the depletionregion formed at the metal-semiconductor interface.
21At low
temperatures, resonant tunneling through individual atoms can
be observed when the gate bias brings the energy level of anionized dopant positioned close to the metal-semiconductorinterface into resonance with the Fermi level.
18
The SBMOSFETs used in this work consist of an n-type
polysilicon gate, a p-doped substrate (5 ×1015cm−3), a 34- ˚A
gate oxide, and 300- ˚A PtSi Schottky barriers.22In previous
research, we explored the nature of transport through B
and Pt dopants.18,23,24Measurements were performed using
standard ac lock-in techniques in a dilution refrigerator witha base temperature of 50 mK. We apply magnetic fields up to4.8 T in the plane parallel to the transport. We have observedantiresonances in five different devices, but, in this paper,we focus on one single device with dimensions of width andlength =3 and 0.5 μm, respectively, in which 10 resonant
dips and 3 resonant peaks were observed.
II. RESULTS
Figure 2(a) shows differential conductance∂Ids
∂Vdsas a
function of gate voltage | Vg|a tVds=0 V and at an applied
magnetic field B=2.7 and 4.7 T. Two resonant peaks and
8 resonant dips, labeled consecutively a–h, are seen here. Atlarger magnetic fields, we observe that the peaks and dipsare shifted either towards (| V
g|>0) or away from (| Vg|<0)
the silicon valence band Ev, but that the lineshapes are not
significantly altered. Figure 2(b) shows a three-dimensional
plot of∂Ids
∂Vdsas a function of B. In order to emphasize the
displacement of the lineshapes with magnetic field, we focus
205415-1 1098-0121/2011/83(20)/205415(8) ©2011 American Physical SocietyL. E. CALVET, J. P. SNYDER, AND W. WERNSDORFER PHYSICAL REVIEW B 83, 205415 (2011)
n+ poly Si
PtSi PtSiSiO2
p -Si
FIG. 1. (Color online) Schematic of the SBMOSFET (not drawn
to scale). The source and drain tunneling contacts consist of themetal silicide PtSi. The gate is formed from a conventional metal-
oxide-semiconductor (MOS) capacitor. The substrate is lightly doped
(5×10
15cm−3) with boron. The device operates in accumulation and
transport is dominated by holes. The device dimensions used in this
paper are width/length =3μm/0.5μm.
on a smaller | Vg| range that includes seven dips (dark lines) and
one resonance (bright white line). Near B=0T ,∂Ids
∂Vdsis strongly
suppressed due to the superconducting gap formed whenthe PtSi electrodes become superconducting at temperaturesbelow 1 K.
25One observes only very small changes in the
conductivity as the magnetic field is changed from 0.5 to4.7 T. It is striking that both the dip and the peak positionsexhibit either positive or negative monotonic displacementswith magnetic field. To explore the nature of this magneticfield dependence, linear fits to the local maximum (peak) orminimum (dips) position as a function of magnetic field areplotted in dashed lines. The resultant slopes for each of theresonances and antiresonances are shown in Table I. In order
to convert | V
g| (V) into energy (eV), we use α=0.155 eV /V,
as discussed in previous research.24
To understand the nature of the lineshapes, Fig. 3plots the
Vdsdependence of the peak (a) and a representative dip (c)
at 4.7 T. We first note that the background conductance is
suppressed at small Vdsdue to the Altshuler-Aronov electron-
electron correction of the tunneling density of states.25We
observe, however, that at Vds=0 V , the resonant peak is
suppressed to a greater extent than the background. Tomake this clearer, in Fig. 3(b), we scale several nonzero V
ds
traces to the background conductance at Vds=0V ,Vg=
−1.951 V . If the density of states correction simply increased
the resonant tunneling proportional to the background, we
expect that, for small Vds, the resonant peaks would overlap.
As|Vds|is increased, the resonance width should increase and
the peak height should diminish due to standard Fermi levelbroadening. We do, indeed, observe that, at V
ds>0.2m V ,
the resonance is subject to the expected broadening; however,between 0 V to 0.2 mV , the peak height and width bothincrease.
Figure 3(c) explores the V
dsdependence of the antireso-
nances. While at Vds=0 V , the lineshape is mostly a dip
below the background current, as Vdsis increased, the dip
turns into an asymmetric lineshape containing both a resonantdip and peak. The most striking result is the reversal ofthe lineshape for negative and positive V
ds. To gain greater
insight, in Fig. 3(d), we plot the evolution of dip c, scaled as
in Fig. 3(b), by the background conductance at Vds=0 and
Vg=− 1.913 V . We observe that the lineshape changes from
a sharp dip to an asymmetric lineshape with a dip and a peak,to finally something that is mostly a small resonant peak. Wenote that the other dips show similar behavior in that, as V
ds
is increased, they transform into an asymmetric lineshape and28
26
24
22
20 ∂I⎯∂Vds (nS)
-1.940 -1.935 -1.930
Vg (V)50
40
30
20
10 ∂I⎯∂Vds (nS)
-2.00 -1.96 -1.92
Vg (V)abcd
efgh
2.7 T
4.7 T
(a)
(b)
4 3 2 1 0
Magnetic Field (T)-1.98
-1.96
-1.94
-1.92
-1.90Vg (V)
abcdefg 45
40
35
30
25
20
15
10
∂I⎯∂Vds (nS)
FIG. 2. (Color online) (a) Differential conductance∂Ids
∂Vdsvs gate
voltage | Vg|a tVsd=0 V and magnetic fields B=2.7 and 4.7 T at
50 mK. There are eight antiresonances, labeled a–h, and two resonantpeaks. These lineshapes are shifted in energy (either to higher or
lower | V
g|) with B, attributed mostly to the Zeeman effect. Although
the magnitude of the differential conductance decreases slightly withfield, there is not a substantial change in the lineshape. The inset
shows a more detailed plot of dip e. (b) Three-dimensional plot of the
∂Ids
∂Vdsshowing the displacement of the lineshapes in (a) as a function
of magnetic field. Here we use a small range of | Vg| for clarity.
The dotted lines are the linear fits obtained by extracting the local
minimum (maximum) near the lineshape and the resultant slopes arereported in Table I.
then into either a very small resonant peak or disappear into
the background current.
205415-2FANO RESONANCE IN ELECTRON TRANSPORT THROUGH ... PHYSICAL REVIEW B 83, 205415 (2011)
TABLE I. Slopes of the magnetic field dependence of the peak or
dip positions for the lineshapes observed in Fig. 2.
Lineshape Slope ( μeV/T) Std. dev. ( μeV/T)
a −96.4 2.39
b −111.8 1.49
c4 8 .8 1.44
d −162.3 5.91
e6 2 .3 0.41
Peak 1 −44.33 0.98
f6 4 .5 0.74
g 106 .8 4.84
h −110.5 1.16
Peak 2 −55.5 0.98
III. DISCUSSION
We argue that the resonant dips are due to quantum
interference. This is a very surprising result: we are able toprobe coherent transport through a dopant atom in a devicewith relatively large dimensions (3 μm width by 0.5 μm
length). Our attempts to explain this effect using other physicalphenomena have proved untenable. For instance, the resonantdips can not be due to a thermoelectric effect because any sucheffects are not observable in an ac measurement in which thein-phase component with the same frequency as the excitationis measured.
26,27These data can also not be explained by Pauli
spin blockade in which two impurities coupled together28,29
or a spin-polarized electrode30,31could result in significant
current reduction. Such an effect would result in rectifyingbehavior as a function of bias voltage direction. While wedo observe changes in the lineshape with bias direction, wedo not report a greater suppression of the resonance for oneparticular polarity. To fully appreciate the experimental datain the previous section, it is first necessary to understand thenature of the resonant states and the shift in energy of thelineshapes with magnetic field.
A. Origin of the resonances and antiresonances and magnetic
field dependence
In previous research, we have explored resonant peaks due
to Pt (Ref. 24) and/or B atoms18,23that are present in the device.
We investigated the Zeeman effect as a function of magneticfield and excited state spectroscopy with V
ds. In Table I,w e
recognize many of the slopes from this previous research.Antiresonance d,for instance, shows a slope indicative of
am
j=3/2 and g=1.86±0.07, consistent with boron
atoms.18,23For the impurity shown in the inset of Figs. 3(c)
and3(d) (dip c), the magnetic field dependence [Fig. 2(b)]i s
−48.825 ±1.44μeV/T. Assuming a 1 /2 spin, this results in
agfactor of 1.68 ±0.05, which is within the experimental
accuracy of the 1.38 ±0.43 gfactor from previous research,24
and thus this dopant is likely to be a double donor of Pt. We
note that the slopes of dips a, b, g, and h and the two peakshave previously been reported as related to Pt atoms.
24The
remaining dips (e and f) are likely due to boron atoms.
Compared with spectroscopy of small microfabricated
quantum dots,32,33the charging energies of typical dopants
in silicon are large. Pt is known to have three charged stateslocated at approximately Ec−0.243 eV (single acceptor),
Ev+0.330 eV (single donor), and Ev+0.1 eV (double
donor), where Ex=c,vis, respectively, the conduction or valence
band.34Defects of Pt involving other atoms such as H and
O have similarly large charging energies between differentstates.
34Researchers have investigated B in bulk samples using
piezo,35electron paramagnetic resonance,36and infrared37
spectroscopies to investigate the energy spectrum of the A0
(located 0.045 eV above Ev) and excited states (located
between 12–2 meV above Ev). To access the second bound
hole state A+, phonon spectroscopy has been used.38This
state is very close to the valence band edge (2 meV) and thusinvestigations are quite limited. Recent research has exploredtheA
+state using transport spectroscopy of a δ-doped silicon
layer.39We have previously investigated the ground state of
B dopants,18,23but are not able to determine the ionization
energy using our experimental method. Another group has
200
150
100
50 ∂I⎯∂Vds (nS)
-1.958 -1.956 -1.954 -1.952 |Vds| > 0 V
|Vds| < 0 V
Vds = 0 V100
80
60
40
20
-1.94 -1.92 -1.90 |Vds| > 0 V
|Vds| < 0 V
Vds = 0 V(a) (c)
70
60
50
40
30
20 ∂I⎯∂Vds (nS)
-1.958 -1.956 -1.954 -1.952
Vg (V)Vds =
0.0 mV
0.1 mV
0.2 mV
0.3 mV
0.4 mV(b)24
22
20
18
16
-1.920 -1.918 -1.916 -1.914 -1.912
Vg (V)Vds =
0.0 mV
0.1 mV
0.2 mV
0.3 mV
0.4 mV(d)
FIG. 3. (Color online) Changes in the lineshape at different Vsd
of a resonant peak (a) and (c) and dip c (b) and (d). In both (a)
and (c), the∂Ids
∂Vdsof the Vds=0 V curve is significantly suppressed
compared to higher bias. This suppression of the background current
is due to electron-electron interactions. In (b) and (d), the lineshapes
are scaled, respectively, by the∂Ids
∂VdsatVds=0V ,Vg=− 1.951 V ,
andVds=0V,Vg=− 1.913 V . However, as shown in (b) and (d),
the lineshapes do not scale in the same way as the background. All
curves are shown at 50 mK and at 4.7 T.
205415-3L. E. CALVET, J. P. SNYDER, AND W. WERNSDORFER PHYSICAL REVIEW B 83, 205415 (2011)
investigated single40,41and coupled acceptors42in nanoscale
silicon-on-insulator (SOI) FET devices and found a muchsmaller ionization energy (10 meV) than the bulk value. Thisis likely due to the position of the dopant close to the SOIinterface.
43Other researchers who have investigated single
donors of small silicon SOI MOSFETs have found valuessimilar to the bulk,
44exceeding the bulk up to 2 ×(Ref. 43)
or less than the bulk down to 0.5 ×.45We estimate that the
charging energy of individual dopants in silicon should be /greaterorequalslant
20 meV , which is at least an order of magnitude larger than thecharging energy of small microfabricated quantum dots.
32,33
We thus see that the energy scale represented in Fig. 2, which is
∼14 meV , is small on the energy scale of the charging energies
of typical dopants in silicon. If the individual resonances andantiresonances reported here are related to single atoms or asingle atom coupled to H or O, the different levels should,therefore, not correspond to levels of the same dopant. Thereare two other possibilities that we briefly consider: coupledatoms and the creation of a small quantum dot based on acluster of dopants.
We can compute an average number of Batoms near the
metal-semiconductor interface (about 1.5 dopants for every1μm of device width) based on the device dimensions and
background doping. Such dopants are spread out randomlyover the entire micron size device width, and it can beexpected that most resonant boron atoms are uncoupledwith each other. However, Pt atoms present another problembecause the number of atoms that have diffused from themetal-semiconductor interface can vary widely
46and TEM
observations show the occasional formation of Pt clusters.47
We thus consider the possibility that many Pt atoms could forma multiatom cluster or a few-atom coupled state.
Forming a quantum dot from a cluster of dopants was
recently considered both experimentally and theoretically bythe Nottingham group.
48,49This research showed that the
deepest quantum well was created by about 10 randomlyplaced Mn atoms in GaAs quantum wells confined in about∼10 nm. Generation of quantum dot formed from a cluster
of atoms resulted in a very distinguishable Fock-Darwin-typeenergy spectrum in magnetic field due to orbital effects offilling additional electrons. This spectrum is taken by assuminga circular confining potential and solving the Schrodingerequation. The eigenenergies are given by
E
n,l=/bracketleftBigg
(2n+|l|+1)¯h/parenleftbigg
ω2
0+ω2
c
4/parenrightbigg1
2
−1
2¯hlωc/bracketrightBigg
,(1)
where nandlare the radial and angular momentum quantum
numbers, ω0is the oscillator frequency, and ωc=eB/m∗
is the cyclotron frequency. This spectrum has some very
well-defined magnetic features; however, depending on theconfinement of the dot, they can be spread out, and dependingon the charging energy, they may become truly evident onlyat high magnetic fields. We now consider the relevant energyscales in our system.
For our system, the ionization energies of the relevant single
atoms are between 50–100 meV below E
v, which would result
in quantum dot clusters with the smallest addition energies∼4–20 meV and, thus, a Fock-Darwin spectra observable
within the 5-T magnetic field dependence of the experiment.In the silicon valence band, the energy scales of the Fock-
Darwin spectra are considerably different from GaAs. First, thecontribution of the cyclotron energy ¯ hω
c/2B≈115μeV/T
(assuming an effective mass m∗=0.5me) is much smaller than
the expected confinement energy ¯ hω0/greaterorequalslant4 meV , so that Eq. ( 1)
can be approximated as En,l≈[(2n+|l|+1)¯hω0−1
2¯hlωc].
For small values of l, the magnetic field dependence is the same
order as the Zeeman energy /Delta1Ez/B=gmjμB≈58μeV/T
forg=2 and mj=1/2. We must therefore include /Delta1Ez
when considering such orbital effects.
The |Vg| range in Fig. 2(b) corresponds to 14 meV . We
can not exclude orbital effects and the presence of clustersfrom these data, however, we can not account for all of theresonance and antiresonances by one cluster of defects forminga quantum dot. Specifically, the peak position versus magneticfield slopes should be larger with increasing filling, but, inTable I, the slopes cluster around ±100 and ±50μeV/T. We
note that dips c and d might be attributed to the l=± 1 states of
a cluster, but in this energy window, we could not attribute anyof the other structures to either the l=0o r±2 states. Precise
assignment here is further complicated because the slope ofdip d could also be attributed to the ground state of a boronacceptor impurity.
18,23
An additional possibility is that the magnetic field orien-
tation is not suitable for observing orbital effects and thatsome of the slopes in Table Iare due to successive spin filling
of a small impurity quantum dot.
32,50Unlike microfabricated
quantum dots in GaAs, quantum dots in silicon are muchless prone to the overall diamagnetic shift observed on thedifferent addition energies. This shift becomes important whenthe charging energy is comparable to the diamagnetic energy
e
2B2r2
8m∗≈e2
4πεsr,
where r is the overlap of the wave function and the electrode.
The∼10×larger effective mass in silicon renders this effect
negligible in our system. We thus can compare the shiftsof each of the energies directly without having to comparepeak spacings. While one expects the typical Pauli filling,the research in microfabricated quantum dots has shownthat deviations are possible, providing evidence for higherspin states. Successive spin filling should follow ±gμ
Band,
therefore, we only consider slopes in Table Iin which the
absolute values are approximately equal. If we interpret dipsa, b, g, and h in this context, we would have successive spinfilling of ( ↓,↓,↑,↓).
Next, we consider the possibility of Pt atoms forming
clusters of a few atoms so that a single defect can resultin several levels. Such clusters were recently considered inan extension of scanning charge accumulation imaging.
51In
transport, other researchers have discussed the characteristicsof donor molecules
52and explored the effect of interacting
impurities.53–55All of these investigations demonstrate that
a coupled impurity state (containing the same species ofimpurity) in which two atoms are close together results ina large additional charging energy associated with the secondstate. This second state is situated at much higher energy,ranging from 1 to 4 ×the binding energy depending on the
distance between the two impurities. In order to observethe effects of interacting impurity on a resonant level, it is
205415-4FANO RESONANCE IN ELECTRON TRANSPORT THROUGH ... PHYSICAL REVIEW B 83, 205415 (2011)
necessary to consider the Vdsdependence and investigate the
nature of excited states. Alternatively, by modifying a perpen-dicular magnetic field, the overlap between the impurities canbe modified. The previous research that explored interactingimpurities in electron transport
53–55focused on the case where
the impurities were separated by large distances. Essentially,if the impurities are close enough together, they are difficultto distinguish from the single impurity case, especially atV
ds=0 V and in parallel magnetic fields. In the next section,
we discuss how the fact that the quantum interference does notchange in magnetic fields up to 5 T implies that the path lengthis quite small ( /lessmuch32 nm). If the interference originates from
two paths through two different impurities, these impurities arenecessarily situated very close together. We can not distinguishif a given resonance or antiresonance in Fig. 2(b) is due to a
coupled or a single impurity given the data reported here.However, if there are coupled impurities, not all the resonancesin Fig. 2can originate from one coupled system because the
transport window is too small.
Our system is unique compared to previous investigations in
that the device contains two known species of dopants: singleB acceptors and double Pt donors. Transport is close to thevalence band and both atoms can be viewed as binding holes.However, the acceptor site is initially charged negatively andbecomes neutral and the Pt site is initially charged positivelyand an additional positive charge is added. If these two speciesare close enough to be coupled, then the occupation of one levelwill result in the other level moving away from the valenceband (to lower energy). Coupling between two states of anacceptor (double donor) would thus result in a decrease of the
ionization of one of the states when the other level becomesoccupied. In this context, two of the levels in Fig. 2could be
related to such coupled atoms.
In summary, we have seen in this section that the levels in
Fig. 2have several possible origins. They can originate from
Pt or Batoms that are isolated or coupled either with each
other or with an atom of the same species or with O or H. It ispossible that two or three of the levels in Fig. 2belong to the
same coupled atoms. Another possibility is that a small clustercontaining a few impurities forms a very small quantum dotand several of the levels belong to the same cluster.
Without additional spectroscopy experiments, one can not
precisely classify the origin of each resonance in Fig. 2.A s
we have shown by comparing the values in Table Iwith our
earlier work,
18,23,24the identity of a given state, as determined
by the change of its peak position at Vds=0 V as a function of
magnetic field, does not determine whether it will be observedas a resonant dip or a resonant peak. What is clear from thisanalysis is thus that the presence of antiresonances due tosingle atoms, coupled atoms, or a small quantum dot based ona few dopants is a robust effect in these devices.
B. Interpretation of the antiresonances as quantum interference
We now consider how an antiresonance might occur. To
simplify the exposition, we assume for the moment that theresonant path is due to a single atom, although, from ourdiscussion above, this may not be true. What is surprising isthat the dip goes below the apparent background current. One
naively expects that the background tunneling current in thedevice is due to transport along the entire width. Thus, if there
is a destructive interference through a resonant level, it couldnot go below this incoherent background current. However, atlow temperatures in a relatively small Schottky barrier like theone we have here, the depletion region is nonuniform. Near adopant level, direct tunneling is significantly enhanced becausethe barrier height and width are modified by the electrostaticpotential of the charge. As a result, the width of the Schottkybarrier is broken up into regions of high conduction, near aresonant atom, and regions of low conduction, where there areno dopant levels. In order to observe interference effects, whenthe equilibrium Fermi level is near resonance with an atomiclevel, the dominant transport mechanism must be coherent.When there is destructive interference, the path through theatom is blocked and the current can be significantly decreasedbecause one of the few paths through the entire barrier widthhas been blocked off.
Evidence for this effect can be found by looking at the
amount by which the antiresonances dip below the backgroundas a function of |V
g|. When transport through a quantum dot
is strictly determined by the Fano resonance, e.g., there are noother transport paths, the distance from the bottom of the dip tozero can be indicative of the coherence of the transport.
5Here
the dip can only go so low until the conductance from otherpaths through the barrier becomes important. However, as |V
g|
is increased, we expect that direct tunneling through the entirebarrier width will increase and, correspondingly, the differencebetween regions of high conductance and low conductancewill decrease. Thus, the general trend of how much the dipsgo below the background should be inversely proportional to|V
g|. Figure 4shows the percent change of the total current for
each of the antiresonances. Indeed, the general trend is that thenumber and strength of the antiresonances decreases as |V
g|is
increased.
Assuming that electron-electron interactions can be
neglected,56coherent transport through the impurity can be
described by the Landauer-Buttiker formalism1in which a
FIG. 4. For each of the antiresonances in Fig. 2(a) at 4.7 T, we
calculate the percentage the differential conductance dips below the
accompanying shoulder. The frequency and percent change of the
differential conductance decreases as |Vg|increases. This provides
evidence that the background current in the channel is increasingly
dominated by incoherent direct tunneling through the barrier rather
than “hot spots” resulting from resonant levels.
205415-5L. E. CALVET, J. P. SNYDER, AND W. WERNSDORFER PHYSICAL REVIEW B 83, 205415 (2011)
scattering transmission coefficient is used to describe the
current. When quantum interference comes into play, thetypically Lorentzian transmission coefficient can be replacedwith the Fano transmission amplitude describing interferencebetween two paths:
7
T(E,Vds)=tnr|2ε+q|2
4ε2+1, (2)
where tnris the transmission probability of the nonresonant
path (likely to be direct tunneling), ε=E−Eres//Gamma1is the
change in energy near the resonance, and qis the asymmetry
factor related to the cotangent of the phase difference δbetween
the two paths and proportional to the transmission probabilityof resonant tunneling t
rtover that for nonresonant tunneling:
q=cotδ∝trt
tnr/Gamma1.
Forq=0, the lineshape exhibits a resonant dip, for large
qthe Lorentzian is recovered, and, at intermediate values,
asymmetric lineshapes are observed.
The data in Fig. 3have a relatively simple interpretation in
this context. For the resonant tunneling peak in Figs. 3(a) and
3(b), we observe that the resonance is initially enhanced as
|Vds| is increased. This is unusual because typically increasing
|Vds| introduces thermal energy that will broaden the resonance
and not result in a more pronounced resonant tunneling. Here,however, tunneling into the semiconductor channel is stronglysuppressed at zero bias;
25thus, introducing a small bias voltage
effectively enhances all transport into the semiconductorchannel. We note that a similar suppression near |V
ds|=0V
has been described theoretically.57Pure resonant
tunneling can be described by a Breit-Wigner lineshapeT=/Gamma1
MD/Gamma1DS/[(E−Eres)2+(/Gamma1MD/2+/Gamma1DS/2)2], where
/Gamma1x=MD,DS are, respectively, the leak rates from the metal
to the dopant and the dopant to the semiconductor. In anSBMOSFET, the leak rates are determined by the depletionpotential resulting from the metal-semiconductor interface andare thus highly asymmetric. As a result, removing the zero biascurrent suppression unequally changes these two parameters.The leak rate closest to the semiconductor will experiencethe largest change. If the dopant is thus very close to themetal, then there will be an enhancement in the total resonanttunneling component. If the total transmission coefficient,which includes the probability for nonresonant and resonanttunneling through the atom, is such that resonant tunneling ismuch larger than the nonresonant component, increasing thebias voltage will result in a stronger Lorentzian peak.
If, however, the transmission coefficient for resonant
tunneling is smaller or comparable to that of the nonresonantcomponent, then the resonance exhibits a Fano lineshape andthe bias voltage modifies its parameters, as in the data inFig. 3(d). Effectively, by changing the bias voltage, one is
tuning one of the leak rates of the resonant tunneling com-ponent, increasing its importance relative to the nonresonantcomponent. Figure 5(a) shows the fits to the Fano expression
for the resonance in Fig. 3(d). Figure 5(b) plots the fitted
expression for the asymmetry parameter qas a function of bias
voltage. Most interestingly, the lineshape goes from having al-most complete destructive interference ( q=0) to constructive
interference ( q=± 1). Because the leak rates are modified bythe bias voltage and not more directly by adjusting a tunnel
barrier,
7,8an additional effect related to the direction of |Vds|
is observed. In Fig. 5(b), the sign of qreverses from positive
to negative with bias voltage direction. The explanation forthis is that the asymmetry parameter qis a measure of the
phase difference between the direct and resonant tunnelingpaths. Thus, applying a bias voltage in the positive or negativedirection is equivalent to shifting the phase by 180
◦.
One surprising feature of these data is that the lineshape
changes form with bias voltage but not with magnetic field.One possibility is that the path length of the interference issmaller than the corresponding magnetic field length we haveaccessed. In order to change the interference by one completecycle, the magnetic field flux /Phi1=BS, where Sis the surface
area of the path, must be of order
h
e. At 5 T, this implies
a maximum surface area of 8 .27×10−16m2or, assuming a
circular path, a diameter of order 32 nm. The fact that wesee no significant change implies that the interference path islikely to be much smaller than this. This diameter is still muchlarger than the typical ∼2-nm Bohr radius for a shallow dopant
level near the valence band.
58
Finally, we consider briefly the origin of the nonresonant
path. There are several possibilities including (1) tunnelingthrough a nearby dopant(s), (2) direct tunneling due to theproximity of the atom to the metallic electrode, (3) enhanceddirect tunneling through surface states present at the metal-
80
60
40
20 ∂I⎯∂Vds (nS)
-1.915 -1.912
Vg (V)-1.5-1.0-0.50.00.51.0q-factor
-0.4 -0.2 0.0 0.2
Vds (mV)
(a) (b)
FIG. 5. (Color online) (a) We fit the Fano lineshape [Eq. ( 2)] to
dip c in Fig. 3at 4.7 T and 50 mK. The value of the Vdsfrom bottom
to top is 0, ±0.1,±0.2,±0.3, and ±0.4 mV . Note that the curves
are NOT offset. The Vds=0 V curve is plotted in black with the fit
indicated in gray, the positive Vdscurves are plotted dark gray with
fits in light gray and the negative Vdscurves are plotted in light gray
with fits in dark gray. (b) Plot of the asymmetry factor qas a function
of bias voltage Vds.A tVds=0 V , we observe an antiresonance,
corresponding to an asymmetry parameter qclose to 0, and indicative
of destructive interference. For Vds>0 V , we find positive qvalues
and, for Vds<0V ,n e g a t i v e qvalues, as might be expected if the
interference is changed by a 180◦phase shift. As |Vds|is increased,
the resonant tunneling component becomes more important and the
interference becomes increasingly constructive, indicated by q=1.
Note that the standard deviation of the q-parameter fit is indicated
by error bars, which demonstrate the robustness of the changes
with |Vds|.
205415-6FANO RESONANCE IN ELECTRON TRANSPORT THROUGH ... PHYSICAL REVIEW B 83, 205415 (2011)
semiconductor surface, and (4) an additional current path
within the atom due to the resulting physical structure withinthe silicon lattice. The very fact that we observe resonanceswith different magnetic field dependences may indicate thatthe dopants do not all exhibit quantum interference of exactlythe same origin and we, therefore, can not exclude anyof these options without further experimental investigations.We mention that, in the first case, the second atom must bein very close proximity because of the very small path lengthimplied by the magnetic field dependence. Two possible waysto address this issue are to explore the temperature dependenceof the interference
8or how the quantum interference of dopant
levels with different magnetic field dependences changes as thesuperconductivity in the PtSi contacts is suppressed.
59
IV . SUMMARY
We have explored antiresonances in the transport through
dopant atoms located near the metal in a Schottky barrierMOSFET. We find that the lineshapes can be explained in
terms of quantum interference via a Fano lineshape. Weobserved significant changes in the lineshapes with appliedbias voltage that are due to changes in the resonant tunnelingcomponent of the interference. Finally, because the lineshapesexhibit very little change in magnetic fields up to 5 T, weargue that the quantum interference path must be very small.The data show that this system is capable of demonstratinginterference on a very small length scale and thus provide anovel test bed for exploring coherence with a small number ofatoms.
ACKNOWLEDGMENTS
The idea of observing Fano resonances in Schottky barrier
MOSFETS originates from research at Yale University withM. Reed and R. Wheeler. L.E.C. thanks M. Jang, K. Kang, E.Hoffman, and H. Linke for invaluable discussions.
*laurie.calvet@u-psud.fr
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205415-8 |
PhysRevB.76.205104.pdf | Coulomb blockade versus coherence in transport through a double junction
Ursula Schröter *and Elke Scheer
Fachbereich Physik, Universität Konstanz, Universitätsstraße 10, 78457 Konstanz, Germany
/H20849Received 18 April 2007; revised manuscript received 23 July 2007; published 13 November 2007 /H20850
We construct a model describing current transport through a superconducting or normal conducting circuit
consisting of two point contacts in series by extending a Green’s functions technique. In between the contactsis a mesoscopically large and bulklike island. The model can, in principle, handle contacts in all transmissionregimes. Coherent interaction throughout the whole system is included in the form of multiple and multipleAndreev reflections extending over both contacts while accounting for charging effects by a changing electro-static potential of the island. Our calculations show that even though the onsets of certain current contributionsare independent of the island charging energy, Coulomb blockade persists, especially in the normal state.Coulomb staircases can still be visible but get smeared out for particular ratios of the charging energy and thegap in the superconducting state. However, as a general trend, we find that including coherent coupling acrossthe island does not very significantly change the shape of the current-voltage curves compared to the incoher-ent results.
DOI: 10.1103/PhysRevB.76.205104 PACS number /H20849s/H20850: 74.50. /H11001r, 73.23.Hk, 74.45. /H11001c, 72.10. /H11002d
I. INTRODUCTION
For electric circuits with microscopic islands, thus suffi-
ciently small capacitances, the quantization of charge mani-fests itself by the phenomenon of Coulomb blockade.
1Trans-
port through quantum dots in superconducting devices hasextensively been studied focusing also on aspects such aseven-odd occupation,
2,3fractional charge,4and pumping.5
Here, we address bulklike metallic islands with negligible
intrinsic level spacing of the electronic states. Unlike in Ref.6by Kouvenhoven et al. , this is mostly not the case for
quantum dots for which then a single-level picture isappropriate.
5,7–9The contacts, although presenting barriers of
the order of the quantum resistance Rk=h/2e2, are not nec-
essarily tunneling junctions but can as well be point contactsand, for example, be arranged with break junctions.
10These
accommodate few transport channels with varying transmis-sion probabilities. This implies that multiple-reflection cor-rections and, in the superconducting state, multiple-chargetransport processes in the form of Andreev reflections playan important role. The transport properties of setups in whichsuch junctions of arbitrary transmission get combined withislands sensitive to single-electron charging are not yet fullyunderstood, the principal question being whether Coulombblockade suppresses multiple Andreev reflections /H20849MARs /H20850or
vice versa.
8,11A charging-energy threshold on the Josephson
current across a quantum dot has recently been predicted.12
We recently presented a model for calculating current-
voltage /H20849IV/H20850characteristics of a double-junction circuit with
a metal island. In Ref. 13, we assumed that coherence was
not maintained across the island from one junction to theother, which should be correct, if the two contacts are farapart. One could, however, place the junctions close to eachother such that quantum coherence is not lost between them.Here, we show how our method can be altered and extendedto this situation. The model from Ref. 13included for each
junction in themselves coherent MAR processes togetherwith changes of the island charge, but we shall refer to it forshort as the “incoherent” model. The model presented nowand keeping coherence between the contacts is called “coher-
ent.”
The setup considered is shown in Fig. 1/H20849a/H20850. An island is
linked to a left and a right electrode by quantum-point con-tacts. Each contact has a capacitance C
i, and the potential of
the neutral island is determined by the divider, made up ofthese two, when a transfer voltage Vis applied between left
and right. The island potential may further be continuouslyaltered, however, by inducing charge via a gate capacitance.At variance to Ref. 13, we here only consider junctions with
a single transport channel each. Our model is made for allthree regions L,I, and Rof the same material, which has gap
(a)
(b)L RI
GI
VUCgC C 1 2 /c81 /c811 2
FIG. 1. /H20849a/H20850Setup: island connected to a left and a right lead
between which a transport voltage Vis applied. Each junction is
characterized by a capacitance Ciand a transmission /H9258i. The island
can further be tuned via a gate capacitance Cgwith voltage U. The
net current Iis the quantity to deduce. /H20849b/H20850Multiple-reflection pro-
cess extending over both junctions. Effectively transported chargesacross each junction are counted by pandk.PHYSICAL REVIEW B 76, 205104 /H208492007 /H20850
1098-0121/2007/76 /H2084920/H20850/205104 /H208496/H20850 ©2007 The American Physical Society 205104-1parameter /H9004for the superconducting state. The latter works
with a BCS density of states, which collapses to a constantdensity of states /H20849DOS /H20850with occupied electron states below
and empty ones above the Fermi energy in the normal con-ducting limit /H9004→0. No spin polarization
7,14,15is considered,
and the system is treated at zero temperature. Furthermore,the island is sufficiently large such that a small number ofexcess electrons or holes do not influence the form of theDOS. The charge determines the Fermi level with respect tothe lead potentials. An example of a multiple-reflection pro-cess extending over both junctions is depicted in Fig. 1/H20849b/H20850.
We include multiple Andeev reflections but neglect Cooper-pair tunneling.
16Herewith, the assumptions for the model are
concluded.
We consider our model as realistic for possible realiza-
tions of mesoscopic systems. However, the interest of thepresent paper is not a direct comparison to experiment but ademonstration of conceptual feasibility. The combination ofGreen’s functions and rate equations allows us to treat coher-ent interaction through the double junctions with bulky is-land for the low,
17intermediate, and high-transmission re-
gimes. In Sec. II, we briefly present the extensions of theformalism needed for the coherent case. A selection of ex-ample calculations of current-voltage characteristics is pre-sented and discussed in Sec. III before summarizing the re-sults in Sec. IV .
II. GREEN’S FUNCTIONS AND RATE EQUATIONS
Maintaining coherence in transport across the island, there
is only one Hamiltonian containing all three reservoirs andboth couplings over junctions,
H=H
L+HI+HR+/H9268L↔I+/H9268I↔R. /H208492.1/H20850
For the transfer Green’s function T, which is a 3 /H110033 matrix
in site space, we have to solve the Dyson equation,
T=/H20849/H92681+/H92682/H20850+/H20849/H92681+/H92682/H20850gT. /H208492.2/H20850
By/H92681and/H92682, we denote couplings across the left and right
junctions, respectively. gis the Green’s function of the un-
coupled sites L,I, and R.Tis transformed to Fourier space
through
Tn1n2/H20849/H9270,/H9270/H11032/H20850=/H20858
k+p=n2−n1/H20885d/H9275e−i/H9275/H9270Tkmpn1n1+k+p/H20849/H9275/H20850
/H11003e−ikA/H9270e−im/H20849n1,k,p/H20850B/H9270e−ipC/H9270ei/H9275/H9270/H11032, /H208492.3/H20850
with AandCbeing the potential differences of the neutral
island to the leads, Bproportional to the charging energy
Ec=e2//H20849C1+C2+Cg/H20850, and k,mandpare integers. Tmediates
between the island state with initial and final excess charges
n2andn1. By building two couplings into the Green’s func-
tion successively,18and following Refs. 13and19to con-
struct Tthrough recursions in k and p for the coherent
model, Eq. /H208492.2/H20850can be solved in two steps. Regarding solely
coupling across the left junction, in a first step, an interme-diate function T
1fulfillingT1=/H92681+/H92681gT1/H208492.4/H20850
is calculated, which then enters the second-step equation,
T=T1+/H92682+T1g/H92682+/H92682gT+T1g/H92682gT, /H208492.5/H20850
where now only the coupling across the right junction gets
newly introduced. Equation /H208492.2/H20850can alternatively be solved
by direct inversion for not too high maximum values of kand
p,
T=/H208511− /H20849/H92681+/H92682/H20850g/H20852−1/H20849/H92681+/H92682/H20850. /H208492.6/H20850
A special procedure to evaluate inverse matrices of the form
/H208491−M/H20850−1from the electronic supplement of Ref. 18has been
adapted to this problem.
The formula giving the rate by which the island charge is
changed between nandn+1 by a transfer through a junction
/H20849channel /H20850expressed with G+−remains the same as in Ref. 13,
however, with G’s now 3 /H110033 matrices in site space, more
terms occur when rewriting it using T. For the right junction,
as an example, we get
2R e/H20858
n/H11032/H20853/H20849/H9268RIn+1n/H11032GIR,+−n/H11032n+1/H20850hh+/H20849GRI,+−nn/H11032/H9268IRn/H11032n/H20850ee/H20854
=2R e /H20853/H20849TRIrgII+−TIRagRRa/H20850hhn+1+/H20849gIIrTIRrgRR+−TRIa/H20850hhn
+/H20849TRLrgLL+−TLRagRRa/H20850hhn+1−/H20849TLRrgRR+−TRLagLLa/H20850hhn+1
+/H20849gRRrTRIrgII+−TIRa/H20850een+/H20849TIRrgRR+−TRIagIIa/H20850een+1
+/H20849gRRrTRLrgLL+−TLRa/H20850een−/H20849gLLrTLRrgRR+−TRLa/H20850een/H20854, /H208492.7/H20850
where norn+1 denotes the outer charge index of a product.
TheTLR/RLfunctions represent an effective coupling between
not directly connected sites.20As all multiple /H20849Andreev /H20850re-
flection processes are broken down into single-charge hop-pings entering the rate terms /H20851Eq. /H208492.7/H20850/H20852, a rate matrix Ris
constructed solely from /H9004n=1 island-charge changes, adding
processes from both junctions. With P/H6023denoting the vector
composed of Pn, the probabilities for the island to have n
excess charges on it, stationary state requires
d
dtP/H6023=R·P/H6023=0 . /H208492.8/H20850
With the Pnobtained from Eq. /H208492.8/H20850, the current /H20849dc/H20850is easily
calculated using charge transport rates for any single junc-tion.
III. RESULTS
In Fig. 2, we show an IVfor the normal conducting case
/H20849with finite Ec/H20850for medium transmissions of the junctions.
The respective result without coherent interaction betweenboth junctions
13is also plotted. In the incoherent model, cur-
rent onset is at eV=2Ec, and further jumps in the derivative
are found at eV=/H208494n+2/H20850Ec— for the symmetric case with
equal junction capacitances. In the coherent model, we ob-
tain nonzero current at lower voltages. However, finite cur-rent at infinitely small voltage merely occurs in the limit ofvanishing E
c, in accordance with the picture of an Ohmic
resistance. Otherwise, a Coulomb blockade regime persists.URSULA SCHRÖTER AND ELKE SCHEER PHYSICAL REVIEW B 76, 205104 /H208492007 /H20850
205104-2The lowest possible voltage threshold for current onset is at
eV=2
3Ecfor maximally open channels /H92581=/H92582=1.0. We re-
gard setups with the same junction transmission amplitudest
1andt2in the incoherent and the coherent model.21Depend-
ing on /H92581and/H92582, the coherent model can yield a greater
current than the incoherent one, but it is also possible for thecoherent curve to drop below the incoherent toward larger
voltages. Both models fall together in the limit of two verysmall transmissions as expected and required for consistency.The current as a function of both the transport and the gatevoltage /H20851Fig. 3/H20849a/H20850/H20852in the coherent model exhibits Coulomb
diamonds like in orthodox theory
1or in the incoherent
model. In the UVplane, the lines marking their edges are not
the same though /H20851Fig. 3/H20849b/H20850/H20852, and in the coherent case, the
slopes are no longer determined by the capacitances alonebut depend on the junction transmissions as well.
3One will,
nevertheless, not be able to tell whether transport is incoher-ent or coherent from the shape of I/H20849V/H20850orI/H20849U/H20850. The fact that
the edges’ first crossing point /H20851see Fig. 3/H20849b/H20850/H20852is not found at
U=0 for the coherent case is of no help, as this would hap-
pen for C
1/HS11005C2in the incoherent case, too. With slightly
different parameters C1,C2,t1, and t2— usually not inde-
pendently known in experiment — it is possible to deducethe same IVfrom both models.
In Figs. 4–6, we show example calculations for the super-
conducting state. Here, also IVs containing processes of the
kind from Fig. 1/H20849b/H20850are compared to the case where coherent
interaction between the two junctions is switched off. In Fig.4, for medium transmission of both junctions and E
cgreater
than/H9004, we see that with full coherence, the current onset is
at slightly lower voltage than eV=2Ec, where it was in the
incoherent model. At higher voltage, we obtain a somewhatlarger current. However, there is no prominent difference intheIV-curves from both models.
In order to provide evidence for direct lead-to-lead trans-
port in the coherent case, we therefore investigate the artifi-cial situation that Andreev reflection /H20849AR /H20850is turned off de-
spite keeping the density of states with the gap from thesuperconducting state. This is easily done in our formalism.With only electron or hole transfers, current onset would beateV=4/H9004/H20849for the symmetric case C
1=C2/H20850in the incoherent
model, whereas the coherent case would give nonvanishingcurrent starting already at eV=2/H9004. The independence of this
contribution of the island charge can be proven by repeatingthe calculation without AR for different gate voltages U. The
onset is always found at eV=2/H9004as long as Udoes not enable
FIG. 2. Normal-state IVcurves for symmetric junction capaci-
tances C1=C2, no gate voltage U=0, and transmissions /H92581=0.7 and
/H92582=0.3. The light gray curve is the result of the model from Ref. 13.
(a)
(b)
FIG. 3. /H20849a/H20850Current Ifrom coherent model drawn as a function
of both transport voltage Vand gate voltage U./H20849b/H20850Lines marking
the edges of the Coulomb diamonds /H20849jumps in dI/dVordI/dU/H20850in
the coherent /H20849black /H20850and the incoherent /H20849gray /H20850model. Parameters
for both /H20849a/H20850and /H20849b/H20850are the same as in Fig. 2.
FIG. 4. Calculated IVfor equal transmissions, C1=C2,U=0 for
Ec/H11022/H9004. Current is zero in calculations with Andreev reflection at
low voltages where no /H20849big/H20850symbols are drawn.COULOMB BLOCKADE VERSUS COHERENCE IN … PHYSICAL REVIEW B 76, 205104 /H208492007 /H20850
205104-3sequential transport at lower voltage. In the full calculation
with AR, except for a slightly shifted onset and a rather smallenhancement in current, there is not much difference be-tween the IV-curves from the coherent and the incoherent
models for any gate voltage U. Back-transport of charges
from the island by MAR across a single junction hinders netcurrent from coherent lead-to-lead coupling.For both junctions of medium transmission and E
cless
than/H9004/H20851Fig. 5/H20849a/H20850/H20852, even the incoherent model produced little
current below eV=2/H9004/H20849forU=0/H20850due to MAR in each single
junction. U/HS110050, of course, shifts the marked current rise
from eV=2/H9004— due to first-order AR here with the double
junction — to lower voltage. The coherent model for all gatevoltages shows a distinct current rise at smaller voltages anda greater current than the incoherent one in the subgap rangebelow roughly eV=4/H9004. Plotting Ifrom the incoherent model
as a function of VandU, we clearly recognize a Coulomb
diamond pattern /H20851Fig. 5/H20849b/H20850/H20852. The steps get smoothed out, but
remain visible, in the coherent model /H20851Fig. 5/H20849c/H20850/H20852. Part of the
current is now carried by coherent transport from lead tolead, for which voltage thresholds do not depend on the is-land charging energy.
In Fig. 6, we show IV-curves, again for equal medium
transmission of both junctions, but still lower charging en-ergy, for three different gate voltages. Here, the increase incurrent of the coherent as compared to the incoherent modelis such that the step structure gets lost. Curves without co-herent coupling between junctions /H20849open symbols /H20850have
some visible steps in the range up to about eV=4/H9004, at least
for gate voltages U=0 and UC
g=e/2, where crossings of
edges of Coulomb diamonds and their subedges from ARoccur. For small E
c, the step structure rather quickly becomes
too weak to be seen toward higher V, even in the incoherent
(a)
(b)
(c)
FIG. 5. Calculated IVfor the same transmissions, as in Fig. 4,
also equals capacitances, but Ec/H11021/H9004, slightly below /H9004/2. /H20849a/H20850IV
curves for Uat the beginning and middle of a period in gate voltage
from both models. /H20849b/H20850Ias a function of VandUfor half a period
in gate voltage from incoherent model. /H20849c/H20850Same as /H20849b/H20850forcoherent
model.
(a)
(b) (c)
FIG. 6. IVfor three different Ufor equal medium transmission
of both junctions, equal capacitances, and Ecabout one-third of /H9004.
All curves are put together in /H20849a/H20850to show where the ones drawn
with full symbols nearly fall together and exceed those drawn withopen symbols. The same curves from the incoherent and the coher-ent model are also shown separately and shifted /H208490.2 units per
curve, all have I=0 for V=0/H20850for different Uin/H20849b/H20850and /H20849c/H20850to see
their individual shapes.URSULA SCHRÖTER AND ELKE SCHEER PHYSICAL REVIEW B 76, 205104 /H208492007 /H20850
205104-4case. The important point, however, is that, in the coherent
model /H20849full symbols /H20850, all IV-curves for different gate volt-
ages almost coincide beginning at a voltage even below eV
=2/H9004. This indicates that a transfer mechanism dominates,
which does not depend on the charging state of the island,namely, direct coherent lead-to-lead transport. The curvesonly split for very small V.
Current contributions from coherent transport through
both junctions, which add to incoherent processes, becomenegligible in the subgap regime if one junction has a consid-erably smaller transmission than the other because the lower-transmission junction then presents a bottleneck and theIV-curves from both models differ less from one another than
in the shown examples. Regardless of the ratio E
c//H9004or the
transmissions, we do not find signs of crossed AR in the IV
characteristics. By expanding terms in Eq. /H208492.7/H20850into series of
powers of /H9268, one can show that counteracting multiple-
reflection corrections are stronger.
IV. SUMMARY
A Green’s function formalism known to quantitatively de-
scribe transport through a single quantum-point contact, es-pecially with medium or high-transmission channels, in com-bination with rate equations has been extended to model twosuch junctions in series. The enclosed island is treated asbulk material, however, sensitive to single-electron charging.Our model further includes a gate electrode. The island po-tential changing with time is not a principal hindrance forkeeping quantum coherence over the whole system, which isthe premise set differently here from our previous work.
13
The two couplings corresponding to the two junctions are ofequal importance and built into the Green’s functions with-out approximation.
From our model calculations applied to the normal con-
ducting state, we find that allowing to maintain coherence intransport across the island does not remove the Coulombblockade regime, although it reduces its voltage range. Char-
acteristic positions in the current-voltage curves like the on-set and sudden changes in slope no longer represent mul-tiples of the island charging energy like in orthodox theory,but depend in a more complex manner also on the junctiontransmissions.
In the superconducting state with coherent coupling be-
tween junctions, it can be proven that transport processesexist, even without Andreev reflection, which require the ap-plied voltage to overcome only once the gap 2 /H9004despite the
bridging of the two junctions. Indications for such directlead-to-lead transport are the smearing out of the Coulombsteps and the independence of current-voltage characteristicsof the gate voltage. When assuming bulk properties for theisland as for the leads, sequential, incoherent transport pro-cesses are always competing. This has for consequence thatcoherence can only cause additional current above the inco-herent model in the subgap regime if the island chargingenergy E
cis less than /H9004. Subgap regime means voltages
below eV=4/H9004for the double junction. A larger ratio Ec//H9004
seems to favor a current increase at higher voltages. Never-theless, apart from the mentioned observations, we have tonote that, compared to the case without coherent couplingacross the island, full coherence does not introduce otherprominent features into the current-voltage characteristics,the qualitative shape of which remains very similar. This willrender the distinction between incoherent and coherent trans-port in an eventual experiment extremely difficult.
As an outlook, we remark that for further improving our
model, one should include Cooper-pair tunneling, spin polar-ization, the possibility to have multichannel junctions in thecoherent case, or the two contacts being only partiallycoupled coherently.
ACKNOWLEDGMENTS
We thank the Deutsche Forschungsgemeinschaft and the
Landesstiftung Baden-Württemberg for financial support.
*ursula.schroeter@uni-konstanz.de
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New York, 1992 /H20850.
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/H208491990 /H20850.21We prefer to indicate /H9258i=4ti2
/H208491+ti2/H208502as equivalent parameters, although
the explicit calculation of, for example, /H20841TRI/H20841=t2
1+t12+t22in the nor-
mal state in the coherent case indicates a different conversion to
renormalized transmission probabilities.URSULA SCHRÖTER AND ELKE SCHEER PHYSICAL REVIEW B 76, 205104 /H208492007 /H20850
205104-6 |
PhysRevB.46.7407.pdf | PHYSICAL REVIEW B VOLUME 46,NUMBER 12 15SEPTEMBER 1992-II
Spectroscopic signatures ofphasetransitions inacharge-density-wave system: 1T-TaSz
B.Dardel,M.Grioni,D.Malterre, P.Weibel, andY.Baer
Institut dePhysique, Uniuersite deNeuch&tel, CH-2000 Neuch&tel, Switzerland
F.Levy
Laboratoire dePhysique Appliquee, EcolePolytechnique Federale, CH-1015 Lausanne, Switzerland
(Received 7April1992)
Photoelectron spectroscopy withhigh-energy resolution hasbeenutilizedtoinvestigate thecharge-
density wave(CDW) in1T-TaSz between 20and360K.Constant binding-energy curvesrevealdiscon-
tinuities inthetemperature dependence ofthephotoelectron spectral functionp(E),anddemonstrate
thatsuddenmodifications oftheelectronic structure, namely, inthevicinityoftheFermilevelEF,mark
thefirst-order CDWtransitions. Inthecommensurate phase,below180K,p(EF)reflectstheformation
ofacorrelation pseudogap. Ondisordered surfaces, however, thelong-range coherence typicalofthe
commensurate phaseislost,andanormalmetallic behavior isrecovered.
I.INTRODUCTION
1T-TaSz hasattracted muchinterest becauseofpecu-
liarphysical properties, unique amongthetransition-
metaldichalcogenides.'Itslayeredstructure consists
ofstrongly boundS-Ta-Splanarbuilding layerscoupled
byweakerforces.Asaresult,severalmacroscopic prop-
erties,liketheelectrical resistivity, areanisotropic and
1T-TaSz isgenerally considered asaquasi-two-
dimensional (2D)material. Themarked2Dcharacter is
exemplified bythecalculated electronic structure and
ithasbeenconfirmed byangle-resolved photoemission ex-
periments8-12Suitable nestingconditions ofthequasi-
2DFermisurfacefavortheappearance (at-550K)ofa
periodic latticedistortion withacomplex charge-density
wave(CDW}."'TheCDWisincommensurate (I)
above-350K,andcommensurate (C)withperiodicity
&13XV'13below180K,whilebetween 350and180K
(between-230and350Kuponheating), itisonthe
average incommensurate, orquasicommensurate (QC},
withcommensurate domains arranged inahexagonal lat-
ticeandseparated bydiscommensurations.
Unlikeotherlayered materials, theproperties of1T-
TaSzpresent anunusual andstrongtemperature depen-
dence.Theelectrical resistivity, forinstance, exhibits
sharpstepsattheI-QCandQC-Ctransitions, andanun-
bounded risebelow-60K,'suggesting largechanges
intheelectronic densityofstates(DOS).Thisobserva-
tioncontrasts withtheusualassumption thatCDW-
relatedeffectsshouldbesmallina2Dsystem, sincea
Peierlsgapcanonlyappearoverlimitedportionsofthe
Fermisurface. Additional mechanisms, actingwithor
besidestheCDW,havetherefore beeninvoked. Low-
temperature resistivity data,suggestive ofvariable range
hopping conduction,''haveprompted amodelpredict-
ingtheoccurrence ofelectron (A.nderson} localization in
therandom fieldcreated byimpurities ordefects. Faze-
kasandTosatti, ontheotherhand,havestressed the
importance oftheelectronic correlations. Theirmodel
maintains thattheQC-Ctransition isaccompanied byaMotttransition, whichcausesasuddenreduction ofthe
electron densityofstatesattheFermilevelandaccounts
fortheobserved resistivity jump.
Photoemission spectroscopy (PES}andinversephoto-
emission spectroscopy (IPES)measurements haveshown
thatthewholevalence bandof1T-TaSz isaffected bythe
CDWandthatcleardifferences intheelectronic density
ofstatescharacterize thevariousCDWphases.'
Theseresultsgenerally support themodelofFazekas and
Tosatti. However, surprisingly, estimates oftheenergy
gap(150—200meV}basedonspectroscopic dataaretwo
ordersofmagnitudes largerthanvaluesdeduced from
transport measurements.'Recently, wehaveshown
thatthisinconsistency actuallyrejectsthedifferent sensi-
tivityofthevarioustechniques tothetwodistinct energy
scalesofthismaterial. PESandIPESdata,infact,
directly display thecharacteristic energyoftheMott
transition, theCoulomb correlation energyU-200 meV,
butamuchsmaller energyscale,oftheorderofafew
meVcontrols thelow-temperature properties ofthema-
terial.Thissmallenergyscale,whichconventional spec-
troscopic measurements cannot reveal, emerges from
measurements performed withstate-of-the-art energy
resolution. Inthispaperwepresenttheresultsofapho-
toemission investigation of1T-TaSz combining high-
energyresolution andanaccurate temperature control
overawiderange(20—370K),andshowthatwiththese
jointcapabilities itispossible toidentify clearfinger-
printsofthestructural phasetransitions.
II.EXPERIMENT
Singlecrystalline samplesof1T-TaSz intheformof
platelets ofapproximately 5X5X0.5mmhavebeen
prepared fromtheelements byreversible chemical reac-
tionwithiodineasatransport agent,asdescribed else-
where. Theelectrical resistance, measured betweenRT
and20Kwithastandard fourpoint technique, presented
sharpstepsat180Koncooling andat230Konwarm-
ing.
467407 1992TheAmerican Physical Society
46
byarapiddeerecreaseofint
sd
issionattheFer
a.hewhole 1ermilevelisv
eow-temperatu
ngangular deurespectrum
p (oexhib-
nsverysmallat ~remainsown)utthein-
sareconsistentatallangles.T n at .hesere-
volutionofp cion
v'ani-
ll),derthfk
p"kcentered'1
elowerHubat80meVis'
bdbbdNj
ngesareobservomaor
eemperature islobt 11t1
ion.sispointinaome
inasubsequent sec-Thesammples,mounte
1 d'tl
Rh-Fresistance couldb
anaccurac f
pepared'situ aceswerereyo+1'.Clean
hbid tp erthan
picallyclean spectrosconitionsthe
erproloneoveraeri
owestterneroteresidua
pature(-20K
evaence-band spectra was
III.RESULTS ANDDISCUSSIONDARDEL, GRIONONI,MALTERRE ,WEIBEL, BAER,ANDLEVY
Near-normal emission
15Vofth 1
bt 20 reasing ternerao-TaSz,meaP
h
oftheieincommensu(356
bandcpK)isre
lliaicedgewhosewidh'
rmifunction atthatte
100,300,d800
F'grbind-
ure,andat230Kth
kB1230Kthe
romthefirst
p
westpeakisaccompaniedA.Thein'ncommensurate- uae-quasicommensuratae-u uratetransition
AlhoughFi.1
dedft}1PESig.1maysue
transitions InF
transition at35g
suredonboth'dpare
siesoftheI-
k'd00
a15%reduction ofth
smalld"'h'1 p ower
reecorrelatedraureeffectsand
icethatthiscorrelationtransition. We
b ,aa6xedbinding enerrgy,
rI)
~M
tI)
0(K)
20
130
224
230
236
246
254
292
I
1.51.00.50=E
ingEnergy (eV)356
FIG.1.ear-normear-normal emission
of1Spotol
gp
commensaincreasin
an356K(sI
0.5 1.0
BindinO=EF
indingEnerg (eV1.5
FIG.2.y)
PESspectrar
einsetshowstemperatureWtransition ternera
auredependence fhyui}tptoemission
sponingtotheepeakinthe356-K.iningenergyof180meV,
-Kspectrum.
SPECTROSCOPIC SIGNATURES OFPHASETRANSITIONS IN...
B.Thequasicommensurate-commensurate transition
Thelargeanomaly observed intheelectrical resistivity
attheQC-Ctransition suggests important modifications
oftheelectronic DOS,whichshouldmanifest themselves
inthephotoemission spectra. Thisisconfirmed bythe
largechanges wehaveobserved intemperature-
dependent scans.InFig.3(a)wepresent, inthetempera-
turerange150—300K,aTCEcurvecollected atabind-
ingenergyof180meV(corresponding tothemainpeak
inthelow-temperature spectra) duringacomplete cool-
ingandwarming cycle300~20~300 K.Weobservea
100%riseofintensity at187Kuponcoolingandacorre-
sponding intensity dropat230Kuponwarming. The
transition at230Kisnotassharpastheoneat187K,
andthemainstepisfollowed byasecondonearound245
10-
e5-b) =OmeV
~Wc0-
g20-0
o15
0
10-
I
150I
200I
250I
300
Temperature (K)
FIG.3.Temperature-dependent constant-energy (TCE)pho-
toemission intensity curves,collected duringacomplete cooling
(opensymbols) andwarming (solidsymbols) cycle,between
roomtemperature and20K:(a)atabinding energyof180meV
and(b)attheFermilevel.Theinsetshowstheelectrical resis-
tivityofIT-TaS& (arbitrary units)inthesametemperature
range.thetemperature dependence ofthephotoelectron signal.
TheinsetofFig.2reproduces suchatemperature-
dependent-constant-energy (TCE)curvecollected, around
thetransition temperature, atthepeak'sbinding energy
(180meV)ofthe356-Kspectrum. Theclearstepindi-
catesthat,inlinewiththefirst-order character oftheI-
QCtransition, thephotoelectron intensity presents a
discontinuity atthecriticaltemperature. Asimilarde-
creaseofthephotoemission intensity atEFsuggests a
reduction oftheelectronic DOSwhichcanbecorrelated
withthestepwise increase oftheelectrical resistivity.
FromFig.2wecanthenconclude thattheI-QCCD&
transition bringsaboutasudden rearrangement ofthe
electronic statesof1T-TaSz overabout 1eV.K.Following thissecondstep,the"warm-up" curvefalls
belowthe"cool-down" curve,untilanewstepat285K
equalsagainthetwocurves.
ThecurveofFig.3(a)reproduces, withtheexception
ofthedoublestepinthe"warm-up" curves,thecharac-
teristictemperature dependence ofbulkparameters. Pre-
viousauthors hadsuggested, onthebasisofspectro-
scopicresults, thatthebacktransformation fromthe
correlated low-temperature phasetothenormalmetallic
stateshouldbeidentified withastructural transition
occurring at285KfromtheT(triclinic} phasetothe
QCphase.Thisconjecture wasratherdisturbing because
itimpliedthatmetallic conductivity wasrecovered-50
KbelowtheMotttransition, whenelectrons arestilllo-
calized. TheresultsofFig.3demonstrate unambiguously
thatthecriticaltemperature coincides withthatdeter-
minedfromtheelectricresistivity. Thestructural transi-
tionat285K,whichmainlyconcerns thecaxis(perpen-
diculartotheTaplanes}, isrevealed inFig.3byasmall
jumpinthewarm-up curve.ThefactthattheTCEbears
asignature ofthistransition actually demonstrates that
thesensitivity ofPESiswelladaptedtostudytheclose
relationship between structural phasetransitions andsub-
tlechanges intheelectronic structure in1T-TaSz.
Ontheotherhand,thepresence oftwostepsinthe
warm-up curvesofFig.3,butnotintheresistivity, calls
foranexplanation. Thelarge,sample-dependent hys-
teresis,indicative oftheinfiuence ofdefectsontheQC-C
transition, andthehighsurface sensitivity (10—20A)of
PES,suggestthatourobservations mightreflectdifferent
pinning mechanisms actinginthebulkandatthesurface.
Inordertoverifywhether thetwostepscouldbeas-
signedtodistinct surface andbulktransitions, wehave
repeated themeasurement withadifferent photonenergy
(40.8eV)andtherefore aslightly different probing depth,
butthenewcurvesexactlyreproduced theresultsofFig.
3.Wemusttherefore conclude thatbothtransitions
occuratthesurface. Intheabsenceofamoresystematic
study,wecanonlyspeculate thatcleavage defectscould
actaspinning centersfortheCDW.Acleaved surface
willinevitably presentbothregions withalowdensityof
defects,insuScient toopposethebulk-driven transition,
andregions wherethedensityofdefectsmightbelarge
enoughtolocallyretardthetransition totheCphase.
ThecurvesofFig.3shouldtherefore beregarded asac-
counting foradistribution ofsuchsurfacedomains, and
theobservation oftwosharpsteps,ratherthanabroad,
continuous transition, wouldthensuggestthatthesesur-
facedefectsareonlyeffective aboveawell-defined thresh-
olddensity. Acombined useofPESandofasurface-
sensitive structural technique likescanning tunneling mi-
croscopy (STM)shouldbeextremely valuable toelucidate
thispoint.
Thetemperature dependence ofthephotoemission sig-
nalattheFermilevel,showninFig.3(b),isverysimilar
tothatofthemainpeak[Fig.3(a)],withsharpedgesat
187Koncoolingandstepsat230andat245Konwarm-
ing.However, therelative intensity variations areabout
tentimesaslarge,andofopposite sign;moreover, the
transition at285Kisnotvisible. Opposite temperature
dependences ofthespectral function atEzandat180
7410 DARDEL, GRIONI, MALTERRE, WEIBEL, BAER,ANDLEVY
M
~~
QJ
0
~~lf)I
130
186
188
190.2eV
T(K)
I I
0,20.10=EF -0.1
Binding Energy (eV)
FIG.4.PESspectraofasmallenergyregionaroundtheFer-
milevelattheQC-CCDWtransition uponcooling. Thetransi-
tionischaracterized byasuddenlossofintensity atEFandby
thegrowthofapeakatabinding energyof180meV.
C.Thelow-temperature correlated statemeVwerealready suggested bytheanalysisofFig.1,
whichindicates atransferofspectral weightfromthe
Fermileveltothemainpeak.Itisquiteinteresting to
observe herethatthistransfer abruptly takesplaceatthe
QC-Ctransition temperature. Thesharp90%intensity
dropat187K,andsimilarobservations madeatdi6'erent
emission angles,markacollapseoftheFermisurfaceex-
plaining therelatedtenfoldincrease inresistivity. These
rapidmodifications canbewellappreciated fromtheraw
spectraofFig.4.Againitmustbestressed thatsincethe
instrumental broadening issmaller thantheintrinsic
thermal broadening, thesespectraprovide afaithful im-
ageofthe(temperature-dependent) spectral function.beendiscussed inarecentpaper. Thephotoemission
intensity atE~,p(EF),decreases linearly withdecreasing
temperature, butarealgapneveropens,evenattempera-
turesmuchlowerthantheresistivity minimum (-60K).
Thisobservation iscrucialtounderstanding thephysical
properties of1T-TaS2, because itindicates thattheFermi
surface doesnotcompletely disappear atthetransition
temperature andthattheQC-Ctransition mustbere-
gardedasagaplessMotttransition, apossibility already
envisaged byFazekas andTosatti. 1T-TaS2 wouldthen
exhibitaweakmetallic character evenatlowtempera-
tureifadisorder-driven localization, whichisboundto
occurinthedeeppseudogap formed bytheoverlapping
tailsoftheHubbard subbands,'didnotprevent metal-
licconduction. Thelow-temperature electrical conduc-
tivityistherefore controlled bytheenergyseparation be-
tweenthemobility edgeandtheFermilevel.Aroughes-
timate, basedonthetemperature oftheresistivity
minimum, yields-5rneVforthischaracteristic ener-
gy
Themodifications ofthePESspectrum intheCphase
arenotlimitedtothecrucialregionaroundtheFermi
level.Changes inthevalence-band emission overmore
than1.5eVcanbeobserved inFig.5,wherewecompare
spectra takenat165and20K.Allthespectral struc-
turesappearsharper andmoreintense inthe20-Kspec-
trum(solidsymbols), whiletheintegrated intensity
remains, withintheexperimental accuracy, constant. In
particular, theprogressive contraction ofthemainpeak
arounditscenteraccounts forthelow-temperature be-
havioroftheTCEsofFig.3.Itistempting tointerpret
themeasured contraction asthespectroscopic conse-
quenceofanincreasing CDWamplitude. Thegrowthof
theCDWbelowtheQC-Ctransition, suggested bythe
temperature dependence oftheHallcoefticient andby
x-ray-photoelectron-spectroscopy dataontheTa4fcore
levels,''shouldinfactleadtoafurtherlocalization of
theconduction electron wavefunctions withinthe13-
ThesuddencollapseoftheFermisurface implied by
theTCEsofFig.3(b)isconsistent withtheprediction,
byFazekas andTosatti, ofaMotttransition taking
placeattheQC-Ccriticaltemperature. Thistransition
corresponds tothelocalization ofelectrons fromanar-
rowbandstraddling theFermilevelintomolecularlike
orbitalsofthestar-shaped 13-atom clusters thatconsti-
tutethefundamental unitsofthedistorted structure.
Theseparation between thecenterofmassoftheoccu-
piedlowerandtheunoccupied upperHubbard subbands,
whichyieldsagoodestimateofthestrengthoftheon-site
Coulomb correlation U,canbedetermined fromPES
(e.g.,Fig.l)andIPES(Ref.24)spectratobeapproxi-
mately200meV.Thiscircumstance hasledanumberof
spectroscopists toconclude that1T-TaS2, inthecom-
mensurate phase,isasemiconductor withagapoftheor-
derof200meV,incontradiction withtheresistivity mea-
surements. TheresultsofFigs.3and4,obtained with
high-energy resolution, show,onthecontrary, thatthe
densityofstatesattheFermilevelissmallbutfinitewell
beyondtheQC-Ctransition temperature. Thispointhas~W
QJ
0
~M
~TE
C
(D001T-TaS2
~T=T=1
I
1.5I I I
1.00.5O=EF
Binding Energy(eV)
FIG.5.Valence-band PESspectraof1T-TaS~ intheCCDW
phase.Thespectral features aresharper andmoreintenseinthe
lowertemperature {20-K}spectrum.
SPECTROSCOPIC SIGNATURES OFPHASETRANSITIONS IN... 7411
atomclusters. Theprogressive development ofamolecu-
larlikesituation wouldnecessarily resultinsharperspec-
tralfeatures. However, additional temperature-
dependent sourcesofbroadening thatcannotbeeasily
quantified, liketheinfluence ofelectron-phonon scatter-
ingonthe(k-dependent) spectral function,'andmore
generaltemperature effectsintheelectronic structure,
cannotbeexcluded.
D.TheefFectofsurfacedisorderCh
~A
QP
0
~HNth
Asalastpoint,weconsider theinfluence ofsurfacedis-
orderontheevolution ofthecharge-density wave.Dis-
order,asadrivingforceforlocalization, hasbeenregard-
edinthepastastheprimary causeofthediverging low-
temperature electrical resistivity in1T-TaSz. Subsequent
measurements onirradiated samples, wherethedensity
ofdefects, andtherefore theamountofdisorder, couldbe
accurately controlled, have,however, demonstrated that
addeddisorder reducesthelow-temperature resistivity by
suppressing thetransition tothecommensurate phase.
Thesuppression oftheQC-Ctransition canbeunder-
stoodastheconsequence oflocalpinningoftheCDW
phasebydefects, whichprevents thedevelopment ofa
coherent low-temperature groundstate.Fromtheprevi-
ousdiscussion, wecanexpect thedisorder-driven
suppression ofthecommensurate phase,andtheappear-
anceofmetallic conduction, tobeaccompanied byvisible
changes intheelectronic DOS,andtherefore inthePES
spectral function.
Inanattempttotestthishypothesis, wehaveexam-
inedsurfaces prepared bycleavage inUHVimmediately
followed byscraping withathin-wired tungsten brush.
Theamount andthenatureofdisorder thusinduced in
thesampleisadmittedly illdefined, butourprocedure
canfindsomejustification initssimplicity and,apos-
teriori,inthestriking resultsreported inFig.6.Therewe
compare thePESspectrum ofanas-cleaved surface with
thespectrum ofthesamesampleafterscraping; similar
curveshavebeenobtained onallsurfaces prepared with
thesameprocedure. Thedisappearance ofthemainpeak
at180meVcoincides witharedistribution ofspectral
weightoverthewholebandandwiththerecoveryofa
clearmetallic edge.Structures at400and900meVare
indicative ofapersisting CDWdistortion, and,overall,
thespectrum ofthe"disordered" surfacecloselyresem-
blesspectracollected intheQCphase,although, because
ofthelowertemperature (20K),theFermiedgeiscon-
siderably sharper. Sincethelow-temperature spectraof
cleaved surfaces exhibitavanishingly smallintensity at
EFatallemission angles, macroscopic effects(e.g.,
misoriented crystallites) canberuledout,andwecan
conclude thattheperturbation actsatamicroscopic level
insuchawaythatametallic phasehasbeenstabilized at
lowtemperature.
Thenatureofthislow-temperature phaseisclarified by
theanalysisoftheTa4fcorelines,whoseshapeishighly
sensitive totheCDW.TheinsetofFig.6showstheTa
4f7/2linefortheas-cleaved andthedisordered surface.
Thesplittingofthe4flineintotwowell-resolved com-I
1.51.00.50=E„
Binding Energy(eV)
FIG.6.Valence-band PESspectraofacleaved 1T-TaS2 sur-
faceat20K(opensymbols) andofthesamesample, mechani-
callydisordered bybrushing (solidsymbols). Theinsetshows
x-ray-photoelectron spectraoftheTa4f7/2corelevelforthe
cleaved (opensymbols) andthedisordered (solidsymbols) sur-
face.
ponents inthespectrum oftheas-cleaved samplereflects
nonequivalent Tasites,anditisconsistent withprevious
measurements oftheCphase.Inthespectrum ofthe
disordered surface, instead, thetwocomponents are
broader, andtheirapparent separation isnoticeably re-
duced.Thisisexactlytheevolution expected foratransi-
tionfromthelong-range orderoftheCphasetothe
domainstructure oftheQCphase.''Ourresultsthere-
foreconfirm thatdisorder inhibits theformation ofthe
coherent staterepresented bytheCphase,andyielda
directimageofthedisorder-stabilized metallic phase.
IV.CONCLUSIONS
Thephysical properties of1T-TaS2 aredetermined by
theintrinsic instability ofitsquasi-2D Fermisurface. We
haveinvestigated therearrangement oftheelectronic
densityofstatesthataccompanies thetransitions between
different CDWphases, andwehaveidentified precise
spectroscopic signatures ofeachphase.Wehaveshown
thatphotoelectron spectroscopy, intheternperature-
dependent constant-energy mode,reveals animpressive
correspondence between structural andelectronic proper-
tiesofthismaterial, whichwasnotfullyrecognized in
previous investigations. Thehigh-energy resolution of
ourmeasurements allowsustodemonstrate theforma-
tionofacorrelation pseudogap inthecornrnensurate
CDWphase.Thisobservation isnotaccessible tomore
conventional experiments wheretheenergyresolution is
typically largerthantheintrinsic breadthofthemetallic
Fermiedgeatthetransition temperature. Thestabiliza-
7412 DARDEL, GRIONI, MALTERRE, WEIBEL, BAER,ANDLEVY 46
tionofalow-temperature metallic groundstatebyde-
fectshasalsobeenbrieflydiscussed. Theseresults
demonstrate thatthebringing intoplayofhighresolution
andofanaccurate temperature controlestablishes photo-
emission asapowerful andpromising toolfortheinvesti-
gationofthesubtlerelationship between thelow-energy
properties ofsolidsandtherelevant electronic states.ACKNOWLEDGMENTS
Thisworkhasbeensupported bytheFondsNational
SuissedelaRecherche Scienti6que. Wegratefully ac-
knowledge discussions withY.Petroff. Manythanksare
duetoC.Paladion andG.Montenegro foracriticalread-
ingofthemanuscript.
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|
PhysRevB.85.014117.pdf | PHYSICAL REVIEW B 85, 014117 (2012)
Hybrid functional calculations of the Al impurity in αquartz: Hole localization and electron
paramagnetic resonance parameters
Roland Gillen*and John Robertson
Department of Engineering, University of Cambridge, Cambridge CB3 0FA, United Kingdom
(Received 7 November 2011; revised manuscript received 16 January 2012; published 27 January 2012)
We performed first-principles calculations based on the supercell and cluster approaches to investigate the
neutral Al impurity in smoky quartz. Electron paramagnetic resonance measurements suggest that the oxygenatoms around the Al center undergo a polaronic distortion that localizes the hole on one of the four oxygen atoms.We find that the screened-exchange hybrid functional successfully describes this localization and improves onstandard local-density approaches or on hybrid functionals that do not include enough exact exchange such asB3LYP. We find a defect level at about 2.5 eV above the valence band maximum, corresponding to a localized holein an O 2 porbital. The calculated values of the gtensor and the hyperfine splittings are in excellent agreement
with experiment.
DOI: 10.1103/PhysRevB.85.014117 PACS number(s): 61 .72.−y, 61.05.Qr, 71 .15.Mb
I. INTRODUCTION
Silicon dioxide is among the most commonly encountered
substances in both daily life and electronic applications. It ishighly abundant in nature in the form of crystalline quartzand can be grown extremely pure by experimental techniques.A prominent defect is the neutral [AlO
4]0center, which has
been identified by its electron paramagnetic resonance (EPR)signature.
1–6Here an Al3+ion substitutes a Si3+ion, leaving an
unpaired electron at one of the four oxygen atoms adjacent tothe Al center. The corresponding localized spin gives rise to anelectron spin resonance (EPR) signal. This defect is believedto cause the smoky coloration of quartz crystals.
2,7On the
theoretical level, this “classical” model has been confirmed bycluster calculations on an unrestricted Hartree-Fock or hybridfunctional level,
8–11yielding a polaronic hole localization
and hyperfine coupling constants with27Al,17O, and29Si
in good agreement with experiments. It was shown that thedefect undergoes local symmetry breaking and Jahn-Tellerreconstruction with one oxygen atom relaxing away from theAl ion.
On the other hand, density functional theory (DFT) cal-
culations predicted that the hole and spin delocalize overall four oxygen atoms and no symmetry breaking betweenthe oxygen sites
12,13occurs. In general, DFT on the level
of the local-density approximation (LDA) or generalizedgradient approximation (GGA) fails to correctly predict thepolaronic hole localization on oxygen, such as in the caseof cation vacancies in ZnO,
14–16HfO 2,17and TiO 2(Ref. 18)
and acceptor impurities in GaN,19In2O3,20or Sn 2O3.20The
observed delocalization of the hole arises from the residualone-electron self-interaction from the Hartree energy, whichis insufficiently canceled by the exchange-correlation termof LDA- and GGA-type functionals, thus promoting artificialdelocalization.
21
Similarly, the self-interaction also contributes to the un-
derestimation of experimental band gaps by predicting va-lence (conduction) bands at too high (low) energies. Indeed,Mauri et al.
22showed that self-interaction-corrected LDA23
calculations, where the contributions due to self-interaction
are explicitly substracted in the energy expression, favor adistorted geometry and trapped hole. Different from LDA- andGGA-type functionals, “exact” Hartree-Fock (HF) exchange
completely cancels self-interaction contributions but resultsin a gross overestimate of the band gap if correlation effectsare not taken into account. A recently popular approach ismixing a fraction of Hartree-Fock exchange into the common(semi)local exchange-correlation functionals, which typicallycan compensate for the shortcomings of LDA and GGA. Itis interesting that some of these “hybrid” functionals, such asB3LYP, do not give the full hole localization in SiO
2.10,24This
suggests that a minimum amount of exact exchange is neededto give this result correctly.
24
In this paper, we investigate the [AlO 4]0center using
periodic boundary conditions and the screened-exchange (sX-LDA) hybrid functional, which includes a screened version ofthe full exact exchange. It is essentially self-interaction freefor all electron spacings less than the screening length andimproves on the predicted electronic band gaps.
25–27We show
that sX-LDA can restore the localization of the polaronic holeon one oxygen atom and yields the correct energy for thedefect level in the band gap. We further provide calculatedEPR parameters, which we find to be in good agreement withexperiment. To the best of our knowledge, calculated Land ´eg
tensors of the [AlO
4]0center have not been reported so far.
II. METHOD
The calculations were performed in the frame of spin-
polarized DFT using the hybrid functional sX-LDA,25,26which
has been recently28implemented in the plane-wave code
CASTEP .29Here the self-energy of an electron in the crystal is
approximated by a combination of a short-range Hartree-Fockexchange-type term,
ρ
ij(r)=φ∗
i(r)φj(r),
EsX
x[φ]∼occ/summationdisplay
i,j/integraldisplay/integraldisplayρij(r)φj(r)e−ks|r−r/prime|ρ∗
ij(r/prime)
|r−r/prime|d3r/primed3r,
and a long-range, local-density-dependent, LDA-type term,
ELDA,LR
xc [ρ]=ELDA
xc[ρ]−EsX,loc
x [ρ].
014117-1 1098-0121/2012/85(1)/014117(5) ©2012 American Physical SocietyROLAND GILLEN AND JOHN ROBERTSON PHYSICAL REVIEW B 85, 014117 (2012)
This range separation into short- and long-range terms is
similar to that in other functionals like the Heyd-Scuseria-Ernzerhof (HSE) version.
30,31In theory, the Thomas-Fermi
wave vector kTF, which depends on the average charge density,
is used as the inverse screening length ks. In practice, the TF
wave vector needs to be chosen carefully in terms of the densityof valence electrons. Here we use a fixed value of 0.76 /bohr,
which works well for spsemiconductors.
The calculations were performed in two steps: In the first
step, we used periodic boundary conditions to model thedefect in the solid by a 2 ×2×2 supercell of the common
(hexagonal) nine-atom unit cell of α-SiO
2with one Al
atom substituting one of the 24 Si atoms and optimized theatomic positions while keeping the lattice constants at theexperimental values. The integrals in reciprocal space wereapproximated by the values at the Baldereschi kpoint for
hexagonal lattices,
32and we used OPIUM33pseudopotentials
with a cutoff energy of 800 eV to model the valence electronsof Si and O.
In the second step, we calculated the EPR parameters with
the quantum chemistry code
ORCA ,34which particularly aims
at the spectroscopic properties of open-shell molecules. Weused a 68-atom cluster, which we obtained by cutting outthe first four atomic shells surrounding the Al center fromthe previously relaxed 72-atom supercell and passivating thedangling bonds of the outmost 12 silicon atoms by hydrogen.The atoms were represented by all-electron def2-TZVP setsfrom the Karlsruhe group,
35except for the oxygen atoms
directly adjacent to the Al center, which were described byBarone’s EPR-III
36basis set, specifically designed for the
calculation of EPR properties. As ORCA does not offer screened
hybrid functionals, we chose to use Becke’s “half-and-half”(BHandHLYP)
37hybrid functional
EBHandHLYP
xc [ψ]=1
2/parenleftbig
EHF
x[ψ]+EB88
x[ρ]/parenrightbig
+ELYP
c[ρ]
in comparison with pure Hartree-Fock exchange instead.
III. RESULTS AND DISCUSSION
Figure 1shows the calculated partial density of states
(PDOS) for a 72-atom supercell of α-SiO 2containing a
single Al center. Clearly, the impurity introduces a singlespin-polarized defect state at an energy of 2.5 eV abovethe valence band maximum, which nicely coincides withthe experimentally observed absorption peak at 2.5 eV forsynthetic quartz crystals.
38The defect state is predominantly
of O 2 pcharacter with small contributions from O 2 sand Si
3pmixed in. We find that the overall magnetic moment of
0.99μBof the system is localized almost exclusively at one
single oxygen atom (O∗) with a dumbbell-shaped defect wave
function (refer to the spin-density plot in Fig. 2). Together with
the localized hole, our sX-LDA approach yields an asymmetricdistortion of the geometry surrounding the Al atom. We findan Al-O* bond length of 1.92 ˚A for the oxygen atom carrying
the localized hole, corresponding to a bond elongation of 13%compared to the other three Al-O bonds (1.69–1.71 ˚A). This
prediction is in reasonable agreement with estimation fromEPR measurements
6(12% bond elongation).-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20
Energy (eV)SiOAl
2p2p 3p
2s
FIG. 1. (Color online) Partial density of states of a 72-atom
supercell of SiO 2:Al from sX-LDA calculations. The dashed lines
represent the energy of the valence band maximum and the conduction
band minimum.
We note that the degree of spin localization and the
geometry depend on the included portion of Hartree-Fockexchange in the functional. Pacchioni et al.
10reported that
the popular B3LYP functional (20% HF exchange) predictsa partial spin localization with spin populations of 0.29 μ
B
on two oxygen atoms and 0.21 μBon the remaining two,
close to the predictions from GGA. Correspondingly, theAl-O bond lengths in this case are very similar. On the otherhand, increasing the fraction of HF exchange quickly leads tosymmetry breaking and a polaronic distortion of the defectgeometry. In a separate calculation, we found a Mullikenpopulation of 0.68 μ
Bon O∗and a bond elongation of 12% for
PBE0 (25% HF exchange). To et al.11found a bond elongation
of 12% and a spin moment of 0.93 μBusing the BB1K
FIG. 2. (Color online) Plot of the spin density of SiO 2:Al around
the Al center. The system undergoes a polaronic distortion, and thespin is localized almost exclusively at one of the four oxygen atoms
adjacent to the Al impurity. Some atoms are hidden by the spin-density
isoplane.
014117-2HYBRID FUNCTIONAL CALCULATIONS OF THE Al ... PHYSICAL REVIEW B 85, 014117 (2012)
TABLE I. Calculated principle values of the gtensor and the hyperfine matrix of the [AlO 4]0center from calculations on an Al(O 4SiH 3)4
cluster and comparisons with previous theoretical calculations from other authors and experiments. Following Ref. 11, the oxygen labels O∗
and O1,2,3refer to the oxygen atom with the hole and the three remaining oxygen atoms adjacent to the aluminium center, respectively. The
values for29Si are given for the Si nucleus with the largest contribution. For UHF, we combined our calculations for the gtensor with the
hyperfine coupling parameters from Ref. 10.
BHandHLYPaBHandHLYPbUHFac(Ref. 10) BB1Kb(Ref. 11)E x p ( R e f s . 4–6)
Spin (in μB)O∗0.93 0 .94 1 .04 0 .93
O1,2,30.01−0.05 0 .01−0.03 <0.01 0 .01−0.06
gtensor g1 2.0031 2 .0028 2 .00235c2.0024±0.0003
g2 2.0093 2 .0091 2 .0087c2.0091±0.0003
g3 2.0412 2 .043 2 .0475c2.0614±0.0003
giso 2.0179 2 .0183 2 .0195c
17O∗hyperfine coupling A1−119.23 −115.46 −128.6 −109.46 −111.0
matrix parameters (in G) A2 22.54 25 .92 11 .62 3 .48 15 .2
A3 22.86 26 .21 3 .62 3 .76 17 .8
Aiso −24.61 −21.11 −34.5 −20.74 −26.0
B1 −94.62 −94.44 −94.1 −88.72 −85.0
B2 47.15 46 .14 7 .04 44 .22 41 .2
B3 47.47 47 .24 48 .14 4 .50 43 .8
27Al hyperfine coupling A1 −6.24 −5.92 −5.1 −7.09 −6.2
matrix parameters (in G) A2 −5.97 −5.79 −5.1 −6.99 −6.1
A3 −4.95 −4.7 −4.2 −5.90 −5.1
Aiso −5.72 −5.47 −4.8 −6.66 −5.8
B1 −0.52 −0.45 −0.3 −0.43 −0.4
B2 −0.25 −0.32 −0.3 −0.33 −0.3
B3 0.77 0 .77 0 .60 .76 0 .7
29Si hyperfine coupling A1 8.75 9 .06 15 .51 0 .11 10 .8
matrix parameters (in G) A2 9.35 9 .73 17 .41 0 .57 11 .4
A3 9.53 10 .04 17 .81 0 .77 11 .6
Aiso 9.21 9 .61 16 .91 0 .48 11 .3
B1 −0.4 −0.55 −1.4 −0.37 −0.5
B2 0.14 0 .12 0 .50 .09 0 .1
B3 0.32 0 .43 0 .90 .29 0 .3
aObtained with an Al(O 4Si4H3)4cluster.
bObtained with an Al(O 4Si4H9)4cluster.
cThis work.
functional (42% HF exchange), whereas Pacchioni et al.10
reported a 14% elongation from their 100% Hartree-Fock
calculations. Our cluster calculations using the BHandHLYPhybrid functional (50% exact exchange) yield spin localizationand defect geometry very similar to our sX-LDA results.
In the next step, we calculated the EPR parameters for the
defect. The EPR spectrum can be modeled by the Hamiltonian
H
eff=α
2S·g·B+/summationdisplay
iS·A·Ii, (1)
where αis the fine-structure constant. The first term of Eq. ( 1)
describes the coupling of the spin momenta Sof the unpaired
electrons with an external magnetic field Bby the gtensor,
g=⎛
⎝g100
0g20
00 g3⎞
⎠.
The second term of Eq. ( 1) represents the hyperfine coupling of
the electron spin with the nuclear spins Iiand can be described
in terms of the hyperfine matrix A, which can be separatedinto its isotropic and anisotropic part B(related to dipolar
interaction),
A=⎛
⎝A100
0A20
00 A3⎞
⎠+AisoI+⎛
⎝B100
0B20
00 B3⎞
⎠.
A problem arises from the fact that the gtensor is not gauge
invariant, so that the results technically depend on the choiceof origin in the calculation. While the origin dependence isusually small as long as sufficiently large basis sets are used,we have checked the sensitivity of the results with respectto the origin. We found that on the GGA level the centerof the electron charge gives practically the same values ascalculations using the fully invariant individual gauges forlocalized orbitals (IGLO)
39procedure (which, however, is
not rigorous for hybrid functionals) and used this point asthe origin for our calculations. Table Ishows the obtained
principle values of the gtensor and the hyperfine matrix in
comparison with previous theoretical results and experiment.In case of the gtensor, our computed values for the two smaller
principle values g
1=2.0028,g2=2.0093 from BHandHLYP
014117-3ROLAND GILLEN AND JOHN ROBERTSON PHYSICAL REVIEW B 85, 014117 (2012)
calculations are in very good agreement with the experimental
estimations g1,e=2.0024 and g2,e=2.0091. The deviations
for the third principal value are larger. We further obtainedg
3=2.043, which, however, is considerably smaller than
g3,e=2.0614 as reported by Schnadt et al.4
We find that both cluster size and spin distribution affect
the entries of the gtensor. Comparing the results from clusters
with 33 atoms and 68 atoms, both within BHandHLYP, wesee a slight improvement of the prediction with increasingcluster size. The dependence on the spin localization is morepronounced, particularly that of the anisotropy of the gtensor.
For PBE0 (and a 33-atom cluster), g
1(=2.0073) and g2
(=2.0094) are overestimated, whereas g=(=2.031) is an even
stronger underestimation than the value from BHandHLYP.On the other hand, pure Hartree-Fock exchange seems toovershoot the geometrical distortion and spin moment at O
∗
but improves on the anisotropy of the principal values; seeTable I.
For the hyperfine coupling, we find a good prediction of
our approach for the investigated
17O,27Al, and29Si nuclei.
This is particularly true for27Al, where our results, being the
golden mean of the reported values of UHF10and BB1K11
calculations, are remarkably close to experiment. For29Si,
our calculations slightly underestimate the experimental valuesand yield similar results to those from To et al. This is almost
entirely due to the underestimation of the isotropic componentofA,A
iso,b y≈2 G, while the (small) dipolar interaction
is reproduced accurately. The dominant contribution of theoxygen atoms to the hyperfine coupling is to be expected fromthe oxygen atom carrying the localized hole. Here the electron-nucleus interaction contains a strong dipolar component andthe experimentally obtained hyperfine matrix shows a stronganisotropy for the three principal axes. The UHF calculationsby Pacchioni et al. underestimated the experimental values
A
1,e=− 111.00 G, A2,e1=15.2 G, and A3,e=17.8 G due
to a too large isotropic contribution. In contrast, the BB1Kcalculations predicted a considerably weaker A
iso, resulting
in a slight underestimation of A1and overestimation of A2
andA3and the opposite behavior for the anisotropic matrix.
We find the hyperfine parameters A1=− 115.46 G, A2=
25.92 G, and A3=26.21 G, which overestimate all three of the
experimental values. As for the other methods, our predictionssuffer from a unfavorable combination of too strong isotropicand anisotropic components, as Table Ishows. The differences
between our parameters and those from Pacchioni et al. arise
from the different A
iso, while the dipolar interaction is close
to identical in both cases. The dipolar contribution depends onthe angular momentum of the unpaired electron and hence is
roughly proportional to the population of the O 2 porbital and
the localization of the spin density. To et al. have reported that
increasing the size of the cluster in their case led to a strongerspin localization and a larger dipolar interaction.
From comparing the predictions for the gtensor and
hyperfine coupling from various methods with experiment,it appears that the gtensor benefits from a strong spin
localization and thus a large Hartree-Fock fraction in thehybrid functional, whereas the hyperfine coupling matrix ismore sensitive to the quality of the long-range correlation.Pure Hartree-Fock exchange gives a good description of thegtensor but performs poorly for the hyperfine coupling.
On the other hand, BHandHLYP (and probably BB1K) ismarginally worse for the gtensor but outperforms HF with
respect to the hyperfine coupling parameters and yields anoverall good description for both quantities. Based on theseobservations, we believe that a screened hybrid functionalwith a strong short-range fraction of exact exchange, such assX-LDA, would be predestinated for the calculation of the EPRparameters of polarons in wide band-gap oxides. This wouldlead to a HF-like spin-localization at O
∗, thus yielding good
predictions for the g-tensor, but dominating local exchange-
correlation interaction in the long-range part, which in turnbenefits the description of the hyperfine parameters.
IV . CONCLUSION
We showed that sufficient inclusion of Hartree-Fock in
hybrid functionals does improve on DFT supercell calcu-lations and can correctly describe the polaronic hole andthe corresponding symmetry distortion of the neutral Alimpurity in αquartz. The observed localization is inher-
ently connected to the reduced self-interaction in hybridfunctionals, which shows by its dependence on the ratio ofHF:DFT exchange. The validity of the presented approachwas further shown by a subsequent calculation of the g
tensor and the hyperfine coupling matrix using a clusterapproximation for the sX-LDA geometry, which was in goodagreement with the conclusions from experimental EPR mea-surements. We suggest that the screened hybrid form wouldimprove on the performance of “standard” nonscreened hybridfunctionals.
ACKNOWLEDGMENT
The authors thank S. J. Clark for useful discussions.
*rg403@cam.ac.uk
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014117-5 |
PhysRevB.77.035118.pdf | Electronic structure of CeRu 2X2„X=Si,Ge …in the paramagnetic phase studied by soft x-ray
ARPES and hard x-ray photoelectron spectroscopy
M. Yano,1A. Sekiyama,1H. Fujiwara,1Y . Amano,1S. Imada,1T. Muro,2M. Yabashi,2,3K. Tamasaku,3A. Higashiya,3
T. Ishikawa,3Y.Ōnuki,4and S. Suga1
1Division of Materials Physics, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
2Japan Synchrotron Radiation Research Institute, Mikazuki, Sayo, Hyogo 679-5198, Japan
3RIKEN, Mikazuki, Sayo, Hyogo 679-5148, Japan
4Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
/H20849Received 10 August 2007; published 15 January 2008 /H20850
Soft and hard x-ray photoelectron spectroscopy has been performed for one of the heavy fermion system
CeRu 2Si2a n da4 f-localized ferromagnet CeRu 2Ge2in the paramagnetic phase. The three-dimensional band
structures and Fermi surface shapes of CeRu 2Si2have been determined by soft x-ray h/H9263-dependent angle-
resolved photoelectron spectroscopy. The differences in the Fermi surface topology and the non-4 felectronic
structures between CeRu 2Si2and CeRu 2Ge2are qualitatively explained by the band-structure calculation for
both 4 fitinerant and localized models, respectively. The Ce valences in CeRu 2X2/H20849X=Si,Ge /H20850at 20 K are
quantitatively estimated by the single impurity Anderson model calculation, where the Ce 3 dhard x-ray
core-level PES and Ce 3 dx-ray absorption spectra have shown stronger hybridization and signature for the
partial 4 fcontribution to the conduction electrons in CeRu 2Si2.
DOI: 10.1103/PhysRevB.77.035118 PACS number /H20849s/H20850: 71.18. /H11001y, 71.27. /H11001a, 79.60. /H11002i
I. INTRODUCTION
Strongly correlated electron systems show many interest-
ing physical properties due to their complicated electronicstructures. Particularly, Ce based compounds have been ex-tensively studied by both experimental and theoretical ap-proaches because they show variety of 4 fstates due to dif-
ferent hybridization strength between the Ce 4 fand valence
electrons. Dominance of the Ruderman-Kittel-Kasuya-Yosida /H20849RKKY /H20850interaction or the Kondo effect is one of the
attractive subjects in this system. Especially, Ce T
2X2/H20849T
=Cu,Ag,Au,Ru, etc. and X= S io rG e /H20850with the tetragonal
ThCr 2Si2structure1/H20849a=b/H110114 Å and c/H1101110 Å /H20850shows various
4felectron behaviors.2–7Among them, the 4 felectrons in
CeRu 2Ge2are localized and ferromagnetically ordered due to
ascendancy of the RKKY interaction below the Curie tem-perature T
C/H110118 K in ambient pressure.8On the other hand, in
the case of CeRu 2Si2, the Ce 4 felectron couples with the
valence electron and makes the so-called Kondo singlet statebelow the Kondo temperature T
K/H1101120 K.9,10CeRu 2Si2is
known as a typical heavy fermion system which has a largevalue of the electronic specific heat coefficient
/H9253
/H11011350 mJ /mol K2/H20849about 20 times larger than that of
CeRu 2Ge2/H20850.11–13Such a difference in 4 felectronic states is
thought to originate from the different hybridizationstrengths between the 4 fand valence electrons. For example,
when high pressure is applied to CeRu
2Ge2, the hybridiza-
tion strength increases and the electronic structures ofCeRu
2Ge2approach to those of CeRu 2Si2.6,14Clarification of
the electronic structures of CeRu 2X2is thus the key to reveal
a connection between the 4 flocalized and itinerant elec-
tronic states. Therefore, complete study of the electronicstates of CeRu
2X2is essential for understanding physics of
the strongly correlated electron systems.
So far, CeRu 2Si2has eagerly been studied in order to
elucidate the mechanism of the metamagnetic transitionwhich occurs at Hm/H110117.7 T.15,16According to the de Haas–
van Alphen /H20849dHvA /H20850studies,17the Fermi surface /H20849FS/H20850of
CeRu 2Si2approaches to that of LaRu 2Si2above Hmdue to
the localization of the 4 felectrons. However, the dHvA re-
sults of CeRu 2Si2cannot fully be explained by the band-
structure calculations under H/H11021Hm.18For example, the
heaviest effective mass FS or the smallest FS signals havenot been observed along the magnetic field direction of/H20855001/H20856. Recently, another experimental method named “soft
x-ray h
/H9263-dependent angle-resolved photoelectron spectros-
copy /H20849ARPES /H20850” has been established to determine three-
dimensional /H208493D/H20850FSs by using energy tunable soft x rays
from third-generation high brilliance synchrotron radiationlight sources.
19In the ARPES studies, we can observe elec-
tronic structures in solids at various temperatures and deter-mine the shapes of FSs which can be compared with theresults from dHvA measurements. dHvA measurements canbe performed under magnetic fields and high pressures butare confined to low temperatures. While dHvA measure-ments probe genuine bulk electronic states, conventionalphotoelectron spectroscopy /H20849PES /H20850measurements have been
believed as a rather surface sensitive technique as far as thephotoelectron kinetic energies are in the range of20–200 eV. However, the bulk sensitivity of the soft x-ray/H20849h
/H9263/H11011800 eV /H20850PES was confirmed by the observation of the
4fPES for Ce compounds.20Soft x-ray PES is currently an
essential approach to reveal electronic states of transitionmetal compounds
21,22and rare earth compounds.23
In addition, more bulk-sensitive PES by using hard x-ray
photoelectron spectroscopy /H20849HAXPES /H20850has become
feasible.24The probing depth of PES depends on the kinetic
energy of the photoelectron. According to TPP-2M formula/H20849the inelastic mean free path formula developed by Tanuma,
Powell, and Penn /H20850,
25a photoelectron inelastic mean free path
/H9261can be estimated as a function of the electron kinetic en-
ergy /H20849EK/H20850. For example, /H9261/H1101119 Å at EK=800 eV and /H9261PHYSICAL REVIEW B 77, 035118 /H208492008 /H20850
1098-0121/2008/77 /H208493/H20850/035118 /H208498/H20850 ©2008 The American Physical Society 035118-1/H11011115 Å at EK=7000 eV for CeRu 2Si2. HAXPES is also
useful in order to obtain bulk-sensitive core-level spectrawith negligible surface contribution. For example, the sur-face spectral weight of the Ce 3 dlevel located at the binding
energy /H20849E
B/H20850of/H11011900 eV can be significantly reduced by
HAXPES.
By virtue of the soft x-ray ARPES experiments, we have
so far clarified 3D FSs of CeRu 2Ge2in the paramagnetic
phase. The results of the ARPES measurements were com-pared with the local-density approximation /H20849LDA /H20850calcula-
tion for LaRu
2Ge2performed on the Ce 4 felectron localized
model.26The difference between the ARPES results and the
calculation for LaRu 2Ge2or dHvA results for CeRu 2Ge2in
the ferromagnetic phase27can be explained by non-
negligible small hybridization between its Ce 4 fand valence
electrons.19We have extended the study to h/H9263-dependent soft
x-ray ARPES, HAXPES, and x-ray absorption spectroscopy/H20849XAS /H20850for CeRu
2Ge2and a heavy fermion system CeRu 2Si2
in order to reveal their electronic structures. The ARPES
results for CeRu 2Si2are compared with those for CeRu 2Ge2
in the paramagnetic phase and the band-structure calculationfor CeRu
2Si2, in which the 4 felectrons are treated as itiner-
ant. The HAXPES and XAS spectra have been analyzed bythe single impurity Anderson model /H20849SIAM /H20850, by which the
clear differences in the mean 4 felectron number and hybrid-
ization strength between CeRu
2Si2and CeRu 2Ge2are con-
firmed. We show the transformation of 3D FSs resultingfrom different hybridization strengths between Ce 4 fand va-
lence elections.
II. METHODS
A. Soft x-ray angle-resolved photoelectron spectroscopy and
Ce 3dx-ray absorption spectroscopy
The CeRu 2X2single crystals were grown by the Czochral-
ski pulling method.28The soft x-ray ARPES and XAS mea-
surements were performed at BL25SU29in SPring-8. A Sci-
enta SES200 analyzer was used covering more than a wholeBrillouin zone along the direction of the slit.
30The energy
resolution was set to /H11011200 meV for FS mappings and
/H11011100 meV for a high resolution measurement. The angular
resolution was ±0.1° and ±0.15° for the perpendicular andparallel directions to the analyzer slit, respectively. Thesevalues correspond to the momentum resolution of±0.025 Å
−1and ±0.038 Å−1ath/H9263=800 eV. The clean sur-
face was obtained by cleaving in situ providing a /H20849001/H20850plane
in the base pressure of /H110113/H1100310−8Pa. All of the ARPES mea-
surements were performed at 20 K. The surface cleanlinesswas confirmed by the absence of the O 1 sphotoelectron sig-
nals. We have measured Pd valence band to determine theFermi level /H20849E
F/H20850and estimate the energy resolution of the
system. The measurements of the valence band and the Si 2 p
core spectra were alternated for the purpose of normalizationof each valence band spectrum. In ARPES measurements,we have first performed the k
z−kxymapping at several h/H9263
and angles. Photon momenta were taken into account to de-
termine the exact value of /H20841kz/H20841.31In order to analyze ARPES
data as functions of the binding energy and momentum, wehave employed both energy distribution curves /H20849EDCs /H20850andmomentum distribution curves /H20849MDCs /H20850. The XAS was mea-
sured by the total electron yield mode whose probing depthis comparable to that of the HAXPES. The energy resolutionwas set to better than 200 meV. The detailed experimentalconditions are given in Ref. 32.
B. Hard x-ray photoelectron spectroscopy
The HAXPES measurements for CeRu 2X2were carried
out at BL19LXU33in SPring-8 with a MB Scientific A1-HE
analyzer. The /H20849001/H20850clean surface was obtained by cleaving
in situ in the pressure of 10−8Pa at the measuring tempera-
ture of 20 K. The photon energy was set to about 8 keV andthe energy resolution was set to about 400 meV for theCe 3dcore-level measurements. We have measured an
evaporated Au valence band spectrum to determine E
F, and
the Ce 4 sand Ru 3 dspectra to estimate the energy loss /H20849in-
cluding plasmon excitation /H20850peak position and its intensity.
Figure 1shows the HAXPES spectra of the Ce 4 sand Ru 3 d
core-levels for CeRu 2Si2and CeRu 2Ge2with the energy res-
olution of about 200 meV. In CeRu 2Si2spectra, the sharp
Ru 3 d5/2and 3 d3/2levels locate at 279.4 and 283.5 eV, re-
spectively. The rather broad Ce 4 sstate lies at 289.5 eV.34
Rather weak and broad energy loss peaks are located about
20.8 eV from the main peak for CeRu 2Si2and about 18.1 eV
for CeRu 2Ge2according to a line shape analysis.36The en-
ergy loss peak position from the main peak and its intensitywere later taken into account in the fitting of the Ce 3 dPES
spectra.
C. Single impurity Anderson model
In order to obtain the information on the bulk 4 felec-
tronic states as well as the mean 4 felectron number /H20849nf/H20850Binding Energy (eV)Intensity (arb. units)CeRu2Si2
20 K
hν= 8175 eV
CeRu2Ge2
20 K
hν= 8170 eV
320Ce 4sRu 3dBGFitExp.Ce 4sRu 3dBGFitExp.
310 300 290 280(a)
(b)
FIG. 1. /H20849Color online /H20850Ce 4sand Ru 3 dcore-level HAXPES
spectra for /H20849a/H20850CeRu 2Si2and /H20849b/H20850CeRu 2Ge2. The spectra were fitted
by Gaussian and Lorentian broadenings of Mahan’s line shape /H20849Ref.
36/H20850for each component and plasmon. The dashed line represents
background /H20849BG/H20850of each spectrum. The energy loss peak appears at
about 20 eV higher binding energy from each main peak as shownby vertical bars.YANO et al. PHYSICAL REVIEW B 77, 035118 /H208492008 /H20850
035118-2from the Ce 3 dcore-level PES and XAS for CeRu 2X2,w e
performed the SIAM calculation based on the1/N
f-expansion method developed by Gunnarsson and
Schönhammer.37Here, Nf/H20849the degeneracy of the Ce 4 flevel /H20850
was set to 14 for simplicity. We calculated the 3 dPES and
XAS spectra to the lowest order in 1 /Nf, where the f0,f1,
andf2configurations were taken into account for the initial
state. We divided the band continuum into N/H20849we set to 21 /H20850
discrete levels following Jo and Kotani.38The configuration
dependence of the hybridization strength was also taken intoaccount and was chosen to be the same as that obtained for
/H9251-Ce by Gunnarsson and Jepsen.39,40The energy dependence
of the hybridization strength was assumed to be constant inthe binding energy range from 0 /H20849E
F/H20850toB/H20849we set to 4 eV /H20850.The multiplet effects were not taken into account for
simplicity.42
III. RESULTS
A. Three-dimensional angle-resolved photoelectron
spectroscopy
Figures 2/H20849a/H20850,2/H20849b/H20850, and 2/H20849a/H11032/H20850, and 2/H20849b/H11032/H20850display the soft
x-ray ARPES results for CeRu 2Si2, indicating existence of
six bands in the region from EFto about 2 eV. These bands
are numbered from 0 to 5 from the higher binding energyside. According to Figs. 2/H20849a/H20850and2/H20849a
/H11032/H20850along the Z-Xdirec-
tion, the energy positions of the bands 2–5 approach the X
point, and therefore a strong intensity peak is observed atabout 0.69 eV. Band 1 has the lowest binging energy at the Z
point and does not cross E
F, while bands 2 and 3 cross EF
near the Zpoint as a merged band due to the limited reso-
lution. Bands 4 and 5 have been observed separately asclearly shown in the expanded MDCs /H20849a
/H11033/H20850and these bands
can be then traced in Fig. 2/H20849a/H11032/H20850. Figures 2/H20849b/H20850and2/H20849b/H11032/H20850along
the/H9003-Xdirection show some bands, whose dispersions are
smaller than those along the Z-Xdirection except for the
band 0. Along the /H9003-Xdirection, bands 0–4 are on the occu-
pied side, while only band 5 has Fermi wave number /H20849kF/H20850
near the /H9003point as shown in the expanded figure /H20849b/H11033/H20850, al-
though the strong intensity of band 4 is overlapped near the
/H9003point. Since the contribution from band 4 is relatively de-
creased when the energy approaches EF, the contribution of
band 5 is confirmed. Band 5 can be then traced in Fig. 2/H20849b/H11032/H20850.
Figure 3shows the h/H9263-dependent ARPES results along
the/H208490,0/H20850-/H20849/H9266/a,/H9266/a/H20850direction. The shape of each band and
kFhave comprehensively been evaluated by both EDCs and
MDCs. As shown in Figs. 3/H20849a/H20850and3/H20849a/H11032/H20850, band 5 crosses EF
near the /H20849/H9266/a,/H9266/a/H20850point. Although the spectral weight de-
rived from band 4 is weak, we can trace band 4 dispersion
crossing EF. Figures from /H20849a/H20850–/H20849c/H20850also indicate that the kFfor
band 4 approaches to /H20849/H9266/a,/H9266/a/H20850point when h/H9263is away
from 745 to 790 eV although the intensity of the band 4 at
h/H9263=760 eV is so weak that kFcannot be determined so ac-
High
Low
Binding Energy (eV)120
1201 20ZZ X
XΓ
ΓX
XΓ(a’)(a)
(b)(b’)
0
01
122
3
34
444
55
Z X
XBinding Energy (eV) Binding Energy (eV)0
00.5
0.3(a”)
2,3
(b”)55
5
FIG. 2. /H20849Color online /H20850ARPES spectra of CeRu 2Si2near EFat
20 K with h/H9263=725 eV. /H20849a/H20850and /H20849b/H20850are the second order differential
images along the Z-Xand/H9003-Xdirections. EDC /H20849a/H11032/H20850and /H20849b/H11032/H20850cover
the same regions of /H20849a/H20850and /H20849b/H20850./H20849a/H11033/H20850is MDCs of the same region as
/H20849a/H20850or/H20849a/H11032/H20850./H20849b/H11033/H20850is the expanded MDCs along the /H9003-Xdirection. The
energy resolutions of /H20849a/H20850and /H20849b/H20850series are set to about 200 and
100 meV, respectively. The dashed lines representing each band areguides to the eye.
11 1
22 233
33
444
4
5555 5
Z
X4
hν= 745 eV
(a)
0
0.5(b) (c)
Binding Energy (eV) Momentum (unit of 1/a)0 10 10 1760 eV 790 eV
(0,0)(0,0)
(0,0)(π/a,π/a)(π/a,π/a)
(π,π)(0,0) (π,π) Binding Energy (eV)
(b’) (a’)
FIG. 3. /H20849Color online /H20850h/H9263-dependent ARPES spectra of CeRu 2Si2along the /H208490,0/H20850-/H20849/H9266/a,/H9266/a/H20850direction at 20 K. The values of kzare/H20849a/H20850
4
132/H9266
cand /H20849b/H208507
132/H9266
cachieved by the photon energies of 745 and 760 eV, respectively. /H20849c/H20850corresponds to the high symmetry line Z-Xachieved
by 790 eV /H20849kz=2/H9266
c/H20850./H20849a/H11032/H20850and /H20849b/H11032/H20850are the expanded MDC figures corresponding to /H20849a/H20850and /H20849b/H20850, respectively. The dashed lines representing
each band are guides to the eye. The thick lines are given for band 4 whose intensities are weaker than others.ELECTRONIC STRUCTURE OF CeRu 2X2/H20849X=… PHYSICAL REVIEW B 77, 035118 /H208492008 /H20850
035118-3curately. Therefore, the position of band 4 is given by the
zones to allow possible experimental ambiguity. Hereafter,the lines of the guides to the eye are drawn to pass throughthe adjacent /H20849in the sense of wavenumber in EDCs and en-
ergy in MDCs /H20850noticeable structures in order to compromise
with the experimental statistics. When the excitation photonenergy is increased from k
z/H110110/H20849h/H9263=725 eV /H20850as/H20849a/H20850→/H20849b/H20850
→/H20849c/H20850, we can recognize that bands 2 and 3 cross EFsequen-
tially. At h/H9263=745 eV /H20849kz=4
132/H9266
c/H20850, bands 1–3 are on the occu-
pied side at the /H208490, 0/H20850point. When kz=7
132/H9266
cwas chosen by
760 eV /H20849b/H20850, only band 3 crosses EFnear the /H208490,0/H20850point,
while bands 1 and 2 are fully occupied. At h/H9263=790 eV /H20849kz
=2/H9266
c/H20850, both bands 2 and 3 cross EFnear the Z/H208490,0,2 /H9266/c/H20850
point, while band 1 is still on the occupied side at the binding
energy of 0.27 eV, indicating that band 1 does not form FS.
We have determined kF’s by means of both EDCs and
MDCs for the FS mapping. We have integrated the intensi-ties of MDCs from E
Fto −0.1 eV as a function of momen-
tum from a slice of the ARPES data. The topology of the FSsthus obtained is displayed in Fig. 4. Figure 4/H20849a/H20850shows a k
x
−kyslice at kz/H110110 obtained by changing the detector angles
and Fig. 4/H20849b/H20850shows a kxy−kzslice including the /H9003-Xaxis
whose kzcorresponds to the excitation photon energies from
715 to 805 eV with 5 eV steps. The kFs estimated from
EDCs and MDCs are plotted by dots on the figure with errorbars. As shown in Fig. 4/H20849a/H20850, there is a small holelike FS
centered in the vicinity of the Zpoint derived from bands 2and 3. The contour of the holelike FS derived from band 4
exists mostly inside the square Brillouin zone centered at theZpoint. Its intensity in the vicinity of E
Fis considerably
small compared with that for CeRu 2Ge2,19suggesting that
the electron correlation in band 4 is larger for CeRu 2Si2than
for CeRu 2Ge2because the intensity of the coherent part of a
band is suppressed by the smaller magnitude of the quasipar-ticle renormalization factor /H20849or coherent factor /H20850due to elec-
tron correlations.
43,44The largest circle shaped electron FS of
band 5, whose obvious contour is clearly seen, centered attheZpoint surrounds the square Brillouin zone. Some inten-
sities derived from band 5 centered at the /H9003point can also be
seen. The strongest intensity in Fig. 4/H20849a/H20850around the /H9003point
is caused by the spectral weight of band 4 near E
Fwhich
does not, however, cross EFnear the /H9003point, as shown in
Fig. 2/H20849b/H11032/H20850. Figure 4/H20849b/H20850shows some outlines of FSs in the
kz−kxyplane. The elliptical contours centered at the Zpoint
derived from bands 2 and 3 can be separately observed as inFig. 3. The prolonged elliptical contour of the FS derived
from band 4 can also be confirmed in Fig. 4/H20849b/H20850along the
in-plane Z-Xdirection. Another contour of FS derived from
band 5, which is symmetric with respect to the k
zaxis of X-X
orZ-/H9003and has a narrow part near the /H9003point, is also seen in
Fig. 4/H20849b/H20850.
From these two slices of the FSs, we suggest rough 3D
shapes of the FSs of CeRu 2Si2in Fig. 5. It was found that
CeRu 2Si2has four FSs derived from bands 2 to 5. Bands 2
Γ
ZX
X 44
555
2,3
ΓZX
X
(π,π) (0,0)725790<110>
hν(eV)<110>(a)
High
Low
(b)Brillouin ZoneZZ
XXΓ
44
5 53
2
FIG. 4. /H20849Color online /H20850FSs slices of CeRu 2Si2at 20 K obtained
by integrating the photoelectron intensity from 0 to −0.1 eV andthe Brillouin zone for the ThCr
2Si2-type structure. The solid lines
represent the corresponding Brillouin zone and the dashed-and-dotted lines represent high symmetry lines. White dots with errorbars represent the estimated k
F. The dashed lines represent the FSs
following the experimentally evaluated kF’s./H20849a/H20850FSs slice in the
kx−kyplane at kz/H110110/H20849h/H9263=725 eV /H20850./H20849b/H20850FSs slice in the kz
/H20849ordinate /H20850-kxy/H20849abscissa /H20850plane. h/H9263=725 and 790 eV correspond to
the/H9003point and the Zpoint, respectively, along the /H9003-Zdirection.
FIG. 5. /H20849Color online /H20850Qualitative 3D FSs images of CeRu 2Si2
at 20 K obtained by h/H9263-dependent soft x-ray ARPES. The shapes of
the 3D FSs were determined by the results in Figs. 4/H20849a/H20850and4/H20849b/H20850.
The obtained holelike FSs derived from bands 2–4 centered at the Z
point are shown in /H20849a/H20850. FSs from bands 2 and 3 are in the FS from
band 4 and FSs of bands 3 and 4 are intentionally opened up here sothat the inner FS can be visible. Band 5 constructs a rather compli-cated FS shape as shown in /H20849b/H20850. The center of /H20849b/H20850corresponds to the
/H9003point.YANO et al. PHYSICAL REVIEW B 77, 035118 /H208492008 /H20850
035118-4and 3 form ellipsoidal shaped holelike FSs prolonged along
thekzdirection centered at the Zpoint. The prolonged length
of the FS of band 2 is shorter than that of band 3, as con-firmed by Figs. 3and4/H20849b/H20850. The FS of band 2 is surrounded
by that of band 3. Band 4 forms a large holelike swelled-diskFS centered at the Zpoint. This FS encompasses both FSs of
bands 2 and 3. These holelike FSs are similar to those of theARPES results for CeRu
2Ge2in the paramagnetic phase, al-
though the size of the FS of CeRu 2Si2derived from band 4 is
smaller than that of CeRu 2Ge2/H20851Figs. 6/H20849a/H20850and6/H20849b/H20850/H20852.19Mean-
while, the shape of the FS formed by band 5 is quantitativelydifferent between CeRu
2Si2and CeRu 2Ge2. Namely, the FS
from band 5 for CeRu 2Si2can be understood as if the small
doughnutlike FS surrounding the /H9003point for CeRu 2Ge2in
the paramagnetic phase expands and touches the cylindri-cally shaped FS along the k
zdirection centered at the Xpoint.
The detailed difference of the band structures betweenCeRu
2Si2and CeRu 2Ge2are shown in Fig. 6and will be
discussed below.
B. 3dcore-level hard x-ray photoelectron spectroscopy and x-
ray absorption spectroscopy
In order to clarify the bulk Ce 4 fstates in CeRu 2X2,w e
have performed the Ce 3 dcore-level HAXPES and Ce 3 d-4f
XAS. Figures 7/H20849a/H20850and7/H20849b/H20850show HAXPES results at 20 K
for CeRu 2Si2and CeRu 2Ge2, respectively. The f0andf2pho-
toelectron emission final state components in CeRu 2Si2spec-
trum are stronger than in the spectrum of CeRu 2Ge2, whose
f0contribution is very small. In Figs. 7/H20849a/H11032/H20850and7/H20849b/H11032/H20850XASresults for CeRu 2X2are shown. A shoulder structure around
887.8 eV can be seen in CeRu 2Si2spectrum, while it is very
weak for CeRu 2Ge2. This shoulder structure is mainly repre-
sented by the f1XAS final state component. These differ-
ences in the spectra between CeRu 2Si2and CeRu 2Ge2reflect
the relatively itinerant 4 fcharacter for CeRu 2Si2.
In order to estimate the 4 felectron number nfandfn/H20849n
=0,1,2 /H20850contributions in the initial state for CeRu 2Si2and
CeRu 2Ge2, we have fitted both 3 dcore-level HAXPES and
XAS spectra by the SIAM calculation with unique parametersets. The optimized parameters in the calculation are the bare4fbinding energy
/H9280f, the 4 f-4fon-site Coulomb repulsive
energy Uff, the 4 f-core-level Coulomb attractive energy Ufc,
and the hybridization strength Vdefined by /H20881Nv/H20849Nis 21 in
this calculation /H20850, where vis the hybridization strength be-
tween the 4 fand one discrete level /H20849same definition as in
Ref. 38/H20850. The mean hybridization strength often used for the
SIAM calculations45defined by /H9004/H11013/H20849/H9266/B/H20850/H208480B/H9267v2/H20849E/H20850dE,
where /H9267v2/H20849E/H20850is the energy dependence of the hybridization
strength between the 4 flevel and continuum valence band,
can be evaluated as /H11011/H20849/H9266/B/H20850V2/H20849Bis 4 eV in this calcula-
tion/H20850.
The results of the SIAM calculation are summarized in
Fig.7and Table I. As shown in Fig. 7, the SIAM calculation
well reproduces the experimental spectra by using the uniqueparameter set for each compound.
46The optimized param-
eters are comparable to those in the NCA calculation for thebulk 4 fphotoemission spectra.
7The estimated nfis very
close to 1 for CeRu 2Ge2, reflecting its localized 4 fcharacter.
Still, there is a tiny amount of “non- f1” contributions in the
initial state for CeRu 2Ge2in the paramagnetic phase at 20 K.
The shift of EFcompared with that for LaRu 2Ge2seen in the
ARPES results for CeRu 2Ge2is thought to be attributable to
these non- f1components. nfis also found to be close to 1 for
CeRu 2Si2. However, the f0contribution is about twice larger
and the f2weight is apparently larger than for CeRu 2Ge2.
The f1contribution in the initial state is less than 0.9 butCeRu2Si2
CeRu2Ge2(0,0)0
0.5
0
0.5
0 1 0 1(c)
(d)(e)
(f)(a)
(b)
112
2
33
3
34455
55
Binding Energy (eV) Momentum (unit of 1/a)(0,0)π/2,
π/2
(π,π) (0,0)Z
Γ XΓ X
X
X
Z()
π/2,
π/2()
π/2,
π/2()π/2,
π/2()Binding Energy (eV)
FIG. 6. /H20849Color online /H20850The expanded EDC- and MDC-displayed
ARPES spectra near EFof CeRu 2X2at 20 K. Upper figures are for
CeRu 2Si2and lower figures are for CeRu 2Ge2in the paramagnetic
phase, respectively. The numbered dashed lines which representrespective bands are guides to the eye. /H20849a/H20850–/H20849d/H20850are EDCs along the
Z-Xdirection at k
z/H110112/H9266/c. The photon energies are /H20851/H20849a/H20850and /H20849c/H20850/H20852
725 eV and /H20851/H20849b/H20850and /H20849d/H20850/H20852755 eV, respectively. /H20849e/H20850and /H20849f/H20850are MDCs
along the /H9003-Xdirection at kz/H110110. The photon energies are /H20849e/H20850
725 eV and /H20849f/H20850820 eV, respectively. The details for CeRu 2Ge2can
be seen in Ref. 19(a)
Binding Energy (eV)940 920 900 880 880 880 890 890
Photon Energy (eV)(a’) (b’)
(b)CeRu2Si2
20 K
CeRu2Si2
20 K
Ce 3 d-4f
XASCeRu2Ge2
20 K
Ce 3 d-4f
XAS
CeRu2Ge2
20 Kf0
3d94f1f1
3d94f2
3d94f33d94f1
3d94f2
3d94f3f2
f0
f1
f2Exp.
Exp.
FitExp.FitFit
BG
Exp.FitBGhν=8175 eV
hν=8170 eV
FIG. 7. /H20849Color online /H20850Ce 3dHAXPES spectra /H20849h/H9263=8175 and
8170 eV /H20850and the fitted results; /H20849a/H20850CeRu 2Si2and /H20849b/H20850CeRu 2Ge2.
The Ce 3 d-4fXAS spectra /H20849Ref. 32/H20850and fitted results; /H20849a/H11032/H20850
CeRu 2Si2and /H20849b/H11032/H20850CeRu 2Ge2. The dots represent the experimental
data and the solid lines in the upper part of each figure are fittedresults. The dashed lines are the background /H20849BG/H20850. The deconvo-
luted f
n/H20849fn+1/H20850final states in the 3 dHAXPES /H20849XAS /H20850spectra com-
ponents are also shown in the bottom of each figure.ELECTRONIC STRUCTURE OF CeRu 2X2/H20849X=… PHYSICAL REVIEW B 77, 035118 /H208492008 /H20850
035118-5considerably larger than those for such strongly valence-
fluctuating systems as Ce T2/H20849T=Fe, Rh, Ni, and Ir /H20850.41,45,48We
conclude that this “in-between” value of the initial f1weight
reflects the heavy fermion behavior and metamagnetic tran-sition /H20849localization of the 4 fstate /H20850under high magnetic fields
for CeRu
2Si2.
IV. DISCUSSION
The dHvA measurements for CeRu 2Si2have shown four
FSs,18from which the effective mass of each FS has been
estimated. The effective mass has been reported as 1.5 m0for
the FS of band 2, 1.6 m0for the FS of band 3, 120 m0for the
FS of band 4, and 10–20 m0for the FS of band 5 /H20849m0mass of
a free electron /H20850.50If a band had the heavy electron character,
the slope of the band dispersion close to EFwould be very
small and spectral weight near EFwould be weak. Thus, the
slope of band 4 is thought to be small from the dHvA resultssince band 4 has the heaviest effective mass. The smallerslope of band 4 near E
Fhas been observed in our ARPES
measurements as revealed in Figs. 2and 3. Furthermore,
intensity of band 4 is weaker than any other observed bandsdue to smaller renormalization factor. The ARPES resultssuggest that the effective mass of band 4 is largest among thewhole bands forming FSs, being consistent with the dHvAresults.
Apart from band 4, most of the band structures of
CeRu
2Si2revealed by the soft x-ray ARPES results resemble
those obtained by the LDA calculation for CeRu 2Si2,49which
treats 4 felectrons as itinerant. The band calculation for
CeRu 2Si2/H20849Ref. 49/H20850is shown in Fig. 8. When we compare
Fig. 2of the ARPES results and Fig. 8of the LDA calcula-
tion, the qualitative consistency between the experimentalresults and the band calculation is recognized. The shape ofeach band along the Z-Xdirection obtained by ARPES is
similar to that by the calculation. The approach of the fourbands 2–5 toward the Xpoint is remarkably similar to thecalculation which predicts the approach of bands 3–5 at the
Xpoint. However, it was experimentally found that band 1
does not cross E
F, as shown in Fig. 6/H20849c/H20850in spite that the band
calculation49,51predicts that the band 1 crosses EFnear the Z
point. Furthermore, the absence of the FS derived from band1 in our ARPES is consistent with the results of dHvA mea-surement for CeRu
2Si2.
Figures 6/H20849c/H20850and6/H20849d/H20850are show the expanded figures of
EDCs near the Zpoint for CeRu 2X2. The peak position of
band 1 at the Zpoint for CeRu 2Si2is located at higher bind-
ing energies than that of CeRu 2Ge2. Band 1 peak position of
CeRu 2Si2at the Zpoint is 0.27 eV, whereas that of
CeRu 2Ge2is about 0.14 eV. Additionally, Figs. 6/H20849a/H20850and6/H20849b/H20850
show the difference between the bottom positions of band 4and band 3 in CeRu
2X2. The position of each band of
CeRu 2Si2lies in higher binding energies than those of
CeRu 2Ge2near the Xpoint. The prominent difference of the
band structures between CeRu 2X2appears for band 5, as
shown in Figs. 6/H20849e/H20850and6/H20849f/H20850. This difference of band 5 ob-
tained from ARPES is in good agreement with the differenceof the band calculations between LaRu
2Ge2and
CeRu 2Si2.26,49Band 5 of LaRu 2Ge2crosses EFthree times in
the region along the /H9003-Xdirection, while that of CeRu 2Si2
crosses only once in the same region. When we take a dif-
ferent perspective, it is possible to think that EFof CeRu 2Si2
is shifted to lower binding energies or to the unoccupied side
compared to CeRu 2Ge2along the /H9003-Xdirection.
These differences of the band position in CeRu 2X2is
roughly understood if EFof CeRu 2Si2is energetically higher
than that of CeRu 2Ge2in the paramagnetic phase. The EF
shift of CeRu 2Si2from CeRu 2Ge2in the paramagnetic phase
is caused by the increased number of electrons contributingto the bands forming the FSs in CeRu
2Si2due to the hybrid-
ization with the 4 felectrons. As confirmed by the SIAM
calculation of the 3 dPES and XAS spectra and by the 3 d-4f
resonance PES, the hybridization is stronger for CeRu 2Si2
than for CeRu 2Ge2indicating the EFshift mentioned above.
For more precise understanding of the difference in the
electronic structures between CeRu 2Si2and CeRu 2Ge2, this
“rigid-band-like” energy shift is not sufficient because theband structures themselves can be modified due to the differ-ent hybridization strengths at different kvalues. Indeed, such
a modification can be seen in Figs. 6/H20849a/H20850–6/H20849d/H20850. For instance,
bands 3–5 are nearly degenerated at the Xpoint in CeRu
2Si2,
whereas these are energetically separated in CeRu 2Ge2. Band
5 dispersion along the /H9003-Xdirection shown in Figs. 6/H20849e/H20850and
6/H20849f/H20850, especially near the /H9003point, is also essentially modified
because the simple EFshift would lead to the shift of kFdue
to band 5 indicated by the bold dotted line in Figs. 6/H20849e/H20850and
6/H20849f/H20850toward the /H9003point for CeRu 2Si2, which is inconsistent
with the experimental results. In this way, the essential dif-TABLE I. Optimized parameters /H20849/H92804f,Uff,Ufc, and Vgiven in units of eV /H20850and estimated f0,f1, and f2
contributions and nfin the initial state for CeRu 2X2by the SIAM calculation.
/H92804f Uff Ufc Vf0f1f2nf
CeRu 2Si2 1.7 7.0 10.6 0.295 0.060 0.894 0.047 0.987
CeRu 2Ge2 1.7 7.0 10.6 0.234 0.028 0.942 0.030 1.002
Γ∆ XYZ0
12Binding Energy (eV)CeRu2Si2
1
0345
2
FIG. 8. /H20849Color online /H20850The band calculation with augmented
plane wave /H20849APW /H20850method for CeRu 2Si2/H20849Ref. 49/H20850along the /H9003-X
andZ-Xdirections within a region between −0.5 and 2 eV. Only
bands from 0 to 5 are displayed.YANO et al. PHYSICAL REVIEW B 77, 035118 /H208492008 /H20850
035118-6ferences in the band structures between CeRu 2Si2and
CeRu 2Ge2are experimentally clarified.
V. CONCLUSION
We have performed bulk-sensitive 3D ARPES Ce 3 d
core-level HAXPES and XAS for a heavy fermion systemCeRu
2Si2a n da4 f-localized system CeRu 2Ge2by using soft
and hard x rays. The detailed band structures and the shapesof the FSs of CeRu
2Si2are revealed and they are found to be
different from those of CeRu 2Ge2. The differences between
them are consistent with the differences between the calcu-lation for 4 felectron itinerant model and localized one. The
FS shapes of CeRu
2X2are consistently understood as the
reflection of the hybridization strength between the Ce 4 f
and valence electrons, which is revealed by the analysisbased on SIAM for the Ce 3 dHAXPES and 3 d-4fXASspectra. Each mean 4 felectron number of CeRu
2X2is quan-
titatively estimated in agreement with the qualitative changesin FSs.
ACKNOWLEDGMENTS
We are grateful to H. Yamagami for fruitful discussions.
We thank J. Yamaguchi, T. Saita, T. Miyamachi, H. Higash-imichi, and Y . Saitoh for supporting the experiments. Thesoft x-ray ARPES was performed under the approval of theJapan Synchrotron Radiation Research Institute /H20849Proposal
Nos. 2004A6009, 2006A1167, and 2007A1005 /H20850. This work
was supported by the Grant-in-Aids for Scientific Research/H2084915G213, 1814007, and 18684015 /H20850of MEXT, Japan, and the
21st Century COE program /H20849G18 /H20850of JSPS, Japan. This work
was also supported by the Asahi Glass Foundation andHyogo Science and Technology Association.
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30The detectable angle along the slit is about 10°. /H20841/H9003-X/H20841distance of
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corresponding to this distance is about 4.2° at h/H9263=800 eV. Thus,
the analyzer can cover whole Brillouin zone.ELECTRONIC STRUCTURE OF CeRu 2X2/H20849X=… PHYSICAL REVIEW B 77, 035118 /H208492008 /H20850
035118-731If an x ray was incident onto a sample at 45° with respect to the
surface normal, for example, this incident photon has the mo-mentum parallel /H20849q
/H20648/H20850and perpendicular /H20849q/H11036/H20850to the surface.
When h/H9263/H11011800 eV, the photon momentum values of both /H20841q/H20648/H20841
and /H20841q/H11036/H20841are about 0.29 Å−1.
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34The Ce 4 sintensity is relatively stronger than that of the Ru 3 d
excitations because the relative /H20849Ce 4s/H20850//H20849Ru 3d/H20850cross section is
large at h/H9263=8180 eV /H20849/H110111.5/H20850compared with that at such con-
ventional soft x-ray excitations as h/H9263=1486 eV /H20849/H110110.1/H20850/H20849Ref.
35/H20850.
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035118-8 |
PhysRevB.72.085423.pdf | Classical field theory of transport of interacting classical particles through one-dimensional
channels
Xiang Xia and Robert J. Silbey *
Theoretical Chemical Physics Group, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139,
USA
/H20849Received 31 January 2005; revised manuscript received 16 May 2005; published 8 August 2005 /H20850
A classical field theory is constructed to describe the interacting classical particles in one-dimensional /H208491D/H20850
channels. We show the intrinsic connection between the classical field theory and the corresponding quantumtheory, i.e., Tomonaga-Luttinger liquid theory, which describes the interacting fermions in 1D. As applications,we calculate the particle density function along the tube axis and the particle current through the channels byexplicitly including the particle-particle and particle-wall interactions and we find quantum-classical correspon-dence in the conductance formula.
DOI: 10.1103/PhysRevB.72.085423 PACS number /H20849s/H20850: 47.60. /H11001i, 03.50.Kk, 05.60.Cd
Particle dynamics in one dimension /H208491D/H20850is of great cur-
rent interest, in both the classical and the quantum mechani-cal regimes. In the latter case, the dimensionality constraintplays an extremely important role; due to the peculiar topol-
ogy of the Fermi surface in one dimension, strongly corre-lated electrons in one dimension are described by theTomonaga-Luttinger model
1,2rather than the usual Fermi liq-
uid model. On the other hand, although there have been anumber of important theoretical studies of the dynamics ofclassical particles, e.g., water molecules and ions in one-dimensional channels,
3–7the role of the dimensionality con-
straint in such systems has not been considered in a funda-mental manner. A comparison of and a possible unification ofthe classical and quantum theories of transport in one-dimensional mesoscopic channels is therefore of both theo-retical and experimental interest.
Molecular dynamics studies of the motion of water mol-
ecules in hydrophobic and/or hydrophilic channels
3,4and
proton transport in carbon nanotubes5have provided evi-
dence for such striking phenomena as rapid transport, burst-like transmission, and particle density oscillations along thetube axis. Contrasting probabilistic models have been sug-gested to describe these effects: a continuous time randomwalk /H20849CTRW /H20850model
6in which a chain of water molecules
moves as a whole, and a single particle sequential hoppingmodel without particle-particle correlation.
7These contrast-
ing views suggest the need for a microscopic theory thatexplicitly includes the particle-particle /H20849PP/H20850interactions as
well as the particle-wall /H20849PW /H20850interactions.
In this work, we construct a classical field theory of inter-
acting classical particles in 1D channels from general con-servation principles and dimensionality constraints. Its rela-tion to the theory of interacting fermions in 1D /H20849Refs. 1 and
8/H20850is established at the Hamiltonian level. As applications,
we obtain the particle density function /H20849PDF /H20850along the tube
axis /H20849thezˆdirection /H20850and an expression for the particle cur-
rent, and compare with earlier results.
Generally in kinetic theory, the PW interaction is ne-
glected and the PP interactions are difficult to treat exactly inphase space, e.g., the Bogoliubov-Born-Green-Kirkwood-Yvon /H20849BBGKY /H20850hierarchy. However, the available phasespace is severely restricted by the physical dimensionality,
which makes some special models exactly solvable.
Consider a system of Nparticles of mass m, confined in a
1D channel of length L, subjected to periodic boundary con-
ditions. Define a simple system by the following interactions:/H208491/H20850hard-sphere exclusion; /H208492/H20850elastic collisions. Such a
simple system is equivalent to a system of free particles, ifthe following conditions hold: /H20849i/H20850spatial dimension d=1; /H20849ii/H20850
identical particles; /H20849iii/H20850/H9004t/H11022
/H9270c,/H9004x/H11022r0where /H9004t,/H9004xare
temporal, spatial resolutions and /H9270c,r0are the collision time
and the effective diameter of the particles, respectively. Theproof is as follows:
/H208491/H20850Collisions: For particles 1 and 2, let p
1,p2be the mo-
menta before the collision and p1/H11032,p2/H11032be the momenta after
the collision. Since d=1, collisions are head on collisions.
Since collisions are elastic in a simple system, p1/H11032=p2,p2/H11032
=p1. Since particles are identical, we can define a set of new
particles or modes which propagate freely on the scales setby conditions /H20849iii/H20850.
/H208492/H20850No collisions: Each particle propagates freely.
The newly defined set of particles /H20849or modes /H20850for a simple
system are many body in nature and they greatly simplify theproblem by maximally utilizing the conservation of particles,as will become clear later. We will refer to this new setsimply as particles or momentum /H20849velocity /H20850modes in the rest
of this paper.
ForNparticles /H20853x
n/H20849t/H20850/H20854, the density of particles /H9267/H20849x,t/H20850and
the current density J/H20849x,t/H20850are
/H9267/H20849x,t/H20850=/H20858
n/H9254/H20849x−xn/H20849t/H20850/H20850, /H208491/H20850
J/H20849x,t/H20850=/H20858
ndxn/H20849t/H20850
dt/H9254/H20849x−xn/H20849t/H20850/H20850. /H208492/H20850
If the Nparticles are subjected to no external forces, the
energy-momentum tensor T/H9262/H9263in the nonrelativistic regime isPHYSICAL REVIEW B 72, 085423 /H208492005 /H20850
1098-0121/2005/72 /H208498/H20850/085423 /H208495/H20850/$23.00 ©2005 The American Physical Society 085423-1T/H9262/H9263/H20849t,x/H20850=/H20858
nPn/H9262/H20849t/H20850/H9254/H20849x−xn/H20849t/H20850/H20850dXn/H9263
dt,/H9262= 0,1, /H208493/H20850
where Xn/H9262/H11013/H20849t,xn/H20849t/H20850/H20850andPn/H9262/H20849t/H20850/H11013/H20849/H9255n/H20849t/H20850,pn/H20849t/H20850/H20850are defined as
two-vectors and two-momenta, respectively, with pn/H20849t/H20850
=mdx n/H20849t/H20850/dt,/H9255n/H20849t/H20850=pn2/H20849t/H20850/2m. For free particles, we have
dPn/H9262/H20849t/H20850/dt=0. The conservation of the total energy and mo-
mentum can be expressed by the continuity equation satisfied
by the energy-momentum tensor
/H11509/H9263T/H9262/H9263=0 . /H208494/H20850
The/H9262=1 component of the Eq. /H208494/H20850corresponds to the con-
servation of total momentum, which can be written in termsofJ/H20849x,t/H20850as
/H11509tJ/H20849x,t/H20850=−/H11509x/H20858
v/H20858
/H20841u/H20841=vu2/H20858
nu/H9254/H20849x−xnu/H20849t/H20850/H20850, /H208495/H20850
where nu/H33528/H20853n/H20841dxn/H20849t/H20850/dt=u/H20854. A summation on vis over all
the velocity modes in the N-particle system. /H20849Particles that
propagate with ± v, by definition, belong to the same velocity
mode v./H20850The conservation of particles is the equation of
motion of the particle density /H9267/H20849x,t/H20850
/H11509t/H9267/H20849x,t/H20850=−/H11509xJ/H20849x,t/H20850. /H208496/H20850
Define the dynamical field /H9278/H20849x,t;v/H20850and its conjugate mo-
mentum density /H9266/H20849x,t;v/H20850such that
/H11509x/H9278/H20849x,t;v/H20850/H11013/H92670−/H20858
nu
/H20841u/H20841=v/H9254/H20849x−xnu/H20849t/H20850/H20850=/H92670−/H9267/H20849x,t;v/H20850,/H208497/H20850
/H9266/H20849x,t;v/H20850/H11013/H6036v
v/H20858
nu
/H20841u/H20841=vu/H9254/H20849x−xnu/H20849t/H20850/H20850, /H208498/H20850
where /H92670is a constant which may be different for different
models; however, the e.o.m. of /H9278/H20849x,t;v/H20850and/H9266/H20849x,t;v/H20850will
not be changed by this constant. /H6036vhas the dimension of
action and it is a constant for each velocity mode v. We can
obtain the e.o.m. of /H9278/H20849x,t;v/H20850and/H9266/H20849x,t;v/H20850immediately
from Eqs. /H208495/H20850and /H208496/H20850
/H11509t/H9278/H20849x,t;v/H20850=v/H9266/H20849x,t;v/H20850//H6036v, /H208499/H20850
/H11509t/H9266/H20849x,t;v/H20850=/H6036vv/H11509x2/H9278/H20849x,t;v/H20850. /H2084910/H20850
Up to a surface term and a constant, the noninteracting field
Hamiltonian for the classical particles in 1D is obtained
H0=/H20858
v/H6036vv
2/H20885
0L
dx/H20853/H92662/H20849x;v/H20850//H6036v2+/H20851/H11509x/H9278/H20849x;v/H20850/H208522/H20854. /H2084911/H20850
In constructing the Hamiltonian /H2084911/H20850, we employ the con-
tinuity equations of the conserved quantities: particles, en-ergy, and momentum. It is worth pointing out that in 1D theconservation of particles is more fundamental than the othertwo due to the unique dimensional and geometric constraint.In addition, the defined free particles /H20849momentum modes /H20850
facilitate the possibility to identify the density current Eq. /H208499/H20850and the axial current Eq. /H2084910/H20850from the conservation of total
particles Eq. /H208496/H20850, upon which we find the dynamical field
/H9278/H20849x,t;v/H20850and its conjugate momentum density /H9266/H20849x,t;v/H20850.
Furthermore, the field Hamiltonian /H2084911/H20850is a general struc-
tural Hamiltonian, which is determined by the conservation
laws and the 1D constraint, yet the definite physical meaningof the fields
/H9278/H20849x,t;v/H20850and field parameters /H92670and/H6036vare still
up to specific models.
By canonical quantization
/H20851/H9278/H20849x;v/H20850,/H9266/H20849y;v/H20850/H20852=i/H6036/H9254/H20849x−y/H20850,
the Hamiltonian /H2084911/H20850is then a collection of massless free
boson scalar fields. If only one of the velocity modes is im-portant, which is the case for the electrons confined in 1D atthe low energy regime, and upon velocity and field renormal-ization due to electron-electron interactions, the Tomonaga-Luttinger Hamiltonian
8,9is obtained. In fact, Dzyaloshinskii
and Larkin’s solution10to the 1D interacting fermion prob-
lem using Ward identities also relies on the realization of theimportant role played by the conservation of charges /H20849par-
ticles /H20850in 1D: in the 1+1 dimensions, the conservation of
axial charges /H20849particles /H20850together with total charges /H20849par-
ticles /H20850determine the low energy structure of theory. This is
directly related to the 1D constraint /H20849disconnected Fermi
surface /H20850.
11
The physical interpretation of the classical field Hamil-
tonian /H2084911/H20850and the origin of the various parameters: /H92670,/H6036v
can be made clear by considering the following 1D lattice
model of Nsites, with occupation number ni/H33528/H208530,1 /H20854,
∀i/H33528N. Without losing generality, examine a subset of all
the particles which propagate with the same velocity ± v.
Define the dynamical variable /H9278˜ito be/H9278˜i/H11013−/H20858j/H33355inj. For 1D
geometry, it is convenient to separate the left /H20849L/H20850and the
right /H20849R/H20850moving particles /H9278i,L/H20849R/H20850/H11013−/H20858j/H33355inj,L/H20849R/H20850,nj,L/H20849R/H20850
/H33528/H208530,1 /H20854with the properties ni,L+ni,R=niandni,Lni,R=0. So
the free part of the Hamiltonian can be written in terms of
these left and right moving particles
H0,v=mv2a/H20858
i,ra/H20873ni,r
a/H20874/H20873ni,r
a/H20874−mv2
2/H20858
ia/H20873ni
a/H20874, /H2084912/H20850
where ais the lattice spacing and r=L,R. The form of the
lattice Hamiltonian is not unique at first glance; however, in1D, since the structure of continuum limit Hamiltonian /H2084911/H20850
isdetermined by conservation laws Eq. /H208495/H20850and Eq. /H208496/H20850, the
corresponding form of lattice Hamiltonian is restricted thatof/H2084912/H20850.
In the continuum limit
/H9278˜i→/H9278˜/H20849x;v/H20850=−/H208480xdy/H9267/H20849y;v/H20850,
/H9278i,L/H20849R/H20850→/H9278L/H20849R/H20850/H20849x;v/H20850=−/H208480xdy/H9267L/H20849R/H20850/H20849y;v/H20850and the Hamiltonian
/H2084912/H20850becomes
H0,v→/H6036vv/H20885
0L
dx/H20853/H20851/H11509x/H9278L/H20849x;v/H20850/H208522+/H20851/H11509x/H9278R/H20849x;v/H20850/H208522/H20854−E0,
/H2084913/H20850
where /H6036v/H11013mvaandE0=/H20849mv2/2/H20850/H208480Ldx/H20851−/H11509x/H9278˜/H20849x;v/H20850/H20852. Defining
the dynamical field /H9278/H20849x;v/H20850and its conjugate momentum
density /H9266/H20849x;v/H20850byX. XIA AND R. J. SILBEY PHYSICAL REVIEW B 72, 085423 /H208492005 /H20850
085423-2/H11509x/H9278/H20849x;v/H20850/H11013/H11509x/H9278˜/H20849x;v/H20850+/H92670=−/H9267/H20849x;v/H20850+/H92670, /H2084914/H20850
/H9266/H20849x;v/H20850/H11013/H6036v/H20851/H9267R/H20849x;v/H20850−/H9267L/H20849x;v/H20850/H20852
=−/H6036v/H20851/H11509x/H9278R/H20849x;v/H20850−/H11509x/H9278L/H20849x;v/H20850/H20852 /H20849 15/H20850
with/H92670=1/2 a=N/2L, we obtain the field Hamiltonian
H0,v=/H6036vv
2/H20885
0L
dx/H20877/H9266/H20849x;v/H208502
/H6036v2+/H20851/H11509x/H9278/H20849x;v/H20850/H208522/H20878. /H2084916/H20850
In this lattice model, /H92670is taken to be a uniform density of
half a particle per site, so − /H11509x/H9278/H20849x,t;v/H20850is equivalent to a spin
−1
2density with +1/2 aspin up /H20849site occupied /H20850and −1/2 a
spin down /H20849site unoccupied /H20850./H6036vis defined such that the defi-
nition of /H9266/H20849x,t/H20850Eq. /H2084915/H20850is consistent with the actual mo-
mentum density. A summation over all the velocity modes
will then give the field Hamiltonian /H2084911/H20850.
We now consider PP /H20849besides the hard-sphere interaction /H20850
and PW interactions. The general form of the two-body PPinteraction in 1D is
H
PP=1
2/H20885
0L
dx dy V PP/H20849x,y/H20850/H9267/H20849x/H20850/H9267/H20849y/H20850=H2+H4. /H2084917/H20850
H2is the forward scattering between the left /H20849L/H20850and the right
/H20849R/H20850branches, while H4is the forward scattering in the same
branch. The PP interaction will generally couple the dynam-
ics of individual particles and thus complicate the micro-
scopic treatment of the transport. To simplify the theoreticaldescription, a standard procedure would be finding the nor-mal modes of the interaction. The many-body velocity modes
defined above are the normal modes of the hard-sphere PPinteraction. These normal modes renormalize only whenthere is also a nonlinear interaction, e.g., the soft part of thePP potential, which generates inelastic scattering processes.Obviously, these processes do not ensure a perfect gas ap-proximation or an isentropic flow. The exact solution to the1D imperfect and nonisentropic flow is a difficult task.
12
Nevertheless, we proceed to propose a simplified modelbased on the generic results observed in molecular dynamicssimulations:
3,6,15/H208491/H20850a threshold energy Eth/H11011kBTexists for
interacting particles to enter the 1D channel, so only a fewactivated modes are responsible for the transport; /H208492/H20850inter-
acting particles transport through a 1D channel is highly col-lective unhindered by the interactions with the walls. A con-certed motion in the channel is observed; /H208493/H20850the time series
of the number of particles transported through a 1D channelfalls into a narrow range /H1101120/ns. These simulation results
suggest a simplified two-parameter model: The first param-eter
v0is the typical velocity mode responsible for the trans-
port through a 1D channel connecting to two fluid reservoirs.In fact, small perturbations to an equilibrium fluid wouldpropagate with the velocity of sound, which are then trans-mitted into a 1D channel. The boundary conditions requirethe pressures and normal velocity components of the inci-dent, reflected, and transmitted waves to be equal at the con-tact regions. The second parameter Kdescribes the PP inter-
action within the mode
v0, which is described below. Even
with these simplifications, we find the theory exhibits richphysical phenomena and qualitatively explained the results
of the simulations.
In 1D and the continuum limit, the PP interaction can be
taken to be local;13we therefore have
H2=−/H6036v0g2
2/H20885
0L
dx/H20875/H92662
/H60362−/H20849/H11509x/H9278/H208502/H20876, /H2084918a /H20850
H4=/H6036v0g4
2/H20885
0L
dx/H20875/H92662
/H60362+/H20849/H11509x/H9278/H208502/H20876, /H2084918b /H20850
where /H6036/H11013mv0a, and g2=g4=Vpp/2mv02for density-density
interactions but can be taken as parameters in a more generalcase. g
2andg4are negative if the PP interaction is attractive
and they are positive if the PP interaction is repulsive. ThePP interaction renormalizes the velocity of the density waveand the fields.
In the nanoscale, the inhomogeneous PW interaction,
which can arise either from the atomic structure of carbonnanotube wall or the complex composition of the cell mem-brane, etc., become important and cannot be neglected. Thegeneral form of the PW interaction is
H
PW=−/H20885
0L
dx V PW/H20849x/H20850/H20849/H11509x/H9278/H20850, /H2084919/H20850
where VPW/H20849x/H20850is equivalent to a spatial varying magnetic
field. The total Hamiltonian is obtained
H=/H6036v
2/H20885
0L
dx/H20875K/H92662
/H60362+1
K/H20849/H11509x/H9278/H208502/H20876−/H20885
0L
dx V PW/H20849x/H20850/H20849/H11509x/H9278/H20850
=H0+HPP+HPW. /H2084920/H20850
The renormalized density wave velocity v=v0/H20881/H208491+g4/H208502−g22.
The PP interaction parameter is K
=/H20881/H208491−g2+g4/H20850//H208491+g2+g4/H20850, with K/H110221 for attractive PP in-
teraction, K/H110211 for repulsive PP interaction, and K=1 for
noninteracting particles. In the case of inhomogeneous inter-action,
vandKare spatial dependent.
The particle density along zˆdirection is obtained by solv-
ing the canonical e.o.m. of the field /H9278/H20849x,t/H20850
/H11509t2/H9278=v2/H11509x2/H9278+vK
/H6036dVPW
dx/H2084921/H20850
subject to periodic boundary condition and initial conditions.
The contribution from initial conditions are time averaged tozero and the steady PDF is
/H9267/H20849x/H20850=/H92670−KV PW/H20849x/H20850
/H6036v=/H9267¯+/H9267s/H20849x/H20850. /H2084922/H20850
The average particle density /H9267¯and the variation density /H9267s/H20849x/H20850
due to channel structure are
/H9267¯=/H92670/H208731−2V¯PW
VPP+mv02/H20874, /H2084923a /H20850CLASSICAL FIELD THEORY OF TRANSPORT OF … PHYSICAL REVIEW B 72, 085423 /H208492005 /H20850
085423-3/H9267s/H20849x/H20850=−2VPW/H20849s/H20850/H20849x/H20850/H92670
VPP+mv02, /H2084923b /H20850
where V¯PW=/H208491/L/H20850/H208480Ldx V PW/H20849x/H20850is the average PW interac-
tion. VPW/H20849s/H20850/H20849x/H20850=VPW/H20849x/H20850−V¯PWis the inhomogeneous part of the
PW interaction and it is directly related to the wall structure.
The intrinsic parameter /H92670of the field theory appeared in the
density function /H9267/H20849x/H20850can be scaled away by defining the
corresponding density /H9267r/H20849x/H20850/H11013/H9267/H20849x/H20850//H9267¯
/H9267r/H20849x/H20850=1−2VPW/H20849s/H20850/H20849x/H20850
/H20849VPP−2V¯PW/H20850+mv02, /H2084924/H20850
which is a good observable. We find /H208491/H20850the structure of the
PDFs /H2084923b /H20850and /H2084924/H20850is determined by the channel wall struc-
ture and composition. We identify the particle density oscil-lation periodicity observed in Ref. 3 and Fig. 3 /H208492.6 Å /H20850of
Ref. 4 as the periodicity of the atomic lattice wall. /H208492/H20850If the
1D channel is connected to reservoirs, the average particledensity /H2084923a /H20850is determined not only by the PW coupling
strength but also by the PP interaction and the particle kineticenergy. Note that the change of particle density due to PWinteraction modification is more sensitive for the attractivePP interaction than for the repulsive PP interaction.
The particle current through the 1D channel is directly
related to the dynamics of the field
/H9278/H20849x,t/H20850
J/H20849x,t/H20850=v0/H9266/H20849x,t/H20850
/H6036=/H11509t/H9278/H20849x,t/H20850. /H2084925/H20850
Consider an external driving field Uex/H20849x,t/H20850=e−i/H9275tU/H20849x/H20850+c.c.
coupled to the system. The corresponding interaction Hamil-
tonian is Hint=−/H208480Ldx U ex/H20849x,t/H20850/H11509x/H9278. The contact regions are
included as part of the 1D channel. In the limit of /H9275→0, the
steady state current is obtained
J/H20849x/H20850= lim
t→/H11009/H20855J/H20849x,t/H20850/H20856=K/H20849/H9004Uex+/H9004VPW/H20850
2/H6036, /H2084926/H20850
where the time average is taken. /H9004Uexand/H9004VPWare the
difference in the external potential and PW interaction be-tween the two contact regions. If the current is caused by thepressure difference, then /H9004U
ex=/H9262L−/H9262R, where /H9262L/Ris the
chemical potential of the left and/or right reservoir. The hy-draulic permeability L
Pis
LP=J/H20849x/H20850
/H9004P=K
2mv0r0/H20873/H11509/H9004/H9262
/H11509P/H20874
T. /H2084927/H20850
The expression of the current /H2084926/H20850is the same as that of the
electric current for a quantum wire, except that for the clas-sical particle current, /H6036depends on lattice spacing /H20849we take
a/H11011r
0/H20850and the bare density wave velocity. The current and
hydraulic permeability explicitly depend on the PP interac-tion, with higher current for the attractive PP interaction/H20849K/H110221/H20850and lower current for the repulsive PP interaction/H20849K/H110211/H20850. In this model, the steady state current does not de-
pend on the detailed PW interaction or the length of the
channel. This is in agreement with the moleculardynamics
14,15and experimental16results. However, if the PW
attractive interaction dominates over kinetic energy, then anactivated sequential transport theory is needed and the cur-rent will depend on the PW interaction strength and channellength.
Extensions of our model to structured particles can be
made straightforwardly. For example, water molecules withdipolar orientation can be mapped to electrons with spins.The classical theory will correspond to the TL liquid theorywith spin degree of freedom, where spin-charge separationwas predicted. Though the Umklapp backscattering is notpresent in the classical model, the hydraulic permeability/H2084927/H20850is determined by the interaction parameter Kresembling
the quantum case. If reservoirs together with the 1D channel
are modeled as an inhomogeneous system as a whole, thepermeability would be determined by the interaction param-eters of the reservoirs instead. So it would be interesting tostudy classically how sound waves of reservoirs are transmit-ted and reflected at the exits and/or entrances in the presenceof interactions. In particular, what is the classical mechanismof the transformation from the many-body reservoir modesto those of the 1D channel as compared to the quantumcase;
17,18what are the functional structures in biological 1D
systems that facilitate the transformation. On the other hand,the effects of classical PP interaction, repulsive or attractive/H20849e.g., hydrogen bonding /H20850, on particle current could provide
an interesting insight into the effects of electron-electron in-teraction on electric conductivity in 1D. In addition to theabove extensions, we note that a general solution of one-dimensional isentropic flow is given by Landau andLifshitz
12in terms of a linear differential equation valid for
imperfect and perfect gases but only easily solvable in thelatter case. It would be interesting to compare our resultsincluding particle-particle interactions with those of the treat-ment of Landau and Lifshitz. This will be considered in asubsequent paper.
In conclusion, we have constructed a classical field theory
for interacting classical particles /H20849structureless /H20850in 1D chan-
nels. The unification of the classical and quantum theories isa direct consequence of the 1D dimensionality constraint andthe conservation laws. Because of these constraints, the den-sity wave and/or particle-hole excitation is generic forparticle-conserved systems in 1D. In our simplified model,the field theoretical calculations showed that both PW andPP interactions are important to the filling process, while PPinteraction determines the steady state transport properties.We hope the simplified model will serve as a basis for ex-tension to more complicated situations.
This work is supported by the National Science Founda-
tion /H20849NSF /H20850under Grant No. CHE0306287.X. XIA AND R. J. SILBEY PHYSICAL REVIEW B 72, 085423 /H208492005 /H20850
085423-4*Corresponding author. Electronic address: silbey@mit.edu
1A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosonization
and Strongly Correlated Systems /H20849Cambridge University Press,
Cambridge, UK, 1998 /H20850.
2A. M. Chang, Rev. Mod. Phys. 75, 1449 /H208492003 /H20850.
3G. Hummer, J. R. Rasaiah, and J. P. Noworyta, Nature /H20849London /H20850
414 /H208498/H20850, 188 /H208492001 /H20850.
4T. W. Allen, S. Kuyucak, and S.-H. Chung, J. Chem. Phys. 111,
7985 /H208491999 /H20850.
5C. Dellago, M. M. Naor, and G. Hummer, Phys. Rev. Lett. 90,
105902 /H208492003 /H20850.
6A. Berezhkovskii and G. Hummer, Phys. Rev. Lett. 89, 064503
/H208492002 /H20850.
7T. Chou, Phys. Rev. Lett. 80,8 5 /H208491998 /H20850.
8F. D. M. Haldane, J. Phys. C 14, 2585 /H208491981 /H20850.
9C. L. Kane and M. P. A. Fisher, Phys. Rev. B 46, 15233 /H208491992 /H20850.
10I. E. Dzyaloshinskii and A. I. Larkin, Sov. Phys. JETP 38, 202
/H208491974 /H20850.
11J. Voit, Rep. Prog. Phys. 57, 977 /H208491994 /H20850.12Due to the fact that 1D flow must be a potential flow, an exact
solution to the 1D isentropic flow of a perfect gas can be found.The normal modes for the hard-sphere PP interaction and thesoundlike e.o.m of the renormalized field Eq. /H2084921/H20850are compat-
ible with that solution. See L. D. Landau and E. M. Lifshitz,Fluid Mechanics /H20849Pergamon Press, New York, 1959 /H20850.
13V. J. Emery, in Highly Conducting One-dimensional Solids , ed-
ited by J. T. Devreese et al. /H20849Plenum, New York, 1979 /H20850.
14A. I. Skoulidas, D. M. Ackerman, J. K. Johnson, and D. S. Sholl,
Phys. Rev. Lett. 89, 185901 /H208492002 /H20850.
15A. Kalra, S. Garde, and G. Hummer, Proc. Natl. Acad. Sci.
U.S.A. 100, 10175 /H208492003 /H20850.
16M. L. Zeidel, S. V. Ambudkar, B. L. Smith, and P. Agre, Bio-
chemistry 31, 7436 /H208491992 /H20850.
17V. V. Ponomarenko and N. Nagaosa, Phys. Rev. Lett. 83, 1822
/H208491999 /H20850.
18V. V. Ponomarenko and N. Nagaosa, Phys. Rev. B 60, 16865
/H208491999 /H20850.CLASSICAL FIELD THEORY OF TRANSPORT OF … PHYSICAL REVIEW B 72, 085423 /H208492005 /H20850
085423-5 |
PhysRevB.83.113306.pdf | PHYSICAL REVIEW B 83, 113306 (2011)
Finite-temperature spintronic transport through Kondo quantum dots: Numerical
renormalization group study
Ireneusz Weymann1,2,*
1Department of Physics, Adam Mickiewicz University, 61-614 Pozna ´n, Poland
2Physics Department, Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience,
Ludwig-Maximilians-Universit ¨at, Theresienstrasse 37, D-80333 Munich, Germany
(Received 18 January 2011; published 21 March 2011)
We study the temperature dependence of the linear conductance and tunnel magnetoresistance of quantum
dots coupled to ferromagnetic leads. Using the numerical renormalization group method, we determine transportproperties for a wide range of temperatures T, ranging from zero through the Kondo temperature T
Kup to
the high-temperature regime. We show that tunnel magnetoresistance in the local moment regime displays anonmonotonic dependence on Tand vanishes when T∼T
K. In addition, we also analyze the spin polarization P
of the linear conductance in the parallel configuration and show that Pis suppressed in the odd electron Coulomb
blockade valley and can be enhanced above the spin polarization of the leads in the even Coulomb valley.
DOI: 10.1103/PhysRevB.83.113306 PACS number(s): 72 .25.Mk, 73 .63.Kv, 85 .75.−d, 73.23.Hk
Transport properties of quantum dots and molecules tunnel-
coupled to external leads have already been a subject ofextensive theoretical and experimental studies for almost twodecades.
1,2Depending on the ratio between relevant system
parameters, such as the charging energy U, temperature T,
and the coupling to electrodes /Gamma1, one can observe a couple
of interesting phenomena ranging from, for example, theCoulomb blockade
1in the weak coupling regime, /Gamma1/T/lessmuch1, to
the Kondo effect3–6in the strong coupling regime, /Gamma1/T/greatermuch1,
provided T/lessorsimilarTK, where TKis the Kondo temperature. The
physics becomes even more exciting when the electrodesare made of a ferromagnetic material. In particular, when aquantum dot is coupled to magnetic leads, transport propertiesof the system strongly depend on the relative orientation of themagnetizations of electrodes—the conductance usually drops
when the magnetic configuration switches from the parallel
to the antiparallel one.
7,8The relative difference between the
currents flowing through the system in the two magneticconfigurations is described by the tunnel magnetoresistance(TMR) effect.
7In addition, in the strong coupling regime, the
spin dependence of tunneling processes may give rise to aneffective exchange field, δε
exch, which can spilt the dot level
and thus suppress the Kondo effect.9–13The exchange field is
thus another relevant energy scale that determines the transportproperties of the system.
Although spin-dependent transport has been analyzed
theoretically in both the weak
14–18and strong19–27coupling
regimes, the behavior of transport in the crossover betweenthese two limits is still rather unexplored. In experiments,crossing over from one limit to the other can be simplyrealized by changing the temperature, and in fact, one oftendoes measurements for parameters where neither /Gamma1/T/greatermuch1
nor/Gamma1/T/lessmuch1 can be assumed. Therefore the goal of this
Brief Report is to analyze the full temperature dependenceof the spin-dependent transport properties of a quantum dotcoupled to ferromagnetic leads. For this purpose, we employthe numerical renormalization group (NRG) method,
28–30with
the recent idea of a complete basis set defined in the discardedstates of the Wilson chain
31that is used to construct the full
density matrix of the system.32,33The model considered consists of a single-level quantum
dot coupled to external ferromagnetic leads whose magne-tizations are assumed to be aligned either in parallel (P)or antiparallel (AP) [see Fig. 1(a)]. The Hamiltonian of
the system is H=H
Leads+HDot+HTun, where HLeads=/summationtext
rσ/integraltextD
−D/epsilon1c†
rσ(/epsilon1)crσ(/epsilon1)d/epsilon1describes the electrons in the left
(r=L) and right ( r=R) leads, c†
rσ(/epsilon1) is the correspond-
ing creation operator, {c†
rσ(/epsilon1),cr/primeσ/prime(/epsilon1/prime)}=δrr/primeδσσ/primeδ(/epsilon1−/epsilon1/prime),
andDis the cutoff energy. The dot Hamiltonian reads
HDot=ε/summationtext
σd†
σdσ+Ud†
↑d↑d†
↓d↓, where d†
σcreates a spin-
σelectron of energy εin the dot level. The tunneling
processes between the dot and leads are described by HTun=/summationtext
rσ√/Gamma1rσ/π/integraltextD
−Dd/epsilon1[c†
rσ(/epsilon1)dσ+d†
σcrσ(/epsilon1)], where /Gamma1rσis the
spin-dependent coupling of the dot level to lead r.I nt h e
parallel configuration, the couplings can be written as /Gamma1P
r↑(↓)=
(1±pr)/Gamma1rfor the spin-up and spin-down electrons, where pr
denotes the spin polarization of lead r, while in the antipar-
allel configuration, they are /Gamma1AP
Lσ=/Gamma1P
Lσand/Gamma1AP
R↑(↓)=(1∓
pR)/Gamma1R. In the following, we assume pL=pR≡pand/Gamma1L=
/Gamma1R≡/Gamma1/2.
The key idea of NRG is logarithmic discretization of
the conduction band, which allows us to resolve transportproperties on energy scales approaching the Fermi energyof the leads in a logarithmic way.
28,29Using NRG, we can
calculate the spin-resolved spectral function of the dot level,A
σ(ω), which allows us to determine the spin-dependent linear
conductance:34
Gσ=e2
h4/Gamma1Lσ/Gamma1Rσ
/Gamma1Lσ+/Gamma1Rσ/integraldisplay
dωπA σ(ω)/parenleftbigg
−∂f(ω)
∂ω/parenrightbigg
,(1)
where f(ω) denotes the Fermi function. For the parallel
and antiparallel magnetic configurations, the conductance is
explicitly given by GP
↑(↓)=e2
h(1±p)π/Gamma1AP
↑(↓)(ω=0), and
GAP
↑(↓)=e2
h(1−p2)π/Gamma1AAP
↑(↓)(ω=0), with AP/AP
σ(ω) being the
spectral function in respective configuration. The change in theconductance when switching from one magnetic configurationto the other is described by the tunnel magnetoresistance
7
TMR=GP/GAP−1, where GP(AP)=/summationtext
σGP(AP)
σ.
113306-1 1098-0121/2011/83(11)/113306(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW B 83, 113306 (2011)
ωU
εU lellarap lellarapitna
εUωUAP PΓL ΓR
ε,U(a)
(b) (c)
FIG. 1. (Color online) (a) Schematic of a quantum dot coupled to
ferromagnetic leads. The leads’ magnetizations can form either par-
allel or antiparallel configurations. The normalized spectral function
AAP/Pin the (b) antiparallel (AP) and (c) parallel (P) configuration
as a function of energy ωand the position of the dot level ε.F o r
parameters, we assumed that U=1.2m e V , /Gamma1=0.1m e V , p=0.5,
andT=0.
When studying the temperature dependence of transport
characteristics, it is relevant to note that there are, in principle,two important energy scales: For a singly occupied dot, 0 >
ε>−U(odd Coulomb valley), T
Ksets the relevant energy
scale,35whereas in the case of an empty, ε>0, or a fully
occupied dot, ε<−U(even Coulomb valley), the coupling /Gamma1
is important. In addition, in the case of ferromagnetic leads,another relevant energy scale emerges: the effective exchangefieldδε
exch.
The normalized spectral functions in the antiparallel and
parallel configurations, AAP/P=/summationtext
r,σπ/Gamma1AP/P
rσAAP/P
σ,i nt h e
low-temperature regime, T//Gamma1/lessmuch1 andT/T K<1, are shown
in Figs. 1(b) and1(c). Experimentally, the position of the
dot level εcan be changed by sweeping the gate voltage.
First of all, we note that in the antiparallel configuration,for left-right symmetric systems, the resultant coupling isthe same for each spin direction, and the system behavesas if coupled to nonmagnetic leads—there is a pronouncedresonance at the Fermi level due to the Kondo effect for aCoulomb valley with an odd electron number [see Fig. 1(b)
for 0>ε>U ]. In the parallel configuration, on the other
hand, the couplings are different for each spin direction, whichresults in spin-dependent renormalization of the dot level.
22
As a consequence, the presence of ferromagnetic leads givesrise to an effective exchange field which splits the dot level,thus suppressing the Kondo resonance. At low temperatures,the exchange field depends on the dot level position as
22
δεexch/similarequal2p/Gamma1/π ln|ε/(ε+U)|. From this formula follows
that the magnitude and sign of δεexchdepend on εand the
exchange field vanishes at the particle-hole symmetry point,ε=−U/2, which is due to the balance between the electron
and hole cotunneling processes driving the Kondo effect.This is presented in Fig. 1(c), where one can see that for
ε/negationslash=−U/2,there is a splitting of the Kondo resonance, while
forε=−U/2,the resonance is restored. We also notice that
the spectral functions display typical Coulomb resonances forε=0 andε=−U[see Figs. 1(b)and1(c)].
G↑P(e2/h) GAP(e2/h) TΓ TΓ
εU εUG↓P(e2/h) GP(e2/h) TΓ TΓ
εU εU
TMR TΓ TΓ
εU εU
(a) (d)
(e) (b)
(c) (f)
FIG. 2. (Color online) The linear conductance in the (a) antipar-
allel and the (b) parallel configurations, GAP/P=/summationtext
σGAP/P
σ;t h e
spin-resolved conductance in the parallel configuration for (d) spin-up
GP
↑and (e) spin-down GP
↓; as well as (c) the TMR and (f) spin
polarization P=(GP
↑−GP
↓)/GPas a function of the temperature T
and the position of the dot level ε. The other parameters are the same
as in Fig. 1.
The behavior of spectral functions is directly reflected in the
linear conductance and thus in the TMR. The relevant transportcharacteristics are shown in Fig. 2as a function of temperature
Tand the dot level position ε. For low temperatures, that
is,T//Gamma1/lessmuch1 and T/T
K<1, in the odd Coulomb valley for
symmetric Anderson model, ε=−U/2, the Kondo effect is
present in both magnetic configurations. The conductance isthen given by G
P=2e2/handGAP=(1−p2)2e2/hfor the
parallel and antiparallel configurations, yielding, for the TMR,
TMRT//Gamma1/lessmuch1
ε=−U/2=p2
1−p2. (2)
On the other hand, when the position is shifted away from
ε=−U/2, the conductance in the parallel configuration drops
and becomes much suppressed due to spin splitting of thedot level caused by the exchange field. The width of theconductance peak at ε=−U/2 is determined by δε
exch, that
is, the conductance becomes suppressed when |δεexch|/greaterorsimilarTK,
where |δεexch|≈TKforε=−U/2±δεwithδε≈U/60.
Because in the antiparallel configuration, there is no exchangefield, the conductance is still given by G
AP=(1−p2)2e2/h
113306-2BRIEF REPORTS PHYSICAL REVIEW B 83, 113306 (2011)
due to the Kondo effect. Consequently, one finds, GAP/greatermuchGP,
and the TMR becomes
TMRT//Gamma1/lessmuch1
ε/negationslash=−U/2→− 1, (3)
irrespective of the magnitude of the leads’ spin polarization
p, provided that the exchange field δεexchis larger than Tand
TK. As can be seen in Fig. 2, the minimum value of TMR is,
however, not precisely −1,but around −0.8. This is related to
the fact that for assumed parameters, the ratio /Gamma1/U is relatively
large, and GPhas tails coming from the Coulomb resonances.
By decreasing /Gamma1/U , that is, for /Gamma1/lessmuchU, the TMR should
finally approach −1.
In the even Coulomb valley ( ε>0o rε<−U), when
T//Gamma1 < 1, transport is mediated by elastic cotunneling pro-
cesses. The conductance is then simply proportional to Gσ∼
/Gamma1Lσ/Gamma1Rσ, from which one finds that GAP∼2(1−p2) and
GP∼(1+p)2+(1−p)2, and
TMRT//Gamma1< 1
ε>0=2p2
1−p2(4)
for the TMR.17All the above TMR values determined for
different dot level positions can be clearly identified in Fig. 2
forT//Gamma1/lessmuch1.
From an application point of view, it is also interesting to
study the temperature dependence of the spin polarization ofthe system, which we define as P=(G
P
↑−GP
↓)/(GP
↑+GP
↓),
where GP
σis the spin-resolved linear conductance in the
parallel configuration. In the antiparallel configuration, due tothe left-right symmetry of the system, the spin polarization iszero. The spin-resolved conductances and the spin polarizationare shown in Figs. 2(d)–2(f). It can be seen that generally,
G
P
↑>GP↓, except for ε=−U/2,where GP
↑≈GP
↓.T h e
difference between GP
↑andGP
↓is directly related to the fact
that there is an exchange-induced splitting of the dot leveland that spin-up electrons are majority electrons in both theleft and right leads. Consequently, one observes an increasedtunneling of spin-up electrons as compared to spin-down ones.By changing the position of the dot level, the spin polarizationstarts increasing when moving away from the particle-hole
symmetry point. In particular, for ε=−U/2,P
T//Gamma1/lessmuch1
ε=−U/2≈0,
whereas in the even Coulomb valley, one finds that PT//Gamma1< 1
ε>0=
2p/(1+p2). Note that in the elastic cotunneling regime, the
spin polarization is larger than the bare spin polarization offerromagnetic leads, and for assumed parameters ( p=0.5),
PT//Gamma1< 1
ε>0=0.8.
When T//Gamma1/greatermuch1 and T/U/greaterorsimilar1, the Coulomb blockade is
smeared out by temperature, and transport is dominated byuncorrelated sequential tunneling, for which the conductanceis proportional to G
σ∼/Gamma1Lσ/Gamma1Rσ/(/Gamma1Lσ+/Gamma1Rσ). By inserting
the spin-dependent couplings in each configuration, onefinds
17TMRT//Gamma1/greatermuch1=p2/(1−p2). On the other hand, the spin
polarization in the high-temperature regime is just given by thespin polarization of the leads, P
T//Gamma1/greatermuch1=p.
The most interesting is probably the crossover regime,
when, by changing the temperature, one can smoothly crossover from, for example, the Coulomb blockade to the Kondoregime, provided that |δε
exch|<TK. The explicit variation of
the linear conductance, TMR, and spin polarization with T
is shown in Fig. 3. First of all, one can see that by lowering10-410-310-210-11001010.00.20.40.60.8
P
T/Γ10-410-310-210-1100-0.9-0.6-0.30.00.30.6
T/ΓTMR
(d)0.00.30.60.91.21.5ε/U=- 1 / 2
ε/U = -59/120
ε/U = -29/60
ε/U=- 7 / 1 5GAP(e2/h)
0.00.40.81.21.62.0
GP(e2/h)ε/U=- 5 / 1 2
ε/U=- 1 / 3
ε/U=0
ε/U=1 / 3(a) (b)
(c)TK
FIG. 3. (Color online) The temperature dependence of the linear
conductance GAP/Pin the (a) antiparallel and (b) parallel configura-
tions as well as (c) the TMR and (d) spin polarization Pfor different
values of the level position ε, as indicated. The vertical dashed line
indicates the Kondo temperature TK,TK//Gamma1/similarequal0.022. Parameters are
as in Fig. 1.
temperature, the Coulomb charging energy Ustarts playing
an important role, leading to the Coulomb blockade effect(see also Fig. 2forT//Gamma1≈1). Transport is then mainly
governed by cotunneling processes, except for resonanceswhere processes of all orders may contribute. For empty ordoubly occupied levels, transport is then spin coherent becauseonly elastic cotunneling is present, as discussed above. Inthe local moment regime, however, both elastic and inelasticcotunneling is relevant, the latter one introducing a mechanismfor spin relaxation due to spin-flip processes. The spin flipsgenerally destroy the spin coherence of cotunneling throughthe system and, as a consequence, both TMR and Pare then
suppressed as compared to the even Coulomb valley [see Fig. 3
forT
K/lessorsimilarT/lessorsimilar/Gamma1].
By lowering the temperature further, in the Coulomb
blockade regime, another energy scale sets in, which isthe Kondo temperature. Once T/lessorsimilarT
K, the spin in the
dot hybridizes with the conduction electrons in the leads,forming a many-body singlet state. The conductance throughthe system increases, then, to its maximum value. This,however, happens only when |δε
exch|<TK; otherwise the
Kondo effect will not develop irrespective of the ratio T//Gamma1 .
The increase of the conductance can be observed in theantiparallel configuration in the whole odd Coulomb blockadevalley, where G
AP=(1−p2)2e2/h,and for ε=−U/2i nt h e
parallel configuration, where GP=2e2/h, that is, precisely in
regimes where |δεexch|<TK[see Figs. 2(a) and2(b)]. On
the other hand, when T≈TK, the correlations leading to the
Kondo effect become smeared by thermal fluctuations, and theconductance drops to approximately half its low-temperaturevalue. Interestingly, in the parallel configuration for ε/negationslash=
−U/2, that is, when the Kondo effect is suppressed by the
exchange field, we observe an increase of linear conductancefor a certain range of temperatures [see Fig. 3(b)], for example,
forε=−7U/15. This in fact happens when the temperature
is of the order of the exchange field, T≈|δε
exch|;i nt h i s
case, finite temperature assists spin-flip cotunneling processes,
113306-3BRIEF REPORTS PHYSICAL REVIEW B 83, 113306 (2011)
leading to the Kondo effect. Furthermore, it also turns out that
forT≈TK,the linear conductance in the antiparallel config-
uration becomes comparable to the conductance in the parallelconfiguration G
AP≈GPand, consequently, TMR vanishes
[see Fig. 3(c)].
The temperature dependence of the spin polarization is
displayed in Fig. 3(d). It can be seen that when moving
away from the particle-hole symmetry point, ε=−U/2, in
the Coulomb blockade regime, Pis constant for T/lessorsimilar|δεexch|
and exhibits a minimum once T≈|δεexch|. Finally, for an even
Coulomb valley (see Fig. 3forε>0), both the TMR and spin
polarization Pare constant for T/lessorsimilar/Gamma1and start decreasing
whenT≈/Gamma1to reach their high-temperature values.
In this work, by employing the numerical renormaliza-
tion group method, we have determined the temperaturedependence of spin-polarized transport characteristics, cross-
ing over from the Kondo to the high-temperature regime. Wehave shown that in certain transport regimes, the TMR takesuniversal values with respect to the spin polarization of theleads. Moreover, in the odd Coulomb valley, the TMR wasfound to display a nonmonotonic dependence on temperature,with a vanishing TMR for T≈T
K. We have also studied the
spin polarization of the linear conductance and showed that atresonance and in the even Coulomb valley, it may be enhancedabove the spin polarization of the leads.
We acknowledge support from the Polish Ministry of
Science and Higher Education through a “Iuventus Plus”research project for years 2010–2011.
*weymann@amu.edu.pl
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35The Kondo temperature in the antiparallel configuration for ε=
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113306-4 |
PhysRevB.91.201303.pdf | RAPID COMMUNICATIONS
PHYSICAL REVIEW B 91, 201303(R) (2015)
Two-step photon absorption in InAs/GaAs quantum-dot superlattice solar cells
T. Kada, S. Asahi, T. Kaizu, Y . Harada, and T. Kita
Department of Electrical and Electronic Engineering, Graduate School of Engineering,
Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan
R. Tamaki, Y . Okada, and K. Miyano
Research Center for Advanced Science and Technology (RCAST), The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan
(Received 12 September 2014; revised manuscript received 18 March 2015; published 18 May 2015)
We studied the two-step photon absorption (TSPA) process in InAs/GaAs quantum-dot superlattice (QDSL)
solar cells. TSPA of subband-gap photons efficiently occurs when electrons are pumped from the valence bandto the states above the inhomogeneously distributed fundamental states of QDSLs. The photoluminescence(PL)-excitation spectrum demonstrates an absorption edge attributed to the higher excited states of the QDSLsin between the InAs wetting layer states and the fundamental states of QDSLs. When the absorption edge of theexcited state was resonantly excited, the superlinear excitation power dependence of the PL intensity demonstratedthat the electron and hole created by the interband transition separately relax into QDSLs. Furthermore, time-resolved PL measurements demonstrated that the electron lifetime is extended by thereby inhibiting recombinationwith holes, enhancing the second subband-gap absorption.
DOI: 10.1103/PhysRevB.91.201303 PACS number(s): 72 .40.+w,78.67.Hc,88.40.fh
In the face of increasing demand for renewable energy
sources to permit our safe and secure ways of life to continue,high-efficiency photovoltaics using p-i-nsemiconductor solar
cells (SCs) are very promising for generating electricalpower by converting solar radiation. The conversion efficiencyof a single-junction SC is limited by several unavoidablelosses, such as transmission loss, thermalization loss, Carnotloss, Boltzmann loss, and emission loss [ 1]. The maximum
conversion efficiency is thereby restrained to be approximately30% [ 2]. Here, a major loss comes from transmission loss
for photons with energy less than the band-gap energy of aSC. The key concept in breaking through the efficiency limitis an energy conversion covering the broad solar radiationspectrum. This can be achieved by multicolor absorption ina multi-energy gap system. Multijunction SCs comprisinga series connection of junctions with different band-gapenergies are one of the promising structures that accomplishthe highest energy conversion efficiency [ 3]. Recently, the
intermediate-band (IB) SC [ 4] with an additional parallel diode
connection has attracted strong interest because of its morerobust operation under a solar spectral variation in comparisonto a multijunction SC [ 5].
Based on theoretical predictions, IBSCs are expected to
realize extremely high conversion efficiency: greater than 60%under the maximum concentration and 48.2% under one-sunirradiation [ 6]. The IBSC with a single IB provides two
additional subband-gap absorptions: valence band (VB) →IB
and IB →conduction band (CB), together with the interband
absorption from VB to CB. These additional transitions pro-duce extra photocurrent without degrading the photovoltagewhen electrons in the IB are optically pumped out to the CB[7,8]. Since the series-connected electron pumping by way of
the IB occurs under a current-matched condition, the opticaltransition strength between the IB and CB should be almostthe same as that between the VB and IB, which generallyrelates to the oscillator strength and the electron densityin the IB [ 9,10]. To enhance the second optical transition
(IB→CB) for a photon flux illuminating a SC surface,zero-dimensional systems such as quantum dots (QDs) have
been investigated [ 11,12]. Despite the use of stacked QDs
for IBSC being strongly anticipated as reviewed in Refs. [ 4]
and [ 8], the key physics practically realizing efficient two-step
photocurrent generation has been unclear. So far, a superlattice(SL) structure has been expected to form minibands playingthe role of the IB. In this work, we focus on a new role ofSL in two-step photocurrent generation; efficient two-stepprocess can be accomplished by extending electron lifetimein InAs/GaAs QDSLs. A long carrier lifetime is, of course,preferable for the second photoexcitation, as it enhances thechance of photoexcitation of electrons in the IB [ 13]. We
studied the two-step photon absorption (TSPA) process viaquantized states formed in IBSCs, including a SL structure ofInAs/GaAs QDs. We carried out photoluminescence (PL) andits excitation (PLE) measurements, as well as the externalquantum efficiency (EQE) measurements under two-colorphotoexcitation for VB →IB and IB →CB. We found that
TSPA efficiently occurs when electrons are pumped into anenergy level located above the inhomogeneously distributedfundamental states of QDSLs. In addition, we conductedthe excitation power dependence of the PL intensity andtime-resolved PL measurements to study separate electron andhole energy relaxation.
An IBSC structure was fabricated on the n
+-GaAs(001)
substrate using solid-source molecular beam epitaxy. AnInAs/GaAs QDSL was included in the intrinsic layer. Anundoped nine-layer stacked InAs/GaAs QD was formed at480
◦C on undoped-GaAs/ n-GaAs(Si: 5 ×1017cm−3)/n+-
GaAs(Si: 1 ×1018cm−3) grown at 550◦C. Nominal thickness
of InAs was 2.0 monolayers (ML) for the first QD layer and1.4 ML for the stacked layers in order to prevent increasing thelateral size [ 14]. The beam-equivalent pressure of the As
2flux
was 1.3×10−3Pa. Nominal GaAs spacer layer thickness was
4 nm, which is thin enough to couple the electronic states alongthe stacking direction and forms a miniband [ 14,15]. Then,
ap
+-GaAs(Be: 1 ×1019cm−3)/p-GaAs(Be: 2 ×1018cm−3)
layer was grown on the top of the SC structure. Metal contacts
1098-0121/2015/91(20)/201303(6) 201303-1 ©2015 American Physical SocietyRAPID COMMUNICATIONS
T. KADA et al. PHYSICAL REVIEW B 91, 201303(R) (2015)
on the top and the bottom surface were Au/Au-Zn and In,
respectively. The in-plane QD density and the total thickness ofthe QDSL was approximately 1 ×10
10QDs/cm2and 38 nm,
respectively. The thickness of the intrinsic layer in the IBSCstructure was 2000 nm. The built-in electric field applied to theQDSLs was expected to be 7 kV /cm, which is low enough to
prevent electric-field induced carrier escape [ 16,17]. Here, the
fundamental state does not form a miniband at temperaturesless than approximately 20 K because the homogeneouslinewidth producing the electronic coupling is smaller than theinhomogeneous distribution of the fundamental energy statesalong the stacking direction [ 14,16].
Photocurrent measurements were carried out at 20 and
300 K. The excitation light source used was a tungstenhalogen lamp passed through a 140-mm monochromator andchopped by an optical chopper ( f=800 Hz). The excitation
power density depending on the wavelength was 6.82–16.7μW/cm
2, which was much smaller than that of the one-
sun solar irradiance. The photocurrent was detected by alock-in amplifier synchronized with the optical chopper. Themeasurements were conducted under short-circuit conditionwithout applying any external bias voltage. Here, the EQEwas defined as an efficiency of the photocurrent productionunder the monochromatic excitation. The EQE spectrumwas obtained by the number of electrons collected as thephotocurrent normalized by the incident photon flux at eachwavelength.
TSPA was demonstrated by measuring a change in the EQE
signal amplitude under two-color excitation using two lightsources [ 18–22]. The primary light source, a monochromated
tungsten halogen lamp, excited electrons from the VB intothe IB. A second infrared (IR) light source in this experimentpumped electrons accumulated in the IB to the CB. The IR lightsource used here was a pulsed laser light with a photon energyof 0.30 eV and repetition rate of 200 kHz. The excitationintensity was 6 .5×10
18photons /(cm2s), which is roughly
equivalent to 60 times the solar irradiance in the 0.4–1.0eV range. The photon energy of 0.30 eV was chosen to beslightly greater than the thermal activation energy of excitonsin QDSLs (0.26 eV). The thermal activation energy of theconfined excitons was examined by means of the temperature
dependence of the PL intensity of the fundamental transitionof QDSLs. /Delta1EQE is defined as the change in the EQE signal
amplitude under the IR irradiation obtained by a lock-inamplifier synchronized with an optical chopper ( f∼1.8k H z )
modulating the IR pulse trains [ 21].
PL and PLE were measured at 9 K. For the excitation, we
used a superluminescent light source. The excitation powerdensity depends on the wavelength was 2.02–2.88 mW /cm
2.
For example, the power density at 940 nm was 2.51 mW /cm2
and the photon-flux density was 1 .19×1016photons /(cm2s).
The PLE spectrum was obtained by the PL intensity normal-ized by the photon flux at each wavelength. The PL signal wasdispersed by a 140-mm single monochromator and detectedby a deep-thermoelectric-cooled InGaAs diode array. Theexcitation power dependence of PL under weak illuminationwas also measured at 9 K while using light-emitting diodeswith peak wavelengths of 940, 850, and 780 nm. Here,the excitation power densities were ∼1–1000 μW/cm
2.I n
addition, the time-resolved PL measurements were performedat 4 K by using a near-infrared streak camera system witha temporal resolution of 20 ps. The light source used was amode-locked Ti:sapphire pulse laser with wavelengths of 900and 800 nm. The pulse duration was 130 fs, and the repetitionrate was 80 MHz.
Figure 1(a) shows EQE spectra obtained at different
temperatures. The absorption edge of the EQE spectrum at20 K shifts according to the temperature dependence of theGaAs band gap and indicates a sharp edge. Figure 1(b) displays
EQE spectra in the subband-gap region. Here, the horizontalaxis is the difference between the photon energy and the GaAsband edge ( E
g=1.52 eV at 20 K and 1.42 eV at 300 K).
EQE at 300 K is higher than at 20 K and shows a tail structurecontinuously extending toward the QDSL states. This can bedominantly attributed to thermally excited electrons from theInAs wetting-layer (WL) states [ 23]. In other words, electrons
are effectively accumulated in the IB at 20 K, which is suitablefor studying TSPA in an IBSC.
A typical /Delta1EQE spectrum measured at 9 K is shown
in Fig. 2(a). The signal amplitude of /Delta1EQE is relatively
FIG. 1. (Color online) EQE spectra measured at 20 and 300 K, (a) entire spectra, and (b) magnified plots in the subband-gap region. The
horizontal axis is the difference between the photon energy and the GaAs band edge ( Eg=1.52 eV at 20 K and 1.42 eV at 300 K).
201303-2RAPID COMMUNICATIONS
TWO-STEP PHOTON ABSORPTION IN InAs/GaAs . . . PHYSICAL REVIEW B 91, 201303(R) (2015)
FIG. 2. (Color online) (a) /Delta1EQE spectrum measured at 9 K. The
IR light source used here was a pulsed laser light with a photon
energy of 0.30 eV and repetition rate of 200 kHz. The excitationintensity was 6 .5×10
18photons /(cm2s). (b) PL and PLE spectra
measured at 9 K. PLE spectrum indicates the PL peak intensity
of the fundamental transitions of QDSLs. Several absorption edgesare indicated by arrows in the PLE spectrum. (c) Comparison of
/Delta1EQE spectra obtained for InAs/GaAs(4 nm) QDSL and uncoupled
InAs/GaAs(30 nm) stacked QDs at 9 K.
large as compared with the result reported in an earlier work
done at room temperature [ 19] because of efficient electron
accumulation in the IB of the low internal electric field SCmaintained at low temperature. The broad, strong /Delta1EQE signal
appearing in the wavelength region from 700 to 800 nm isattributed to a photocurrent generated by IR photoexcitationof electrons in the IB pumped by the interband transitionin GaAs. This indicates that part of the electrons pumpedby the interband transition in the GaAs layer are trapped byQDs before being extracted toward the n-side electrode. For
wavelengths shorter than 700 nm, /Delta1EQE signal decreases
drastically with the wavelength. Because of the shallowpenetration depth for the shorter wavelength light, electronsexcited in the p-layer easily recombine with holes in the VB
before drifting toward the QDSLs. As a result, the electrondensity becomes low in the IB and the IR photoexcitation
becomes inefficient. Below the absorption edge of GaAs(820 nm), the /Delta1EQE spectrum shows a tail structure that arises
from the TSPA process. A small edge at approximately 880 nmcan be attributed to the absorption edge of the InAs WL.The tail structure of /Delta1EQE extends further toward the longer
wavelength side. These features are consistent with structuresobserved in a PLE spectrum discussed later. Since the IRexcitation power dependence of the /Delta1EQE signal intensity
yields a linear dependence, nonlinear two-photon absorptionof the IR photons is negligible in this experiment.
Figure 2(b) shows PL and PLE spectra measured at 9 K.
An inhomogeneously broad PL signal appears at 1054 nm,which is attributed to the fundamental transitions of QDSLs.The PLE signal intensity detected at 1054 nm is graduallyincreases in wavelengths shorter than 1000 nm. The PLEspectrum corresponds to the optical absorption profile. Severalabsorption edges indicated by arrows were confirmed. Theclear edge at 820 nm is due to the GaAs band edge. The edgeobserved at approximately 880 nm corresponds to absorptionof the InAs WL. These two structures were also observed in the/Delta1EQE spectrum in Fig. 2(a). In the longer wavelength region, a
weak absorption edge was found at approximately 950 nm. Toclarify the origin of this, we examined the PL peak wavelengthas a function of the excitation wavelength. We found thatthe PL peak starts shifting with the excitation wavelength inthe region longer than 950 nm when the inhomogeneouslydistributed fundamental QDSL states are directly excited asreported in Ref. [ 24]. The critical wavelength showing the
peak shift produced the edge structure at 950 nm in thePLE spectrum. Therefore, this absorption edge located abovethe inhomogeneously distributed fundamental states can beattributed to the higher excited states of the QDSLs.
Figure 2(c) compares the /Delta1EQE spectrum obtained for
QDSLs of InAs/GaAs(4 nm) with that for uncoupled ten-stacked QDs of InAs/GaAs(30 nm). The thick GaAs spacerlayer of 30 nm in the reference SC diminishes the interdotcoupling, and miniband is not formed. Hence the comparisonshows a clear and significant proof of the SL effect onthe SC performance. It is noted that /Delta1EQE of QDSL is
dramatically enhanced in the wavelength region correspondingto the excited states. This is a unique feature appeared inQDSL. /Delta1EQE is proportional to the intersubband absorption
coefficient between IB and CB. The intersubband absorptioncoefficient is also proportional to the state filling of IB given bythe Fermi-Dirac distribution [ 10]. Next, we discuss influence
of carrier dynamics on the state filling.
We measured the excitation power dependence of the PL
intensity. The excitation wavelengths, 940, 850, and 780 nm,were chosen to excite the absorption edge at 950 nm, theInAs WL states, and GaAs barrier, respectively. Here, wecarefully dealt with the excited carrier density. The carrierdensities excited at each wavelength will be different forthe same power density because of the different absorptioncoefficient. According to the PLE spectrum correspondingto the optical absorption profile, we estimated the relativeabsorption coefficient. Figure 3shows the integrated intensity
of the QDSL fundamental state PL as a function of theexcitation power density in the same excited carrier density.The results exhibit different slopes depending on the excitation
201303-3RAPID COMMUNICATIONS
T. KADA et al. PHYSICAL REVIEW B 91, 201303(R) (2015)
FIG. 3. (Color online) Integrated PL intensity from the funda-
mental transitions as a function of the excitation power. The excitation
wavelengths, 940, 850, and 780 nm, were chosen to excite theabsorption edge at 950 nm, the InAs WL states, and GaAs barrier,
respectively. Here, the excitation power density was carefully chosen
by taking into account the relative absorption coefficient estimatedfrom the PLE spectrum. Dotted lines are fitting of the power
dependence.
wavelength. It is noted that the power dependence when se-
lectively exciting the absorption edge at 950 nm demonstratesa superlinear relation, though the slopes for the excitationsat the WL and GaAs states were linear. The superlineardependence indicates that the excited electron and hole areseparately relaxing into the QDSL states [ 25–27]. According
to the excitation power dependence of the PL under a higherexcitation (not shown here), the first excited states of theQDSLs correspond to the tail structure appeared at the shorterwavelength side ( ∼1000 nm) of the peak. Thus the energy
distribution of the first excited states overlaps with that of thefundamental states. Such energy overlapping will reduce theexcitation cross section for the excited states. Thereby, thehigher excited states appearing above the inhomogeneouslydistributed fundamental states play an important role in theefficient spatial carrier separation in the miniband. On theother hand, when excited above the GaAs barrier, the short-wavelength excitations generate carriers dominantly in thethick GaAs layer deposited above ( p-side) the QDSL layer.
Recombination of diffused electrons and holes in QDs causesthe linear excitation power dependence as observed at the780-nm excitation.
When the adjacent QD states along the stacking direction
are distributed within the homogeneous band, electronic
coupling occurs and, thereby, the miniband is formed. Thehomogeneous linewidth of the excited state resonance ismore than one order of magnitude larger than that of thefundamental state resonance because of the weaker electronconfinement in the excited state [ 28]. Such wide homogeneous
FIG. 4. (Color online) (a) Time-resolved PL intensity from the
fundamental transitions. The excitation wavelengths, 900 and 800 nm,
were chosen to excite the absorption edge at 950 nm and GaAs
barrier, respectively. PL decay curves at different electric fields weremeasured at the 900-nm excitation. Solid lines indicate best fitting
curves, where the coefficients of determination, R
2values, were 0.936,
0.943, 0.974, and 0.968 for 7 kV /cm at the 800-nm excitation, and 7,
2 0 ,a n d3 0k V /cm at the 900-nm excitation, respectively. (b) Analyzed
decay times as a function of the detection-time delay in the stretched
exponential decay profile. (c) Excitation power density dependenceof the decay time. The excitation power densities were chosen by
taking into account the relative absorption coefficient estimated from
the PLE spectrum.
linewidth of the excited state easily causes miniband formation
even in the inhomogeneous energy distribution along thestacking direction, despite the fact that the miniband is notformed in the fundamental state at the low temperature below
201303-4RAPID COMMUNICATIONS
TWO-STEP PHOTON ABSORPTION IN InAs/GaAs . . . PHYSICAL REVIEW B 91, 201303(R) (2015)
∼20 K. This enables effective carrier transport of electrons
and holes in the opposite direction in the miniband of thehigher excited states under the influence of the internal electricfield. Independent energy relaxation of electrons and holes intothe fundamental states located along the stacking directionreduces the recombination rate, causing the electron lifetimeto become long, which enhances the chances of intersubbandphotoexcitation.
To confirm the separate electron-hole energy relaxation
we studied the radiative recombination lifetime by the time-resolved PL measurements. Figure 4(a) shows the time-
resolved PL intensity from the fundamental transitions. Theexcitation power densities were 18.5(128) mW /cm
2for the
800(900)-nm excitation, which was provided to maintainthe same excited carrier density. The result of the 800-nmexcitation ( ∼1.55 eV) which is above the GaAs band gap
dominantly exhibited a single exponential decay with thedecay time of 1.1 ns. The decay time slightly increases withthe delay. On the other hand, when excited at 900 nm, thisslow decay component turns significant. This slow decaycomponent yields to a stretched exponential profile given by afractional power law representing a continuous distribution
of lifetimes [ 29]. The solid line in Fig. 4(a) indicates the
fitting curve. Figure 4(b) shows analyzed decay times as
a function of the detection-time delay. Though the rapiddecay component coincides with the decay obtained at the800-nm excitation, the slow decay component shows the timeconstant of 2.7 ns at the delay of 6 ns. According to ourearlier work [ 16], the PL decay time of the fundamental
state exhibiting the Stark localization under an extremelystrong electric field of approximately 200 kV /cm was almost
the same as that of conventional single-layer QDs. This isdue to the short QD height of approximately 5 nm whichnegligibly diminishes the spatial carrier separation. Thus thisslow decay component can be attributed to recombination ofspatially separated electron and hole in the miniband of the
excited states. The /Delta1EQE enhancement in Fig. 2(c) arises from
longer carrier lifetime taking over the significant stretchedexponential profile. Furthermore, this slow decay componentdepends on the internal electric field by applying dc biasvoltage at 0, −2.6, and −4.6Vf o r F=7, 20, and 30 kV /cm,
respectively. With increasing the electric field, the decay timebecomes longer as shown in Fig. 4(b). Since the strong electric
field causes significant carrier separation, the slower decay atthe higher electric field convincingly supports our discussion.In addition, we measured the excitation power dependence of
the decay time in the same excited carrier density as shown
in Fig. 4(c). The decay time excited at 900 nm decreases with
the excited carrier density, while one at 800 nm was almostconstant. This implies that high density electrons and holesexcited by the high-power excitation uniformly occupy thefundamental state of different QDs.
The extension of the electron lifetime reflects on /Delta1EQE
being proportional to the intersubband absorption coefficientdetermined by the state filling of IB. The slow electronlifetime increases the electron density in IB, which lifts thequasi-Fermi level and increases the filling factor. Thereby,/Delta1EQE is enhanced. The slower decay component at the 900-
nm excitation was approximately 2.1 times longer than that at800 nm at the delay of 6 ns, which is further extended accordingto the stretched exponential behavior of the slow decaycomponent. /Delta1EQE was enhanced approximately 10 times
greater at 950 nm by the SL structure. This result indicatesthat the filling factor changed by the extended electron lifetimeremarkably increased the intersubband absorption coefficient.Besides, it is noted that the excitation power density used in thetime-resolved measurements is approximately nine orders ofmagnitude stronger than that used for the EQE measurements,because the electron lifetime decreases with increasing theexcitation power density.
In summary, we studied a key mechanism enhancing
TSPA in an IBSC that included InAs/GaAs QDSLs. TSPAof subband-gap photons remarkably occurs when pumpingelectrons from the VB to an absorption edge located above the
inhomogeneously distributed fundamental states of QDSLs.
The excitation power dependence of the PL intensity and thetime-resolved PL demonstrated that electrons and holes sepa-rately relax into QDSLs when directly excited at the absorptionedge attributed to the higher excited states of the QDSLs. Thisextends electron lifetime by inhibiting recombination withholes, thereby enhancing the second subband-gap absorption.These results demonstrate an important role of IB electronlifetime in improving the conversion efficiency of IBSCs.
This work has been partially supported by the Incorporated
Administrative Agency New Energy and Industrial Technol-ogy Development Organization (NEDO), and Ministry ofEconomy, Trade and Industry (METI), Japan. The authorswould like to thank S. Naitoh of the University of Tokyo forfabrication of the reference QD sample.
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201303-6 |
PhysRevB.72.195323.pdf | Observation of substitutional and interstitial phosphorus on clean Si „100 …-„2Ã1…with scanning
tunneling microscopy
Geoffrey W. Brown, *Blas P. Uberuaga, Holger Grube, and Marilyn E. Hawley
Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Steven R. Schofield,†Neil J. Curson, Michelle Y. Simmons, and Robert G. Clark
Centre for Quantum Computer Technology, School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
/H20849Received 3 June 2005; revised manuscript received 11 August 2005; published 14 November 2005 /H20850
We have used scanning tunneling microscopy to identify phosphorus that is present at the clean silicon
/H20849100/H20850-/H208492/H110031/H20850surface as a result of the thermal cycling necessary for preparation of samples cut from heavily
doped wafers. Substitutional phosphorus is observed in top layer sites as buckled Si-P heterodimers. We alsoobserve a second type of feature that appears as a single depressed dimer site. Within this site, the atoms appearas a pair of protrusions in the empty states and a single protrusion in the filled states. These properties are notconsistent with known adsorbate signatures or previously reported observations of P-P dimers on the /H20849100/H20850
-/H208492/H110031/H20850surface. The lack of other impurity sources suggests that they are due to either phosphorus or silicon.
The symmetry of the features and their magnitude are consistent with one of those elements residing in aninterstitial site just below the top layer of atoms. To identify the type of interstitial, we performed densityfunctional theory calculations for both phosphorus and silicon located below a surface dimer. The resultingcharge density plots and simulated STM images are consistent with interstitial phosphorus and not interstitialsilicon.
DOI: 10.1103/PhysRevB.72.195323 PACS number /H20849s/H20850: 68.37.Ef, 68.47.Fg, 61.72.Ji
I. INTRODUCTION
The behavior of phosphorus impurities near silicon /H20849100/H20850
is important because of its relevance to electronic deviceprocessing issues. Detailed, atomic-resolution informationabout this system has been obtained in the past using scan-ning tunneling microscopy /H20849STM /H20850.
1–6STM studies of
Si/H20849100/H20850-/H208492/H110031/H20850dosed with PH 3and subsequently annealed
to remove hydrogen have elucidated the basic structures of
substitutional P and substitutional P-P dimers in thesesurfaces.
1–3Those results provided evidence for surface seg-
regation of P-P dimers from bulk-doped crystals4,6and both
P and P-P from epitaxially grown /H9254-doped layers.3,5The seg-
regation and presence of these species is expected since bothare known to play a role in bulk diffusion of phosphorus insilicon.
7,8The lack of observation of substitutional P in the
bulk-doped crystal work is puzzling since single substitu-tional phosphorus should diffuse more readily than P-Pdimers. In addition, there was no report of interstitial phos-phorus, which should be present at some level since it alsoplays a role in bulk diffusion of phosphorus in silicon. In thispaper we report experiments on bulk-doped crystals in whichwe have observed substitutional P in the Si /H20849100/H20850-/H208492/H110031/H20850sur-
face and a second type of feature that we identify as intersti-
tial phosphorus just below the top surface layer of atoms.
II. EXPERIMENT AND SAMPLE PREPARATION
Sample preparation and STM experiments were carried
out in an ultrahigh vacuum system with a base pressureof 2/H1100310
−11Torr. The silicon samples were rectangles
/H208491.5 mm /H1100310 mm /H20850cut from 0.5-mm-thick /H20849100/H20850-oriented
prime grade wafers that were doped with phosphorus at/H333561/H110031019/cm3. The samples were handled with ceramic
tools and mounted in tantalum/molybdenum/ceramic sampleholders to avoid contamination with iron, nickel, or otherdetrimental transition metals. The samples were prepared byoutgassing overnight at 600 °C, followed by a 2 min flash at/H110111250 °C, a 30 min to 1 h postanneal at /H11011900 °C, and
then slow cooling /H20849/H110114 °C/sec /H20850to room temperature. This
procedure was necessary to reduce the number of multidimer
vacancy defects on the surface of these highly dopedsamples. For comparison, a second set of samples was pre-pared using the standard procedure for Si /H20849100/H20850-/H208492/H110031/H20850,
9
which does not include the postanneal. During the flash and
subsequent heating steps, the chamber pressure remained be-low 2/H1100310
−10Torr. Electrochemically etched platinum iri-
dium alloy /H2084990/10 /H20850tips were used for imaging in constant
current mode at room temperature.
Figure 1 shows an example of a surface prepared by the
standard flash technique, that is, without the postanneal. Asshown, even with careful attention to match the temperaturesused for lower doped samples, the resulting surface is verydefective, having a high density of multidimer vacancies thathave lined up in the direction perpendicular to the
/H9266-bonded
dimer rows on each terrace. This is reported to arise from acombination of stresses induced by the 2 /H110031 reconstruction
and P-P dimers at the surface.
4
In contrast, Fig. 2 shows the surface of a sample from the
same wafer that has been prepared using a 30-min-long post-anneal as described above. This process is known to allowdesorption of P from the surface layer
10and should also al-
low rearrangement of the remaining surface vacancies. Herethe total density of multidimer vacancy defects is lower andthe tendency for the remaining defects to line up has dimin-ished. The fact that the line defects can be eliminated with aPHYSICAL REVIEW B 72, 195323 /H208492005 /H20850
1098-0121/2005/72 /H2084919/H20850/195323 /H208495/H20850/$23.00 ©2005 The American Physical Society 195323-1postanneal is evidence that they are not transition-metal-
induced features. Vacancy line defects induced by metal con-tamination reportedly cannot be eliminated from the surfaceby annealing.
11Furthermore, the fact that the terrace widths
are similar in both cases implies that the primary source ofstress in Fig. 1 is the near surface P.
1The line defects in Fig.
1 are, therefore, related to the high doping level /H20849although
the high number of defects present suggests that each one isnot associated with an individual phosphorus atom /H20850.
III. RESULTS AND DISCUSSION
In addition to vacancies, both Figs. 1 and 2 have examples
of two other types of features. In Fig. 2, these are in the areasmarked by the circles and can be described initially as“bright” and “dark” features. The empty-state image /H20851Fig.
2/H20849b/H20850/H20852more clearly illustrates their difference with respect to
common vacancies, to “ C”-type defects identified in early
studies of this surface,
12and to split-off dimer defects13that
are all observed in other areas of the image. In general, thebright features are smaller in height and extension than the C
defects while the dark features are not as deep as vacancies.
A higher resolution example of one of the asymmetric
features from a different sample is shown in the circle in theimages in Fig. 3. The feature is asymmetric with respect tothe mirror plane of the dimer row and several neighboringdimers in the row are buckled. As noted above, the featurealso has a much smaller protrusion in the empty states thando the very obvious Ctype defects above and below it in the
image. Sections through the empty state density of the cen-tral part of the feature show it to be /H110110.5 Å tall at this
sample bias compared to the /H110111 Å tall Cdefect protrusions.
The density of these features is typically less than 1 per150 nm /H11003150 nm area on the postannealed samples,but there is some variation within and between samples.
This may be due to the uncertainty in the doping level/H20849/H110113/H1100310
19/cm3/H20850.
The symmetry, height, and buckled neighboring dimers of
these features are all consistent with what has been observedfor P atoms incorporated into Si /H20849100/H20850-/H208492/H110031/H20850after a dose-
anneal cycle with phosphine.
1–3When the P replaces a Si
atom in the top layer, the resulting heterodimer buckles andinduces buckling of neighbors along the same row. The as-signment of Si-P to these features in our images is also mo-tivated by the observations that phosphorus is expected to bethe most common impurity in the sample and that the samplepreparation and appearance of these features are also incon-sistent with hydrogen or other residual gas impurities in thechamber. Monoatomic hydrogen should not be present sincethe postanneal temperature is too high for it to remain on thesurface. Other gas phase species can also be ruled out be-cause they should occur at lower density, if at all, based ontheir partial pressures in residual gas analysis. We also do notobserve the density of the features to increase in time. One
FIG. 1. Filled-state image of a Si /H20849100/H20850-/H208492/H110031/H20850surface prepared
with the standard flash-anneal technique on a sample cut from ahighly doped wafer. Stress due to phosphorus near the surface leadsto the vacancy line defects. Several phosphorus-induced featuresare visible in the clean areas. The image was acquired at −1.25 Vsample bias and 400 pA tunnel current. The image area is350 Å/H11003350 Å.
FIG. 2. /H20849a/H20850Filled- and /H20849b/H20850empty-state images of the same area
o naS i /H20849100/H20850-/H208492/H110031/H20850surface prepared with the standard flash to
/H110111250 °C and a 30 min postanneal at /H11011900 °C. The vacancy line
defects have diminished due to desorption of phosphorus. Thecircles mark remaining features that appear to be phosphorus in-duced. The tunneling conditions were −1.2 V for the filled states,+1.2 V for empty states and 150 pA tunnel current in both cases.The image area is 250 Å /H11003250 Å.BROWN et al. PHYSICAL REVIEW B 72, 195323 /H208492005 /H20850
195323-2final possibility is that the structures are due to silicon; how-
ever, silicon monomers and monovacancies are not stable onthese surfaces at room temperature. Given these observa-tions, we identify these bright features as Si-P in the toplayer dimer rows resulting from P dopants exposed at, ordiffusing to, the surface during the flash or postanneal. This
is an interesting result since the P in the heterodimers in ourimages originates from bulk impurities and not from gasphase dosing of phosphorus-bearing molecules.
An example of one of the dark features is shown in the
center of the images in Fig. 4. The specifics of its appearanceare weakly tip dependent with small variations in height orsymmetry observed from example to example. In general,the features are described as a depressed dimer in both filledand empty states. Specifically in Fig. 4, the depressed dimerin the filled states appears with narrower lateral extent thanothers on the surface. /H20849We have observed the same details in
images in which the tip is able to resolve the dimers in thesurface defects, although that is not the case in this image. /H20850
In the empty-states image the feature appears as a depressedand resolved dimer /H20849“resolved” implying its appearance as
two protrusions /H20850. The adjacent dimers in the same row are
also resolved. Slight buckling is also present in the depresseddimer, although the lack of buckling in the filled-states imagemay indicate a tip asymmetry instead. The density of thesefeatures is similar to that of the bright features. Finally, thefeatures appear to be charge neutral. We have previouslyobserved charged vacancies on these surfaces
14and
charged phosphorus atoms in substitutional sites beneathSi/H20849100/H20850-/H208492/H110031/H20850and see a band bending-induced enhance-
ment around either type of feature.
15That enhancement is
not present for the dark features here.
These dark features must also be due to phosphorus or
silicon based on the arguments given above concerning otheradsorbates or contaminants. These do not appear to be sub-stitutional P-P dimers as seen in another study on hydrogencovered Si /H20849100/H20850.
6P-P dimers usually appear strongly de-
pressed and resolved in filled state images.1,4We do not see
this, even in cases where other features on the surface /H20849de-
fects and step edges /H20850do show resolved dimers. Previous
FIG. 3. /H20849a/H20850Filled- and /H20849b/H20850empty-state images from a highly
doped sample with one of the “bright” features marked by thecircle. A Ctype defect combined with a vacancy is directly below it
in the image. The asymmetric dimer and its neighboring structureare consistent with previous observations of Si-P heterodimers. Thetunneling conditions were −1.4 V for the filled states, +1.2 V forempty states, and 150 pA tunnel current in both cases. The imagearea is 120 Å /H11003120 Å.
FIG. 4. /H20849a/H20850Filled- and /H20849b/H20850empty-state images of one of the
“dark” features observed on the highly doped samples. The tunnel-ing conditions were −1.4 V for the filled states, +1.0 V for emptystates, and 150 pA tunnel current in both cases. The image area is120 Å/H11003120 Å.OBSERVATION OF SUBSTITUTIONAL AND … PHYSICAL REVIEW B 72, 195323 /H208492005 /H20850
195323-3work also reported little corrugation in the empty states im-
ages while we consistently image structure in the depressionand well resolved adjacent dimers.
At a qualitative level, the features we observe are consis-
tent with what might be expected for an atom in the intersti-tial space just below a top-surface Si dimer. The structureconsists primarily of a single perturbed dimer and does notinduce large surface asymmetries. In addition, it creates adepression at the surface in both biases, as might be expectedfor an interstitial that forms bonds beneath the top layer withthe atoms around it. These bonds can cause the top layeratoms to be physically displaced downwards. In addition,bonds created below the surface will lessen the filled- andempty-state orbital density protruding outward from the sur-face.
The presence of interstitial phosphorus and silicon is ex-
pected since both are known to play a role in impurity dif-fusion in bulk silicon.
7,8,16At elevated temperatures, intersti-
tials can be produced and migrate through the crystal. Inaddition, under certain circumstances, such as high doping,simulations show that phosphorus interstitials can be a majorconstituent of the phosphorus species.
17
To determine whether the features are due to interstitial
phosphorus or silicon, we performed density functionaltheory calculations with the Vienna Ab initio Simulation
Package /H20849VASP /H20850.
18–21A plane wave basis set with an energy
cutoff of 255 eV, appropriate for the projector augmentedwave pseudopotentials,
22,23was used. Si surfaces were
modeled by eight layers and a vacuum spacing of 10 Å. A2/H110032/H110031 Monkhorst-Pack kpoint mesh
24was used for sam-
pling the Brillouin zone. Initially, the minimum energy struc-tures and total charge densities for interstitial phosphorus andsilicon were calculated. Stable configurations occurred wheneither atom was located between the third and fourth layer ofthe surface with phosphorus located nearly directly underone of the surface dimer atoms and silicon located closer tothe center, but still in an asymmetric position. Since the sur-face dimers can buckle in either direction, there were there-
fore two stable configurations for each type of atom. Figure 5schematically illustrates the locations.
To compare with STM data, the charge densities for elec-
trons between 0 eV /H20849the Fermi level /H20850and −3 eV and be-
tween 0 eV and −2 eV were considered. Either of these win-dows roughly corresponds to the states sampled by the tip inthe filled-state imaging process. For each window, the chargedensities for both interstitial configurations were averagedtogether to account for the STM tip averaging over the twostable positions. From this data, the method of Tersoff andHamann
25was used to derive simulated STM images. Once
the constant charge density surface was obtained, it was con-voluted with box and Gaussian smoothing functions to ac-count for a realistic tip size and shape. The results for the0 to −3 eV window are shown in Fig. 6. The results for the0 to −2 eV window were very similar. In each case, thephosphorus interstitial induces a narrowed and depresseddimer and is therefore consistent with the “dark” features inthe STM images in Fig. 2. The silicon interstitial induces anenhancement of the top dimer, inconsistent with observa-tions.
Finally, we calculated the barrier for motion between the
two interstitial phosphorus configurations using the dimermethod.
26The barrier was found to be about 0.7 eV, which,
assuming a standard prefactor of 1013/s, would imply that
the structure flips back and forth every 50 ms at 300 K. Thestructure appears symmetric since the relatively slow STMimaging process shows the average of the two configura-tions.
IV. CONCLUSION
In summary, we have identified phosphorus segregated at
the surface of Si /H20849100/H20850-/H208492/H110031/H20850as a result of the thermal cy-
cling necessary to prepare the samples. Single phosphorus
atoms are found in substitutional sites at the surface and weobserve the effect of interstitial phosphorus just below thedimer rows. Both features represent single P species in thesilicon crystal that have been frozen in at the end of ourpostanneal step. Since phosphorus primarily desorbs as P
2at
these temperatures,10these features represent impurities that
had not encountered another of their kind by the time thetemperature was lowered. This may explain why we do not
FIG. 5. Schematic diagrams of stable interstitial positions with
respect to the surface dimer. The top surface /H20849lower diagram /H20850and
side views /H20849upper diagrams /H20850are shown. The empty circles are top
layer atoms in dimer rows. The black circles are subsurface atomswhose size diminishes with increasing depth. The gray circles indi-cate the two possible stable interstitial phosphorus positions. Forinterstitial silicon, the gray circles would be slightly closer together.
FIG. 6. /H20849Left /H20850Simulated filled-state STM image with phos-
phorus interstitial located below the center surface dimer. /H20849Right /H20850
Simulated filled-state STM image with silicon interstitial locatedbelow the center surface dimer. In each case, the charge density wasaveraged over both possible stable interstitial positions before theSTM image was generated.BROWN et al. PHYSICAL REVIEW B 72, 195323 /H208492005 /H20850
195323-4observe P-P dimers in our images. This work also demon-
strates an improved method of preparing highly phosphorus-doped silicon samples for scanning tunneling microscopystudies.ACKNOWLEDGMENT
This work was supported by the U.S. Department of En-
ergy under Contract No. W-7405-ENG-36.
*Electronic address: geoffb@lanl.gov
†Now at School of Mathematical and Physical Sciences, University
of Newcastle, Callaghan, NSW 2308, Australia.
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195323-5 |
PhysRevB.71.115434.pdf | Structure of the hydrogen-stabilized MgO 111-1ˆ1polar surface:
Integrated experimental and theoretical studies
V. K. Lazarov,1R. Plass,1,*H-C. Poon,1D. K. Saldin,1M. Weinert,1S. A. Chambers,2and M. Gajdardziska-Josifovska1,†
1Department of Physics and Laboratory for Surface Studies, University of Wisconsin Milwaukee, P. O. Box 413,
Milwaukee, Wisconsin 53201, USA
2Fundamental Science Directorate, Pacific Northwest National Laboratory, P. O. Box 999, Richland, Washington 99352, USA
sReceived 20 July 2004; revised manuscript received 19 October 2004; published 31 March 2005 d
The surface structure of MgO s111d-s131dbulk and thinned single crystals have been investigated by
transmission and reflection high-energy electron diffraction, low-energy electron diffraction sLEED d, and x-ray
photoelectron and Auger electron diffraction. The s131dpolar surface periodicity is observed both after
800 °C annealing in air and also after oxygen plasma cleaning and annealing in ultrahigh vacuum. The x-rayphotoelectron spectroscopy and diffraction results were analyzed by simulations based on path-reversed LEEDtheory and by first-principles calculations to help distinguish between different mechanisms for the stabiliza-tionofthisextremelypolaroxidesurface: s1dstabilizationbyadsorptionofahydrogenmonolayer;maintaining
the insulating nature of the surface and s2dstabilization of the clean O or Mg terminated 1 31 surface by
interlayer relaxations and two-dimensional surface metallization. The analysis favors stabilization by a singleOH layer, where hydrogen sits on top of the O ions with O-H bond distance of 0.98Å. The in-plane O and Mgpositions fit regular rocksalt sites, the distance between the topmost O and Mg plane is 1.04 Å, contracted by,14% with respect to bulk MgO distance of 1.21 Å, while the interlayer separation of the deeper layers is
close to that of bulk, contracted by less than 1%. The presence of a monolayer of H associated with theterminal layer of oxygen reduces significantly the surface dipole and stabilizes the surface.
DOI: 10.1103/PhysRevB.71.115434 PACS number ssd: 68.47.Gh, 61.14.Qp, 68.49.Uv, 73.20.At
I. INTRODUCTION
The stability of polar oxide surfaces has long been a prob-
lematic question, as discussed in books on oxide surfaces1,2
and recent reviews of polar oxide surfaces.3,4In the standard
model, bulk-terminated polar oxide surfaces have divergingssurface denergies because alternating layers of oppositely
charged ions produce a diverging electric dipole momentperpendicular to the surface, giving rise to the so-called “po-lar surface instability” problem. The rocksalt structure is of-ten the most stable form for highly ionic solids, and has beenthe prototype for experimental and theoretical studies of neu-tral and polar oxide surfaces. The structure consists of twointerpenetrating fcc lattices of oxygen anions and metal cat-ions in a +2 oxidation state, making these oxides even moreionic than the prototypical ionic NaCl. The polar MgO s111d
surface, consisting of equidistant alternating layers of oxy-gen and magnesium, in particular, has been an important testcase of the polar instability problem.
In the 1970’s, the polar oxide surface problem was con-
sidered settled, with a consensus between theory
5and
experiment6,7that clean MgO s111dpolar surfaces do not ex-
ist, but facet into neutral h100jplanes, and thereby making
the surface energy finite. The first scanning electron micros-copy sSEM dand low-energy electron diffraction sLEED d
study of MgO s111dsurfaces reported thermal faceting into
three-sided pyramids.
6The faces of these micropyramids
were interpreted as the neutral h100jsurfaces suggested by
theory, although there were no direct measurements of theactual facet angles due to charging effects in LEED and theinability to quantify topography from single SEM images.Discovery of reconstruction-stabilized MgO s111dpolar oxidesurfaces by reflection high-energy electron diffraction
sRHEED dand reflection electron microscopy sREM d
8,9ne-
cessitated a revisiting of the faceting model. Plass et al.10
showed that acid etching, rather than thermal annealing, is
responsible for the reported faceting of the MgO s111dinto
three sided pyramids. Atomic force microscopy sAFM dand
tomographic SEM measurement found the facets to be closeto the higher index h332jplanes instead of the neutral h100j
planes.
4,10Faceting as a stabilization mechanism requires
mass transport that one would expect to happen at higherannealing temperatures. However, instead of inducing or pro-moting faceting, increasingly higher temperature annealingwas found to have no effect on the faceting at low to mod-erate temperatures, and to erase the faceting at high tempera-tures, replacing it with reconstruction-stabilized flat s111d
terraces with air stable reconstructions. The structure of the
MgO s111ds
˛33˛3dR30°, s232d, and s2˛332˛3dR30° re-
constructions were studied by direct methods applied to
transmission electron diffraction to find terminations givenby periodic arrangements of oxygen trimers and single oxy-gen atoms.
11The s232dreconstruction, also reported for
NiOs111dsRefs. 12–14 dand FeO s111d,15fits well with the
simple electrostatic prediction as the most favorable solutionfors111drocksalt oxide surfaces. The idea of an octopole as
the smallest neutral building block of the rocksalt structure isthe basis of these models.
16–18The simple electrostatic octo-
polar model has been questioned by recent grazing x-raydiffraction experiments and first-principle calculations forthe model MgO s111d-s232dsurface, finding marked depen-
dence of the s232dstructure on thermodynamic parameters
such as oxygen partial pressure and temperature. For ex-
ample, Mg-covered terminations with peculiar insulatingPHYSICAL REVIEW B 71, 115434 s2005 d
1098-0121/2005/71 s11d/115434 s9d/$23.00 ©2005 The American Physical Society 115434-1electronic structure were favored in O-poor conditions, and
the O-terminated octopole was favored in O-rich environ-
ments. Similarly, the remaining two reconstructions s˛3
3˛3dR30° and s2˛332˛3dR30° cannot be explained by the
electrostatic models; recent work19suggests that the
MgO s111ds˛33˛3dsurface may have vacancies in the top
Mg layer.
Within the ionic picture, the existence of a bulk termi-
nated MgO s111ds131dsurface is not allowed,16,17although
this surface might be stabilized by the adsorption of foreign
species. For example, hydrogen adsorption on an oxygen-terminated MgO s111dsurface would satisfy the “sigma-half”
criterion
3sas would OH absorption on Mg terminated sur-
faced, and the huge electric fields/dipoles across the speci-
men would be canceled.20
As was discussed above, experimental data rule out face-
ting as a stabilization mechanism, and air-stable reconstruc-tions at temperatures higher than 1200 °C stabilize theMgO s111dsurface. However, the structure of MgO s111dat
lower temperature s,950°C dis still an open question. First-
principles calculations
21,22suggest a metallic character for
MgO s111d, and both quantum-mechanical calculations and
classical selectrostatic dtheory predict hydroxylation as a
mechanism that will cancel polarity. Refson et al.20argue
that the surface energy of hydroxylated MgO s111dwould be
smaller than MgO s100d, the most stable surface in MgO. In
this paper we report the experimental observation of theMgO s111d-s131dstructure on bulk and thinned single crys-
tals, and analyze the data to distinguish between the pro-
posed stabilization models.
II. EXPERIMENTALAND THEORETICAL METHODS
A.Ex situexperiments
Disks cut from single-crystal MgO s111dslabs were me-
chanically polished and dimpled, then etched to perforationin hot concentrated nitric acid, washed with water, sputteredwith 5 keV Ar ions, and furnace annealed in air to 800 °C.The samples were transported through air to a HitachiH9000-NAR high-resolution transmission electron micro-scope where the electron transparent regions were investi-gated by bright field imaging sBF-TEM dand by transmission
high-energy electron diffraction sTHEED d. Similar prepara-
tions, but higher temperature anneals in vacuum and in airwere used in our prior studies that resulted in reconstructionstabilized MgO s111dsurfaces.
8–11,13
B.In situexperiments
Two MgO single crystal slabs cut along the s111dplane
were mechanically polished and cleaned with acetone andmethanol solvents. One crystal was etched in hot concen-trated nitric acid, washed with water, and furnace annealed inair to 800 °C. The second crystal was kept in the as-polishedstate. Once in ultrahigh vacuum, both crystals were furthercleaned with oxygen plasma at room temperature, followedby annealing at 800 °C in UHV by electron beam bombard-ment on the back side of the Mo plate on which the MgOcrystals were mounted. The structure of the top surface wasstudied by reflection high-energy electron diffraction
sRHEED d, low-energy electron diffraction sLEED d, and by
x-ray photoelectron andAuger electron diffraction sXPD d.A
Gammadata/Scienta SES 200 photoelectron spectrometersXPSdwith monochromatic Al K
ax-ray source was used to
study the chemical composition of the MgO s111dsurface,
and a computerized two-axis sample goniometer was em-ployed to record the polar and azimuthal variations on the O1sand Mg KLL emission intensities. A flood gun was used
to minimize the effects of surface charging. Details about theexperimental system can be found elsewhere.
23
The preparations used for the MgO s111dsurface in this
work were also used in our recent studies of polar oxideinterfaces formed by epitaxial growth of Fe
3O4film in a
polar s111dorientation.24Such preparations have been used
extensively in the same system for preparation of clean neu-tral MgO s100dsurfaces without any detectable surface
hydroxides.
25The oxygen pressure in the chamber during
plasma cleaning was ,2310−5Torr, as measured with an
ion gauge not directly in the activated oxygen beam. Room-temperature exposure to activated oxygen for tens of minutesresults in the removal of adventitious carbon, as judged byXPS. The only surface contamination present sother than H d
was a trace of F. The amount of F on the surface was about0.15–0.2 ML, compared to ,1 ML of OH. Sputtering was
not used to avoid further damage to the polar surface. Sub-sequent annealing in high vacuum restored the crystallo-graphic order in the surface lost during the mechanical pol-ishing, as judged by significant improvements in theappearance of the LEED and RHEED patterns.
C. Theoretical analysis
Three different theoretical approaches have been used to
analyze the x-ray photoelectron and Auger electron diffrac-tion data: single and multiple scattering XPD in clustergeometry
26and multiple scattering XPD in a slab
geometry.27The results presented in this paper were obtained
using the path-reversed photoelectron diffractionalgorithm
28,29based on the reciprocity theorem in a slab ge-
ometry. In this approach the wave amplitudes at differentemitters can be found by propagating the electron backwardfrom the detector to the emitter. The photoelectron intensityis then evaluated by multiplying these amplitudes by theatomic photoelectron matrix elements. InAuger electron dif-fraction, the matrix elements are replaced by atomic Augermatrix elements. Since the latter are not as easy to evaluatenumerically, we have assumed isotropic emitters for simplic-ity. This approximation is not expected to cause significanterrors in the range of electron energies considered.
The MgO s111dcrystal was divided into layers consisting
either of Mg or O atoms with alternating stacking. The ter-minating layer was varied to be Mg, O, and OH, as proposedpreviously. Variations that move the topmost H and O layersto nonstandard lattice sites were also explored, as weredouble and triple OH layers. The layer diffraction matriceswere found using a conventional LEED package.
30Propaga-
tion of the electron from the detector to the emitter, withmultiple scattering among different layers, was performed byLAZAROV et al. PHYSICAL REVIEW B 71, 115434 s2005 d
115434-2the renormalized forward scattering sRFSdalgorithm.30,31
Both the atomic matrix elements and phase shifts were cal-
culated from a potential constructed using the program
MUFPOT.31
Rfactor analysis was initially performed for interlayer
spacing optimization for atomic models with and withouthydrogen. The weak scattering of the oxygen photoelectronsand magnesium Auger electrons by hydrogen ions producessubtle differences between the calculated model XPD pat-terns, reducing the sensitivity of model refinements. To over-come this limitation, first-principles density functional theorysDFT dcalculations were used to calculate optimized atomic
positions for each model. Then the path-reversed algorithmwas used to calculate XPD patterns for these positions, andanRfactor analysis was performed to compare with the ex-
perimental XPD data, without further structural optimiza-tions.
The DFT equations were solved using the full-potential
linearized augmented plane wave sFLAPW dmethod
32,33as
implemented in the FLAIRcode. Ground-state atomic geom-
etries for all surface terminations were obtained by minimiz-ing the forces and total energies. The calculated MgO bulklattice constants for both the local density approximationsLDA dvalue of 4.194 Å and the generalized gradient ap-
proximation sGGA dvalue of 4.266 Å are in good agreement
with the experimental MgO bulk lattice constant of 4.217 Å;we choose to use LDA in this study because of the slightlybetter agreement with experiment.
The surfaces were modeled by repeated slabs of between
13 and 39 layers. Calculations of polar surfaces in a slabgeometry required particular care due to the long-range Cou-lomb interactions. One way to avoid this problem is to con-struct slabs that are symmetrically terminated fi.e., with odd
numbers of close packed s111dlayers g. We have tested that
our reported results are essentially unchanged with respect toincreasing the number of layers, the “vacuum” region, theBrillouin zone sampling, and the basis size cutoffs. The rela-tively large number of layers fcompared to four or five layers
used in prior MgO s111dcalculations gwas selected to repro-
duce the bulk interplanar distances in the middle of the slab,to give better quantitatively reliable calculated surface inter-planar spacings, as well as allowing for a clearer separationbetween surface and bulk contributions.
III. EXPERIMENTAL RESULTS
A.Ex situTHEED experiments
Figure 1 sadshows a selected area THEED pattern with
pronounced s131dspots. The bright field TEM images from
the same area fFig. 1 sbdgreveal terrace and step structures.
The terraces must have s111dcharacter to allow the observa-
tion of the surface specific s111d-131 spots findicated with
arrows in Fig. 1 sadg, which are forbidden for bulk MgO,
having fractional Muller indices of1
3s422dtype. These
s111d131 reflections are not commonly observed in electron
diffraction data from MgO powder or bulk samples that tend
to be terminated by neutral s100dtype surfaces. To the best
of our knowledge this is the first observation of rocksalts111d-131 THEED patterns. We have observed these reflec-
tions, along with additional reconstruction specific reflec-
tions, in our earlier THEED patterns from MgO s111dsur-
faces with s˛33˛3dR30°, s232dand s2˛332˛3dR30°
structures obtained at much higher annealing temperatures.11
The forbidden1
3s422dreflections were first reported in
THEED patterns from Au foils terminated with s111d-131
surfaces and identified as surface truncation spots.34These
bulk forbidden reflections remain extinguished when thespecimen has a complete number of bulk unit cells in thepropagation direction of the incident beam fi.e., thickness
=3N, whereNis the number of close packed s111dAu
plane g, but become visible for incomplete number of bulk
unit cells si.e., sample thickness of 3 N±1 atomic layers d.The
same analysis and conclusions can be applied to our obser-vation of the MgO s111d-131 reflections, but Nhere be-
comes the number of Mg-O s111dbilayers.
We have previously used the surface reconstruction spot
intensity from THEED to propose structures for the recon-structed MgO s111dsurfaces,
11but the 1 31 reflections were
omitted from these analyses because they contain contribu-tions from incomplete numbers of unit cells in the electronbeam direction. THEED is a powerful method for findingin-plane positions, but not for out-of-plane relaxations.
35For
this reason we undertook further diffraction andspectroscopy experiments to gain information about theMgO s111d-131 surface structure and composition under
controlled vacuum conditions.
B.In situLEED and RHEED experiments
Upon oxygen plasma cleaning and annealing in UHV,
both MgO s111dsingle crystal samples displayed s131d
LEED sFig. 2 dand RHEED sFig. 3 dpatterns, in agreement
with the THEED observations after air anneals. RHEEDfrom the acid-etched and annealed crystals displayed sometransmission contributions, as expected from our previousobservations of vicinal faceting of MgO surfaces upon acidetching.
10,36The LEED patterns shown in Fig. 2 are from a
polished, plasma-cleaned, and UHV-annealed crystal. TheseLEED patterns show a 1 31 surface structure without three-
fold splitting which had been observed previously on facetedMgO s111dsurfaces.
6At present we do not have experimental
LEEDI-Vcurves from these surfaces. Instead, we use angle-
FIG. 1. THEED sadand bright field TEM sbdof an electron
transparent MgO s111dsample annealed at 800 °C in air. Reflections
marked by arrows in saddenote a 1 31 surface structure.STRUCTURE OF THE HYDROGEN-STABILIZED … PHYSICAL REVIEW B 71, 115434 s2005 d
115434-3resolved x-ray photoelectron core-level spectroscopy, and
scanned angle x-ray photoelectron and Auger electron dif-fraction sXPD/AED dto probe the MgO s111d-s131dsurface
composition and structure.
C.In situXPS/XPD experiments
Scanned-angle XPD/AED at low take-off angles enhances
the surface sensitivity of the technique. We have thus mea-sured O 1 sand MgKLLazimuthal angular distributions at
take-off angles ranging from 7° to 16°, as well as polar scansin high-symmetry azimuths fFig. 4 sadg. The s131dsurface
termination is readily determined from qualitative consider-
ations of the polar scans, at least for this particular crystaltype and surface orientation. The O 1 s/MgKLLintensity
ratio is expected to increase sdecrease das grazing emission is
approached if the surface is terminated with O sMgd, due
simply to inelastic attenuation of outgoing photoelectrons.We have found that this ratio increases at low take-off anglesin all low-symmetry azimuths, as seen in Fig. 4 sbdfor polar
angles smaller than 15°, favoring the O and OH terminationmodels over the Mg termination model. A detailed theoreti-cal analysis of the XPD data is presented in Sec. IV, whichfavors the OH termination.
High-energy-resolution O 1 sspectra obtained at normal
emission s
u=90° dand grazing emission su=10° dreveal a
surface specific peak that is shifted by ,2 eV towards higher
binding energy, as shown in Fig. 5.This peak can be ascribedto the presence of a terminal layer of OH based on the nearlyidentical chemical shift measured for dissociative chemisorp-tion of water on MgO s001d.
37However, the observed O 1 s
shift on the polar surface cannot be used to unambiguouslyexclude the O-terminated model without independent knowl-edge of the magnitude and shift direction that would be pro-duced by the predicted two-dimensional s2Ddmetallization
and interlayer relaxation. The DFT calculations of the O 1 s
and 2slevels discussed in Sec. IV confirm the OH termina-
tion interpretation, allowing us to label the shifted peak inFig. 5 as an OH peak. The angular dependence of the OHpeak intensity, relative to that of lattice oxygen, shows thatthe amount of OH is equal to ,1M Li ft h eO1 sphotoelec-
tron escape length is assumed to be 30–35 Å, which is notunreasonable for a wide band gap insulator such as MgO.
IV. THEORETICALANALYSIS
A. Atomic structure from DFT and XPD/AED
Several structural models were constructed for the
MgO s111d-s131dsurface, as summarized in Table I. For all
models, the in-plane and out-of-plane atomic positions were
FIG. 2. LEED patterns for the MgO s111d-131 surface obtained
by oxygen plasma cleaning and annealing at 800 °C in UHV. A131 surface periodicity is evident at all three energies.
FIG. 3. RHEED patterns of MgO s111d-s131dsurface as pre-
pared in Fig. 2.The upper and lower panels show the patterns alongthef11-2gandf1-10gazimuths, respectively.LAZAROV et al. PHYSICAL REVIEW B 71, 115434 s2005 d
115434-4optimized by minimization of the atomic forces in the DFT
total energy calculations. In Table I we report only the firstthree important interplanar distances between the O and Mgclose-packed s111dplanes:d
1is the distance between the two
topmost planes se.g., between the first O and the second Mg
plane for O-terminated and OH-terminated surfaces, or be-tween the first Mg and second O plane for the Mg-terminatedsurface d,d
2is the distance between the second and third
planes, and d3between the third and fourth planes.The first two models are for clean Mg- and O-terminated
surfaces. The interlayer separations for the Mg-terminatedsurface are close to the bulk value, the first layer is expandedby 3.6%, while the second and third layer are contracted by0.3 and 0.2 %, respectively. In contrast, the cleanO-terminated surface layer contracts by 29.4% from the bulkvalue, the second Mg layer expands by 9.4%, and the third Olayer is contracted by 6.3%.
The H-terminated model, with an H on top of every O and
the O in regular rocksalt stacking sites, has an H-O distanceof 0.98 Å. Total energy calculations favor H on top of Oinstead of the hollow site by 0.16 eV. The OH bond length isexpanded by 7.2% with respect to the bulk OH distance inMgsOHd
2. The first O-Mg distance of 1.041 Å is expanded
by 8.6% from the corresponding bulk Mg sOHd2distance of
0.958 Å, and contracted by 14.4% from the bulk MgO dis-
tance of 1.217 Å. The second and third interlayer distancesare close to the bulk oxide value, displaying contractions of0.8 and 0.5 %.
The next three models with two OH groups cover the
options of having H between the second and third layer inaddition to its presence as a top layer. Three models were
considered: sidH on top of the first O layer and H beneath
the O of the second layer, siidH on top of the first O layer
and H in a hollow site between the second and third layers,andsiiidH on top of the first O layer and half a monolayer of
H beneath the second O layer. For all three cases, the top-most H sas in the case with a single OH model dinduced a
contraction for the first sO-Mg dinterlayer distance in the
11–14 % range. The second Mg-O interplanar distance iscontracted by 2.1% in case sid, but expanded by 3.1 and
18.1 % for cases siidandsiiid. Finally the third O-Mg inter-
planar distance was expanded in all three cases, by 51.6,72.9, and 59.4 % with respect to bulk MgO interplanar dis-tances, respectively. It is not unexpected that the presence ofan H plane between the second and third layer should yieldsignificantly enlarged the O-Mg distance.
The last model considered has three OH layers whose
structure is based on the Mg sOHd
2sbrucite dstructure. The
FIG. 4. sadXPD polar scans in three high
symmetry azimuths sf=0°, 30°, and 60° dshow-
ing the Mg KLLAuger sbold line dand the O1 s
photoelectron sthin line dintegrated intensity as
function of take off angle. sbdO1s/MgKLLin-
tensity ratio vs polar angle for same azimuths.This ratio increases at low take-off angles si.e.,
polar angles smaller than 15° dindicating that the
terminating layer is oxygen rich.
FIG. 5. XPS of O1 speak for normal su=90° dand grazing
take-off angles showing surface specific shift toward higher bindingenergy indicative of OH termination of the UHV annealedMgO s111dsurface.STRUCTURE OF THE HYDROGEN-STABILIZED … PHYSICAL REVIEW B 71, 115434 s2005 d
115434-5first O-Mg plane distance in brucitelike termination of
MgO s111dhad a contraction of 2.5%, while the second
Mg-O planar distance is expanded by 2.8%. The third inter-planar O-Mg distance in brucitelike models is distinct fromthe previous models, due to the missing Mg plane beingreplaced by the double H layers. Thus the third planar dis-tance was determined to be 130.9% larger than bulk MgO.
The three topmost planar DFT optimized distances, ex-
cluding the H planes because of their negligible scatteringpower, were used for XPD calculations. A quantitative com-parison between the theoretical and the experimental XPDresults is provided by Pendry’s RfactorR
P.38TheRfactor
analysis, given in Table I, shows that the best fit to the ex-perimental data is given by the OH-terminated surface, witha single monolayer of H on top of an oxygen layer in aregular rocksalt lattice site. However, the difference in R
factors is rather small, especially between the O-terminatedand the single OH-terminated models, necessitating furtheranalysis of the XPD line shapes.
Figure 6 shows the calculated Mg KLLAuger electron
diffraction intensities and the calculated O 1 sphotoelectron
diffraction intensities as a function of polar angle along azi-muth f110g, for the clean Mg- and O-terminated surfaces and
for the OH-terminated structural models obtained from thetotal energy calculations. Since hydrogen atoms have negli-gible cross section in the medium energy range, theH-terminated and O-terminated surfaces have the samestacking sequence and differ only in their surface relaxationsas obtained by the total energy calculation. The experimentaldata are also shown. The prominent peaks at 54.7° swith
respect to surface normal dare due to both the focusing
39,40
and defocusing41effects along an internuclear axis joining
the atoms in adjacent atomic layers. Therefore, this azimuthis expected to have the greatest sensitivity to the surfacestructure.
In Fig. 7 we compare the azimuthal dependence of the
calculated ratio of the Mg KLLto O 1sintensities and the
corresponding experimental data for electrons exiting at aglancing angle of 7° with respect to the surface. If we ignorethe difference in scattering factors of the two atomic speciesand the relatively small difference in electron energy, both
the Auger and photoelectron emitters have the same struc-tural environment in the bulk. Taking the ratio would en-hance the difference that is mainly due to the structure nearTABLE I. The topmost Mg-O or O-O distances used for XPD calculation. Pendry’s Rfactor sRPd
averaged over Mg KLLand O1 sspectra for electron intensities as a function of polar angles along the f110g
are presented.
Model d1sÅdd2sÅdd3sÅdRPsÅd
Mg/O/MgMg terminated 1.254 1.207 1.208 0.24
O/Mg/O O-terminated 0.855 1.325 1.135 0.21
H/O/Mg/ flOH terminated 1.041 1.206 1.209 0.19
H/O/Mg/ flOH terminated with stacking fault 1.067 1.276 1.215 0.26
H/O/Mg/O/H/Mg/OOHontopwitha
monolayer of H in second O plane1.050 1.185 1.836 0.35
H/O/Mg/O/H/Mg/OOHontopwitha
monolayer of H in hollow beneath second O plane1.078 1.248 2.093 0.37
H/O/Mg/O/H/Mg/OOHontopwitha halve
monolayer of H in second O plane1.047 1.429 1.929 0.44
H/O/Mg/O/H/H/O Brucite with OH on top 1.118 1.245 2.795 0.41
FIG. 6. Calculated Mg KLLAuger intensities and O1 sphoto-
electron intensities as function of polar angle smeasured from sur-
face plane dalong the f110gazimuth, assuming Mg-, O-, and OH-
terminated MgO s111d-s131dstructural models with distances
obtained from the total energy calculations. The corresponding ex-perimental results are shown as bold lines.LAZAROV et al. PHYSICAL REVIEW B 71, 115434 s2005 d
115434-6the surface. The different models do show considerable sen-
sitivity to the surface structure. As seen from the figure, theMg-terminated model can be excluded. For the OH- andO-terminated models, the relative intensities of peaks favorthe former model. We also note that there are shifts betweenthe calculated and experiment peaks which could be due touncertainty in sample alignment.
B. Electronic structure from DFT calculations
In addition to structural properties, the FLAPW calcula-
tions provide information about the electronic properties ofthe systems. The local density of states sLDOS d, shown in
Fig. 8, show the valence band and the deep-lying O 2 sstates,
and provide unique signatures for the preferred surface ter-mination. Figures 8 sad–8scdshows the LDOS of the top Mg
and O layers for the Mg, O, and OH terminations. The bulkLDOS for the Mg and O are also presented for reference inFig. 8 sdd. For both O and Mg termination, the top layers
have states around the Fermi level, indicative of the pro-posed 2D metallic character of both surfaces. The DOS ofthe top O layer are derived from the O valence band, while inthe Mg terminations the states around the Fermi energy inthe middle of the gap are derived mainly from the conduc-tion Mg band. In contrast, the OH-terminated surface doesnot exhibit metallic spartially occupied dstates, but the sur-
face band gap is reduced by 70% with respect to bulk MgObecause of the H-induced surface states in the gap. Experi-mental recording of valence band photoelectron spectra isprecluded by the long counting times sseveral hours dduring
which charging causes slow drifts in the binding energy scalethat are hard to compensate by flood guns. UPS was furthercomplicated by charging because the He source is so much
brighter than the x-ray source.
For the OH termination, the O-derived 2 sstates at the
surface display a ,2 eV shift towards higher binding energy
fFig. 8 sadg, and the 1 sstates are shifted by ,1e V snot
shown d. These shifts are in the same direction as the experi-
mental shift of 2 eV seen in the XPS spectra in Fig. 5. Forthe O termination, the calculations a −1 eV shift of the 2 s
surface state and −0.2 eV of the 1 sstate, both towards lower
binding energy and in the oppositedirection from the experi-
ment. The shifts of the O 1 sand 2sstates for the Mg-
terminated slabs are essentially zero, in contrast to the ex-perimentally observed shifts. These trends can also be seenin the dispersion of the surface and bulk bands for the threemodels, as shown to the right in Fig. 8. The different termi-nations also will affect the surface dipole, and hence thework functions of the system. The H changes the densitysignificantly in the surface region, and thereby reducing thesurface dipole of the O-terminated surface by several eV,leading to a work function for the OH-terminated surface of,3.6 eV.
FIG. 7. Calculated azimuthal Mg KLL/O1sintensity ratios for
the models shown in Fig. 6, and the corresponding experimentaldata for electrons exiting at a glancing angle of 7° with respect tothe surface. The best fit is obtained for the OH-terminated surfacestructure.
FIG. 8. Local density of states for the topmost O and Mg atomic
planes for slabs terminated with sadOH, sbdO, and scdMg, and sdd
for bulk MgO. The dashed vertical lines represent the Fermi levels,showing the insulating nature of the OH termination, and the me-tallic nature of the O and Mg terminations. sThe “bulk” atoms of
the different calculations are used to align the energies. dCompari-
son with the position of bulk O2 slevel sdashed-dotted line dshows
a shift to higher slower dbinding energy for OH sOdtermination.
The band structure for the three terminations is shown on the rightfor the OH stopd,Osmiddle d, and Mg sbottom dterminations.STRUCTURE OF THE HYDROGEN-STABILIZED … PHYSICAL REVIEW B 71, 115434 s2005 d
115434-7The above calculations also allow for spin polarization.
While the oxygen-terminated surface has a calculated mag-netic moment, this moment is quenched when hydrogen isadsorbed. The changes in the calculated structural propertiesfor the magnetic O-terminated surface are small compared tothe scale needed for noticeable differences in the Rfactor.
Regarding the possibly more significant changes in the elec-tronic properties with surface magnetism, especially with re-gard to the shifts in the density of states, the basic physicsand conclusions are the same as for a nonmagnetic surface:For the magnetic O-terminated surface, the sfully occupied d
majority DOS were found to be aligned with the bulk ODOS, while the minority DOS are again shifted towardslower binding energy swith the shift large for magnetic O d,
opposite to the shift for OH termination.
V. DISCUSSION AND CONCLUSIONS
Our experimental and theoretical results favor the OH-
terminated model over the stoichiometric O-terminated orMg-terminated models for this polar oxide surface, in agree-ment with prior theoretical predictions of dipole moment andsurface energy minimization provided by the OHtermination.
19,42,43The interlayer separations obtained in our
DFT calculations and XPD experiments are in close agree-ment with recently published DFT results for this surface.
43
In addition, we have considered new models with double andtriple OH layers in a Mg sOHd
2configuration. These models
generate poorer fits to the experimental data; however, they
could be used in modeling of the initial steps of bulk hy-droxylation studied recently by environmental electronmicroscopy.
44
Hydrogen is difficult to detect in electron diffraction ex-
periments due to its small scattering factor. Our XPD resultsindicate that the OH termination with the first O-Mg contrac-tion of 14.4% presents only a slightly better fit to the experi-ments than the O termination with contraction of 29.4%.However, it is important to note that our DFT calculationsindicate that the O core levels can be used to easily discrimi-nate between the O and the OH termination, because theygive chemical shifts in opposite directions.
Our results pointing to H stabilization of the MgO s111d
surface suggests that an unreconstructed stoichiometric s111d
surface remains as yet unobserved. It is notable that the hy-drogen termination is present even upon oxygen plasmacleaning of the native MgO s111dsurface followed by 800 °C
annealing under ultrahigh vacuum s,10
−11Torrd. The same
preparation conditions have been found to yield hydrogen-
free neutral MgO s100d-s131dsurfaces. The source of the
surface hydrogen in the MgO s111dexperiments is not fully
understood. It could be present on the native s111dsurface
via interaction with atmospheric water, as suggested by thetheoretical study of Refson et al.
20that showed the hydroxy-
lated MgO s111dface to be more stable than the s100dorien-
tation. These authors have argued that the interaction withwater drives the frequent occurrence of s111dterminations on
geological MgO spericlase dsamples. Once formed, the OH
termination would provide a very effective stabilization ofthe polar surface and might be present at the native surfaces,as also suggested for
a−Al2O3s0001 d.45Indeed, our recent
follow up XPS experiments with air exposed unreconstructed
MgO s111dsurfaces found a strong OH signal that is reduced
upon annealing in UHV, but not eliminated.46At this time we
cannot determine if this native hydroxide termination is im-pervious to the oxygen plasma cleaning and UHV annealingthat were used in the present work, or if the hydrogen termi-nation is removed and then reformed upon cooling by ad-
sorption of atomic hydrogen or by dissociative chemisorp-tion of H
2or H2O from the UHV ambient. These results
suggest that polar oxide surfaces may be of interest for split-ting water and/or hydrogen storage.
Apart from the effects of the annealing temperature and
pressures, the crystal thickness in the polar direction alsomerits discussion. In our study we find the MgO s111d-s1
31d-OH structure on both macroscopic single crystals and
on thinned unsupported single crystals of thickness ranging
from few nanometers to several hundred micrometers. Allprior reports of the s111d-s131d-OH surface structure have
been for ultrathin films of NiO, CoO, and FeO grown on
metal substrates, using spectroscopic methods to determinethe presence of hydrogen, and diffraction methods to docu-ment the 1 31 periodicity.
3To the best of our knowledge
there are no reported surface structure determinations for thes111d-s131d-OH atomic positions in these thin films. A re-
versible s131d-OH tops232dreconstruction has been re-
ported for 1–2 nm thin films of NiO s111don Ni s111dupon
heating in UHV and readsorption of water.
47However, the
single crystal NiO s111d-s232dsurfaces were not found to
transform to a s131d-OH structure upon water exposure un-
der conditions that readily transform the thin films.14In this
sense our elucidation of the MgO s111d-s131d-OH surface
structure marks a first such finding on a bulk rocksalt oxide
crystal.
A few reports of unreconstructed surfaces exist for ultra-
thin oxide films grown in the polar direction on metal sub-strates. A 14.8% contraction of the O-Ni interplanar separa-tion was reported for O-terminated NiO s111dfilms on
Nis100dsubstrates from LEED,
48but hydrogen was not in-
cluded in the models. This result has not been reproduced inother studies of NiO s111dfilms. A bulk-terminated 1 31
structure has also been reported for a single bilayer ofFeOs111don Pt s111d,
49and for a single trilayer of NaCl s111d
on Al s111d,50but such ultrathin polar surface structures on
macroscopic metal substrates are expected to have differentstabilization behavior. Most recently Kiguchi et al.
51have
shown that a rather flat unreconstructed polar MgO s111d-
s131dcould be grown by alternate adsorption of Mg and O 2
on Ag s111d. While the absence of hydrogen has not been
directly proven, electron energy loss spectroscopy sEELS d
and ultraviolet photoelectron spectroscopy sUPSdresults in-
dicate that the surface is semiconducting or metallic, in con-trast to the insulating nature expected for the OH termina-tion.
In conclusion, our combined experimental and theoretical
study shows that hydrogen stabilizes the MgO s111dpolar
oxide surface. In addition, we report on the structural deter-mination of a s111d-s131d-OH surface for a polar rocksalt
crystal.LAZAROV et al. PHYSICAL REVIEW B 71, 115434 s2005 d
115434-8ACKNOWLEDGMENTS
This study was sponsored by the National Science Foun-
dation sGrant No. NSF/DMR-95531489 d, the Research Cor-
poration sGrant No. RA0331 d, and the U.S. Department of
Energy sGrant No. DE-FG02-84ER45076 d. Thein situex-periments were performed in the Environmental Molecular
Sciences Laboratory, a national scientific user facility spon-
sored by the Department of Energy’s Office of Biologicaland Environmental Research located at Pacific NorthwestNational Laboratory. We would like to thank Mark Pauli forcluster XPD calculations in the early stages of this study.
*Current address: Sandia National Labs, M.S. 1310, P.O. Box 5800,
Albuquerque, NM 87185, USA.
†Electronic address mgj@uwm.edu
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115434-9 |
PhysRevB.81.012502.pdf | Josephson effect for SU(4) carbon-nanotube quantum dots
A. Zazunov,1A. Levy Yeyati,2and R. Egger1
1Institut für Theoretische Physik, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany
2Departamento de Física Teórica de la Materia Condensada C-V , Universidad Autónoma de Madrid, E-28049 Madrid, Spain
/H20849Received 24 November 2009; published 8 January 2010 /H20850
We present the theory of the Josephson effect in nanotube dots where an SU /H208494/H20850symmetry can be realized.
We find a remarkably rich phase diagram that significantly differs from the SU /H208492/H20850case. In particular,
/H9266-junction behavior is largely suppressed. We analytically obtain the Josephson current in various parameter
regions: /H20849i/H20850in the Kondo regime covering the full crossover from SU /H208494/H20850to SU /H208492/H20850,/H20849ii/H20850for weak tunnel
couplings, and /H20849iii/H20850for a large BCS gap. The transition between these regions is studied numerically.
DOI: 10.1103/PhysRevB.81.012502 PACS number /H20849s/H20850: 74.50. /H11001r, 73.63. /H11002b, 74.78.Na
I. INTRODUCTION
Several experimental groups have recently started to
study the Josephson effect in ultrasmall nanostructures,1
where the supercurrent can be tuned via the gate voltagedependence of the electronic levels of the nanostructure. Animportant system class where supercurrents have been suc-cessfully observed
2is provided by carbon-nanotube /H20849CNT /H20850
quantum dots. In many cases, the experimental results com-pare quite well to predictions based on modeling the CNTdot as a spin-degenerate electronic level with SU /H208492/H20850spin
symmetry, where the presence of a repulsive on-dot chargingenergy Umay allow for a /H20849normal-state /H20850Kondo effect. De-
pending on the ratio T
K//H9004, where /H9004is the energy gap in the
superconducting electrodes and TKthe Kondo temperature,
theory3–8predicts a transition between a unitary /H20849maximum /H20850
Josephson current for /H9004/H11270TK, possible thanks to the survival
of the Kondo resonance in that limit, and a /H9266-junction re-
gime for /H9004/H11271TK, where the critical current is small and nega-
tive, i.e., the junction free-energy F/H20849/H9272/H20850has a minimum at
phase difference /H9272=/H9266as opposed to the more common
0-junction behavior.
Recent progress has paved the way for the fabrication of
very clean CNTs, resulting in a generation of quantum trans-port experiments and thereby revealing interesting physics,e.g., spin-orbit coupling effects
9or incipient Wigner crystal
behavior.10In ultraclean CNTs, the orbital degree of freedom
/H20849/H9251=/H11006/H20850reflecting clockwise and anticlockwise motion
around the CNT circumference /H20849i.e., the two Kpoints /H20850is
approximately conserved when electrons enter or leave thedot.
11Due to the combined presence of this orbital “pseu-
dospin” /H20849denoted in the following by T/H20850and the true elec-
tronic spin /H20849S/H20850, an enlarged SU /H208494/H20850symmetry group can be
realized. In addition, a purely orbital SU /H208492/H20850symmetry arises
when a Zeeman field is applied. Experimental support forthis scenario has already been published
12/H20849for the case of
semiconductor dots, see Ref. 13/H20850and several aspects have
been addressed theoretically.11,14In particular, the SU /H208494/H20850
Kondo regime is characterized by an enhanced Kondo tem-perature and exotic local Fermi liquid behavior, where theKondo resonance is asymmetric with respect to the Fermilevel. However, so far both experiment and theory have onlystudied the case of normal conducting leads, where conven-tional linear response transport measurements cannot reliablydistinguish the SU /H208494/H20850from the SU /H208492/H20850scenario.
14Here we
provide the first theoretical study of the Josephson effect forinteracting quantum dots with /H20849approximate /H20850SU/H208494/H20850symme-
try, and find drastic differences compared to the standardSU/H208492/H20850picture. In the Kondo limit, a qualitatively different
current-phase relation /H20849CPR /H20850is found, with the critical cur-
rent smaller by a factor /H110150.59. The usual
/H9266-junction behav-
ior is largely suppressed, but unconventional phases do ap-pear and time-reversal symmetry can be spontaneouslybroken. Our predictions can be tested using state-of-the-artexperimental setups, and offer clear signatures of the SU /H208494/H20850
symmetry in very clean CNT quantum dots.
II. MODEL AND FORMAL SOLUTION
We study a quantum dot /H20849Hd/H20850contacted via a standard
tunneling Hamiltonian /H20849Ht/H20850to two identical superconducting
electrodes /H20849HL/R/H20850,H=Hd+Ht+HL+HR. We assume that the
dot has a spin- and orbital-degenerate electronic level /H9280/H9251/H9268
=/H9280with identical intra- and inter-orbital charging energy U,15
Hd=/H9280nˆ+Unˆ/H20849nˆ−1/H20850/2 with nˆ=/H20858/H9251/H9268d/H9251/H9268†d/H9251/H9268, where d/H9251/H9268†creates
a dot electron with spin /H9268=↑,↓=/H11006and orbital pseudospin
projection /H9251. Since the /H9251=/H11006states are related by time-
reversal symmetry /H20849clockwise and anticlockwise states are
exchanged /H20850, we take the lead Hamiltonian as
Hj=/H20858
k/H9251/H9268/H9264kcjk/H9251/H9268†cjk/H9251/H9268+/H20858
k/H9251/H20849/H9004e/H11007i/H20849/H9272/2/H20850cjk/H9251↑†cj,−k,−/H9251,↓†+ H.c. /H20850,
where cjk/H9251/H9268†creates an electron with wave vector kin lead
j=L/R, and /H9264kis the single-particle energy. The tunneling
Hamiltonian is Ht=/H20858jk/H9268,/H9251/H9251/H11032/H20849t/H9254/H9251/H9251/H11032+t˜/H9254/H9251,−/H9251/H11032/H20850cjk/H9251/H9268†d/H9251/H11032/H9268+H.c.,
where t/H20849t˜/H20850describes orbital /H20849non/H20850conserving tunneling pro-
cesses. Following standard steps,4the noninteracting lead
fermions can now be integrated out. The partition functionZ/H20849
/H9272/H20850=e−/H9252F/H20849/H9272/H20850at inverse temperature /H9252then reads /H20849we often
sete=/H6036=1/H20850
Z/H20849/H9272/H20850=T r d/H20849e−/H9252HdTe−/H208480/H9252d/H9270d/H9270/H11032D†/H20849/H9270/H20850/H9018/H20849/H9270−/H9270/H11032/H20850D/H20849/H9270/H11032/H20850/H20850, /H208491/H20850
where the trace extends over the dot Hilbert space, Tdenotes
time ordering, and we use the Nambu bispinor D
=/H20849de↑,de↓†,do↑,do↓†/H20850with even/odd linear combinations of the
orbital states, de/H9268=/H20849d+,/H9268+d−,/H9268/H20850//H208812 and do/H9268=/H9268/H20849d+,/H9268
−d−,/H9268/H20850//H208812. In this basis, the self-energy /H9018/H20849/H9270/H20850representingPHYSICAL REVIEW B 81, 012502 /H208492010 /H20850
1098-0121/2010/81 /H208491/H20850/012502 /H208494/H20850 ©2010 The American Physical Society 012502-1the BCS leads is diagonal in orbital space. With the orbital
mixing angle /H9258=2 tan−1/H20849t˜/t/H20850and the normal-state density of
states/H92630=2/H20858k/H9254/H20849/H9264k/H20850, the even/odd channels are characterized
by the hybridization widths /H9003/H9263=e,o=/H208491/H11006sin/H9258/H20850/H9003with/H9003
=/H9266/H92630/H20849t2+t˜2/H20850. In what follows, we study the zero-temperature
limit and assume the wide-band limit1for the leads. The
Fourier transformed self-energy is then expressed in terms ofthe 2/H110032 Nambu matrices
/H9018
/H9263=e,o/H20849/H9275/H20850=/H9003/H9263
/H20881/H92752+/H90042/H20898−i/H9275/H9004cos/H9272
2
/H9004cos/H9272
2−i/H9275/H20899.
The result /H208491/H20850will now be examined in several limits. We
start with the strong-correlation limit U→/H11009, and later ad-
dress the case of finite U. Note that Eq. /H208491/H20850for/H9258=0 corre-
sponds to the SU /H208494/H20850symmetric case while for /H9258=/H9266/2 there
is only one conducting channel with nonzero transmissionwhich, under certain conditions, corresponds to the usualSU/H208492/H20850model.
III. DEEP KONDO LIMIT
Let us first discuss the Kondo limit TK/H11271/H9004 in the quarter-
filled case, /H9280/H110210 and /H20855nˆ/H20856/H110151. The Kondo temperature is
given by TK=Dexp/H20849/H9266/H9280/4/H9003/H2085011with bandwidth D. As in the
SU/H208492/H20850case,3the Josephson current at T=0 can be computed
from local Fermi-liquid theory, either using phase shift argu-ments or an equivalent mean-field slave-boson treatment.
5
The latter approach yields the self-consistent dot level /H9280˜and
thereby the transmission probability for channel /H9263=e,o,11
T/H9263=/H208491/H11006sin/H9258/H208502TK2
/H9280˜2+/H208491/H11006sin/H9258/H208502TK2,/H9280˜
TK=/H208491 − sin /H9258/H20850/H20849sin/H9258+1/H20850/4
/H208491 + sin /H9258/H20850/H20849sin/H9258−1/H20850/4.
/H208492/H20850
In the SU /H208494/H20850case /H20849/H9258=0/H20850, we have Te=To=1 /2, while the
SU/H208492/H20850limit /H20849/H9258=/H9266/2/H20850has a decoupled odd channel, Te=1
andTo=0. The CPR covering the crossover from the SU /H208494/H20850
to the SU /H208492/H20850Kondo regime then follows as
I/H20849/H9272/H20850=e/H9004
2/H6036/H20858
/H9263=e,oT/H9263sin/H9272
/H208811−T/H9263sin2/H9272
2. /H208493/H20850
The known SU /H208492/H20850result3is recovered for /H9258=/H9266/2. The SU /H208494/H20850
CPR has a completely different shape, as shown in Fig. 1.
We note that the critical current Ic=max /H20851I/H20849/H9272/H20850/H20852is suppressed
by the factor 2− /H208812/H110150.59 relative to the unitary limit e/H9004//H6036
reached for the SU /H208492/H20850dot. The Josephson current in the deep
Kondo regime is thus very sensitive to the SU /H208494/H20850vs SU /H208492/H20850
symmetry.
IV. PERTURBATION THEORY IN /H9003
Next we address the opposite limit of very small /H9003/H11270/H9004 ,
where lowest-order perturbation theory in /H9003applies. After
some algebra, Eq. /H208491/H20850for/H9258=0 yields the CPR of a tunneljunction, I/H20849/H9272/H20850=Icsin/H20849/H9272/H20850, where the critical current is
Ic=/H208514/H9008/H20849/H9280/H20850−/H9008/H20849−/H9280/H20850/H20852F/H20849/H20841/H9280/H20841//H9004/H20850I0, /H208494/H20850
with the Heaviside function /H9008, the current scale I0
=/H9004/H20849/H9003//H9266/H9004/H208502, and /H20849see also Ref. 16/H20850
F/H20849x/H20850=/H20849/H9266/2/H208502/H208491−x/H20850− arccos2x
2x/H208491−x2/H20850.
In this U→/H11009limit, the dot contains one electron for /H20849finite /H20850
/H9280/H110210, and thus we have spin S=1 /2. Equation /H208494/H20850shows that
such a magnetic junction displays a /H9266phase. For the SU /H208494/H20850
case, the ratio Ic/H20849−/H20841/H9280/H20841/H20850/Ic/H20849/H20841/H9280/H20841/H20850=−1 /4 is twice smaller than in
the SU /H208492/H20850case, i.e., /H9266-junction behavior tends to be sup-
pressed. This tendency is also confirmed for U/H11270/H9004 /H20849see be-
low/H20850, where the /H9266phase is in fact essentially absent. The
factor 1/4 can be understood in simple terms by counting thenumber of possible processes leading to a Cooper pair trans-fer through the dot.
17,18When /H9280/H110220, there are four possibili-
ties corresponding to the quantum numbers /H20849/H9251,/H9268/H20850of the first
electron entering the dot. However, for /H9280/H110210 there is only
one possibility since an electron already occupies the dot andthen only one specific choice of /H20849
/H9251,/H9268/H20850allows for Cooper
pair tunneling. This argument is readily generalized to theSU/H208492N/H20850case, where the above ratio of critical currents is
obtained as −1 /2N.
V. EFFECTIVE HAMILTONIAN FOR /H9004\/H11557
The partition function /H208491/H20850simplifies considerably when /H9004
exceeds all other energy scales of interest. Then the dynam-ics is always confined to the subgap region /H20849Andreev states /H20850
and quasiparticle tunneling processes from the leads /H20849con-
tinuum states /H20850are negligible. In particular, this allows to
study the case U/H11270/H9004. In fact, for /H9004→/H11009, with the Cooper
pair operators b
1†=de↑†do↓†and b2†=do↑†de↓†, Eq. /H208491/H20850is equiva-
lently described by the effective dot Hamiltonian
H/H11009=Hd+ cos /H20849/H9272/2/H20850/H20851/H9003eb1+/H9003ob2+ H.c. /H20852. /H208495/H20850
The resulting Hilbert space can be decomposed into three
decoupled sectors19according to spin Sand orbital pseu-0 0.2 0.4 0.6 0.8 1
ϕ/π00.20.40.60.81I(ϕ)/Icθ=0
θ=π/4
θ=π/2
FIG. 1. /H20849Color online /H20850Josephson CPR in the Kondo limit for
various /H9258. The SU /H208494/H20850case corresponds to /H9258=0, the SU /H208492/H20850case to
/H9258=/H9266/2. The supercurrent is given in units of the unitary limit Ic
=e/H9004//H6036.BRIEF REPORTS PHYSICAL REVIEW B 81, 012502 /H208492010 /H20850
012502-2dospin T/H20849notice that these quantities are localized on the dot
for/H9004→/H11009/H20850. The ground-state energy Eg/H20849/H9272/H20850=min /H20849E/H20849S,T/H20850/H20850then
determines the Josephson current I/H20849/H9272/H20850=2/H11509/H9272Eg/H20849/H9272/H20850./H20849i/H20850The
/H20849S,T/H20850=0 sector is spanned by the four states
/H20853/H208410/H20856,b1†/H208410/H20856,b2†/H208410/H20856,b1†b2†/H208410/H20856/H20854, where /H208410/H20856is the empty dot state.
The matrix representation reads
H/H20849S,T/H20850=0=/H208980 /H9003ecos/H9272
2/H9003ocos/H9272
20
/H9003ecos/H9272
2E2 0 /H9003ocos/H9272
2
/H9003ocos/H9272
20 E2/H9003ecos/H9272
2
0 /H9003ocos/H9272
2/H9003ecos/H9272
2E4/H20899,
with the eigenenergies En=/H9280n+Un/H20849n−1/H20850/2 of the decoupled
dot. The lowest energy E/H20849S,T/H20850=0=E2+zthen follows from the
smallest root of the quartic equation /H20863/H11006/H20851z2−2zU
−/H20849/H9003e/H11006/H9003 o/H208502cos2/H9272
2/H20852=/H20849E4z/2/H208502./H20849ii/H20850The /H20849S,T/H20850=1 /2 sector can
be decomposed into four subspaces with one or three elec-trons according to S
z=/H110061/2 and Cooper pair channel /H9263
=e,o. The Hamiltonian is
H/H20849S,T/H20850=1 /2/H20849/H9263/H20850=/H20873E1/H9003/H9263cos/H9272
2
/H9003/H9263cos/H9272
2E3/H20874,
where H/H20849S,T/H20850=1 /2/H20849e/H20850operates in the subspace spanned by
/H20853do↑†/H208410/H20856,b1†do↑†/H208410/H20856/H20854forSz=+1 /2, and /H20853de↓†/H208410/H20856,b1†de↓†/H208410/H20856/H20854forSz
=−1 /2./H20849Similarly, the subspaces corresponding to H/H20849S,T/H20850=1 /2/H20849o/H20850
are obtained by letting d/H9263/H9268†→d/H9263,−/H9268†and b1†→b2†./H20850With/H9003e
/H11350/H9003 o, the lowest energy is E/H20849S,T/H20850=1 /2=/H20849E1+E3−/H20851/H20849E3−E1/H208502
+4/H9003e2cos2/H20849/H9272/2/H20850/H208521/2/H20850/2./H20849iii/H20850Finally, the /H20849S,T/H20850=/H208491,0/H20850sector
is spanned by the two uncoupled two-particle states
de,/H9268†do,/H9268†/H208410/H20856, with /H9272-independent energy ES=1,T=0=E2. In addi-
tion, there are two decoupled /H20849S,T/H20850=/H208490,1/H20850states d/H9263↑†d/H9263↓†/H208410/H20856
with the same energy E2. In the limit /H9004→/H11009, this /H20849S,T/H20850=1
sector is energetically unfavorable except possibly at /H9272=/H9266.VI. PHASE DIAGRAM FOR /H9004š/H9003
Next we discuss the resulting phase diagram in the SU /H208494/H20850
limit /H20849/H9258=0/H20850. The result for /H9004→/H11009is shown in Fig. 2in the
U−/H9280plane. The phases are classified according to the three
sectors defined above.20The reported phases are specific for
the SU /H208494/H20850symmetry and are qualitatively different from the
standard SU /H208492/H20850case. We observe that the /H9272-dependence of
the/H9004→/H11009ground-state energy implies 0-junction behavior
for both S=0 and S=1 /2. While the magnetic S=1 /2 sector
often represents a /H9266-junction,3,4,6in multilevel dots there is
no direct connection between the spin and the sign of theJosephson coupling.
18The/H9266-phase found under perturbation
theory /H20851Eq. /H208494/H20850for/H9280/H110210/H20852is in fact restricted to the regime
U/H11271/H9004, while for U/H11270/H9004, the S=1 /2 state displays a 0-phase.
In the intermediate regime one should therefore observe acrossover between those two behaviors. Interestingly, thereare parameter regions with a spin/pseudospin transition as
/H9272
varies. For instance, the “black” regions in Fig. 2correspond
to a mixed state with /H20849S,T/H20850=0 at /H9272=0 and /H20849S,T/H20850=1 /2a t/H9272
=/H9266, while for the “blue” region, the ground state is in the
/H20849S,T/H20850=0 sector except at /H9272=/H9266where it crosses to the
/H20849S,T/H20850=1 sector.
We find that these phases are also observable at finite /H9004
/H11407/H9003, where we have employed two complementary ap-
proaches. First, a full numerical solution is possible whenapproximating each electrode by a single site /H20849zero-
bandwidth limit /H20850, which can provide a satisfactory, albeit not
quantitative, understanding of the phase diagram.
8Second,
one can go beyond the above /H9004→/H11009limit by including co-
tunneling processes in a systematic way. Both approachesgive essentially the same results, and here we only showresults from the single-site model. As can be observed in Fig.0 1 02 03 04 0
-ε/Γ05101520 U/Γ
FIG. 2. /H20849Color online /H20850Phase diagram for /H9004→/H11009. White regions
correspond to /H20849S,T/H20850=0, and green regions to /H20849S,T/H20850=1 /2. In the
black regions, the ground state has /H20849S,T/H20850=0 for /H9272=0 and /H20849S,T/H20850
=1 /2 for/H9272=/H9266. For the blue region, we have /H20849S,T/H20850=0 at /H9272=0 and
/H20849S,T/H20850=1 at /H9272=/H9266./H20849Ref. 20/H208500 0.5 1
ϕ/π-0.500.5I(ϕ)/Ι0
0 5 10 15 20
-ε/Γ05101520 U/ΓU/Γ=10.0
10.5
11.0
11.5(b)(a)
FIG. 3. /H20849Color online /H20850/H20849a/H20850Same as Fig. 2but for /H9004=10/H9003within
the zero-bandwidth limit for the leads /H20849see text /H20850. Although the /H9004
→/H11009phase diagram is basically reproduced, for finite /H9004, the
/H20849S,T/H20850=1 /2 phase /H20849green /H20850exhibits a crossover from 0- to /H9266-junction
behavior for U/H11229/H9004, as illustrated in panel /H20849b/H20850, where the CPR is
shown for /H9280//H9003=−5 and several U; the current is normalized to I0,
see Eq. /H208494/H20850. Moreover, a phase with /H20849S,T/H20850=0 at /H9272=0 and /H20849S,T/H20850
=1 /2a t/H9272=/H9266appears, where /H20849contrary to the “black” phase /H20850/H9272
=/H9266corresponds to the lowest energy /H20849/H9266/H11032behavior /H20850, indicated in red
/H20851within the dashed ellipses in panel /H20849a/H20850/H20852.BRIEF REPORTS PHYSICAL REVIEW B 81, 012502 /H208492010 /H20850
012502-33/H20849a/H20850, the overall features of the /H9004→/H11009phase diagram are
reproduced for finite /H9004, with somewhat shifted boundaries
between the different regions. In particular, in the “green”/H20849S=1 /2/H20850regime, this calculation captures the mentioned
transition from a 0-junction at /H9004/H11271Uto a
/H9266junction at /H9004
/H11270U, as illustrated in Fig. 3/H20849b/H20850. Consequently, for finite /H9004,
the “black” phase may now have lowest energy at /H9272=/H9266,
implying the /H9266/H11032phase4,6,8indicated in “red” in Fig. 3/H20849a/H20850.
Finally, for the junctions with U//H9003=10.5 and 11 in Fig. 3/H20849b/H20850,
the ground state is realized at phase difference 0 /H11021/H9272/H11021/H9266,
which implies that time-reversal symmetry is spontaneouslybroken here.
To conclude, we have studied the Josephson current in
SU/H208494/H20850symmetric quantum dots, including the crossover to
the standard SU /H208492/H20850symmetric case. Contrary to normal-state
transport, the supercurrent is very sensitive to the symmetrygroup and should allow to observe clear signatures of theSU/H208494/H20850state in ultraclean CNT dots. In particular, the
/H9266phase
is largely suppressed, the CPR in the Kondo limit has a dis-tinctly different shape and a smaller critical current, and thephase diagram turns out to be quite rich. In addition, follow-ing Ref. 21, we expect a strongly reduced thermal noise in
the deep SU /H208494/H20850Kondo regime since /H20851in contrast to the SU /H208492/H20850
case /H20852there are two channels with imperfect transmission.
Future theoretical work is needed to give a quantitative un-derstanding of the crossover between the various regimesdiscussed above.
ACKNOWLEDGMENTS
This work was supported by the SFB TR/12 of the DFG,
the EU network HYSWITCH, the ESF network INSTANS,and by the Spanish MICINN under Contracts No. FIS2005-06255 and No. FIS2008-04209.
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012502-4 |
PhysRevB.97.125422.pdf | PHYSICAL REVIEW B 97, 125422 (2018)
Near-field three-terminal thermoelectric heat engine
Jian-Hua Jiang1and Yoseph Imry2
1College of Physics, Optoelectronics and Energy, & Collaborative Innovation Center of Suzhou Nano Science and Technology,
Soochow University, 1 Shizi Street, Suzhou 215006, China
2Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
(Received 17 November 2017; revised manuscript received 21 January 2018; published 19 March 2018)
We propose a near-field inelastic thermoelectric heat engine where quantum dots are used to effectively rectify
the charge flow of photocarriers. The device converts near-field heat radiation into useful electrical power.Heat absorption and inelastic transport can be enhanced by introducing two continuous spectra separated byan energy gap. The thermoelectric transport properties of the heat engine are studied in the linear-responseregime. Using a small band-gap semiconductor as the absorption material, we show that the device achievesvery large thermopower and thermoelectric figure of merit, as well as considerable power factor. By analyzingthermal-photocarrier generation and conduction, we reveal that the Seebeck coefficient and the figure of merithave oscillatory dependence on the thickness of the vacuum gap. Meanwhile, the power factor, the charge, andthermal conductivity are significantly improved by near-field radiation. Conditions and guiding principles forpowerful and efficient thermoelectric heat engines are discussed in details.
DOI: 10.1103/PhysRevB.97.125422
I. INTRODUCTION
Thermoelectric phenomena are at the heart of transport
study of material properties which reveals important informa-tion of microscopic quasiparticle processes [ 1,2]. Being able to
generate electrical power using a temperature gradient or coolan object using electrical power without mechanical motionor chemical processes, thermoelectricity has been one of themain branches in renewable energy research for decades [ 3–6].
The main challenges here are to improve the energy efficiencyand power density, using low-cost and environment-friendlymaterials and designs.
Mahan and Sofo pointed out that the thermoelectric figure
of merit can be written as [ 4]
ZT=σS
2T
κe+κp=/angbracketleftE−μ/angbracketright2
Var (E−μ)+/Lambda12,/Lambda1=e2κpT/σ, (1)
where σis the electrical conductivity, Sis the Seebeck coeffi-
cient,Tis the equilibrium temperature, κeis the electronic heat
conductivity, and κpis the heat conductivity due to phonons.
The last equality rewrites the figure of merit in microscopicquantities. The key is to assume an energy-dependent conduc-tivityσ(E), and the average here is defined as [ 4]
/angbracketleft.../angbracketright=/integraltext
dEσ (E)(−∂
Ef(E)).../integraltext
dEσ (E)(−∂Ef(E)), (2)
where f(E)=[exp(E−μ
kBT)+1]−1is the Fermi distribution
function with μbeing the equilibrium chemical potential.
The symbol “Var” in Eq. ( 1) denotes the variance, Var( E−
μ)=/angbracketleft(E−μ)2/angbracketright−/angbracketleftE−μ/angbracketright2. Equation ( 1) shows that with-
out phonon heat conductivity, the thermoelectric figure ofmerit is proportional to the square of the average of electronicheatE−μover its variance. Based on these observations,Mahan and Sofo [ 4] proposed to achieve high figure of
merit by using materials with narrow-bandwidth conductionband (such as f-electron bands) to reduce the variance of
electronic heat E−μ, while keeping a decent average value.
However, in reality, narrow-band semiconductors have verysmall electrical conductivity, hence the factor /Lambda1is significantly
increased, which instead results in much reduced figure ofmerit [ 7,8]. Meanwhile, the power obtained is also small if the
electrical conductivity is small [ 8–11]. These factors reveal
that the intrinsic entanglement between heat and electricalconductivity, as well as the parasitic phonon heat conduction,impede the improvement of thermoelectric energy efficiency.
One approach to improve thermoelectric performance is
to use inelastic transport processes, where electrical and heatcurrents are carried by different quasiparticles and separatedspatially. By inelastic transport processes we mean that thetransported electrons exchange energy with another degree offreedom (characterized by some collective excitations, e.g.,phonons) during the transport processes. Such effects alsotake place in conventional thermoelectric materials, but playonly a minor role since the elastic transport processes arealways dominant. When, e.g., elastic transport processes aresuppressed by an energy barrier, the inelastic processes becomedominant. A typical setup for inelastic thermoelectricity is athree-terminal device, where the third terminal supplies thecollective excitations (which can often be described by somekind of bosonic quasiparticles) from a thermal reservoir (e.g.,see Fig. 1). It has been shown that the inelastic thermoelectric
figure of merit is given by [ 12–14]
Z
inT=/angbracketleft¯hω/angbracketright2
Var ( ¯hω)+/Lambda1in, (3)
where ¯ hωdenotes the energy of bosons that assist the in-
elastic transport and /Lambda1inmeasures the useless parasitic heat
2469-9950/2018/97(12)/125422(11) 125422-1 ©2018 American Physical SocietyJIAN-HUA JIANG AND YOSEPH IMRY PHYSICAL REVIEW B 97, 125422 (2018)QD
QDSource Drain, ,
,,Thermal bath
ℎ(a)
QDs
sourcedrain
absorber
emitter(b)
FIG. 1. (a) Schematic of near-field three-terminal thermoelectric
heat engine. A hot thermal reservoir of temperature Thinjects heat
flux into the device through near-field heat radiation. The device
is held at a lower temperature Tc. The absorption of the heat
radiation is realized by photon-assisted transitions between the two
continua. As a result, the upper and lower continua have different
chemical potentials μeandμv, respectively. The source and drain
have different electrochemical potentials, denoted as μSandμD,
separately. Extracting the photothermal carriers through the two QD
layers can lead to useful electrical power output. The typical energyof QDs in the left (right) layer is E
/lscript(Er). The energy levels of the left
(right) QD layer match the lower (upper) continuum in the absorption
material to induce resonant tunneling. (b) A possible setup for thethree-terminal near-field heat engine. The emitter is a heat source of
temperature T
hwhich is separated from the device by a vacuum gap
of thickness d. The device is held at a lower temperature Tcwhich
consists of the source, drain, and absorber layers. These three parts
are divided by two layers of quantum-dot arrays.
conduction. The above average is defined as
/angbracketleft¯hω/angbracketright=/integraltext
¯hdωG in(¯hω)¯hω/integraltext
¯hdωG in(¯hω), (4)
where the boson energy-dependent conductance Gin(¯hω)a l -
ready includes electron and boson distribution functions. Toobtain high figure of merit through inelastic thermoelectricity,a narrow bandwidth of the bosons is needed, which, however,does not conflict with a high electrical conductivity. The
above analysis thus opens the possibility of realizing high-performance thermoelectric materials using inelastic thermo-electric transport. In the past years, studies have revealed theadvantages of inelastic thermoelectric materials and devices[15–50].
In this work, we propose a type of near-field inelastic
three-terminal (NFI3T) quantum-dot (QD) heat engine. Thestructure and working principle of the device are schematicallyillustrated in Fig. 1. The near-field enhanced heat flux, in
the form of infrared photons, is injected into the device,absorbed, and converted into useful electrical power. Theunderlying mechanisms are the photocarrier generation in thecentral region and hot-carrier extraction through two QDstunneling layers at the left and right sides. This device takes thefollowing advantages of near-field radiations: First, the near-field radiation can strongly enhance heat transfer across thevacuum gap [ 51] (for recent experiments, see Refs. [ 52–58])
and thus leads to significant heat flux injection. The injectedinfrared photons with energy greater than the energy gap inthe absorption material E
glead to generation of photocarriers
and electrical power output. The larger the injected heat flux is,the larger is the output power. Second, unlike phonon-assistedinterband transitions, photon-assisted interband transition isnot limited by the small phonon frequency and can work forlarger band gaps due to the continuous photon spectrum.
In order to elaborate on the design principles, thermoelectric
transport properties, and merits of the NFI3T heat engine,
this paper is organized as follows: In Sec. II, we introduce
the working principles of the NFI3T heat engine. In Sec. III,
we obtain the transport equations from the linear-responsetheory. In Sec. IV, we study the transport properties and
thermoelectric energy conversion from microscopic theory. InSec. V, we discuss the performance of the NFI3T heat engine.
We conclude and give an outlook in Sec. VI. Our study here is
mainly focused on the linear-response regime where a directcomparison with conventional thermoelectricity is available.
II. WORKING PRINCIPLES OF THE NEAR-FIELD
THREE-TERMINAL HEAT ENGINE
The system we consider is schematically illustrated in
Fig. 1(a). At the center of the engine, there are two con-
tinuous spectra of electrons which can be realized by thevalence and conduction bands in semiconductors, or by theminibands in superlattices, or anything alike. The equilibriumchemical potential is set in-between the two continua. Byabsorbing infrared photons from the thermal bath, electronsin the lower continuum jump into the upper continuum. Theexcess electrons occupying the upper continuum will flowinto the drain terminal, while the photon-generated holesin the lower continuum will be refilled by electrons fromthe source. Since the lower continuum is strongly coupledwith the source, whereas the upper continuum is stronglycoupled with the drain, the hot photocarrier diffusion leads toa directional electrical current. Such selective strong couplingbetween the lower (upper) continuum and the source (drain)can be realized using resonant-tunneling through QDs. This isbecause the lowest-energy level and the level spacing in QDscan be controlled by the material and the quantum confinement
125422-2NEAR-FIELD THREE-TERMINAL THERMOELECTRIC HEAT … PHYSICAL REVIEW B 97, 125422 (2018)
effect (both conduction and valence band energy levels are
usable). To improve the harvesting of heat from the thermalterminal (a reservoir with high temperature T
h), we utilize the
near-field thermal radiation across the vacuum gap betweenthe thermal emitter (i.e., the thermal terminal) and the device.It was known a long time ago [ 51] that when two objects
are spaced closely, the evanescent electromagnetic waves candramatically enhance the heat transfer between them. In recentdecades, near-field heat transfer across nanoscale vacuum gapswas realized and measured in different experimental systems[51–58]. Spectral tailoring of near-field radiation is proposed
by using surface electromagnetic waves [ 59–67], meanwhile
the light-matter interaction can also be strongly enhancedthrough these surface waves [ 59–68]. In the NFI3T heat engine,
the use of near-field heat transfer has several advantages. Oneof them is to enhance the heat injection flux so that the outputpower of the heat engine can be considerably increased. On theother hand, since near-field heat transfer is determined by theoptical properties of the emitters and receivers, the near-fieldheat transfer becomes more controllable than heat diffusion inall-solid devices. For instance, the parasitic heat conductioncan be reduced by manipulating the infrared thermal radiationusing optical materials or metamaterials. The resonant tun-neling between the lower (upper) continuum and the source(drain) can be made efficient, if there are many QDs withenergies matching the lower (upper) continuum (e.g., using QDarrays). We remark that the NFI3T heat engine proposed here
is different from the near-field thermophotovoltaic devices in
several aspects. First, thermophotovoltaic devices work in thetemperature range of 1000 K <T
h<2000 K, while our device
is designed for applications with Th<800 K which fits most
of the industrial and daily-life renewable energy demands forwaste heat harvesting. Besides, the transport mechanism hereis different from the thermophotovoltaic devices. Instead ofthep-njunction-type transport in thermophotovoltaic devices,
here we have the resonant tunneling through the QDs as oneof the main transport mechanisms.
From electronic aspects, the proposed device has the fol-
lowing merit, compared to the inelastic thermoelectric devicesin the existing studies [ 13]. The use of the two continua can
significantly increase the photon absorption rate, comparedto previous devices where direct photon-assisted (or phonon-assisted) hopping between QDs are considered [ 12,19,20,38]
(i.e., without the continua in the center). This improvementcomes mainly from two aspects: First, by introducing the twocontinua in the center, the electronic densities of states areconsiderably increased. Second, we notice that the direct tran-sition matrix element between two QDs assisted by photons issuppressed by an exponential factor ∼exp(−2d/ξ) where dis
the distance between the QDs and ξis the localization length
of the QD wave functions. In contrast, the interband transitionmatrix element between the two continua does not have suchexponential suppression. Thus, with the help of the continua,the photon-assisted charge transfer between the two QD layersis significantly increased.
III. LINEAR-RESPONSE THEORY
In the linear-response regime, we can characterize thermo-
electric transport in the device using a simple current model. Asshown in Refs. [ 52,59], in near-field heat transfer, the tempera-
ture gradients in the receiving material due to inhomogeneousheating is negligible. For simplicity, we assume that electronsin the two continua give energy to phonons and quickly reachthermal equilibrium with the substrate. Hence, the electronictemperatures in the two continua and in the source and thedrain are assumed to be the same, which is denoted as T
c.
However, since the thermal radiation is continuously pumpingelectrons from the valence band to the conduction band,the “pseudoelectrochemical” potentials of these two continuaare different from the equilibrium chemical potential. Wedenote the “pseudoelectrochemical” potential of the valence(conduction) band as μ
v(μe). Consider the three processes
which are described by three electrical currents: the electronflow from the source to the valence band I
S,v; the electron
transition between the valence band and the conduction band,described as a current via I
v,e=e/Gamma1ve, where /Gamma1veis the flux
of photocarriers and e< 0 is the charge of an electron; the
electron flow from the conduction band to the drain Ie,D.I n
linear response, these currents are calculated as
IS,v=G/lscript(μS−μv)/e, (5a)
Iv,e=Gve(μv−μe)/e+Lve(Th−Tc)/T, (5b)
Ie,D=Gr(μe−μD)/e, (5c)
where G/lscriptandGrare the electrical conductances of the left and
right QD layers, respectively. Here, Gveis the conductance
describing the charge transfer between the two continua,whileL
vedescribes the associated Seebeck effect due to heat
absorption. μSandμDare the electrochemical potentials of the
source and the drain, separately. Because the central regionis assumed to have the same temperature as the source andthe drain, the Seebeck effects in ( 5a) and ( 5c) are neglected.
Note that the interband transition current I
veis driven by two
“forces”: the electrochemical potential difference of the twocontinua and the temperature difference between the thermalbath and the device.
In the next section, we will present a microscopic the-
ory which expresses the above transport coefficients usingmicroscopic quantities. For this moment, we focus on thephenomenological transport properties of the NFI3T heatengine. The three currents in Eq. ( 5) must be the same to
conserve the total charge:
I
S,v=Iv,e=Ie,D. (6)
These equations determine both μvandμeonceμS,μd,Tc, and
Thare set. In the following, we denote the equilibrium chemical
potential and temperature as μandT, respectively. Besides,
μS−μ=− (μD−μ)=/Delta1μ/ 2 with /Delta1μ≡μS−μD, while
Th=T+/Delta1T/ 2 and Tc=T−/Delta1T/ 2. The solution of the
above equations yields
μe=(GveG/lscript−GveGr−GlGr)/Delta1μ/ 2+eG/lscriptLve/Delta1T/T
GlGr+GveG/lscript+GveGr,
μv=G/lscript−Gr
G/lscript/Delta1μ
2−Gr
G/lscriptμe. (7)
125422-3JIAN-HUA JIANG AND YOSEPH IMRY PHYSICAL REVIEW B 97, 125422 (2018)
Inserting these solutions, we obtain the following simple
results:/parenleftbigg
Ie
IQ/parenrightbigg
=/parenleftbigg
GeffLeff
LeffKve/parenrightbigg/parenleftbigg
/Delta1μ/e
/Delta1T/T/parenrightbigg
. (8)
Here, Ieis the electrical current between the source and the
drain, while IQis the heat flux transferred from the thermal
terminal to the device. Geffis the total conductance, and Leff
describes the Seebeck effect due to near-field heat transfer.
They are given by
Geff=/parenleftbig
G−1
/lscript+G−1
r+G−1
ve/parenrightbig−1, (9a)
Leff=Geff
GveLve, (9b)
andKveis the heat conductance associated with the heat
transfer between the thermal terminal and the device.
The above tells us that the figure of merit of our inelastic
thermoelectric device is
ZT=L2
eff
GeffKtot−L2
eff, (10)
where we have included the undesirable parasitic heat con-
ductance Kpardue to hot-carrier heating and other dissipation
effects via Ktot=Kve+Kpar. The parasitic heat conduction
will be analyzed in details in Sec. V.
IV . MICROSCOPIC THEORY
Here, we present a microscopic theory for the thermoelec-
tric transport in the NFI3T heat engines. The Hamiltonian ofthe system is
H=H
SD+HQD+HC+Htun+He-ph. (11)
Here, HSD,HQD, andHCare the Hamiltonian of the source
and drain, the QDs, and the two central continua, separately.H
tundescribes tunneling through the QDs, whereas He-ph
describes the optical transition between the two continua. The
Hamiltonian for the source and drain is
HSD=/summationdisplay
/vectorq(ES,/vectorqc†
S,/vectorqcS,/vectorq+ED,/vectorqc†
D,/vectorqcD,/vectorq), (12)
where /vectorqis the wave vector of electrons. The Hamiltonian of
the QDs is
HQD=/summationdisplay
j=/lscript,rEjd†
jdj, (13)
where j=/lscript,rdenotes the left and right dot, respectively. We
first consider the case where only one (two if spin degeneracyis included) level in each QD is relevant for the transport. TheHamiltonian for the two central continua is
H
C=/summationdisplay
/vectorq(Ev,/vectorqc†
v,/vectorqcv,/vectorq+Ee,/vectorqc†
e,/vectorqce,/vectorq). (14)
The tunnel coupling through the QDs is given by
Htun=/summationdisplay
/vectorq(JS,/vectorqc†
S,/vectorqd/lscript+JD,/vectorqc†
D,/vectorqdr
+Jv,/vectorqc†
v,/vectorqd/lscript+Je,/vectorqc†
e,/vectorqdr)+H.c. (15)The coupling coefficients J’s determine the tunnel rates
through the Fermi golden rule:
/Gamma1S,/lscript=2π
¯h/summationdisplay
/vectorq|JS,/vectorq|2δ(E/lscript−ES,/vectorq), (16a)
/Gamma1D,r=2π
¯h/summationdisplay
/vectorq|JD,/vectorq|2δ(Er−ED,/vectorq), (16b)
/Gamma1v,/lscript=2π
¯h/summationdisplay
/vectorq|Jv,/vectorq|2δ(E/lscript−Ev,/vectorq), (16c)
/Gamma1e,r=2π
¯h/summationdisplay
/vectorq|Je,/vectorq|2δ(Er−Ee,/vectorq). (16d)
The electrical currents through the QDs are given by ( e< 0i s
the electron charge)
IS,v=2e
h/integraldisplay
dET/lscript(E)[fS(E)−fv(E)], (17a)
Ie,D=2e
h/integraldisplay
dETr(E)[fe(E)−fD(E)]. (17b)
Here, the factor of 2 comes from the spin degeneracy (we
consider around room temperature or above, where Coulombblockade is negligible). eis the electronic charge. f
S,fv,fe,
andfDare the electron distribution functions in the source,
valence band, conduction band, and drain, separately. Thetransmission functions are given by
T
/lscript(E)=¯h2/Gamma1S,/lscript/Gamma1v,/lscript
(E−E/lscript)2+¯h2
4(/Gamma1S,/lscript+/Gamma1v,/lscript)2, (18a)
Tr(E)=¯h2/Gamma1D,r/Gamma1e,r
(E−Er)2+¯h2
4(/Gamma1D,r+/Gamma1e,r)2. (18b)
The linear-response conductance is given by
Gj=2e2
hkBT/integraldisplay
dETj(E)f(E)[1−f(E)],j=/lscript,r. (19)
We now consider the photon-assisted transitions in the
center. The Hamiltonian governing such transitions is givenby
H
e-ph=/summationdisplay
/vectorq,/vectork,τg/vectork,τ√
Vc†
e,/vectorq+/vectorkcv,/vectorqa/vectork,τ+H.c., (20)
where g/vectorkis the electron-photon interaction strength, the oper-
atora/vectork,τ(τ=s,pdenotes the sandppolarized light) annihi-
lates an infrared photon with polarization τ.Vis the volume of
the photonic system. The electron-photon interaction strengthis determined by the interband dipole matrix elements andthe photon frequency ω
/vectork. For instance, the electron-photon
interaction in a direct-gap semiconductor is
g/vectork,τ=i/radicalBigg
¯hω/vectork
2ε0εrdcv, (21)
where ε0andεrare the vacuum and relative permittivity, re-
spectively, and dcvis the interband dipole matrix element. The
125422-4NEAR-FIELD THREE-TERMINAL THERMOELECTRIC HEAT … PHYSICAL REVIEW B 97, 125422 (2018)
Fermi golden rule determines the electrical current generated
by interband transitions,
Ive=2πe
¯h/summationdisplay
/vectork,/vectorq,τ|g/vectork,τ|2
Vδ(Ee,/vectork+/vectorq−Ev,/vectorq−¯hω/vectork)
×{fv(Ev,/vectorq)[1−fe(Ee,/vectork+/vectorq)]˜N/vectork,τ
−fe(Ee,/vectork+/vectorq)[1−fv(Ev,/vectorq)](˜N/vectork,τ+1)}, (22)
where fvandfeare the electronic distribution functions for the
lower and upper continua, respectively. The nonequilibriumphoton distribution
˜N
/vectork,τ=N0(ω/vectork,Tc)+δN/vectork,τ,
δN/vectork,τ=[N0(ω/vectork,Th)−N0(ω/vectork,Tc)]Tτ(ω/vectork,k/bardbl,d) (23)
consists of the equilibrium photons in the absorption material
N0(ω/vectork,Tc)=1/[exp(¯hω/vectork
kBTc)−1] and the hot photons tunneled
from the thermal emitter. The probability of encountering suchhot photons is determined by the product of their distributionand the tunneling probability. The photon tunneling probabilitybetween the emitter and the absorber across a planar vacuumgap is a function of photon frequency ω
/vectork, the amplitude of the
wave vector parallel to the planar interface k/bardbl=|/vectork/bardbl|, and the
thickness of the gap d[51,62]
Tτ(ω/vectork,k/bardbl,d)=⎧
⎪⎨
⎪⎩(1−|rτ
01|2)(1−|rτ
02|2)
|1−rτ
01rτ
02ei2k0zd|2,ifk/bardbl/lessorequalslantω/c
4Im(rτ
01)Im(rτ
02)e−2β0zd
|1−rτ
01rτ
02e−2β0zd|2,otherwise .(24)
Here, rτ
01(rτ
02) is the Fresnel reflection coefficient for the
interface between the vacuum (denoted as “0”) and the emitter
(absorber) [denoted as “1” (“2”)]. k0
z=√
(ω/c)2−k2
/bardblis the
wave vector perpendicular to the planar interfaces in the vac-uum. For k
/bardbl>ω / c , the perpendicular wave vector in the
vacuum is imaginary iβ0
z=i√
k2
/bardbl−(ω/c)2, where photon
tunneling is dominated by evanescent waves. For isotropicelectromagnetic media, the Fresnel coefficients are given by[62]
r
s
0j=k0
z−kj
z
k0z+kj
z, (25a)
rp
0j=εjk0
z−kj
z
εjk0z+kj
z,j=1,2 (25b)
where kj
z=√
εj(ω/c)2−k2
/bardblandεj(j=0,1,2) are the (com-
plex) wave vector along the zdirection and the relative permit-
tivity in the vacuum, emitter, and the absorber, separately.
From the above equations, we obtain the linear thermoelec-
tric transport coefficients
Gve=e2
kBT/integraldisplay
dω/Gamma1 0(ω), (26a)
Lve=e
kBT/integraldisplay
dω/Gamma1 0(ω)¯hω, (26b)
Kve=1
kBT/integraldisplay
dω/Gamma1 0(ω)¯h2ω2, (26c)where
/Gamma10(ω)=2πνphFnf(ω)/summationdisplay
/vectorq|g(ω)|2δ(Ee,/vectorq−Ev,/vectorq−¯hω)
×f0(Ev,/vectorq,T)[1−f0(Ee,/vectorq,T)]N0(ω,T).(27)
The superscript 0 in the above stands for the equilibrium dis-
tribution, and f0(Ev,/vectorq,T)=1/[exp(Ev,/vectorq−μ
kBT)+1].|g(ω)|2=
¯hωd2
cv
2ε0εr,νph=ω2ε3/2
r
¯hπ2c3is the photon density of states, and the factor
Fnf(ω)=1
νph1
V/summationdisplay
/vectork,τδ(¯hω−¯hω/vectork)Tτ(ω,k/bardbl,d)
=1
4/integraldisplay1
0xkdxk/radicalBig
1−x2
k/summationdisplay
τTτ(ω,xknω/c,d ), (28)
where xk=k/bardbl/(nω/c ). In the above, we have used the fact
that the wave vector of photons is much smaller than thatof electrons, thus /vectorq+/vectork/similarequal/vectorq. In generic situations, the above
transition rate also depends on the position (e.g., if the localdensity of states of photon is not uniform due to, say, surfaceplasmon polaritons), and an integration over the whole centralregion is needed. For simplicity, we consider for now onlypropagating photons in the absorption material. The Seebeckcoefficient of the NFI3T heat engine is
S=L
eff
TG eff=/angbracketleft¯hω/angbracketright
eT, (29)
where the average is defined as
/angbracketleft.../angbracketright=/integraltext
dω/Gamma1 0(ω).../integraltext
dω/Gamma1 0(ω). (30)
We can then express the figure of merit in microscopic
quantities:
ZT=/angbracketleft¯hω/angbracketright2
α/angbracketleft¯h2ω2/angbracketright−/angbracketleft ¯hω/angbracketright2+/Lambda1nf, (31)
where α=Gve
Geff, and /Lambda1nf=e2Kpara/Gvecharacterizes the
parasitic heat conductance Kpara that does not contribute
to thermoelectric energy conversion. Although αis always
greater than unity, in this work we shall focus on the regimewithα/similarequal1, which can be realized by using highly conducting
layers on the left and right sides of the absorption material.Aside from high-density quantum-dot layers, quantum wellsor doped semiconductors can also be used to attain highconductivity between the source (drain) and the absorptionmaterial. In addition, at this stage we shall not quantify theparasitic heat conduction that does not contribute to thermo-electric energy conversion, i.e., we set /Lambda1
nf=0 and hence
the heat current IQcounts only absorbed infrared photons.
We will discuss briefly possible parasitic heat conductionmechanisms in Sec. VC and the effects of parasitic heat
conduction on thermoelectric performance. We remark that bysetting /Lambda1
nf=0 and considering only photons with ¯ hω > E g,
we assume 100% recycling of low-frequency ¯ hω < E gand
unabsorbed photons (e.g., they can be reflected back to theemitter by a “back-side reflector” and be thermalized withother quasiparticles in the emitter again). Therefore, the only
125422-5JIAN-HUA JIANG AND YOSEPH IMRY PHYSICAL REVIEW B 97, 125422 (2018)
way that heat leaves the hot bath is by exciting electrons from
the lower to the upper band via light-matter interaction.
We now estimate the conductances G/lscript,Gr, andGve.T h e
tunneling conductances are estimated as
G/lscript/similarequal2e2
hnQDf0
/lscript/parenleftbig
1−f0
/lscript/parenrightbig/Gamma1/lscript
kBT, (32)
Gr/similarequal2e2
hnQDf0
r/parenleftbig
1−f0
r/parenrightbig/Gamma1/lscript
kBT, (33)
when/Gamma1is smaller than kBT. Here, f0
/lscript=nF(E/lscript,T) andf0
r=
nF(Er,T) where nFis the equilibrium Fermi distribution, and
nQDis the density of QDs which can be as large as 1015m−2
(see, e.g., Ref. [ 69]). These conductances can reach about 100
S/m. If the thickness of the QD layer is 10 nm, the conductance
is about 1010S.
The conductance associated with the interband transition
can be roughly estimated as
Gve∼e2δ/prime
kBTρve|g|2Fnfνph(/angbracketleft¯hω/angbracketright)N0(/angbracketleft¯hω/angbracketright,T). (34)
Here, δ/prime=min(kBT,δve,δph) where δveis the bandwidth of
the continua, δphis the bandwidth of the near-field photons,
ρvcis the joint electronic density of states for the lower
and upper continua, and /angbracketleft¯hω/angbracketrightis the average photon energy
for photon-assisted interband transitions. We emphasize that,in comparison with photon-assisted hopping between QDsconsidered before [ 19,20], here the conductance is much
enhanced because of the following reasons: (1) As statedbefore, the electron-photon interaction is not reduced by the“form factor” of the QDs as [ 31]∼exp(−2d/ξ). (2) The joint
density of states of the two continua can be larger than that ofQD ensembles.
V . THERMOELECTRIC PERFORMANCE
A. General analysis
In a simple picture, each infrared photon (say, with energy
¯hω=Eg) absorbed by the NFI3T device is used to move an
electron from the source to the drain. The energy quantum forthe absorbed photon is ¯ hω/greaterorsimilarE
g, while the energy increment of
the electron is |eV|, if there is a voltage bias V=(μS−μD)/e
(μD>μS) between the source and the drain. We then define
a “microscopic” energy efficiency
ηmic=|eV|
Eg, (35)
which characterizes the ratio of the output energy over the
input energy for each photon absorption processes. At firstsight, it seems that the efficiency can be improved by applyinga larger voltage bias. However, when detailed-balance effectsare taken into account, the efficiency is reduced. First, thebackflow of the electron and the emission of photon byrecombination of carriers are also possible. The net currentfrom source to drain is determined by thermodynamic balanceof the forward (electricity generation) and backward (carrierrecombination and photon generation) processes. Besides, thephoton source is not monochromatic, but follows the Bose-Einstein distribution. Since the increase of photon energyreduces the efficiency, the broad energy distribution of photons 0 0.5 1 1.5
-1.2 -1-0.8 -0.6 -0.4 -0.2 0efficiency
eVT/(EgΔT)(a)η / ηCηmic / ηC
Carnot
0 0.5 1 1.5
-1 -0.8 -0.6 -0.4 -0.2 0efficiency
eVT/(EgΔT)(b)
η / ηCηmic / ηC
Carnot
FIG. 2. Energy efficiency of the NFI3T heat engine vs the voltage
biasVfor (a) α=1,β=0.5, and /angbracketleftxω/angbracketright=1.2, and (b) α=1,β=
0.002, and /angbracketleftxω/angbracketright=1. The calculation of the efficiencies are according
to Eqs. ( 37)a n d( 38). The energy efficiencies are scaled by the Carnot
efficiency.
also reduces the efficiency. Also, there are photons that are
not absorbed (reflected or transmitted). The first and the thirdmechanisms are the main factors that reduce the efficiency.
To give a simple quantitative picture, we show how the effi-
ciency varies with the voltage bias. To characterize the thermo-electric transport coefficients, we introduce the dimensionlessparameters u≡eVT/E
g/Delta1T,/angbracketleft¯h2ω2/angbracketright≡(1+β)/angbracketleft¯hω/angbracketright2,xω≡
¯hω/E g.U s i n gE q .( 26), the thermoelectric transport can be
written as
/parenleftbigg
Ie
IQ/parenrightbigg
=Geff/parenleftbigg1 /angbracketleft¯hω/angbracketright/e
/angbracketleft¯hω/angbracketright/e α /angbracketleft¯h2ω2/angbracketright/e2/parenrightbigg/parenleftbigg
V
/Delta1T/T/parenrightbigg
,(36)
where Geffis the total conductance of the device. The energy
efficiency can then be written as
η
ηC=−u(u+/angbracketleftxω/angbracketright)
u/angbracketleftxω/angbracketright+α(1+β)/angbracketleftxω/angbracketright2. (37)
The energy efficiency of the heat engine is always smaller than
the Carnot efficiency ηC≡/Delta1T/T sinceα> 1 andβ> 0. The
working region of the heat engine is −/angbracketleftxω/angbracketright<u< 0. On the
other hand, we have
ηmic
ηC=−u, (38a)
η
ηmic=u+/angbracketleftxω/angbracketright
u/angbracketleftxω/angbracketright+α(1+β)/angbracketleftxω/angbracketright2. (38b)
In Fig. 2, we study two examples: (i) α=1,β=0.5,/angbracketleftxω/angbracketright=
1.2 [Fig. 2(a)], and (ii) α=1,β=0.002,/angbracketleftxω/angbracketright=1 [Fig. 2(b)].
The first example is closer to realistic devices where theoptimal efficiency is considerably smaller than the Carnotefficiency, due to dissipations. From Fig. 2(a) one can see that
the efficiency initially increases with the voltage. However, thebackflow of electrons becomes more and more important as thevoltage increases. After an optimal value, the efficiency thendecreases with the voltage. Consistent with the second law ofthermodynamics [ 34], the optimal efficiency is always smaller
than the Carnot efficiency. The second example has a muchsmaller dissipation (in fact, it is very close to the reversiblelimit). In a quite large range, the efficiency indeed increaseswith the voltage and is almost close to the microscopicefficiency η
mic[see Fig. 2(b)]. The ratioη
ηmicis always a
125422-6NEAR-FIELD THREE-TERMINAL THERMOELECTRIC HEAT … PHYSICAL REVIEW B 97, 125422 (2018)
decreasing function of |u|=|eVT/E g/Delta1T|. However, as u
reaches close to 1, the energy efficiency rapidly reduces fromthe microscopic efficiency η
mic, staying smaller than the Carnot
efficiency. We remark that with perfect energy filters, a heatengine that reaches the Carnot efficiency can be obtained[8,9,12,19,26,38].
B. Calculation of thermoelectric transport and performance
using InSb as absorption material
We now present a more concrete calculation of the electrical
conductivity, thermopower, the figure of merit, and the powerfactor for a particular NFI3T heat engine where the twocontinua in the center are the valence and conduction bands of anarrow-band-gap semiconductor, InSb. The band gap of InSb is0.17 eV . To avoid further complication, we do not consider thesurface electromagnetic waves and temperature dependence ofthe band gap. In this situation, the electron-photon interactioncoefficient is given by
|g(ω)|
2=¯hωd2
cv
2ε0εr, (39)
where dcv=8.8×10−28C m is the interband dipole matrix
element in InSb [ 70],ε0is the vacuum permittivity, εr=15.7
is the (high-frequency) dielectric constant, and cis the speed of
light in vacuum. The dispersions of the conduction and valencebands are given by
E
v,/vectork=−¯h2k2
2mv,Ee,/vectork=Eg+¯h2k2
2me, (40)
where the effective masses are me=0.0135m0andmv=
0.43m0withm0being the mass of free electron. We study
at h i nfi l mo f lab=10−6m thickness sandwiched by the
front (source) and back (drain) gates. The thermoelectricefficiency and power are calculated using the above equations(for details, see Appendix A). The thermoelectric transport
coefficients are calculated using Eqs. ( 9) and ( 26). We also take
into account the conductance of the conduction and valenceelectrons using the formulas for conductivity σ
e=n|e|μe
andσv=p|e|μp. Here, nandpare the electron and hole
densities, respectively, while μeandμpare the electron and
hole mobilities, respectively. The electron and hole densitiesare calculated using the Fermi distribution of electrons once thesteady-state chemical potentials and temperatures are given.We use the mobilities of electrons and holes from empiricaltemperature dependencies obtained from experimental dataon intrinsic InSb [ 71,72]:μ
p=5.4×102/T1.45m2/Vs and
μe=3.5×104/T1.5m2/Vs where the temperature Tis in
units of Kelvin. The total conductance is given by G−1
eff=
G−1
/lscript+G−1
v+G−1
ve+G−1
e+G−1
r, where Ge=σe/laband
Gv=σv/labare the conductances for the upper and lower
continua, respectively, and the conductances of the left andright QD layers are taken as G
/lscript=Gr=1010S for 10-nm
layers [as from the estimation after Eq. ( 33)], respectively. We
find that the total conductance is in fact mainly limited by theoptical absorption G
ve. Thus, the conductances of the other
parts are not important and αis close to unity. For simplicity, we
assume that the emitter has the same permittivity of εr=15.7
as the absorber. The temperature range considered here isbetween 100 to 600 K, while the melting temperature of InSb-0.5 0 0.5 1 1.5
-12 -8 -4 0Heat, charge flux
Voltage (mV)(a)
IQ (104 W/m2)
Ie (104 A/m2)
0 0.2 0.4 0.6 0.8 1
-8 -4 0power, efficiency
Voltage (mV)(b)
P/Pmaxη/ηC
FIG. 3. (a) Heat and charge fluxes in the NFI3T heat engine as
functions of the voltage ( μS−μD)/eforTc=500 K, μ=0.1e V ,
and/Delta1T=20 K.IQis the absorbed heat flux, while Ieis the induced
charge current. (b) The output power P=−IeVand the energy
efficiency ηof the heat engine for the same conditions. Here, the
maximum output power for this condition is Pmax=33.7W/m2.T h e
two electronic continua in the center are the valence and conductionbands of InSb, which has a direct band gap of E
g=0.17 eV . The
thickness of the vacuum gap is d=100 nm.
is about 800 K. We focus on small voltage and temperature
biases, while linear-response theory is applicable.
In Fig. 3we plot the heat and charge fluxes, the output
electrical power, and the energy efficiency as functions of theapplied voltage for a NFI3T heat engine with equilibriumchemical potential μ=0.1 eV . The temperatures are T
h=
520 K and Tc=500 K. Using InSb as the light absorption
material, we find that the optimal efficiency can reach to 60% ofthe Carnot efficiency. The maximum output power is as large as34 W/m
2. The absorbed heat flux reaches 0 .38×104W/m2.
The Seebeck coefficient S=/angbracketleft¯hω/angbracketright/eT as a function of the
chemical potential and temperature is plotted in Fig. 4(a).I ti s
seen that the Seebeck coefficient does not vary considerablywith the chemical potential, which is a characteristic of theinelastic thermoelectric effect, since the average energy /angbracketleft¯hω/angbracketright
is mainly limited by the band gap E
gand the temperature. The
variance of the energy Var(¯ hω)≡/angbracketleft¯h2ω2/angbracketright−/angbracketleft ¯hω/angbracketright2is plotted
in Fig. 4(b). From the figure one can see that the variance
of the energy is close to ( kBT)2, which is consistent with
the physical intuition. We remark that for the whole temper-ature and chemical potential ranges, the variance Var(¯ hω)/lessorsimilar
4(k
BT)2is much smaller than the square of the average energy
/angbracketleft¯hω/angbracketright2∼100(kBT)2. Particularly for low temperatures, the
ratio/angbracketleft¯hω/angbracketright2/Var ( ¯hω) can be as large as ∼200. From this
observation, we predict that the figure of merit of the NFI3Theat engine can be very large.
Figure 4(c) shows that for various temperatures, the average
excess kinetic energy of photocarriers is around 2 k
BT.T h ea v -
erage of energy of the absorbed photons is mainly determinedby the band gap E
g. Since Egis much larger than the thermal
energy kBT[also shown in Fig. 4(c)], the Seebeck coefficient
is very large. To give a straightforward understanding of theenergy dependence of the photon absorption rate, we plot/Gamma1
0(¯hω) for two temperatures T=300 and 600 K, respectively,
withμ=0.1 eV , in Fig. 4(d). The sharp peaks in the figure
manifest efficient energy filtering in the inelastic transportprocesses.
125422-7JIAN-HUA JIANG AND YOSEPH IMRY PHYSICAL REVIEW B 97, 125422 (2018)
(a) S (mV/K)
1 2 3 4 5 6
T (100K) 0 0.1 0.2μ (eV)
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2(b) Var(−hω)/(kBT)2
1 2 3 4 5 6
T (100K) 0 0.1 0.2μ (eV)
1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4
0 0.5 1 1.5 2 2.5
1 2 3 4 5 6Energy
T (100K)(c)
(<−hω>-Eg)/kBT
Eg / 10 kBT
0 0.2 0.4 0.6 0.8 1
0 2 4 6 8 10Γ0 (normalized)
(−hω-Eg)/kBT(d)
T=300K
600K
FIG. 4. (a) Seebeck coefficient S(in units of mV/K) and (b)
the variance of the photon energy Var(¯ hω)/k2
BT2for the inelastic
thermoelectric transport as functions of the chemical potential μand
the temperature T. The average is weighted by the conductance for
each channel as in Eq. ( 26). The interband transition rate is calculated
using Eqs. ( 27), (39), and ( 40). The energy zero is set to be the band
edge of the valence band of InSb. (c) The typical energy scales in
the inelastic thermoelectric transport versus the temperature T:t h e
average excess kinetic energy /angbracketleft¯hω/angbracketright−Egmeasured in kBTand the
ratioEg/kBT. Note that to scale the two together, the latter is divided
by a factor of 10 (i.e., the band gap is much larger than kBT). (d)
The normalized function /Gamma10(ω) as a function of the excess kinetic
energy ¯ hω−Eg(“normalized” in the sense that the maximal values
in the figure is 1) for two different temperatures T=300 and 600 K,
respectively. The equilibrium chemical potential for (c) and (d) isμ=0.1 eV . The thickness of the vacuum gap is d=100 nm.
To demonstrate the near-field effect, we calculate the heat
flux, the power factor, Seebeck coefficient, and thermoelectricfigure of merit as functions of the vacuum gap d. The results
are presented in Fig. 5where the temperature and chemical
10-310-210-1100
101102103104Heat flux, Power-factor
Vacuum gap d (nm)T = 500 K(a)
IQ (105 W/m2)
Pnf (μW/K2m)
4 6 8 10 12 14 16 18
101102103104S, ZT
Vacuum gap d (nm)T = 500 K(b)
S (100 μV/K)
ZT
FIG. 5. (a) Absorbed heat flux IQand power factor Pnf=σeffS2
as functions of the vacuum gap d. (b) Seebeck coefficient Sand
thermoelectric figure of merit ZTas functions of the vacuum gap d.
Equilibrium temperature T=500 K and chemical potential μ=0.1
eV . The heat flux is calculated at zero voltage bias and /Delta1T=20 K.potential are set as T=500 K and μ=0.1 eV , respectively.
We find that the heat flux and the power factor are significantlyenhanced by the near-field effect when the vacuum gap dis
decreased from 10 μm to 10 nm. This result also indicates that
the heat and charge conductivity are significantly improved bythe near-field effect. The Seebeck coefficient and the figureof merit, however, exhibit oscillatory dependencies on thevacuum gap d. The oscillation of thermoelectric figure of
merit originates from the oscillation of the Seebeck coefficient/angbracketleft¯hω/angbracketright/eT and the variance of photon energy Var(¯ hω). These
oscillations are commonly found in near-field photon transmis-sion since the interference effects that strongly modulate thephoton transmission have periodic dependence on the vacuumgapd,a ss h o w ni nE q .( 24).
C. Parasitic heat conduction and optimization
of thermoelectric performance
Here, parasitic heat conduction represents the part of heat
radiation absorbed by the device but not contributed to thermo-electric energy conversion. Aside from the unabsorbed photonsfrom the thermal terminal (which can be reflected back byplacing a mirror on the back side of the device), there areother mechanisms for parasitic heat conduction. First, viaphoton–optical-phonon interactions, photons can directly giveenergy to phonons. Such photon-phonon conversion leads tounwanted heating of the device. Second, photocarriers can giveenergy to phonons by nonradiative recombinations (e.g., viaShockley-Hall-Read mechanism). Third, hot carriers can emitphonons and give energy to the lattice. We suspect that thefirst mechanism is rather weak, while the second mechanismdepends on the density of impurities and disorder in the absorp-tion material. The third mechanism is usually considered asone of the main mechanisms for parasitic heat conduction andreduction of efficiency. We remark that the third mechanism hasalready been taken into account in our calculation. Accordingto Fig. 4(b), such parasitic heat conduction leads only to small
parasitic heat /lessorsimilar2k
BTper photocarrier, which is the main
reason for the high figure of merit of the NFI3T heat engine.
While quantifying the parasitic heat conduction needs
material and device details, which is rather hard to achievein theory, we study the effects of parasitic heat conductionthrough the parametrization K
para=Gve/Lambda1nf/e2, where Kpara
is the heat conductance and /Lambda1nfis a parameter of the dimension
energy square. We calculate the figure of merit for two caseswith parasitic heat conduction parameters /Lambda1
nf=0.2E2
gand
/Lambda1nf=1.2E2
g. Since Egis much larger than the thermal energy
kBT, the latter case corresponds to very strong parasitic heat
conduction.
We find that for the case with smaller parasitic heat
conduction, the optimal figure of merit is reached around350 K for electron-doped InSb [see Fig. 6(a)]. The optimal
figure of merit can be as large as ZT > 7. For the case with
/Lambda1
nf=1.2E2
gthe optimal figure of merit is reached around
600 K for electron-doped InSb with a large value of ZT > 2
within the range of calculation [see Fig. 6(b)]. From these
results, one can see that the optimization of thermoelectricperformance strongly depends on the parasitic heat conductionwhich needs to be examined in the future. Figures 6(c) and
6(d) give the dependencies of the power factor P
nf=σeffS2
125422-8NEAR-FIELD THREE-TERMINAL THERMOELECTRIC HEAT … PHYSICAL REVIEW B 97, 125422 (2018)
(a) ZT
1 2 3 4 5 6
T (100K) 0 0.1 0.2μ (eV)
5.6 5.8 6 6.2 6.4 6.6 6.8 7 7.2 7.4(b) ZT
1 2 3 4 5 6
T (100K) 0 0.1 0.2μ (eV)
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
(c) Pnf (mW/K2m)
1 2 3 4 5 6
T (100K) 0 0.1 0.2μ (eV)
10-1110-1010-910-810-710-610-510-410-3
10-910-810-710-610-510-410-310-210-1100101
1 2 3 4 5 6σeff (S/m)
T (100K)(d)
μ = 0
0.2 eV
FIG. 6. Thermoelectric figure of merit, power factor, and electri-
cal conductivity of the NFI3T heat engine using InSb as the absorptionmaterial. (a), (b) Thermoelectric figure of merit for the parameters
/Lambda1
nf/E2
g=0.2 and 1.2, respectively. (c) The power factor Pnfand (d)
the electrical conductivity σeffof the NFI3T heat engine. The vacuum
gap is d=100 nm.
and the charge conductivity σeff(determined from the total
conductance Geffand the cross section Aand the thickness
lof the device, via σeff=lGeff/A) on the temperature and
chemical potential, respectively. Both of them increase rapidlywith the temperature because the thermal distributions of theinfrared photons with ¯ hω > E
gare significantly increased at
elevated temperatures. At high temperatures, the power factor
and the charge conductivity can be considerable. Roughly, a
good balance of the figure of merit and the power factor can befound in the region of E
g/7kB<T <E g/2kB. In Appendix B,
we include the temperature dependence of the band gap ofInSb and study the temperature and doping dependencies ofthermoelectric figure of merit.
VI. CONCLUSION AND OUTLOOK
We propose a powerful and efficient inelastic thermoelec-
tric QD heat engine based on near-field enhanced radiativeheat transfer. By introducing two continua (separated by aspectral gap E
g) which are connected to the left and right
QD layers, respectively, we introduce a ratchet mechanismthrough photocarrier generation and conduction in the infraredregime. The only way that heat leaves the hot bath is by excitingelectrons from the lower to the upper continuum. The injectedthermal radiation then induces a directional charge flow andgenerates electrical power, manifesting as a three-terminalthermoelectric effect. Introducing the two continua and thenear-field radiative heat transfer substantially increases theinelastic transition rates and hence the output power. Com-pared with previous designs of direct photon-assisted hoppingbetween QDs, the thermoelectric heat engine proposed here ismuch more powerful. Using a narrow-band-gap semiconductorInSb as the absorption material, we show that the photocurrentand the output power can be considerably large. The Seebeck
coefficient and figure of merit for the proposed heat engineare significantly large, even when parasitic heat conduction(i.e., conduction of heat through unused photons) is takeninto account. Near-field effects can improve the absorbed heatflux by nearly two orders of magnitude, leading to strongenhancement of output power.
Roughly, around the regime with E
g/6kB<T <E g/2kB
gives a good balance of thermoelectric figure of merit and
power factor. Specific optimization depends rather on thetransport details and material properties of the heat engine.However, there are several guiding principles in the searchof promising materials for near-field inelastic thermoelectricheat engine. Aside from a suitable band gap, the interbandtransitions must be efficient which can be realized by a largedipole matrix element or a large joint density of states for thelower and upper bands. The near-field heat transfer must beoptimized by utilizing advanced optical structures and mate-rials, such as hyperbolic metamaterials. Using optical means,near-field heat radiation can be controlled more effectively,compared to heat diffusion, which provides opportunities forhigh-performance thermoelectric devices. Although the needfor a submicron vacuum gap is technologically challenging,recent studies have shown that near-field radiative heat transfercan also be realized without a vacuum gap [ 74,75], opening
opportunities for future thermoelectric technologies based oninelastic transport mechanisms. Our study presents analog
and comparison between photon-induced inelastic thermoelec-
tricity to conventional thermoelectricity, which already givespromising results in the linear-response regime and will serveas the foundations for future studies.
ACKNOWLEDGMENTS
J.H.J. acknowledges financial support from the National
Natural Science Foundation of China (Grant No. 11675116)and the Soochow University. He also thanks Weizmann In-stitute of Science for hospitality and M.-H. Lu for helpfuldiscussions. Y .I. acknowledges financial support from the US-Israel Binational Science Foundation (BSF) and the WeizmannInstitute of Science.
APPENDIX A: COMPUTING INTERBAND
TRANSITION RATES
The transition rate in Eq. ( 27) is the central quantity of
interest. We show below how it can be calculated numerically.Consider a thin film of the absorbing material with area Aand
thickness l
ab, then
/Gamma10(ω)=2πνph(ω)Fnf(ω)/summationdisplay
/vectorq|g(ω)|2δ(Ee,/vectorq−Ev,/vectorq−¯hω)
×f0(Ev,/vectorq,T)[1−f0(Ee,/vectorq,T)]N0(ω,T). (A1)
For parabolic bands with dispersions given in Eq. ( 40), energy
conservation gives
Ee,q=Eg+mev(¯hω−Eg)
me,E v,q=−mev(¯hω−Eg)
mv,
125422-9JIAN-HUA JIANG AND YOSEPH IMRY PHYSICAL REVIEW B 97, 125422 (2018)
where mev=(m−1
e+m−1
v)−1. The integral over qcan be
carried out analytically:
/Gamma10(ω)=mev
π¯h3νph(ω)Fnf(ω)|g(ω)|2Alab/radicalbig
2mev(¯hω−Eg)
×f0(Ev,q)[1−f0(Ee,q)]N0(ω,T). (A2)
The integral over ωcan be carried out numerically from Eg/¯h
to a sufficiently large energy cutoff.
APPENDIX B: EFFECTS OF TEMPERATURE
DEPENDENCE OF THE INSB BAND GAP
Here, we study the effects of temperature-dependent band
gap of the absorption material, InSb, on thermoelectric figureof merit. The temperature dependence of the band gap is givenby the empirical law of E
g=0.24−6×10−4T2/(T+500)
as found in Ref. [ 73] for the temperature range 0 <T < 300 K.
Here, Egis in units of eV , while the temperature Tis in
units of Kelvin. We assume this temperature dependenceis approximately applicable for the range of interest 100 <
T< 500 K. With such temperature dependence taken into
account, we recalculate the temperature and chemical-potentialdependencies of thermoelectric figure of merit. The results for/Lambda1
nf=0.2E2
gand 1.2E2
gare presented in Figs. 7(a) and7(b),
respectively. We find that the optimal conditions for the figureof merit are modified for both cases. Besides, the optimalfigure of merit for /Lambda1
nf=0.2E2
gis increased to 11, whereas
the optimal figure of merit for /Lambda1nf=1.2E2
gis reduced to 1.9.
The power factor and the charge conductivity are optimal in thehigh-temperature regime, whereas the figure of merit is optimalin the low-temperature regime. A balanced optimization ofboth the figure of merit and the power factor can be found
(a) ZT
1 2 3 4 5
T (100K) 0 0.1 0.2μ (eV)
3 4 5 6 7 8 9 10 11(b) ZT
1 2 3 4 5
T (100K) 0 0.1 0.2μ (eV)
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
(c) Pnf (mW/K2m)
1 2 3 4 5
T (100K) 0 0.1 0.2μ (eV)
10-1410-1310-1210-1110-1010-910-810-710-610-510-410-3(d)σeff (S/m)
1 2 3 4 5
T (100K) 0 0.1 0.2μ (eV)
10-1210-1010-810-610-410-2100102
FIG. 7. (a), (b) Thermoelectric figure of merit as functions of the
chemical potential and temperature, for the parameters /Lambda1nf/E2
g=
0.2 and 1.2, respectively. (c), (d) Thermoelectric power factor Pnf
and charge conductivity σeffas functions of chemical potential and
temperature, respectively. The vacuum gap is d=100 nm. The
temperature dependence of the band gap Egof InSb is taken into
account.
in the region of Eg/6kB<T <E g/2kB. Our calculation here
demonstrates again that optimization of thermoelectric perfor-mance depends on the transport details of the heat engine.
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125422-11 |
PhysRevB.88.085430.pdf | PHYSICAL REVIEW B 88, 085430 (2013)
Understanding electron behavior in strained graphene as a reciprocal space distortion
M. Oliva-Leyva*and Gerardo G. Naumis†
Departamento de F ´ısica-Qu ´ımica, Instituto de F ´ısica, Universidad Nacional Aut ´onoma de M ´exico (UNAM), Apartado Postal 20-364, 01000
M´exico, Distrito Federal, M ´exico
(Received 24 April 2013; revised manuscript received 10 June 2013; published 26 August 2013)
The behavior of electrons in strained graphene is usually described using effective pseudomagnetic fields in a
Dirac equation. Here we consider the particular case of a spatially constant strain. Our results indicate that latticecorrections are easily understood using a strained reciprocal space, in which the whole energy dispersion is simplyshifted and deformed. This leads to a directional-dependent Fermi velocity without producing pseudomagneticfields. The corrections due to atomic wave function overlap changes tend to compensate such effects. Also, theanalytical expressions for the shift of the Dirac points, which do not coincide with the Kpoints of the renormalized
reciprocal lattice, as well as the corresponding Dirac equation are found. In view of the former results, we discussthe range of applicability of the usual approach of considering pseudomagnetic fields in a Dirac equation derivedfrom the old Dirac points of the unstrained lattice or around the Kpoints of the renormalized reciprocal lattice.
Such considerations are important if a comparison is desired with experiments or numerical simulations.
DOI: 10.1103/PhysRevB.88.085430 PACS number(s): 73 .22.Pr, 81.05.ue, 72.80.Vp
I. INTRODUCTION
Since the experimental observation of graphene,1at w o -
dimensional form of carbon, there have been many theoreticaland experimental studies to understand and take advantage ofits surprising properties.
2–5Among its most interesting fea-
tures, one can cite the peculiar interplay between its electronicand its mechanical properties. Graphene can withstand elasticdeformations up to 20%, much more than in any other crystal.
6
Needless to say, this long interval of elastic response results instrong changes in the electronic structure, which offers a newdirection of exploration in electronics: strain engineering.
7–10
The prospect is to explore mechanical deformations as a
tool for controlling electrical transport in graphene devices: atechnological challenge owing to the counterintuitive behaviorof electrons as massless Dirac fermions.
11
The most popular model proposed in the literature for
studying the concept of strain engineering is based ona combination of a tight-binding (TB) description of theelectrons and linear elasticity theory.
12–15In this approach,
where the absence of electron-electron interactions is assumed,the electronic implications of lattice deformations are capturedby means of a pseudovector potential A, which is related to
the strain tensor /epsilon1by
15
Ax=β
2a(/epsilon1xx−/epsilon1yy),A y=−β
2a(2/epsilon1xy), (1)
where a≈1.42˚A is the unstrained carbon-carbon distance
[see Fig. 1(a)] and β≈3 modulates the variation of the
hopping energy tof the TB model with the changes in
the intercarbon distance due to lattice deformations.4,7Note
that the xaxis is selected parallel to the zigzag direction.
Thisβ-dependent pseudovector potential gives a coupling of
the pseudomagnetic field ( B=∇×A) with the electronic
density. The idea of pseudomagnetic fields has been key inthe understanding of the pseudo-Landau-level experimentalobservations made in strained graphene, which had beentheoretically predicted earlier.
16,17
In recent work, the standard description of the strain-
induced vector field has been supplemented with the explicitinclusion of the local deformation of the lattice vectors.18–22
After accounting for the actual atomic positions to the TB
Hamiltonian, Kitt et al. proposed an extra pseudovector
potential which is βindependent and different at each of the
strained Dirac points.18The possible physical relevance of the
extraβ-independent term, predicted in the work of Kitt et al. ,
was discussed by de Juan et al. within the TB approach.19They
also obtained an extra β-independent pseudovector potential
but with zero curl. Therefore, they concluded that in strained
graphene no β-independent pseudomagnetic field exists.19,20
The controversy created by Kitt et al. has also been solved
in Refs. 21and22, where the concept of renormalization of
the reciprocal space was a core and novel idea developedto meet that end. However, as has been documented inRef. 7, the positions of the energy minima and maxima (Dirac
points) do not coincide with the high-symmetry points atthe corners of the renormalized Brillouin zone [e.g., the K
point in Fig. 1(c)]. This last statement motivates us to seek
the effective Hamiltonian around the Dirac points using suchrenormalization, since, as shown here, this is essential tounderstand the experimental data.
In this paper we analyze the most simple case, a spatially
uniform strain. The reason is that such a case must be containedas a limiting case in any of the general theories, and at thesame time, as shown here, it can be solved exactly. Thus, itis an important benchmark tool to compare and discriminatethe goodness of previous approaches. For example, this leadsto a simple explanation for the lattice correction terms anddirection-dependent Fermi velocity, since both are due to theeffects of strain in reciprocal space. Hopefully, this will help toderive the consequences of lattice corrections of flexural modesor curved graphene. The layout of this work is the following.In Sec. IIwe present the model and find the corresponding
energy dispersion surface. In Sec. III, we discuss the properties
of the energy dispersion and find the analytical expressionsfor the shift of the Dirac points and the strained Dirac
Hamiltonian. Section IVdeals with the problem of how
the usual pseudomagnetic-field approach needs to be addedwith some requirements in order to compare experiments andsimulations. Finally, in Sec. V, our conclusions are given.
085430-1 1098-0121/2013/88(8)/085430(7) ©2013 American Physical SocietyM. OLIV A-LEYV A AND GERARDO G. NAUMIS PHYSICAL REVIEW B 88, 085430 (2013)
II. MODEL: ELASTICITY AND TIGHT BINDING
We are interested in uniform planar strain situations; i.e., the
components of two-dimensional strain tensor /epsilon1a r ea s s u m e dt o
be position independent. In this case, the displacement vector
u(x)i sg i v e nb y u(x)=/epsilon1·x, and therefore, the actual position
of an atom x/prime=x+u(x) can be written as x/prime=(I+/epsilon1)·x,
Ibeing the 2 ×2 identity matrix. In general, if rrepresents a
general vector in the unstrained graphene lattice, its strainedcounterpart is given by the relationship r
/prime=(I+/epsilon1)·r.
We investigate the electronic implications of strain by
means of the nearest-neighbor TB Hamiltonian,
H=−/summationdisplay
x/prime,ntx/prime,na†
x/primebx/prime+δ/primen+H.c., (2)
where x/primeruns over all sites of the deformed Asublattice and
δ/prime
nare the three nearest-neighbor vectors. The operators a†
x/prime
andbx/prime+δ/primencorrespond to creating and annihilating electrons
on sublattices AandB, at sites x/primeandx/prime+δ/prime
n, respectively.
The dispersion relation arises upon writing Eq. (2)in the
momentum space. For this purpose, we replace the cre-ation/annihilation operators with their Fourier expansions,
23
a†
x/prime=1√
N/summationdisplay
k1eik1·(x+u(x))a†
k1, (3a)
bx/prime+δ/primen=1√
N/summationdisplay
k2e−ik2·(x+δn+u(x+δn))bk2, (3b)
where Nis the number of elementary cells. In Eq. (2)we have
written the hopping integral tx/prime,nas position dependent, but in
the considered case (uniform strain), it does not depend on theposition, only on the direction: t
x/prime,n=tn.
Under these considerations, calculation of the Hamiltonian
inkspace is fairly straightforward; Hbecomes
H=−/summationdisplay
k,ntne−ik·(I+/epsilon1)·δna†
kbk+H.c. (4)
From this equation, it follows that the dispersion relation
of graphene under spatially uniform strain is
E(k)=±/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
ntne−ik·(I+/epsilon1)·δn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (5)
which is a closed expression for the energy. This equation
provides a benchmark tool case for testing any Hamiltonianconcerning strain in graphene and suggests the procedure thatis developed in the following section. If in Eq. (5), we define
an auxiliary reciprocal vector k
∗=(I+/epsilon1)·k, the dispersion
relationship is almost equal to the case in unstrained graphene,except for the different values of t
nas a function of n. When
such hopping changes are not considered, as explained in thefollowing section, one gets that
E(k)=±/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay
nt0e−ik∗·δn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (6)
which is exactly the same Hamiltonian as for unstrained
graphene, but now with kreplaced with k∗. Here, no
approximations are used and the spectrum can be obtainedfor all values of k∗. If this Hamiltonian develops around the
corresponding Dirac point, it is obvious that the same DiracHamiltonian observed in unstrained graphene will appear(see below), with kreplaced with k
∗. This suggests doing
a renormalization of the reciprocal space as performed inthe next section, a result that was also found in Refs. 21
and22. Furthermore, Eq. (5)can be numerically evaluated to
test any effective Hamiltonian obtained by developing aroundparticular points in kspace.
It is important to remark that in the general case of a nonuni-
form strain, δ
/prime
nare not given by δ/prime
n=(I+/epsilon1)·δn. In this case,
/epsilon1(x) needs to be replaced by the displacement gradient tensor
∇u.20,24See in Ref. 20how the use of δ/prime
n=(I+∇u)·δn
allowed Kitt et al. to solve the controversy concerning whether
or not lattice corrections produce pseudovector potentials.
III. ENERGY SPECTRUM OF STRAINED GRAPHENE
The variation of the hopping energy tnwith the
changes in the intercarbon distance fulfills a physicallyaccurate exponential decay t
n=t0exp[−β(|δ/prime
n|/a−1)],
witht0/similarequal2.7 eV being the equilibrium hopping energy.4,25
Nevertheless, for the sake of comparison with previous work,
we consider first order in strain,
tn/similarequalt0/parenleftbigg
1−β
a2δn·/epsilon1·δn/parenrightbigg
. (7)
Defining the three nearest-neighbor vectors as depicted in
Fig. 1,
δ1=a
2(√
3,1),δ2=a
2(√
3,1),δ3=a(0,−1),(8)
FIG. 1. (Color online) (a) Unstrained graphene lattice showing
the vectors δithat point to the neighbors of type Asites, (b) the
same lattice under a uniform stress, and (c) the first Brillouin zoneof the reciprocal lattice for unstrained (dashed lines) and strained
(solid lines) graphene. Note how the reciprocal lattice is contracted
in the direction where the lattice is stretched, and the change of the
K
0symmetry point into K. (d) How the distortion of the reciprocal
lattice transforms the original Dirac cone (left) into a distorted one
(right) with a directional-dependent Fermi velocity.
085430-2UNDERSTANDING ELECTRON BEHA VIOR IN STRAINED ... PHYSICAL REVIEW B 88, 085430 (2013)
and plugging Eq. (7)into Eq. (5), one gets the following
expression for the dispersion relation,
E(k)=±t0/radicalbig
3+f(k∗)−β(3Tr(/epsilon1)+f/epsilon1(k∗))+β2f/epsilon12(k∗),
(9)
where f(k∗) has exactly the same functional form as its
unstrained graphene counterpart,4
f(k∗)=2 cos(√
3k∗
xa)+4 cos/parenleftbigg√
3k∗
xa
2/parenrightbigg
cos/parenleftbigg3k∗
ya
2/parenrightbigg
,
(10)
but now evaluated at different points of reciprocal space, since
herek∗=(k∗
x,k∗
y) is given by the transformation,
k∗=(I+/epsilon1)·k. (11)
This last equation is very important. It provides a mapping
of the original reciprocal space into a new distorted one. Aswe will see, this mapping and the fact that f(k
∗) is equal to its
undistorted counterpart lead to pure geometrical effects thatonly very recently have been identified.
18–22The other terms
depend on the same distortion but contain hopping corrections.These terms are explicitly detailed in Appendix A.f
/epsilon1(k∗)
contains the modification of the spectrum due to first order inβ, while f
/epsilon12(k∗) is the second-order correction in β.
A. Hypothetical case: β=0
Several important consequences follow from these equa-
tions. First, one can observe that in the case of deforming thelattice without changing the hopping parameters, i.e., if onedeforms the lattice keeping β=0,E(k) is simplified to
E(k)=±t
0/radicalbig
3+f(k∗). (12)
This corresponds to the same dispersion relationship ob-
served in graphene, but now with different reciprocal vectors,which are obtained by applying strain to the original reciprocalvectors. In other words, the space is strained, while theeigenvalues remain the same. As a consequence, the Dirac conechanges its shape due to this lattice deformation, as illustratedin Fig. 1. This is exactly the result that we would obtain if a
diagonalization of the TB Hamiltonian were performed usinga computer. Since β=0 and the connectivity of the lattice is
not changed, the eigenvalues of the Hamiltonian must remainequal to the undistorted case. Only when a plot is made against
the wave vectors does the cone turn out to be distorted, as
shown in Fig. 1(d). For example, the brick wall lattice has the
same connectivity as graphene, and thus the spectrum mustbe the same. However, only when the spectrum is plottedin reciprocal space does the energy-momentum relationshipappear distorted.
Also, the case β=0 allows us to appreciate a subtle point.
Since the spectrum is the same as in unstrained graphene, it iseasy to see that the Ksymmetry points of the distorted lattice
coincide with the Dirac point of the new relationship givenby Eq. (12). In other words, the condition E(K
D)=0, which
defines the KDDirac points, corresponds to KD=K, where
Kis the image of point K0under the mapping K=(I+
/epsilon1)−1·K0. Thus, for β=0 it makes sense to develop the TB
Hamiltonian around the original Dirac points, as k=KD+q,with|q|/lessmuch| KD|. It is easy to show that the pure geometrical
distortion allows us to write the Dirac Hamiltonian as (seeAppendix B)
H=v
0σ·(I+/epsilon1)·q=v0σ/prime·q, (13)
qbeing the momentum measured relative to the Dirac points;
v0=3t0a/2, the Fermi velocity for the undeformed lattice;
σ=(σx,σy), the two Pauli matrices; and σ/prime=(I+/epsilon1)·σ
turns out to be the spinorial connection.26From this equation
follows a direction-dependent Fermi velocity for strainedgraphene, which has also been found in other work.
7,19
Furthermore, a direction-dependent velocity appears simply
by looking at the isoenergetic curves of Eq. (12) around K.I n
this case one obtains
E(K+q)2=(v0σ·(I+/epsilon1)·q)2, (14)
therefore, the isoenergetic curves around Kare rotated
ellipses, as depicted in Fig. 2(b). This figure was made for
a zigzag uniaxial strain of 5%, i.e., /epsilon1xx=0.05,/epsilon1xy=0, and
/epsilon1yy=−ν/epsilon1xx,νbeing the Poisson ratio, which is very low
for graphene, ν∼0.1–0.15, according to some theoretical
estimations.27,28
At this point, one can conclude that the basic mechanism
behind the anisotropic Fermi velocity is the distortion of thereciprocal space. This distortion gives a simple interpretationto the resulting geometric crystal frame terms that appears inthe covariant version of the equations.
19Clearly, there are not
associated pseudomagnetic fields.7,19
B. Actual case: β/negationslash=0
Let us now consider the case in which the space is distorted
and the hopping is changed, i.e., β/negationslash=0. Here, we have two
effects. Again, one has the pure geometrical distortion dueto the strain of the reciprocal space, but at the same time,there is a change in the spectrum. The latter effect is the onlyone observed when a diagonalization of the Hamiltonian isperformed in a computer for a finite number of atoms.
In Fig. 2, a comparison of case β=0 versus case β/negationslash=0i s
presented for E(k). As can be seen, the effect of β/negationslash=0i st o
distort the β=0 case in such a way that it tends to compensate
the strain of the reciprocal space; i.e., the ellipses are rotatedbyπ/2 for a realistic value of β. The physical reason for this
occurrence is that a stretched direction in real space shirks inreciprocal space, resulting in a higher Fermi velocity, whilein the same direction, the orbital overlap decreases since thedistance between atoms increases (see Fig. 1). This tends to
reduce the Fermi velocity. As a result, lattice distortion andhopping changes tend to compensate. This fact can also beseen in the movement of the Dirac points. From Fig. 2, one
can see that the Dirac points for β/negationslash=0 are closer to the original
ones than their β=0 counterparts.
The qualitative results discussed above, and depicted in
Fig. 2, can be understood by finding analytical expressions
forK
DandH. The position of KDcan be obtained from the
condition E(KD)=0. Up to first order in strain, we obtained
thatKDis given as
KD/similarequal(I+/epsilon1)−1·(K0+ξA)/similarequalK+ξA, (15)
085430-3M. OLIV A-LEYV A AND GERARDO G. NAUMIS PHYSICAL REVIEW B 88, 085430 (2013)
FIG. 2. (Color online) Isoenergetic curves obtained from the energy dispersion given by Eq. (9). A blowup is presented around the Dirac
points KDfor each surface. (a) Unstrained graphene; (b) strained graphene with β=0,/epsilon1xx=0.05,/epsilon1xy=0,/epsilon1yy=−ν/epsilon1xx; and (c) strained
graphene with β≈3 and the same strain tensor as in (b). Note how although the strain tensor is the same in (b) and (c), the ellipses are rotated
byπ/2, since the reciprocal space deformation and hooping effects tend to compensate.
with Adefined by Eq. (1)andξthe valley index of K0.23The
previous equation confirms the remark that the Dirac pointsforβ/negationslash=0 do not coincide with the Khigh-symmetry points of
the strained Brillouin zone. The shift, which is only producedbyβ, is given by the pseudovector potential and do not depend
onK
0.
Furthermore, once the points KDare known, it is possible
to obtain a new Dirac Hamiltonian considering the lattice cor-
rection and orbital overlap changes. To do this, we developed
Eq.(4)around the Dirac points using Eq. (15) and derived that
(see Appendix C)
H=v0σ·(I+/epsilon1−β/epsilon1)·q, (16)
which is a general version of Eq. (13), since βeffects are
included. Note that the isoenergetic curves around KDremain
ellipses, as depicted in Fig. 2(c), but with different values of
the semiaxes owing to the βcorrections. Equation (16) clearly
shows the tendency of βto cancel the lattice corrections.
Let us make two important remarks about Eqs. (15)
and (16), which are among the main contributions of this
paper. First, these equations are a generalization of analogousexpressions to the case of graphene under uniaxial strain whichwere inherited from studies on deformed carbon nanotubes.
29
Similar expressions were also found for a particular caseof distortion without shear. Thus, our generalization can bereduced to other special cases for which the results areknown,
7,29,30and coincides with the exact solvable case for
β=0. Such limiting cases allow us to check in different ways
the validity of the presented results. Second, Eq. (16) cannot be
derived from the theory of the strain-induced pseudomagneticfield. Namely, the effective Dirac Hamiltonian obtained by
this theory (as for example in Ref. [ 19]) does not reduce to our
Eq.(16) for the case of uniform strain, for reasons explained
in the following section.
IV . EXPERIMENTAL OBSERVATION OF
PSEUDOMAGNETIC FIELDS
From the point of view developed in the previous section,
it is clear that, basically, the Dirac cone is translated anddistorted. As a result, if one tries to derive an effectiveDirac equation using K
0as starting point to develop E(k)a s
k=K0+q, the resulting energy can be quite far away from
the Fermi energy, as shown in Fig. 3. This poses a problem
that has been overlooked in the usual treatment of strain ingraphene using pseudomagnetic fields in the Dirac equations.
In general, if Eq. (9)is developed around a general point in
reciprocal space given by K
G, we get
E2(KG+q)/similarequalE2(KG)+∇E2(KG)
·q+1
2q·∇∇E2(KG)·q, (17)
where ∇E2(KG) is the Jacobian vector and ∇∇E2(KG)t h e
Hessian matrix of E2(k), which are evaluated at k=KG.
In the usual procedure KG=K0. However, E2(K0)/negationslash=0
and∇E2(KG)/negationslash=0. This produces an energy shift and a
q-dependent term, observed in other approaches,19which
complicates the description of the dynamics somehow.
This also poses an issue concerning the experimental
possibility of observing the pseudomagnetic fields. Since the
085430-4UNDERSTANDING ELECTRON BEHA VIOR IN STRAINED ... PHYSICAL REVIEW B 88, 085430 (2013)
FIG. 3. (Color online) The Dirac cone in unstrained (dashed line)
and strained (solid line) graphene and the experimental observationof electron behavior for a probe that shifts the chemical potential
(μ) with respect to the Fermi energy. The shaded (green) rectangle
indicates the width of the thermal selector due to the Fermi-Diracdistribution. The effective Dirac equation with pseudomagnetic fields
can be obtained by developing around the original K
0points or in
the Dirac points KDof the strained lattice. For μ=0, only the latter
approach will work for low temperatures.
energy evaluated at the original Dirac point E(K0)i sd i f f e r e n t
from 0, the Fermi energy does not fall at this point, as weillustrate in Fig. 3. In general, if an experiment is performed
at temperature T, and the chemical potential μis shifted by a
field, the condition to observe the pseudomagnetic fields in theusual derivation around the original Dirac point must satisfy
|E(K
0)−μ|/lessorequalslantkBT, (18)
since the difference between E(K0) and μmust be less
than a zone defined from the derivative of the Fermi-Diracdistribution against the energy, as explained in Fig. 3using the
rectangle around μ.A sT→0, the derivative is a δfunction
centered around the Fermi energy, and the pseudomagneticfields calculated from K
0are usually far from the region of
validity. For example, even a zigzag uniaxial strain of 1% willproduce a E(K
0)/greaterorequalslant27 meV , which is much higher than the
thermal width of kBT≈8.6 meV , obtained at T=10 K. This
breaks the approximation of using pseudomagnetic fields in aDirac equation unless a very well-defined field is used.
The option is to have a better description of the energy
dispersion near the Fermi energy, by developing Eq. (17)
around the true Dirac points of the strained lattice, i.e., bysetting K
G=KD, for which the corresponding energies fall
at the Fermi level. In this case,
E2(KD+q)/similarequal1
2q·∇∇E2(KD)·q, (19)
sinceE2(KD)=0 and ∇E2(KD)=0. Now one obtains an
energy dispersion which corresponds to a distorted cone, with
a direction-dependent Fermi velocity given by the elementsof the Hessian of E
2(k) evaluated at KD. This result is the
same as the one obtained from the Dirac Hamiltonian givenby Eq. (16).V . CONCLUSIONS
In conclusion, we have analyzed the case of a spatially
uniform strain in graphene. The lattice correction termsare simply an effect of the strained reciprocal space. As aconsequence, the Dirac cones are deformed and translated.No pseudomagnetic fields are associated with such terms, ashas been recently discussed.
19–21When hopping changes are
considered, there is an extra deformation of the cone that tendsto cancel the effect of the reciprocal space strain. The newDirac points of the strained Hamiltonian do not coincide withtheKsymmetry points of the strained reciprocal lattice. Due
to this fact, the effective Dirac equation can be obtained bydeveloping around either the old or the new Dirac points.If the old points are chosen, as is usual in the grapheneliterature, there is a restriction to observe the dynamicsproduced by the calculated pseudomagnetic fields since onlyfor very high temperatures or carefully designed probes is itpossible to make a comparison with the usual theory. If the newDirac points are used, we have shown that it is possible to finda very simple modification of the Dirac equation. In computersimulations, it is also important to distinguish between latticedistortion effects and connectivity matrix. Some of these issuescan explain the differences between theory and simulations ingraphene.
31
Finally, it is worth mentioning that although we only treated
a particular case, the ideas and lessons obtained from thisstudy can be translated to general cases, as we will show inforthcoming work.
ACKNOWLEDGMENTS
We are grateful to V . M. Pereira and B. B Goldberg for help-
ful discussions. This work was supported by UNAM-DGAPA-PAPIIT, Project No. IN- 102513. M.O.L acknowledges supportfrom CONACYT (Mexico).
APPENDIX A
In this section, we provide explicit expressions for the last
terms in Eq. (9):
f/epsilon1(k∗)=(3/epsilon1xx+/epsilon1yy) cos(√
3k∗
xa)
+(3/epsilon1xx+5/epsilon1yy) cos/parenleftbigg√
3k∗
xa
2/parenrightbigg
cos/parenleftbigg3k∗
ya
2/parenrightbigg
−2√
3/epsilon1xysin/parenleftbigg√
3k∗
xa
2/parenrightbigg
sin/parenleftbigg3k∗
ya
2/parenrightbigg
,
f/epsilon12(k∗)=1
8/parenleftbig
9/epsilon12
xx+6/epsilon1xx/epsilon1yy+9/epsilon12
yy+12/epsilon12
xy
+/parenleftbig
(3/epsilon1xx+/epsilon1yy)2−12/epsilon12
xy/parenrightbig
cos(√
3k∗
x)
+8/epsilon1yy(3/epsilon1xx+/epsilon1yy) cos(√
3k∗
x/2) cos(3 k∗
y/2)
−16√
3/epsilon1yy/epsilon1xysin(√
3k∗
x/2) sin(3 k∗
y/2)/parenrightbig
.
APPENDIX B
For the case β=0, the hopping integral does not depend
on the direction, tn=t0, and consequently, the Hamiltonian
085430-5M. OLIV A-LEYV A AND GERARDO G. NAUMIS PHYSICAL REVIEW B 88, 085430 (2013)
TB given by Eq. (4)reduces to
H=−t0/summationdisplay
k,ne−ik·(I+/epsilon1)·δna†
kbk+H.c.
The closed dispersion relation derived from this Hamiltonian
has the form
E(k)=±t0/radicalbig
3+f(k∗),
where f(k∗)i sg i v e nb yE q . (10). As discussed in Sec. II,
the condition E(KD)=0, which defines the KDDirac points,
corresponds to KD=K, where Kis the image of point K0
under the mapping K=(I+/epsilon1)−1·K0. Thus, for β=0i t
makes sense to develop the TB Hamiltonian around the originalDirac points, as k=K+q, with |q|/lessmuch| K|:
E(K+q)=±t
0/radicalbig
3+f((I+/epsilon1)·((I+/epsilon1)−1·K0+q))
=±t0/radicalbig
3+f(K0+(I+/epsilon1)·q))
=±t0/radicalbig
3+f(K0+q∗)),q∗=(I+/epsilon1)·q
/similarequal±v0|q∗|
/similarequal±v0|(I+/epsilon1)·q|.
In the next section we provide a more general proof of
Eq. (13). At this point, is clear that the case β=0i sa
benchmark tool for any effective Hamiltonian, since it canbe solved without using any approximation.
APPENDIX C
We start with the Hamiltonian in momentum space of
strained graphene,
H=−3/summationdisplay
n=1tn/parenleftbigg
0 e−ik·(I+/epsilon1)·δn
eik·(I+/epsilon1)·δn 0/parenrightbigg
, (C1)
where tnis given by Eq. (7). Now, let us develop this
Hamiltonian around an original Dirac point KD, which is
defined by KD=K+A. Expanding k=KD+qwe get
H=−3/summationdisplay
n=1tn/parenleftbigg
0 e−i(KD+q)·(I+/epsilon1)·δn
ei(KD+q)·(I+/epsilon1)·δn 0/parenrightbigg
,(C2)butKD·(I+/epsilon1)·δn=(K0+A)·δn, and to first order in q
and/epsilon1we may write
H/similarequal−3/summationdisplay
n=1tn/parenleftbigg
0 e−iK0·δn
eiK0·δn 0/parenrightbigg
×(1−iσ3A·δn)(1−iσ3q·(I+/epsilon1)·δn); (C3)
note that Ais an expression in the first order of strain. Using
the identity
/parenleftbigg
0 e−iK0·δn
eiK0·δn 0/parenrightbigg
=iσ·δn
aσ3, (C4)
σ=(σx,σy) being the two Pauli matrices, the Hamiltonian
becomes
H/similarequal−t03/summationdisplay
n=1/parenleftbigg
1−β
a2δn·/epsilon1·δn/parenrightbigg/parenleftbigg
iσ·δn
aσ3/parenrightbigg
(1−iσ3A·δn
−iσ3q·(I+/epsilon1)·δn−(A·δn)(q·δn))
/similarequal−t03/summationdisplay
n=1/parenleftbigg
iσ·δn
aσ3/parenrightbigg/parenleftbigg
1−iσ3q·(I+/epsilon1)·δn
−(A·δn)(q·δn)+iβ
a2σ3(δn·/epsilon1·δn)(q·δn)
−iσ3A·δn−β
a2δn·/epsilon1·δn/parenrightbigg
/similarequalv0σ·(I+/epsilon1)q−v0σ·β
4(2/epsilon1−Tr(/epsilon1)I)
·q−v0σ·β
4(2/epsilon1+Tr(/epsilon1)I)·q
/similarequalv0σ·(I+/epsilon1−β/epsilon1)·q. (C5)
This is our Eq. (16), which also reproduces Eq. (13) forβ=0;
therefore, this section can be taken as a proof of both equations,Eq.(13) and Eq. (16). It is important to emphasize that in this
proof we assumed that the valley index of K
0isξ=1. For the
caseξ=− 1, the proof is analogous, and the Hamiltonian is
H/similarequalv0σ∗·(I+/epsilon1−β/epsilon1)·q, (C6)
with σ∗=(σx,−σy).
*moliva@fisica.unam.mx
†naumis@fisica.unam.mx
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PhysRevB.95.100201.pdf | RAPID COMMUNICATIONS
PHYSICAL REVIEW B 95, 100201(R) (2017)
Fractional quantum Hall effect in strained graphene:
Stability of Laughlin states in disordered pseudomagnetic fields
Andrey A. Bagrov,*Alessandro Principi,†and Mikhail I. Katsnelson‡
Institute for Molecules and Materials, Radboud University, Heijndaalseweg 135, 6525 AJ, Nijmegen, The Netherlands
(Received 23 November 2016; published 14 March 2017)
We address the question of the stability of the fractional quantum Hall effect in the presence of pseudomagnetic
disorder generated by mechanical deformations of a graphene sheet. Neglecting the potential disorder and takinginto account only strain-induced random pseudomagnetic fields, it is possible to write down a Laughlin-like trialground-state wave function explicitly. Exploiting the Laughlin plasma analogy, we demonstrate that in the caseof fluctuating pseudomagnetic fluxes of a relatively small amplitude, the fractional quantum Hall effect is alwaysstable upon the deformations. By contrast, in the case of bubble-induced pseudomagnetic fields in graphene ona substrate (a small number of large fluxes) the disorder can be strong enough to cause a glass transition in thecorresponding classical Coulomb plasma, resulting in the destruction of the fractional quantum Hall regime andin a quantum phase transition to a nonergodic state of the lowest Landau level.
DOI: 10.1103/PhysRevB.95.100201
I. INTRODUCTION
Massless Dirac fermions were discovered [ 1,2] in graphene
via the observation of an unusual (“half-integer”) quantumHall effect (QHE) [ 1–7] which is a manifestation of the
existence of a topologically protected zero-energy Landaulevel [ 1,4,7]. This means that this level is not broadened
by any inhomogeneity of the magnetic field. It was realizedvery soon after this discovery [ 8] that inhomogeneities of the
effective magnetic field are unavoidable in graphene, due tothe effect of pseudomagnetic fields induced by strain (for areview, see Refs. [ 7,9,10]). In earlier works [ 8,11] random
pseudomagnetic fields created by defects (such as intrinsic andextrinsic ripples) were considered. Later it was theoreticallypredicted [ 12,13] and experimentally confirmed [ 14] that
(pseudo) Landau level quantization and the valley quantumHall effect can be created in graphene by external smoothdeformation with a trigonal symmetry, and that effective fieldsas high as hundreds of teslas may be easily reached in this way(an order of magnitude stronger than what may be observed inconventional high-field magnetic laboratories).
The fractional quantum Hall effect (FQHE) has been
experimentally discovered in graphene and its observationwas reported in Refs. [ 15,16]. A very natural and interesting
question is whether or not this state is also protected, to someextent, with respect to inhomogeneities of the pseudomagneticfield. Here, we answer this question within a framework of amodel with random pseudomagnetic fields but assuming theabsence of potential disorder. In terms of deformations thismeans strong shear deformations and no dilatation [ 7]. It was
shown recently [ 17] that at least in some graphene samples
random strain-induced pseudomagnetic fields are indeed themain source of electron scattering and therefore the model maybe quite realistic.
*a.bagrov@science.ru.nl
†a.principi@science.ru.nl
‡m.katsnelson@science.ru.nlII. THE MODEL OF PSEUDOMAGNETIC DISORDER
It was shown experimentally [ 17] that in many cases the
sources of the pseudomagnetic field can be considered asrandomly distributed centers of deformations. The typical sizeof these centers is much smaller than the distances betweenthem, so we can effectively treat them as pointlike objects.The quenched pseudomagnetic disorder therefore emerges inthe form of a set of highly localized fluxes, and the problem
reduces to the study of the properties of a correlated electron
gas in the presence of a random flux distribution coexistingwith a homogeneous background magnetic field.
Disorder of this geometrical structure appears in suspended
graphene due to ripples [ 11]. Another system where it can
be observed is a graphene sheet put on a substrate, e.g., aplatinum surface [ 14]. When epitaxial graphene is grown on
such a substrate, bubbles of a characteristic width of about
10 nm and height 2 nm tend to form. In this case, the number
of fluxes is much smaller and the value of each flux is muchlarger than in the case of rippled graphene.
In what follows, we neglect intervalley scattering and model
graphene as two independent massless-Dirac-fermion systems,associated with the valleys KandK
/primeat the corners of its
hexagonal Brillouin zone. The two valleys differ by the signof the strain-induced pseudomagnetic field they experience.A trial wave function for the zero-energy Landau level of asystem of massless Dirac fermions in the presence of localizedmagnetic fluxes can be constructed using the Aharonov-Cashersolution [ 18]. We briefly recall this construction in Ref. [ 19]
(for a more recent detailed study of a single magnetic fluxeffect on spectrum of states in graphene see [ 20]). Following
this prescription, we derive a wave function that can be viewedas a square root of the partition function of a two-dimensional(2D) Coulomb plasma evolving in a background of randomlydistributed (quenched) point charges, i.e., Z=/integraltext
d/vectorz|ψ(/vectorz)|
2=/integraltext
d/vectorze−H/m, where
H=− 2m2N/summationdisplay
k<lln|zk−zl|+m
2N/summationdisplay
n|zn|2
+2mN/summationdisplay
iN/Phi1/summationdisplay
j/Phi1j
/Phi10ln|zi−˜zj|. (1)
2469-9950/2017/95(10)/100201(5) 100201-1 ©2017 American Physical SocietyRAPID COMMUNICATIONS
BAGROV , PRINCIPI, AND KATSNELSON PHYSICAL REVIEW B 95, 100201(R) (2017)
Here, /Phi10is the magnetic flux quantum, and mis the inverse
filling factor of the lowest Landau level. This classicalHamiltonian is the central object of our study. For the sakeof simplicity, we assume that all pseudomagnetic fluxes havethe same value, /Phi1
j=/Phi1.
A few comments are now in order. While the two afore-
mentioned physical realizations of disorder can be describedby the same formal model, they are fairly different on aphenomenological level. First of all, inhomogeneities of thepseudomagnetic field in rippled graphene are normally notvery strong, with deviations |δB|∼1 T on a length scale
ofl∼1n m [ 8], resulting in very moderate flux values,
/Phi1/similarequal10
−3/Phi10. In contrast, nanobubbles observed in graphene
on a metal substrate lead to very strong pseudomagnetic fields,of the order of |δB|∼300 T, localized within regions of a
characteristic size ∼5–10 nm [ 14]. Hence, the corresponding
fluxes are of the order of /Phi1∼10–15 /Phi1
0. Another fundamental
difference is that out-of-plane deformations of freely sus-pended graphene (ripples) can occur in both directions, hencethe signs of the /Phi1
jfluxes are randomly distributed, and each
of them acts as a local repulsive or attractive potential on theparticles of the associated classical plasma. On the other hand,the fluxes due to nanobubbles are all of the same sign, makingthe potential landscape for the Laughlin plasma either purelyrepulsive or purely attractive (depending on the valley). Inwhat follows, we mainly perform computations for the second
case and demonstrate that a transition of the liquid plasma to a
glass state is possible. The first case turns out to be more trivialdue to the weakness of the disorder, and it will be clear that thequantum Hall effect is insensitive to deformations of this kind.
III. DISCUSSION OF THE MODEL
Since pseudomagnetic fields preserve the time-reversal
symmetry [ 7], the same bubble deformation of the graphene
sheet induces a flux codirected with the background magneticfield in one valley (which results in an attractive potentialfor the classical plasma), and oppositely directed in the othervalley. In the latter case it enters as a quenched repulsivepotential in the action of the classical plasma. Great care shouldbe taken here: Since we allow for strong variations of thepseudomagnetic field, to remain within the regime of validityof the model we need to make sure that these inhomogeneitiesnever lead to mixing between Landau levels (LLs). In thevalley where pseudomagnetic fluxes are codirected with thebackground magnetic field the problem is not expected tooccur: While the lowest Landau level (LLL) is protectedupon variations of the magnetic field, the gap between it andthe next LL is bounded from below by the correspondinghomogeneous value. In the other valley the situation ispotentially more dangerous, as the presence of oppositelydirected large magnetic fluxes may imply the existence ofzero-magnetic-field lines in the sample and, possibly, regionswhere LLs merge. However, since the corresponding regionsact on classical particles in the Laughlin plasma as strongrepulsive potentials, on the quantum level we expect the systemto avoid occupying states within these domains, and the effectof level mixing should be mild.
Another potentially problematic aspect related to the exis-
tence of B=0 lines is the possibility of percolation throughthe bulk of the sample that destroys the quantization of the
Hall conductivity [ 21]. But since we consider only the case
of a low density of fluxes, and the distance between any twofluxes is much bigger than their characteristic size, such a lineis an isolated loop circumventing a flux, and there is no chanceof percolation.
IV . PLASMA STATIC STRUCTURE FACTOR
Having defined the model, we can numerically calculate the
static structure factor of the Coulomb plasma in the presenceof a number of static (disorder) charges by means of the replicaOrnstein-Zernike equations derived for a partly quenchedtwo-component fluid in Ref. [ 22]. Referring the reader to
the original paper for a detailed discussion, hereafter we justbriefly quote the idea. In this language, the potential landscapeprovided by pseudomagnetic fluxes can be implemented as afrozen “liquid” whose direct ( c
00) and full ( h00) pair correlation
functions (and thus the static structure factor S00=1+ρ0h00)
are fixed. The correlation functions hij,cijof the annealed
component can be obtained by solving the replicated systemof equations
h
01=c01+ρ0c00⊗h01+ρ1c01⊗(h11−h12),
h11=c11+ρ0c01⊗h01+ρ1c11⊗h11−ρ1c12⊗h12,
h12=c12+ρ0c01⊗h01+ρ1c11⊗h12
+ρ1c12⊗h11−2ρ1c12⊗h12, (2)
where the ⊗symbol stands for convolution f⊗g=/integraltext
f(r−
r/prime)g(r/prime)d2r/prime. Here, the index 0 denotes the quenched com-
ponent (i.e., the magnetic fluxes), while 1 and 2 refer totwo replicas of the annealed one (the electron liquid). Theseequations are to be supplemented by the closure relations. Weuse the hypernetted chain closure [ 23,24], which has been
proven to give very accurate results for quantum Hall plasmas,
h
ij(r)=exp[hij(r)−cij(r)−βvij(r)]−1, (3)
where βvij(r) are radially symmetric interaction potentials
(β=mis the inverse temperature of the classical plasma), and
replicas are required not to interact with each other directly, i.e.,v
12(r)=0. Here, ρ0is the density of pseudomagnetic fluxes
which can be estimated as follows. The typical backgroundmagnetic field which is required to develop a ν=
1
3FQHE
state is B0/similarequal15 T [ 16], hence lB/similarequal6 nm. In experiments on
graphene on a platinum substrate, a density of nanobubbles ofabout 5 per 2500 nm
2[14] was observed, which in rescaled
units would correspond to ρ0/similarequal0.07lB−2. The particle density
of itinerant electrons is instead fixed to ρ1=1/(2πl2
Bm)i na
Laughlin state with the corresponding filling factor ν=1/m
(in our calculations we fix ν=1
3). Hereafter we set the
magnetic length lB=1 for convenience.
Before proceeding with solving ( 2)w eh a v et ofi xt h es t a t i c
structure factor S00(q) of the pseudomagnetic disorder. While
the latter is not explicitly known at all momentum scales, thereare two qualitatively distinct cases that correspond to differentbehavior at small wave vectors. Inhomogeneities of thepseudomagnetic field can be long- or short-range correlated.In the former case the correlator of pseudomagnetic vectorpotential /angbracketleft|A
q|2/angbracketrightbehaves as 1 /q2forq→0[11,17], resulting
100201-2RAPID COMMUNICATIONS
FRACTIONAL QUANTUM HALL EFFECT IN STRAINED . . . PHYSICAL REVIEW B 95, 100201(R) (2017)
FIG. 1. Static structure factor S00(q) of short-range (blue) and
long-range (red) correlated pseudomagnetic disorder at packing ratio
η=0.07.
in the correlator of pseudomagnetic fields S00(q)∝q2/angbracketleft|Aq|2/angbracketright
to approach a nonvanishing constant at q=0. This kind
of pseudomagnetic disorder is relevant for ripple-scattering-dominated electronic transport [ 7,11,17]. At the same time,
short-range disorder with /angbracketleft|A
q|2/angbracketrightapproaching a constant at
q→0 should always exist but does not lead to any appreciable
contribution to the electron mobility at zero magnetic field.However, as we will see in what follows, it can substantially
effect the FQHE. In this case, S
00(q)∝q2atq→0.
A natural way to generate such model structure factors is
to imagine for a second that, prior to being quenched, thefluxes themselves were released to anneal as a liquid. If weassume that they interact with each other via a two-dimensionalCoulomb potential, we will end up with a structure factorcorresponding to short-range disorder (represented by the bluecurve in Fig. 1). If the “annealing” potential is instead taken to
be of the hard-sphere type, we obtain a model of long-rangedisorder (the red curve in Fig. 1). Since real correlation
functions of pseudomagnetic fields are unknown (and maybe very different for different samples and substrates) weuse those obtained from these two models. This is enoughto demonstrate a qualitative difference between short-rangeand long-range correlated disorder.
The solutions of the replica Ornstein-Zernike equations in
the two aforementioned cases are shown in Figs. 2and 3,
respectively. We can see that the two types of pseudomagneticdisorder lead to very different physical effects. The structurefactor S
11(q) of a quantum Hall plasma in the presence of a
short-range disorder remains vanishing at q→0 regardless
of the strength of the pseudomagnetic fluxes. We can alsocheck that the incompressibility sum rule is always satisfied:
1
2π/integraltext
[g(r)−1]d2r=− 1[25]. On the other hand, long-range
correlations in the pseudomagnetic disorder already at a smallstrength /Phi1lead to a change in the small- qbehavior of the
QHE plasma structure factor, making it compressible and thusdestroying the quantum Hall effect.
The high peak in S
11(q) for a strong enough magnetic
disorder is a precursor to a glass phase transition of theLaughlin plasma. The latter can be interpreted as a breakdownof ergodicity of the corresponding quantum ground state.To show this, we use the mode coupling theory [ 26,27].
Details of this approach are given in Ref. [ 28]. The criterionFIG. 2. Static structure factors S(q) and pair correlation functions
g(r) (inset) of a Laughlin plasma in the presence of short-range
correlated magnetic disorder: the cases of attractive (solid) and
repulsive (dashed) flux potentials.
that distingusihes between ergodic and glassy phases is
the behavior of the density relaxation function at a largetime,φ(q)≡/angbracketleftρ
q(t→∞ )ρ−q(0)/angbracketright/S11(q), where the notion
of average uses the static liquid structure factor as an input.If¯φ(q) is nonzero, the plasma is in a glassy phase. For
the studied model of background structure factor and at thegiven density of defects ρ
0/similarequal0.07l−2
B, the glass transition
occurs when the amplitude of fluxes exceeds /Phi1∼12–13 /Phi10,
i.e., when the fluxes are about 300 T across 100 nm2.T h i s
critical value might be perceived as very large, but stillexperimentally attainable, especially in light of the recentdiscovery of ultrahigh ( ∼1000 T) pseudomagnetic fields [ 29].
FIG. 3. Static structure factors S(q) and pair correlation functions
g(r) (inset) of a Laughlin plasma in the presence of long-range
correlated magnetic disorder: the cases of attractive (solid) andrepulsive (dashed) flux potentials.
100201-3RAPID COMMUNICATIONS
BAGROV , PRINCIPI, AND KATSNELSON PHYSICAL REVIEW B 95, 100201(R) (2017)
We anticipate, from a calculation performed for single-
sign pseudomagnetic disorder (corresponding to graphene on asubstrate), that no glass should be expected in rippled graphenewhere the disorder is smooth. Although we cannot exclude thatthe interplay of positive and negative magnetic fluxes can inprinciple slightly enhance the phase transition, the flux valuesare much smaller (by several orders of magnitude) than what isrequired for it to occur. Moreover, even if the disorder is longranged, /Phi1/similarequal10
−3/Phi10fluxes cannot lead to an experimentally
relevant shift of the infrared asymptotic value of S(q→0),
and will not destroy the quantum Hall effect.
V . DISCUSSION OF THE RESULTS
We have considered two models of pseudomagnetic disor-
der, namely, with long- and short-range correlations. Long-range correlated disorder occurs in the presence of ripples,and is characterized by a structure factor which does notvanish in the limit q→0[7]. Its effect on the fractional
quantum Hall state turns out to be dramatic. Even for smallconcentrations of not very strong disorder, the structure factorof the annealed component (i.e., the electron liquid) does notvanish in the limit of q→0. This in turn implies that the liquid
becomes compressible and the fractional quantum Hall stateis completely destroyed [ 23].
On the other hand, the short-range disorder, whose
quenched structure factor resembles that of a normal liquid,has a more subtle effect on the fractionalized state, whichneeds to be discussed in more detail. It is well known thatinhomogeneous ground states can be realized in fractionalquantum Hall systems at filling factors between those corre-sponding to incompressible states [ 30–37]. Striped or bubble
phases have been predicted to have energies of the orderof the homogeneous (liquid) ground state, and to appear atintermediate filling factors whenever long-range interactionsare present [ 30,35,36]. Striped phases are especially favored
by nonisotropic or dipolarlike interactions (such as, e.g.,those resulting from the screening of the Coulomb potentialby neighboring metal plates) [ 30]. A characteristic hallmark
of such phases is, for example, a nonhomogeneous Hallconductivity [ 38–42]. In clean systems and at low temperature
such phases usually exhibit a long-range order.
An incompressible state in the presence of quenched mag-
netic disorder will exhibit a somewhat similar phenomenology.Local fluctuations of the filling factor will result in theformation of chaotic patterns. Although the pattern appearsto be, at a first sight, completely random, a careful study canreveal the hidden typical length scale, corresponding to a sharpmaximum of the structure factor. Such patterns can, in general,
be thought of as stripe or bubble glass phases [ 43,44]. Their
origin is due to the fact that the system finds itself frustrated bythe presence of disorder, and wants to modulate with a periodcorresponding to the typical length scale but in all possibledirections at the same time [ 43,44].
We predict a phase transition as a function of the strength
of the magnetic disorder. At all strengths the structure factorwill exhibit a liquidlike behavior at a small momentum [i.e.,S(q)→q
2in the limit of q→0]. A sharp peak develops
at finite wave vectors, and grows with the strength of thequenched magnetic disorder /Phi1//Phi1
0. This structure of S(q)
is crucial and leads to the emergence of a glassy behavior.The latter is revealed by the presence of nondecaying densityfluctuations, encoded in the finiteness of the long-time partof the dynamical structure factor. Below the critical value ofthe strength of the quenched magnetic disorder, S(q,t)→0
at large time. However, above this critical value of /Phi1//Phi1
0it
remains finite, signaling that density fluctuations do not decayover time. This behavior is typical for frozen systems.
From the behavior of the classical plasma associated with
the deformed Laughlin wave function in the presence ofquenched magnetic disorder, we infer that there is a phasetransition analogous to the formation of a Wigner crystal at asmall filling fraction. In our case the transition is driven not bythe small density but by the presence of magnetic disorder and,
depending on its the strength, can occur at all filling fractions.
Whether the quantum electron liquid itself behaves noner-
godically can be tested by means of scanning single-electrontransistor experiments in combination with thermal cycling.The chaotic charge texture we predict is similar to electron-hole puddles observed in graphene in the presence of potentialdisorder and can be read off with the same experimentaltechniques [ 45]. However, in the case of an inhomogeneous
pseudomagnetic field we do not expect the texture to reproducethe geometric pattern of the underlying quenched disordermatrix. That is, if we heat up a sample and cool it down againfor a number of times, the texture should be unique at everyiteration. If this prediction is confirmed, it would mean thatthe quantum liquid indeed exhibits aging [ 46]. In this case the
excited thermal states over such a vacuum can be thought ofas an example of many-body localization [ 47].
ACKNOWLEDGMENT
The authors acknowledge support from the ERC Advanced
Grant No. 338957 FEMTO/NANO and from the NWO via theSpinoza Prize.
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100201-5 |
PhysRevB.79.205322.pdf | Efficient wave-function matching approach for quantum transport calculations
Hans Henrik B. Sørensen and Per Christian Hansen
Informatics and Mathematical Modelling, Technical University of Denmark, Bldg. 321, DK-2800 Lyngby, Denmark
Dan Erik Petersen, Stig Skelboe, and Kurt Stokbro
Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen, Denmark
/H20849Received 5 May 2008; published 29 May 2009 /H20850
The wave-function matching /H20849WFM /H20850technique has recently been developed for the calculation of electronic
transport in quantum two-probe systems. In terms of efficiency it is comparable to the widely used Green’sfunction approach. The WFM formalism presented so far requires the evaluation of all the propagating andevanescent bulk modes of the left and right electrodes in order to obtain the correct coupling between deviceand electrode regions. In this paper we will describe a modified WFM approach that allows for the exclusionof the vast majority of the evanescent modes in all parts of the calculation. This approach makes it feasible toapply iterative techniques to efficiently determine the few required bulk modes, which allows for a significantreduction of the computational expense of the WFM method. We illustrate the efficiency of the method on acarbon nanotube field-effect-transistor device displaying band-to-band tunneling and modeled within the semi-empirical extended Hückel theory framework.
DOI: 10.1103/PhysRevB.79.205322 PACS number /H20849s/H20850: 73.40. /H11002c, 73.63. /H11002b, 72.10. /H11002d, 85.35.Kt
I. INTRODUCTION
Quantum transport simulations have become an important
theoretical tool for investigating the electrical properties ofnanoscale systems.
1–5The basis for the approach is the
Landauer-Büttiker picture of coherent transport, where theelectrical properties of a nanoscale constriction are describedby the transmission coefficients of a number of one-electronmodes propagating coherently through the constriction. Theapproach has been used successfully to describe the electricalproperties of a wide range of nanoscale systems, includingatomic wires, molecules, and interfaces.
6–15In order to apply
the method to semiconductor device simulation, it is neces-sary to handle systems comprising many thousand atoms,and this will require new efficient algorithms for calculatingthe transmission coefficient.
Our main purpose in this paper is to give details of a
method we have developed based on the wave-functionmatching /H20849WFM /H20850technique,
16–18which is suitable for study-
ing electronic transport in large-scale atomic two-probe sys-tems, such as large carbon nanotubes or nanowire configura-tions.
We adopt the many-channel formulation of Landauer and
Büttiker to describe electron transport in nanoscale two-probe systems composed of a left and a right electrode at-tached to a central device /H20849see Fig. 1/H20850. In this formulation,
the conduction Gof incident electrons through the device is
intuitively given in terms of transmission and reflection ma-trices, tandr, that satisfy the unitarity condition t
†t+r†r
=1in the case of elastic scattering. The matrix element tijis
the probability amplitude of an incident electron in a mode i
in the left electrode being scattered into a mode jin the right
electrode, and correspondingly rikis the probability of it be-
ing reflected back into mode kin the left electrode. This
simple interpretation yields the Landauer-Büttiker formula3G=2e2
hTr/H20851t†t/H20852, /H208491/H20850
which holds in the limit of infinitesimal voltage bias and zero
temperature.
To our knowledge, the WFM schemes presented so far in
the literature require the evaluation of all the Bloch and eva-nescent bulk modes of the left and right electrodes in order toobtain the correct coupling between device and electrode re-gions. The reason for this is that the complete set of bulkmodes is needed to be able to represent the proper reflectedand transmitted wave functions. In this paper we will de-scribe a modified WFM approach that allows for the exclu-sion of the vast majority of the evanescent modes in all partsof the calculation. The primary modification can be picturedas a simple extension of the central region with a few prin-cipal electrode layers. In this manner, it becomes advanta-geous to apply iterative techniques for obtaining the rela-tively few Bloch modes and slowly decaying evanescentmodes that are required. We have recently developed such aniterative method in Ref. 19, which allows for an order of
magnitude reduction of the computational expense of theWFM method in practice.
In this work, the proper analysis of the modified WFM
approach is presented. The accuracy of the method is inves-
Device Bulk electrode Bulk electrode
FIG. 1. /H20849Color online /H20850Schematic illustration of a nanoscale two-
probe system in which a device is sandwiched between two semi-infinite bulk electrodes.PHYSICAL REVIEW B 79, 205322 /H208492009 /H20850
1098-0121/2009/79 /H2084920/H20850/205322 /H2084910/H20850 ©2009 The American Physical Society 205322-1tigated and appropriate error estimates are developed. As an
illustration of the applicability of our WFM scheme we con-sider a 1440 atom carbon nanotube field-effect transistor/H20849CNTFET /H20850device of 14 nm in length. We calculate the zero-
bias transmission curves of the device under various gatevoltages and reproduce previously established characteristicsof band-to-band tunneling.
20We compare directly the results
of the modified WFM method to those of the standard WFMmethod for quantitative verification of the calculations.
The rest of the paper is organized as follows. The WFM
formalism that is used to obtain tandris introduced in Sec.
II. In Sec. IIIwe present our method to effectively exclude
the rapidly decaying evanescent modes from the two-probetransport calculations. Numerical results are presented inSec. IVand the paper ends with a short summary and out-
look.
II. FORMALISM
In this section we give a minimal review of the formalism
and notation that is used in the current work in order todetermine the transmission and reflection matrices tandr.
This WFM technique has several attractive features com-pared to the widely used and mathematically equivalentGreen’s function approach.
1,2Most importantly, the transpar-
ent Landauer picture of electrons scattering via the centralregion between Bloch modes of the electrodes is retainedthroughout the calculation. Moreover, WFM allows one toconsider the significance of each available mode individuallyin order to achieve more efficient numerical procedures toobtain tandr.
A. Wave-function matching
The WFM method is based on direct matching of the bulk
modes in the left and right electrodes to the scattering wavefunction of the central region. For the most part this involvestwo major tasks: obtaining the bulk electrode modes andsolving a system of linear equations. The bulk electrodemodes can be characterized as either propagating or evanes-cent /H20849exponentially decaying /H20850modes but only the propagat-
ing modes contribute to Gin Eq. /H208491/H20850. We may write G
=/H208492e
2/h/H20850T, where
T=/H20858
kk/H11032/H20841tkk/H11032/H208412/H208492/H20850
is the total transmission and the sum is limited to propagat-
ing modes kandk/H11032in the left and right electrodes, respec-
tively. Notice, however, that the evanescent modes are stillneeded in order to obtain the correct matrix elements t
kk/H11032.W e
will discuss this matter in Sec. III C .
We assume a tight-binding setup for the two-probe sys-
tems in which the infinite structure is divided into principallayers numbered i=−/H11009,...,/H11009and composed of a finite cen-
tral /H20849C/H20850region containing the device and two semi-infinite
left/H20849L/H20850and right /H20849R/H20850electrode regions /H20849see Fig. 2/H20850. The wave
function is
/H9274i/H20849x/H20850=/H20858jmici,j/H9273i,j/H20849x−Xi,j/H20850in layer i, where /H9273i,jde-
notes localized nonorthogonal atomic orbitals and Xi,jare the
positions of the miorbitals in layer i. We represent /H9274i/H20849x/H20850bya column vector of the expansion coefficients, given by /H9274i
=/H20851ci,1,..., ci,mi/H20852T, and write the wave function /H9274extending
over the entire system as /H9274=/H20851/H9274−/H11009T,...,/H9274/H11009T/H20852T. We also assume
that the border layers 1 and nof the central region are always
identical to a layer of the connecting electrodes.
We refer the reader to Refs. 16–18and21for details on
how to employ WFM to our setup. Here and in the rest ofthis paper, we will use the following notation for the key
elements. The matrices /H9021
L/H11006=/H20851/H9278L,1/H11006,...,/H9278L,mL/H11006/H20852contain in
their columns the full set of mLleft-going /H20849−/H20850andmLright-
going /H20849+/H20850bulk modes /H9278L,k/H11006of the left electrode, and the
diagonal matrices /H9011L/H11006=diag /H20851/H9261L,1/H11006,/H9261L,2/H11006,...,/H9261L,mL/H11006/H20852hold the
corresponding Bloch factors.22If trivial modes with /H20841/H9278L,k+/H20841
=0 or /H20841/H9278L,k−/H20841=/H11009occur they are simply rejected. We assume
that all the evanescent bulk modes are /H20849state- /H20850normalized
/H9278L,k/H11006†/H9278L,k/H11006=1, while all the Bloch bulk modes are flux
normalized23/H9278L,k/H11006†/H9278L,k/H11006=dL/vL,k/H11006, where vL,k/H11006are the group
velocities15,24anddLis the layer thickness. Similarly for the
right electrode the matrices /H9021R/H11006and/H9011R/H11006are formed.
We also introduce the Bloch matrices17BL/H11006
=/H9021L/H11006/H9011L/H11006/H20849/H9021L/H11006/H20850−1and BR/H11006=/H9021R/H11006/H9011R/H11006/H20849/H9021R/H11006/H20850−1, which propagate
the layer wave functions in the bulk electrode
/H9274j/H11006=/H20849B/H11006/H20850j−i/H9274i/H11006, /H208493/H20850
where subscript Lis implied for the left electrode /H20849i,j/H113491/H20850
andRfor the right electrode /H20849i,j/H11350n/H20850. Notice that the first
central region layer is defined for layer 1 and not layer 0, asis the case in Ref. 18.
As explicitly shown in Refs. 16–18, by fixing the layer
wave functions coming into the Cregion /H20849e.g., in our case
/H92741+=/H9261L,k+/H9278L,k+and/H9274n−=0/H20850and matching the layer wave func-
tions across the Cregion boundaries, the system of linear
equations for the central region wave function /H9274Ccan be
written as
/H20849ESC−HC−/H9018L−/H9018R/H20850/H9274C=b, /H208494/H20850
where Eis the energy, SCthe overlap, and HCthe Hamil-
tonian matrix of the central region. In the following we dis-cuss the terms, /H9018
L,/H9018R, and b, which arise from matching the
boundary conditions with the electrode modes.LCR
/bracehtipdownleft
/bracehtipupright/bracehtipupleft
/bracehtipdownright/bracehtipdownleft
/bracehtipupright/bracehtipupleft
/bracehtipdownright/bracehtipdownleft
/bracehtipupright/bracehtipupleft
/bracehtipdownright
ψ0ψ1ψ2 ψnψn+1 ψn−1In-
coming
Re-
flectedTrans-
mitted
Zero···
FIG. 2. Schematic representation of WFM applied to layered
two-probe systems, where the central device region, consisting oflayers i=1,..., n, is attached to left and right semi-infinite elec-
trodes. The incoming propagating mode from the left electrode isscattered in the central region and ends up as reflected and trans-mitted superpositions of propagating and evanescent modes.SØRENSEN et al. PHYSICAL REVIEW B 79, 205322 /H208492009 /H20850
205322-2The self-energy matrices, /H9018Land/H9018R, arise from matching
with the outgoing left and right electrode modes. They onlyhave nonzero terms in the upper left and lower right cornerblocks, respectively, and these elements can be calculated interms of the Bloch matrices
16,17
/H20851/H9018L/H208521,1=H0,1†/H20851H1+H0,1†/H20849BL−/H20850−1/H20852−1H0,1 /H208495/H20850
and
/H20851/H9018R/H20852n,n=Hn,n+1/H20849Hn+Hn,n+1BR+/H20850−1Hn,n+1†, /H208496/H20850
where we have introduced the overline notation Hi/H11013ESi
−HiandHi,j/H11013ESi,j−Hi,j. For the current setup, these matri-
ces are identical to the self-energy matrices introduced in the
Green’s function formalism1/H20849to within an infinitesimal
imaginary shift of E/H20850and may be evaluated by well-known
recursive techniques25,26or constructed directly from the
electrode modes using Eq. /H208496/H20850.
The source term barises from the incoming mode. As-
suming an incoming mode from the left, we have b
=/H20851b1T,0T,..., 0T/H20852Tspecified by the expression
b1=−/H20849H0,1†+/H20851/H9018L/H208521,1BL+/H20850/H92740, /H208497/H20850
where /H92740is the incoming wave function.
For notational simplicity in the following sections, we
leave out the implied subscripts LorR, indicating the left or
right electrode, whenever the formalism is the same for both/H20849e.g., for symbols m,/H9261
k,/H9278k,/H9021/H11006,/H9011/H11006,B/H11006,/H9018, etc. /H20850.
B. Transmission and reflection coefficients
As a final step we want to determine the tandrmatrices
from the boundary wave functions /H92741and/H9274nthat have been
obtained by solving Eq. /H208494/H20850. When the incoming wave /H92740is
specified to be the kth right-going mode /H9278L,k+of the left elec-
trode, then /H9274nwill be the superposition of outgoing right-
transmitted waves. The kth column of the transmission ma-
trixtkis defined as the corresponding expansion coefficientsin right electrode modes and can be evaluated by solving
/H9021R+tk=/H9274n, /H208498/H20850
where /H9021R+is the mR/H11003mRcolumn matrix holding the right-
going bulk modes of the right electrode /H20849and here assumed to
be nonsingular /H20850. Similarly the kth column of the reflection
matrix rkis given by
/H9021L−rk=/H92741−/H9261L,k+/H9278L,k+, /H208499/H20850
where /H9021L−holds the left-going bulk modes of the left elec-
trode. The flux normalization ensures that t†t+r†r=1.
III. EXCLUDING EV ANESCENT MODES
The most time-consuming task of the WFM method is
often to determine the electrode modes, which requires solv-ing a quadratic eigenvalue problem.
16As examples, see the
profiling results listed in Table I, where we have used the
method to compute tand rfor a selection of two-probe
systems.27The CPU timings show that to determine the elec-
trode modes by employing the state-of-the-art LAPACK eigen-
solver DGEEV is, in general, much more expensive than to
solve the system of linear equations in Eq. /H208494/H20850. We expect
this trend to hold for larger systems as well. Therefore, in theattempt to model significantly larger devices /H20849thousands of
atoms /H20850, it is of essential interest to reduce the numerical cost
of the electrode modes calculation. We argue that a compu-tationally reasonable approach is to limit the number of elec-trode modes taken into account, e.g., by excluding the leastimportant evanescent modes. In this section, a proper tech-nique to do this in a rigorous and systematic fashion is pre-sented.
A. Decay of evanescent modes
The procedure to determine the Bloch factors /H9261kand non-
trivial modes /H9278kof an ideal electrode and subsequently char-TABLE I. CPU times in seconds when using WFM for calculating tandrat 20 different energies inside
E/H33528/H20851−2 eV;2 eV /H20852for various two-probe systems. The numbers of atoms in the central region /H20849electrode unit
cell/H20850are indicated. The four rightmost columns show the CPU times spent for computing the electrode bulk
modes with DGEEV and in this work vs solving the central region linear systems in Eq. /H208494/H20850and the system
with two extra principal layers on each side.
System Atoms Equation /H208494/H20850Equation /H208494/H20850
/H20849l=2/H20850 DGEEV This work
Fe-MgO-Fe 27 /H208496/H20850 0.8 0.9 1.3 1.1
Al-C/H110037-Al 74 /H2084918/H20850 0.4 0.6 3.6 1.6
Au-DTB-Au 102 /H2084927/H20850 8.1 13.5 91.0 28.2
Au-CNT /H208498,0/H20850/H110031-Au 140 /H2084927/H20850 11.4 16.6 77.6 17.1
Au-CNT /H208498,0/H20850/H110035-Au 268 /H2084927/H20850 45.3 50.3 83.6 17.8
CNT /H208498,0/H20850-CNT /H208498,0/H20850 192/H2084964/H20850 7.0 11.9 129.0 19.4
CNT /H208494,4/H20850-CNT /H208498,0/H20850 256/H2084964/H2084164/H20850 7.2 12.4 121.5 21.0
CNT /H208495,0/H20850-CNT /H2084910,0 /H20850 300/H2084940/H2084180/H20850 24.7 31.5 113.3 22.6
CNT /H2084918,0 /H20850-CNT /H2084918,0 /H20850 576/H20849144/H20850 172.2 225.5 1362.2 253.3
CNTFET /H20849see Fig. 6/H20850 1440 /H20849160/H20850 259.8 286.9 4633.0 372.3EFFICIENT WA VE-FUNCTION MATCHING APPROACH FOR … PHYSICAL REVIEW B 79, 205322 /H208492009 /H20850
205322-3acterize these as right-going /H20849+/H20850or left-going /H20849−/H20850is well
described in the literature.16–18,28We note that only the ob-
tained propagating modes with /H20841/H9261k/H20841=1 are able to carry
charge deeply into the electrodes and thus enter the Landauerexpression in Eq. /H208492/H20850. The evanescent modes with /H20841/H9261
k/H20841/HS110051, on
the other hand, decay exponentially but can still contribute tothe current in a two-probe system, as the “tails” may reachacross the central region boundaries.
Consider a typical example of an electrode modes evalu-
ation. We look at a gold electrode with 27 atoms in the unitcell represented by 9 /H20849sp
3d5/H20850orbitals for each Au atom. Such
a system results in 243 right-going and 243 left-goingmodes. Figure 3/H20849a/H20850shows the positions in the complex plane
of the Bloch factors corresponding to the right-going modes/H20849i.e., /H20841/H9261
k/H20841/H113491/H20850for energy E=−1.5 eV. We see that there are
exactly three propagating modes which have Bloch factorslocated on the unit circle. The remaining modes are evanes-cent, of which many have Bloch factors with small magni-tude very close to the origin.
Figure 3/H20849b/H20850illustrates how the 243 left-going modes
would propagate through ten successive gold electrode unitcells. The figure shows that the amplitudes of the threepropagating modes are unchanged, while the evanescentmodes are decaying exponentially. In particular, we note thatthe evanescent modes with Bloch factors of small magnitude
are very rapidly decaying and vanishing in comparison to thepropagating modes after only a few layers. In the following,we will exploit this observation and attempt to exclude suchevanescent modes from the WFM calculation altogether. For-mally this can be accomplished if only the electrode modes
/H9278kwith Bloch factors /H9261ksatisfying
/H9261min/H11349/H20841/H9261k/H20841/H11349/H9261min−1/H2084910/H20850
are computed and subsequently taken into account, for a rea-
sonable choice of 0 /H11021/H9261 min/H110211. Equation /H2084910/H20850is adopted as
the key relation to select a particular subset of the availableelectrode modes /H20849as recently suggested in Ref. 17/H20850.
B. Extra electrode layers
We will denote the mode, Bloch, and self-energy matrices
from which the rapidly decaying evanescent modes are ex-
cluded with a tilde, i.e., as /H9021˜/H11006,B˜/H11006, and/H9018˜. The mode ma-
trices holding the excluded modes are denoted by a math-
ring accent /H9021˚/H11006, so that
/H9021/H11006=/H20851/H9021˜/H11006,/H9021˚/H11006/H20852/H20849 11/H20850
is the assumed splitting of the full set. All expressions to
evaluate the Bloch and self-energy matrices are unchanged
as given in Sec. II/H20851now /H20849/H9021˜/H11006/H20850−1merely represents the
pseudoinverses of/H9021˜/H11006/H20852. However, since the column spaces of
/H9021˜/H11006are not complete, there is no longer any guarantee that
WFM can be performed so that the resulting self-energy ma-
trices and, in turn, the solution /H9274C=/H20851/H92741T,...,/H9274nT/H20852Tof the lin-
ear system in Eq. /H208494/H20850, are correct. In addition, it is clear that
errors can occur in the calculation of tandrfrom Eqs. /H208498/H20850
and /H208499/H20850because the boundary wave functions /H92741and/H9274n
might not be fully represented in the reduced sets /H9021˜
R+and
/H9021˜
L−.
In order to diminish the errors introduced by excluding
evanescent modes, we propose to insert additional electrodelayers in the central region /H20849see Fig. 4/H20850. As illustrated in Sec.
III A , this would quickly reduce the imprint of the rapidly
decaying evanescent modes in the boundary layer wave
functions
/H9274˜1and/H9274˜n, which means that the critical compo-
nents outside the column spaces /H9021˜/H11006become negligible at an
exponential rate in terms of the number of additional layers.We emphasize that the inserted layers may be “fictitious” inthe sense that they can be accommodated by simple block-0.20.40.60.81
Re{λ}Im{λ}
0π
2
π
3
2π(a)
1.0
0.5
0.0
-0.5-1.001 23456789 1 0Re{λl}
l(layers )(b)
FIG. 3. /H20849Color online /H20850/H20849a/H20850Positions of the Bloch factors
/H9261k/H20849/H20841/H9261k/H20841/H113491/H20850obtained for a bulk Au /H20849111/H20850electrode with 27 atoms per
unit cell at E=−1.5 eV. /H20849b/H20850Amplitudes of the corresponding nor-
malized electrode modes /H9278kmoving through ten layers of the ideal
bulk electrode. A total of 243 modes are shown of which three arepropagating /H20849colored/dashed /H20850and the rest are evanescent
/H20849circles/black /H20850.LCR
z
}|
{z
}|
{z
}|
{
ψ0ψ(l)
1ψ(0)
1ψ2 ψ(l)
n ψ(0)
n ψn+1 ψn−1 ··· ··· ···
|
{z
}|
{z
}
lextra layers lextra layers
FIG. 4. Two-probe system in which the Cregion boundaries are
expanded by lextra electrode layers.SØRENSEN et al. PHYSICAL REVIEW B 79, 205322 /H208492009 /H20850
205322-4Gaussian eliminations prior to the solving of Eq. /H208494/H20850for the
original system.
The above statements are confirmed by the following
analysis. We expand the electrode wave functions in the cor-responding complete set of bulk modes
/H9274i/H11006=/H9021/H11006ai/H11006=/H20851/H9021˜/H11006,/H9021˚/H11006/H20852/H20875a˜i/H11006
a˚i/H11006/H20876, /H2084912/H20850
where ai/H11006=/H20851a˜i/H11006T,a˚i/H11006T/H20852Tare vectors that contain the expansion
coefficients. In the particular case, where lextra electrode
layers are inserted and the border layers of the Cregion are
identical to the connecting electrode layers, the electrodewave functions entering the matching boundary equationswill be
/H92741/H20849l/H20850−=/H20849BL−/H20850−l/H92741−=/H20851/H9021˜
L−,/H9021˚
L−/H20852/H20875/H20849/H9011˜
L−/H20850−la˜1−
/H20849/H9011˚
L−/H20850−la˚1−/H20876 /H2084913/H20850
and
/H9274n/H20849l/H20850+=/H20849BR/H11006/H20850l/H9274n+=/H20851/H9021˜
R+,/H9021˚
R+/H20852/H20875/H20849/H9011˜
R+/H20850la˜n+
/H20849/H9011˚
R+/H20850la˚n+/H20876 /H2084914/H20850
using the definition B/H11006=/H9021/H11006/H9011/H11006/H20849/H9021/H11006/H20850−1. This shows that the
critical components outside the column spaces of /H9021˜
L/H11006and
/H9021˜
R/H11006are given by coefficients /H20849/H9011˚
L−/H20850−la˚1−and /H20849/H9011˚
R+/H20850la˚n+, respec-
tively. If this set only consists of the most rapidly decayingof the evanescent modes according to Eq. /H2084910/H20850, that is, /H20841/H9261
k/H20841
/H11022/H9261min−1for the diagonal elements of /H9011˚
L−and /H20841/H9261k/H20841/H11021/H9261 minfor the
diagonal elements of /H9011˚
R+, where /H9261minis less than 1, these
coefficients always decrease as a function of l.
We conclude that WFM with the reduced set of modes
approaches the exact case if additional electrode layers are
inserted and the solution /H9274˜Cobtained from Eq. /H208494/H20850ap-
proaches the correct solution /H9274Caccordingly.
C. Accuracy
As pointed out above, the exclusion of some of the eva-
nescent modes from the mode matrices /H9021/H11006will introduce
errors because the column spaces in /H9021˜/H11006are incomplete. In
this section we will estimate how this will influence the ac-curacy of the calculated transmission and reflection coeffi-cients in terms of the parameter /H9261
minand the number lof
extra electrode layers.
Consider first the accuracy of the transmission matrix tin
the case of the extended two-probe system in Fig. 4. For a
specific incoming mode k, we compare the correct result ob-
tained with the complete set of modes /H20851cf. Eq. /H208498/H20850/H20852,
tk=/H20875t˜k
t˚k/H20876=/H20851/H9021˜
R+,/H9021˚
R+/H20852−1/H9274n/H20849l/H20850+, /H2084915/H20850
to the result obtained with the reduced mode matrix /H20849denoted
by a prime /H20850,tk/H11032=/H20875t˜k/H11032
0˚/H11032/H20876=/H20851/H9021˜
R+,0˚/H20852−1/H9274n/H20849l/H20850+, /H2084916/H20850
where 0˚/H11032represents the zero vector of size m˚Rand0˚the zero
matrix of size mR/H11003m˚R.
The important coefficients in tkandtk/H11032for transmission
calculations are the ones representing the Bloch modeswhich enter the Landauer-Büttiker formula in Eq. /H208492/H20850. Since
these are never excluded they will always be located within
the first m
˜Relements, i.e., in t˜kandt˜k/H11032. It then suffices to
compare these parts of the transmission matrix which we cando as follows.
From the properties of the pseudoinverse we are able to
write the relation
/H20849/H9021˜
R+/H20850−1/H20851/H9021˜
R+,/H9021˚
R+/H20852=/H20851I˜,/H20849/H9021˜
R+/H20850−1/H9021˚
R+/H20852, /H2084917/H20850
where I˜is the identity matrix of order equal to the number of
included modes m˜R. Using the expression in Eq. /H2084914/H20850it then
follows that
t˜k=/H20849/H9011˜
R+/H20850la˜n+ /H2084918/H20850
and
t˜k/H11032=t˜k+/H20849/H9021˜
R+/H20850−1/H9021˚
R+/H20849/H9011˚
R+/H20850la˚n+, /H2084919/H20850
where the t˜k/H11032expression clearly corresponds to the correct
coefficients t˜kplus an error term.
We have already established in Sec. III B that the /H20849/H9011˚
R+/H20850la˚n+
factor in the error term will decrease as a function of l.W e
now show that the other term, /H20849/H9021˜
R+/H20850−1/H9021˚
R+is independent of l,
and consequently, that the error term in Eq. /H2084919/H20850must de-
crease as a function of l. To this end we look at the two-norm
of/H20849/H9021˜
R+/H20850−1/H9021˚
R+, which satisfies
/H20648/H20849/H9021˜
R+/H20850−1/H9021˚
R+/H206482/H11349m˚R1/2/H20648/H20849/H9021˜
R+/H20850−1/H206482, /H2084920/H20850
since /H20648/H9021˚
R+/H206482/H11349m˚R1/2when all evanescent modes are assumed
to be normalized. The norm /H20648/H20849/H9021˜
R+/H20850−1/H206482can be readily evalu-
ated and depends on the set of modes included via the pa-rameter /H9261
minbut not on l. Thus, we conclude that the only
term of Eq. /H2084919/H20850which depend on lis/H20849/H9011˚
R+/H20850la˚n+, and the error
is therefore decreasing as function of l.
Writing Eq. /H2084919/H20850ast˜k/H11032=t˜k+/H9280˜k, where /H9280˜kholds the errors
on the coefficients of the kth column, we further obtain that
the total transmission T/H11032can be expressed as
T/H11032=T+/H20858
kk/H11032/H20849t˜kk/H11032/H11569/H9280˜kk/H11032+/H9280˜kk/H11032/H11569t˜kk/H11032+/H20841/H9280˜kk/H11032/H208412/H20850, /H2084921/H20850
where Tis the exact result and the summation is over the
Bloch modes kandk/H11032in the left and right electrodes, respec-
tively.
For a first-order estimate of the error term in Eq. /H2084921/H20850we
consider the worst case approximation, where all diagonal
elements of /H9011˚
R+are equal to the maximum range /H9261minof Eq.
/H2084910/H20850. This makes all elements /H9280˜kk/H11032proportional to /H9261minland
we arrive at the simple relationEFFICIENT WA VE-FUNCTION MATCHING APPROACH FOR … PHYSICAL REVIEW B 79, 205322 /H208492009 /H20850
205322-5/H20841T/H11032−T/H20841/H11011/H9261minl+O/H20851/H20849/H9261minl/H208502/H20852, /H2084922/H20850
which shows that the error decreases exponentially in terms
of the number of extra layers l.
For a higher-order estimate of the error, we directly moni-
tor the error arising on the boundary conditions in terms of
the coefficient vectors b˜L,k/H11013/H20849/H9021˜
R+/H20850−1/H20849/H92741/H20849l/H20850+−/H9261L,k+/H9278L,k+/H20850and
b˜R,k/H11013/H20849/H9021˜
R−/H20850−1/H9274n/H20849l/H20850−, where /H92741/H20849l/H20850+and/H9274n/H20849l/H20850−are given by solving
Eq. /H208494/H20850. When the boundary conditions are exactly satisfied,
we have /H20841b˜L,k/H20841=0 and /H20841b˜R,k/H20841=0. In the case where the bound-
ary conditions are not exactly satisfied, b˜R,krepresents the
error on the left-going components within the right boundary
layer in the same way that /H9280˜krepresents the error on the
right-going /H20849transmitted /H20850components. We would therefore
expect the same orders of magnitude of /H20841b˜R,k/H20841and /H20841/H9280˜k/H20841in an
actual calculation for a given mode k. This suggests the fol-
lowing error estimate from Eq. /H2084921/H20850:
/H20841T/H11032−T/H20841/H11349/H20858
k/H208492/H20841t˜k/H20841/H20841/H9280˜k/H20841+/H20841/H9280˜k/H208412/H20850/H11011/H20858
k/H208492/H20841t˜k/H20841/H20841b˜R,k/H20841+/H20841b˜R,k/H208412/H20850,
/H2084923/H20850
where all the vector norms /H20849e.g., /H20841t˜k/H208412=/H20858k/H11032/H20841t˜kk/H11032/H208412/H20850are assumed
to be taken over the elements corresponding to Bloch bulk
modes k/H11032only.
Finally, we note without explicit derivation that similar
arguments for the reflection matrix with columns r˜k/H11032
=/H20849/H9021˜
L−/H20850−1/H20849/H92741/H20849l/H20850−−/H9261L,k+/H9278L,k+/H20850and the total reflection coefficient
R/H11032result in the same accuracy expressions for /H20841R/H11032−R/H20841if we
substitute t˜k→r˜kandb˜R,k→b˜L,kin Eqs. /H2084922/H20850and /H2084923/H20850.D. Example
To end this section, we exemplify the previous discussion
quantitatively by looking at the Au /H20849111/H20850electrode described
earlier and assuming a 128-atom /H208494 unit cells /H20850device of zig-
zag /H208498,0/H20850carbon nanotube /H20849CNT /H20850sandwiched between the
gold electrodes /H20849see the configuration in Fig. 1/H20850. For energy
E=−1.5 eV, we have calculated the deviation between the
total transmission obtained when all bulk modes are takeninto account /H20849T/H20850and when some evanescent modes are ex-
cluded /H20849T
/H11032/H20850as specified with different settings of /H9261min. De-
viations are also determined for the corresponding total re-
flection coefficients /H20849RandR/H11032/H20850. Figure 5shows the results as
a function of l, together with the estimate /H9261minlof Eq. /H2084922/H20850
and the estimate of Eq. /H2084923/H20850both for the transmission and
reflection coefficients, where the higher order terms havebeen neglected.
We observe that the absolute error in the obtained trans-
mission coefficients /H20849red curves /H20850and reflection coefficients
/H20849blue curves /H20850is generally decreasing as a function of l, fol-
lowing the same convergence rate as /H9261
minl/H20849dashed line /H20850.
Looking closer at results for neighbor lvalues, we see that
the errors initially exhibit wavelike oscillations. This is di-rectly related to the wave form of the evanescent modes thathave been excluded /H20851see the propagation of the slowest de-
caying black curves in Fig. 3/H20849b/H20850/H20852. In other words, although
the norm of the errors /H20841
/H9280˜k/H20841are decreasing as a function of l,
the specific error /H9280˜kk/H11032on a given /H20849large /H20850coefficient of t˜kk/H11032/H11032or
r˜kk/H11032/H11032may increase, which means that the overall error term in
Eq. /H2084921/H20850can go up. Fortunately this is only a local phenom-
enon with the global trend being rapidly decreasing errors.
Consider also the quality of the simple accuracy estimate
of/H9261minland the estimates expressed by Eq. /H2084923/H20850for the trans-10−610−610−6
10−410−410−4
10−210−210−2
100100100
0123456789 1 0Error Error Error
l(layers)λmin=0.5
λmin=0.3
λmin=0.1
(λmin)l|˜T−T|
|˜R−R|
Eq. (23; ˜bR,k)
Eq. (23; ˜bL,k)FIG. 5. /H20849Color online /H20850Error /H20849absolute /H20850in the
calculated total transmission /H20849circles/solid red
lines /H20850and reflection /H20849squares/solid blue lines /H20850co-
efficients T/H11032andR/H11032as a function of l. The panels
show the cases of /H9261minset to 0.5, 0.3, and 0.1,
which corresponds to 3, 14, and 31 Au bulkmodes /H20849out of 243, see Fig. 3/H20850taken into account,
respectively. Dashed line indicates the first-ordererror estimate /H9261
minl. The upward-pointing and
downward-pointing triangles /H20849green and yellow
lines /H20850show error estimates obtained from Eq.
/H2084923/H20850.SØRENSEN et al. PHYSICAL REVIEW B 79, 205322 /H208492009 /H20850
205322-6mission coefficients /H20849green curves /H20850and reflection coefficients
/H20849yellow curves /H20850, respectively. For relatively large /H9261minall es-
timates are very good. However, for smaller values of /H9261min,
only the latter two retain a high quality while the /H9261minlesti-
mate tends to be overly pessimistic. It is important to remem-ber that these estimates are by no means strict conditions butin practice give very reasonable estimates of the accuracy.
We note in passing that the results in the top panel of Fig.
5correspond to using only the propagating Bloch modes in
the transmission calculation. Still we are able to compute T
andRto an absolute accuracy of three digits by inserting 2
/H110035 extra electrode layers in the two-probe system. This is
quite remarkable and shows promise for large-scale systems,e.g., with nanowire electrodes, for which the total number ofevanescent modes available becomes exceedingly great.
IV . APPLICATION
In this section we will apply the developed method to a
nanodevice consisting of a CNT stretched between to twometal electrodes and controlled by three gates. The setup isinspired by Appenzeller et al.
20and we expect this particular
arrangement to be able to display so-called band-to-band/H20849BTB /H20850tunneling, where one observes gate-induced tunneling
from the valence band into the conduction band of a semi-conducting CNT and vice versa.
We show the configuration of the two-probe system in
Fig.6. The device configuration contains ten principal layers
of a CNT /H208498,4/H20850, having 112 atoms in each layer. The diameter
of the tube and the thickness of the principal layer are 8.3 Åand 11.3 Å, respectively. The electrodes consist of CNT /H208498,4/H20850
resting on a thin surface of Li, where the lattice constant ofthe Li layers is stretched to fit the layer thickness of the CNT.The central region of the two-probe system comprises a totalof 1440 atoms. An arrangement of rectangular gates is posi-tioned below the carbon nanotube as indicated on the figure.In the plane of the illustration /H20849length/H11003height /H20850the dimen-
sions are as follows: dielectric 106 /H110035Å
2; gate A 20
/H110035Å2; gate B 50 /H110035Å2. We set /H9280=4 for the dielectric
constant of the dielectric in order to simulate SiO 2or Al 2O3
oxides. All the regions are centered with respect to the elec-trodes so that the complete setup has mirror symmetry in thelength direction. In the direction perpendicular to the illus-tration the configuration is assumed repeated every 19.5 Åas a supercell.
We have obtained the density matrix of the BTB device
by combining the nonequilibrium Green’s function formal-ism with a semiempirical extended Hückel model /H20849EHT /H20850us-
ing the parameterization of Hoffmann.
29From the density
matrix we calculate Mulliken populations on each atom andrepresent the total density of the system as a superposition of
Gaussian distributions on each atom properly weighted bythe Mulliken population. The width of the Gaussian is cho-sen to be consistent with CNDO parameters.
30The electro-
static interaction between the charge distribution and the di-electrics and gates is subsequently calculated. The Hartree-type term is then included in the Hamiltonian and thecombined set of equations are solved self-consistently. Theresulting self-consistent EHT model is closely related to thework of Ref. 30, and a detailed description of the model will
be presented elsewhere.
31
In order to adjust the charge transfer between the CNT
and the Li electrodes we add the term /H9254/H9280S to the Li param-
eters. With an appropriate adjusted value of /H9254/H9280, the carbon
nanotube becomes n-type doped. We adjust the value such
that the average charge transfer from Li to the nanotube atself-consistency is 0.002 eper carbon atom in the electrode.
The Fermi energy is then located at −4.29 eV, which is 0.07eV below the conduction band of the CNT /H208498,4/H20850.
In the following we fix V
gate A =−2.0 eV and vary the
gate B potentials in the range /H20851−2–4 eV /H20852. Note that we re-
port the gate potentials as an external potential on the elec-trons, and to translate the values into a gate potential of unitvolts the values must be divided with − e.
In the left part of Fig. 7we present the total self-
consistent potential induced by the three gates on the carbonatoms in the CNT over the full extension of the device. Foreach configuration of the gate potentials the electrostatic po-tential is shown twice, i.e., by two curves with the samecolor displaced relative to each other with the energies of thevalence-band and conduction-band edges, respectively. Inthis way the curves not only represent the electrostatic po-tential of the device but also the position of the valence- andconduction-band edges.
Along with this, in the right part of Fig. 7, we show the
corresponding transmission spectrum T/H20849E/H20850for four gate po-
tentials V
gate B =−2.0, 1.0, 2.0, and 4.0 eV . When Vgate B
=−2.0 eV the nanotube is largely unperturbed by the gate
and the transmission coefficient is close to an ideal /H208498,4/H20850
CNT. We note that this is in agreement with ab initio calcu-
lations by Nardelli et al. ,32which found that a two terminal
/H208495,5/H20850CNT device in a similar contact geometry showed a
nearly ideal conductance spectrum. In addition, the calcu-lated band gap of the /H208498,4/H20850nanotube is 0.81 eV , which is in
good agreement with the value of 0.96 eV obtained from ab
initio density-functional calculations in the generalized gra-
dient approximation.
33
From Fig. 7we see how the bands are shifted upwards by
an increasing amount as the gate B potential is turned up. Tobegin with, e.g., for V
gate B =1 eV, this results in lower con-
duction since the conduction band bends away from the
FIG. 6. /H20849Color online /H20850Schematic illustration of a carbon nanotube /H208498,4/H20850band-to-band tunneling device. The carbon nanotube is posi-
tioned on Li surfaces next to an arrangement of three gates.EFFICIENT WA VE-FUNCTION MATCHING APPROACH FOR … PHYSICAL REVIEW B 79, 205322 /H208492009 /H20850
205322-7Fermi-level and the Fermi-energy electrons need to tunnel
through the central region. When the gate voltage is atV
gate B =2 eV, the valence band almost reaches the conduc-
tion band in which case BTB tunneling becomes possible.By increasing the gate voltage further, more bands becomeavailable for BTB tunneling and the effect is visible as asteady increase in the calculated transmission T/H20849E/H20850just
above the Fermi level.
The results for the Fermi-level transmission T/H20849E
F/H20850corre-
sponding to the T=0 K unit conduction G0are displayed
with the black curve in Fig. 8. It shows an initial conduc-
tance for Vgate B =−2.0 V of the order of one, a subsequent
drop by 4 orders of magnitude around Vgate B =2.0 V, and a
final increase of 1 order of magnitude toward Vgate B
=4.0 V. We also display the results for the room-
temperature T=300 K conductance /H20849red curve /H20850, which can
be obtained from
G=/H20885dET /H20849E/H20850e/H20849E−EF/H20850/kBT
/H208491+e/H20849E−EF/H20850/kBT/H208502. /H2084924/H20850
The two conduction curves are similar, showing that the de-
vice is operating in the tunneling regime rather than the ther-mal emission regime.We next briefly comment on the comparison of the simu-
lation to the experiment of Appenzeller et al.
20In both cases
the conduction curves have two branches, which we denotefield emission /H20849FE/H20850and BTB. Initially, the conduction de-
creases with applied gate potential due to the formation of abarrier in the central region: this is the FE regime. For largerbiases the conduction increases again due to BTB tunneling,this is the BTB regime. The experimental device displaysthermal emission conduction and shows a correspondingsubthreshold slope, S,o f k
BTln/H2084910/H20850/e/H1101560 mV /dec in the
FE regime. The theoretical device, on the other hand, dis-plays tunneling conduction and has S/H11015500 mV /dec in the
FE regime. In the BTB regime, the theoretical device has S
/H110152000 mV /dec, while the experimental device shows S
/H1101540 mV /dec.
The very different behavior is due to the short channel
length of the theoretical device. The central barrier has alength of /H110155 nm and at this length the electron can still
tunnel through the barrier. We see that the short channellength not only affects the subthreshold slope of the FE re-gime, but also strongly influences the BTB regime. Worksare in progress for a parallel implementation of the method-ology, which will make it feasible to simulate larger systemsand thereby investigate the transition from the tunneling tothe thermal emission regime.
All the above results have been calculated with the modi-
fied WFM method using parameters /H9261
min=0.1 and l=1.
Thus, the results present a nontrivial application of themethod. To verify the transmission results in Fig. 7we
present a comparison to the standard WFM method in Fig. 9.
The figure shows that the transmissions curves are identicalto about three significant digits. The CPU time required forcalculating a complete transmission spectrum for Fig. 7is
/H20849/H110113h/H20850, while the corresponding calculation presented in
Fig. 9with the standard WFM method took /H20849/H1101135 h /H20850. Thus,
the overall time saving achieved with the method was there-fore more than an order of magnitude. The results in Table I
indicate that similar time savings can be expected for othersystems with nontrivial electrodes.−5−4−3
10−410−310−210−1100101 -5.5-4.5-3.5
0 2 6 8 10 12 4T(E) Channel length [nm ]Energy [eV]
VGate−B=−2.0V
VGate−B=1 .0V
VGate−B=2 .0V
VGate−B=4 .0V
FIG. 7. /H20849Color online /H20850/H20849Left panel /H20850Representation of the electrostatic induced shift of the valence- and conduction-band edges along the
length of the device for gate potentials Vgate B =−2.0, 1.0, 2.0, and 4.0 eV . /H20849Right panel /H20850The corresponding transmission spectrum. Dotted
line shows the position of the Fermi level and the solid line shows the transmission coefficient for an ideal CNT /H208498,4/H20850.
010−410−310−210−1100101
-2 -1 1 2 3T=0 K
T = 300K
4
Gate potential [ eV]Conduction [G 0]
FIG. 8. /H20849Color online /H20850Conduction in units of the conductance
quantum G0as a function of the gate B potential. In the calculations
we use a dielectric constant of 4, Vgate A =−2.0 eV, and vary Vgate B
from −2.0 to 4.0 eV as indicated.SØRENSEN et al. PHYSICAL REVIEW B 79, 205322 /H208492009 /H20850
205322-8V . SUMMARY
We have developed an efficient approach for calculating
quantum transport in nanoscale systems based on the WFMscheme originally proposed by Ando.
16In the standard
implementation of the WFM method for two-probe systems,all bulk modes of the electrodes are required in order torepresent the transmitted and reflected waves in a completebasis. By extending the central region of the two-probe sys-
tem with extra electrode principal layers, we are able to ex-clude the vast majority of the evanescent bulk modes fromthe calculation altogether. Our final algorithm is thereforehighly efficient, and most importantly, errors and accuracycan be closely monitored.
We have applied the developed WFM algorithm to a
CNTFET in order to study the mechanisms of band-to-bandtunneling. The setup was inspired by Ref. 20, and the calcu-
lations display features that are also observed in the experi-ment. However, due to the short channel length the theoret-ical device operates in the tunneling regime, while theexperimental device operates in the thermal emission regime.
By measuring the CPU times for calculating transmission
spectra of the CNTFET two-probe system and comparing tothe cost of the standard WFM method we have observed aspeed up of more than a factor of 10. We see similar speedup for other nontrivial systems. We therefore believe that thisis an ideal method to be used with ab initio transport
schemes for large-scale simulations.
ACKNOWLEDGMENTS
This work was supported by the Danish Council for Stra-
tegic Research /H20849NABIIT /H20850under Grant No. 2106–04–0017,
“Parallel Algorithms for Computational Nano-Science.”
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mark, 2008.
22Bloch’s theorem /H20849Ref. 24/H20850/H9274i=/H9261k/H9274i−1for the ideal electrodes
defines the phase factors /H9261k/H11013eıqkd, where qkis the complex
wave number and dis the layer thickness, which are referred to
as Bloch factors throughout this paper.
23When using the Landauer formula in Eq. /H208491/H20850it is assumed that
the electrode Bloch modes carry unit current in the conductiondirection. This can be conveniently accommodated by flux nor-malizing the Bloch modes, i.e.,
/H9278L,k/H11006→/H20849dL/vL,k/H11006/H208501/2/H9278L,k/H11006,i nt h e
case of the left electrode /H20849Ref. 34/H20850.
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27We should point out that the metallic electrodes in the two-probe
systems considered in Table Ican be fully described by much
smaller unit cells than indicated /H20849often only a few atoms areT(E)
Energy [eV]T
˜T
|˜T−T|
10−1010−910−810−710−610−510−410−310−210−1100101
-5.5 -5 -4.5 -4 -3. 5
FIG. 9. /H20849Color online /H20850Transmission coefficients TandT˜calcu-
lated with the standard WFM method /H20849black solid /H20850and the method
of this work /H20849red dashed /H20850, respectively, and the difference /H20841T˜−T/H20841
/H20849blue line /H20850as a function of energy Ein the Vgate B =2 V case.EFFICIENT WA VE-FUNCTION MATCHING APPROACH FOR … PHYSICAL REVIEW B 79, 205322 /H208492009 /H20850
205322-9needed /H20850and therefore the time spent on computing the bulk
modes can be vastly reduced in these specific cases. For a gen-eral method, however, which supports CNTs, nanowires, etc., aselectrodes, the timings are appropriate for showing the overalltrend in the computational costs.
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S. Datta, J. Chem. Phys. 123, 064707 /H208492005 /H20850.
31K. Stokbro /H20849unpublished /H20850.
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205322-10 |
PhysRevB.82.115314.pdf | Three-terminal thermoelectric transport through a molecular junction
O. Entin-Wohlman,1,2,*,†Y . Imry,3and A. Aharony1,†
1Department of Physics and the Ilse Katz Center for Meso- and Nano-Scale Science and Technology, Ben Gurion University,
Beer Sheva 84105, Israel
2Albert Einstein Minerva Center for Theoretical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
3Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
/H20849Received 27 May 2010; revised manuscript received 31 August 2010; published 20 September 2010 /H20850
The thermoelectric transport through a molecular bridge is discussed with an emphasis on the effects of
inelastic processes of the transport electrons caused by the coupling to the vibrational modes of the molecule.In particular, it is found that when the molecule is strongly coupled to a thermal bath of its own, which may beat a temperature different from those of the electronic reservoirs, a heat current between the molecule and theelectrons can be converted into an electric current. Expressions for the transport coefficients governing thisconversion and similar ones are derived, and a possible scenario for increasing their magnitudes is outlined.
DOI: 10.1103/PhysRevB.82.115314 PACS number /H20849s/H20850: 85.65. /H11001h, 73.63.Kv, 65.80. /H11002g
I. INTRODUCTION
The investigation of thermoelectric phenomena in nano-
scale devices at low temperatures has several interesting as-pects. From the practical point of view, it is important tounderstand the heat flow and the dissipation because the heatgenerated by electric potentials used to switch-on transportcurrents inevitably induces decoherence in the quantumfunctioning of the device and also leads to dissipation. Inbulk conductors, thermoelectric transport necessitates anasymmetry between holes and electrons, which is usuallysmall. In mesoscopic structures, this asymmetry may befairly high and can be also controlled experimentally. Onewould hence like to have a full picture of the symmetries andthe inter-relations dominating the various transport coeffi-cients of a small mesoscopic system, in particular, the effectsof inelastic processes. Indeed, when transport is through amolecular bridge, the tunneling electrons may undergo in-elastic collisions with the vibrational modes even in thelinear-response regime. This is because at finite tempera-tures, the transport electrons may excite or deexcite thephonons and thus exchange energy with them. These inelas-tic processes modify the electronic transport coefficients,leading to the question of what, if any, are the analogs of the/H20849bulk /H20850Onsager-Casimir relations. Another intriguing issue is
the possibility to convert heat from the vibrations into anelectric current between the electronic reservoirs or viceversa.
Early studies of thermoelectric transport coefficients of
microstructures were based on the Landauer approach
1–4
which was also extended to include mesoscopic
superconductors.5Once feasibility of measuring thermal and
thermoelectric transport in atomic-scale samples had beenestablished,
6mesoscopic thermoelectric phenomena, e.g.,
peaks in the thermopower of a point contact orchestratedwith the transitions among plateaux of the quantizedconductance,
7or oscillations /H20849as a function of a gate voltage /H20850
in the same coefficient measured on a quantum dot,8–10were
detected and analyzed.2,11The thermopower measured on
nanotubes was found to be unexpectedly high, and this wasattributed to a broken electron-hole symmetry.
12,13Similarly,nanotubes exhibited enhanced thermal conductivity,14as did
also silicon nanowires.15The dependence of the thermoelec-
tric response on the length of the atomic chain has beenrecently computed within density-functional theory.
16Being
based on the Landauer approach, the above-mentioned theo-retical studies mainly focused on elastic processes of thetransport electrons. Later on, effects of inelastic electron-electron processes and electronic correlations /H20849increasingly
important at lower temperatures /H20850, as well as that of an ap-
plied magnetic field, on the thermopower produced in large
17
and single-level18quantum dots, and also in quantum wires19
were considered. The effect of attractive electronic interac-tions on the thermopower was considered in Ref. 20.
Inelasticity of electronic processes should play a signifi-
cant role in thermoelectric transport through molecularbridges, also in the nonlinear regime.
21Indeed, a density-
functional computation of the nonlinear differential conduc-tance of gold wires attributed changes in the I-Vcharacter-
istics to phonon heating,
22,23and the thermopower coefficient
was proposed as a tool to monitor the excitation spectrum ofa molecule forming the junction between two leads.
24,25It
was suggested that the Seebeck effect in such bridges can beused for converting heat into electric energy,
26and to deter-
mine the location of the Fermi level of the transport electronsrelative to the molecular levels, and also the sign of thedominant charge carriers, either for a molecularconductor
27–29or for an atomic chain.30,31This was con-
firmed experimentally: the Seebeck coefficient as measuredby scanning tunnel microscope yielded that in the case of thebenzenedithiol family sandwiched between two gold elec-trodes the charge carriers are holes passing through the high-est occupied molecular orbital, whose location with respectto the metal Fermi level was determined from the magnitudeof the coefficient.
32
Inelastic electron-vibration interactions are not included
in several of the theoretical studies devoted to molecularjunctions /H20849see, e.g., Refs. 16,27, and 30/H20850or are treated at
off-resonance conditions, expanding them in the molecular-lead coupling.
24When these interactions are ignored, the
transport coefficients have the same functional form as inbulk conductors, with the energy-dependent transmission co-efficient and its derivative replacing the conductivity.
16Not-PHYSICAL REVIEW B 82, 115314 /H208492010 /H20850
1098-0121/2010/82 /H2084911/H20850/115314 /H208499/H20850 ©2010 The American Physical Society 115314-1withstanding the relative smallness, often, of the inelastic
corrections to the thermoelectric transport, their study is stillof interest because of fundamental questions related to thesymmetries of the conventional transport coefficients, andsince they give rise to additional coefficients connecting theheat transport in between the electrons and the phonons.
Here we study the heat and charge transport in a small
mesoscopic /H20849or nanometric /H20850system depicted schematically
in Fig. 1: a molecule attached to two electronic /H20849no phonons /H20850
reservoirs, held in general at different temperatures, T
L,R, and
at different chemical potentials, /H9262L,R. We distinguish be-
tween two /H20849extreme /H20850situations. In the first, the molecule is
“floating” and is attached solely to the leads; then the vibra-tion population is determined by the transport electronsalone. In that case, the system is a two-terminal junction. Inthe second case, the molecule is coupled to its own /H20849typically
a phonon /H20850heat bath which is kept at the temperature T
P,
making the system a three-terminal one. It is then assumedimplicitly that the coupling of the molecule to that heat bathlargely exceeds its coupling to the transport electrons. Thelatter is determined by our small parameter, the couplingbetween the molecule vibrations and the transport electrons,
/H9253. Thus, we assume that the relaxation time due to the cou-
pling to the heat bath, /H9270P, is short on the scale /H90032//H20849/H92532/H92750/H20850/H20851see
Eq. /H20849A18 /H20850in the Appendix /H20852,/H9003being the level width on the
molecule, due to the coupling with the leads and /H92750is the
frequency of the vibrations. /H6036//H9270Pmay still be very small on
all other physical scales, such as /H6036/H92750and/H9003. The phonon
bath may be realized simply by an electronically insulatinghard substrate /H20849assuming that the large Kapitsa-type phonon
thermal resistance between the lead and the sample is largeenough to sufficiently reduce the thermal contact of the mol-ecule to the substrate via the leads /H20850, or a piece of such ma-
terial touching the junction, each of those held at a tempera-ture T
P. A vacuum gap between the two separate substrates
for the two leads would be ideal. However, with presentfabrication technology, this appears possible for a quantumdot but not for a small molecule.
The consideration of the entropy production of such a
three-terminal system is quite illuminating. Using the ther-modynamic identity TdS=dE−
/H9262dN,33one finds that the dis-
sipation at the left /H20849right /H20850reservoir leads toS˙L/H20849R/H20850=1
TL/H20849R/H20850/H20849E˙L/H20849R/H20850−/H9262L/H20849R/H20850N˙L/H20849R/H20850/H20850. /H208491/H20850
Here, − E˙L/H20849R/H20850is energy current emerging from the left /H20849right /H20850
reservoir while − N˙L/H20849R/H20850is the particle current leaving the left
/H20849right /H20850reservoir. Adding to Eq. /H208491/H20850the entropy production of
the phonon heat bath, S˙P=E˙P/TP, where − E˙Pis the energy
current leaving that bath, yields the total dissipation of thesystem,
S˙
P+S˙L+S˙R=E˙P
TP+1
TL/H20849E˙L−/H9262LN˙L/H20850+1
TR/H20849E˙R−/H9262RN˙R/H20850.
/H208492/H20850
Charge conservation implies that
N˙L+N˙R=0 /H208493/H20850
while energy conservation requires
E˙L+E˙R+E˙P=0 . /H208494/H20850
In the linear-response regime all three temperatures /H20849see Fig.
1/H20850are only slightly different,
TL/H20849R/H20850=T/H11006/H9004T
2,
TP=T+/H9004TP, /H208495/H20850
and the chemical potentials differ by a small amount,
/H9262L/H20849R/H20850=/H9262/H11006/H9004/H9262
2. /H208496/H20850
Expanding Eq. /H208492/H20850and using Eqs. /H208493/H20850and /H208494/H20850yields
S˙P+S˙L+S˙R=/H9004TP
T2/H20849−E˙P/H20850+/H9004/H9262/e
TI+/H9004T
T2IQ, /H208497/H20850
where Iis the net charge current flowing from the left reser-
voir to the right one,
I=−e
2/H20849N˙L−N˙R/H20850/H20849 8/H20850
while IQis the net heat current carried by the electrons,
IQ=IE−/H20849/H9262/e/H20850Iwith IE=−1
2/H20849E˙L−E˙R/H20850. /H208499/H20850
Finally, the heat current flowing from the phonon bath to the
quantum system is simply given from the condition of en-ergy conservation,
−E˙
P=E˙L+E˙R. /H2084910/H20850
Thus, the entropy production of our three-terminal system is
a simple example of the general expressions for linear trans-port, consistent with the Onsager theory.
34
Since our molecular bridge is not necessarily at equilib-
rium within the transport process, it exchanges energy withthe phonons of the phonon reservoir by going up/down in theTp
TL
ΜLTR
ΜR
FIG. 1. /H20849Color online /H20850A three-terminal system, modeled by a
resonant level attached to two electronic reservoirs, having differentchemical potentials and temperatures
/H9262L,Rand TL,R, respectively.
An electron residing on the level interacts with its vibrationalmodes. The population of these phonons can be determined by thetransport electrons /H20849a “floating molecule” /H20850or by a coupling to a
phonon source kept at temperature T
P.ENTIN-WOHLMAN, IMRY , AND AHARONY PHYSICAL REVIEW B 82, 115314 /H208492010 /H20850
115314-2vibrational ladder with absorbing/emitting a phonon in the
bath. This is the physical origin of the current − E˙P.O nt h e
other hand, when the molecule is floating, then − E˙Pvanishes.
Since − E˙Pis proportional to the rate of change in the vibra-
tional level population on the dot /H20849see the Appendix for de-
tails /H20850this in turn will determine the vibration population that
will adjust itself according to the temperature and chemicalpotential differences applied to the electrons. In this situationour device becomes a two-terminal one, and the energy cur-rent carried by the electrons is conserved.
In Sec. II, we outline our model, and give explicit expres-
sions for all three currents I,I
Q, and − E˙P, and in Sec. IIIwe
discuss them in the linear-response regime. In particular, wefind there that by the three-terminal junction, one may con-vert the heat current from the phonon bath into electric andheat currents carried by the electrons even at zero bias volt-age and when T
L=TR. In Sec. IV, we discuss the necessary
conditions for this conversion to be established, i.e., the junc-tion couplings to the electronic reservoirs should not be spa-tially symmetric and should not depend on the energy in anidentical manner. We show that an opposite dependence onenergy of the couplings to two electron reservoirs will tendto maximize the new transport coefficients we find.
II. CURRENTS
In our analysis, the molecular bridge is replaced by a
single resonant level; when a transport electron resides onthe level, it interacts /H20849linearly /H20850with the phonons. Such a
model, which neglects effects of spin and electronic correla-tions, is applicable in the Coulomb-blockade regime for low-energy molecular levels. /H20849We also ignore the possibility of
the Kondo effect to develop, namely, the average tempera-ture of the system should exceed the Kondo temperature. /H20850
The model Hamiltonian /H20849see Fig. 1/H20850is thus
H=H
L+HR+Hdot+Hcoup, /H2084911/H20850
in which HL/H20849R/H20850is the Hamiltonian of the left /H20849right /H20850lead,
HL/H20849R/H20850=/H20858
k/H20849p/H20850/H9280k/H20849p/H20850ck/H20849p/H20850†ck/H20849p/H20850 /H2084912/H20850
/H20851using k/H20849p/H20850for the left /H20849right /H20850lead /H20852. The Hamiltonian of the
bridge, which includes the electron-phonon interaction, is
Hdot=/H92800c0†c0+/H92750/H20873b†b+1
2/H20874+/H9253/H20849b+b†/H20850c0†c0, /H2084913/H20850
where /H92750is the frequency of the harmonic oscillator and /H9253is
its coupling to the transport electrons. /H20849We use units in which
/H6036=1. /H20850Finally, the coupling between the dot and the leads is
described by
Hcoup=/H20858
k/H20849Vkck†c0+ H.c. /H20850+/H20858
p/H20849Vpcp†c0+ H.c. /H20850. /H2084914/H20850
The operators c0†,ck†, and cp†/H20849c0,ck, and cp/H20850create /H20849destroy /H20850
an electron on the dot, on the left lead, and on the right lead,respectively, while b
†/H20849b/H20850creates /H20849destroys /H20850an excitation of
the harmonic oscillator, of frequency /H92750. The electron distri-
butions of the leads, fLand fR, are given byfL/H20849R/H20850/H20849/H9275/H20850=/H208491 + exp /H20851/H9252L/H20849R/H20850/H20849/H9275−/H9262L/H20849R/H20850/H20850/H20852/H20850−1, /H2084915/H20850
where /H9252L/H20849R/H20850=1 //H20849kBTL/H20849R/H20850/H20850.
The couplings of the leads to the resonance level broadens
it, such that
/H9003L/H20849R/H20850/H20849/H9275/H20850=2/H9266/H20858
k/H20849p/H20850/H20841Vk/H20849p/H20850/H208412/H9254/H20849/H9275−/H9280k/H20849p/H20850/H20850/H20849 16/H20850
are the partial widths brought about by the left and the right
leads. These couplings are treated to all orders, encompass-ing the case in which the transport electrons excite effec-tively the phonons /H20849the dwell time of the electrons on the
junction largely exceeds the response time of the oscillator,
about
/H92750−1/H20850, and also the inverse situation. Strictly speaking,
the Hamiltonian /H2084911/H20850pertains to a floating molecule, which is
not coupled to a heat bath of its own; however, the analysispresented above in Sec. Ienables us to consider the three-
terminal case /H20849see Fig. 1/H20850as well.
The explicit calculation of the currents is carried out up to
second order in the electron-phonon coupling, using theKeldysh technique,
35and the details are given in the Appen-
dix. We find that the charge current consists of two terms,which can be related to elastic and inelastic transitions of theelectrons through the junction /H20851the first and the second terms
in Eq. /H2084917/H20850, respectively /H20852
I=e/H20885d/H9275
2/H9266/H20841G00r/H20849/H9275/H20850/H208412/H9003L/H20849/H9275/H20850/H9003R/H20849/H9275/H20850/H20851fL/H20849/H9275/H20850−fR/H20849/H9275/H20850/H20852
+e/H92532/H20885d/H9275
2/H9266/H20841G00r/H20849/H9275+/H20850/H208412/H20841G00r/H20849/H9275−/H20850/H208412/H11003/H20851/H9003R/H20849/H9275+/H20850/H9003L/H20849/H9275−/H20850FRL/H20849/H9275/H20850
−L↔R/H20852, /H2084917/H20850
where we have introduced the abbreviations
/H9275/H11006=/H9275/H11006/H92750
2. /H2084918/H20850
Here, G00is the Green’s function of the dot, given by Eq.
/H20849A5 /H20850, and G00is its counterpart when the coupling to the
phonons is ignored, i.e.,
/H20841G00r/H20849/H9275/H20850/H208412=/H208791
/H9275−/H92800+i/H9003/H20849/H9275/H20850/2/H208792
/H2084919/H20850
represents the bare Breit-Wigner resonance on the dot, with
/H9003/H20849/H9275/H20850=/H9003L/H20849/H9275/H20850+/H9003R/H20849/H9275/H20850. /H2084920/H20850
Finally,
F/H9251/H9251/H11032/H20849/H9275/H20850=N/H208511−f/H9251/H20849/H9275+/H20850/H20852f/H9251/H11032/H20849/H9275−/H20850−/H208511+N/H20852/H208511−f/H9251/H11032/H20849/H9275−/H20850/H20852f/H9251/H20849/H9275+/H20850
/H2084921/H20850
embodies the populations of the electrons /H20849fL,R/H20850and the
phonons /H20849N/H20850. Note that the latter population is notnecessar-
ily given by the Bose-Einstein distribution; this is the caseonly when the molecule is strongly coupled to a heat bath ofits own /H20849this distribution is denoted below by N
T/H20850. In the case
of the floating molecule, the population Nis determined by
the transport electrons as explained below and in the Appen-dix.THREE-TERMINAL THERMOELECTRIC TRANSPORT … PHYSICAL REVIEW B 82, 115314 /H208492010 /H20850
115314-3The energy current carried by the electrons, IE,/H20851see Eq.
/H208499/H20850/H20852is shown in the Appendix to be
IE=/H20885d/H9275
2/H9266/H20841G00r/H20849/H9275/H20850/H208412/H9275/H9003L/H20849/H9275/H20850/H9003R/H20849/H9275/H20850/H20851fL/H20849/H9275/H20850−fR/H20849/H9275/H20850/H20852
+/H92532/H20885d/H9275
2/H9266/H20841G00r/H20849/H9275+/H20850/H208412/H20841G00r/H20849/H9275−/H20850/H208412
/H11003/H20877/H92750
2/H20851/H9003R/H20849/H9275+/H20850/H9003R/H20849/H9275−/H20850FRR/H20849/H9275/H20850−/H20849R→L/H20850/H20852
+/H9275/H20851/H9003R/H20849/H9275+/H20850/H9003L/H20849/H9275−/H20850FRL/H20849/H9275/H20850−/H20849L↔R/H20850/H20852/H20878, /H2084922/H20850
where again the first and second terms pertain to the elastic
and inelastic contributions to the electronic energy current.The energy current carried by the phonons /H20851see Eqs. /H208494/H20850and
/H2084910/H20850/H20852is
−E˙
P=/H20885d/H9275
2/H9266/H20841G00r/H20849/H9275+/H20850/H208412/H20841G00r/H20849/H9275−/H20850/H208412
/H11003/H92532/H92750/H20858
/H9251,/H9251/H11032=L,R/H9003/H9251/H20849/H9275+/H20850/H9003/H9251/H11032/H20849/H9275−/H20850F/H9251/H9251/H11032/H20849/H9275/H20850. /H2084923/H20850
In the next section, we examine these currents in the linear-
response regime.
III. LINEAR-RESPONSE REGIME
The temperatures and the chemical potentials of the three-
terminal junction are given by Eqs. /H208495/H20850and /H208496/H20850. In the linear-
response regime, one expands the currents /H20851see Eqs.
/H208498/H20850–/H2084910/H20850/H20852to first order in /H9004/H9262,/H9004T, and/H9004TP. In order to ex-
press the resulting transport coefficients in a convenientform, we note that all integrals resulting from the elasticprocesses include the function,
F
el/H20849/H9275/H20850=/H9252f/H20849/H9275/H20850/H208511−f/H20849/H9275/H20850/H20852/H20841G00r/H20849/H9275/H20850/H208412, /H2084924/H20850
where f/H20849/H9275/H20850is the thermal-equilibrium Fermi distribution of
temperature T, and /H9252=1 //H20849kBT/H20850. The transport coefficients
coming from the inelastic processes include in their integralforms the function,
F
inel/H20849/H9275/H20850=/H92532/H20841G00r/H20849/H9275+/H20850/H208412/H20841G00r/H20849/H9275−/H20850/H208412NT/H9252f/H20849/H9275−/H20850/H208511−f/H20849/H9275+/H20850/H20852,
/H2084925/H20850
where NTis the thermal-equilibrium Bose distribution func-
tion of temperature T.
The relations between the currents and the driving forces
in the linear-response regime can be written in the matrixform
/H20900I
IQ
−E˙P/H20901=M/H20900/H9004/H9262/e
/H9004T/T
/H9004TP/T/H20901, /H2084926/H20850
where the matrix of the transport coefficients, M,i sM=/H20900GK XP
KK2+K2PX˜P
XPX˜P CP/H20901. /H2084927/H20850
Let us first describe the conventional transport coefficients,
pertaining to the transport by the electrons. In Eq. /H2084927/H20850,Gis
the electrical conductance,
G=Gel+Ginel, /H2084928/H20850
which consists of the contribution of elastic processes,
Gel=e2
2/H9266/H20885d/H9275Fel/H20849/H9275/H20850/H9003L/H20849/H9275/H20850/H9003R/H20849/H9275/H20850, /H2084929/H20850
and the contribution of the inelastic ones
Ginel=e2
2/H9266/H20885d/H9275Finel/H20849/H9275/H20850/H20851/H9003L/H20849/H9275+/H20850/H9003R/H20849/H9275−/H20850+/H9003L/H20849/H9275−/H20850/H9003R/H20849/H9275+/H20850/H20852.
/H2084930/H20850
Clearly, Eq. /H2084930/H20850corresponds to the two inelastic processes
by which the transport electron excites or deexcites the pho-non upon moving between the reservoirs. The transport co-efficient yielding the thermopower and the Seebeck effect, K,
and the one giving the main contribution to the electric ther-mal conductance, K
2, also consist of two contributions each,
K=Kel+Kinel,
K2=K2el+K2inel/H2084931/H20850
with
Kel=e
2/H9266/H20885d/H9275Fel/H20849/H9275/H20850/H20849/H9275−/H9262/H20850/H9003L/H20849/H9275/H20850/H9003R/H20849/H9275/H20850,
Kinel=e
2/H9266/H20885d/H9275Finel/H20849/H9275/H20850/H20849/H9275−/H9262/H20850/H20851/H9003L/H20849/H9275+/H20850/H9003R/H20849/H9275−/H20850
+/H9003L/H20849/H9275−/H20850/H9003R/H20849/H9275+/H20850/H20852, /H2084932/H20850
and
K2el=1
2/H9266/H20885d/H9275Fel/H20849/H9275/H20850/H20849/H9275−/H9262/H208502/H9003L/H20849/H9275/H20850/H9003R/H20849/H9275/H20850,
K2inel=1
2/H9266/H20885d/H9275Finel/H20849/H9275/H20850/H20849/H9275−/H9262/H208502/H20851/H9003L/H20849/H9275+/H20850/H9003R/H20849/H9275−/H20850
+/H9003L/H20849/H9275−/H20850/H9003R/H20849/H9275+/H20850/H20852. /H2084933/H20850
All other coefficients appearing in Eq. /H2084927/H20850result from the
inelastic processes. One of them, K2P, just augments the /H20849con-
ventional /H20850ratio K2between the heat current carried by the
electrons and the temperature gradient /H9004Tacross the junc-
tion,ENTIN-WOHLMAN, IMRY , AND AHARONY PHYSICAL REVIEW B 82, 115314 /H208492010 /H20850
115314-4K2P=/H927502
8/H9266/H20885d/H9275Finel/H20849/H9275/H20850/H20851/H9003R/H20849/H9275+/H20850/H9003L/H20849/H9275−/H20850+/H9003L/H20849/H9275+/H20850/H9003R/H20849/H9275−/H20850/H20852.
/H2084934/H20850
It therefore follows that the electron-phonon interaction just
renormalizes slightly the conventional transport coefficientsof the two-terminal single-dot junction but does not lead tonovel effects /H20849see also Sec. IVbelow /H20850.
On the other hand, keeping the phonon bath to which the
molecule is attached at a temperature different from those ofthe electron reservoirs leads to new thermoelectric effects.We find that there is an electric current flowing in responseto the temperature difference /H9004T
Pwith the phonon bath, with
the novel transport coefficient,
XP=e/H92750
2/H9266/H20885d/H9275Finel/H20849/H9275/H20850/H20851/H9003R/H20849/H9275+/H20850/H9003L/H20849/H9275−/H20850−/H9003L/H20849/H9275+/H20850/H9003R/H20849/H9275−/H20850/H20852.
/H2084935/H20850
The same coefficient controls the heat current between the
junction and the phonon bath in response to the chemicalpotential difference between the electronic reservoirs. Like-wise, there is a heat current flowing between the electronicreservoirs in response to /H9004T
P, which is governed by a coef-
ficient analogous to Eq. /H2084935/H20850,
XP=/H92750
2/H9266/H20885d/H9275Finel/H20849/H9275/H20850/H20877/H20849/H9275−/H9262/H20850/H20851/H9003R/H20849/H9275+/H20850/H9003L/H20849/H9275−/H20850
−/H9003L/H20849/H9275+/H20850/H9003R/H20849/H9275−/H20850/H20852+/H92750
2/H20851/H9003R/H20849/H9275+/H20850/H9003R/H20849/H9275−/H20850
−/H9003L/H20849/H9275+/H20850/H9003L/H20849/H9275−/H20850/H20852/H20878 /H2084936/H20850
with the same coefficient governing the heat current from the
phonon reservoir in response to the electronic temperaturedifference /H9004T. Thus, the matrix of coefficients Mobeys the
Onsager symmetry relations also in the three-terminal situa-tion with the two types of carriers and their interaction.
Finally, the coefficient C
Pgives the response of the heat
current carried by the phonons to the temperature difference/H9004T
P,
CP=/H927502
2/H9266/H20885d/H9275Finel/H20849/H9275/H20850/H9003/H20849/H9275+/H20850/H9003/H20849/H9275−/H20850, /H2084937/H20850
where we have used Eq. /H2084920/H20850.
IV. DISCUSSION
Using a simple model, we have considered the thermo-
electric and thermal transport of electrons through a molecu-lar bridge, in particular, the subtle effects of the inelasticelectron-vibrational mode processes. Of a paramount impor-tance is the mechanism by which the vibration population isdetermined.
When the molecule is not attached to any heat bath, the
phonon population is determined by the voltage and the tem-perature difference across the junction. We show in the Ap-pendix that in this case /H20851see Eq. /H20849A16 /H20850/H20852the heat current be-
tween the vibrations and the transport electrons is
E˙
P=/H92750dN
dt, /H2084938/H20850
where Ndenotes the vibrational mode population. At steady
state that population does not vary with time, and conse-quently the heat current between the molecule and the junc-tion vanishes. This requirement, in turn, fixes /H9004T
Pin terms
of/H9004/H9262and/H9004T, and consequently determines the vibration
population /H20851see Fig. 1and Eqs. /H208495/H20850and /H208496/H20850/H20852. In other words,
the requirement that − E˙P=0 yields
XP/H9004/H9262
e+X˜P/H9004T
T=−CP/H9004TP
T, /H2084939/H20850
and hence transforms the three-terminal junction into a two-
terminal one with
/H20875I
IQ/H20876=/H20875G−/H20849XP/H208502/CPK−XPX˜P/CP
K−XPX˜P/CPK2+K2P−/H20849X˜P/H208502/CP/H20876/H20875/H9004/H9262/e
/H9004T/T/H20876.
/H2084940/H20850
In this situation we find that the inelastic processes modify
the transport coefficients but do not give rise to any intrigu-ing effects.
On the other hand, when the molecule is attached
/H20849strongly /H20850to its own thermal bath, see Fig. 1, such that the
system becomes a three-terminal junction, the vibrationalmodes and the transport electrons may exchange heat, and atemperature difference between the phonons and the trans-port electrons can induce an electron current between theelectronic reservoirs. Likewise, a voltage between the lattercan induce a heat current to the phonons. These two newtransport coefficients, having two types of carriers and in-cluding inelastic processes, are related by Onsager symme-try. This situation is characterized by the appearance of newtransport coefficients that result solely from the inelastictransport processes /H20851see Eqs. /H2084926/H20850and /H2084927/H20850/H20852, and requires the
breaking of spatial symmetry between the two sides of thejunction, /H9003
L/HS11005/H9003R. Note, in particular, the change in the rela-
tive sign of the combinations /H9003R/H20849/H9275+/H20850/H9003L/H20849/H9275−/H20850and
/H9003L/H20849/H9275+/H20850/H9003R/H20849/H9275−/H20850between the expressions for the usual thermo-
electric coefficients, Eqs. /H2084932/H20850–/H2084934/H20850, and the new three-
terminal ones, Eqs. /H2084935/H20850and /H2084936/H20850. This change occurs be-
cause the latter expressions are for the heat currents fromeach lead to the phonons and not between the two leads. Theanalysis of the above combinations of the /H9003’s can tell us how
to maximize the new, three-terminal, thermoelectric coeffi-cients. Usually the
/H9275dependence of the resonance widths /H9003’s
is not too strong. Let us then expand them around the run-ning
/H9275,
/H9003L/H20849/H9275/H11032/H20850=/H9003L/H20849/H9275/H20850+AL/H20849/H9275/H11032−/H9275/H20850+¯
with an analogous expansion for /H9003R. The crucial quantity is
the one in parentheses on the right-hand side of Eq. /H2084935/H20850.T o
order/H92750, it givesTHREE-TERMINAL THERMOELECTRIC TRANSPORT … PHYSICAL REVIEW B 82, 115314 /H208492010 /H20850
115314-5/H92750/H20851AR/H9003L/H20849/H9275/H20850−AL/H9003R/H20849/H9275/H20850/H20852. /H2084941/H20850
To increase the usual thermopower, we want the transmission
to depend strongly on energy. In our case, to make the twoterms in Eq. /H2084941/H20850add and not tend to cancel, we also want /H9003
L
and/H9003Rto have opposite dependencies on the frequency. One
way to effect this is to have a lead with an electron-bandmaterial on the left lead, and one with a hole-band materialon the right lead. This will however decrease the values ofthe usual two-terminal thermal and thermoelectric coeffi-cients. Hence, more down-to-earth estimates of the new ther-moelectric coefficients require realistic descriptions of themolecular bridge, which will depend on the type of mol-ecules involved and other parameters of the system.
In order to elucidate the above considerations, we com-
pute the coefficient governing the conversion of heat fromthe phonon bath into a voltage difference across the bridge,
S
P/H11013eXP
TG, /H2084942/H20850
where XPis given by Eq. /H2084935/H20850./H20849This definition follows the
conventional one for the thermopower. /H20850Let us assume that
the left reservoir is represented by an electron band, such thatthe partial width it causes to the resonant level is given by
/H9003
L/H20849/H9275/H20850=/H9003L/H20881/H9275−/H9275c
/H9275v−/H9275c/H2084943/H20850
while the right reservoir is modeled by a hole band with
/H9003R/H20849/H9275/H20850=/H9003R/H20881/H9275v−/H9275
/H9275v−/H9275c. /H2084944/H20850
Here,/H9275cis the bottom of the conductance band /H20849on the left
side of the junction /H20850while/H9275vis the ceiling of the hole band
/H20849on the right one /H20850. The energy integration determining the
various transport coefficients is therefore limited to the re-gion
/H9275c/H11349/H9275/H11349/H9275v./H20849For convenience, we normalize the /H9003’s by
the full bandwidth, /H9275v−/H9275c./H20850
Measuring all energies appearing in the explicit expres-
sions in units of the temperature /H9252−1, and choosing /H9262=/H92800
=0 for simplicity, we obtained the curves shown in Figs. 2.
One observes that the magnitude of the effect is nonmono-tonic in the value of the vibration frequency
/H92750: being theoutcome of inelastic processes, it vanishes at /H92750=0, and also
at/H9252/H92750/H112711, since then the vibrational level population be-
comes very small. It increases for values /H9003L/H20849R/H20850which are
smaller than the temperature, and also increases when /H9003L
/HS11005/H9003R, because then the electric conductance Gbecomes
smaller.
ACKNOWLEDGMENTS
We thank Achim Rosch and Peter Wölfle for illuminating
discussions. This work was supported by the German FederalMinistry of Education and Research /H20849BMBF /H20850within the
framework of the German-Israeli project cooperation /H20849DIP /H20850,
by the U.S.-Israel Binational Science Foundation /H20849BSF /H20850,b y
the Israel Science Foundation /H20849ISF /H20850, and by its Converging
Technologies Program. We thank the /H20849anonymous /H20850referee
for pointing out the issues of the vibration relaxation by cou-pling to the electrons of the leads and the resulting thermalcontact to a substrate.
APPENDIX: DETAILS OF THE CURRENTS’
CALCULATION
The particle and the energy currents can be expressed in
terms of the electronic Keldysh Green’s functions, in particu-lar, the Green’s function G
00on the dot. To this end, we write
the particle current emerging from the left /H20849right /H20850reservoir in
the form
N˙L/H20849R/H20850=d
dt/H20883/H20858
k/H20849p/H20850ck/H20849p/H20850†ck/H20849p/H20850/H20884=/H20885d/H9275
2/H9266IL/H20849R/H20850/H20849/H9275/H20850. /H20849A1 /H20850
Likewise, the energy current emerging from the left /H20849right /H20850
reservoir can be shown to be given by
E˙L/H20849R/H20850=d
dt/H20883/H20858
k/H20849p/H20850/H9280k/H20849p/H20850ck/H20849p/H20850†ck/H20849p/H20850/H20884=/H20885d/H9275
2/H9266/H9275IL/H20849R/H20850/H20849/H9275/H20850./H20849A2 /H20850
The Green’s-function calculation yields
IL/H20849R/H20850/H20849/H9275/H20850=−i/H9003L/H20849R/H20850/H20849/H9275/H20850/H20853G00/H11021/H20849/H9275/H20850−fL/H20849R/H20850/H20849/H9275/H20850/H20851G00a/H20849/H9275/H20850−G00r/H20849/H9275/H20850/H20852/H20854,
/H20849A3 /H20850
where the superscripts /H11021,a, and r, denote the lesser, ad-
vanced, and retarded Green’s function, respectively. The0 2 4 6 8 10/Minus0.035/Minus0.030/Minus0.025/Minus0.020/Minus0.015/Minus0.010/Minus0.0050.000
ΒΩ0SP/Slash1/LParen1ΒΓ/RParen12
0 2 4 6 8 10/Minus0.04/Minus0.03/Minus0.02/Minus0.010.00
ΒΩ0SP/Slash1/LParen1ΒΓ/RParen12
FIG. 2. /H20849Color online /H20850The coefficient SP, Eq. /H2084942/H20850, as a function of /H9252/H92750for/H9252/H9003L=0.2 /H20849thin line /H20850,/H9252/H9003L=1 /H20849dashed line /H20850, and/H9252/H9003L=5
/H20849dotted line /H20850. Left panel, /H9003R=/H9003L, right panel /H9003R=0.7/H9003L/H20851see Eqs. /H2084943/H20850and /H2084944/H20850/H20852. The total bandwidth is determined by /H9252/H9275c=−/H9252/H9275v=100.ENTIN-WOHLMAN, IMRY , AND AHARONY PHYSICAL REVIEW B 82, 115314 /H208492010 /H20850
115314-6Fermi distributions, fL/H20849R/H20850, are given in Eq. /H2084915/H20850, and the par-
tial widths of the resonance level, /H9003L/H20849R/H20850,i nE q . /H2084916/H20850.
The Green’s function on the dot is calculated up to second
order in the electron-phonon coupling /H9253.36One finds
G00/H11021/H20849/H9275/H20850=G00r/H20849/H9275/H20850/H20851/H9018P/H11021/H20849/H9275/H20850+/H9018l/H11021/H20849/H9275/H20850/H20852G00a/H20849/H9275/H20850, /H20849A4 /H20850
where
G00r/H20849/H9275/H20850=/H20851/H9275−/H92800−/H9018lr/H20849/H9275/H20850−/H9254/H9280P−/H9018Pr/H20849/H9275/H20850/H20852−1. /H20849A5 /H20850
Here,
/H9254/H9280P=2i/H92532
/H92750/H20885d/H9275
2/H9266G00/H11021/H20849/H9275/H20850/H20849 A6 /H20850
is the polaron energy shift, where G00/H11021/H20849/H9275/H20850is the lesser Green’s
function on the dot in the absence of the coupling with theoscillator,
G
00/H11021/H20849/H9275/H20850=i/H9003L/H20849/H9275/H20850fL/H20849/H9275/H20850+/H9003R/H20849/H9275/H20850fR/H20849/H9275/H20850
/H20849/H9275−/H92800/H208502+/H20851/H9003/H20849/H9275/H20850/2/H208522/H20849A7 /H20850
with/H9003=/H9003L+/H9003R. Here we have ignored a possible shift in the
resonance energy due to the coupling with the leads since itis not expected to play a significant role.
As is seen from Eqs. /H20849A4 /H20850and /H20849A5 /H20850, the self-energy on the
dot includes two contributions. The first, /H9018
l, is due to the
coupling with the leads,
/H9018lr/H20849/H9275/H20850=−i
2/H20851/H9003L/H20849/H9275/H20850+/H9003R/H20849/H9275/H20850/H20852,
/H9018l/H11021/H20849/H9275/H20850=i/H20851/H9003L/H20849/H9275/H20850fL/H20849/H9275/H20850+/H9003R/H20849/H9275/H20850fR/H20849/H9275/H20850/H20852. /H20849A8 /H20850
The second contribution to the self-energy results from the
interaction with the phonons, and in second order in /H9253reads
/H9018Pr/H20849/H9275/H20850=i/H92532/H20885d/H9275/H11032
2/H9266/H20875/H208491+N/H20850G00/H11022/H20849/H9275/H11032/H20850−NG00/H11021/H20849/H9275/H11032/H20850
/H9275−/H92750−/H9275/H11032+i0+
+NG00/H11022/H20849/H9275/H11032/H20850−/H208491+N/H20850G00/H11021/H20849/H9275/H11032/H20850
/H9275+/H92750−/H9275/H11032+i0+/H20876 /H20849A9 /H20850
and
/H9018P/H11021/H20849/H9275/H20850=/H92532/H20851NG00/H11021/H20849/H9275−/H92750/H20850+/H208491+N/H20850G00/H11021/H20849/H9275+/H92750/H20850/H20852,
/H20849A10 /H20850
where Ndenotes the phonon population. The lesser Green’s
function G/H11021is given in Eq. /H20849A7 /H20850, and the greater one, G/H11022,i s
given by the same expression with the distributions fL,Rre-
placed by fL,R−1.
Inserting the expressions for the Green’s function G00into
Eq. /H20849A3 /H20850, one finds that IL/H20849R/H20850/H20849/H9275/H20850can be written as a sum of
two terms, one arising from the elastic transitions of thetransport electrons and the other coming from the inelasticones,
I
L/H20849R/H20850/H20849/H9275/H20850=IL/H20849R/H20850el/H20849/H9275/H20850+IL/H20849R/H20850inel/H20849/H9275/H20850. /H20849A11 /H20850
The elastic-process contribution isILel/H20849/H9275/H20850=/H20841G00r/H20849/H9275/H20850/H208412/H9003L/H20849/H9275/H20850/H9003R/H20849/H9275/H20850/H20851fR/H20849/H9275/H20850−fL/H20849/H9275/H20850/H20852 /H20849A12 /H20850
while the inelastic one is proportional to the strength of the
electron-phonon coupling,
ILinel/H20849/H9275/H20850=/H92532/H9003L/H20849/H9275/H20850/H20841G00r/H20849/H9275/H20850/H208412/H20841G00r/H20849/H9275−/H92750/H20850/H208412/H20858
/H9251=L,R/H9003/H9251/H20849/H9275−/H92750/H20850
/H11003/H20853Nf/H9251/H20849/H9275−/H92750/H20850/H208511−fL/H20849/H9275/H20850/H20852−/H208491+N/H20850fL/H20849/H9275/H20850/H208511
−f/H9251/H20849/H9275−/H92750/H20850/H20852/H20854−/H92532/H9003L/H20849/H9275/H20850/H20841G00r/H20849/H9275/H20850/H208412/H20841G00r/H20849/H9275+/H92750/H20850/H208412
/H11003/H20858
/H9251=L,R/H9003/H9251/H20849/H9275+/H92750/H20850/H20853NfL/H20849/H9275/H20850/H208511−f/H9251/H20849/H9275+/H92750/H20850/H20852−/H208491
+N/H20850f/H9251/H20849/H9275+/H92750/H20850/H208511−fL/H20849/H9275/H20850/H20852/H20854, /H20849A13 /H20850
where G00r, the retarded Green’s function in the absence of
the coupling to the vibrational modes, is given in Eq. /H2084919/H20850.
Since IRis obtained from Eqs. /H20849A12 /H20850and /H20849A13 /H20850upon inter-
changing L with R, it is easy to see that the elastic-processparts of both the particle and the energy currents are con-
served /H20851this is so because I
Lel/H20849/H9275/H20850+IRel/H20849/H9275/H20850=0/H20852. The consider-
ation of the inelastic-process part is a bit more delicate. Bychanging the integration variables, one finds that
/H20885d/H9275
2/H9266/H9275sILinel/H20849/H9275/H20850=/H92532/H20885d/H9275
2/H9266/H20841G00r/H20849/H9275+/H20850/H208412/H20841G00r/H20849/H9275−/H20850/H208412
/H11003/H20851/H9003L/H20849/H9275+/H20850/H9003L/H20849/H9275−/H20850/H20849/H9275+s−/H9275−s/H20850FLL/H20849/H9275/H20850
+/H9275+s/H9003L/H20849/H9275+/H20850/H9003R/H20849/H9275−/H20850FLR/H20849/H9275/H20850
−/H9275−s/H9003R/H20849/H9275+/H20850/H9003L/H20849/H9275−/H20850FRL/H20849/H9275/H20850/H20852,s=0 o r 1 ,
/H20849A14 /H20850
where F/H9251/H9251/H11032is given in Eq. /H2084921/H20850. Here, /H9275/H11006/H11013/H9275/H11006/H92750/2.
Hence, the inelastic-process parts of the particle current /H20849for
which s=0/H20850are also conserved, i.e., /H20848/H20849d/H9275/2/H9266/H20850/H20851ILinel/H20849/H9275/H20850
+IRinel/H20849/H9275/H20850/H20852=0. Using Eqs. /H20849A1 /H20850,/H20849A12 /H20850, and /H20849A14 /H20850in Eq. /H208498/H20850
produces Eq. /H2084917/H20850for the charge current.
On the other hand, the energy current carried by the elec-
trons alone is not conserved since /H20851using Eq. /H20849A14 /H20850with s
=1/H20852
/H20885d/H9275
2/H9266/H9275/H20851ILinel/H20849/H9275/H20850+IRinel/H20849/H9275/H20850/H20852
=/H92750/H92532/H20885d/H9275
2/H9266/H20841G00r/H20849/H9275+/H20850/H208412
/H11003/H20841G00r/H20849/H9275−/H20850/H208412/H20858
/H9251,/H9251/H11032=L,R/H9003/H9251/H20849/H9275+/H20850/H9003/H9251/H11032/H20849/H9275−/H20850F/H9251/H9251/H11032/H20849/H9275/H20850.
/H20849A15 /H20850
This result, in conjunction with Eqs. /H2084910/H20850and /H20849A2 /H20850, leads to
Eq. /H2084923/H20850. Finally, the net energy current carried by the elec-
trons /H20851see Eq. /H208499/H20850/H20852is obtained by using Eq. /H20849A14 /H20850in Eq. /H20849A2 /H20850
/H20849and the corresponding equation for E˙R/H20850. This yields Eq.
/H2084922/H20850.THREE-TERMINAL THERMOELECTRIC TRANSPORT … PHYSICAL REVIEW B 82, 115314 /H208492010 /H20850
115314-7In the case of the floating molecule, which is not coupled
to a heat bath of its own, it is straightforward to show /H20851using
the Hamiltonian /H2084911/H20850/H20852that the rate of change in the phonon
population36is given by minus the right-hand side of Eq.
/H20849A15 /H20850divided by /H92750; i.e.,
/H20885d/H9275
2/H9266/H9275/H20851ILinel/H20849/H9275/H20850+IRinel/H20849/H9275/H20850/H20852+/H92750dN
dt=0 , /H20849A16 /H20850
yielding Eq. /H208494/H20850for the energy conservation with E˙P
=/H92750dN /dt. However, the phonon population of a floating
molecule will arrange itself according to the chemical poten-tials and the temperatures of the electronic reservoirs.
36Con-
sequently at steady state dN /dtwill vanish, implying that
E˙L+E˙R=0, such that the energy current of the electrons is
conserved.
Finally we estimate the rate of decay of the vibration
population due to the coupling with the electrons in theleads, when the latter are at thermal equilibrium. In diagram-matic language that rate is given by dressing the “phonon”line with an electron bubble, i.e.,
−dN
dt=/H92532/H20885d/H9275
2/H9266/H9003/H20849/H9275+/H20850
2/H20841G00a/H20849/H9275+/H20850/H208412/H9003/H20849/H9275−/H20850
2/H20841G00a/H20849/H9275−/H20850/H208412
/H11003/H20853Nf/H20849/H9275−/H20850/H208511−f/H20849/H9275+/H20850/H20852−/H208491+N/H20850f/H20849/H9275+/H20850/H208511−f/H20849/H9275−/H20850/H20852/H20854.
/H20849A17 /H20850
At zero temperature, the last factor in the integrand limits it
to the range /H20841/H9275−/H9262/H20841/H11349/H92750/2, leading to the rate/H20849/H92750//H208512/H9266/H20852/H20850/H20849/H9253//H9003/H208502/H20849A18 /H20850
when/H9003/H11271/H92750, as mentioned in Sec. I. In general, the rate is a
nonmonotonic function of the ratio /H92750//H9003, reaching a maxi-
mal value when these two energies are comparable.
The result of Eq. /H20849A18 /H20850can be qualitatively obtained in a
more elementary, but equivalent, fashion using third-orderperturbation theory. Consider the decay of the excited vibra-tional mode with a T=0 Fermi gas on each lead /H20849having a
density of states
/H92630/H20850. For simplicity we take the resonant case
and assume /H9003/H11271/H92750to start with. For the decay with the left
lead, the first intermediate state has an electron from that
lead go into the molecule with an amplitude VL/H11569and an ef-
fective energy denominator /H9003L/2/H20849due to being on reso-
nance /H20850, in the second intermediate state, the vibration is de-
excited /H20849amplitude /H9253/H20850and the electrons stay put. For /H9003
/H11271/H92750, the energy denominator is again approximately /H9003L/2.
In the final state, the electron goes back to the same lead,with an energy
/H92750higher that the one it started from, and
with an amplitude VL. The total amplitude for this process is
VL/H11569/H9253VL//H20849/H9003L/2/H208502. Finally, taking the absolute square of the
amplitude and multiplying by 2 /H9266/H92630, we get the golden-rule
rate for this decay. Multiplying by the number of such pro-cesses,
/H92630/H92750, and summing over the /H20849equivalent /H20850leads, we
get Eq. /H20849A18 /H20850in order of magnitude. Clearly, in the opposite
case/H9003/H11270/H92750, one/H9003in the denominator is replaced by /H92750.A
surprising feature of this result, which must be pointed out, isits decrease with /H9003, the rate to get into/from the molecule
from/into the leads. This is quite counterintuitive but it iswhat the quantum-mechanical calculation tells us. Formally,this is due to Green’s function at the resonance having i/H9003/2
as its denominator, meaning physically that the width of theresonance sets the scale for the “closest approach” to it.
*oraentin@bgu.ac.il
†Also at Tel Aviv University, Tel Aviv 69978, Israel.
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115314-9 |
PhysRevB.70.195336.pdf | Low-temperature mobility of holes in Si/SiGe p-channel heterostructures
Doan Nhat Quang *
Department of Physics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan
Vu Ngoc Tuoc, Tran Doan Huan,†and Pham Nam Phong
Institute of Engineering Physics, Hanoi University of Technology, 1 Dai Co Viet Road, Hanoi, Vietnam
(Received 31 December 2003; published 23 November 2004 )
We present a theory of the low-temperature mobility of holes in strained SiGe layers of Si/SiGe p-channel
heterostructures. Our theory must not be based on the unclear concept of interface impurity charges assumedin the previous calculations, but takes adequate account of the random deformation potential and randompiezoelectric field. These appear as effects arising from both lattice mismatch and interface roughness. It isproved that deformation potential scattering may be predominant over the well-known scattering mechanismssuch as background doping, alloy disorder, and surface roughness for a Ge content x*0.2, while piezoelectric
scattering is comparable thereto for x*0.4. Our theory turns out to be successful in providing a good quan-
titative explanation of recent experimental findings not only about the low value of the hole mobility but alsoits dependence on carrier density as well as its decrease with Ge content.
DOI: 10.1103/PhysRevB.70.195336 PACS number (s): 73.50.Bk, 73.63.Hs
I. INTRODUCTION
Recently, there has been considerable interest in the incor-
poration of a strained SiGe layer into p-channel metal-oxide-
semiconductor structures, to give increases in hole mobilityand high-field drift velocity.
1–3However, low-temperature
measurements reveal the following striking phenomena.
First, the hole mobilities are significantly less than the elec-tron ones. For the two-dimensional electron gas (2DEG )in a
Sin-channel, peak low-temperature mobilities were reported
to be ,4310
5cm2/Vs,3while for the two-dimensional
hole gas (2DHG )in a strained SiGe p-channel, the best mo-
bilities reported not exceeding ,23104cm2/Vs.4–6Sec-
ond, the hole mobility is degraded when increasing the Gecontent despite a reduction in effective hole mass.
3,7–9
It has been shown9–13that all so-far known scattering
mechanisms such as impurity doping, alloy disorder, and sur-face roughness are unable to account for the earlier experi-mental data. Therefore, several authors had to invoke theconcept of interface charges as a key scattering source at lowtemperatures. This enables a somewhat satisfactory descrip-tion of the 2DHG mobility in different strained Si/SiGe het-erostructures with a suitable choice of the interface chargedensity as a fitting parameter.
Nevertheless, there are several drawbacks in the previous
theories. First, it was indicated
3,9,10that the nature of inter-
face charges has been, to date, quite unclear. It was supposedthat they can originate from impurity contamination of epi-taxial layers during and after growth.The areal impurity den-sity for such an unintentional doping has to be claimedhigh,
9–13up to ,1011cm−2. The mechanisms for trapping
and charging impurities at the heterointerface Si/SiGe arealso not clarified.
Second, the decrease observed
3,8in the low-temperature
2DHG mobility of strained SiGe layers when increasing theGe content is also still unclear, since alloy disorder wasdemonstrated
1,7,9–13likely not to be a dominant scattering
source at a low carrier density, e.g., of ,1011cm−2. Plewsand co-workers14have obtained strong experimental evi-
dence that for Si/SiGe/Si systems screening of any scatter-ing potential, whatever its nature, is important, especially, atlow values of temperature, carrier density, and Ge content.As a result, with screening included, one cannot fit the ob-served data simply on the basis of alloy disorder scatteringalone, even by taking an unjustifiably large value of the alloypotential.
1,9,13
Third, it was proved15–19that interface roughness gives
rise to random variations in all components of the strain fieldin actual lattice-mismatched heterostructures. As a result,Feenstra and Lutz
16found that for an n-channel Si/SiGe sys-
tem these fluctuations cause a random nonuniform shift ofthe conduction band edge. This implies a random deforma-tion potential acting on electrons as a source of scattering,which yields much better agreement with experimental dataabout the 2DEG mobility
20than surface roughness scattering
does.The existing calculations9,11,12of the 2DHG mobility in
ap-channel Si/SiGe system have been carried out with an
extension of the idea of Feenstra and Lutz to the valenceband edge, based on the assumption that the deformation
potential for holes is almost identical to that for electrons.Asseen later, this is in fact invalid.
Finally, in the last years some experimental evidences for
piezoelectricity of strained SiGe layers in Si/SiGe systemshave been found.
21–24Further, interface roughness was
shown17–19to induce a fluctuating density of piezoelectric
charges. Scattering by them is to be included in a full treat-ment of the hole mobility.
Thus, the goal of this paper is to present a theory of the
low-temperature 2DHG mobility in strained SiGe layers ofSi/SiGep-channel heterostructures. Our theory is to be de-
veloped for explaining the experimental data recently re-ported in Refs. 3, 8, 11, and 12. Moreover, the theory mustnot be based on the unclear concept of interface impuritycharges, but adequately include the possible sources of scat-tering: alloy disorder, surface roughness, deformation poten-tial, and piezoelectric charges. In particular, the deformationPHYSICAL REVIEW B 70, 195336 (2004 )
1098-0121/2004/70 (19)/195336 (10)/$22.50 ©2004 The American Physical Society 70195336-1potential for holes must be rigorously derived.
The paper is organized as follows. In Sec. II, we formu-
late our model and basic equations used to calculate thedisorder-limited 2DHG mobility, taking explicitly into ac-count the finiteness of the potential barrier height. In Sec. III,the autocorrelation functions for diverse scattering mecha-nisms are derived. Section IV is devoted to numerical resultsand comparison with experiment. Finally, a summary in Sec.V concludes the paper.
II. BASIC FORMULATION
A. Finitely deep triangular quantum well
We are dealing with a 2DHG in a strained SiGe layer as
the conduction channel in a Si/SiGe/Si sandwich configura-tion. This may be the 2DHG located near the upper interfaceof the strained alloy layer in a gated oxide Si/SiGeheterostructure,
9–11,13or near its backinterface in a p-channel
field-effect structure.12
It is well known25that scattering by a random field is
specified by its autocorrelation function in wave vector spacekuUsqdu
2l. Hereafter, the angular brackets stand for an en-
semble average. Usqdis a 2D Fourier transform of the ran-
dom potential averaged with the envelope wave function of a
2D subband
Usqd=E
−‘+‘
dzuzszdu2Usq,zd. s1d
As usual,9–13for the holes confined in the SiGe layer, we
assume a triangular quantum well (QW)located along the
growth direction, e.g., [001]chosen as the zaxis, where z
=0 defines the Si/SiGe interface plane.
The band structure for holes in a strained SiGe layer
grown pseudomorphically on a relaxed (001)Si substrate is
calculated in Refs. 26–28, including both the strain andquantum confinement effects. It is found
1,26–28that the top
valence band edge is formed by the lowest heavy-hole (HH1 )
subband, and its energy separation from the first excited sub-band is large compared to the Fermi level at a rather low holedensity. At very low temperatures, the carriers are then as-sumed to primarily occupy the lowest heavy-hole subband.
It was indicated
29–34that the potential barrier height in
semiconductor heterostructures may play an important rolein certain phenomena. For Si/SiGe triangular QWs whichwe will be dealing with, the barrier height is rather small (a
few 0.1 eV ). Further, Poisson-Schrödinger simulations
showed
11that the envelope wave function has significant am-
plitude in the Si barrier layer.Therefore, we must, in general,adopt the realistic model of finitely deep wells.
It has been pointed out
29–31that for a finitely deep trian-
gular QW, the lowest subband may be very well described bya modified Fang-Howard wave function, proposed byAndo
29
zszd=HAk1/2expskz/2dforz,0,
Bk1/2skz+cdexps−kz/2dforz.0,J s2d
in which A,B,c,k, and kare variational parameters to be
determined. Here kandkare half the wave numbers in thewell and the barrier, respectively. A,B, andcare dimension-
less parameters given in terms of kandkthrough boundary
conditions at the interface plane z=0 and the normalization.
These read as30,31
Ak1/2=Bk1/2c,
Ak3/2/2=Bk3/2s1−c/2d,
A2+B2sc2+2c+2d=1. s3d
The energy of the ground-state subband is calculated as a
function of the wave numbers kand k. This involves as
parameters the potential barrier height V0, depletion charge
densityNd, and sheet hole density pssuch that30,31
E0sk,kd=−"2
8mzfB2k2sc2−2c−2d+A2k2g+V0A2
+4pe2Nd
«LFB2
ksc2+4c+6d−A2
kG
+4pe2ps
«LFB4
4ks2c4+12c3+34c2+50c+33d
+A4
2k−A2
kG, s4d
wheremz=0.28memeans the effective heavy-hole mass in
the growth direction, and «Lis the dielectric constant of the
SiGe layer. The effects of image charges are neglected sincethey are small for carriers in the SiGe channel.
9,13The wave
numberskandkin turn are fixed so as to minimize the total
energy per electron Esk,kdnumerically.29–31
In the limiting case of V0!‘, we have A=0,B=1/˛2,
c=0, and k!‘, so that Eq. (4)reproduces the lowest-
subband energy for an infinitely deep well described by thestandard Fang-Howard wave function.
25
B. Low-temperature hole mobility in a single-subband model
As mentioned earlier, in this paper we are to focus our
attention on the explanation of the experimental data com-piled in Refs. 3, 8, 11, and 12, where the transport was mea-sured for 2DHGs at very low temperature and rather lowcarrier density. For the purposes of examining the contribu-tions from various scattering mechanisms and find which aredominant, we will adopt a somewhat simplified model, butone that is accurate enough to capture the features of interest.It has been shown
1,9–13,28,35that for the case in question it is
a good approximation to take into consideration merely in-trasubband scattering within the lowest heavy-hole subbandHH1, ignoring intersubband scattering. Further, it is found
28
that this subband is isotropic and parabolic over a relativelylarge range of the 2D wave vector.
As a result, the zero-temperature mobility is determined
via the momentum relaxation time
m=et/m*, s5d
withm* as an effective in-plane mass.
In what follows, we will, for simplicity, ignore the mul-
tiple scattering effects.35,36Within the linear transport theory,QUANG et al. PHYSICAL REVIEW B 70, 195336 (2004 )
195336-2the inverse relaxation time for zero temperature is expressed
in terms of the autocorrelation function for disorder28,37,38
1
t=1
s2pd2"EFE
02kF
dqE
02p
duq2
s4kF2−q2d1/2kuUsqdu2l
«2sqd,
s6d
whereq=sq,uddenotes a 2D wave vector in the x-yplane
given in polar coordinates, EF="2kF2/2m* is the Fermi en-
ergy, and kFthe Fermi wave number fixed by the hole den-
sity:kF=˛2pps. The angle integral appears in Eq. (6)since
the autocorrelation function of a random field may have adirectional dependence as seen in Eqs. (27)and(30)later.
The dielectric function «sqdin Eq. (6)allows for the
screening of a scattering potential by the 2DHG, which is, as
quoted before, an important effect in the system under con-sideration. Within the random phase approximation, this isgiven at zero temperature by
25
«sqd=1+qTF
qFSsq/kdf1−Gsqdgforqł2kF, s7d
withqTF=2m*e2/«L"2the inverse 2D Thomas-Fermi
screening length.
The screening form factor FSsq/kdin Eq. (7)accounts for
the extension of hole states along the growth direction, de-
fined by
FSsq/kd=E
−‘+‘
dzE
−‘+‘
dz8uzszdu2uzsz8du2e−quz−z8u. s8d
By means of Eq. (2)for the lowest-subband wave func-
tion, this is expressed as a function of the dimensionlesswave numbers in the 2DHG plane t=q/kand the barrier
layera=
k/kby31
FSstd=A4a
t+a+2A2B2a2+2cst+1d+c2st+1d2
st+adst+1d3
+B4
2st+1d3f2sc4+4c3+8c2+8c+4d+ts4c4+12c3
+18c2+18c+9d+t2s2c4+4c3+6c2+6c+3dg.s9d
ForV0!‘, Eq. (9)reproduces the well-known formula
for the screening form factor.25
Finally, the function Gsqdappears in Eq. (7)to allows for
the local field corrections associated with the many-body in-
teraction in the 2DHG. Within Hubbard’s approximation, inwhich merely the exchange effect is included, it holds
39
Gsqd=q
2sq2+kF2d1/2. s10d
At very low (zero)temperatures the holes in a strained
SiGe layer are expected to experience the following possiblescattering mechanisms: (i)alloy disorder due to random fluc-
tuations in the constituent, (ii)surface roughness due to ran-
dom fluctuations in the position of the potential barrier, (iii)
deformation potential, and (iv)piezoelectric charges.The lat-
ter two are random effects arising from combination of lat-tice mismatch and interface roughness (not confused withrelevant scatterings due to acoustic waves ).
17–19The total
relaxation time is then determined by
1
ttot=1
tAD+1
tSR+1
tDP+1
tPE. s11d
III. AUTOCORRELATION FUNCTIONS
FOR SCATTERING MECHANISMS
A. Alloy disorder
As evidently seen from Eq. (6), in our calculation of the
disorder-limited mobility the autocorrelation function inwave vector space kuUsqdu
2ltakes a key role. Thus, we ought
to specify it for the earlier-mentioned sources of scattering.
For the 2DHG located on the side of a SiGe layer, the
autocorrelation function for alloy disorder scattering is sup-plied in the form
29,30
kuUADsqdu2l=xs1−xdual2V0E
0L
dzz4szd, s12d
wherexdenotes the Ge content, ualis the alloy potential, L
the SiGe layer thickness. The volume occupied by one alloy
atom is given by V0=aal3sxd/8, withaalsxdthe lattice constant
of the alloy.
By means of Eq. (2)for the lowest-subband wave func-
tion, this is rewritten in terms of the dimensionless wavenumber in the well b=kLas follows:
kuU
ADsqdu2l=xs1−xdual2V0B4b2
Lfc4p0s2bd+4c3p1s2bd
+6c2p2s2bd+4cp3s2bd+p4s2bdg. s13d
Hereafter, we have introduced auxiliary functions plsvdsl
=0−4 dof the variables vandb, defined by
plsvd=bl
vl+1S1−e−vo
j=0lvj
j!D, s14d
withlan integer.
In the limiting case of infinitely deep QWs sB=1/˛2,c
=0din which the SiGe thickness is so large that the hole state
is localized essentially within the SiGe layer sb.1d, Eq. (13)
reproduces the autocorrelation function for alloy disorder
employed previously.9,12In the opposite case of a thin SiGe
layer sb,1d, the hole state overlaps merely in part with the
alloy, so that the probability of alloy disorder scattering be-
comes smaller.
B. Surface roughness
We are now treating scattering of confined charge carriers
from a rough potential barrier of a finite height V0. The scat-
tering potential is due to fluctuations in the position of thebarrier.The average scattering potential in wave vector spaceis fixed by the value of the envelope wave function at thebarrier plane sz=0daccording to
25
USRsqd=V0uzs0du2Dq, s15d
where Dqis a Fourier transform of the interface profile.LOW-TEMPERATURE MOBILITY OF HOLES IN PHYSICAL REVIEW B 70, 195336 (2004 )
195336-3To estimate the average potential for surface scattering
specified by Eq. (15)with the use of a variational wave func-
tion, we are to adopt the following relation:
V0uzs0du2=E
0‘
dzuzszdu2]V
]z. s16d
Here,Vszdmeans the Hartree potential induced by the deple-
tion charge density and the hole density distribution, given
by(Ref. 29 ):
Vszd=4pe2
«LNdz+4pe2
«LpsE
0z
dz8E
z8‘
dz9uzsz9du2.s17d
It should be remarked that Eq. (16)is similar to the rela-
tion due to Matsumoto and Uemura.25However, the former
involves, on the left-hand side, the wave function at the bar-rier plane
zs0d, whereas the latter its derivative dzs0d/dz.
Accordingly, the former is exact and applicable for any value
ofV0, whereas the latter is approximate and applicable
merely for large enough V0.
Upon inserting the lowest-subband wave function from
Eq.(2)into Eqs. (16)and(17), we have
V0uzs0du2=4pe2
«LB2FNdsc2+2c+2d
+ps
2B2sc4+4c3+8c2+8c+4dG. s18d
With the help of Eqs. (15)and(18), we are able to obtain
the autocorrelation function for surface roughness scatteringin finitely deep triangular QWs in the form
kuU
SRsqdu2l=S4pe2
«LD2
B4FNdsc2+2c+2d+ps
2B2sc4+4c3
+8c2+8c+4dG2
kuDqu2l. s19d
ForV0!‘, this reproduces the probability for surface
roughness scattering in an infinitely deep well.12,13,25
Thus, the obtained autocorrelation function depends on
the spectral distribution of the interface profile. For simplic-ity, this has usually been chosen in a Gaussian form.
25How-
ever, no real justification has been provided for this assump-tion. For the case of Si/SiGe heterostructures, Feenstra andco-workers
20measured the surface morphology by means of
atomic force microscopy and indicated that the Fourier spec-trum for the surface roughness contains three distinct com-ponents, each described better by a power-law distribution
kuD
qu2l=pD2L2
s1+q2L2/4ndn+1. s20d
Here Dis the roughness amplitude, Lis a correlation length,
andnis an exponent specifying the falloff of the distribution
at large wave numbers.
C. Deformation potential
Next, we turn to the study of scattering mechanisms
which appear as a result of combination of the lattice mis-match and surface roughness effects. In what follows, we are
concerned with a SiGe layer grown pseudomorphically on a(001)Si substrate.
It is well known
40,41that if the Si/SiGe interface is ideal,
i.e., absolutely flat, the strain field in the SiGe layer is uni-form and has vanishing off-diagonal components. Its in-plane component is defined in terms of the lattice constantsof the Si and alloy layers by
eisxd=aSi−aalsxd
aalsxd, s21d
where the Ge content xdependence is explicitly indicated.
The strain in the alloy is demonstrated to bring about a shiftof the band edges of its conduction and valence bands.
42–46
As mentioned before, because of interface roughness the
strain field in the SiGe layer is subjected to random varia-tions and its off-diagonal components become nonzero.
15–19
These fluctuations in turn give rise to a random nonuniformshift of the band edges. This means that the electrons in theconduction band and the holes in the valence one must ex-perience a random deformation potential. In the existing cal-culations of the 2DHG mobility it has been assumed
9,11,12
that the perturbating potential for holes in the SiGe layer isalmost identical to that for electrons, having one and thesame shape described by Eq. (22)later, only with a different
value of the coupling constant J
u.
Nevertheless, this assumption is invalid. Indeed, with the
use of the strain Hamiltonian for a semiconductor crystal ofcubic symmetry, it has been proved (see, e.g., Refs. 42–46 )
that the impacts of the strain field on electrons and on holesare quite different. This field is calculated within a simpleapproach to cubic symmetry,
18in which the deviation from
isotropy is taken into account in terms of an anisotropyratio.
47The volume dilation is then found unaffected by
strain fluctuations, being uniform in space. As a result, thedeformation potential for electrons in the conduction band isfixed by a single diagonal component of the strainfield
16,42–44
UDPscd=Juezz, s22d
while that for holes in the valence band is fixed by all its
components43–46
fUDPsvdg2=bs2
2fsexx−eyyd2+seyy−ezzd2+sezz−exxd2g
+ds2fexy2+eyz2+ezx2g, s23d
withbsanddsas shear deformation potential constants. Here
eijdenote the roughness-induced variations in the strain field
components. Therefore, in our calculation of the 2DHG mo-bility limited by deformation potential scattering, we willadopt Eq. (23)rather than Eq. (22)assumed previously.
9,11,12
Upon putting the strain fluctuations eijderived in Refs. 17
and 18 into Eq. (23), we readily get a 2D Fourier transform
for the perturbating potential for holes in the SiGe layer asfollows:QUANG et al. PHYSICAL REVIEW B 70, 195336 (2004 )
195336-4UDPsvdsq,zd=ae i
2qDqe−qzF3
2fbssK+1dg2s1+sin4u+cos4ud
+SdsG
4c44D2
s1+sin2ucos2udG1/2
, s24d
for 0 łzłLand is zero elsewhere, with Lthe SiGe layer
thickness, and q=sq,uda 2D wave vector in polar coordi-
nates. Here ais the anisotropy ratio of the alloy
a=2c44
c11−c12; s25d
KandGare its elastic constants
K=2c12
c11,G=2sK+1dsc11−c12d, s26d
withc11,c12, andc44as its elastic stiffness constants.
Upon averaging Eq. (24)by means of the lowest-subband
wave function from Eq. (2), we may represent the autocor-
relation function for deformation potential scattering of holesin terms of the dimensionless variables t=q/kandb=kLby
kuU
DPsvdsqdu2l=SB2b2ae i
2LD2
t2fc2p0sb+btd+2cp1sb+btd
+p2sb+btdg2F3
2fbssK+1dg2s1+sin4u
+cos4ud+SdsG
4c44D2
s1+sin2ucos2udGkuDqu2l,
s27d
in which plsvdsl=0−2 dare functions given by Eq. (14)with
the variable v=b+bt.
D. Piezoelectric charges
As already mentioned in Sec. I, some experimental evi-
dences for piezoelectricity of the strained SiGe layer in aSi/SiGe heterostructure have been found.
21–24However, if
the Si/SiGe interface is ideal, the strain field in the alloy hasvanishing off-diagonal components, so that the SiGe layerexhibits neither a piezoelectric polarization nor any piezo-electric field.
In fact, because of surface roughness the off-diagonal
components of the strain field in the SiGe layer of an actualSi/SiGe system become nonzero and randomlyfluctuating.
15–19Therefore, they induce a piezoelectric polar-
ization and a corresponding fluctuating density of piezoelec-tric charges, which are bulklike distributed in a rather narrowregion inside of the alloy and near the Si/SiGe interface.
17–19
These charges in turn create a random piezoelectric field.
The potential energy for a hole of charge ein this field is
described by a 2D Fourier transform as follows:17,18
UPEsq,zd=3pee14Gae i
4«Lc44qDqFPEsq,z;Ldsin2 u,s28d
wheree14is the piezoelectric constant of the strained SiGe
layer. The form factor for the piezoelectric potential in Eq.(28)is supplied by
18FPEsq,z;Ld
=1
2q5eqzs1−e−2qLdforz,0,
e−qzs1+2qzd−e−qs2L−zd, for 0 łzłL,
2qLe−qzforz.L. 6
s29d
Upon averaging Eqs. (28)and (29)by means of the
lowest-subband wave function from Eq. (2), we arrive at the
autocorrelation function for piezoelectric scattering in theform
kuU
PEsqdu2l=S3pee14Gae i
8«Lc44D2
FPE2sq/kdsin22ukuDqu2l.
s30d
Here the weighted piezoelectric form factor is defined as a
function of t=q/kby
FPEstd=A2a
t+as1−e−2btd+B2bH2c2t
t+1+4ct
st+1d2+4t
st+1d3
+c2s1−2btdp0sb+btd+2cs1+ct−2btdp1sb+btd
+s1+4ct−2btdp2sb+btd+2tp3sb+btd
−e−2btfc2p0sb−btd+2cp1sb−btd+p2sb−btdgJ,
s31d
where as before a=k/k, andplsvdsl=0−3 dare given by Eq.
(14)with the variables v=b±bt.
Thus, within the realistic model of finitely deep triangular
QWs described by the modified Fang-Howard wave function(2), we may rigorously derive the autocorrelation functions
in an analytic form for the scattering mechanisms of interest.These are supplied by Eqs. (13),(19),(27), and (30)for alloy
disorder, surface roughness, deformation potential, and pi-ezoelectric charges, respectively. It is to be noted that thelatter two show up in a dependence not only on the magni-tude of the wave vector but its polar angle as well.
IV. RESULTS AND DISCUSSIONS
A. Choice of input parameters
In this section, we are trying to apply the foregoing theory
to explain the experimental data3,8,11,12about the low-
temperature transport of holes located near an interface ofthe strained SiGe layer as the conduction channel in aSi/SiGe/Si sandwich configuration.
For numerical results, we have to specify parameters ap-
pearing in the theory as input. As always with mobility cal-culations, one has many adjustable fitting parameters. In or-der that the justification of our theory is so independent aspossible of the choice of fitting parameters, these are to bededuced from measurements of other physical propertiesthan the hole mobility.
The lattice constants, elastic stiffness constants, dielectric
constants, and shear deformation potentials for Si and Ge areLOW-TEMPERATURE MOBILITY OF HOLES IN PHYSICAL REVIEW B 70, 195336 (2004 )
195336-5taken from Refs. 44 and 48 and listed in Table I. The corre-
sponding constants for the alloy are estimated according tothe virtual crystal approximation
49except the lattice constant
given by an empirical rule, Eq. (32)later. The potential bar-
rier height of triangular QWs for holes is the valence bandoffset between the Si and SiGe layers, which increases nearlylinearly with the Ge content xasV
0=0.74xeV.1
For thexdependence of the lattice constant of a SiGe
alloy, we use the experimental data, approximated analyti-cally by
50,51
aalsxd=aSis1−xd+aGex−gxs1−xd, s32d
with a parameter g=1.88 310−2Å.
It is to be noted1that the built-in biaxial compressive
strain in a SiGe layer produces a reduction in its in-planeheavy-hole mass, thus leading to an increase of the 2DHGmobilities. The xdependence of the mass is given by a linear
interpolation to fit to the experimental data
52–54
m*sxd/me=0.44−0.42 x. s33d
We are now concerned with choosing the coupling con-
stants of interest. The alloy potential was taken as equal tou
al=0.6 eV.12,13,52The value of the piezoelectric constant
e14=1.6 310−2C/m2was extracted from power loss
measurement21for a SiGe alloy with a Ge content x=0.2.
Furthermore, in view of the fact that piezoelectricity of theSiGe alloy in a lattice-mismatched structure is induced bystrain, for a crude estimate we scale the relevant couplingconstant by the strain ratio.Then, for the xdependence of the
piezoelectric constant it holds
e
14sxd=1.6 310−2eisxd
eis0.2dsC/m2d. s34d
It should be kept in mind that all scattering mechanisms
under consideration depend, in general, strongly on the Gecontent because not only the effective mass but, as seen ear-lier, the lattice mismatch and the piezoelectric constant de-pend on it.
Next, we turn to the characteristics of the interface profile.
It has recently been shown
16,20that for a Si/SiGe hetero-
structure, the most important component of the surfaceroughness is connected with elastic strain relaxation in thechannel layer. Moreover, the experimental data about theelectron mobility in Si/SiGe n-channel heterostructures
suggested
16,20that the correlation length of this component is
L&300 Å, its roughness amplitude D=5–15Å,anditsex-
ponent of the power-law distribution n&4, varying remark-
ably from device to device. For our numerical calculations,we take L=290 Å, D=15.5 Å, and n=4. The somewhat
large value of the roughness amplitude may be explained interms of the segregation and clustering effects, which are
indicated
55,56to be the main reasons for roughening of the
interfaces in a strained layer.
It should be noted that our choice of the interface profile
characteristics satisfies the condition
D/L!1. s35d
This is claimed16in order that the theory of roughness-
induced fluctuations in strain15–19and, hence, that of defor-
mation potential and piezoelectric scatterings presented inSec. III, are justified.
B. Numerical results and comparison with experiment
By means of Eqs. (5)and (6), we have calculated the
low-temperature 2DHG mobilities limited by different scat-tering mechanisms: alloy disorder
mAD, surface roughness
mSR, deformation potential mDP, piezoelectric charges mPE,
and overall mobility mtot; employing Eqs. (13),(19),(27),
(30), and (11), respectively. For device applications, one is
interested in their variation with hole density psand Ge con-
tentx. The theoretical results are to be compared with recent
experimental data.3,8,11,12
As a first illustration, we are dealing with the Si/SiGe
sample studied in Ref. 12 This is specified by a fixed Gecontentx=0.2 and a depletion charge
10Nd,531011cm−2,
while the hole density psis varying.
We need to estimate the effect due to the finiteness of the
potential barrier height on the hole mobilities. This is to bemeasured by the ratio between the values of a partial mobil-ity calculated with a finite and an infinite barrier
Q=
mfin/minfin. s36d
The mobility ratios for the various scattering sources are de-
picted in Fig. 1 versus sheet hole density ranging from ps
=131011–5.5 31011cm−2for a SiGe layer thickness L
=200 Å.
The partial 2DHG mobilities of the sample under study
are plotted versus hole density from ps=1.5 31011–5.5
31011cm−2in Fig. 2, where the 4 K experimental data re-
ported in Ref. 12 is reproduced for a comparison. In addition,these are plotted in Fig. 3 versus alloy layer thickness fromL=20–90 Å for a hole density p
s=231011cm−2.
From the lines thus obtained we may draw the following
conclusions.
(i)Figure 1 reveals that the model of infinitely deep tri-
angular QWs overestimates surface roughness and piezoelec-tric scatterings: Q
SR,QPE.1, while this underestimates alloy
disorderanddeformationpotentialscatterings: QAD,QDP,1.
The finite-barrier effect is found to be small compared withTABLE I. Material parameters used: aas the lattice constant (Å),cijas the elastic stiffness constants
s1010Pad,«Las the dielectric constant, and bsanddsas the shear deformation potential constants (eV).
Material ac 11 c12 c44 «L bs ds
Si 5.430 16.6 6.39 7.96 11.7 −2.35 −5.32
Ge 5.658 12.85 4.83 6.80 15.8 −2.55 −5.50QUANG et al. PHYSICAL REVIEW B 70, 195336 (2004 )
195336-6the one in the case of square QWs,18where the mobility
ratios may become very large, e.g., QSR*10 for a narrow
square well of a thickness &100 Å. The overestimation of
surface scattering is distinct from the earlier statement29con-
cerning GaAs/AlGaAs triangular QWs that surface scatter-ing is independent of their barrier height because the over-lapping of the envelope wave function with the barrier wasneglected.
(ii)It is clearly seen from Fig. 2 that the calculated overall
mobility
mtotspsdalmost coincides with the 2DHG mobility
obtained experimentally12in the region of carrier densitiesused. This exhibits a significant increase when raising the
hole density from ps=1.5 31011–5.5 31011cm−2. The func-
tions mSRspsd,mDPspsd, and mPEspsdare found to increase
with a rise of ps, whereas mADspsdto decrease. The sharp
contrast between the variation tendencies with psof the ex-
perimental data mexptspsdandmADspsdimplies that in terms of
alloy disorder alone one cannot explain the hole density de-
pendence of the measured mobility. Moreover, it has beenpointed out that in terms of alloy disorder one cannot under-stand even qualitatively its dependence on growthtemperature
1,2,7and Si cap thickness.2
It is worthy to recall9–13that with screening included all
the so far-known scattering mechanisms (impurity doping,
alloy disorder, and surface roughness )are unable to explain
the observed data about the 2DHG mobility in strained SiGealloys. Therefore, the existing theories had to invoke the un-clear concept of interface charged impurities with a high fit-ting density, up to ,10
11cm−2, which is equivalent to an
intentional doping at an intermediate level.
Moreover, in several cases the interface had to be as-
sumed to be quite rough with a small exponent of the power-law distribution, a large ratio between the roughness ampli-tude and correlation length and a small value of the latter,e.g.,n,2,D/L,1,L,7Å (Ref. 11 ), andn,1,D/L
,0.5, L,19 Å (Ref. 12 ). With these large values of D/L
the theory of deformation potential scattering adopted in theearlier calculations
9,11,12may fail to be valid. In addition, in
the case of short correlation lengths, the functions mSRspsd
and mDPspsdwere found to decrease with a rise of ps,1,9
which is in opposite to our result. However, such surface
morphologies seem to be suspect.3
(iii)An examination of the different lines in Fig. 2 indi-
cates that for low carrier densities ps,431011cm−2, surface
roughness and deformation potential scatterings are domi-nant mechanisms, whereas alloy disorder one is less relevant,which is in accordance with the previous theories.
1,7,9,10,13
For higher densities ps*431011cm−2the latter is compa-
rable with the former two. Furthermore, piezoelectric scatter-ing is found to be negligibly weak at a rather low Ge contentsx=0.2 d.
FIG. 1. Ratio Q=mfin/minfinbetween the 2DHG mobilities, cal-
culated with a finite and an infinite potential barrier, vs sheet holedensityp
sfor various scattering mechanisms: alloy disorder QAD,
surface roughness QSR, deformation potential QDP, and piezoelec-
tric charges QPE. The triangular QW is made from Si/Si 0.8Ge0.2
with a barrier height V0=0.148 eV and a SiGe layer thickness L
=200 Å.
FIG. 2. Different 2DHG mobilities of the Si/Si 0.8Ge0.2QW in
Fig. 1 vs hole density ps. The solid lines show the calculated mo-
bilities limited by: alloy disorder mAD, surface roughness mSR, de-
formation potential mDP, piezoelectric charges mPE, and overall mtot.
The 4 K experimental data reported in Ref. 12 are marked bysquares.
FIG. 3. Different 2DHG mobilities of the Si/Si 0.8Ge0.2QW in
Fig. 1 vs SiGe thickness L. The interpretation is the same as in Fig.
2.LOW-TEMPERATURE MOBILITY OF HOLES IN PHYSICAL REVIEW B 70, 195336 (2004 )
195336-7(iv)It follows from Fig. 3 that the hole mobilities depend
weakly on the SiGe layer thickness. For L*60 Å, they are
almost independent thereof.
Next, we turn to treating the Ge content dependence of
the partial 2DHG mobilities in a Si/Si 1−xGextriangular QW.
These are plotted versus Ge content varying from x=0.05 to
0.5 for a fixed carrier density ps=231011cm−2and a deple-
tion charge10Nd,531010cm−2in Fig. 4, where the 4 K
experimental data3,8,11are reproduced for a comparison.
There is also presented the hole mobility reported in Ref. 3,which was calculated simply on the basis of alloy disorderscattering alone by neglecting screening and taking a largervalue of the alloy potential: u
al=0.74 eV. As quoted earlier,
the SiGe layer thickness is of minor importance, so we maychose some value, say L=200 Å.
From the solid lines obtained in Fig. 4 we may draw the
following conclusions.
(i)The calculated overall mobility
mtotsxdoffers a good
quantitative description of the pronounced monotonic de-
crease of the experimentally observed 2DHG mobility whenraising Ge content.
3,7,9It is to be noted that most earlier
theoretical studies27,57predicted, in contrast, an increase of
the hole mobility with higher x, based on the strain-induced
reduction in the effective hole mass.
(ii)The functions mDPsxdandmPEsxdshow up in a fast
monotonic decrease, whereas mADsxdandmSRsxdin a mini-
mum. It is seen from Eqs. (27),(30), and (34)that the prob-
abilities for deformation potential and piezoelectric scatter-ings depend quadratically on the lattice mismatch
eisxd, and
the latter also depends quadratically on the piezoelectric con-
stante14sxd. Since eisxdande14sxdincrease when raising x,
the fast increase in the scattering probabilities overwhelms
the reduction in the effective hole mass with higher x, thusleading to an overall decrease of mDPsxdandmPEsxd.I na d -
dition, the distinction between the variation tendencies with x
of the experimental data mexptsxdandmADsxdimplies that in
terms of alloy disorder alone one cannot explain the Ge con-
tent dependence of the measured mobility.
(iii)Surface roughness scattering is found to be most im-
portant at a very low Ge content x,0.05, while the defor-
mation potential scattering to dominate the overall hole mo-bility
mtotsxdforx.0.1, leading to its decrease with higher x.
As mentioned earlier, the effect due to piezoelectric charges
increases very rapidly with a rise of x. So, this is negligibly
small for x&0.3, however, this becomes comparable with
alloy disorder and surface roughness scatterings for x*0.4,
and with the deformation potential one at a higher x,0.5.
It is worth mentioning that the roughness amplitude is
expected55to be increased with strain and, hence, with Ge
contentx. Therefore, with an increase of xthe roughness
amplitude-dependent scattering mechanisms such as surfaceroughness, deformation potential and piezoelectric chargesbecome more important compared to alloy disorder one.
(iv)An inspection of the dashed line in Fig. 4 indicates
that the 2DHG mobility calculation in Ref. 3, based on alloydisorder scattering alone, results in a shallow minimum atx,0.4 and, hence, is unable to supply a satisfactory descrip-
tion of the observed data. In particular, for x=0.5 this calcu-
lation even without screening and with a large alloy potential
su
al=0.74 eV dgives: mADunscr=6.7 3103cm2/Vs, which is
found too large to explain the 4 K experimental data reported
in Ref. 11, which our theory may give: mtot,mexpt=1.2
3103cm2/Vs. The situation becomes much worse when
screening included: mADscr=3.5 3104cm2/Vs.
C. Validity of the single-subband model
To end this section, we verify the validity of the assump-
tions made in our hole mobility calculation.
First, it should be emphasized that our theory of hole
transport is to be developed for 2DHGs studied experimen-tally in Refs. 3, 8, 11, and 12; namely at very low tempera-ture s&4Kdand rather low carrier density s&5
310
11cm−2d, which correspond to small values of the ener-
gies of interest: kBT&0.3 meV and EF&4 meV. At the so
low temperatures phonon scattering is obviously negligiblyweak.
11,13,28
As quoted before, it follows from the band structure
calculation1,26–28for a strained SiGe layer grown on relaxed
(001)Si that the ground-state subband is the lowest heavy-
hole HH1. In addition, Laikhtman and Kiehl28have shown
that the heavy-hole–light-hole band splitting HH1–LH1 andthe subband splitting HH1–HH2 due to the strain and quan-tum confinement effects are both larger than 70 meV. Thesesplittings turn out to be at least one order of magnitudegreater than the thermal and Fermi energies. Therefore, theassumption of intraband scattering within the lowest heavy-hole subband HH1 is firmly confirmed.
1,9–13,28,35Further,
Leadley and co-workers11indicate that the single-subband
model is still a reasonable approximation for strainedSi
0.5Ge0.5at higher temperature s300 K dand higher carrier
density s331012cm−2d.
FIG. 4. Different 2DHG mobilities of the Si/Si 1−xGextriangular
QW vs Ge content xunder a SiGe layer thickness L=200 Å and a
hole density ps=231011cm−2. The interpretation is the same as in
Fig. 2. The dashed line refers to the calculation based on alloydisorder alone without screening and with an alloy potential u
al
=0.74 eV. The 4 K experimental data reported in Refs. 3 and 8 are
marked by filled squares, and in Ref. 11 by open one. [Note the
ordinate axis scale is distinct from that in Figs. 2 and 3. ]QUANG et al. PHYSICAL REVIEW B 70, 195336 (2004 )
195336-8It should be remarked that the earlier scattering model is
to be distinguished from the one employed in thecalculation
58–60of the hole mobility in a strained Si layer at
high temperature s,300 K dand high carrier density s,5
31012cm−2d. In this case, phonon scattering is important.
The band structure calculation due to Fischetti and
co-workers60reveals that the ground-state subband is the
lowest heavy-hole HH1 or light-hole LH1 for compressive ortensile stress, respectively. Moreover, the subband splittingsare found to be of the order of the thermal and Fermi ener-gies. Therefore, a multisubband model is mandatory.
Further, the existing band structure calculation
28also
shows that for a strained SiGe layer on relaxed (001)Si, the
subbands are strongly nonparabolic and anisotropic exceptfor the lowest heavy-hole one. The subband HH1 is isotropicand parabolic over a relatively large range of the 2D wavevector, which corresponds to a large range of the carrier den-sity, up to 10
13cm−2.
At last, based on the assumption of intraband scattering
within the isotropic parabolic subband HH1, we have beensuccessful in the quantitative explanation of the observeddependences of the hole mobility on Ge content
3,8,11as well
as carrier density.12This success provides a real justification
for the adopted model.
V. SUMMARY
In this paper we have presented a theory of the low-
temperature mobility of holes in strained SiGe layers ofSi/SiGep-channel heterostructures, getting rid of the unclear
concept of interface impurity charges. Instead, we took ad-equate account of the random deformation potential and ran-dom piezoelectric field in actual strained SiGe layers.
We have proved that the 2DHG transport in an undoped
systems is, in general, governed by the following scatteringmechanisms: alloy disorder, surface roughness, deformation
potential, and piezoelectric charges. The latter two arise as acombined effect from lattice mismatch and interface rough-ness.
Scatterings by deformation potential and piezoelectric
charges rapidly increase when raising the Ge content x.Thus,
the former may become dominant for x*0.2, whereas the
latter may be one of the principal processes limiting the holemobility for x*0.4.
In the regions of hole density and Ge content in use, sur-
face roughness and deformation potential scatterings arefound to be most important. In combination with the othersources of scattering, these enable a good quantitative expla-nation of the recent experimental findings both about thedependence of the low-temperature 2DHG mobility on holedensity and its decrease with Ge content as well.
It is worth mentioning the counteracting strain effects in
Si/SiGep-channel heterostructures. The strain leads to the
lifting of the degeneracy of the valence band of SiGe and,hence, to a suppression of intersubband scattering and a re-duction of the effective mass, so enhancing the hole mobility.On the other hand, the roughness-induced strain fluctuationsgive rise to new scattering sources, viz. deformation poten-tial and piezoelectric charges, so reducing the mobility.
ACKNOWLEDGMENTS
The authors would like to thank Professor M. Saitoh, De-
partment of Physics, Osaka University, Japan for valuablediscussion and ProfessorT. E.Whall, Department of Physics,University of Warwick, UK for useful communications. Oneof the authors (D.N.Q. )acknowledges support from the Ja-
pan Society for the Promotion of Science, under which thiswork was done.
*Present address: Center forTheoretical Physics,VietnameseAcad-
emy of Science and Technology, P.O. Box 429, Boho, Hanoi10000, Vietnam.
†Present address: Department of Physics, Florida State University,
Tallahassee, Florida 32306-4350.
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195336-10 |
PhysRevB.60.11439.pdf | Low-temperature spin dynamics of doped manganites: Roles of Mn t2g,M neg, and O 2 pstates
Priya Mahadevan
JRCAT-Angstrom Technology Partnership, 1-1-4 Higashi, Tsukuba, Ibaraki 305-0046, Japan
I. V. Solovyev
JRCAT-Angstrom Technology Partnership, 1-1-4 Higashi, Tsukuba, Ibaraki 305-0046, Japan
and Institute of Metal Physics, Russian Academy of Sciences, Ekaterinburg GSP-170, Russia
K. Terakura
JRCAT-NAIR, 1-1-4 Higashi, Tsukuba, Ibaraki 305-8562, Japan
and Institute of Industrial Science, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106-8558, Japan
~Received 26 April 1999 !
The low-temperature spin dynamics of doped manganites have been analyzed within a tight-binding model,
the parameters of which are estimated by mapping the results of ab initiodensity-functional calculations onto
the model. This approach is found to provide a good description of the spin dynamics of the doped manganites,observed earlier within the ab initiocalculations. Our analysis not only provides some insight into the roles of
thee
gand thet2gstates but also indicates that the oxygen pstates play an important role in the spin dynamics.
This may cast doubt on the adaptability of the conventional model Hamiltonian approaches to the analysis ofspin dynamics of doped manganites. @S0163-1829 ~99!06839-3 #
There has been resurgence of interest in transition-metal
oxides with the perovskite structure owing to their widerange of electronic and magnetic properties. Among them,the hole-doped manganites
1have been occupying a special
position: they exhibit dramatic phenomena like colossalmagnetoresistance and are being intensively studied with
prospect for technological applications. LaMnO
3, the parent
material of the manganites, is an antiferromagnetic insulator.
Upon sufficient doping ( x;0.15) with divalent ions ~such as
Sr, Ca !, the system is driven metallic. The holes are allowed
to move only if adjacent spins are parallel, which results in adramatic increase in the conductivity when the spins orderferromagnetically, an effect that can be induced by applyinga magnetic field or by lowering the temperature below the
Curie temperature T
c. Thus, the carrier mobility is inti-
mately related to the underlying magnetic state of the sys-tem, and there have been considerable efforts in recent timesto identify the interactions that control the magnetoresistiveproperties. An approach in this direction has been to analyzethe spin dynamics of the doped manganites.
Early experiments on La
0.7Pb0.3MnO3~Ref. 2 !indicated
that the spin-wave dispersion v(q) in the doped manganites
could be interpreted in terms of a conventional Heisenbergferromagnet with only the nearest-neighbor exchange cou-pling. This behavior is consistent with the double-exchangelimit of the one-band ferromagnetic Kondo lattice model,
3
implying that conduction egelectrons move in a tight-
binding band with one orbital per site and interact with lo-
calizedt2gspins via the large intra-atomic exchange JH.
However, more recent experiments4on other doped manga-
nites have found strong deviation of v(q) from the simple
cosinelike behavior expected from a nearest-neighborHeisenberg model. Farther-neighbor interactions in additionto the nearest-neighbor one had to be taken into account to
reproduce softening of the dispersions for the wave vector qapproaching the zone boundary. The one-band models could
not explain the observed zone-boundary softening, evenqualitatively.
3,5It was then suggested that additional degrees
of freedom, probably the lattice degrees of freedom, mayplay an important role in the spin dynamics of these materi-als. Recently,
6it was shown that the softening at the zone
boundary has a purely electronic origin, and could be ex-plained within the framework of ab initio density-functional
band calculations, in the local-spin-density approximation~LSDA !.
The previous work
6also carried out a perturbative analy-
sis of the exchange interaction strengths within a tight-
binding model considering the double degeneracy of eglev-
els on the Mn site. It was argued that the degeneracy of eg
orbitals plays important roles, and simply by taking into ac-
count the proper structure of the kinetic hopping between
nearest-neighbor eglevels one can, to a large extent, under-
stand the behavior of two strongest interactions, J1andJ4,i n
the half-metallic regime. Here, Jkcorresponds to the ex-
change interaction between the kth neighbor atoms,7as de-
fined later by Eq. ~2!. Furthermore, it was pointed out that a
realistic model including the oxygen porbitals and the Mn
t2gorbitals could indeed modify the quantitative aspects of
the results. While the fully filled majority-spin t2gorbitals
could contribute an antiferromagnetic superexchange compo-
nent toJ1, the partially filled minority spin t2gorbitals could,
as suggested by the ab initio band structure calculations,
contribute a ferromagnetic double-exchange component.
Further, J2was found to increase quite strongly in the
eg-only model, unlike the weak dependence seen in the ab
initioresults. The previous work6suggested that the oxygen
pbands could modify J2considerably as the Mn 3 d-O 2p
energy separation is comparable with the exchange splittingof the majority- and minority-spin orbitals. In the light ofthese observations, we have attempted to make further quan-PHYSICAL REVIEW B 15 OCTOBER 1999-II VOLUME 60, NUMBER 16
PRB 60 0163-1829/99/60 ~16!/11439 ~5!/$15.00 11 439 ©1999 The American Physical Societytitative analysis to understand the origin of the observed
zone-boundary softening. This has been done by mappingthe results of the ab initioband structure calculations onto a
tight-binding model that gives us flexibility of constructingsimpler models and analyzing the contributions to the ob-served softening.
The band structure for hypothetical cubic ferromagnetic
LaMnO
3with the lattice parameter of 3.934 Å, calculated
within the linear-muffin-tin orbital method with the atomicsphere approximation ~LMTO-ASA !, was mapped onto a
nearest-neighbor tight-binding model
8that had been found to
give a good description of the electronic structure of the
transition-metal oxides of the form La MO3, where M
5Ti-Ni. The tight-binding Hamiltonian consists of the bare
energies of the transition metal d(ed) and the oxygen p(ep)
states and hopping interactions between the orbitals onneighboring atoms. The nearest-neighbor hopping interac-tions were expressed in terms of the four Slater-Koster pa-
rameters, namely pp
s,ppp,pds, andpdp. Note that no
directd-dhopping was taken into account. While p-dcova-
lency effects lift the degeneracy of the dorbitals, an addi-
tional interaction sdsbetween the transition metal dand
oxygen 2 sorbitals was required to lift the degeneracy at the
Gpoint.9The energy of the oxygen 2 slevel was fixed at
220 eV. In order to obtain the magnetic ground state within
the single-particle tight-binding model, we have introduced
an extra parameter ( epol) that is the bare energy difference
between the up- and down-spin delectrons at the same site.
An additional splitting, ( epol8) was introduced between the
up- and down-spin dorbitals of egsymmetry.10The param-
eters entering the tight-binding Hamiltonian were determinedby the least-squares fitting of the energies obtained fromtight-binding calculations at several kpoints to those ob-
tained from the LMTO calculations. It should be noted that
the deep-lying oxygen 2 sbands were not involved in the
fitting. The extracted parameters are sd
s521.57 eV, pps
50.91 eV, ppp520.23 eV, pds522.02 eV, pdp
51.0 eV, ed2ep50.48 eV, epol53.2 eV, and epol8
50.3 eV being consistent with the earlier estimate for the
system.8
The frozen spin spiral approximation,11where the orien-
tation of the magnetic moment at each atomic site is spirally
modulated by the wave vector q, was used to calculate the
exchange interaction Jqdefined by
Jq5(
kJkexp@iqRk#, ~1!
whereRiis the position vector of ith Mn atom. Jkis thekth
neighbor exchange interaction appearing in the HeisenbergHamiltonian given by
E
@$ei%#521
2(
ikJkeiei1k, ~2!
witheidenoting the direction of the magnetic moment at the
sitei. By using the local force theorem,12the changes in the
single-particle energy could be related to the exchange inter-action by mapping onto the Heisenberg model as definedabove. A rigid band picture was adopted to simulate the dop-ing effects.
13Simplified models were constructed to eluci-date the mechanism of zone-boundary softening of the spin
wave, and comparison was made with the results from theLMTO calculations whenever possible to ensure that thepresent result is not an artifact of a particular parameter set.
In Fig. 1 ~a!we show the LMTO results for the spin dis-
persion
v(q;x) along the symmetry directions GX,XM, and
MRcalculated for several doping values x. In the small q
region, the spin excitations have a weak dependence on dop-
ing as is evident from the result along the GXdirection.
However, sufficiently away from the Gpoint, the results be-
come a strong function of the concentration x. Considering
the result for x50.4 along the GXdirection, we see that the
spin dispersion is almost flat from midway to the zone
boundary. The experimental result for Pr 0.63Sr0.37MnO3~Ref.
4!along GXshows very similar behavior. The results of the
tight-binding model, calculated by using the parameters ex-tracted by fitting the ab initio band structure are shown in
Fig. 1 ~b!. This model calculation is called model Ain order
to distinguish from other models discussed later. Model Ais
seen to provide a good description of the energetics of thespin dynamics observed within the ab initio approach. The
results in Fig. 1 suggest that the difference in energy betweenthe ferromagnetic ground state and the various commensu-rate antiferromagnetic ~AF!spin configurations such as those
defined at the X(A-type AF !,M(C-type AF !, andR(G-type
AF!points decreases with doping.
The Fourier transform of J
qgives us the real space ex-
change integrals Jias shown in Fig. 2. Dominant interactions
are all confined within the linear -Mn-O-Mn- chains par-
allel to ^001&as was pointed out already.6They are Jiwith
i51, 4, and 8. J2is the interaction for the pairs along ^110&
and takes relatively small values partly because of the can-
cellation between the contribution from the Mn dbands and
that from O pstates. These results are consistent with the
analysis of the experimental results4that required finite J4
andJ8to be included in the Heisenberg Hamiltonian in order
to reproduce the experimentally observed spin-wave disper-sions. In order to understand the behavior of the spin-wave
dispersion
v(q;x) in terms of Ji, the following expressions
forqixwill be useful.
\v~qx!.2@~J114J2!sin21
2qxa
1J4sin2qxa1J8sin23
2qxa#, ~3!
whereais the lattice constant of the cubic unit cell. For
qxa!1, the above expression reduces to
\v~qx!.1
2@J114J214J419J8#~qxa!2, ~4!
which helps us understand the weak dependence of the low-
energy excitations on the concentration x. The large prefac-
tors forJ4andJ8indicate that modest changes in J4andJ8
are sufficient to offset the large changes in J1found within
our model. At the Xpoint, Eq. ~3!reduces to
\vSqx5p
aD.2J118J212J8. ~5!
As the dependence of J2onxis weak, the changes at the
zone boundary are driven by J1andJ8. Unlike in the low q
regime, the prefactors of J1andJ8are equal in this case.11 440 PRB 60 PRIYA MAHADEVAN, I. V. SOLOVYEV, AND K. TERAKURASince the decrease in J1is much larger than the increase in
J8, the energy at the Xpoint decreases as the hole concen-
tration is increased. Another useful information about theflattening of the dispersion beyond half way to the zoneboundary is given by comparing Eq. ~5!with
\
vSqx5p
2aD.J114J212J41J8. ~6!
The energies at qx5p/aandqx5p/2aare comparable
whenJ1;2J4.
We constructed simpler models to make quantitative esti-
mates for the contribution from the t2gelectrons and that
from theegelectrons. Model B(C) includes eg(t2g) orbitals
on the Mn atoms and all porbitals on the oxygens. As the
density of states ~DOS!obtained within model Asuggests
partial occupancy of the minority spin t2gbands with
;0.175 electrons even for the undoped case, the Fermi en-
ergy (EF) of model Cin the undoped case was adjusted so
that the minority-spin t2gbands had 0.175 electrons. As a
consequence, the number of holes in the majority-spin eg
states of model Bin the undoped case should be larger by
0.175 than the case when the minority-spin t2gstates are not
occupied. The dpartial DOS for models BandCalong with
the result for model A, which considers all dorbitals on the
Mn atom, are shown in Fig. 3. The reduced models ( Band
C) are found to give a good description of the respective d
partial DOS of egandt2gsymmetry within model A. Further
justification for the treatment of the contributions from eg
andt2gstates separately is given by the fact that the domi-
nant contributions, ( Ji,i51,4,8), are all for the pairs along^100&for which there is no mixing of the two states in the
exchange coupling. The spin-wave dispersion was calculatedin the reduced models separately. The dispersion along the
GXdirection for model Bis shown in the panel b@inset of
Fig. 3 ~a!#.ydenotes the number of doped e
gholes with
FIG. 2. The doping dependence of the exchange couplings J1,
J2,J4, andJ8between atoms at ( a0 0), (aa0), (2a0 0),
and (3a0 0), where ais the lattice parameter.
FIG. 1. The spin-wave disper-
sions obtained by ~a!LMTO cal-
culations and ~b!the tight-binding
approach ~modelA) along the
symmetry directions GX,XM,
andMRshown as a function of
doping.PRB 60 11 441 LOW-TEMPERATURE SPIN DYNAMICS OF DOPE D...reference to the half-filled majority-spin egband. ~Equiva-
lently, 1 2yis the number of electrons in the egband. !xin
the parentheses indicates the hole concentration in model A
being equivalent to doping of divalent atoms. By considering
the above situation, y50.175 corresponds to the case of un-
doped LaMnO 3, i.e.,x50. As the number of holes increases
fromy50.175, the spin-wave energy at the Xpoint steadily
decreases. In model C, on the other hand, the dominant con-
tribution from the t2gstates to the exchange coupling is an-
tiferromagnetic superexchange. The negative spin-wave en-ergy for all q@Fig. 3 ~d!#is consistent with this expectation.
However, small occupation of the minority spin t
2gstates
produces a ferromagnetic double-exchange contribution. zin
Fig. 3 ~d!denotes the number of electrons in the minority-
spint2gstates. Clearly, doping of divalent elements reduces
zso that the double-exchange contribution diminishes rap-
idly as is clearly seen in the zdependence of the spin-wave
dispersion in Fig. 3 ~d!. Here again the corresponding valueofxin model Aare indicated in brackets. The contribution
from the egstates @Fig. 3 ~b!#and that from the t2gstates
@Fig. 3 ~d!#for the common xvalue are added and the result-
ant spin-wave dispersion shown in Fig. 3 ~e!agrees very well
with the one in Fig. 1 ~b!~modelA). This analysis suggests
that the main source of the zone-boundary softening of the
spin-wave dispersion by doping of divalent atoms for x
,0.3 is the reduction in the ferromagnetic double exchange
of thet2gelectrons. On the other hand, in the doping range
ofx.0.3, thet2gstates may simply act as a source of anti-
ferromagnetic superexchange and further softening and flat-
tening of the spin-wave dispersion comes from the egstates.
The doped holes within our model have considerable oxy-
genpcharacter, and the earlier results6suggested that the
itinerant oxygen band could modify the various exchangeinteraction strengths. It was pointed out that the role of oxy-genpstates in the superexchange interaction is not only to
mediate the d-dtransfer but also to make a direct additional
contribution.
14,15However, in this treatment the banding ef-
fect of oxygen pstates was neglected. This assumption will
not be justified for quantitative arguments if the pband width
is comparable to the p-denergy separation, which is the case
in our systems. In order to obtain information about the roleof the oxygen pband, we made further simplification in the
modelBthat the hopping between oxygen atoms was ne-
glected, i.e., pp
s5ppp50~modelD). As the neglect of
the hopping between the oxygen atoms could reduce the eg
bandwidth, the spin-wave dispersions were calculated for
FIG. 4. The dependence of the spin-wave energies on the eg
hole doping yalong GXwithin model D. The hopping between
oxygen atoms and the t2gorbitals on the Mn atom have been left
out of the model. ~a!pds522.02 eV and ~b!pds522.25 eV.
FIG. 3. The ~a!minority-spin and ~c!majority-spin dpartial
density of states within models A,B, andC. The spin-wave disper-
sions along GXas a function of doping within models ~b!Band~d!
Care shown along with ~e!the combined contributions of models B
andC.yrefers to the hole concentration in the majority-spin eg
band with reference to its half-filled case. zis the electron concen-
tration in the minority-spin t2gband.xis the net concentration of
the doped holes and is given by x5y2z.11 442 PRB 60 PRIYA MAHADEVAN, I. V. SOLOVYEV, AND K. TERAKURAtwo values of pds—the value ( 22.02 eV !estimated already
by the fitting @results are shown in Fig. 4 ~a!#and an in-
creased value of 22.25 eV @results are shown in Fig. 4 ~b!#.
The dispersions shown in Figs. 4 ~a!and 4 ~b!are qualita-
tively similar to each other. In both cases, the spin-waveenergy at the Xpoint increases with doping being in contra-
diction to the behavior observed in Fig. 1 and Fig. 3 ~b!.Since this model does not take into account the antiferromag-
netic superexchange contributions coming from the t
2gde-
grees of freedom and affecting primarily the nearest-neighbor magnetic interactions, the ferromagnetic coupling
J
1remains to be the strongest interaction in the system. In
such a situation, the form of the spin-wave dispersion isclose to the cosinelike.
To analyze further the role played by the interoxygen
hopping, the exchange interactions
$Ji%were obtained for the
cases corresponding to Fig. 3 ~b!and Fig. 4. The results are
shown in Fig. 5. As is expected, the behavior of J1is not
affected so much by the p-phopping. By comparing the
results of model Ain Fig. 2 with those of model Bin Fig. 5,
we see that the doping dependence of J4andJ8comes pri-
marily from the egelectrons. Neglect of the p-phopping
~modelD) strongly suppresses J4andJ8forx>0.2, and
zone-boundary softening of the spin-wave dispersion be-
come less pronounced. On the other hand, increase of J2
with hole doping is enhanced by neglecting the p-phopping.
The energy at Xpoint increases in Fig. 4 ~modelD) with
hole doping, because the variation in J1in this case is not
enough to offset the sharp increase in J2.
In summary, we have analyzed the low-temperature spin
dynamics of the doped manganites with tight-binding mod-
els. Our results provide some insight into the roles of the t2g
and theegstates and also suggest that the channel of hopping
between the oxygen atoms strongly modifies the exchangeinteractions. Thus for the correct quantitative and sometimeseven qualitative description, the simplifications made by
model Hamiltonian approaches that consider only the e
gor-
bitals are questionable.
We thank Professor D.D. Sarma for useful discussions.
Part of the programs used here were developed in ProfessorSarma’s group. The present work was partly supported byNEDO.
1R. von Helmolt, J. Wecker, B. Holzaphel, L. Schultz, and K.
Samwer, Phys. Rev. Lett. 71, 2331 ~1993!; A. Asamitsu, Y.
Moritomo, Y. Tomioka, T. Arima, and Y. Tokura, Nature
~London !373, 407 ~1995!.
2T. G. Perring, G. Aeppli, S. M. Hayden, S. A. Carter, J. P.
Remeika, and S. W. Cheong, Phys. Rev. Lett. 77, 711 ~1996!.
3V. Yu. Irkhin and M. I. Katsnelson, Zh. E´ksp. Teor. Fiz. 88, 522
~1985!@Sov. Phys. JETP 61, 306 ~1985!#; N. Furukawa, J. Phys.
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and H. A. Mook, Phys. Rev. Lett. 80, 1316 ~1998!.
5J. Zang, H. Ro ¨der, A. R. Bishop, and S. A. Trugman, J. Phys.:
Condens. Matter 9, L157 ~1997!; T. K. Kaplan and S. D. Mah-
anti,ibid.9, L291 ~1997!; J. Loos and H. Fehske, Physica B
259-261, 801 ~1999!.
6I. V. Solovyev and K. Terakura, Phys. Rev. Lett. 82, 2959 ~1999!.
7J1,J2,J3,J4, andJ8are the interactions in the cubic lattice
corresponding to the interatomic vectors @0,0,a#,@a,a,0#,
@a,a,a#,@0,0,2a#, and @0,0,3a#, whereais the cubic lattice
constant.
8Priya Mahadevan, N. Shanthi, and D. D. Sarma, Phys. Rev. B 5411 199 ~1996!; Priya Mahadevan, N. Shanthi, and D. D. Sarma,
J. Phys.: Condens. Matter 9, 3129 ~1997!.
9L. F. Mattheiss, Phys. Rev. B 2, 3918 ~1970!.
10The parameter epol8is anyway small and turns out not to be im-
portant in the following calculations.
11L. M. Sandratskii, Phys. Status Solidi B 135, 167 ~1986!; Adv.
Phys.47,9 1~1998!.
12A. I. Liechtenstein, M. I. Katnelson, V. P. Antropov, and V. A.
Gubanov, J. Magn. Magn. Mater. 67,6 5~1987!, and references
therein; I. V. Solovyev and K. Terakura, Phys. Rev. B 58,1 5
496~1998!.
13I.e., we start with the undoped cubic ferromagnetic LaMnO3,
whose electronic structure has been obtained self-consistently,and shift the position of the Fermi level in order to adjust therequired hole concentration x, without an additional self-
consistency. Note that this is different from the procedure usedin Ref. 6, where the electronic structure of the hypotheticalvirtual-crystal alloy was calculated self-consistently in theframework of LSDA for each concentration x.
14T. Oguchi, K. Terakura, and A. R. Williams, Phys. Rev. B 28,
6443 ~1983!.
15J. Zaanen and G. A. Sawatzky, Can. J. Phys. 65, 1262 ~1987!.
FIG. 5. The variation of the exchange couplings J1,J2,J4, and
J8with theeghole doping y. Open circles are for the case ~model
B) including the hopping between oxygen atoms and pds
522.02 eV. Open and filled squares are for the cases ~modelD)
without the hopping between oxygen atoms: pds522.02 eV
~open squares !andpds522.25 eV ~filled squares !. Thet2gorbit-
als on the Mn atom have been left out of the basis set.PRB 60 11 443 LOW-TEMPERATURE SPIN DYNAMICS OF DOPE D... |
PhysRevB.84.144505.pdf | PHYSICAL REVIEW B 84, 144505 (2011)
Brownian refrigeration by hybrid tunnel junctions
J. T. Peltonen,1M. Helle,1,2A. V . Timofeev,1,3P. Solinas,4F. W. J. Hekking,5and J. P. Pekola1
1Low Temperature Laboratory, Aalto University, P .O. Box 13500, FIN-00076 AALTO, Finland
2Nokia Research Center, P .O. Box 407, FIN-00045 NOKIA GROUP , It ¨amerenkatu 11-13, FIN-00180 Helsinki, Finland
3VTT Technical Research Centre of Finland, P .O. Box 1000, FIN-02044 VTT, Espoo, Finland
4Department of Applied Physics/COMP , Aalto University, P .O. Box 14100, FIN-00076 AALTO, Finland
5Laboratoire de Physique et Mod ´elisation des Milieux Condens ´es, C.N.R.S. and Universit ´e Joseph Fourier,
B.P . 166, F-38042 Grenoble Cedex 9, France
(Received 19 April 2011; revised manuscript received 27 July 2011; published 4 October 2011)
V oltage fluctuations generated in a hot resistor can cause extraction of heat from a colder normal metal electrode
of a hybrid tunnel junction between a normal metal and a superconductor. We extend the analysis presented inP h y s .R e v .L e t t . 98, 210604 (2007) of this heat rectifying system, bearing resemblance to a Maxwell’s demon.
Explicit analytic calculations show that the entropy of the total system is always increasing. We then consider asingle-electron transistor configuration with two hybrid junctions in series, and show how the cooling is influencedby charging effects. We analyze also the cooling effect from nonequilibrium fluctuations instead of thermal noise,focusing on the shot noise generated in another tunnel junction. We conclude by discussing limitations for anexperimental observation of the effect.
DOI: 10.1103/PhysRevB.84.144505 PACS number(s): 05 .40.−a, 07.20.Pe, 73 .40.Gk
I. INTRODUCTION
Thermal ratchets and related devices invoke unidirectional
flow of particles by a stochastic drive originating from fluctu-ations of a heat bath.
1–8Analogously, thermal fluctuations can
induce heat flow directed from cold to hot, which constitutes
the principle of Brownian refrigeration. In recent literature,
one can find two examples of a Brownian refrigerator.9,10The
first one9employs the idea of Feynman’s ratchet and pawl,
and demonstrates that a Brownian refrigerator can work inprinciple, whereas the second refrigerator
10relies on well-
characterized properties of hybrid metallic tunnel junctionsand presents thus an illustrative and concrete example of
refrigeration by thermal noise.
In Ref. 10, it was demonstrated that thermal noise generated
by a hot resistor (resistance R, temperature T
R) can, under
proper conditions, extract heat from a cold normal metal(N) at temperature T
Nin contact with a superconductor (S)
at temperature TSvia environment-activated tunneling of
electrons through a thin insulating barrier (I). At first sight,such an NIS junction seems to violate the second law ofthermodynamics and act as Maxwell’s demon,
11allowing only
hot particles to tunnel out from the cold normal metal. Thisprocess would lead to a decrease of entropy if the systemwas isolated. Yet the demon needs to exchange energy withthe surroundings in order to function properly. Thereby, thenet entropy of the whole system is always increasing. It is,however, interesting that one can exploit thermal fluctuationsin refrigeration. In general, high-frequency properties of theelectrical environment close to small tunnel junctions havebeen known for a long time to be important in determiningthe particle tunneling rates and hence the current-voltagecharacteristic in such systems.
12–15On the other hand, their
influence on thermal transport has received less attention,motivating the study of heat currents in different electricalenvironments.
Cooling by electron tunneling is possible in a hybrid tunnel
structure where one of the conductors facing the tunnel barrier
has a hard gap in its quasiparticle density of states. An ordinarylow-temperature Bardeen-Cooper-Schrieffer (BCS) supercon-
ductor, such as aluminum, is an ideal choice for this. In
principle, though not experimentally verified, a semiconductor
with a suitable energy gap could also be a choice. The other
conductor can be a superconductor with smaller energy gap,
16a
normal metal, or a heavily doped, metallic semiconductor.17A
hybrid NIS junction, or a contact of any type described above,
can be characterized as a Brownian refrigerator under proper
external conditions: the most energetic electrons are allowed
to pass through the junction, whereas the low-energy electrons
are forbidden to tunnel. This feature makes the hybrid junctionsunique, well-characterized building blocks for energy filtering
purposes.
Cooling of electrons in the N electrode is well understood
in ordinary NIS junctions biased by a constant voltage,
18
and it is utilized in practical electronic microrefrigerators.19,20
Recently, electronic cooling of a two-dimensional electron
gas has also been demonstrated,21based on energy-dependent
tunneling through two quantum dots in series. In the case ofan NIS junction subject to a noisy environment consisting of ahot resistor, the voltage fluctuations allow the most energeticelectrons to tunnel from the cold normal metal, even under zerovoltage bias across the junction. Figure 1shows a schematic
representation of the system. The phenomenon is analogous to
photon-assisted tunneling
22,23with a stochastic source. The
cooling is observed in a certain temperature range of theenvironment, T
R>T N, where the distribution of thermal noise
is suitable to excite hot electrons to tunnel through the NISjunction to the superconductor side. When the temperature T
R
is further increased, the fluctuating voltage of the hot resistor
starts to extract also cold electrons from the normal metal,resulting eventually in heating of the island. The heat flow isnontrivial also when the resistor is at a lower temperature thanthe normal metal ( T
N>T R): heat flows into the hot normal
metal, and the superconductor side tends to cool down. Thusthe reversal of the temperature bias reverses the heat fluxes.Such a reversed heat flow cannot be realized in a conventionalvoltage-biased NIS refrigerator, for instance, by changing thepolarity of the voltage bias.
18
144505-1 1098-0121/2011/84(14)/144505(14) ©2011 American Physical SocietyJ. T. PELTONEN et al. PHYSICAL REVIEW B 84, 144505 (2011)
FIG. 1. (Color online) (a) Illustration of an environment-assisted
tunneling event in an NIS junction. V oltage fluctuations generated bythe electromagnetic environment of the junction provide the energy
E
/prime−Efor the electron at Eto tunnel to available states at E/primeon
the superconductor side above the energy gap. For positive E/prime−E,
P(E/prime−E) is the probability density for the quasiparticle to emit
energy E/prime−Eto the environment, whereas P(E−E/prime) describes the
probability to absorb the energy E/prime−E. Removal of high-energy
electrons above the Fermi energy EF(E/greaterorequalslant0, marked with the dotted
line) results in refrigeration of the N island. Vdenotes a constant dc
bias voltage across the junction. For the Brownian refrigeration effect,V=0, and only a fluctuating voltage over the junction is present.
(b) Electrical diagram of the resistor (resistance R, temperature T
R)
and the NIS junction (normal-state tunnel resistance RT; temperatures
TNandTSfor the normal metal N and superconductor S, respectively).
The parallel capacitance Cincludes the junction capacitance and a
possible shunt capacitor. The N side of the junction can be connectedto the resistor via a superconducting line with a direct NS contact.
(c) Thermal diagram of the system. The NIS tunnel junction acts as
a Brownian refrigerator (BR) between the normal metal island andthe superconducting electrode. ˙Q
N,˙QS,a n d ˙QRdenote heat flows
in the system. Heat is carried by tunneling electrons in ˙QNand ˙QS,
whereas the resistor is coupled to the NIS junction only via voltagefluctuations, and the heat exchange can be described in terms of
photonic coupling. P
extdenotes the externally applied power needed
to raise TRoverTN. The electrons in the resistor, superconductor, and
the normal metal island are assumed to be thermally coupled to the
lattice phonons, described as a heat bath at temperature T0.
The resistor and the junction can be connected by super-
conducting lines that efficiently suppress the normal electronicthermal conductance. Alternatively, the coupling can becapacitive instead of a direct galvanic connection, allowing oneto neglect the remaining quasiparticle thermal conductance.
24
In both cases, the N electrode of the junction can be connectedto the superconducting line via a direct metal-to-metal SNcontact, which provides perfect electrical transmission but,due to Andreev reflection, exponentially suppresses heat flowat temperatures below the superconductor energy gap.
18,25The
size of the normal metal island is assumed to be small enough(small resistance compared to the tunnel resistance) to ignorethe direct Joule heating by the voltage fluctuations. One shouldfurther keep in mind that, in an on-chip realization, the twosubsystems, i.e., the NIS junction and the resistor typically inthe form of a thin strip of resistive metal such as chromium,are connected through substrate phonons. However, with a
careful design and with low substrate temperature, unwantedheat flow via electron-phonon coupling from the resistor to thejunction can be reduced to a sufficiently low level in a practicalrealization of the device.
The text is organized as follows. In Sec. II, we first expand
the analysis presented in Ref. 10of a single hybrid junction
exposed to the noise of a hot resistor. In particular, we give atransparent picture of the mechanism of Brownian refrigera-tion in this system and we make a systematic analysis in termsof different parameters affecting the cooling performance.In Sec. III, we present quantitative considerations of entropy
production in the system. We move on to Sec. IVto analyze a
single-electron transistor (SET) configuration, consisting of adouble junction SINIS refrigerator subjected to thermal noise;here, charging effects of the small N island become relevant,and the heat currents can be controlled by a capacitivelycoupled gate electrode. In Sec. V, we discuss briefly more
general, non-ohmic dissipative environments. Section VI
considers the refrigeration by nonequilibrium fluctuations,e.g., by shot noise generated in another voltage-biased tunneljunction, instead of the thermal noise in an ohmic resistor.Finally, in Sec. VII, we discuss practical aspects toward an
experimental realization of the Brownian refrigeration device.
II. A HYBRID TUNNEL JUNCTION
The operation principle of the Brownian tunnel junction
refrigerator is illustrated in Fig. 1(a), showing how an electron
in the normal metal can absorb energy E/prime−Eand tunnel
into an available quasiparticle state above the energy gap /Delta1
in the superconductor. Figures 1(b) and 1(c) display electric
and thermal diagrams of the system, respectively. To calculateheat flows in the combined system of the NIS junctionand the resistor, we utilize the standard P(E) theory (for a
review, see Ref. 26) describing a tunnel junction embedded
in a general electromagnetic environment.
27This circuit is
characterized by a frequency-dependent impedance Z(ω)a t
temperature TRin parallel to the junction. To illustrate the
effects of the environment, we mainly deal with the specialcase of a resistive environment with Z(ω)≡Rfrequency-
independent in the relevant range. The theory is perturbativein the tunnel conductance, and we assume a normal-statetunneling resistance R
T/greatermuchRK, where RK≡h/e2/similarequal26 k/Omega1is
the resistance quantum.
A. Heat fluxes for a single junction in a dissipative environment
We start by writing down the heat fluxes associated to
quasiparticle tunneling in a general hybrid junction biasedby a constant voltage V, with normalized density of states
(DOS) n
i(E) in each electrode ( i=1,2). We assume that the
two conductors are at (quasi)equilibrium, i.e., their energydistribution functions obey the Fermi-Dirac form f
i(E)=
1/[1+exp(βiE)] with the inverse temperature βi=(kBTi)−1.
Here, importantly, the temperatures Tineed not be equal,
and the energies are measured with respect to the Fermilevel. In general, the electrode temperatures are determinedconsistently by the various heat fluxes in the complete system,usually via coupling to the lattice phonons.
144505-2BROWNIAN REFRIGERATION BY HYBRID TUNNEL JUNCTIONS PHYSICAL REVIEW B 84, 144505 (2011)
The net heat flux out of electrode iis given by
˙Qi=1
e2RT/integraldisplay∞
−∞/integraldisplay∞
−∞dEdE/primeni(E)Efi(E)P(E−E/prime)
×{nj(E/prime+eV)[1−fj(E/prime+eV)].
+nj(E/prime−eV)[1−fj(E/prime−eV)]}, (1)
which assumes the symmetries ni(E)=ni(−E) and
fi(−E)=1−fi(E). In the case of Brownian refrigeration at
V=0, the heat transport at Ti=Tjis only due to fluctuations
in the environment. Equation ( 1) simplifies to10
˙Qi=2
e2RT/integraldisplay∞
−∞/integraldisplay∞
−∞dEdE/primen1(E)n2(E/prime)Ei
×f1(E)[1−f2(E/prime)]P(E−E/prime), (2)
withE1=EandE2=−E/prime, giving the heat extracted from
electrode i. On the other hand, Ei=E/prime−Efor heat extracted
from the environment, manifesting the conservation of energy.
The function P(E) is obtained as the Fourier transform
P(E)=1
2π¯h/integraldisplay∞
−∞dtexp[J(t)+iEt/ ¯h], (3)
with the phase-phase correlation function J(t) defined as
J(t)=/angbracketleftϕ(t)ϕ(0)/angbracketright−/angbracketleftϕ(0)ϕ(0)/angbracketright
=1
2π/integraldisplay∞
−∞dωS ϕ(ω)[e−iωt−1]. (4)
Here, Sϕ(ω) is the spectral density of the phase fluctuations
ϕ(t) across the junction, i.e., the average value of ϕ(t) satisfies
/angbracketleftϕ(t)/angbracketright=0. For a given Z(ω) and a temperature kBTR=β−1
Rof
the environment, the uniquely defined P(E) can be interpreted
as the probability density per unit energy for the tunnelingparticle to exchange energy Ewith the environment,
26with
E> 0 corresponding to emission and E< 0 to absorption.
The function J(t)i nE q .( 4) can then be written as
J(t)=2/integraldisplay∞
0dω
ωRe[Zt(ω)]
RK{coth(βR¯hω/2)
×[cos(ωt)−1]−isin(ωt)}. (5)
Here, Zt(ω)=1/[iωC+Z−1(ω)] is the total impedance as
seen from the tunnel junction, i.e., a parallel combination ofthe “external” impedance Z(ω) and the junction capacitance
C. Inserting J(t) from Eq. ( 5) into Eq. ( 3), one importantly
finds that P(E)i s( 1 )p o s i t i v ef o ra l l E, (2) normalized to unity,
and (3) satisfies detailed balance P(−E)=exp(−β
RE)P(E).
To relate P(E) andJ(t) to more physical quantities, we use
the fundamental defining relation between the phase ϕ(t)
and the voltage fluctuation δV(t) across the junction. We
haveϕ(t)=(e/¯h)/integraltextt
−∞dt/primeδV(t/prime), from which it follows that
Sϕ(ω) is connected to the voltage noise spectral density SV(ω)
at the junction via Sϕ(ω)=(e/¯h)2SV(ω)/ω2. Furthermore,
P(E) is well approximated in the limit πR/R K/greatermuchβRECby a
Gaussian of width s=√2ECkBTRcentered at EC≡e2/(2C),
the elementary charging energy of the junction.26Lowering R
transforms P(E) toward a delta function at E=0.
B. Results for an NIS junction
The main result of Sec. II A,E q .( 2), applies to a generic
tunnel junction between conductors 1 and 2. An importantspecial case is an NIS junction, where a BCS density of states
with energy gap /Delta1in S and approximately constant DOS in
N near EFmake this system a particularly important example.
In the following, we will consider the heat flows for an NISjunction with n
N(E)≡1, and a smeared BCS DOS
nS(E)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleRe/bracketleftBigg
E+iγ
/radicalbig
(E+iγ)2−/Delta12/bracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(6)
in the superconductor. Here, the small parameter γdescribes
the finite lifetime broadening of the ideally diverging BCSDOS at the gap edges.
28In all the numerical calculations to
follow, we assume /Delta1=200μeV (aluminum) and γ=1×
10−5/Delta1, unless noted otherwise. We limit to low temperatures
so that the temperature dependence of /Delta1can be neglected.
Assuming electrode 1 (2) to be of N (S) type in Eq. ( 2), we
find explicitly
˙QN=2
e2RT/integraldisplay∞
−∞/integraldisplay∞
−∞dEdE/primenS(E/prime)E
×fN(E)[1−fS(E/prime)]P(E−E/prime)( 7 )
and
˙QS=2
e2RT/integraldisplay∞
−∞/integraldisplay∞
−∞dEdE/primenS(E/prime)(−E/prime)
×fN(E)[1−fS(E/prime)]P(E−E/prime)( 8 )
for the heat extracted from N and S, respectively. In Fig. 2,
we compare the numerically calculated cooling powers ˙QNfor
R=10RKandR=0.5RKas a function of TR/TNat various
charging energies EC, i.e., capacitances C. The temperatures
are fixed to kBTN=kBTS=0.1/Delta1. Looking at the qualitative
behavior of ˙QN, we notice that ˙QN>0 in a large temperature
rangeTN<T R<Tmax
R, indicating refrigeration of the normal
FIG. 2. (Color online) Cooling power ˙QNfrom Eq. ( 7)f o r
R=10RK(dashed lines) and R=0.5RK(solid lines) at various
values of /Delta1/E C. Notice that for /Delta1>E Cbetter cooling power is
obtained with large R, whereas for /Delta1/lessorsimilarECthe larger cooling power
is found with R=0.5RK.T h e R=10RKcurves fall below the
R=0.5RKones around /Delta1≈1.5EC. The dash-dotted lines show the
analytical approximation discussed in Appendix A, valid for R/greatermuchRK
andkBTN/lessmuch√2ECkBTR, and capturing most of the cooling effect.
144505-3J. T. PELTONEN et al. PHYSICAL REVIEW B 84, 144505 (2011)
FIG. 3. (Color online) Maximum cooling power ˙Qopt
N(top) and
the corresponding optimum resistor temperature Topt
R(bottom) as
a function of RandECatkBTN=kBTS=0.1/Delta1.A tfi x e d R,Topt
R
increases approximately linearly as a function of /Delta1/E C, and starts to
become independent of RatR/greaterorsimilarRK.
metal. The maximum cooling power, ˙Qopt
N, depends on R
in a nontrivial manner, whereas the corresponding optimum
resistor temperature Topt
R/TN/similarequal/Delta1/E Cis sensitive mainly to
the capacitance. We notice further that for /Delta1>E Cbetter cool-
ing power is obtained with large environmental resistances,whereas for /Delta1/lessorsimilarE
Cthe larger cooling power is found with
R=0.5RK. Comparing the ˙Qopt
Nvalues, the R=10RKcurves
fall below the R=0.5RKones around /Delta1≈1.5EC. Above a
certain circuit-dependent temperature Tmax
R, the N island tends
to heat up ( ˙QN<0), which happens nontrivially also in the
regime TR<T N, i.e., heat flows into the “hot” normal metal
island.
In Fig. 3, we plot the maximum cooling power and the
corresponding optimum resistor temperature as a function ofRandE
C. As evident from Fig. 2, for small junctions with
EC/greaterorsimilar/Delta1, the cooling power is maximized at finite values of
R, and at large Rthe power ˙Qopt
Nis very small for EC/greaterorsimilar2/Delta1.
To further assess the efficiency of the refrigeration at Topt
R,
we consider the ratio η=˙QN/˙QR=− ˙QN/(˙QN+˙QS). In a
resistor with no other relaxation mechanisms but the couplingto the NIS junction, this quantity gives the ratio of thecooling power of N scaled by the power injected into theresistor under steady-state conditions ( ˙Q
R=Pext). However,
in any experimental realization, ˙QR/lessmuchPext, which leads to
very low overall efficiency. The temperature dependence ofηat the maximum power is shown in Fig. 4for various
charging energies at a fixed R/R
K=10. We have neglected
the temperature dependence of /Delta1. Similar to Fig. 3,t h e
dependence of ηonRis weak for R/greaterorsimilarRK. The behavior
ofηbears close resemblance to the coefficient of performanceFIG. 4. Cooling efficiency η=˙QN/˙QRat the optimum resistor
temperature Topt
Ras a function of TN=TS. The curves from top to
bottom correspond to /Delta1/E C=20,10,5,2,and 1, while R/R K=10
remains fixed.
η0=˙Q(V)/[I(V)V] of an ordinary voltage-biased NIS junc-
tion in a low-impedance environment.18ForTR>T N,˙Qopt
N
goes through a maximum at a certain TN, whereas the
corresponding −˙QSincreases monotonously. The saturation
of the maximum value of ηtoward large /Delta1/E Cis related to the
saturation of ˙Qopt
N.F o rEC/lessmuchkBTN/lessmuch/Delta1, an analysis of the
tunneling rates and the associated heat flows at the optimumpoint shows that each tunneling electron removes an averageenergy /similarequalk
BTNfrom the N island. The energy deposited in the
S electrode is /similarequal/Delta1on the average, so that the efficiency is
approximately η/similarequalkBTN//Delta1.
In Ref. 10, two analytical approximations were derived for
˙QN, assuming an idealized high-impedance environment with
R/greatermuchRKat a high enough temperature TRto utilize a Gaussian
P(E). The first of these results was based on replacing
the Fermi functions by their exponential tails, valid at lowtemperatures k
BTN,kBTS/lessmuch/Delta1. For the second approximation,
the quadratic exponent of P(E−E/prime) was linearized around
E=0, whereas the correct form of fN(E) was retained,
resulting in a reasonable result for a wide range of TR/TN.I n
Appendix A, we present another approximation valid at R/greatermuch
RKandkBTN/lessmuchs, shown in Fig. 2as the dash-dotted lines.
This is based on first performing a Sommerfeld expansion oftheEintegral in Eq. ( 7) in terms of k
BTN/s, and treating the
remaining integral over E/primeas in the second approximation in
Ref. 10.
Since the S DOS is strongly peaked at the gap edge as
evident from Eq. ( 6), electrons tunneling out of N end up
mainly at energies near this threshold. Therefore, to understandqualitatively the behavior of ˙Q
Nin Fig. 2, we may evaluate the
integrand in Eq. ( 7) only at superconductor energies E/prime=±/Delta1.
Looking at the dimensionless quantities,
F(±/Delta1)=/integraldisplay∞
−∞dEEf N(E)P(E∓/Delta1), (9)
we find that the cooling power depends on the overlap of the tail
of the Fermi function and P(E∓/Delta1). At high temperatures,
TR>Tmax
R,P(E−/Delta1) is broad, and negative contributions
fromE< 0 outweigh those from E> 0. This corresponds to
low-energy electrons from below the Fermi level tunneling tothe gap edge in S. As a result, the dominant quantity F(/Delta1) and
therefore ˙Q
Nturn negative. At TN/lessorsimilarTR<Tmax
R, the positive
contributions outweigh the negative ones, resulting in a net
144505-4BROWNIAN REFRIGERATION BY HYBRID TUNNEL JUNCTIONS PHYSICAL REVIEW B 84, 144505 (2011)
cooling effect. Finally, at TR/lessmuchTN,TS,P(E) is very sharp, and
mainly E< 0i nF(−/Delta1) contribute via P(E+/Delta1), leading to
˙QN<0.
III. ENTROPY FLOW
In the previous section, we saw that heat can flow out
of the N electrode when the resistor is held at temperatureT
R>T N. Similarly, the S tends to cool for TR<T N.H e r e
we extend the analysis of Ref. 10, showing explicitly that
the system obeys the second law of thermodynamics despitethe counterintuitive heat fluxes. We consider the total entropyproduction for a single NIS junction in an arbitrary equilibriumenvironment (“resistor”) obeying detailed balance, showingexplicitly that it is always increasing. In the following, weassume the NIS junction and the resistor to form an isolatedsystem and ignore couplings to the phonon bath. Let ˙Sbe
the rate of entropy production in the system composed of N,S, and R, at temperatures T
N,TS, and TR, respectively. In
general, the energy conservation ˙QN+˙QS+˙QR=0 holds,
as discussed after Eq. ( 2). In addition, we have the definition
˙S=− ˙QN/TN−˙QS/TS−˙QR/TR. We consider the general
case of three unequal temperatures kBTN=β−1
N,kBTS=β−1
S,
andkBTR=β−1
R. The above results can be combined to yield
˙S/k B=(βR−βN)˙QN+(βR−βS)˙QS. We find
˙S=2kB
e2RT/integraldisplay∞
0dE/primeP(E/prime)/integraldisplay∞
0dEn S(E)
×{(βR−βN)E/prime{fN(E+E/prime)−e−βRE/primefN(E−E/prime)
+fS(E)(1−e−βRE/prime)[1−fN(E+E/prime)−fN(E−E/prime)]}
+(βS−βN)E[fN(E+E/prime)+e−βRE/primefN(E−E/prime)
+fS(E){(1−e−βRE/prime)[fN(E−E/prime)−fN(E+E/prime)]
−1−e−βRE/prime}]}. (10)
Here, we utilized the detailed balance of P(E/prime), and the
symmetry nS(−E)=nS(E) of the S DOS. This equation
should hold for any form of positive P(E/prime) and (symmetric)
nS(E). In order to show that ˙S> 0, we have therefore to
demonstrate that the integrand Ion the last five lines in Eq. ( 10)
is positive for any value of E,E/prime,βN,βS, andβR.
In the following, we assume the distribution functions in N
and S to be of the equilibrium form fi(E)=1/(1+eβiE).
After straightforward manipulations (see Appendix Bfor
details), the last five lines in Eq. ( 10) transform into
I=NS(eX−1)X+NA(eY−1)Y. (11)
Here, the quantities
NS=e−βRE/primeeβN(E+E/prime)
(1+eβSE)(1+eβN(E+E/prime)), (12)
NA=eβN(E−E/prime)
(eβN(E−E/prime)+1)(1+eβSE)(13)
are always positive. We also defined the combinations
X=(βS−βN)E+(βR−βN)E/primeandY=(βS−βN)E−
(βR−βN)E/prime. Now, for any value of XandY, the functions
(eX−1)Xand (eY−1)Yin Eq. ( 11) are positive or zero.
Thus, since NSandNAare always positive, we find that
I/greaterorequalslant0 for any value of XandY, and hence ˙S/greaterorequalslant0 always.FIG. 5. (a)Hybrid single-electron transistor in the presence of an
environment, modeled as an impedance Z(ω) in series with the bias
voltage source V. A gate voltage Vgis coupled capacitively to the
Ni s l a n dv i a Cg. The arrows define tunneling rates and heat fluxes
for each of the two NIS junctions, with tunneling resistance RT,iand
capacitance Ci(i=1,2).(b)Transformation of the environment seen
from junction 1 into an effective single junction circuit.
Furthermore, at the special point βR=βN=βS,w eh a v e
˙QN=˙QS=0. Differentiating ˙Sthen gives∂˙S
∂βi=0, and we
conclude that this point yields a local extremum (minimum)
of˙S, and at this point ˙S=0 always. We have therefore shown
that the entropy of the system is increasing for arbitrary valuesofT
N,TS, andTS. The procedure can be generalized to include
a phonon bath at temperature T0, in which case one considers
the total system of N, S, R, and the phonons.
IV . NOISE COOLING IN TWO-JUNCTION SINIS WITH
COULOMB INTERACTION
In this section, we analyze the refrigeration effect combined
with charging effects in a double junction SINIS configuration,i.e., a hybrid single-electron transistor (SET) with a small Nisland connected to S leads via two tunnel junctions of the NIStype. Figure 5(a) shows such a SINIS structure coupled to a
general environment Z(ω), and the various tunneling rates in
the system. The two junctions are assumed to be characterizedby resistances R
T,iand capacitances Ci(i=1,2). We assume
charge equilibrium to be reached before each tunneling event,so that the state of the system can be characterized by n,t h e
number of excess electrons on the island. The allowed values ofncan be controlled by the gate voltage V
gcoupled capacitively
to the island via Cg. We assume the gate capacitance Cgto be
much smaller than the junction capacitances, but the voltage Vg
to be large enough so that the only effect of the gate is an offset
ng=CgVg/eto the island charge. Following Refs. 26,29,
and 30, it is straightforward to calculate numerically the net
heat flux ˙Qout of the island in terms of the heat fluxes
˙Q±
i,nthrough junction iwith the island in state n.T h i s
is accomplished by solving a steady-state master equationthat gives the occupation probability of each charge state n,
determined by the tunneling rates /Gamma1
±
i,n.
The difference to the case of a single junction in an envi-
ronment becomes evident in Fig. 5(b). We neglect cotunneling
effects and assume the tunneling events to be uncorrelated,so that the other junction can be viewed simply as a seriescapacitor. Concentrating on tunneling in junction 1, the upperhalf of Fig. 5(b) displays the circuit of Fig. 5(a) as seen
from junction 1. It can be transformed
26into an equivalent
144505-5J. T. PELTONEN et al. PHYSICAL REVIEW B 84, 144505 (2011)
FIG. 6. (Color online) Noise-induced cooling power ˙Qin a SINIS
structure with /Delta1/E /Sigma1=2. Solid lines correspond to ng=0.5 (“gate
open”), and dashed lines to ng=0 (“gate closed”). Dash-dotted lines
indicate the cooling power of a single NIS junction, showing the
influence of the effective circuit parameters C∗(in general, optimum
cooling shifts toward larger TRin the SINIS) and R∗(cooling power
per junction is reduced in the SINIS for large /Delta1/E /Sigma1and increased
for small /Delta1/E /Sigma1).
single junction circuit shown in the lower half, consisting
first of an effective impedance κ2
1Zt(ω), where Zt(ω)i sa s
in Eq. ( 5), but defined in terms of the series capacitance
˜C=C1C2/(C1+C2), i.e., Zt(ω)=1/[iω˜C+Z−1(ω)]. The
reduction factors κi=˜C/C i<1(i=1,2) show the weak-
ened effect of the external impedance Z(ω) due to shielding by
the second junction capacitance. In addition, the transformedcircuit contains a capacitance C
1+C2and a voltage source
with voltage κ1V. The series capacitance does not influence
the real part of the total external impedance, and for Brownianrefrigeration we consider only V=0 in the end. The circuit
for junction 2 is identical, except κ
1is replaced by κ2and
the voltage Vis inverted. Apart from charging effects, in the
important special case of Z(ω)=Rand identical junctions
(RT,1=RT,2=RT,C1=C2=C), we can directly apply
the analysis of Sec. IIto the double-junction system if the
resistance is replaced by R∗=R/4 and the capacitance by
C∗=2C.
Figure 6displays the total cooling power ˙Qout of the N
island for a SINIS structure with /Delta1/E /Sigma1=2. The curves corre-
spond to various values of the resistance R/R Kand the extreme
values of the gate charge ng. Here, E/Sigma1=e2/(2C/Sigma1) denotes
the charging energy of the two-junction system with the totalcapacitance C
/Sigma1=C1+C2. We assume a symmetric structure
withRT,1=RT,2=RTandC1=C2=C. As expected, in a
SINIS with large junctions ( /Delta1/E /Sigma1/greaterorsimilar10), the charging effects
do not affect the cooling power. In contrast, with smallerjunctions ( /Delta1/E
/Sigma1/lessorsimilar2a si nF i g . 6) the cooling power depends
strongly on the gate charge ng. As a consequence of rescaling
the circuit parameters in the SINIS configuration, bettercooling power per junction is achieved, in general, with a singleNIS junction when compared to SINIS with two junctions ofthe same size. However, with small junctions ( /Delta1/E
/Sigma1/lessorsimilar2),
greater cooling power can be reached in the SINIS circuit.Interestingly, in the “gate closed” position ( n
g=0, maximumFIG. 7. Gate modulation of the maximum cooling power for four
different charging energies at R∗/R K=1a n dkBTN=kBTS=0.1/Delta1.
Coulomb blockade in a voltage-biased SET), we find nontrivial
solutions for the heat fluxes for small junctions. In Fig. 6,t h e
gate voltage is seen to reverse the heat fluxes at TR<T N
instead of only suppressing them close to zero in the “gate
closed” position. Single-electron effects in zero voltage-biasrefrigeration in an NIS junction are discussed also in Ref. 31.
There, the influence of a deterministic radio-frequency signalapplied to the gate was analyzed, assuming negligible effectfrom the environment [ P(E)=δ(E)]. With ultrasmall tunnel
junctions in general, the electronic refrigeration is sensitive tosingle-electron effects.
Figure 7emphasizes the gate dependence of ˙Q, already
evident in Fig. 6. The gate-dependent maximum cooling
power is shown for symmetric SINIS structures with /Delta1/E
/Sigma1=
4,3,2,and 1, assuming fixed R∗=RKandkBTN=kBTS=
0.1/Delta1.
V . OTHER TYPES OF DISSIPATIVE ENVIRONMENTS
Up to this point, the environment parallel to the junction
capacitance was assumed to be purely ohmic with Z(ω)=R
independent of frequency. In this section, we analyze threeexamples of frequency-dependent Z(ω). These include a
lumped inductance in series with the hot resistor, a distributedmodel treating the resistor as an RLC transmission line, andfinally a lumped resistor connected to the junction via a losslessLC transmission line.
FIG. 8. (Color online) Models for non-ohmic junction environ-
ments. (a)Junction environment formed by an inductance Lin series
with the resistance R.(b)Symmetric distributed model showing the
transformation to two standard transmission lines.
144505-6BROWNIAN REFRIGERATION BY HYBRID TUNNEL JUNCTIONS PHYSICAL REVIEW B 84, 144505 (2011)
FIG. 9. Series inductance: (a)P(E)a tTR=5TNfor ten evenly
spaced values of Qbetween 0 and 1, with the thick black line denoting
the case Q=0 (pure RC circuit). (b)Cooling power ˙QNas a function
ofTRatR=RK,/Delta1/E C=5, and kBTN//Delta1=0.1f o rt h es a m ev a l u e s
ofQas in (a).
A. Series inductance
If an inductance Lconnects the environmental resistance
Rto the junction capacitance Cas in Fig. 8(a), the total
impedance is given by
Zt(ω)
RK=R
RK1+iQ2(ω/ω R)
1+i(ω/ω R)−Q2(ω/ω R)2, (14)
where Q=ωR/ωLis the quality factor with ωL=1/√
LC
andωR=1/(RC). Numerically calculated finite- Qcooling
powers ˙QNforR=RKand/Delta1/E C=5a r es h o w ni nF i g . 9(b).
The series inductance filters out part of the high-frequency tailof the noise spectrum, thereby enhancing the cooling effect.However, the quality factor can be written in the form Q=√
(L/1n H )/[√(C/1f F ) (R/1k/Omega1)]. For typical experimental
values of C/similarequal1 fF and R/similarequalRK, it then becomes evident
that most typical on-chip inductances L/lessmuch1μH will result
inQ/lessmuch1, and the RC circuit of Sec. II B is an adequate
description of the system.
B. Lossy transmission line
To model an on-chip resistor taking also stray capacitance
into account, we characterize it in terms of a resistance,capacitance, and inductance per unit length, denoted by R
0,
C0, andL0, respectively. The top half of Fig. 8(b) sketches
a distributed model of the NIS junction environment, andthe bottom half shows the transformation to two standardtwo-port RLC transmission lines in series, each of length land
terminated by an impedance Z
L/2. For a single transmissionFIG. 10. RLC transmission line: (a)P(E)a tTR=5TN,a n d
(b)cooling power ˙QNout of the N electrode for an NIS junction
coupled to an RLC transmission line. The different curves correspondto the indicated values of the line length lat the fixed inductance and
capacitance per unit length C
0andL0. The solid black line shows ˙QN
for a lumped RC environment with R=2RK.
line terminated by a load impedance ZL/2 at the position x=l,
the impedance at x=0 reads
Z(ω)=Z0e2ikl−λ
e2ikl+λ, (15)
with the wave number k=/radicalbig
−iωR 0C0+ω2L0C0, the char-
acteristic impedance Z0=√(R0+iωL 0)/(iωC 0),and the
reflection coefficient λ=(Z0−ZL/2)/(Z0+ZL/2). Here,
Z0gives the impedance of a semi-infinite transmission line. In
Fig. 10(a) we plot P(E)a tTR=5TNand in Fig. 10(b) ˙QNas
a function of TRfor a single NIS junction, assuming ZL=0.
Each of the two transmission lines is described by a fixedL
0=1.25μH/m and C0=25 pF/m, whereas R0=RK/lis
changing as lvaries from 25 to 250 μm. The values of L0and
C0are feasible for a resistor consisting of a thin and narrow
strip of a resistive metal or alloy. Figure 10(b) illustrates how
the nonzero stray capacitance reduces the cooling power. Onthe other hand, the distributed inductance can be neglected,and the results are almost indistinguishable from those of anRC transmission line.
C. Lossless transmission line
Instead of distributing the resistance Ralong the transmis-
sion line, here we calculate ˙QNfor a lossless LC line with
R0=0 andZL=RK. Figure 11(a) shows P(E)a tTR=5TN
144505-7J. T. PELTONEN et al. PHYSICAL REVIEW B 84, 144505 (2011)
FIG. 11. LC transmission line: (a)P(E)a tTR=5TN,a n d
(b)cooling power ˙QNout of the N electrode for an NIS junction
coupled to an LC transmission line terminated by a lumped resistor
R. The different curves correspond to the indicated values of the line
length lat the fixed inductance and capacitance per unit length C0and
L0. The thick black line shows ˙QNfor a lumped RC environment.
and Fig. 11(b) ˙QNas a function of TRfor a single NIS
junction. Again, each of the two transmission lines of length l
is described by a fixed L0=1.25μH/m and C0=25 pF/m,
whereas now R0=0. In contrast to the RLC transmission line
in Sec. VB,˙QNis maximized at a certain length lwhen the
inductance filters the high-frequency fluctuations, but the straycapacitance does not yet shunt them. The side peaks at E> 0
visible in P(E) occur around energies corresponding to the
frequencies at which Re[ Z
t(ω)] has a local maximum, and
their sum frequencies.
VI. COOLING BY SHOT NOISE
So far, the analysis has been limited to equilibrium
fluctuations as the origin of the noise-induced cooling powerout from the normal metal electrode. In this section, we expandthe treatment to include a special case of nonequilibriumfluctuations: we focus on the system consisting of an NISjunction coupled to the shot noise generated by another,on-chip, voltage-biased tunnel junction. To be more specific,we analyze the circuit illustrated in Fig. 12, where the
NIS junction is coupled capacitively (via on-chip couplingcapacitor of capacitance C
C) to two sources of shot noise,
tunnel junctions A and B. The former is again characterizedby the tunnel resistance R
T, capacitance C, and temperatures
TNandTS, whereas the corresponding values for the latterFIG. 12. Circuit for studying shot-noise-induced cooling in a hy-
brid tunnel junction: an NIS junction of resistance RTand capacitance
Cis coupled capacitively through CCto two voltage-biased NIN
junctions A and B that generate shot noise. The noise is described
by the current fluctuations δIi, and the junction parameters include
the resistance Ri, capacitance Ci, bias voltage Vi, and temperature Ti
(i=A,B).
two read RAandRB,CAandCB, andTAandTB, respectively.
Junctions A and B in series are biased by a constant voltageV
N, producing an average current INas well as the current
fluctuations δIAandδIB. V oltages across individual junctions
are denoted by VAandVB. The two noise source junctions are
shunted by impedance ZD, e.g., a large capacitance CD.I nt h e
following, either ZDor the voltage-biasing circuit itself are
assumed to act effectively as a short at the relevant frequencies.For most of the discussion to follow, we limit for simplicity tofully normal NIN junctions as the noise generators, althoughsome of the results apply to any type of hybrid tunnel junctions.
With nonequilibrium fluctuations present, the tunneling
rates across the NIS junction can in general no longer bewritten in terms of a single function P(E) defined by Eq. ( 3).
Instead, we find
/Gamma1
+=1
e2RT/integraldisplay∞
−∞/integraldisplay∞
−∞dEdE/primeni(E)nj(E/prime+eV)
×fi(E)[1−fj(E/prime+eV)]P+(E−E/prime) (16)
for the forward tunneling rate from electrode itoj. Analo-
gously, the backward rate reads
/Gamma1−=1
e2RT/integraldisplay∞
−∞/integraldisplay∞
−∞dEdE/primeni(E)nj(E/prime+eV)
×[1−fi(E)]fj(E/prime+eV)P−(E/prime−E). (17)
Here, the functions
P±(E)=1
2π¯h/integraldisplay∞
−∞dt eiEt/ ¯h/angbracketlefte±iϕ(t)e∓iϕ(0)/angbracketright (18)
have a similar interpretation to the P(E)o fE q .( 3)v a l i df o ra n
equilibrium environment of the NIS junction in terms of energyabsorption and emission.
32However, for non-Gaussian noise,
they are not necessarily equal to each other, and the statisticalaveraging over the environment is hard to perform. Anextension of the P(E) theory to a nonequilibrium environment
with possibly nonzero higher cumulants is considered alsoin Ref. 33. In the following, we limit ourselves to effects
arising from the second cumulant of the shot noise, and setP
±(E)=P(E). This corresponds to performing a cumulant
expansion of the quantities /angbracketlefte±iϕ(t)e∓iϕ(0)/angbracketrightand keeping only
the first nonvanishing terms, which is justified if the expansionis converging quickly. Following Ref. 32and assuming the
144505-8BROWNIAN REFRIGERATION BY HYBRID TUNNEL JUNCTIONS PHYSICAL REVIEW B 84, 144505 (2011)
circuit cutoff frequency ωCto be smaller than the intrinsic
frequency scales of the cumulants, we can estimate that therequirement R
eff/RK<1 should be satisfied for fast decay of
the higher-order terms. Here, Reffis the effective noise source
resistance seen by the NIS junction. It depends on the intrinsicresistances R
AandRBas well as the various capacitances in
Fig. 12, as will be shown below.
Assuming weak effects from the higher-order phase corre-
lations, P(E) can be written as in Eqs. ( 3) and ( 4)i nt e r m s
of the spectral density of the phase fluctuations across thejunction, and the problem reduces to specifying this quantityin the presence of shot noise. We start by analyzing thecircuit of Fig. 12to arrive at a relation connecting the voltage
fluctuation δV(ω) across the NIS junction to the intrinsic
current fluctuations δI
A(ω) and δIB(ω) of the two source
junctions. Assuming ZDto be negligibly small, we obtain
δV(ω)=ZT(ω)[δIA(ω)−δIB(ω)]with the transimpedance
ZT(ω)=Reff/(1−iωR effCeff). Here, the effective resistance
Reffand effective capacitance Ceffare related to parameters of
the circuit elements by
Reff=CC
CC+CRAB,with RAB=RARB
RA+RB, (19)
Ceff=CA+CB+CA+CB+CC
CCC. (20)For stationary and uncorrelated fluctuations δIA/B(ω), the
spectral density SV(ω) of voltage noise δV(ω)a tt h eN I S
junction is then related to the spectral densities SI,A/B(ω)o f
δIA/B(ω)v i a
SV(ω)=|ZT(ω)|2[SI,A(ω)+SI,B(ω)]. (21)
Based on this relation, the phase noise spectral density is
given by
Sϕ(ω)=/parenleftbigge
¯h/parenrightbigg21
ω2|ZT(ω)|2[SI,A(ω)+SI,B(ω)].(22)
The remaining task to obtain the correlation function J(t) from
Eq. ( 4) and using it to calculate P(E)f r o mE q .( 3) reduces
hence to specifying the intrinsic current noise spectral densitiesS
I,A/B(ω) appearing in Eq. ( 22). For tunnel junction A, one
finds
SI,A(ω)=eIA
qp(¯hω/e+VA)
1−exp/parenleftbig
−¯hω+eVA
kBTA/parenrightbig+eIA
qp(¯hω/e−VA)
1−exp/parenleftbig
−¯hω−eVA
kBTA/parenrightbig,
(23)
where IA
qp(VA) denotes the dc quasiparticle current through the
junction at the bias voltage VA, andTAdenotes its equilibrium
temperature.34A similar result holds for junction B. For NIN
noise sources, Eq. ( 23) is identical to an expression for SI,A(ω)
derived from a scattering matrix calculation:35
SI,A(ω)=¯hω
RA[coth (βA¯hω/2)+1]+FA
RAeVAsinh (βAeVA)−2¯hωcoth (βA¯hω/2)sinh2(βAeVA/2)
cosh (βAeVA)−cosh (βA¯hω), (24)
withβA=1/(kBTA), andFAdenoting the (second-order) Fano
factor of the junction. Here we identify the two independentnoise sources S
I,A(ω)=Seq
I,A(ω)+Sshot
I,A(ω), where Seq
I,A(ω)=
(¯hω/R A)[coth (βA¯hω/2)+1]is the equilibrium, i.e., zero-
bias contribution to the spectral density, and the shot-noisepart is defined as the second term in Eq. ( 24). It is worth
noting that inserting the equilibrium current noise for a singleresistor into Eq. ( 22) and using this phase spectral density to
calculate J(t) from Eq. ( 4), one recovers the equilibrium result
of Eq. ( 5). We define S
ϕ(ω)=Seq
ϕ(ω)+Sshot
ϕ(ω) with
Seq/shot
ϕ (ω)=/parenleftbigge
¯h/parenrightbigg2|ZT(ω)|2
ω2/bracketleftbig
Seq/shot
I,A(ω)+Seq/shot
I,B(ω)/bracketrightbig
.(25)
Similarly, J(t)=Jeq(t)+Jshot(t), with
Jeq/shot(t)=1
2π/integraldisplay∞
−∞dωSeq/shot
ϕ (ω)[e−iωt−1].(26)
Starting with Jeq(t), we have explicitly
Jeq(t)=/parenleftbiggReff
RA/parenrightbigg
JRC(t;Reff,Ceff,TA)
+/parenleftbiggReff
RB/parenrightbigg
JRC(t;Reff,Ceff,TB), (27)
where JRC(t;R,C,T R) denotes the equilibrium J(t)o fE q .( 5)
for a resistance Rat temperature TRin parallel with thejunction capacitance C. On the other hand, since Sshot
I,A/B(ω)
are symmetric in ω, the shot-noise contribution reads
Jshot(t)=2
RK/integraldisplay∞
0dω|ZT(ω)|2
¯hω2[cosωt−1]
×/bracketleftbig
Sshot
I,A(ω)+Sshot
I,B(ω)/bracketrightbig
. (28)
To proceed, we assume the conditions βA/B¯h/(2ReffCeff)/lessmuch1
to hold, which is reasonable at typical experimental tem-peratures for typical values R
eff/greaterorsimilar10 k/Omega1andCeff/greaterorsimilar1f F .
Then, Sshot
I,A/B(ω) are essentially frequency-independent up
to the circuit cutoff frequency ωC=1/(ReffCeff), and we
approximate
Jshot(t)/similarequal2
RKSshot
I,A(0)+Sshot
I,B(0)
¯h/integraldisplay∞
0dω|ZT(ω)|2
ω2[cosωt−1]
=ρ
2R2
effCeff
¯h/bracketleftbig
Sshot
I,A(0)+Sshot
I,B(0)/bracketrightbig
(1−|τ|−e−|τ|).
(29)
Here,ρ=2πR eff/RKandτ=t/(ReffCeff). Assuming further
thatβA/BeVA/B/greatermuch1, we recover the usual result Sshot
I,A/B(0)/similarequal
eFA/BIN, withIN=VA/RA=VB/RB. Under these conditions
144505-9J. T. PELTONEN et al. PHYSICAL REVIEW B 84, 144505 (2011)
FIG. 13. Examples of the shot-noise-induced cooling power ˙QN
for various noise source resistances RA=RB, as a function of the
average current IN. Black curves correspond to kBT=0.12/Delta1,a n d
gray ones to kBT=0.1/Delta1. Other parameters were kept fixed at C=
2f F ,CA=CB=0.5f F ,a n d CC=10 fF.
at long times |τ|/greatermuch 1, the behavior of J(t) approaches
J∞(τ)=−ρ|τ|ReffCeff
¯h/bracketleftbigg/parenleftbiggReff
RA/parenrightbigg
kBTA
+/parenleftbiggReff
RB/parenrightbigg
kBTB+1
2ReffeIN(FA+FB)/bracketrightbigg
.(30)
Comparing to the equilibrium value −ρ|τ|ReffCeffkBTeff/¯hfor
anRC environment formed by CeffandReffat a temperature
Teff, we can define an effective temperature Teffvia36
Teff=ReffeIN(FA+FB)
2kB+/parenleftbiggReff
RA/parenrightbigg
TA+/parenleftbiggReff
RB/parenrightbigg
TB.(31)
It is noteworthy that reaching Teff/greaterorsimilar1 K requires a power
input of only 1–10 pW for RAandRBin the range of tens
of k/Omega1s, instead of 0 .1–1 nW often needed to heat up an on-
chip thin-film resistor. To illustrate the cooling effect in thepresence of shot noise, Fig. 13plots ˙Q
Nas a function of
the average current INthrough the NIN noise sources. For
simplicity, we assume TN=TS=TA=TB=T. The different
curves correspond to different resistances RA=RB, whereas
the other circuit parameters were fixed to the shown values.The result is qualitatively similar to cooling induced by thermalfluctuations, but T
Ris replaced by IN.
VII. CONSIDERATIONS FOR AN EXPERIMENTAL
OBSERV ATION
A. Coupling of the NIS junction and the resistor
In an experimental realization of the hot resistor coupled
to an NIS junction, an average heating current IRis passed
through the resistor with the help of a biasing circuit. Theresistor is connected to external leads and, in addition, onemust prevent the average current I
Rfrom flowing through the
NIS junction. Instead of the schematic in Fig. 1(b), here we take
the circuit in Fig. 14as a more realistic starting point. Current
fluctuations δI(ω) generated in the resistor are transformed
into voltage fluctuations in the circuit, and coupled capacitivelyFIG. 14. (Color online) Practical coupling scheme for the noise-
generating resistor of resistance Rand the NIS junction of capacitance
Cand tunneling resistance RT. See the text for details.
via capacitors CCto the junction, whereas the average current
IRis blocked. The resistances RB/lessorsimilarRin the bias leads should
be located on-chip close to the resistor R, to prevent most of the
fluctuations δI(ω) from being shunted in the external biasing
circuit. In Fig. 14, this circuit is represented by the series lead
impedances ZSand the shunting impedance ZD, the latter of
which can consist of a purposely fabricated capacitor CD.I n
the following, we assume either ZSorZDto act as a short at the
frequencies of interest, so that looking from the resistor R,t h e
bias circuit appears as a resistance 2 RB. Such high impedance
bias leads with good shunting are important to create a well-defined electrical environment for the junction, formed ideallyonly by on-chip circuit elements.
34,37,38Propagation of the
current fluctuations δI(ω) andδIB(ω) to voltage fluctuations
δV(ω) across the junction in an arbitrary circuit can be
described systematically in terms of Langevin equations, in amanner similar to Ref. 32. The approach is valid at frequencies
ωlow enough for the corresponding wavelengths to exceed
the typical circuit dimensions. Analogously to Sec. VI,w e
obtain
S
V(ω)=R2
eff
1+(ωR effC)2/bracketleftbigg
SI(ω)+SIB,1(ω)
4+SIB,2(ω)
4/bracketrightbigg
.
(32)
Here, the effective resistance Reff is given by
Reff=R/bardblCC/(CC+2C) with R−1
/bardbl=R−1+(2RB)−1.
Remarkably, assuming the equilibrium noise SI(ω)=
(¯hω/R )[coth( βR¯hω/2)+1] and neglecting SIB(ω), we can still
employ the simple model of an RC environment, provided wereplace RbyR
effand scale J(t)b yReff/R. On the other hand,
ifRB=Rand all the resistors are at the same temperature, also
in this case Rcan simply be replaced by ReffandJ(t) scaled
by 3Reff/2R. Finally, in the limit of CC/greatermuchCandRB/greatermuchR,w e
haveReff→R, and an RC environment is again recovered.
B. Absorption of photons by the N electrode
If the resistance RNof the N electrode to be refrigerated
is not negligibly small, there is an additional, counteractingheat flow. This direct photonic heat flow P
phfrom the hot
resistor toward the colder N island via the junction capacitancediminishes the observable temperature reduction from thecooling power ˙Q
N. Assuming the resistor RatTRto be coupled
to the N island (resistance RN, temperature TN) via a reactive
144505-10BROWNIAN REFRIGERATION BY HYBRID TUNNEL JUNCTIONS PHYSICAL REVIEW B 84, 144505 (2011)
FIG. 15. Photon absorption power Pphdue to a finite RNcompared
to the cooling power ˙QN(gray dashed line). The curves from bottom
to top were calculated with the indicated values of Cph
Cranging from
1 to 100 fF, whereas other parameters were fixed to the shown values.
impedance ZC(ω), the photonic power reads39–41
Pph=/integraldisplay∞
0dω
2π¯hωT(ω)[˜nR(ω)−˜nN(ω)]. (33)
Here, T(ω)=4RR N/|R+RN+ZC(ω)|2can be viewed as
a transmission coefficient for photons, whereas ˜ni(ω)=
1/[exp(βi¯hω)−1],i=(R,N) denote Bose occupation fac-
tors of the two resistors. For direct coupling [ ZC(ω)≡0],
integration of Eq. ( 33) yields
P0
ph=4RR N
(R+RN)2k2
B
π¯hπ2
6TR+TN
2(TR−TN), (34)
demonstrating the quantized photon heat conductance.40To
analyze photon absorption by RNin the Brownian refrigerationscheme, we assume capacitive coupling [ ZC(ω)=1/(iωCph
C)]
with the effective coupling capacitance Cph
Carising from the
junction and stray capacitances. In Fig. 15, we compare the
cooling power ˙QNand power Pphby which the island is heated
due to the finite resistance RN. Assuming realistic experimental
values R=RK,RN=5/Omega1,RT=20 k/Omega1,/Delta1/E C=5(C/similarequal
2f F ) , /Delta1=200μeV, and kBTN=kBTS=0.1/Delta1, the curves
from bottom to top correspond to values of Cph
Cbetween 1
and 100 fF. For the majority of temperatures TRcontained in
Fig. 15,E q .( 33) yields values very close to P0
ph. We can
conclude that the photonic heat flow constitutes a sizableeffect that cannot be neglected in a wide range of T
R.Al a r g e
mismatch between the resistances RNandRis essential for
diminishing this heat flow compared to ˙QNarising from the
environment-assisted quasiparticle tunneling.
C. Heat balance
In this section, we consider how to observe the cooling
power ˙QN. In a typical on-chip configuration with a low-
temperature superconductor such as aluminum or titaniumwith/Delta1/lessmuch1 meV, and small NIS tunnel junctions with
C/similarequal1 fF and R
T/greaterorsimilar10 k/Omega1, the magnitude of ˙QNbecomes
evident by writing the prefactor /Delta12/(e2RT)i nt h ef o r m
(/Delta1/100μeV)2/(RT/10 k/Omega1)×1 pW. Similarly, we can write
/Delta1/E C/similarequal(/Delta1/μ eV)×(C/fF)/80. The heat flow ˙QNcan be
detected as a change in the electronic temperature TNof an N
electrode of finite size. To calculate the change in TNdue to
˙QN, we analyze the steady-state heat balance equations
Pext−/Sigma1R/Omega1R/parenleftbig
T5
R−T5
0/parenrightbig
+˙QS+˙QN−Pph=0,(35)
/Sigma1N/Omega1N/parenleftbig
T5
0−T5
N/parenrightbig
−˙QN−˙Qtherm+P0+Pph=0,(36)
FIG. 16. (Color online) Heat balance of the Brownian NIS refrigerator. (a)Contour plot of the temperature drop TN−TN,0of the N island
as a function of the environment (resistor) and bath temperatures TRandT0, obtained as a solution of the steady-state heat balance equation of
the N island, Eq. ( 36). The temperature TNis compared to TN,0,d e fi n e da s TNatTR=T0. The calculation assumes /Delta1=200μeV,R=RK,
/Delta1/E C=5,RT=20 k/Omega1,/Omega1N=2×10−21m3,/Sigma1N=2×109WK−5m−3,a n dRN=5/Omega1. In addition, we assume bias resistors with RB=R,
and that all three resistors are heated uniformly to TR. The coupling capacitance CCis set to 100 C, and we include the constant parasitic
heating P0=1 fW, the cooling power ˙Qtherm of an NIS junction thermometer of resistance RT,therm=20RTbiased at eV=0.6/Delta1[Eq. ( 1) with
P(E)=δ(E)], and finally the N island photon absorption PphfromReffvia an effective capacitance Cph
C=2 fF. The value of TRat each T0
that results in the minimum TNis indicated by the thick black line, while the black dot shows the point of optimum TRandT0.(b)Influence of
various parameters on the minimum temperature of the island as a function of T0. The reference curve corresponds to the optimum line in (a).
144505-11J. T. PELTONEN et al. PHYSICAL REVIEW B 84, 144505 (2011)
describing the coupled system of an N island, the resistor, and
their phonon systems. The phonon temperatures are assumedto equal the bath temperature T
0, thereby neglecting any
phonon cooling or heating. In addition, we assume the Selectrodes to be well thermalized with the phonons, so thatT
S=T0. Equation ( 35) gives the externally applied power
Pextrequired to heat the on-chip resistor with volume /Omega1Rand
electron-phonon coupling constant /Sigma1RtoTR.˙QN+˙QSfrom
Eqs. ( 7) and ( 8) gives the heat absorbed by the resistor in the
environment-assisted tunneling in the NIS junction. Finally,P
phfrom Eq. ( 33) denotes the heat flow via photonic coupling
between the resistor Rand the N island of finite resistance
RN. Equation ( 35) assumes that any heat conduction into the
resistor biasing leads can be neglected, so that the resistorheats up uniformly to T
R. In case of transparent NS contacts
these heat flows are strongly suppressed due to Andreevreflection at low temperatures. If the resistor and island aregalvanically coupled, the heat flows can become notable attemperatures T
R/similarequalTC/greatermuchT0,24often required to maximize the
cooling power ˙QN, necessitating a capacitive coupling.
Moving on to Eq. ( 36), its solution gives the temperature
TNof the N island of volume /Omega1Nand electron-phonon
coupling /Sigma1Nin response to the cooling power ˙QN.T h et e r m
˙Qtherm includes the heat flow due to an NIS thermometer
junction placed on the N island. Finally, P0/similarequal1f W i s a
constant phenomenological residual power that takes intoaccount the unavoidable heating of the small island due toexternal noise caused by nonideal filtering of the leads tothe external measurement circuit. In Fig. 16(a) we show the
result of solving Eq. ( 36)f o rT
Nat given TRandT0in case
of refrigeration by a single NIS junction, with experimentallyrealistic parameters. We assume aluminum with /Delta1=200μeV
and transition temperature T
C/similarequal1.5 K as the superconductor,
and copper with /Sigma1N=2×109WK−5m−3as the normal
metal, whence the junction is of the type Al-AlOx-Cu. Themaximum cooling of approximately 3% corresponds to over10 mK, which is straightforward to detect by a standard NISthermometer.
18
In Fig. 16(b) we plot examples of how the minimum
temperature TN,minis affected by changes in the various
parameters, with the reference curve corresponding to theoptimum black line in Fig. 16(a) . Reducing the island
volume /Sigma1
Nor the junction resistance RTwill lead to a clear
enhancement of the cooling effect. A thermometer junctionwith smaller resistance will slightly diminish the temperaturedrop due to increased self-cooling. Reducing /Delta1has the
largest effect: although ˙Q
Ndecreases with decreasing /Delta1,t h e
optimum bath temperature is also lower, and the counteractingelectron-phonon heat flow has decreased even more due toits strong temperature dependence. Reducing E
Chas only
a minor influence on TN, although Topt
Ris strongly affected.
As noted in Ref. 10, increasing Cand thus reducing EC
would lead to a slight enhancement of the effect due to
better filtering (lower ωR) of the voltage fluctuations with
the highest frequencies. Choosing Cbecomes a tradeoff as
this is at the cost of higher temperatures TRrequired for the
maximum effect. Most of the curves were calculated withR=R
K. A larger Rwill lead to somewhat increased cooling,
but the power Pextrequired to heat up the resistor will also
be higher. According to our preliminary measurements and inagreement with earlier experiments,42Pext/similarequal100 pW–1 nW
applied to an on-chip thin-film chromium resistor can resultin parasitic heating of the N island via substrate phonons toan extent clearly exceeding any heat extraction ˙Q
Ndue to
Brownian refrigeration. Therefore, it is important to minimizethe resistor volume even at the cost of lower resistance, as longasR/greaterorsimilarR
K. Suspending the resistor would be advantageous
but result in a more complicated fabrication process. Finally,we note that instead of heating the resistor ( T
R/greatermuchT0,TN), it
could be cooled ( TR<T 0,TN) with NIS junctions, and one
could observe the cooling of a small S electrode predicted byEq. ( 8), although the power is generally considerably smaller
than ˙Q
Nin the case of TR>T N. Moreover, probing the S
temperature is not as straightforward, and the effect may bemasked by direct photonic cooling of the S electrode.
24
VIII. SUMMARY AND CONCLUSIONS
In summary, we have analyzed Brownian refrigeration in a
tunnel junction between a normal metal and a superconductor,where thermal noise generated in a hot resistor can cause heatextraction from the cold normal metal. The net entropy of thewhole system was shown to be always increasing for a generalequilibrium environment. It is, however, interesting that onecan exploit thermal fluctuations in cooling.
We considered the heat extraction in a single NIS junction,
and in a two-junction hybrid single-electron transistor, in aregime where charging effects become important. If phononheating is kept at a sufficiently low level, the effect can berealized straightforwardly in an on-chip configuration usingstandard fabrication techniques. Under realistic values for thecircuit parameters, the cooling power is expected to resultin a sizable drop of the electronic temperature of a smallnormal metal island. More generally, our results demonstratethe importance of the electromagnetic environment in ananalysis of not only electric, but also of the less studied,environmentally assisted heat transport in tunnel junctions.
ACKNOWLEDGMENTS
We acknowledge financial support from the EU FP7
projects “GEOMDISS” and “SOLID.” We thank V . Maisi,M. Meschke, M. M ¨ott¨onen, and O.-P. Saira for useful discus-
sions. J.T.P. acknowledges financial support from the FinnishAcademy of Science and Letters.
APPENDIX A: ANALYTICAL APPROXIMATION FOR ˙QN
OF AN NIS JUNCTION
Assuming a Gaussian P(E) of width s=√2ECkBTRand
center EC, we present an approximation for ˙QNbased mainly
on the Sommerfeld expansion of fN(E). First, we rewrite
Eq. ( 7)a s
˙QN=2
e2RT/integraldisplay∞
0dE/primenS(E/prime)
×{F(E/prime)−fS(E/prime)[F(E/prime)−F(−E/prime)]},(A1)
with the function F(E/prime) defined in Eq. ( 9). At low tem-
peratures kBTN/lessmuchs, we identify H(E)=EP(E−E/prime) and
144505-12BROWNIAN REFRIGERATION BY HYBRID TUNNEL JUNCTIONS PHYSICAL REVIEW B 84, 144505 (2011)
approximate F(E/prime) by the first three terms in its Sommerfeld
expansion:
F(E/prime)=/integraldisplay∞
−∞dEf N(E)H(E)
/similarequal/integraldisplay0
−∞dEH (E)+π2
6(kBTN)2dH(E)
dE/vextendsingle/vextendsingle/vextendsingle/vextendsingle
E=0
+7π4
360(kBTN)4d3H(E)
dE3/vextendsingle/vextendsingle/vextendsingle/vextendsingle
E=0. (A2)
In terms of the dimensionless variable y=(E/prime+EC)/sand
the dimensionless temperature a=kBTN/s, this can be written
explicitly as
F(E/prime)/similarequalse−y2
2
√
2π/bracketleftbigg
−1+π2
6a2+7π4
120a4/bracketrightbigg
+1
2yserfc/parenleftbiggy√
2/parenrightbigg
,
(A3)
where erfc denotes the complementary error function. Next,
assuming a perfect BCS DOS with γ=0, we write Eq. ( A1)
in terms of x=E/prime//Delta1as
˙QN=2/Delta1
e2RT/integraldisplay∞
1dxx√
x2−1{F(x)−fS(x)[F(x)−F(−x)]}.
(A4)
To obtain a closed form expression for ˙QN, further ap-
proximations are still needed. We consider low temperaturesk
BTS/lessmuch/Delta1, where most contributions to ˙QNin Eq. ( A1)
come from energies E/prime/similarequal/Delta1, and we approximate the DOS
atx/similarequal1b yx/√
x2−1/similarequal1/√2(x−1). At kBTS/lessmuch/Delta1,t h e
terms in Eq. ( A4) containing fS(E/prime) can be neglected in a first
approximation. This requires fS(E/prime) to have decayed close to
zero at energies E/prime/similarequal/Delta1. Combining Eqs. ( A3) and ( A4) then
yields
/integraldisplay∞
1dxF(x)√2(x−1)
=s√π
960e−d/radicalbig
1+g{2(160 +7π4a4)
×[−I−1/4(d)+2dI−3/4(d)−2dI−5/4(d)]
+[−160(1 +4d)+40π2a2+7π4a4(−1+4d)]
×[I−1/4(d)−I1/4(d)]}, (A5)
where Iν(z) denote modified Bessel functions of the first kind,
of fractional order νand argument z, and we introduced the
quantity d=(1+g)2/4r2withg=EC//Delta1andr=s//Delta1.I f
d/greatermuch1o rd/lessmuch1, Eq. ( A5) can be further simplified with
asymptotic expansions of Iν(z), but we do not present them
here, since d/similarequal1 for typical experimental parameters. To
include the effect of a finite but small TS,w eh a v et oi n t e g r a t e
also the term containing fS(x)[F(x)−F(−x)] in Eq. ( A4). We
approximate fS(E/prime)/similarequalexp/parenleftbig
−E/prime/kBTS/parenrightbig
, valid at kBTS/lessmuch/Delta1
around E/prime/similarequal/Delta1. To get a rough estimate, we impose the
further limitation /Delta1/greatermuchEC, whence we can directly replace
ybyx/r in Eq. ( A3), and make the major simplificationF(x)−F(−x)/similarequalxs/r . We arrive at
/integraldisplay∞
1dxx√
x2−1fS(x)[F(x)−F(−x)]
/similarequal/integraldisplay∞
1dx1√2(x−1)exp(−hx)xs
r=/radicalbiggπ
2s
r1+2h
2h3/2e−h,
(A6)
withh=/Delta1/k BTS. Typically h/greatermuch1, and Eq. ( A6)g i v e sa
negligibly small correction compared to neglected higher-order terms in Eq. ( A2).
APPENDIX B: POSITIVITY OF THE ENTROPY
PRODUCTION RATE
Here we fill in details on how to manipulate the integrand
Ion the last five lines of Eq. ( 10) into the form of Eq. ( 11).
We start by writing
I=(βS−βN)E[/Delta11+fS(E)/Delta12]
+(βR−βN)E/prime[/Delta13+fS(E)/Delta14], (B1)
where the quantities /Delta1ican be identified as
/Delta11=fN(E+E/prime)+e−βRE/primefN(E−E/prime),
/Delta12=[−1−fN(E+E/prime)+fN(E−E/prime)]
−e−βRE/prime[1−fN(E+E/prime)+fN(E−E/prime)],
/Delta13=fN(E+E/prime)−e−βRE/primefN(E−E/prime),
/Delta14=[1−fN(E+E/prime)−fN(E−E/prime)]
−e−βRE/prime[1−fN(E+E/prime)−fN(E−E/prime)].(B2)
Introducing the symmetric and antisymmetric combinations
S1andA1via
S1=1
2(/Delta11+/Delta13)=fN(E+E/prime),
(B3)
A1=1
2(/Delta11−/Delta13)=e−βRE/primefN(E−E/prime),
and similarly S2andA2as
S2=1
2(/Delta12+/Delta14)=−e−βRE/prime−(1−e−βRE/prime)fN(E+E/prime),
(B4)
A2=1
2(/Delta12−/Delta14)=− 1+(1−e−βRE/prime)fN(E−E/prime),
we have /Delta11=S1+A1,/Delta13=S1−A1,/Delta12=S2+A2, and
/Delta14=S2−A2. Notice that in this way we have separated the
fN(E+E/prime), which appears in the S terms, from the fN(E−
E/prime) appearing in the A terms. We find
/Delta11+fS(E)/Delta12=[S1+fS(E)S2]+[A1+fS(E)A2]
=S+A,
/Delta13+fS(E)/Delta14=[S1+fS(E)S2]−[A1+fS(E)A2]
=S−A, (B5)
withS=S1+fS(E)S2andA=A1+fS(E)A2. Inserting
this into Eq. ( B1) yields
I=(βS−βN)E[S+A]+(βR−βN)E/prime[S−A]
=A[(βS−βN)E−(βR−βN)E/prime]
+S[(βS−βN)E+(βR−βN)E/prime]. (B6)
144505-13J. T. PELTONEN et al. PHYSICAL REVIEW B 84, 144505 (2011)
The following step is to write explicitly SandA, yielding
S=e−βRE/prime[fN(E+E/prime)(1+eβRE/prime+βSE)−1]
1+eβSE,(B7)
A=fN(E−E/prime)(eβSE−βRE/prime+1)−1
1+eβSE. (B8)Finally, inserting the explicit equilibrium forms of fN(E±
E/prime)g i v e s S=NS(eX−1) and A=NA(eY−1), where the
positive quantities NSandNAare defined by Eqs. ( 12) and
(13), respectively. Similarly, X=(βS−βN)E+(βR−βN)E/prime
andY=(βS−βN)E−(βR−βN)E/prime. Putting everything to-
gether, we arrive at I=NS(eX−1)X+NA/parenleftbig
eY−1/parenrightbig
Y,
which is Eq. ( 11) in Sec. III.
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144505-14 |
PhysRevB.72.085221.pdf | Momentum selectivity and anisotropy effects in the nitrogen K-edge resonant inelastic x-ray
scattering from GaN
V . N. Strocov,1,*T. Schmitt,2,†J.-E. Rubensson,2P. Blaha,3T. Paskova,4and P. O. Nilsson5
1Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
2Department of Physics, Uppsala University, Box 530, S-75121 Uppsala, Sweden
3Institut für Materialchemie, Technische Universität Wien, A-1060 Wien, Austria
4Linköping University, S-581 83 Linköping, Sweden
5Chalmers University of Technology, S-412 96 Göteborg, Sweden
/H20849Received 6 December 2004; revised manuscript received 14 March 2005; published 31 August 2005 /H20850
High-resolution soft x-ray emission and absorption spectra near the N K-edge of wurtzite GaN are presented.
The experimental data are interpreted in terms of full-potential electronic structure calculations. The absorptionspectra, compared with calculations including core hole screening, indicate partial core hole screening in theabsorption process. The resonantly excited x-ray emission spectra show pronounced dispersion of spectralstructures, which is attributed to effects of momentum conservation in the resonant inelastic x-ray scattering/H20849RIXS /H20850process. In development of GaN based optoelectronics, momentum selectivity in RIXS can be utilized
to control development of band structure in GaN nanostructures.
DOI: 10.1103/PhysRevB.72.085221 PACS number /H20849s/H20850: 78.70.En, 78.55.Cr, 78.70.Dm, 71.20.Nr
I. INTRODUCTION
Soft-x-ray emission /H20849SXE /H20850and absorption /H20849SXA /H20850spec-
troscopies allow investigation of the electronic structure ofthe valence band /H20849VB/H20850and conduction band /H20849CB/H20850, respec-
tively, with elemental specificity and large probing depth upto 3000 Å. In particular, the electronic structure of buriedquantum dots, interfaces
1and even isolated impurities2can
be accessed by these techniques /H20849see further references in a
review by Kotani and Shin /H20850.3
In general, the SXE/SXA spectroscopies characterize the
electron partial density of states /H20849PDOS /H20850resolved in their
elemental and orbital character, but averaged over the wavevectors k. For weakly correlated semiconductor systems,
however, certain k-selectivity appears in resonant inelastic
x-ray scattering /H20849RIXS /H20850where the core electron is excited
within a few eV above the absorption threshold.
3In this case
the absorption and emission events are coupled in a fast co-herent scattering process, in which the full momentum isconserved as
q
in−qout=ke+kh,
where qinandqoutare wave vectors of the absorbed and
emitted photons, and keandkhare those of the conduction
electron and valence hole, respectively. Dependence of theresonant SXE spectra on the excitation energy h
/H9263exreflects
then the VB and CB dispersions E/H20849k/H20850. The kselectivity of
RIXS has been demonstrated for a number of semiconduc-
tors such as Si and SiC, and even for the graphitesemimetal.
3–5
GaN is a prototype nitride compound representative of a
large family of binary and ternary group III nitrides, having awide range of optoelectronic device applications coveringthe entire region from 200 to 1200 nm. Presently commercialblue laser diodes are available, and lasers working in theblue-violet spectral region are realized for both pulsed andcontinuous wave operations. Further progress can be ex-pected in low dimensional nitride devices like quantum dot
lasers or single photon emitters. The current strong interest innanostructure systems like self-assembled GaN quantumdots in AlN or AlGaN matrix is motivated by their potential
to achieve lasers with longer lifetimes, higher gains, lowerthreshold currents and improved temperature characteristics.
6
Progress in fabrication of such devices is closely linked witha better knowledge about the electronic properties of the bulknitride compounds as well as nitride based nanostructures.
Spectroscopic investigations of the GaN electronic struc-
ture are far from being exhaustive. In particular, itsk-resolved studies with conventional angle-resolved photo-
electron spectroscopy /H20849ARPES /H20850are burdened by surface ef-
fects sensitive to the sample growth and surface preparationprocedures /H20849see, e.g., Refs. 7 and 8 /H20850. Moreover, small photo-
electron escape depth principally limits applications of thistechnique to the nanostructure systems buried in a matrix ofother material. The soft-x-ray spectroscopies with their largeprobing depth and elemental specificity are ideally suited forthis purpose. In application to GaN, Stagarescu et al.
9and
Lawniczak-Jablonska et al.10used the SXE and SXA spec-
troscopies to study the occupied and unoccupied k-integrated
PDOS, respectively. Eisebitt et al.11found pronounced
changes in the N K-edge SXE spectral shape occurring near
the absorption threshold, which they interpreted in terms ofmomentum conservation. However, a systematic study ofRIXS in GaN and its connection to the band structure ispresently missing.
Here, we present a high-resolution SXE/SXA study per-
formed on wurtzite GaN near the N K-edge /H208491score level /H20850.
Compared to our pilot study in Ref. 12, the present resultsare based on experimental data obtained for a larger set ofexcitation energies at higher resolution and varied scatteringgeometries, and receive extended theoretical analysis. Weunambiguously identify effects of momentum conservationin the RIXS process. The experimental results are supportedby state-of-art band structure calculations.PHYSICAL REVIEW B 72, 085221 /H208492005 /H20850
1098-0121/2005/72 /H208498/H20850/085221 /H208499/H20850/$23.00 ©2005 The American Physical Society 085221-1II. EXPERIMENT
We have studied wurtzite GaN with the /H208490001 /H20850surface
orientation. The sample was a 30 /H9262m thick GaN film grown
on a sapphire substrate using hydride vapour phase epitaxyin the optimum growth window.
13High crystalline quality of
the sample was confirmed by high resolution x-ray diffrac-tion rocking curve and low temperature photoluminescence/H20849LT-PL /H20850measurements. The FWHM values of the
/H9275- and
2/H9258-/H9275radial scans for the symmetric /H20849002/H20850and asymmetric
/H20849102/H20850,/H20849104/H20850, and /H20849114/H20850reflections approached the best values
reported for thick nitride films grown on foreign substrate,and indicate relatively low mosaicity without domain forma-tion achieved at this thickness. The LT-PL spectrum of thesample shows well resolved exciton-related peaks in the nearband edge region with FWHM less than 2 meV , confirmingthe high crystalline and optical quality of the film.
The SXE/SXA experimental data were taken near the N
K-edge /H208491score level /H20850at/H11011400 eV. The experiments were
performed at MAX-lab, Sweden, at the undulator beamlineI511-3 equipped with an SX-700 type plane gratingmonochromator,
14and a high-resolution Rowland-mount
grazing incidence SXE spectrometer.15The incident radiation
had linear p-polarization /H20849Evector in the incidence plane /H20850.
The sample was positioned at a near-grazing /H20849NG/H20850incidence
geometry with a grazing angle of /H1101120° relative to the inci-
dent beam.
The SXA data were measured with a monochromator
resolution of 0.2 eV , which is comparable to the N 1 score
level intrinsic width /H110110.1 eV.10The absorption signal was
recorded in total electron yield mode and normalized to thephotocurrent from a gold mesh introduced into the synchro-tron radiation beam upstream the experimental chamber.
The experimental SXA spectrum is shown in Fig. 1 /H20849right
spectrum on top /H20850. In all figures the experimental data are
represented in two energy scales: the one on top of the panelsreflects the photon energies, whereas the one in the bottomgives the energies relative to the valence band maximum/H20849VBM /H20850determined by aligning the leading edges of the ex-
perimental and theoretical SXE spectra /H20849see below /H20850.
The SXA spectrum shows significant angle dependence
due to high anisotropy of GaN /H20849Refs. 10 and 16 /H20850/H20849see below /H20850.
The spectrum in Fig. 1 measured at a NG-incidence geom-etry is in good agreement with the previous data
10,16mea-
sured under similar conditions, although with slight differ-ences in the peak amplitudes due to a somewhat differentincidence angle.
The SXE measurements were performed with the spec-
trometer installed in the incidence plane at a scattering angleof 90° relative to the incident beam. With our NG-incidencegeometry, the SXE spectra were therefore measured at near-normal /H20849NN/H20850emission geometry with an emission angle of
/H1101120° relative to the surface normal. The spectrometer was
operated in 1st diffraction order with a spherical grating of 5m radius and 1200 lines/mm groove density, providing aresolution around 0.2 eV .
The energy scale of the spectrometer was set using the
tabulated characteristic x-ray line energies of the Co L
/H9251and
L/H9252lines from Ref. 17 recorded from a pure Co foil in 2nd
diffraction order. Based on the elastic peaks in the SXE spec-tra, this energy scale was then used to calibrate the mono-
chromator relative to the spectrometer energy scale. This en-sured the residual misfit between the spectrometer andmonochromator energy scales below 0.2 eV . For further ex-perimental details see Ref. 2.
The experimental off-resonant SXE spectrum, taken at an
excitation energy h
/H9263exof 428.7 eV well above the absorption
threshold, is shown in Fig. 1 /H20849top left spectrum /H20850. It is in good
agreement with previous off-resonant data.9Taken at h/H9263ex
well above the N K-edge, this spectrum will further be used
as a reference for the resonant series.
The experimental resonant SXE spectra are shown in Fig.
2 in comparison with the reference off-resonant one /H20849at the
FIG. 1. /H20849Top/H20850Experimental N 1 soff-resonant SXE spectrum
/H20849h/H9263ex=428.7 eV /H20850and SXA spectrum. The energy scale on top of the
panel reflects the photon energies, whereas in the one in the bottomthe energies are relative to VBM. With our near-grazing incidenceexperimental geometry, the SXE and SXA spectra reflect mainly theNp
xyPDOS in the VB and N pzPDOS in the CB; /H20849gray curves
below /H20850theoretical results excluding and including the core hole, and
/H20849superimposed black curves /H20850empirical simulation of the lifetime
and instrumental broadening effects, scaled /H110032;/H20849chart between the
experimental and theoretical curves /H20850experimental energies of the
SXE and three leading SXA peaks compared with the no-hole the-oretical ones.STROCOV et al. PHYSICAL REVIEW B 72, 085221 /H208492005 /H20850
085221-2bottom /H20850. The corresponding h/H9263exin relation to the SXA
structures /H20849see inset of Fig. 2 /H20850are indicated on the right. The
resonant spectra were measured with a monochromator reso-lution better than 0.2 eV , matching that of the spectrometer.The resonant SXE series shows clear dispersions and line-shape changes of both spectral peaks in the VB, with theirdispersion ranges being about 0.3 eV for the lower-energypeak, and 1.0 eV for the higher-energy one. This identifiespronounced RIXS effects in GaN.
III. COMPUTATIONS
Our computations of the band structure E/H20849k/H20850and PDOS
were performed within the standard DFT formalism. The
electron exchange-correlation was described within the gen-eralized gradient approximation /H20849GGA /H20850.
18The calculations
employed a full-potential LAPW + Local Orbitalsmethod
19,20implemented in the WIEN2k package.21This computational framework was extended to evaluate
the SXA and off-resonant SXE spectra. The SXE calcula-tions employed “the final-state rule,” where the core hole isfilled.
22The SXA calculations, in the first step, were per-
formed within the “initial-state approximation,”23where the
core hole is completely neglected /H20849for the core hole effects
see below /H20850. Within these approaches the SXE/SXA spectra
appear as the element and orbital projected PDOS, multipliedwith the energy dependent dipole matrix elements betweenthe core and the corresponding VB/CB states.
24In our cal-
culations the wave function integration to evaluate the matrixelements was restricted to the atomic spheres /H20849defined in the
APW + LO method /H20850where the core level wave functions are
localized. The dipole matrix elements usually appear as quitesmooth functions of energy and do not much alter the corre-sponding PDOS. /H20849This is different when one has a p-type
core hole, e.g., an L
2,3spectrum, where the contributions of
thep→sandp→dchannels crucially depend on the ratio of
the corresponding matrix elements. /H20850
Due to the anisotropy of GaN, the PDOS and SXE/SXA
computations were performed for the N pzand N pxyorbitals
separately. The SXE/SXA spectra relevant for our experi-mental geometry were obtained by summation of the N p
z
and N pxyspectra weighted by relative squared amplitudes of
theE/H20648candE/H11036ccomponents of the Evector /H20849see below /H20850.
The calculated SXE/SXA spectra are shown in Fig. 1 in
comparison with the experiment. The smooth curves includean empirical simulation of the energy dependent lifetimes ofthe valence band hole and conduction band electron. Thiswas achieved by convolution of the calculated spectra with a
Lorentzian, whose full width was heuristically taken to de-pend on energy in proportion to the energy distance from theFermi level /H20849taken in the middle of the band gap /H20850as/H9003
=
/H9251/H20841E−EF/H20841. Additional broadening due to the N 1 score level
width of /H110110.1 eV was insignificant. The experimental reso-
lution was also included into this simulation by additionalGaussian convolution. By comparison of the simulated spec-tral broadening with the experiment, we estimated the pro-portionality coefficient to be
/H9251/H110110.1. This gives a rough es-
timate for the energy dependence of lifetimes through the VBand CB. The simulated SXE spectrum was used to set theVBM energy position in the experimental SXE spectra byaligning the spectral leading edges.
The theoretical SXE/SXA spectra in Fig. 1 demonstrate,
on the whole, convincing agreement with the experiment.Even such a subtle detail as a structure on the low-energyslope of the dominant SXE peak is well reproduced. Theagreement extends up to 30 eV above the absorption thresh-old. Some energy shifts between the theoretical and experi-ment spectral structures are due to excited-state self-energycorrections, and relative amplitude disagreements in SXAcan be attributed to core hole effects /H20849see below /H20850.
Our computational results are a major improvement com-
pared to the previous LMTO calculations from Ref. 16,which were restricted by the use of the muffin-tin /H20849MT/H20850po-
tential. This illustrates the importance of full-potential effectsin covalent materials such as GaN. Further computationalresults are presented in the section below.
FIG. 2. Experimental N 1 sresonant SXE series in comparison
with the off-resonant spectrum /H20849the bottom curve /H20850, all normalized to
the same peak value /H20849relative to the foot background /H20850. The indicated
excitation energies h/H9263exare also marked at the SXA spectrum /H20849in-
set/H20850. The off-resonant energies of the two spectral peaks are marked
by dashed lines. The excitation energy dependent dispersion andline shape changes of the SXE structures are due to momentumconservation in the RIXS process.MOMENTUM SELECTIVITY AND ANISOTROPY EFFECTS … PHYSICAL REVIEW B 72, 085221 /H208492005 /H20850
085221-3IV. RESULTS AND DISCUSSION
A. SXA and off-resonant SXE
Overview of the experimental spectra
The experimental N 1 sSXA spectrum in Fig. 1 reflects
thep-projected PDOS in the N core region through the CB,
This is because the dipole selection rules require that theorbital angular momentum quantum number lchanges by ±1
in the process of photon absorption or emission. In fact, dueto predominance of the antibonding 2 pstates in the CB of
GaN this PDOS is almost equivalent to the total density ofstates /H20849DOS /H20850. With the fundamental band gap of GaN being
3.4 eV , the SXA spectral onset is placed /H110111 eV above the
conduction band minimum /H20849CBM /H20850, where the PDOS van-
ishes /H20849see below /H20850.
The experimental SXE spectrum in Fig. 1 reflects, by the
same selection rules, the p-projected PDOS in the N core
region for the occupied states. In the photon energy regionfrom 387 to 395 eV , the spectrum reflects the VB composedmostly of the bonding N 2 pstates. It shows here two peaks,
which are further referred to as Afor the main one and Bfor
the minor one at lower energy.
A small peak in the SXE spectrum near 377.5 eV was first
noticed in Ref. 9 /H20849although we find it 1.6 eV higher in en-
ergy, which may be traced back to different x-ray lines usedin calibration /H20850. It was interpreted as due to hybridization of
the Ga 3 dsemicore states with N 2 pstates. The Ga 3 dorigin
of this SXE peak is consistent with the binding energy posi-tion of the Ga 3 dpeak in the ARPES spectra of GaN by
Dhesi et al.
6It should be noted, however, that the hybridiza-
tion with N 2 pstates is not crucial for the Ga 3 dstates to
appear in the N 1 sSXE spectrum: If they merely protrude
into the N core region without hybridization, their angularmomentum expansion from the N site will always containsome pcomponent. This picture is confirmed by our band
structure computations, which indicate negligible hybridiza-tion of the Ga 3 da n dN2 pstates. Interestingly, the onset of
the Ga 3 dderived structure with h
/H9263ex/H20849Fig. 2 /H20850seems to be
slightly delayed compared to the main N 2 pregion.
On the high-energy side of the experimental Ga 3 dpeak
we can distinguish a weak but statistically significant pla-teau, extending by /H110115 eV to higher energy. It is not repro-
duced by our calculations restricted by the dipole approxi-mation. In an ARPES study of GaN, Dhesi et al.
6also
observed a weak structure on the high-energy side of the Ga3dpeak, and interpreted it as due to the N 2 sstates. Al-
though in SXE these states are forbidden by the dipole se-lection rules, some intensity can in principle appear throughnon-dipole transitions.
Anisotropy effects
In GaN the VB and CB are dominated by the N 2 pzand
2pxyorbitals, oriented, parallel and perpendicular to the
/H208490001 /H20850surface normal c, respectively. The selection rules in
SXA are such that the excitation cross section into a particu-larporbital is maximized if the incident Evector is oriented
parallel to the orbital axis, and vanishes if perpendicular.
25
Therefore, the E/H20648ccomponent of the Evector excites the pz
orbitals, and the E/H11036ccomponent the pxyones. For our ex-perimental SXA spectrum taken with a grazing incidence
angle of 20°, the ratio between /H20849E/H20648c/H208502and /H20849E/H11036c/H208502, repre-
senting the relative weight of the N pzand N pxycontribu-
tions to the SXA spectrum, is about 7.5:1. Therefore, ourSXA spectrum taken at a NG-incidence geometry reflectsmainly the unoccupied N p
zorbitals. The same selection
rules apply to SXE. For our experimental SXE spectrum theratio between the N p
xand N pxycontributions of about
1:7.5, is reverse to the SXA case. Therefore, our SXE spec-trum taken at a NN-emission geometry reflects mainly theoccupied N p
xyorbitals.
In highly anisotropic wurtzite GaN, the PDOS associated
with the N pzand N pxyorbitals is significantly different.
This is illustrated by computations in Fig. 3. The SXE/SXAspectra are seen to be essentially the PDOS multiplied by thedipole matrix element with weak energy dependence. The
FIG. 3. /H20849Top curves /H20850Theoretical N pxyPDOS with the corre-
sponding normal emission SXE spectrum, and normal incidenceSXA spectrum /H20849excluding and including core hole /H20850;/H20849bottom /H20850Np
z
PDOS with the corresponding grazing emission SXE and grazing
incidence SXA spectra. Difference between the N pxyand N pz
PDOS results in angle dependence of the SXE/SXA spectra. The
lifetime and instrumental broadening is simulated as in Fig. 1.STROCOV et al. PHYSICAL REVIEW B 72, 085221 /H208492005 /H20850
085221-4difference of the N pzand N pxyPDOS, replicated by differ-
ences of the corresponding grazing and normal incidence oremission spectra, gives rise to the SXE/SXA angle depen-dence. Dramatic differences of the N p
zand N pxyPDOS in
the CB result in a particularly strong angle dependence of theSXA spectra.
10,16The theoretical SXE/SXA spectra in Fig. 1,
corresponding to our experimental geometry, were obtainedas a superposition of the N p
zand N pxyspectra weighted
according to the above /H20849E/H20648c/H208502to/H20849E/H11036c/H208502ratios.
Core hole effects in SXA
The core hole generated in the SXA process in fact dy-
namically interacts with the valence electrons and modifiesthe crystal potential around it where the conduction electronis placed /H20849see, e.g., Ref. 26 /H20850. Of the static approximations to
this process,
22,23one limit is the “initial-state approxima-
tion,” where the core hole is completely ignored. It was im-plied by the above SXA calculations. The other limit is the“final-state rule,” which considers a crystal potential result-ing from static screening of the core hole by valence elec-trons.
Our SXA computations were further extended to include
the core hole within the “final-state rule.” A self-consistent-field /H20849SCF /H20850potential was generated in a supercell with the
core hole on a probe atom. The computations employed a2/H110032/H110031 supercell /H2084916 atoms/cell /H20850with one 1 score hole on
one of the N atoms. The missing core electron was added to
the valence electrons. During the SCF-cycle all states /H20849of the
atom where the core hole resides and of all neighbor atoms /H20850
were allowed to respond to this core hole /H20849a larger “effective
nuclear charge” /H20850and participate in the screening.
The N p
zand N pxySXA spectra calculated within the
“final-state rule” are shown in Fig. 3, and their superpositioncorresponding to our experimental geometry in Fig. 1. Com-pared to the “initial-state approximation,” the spectral struc-tures slightly shift to lower energies. This reflects excitoniccoupling between the screened core hole and conductionelectron. The shifts show non-monotonous energy depen-dence, which identifies dependence of the core hole effectson the character of the CB states. Relative amplitudes of thespectral structures are changed dramatically. In particular, theamplitude of the first SXA peak in the “initial-state approxi-mation” appears underestimated compared to the experiment,whereas the screened core hole results in an overestimatedamplitude. The opposite holds for the third SXA peak. Theexperimental amplitudes appear somewhere between the“initial-state” and “final-state” limits, which suggests partialcore hole screening,
22implying significant mobility of the
valence charge in GaN. Previously it was shown that “opti-mization” of the theoretical spectra by choosing a partialcore hole /H20849e.g., half a 1 selectron /H20850may lead to better agree-
ment with experiment.
22However, here we refrained from
such an approach because of its empirical character.
Excited-state self-energy corrections
In SXE, as far as the “final-state rule” originally derived
for simple metals22holds, the spectral structures reflect the
quasi-particle energy levels of the valence hole in the N-1electron system left behind after the radiative deexcitation.
Their shifts from the DFT theoretical energy levels are theself-energy corrections, /H9004/H9018, appearing due to difference in
the exchange-correlation potential between the excited andground state. Difference of our experimental and theoreticalpeak energies /H20849see the chart in Fig. 1 /H20850yields /H9004/H9018values of
−0.1,−0.5, and −4.6 eV for the A,B, and Ga 3 dderived
peaks, respectively. As expected, /H9004/H9018increases when going
away from the Fermi level, and is particularly large for theGa 3 dderived peak due to the localized semicore character
of this state.
In SXA, the /H9004/H9018corrections reflecting quasi-particle en-
ergy levels of the conduction electron in N+1 electron sys-
tem are distorted by excitonic interaction with the partiallyscreened core hole. The interaction energy can be estimated,roughly, as half the energy difference between the SXApeaks calculated with and without the core hole. With such acorrection, the shifts between the experimental and theoreti-cal peaks in Fig. 1 yield /H9004/H9018 values of +0.9, +1.4, and
+1.1 eV for the three dominant SXA peaks, in the order ofincrease in energy. Their sign is consistent with the band gapproblem of the DFT, which neglects the exchange-correlationdiscontinuity for excitations across the band gap. However,their magnitude shows nonmonotonous energy dependenceand, moreover, appears reduced compared to the +1.5 eVshift in the CBM /H20849difference between the experimental opti-
cal band gap of 3.4 eV and the DFT one of 1.9 eV /H20850. Similarly
to the core hole effects, such an anomalous behavior of /H9004/H9018
may indicate dependence of the self-energy effects on thecharacter of the CB states.
27
B. RIXS
Energy dependence of the coherent fraction
Changes in the experimental resonant SXE series within
the VB region are emphasized in difference spectra shown inFig. 4 /H20849top/H20850. These spectra are obtained from the curves in
Fig. 2 by subtracting the off-resonant reference spectrum.The latter was scaled, for each spectrum, to set the differencespectrum positive and having its minimal value equal tozero. To reduce the influence of noise on this procedure, thescaling coefficient was determined from the spectra, whichhad been denoised with Gaussian smoothing.
The difference spectra in Fig. 4 reflect the coherent frac-
tion in the resonant SXE spectra: If the off-resonantspectrum—by virtue of large phase space available at highexcitation energies for electron-phonon /H20849e-ph/H20850and electron-
electron /H20849e-e/H20850interactions with the conduction electron in the
intermediate state, which results in effective averaging over
kspace—represents the incoherent fraction, the positive in-
tensity remaining after its subtraction is the coherentfraction.
4The difference spectra show two dispersing peaks.
The relative weight of the coherent fraction in the total
SXE spectra within the VB as a function of h/H9263exis shown in
Fig. 5. They were evaluated as integrals of the differencespectra from Fig. 4 over the VB energy interval from 387.4to 395.4 eV , divided by corresponding integrals of the totalspectra from Fig. 2. The h
/H9263exdependence of the latter is in
fact equivalent to the SXA spectrum, also shown in Fig. 5,MOMENTUM SELECTIVITY AND ANISOTROPY EFFECTS … PHYSICAL REVIEW B 72, 085221 /H208492005 /H20850
085221-5because the N 1 score hole can be radiatively filled only
from the N 2 pVB states /H20849the contribution from the Ga 3 d
derived states is negligible /H20850.
The coherent fraction in Fig. 5 vanishes at higher energies
because the strength of the e-phand e-einteraction, in-
creases with energy due to increase of the phase space avail-able for them. However, below h
/H9263exof 405 eV our experi-
mental data demonstrate, surprisingly, that the coherentfraction weight behaves non-monotonously and even in-
creases with energy. This fact is beyond the above simplephase space arguments. Evidently, the low-energy conduc-
tion band states in GaN change their character in this energyregion in such a way that their susceptibility to the incoher-ente-phande-escattering varies. Tentatively, we attribute
this effect to the anisotropy of the material. As the coherentfraction is larger in the region where the CB layer-projectedDOS is dominated by N p
xystates, it suggests that the inco-
herent scattering is in some way less pronounced within theatomic layers than between the layers. Interestingly, the co-herent fraction remains large up to rather high excitation en-ergies.
k-selectivity effects
Our experimental SXE and SXA spectra, taken in a NN-
emission/NG-incidence geometry, are related primarily to theVB states having the N p
xycharacter and CB states having N
pzcharacter. Figure 4 /H20849bottom /H20850shows the calculated E/H20849k/H20850
where the N 2 pxyweights of the VB states and N 2 pzof the
CB states are indicated. As a link between the experimentalspectra and calculated E/H20849k/H20850, Fig. 4 /H20849middle /H20850reproduces the
calculated off-resonant N p
xySXE and N pzSXA spectra
from Fig. 3, which are essentially equivalent to the corre-sponding PDOS. E/H20849k/H20850of GaN appears rather involved, fea-
turing a multitude of critical points /H20849including those from
non-symmetry Brillouin zone directions /H20850with energy separa-
tion smaller than the lifetime broadening. The peaks in theoff-resonant SXE /H20849and SXA /H20850spectra can therefore not be
related to certain critical points. Such a relation appears onlyfor the resonant SXE spectra due to k-selectivity effects in
RIXS.
The following k-selectivity effects are identified in the
resonant SXE series in comparison with the calculated E/H20849k/H20850.
/H208491/H20850When h
/H9263exis tuned to the absorption onset at 398.4
eV , the SXE spectrum /H20849Fig. 2 /H20850shows a peak splitting off on
the high energy side of the main peak. This effect is relatedto the VBM at the /H9003point: At the absorption onset, as seen
FIG. 4. /H20849Top/H20850Differential resonant SXE spectra, representing
the coherent spectral fraction, normalized to the same intensitymaximum. The indicated excitation energies h
/H9263exare also marked at
the SXA spectrum. The energies of the two off-resonant SXE spec-tral peaks are marked by dashed lines; /H20849middle /H20850calculated N p
xy
derived SXE and N pzderived SXA spectra; /H20849bottom /H20850calculated
E/H20849k/H20850. The weights of the N 2 pxyVB states and N 2 pzCB states are
indicated by the radii of the circles. Dispersion of the resonant SXEspectral structures is related primarily to that of the N 2 p
xyvalence
states and N 2 pzconduction states.
FIG. 5. Relative weight of the coherent SXE fraction as a func-
tion of excitation energy /H20849the thin line connecting the dots is a guide
for the eye /H20850compared with the SXA spectrum /H20849dashed line , arbi-
trarily scaled /H20850reflecting the total SXE intensity. Anomalous behav-
ior of the coherent fraction at low h/H9263exreflects changes in character
of the conduction states.STROCOV et al. PHYSICAL REVIEW B 72, 085221 /H208492005 /H20850
085221-6by comparison of the calculated SXA spectrum to E/H20849k/H20850, the
excited conduction electrons appear already in some 1 eV
above the CBM /H20849the onset is delayed relative to the CBM
because the lowest CB states, depleted in the N pcharacter
and having rather steep dispersions, deliver too smallPDOS /H20850. The lowest conduction band puts the corresponding
k-vectors in some 0.3 Å
−1from the /H9003point near the /H9003A
direction, where this band acquires significant N pzcharacter.
The photon momentum transfer /H9004q=qout−qinin our experi-
ment has almost the same value 0.28 Å−1, and is also di-
rected close to /H9003A. Therefore, as indicated in the E/H20849k/H20850panel,
thek-conserving RIXS process couples the excited N pzcon-
duction electrons to the N pxyvalence holes in the VBM at
the/H9003point, which results in a coherent emission peak from
the VBM appearing on the high energy side of the mainpeak.
The VB bottom is placed in the same /H9003point as the VBM.
Therefore, at the absorption onset it also gives coherentemission, most clearly seen in the difference spectra /H20849Fig. 4 /H20850
as an intensity enhancement near the binding energy of−8.2 eV. Taking self-energy effects into account, this figureis consistent with the theoretical VB bottom at −7.1 eV.
It should be noted that due to relatively high photon en-
ergies at the N K-edge the proper interpretation of RIXS
requires that non-negligible photon momentum transfer istaken into account.
/H208492/H20850Ash
/H9263exincreases, the main peak in the difference
spectra disperses to lower energies. In the total SXE spectra,this makes the main peak disperse towards the off-resonantposition A. This effect is related to the band dispersion
around the /H9003point: With increase of h
/H9263ex, the conduction
band electron k-vector moves away from /H9003, following the
2pzconduction band along the /H9003AH line. Reflecting the 2 pxy
heavy-hole valence bands along /H9003AH /H20849the light-hole band
does not contribute due to its pzcharacter in this region of
thekspace /H20850, the main coherent peak disperses then to lower
energies. Its dispersion range towards h/H9263ex/H11011401 eV, where
it passes the off-resonant position, is /H110110.9 eV. This figure
well matches the theoretical CB and VB dispersions. We donot detect any dispersion renormalization due to core holeeffects. It should be noted that the GaN anisotropy is crucialin this picture, allowing us to disentangle various bands inE/H20849k/H20850based on the orbital orientation selectivity.
In principle, pushing h
/H9263exbelow the 398.4 eV shifts the
conduction electron towards the CBM, and the valence hole,coupled by the /H9004qmomentum transfer, away from the VBM.
This should result in shifting of the main coherent peak againto lower energies. However, the experiment in this h
/H9263exre-
gion is difficult due to vanishing absorption.
/H208493/H20850With h/H9263exvaried near 401 eV , the difference spectra
display a minor coherent peak dispersing below the incoher-ent one Bat −6.5 eV. Based on the N p
zdispersion in the
CB, this can be associated with the N pxydispersion in the M
valley near the VB bottom.
/H208494/H20850Upon further increase of h/H9263exabove 401 eV , the main
coherent peak moves below the incoherent one A. Its energy
position can be associated with an integral effect of a fewcritical points in /H110112 eV below the VBM such as that in the
M point.
RIXS phenomena have also been observed for isolated N
impurities in GaAsN.
2These phenomena were interpreted interms of remnant k-conservation using spectral
decomposition28of localized N wave functions over the
Bloch waves of bulk GaAs.
Emission angle dependence
The above discussion involving the anisotropy effects
suggests that the resonant SXE spectra should show certaindependence on the emission angle. To check this, we rotatedthe sample towards a NN-incidence or NG-emission geom-etry with a grazing emission angle of 20°, opposite to theexperimental geometry above. In this case the absorption ismainly due to the unoccupied N p
xyorbitals and emission
due to the occupied N pzstates. However, the SXE signal in
this geometry gets strongly suppressed due to the self-absorption as well as the larger distance between the spot andspectrometer slits. We have therefore measured only one rep-resentative resonant SXE spectrum with h
/H9263ex=399.3 eV not
far from the absorption threshold.
The experimental NG-emission spectrum is shown in Fig.
6 compared with the NN-emission one from Fig. 2 measuredat the same h
/H9263ex. Their difference is plotted below. The spec-
tra show statistically significant differences, reflecting thedifference between the N p
xyand N pzstates. In particular, in
the NG-emission spectrum the spectral maximum is shiftedby/H110110.15 eV to deeper energies. This reflects more steep
dispersion of the valence N p
zstates away from the /H9003-point
compared to the N pxystates.
Outlook: Nitride nanostructures
A number of recent investigations in the field of nitride
compounds optoelectronics are focused on nanostructuressuch as Ga /H20849Al/H20850N quantum dots /H20849for a recent review, see Ref.
6/H20850. It is expected that quantum confinement effects and
FIG. 6. Resonant SXE spectrum measured with h/H9263ex
=399.3 eV at a NG-emission geometry /H20849top curve /H20850compared to its
NN-emission counterpart from Fig. 2 normalized to the same peakvalue /H20849below /H20850, and their difference /H20849bottom /H20850. The difference be-
tween the spectra is due to the difference between the N p
xyand N
pzstates.MOMENTUM SELECTIVITY AND ANISOTROPY EFFECTS … PHYSICAL REVIEW B 72, 085221 /H208492005 /H20850
085221-7smaller defect densities achieved in nanostructures would
lead to higher optical efficiency compared to bulk materials.They are also considered as potential UV-light emitters. Ofparticular interest are Ga /H20849Al/H20850N nanocolumns, which can be
grown with a diameter of 30–150 nm and heights up to1.6
/H9262m on different substrates such as Si /H20849111/H20850or Al 2O3
/H20849with or without buffer layer, typically AlN having a larger
band gap /H20850.29Soft-x-ray spectroscopies with their large prob-
ing depth /H20849up to 300 nm /H20850and elemental specificity are ide-
ally suited to investigate the electronic structure of suchnanostructures. Grazing incidence experimental geometrycan be used to stay away from the N signal of the bufferlayer. The k-selectivity in RIXS from GaN will allow for
characterization of the band structure in such nanostructureswith resolution in k-space /H20849with description in terms of k
applicable to the confined electron states in the sense of spec-tral decomposition of wave functions
28over the Bloch waves
of bulk GaN /H20850and its development with diameter and heights
of nanocolumns. Other interesting objects, although not yetpractically realized, are GaN nanotubes.
30They are predicted
to behave as direct band gap semiconductors when zigzagshaped, and indirect when armchair shaped. RIXS can beused to discriminate direct vs. indirect character of such
nanostructures.V. SUMMARY AND CONCLUSIONS
High-resolution SXE/SXA experimental data on wurtzite
GaN near the N Kedge are presented. The measurements are
supported by full-potential calculations extended to the corehole screening.
The obtained results identify, in particular the following:
/H208491/H20850partial core hole screening in the SXA process; /H208492/H20850effects
of the GaN anisotropy in the SXE/SXA processes. Our ex-perimental geometry invoked primarily the p
xystates in the
VB and pzstates in the CB; /H208493/H20850pronounced dispersions of
the resonant SXE structures, identifying the effects of mo-mentum conservation and k-selectivity in the RIXS process;
/H208494/H20850nonmonotonous behavior of the coherent SXE fraction,
reflecting different effects of the involved CB states in thee-eande-phscattering.
Thekselectivity and anisotropy effects in RIXS can be
utilized as an advanced tool to control development of bandstructure in GaN based nanostructures, allowing optimizationof the shape, size and density of the nanostructure organiza-tion for particular applications.
*Corresponding author. Email address: vladimir.strocov@psi.ch
†Corresponding author. Present address: Swiss Light Source, Paul
Scherrer Institute, CH-5232 Villigen PSI, Switzerland. Email ad-dress: thorsten.schmitt@psi.ch
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085221-9 |
PhysRevB.77.104520.pdf | Gap structure in noncentrosymmetric superconductors
K. V. Samokhin1and V. P. Mineev2
1Department of Physics, Brock University, St. Catharines, Ontario, Canada L2S 3A1
2Commissariat à l’Energie Atomique, DSM/DRFMC/SPSMS, 38054 Grenoble, France
/H20849Received 9 November 2007; revised manuscript received 15 January 2008; published 18 March 2008 /H20850
Gap structure in noncentrosymmetric superconductors with spin-orbit band splitting is studied using a
microscopic model of pairing mediated by phonons and/or spin fluctuations. The general form of pairinginteraction in the band representation is derived, which includes both the intraband and interband pairing terms.In the case of isotropic interaction /H20849in particular, for a BCS-contact interaction /H20850, the interband pairing terms
vanish identically at any magnitude of the band splitting. The effects of pairing interaction anisotropy areanalyzed in detail for a metal of cubic symmetry with strong spin-orbit coupling. It is shown that if phononsare dominant then the gaps in two bands are isotropic, nodeless, and have in general different amplitudes.Applications to the Li
2/H20849Pd1−x,Ptx/H208503B family of noncentrosymmetric superconductors are discussed.
DOI: 10.1103/PhysRevB.77.104520 PACS number /H20849s/H20850: 74.20.Fg, 74.20.Rp, 74.90. /H11001n
I. INTRODUCTION
Superconducting materials without inversion symmetry
have recently become a subject of considerable interest, bothexperimental and theoretical. Starting from CePt
3Si/H20849Ref. 1/H20850,
the list of noncentrosymmetric superconductors has grown toinclude UIr /H20849Ref. 2/H20850, CeRhSi
3/H20849Ref. 3/H20850, CeIrSi 3/H20849Ref. 4/H20850,
Y2C3/H20849Ref. 5/H20850,L i 2/H20849Pd1−x,Ptx/H208503B/H20849Ref. 6/H20850,K O s 2O6/H20849Ref. 7/H20850,
and other compounds. In most cases, the fundamental ques-
tions about the gap symmetry and the pairing mechanismremain unresolved.
The spin-orbit /H20849SO/H20850coupling of electrons with a noncen-
trosymmetric crystal lattice lifts spin degeneracy of the elec-tron energy bands almost everywhere, which has importantconsequences for superconductivity: In the limit of strongSO coupling, the Cooper pairing between the electrons withopposite momenta occurs only if they are from the samenondegenerate band. This scenario is realized in CePt
3Si,
where the SO band splitting exceeds the critical temperatureby orders of magnitude.
8The same is likely to be the case in
other materials, for instance in Li 2/H20849Pd1−x,Ptx/H208503B; see Ref. 9.
The pairing interaction between electrons is most natu-
rally introduced using the exact band states,8,10–12which take
into account all the effects of the crystal lattice potential andthe SO coupling, see Sec. II. In the strong SO coupling limit,the order parameter is represented by a set of complex func-tions, one for each band, which makes the theory of noncen-trosymmetric superconductors similar to that of usual multi-band superconductors, see Ref. 13. An alternative approach
based on the representation of the pairing interaction in termsof the pure spinor states unaffected by the SO coupling wasdeveloped in Refs. 14and15.
In a phenomenological multiband pairing Hamiltonian,
the relative strength of pairing in different bands can be ar-bitrary. In this article, we go beyond the phenomenologicaldescription and study the gap structure in noncentrosymmet-ric superconductors under some fairly general assumptionsabout the microscopic mechanism of pairing. Specifically,we consider the interaction mediated by bosonic excitations/H20849phonons and/or spin fluctuations /H20850. Starting with a micro-
scopic expression for a momentum and frequency dependentpairing interaction, we derive the general form of the pairing
interaction in the band representation, which contains boththe intraband and interband pairing terms. The latter isshown to vanish identically in the case of isotropic BCS-contact interaction for any magnitude of the SO band split-ting; see Sec. III. In general, the interband pairing is absentonly in the limit of large band splitting; see Sec. IV.
In Sec. V, we present a detailed analysis of the possible
gap structures in noncentrosymmetric superconductors of cu-bic symmetry, in a model which includes both the phononand spin-fluctuation mediated interactions. The Conclusioncontains a discussion of our results in the context ofLi
2/H20849Pt1−x,Ptx/H208503B experiments.
II. BASIC DEFINITIONS
The Hamiltonian of noninteracting electrons in a noncen-
trosymmetric crystal has the following form:
H0=/H20858
k/H20851/H92800/H20849k/H20850/H9254/H9251/H9252+/H9253/H20849k/H20850/H9268/H9251/H9252/H20852ak/H9251†ak/H9252=/H20858
k/H20858
/H9261=/H11006/H9264/H9261/H20849k/H20850ck/H9261†ck/H9261,
/H208491/H20850
where /H9251,/H9252=↑,↓are spin indices, /H9268are the Pauli matrices,
/H9264/H9261/H20849k/H20850=/H92800/H20849k/H20850+/H9261/H20841/H9253/H20849k/H20850/H20841are the band dispersion functions, and
the sum over kis restricted to the first Brillouin zone. In Eq.
/H208491/H20850and everywhere below, summation over repeated spin
indices is implied, while summation over the band indices isalways shown explicitly. The SO coupling of electrons withthe crystal lattice is described by the pseudovector
/H9253/H20849k/H20850,
which satisfies /H9253/H20849−k/H20850=−/H9253/H20849k/H20850and /H20849g/H9253/H20850/H20849g−1k/H20850=/H9253/H20849k/H20850, where g
is any operation from the point group Gof the crystal; see
the examples below.
The Hamiltonian in the first line of Eq. /H208491/H20850is diagonalized
by the following transformation:
ak/H9251=/H20858
/H9261=/H11006u/H9251/H9261/H20849k/H20850ck/H9261, /H208492/H20850
with the coefficientsPHYSICAL REVIEW B 77, 104520 /H208492008 /H20850
1098-0121/2008/77 /H2084910/H20850/104520 /H208499/H20850 ©2008 The American Physical Society 104520-1u↑/H9261/H20849k/H20850=/H20881/H20841/H9253/H20841+/H9261/H9253z
2/H20841/H9253/H20841,
u↓/H9261/H20849k/H20850=/H9261/H9253x+i/H9253y
/H208812/H20841/H9253/H20841/H20849/H20841/H9253/H20841+/H9261/H9253z/H20850/H208493/H20850
forming a unitary matrix uˆ/H20849k/H20850. The Fermi surfaces defined by
the equations /H9264/H11006/H20849k/H20850=0 are split, except for the points or lines
where /H9253/H20849k/H20850=0. The band dispersion functions /H9264/H9261/H20849k/H20850are in-
variant with respect to all operations from G, and also even
inkdue to time reversal symmetry: The states /H20841k,/H9261/H20856and
K/H20841k,/H9261/H20856belong to kand − k, respectively, and have the same
energy. Here K=i/H9268ˆ2K0is the time reversal operation, and
K0is the complex conjugation. One can write K/H20841k,/H9261/H20856
=t/H9261/H20849k/H20850/H20841−k,/H9261/H20856, where t/H9261/H20849k/H20850=−t/H9261/H20849−k/H20850is a nontrivial phase
factor.10,11For the eigenstates defined by expressions /H208493/H20850we
obtain
t/H9261/H20849k/H20850=/H9261/H9253x/H20849k/H20850−i/H9253y/H20849k/H20850
/H20881/H9253x2/H20849k/H20850+/H9253y2/H20849k/H20850. /H208494/H20850
The momentum dependence of the SO coupling is deter-
mined by the crystal symmetry. For the cubic group G=O,
which describes the point symmetry of Li 2/H20849Pd1−x,Ptx/H208503B, the
simplest form compatible with the symmetry requirements is
/H9253/H20849k/H20850=/H92530k, /H208495/H20850
where /H92530is a constant. For the point groups containing im-
proper elements, i.e., reflections and rotation-reflections, ex-pressions become more complicated. In the case of the fulltetrahedral group G=T
d, which is relevant for Y 2C3and pos-
sibly KOs 2O6, one has
/H9253/H20849k/H20850=/H92530/H20851kx/H20849ky2−kz2/H20850xˆ+ky/H20849kz2−kx2/H20850yˆ+kz/H20849kx2−ky2/H20850zˆ/H20852./H208496/H20850
This is also known as the Dresselhaus interaction,16and was
originally proposed to describe the SO coupling in bulksemiconductors of zinc-blende structure. For the tetragonalgroup G=C
4v, which is relevant for CePt 3Si, CeRhSi 3and
CeIrSi 3, the SO coupling is given by
/H9253/H20849k/H20850=/H9253/H11036/H20849kyxˆ−kxyˆ/H20850+/H9253/H20648kxkykz/H20849kx2−ky2/H20850zˆ. /H208497/H20850
In the purely two-dimensional case, setting /H9253/H20648=0 one recov-
ers the Rashba interaction,17which is often used to describe
the effects of the absence of mirror symmetry in semicon-ductor quantum wells.
Now let us take into account an attractive interaction be-
tween electrons in the Cooper channel, using the basis of theexact eigenstates of the noninteracting problem. The mostgeneral form of the interaction Hamiltonian the band repre-sentation is
H
int=1
2V/H20858
kk/H11032q/H20858
/H92611,2,3,4V/H92611/H92612/H92613/H92614/H20849k,k/H11032;q/H20850
/H11003ck+q,/H92611†c−k,/H92612†c−k/H11032,/H92613ck/H11032+q,/H92614. /H208498/H20850
We assume that the qdependence of the pairing interaction is
neglected /H20849see the next section /H20850. The terms with /H92611=/H92612and
/H92613=/H92614describe intraband pairing and the scattering of theCooper pairs from one band to the other, while the remaining
terms describe pairing of electrons from different bands. Theabove Hamiltonian can be considerably simplified in the ab-sence of the interband pairing, which is the case if the SOsplitting of the bands, E
SO, is large compared with all energy
scales associated with superconductivity. Since the pairinginteraction is effective only inside the shells of width
/H9275c/H20849the
cutoff energy /H20850in the vicinity of the Fermi surfaces, one can
set/H92611=/H92612=/H9261and/H92613=/H92614=/H9261/H11032, and obtain
Hint=1
2V/H20858
kk/H11032q/H20858
/H9261/H9261/H11032V/H9261/H9261/H11032/H20849k,k/H11032/H20850ck+q,/H9261†c−k,/H9261†c−k/H11032,/H9261/H11032ck/H11032+q,/H9261/H11032,/H208499/H20850
where
V/H9261/H9261/H11032/H20849k,k/H11032/H20850=t/H9261/H20849k/H20850t/H9261/H11032*/H20849k/H11032/H20850V˜/H9261/H9261/H11032/H20849k,k/H11032/H20850. /H2084910/H20850
The pairing amplitudes V˜/H9261/H9261/H11032are even in both kandk/H11032/H20849due
to the anticommutation of fermionic operators /H20850and also in-
variant under the point group operations: V˜/H9261/H9261/H11032/H20849g−1k,g−1k/H11032/H20850
=V˜/H9261/H9261/H11032/H20849k,k/H11032/H20850.18
In the case of large SO band splitting, the order parameter
has only intraband components. It is uniform /H20849in the absence
of external fields /H20850and can be represented in the form
/H9004/H9261/H20849k/H20850=t/H9261/H20849k/H20850/H9004˜/H9261/H20849k/H20850. The gap functions /H9004˜/H9261transform accord-
ing to one of the even irreducible representations of the point
group and satisfy the following equations:
/H9004˜/H9261/H20849k/H20850=−T/H20858
n/H20858
/H9261/H11032/H20885d3k/H11032
/H208492/H9266/H208503V˜/H9261/H9261/H11032/H20849k,k/H11032/H20850
/H11003/H9004˜/H9261/H11032/H20849k/H11032/H20850
/H9275n2+/H9264/H9261/H110322/H20849k/H11032/H20850+/H20841/H9004˜/H9261/H11032/H20849k/H11032/H20850/H208412. /H2084911/H20850
The expression on the right-hand side converges due to the
energy cutoff at /H9275c.
III. BCS MODEL
Let us calculate the pairing amplitudes and the gap func-
tions in a simple BCS-like model, in which the attractiveinteraction is both instantaneous in time and local in space:
H
int=−V/H20885d3r/H9274↑†/H20849r/H20850/H9274↓†/H20849r/H20850/H9274↓/H20849r/H20850/H9274↑/H20849r/H20850
=−V
4/H20885d3r/H20849i/H92682/H20850/H9251/H9252/H20849i/H92682/H20850/H9253/H9254†/H9274/H9251†/H20849r/H20850/H9274/H9252†/H20849r/H20850/H9274/H9253/H20849r/H20850/H9274/H9254/H20849r/H20850,
/H2084912/H20850
where V/H110220. Using the band representation of the field op-
erators,
/H9274/H9251/H20849r/H20850=1
/H20881V/H20858
k,/H9261u/H9251/H9261/H20849k/H20850eikrck/H9261, /H2084913/H20850
we obtain the pairing Hamiltonian in the form /H208498/H20850withK. V. SAMOKHIN AND V. P. MINEEV PHYSICAL REVIEW B 77, 104520 /H208492008 /H20850
104520-2V/H92611/H92612/H92613/H92614/H20849k,k/H11032/H20850=−V
2/H20849i/H92682/H20850/H9251/H9252/H20849i/H92682/H20850/H9253/H9254†u/H9251/H92611*/H20849k/H20850u/H9252/H92612*
/H11003/H20849−k/H20850u/H9253/H92613/H20849−k/H11032/H20850u/H9254/H92614/H20849k/H11032/H20850.
Here we neglected the difference between u/H9251/H9261/H20849/H11006k+q/H20850and
u/H9251/H9261/H20849/H11006k/H20850, which is O/H20849q/kF/H20850. In conventional centrosymmet-
ric superconductors, we have q/kF/H11011/H20849/H9264kF/H20850−1/H112701/H20849/H9264is the cor-
relation length /H20850. In the noncentrosymmetric case, the above
estimate might not work and the qdependence of the pairing
interaction might be more important, leading, for instance, tothe Lifshitz invariants in the free energy
8,19and a spatial
modulation of the order parameter even in the absence ofexternal fields. We leave this issues to a separate publication.
Using the identities
u
/H9251/H9261/H20849−k/H20850=t/H9261*/H20849k/H20850/H20849i/H92682/H20850/H9251/H9252u/H9252/H9261*/H20849k/H20850, /H2084914/H20850
and also the unitarity of the matrix uˆ/H20849k/H20850, we obtain for the
pairing potential:
V/H92611/H92612/H92613/H92614/H20849k,k/H11032/H20850=−V
2t/H92612/H20849k/H20850t/H92613*/H20849k/H11032/H20850/H9254/H92611/H92612/H9254/H92613/H92614. /H2084915/H20850
Therefore, interband pairing is absent in the BCS model for
any strength of the SO coupling. Comparing this expressionwith Eq. /H2084910/H20850, one can see that both the intraband pairing and
the pair scattering between the bands are characterized by the
same coupling constant: V˜/H9261/H9261/H11032/H20849k,k/H11032/H20850=−V/2. The pairing sym-
metry is isotropic, and it follows from Eqs. /H2084911/H20850that the gap
functions are the same in both bands: /H9004˜+/H20849k/H20850=/H9004˜−/H20849k/H20850=/H9257. This
is not surprising, since the local interaction /H2084912/H20850cannot lead
to any kdependence of the gaps.
The critical temperature is given by Tc
=/H208492eC//H9266/H20850/H9275ce−1 /NFV, where C/H112290.577 is Euler’s constant, NF
=/H20849N++N−/H20850/2, and N/H9261is the Fermi-level density of states in
the/H9261th band. Although this has the usual BCS form, the
superconductivity is non-BCS, because the order parameterresides in two nondegenerate bands, with T
cand/H9257indepen-
dent of the band splitting and the difference between N+and
N−. One can show that both the critical temperature and the
gap magnitude are not affected by isotropic scalarimpurities.
20
IV. INTERACTION MEDIATED BY BOSONIC
EXCITATIONS
Now we investigate a more general model, in which the
pairing is assumed to be due to the exchange of somebosonic excitations. We consider two types of excitations:Scalar /H20849phonons /H20850, which couple to the electron density
/H9267/H20849r/H20850
=/H9274/H9251†/H20849r/H20850/H9274/H9251/H20849r/H20850, and pseudovector /H20849spin fluctuations /H20850, which
couple to the electron spin density s/H20849r/H20850=/H9274/H9251†/H20849r/H20850/H9268/H9251/H9252/H9274/H9252/H20849r/H20850. Us-
ing the standard functional-integral representation of the par-
tition function of the system, we obtain the following term inthe fermionic action describing an effective two-particle in-teraction between electrons:S
int=gph2
2/H20885dx dx /H11032/H9267/H20849x/H20850D/H20849x−x/H11032/H20850/H9267/H20849x/H11032/H20850
+gsf2
2/H20885dx dx /H11032si/H20849x/H20850Dij/H20849x−x/H11032/H20850sj/H20849x/H11032/H20850, /H2084916/H20850
where x=/H20849r,/H9270/H20850is a shorthand notation for the coordinates in
real space and the Matsubara time, /H20848dx/H20849.../H20850=/H20848dr/H208480/H9252d/H9270/H20849.../H20850,
gphand gsfare the coupling constants of electrons with
phonons and spin fluctuations, while D/H20849x−x/H11032/H20850andDij/H20849x
−x/H11032/H20850are the phonon and spin-fluctuation propagators, re-
spectively. The spin fluctuations can be associated either
with the localized spins, if such are present in the system, orwith the collective spin excitations of the itinerant electrons/H20849paramagnons /H20850.
21In the latter case, Dij/H20849x−x/H11032/H20850can be ex-
pressed in terms of the electron dynamical spin susceptibility
/H9273ij/H20849q,/H9275/H20850. In general, the interaction /H2084916/H20850is nonlocal both in
space and time. The BCS-contact Hamiltonian /H2084912/H20850is recov-
ered when the spin fluctuations are neglected and gph2D/H20849r,/H9270/H20850
is replaced by - V/H9254/H20849r/H20850/H9254/H20849/H9270/H20850.
In the momentum-frequency representation, Eq. /H2084916/H20850
yields the following pairing action:
Sint=1
2/H9024/H20858
kk/H11032q/H20851gph2D/H20849k−k/H11032/H20850/H9254/H9251/H9254/H9254/H9252/H9253+gsf2Dij/H20849k−k/H11032/H20850/H9268/H9251/H9254i/H9268/H9252/H9253j/H20852
/H11003a¯/H9251/H20849k+q/H20850a¯/H9252/H20849−k/H20850a/H9253/H20849−k/H11032/H20850a/H9254/H20849k/H11032+q/H20850, /H2084917/H20850
where /H9024=/H9252Vis the space-time volume, a¯/H9251/H20849k/H20850anda/H9251/H20849k/H20850are
Grassmann fields, k=/H20849k,/H9275n/H20850,q=/H20849q,/H9263m/H20850, and/H9275n=/H208492n+1/H20850/H9266T
and/H9263m=2m/H9266Tare the fermionic and bosonic Matsubara fre-
quencies, respectively. We assume that the conditions of theMigdal theorem are fulfilled, and also neglect the frequencyrenormalization, which corresponds to the weak-couplinglimit of the Eliashberg theory. The theory developed belowshould work, at least qualitatively, even for such materials asCePt
3Si, in which strong electron correlations are responsible
for a heavy-fermion behavior and the above assumptionsmight be inapplicable.
The phonon propagator is real and even in both frequency
and momentum, D/H20849k−k
/H11032/H20850=D/H20849k/H11032−k/H20850, and can therefore be
written as follows:
D/H20849k−k/H11032/H20850=Dg/H20849k,k/H11032/H20850+Du/H20849k,k/H11032/H20850, /H2084918/H20850
where the first term on the right-hand side, Dg/H20849k,k/H11032/H20850
=/H20851D/H20849k−k/H11032/H20850+D/H20849k+k/H11032/H20850/H20852/2, is even in both kandk/H11032, while the
second term, Du/H20849k,k/H11032/H20850=/H20851D/H20849k−k/H11032/H20850−D/H20849k+k/H11032/H20850/H20852/2, is odd in
both kandk/H11032.
The spin-fluctuation propagator satisfies Dij/H20849k−k/H11032/H20850
=Dji/H20849k/H11032−k/H20850and can be broken up into the symmetric and
antisymmetric in ijparts. Representing the latter in terms of
a dual vector R, we obtain
Dij/H20849k−k/H11032/H20850=Dijg/H20849k,k/H11032/H20850+Diju/H20849k,k/H11032/H20850+ieijlRl/H20849k−k/H11032/H20850,/H2084919/H20850
where the first /H20849second /H20850term on the right-hand side is an
even /H20849odd/H20850function of kand k/H11032, while Ri/H20849k−k/H11032/H20850=−Ri/H20849k/H11032
−k/H20850. The antisymmetric component of the spin-fluctuation
propagator is associated with the Dzyaloshinskii-Moriya
interaction.22It is absent in the centrosymmetric case, due toGAP STRUCTURE IN NONCENTROSYMMETRIC … PHYSICAL REVIEW B 77, 104520 /H208492008 /H20850
104520-3the additional symmetry Dij/H20849k−k/H11032/H20850=Dij/H20849k/H11032−k/H20850.
After some straightforward algebra /H20849see the Appendix /H20850,
the action /H2084917/H20850takes the following form:
Sint=1
2/H9024/H20858
kk/H11032qV/H9251/H9252/H9253/H9254/H20849k,k/H11032/H20850a¯/H9251/H20849k+q/H20850a¯/H9252/H20849−k/H20850
/H11003a/H9253/H20849−k/H11032/H20850a/H9254/H20849k/H11032+q/H20850, /H2084920/H20850
where the pairing interaction is represented as a sum of the
k-even, k-odd, and mixed-parity terms: V=Vg+Vu+Vm. The
even contribution is
V/H9251/H9252/H9253/H9254g/H20849k,k/H11032/H20850=vg/H20849k,k/H11032/H20850/H20849i/H92682/H20850/H9251/H9252/H20849i/H92682/H20850/H9253/H9254†, /H2084921/H20850
where
vg/H20849k,k/H11032/H20850=1
2/H20851gph2Dg/H20849k,k/H11032/H20850−gsf2trDˆg/H20849k,k/H11032/H20850/H20852. /H2084922/H20850
The odd contribution is
V/H9251/H9252/H9253/H9254u/H20849k,k/H11032/H20850=vu,ij/H20849k,k/H11032/H20850/H20849i/H9268i/H92682/H20850/H9251/H9252/H20849i/H9268j/H92682/H20850/H9253/H9254†, /H2084923/H20850
where
vu,ij/H20849k,k/H11032/H20850=1
2/H20851gph2Du/H20849k,k/H11032/H20850+gsf2trDˆu/H20849k,k/H11032/H20850/H20852/H9254ij−gsf2Diju/H20849k,k/H11032/H20850.
/H2084924/H20850
Finally, the mixed-parity contribution is
V/H9251/H9252/H9253/H9254m/H20849k,k/H11032/H20850=vm,i/H20849k,k/H11032/H20850/H20849i/H9268i/H92682/H20850/H9251/H9252/H20849i/H92682/H20850/H9253/H9254†+vm,i/H20849k/H11032,k/H20850
/H11003/H20849i/H92682/H20850/H9251/H9252/H20849i/H9268i/H92682/H20850/H9253/H9254†, /H2084925/H20850
where
vm,i/H20849k,k/H11032/H20850=gsf2
2/H20851Ri/H20849k−k/H11032/H20850+Ri/H20849k+k/H11032/H20850/H20852. /H2084926/H20850
The first term on the right-hand side of Eq. /H2084925/H20850is odd in k
and even in k/H11032, while the second term is even in kand odd in
k/H11032.
We would like to note that expressions /H2084921/H20850,/H2084923/H20850, and
/H2084925/H20850have completely general form in the sense that they do
not rely on our assumptions about boson-mediated interac-tions and exhaust all possible spin structures of the pairingamplitude. Under the point group operations g, the coeffi-
cients
vg,vu,ij, and vmtransform like a scalar, a second-rank
tensor, and a pseudovector, respectively, and satisfy the in-variance conditions
vg/H20849g−1k,/H9275n;g−1k/H11032,/H9275n/H11032/H20850
=vg/H20849k,/H9275n;k/H11032,/H9275n/H11032/H20850, etc. By analogy with the theory of super-
conductivity in centrosymmetric compounds, see, e.g., Ref.
24, Eqs. /H2084921/H20850and /H2084923/H20850correspond to spin-singlet and spin-
triplet pairing channels respectively, while Eq. /H2084925/H20850describes
singlet-triplet mixing. The possibility of singlet-triplet mix-ing due to the Dzyaloshinskii-Moriya interaction in the staticcase was pointed out in Ref. 23.
Next, we use Eqs. /H208492/H20850to transform the pairing action into
the band representation. Using identities /H2084914/H20850, we obtain the
transformation rules for the pair creation operators in thespin-singlet and spin-triplet channels/H20849i
/H92682/H20850/H9251/H9252a¯/H9251/H20849k+q/H20850a¯/H9252/H20849−k/H20850=−/H20858
/H92611,2t/H92612/H20849k/H20850/H9254/H92611/H92612c¯/H92611/H20849k+q/H20850c¯/H92612/H20849−k/H20850,
/H20849i/H9268/H92682/H20850/H9251/H9252a¯/H9251/H20849k+q/H20850a¯/H9252/H20849−k/H20850=−/H20858
/H92611,2t/H92612/H20849k/H20850/H9270/H92611/H92612/H20849k/H20850
/H11003c¯/H92611/H20849k+q/H20850c¯/H92612/H20849−k/H20850,
where
/H9270ˆi/H20849k/H20850=uˆ†/H20849k/H20850/H9268ˆiuˆ/H20849k/H20850. /H2084927/H20850
Inserting these in Eq. /H2084920/H20850, we obtain
Sint=1
2/H9024/H20858
kk/H11032q/H20858
/H92611,2,3,4V/H92611/H92612/H92613/H92614/H20849k,k/H11032/H20850
/H11003c¯/H92611/H20849k+q/H20850c¯/H92612/H20849−k/H20850c/H92613/H20849−k/H11032/H20850c/H92614/H20849k/H11032+q/H20850, /H2084928/H20850
where
V/H92611/H92612/H92613/H92614/H20849k,k/H11032/H20850=t/H92612/H20849k/H20850t/H92613*/H20849k/H11032/H20850V˜/H92611/H92612/H92613/H92614/H20849k,k/H11032/H20850, /H2084929/H20850
and
V˜/H92611/H92612/H92613/H92614/H20849k,k/H11032/H20850=vg/H20849k,k/H11032/H20850/H9254/H92611/H92612/H9254/H92613/H92614
+vu,ij/H20849k,k/H11032/H20850/H9270i,/H92611/H92612/H20849k/H20850/H9270j,/H92613/H92614/H20849k/H11032/H20850
+vm,i/H20849k,k/H11032/H20850/H9270i,/H92611/H92612/H20849k/H20850/H9254/H92613/H92614
+vm,i/H20849k/H11032,k/H20850/H9254/H92611/H92612/H9270i,/H92613/H92614/H20849k/H11032/H20850. /H2084930/H20850
The pairing amplitudes satisfy the following symmetry prop-
erties:
V˜/H92612/H92611/H92613/H92614/H20849−k,k/H11032/H20850=/H92611/H92612V˜/H92611/H92612/H92613/H92614/H20849k,k/H11032/H20850,
V˜/H92611/H92612/H92614/H92613/H20849k,−k/H11032/H20850=/H92613/H92614V˜/H92611/H92612/H92613/H92614/H20849k,k/H11032/H20850.
To obtain these, we used the anticommutation of the Grass-
mann fields in Eq. /H2084928/H20850and also the expressions /H208494/H20850for the
phase factors in Eq. /H2084929/H20850.
It follows from Eq. /H2084930/H20850that, in general, all possible chan-
nels are present in the pairing interaction, including inter-band pairing. The latter is absent, for any magnitude of theSO band splitting, if the odd harmonics of the bosonic propa-gators are negligible, so that
vu,ij/H20849k,k/H11032/H20850=0 and vm,i/H20849k,k/H11032/H20850=0.
This happens, in particular, for a fully isotropic interaction,
in which case vg/H20849k,k/H11032/H20850=vg/H20849/H9275n,/H9275n/H11032/H20850.
We are particularly interested in the limit of large SO
band splitting, which is relevant for the majority of noncen-trosymmetric superconducting materials. In this limit, we set/H9261
1=/H92612=/H9261and/H92613=/H92614=/H9261/H11032in Eq. /H2084930/H20850/H20849the case of arbitrary
band splitting, with both intra- and interband components ofthe order parameter present, will be considered in a separate
publication /H20850. Since
/H9270/H9261/H9261=/H9261/H9253ˆ/H20849k/H20850, the pairing action becomesK. V. SAMOKHIN AND V. P. MINEEV PHYSICAL REVIEW B 77, 104520 /H208492008 /H20850
104520-4Sint=1
2/H9024/H20858
kk/H11032q/H20858
/H9261/H9261/H11032t/H9261/H20849k/H20850t/H9261/H11032*/H20849k/H11032/H20850V˜/H9261/H9261/H11032/H20849k,k/H11032/H20850
/H11003c¯/H9261/H20849k+q/H20850c¯/H9261/H20849−k/H20850c/H9261/H11032/H20849−k/H11032/H20850c/H9261/H11032/H20849k/H11032+q/H20850, /H2084931/H20850
where
V˜/H9261/H9261/H11032/H20849k,k/H11032/H20850=vg/H20849k,k/H11032/H20850+/H9261/H9261/H11032vu,ij/H20849k,k/H11032/H20850/H9253ˆi/H20849k/H20850/H9253ˆj/H20849k/H11032/H20850
+/H9261vm/H20849k,k/H11032/H20850/H9253ˆ/H20849k/H20850+/H9261/H11032vm/H20849k/H11032,k/H20850/H9253ˆ/H20849k/H11032/H20850./H2084932/H20850
This expression, together with Eqs. /H2084922/H20850,/H2084924/H20850, and /H2084926/H20850re-
lates the amplitudes of the intraband pairing and the inter-band pair scattering to the bosonic excitation spectra. Note
that V˜/H9261/H9261/H11032/H20849k,k/H11032/H20850is even in both kandk/H11032. Treating the inter-
action /H2084931/H20850in the mean-field approximation, see, e.g., Ref.
24, one introduces the order parameters /H9004/H9261/H20849k/H20850
=t/H9261/H20849k/H20850/H9004˜/H9261/H20849k,/H9275n/H20850, where, due to the symmetry of the pairing
amplitudes, /H9004˜/H9261/H20849−k,−/H9275n/H20850=/H9004˜/H9261/H20849k,/H9275n/H20850.
A. Weak coupling model
In order to make progress, we approximate the frequency
dependence of the pairing amplitudes by an anisotropic“square-well” model:
25
V˜/H9261/H9261/H11032/H20849k,k/H11032/H20850=V˜/H9261/H9261/H11032/H20849k,k/H11032/H20850/H9258/H20849/H9275c−/H20841/H9275n/H20841/H20850/H9258/H20849/H9275c−/H20841/H9275n/H11032/H20841/H20850,/H2084933/H20850
where /H9258/H20849x/H20850is the step function, /H9275cis the frequency cutoff,
andV˜/H9261/H9261/H11032/H20849k,k/H11032/H20850depend on the directions of kandk/H11032near the
corresponding Fermi surfaces. The approximation /H2084933/H20850has
been used both for conventional phononic pairing interaction/H20851see Ref. 25/H20852, and also for spin-fluctuation mediated interac-
tion /H20851see Ref. 26/H20852. The “square-well” decomposition also
holds for the gap functions: /H9004˜/H9261/H20849k,/H9275n/H20850=/H9004˜/H9261/H20849k/H20850/H9258/H20849/H9275c−/H20841/H9275n/H20841/H20850,s o
that the energy of quasiparticle excitations in the /H9261th band is
given by
E/H9261/H20849k/H20850=/H20881/H9264/H92612/H20849k/H20850+/H20841/H9004˜/H9261/H20849k/H20850/H208412. /H2084934/H20850
The gap functions satisfy Eqs. /H2084911/H20850, in which the Matsubara
sum is cut off at /H9275c.
The pairing amplitude given by the matrix Eq. /H2084932/H20850is
invariant under all operations from the crystal point group G,
therefore V˜/H9261/H9261/H11032/H20849g−1k,g−1k/H11032/H20850=V˜/H9261/H9261/H11032/H20849k,k/H11032/H20850. Therefore, the mo-
mentum dependence of each matrix element can be repre-
sented as a sum of the products of the basis functions ofirreducible representations of G. In general, the basis func-
tions are different for each matrix element. Neglecting thiscomplication the pairing amplitude can be factorized as fol-lows:
V
˜/H9261/H9261/H11032/H20849k,k/H11032/H20850=−/H20858
aV/H9261/H9261/H11032a/H20858
i=1da
/H9278a,i/H20849k/H20850/H9278a,i*/H20849k/H11032/H20850, /H2084935/H20850
where alabels the irreducible representations /H20849of dimension-
ality da/H20850ofG, which correspond to pairing channels of dif-
ferent symmetry, with /H9278a,i/H20849k/H20850being the even basis
functions.24The coupling constants V/H9261/H9261/H11032aform a Hermitianmatrix, which becomes real symmetric if the basis functions
are real. Keeping only the irreducible representation /H9003which
corresponds to the maximum critical temperature, the gapfunctions take the form
/H9004˜/H9261/H20849k/H20850=/H20858
i=1d/H9003
/H9257/H9261,i/H9278i/H20849k/H20850, /H2084936/H20850
and/H9257/H9261,iare the superconducting order parameter compo-
nents in the /H9261th band. The basis functions are assumed to
satisfy the following orthogonality conditions:/H20855
/H9278i*/H20849k/H20850/H9278j/H20849k/H20850/H20856/H9261=/H9254ij, where the angular brackets denote the av-
eraging over the /H9261th Fermi surface.
Linearizing the gap equations /H2084911/H20850we obtain the follow-
ing expression for the critical temperature:
Tc=2eC
/H9266/H9275ce−1 /g, /H2084937/H20850
where
g=g+++g−−
2+/H20881/H20873g++−g−−
2/H208742
+g+−g−+ /H2084938/H20850
is the effective coupling constant, and
g/H9261/H9261/H11032=V/H9261/H9261/H11032N/H9261/H11032. /H2084939/H20850
While the critical temperature is the same for all d/H9003compo-
nents of /H9257/H9261, the gap structure in the superconducting state
below Tc, see Eq. /H2084936/H20850, is determined by the nonlinear terms
in the free energy, which essentially depend on the symmetryof the dominant pairing channel.
V. PAIRING SYMMETRY IN A CUBIC CRYSTAL
In the case of isotropic pairing interaction, one can write
vg/H20849k,k/H11032/H20850=vg/H20849/H9275n,/H9275n/H11032/H20850=−Vg/H9258/H20849/H9275c−/H20841/H9275n/H20841/H20850/H9258/H20849/H9275c−/H20841/H9275n/H11032/H20841/H20850in the
square-well approximation. In this way, one recovers the
BCS model of Sec. III, with V=2Vgand the same isotropic
gaps in both bands.
To illustrate the effects of the interaction anisotropy on
the gap structure, let us consider the following example. In acubic crystal with G=O, the SO coupling can be described
by
/H9253/H20849k/H20850=/H92530k. This model is applicable to the
Li2/H20849Pd1−x,Ptx/H208503B family of noncentrosymmetric compounds.
The attractive interaction in these materials is likely medi-
ated by phonons,9,27therefore we neglect spin fluctuations by
setting gsf=0 in expressions /H2084922/H20850,/H2084924/H20850, and /H2084926/H20850. Then,
vg/H20849k,k/H11032/H20850=/H20849gph2/2/H20850Dg/H20849k,k/H11032/H20850,vu,ij/H20849k,k/H11032/H20850=/H20849gph2/2/H20850Du/H20849k,k/H11032/H20850/H9254ij,
andvm,i/H20849k,k/H11032/H20850=0. Using the square-well approximation, one
hasGAP STRUCTURE IN NONCENTROSYMMETRIC … PHYSICAL REVIEW B 77, 104520 /H208492008 /H20850
104520-5vg/H20849k,k/H11032/H20850=vg/H20849k,k/H11032/H20850/H9258/H20849/H9275c−/H20841/H9275n/H20841/H20850/H9258/H20849/H9275c−/H20841/H9275n/H11032/H20841/H20850,
vu,ij/H20849k,k/H11032/H20850=vu,ij/H20849k,k/H11032/H20850/H9258/H20849/H9275c−/H20841/H9275n/H20841/H20850/H9258/H20849/H9275c−/H20841/H9275n/H11032/H20841/H20850,
with the momentum dependence inherited from the phonon
propagator. Assuming a spherical Fermi surface and keepingonly the sand pharmonics in the phonon propagator, we
obtain
vg/H20849k,k/H11032/H20850=−Vg,
vu,ij/H20849k,k/H11032/H20850=−Vu/H20849kˆkˆ/H11032/H20850/H9254ij,
vm,i/H20849k,k/H11032/H20850=0 , /H2084940/H20850
where VgandVuare constants. Note that this interaction is
the same as the one considered phenomenologically by Edel-stein in Ref. 14. From Eq. /H2084932/H20850we obtain the pairing ampli-
tudes in the band representation as follows:
V˜/H9261/H9261/H11032/H20849k,k/H11032/H20850=−Vg−/H9261/H9261/H11032Vu/H20849kˆkˆ/H11032/H208502. /H2084941/H20850
The components of the symmetric tensor kˆikˆjtransform ac-
cording to the representation A1+E+F2, where A1,E, and F2
are respectively one-, two-, and three-dimensional irreduc-
ible representations of the cubic group O/H20849the notations are
the same as in Ref. 28/H20850. Therefore there are three pairing
channels in the expansion /H2084935/H20850, with the following basis
functions and coupling constants:
V/H9261/H9261/H11032A1=Vg+1
3/H9261/H9261/H11032Vu,/H9278A1/H20849k/H20850=1 ,
V/H9261/H9261/H11032E=2
15/H9261/H9261/H11032Vu,/H9278E/H20849k/H20850/H11008/H20849kˆ
x2+/H9275kˆ
y2+/H9275*kˆ
z2,kˆ
x2+/H9275*kˆ
y2+/H9275kˆ
z2/H20850,
V/H9261/H9261/H11032F2=2
15/H9261/H9261/H11032Vu,/H9278F2/H20849k/H20850/H11008/H20849kˆykˆz,kˆzkˆx,kˆxkˆy/H20850, /H2084942/H20850
where /H9275=exp /H208492/H9266i/3/H20850.
Since phonons typically lead to a local attraction and can-
not give rise to a substantial kdependence of the interaction,
we expect that the A1pairing channel dominates. Then the
gap functions in the two bands /H20851Eqs. /H2084936/H20850/H20852are isotropic:
/H9004˜/H9261/H20849k/H20850=/H9257/H9261, and satisfy the equations
/H9257/H9261=/H20858
/H9261/H11032g/H9261/H9261/H11032/H9266T/H20858
n/H9257/H9261/H11032/H20881/H9275n2+/H9257/H9261/H110322, /H2084943/H20850
where g/H9261/H9261/H11032=V/H9261/H9261/H11032A1N/H9261/H11032. The critical temperature is given by
Eq. /H2084937/H20850. The gap magnitudes are not necessarily equal: For
instance, in the vicinity of Tcwe find the following expres-
sion for the gap variation between the bands:
r/H11013/H9257+−/H9257−
/H9257++/H9257−=g++−g−−−2g−++/H20881D
g++−g−−+2g−++/H20881D, /H2084944/H20850
where D=/H20881/H20849g++−g−−/H208502+4g+−g−+. Assuming that N+−N−is
small and that Vg/H11271Vu, we haver/H11229Vu
6VgN+−N−
NF. /H2084945/H20850
Thus the gaps are different only if an appreciable p-wave
harmonic is present in the phonon-mediated interaction and
the SO coupling is sufficiently strong to create a considerabledifference between the densities of states in the two bands.
The coupling strengths being the same in both bands is
not a generic situation. In the spirit of the standard model oftwo-band superconductivity,
13it is possible that the coupling
constants corresponding to the intraband pairing channelsand the interband pair scattering are all different. To obtainthis we consider a generalization of the model /H2084940/H20850which
includes, along with phonons, also a contribution from spinfluctuations. In the absence of detailed information about thephonon and spin-fluctuation spectra in real noncentrosym-metric materials, in particular in Li
2/H20849Pd1−x,Ptx/H208503B, we use the
model which includes only the lowest angular harmonics
consistent with the symmetry requirements:
vg/H20849k,k/H11032/H20850=−Vg,
vu,ij/H20849k,k/H11032/H20850=−Vu/H20849kˆkˆ/H11032/H20850/H9254ij−Vu/H11032kˆikˆ
j/H11032,
vm,i/H20849k,k/H11032/H20850=−Vmkˆi. /H2084946/H20850
Here the coefficients VgandVuare, in general, different from
those in the model /H2084940/H20850. In the band representation, the pair-
ing amplitudes become
V˜/H9261/H9261/H11032/H20849k,k/H11032/H20850=−Vg−/H9261/H9261/H11032/H20851Vu/H20849kˆkˆ/H11032/H208502+Vu/H11032/H20852−/H20849/H9261+/H9261/H11032/H20850Vm.
/H2084947/H20850
There are three pairing channels, corresponding to the A1,E,
andF2representations; see Eqs. /H2084942/H20850. The coupling constants
in the A1channel now have the following form:
V/H9261/H9261/H11032A1=Vg+1
3/H9261/H9261/H11032Vu+/H9261/H9261/H11032Vu/H11032+/H20849/H9261+/H9261/H11032/H20850Vm. /H2084948/H20850
The gap functions are isotropic: /H9004˜/H9261/H20849k/H20850=/H9257/H9261, where the /H9257/H9261s
are found from Eqs. /H2084943/H20850. The difference from the previous
case is that now /H9257+/HS11005/H9257−even if the density of states varia-
tion between the bands is negligible, i.e., N+=N−=NF. As-
suming that Vmis smaller than the other constants /H20849i.e., the
singlet-triplet mixing due to the Dzyaloshinskii-Moriya inter-action is weak /H20850, we obtain from Eq. /H2084944/H20850that
r/H11229V
m
Vg−Vu/3−Vu/H11032/H2084949/H20850
near the critical temperature.
Finally let us consider the case of p-wave interaction
dominating, which leads to an anisotropic pairing of the F2
symmetry. This happens if Vuis large enough, and the de-
generacy between the F2andEchannels is lifted, e.g., by the
Fermi surface anisotropy. The order parameter has the fol-lowing form:K. V. SAMOKHIN AND V. P. MINEEV PHYSICAL REVIEW B 77, 104520 /H208492008 /H20850
104520-6/H9004˜/H9261/H20849k/H20850=/H9261/H20849/H92571kˆykˆz+/H92572kˆzkˆx+/H92573kˆxkˆy/H20850. /H2084950/H20850
The symmetry of the gap, in particular the location of the
nodes, depends on the relation between the components of /H9257.
There are four stable states of a three-dimensional order pa-rameter in a cubic crystal:
29/H20849i/H20850/H9257=/H92570/H208491,0,0 /H20850, with two lines
of nodes at kz=0 and ky=0; /H20849ii/H20850/H9257=/H92570/H208491,i,0/H20850, with a line of
nodes at kz=0, and also point nodes at kx=ky=0; /H20849iii/H20850/H9257
=/H92570/H208491,1,1 /H20850, with two lines of nodes at the intersection of the
planes kˆx+kˆy+kˆz=/H110061 with the Fermi surface, and also point
nodes at kx=ky=0, ky=kz=0, and kz=kx=0; and /H20849iv/H20850/H9257
=/H92570/H208491,/H9275,/H92752/H20850, with point nodes at kx=ky=0, ky=kz=0, kz
=kx=0, and kx=ky=kz. For the first three states one would
have cV/H20849T/H20850/H11008T2at low temperatures,30while for the last one
cV/H20849T/H20850/H11008T3.
It is instructive to interpret our results using the spin rep-
resentation of the order parameter:
/H9004/H9251/H9252/H20849k/H20850=/H9274/H20849k/H20850/H20849i/H9268ˆ2/H20850/H9251/H9252+d/H20849k/H20850/H20849i/H9268ˆ/H9268ˆ2/H20850/H9251/H9252, /H2084951/H20850
where
/H9274/H20849k/H20850=−/H9004˜+/H20849k/H20850+/H9004˜−/H20849k/H20850
2/H2084952/H20850
is the spin-singlet component, and
d/H20849k/H20850=−/H9004˜+/H20849k/H20850−/H9004˜−/H20849k/H20850
2/H9253ˆ/H20849k/H20850/H20849 53/H20850
is the spin-triplet component.12,31The relative strength of the
triplet and singlet order parameters is controlled by the dif-ference between
/H9257+and/H9257−:/H20841d/H20841//H20841/H9274/H20841=r; see Eq. /H2084944/H20850. In agree-
ment with Ref. 15, only the component of d/H20849k/H20850which is
parallel to /H9253ˆ/H20849k/H20850survives /H20849is “protected” /H20850in the limit of large
SO band splitting. However, in the case of a weakly aniso-
tropic phonon-dominated interaction, it follows from expres-sion /H2084945/H20850that the triplet component is negligibly small. In the
opposite case, when the interaction is strongest in the p-wave
channel, one obtains from Eq. /H2084942/H20850that
/H9274/H20849k/H20850=0, i.e., the
pairing is purely triplet.
VI. CONCLUSIONS AND DISCUSSION
We have studied the pairing symmetry in noncentrosym-
metric superconductors with SO splitting of the electronbands. The pairing interaction is derived using a microscopicmodel which includes both phonons and spin fluctuations.The interband pairing is shown to be absent for any strengthof the SO coupling, if the interaction anisotropy is negligible.We have analyzed possible gap structures in the strong SOcoupling limit with only intraband pairing and interband pairscattering present, using a cubic system as an example. Ifphonons are dominant, then the superconducting gaps in bothbands are isotropic and nodeless /H20849barring accidental zeros of
the basis function of the unity representation /H20850, but do not
necessarily have the same magnitude.
Let us discuss the application of our results to the non-
centrosymmetric compounds Li
2/H20849Pd1−x,Ptx/H208503B, where x
ranges from 0 to 1 /H20849Ref. 6/H20850. The critical temperature variesfrom 7–8 K for x=0 to 2.2–2.8 K for x=1. The electronic
band structure also exhibits considerable variation: The SOband splitting in Li
2Pd3B is as large as 30 meV, while in
Li2Pt3B it reaches 200 meV /H20849Ref. 9/H20850, which in both cases is
much larger than Tc. Due to the absence of strong correlation
effects and magnetic order, these materials provide a conve-nient testing ground for theories of noncentrosymmetric su-perconductivity. Superconducting pairing in Li
2Pd3B is due
to the exchange of phonons, and the monotonic, almost lin-ear, dependence of T
con the doping level x/H20849Ref. 6/H20850suggests
that it remains phononic for all xf r o m0t o1 .9,27
Experimental data on the magnetic penetration depth,32
the electronic specific heat,33and the NMR characteristics,34
all seem to agree that Li 2Pd3B is a conventional BCS-like
superconductor with no gap nodes. In contrast, the gap struc-ture in Li
2Pt3B is still a subject of intensive debates. While
earlier experiments, see Refs.32–34, suggested the presence of
lines of nodes in the gap, the recent /H9262SR and specific heat
data35have found no evidence of those. Moreover, according
to Ref. 35, the whole Li 2/H20849Pd1−x,Ptx/H208503B family of compounds
are single-gap isotropic superconductors. This conclusion is
consistent with our results, see Sec. V. Indeed, assuming thatthe pairing interaction in Li
2/H20849Pd1−x,Ptx/H208503B is phononic and
therefore only weakly anisotropic for all x, we obtain that the
A1channel always dominates, giving rise to nodeless isotro-
pic gaps of essentially equal magnitudes in both bands. Inorder to create a noticeable difference between the gap mag-nitudes, see Eq. /H2084945/H20850, the interaction anisotropy would have
to be very strong: Since /H20849N
+−N−/H20850/NF/H11011ESO //H9280Fand varies
from 0.03 in Li 2Pd3B to 0.2 in Li 2Pt3B, the strength of the
p-wave harmonic must be at least an order of magnitude
larger than that of the s-wave harmonic, which is highly
unlikely for a phonon-mediated interaction.
ACKNOWLEDGMENTS
We thank B. Mitrovi ćand S. Bose for useful discussions.
The financial support from the Natural Sciences and Engi-neering Research Council of Canada /H20849K.S. /H20850is gratefully ac-
knowledged.
APPENDIX: DERIVATION OF EQS. ( 21)–(26)
Let us start from Eq. /H2084917/H20850, in which we substitute expres-
sions /H2084918/H20850and /H2084919/H20850:
Sint=1
2/H9024/H20858
kk/H11032q/H20853gph2/H20851Dg/H20849k,k/H11032/H20850+Du/H20849k,k/H11032/H20850/H20852/H9254/H9251/H9254/H9254/H9252/H9253+gsf2/H20851Dijg/H20849k,k/H11032/H20850
+Diju/H20849k,k/H11032/H20850+ieijlRl/H20849k−k/H11032/H20850/H20852/H9268/H9251/H9254i/H9268/H9252/H9253j/H20854
/H11003a¯/H9251/H20849k+q/H20850a¯/H9252/H20849−k/H20850a/H9253/H20849−k/H11032/H20850a/H9254/H20849k/H11032+q/H20850. /H20849A1/H20850
The qdependence of the fermionic fields plays no role in the
algebraic transformations below; hence we use a shorter ex-pression on the right-hand side:
S
int→1
2/H9024/H20849Ig+Iu+Im/H20850, /H20849A2/H20850
whereGAP STRUCTURE IN NONCENTROSYMMETRIC … PHYSICAL REVIEW B 77, 104520 /H208492008 /H20850
104520-7Ig=1
4/H20858
kk/H11032/H20851gph2Dg/H20849k,k/H11032/H20850/H9254/H9251/H9254/H9254/H9252/H9253+gsf2Dijg/H20849k,k/H11032/H20850/H9268/H9251/H9254i/H9268/H9252/H9253j/H20852
/H11003/H20851a¯/H9251/H20849k/H20850a¯/H9252/H20849−k/H20850−a¯/H9252/H20849k/H20850a¯/H9251/H20849−k/H20850/H20852
/H11003/H20851a/H9253/H20849−k/H11032/H20850a/H9254/H20849k/H11032/H20850−a/H9254/H20849−k/H11032/H20850a/H9253/H20849k/H11032/H20850/H20852,
Iu=1
4/H20858
kk/H11032/H20851gph2Du/H20849k,k/H11032/H20850/H9254/H9251/H9254/H9254/H9252/H9253+gsf2Diju/H20849k,k/H11032/H20850/H9268/H9251/H9254i/H9268/H9252/H9253j/H20852
/H11003/H20851a¯/H9251/H20849k/H20850a¯/H9252/H20849−k/H20850+a¯/H9252/H20849k/H20850a¯/H9251/H20849−k/H20850/H20852/H11003/H20851a/H9253/H20849−k/H11032/H20850a/H9254/H20849k/H11032/H20850
+a/H9254/H20849−k/H11032/H20850a/H9253/H20849k/H11032/H20850/H20852,
Im=1
8ieijlgsf2/H20858
kk/H11032/H20851/H20851Rl/H20849k−k/H11032/H20850+Rl/H20849k+k/H11032/H20850/H20852/H9268/H9251/H9254i/H9268/H9252/H9253j/H20852
/H11003/H20851a¯/H9251/H20849k/H20850a¯/H9252/H20849−k/H20850+a¯/H9252/H20849k/H20850a¯/H9251/H20849−k/H20850/H20852/H11003/H20851a/H9253/H20849−k/H11032/H20850a/H9254/H20849k/H11032/H20850
−a/H9254/H20849−k/H11032/H20850a/H9253/H20849k/H11032/H20850/H20852+1
8ieijlgsf2/H20858
kk/H11032/H20851/H20851Rl/H20849k−k/H11032/H20850
−Rl/H20849k+k/H11032/H20850/H20852/H9268/H9251/H9254i/H9268/H9252/H9253j/H20852/H11003/H20851a¯/H9251/H20849k/H20850a¯/H9252/H20849−k/H20850−a¯/H9252/H20849k/H20850a¯/H9251/H20849−k/H20850/H20852
/H11003/H20851a/H9253/H20849−k/H11032/H20850a/H9254/H20849k/H11032/H20850+a/H9254/H20849−k/H11032/H20850a/H9253/H20849k/H11032/H20850/H20852.
The even in kcombinations of the fermionic fields can be
represented as follows:
a¯/H9251/H20849k/H20850a¯/H9252/H20849−k/H20850−a¯/H9252/H20849k/H20850a¯/H9251/H20849−k/H20850
=−/H20849i/H92682/H20850/H9251/H9252†/H20849i/H92682/H20850/H9262/H9263a¯/H9262/H20849k/H20850a¯/H9263/H20849−k/H20850,a/H9253/H20849−k/H11032/H20850a/H9254/H20849k/H11032/H20850−a/H9254/H20849−k/H11032/H20850a/H9253/H20849k/H11032/H20850
=−/H20849i/H92682/H20850/H9253/H9254/H20849i/H92682/H20850/H9267/H9268†a/H9267/H20849−k/H11032/H20850a/H9268/H20849k/H11032/H20850, /H20849A3/H20850
while the odd combinations have the form
a¯/H9251/H20849k/H20850a¯/H9252/H20849−k/H20850+a¯/H9252/H20849k/H20850a¯/H9251/H20849−k/H20850
=/H20849i/H9268i/H92682/H20850/H9251/H9252†/H20849i/H9268i/H92682/H20850/H9262/H9263a¯/H9262/H20849k/H20850a¯/H9263/H20849−k/H20850,
a/H9253/H20849−k/H11032/H20850a/H9254/H20849k/H11032/H20850+a/H9254/H20849−k/H11032/H20850a/H9253/H20849k/H11032/H20850
=/H20849i/H9268i/H92682/H20850/H9253/H9254/H20849i/H9268i/H92682/H20850/H9267/H9268†a/H9267/H20849−k/H11032/H20850a/H9268/H20849k/H11032/H20850. /H20849A4/H20850
Using the matrix identities
/H9254/H9251/H9254/H9254/H9252/H9253/H20849i/H92682/H20850/H9251/H9252†/H20849i/H92682/H20850/H9253/H9254=2 ,
/H20849/H9268i/H20850/H9251/H9254/H20849/H9268j/H20850/H9252/H9253/H20849i/H92682/H20850/H9251/H9252†/H20849i/H92682/H20850/H9253/H9254=−2/H9254ij,
/H9254/H9251/H9254/H9254/H9252/H9253/H20849i/H9268i/H92682/H20850/H9251/H9252†/H20849i/H9268j/H92682/H20850/H9253/H9254=2/H9254ij,
/H20849/H9268i/H20850/H9251/H9254/H20849/H9268j/H20850/H9252/H9253/H20849i/H9268m/H92682/H20850/H9251/H9252†/H20849i/H9268n/H92682/H20850/H9253/H9254=2/H20849/H9254ij/H9254mn−/H9254im/H9254jn−/H9254in/H9254jm/H20850,
/H20849/H9268i/H20850/H9251/H9254/H20849/H9268j/H20850/H9252/H9253/H20849i/H9268m/H92682/H20850/H9251/H9252†/H20849i/H92682/H20850/H9253/H9254=2ieijm,
/H20849/H9268i/H20850/H9251/H9254/H20849/H9268j/H20850/H9252/H9253/H20849i/H92682/H20850/H9251/H9252†/H20849i/H9268m/H92682/H20850/H9253/H9254=−2 ieijm, /H20849A5/H20850
we arrive at Eqs. /H2084921/H20850–/H2084926/H20850.
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104520-9 |
PhysRevB.80.155124.pdf | Calculations of ZnO properties using the Heyd-Scuseria-Ernzerhof screened hybrid density
functional
Jan Wróbel *and Krzysztof J. Kurzydłowski
Faculty of Materials Science and Engineering, Warsaw University of Technology, ul. Wołoska 141, 02-507 Warszawa, Poland
Kerstin Hummer and Georg Kresse
Faculty of Physics, Center for Computational Materials Science, University of Vienna, Sensengasse 8, A-1090 Wien, Austria
Jacek Piechota
Interdisciplinary Centre for Materials Modelling, University of Warsaw, ul. Pawi ńskiego 5a, 02-106 Warsaw, Poland
/H20849Received 24 August 2009; published 15 October 2009 /H20850
The Heyd-Scuseria-Ernzerhof /H20849HSE /H20850screened hybrid density functional has been proven to yield lattice
constants and energy gaps of semiconductors in better agreement with experiment than standard local andsemilocal exchange correlation functionals. The latter underestimate the band gaps of many semiconductorsseverely, i.e., in the case of ZnO the underestimation amounts to 75% of the experimental value. In this work,we report on the structure optimization and the study of the electronic band gap of ZnO in the wurtzite phaseperformed within density-functional theory using the semilocal Perdew-Burke-Ernzerhof as well as the HSEfunctional. Furthermore, the phonon-dispersion relations of ZnO and the dielectric and piezoelectric propertiesare calculated with both functionals and are compared to experimental findings.
DOI: 10.1103/PhysRevB.80.155124 PACS number /H20849s/H20850: 71.15.Mb, 71.20.Nr, 63.20. /H11002e
I. INTRODUCTION
In recent years, ZnO has attracted increasing interest,
since it is a good candidate for application in electronic de-vices, either as active material or as a substrate for thegrowth of other semiconductors such as GaN and SiC.
1It is
possibly ideally suited for blue/UV light-emitting diodes orlaser diodes for the next generation data-storage systems,once n- and p-doped ZnO can be produced reliably and
reproducably.
1,2At ambient conditions, ZnO crystallizes in
the wurtzite structure and is a wide-gap semiconductor witha direct electronic band gap of 3.2–3.4 eV .
3,4Since, ZnO has
an exciton binding energy of 60 meV ,5,6which is much
higher than that of GaN /H20849between 21 and 25 meV /H20850, devices
made from ZnO can operate at high temperatures more effi-ciently. Furthermore, ZnO is found to be significantly morestable against radiation than Si, GaAs, and GaN,
7which is an
important property that prevents wearing out during fieldemission. Another advantage is that ZnO is cheaper in fabri-cation than GaN, also for the production of thin-film materi-als using metal-organic chemical-vapor deposition, becausethere is no need for handling the toxic ammonia associatedwith GaN fabrication. Substrates for thin-film growth arealso cheaper for ZnO than for GaN, and ZnO offers the pos-sibility to grow nanostructures, e.g., for developing nano-scale optoelectronics, from solution instead of the gasphase.
8
The properties of ZnO have been extensively investigated
by many experimental techniques as well as theoreticalmethods /H20849see, for example, review papers Refs. 1,2, and 9/H20850.
However, there is still a need for theoretical studies in orderto gain fundamental knowledge of its properties necessary todevelop ZnO-based materials for novel applications, i.e., inspintronics
10or as transparent conducting oxides. With mod-
ern computational techniques, properties such as the elec-tronic structure and the lattice dynamics can be explored on
the quantum-mechanical level. In particular, the latter deter-mine the thermodynamic properties that are important in the
development of high-quality optoelectronic devices.
The most popular methods for calculating the structural,
electronic, and vibrational properties of extended systems arethose based on density-functional theory /H20849DFT /H20850. However,
the standard approximations to the exchange-correlation/H20849XC/H20850energy, i.e., the local-density approximation /H20849LDA /H20850as
well as the generalized gradient approximation /H20849GGA /H20850, often
fail to describe systems with strongly localized dorfelec-
trons. The main reason for this failure is that both, LDA andGGA, are jellium-based XC energy functionals that sufferfrom /H20849i/H20850an incomplete cancellation of the artificial Hartree
self-interaction and /H20849ii/H20850the lack of the integer discontinuity
in the exchange and correlation energy upon adding an elec-tron. As a direct consequence, for semiconductors and insu-lators, the Kohn-Sham single-particle eigenvalue band gapsignificantly underestimates the measured quasiparticle bandgap. Furthermore, these methods underestimate the bindingenergy of localized d/H20849f/H20850states due to /H20849i/H20850. Consequently, they
predict d/H20849f/H20850states to be much too delocalized and overesti-
mate their hybridization with the anion p-derived valence
states. Both these shortcomings of LDA/GGA are particu-larly evident for ZnO, i.e., the calculated dielectric screeningis too high compared to experiment /H20851/H9255
LDA/H110155 versus /H9255EXPT
/H110153.7 /H20849Ref. 11/H20850/H20852and the calculated energy gap is severely
underestimated /H20851EgLDA=0.7–0.8 eV versus EgEXPT
=3.2–3.4 eV /H20849Refs. 3and 4/H20850/H20852. As we will show this has
serious consequences for many materials properties.
An alternative to conventional /H20849semi /H20850local XC function-
als are hybrid density functionals.12These functionals are
characterized by admixing a certain amount of exact, nonlo-cal HF exchange energy to the /H20849semi /H20850local /H20849GGA /H20850LDA
exchange energy. The DFT-GGA/LDA correlation energy isPHYSICAL REVIEW B 80, 155124 /H208492009 /H20850
1098-0121/2009/80 /H2084915/H20850/155124 /H208498/H20850 ©2009 The American Physical Society 155124-1straightforwardly added. The fraction of HF exchange is usu-
ally1
4, which is justified by the adiabatic connection
theorem.13Hybrid density functionals can be divided into
two groups, i.e., /H20849i/H20850the Perdew-Burke-Ernzerhof /H20849PBE /H20850
-based ones and /H20849ii/H20850the semiempirical /H20849three-parameter /H20850
functionals /H20849B3LYP and B3PW91 /H20850/H20849Ref. 12/H20850that have been
extensively applied in quantum chemistry. In contrast to thelatter, the former are nonempirical, in the sense that the num-ber of parameters is identical to those in the parent function.The PBE0 /H20849Ref. 14/H20850and the recently developed Heyd-
Scuseria-Ernzerhof /H20849HSE /H20850functional
15,16belong to this cat-
egory. The main advantage of HSE is the separation of theexact HF exchange into a short-range /H20849SR/H20850and a long-range
/H20849LR/H20850part to avoid the expensive computation of the slowly
decaying exchange interactions. The LR part of the HF ex-change is replaced by the corresponding density-functionalcounterpart. The separation is accomplished through a de-composition of the Coulomb kernel
1
r=S/H9262/H20849r/H20850+L/H9262/H20849r/H20850=1 − erf /H20849/H9262r/H20850
r+erf/H20849/H9262r/H20850
r, /H208491/H20850
where the screening parameter /H9262defines the range separa-
tion. This enables the wide and routine application of theHSE hybrid functional to condensed-matter systems. The ex-pression for the HSE exchange-correlation energy is givenby
E
xcHSE=1
4ExHF,SR/H20849/H9262/H20850+3
4ExPBE,SR/H20849/H9262/H20850+3
4ExPBE,LR+EcPBE.
/H208492/H20850
For/H9262=0, HSE reduces to the hybrid functional PBE0,
whereas for /H9262→/H11009, HSE becomes identical to PBE. HSE
with a finite value of /H9262can be regarded as an interpolation
between these two limits. It has been found that the optimalvalue for the screening parameter
/H9262is 0.207 Å−1, yielding
almost identical total energies as the PBE0 functional. Thisparticular HSE functional is called HSE06 in the literatureand used throughout this work.
16
The HSE functional has been proven to yield results in
good agreement with experiment for a wide range of solidsincluding metals, semiconductors, and insulators as well asmolecules.
16–22The main purpose of the work presented
herein, is to investigate ground-state properties of ZnO andto access to what extent HSE06 improves upon PBE GGA.
23
This includes the electronic structure, which is important forthe development of optoelectronic devices, and the vibra-tional properties that determine the thermodynamic proper-ties of this material, i.e., the internal energy, the thermalconductivity, the entropy, and the Gibbs free energy, as wellas the dielectric polarizability and piezoelectric tensors.Moreover, this work constitutes an extension of our previousinterest in the ZnO system.
24–27
The remaining part of the paper is organized as follows.
Section IIdescribes the computational methodology used in
this study. Section IIIpresents the results of the calculations
as well as their discussion. Finally, conclusions are given inSec. IV.II. COMPUTATIONAL DETAILS
The calculations were performed within the projector-
augmented wave /H20849PAW /H20850method /H20849Refs. 28and29/H20850as imple-
mented in the V ASP 5.1 package.30The structure optimiza-
tion, the band structure, and the phonon calculations wereperformed using the PBE functional
23as well as the screened
hybrid density functional HSE06,16in the following simply
abbreviated as HSE. More details on the parameters used forgenerating the PAW potentials are given in Table I. The
Gaussian smearing method was chosen with a smearingwidth of 0.1 eV . The sampling of the Brillouin zone wasperformed using a Monkhorst-Pack scheme.
31The applied k
mesh for the structure optimization was 8 /H110038/H110036 corre-
sponding to a kspacing of 0.4 Å−1. For the structure opti-
mization, the internal degrees were relaxed at each volume,and the volume dependence of the total energy was fitted toa Murnaghan equation of state. The band structures E/H20849k/H20850
were computed on a discrete kmesh following high-
symmetry directions in the Brillouin zone.
The calculations of the phonon-dispersion relations
/H9275/H20849q/H20850
were performed with the direct method,32which uses the
Hellmann-Feynmann forces calculated for a supercell. Thesymmetry inequivalent atoms were displaced by + /
−0.015 Å. This method is also referred to as supercell orfrozen phonon approach. Convergence tests with respect tothekmesh on the primitive cell show that an accuracy of 0.1
THz for the highest optical zone-center phonon frequency isachieved with a 3 /H110033/H110032kmesh. For this reason, 2 /H110032
/H110032 supercells in combination with 2 /H110032/H110031kmeshes have
been employed for calculating the phonon-dispersion curve.In the supercell approach, only phonon frequencies
/H9275/H20849q/H20850for
qvectors that are commensurate with the supercell are ob-
tained exactly, and /H9275/H20849q/H20850is interpolated for all remaining q
vectors. Thus, the larger the supercell, the more accurate dis-persion relations are obtained. However, for large supercellsHSE calculations are computationally not yet feasible.Therefore and for the sake of comparison, the 2 /H110032/H110032 su-
percell has been employed for all phonon calculations pre-sented in this work.
The dielectric and piezoelectric properties were deter-
mined using a 24 /H1100324/H1100316kmesh for PBE and 10 /H1100310
/H1100310kpoints for HSE. The HSE results converge rapidly
with the number of kpoints, whereas the PBE results require
a very accurate sampling since the combination of the smallDFT band gap with the strong O pZnstransition at the /H9003
point causes very slow convergence.TABLE I. Core radii rcand energy cutoffs Ecutfor the PAW
potentials. Nonlocal projectors were generated for the states listedin the column valence. As local PAW potential a pseudopotentialwas generated for the states indicated in the column local.
Valence Localr
c
/H20849a.u./H20850Ecut
/H20849eV/H20850
Zn 3 d104s24p04f 2.3 277
O2 s22p43d 1.5/H20849s/H20850/1.8/H20849p/H20850 300WRÓBEL et al. PHYSICAL REVIEW B 80, 155124 /H208492009 /H20850
155124-2III. RESULTS AND DISCUSSION
A. Structure optimization
The total energy versus volume data for the wurtzite
phase of ZnO computed using the PBE and the HSE func-tional are shown in Fig. 1. The lattice parameters and bulk
moduli evaluated from the Murnaghan fits are summarized inTable IItogether with experimental findings at room
temperature.
33,34Due to the usual underbinding in solids, the
PBE functional yields overestimated lattice parameters /H20849a
=3.292 Å and c=5.306 Å /H20850resulting in an underestimation
of the bulk modulus /H20849128 GPa /H20850. In contrast to PBE, the HSE
functional corrects the overestimation /H20849underestimation /H20850of
the lattice parameters /H20849bulk modulus /H20850and provides results
that are in much better agreement with the experimental find-ings. The HSE results for the lattice constants are a
=3.253 Å and c=5.254 Å, respectively, whereas a bulk
modulus of 144 GPa is obtained. In comparison to Ref. 27
/H20849a=3.261 Å, c=5.225 Å, and c/a=1.602 /H20850, a larger cvalue
is obtained in this work resulting in a larger c/aratio of
1.615. This small discrepancy was traced back to minor dif-ferences in the computational settings. Convergence testswith respect to the energy cutoff and the oxygen pseudopo-tential revealed that the values given in Ref. 27were not
entirely converged, and therefore the present values super-sede the previous ones.B. Band structures
The density of states /H20849DOS /H20850and band structure of ZnO
obtained using the PBE functional are shown in Fig. 2. The
band gap is direct, i.e., the valence-band maximum /H20849VBM /H20850
as well as the conduction-band minimum /H20849CBM /H20850are located
at the /H9003point. As in previously reported bulk calculations,27
the band gap is severely underestimated and amounts to 0.77
eV . As evident from the band structure, the valence bandcontains two regions. The energetically lower one originatesfrom the Zn 3 dstates, whereas the energetically higher one
is composed mainly of O 2 porbitals making up the VBM.
The conduction-band manifold originates from Zn 4 sand
Zn 4 pstates strongly hybridized with O 2 porbitals. The er-
roneously high lying and too much delocalized Zn 3 dbands
hybridize too strongly with the O 2 pbands. For this reason,
an unambiguous determination of the bandwidth is not pos-sible. As a consequence of this strong hybridization, the bandgap is significantly underestimated. Summarizing, DFT-PBEdoes not reproduce quantitatively the experimental bandstructure but yields qualitatively correct band characters.
The band structure calculated using the HSE functional,
which is shown in Fig. 3, is qualitatively similar to that of
DFT-PBE. The VBM and CBM are located at the /H9003point. In
contrast to PBE, HSE yields also the details of the bandstructure in good agreement with experiment /H20849Table III/H20850.
First, the Zn 3 dbands are less strongly hybridized with the
Znsstates, and they appear between 5.0 and 7.2 eV , whereas
in the PBE case they are located between 2 and 6 eV /H20849Fig.3/H20850.
As a consequence, the interaction between O 2 pand Zn 3 d
states is less pronounced and the band gap deduced from theHSE band structure is 2.46 eV , which is much closer to theexperimental value of 3.4 eV .
3,4The HSE hybrid density
functional gives also much better agreement with the experi-mental values for the width of the valence p-band and the
d-band position as can be concluded from Table III.
C. Phonons
Since ZnO in the wurtzite structure has four atoms in the
primitive unit cell, a total of 12 phonon branches exist: onelongitudinal-acoustic /H20849LA/H20850, two transverse-acoustic /H20849TA/H20850,
three longitudinal-optical /H20849LO/H20850, and six transverse-optical
/H20849TO/H20850branches that constitute the highest frequency modes.
The latter correspond mainly to the internal vibrations of thelighter oxygen atoms, whereas the lower-frequency modesstem from the vibrations of the heavier zinc atoms. Due to44 46 48 50 52 54
Unit cell volume (Å3)-18.2-18.1-18.0-17.9Energy PBE (eV)PBE
44 46 48 50 52 54-24.6-24.5-24.4-24.3-24.2
Ener gy HSE (eV)HSE
FIG. 1. /H20849Color online /H20850Total energy versus unit-cell volume of
ZnO calculated with the PBE /H20849circles /H20850and the HSE /H20849squares /H20850func-
tional. The experimental equilibrium volume at ambient conditionsis indicated by the vertical dotted line.
TABLE II. Lattice parameters, c/aratio, volume per formula unit, and bulk modulus of ZnO calculated
using the PBE and HSE functional in comparison to experiment.
FunctionalLattice parameters
V olume per formula unit
/H20849Å3/H20850Bulk modulus
/H20849GPa /H20850a
/H20849Å/H20850c
/H20849Å/H20850 c/a
PBE 3.292 5.306 1.612 24.90 128.17
HSE 3.253 5.254 1.615 24.07 143.82Expt.
a3.250 5.207 1.602 23.82 183.0,142.6
aReferences 33and34.CALCULATIONS OF ZnO PROPERTIES USING THE … PHYSICAL REVIEW B 80, 155124 /H208492009 /H20850
155124-3the significant difference in the atomic weights, the six TO
branches are separated from the lower branches by a phonongap.
36This structure is clearly observed in the Figs. 4–6
showing the phonon-dispersion relations /H9275/H20849q/H20850obtained
within DFT-PBE and HSE.
When investigating lattice dynamics using ab initio meth-
ods, the lattice parameters that are applied in the calculations
are exceedingly important.38In particular, when comparing
different XC functionals, the effect introduced by performingthe calculations at different optimized theoretical volumes,which strongly depend on the applied functional, has to beconsidered carefully. To disentangle this effect from thechanges introduced by the functionals at one specific vol-ume, Figs. 4–6show the results of the calculations using the
experimental volume at ambient conditions and at the opti-mized crystal volumes, as listed in Table II, for the PBE and
HSE functional, respectively.In the case of PBE /H20849Fig. 4/H20850, the phonon frequencies ob-
tained at the two different crystal volumes differ significantlyfor the high-frequency optical branches, but agree reasonablywell for the low-frequency branches, in particular, for theacoustic modes. Obviously, the volume effect is most pro-nounced in the high-frequency modes. This can be under-stood by realizing that DFT-PBE severely overestimates thecrystal volume resulting in much too weak nearest-neighborforce constants and a significantly underestimation of theoptic modes.
Concerning the phonon dispersion of the lower six modes,
DFT-PBE achieves reasonable agreement with experiment.In contrast to PBE, the HSE calculation yields lattice param-eters in much better agreement with experiment leading to aless pronounced volume effect in the high-lying opticalmodes and a vanishing volume effect in the branches belowthe phonon gap. Both, the absolute values of the phononfrequencies as well as the dispersion of the phonon modesare in better agreement with experimental findings. This ismade more clear by comparing the phonon-dispersion curves
FIG. 2. /H20849Color online /H20850/H20849left/H20850Density of states and /H20849right /H20850band structure of ZnO obtained using the PBE functional. The character of the
bands is indicated by the color /H20851Zn/H20849d/H20850circles, O /H20849p/H20850squares /H20852, whereas the size of the symbols corresponds to the partial occupancies. The
Fermi level is indicated by the horizontal line.
FIG. 3. /H20849Color online /H20850Band structure of ZnO obtained using the
HSE functional. The dbands /H20849circles /H20850are indicated by the red color,
whereas the green color indicates the pbands /H20849squares /H20850. The size of
the symbols corresponds to the partial occupancies. The Fermi levelis indicated by the horizontal line.TABLE III. Energy gap Eg, width of the 3 d-band W3d, and the
Zn 3 d-band position E3din/H20849eV/H20850, calculated within DFT-PBE and
HSE compared to experimental data /H20849Refs. 3,4, and 35/H20850as well as
previously reported PBE and HSE calculations /H20849Ref. 27/H20850. Theoreti-
cally, the dband position was determined as the centre of gravity of
the occupied part of the dband. We note that this is not necessarily
compatible with the experimental determination of the dband
position.
PBE PBEaHSE HSEaExpt.b
Ed/H20849eV/H20850 0.8 0.7 2.5 2.5 3.2–3.4
W3d/H20849eV/H20850 3.9 2.2
E3d/H20849eV/H20850 −4.8 −4.8 −6.0 −5.8 − /H208497.5–7.8 /H20850
aReference 27.
bReferences 3,4, and 35.WRÓBEL et al. PHYSICAL REVIEW B 80, 155124 /H208492009 /H20850
155124-4calculated within DFT-PBE and HSE at the same volume
/H20849experimentally observed equilibrium volume at ambient
conditions /H20850in Fig. 6and the corresponding phonon frequen-
cies at high-symmetry points summarized in Table IV.
The phonon-dispersion relations depicted in Fig. 6are
qualitatively similar but differ quantitatively. This is mostpronounced in the case of the highest optical zone-centeredmode, e.g., PBE yields 16.3 THz compared to the experi-mental value of 17.8 THz.
37The analysis of Table IVand
Fig. 6reveals that within HSE both, the acoustic and optical
branches are shifted towards higher frequencies. Conse-quently, whenever PBE significantly underestimates the fre-quencies compared to experiment, the HSE functionalclearly improves upon PBE. This is indicated by /H20849/H11569/H20850in the
last column of Table IV. However, when the PBE frequen-
cies are already close to the experimental values, the HSEfunctional leads to an overestimation of the frequencies/H20851marked by /H20849/H11569/H11569/H20850in Table IV/H20852. Unfortunately, the experimen-
tal uncertainties for the available data are 0.2 THz and thereare too few data points available to allow for a more detailedcomparison of the full phonon spectrum. Nevertheless, theoverall agreement with experiment is visually better for HSEthan for DFT-PBE with the most obvious improvement being
related to a stiffening of the phonon frequencies, that is par-ticularly pronounced for the optical modes.
D. Dielectric and piezoelectric properties
Dielectric properties are particularly strongly affected by
the choice of the functional, and we have demonstrated inRef. 22that the effect is very pronounced for ZnO. Table V
summarizes the calculated dielectric properties. All proper-ties were calculated at the experimental volume but the vol-ume dependence is not very pronounced. For comparisonwith previous work, we have included the LDA values fromRef. 44. Overall the agreement with these values is good, but
it is emphasized that the previous values were certainly notentirely k-point converged, resulting in too large dielectric
constants, too large Born effective charges, and slightly toolarge ionic contributions to the piezoelectric tensor. In gen-eral our present LDA values are closer to experiment than theprevious ones /H20849note for instance the large anisotropy of the
static-ion-clamped dielectric tensor in the previous calcula-tions /H20850.
In LDA and PBE, the too small one-electron band gap has
a dramatic effect on the dielectric constants, which are sig-nificantly overestimated. Remarkably, not only the electroniccontributions but also the ionic contributions to the macro-scopic dielectric tensor are too large in LDA. On the otherhand, PBE yields good ionic contributions but still overesti-mates the electronic contributions to the polarizability. Over-all the HSE functional gives the best account of the dielectricproperties, although somewhat underestimating the ioniccontributions, in particular, in the cdirection /H20849dir. 33 /H20850. The
Born effective charges are only little influenced by the choiceof the functional and all values point toward a mostly ionicbehavior regardless of the functional. For the piezoelectricconstants, again little differences are found between the threefunctionals. The electronic contribution gradually decreasesfrom LDA, over PBE to HSE, similar to the electronic con-tributions to the polarizability, although the decrease is notvery pronounced in the piezoelectric tensor. Inclusion of theionic contributions to the piezoelectric tensors yields very0246810121416182022
A H L AΓ K M ΓFrequency (THz )
Wave VectorPBE (experimental volume)
PBE (optimized volume)
FIG. 4. /H20849Color online /H20850PBE phonon-dispersion curves of ZnO
calculated at the PBE optimized volume /H20849dotted line /H20850as well as at
the experimental equilibrium volume /H20849full line /H20850. The black dots cor-
respond to the experimental values given in Ref. 37.
0246810121416182022
A H L AΓ K M ΓFrequency (THz )
Wave VectorHSE (experimental volume)
HSE (optimized volume)
FIG. 5. /H20849Color online /H20850HSE phonon-dispersion curves of ZnO
calculated at the HSE optimized volume /H20849dotted line /H20850as well as at
the experimental equilibrium volume /H20849full line /H20850. The black dots cor-
respond to the experimental values given in Ref. 37.0246810121416182022
A H L AΓ K M ΓFrequency (THz )
Wave VectorHSE (experimental volume)
PBE (experimental volume)
FIG. 6. /H20849Color online /H20850Phonon-dispersion curves of ZnO calcu-
lated using PBE /H20849blue lines /H20850and the HSE functional /H20849red lines /H20850. The
crystal volume has been fixed to the experimental value. The blackdots correspond to the experimental values given in Ref. 37.CALCULATIONS OF ZnO PROPERTIES USING THE … PHYSICAL REVIEW B 80, 155124 /H208492009 /H20850
155124-5good agreement with experiment, in particular, for the HSE
functional.
IV. CONCLUSIONS
We have studied the geometry, the electronic band struc-
ture, the phonon-dispersion relations, and the dielectric prop-erties of ZnO using the PBE as well as the HSE hybridfunctional. We have shown that the HSE functional givesmuch better estimates for all considered ZnO properties thanthe widely utilized semilocal PBE functional. The HSE op-timized crystal structure agrees within 1% with the experi-mentally observed lattice parameters resulting in a bulkmodulus of 144 GPa in reasonable agreement with experi-ment /H20849143–183 GPa /H20850. The band gap calculated using the
HSE functional is 2.5 eV and much closer to the experimen-tal value of 3.2–3.4 eV than the PBE result of 0.8 eV . Thereason for this improvement is a better description of theZn 3 dand O 2 pstates within HSE, e.g., the Zn 3 dbands aremore localized and energetically deeper resulting in less hy-
bridization with the O 2 pstates. We believe that the remain-
ing error is related to the fact that the dstates are still too
shallow using the HSE functional, and more nonlocal ex-change on the tightly bound delectrons and O 2 pstates
would be required to remedy the remaining error. This is alsosupported by the observation that the required amount ofnonlocal exchange depends on the screening properties: withstatic dielectric constants around
/H9280=3.7, GW-type methods
would include more nonlocal exchange /H208511//H9280/H20849q/H20850/H20852than con-
ventional hybrid functionals do at any wavelength. ThusZnO is a material where everything points towards the need
to include a little bit more nonlocal exchange than hybridfunctionals based on the 1/4 rule /H20849see also Ref. 27/H20850. Despite
this observation and despite the remaining underestimationof the band gap, the HSE functional predicts excellentground-state properties.
Furthermore, we have shown that the chosen volume for
the calculations /H20849theoretical versus experimental volume /H20850
significantly affects the phonon frequencies, in particular, inTABLE IV . Phonon frequencies /H20849in THz /H20850of selected modes at high-symmetry points in the Brillouin zone
evaluated at the experimental volume.
DFT-PBE HSE LDAaExpt.bDiscrepancy
/H9003 2.58 2.99 2.71 2.97–3.03 /H11569
7.70 8.03 7.73 n.a.
11.70 11.98 11.99 11.37–11.39
12.10 12.35 12.60,12.76 11.20–12.68 /H11569
13.06 13.29 13.59,13.76 13.10–13.31 /H11569
16.29 17.08 17.13 17.47–17.85 /H11569
M 2.98 2.85 2.60 n.a.
3.52 3.82 3.54 3.15 /H11569/H11569
4.03 4.25 3.98 4.11 /H11569/H11569
4.83 5.29 4.86 5.17 /H11569
6.92 7.28 7.19 n.a.7.59 7.94 7.90 n.a.
13.45 13.36 13.81 n.a.13.47 13.68 14.20 n.a.14.58 14.67 15.08 n.a.14.80 15.34 15.57 n.a.15.88 16.54 16.80 n.a.16.23 16.97 16.80 n.a.
A 2.00 2.34 2.04 2.41 /H11569
5.53 5.55 5.63 5.03,5.59 /H11569
12.58 12.84 13.09,13.26 n.a.16.48 17.47 17.18 n.a.
M 3.04 3.39 3.20 n.a.
3.25 3.73 3.48 n.a.7.95 8.25 8.12 n.a.
13.82 14.05 14.42 n.a.13.85 14.10 14.59 n.a.16.29 17.03 16.91 n.a.
aReference 39.
bReferences 37and39–43.WRÓBEL et al. PHYSICAL REVIEW B 80, 155124 /H208492009 /H20850
155124-6the case of the high-frequency modes. For PBE, the differ-
ence between using the experimental and the theoretical vol-ume is particularly drastic since PBE strongly overestimatesthe lattice parameters. However, even if the phonons are con-sistently evaluated at the experimental volume, HSE yieldssignificantly higher phonon frequencies than PBE in better
agreement with available experimental data. The dielectricproperties are also clearly described best by the HSE func-
tional, which yields dielectric constants in almost perfectagreement with experiment /H20849except for the ionic contribution
in the cdirection /H20850. Also piezoelectric constants are well de-
scribed by the HSE functional. This indicates that the HSEfunctional is a judicious choice for the prediction of materi-
als properties of semiconductors and insulators.
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calculations were performed at the experimental volume with the internal parameters fully relaxed using thecorresponding functional.
Dir. LDA
aLDA PBE HSE06 Expt.a
/H9280/H1100911 5.76 5.14 4.97 3.66 3.70
/H9280/H1100933 5.12 5.16 5.00 3.73 3.78
/H92800−/H9280/H1100911 4.55 4.70 4.17 4.0 4.07
/H92800−/H9280/H1100933 5.15 5.59 4.92 4.6 5.13
/H9280011 10.31 9.84 9.14 7.72 7.77
/H9280033 10.27 10.75 9.92 8.37 8.91
Z 11 2.15 2.08 2.08 2.06
Z 33 2.16 2.13 2.13 2.11
e/H20849C/m2/H20850 31 0.37 0.39 0.38 0.38
e/H20849C/m2/H20850 33 −0.78 −0.80 −0.79 −0.76
e/H20849C/m2/H20850 12 0.39 0.41 0.41 0.40
d/H20849C/m2/H20850 31 −0.55 −0.50 −0.63 −0.48 −0.62
d/H20849C/m2/H20850 33 1.19 1.10 1.32 0.95 0.96
d/H20849C/m2/H20850 12 −0.46 −0.37 −0.47 −0.37 −0.37
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PhysRevB.84.100101.pdf | RAPID COMMUNICATIONS
PHYSICAL REVIEW B 84, 100101(R) (2011)
Bulk structures of PtO and PtO 2from density functional calculations
Ricardo K. Nomiyama,1Maur ´ıcio J. Piotrowski,2and Juarez L. F. Da Silva3
1Instituto de Qu ´ımica de S ˜ao Carlos, Universidade de S ˜ao Paulo, Caixa Postal 780, 13560-970, S ˜ao Carlos, SP , Brazil
2Departamento de F ´ısica, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil
3Instituto de F ´ısica de S ˜ao Carlos, Universidade de S ˜ao Paulo, Caixa Postal 369, 13560-970, S ˜ao Carlos, SP , Brazil
(Received 6 May 2011; revised manuscript received 23 July 2011; published 14 September 2011)
Platinum plays an important role in catalysis and electrochemistry, and it is known that the direct interaction of
oxygen with Pt surfaces can lead to the formation of platinum oxides (PtO x), which can affect the reactivity. To
contribute to the atomistic understanding of the atomic structure of PtO x, we report a density functional theory
study of the atomic structure of bulk PtO x(1/lessorequalslantx/lessorequalslant2). From our calculations, we identified a lowest-energy
structure (GeS type, space group Pnma ) for PtO, which is 0.181 eV lower in energy than the structure suggested
by W. J. Moore and L. Pauling [ J. Am. Chem. Soc. 63, 1392 (1941) ] (PtS type). Furthermore, two atomic
structures were identified for PtO 2, which are almost degenerate in energy with the lowest-energy structure
reported so far for PtO 2(CaCl 2type). Based on our results and analysis, we suggest that Pt and O atoms tend to
form octahedron motifs in PtO xeven at lower O composition by the formation of Pt-Pt bonds.
DOI: 10.1103/PhysRevB.84.100101 PACS number(s): 71 .15.Mb, 71 .15.Nc, 71 .20.−b
Introduction . Platinum is one of the most used chemical
elements in catalysis1and electrochemistry,2e.g., a key
component in three-way catalysts3(TWC) and fuel cells.4
Furthermore, Pt is the base for a wide range of nanoalloys,1,5
which are expected to play an important role in catalysis.
It is well known that oxygen is one of the most commonelements that interacts with Pt surfaces in catalyst devices,which play a crucial role in the formation of platinum oxide
(PtO
x) layers,1,5–10which can affect the reactivity of Pt-base
catalysts. Thus, over time several experimental and theoreticalstudies have been performed to elucidate the mechanismsthat drive the formation of thin PtO
xlayers as well as the
atomic structure and stability of the bulk PtO xphases (e.g., see
Refs. 8–13). Although several studies have been performed,
we will show below that a few doubts remain even for the
bulk PtO xstructures, which clearly compromise an atomistic
understanding of the formation of thin PtO xlayers.
Several x-ray diffraction (XRD) studies have been per-
formed for PtO x, and several structural models have been
known for PtO,14Pt3O4(hereafter called PtO 1.33),15,16and
PtO 2.17–22Moore and Pauling reported a tetragonal structure
(PtS type, space group P42/mmc ) for PtO,14which contains
2 formula units (f.u.) per unit cell, in which Pt and O arefourfold; i.e., there is no formation of Pt octahedra in PtO.
Two model structures have been suggested for PtO
1.33,15,16and
first-principles calculations11have suggested that the Galloni
and Roffo model16is incorrect; i.e., the structure proposed
by Muller and Roy (simple cubic lattice with space groupPm-3n) yields the lowest energy. Several structures have been
reported for PtO
2, e.g., β-PtO 2(CaCl 2type, space group
Pnnm ),17–19,21,22α-PtO 2(CdI 2type, space group P-3m1),21
andβ/prime-PtO 2(rutile type, space group P42/mnm ).20
The XRD PtO xstructures are shown in Fig. 1. It can
be seen that the Pt and O atoms form octahedra for PtO 2,
i.e., Pt is sixfold, while O is threefold, and hence, using theelectronic octet rule, an oxidation state of +4 can be assumed
for Pt. For PtO
1.33, the Pt atoms are surrounded by four O
atoms with Pt-O bond lengths as in PtO 2and two Pt atoms
at 2.83 ˚A, i.e., close to the Pt-Pt distance in bulk Pt (2.81 ˚A),which indicates the formation of Pt-Pt bonding in PtO 1.33.
The Pt-Pt bonding is formed due to the reduction of the Ocomposition, and hence, we would expect also the formation ofPt-Pt bonds in PtO; however, this is not the case. For example,the Pt-Pt distance in the PtS-type structure is about 3.16 ˚A
(the experimental value is 3 .04±0.03).
14Thus, the structure
suggested by Moore and Pauling about 70 years ago, whichhas been used in several first-principles studies,
11–13,23–26is
not consistent with the trends observed in the PtO 1.33and PtO 2
structures.
Thus, in order to investigate these questions, which have
great importance due to the high occurrence of PtO xin Pt-
base materials, we will perform a theoretical first-principlesinvestigation of the atomic structure of PtO
x, in particular, for
x=1.0,4/3,2.0. From our calculations, we identified a lower-
energy structure for PtO, which is about 0.181 eV /f.u. lower in
energy than the well-known Moore-Pauling structure, and thePt atoms are sixfold coordinated. Furthermore, we identifiedtwo structures for PtO
2, which are almost degenerate in energy
with the CaCl 2structure. Therefore, our results indicate an
energetic preference for Pt sixfold arrangements even for lowO concentration such as in PtO. Thus, for low O concentration,we can expect the formation of Pt-Pt bonding.
Computational Approach . Our calculations are based on
spin-polarized density functional theory (DFT) within thePerdew-Burke-Erzenholf
27(PBE) formulation for the gen-
eralized gradient approximation (GGA). The Kohn-Shamequations are solved with the projected augmented wave
28,29
(PAW) method, as implemented in the Vienna Ab-initio
Simulation Package ( V ASP ).30,31The equilibrium volumes for
all PtO xcompositions were obtained by minimizing the atomic
forces and stress tensors using a plane-wave cutoff energy of800 eV , while 400 eV was used for the density of states (DOS).For the Brillouin zone integration of PtO in the tetragonalPtS-type structure, we employed a kmesh of 8 ×8×5 and
24×24×14 for the stress tensor and DOS calculations,
respectively. The same k-point densities were used for all bulk
PtO
xcalculations.
Atomic model structures . To perform our DFT-PBE inves-
tigation, we selected a large set of trial bulk structures for
100101-1 1098-0121/2011/84(10)/100101(4) ©2011 American Physical SocietyRAPID COMMUNICATIONS
NOMIYAMA, PIOTROWSKI, AND DA SILV A PHYSICAL REVIEW B 84, 100101(R) (2011)
2 PtO2 PtO1.33 PtO PtO
2 CaCl −type CdI2−type PbCl2−typePtO2 β− PtO2 α− PtO2 β−’
OrthorhombicZrO −type
GeS−type
Orthorhombic
PnmaTetragonal
P42/mmcCubic
Pm−3nOrthorhombic
PnnmOrthorhombic
PnmaTrigonal
P−3m1 PbcnRutile−type
Tetragonal
P42/mnm Pnma
−0.685 eV −0.504 eV −1.020 eV −1.424 eV −1.418 eV −1.412 eV −1.335 eV −1.294 eVPtS−type Simple cubic2PtO
FIG. 1. (Color online) Atomic structures of the PtO, PtO 1.33,a n dP t O 2compositions, in which the Pt and O atoms are indicated by gray
(large) and red (small) balls. The unit cells are indicated by solid black lines. The structure type, crystalline system, space group, and formation
energy /Delta1H are indicated below the unit cells.
PtO, PtO 1.33, and PtO 2, in which all reported XRD structures
were included. The trial structures were selected from severaloxides, sulfides, chlorides, and fluoride compounds, takinginto account a wide diversity of Pt and O coordination envi-ronments, e.g., from tetrahedron to octahedron, and differentoxidation states, e.g., +4 and+2.
Lowest-energy PtO
xstructures . The lowest-energy bulk
PtOxstructures ( Etrial
tot−Elowest
tot/lessorequalslant0.35 eV/f.u.) are shown in
Fig. 1, i.e., two for PtO, one for PtO 1.33, and five for PtO 2,
while the equilibrium lattice constants ( a0,b0,c0), internal
parameters (Pt and O internal positions), and bond-lengthdistances (Pt-O, Pt-Pt) are summarized in Table I.
We found a lowest-energy structure for PtO, which is
0.181 eV /f.u. lower in energy than the tetragonal structure sug-
gested by Moore and Pauling
14and employed in a large number
of first-principles studies over the last 20 years (e.g., seeRefs. 11–13,23–26). Our lowest-energy structure was based
on the GeS structure, and it has 4 f.u. per unit cell and spacegroup Pnma . In our structure, every Pt atom is surrounded by
six atoms, namely, four O atoms at 2.15 ˚A and two Pt atoms
at 2.57 ˚A, i.e., the octahedra are distorted. In the Moore and
Pauling structure,
14which can be easily derived from the rutile
PtO 2structure by removing O atoms, the Pt atoms bind with
only four O atoms, and the Pt-Pt distance is about 3.16 ˚A. We
would like to point out that Pt-Pt is 2.81 ˚A for the crystalline
Pt face-centered cubic (fcc) structure and 2.33 ˚Af o rt h eP t 2molecule; i.e., the Pt-Pt distance in our proposed PtO structure
is in between both systems, while in the Moore and Paulingstructure it is about 12% larger than in the bulk Pt. Thus, ourresults and analysis indicate clearly that PtO adopts a structurein which the formation of octahedron motifs is maximized,which requires the formation of Pt-Pt bonds due to the lowerO composition.
For PtO
1.33, our approach could not identify a lowest-
energy structure even though several trial configurations werecalculated. In agreement with previous DFT calculations,
11
we found that the structure proposed by Galloni and Roffo16
is 2.48 eV higher in energy than the XRD structure proposedby Muller and Roy,
15which is shown in Fig. 1. Our closest
energy structure is about 1.0 eV /f.u. higher than the structure
proposed by Muller and Roy. In the PtO 1.33structure, the Pt
atoms are surrounded by four O atoms located at the sameplane with Pt-O =2.00 ˚A and two Pt atoms at 2.83 ˚A. It
can be noticed that the Pt-Pt distance is larger than in the GeSstructure for PtO; however, it has about the same value as Pt-Ptin bulk Pt.
For PtO
2, we identified two structures (PbCl 2type and ZrO 2
type) with total energies between the energy limits defined bythe distorted rutile structure (CaCl
2type) and the rutile-type
structure (Fig. 1). The energy differences among the PtO 2
structures are relatively small, e.g., the CaCl 2type is only
0.130 eV /f.u. lower in energy than the rutile type. Therefore,
TABLE I. Equilibrium lattice parameters, internal parameters, and Pt-O and Pt-Pt bond lengths in the PtO xcompounds. The unit cells,
space group, and heat of formation with respect the bulk Pt and molecular oxygen are shown in Fig. 1.
x Structure a0(˚A) b0(˚A) c0(˚A) Pt O Pt-O Pt-Pt
1.00 PtS type 3.16 3.16 5.37 (0, 1 /2, 0) (0, 0, 1 /4) 2.07 3.16
GeS type 6.85 3.40 4.33 (0.5696, 3 /4, 0.6945) (0.3363, 3 /4, 0.9992) 2.15 2.57
1.33 simple cubic 5.67, 5.59a(1/2, 3/4, 0) (3 /4, 3/4, 1/4) 2.00 2.83
2.00 rutile type 4.60 3.24 (0, 0, 0) (0.6906, 06906, 0) 2.03 3.24
ZrO 2type 4.56 5.61 5.24 (0, 0.1489, 1 /4) (0.2510, 0.4011, 0.0695) 2.04 3.11
CdI 2type 3.16 4.91 (0, 0, 0) (0.3333, 0.6667, 0.1932) 2.06 3.16
PbCl 2type 9.62 4.67 3.16 (0.3657, 1 /4, 0.9107) (0.2526, 1 /4, 0.5395) 2.05 3.59
(0.9865, 3 /4, 0.7283)
CaCl 2type 4.61, 4.48b4.55, 4.54b3.19, 3.14b(0, 0, 0) (0.2594, 0.3622, 0) 2.04 3.19
aExperiment; Ref. 15.
bExperiment; Ref. 22.
100101-2RAPID COMMUNICATIONS
BULK STRUCTURES OF PtO AND PtO 2FROM ... PHYSICAL REVIEW B 84, 100101(R) (2011)
it is very impressive that three different bulk structures exist
in this narrow energy window, which can be considered to bean indication of a complex phase diagram. The PbCl
2type is
only 0.006 eV /f.u. (2 meV /atom) higher than the CaCl 2type,
which is almost close to the limit of accuracy in DFT-PBEcalculations, and hence, we can assume that both are almostdegenerated in energy. The CdI
2and rutile structures have been
observed experimentally,8,20and hence, we expect that our
suggestions (PbCl 2type and ZrO 2type) can also be confirmed
in the future by XRD experiments.
In the five PtO 2structures shown in Fig. 1, the Pt and O
atoms form nearly perfect octahedra motifs; i.e., the Pt atomsare surrounded by six O atoms. The Pt-O distances are from2.03 to 2.06 ˚A, while the smallest Pt-Pt distances in the five
structures are spread from 3.11 to 3.59 ˚A. The PbCl
2structure
is characterized by holes, which is not so unexpected as theCdI
2structure is formed by the stacking of two PtO layers,
which have motivated DFT calculations for PtO 2nanotubes.12
It is important to notice that all the O atoms in the five structuresare threefold, and hence, an oxidation state of +4 is assumed
for Pt in PtO
2, which is supported by the electronic octet rule.
Octahedron orientations . For all the lowest-energy PtO x
structures (GeS type for PtO, Muller and Roy for PtO 1.33, and
CaCl 2type for PtO 2), the Pt and O atoms form octahedra
motifs due to the formation of Pt-Pt bonds for lower Ocompositions, e.g., x=1.0,4/3. We noticed that the octahedra
are not oriented along of the same direction; i.e., the octahedra
form arrays in which about half of them are oriented inopposite direction. This trend is observed for all discussed
bulk configurations, except for the PtS structure suggested
by Moore and Pauling
14and the CdI 2-type structure for PtO 2.
Thus, our results and analysis suggest that octahedra structures
might be the most frequent motif in thin films and PtO xlayers,
and its orientation is important.
Formation energy . The formation energies with respect to
the bulk Pt and molecular oxygen ( /Delta1H=EPtO x
tot−EPt/atom
tot −
xEO2/atom
tot ) are summarized in Fig. 1. It can be seen that the
stability of the PtO xincreases with O composition, which
correlates with an increased number of O atoms surroundingthe Pt atoms. Our results are in good agreement with previousDFT calculations.
12
Density of states . The total and local density of states
(LDOS) are shown in Fig. 2. For PtO, we can see clearly
that the GeS structure yields an energy separation betweenthe unoccupied and occupied states of about 0.30 eV for PtO,while the Moore and Pauling structure (PtS type) yields ametallic solution for PtO. Previous DFT studies
12found also a
metallic solution for the PtS structure, which was attributedto the metallic character induced by the formation of thePt-Pt bond in the PtS structure; however, our results do notsupport this argument. For example, we found an energy gapof about 0.30 eV with a Pt-Pt bond length of 2.57 ˚Ai nt h eG e S
structure. We would like to mention that previous DFT +U
(U=9.0 eV for Pt dstates) and hybrid DFT calculations
found an energy separation between the unoccupied andthe occupied states at the Fermi level in the Moore-Paulingstructure.
12,25
For PtO 1.33, DFT-PBE yields also a metallic solution, which
has also been reported by previous studies, and DFT +U
cannot open a gap.12For PtO 2, we obtained an energy gap for
LDOSTotal
Pt
OLDOS LDOS
-8 -4 0 4
Energy (eV)LDOS
-8 -4 0 4
Energy (eV)PtO GeS-type PtO Moore-Pauling
PtO1.33 Muller-Roy PtO2 CaCl2-type
PtO2 PbCl2-type PtO2 CdI2-type
PtO2 ZrO2-type PtO2 Rutile-type
FIG. 2. (Color online) Total and local density of states (LDOS) for
PtOxfor the structures shown in Fig. 1in arbitrary units. The black
(solid), red (dotted), and blue (dashed) lines are the total DOS, Pt
LDOS, and O LDOS, respectively. The vertical dashed lines indicate
the Fermi level (zero energy).
all structures, except for the rutile-type structure; i.e., it shows
clearly that the distortion in the rutile-type structure opens anenergy gap in the CaCl
2-type structure. We observed that the
PbCl 2and CdI 2structures have larger energy gaps between
the unoccupied and the occupied states than CaCl 2, which can
be explained by the weak interaction between the PtO 2layers
in CdI 2and holes in PbCl 2.
Electronic octet rule and oxidation state . Except for the
GeS-type structure, the O atoms bind with three Pt atoms.Thus, in order to satisfy the electronic octet rule, an oxidationstate of +4 is required for the Pt atoms in all PtO
2structures,
which is consistent with previous studies. Following theelectronic octet rule, an oxidation state of +2 has also
been assumed for the Moore and Pauling structure for PtO;however, the same conclusion does not apply for the GeS-typestructure, which is lower in energy than the Moore Paulingstructure. For example, there is no Pt-Pt bonding in the Mooreand Pauling structure; however, there is a very short Pt-Ptbonding in the GeS-type structure; i.e., the Pt atoms are notfourfold in the GeS-type structure, but sixfold. Therefore, anoxidation state of +2 cannot satisfy the electronic octet rule
in the GeS-type structure, and hence, we expect an oxidationstate between +2 and+4.
Summary . In this work, we identified a lowest-energy
structure for PtO (GeS type), which is 0.181 eV /f.u. lower
100101-3RAPID COMMUNICATIONS
NOMIYAMA, PIOTROWSKI, AND DA SILV A PHYSICAL REVIEW B 84, 100101(R) (2011)
than the structure proposed by Moore and Pauling (PtS type)
about 70 years ago.14Furthermore, we found two structures
for PtO 2(PbCl 2type and ZrO 2type) that are degenerated in
energy with the CaCl 2-type structure.17–19,21,22The identified
structures provide insights for obtaining a better understandingof the mechanisms that drive the formation of the PtO
xbulk.
For example, our analysis indicates that Pt atoms maximizesixfold environments (octahedron) with O and Pt atoms inPtO
x. The Pt-Pt bonds are formed only at low O composition,such as in PtO and PtO 1.33, and for the particular case of PtO,
DFT-PBE yields a band gap of about 0.30 eV even with theformation of the Pt-Pt bonding with 2.57 ˚A (smaller than Pt-Pt
in bulk Pt).
R. K. Nomiyama and J. L. F. Da Silva thank the S ˜ao
Paulo Science Foundation (FAPESP) for financial supportand computational resources, and M. J Piotrowski thanks theBrazilian financial agency CAPES for financial support.
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100101-4 |
PhysRevB.87.104107.pdf | PHYSICAL REVIEW B 87, 104107 (2013)
Stability maps to predict anomalous ductility in B2 materials
Ruoshi Sun1,2and D. D. Johnson1,3,4,*
1Materials Science and Engineering, University of Illinois at Urbana-Champaign, 1304 West Green Street, Urbana, Illinois 61801, USA
2Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge,
Massachusetts 02139, USA
3The Ames Laboratory, US Department of Energy, 311 TASF , Iowa State University, Ames, Iowa 50011-3020, USA
4Materials Science and Engineering, Iowa State University, Ames, Iowa 50011-2300, USA
(Received 15 January 2013; revised manuscript received 5 March 2013; published 19 March 2013)
While most B2 materials are brittle, a new class of B2 (rare-earth) intermetallic compounds is observed to
have large ductility. We analytically derive a necessary condition for ductility (dislocation motion) involving
/angbracketleft111/angbracketrightversus /angbracketleft001/angbracketrightslip and the relative stability of various planar defects that must form. We present a sufficient
condition for antiphase boundary bistability on {1¯10}and{11¯2}planes that allows multiple slip systems. From
these energy-based criteria, we construct two stability maps for B2 ductility that use only dimensionless ratios ofelastic constants and defect energies, calculated via density functional theory. These two conditions fully explainand predict enhanced ductility (or lack thereof) for B2 systems. In the 23 systems studied, the ductility of YAg,ScAg, ScAu, and ScPd, ductile-to-brittle crossover for other rare-earth B2 compounds, and brittleness of allclassic B2 alloys and ionic compounds are correctly predicted.
DOI: 10.1103/PhysRevB.87.104107 PACS number(s): 62 .20.fk, 61.72.Nn, 81 .05.Bx, 71 .20.−b
I. INTRODUCTION
In 2003, Gschneidner et al. discovered a family of duc-
tile rare-earth/transition-metal (RM) intermetallic compoundswith the body-centered-cubic-based B2 (or CsCl) crystalstructure.
1The current list of known ductile B2 compounds
can be found in Ref. 2. In contrast to the brittleness of
classic B2 alloys, the ductility of YAg is comparable toface-centered-cubic Al, and YCu is half that of YAg. Asline compounds are usually brittle,
3the reason for anomalous
ductility in RM compounds remains open. Moreover, someRM compounds, such as (Tb
0.88Dy0.12)Zn (Ref. 4) and YMg,5
are brittle. Hence, the questions: Why are they different? Can
anomalous ductility be predicted on a system-dependent basis?
Much work has been done in determining the dominant slip
systems of the B2 alloys. As discussed in the review articles byYamaguchi and Umakoshi
6and Baker,7/angbracketleft111/angbracketrightand/angbracketleft001/angbracketrightare
the two main observed slip directions for dislocation motion inB2 materials (Fig. 1). Yet no previous theories have attempted
to predict B2 ductility because all the known alloys and ioniccompounds are brittle. For example, polycrystalline NiAl hasonly a 2% elongation upon fracture.
3Baker concluded that
limited ductility is associated with /angbracketleft001/angbracketrightslip, and brittleness
with/angbracketleft111/angbracketrightslip.7While off-stoichiometric B2 alloys exhibit
improved ductility, yield strength is sacrificed, which is notuseful for practical purposes.
3In contrast to B2 alloys, this
new class of RM compounds has an exact stoichiometry, andthe compounds are nearly elastically isotropic.
1
Following the discovery of the RM compounds, there have
been several experimental4,8,9and theoretical2,10–12studies.
Morris et al. have hypothesized that the enhanced ductility
in the Y-based compounds is due to the competing structuralstability of B33 and B27 phases, obtained by introducing aperiodic array of
a
2/angbracketleft001/angbracketright{1¯10}superintrinsic stacking faults
(SISFs) to the B2 lattice.12However, the fact that not all the
RM-B2 compounds are ductile highlights the complicationswith classification of the slip modes and the prediction oftheir ductility. More recently, Gschneidner et al. established acorrelation between the absence of delectrons and measured
ductility.
2However, a direct explanation from the perspective
of ductility involving dislocation motion and defect ener-getics is lacking. Such a theory permits prediction, as wellas correlations to specific electronic features, to be made,while relating observed ductility measures to features in theelectronic structure is fruitful but not a theory.
To address the atypical ductility possessed by some RM
compounds and the unresolved issue of predicting /angbracketleft111/angbracketright
versus /angbracketleft001/angbracketrightslip, we provide a quantitative explanation from
mesoscale dislocation mechanics using energy-based stabilitycriteria, whose parameters can be calculated from densityfunctional theory (DFT). In short, we present a predictivetheory for ductility in ideal B2 compounds. From stabilitycriteria derived for the B2 structure, we provide a necessary
and sufficient condition for increased ductility in B2 systems,
which are displayed in terms of predictive dimensionless maps.We apply these maps to three types of B2 materials: (1) Y-basedand Sc-based compounds (YAg, YCu, YIn, YRh, YMg, YZn,ScAg, ScAu, ScCu, ScPd, ScPt, ScRh, and ScRu), (2) classicalloys (NiAl, FeAl, AuCd, AuZn, CuZn, and AgMg), and (3)ionic compounds (CsCl, CsI, TlBr, and TlCl). Any proposedB2 compound can be added to the map to predict its relativeductility. The possibilities can be narrowed using only theZener anisotropy ratio.
II. BACKGROUND
For dislocation-mediated deformation, both elastic
anisotropy and planar defect energies (e.g., antiphase bound-ariesγ
hkl
APBor stacking faults γhkl
SF) in the competing Miller-
indexed (hkl)slip planes are relevant. Clearly, the more elas-
tically isotropic a system, the easier for dislocation movementto other slip planes under shear. As noted, the new ductile B2systems are nearly elastically isotropic,
1with Zener anisotropy
ratioAclose to 1, where
A=2c44
c11−c12. (1)
104107-1 1098-0121/2013/87(10)/104107(12) ©2013 American Physical SocietyRUOSHI SUN AND D. D. JOHNSON PHYSICAL REVIEW B 87, 104107 (2013)
(a) (b)
FIG. 1. For B2 systems, the (a) [001] and [111] slip in the
(1¯10) plane and (b) [ ¯1¯11] slip in the (11 ¯2) plane. (a) Perfect /angbracketleft111/angbracketright
superdislocation can dissociate [Eq. (7a)] into perfect /angbracketleft110/angbracketrightand/angbracketleft001/angbracketright
dislocations (dashed lines), with possible dissociation of /angbracketleft110/angbracketrightinto
/angbracketleft100/angbracketrightand/angbracketleft010/angbracketright(grey dashed lines).
Here, the cij’s are the cubic elastic constants, and, in particular,
c44is the shear modulus. For a B2 lattice constant of a,t h e
product of c44ahas units of γ(in mJ/m2). For an energy-based
criterion for ductility under shear, dimensionless ratios arerelevant, reflecting relative energies; two ratios associated withthe energetics of slip directions and defect formation, makingthe simple maps, are
C=γ
1¯10
APB/slashbig
c44a, (2)
δ=γ1¯10
SF/slashbig
γ1¯10
APB. (3)
These quantities can be obtained via DFT calculations.
In a recent study of the L1 2binaries and pseudobinaries, the
occurrence/loss of the yield-stress anomaly was predicted13in
all systems studied by considering the necessary condition for
the stability of APB versus SISF and a sufficient condition
for the stability of APB(111) versus (100) for cross-slip ofscrew dislocation segments. The APB and SISF energies, aswell as c
ij, were obtained using DFT. The necessary and
sufficient conditions were derived, respectively, by Paidar,Pope, and Yamaguchi
14(modified by Liu et al.13) and Saada
and Veyssiere.15The resulting stability map is applicable to any
L12material. Here, we adopt a similar approach. We construct
two maps based on energy-stability criteria for competing slipmodes in B2 structures that fully explain and predict enhancedductility (or lack thereof) in B2 systems.
III. METHODS
DFT16,17calculations were performed to obtain required
parameters of the theory. We employed the Vienna Ab InitioSimulation Package (
V ASP )18–21that uses pseudopotentials
with a projector augmented wave (PAW) basis.22,23We
adopted the generalized gradient approximation (GGA) tothe exchange-correlation functional.
24,25The lattice aand
elastic cijconstants were calculated for two-atom B2 cells
with 20 ×20×20k-point meshes.26Total energies (forces)
were converged below 0.1 meV /cell (1 meV /˚A). Due to errors
in GGA functionals, afor metals are overestimated ( atheory>
aexptup to 1%), which affect the values cijandγbecause the
defect planes are farther apart and lowers the defect energy,giving material-dependent errors. Therefore, we used a
expt
(if known) to create the maps to remove nonsystematic errors
from calculated quantities. (The value of aexptis unknown forYIn.) Note that removing such nonsystematic error is critical
for predicting quantitatively other deformation processes, suchas twinning,
27,28because the atomic planes away from the
planar defects are separated by geometric multiples of aexpt
(the same distances as in experiment), but relaxation around
the defects plane are included, so DFT provides a more correctshear surface energy.
The planar defect energies of APB {1¯10}and SF {1¯10}were
calculated for 32-atom unit cells having at least 12 ×12×2
kpoints. The APB {11¯2}was calculated using a 24-atom unit
cell and 8 ×8×4kpoints. Examples of the various unit cells
are shown in Fig. 2.T h ek-point meshes were chosen such that
the reciprocal axes had a similar density of kpoints. The aspect
ratio in kspace, then, was roughly the reciprocal of that in real
space. APBs and SFs on the {1¯10}plane were separated by
[110][110]
[001](a) (b)
(c)
[111][112]
[110]
FIG. 2. Unit cells used for (a) APB {1¯10},( b )S F {1¯10},a n d
(c) APB {11¯2}. In (a) and (b), solid and open circles represent different
atomic species. In (c), APB {11¯2}is projected onto the (1 ¯10) plane.
Solid symbols represent atoms in the plane; open symbols represent
atomsa
2[1¯10] behind the plane. The two defect planes per unit cell
are represented by dashed lines.
104107-2STABILITY MAPS TO PREDICT ANOMALOUS DUCTILITY ... PHYSICAL REVIEW B 87, 104107 (2013)
eight layers of atoms, whereas APB {11¯2}’s were separated by
six layers. To remove errors for k-point meshes, the perfect cell
had the same number of atoms and cell shape as the defectivecell. (Note that defect planes may shift off their ideal latticepositions, an effect for which we have not accounted in theDFT results.) Defect energies were computed from
γ=E
defect−Eperfect
m/bardblT1×T2/bardbl(4)
formdefect planes per unit cell. Each unit cell contained
two defect planes, so that orthogonal translation vectors T1,2,3
could be used as coordinate axes along the defect plane. In
defective cells, two layers of atoms on each side of the defectplane were relaxed along T
3, with the cell shape and volume
fixed to remove systematic errors. Specifically, the translationvectors were
1
a⎡
⎢⎣T1
T2
T3⎤
⎥⎦=⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩⎡
⎢⎣¯110
001
880⎤
⎥⎦APB and SF {1¯10}
⎡
⎢⎣¯110
¯1¯11
224⎤
⎥⎦APB{11¯2}.(5)
Following Mehl et al. ,
29we obtained the necessary elastic
constants cijby solving for c44,c/prime=(c11−c12)/2, and
bulk modulus B=(c11+2c12)/3 through appropriate lattice
distortion, where the Zener ratio A=c44/c/primeandG=c44.T h e
total energy for each strain distortion in B2 is proportional to/epsilon1
2, with O(/epsilon14) error, giving more accurate coefficients. The
Poisson ratio in Table Iis computed from
ν=3B−2G
6B+2G. (6)
We provide in Table Ithe DFT and known observed values
of all required quantities that are necessary in the maps thatindicate enhanced ductility using elastic properties and relativedefect energies.
IV . NECESSARY AND SUFFICIENT CONDITIONS
FOR B2 DUCTILITY
We now derive the two conditions for ductility, applying
them in Sec. V. Multiple slip can occur via formation of
/angbracketleft111/angbracketrightAPBs on the {1¯10}and{11¯2}planes. (For simplicity,
thea
2/angbracketleft111/angbracketright{1¯10}anda
2/angbracketleft111/angbracketright{11¯2}APBs anda
2/angbracketleft001/angbracketright{1¯10}
SFs are denoted as APB {1¯10},A P B {11¯2}, and SF {1¯10}.) It
is necessary, then, that the /angbracketleft111/angbracketrightAPBs have to be more
energetically favorable than the /angbracketleft001/angbracketrightSFs. To predict /angbracketleft111/angbracketright
versus /angbracketleft001/angbracketrightslip, Rachinger and Cottrell44gave a simple
criterion in terms of width of APB: If wAPB/a/greatermuch1, then /angbracketleft111/angbracketright
slip is favorable; else if wAPB/a≈1, then /angbracketleft001/angbracketrightis favorable.
We have derived a more quantitative necessary condition45
(see Appendix A) in light of Paidar, Pope, and Yamaguchi’s
work in L1 2systems.14
Saada and Veyssiere39investigated the sufficient condition
for cross-slip of a /angbracketleft111/angbracketrightscrew superdislocation on {1¯10}and
{11¯2}planes that leads to multiple-slip systems. The possible
dissociation mechanisms for a /angbracketleft111/angbracketrightscrew superdislocationare
a/angbracketleft111/angbracketright→a/angbracketleft110/angbracketright+a/angbracketleft001/angbracketright
→a/angbracketleft100/angbracketright+a/angbracketleft010/angbracketright+a/angbracketleft001/angbracketright, (7a)
a/angbracketleft001/angbracketright→a
2/angbracketleft001/angbracketright+SF+a
2/angbracketleft001/angbracketright, (7b)
a/angbracketleft111/angbracketright→a
2/angbracketleft111/angbracketright+APB+a
2/angbracketleft111/angbracketright. (7c)
In Eq. (7a),t h e/angbracketleft111/angbracketrightscrew dislocation can further dissociate
into perfect dislocations along the cube edges (Fig. 1); hence,
there are no APBs or SFs. Equation (7b) involves formation
of SFs.
A. Necessary condition
As described above, there are two criteria that must be
met simultaneously that provide the necessary condition for
ductility: /angbracketleft001/angbracketrightshould be the dominant slip direction, yet
/angbracketleft111/angbracketrightslip should also be possible with formation of /angbracketleft111/angbracketright
APBs; see Fig. 1. An overview of the derivation is provided in
Appendix A.
On purely energetic grounds, for B2 materials to possess
multiple slip during plastic flow, /angbracketleft001/angbracketright{1¯10}slip must be more
favorable than /angbracketleft111/angbracketright{1¯10}slip via APB {1¯10}formation, which
occurs45(see Appendix A)i f
wAPB/lessorequalslantkea∼5.9aor (8a)
lnC/greaterorequalslant−3.9. (8b)
Equation (8)justifies the criterion imposed by Rachinger and
Cottrell44and gives a fixed measure across B2 systems. The
second form is useful for presenting the maps.
Now, to have enhanced ductility, both /angbracketleft001/angbracketrightas the dominant
slip direction, and /angbracketleft111/angbracketrightslip also possible by formation of
/angbracketleft111/angbracketrightAPBs, the APBs must be more energetically favorable
than SFs. The key necessary condition45(see Appendix A),
using Eqs. (2)and(3),i s
δ>0.119C−1/4or (9a)
lnδ>−2.132−1
4lnC, (9b)
where the second form is easier for plotting the maps. Together
Eqs. (8)and(9)constitute the map for B2 systems that will
have both dominant /angbracketleft001/angbracketrightslip and /angbracketleft111/angbracketrightslip due to formation
of APBs.
For our generic map and necessary conditions, as a standard
simplification, we used a Poisson ratio of 1 /3 (not values in
Table I), which yields integer coefficients related to ν.W ea l s o
used an effective dislocation interaction range of twice the corewidth ( r=2r
0=2ka; see Appendix A). While ν=1/3 and
the estimated ksimplifies the algebra for the maps, the reader
should appreciate that the exact borders for each materialcan be shifted by the actual values—the price for a genericmap; hence, borderline cases should be assumed possiblyrelevant. Also, differences in various DFT calculations couldalter locations in the maps, as we show explicitly.
B. Sufficient condition
As noted earlier, a ductile B2 material can have multiple
slip only if APBs have bistable existence on both {1¯10}and
104107-3RUOSHI SUN AND D. D. JOHNSON PHYSICAL REVIEW B 87, 104107 (2013)
TABLE I. Calculated and observed B2 lattice constants ( ain˚A), bulk modulus ( Bin GPa), and elastic constants ( cijin GPa). A,ν,a n dM
are defined in the main text. In the first (second) row of each system, DFT calculations were performed at aDFT(aexpt). References for aexptare
given in the second row. In the third row, B,c/prime,A,ν,a n dMwere derived from the experimental elastic constants. Dashes indicate that no data
are available.
Material aBc/primec11 c12 c44 Aν M Ref.
YAg 3.646 68.3 22.3 98.0 53.4 35.0 1.57 0.281 1.02
3.619 75.1 22.6 105.2 60.0 37.8 1.67 0.284 1.02 10
3.619 70.1 24.2 102.4 54.0 37.2 1.54 0.276 1.02 10
YCu 3.485 71.3 34.4 117.2 48.4 36.4 1.06 0.282 1.00
3.477 73.4 34.6 119.6 50.3 37.2 1.08 0.283 1.00 10
3.477 70.1 32.5 113.4 48.4 32.3 0.99 0.300 1.00 10
YIn 3.769 57.3 6.02 65.3 53.3 43.4 7.20 0.198 1.35
––
YRh 3.442 113.3 38.4 164.5 87.7 36.6 0.95 0.354 1.00
3.407 121.2 38.0 171.8 95.9 40.4 1.06 0.350 1.00 30
–
YMg 3.798 41.2 8.62 52.7 35.5 39.6 4.60 0.136 1.19
3.806 40.4 8.55 51.8 34.7 39.1 4.57 0.134 1.18 31
–
YZn 3.578 62.6 22.6 92.7 47.5 43.2 1.92 0.219 1.03
3.577 62.8 22.6 92.9 47.7 43.3 1.92 0.219 1.03 32
–
ScAg 3.422 87.7 20.4 114.9 74.1 47.1 2.31 0.272 1.06
3.412 91.2 20.7 118.8 77.4 48.5 2.34 0.274 1.06 33
–
ScAu 3.393 111.9 25.3 145.7 95.0 47.4 1.87 0.314 1.03
3.369 123.7 26.1 158.5 106.3 51.8 1.98 0.316 1.04 34
–
ScCu 3.245 96.1 37.5 146.2 71.1 54.8 1.46 0.261 1.01
3.257 91.5 36.8 140.4 67.1 52.8 1.44 0.258 1.01 35
–
ScPd 3.301 119.9 32.8 163.5 98.0 42.8 1.31 0.341 1.01
3.283 128.8 33.2 173.1 106.6 45.5 1.37 0.342 1.01 33
–
ScPt 3.293 146.9 28.7 185.1 127.8 49.8 1.74 0.348 1.03
3.268 162.8 28.4 200.7 143.8 54.9 1.93 0.348 1.04 36
–
ScRh 3.218 149.6 62.1 232.4 108.3 51.8 0.84 0.345 1.00
3.206 157.3 63.5 242.0 114.9 54.0 0.85 0.346 1.00 33
–
ScRu 3.201 152.5 77.1 255.4 101.1 40.4 0.52 0.378 1.04
3.203 151.2 76.8 253.6 100.0 40.0 0.52 0.378 1.04 33
–
NiAl 2.895 159.4 38.4 210.5 133.8 112.8 2.94 0.214 1.10
2.886 166.0 39.5 218.7 139.6 116.5 2.95 0.216 1.10 37
166.0 34.2 211.6 143.2 112.1 3.28 0.224 1.12 38
FeAl 2.879 161.3 52.8 231.6 126.1 130.2 2.47 0.182 1.06
2.909 156.4 51.7 225.4 121.9 123.5 2.39 0.187 1.06 37
136.1 33.7 181.1 113.7 127.1 3.77 0.144 1.14 38
AuCd 3.398 93.0 1.31 94.7 92.1 37.4 28.8 0.323 2.41
3.323 130.8 1.58 132.9 129.8 56.1 36.2 0.312 2.66 37
85 3.5 90 83 44 12.6 0.279 1.68 39
AuZn 3.195 116.9 5.72 124.5 113.1 42.8 7.47 0.337 1.42
3.149 145.4 7.31 155.2 140.5 55.7 7.62 0.330 1.42 40
3.149 131.5 7.73 141.8 126.3 54.52 7.04 0.318 1.38 40
104107-4STABILITY MAPS TO PREDICT ANOMALOUS DUCTILITY ... PHYSICAL REVIEW B 87, 104107 (2013)
TABLE I. ( Continued. )
Material aBc/primec11 c12 c44 Aν M Ref.
CuZn 2.969 113.8 7.67 124.0 108.7 78.6 10.2 0.219 1.53
2.954 122.6 8.00 133.3 117.3 83.5 10.4 0.222 1.54 37
116.2 9.70 129.1 109.7 82.4 8.50 0.213 1.43 41
AgMg 3.331 65.9 13.4 83.8 57.0 47.1 3.51 0.212 1.13
3.314 70.9 14.2 89.8 61.5 49.7 3.51 0.216 1.13 37
65.6 13.7 83.8 56.4 47.6 3.46 0.208 1.13 38
CsCl 4.196 15.0 11.0 29.6 7.65 5.24 0.48 0.344 1.04
4.120 19.3 11.6 34.7 11.6 8.03 0.69 0.317 1.01 42
18.2 14.0 36.6 8.82 8.04 0.58 0.307 1.02 38
CsI 4.656 10.0 7.01 19.4 5.37 4.20 0.60 0.316 1.02
4.567 13.2 7.54 23.3 8.21 6.56 0.87 0.287 1.00 42
12.7 8.95 24.6 6.70 6.24 0.70 0.289 1.01 38
TlBr 4.011 22.1 12.6 38.9 13.7 6.21 0.49 0.372 1.04
3.986 24.4 12.7 41.3 15.9 7.51 0.59 0.360 1.02 43
22.4 11.2 37.3 14.0 7.48 0.67 0.350 1.01 38
TlCl 3.855 25.6 14.5 44.9 16.0 7.11 0.49 0.373 1.04
3.842 27.0 14.5 46.4 17.3 7.86 0.54 0.367 1.03 43
23.6 12.4 40.1 15.3 7.60 0.61 0.355 1.02 38
{11¯2}planes (see Fig. 1)—this is the sufficient condition.
Following Head46and Hirth and Lothe,47the dissociation of
/angbracketleft111/angbracketrightB2 screw superdislocations was analyzed by Saada and
Veyssiere,39and expanded on by Sun,48in terms of the relative
energetics of those planes,
λ≡γ11¯2
APB/slashbig
γ1¯10
APB, (10)
and a ratio of Sijelements
M=/radicalBigg
S11S44
S11S44−S2
15. (11)
See Ref. 47for the full derivation of Mand the definition of
Sij, which are obtained by a rotation of the cijfrom cubic axes
to the/angbracketleft111/angbracketrightaxis, as we also present in Appendix B. To satisfy
the sufficient condition, expressed in terms of a dimensionlessmap (λversus M), theλmust be strictly bounded as
48
√
3
2/lessorequalslantλ/lessorequalslant2√
3. (12)
Within this bound, Sun found48that both slip directions are
active, but (1 ¯10) is dominant for M1/3<λ< 2/√
3 while
(11¯2) is dominant for√
3/2<λ<M1/3.
C. Prediction from combined maps
As the major results, we now have the necessary and
sufficient conditions for enhanced B2 ductility. First, our morequantitative “Rachinger-Cottrell” criterion, Eq. (8),i su s e dt o
predict dominant /angbracketleft001/angbracketrightslip. Then, on the same map, Eq. (9)
compares the relative stability of APB {1¯10}and SF {1¯10}so
that, if APBs are favorable, the systems possessing both /angbracketleft001/angbracketright
and/angbracketleft111/angbracketrightslip directions satisfy the necessary condition for
ductility. Second, multiple slip in {1¯10}and{11¯2}via APB
formation is governed by the sufficient condition [Eq. (12)]i n
a second map. Practically, if the necessary condition is fulfilled(the first map), then the sufficient condition is checked (the
second map) for whether the material possesses bistabilityof APBs, and, hence, multiple slip can occur for enhancedductility. We now use these to predict ductility for several B2systems.
V. R E S U LT S
A. Necessary condition
A necessary condition map is constructed in Fig. 3.T h e
dimensionless ratios Candδare defined in Eqs. (2)and(3),
respectively. From Eq. (10) in Appendix A, it is shown that
-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.0
-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5ln δ
ln C<111> APB
<001> SF<111> slip <001> slip
YAg
YCuYIn
YRhYMg
YZn
ScAg
ScAuScCu
ScPd
ScPtScRhScRu
NiAlFeAl
AuZnCuZn
AgMg
CsClCsI
TlBr
TlCl
FIG. 3. Necessary condition (ln δversus ln Cmap) for preferred
slip and APB/SF stability. To the right (left) of the vertical line,
/angbracketleft001/angbracketright(/angbracketleft111/angbracketright) slip is more favorable. Above the slanted line, APBs
are more stable than SFs. Systems in the upper-right region satisfy
the necessary condition for ductility. Note the data shown lie roughly
along a line of slope slightly less than −1.
104107-5RUOSHI SUN AND D. D. JOHNSON PHYSICAL REVIEW B 87, 104107 (2013)
Eq. (8)can be expressed in terms of C. This condition
translates to the vertical line drawn in Fig. 3, where /angbracketleft001/angbracketright
slip (/angbracketleft111/angbracketrightslip) is more favorable for systems lying to the
right (left). Equation (9)translates to the slanted line in
Fig. 3. Therefore, the necessary condition is satisfied in the
upper-right region defined by these two lines, in which /angbracketleft001/angbracketright
is the dominant slip direction and formation of APBs, ratherthan SFs, is preferred. We note that the data in Fig. 3lie along
a line of slope slightly less than −1.
Systems favoring /angbracketleft001/angbracketrightslip include all the ionic com-
pounds, all the Y-based compounds except for YMg, all theSc-based compounds except for ScRu, and NiAl. Notably,ScPt is ductile
2but just barely predicted to be brittle from
our analytic necessary conditions, a result that does dependon the accuracy of the DFT-derived inputs, or the underlyingsimplifications. For example, ScPt in Fig. 3does satisfy the
necessary conditions if k≈4 (rather than the k=2.17 we
assumed for all alloys, but it is in the acceptable range; seeAppendix A). Fork≈4, the APB versus SF slanted line shifts
slightly down but parallel to the k=2.17 line and ScPt falls
in the upper-right region.
Ionic compounds are expected to exhibit /angbracketleft001/angbracketrightslip because
ions encounter lower charge repulsion as they slip along cubicedges rather than cube diagonals. CuZn and FeAl fall on theleft-hand side of the vertical line, showing that /angbracketleft111/angbracketrightslip
is favorable, agreeing with their observed exclusive /angbracketleft111/angbracketright
slip.
44,49–52Interestingly, both /angbracketleft111/angbracketrightslip44and/angbracketleft001/angbracketrightslip53
have been observed in AgMg, with a transition from /angbracketleft111/angbracketright
to/angbracketleft001/angbracketrightslip at low temperatures.6For AuZn ( A∼7.5),
the predicted /angbracketleft111/angbracketrightslip does not agree with the reported
/angbracketleft001/angbracketrightslip44,53and further investigation is required. From
Table Ithe largest errors in our calculated elastic constants
are found in AuCd; we have omitted it from our ductilitymaps. The discrepancies observed in the two Au compoundsmay be caused by the usage of nonrelativistic pseudopo-tentials. It has been shown that relativistic effects play an
important role in the bonding of Au clusters
54and may also
pertain to the study of bulk compounds. For all the othersystems, our simple model predicts /angbracketleft001/angbracketrightversus /angbracketleft111/angbracketrightslip
accurately.
Figure 3also compares the relative stability of SF {1¯10}
and APB {1¯10}. For the ionic compounds, SFs are more stable
than APBs. Charge repulsion in the APB is much higher thanthat in the SF. (See Table IIfor the calculated planar defect
energies.) On the other hand, APBs are relatively stable in boththe Y-based compounds and the classic B2 alloys.
Systems in the upper-right region, as defined by the vertical
and slanted lines, of Fig. 3satisfy the necessary condition for
ductility. In this region, /angbracketleft001/angbracketrightslip is favorable but /angbracketleft111/angbracketrightAPBs
are stable, which means that
a
2/angbracketleft111/angbracketrightpartial dislocations can
coexist with the /angbracketleft111/angbracketright-dissociated perfect /angbracketleft001/angbracketrightdislocations.
Indeed, it has been reported59that/angbracketleft111/angbracketrightdislocations are
metastable in NiAl and that they have been observed in theY-based compounds.
1
The first central finding is that the B2 stability map in
Fig.3identifies candidates for multiple slip, and only a subset
of Y- and Sc-based systems and some others qualify, namely,YAg, YCu, YIn, YRh, YZn, ScAg, ScAu, ScCu, ScPd, (ScPt),ScRh, NiAl, CsI, and (CsCl), where the parentheses reflect aborderline case that should be checked.
B. Sufficient condition
The necessary condition alone cannot predict ductility. The
sufficient condition (Fig. 4)—whether the APBs are bistable
on{1¯10}and{11¯2}planes—must be verified. Dimensionless
ratiosλandMare defined in Eqs. (10) and(11), respectively.
The condition for bistability of APBs is satisfied in regionsII and III, according to Eq. (12). In region II, the {1¯10}APB
has lower energy, and vice versa in region III (see Fig. 4).
0.91.01.11.21.31.41.5
1.0 1.1 1.2 1.3 1.4 1.5λ
MYAgYCuYIn
YRh ScAgScAuScCu
ScPdScRhYZn
NiAlCsI
YMgScPtFeAlAuZnCuZn
AgMgCsCl
TlBrTlCl
YAgYCuI
II
III
FIG. 4. Left: Sufficient condition ( λversus M) map for multiple slip systems [Eq. (12)] that occurs if√
3/2<λ< 2/√
3. Materials not
satisfying the necessary condition are marked by open diamonds. ScPt, being the borderline case in Fig. 3, is marked by a half-filled diamond.
DFT values of YCu and YAg reflecting energies from Ref. 10are indicated by open circles. Right: Schematics showing the relative energy of
the slip systems (after Ref. 48). Below λ=√
3/2≈0.866, only {11¯2}slip is favored.
104107-6STABILITY MAPS TO PREDICT ANOMALOUS DUCTILITY ... PHYSICAL REVIEW B 87, 104107 (2013)
TABLE II. Calculated B2 APB and SF energies (in mJ /m2). The dimensionless parameters C,δ,λ,a n dMare defined in Eqs. (2),(3),(10),
and(11), respectively. Calculated results are listed in the first row of each system. Other calculated results of γAPBandγSFare provided, if
available.
Material γ1¯10
APB γ11¯2
APB λw1¯10
APB/a C Ref. γ1¯10
SF δ Ref.
YAg 641 732 1.14 2.55 0.0468 364 0.569
745 680 0.91 2.16 0.0553 10 305 0.409 55
YCu 757 931 1.23 2.04 0.0585 322 0.425
1030 1090 1.06 1.30 0.0917 10 270 0.262 55
YIn 366 549 1.50 5.33 0.0224 636 1.740
480 55
YRh 1270 1390 1.10 1.29 0.0924 626 0.493
430 55
YMg 277 259 0.93 6.41 0.0186 714 2.58
YZn 558 700 1.25 3.31 0.0360 536 0.961
ScAg 548 598 1.09 3.60 0.0331 437 0.797
ScAu 805 898 1.12 2.59 0.0461 333 0.414ScCu 713 830 1.16 2.88 0.0415 406 0.569
ScPd 832 908 1.09 2.14 0.0557 266 0.320
ScPt 1042 1139 1.09 2.05 0.0581 222 0.213ScRh 1296 1392 1.07 1.59 0.0749 1069 0.825
ScRu 135 506 3.74 11.3 0.0105 1213 8.967
NiAl 777 971 1.25 5.17 0.0231 1379 1.77
815 995 1.22 4.74 0.0252 10 1290 1.58 10
810 990 1.22 4.93 0.0250 56
FeAl 348 403 1.16 12.3 0.0097 1248 3.59
300 820 2.73 14.7 0.0081 56
AuCd 187 223 1.19 11.9 0.0101 639 3.41
AuZn 247 303 1.22 8.47 0.0141 636 2.58CuZn 98 124 1.27 30.2 0.0040 1027 10.5
50 37 0.74 (1.09) 58.1 0.0021 57
a(58a)
AgMg 254 311 1.22 7.73 0.0154 655 2.58
CsCl 496 659 1.33 0.80 0.1500 85 0.172
CsI 357 459 1.29 1.00 0.1191 78 0.219
TlBr 320 397 1.24 1.12 0.1069 49 0.154TlCl 369 458 1.24 0.98 0.1221 51 0.137
aExperiment.
In regions I ( λ> 2/√
3) and IV ( λ<√
3/2, not shown),
respectively, {11¯2}and{1¯10}APBs are unstable. ScRu is
not shown as its λvalue falls out of range (Table II). ScCu
satisfies the necessary condition and just barely does not satisfythe sufficient conditions; this borderline case is sensitive toDFT approximations. For example, a 1 .4% increase in the
γ
1¯10
APB, i.e., from 713 to 723 mJ /m2, in Table IIwould put
ScCu below the bistability line; hence, we include ScCu asductile. The compounds that satisfy both the necessary andsufficient conditions are YAg, YRh, ScAu, ScAg, (ScPt), ScPd,(ScCu), and ScRh. For these B2 materials we predict enhancedductility; all other compounds are predicted to be brittle.
Out of all the B2 systems, only YAg and YCu have been
examined in other DFT calculations.
10There are two notable
things: (1) elastic constants in Table Ifrom our and Morris
et al. ’s results are the same and agree with experiment, but (2)
the defect energies are significantly different in Table II.W e
have been unable to reproduce their defect energies for YCu,and for YAg the values are similar but swapped, changing therelative energies λin Fig. 4. If Morris et al. ’sλvalues forYCu and YAg are plotted in Fig. 4, their locations both shift
downward, with YCu (YAg) now in region II (III), and both(not just YAg) satisfy bistability explaining enhanced ductility.
The second central finding is that the sufficient condition
shown in Fig. 4identifies B2 materials that can exhibit multiple
slip. Systems that do not satisfy the necessary condition areincluded for comparison. Only YAg, [YCu], YRh, ScAu,ScAg, (ScPt), ScPd, (ScCu), and ScRh possess {1¯10}and
{11¯2}bistability, while other candidates lie away from the
bistability region. Bracketed YCu reflects the unresolved DFTvalues. The bistability of APBs explains the observation ofmany/angbracketleft111/angbracketrightdislocations in the ductile Y-based
1and Sc-based
compounds, even though /angbracketleft001/angbracketrightis the dominant slip direction.
VI. DISCUSSION
The systems that satisfy the necessary conditions (Fig. 3
showing dominant /angbracketleft001/angbracketrightslip existing with /angbracketleft111/angbracketrightslip and stable
APBs, not SFs) and the sufficient condition (Fig. 4, regions II
and III, showing APBs having bistable slip) are predicted to
104107-7RUOSHI SUN AND D. D. JOHNSON PHYSICAL REVIEW B 87, 104107 (2013)
have significant enhanced ductility unexpected in B2 systems.
Borderline cases (if using DFT inputs) should be carefullyaddressed. It happens that none of the elastically anisotropic B2materials satisfy both the necessary and sufficient conditions,explaining the observation of brittleness in all the classic B2alloys, which are anisotropic.
7The ductile materials are all
nearly isotropic. Hence, elastic isotropy ( A∼1) should serve
as an indicator for enhanced ductility. It is, however, not aquantitative indicator because YCu is more isotropic than YAg,but YCu is less ductile.
ScRu is predicted to be brittle, which agrees with
experiment.
2ScAg, ScAu, and ScPd are correctly predicted
to be ductile. ScPt is ductile2but is a borderline case barely
not satisfying the necessary condition. And, if we take ScPtto satisfy the necessary condition, then it is ductile becausethe sufficient condition is also satisfied. (If ScPt is taken tosatisfy necessary condition, then so too should CsCl, but itdoes not satisfy the sufficient condition.) The Rh compoundsYRh and ScRh are brittle
2but predicted to be ductile. Further
investigation in this chemical space is required to understandthe source of the discrepancy. As noted, the neglect of a shift ofthe defect planes from the ideal position, the sensitivity to thespecific DFT exchange-correlation functional, or the neglectof other defect formations may change these cases.
Not all Y-based compounds are predicted to be ductile:
B2 YIn does not satisfy the sufficient condition (similar toAuZn, A∼7.5, which means that it is not very isotropically
elastic); YMg does not satisfy the necessary condition, so itis brittle, as found experimentally.
5YIn has been observed
to form a B2 phase.60,61However, YIn has been reported to
crystallize also into a tetragonal phase,62which is ductile.
Our DFT calculations (unpublished) show that YIn has ashallow energy trough versus c/a making it susceptible to
c/adistortion depending on sample treatment. Thus, while the
B2 YIn is brittle from our theory for B2 ductility (using ourDFT results), if c/adistortion occurs, a more general ductility
criterion for the dislocation-defect reactions should be derivedaccounting for c/adependence.
Finally, regarding correlation of measured ductility with
d-electron density of states (DOS), we note that our theory
addresses the ductility criterion based on defect energies andelastic constants that inherently reflect the bonding representedwithin DFT, as did the theory of Liu et al. on yield-strength
anomalies in L1
2compounds.13A similar approach for quan-
titative prediction of twinning in elements and solid-solutionalloys (based on the interacting dislocation and planar-defectarrays in a twin nucleus) also reflect bonding, which canbe correlated directly to the electronic structure.
28As noted
above, a significant (but not absolute) correlation of calculatedd-electron DOS at the Fermi energy was cited for B2 alloys that
were measured to have little to no ductility.
2Importantly, from
our maps for the necessary and sufficient conditions (using ourDFT results), we can predict ductility and, if desired, attemptto correlate behavior with the DFT-derived DOS.
In Fig. 5, we show the DOS for B2 ScRu (brittle), ScRh
(predicted ductile, observed brittle), and ScPd and ScAg (bothductile) in order of increasing electron-per-atom ratio, ore/a. ScRu has the largest d-state DOS at the Fermi level
and correlates with the predicted/observed lack of ductility.ScRh, with its extra electron over ScRu, is in low- d-state DOSbetween bonding and antibonding dstates, which suggests
a crossover in bonding behavior—not incompatible with thepresent results taken in toto . Both ScPd and ScAg are ductile,
with the Fermi energy beginning to climb into higher d-band
DOS and the bonding peak of the d-band DOS falling farther
below the Fermi energy. These results appear to agree withthe results of Gschneidner et al. ,
2but their DOS contains no
detail to make any direct comparison. They argue only that abroad d-band DOS at the Fermi energy explains the lack of
ductility, whereas, for ScPd and ScAg, this d-band feature is
farther below the Fermi energy and accounts for ductility; howit does so is not explained. Of course, that the Fermi energyis entering the DOS with antibonding character should makeductility, i.e., defect formation, more energetically favorable,as inherently represented in the present theory. We cancorrelate our predicted brittleness, brittle-to-ductile crossover,and ductility with the change in DOS features. Thus, we agreethat the magnitude of the d-band DOS at the Fermi energy
can be correlated with ductility if already known, but sucha correlation by itself is not predictive theory. Indeed, thechange in DOS under shear is more relevant, as is knownfor aluminum;
63for example, sbonds under shear become
very directional, giving rise to a large stacking fault energy,as observed. Hence, investigating the behavior of the chargedensity under shear may be more fruitful than DOS.
Finally, for completeness, we note that the underlying
theory for higher ductility is generally more complicated than
the simple dissociations present here as a starting analysis. Thatis, there are key factors determining slip systems, e.g., elasticanisotropy, the “correct” vectors of possible faults in (101)planes (not always corresponding to the usually presumedAPBs), and the energies of these faults. So, it is possiblethat the splittings could be different from the usual APBsconsidered here. As a starting point, if all B2 systems aretreated equally, we assume that there are well-defined APBs
on{1¯10}and{11¯2}planes with the displacement vector
1
2/angbracketleft111/angbracketright
and also a stacking fault on {1¯10}planes with the vector
1
2/angbracketleft001/angbracketright. However, the existence of such metastable faults is
by no means guaranteed—indeed they are system specific.The metastability of such faults is the crucial condition forany further considerations employing standard (anisotropic)dislocation theory. The symmetry does not provide a guaranteeof the stability of these faults. On {1¯10}planes there may
be metastable
1
2/angbracketleft111/angbracketrightfaults in some materials, e.g., CuZn
or FeTi.64However, the vectors corresponding to metastable
faults may differ from1
2/angbracketleft111/angbracketright; for example, simulations using
empirical potentials found in NiAl that the APB with1
2/angbracketleft111/angbracketright
is not actually stable but other faults on {1¯10}planes existed.65
The point is that vectors of the faults on {1¯10}planes vary
from material to material and are by no means the same in allB2 compounds.
Nonetheless, the present stability analysis provides a very
rapid analysis to identify and reduce the number of candidateanomalously ductile B2 systems, and, from which, one canconsider other, more atypical, instabilities. For future work, wecan investigate the metastability of the planar defects addressedin the present simplest theory, since they may be unstable withrespect to other defects on the activated slip planes. If so,the dissociation mechanisms considered here would then be
104107-8STABILITY MAPS TO PREDICT ANOMALOUS DUCTILITY ... PHYSICAL REVIEW B 87, 104107 (2013)
E−EF
(a) ScRu
E−EF
(b) ScRh
E−EF
s
p
d
(c) ScPdE−EF
(d) ScAg
s
p
d
s
p
d
s
p
d
FIG. 5. (Color online) Electronic DOS [relative to Fermi energy EF(ineV), in arbitrary units] for B2 Sc-based alloys in order of (a) Ru,
(b) Rh, (c) Pd, and (d) Ag, i.e., increasing e/a. ScRu is predicted/observed to be brittle, ScPd and ScAg are predicted/observed to be ductile,
and, at the crossover, ScRh is predicted to be ductile but observed to be brittle.
altered, possibly changing the maps and predictions, which,
nonetheless, already appear highly accurate.
VII. CONCLUSIONS
Through solely energy-based criteria for ductility (dis-
location and defect formation), we have addressed /angbracketleft001/angbracketright
versus /angbracketleft111/angbracketrightslip, the relative stability of APBs and SFs,
and the bistability of APB {1¯10}and{11¯2}, which are the
dominant slip modes and defects in B2 systems that canlead to enhanced ductility. Through these criteria, we haveprovided a set of stability maps requiring only ratios ofdefect energies and/or elastic constants, obtained here fromDFT calculations. For design, these maps determine ap r i o r i
whether a B2 material is brittle or ductile and indicatetypical versus enhanced ductility. These maps explain andpredict the enhanced ductility observed (or lack thereof) inRM intermetallic compounds. One may consider temperatureeffects, point defects, or disorder to modify the maps forsystem-specific predictions, as well as more system-specificsuperdislocation reactions that we did not consider.We have examined 23 B2 materials, some of which show
dramatically enhanced ductility, comparable to fcc aluminum.For B2 materials, /angbracketleft001/angbracketrightslip is more favorable than /angbracketleft111/angbracketrightif the
width of APB {1¯10}is less than 6 a. To summarize our results
from the B2 stability maps:
(1) For ionic compounds, only /angbracketleft001/angbracketrightslip is possible,
as the necessary condition for ductility is not satisfied. Ifthe borderline CsI were assumed to satisfy the necessarycondition, the lack of APB bistability would account for itsbrittleness.
(2) For classic B2 alloys, all but NiAl fail the necessary
condition. Again, APBs of NiAl do not possess bistability(multiple slip) so there is no increased ductility.
(3) For Y- and Sc-based compounds, YAg, YRh, ScAg,
ScAu, (ScPt), ScPd, (ScCu), and ScRh satisfy both conditionsfor multiple-slip systems. Thus, we predict them to exhibithigh ductility (observed for YAg). YIn and YMg do not satisfyeither condition, so they are predicted to be brittle (observedin YMg), while B2 YIn has competing tetragonal distortionsthat will affect prediction.
(4) We predicted some systems, such as ScRh, that are duc-
tile but brittle; these appear at a crossover, e.g., between ScRu
104107-9RUOSHI SUN AND D. D. JOHNSON PHYSICAL REVIEW B 87, 104107 (2013)
(brittle) and ScPd (ductile) correctly predicted, suggesting the
more detailed dislocation reaction or computational detailsmay be at issue, not the general theory.
Overall the results are in very good agreement with
experiment and, if desired, can be correlated directly with theunderlying electronic-structure details, as done with other sim-ilar theories for yield-strength anomalies in L1
2compounds or
twinning in fcc metals, because the theory inherently containsall the bonding information within the defect energy and elasticconstants that are needed.
In closing, an energy-based mesoscale dislocation analysis
combined with first-principles calculations accurately char-acterizes permitted slip modes in B2 systems and predictsenhanced ductility due to coexistence of /angbracketleft001/angbracketrightslip and
/angbracketleft111/angbracketrightAPBs, and bistability of APBs on {1¯10}and{11¯2}
planes. The Zener anisotropy ratio can be used to screencandidates for further investigation via these stability maps.Given that ductility is such a complex phenomenon at theatomistic level, it is remarkable that our ductility model (withsome simplifying assumptions) and associated necessary andsufficient conditions give fairly accurate predictions by simplyconsidering dissociation energies of a single superdislocationand the planar energies of the resulting defects.
ACKNOWLEDGMENTS
We thank Karl Gschneidner, Jr. and James Morris for
sharing their results, respectively, on YMg and B33 versus B27stability. We thank Vaclav Vitek for discussions on complexi-ties for more general theory. Funding was from the Departmentof Energy, Basic Energy Sciences, Division of MaterialsScience and Engineering (Grant No. DEFG02-03ER46026)and “materials discovery” seed funding in Ames. The researchwas performed at the Ames Laboratory. The Ames Laboratoryis operated for the US Department of Energy by Iowa StateUniversity under Contract No. DE-AC02-07CH11358. Inpreparation for this work, RS did an undergraduate summerREU with DDJ at Illinois supported partially by the NationalScience Foundation (Grant No. DMR-07-05089).
APPENDIX A: DERIV ATION OF
THE NECESSARY CONDITIONS
We outline the derivation of Eq. (8), i.e.,wAPB/lessorequalslant5.9a.W e
also provide computed lattice constants, elastic constants, andAPB and SF planar defect energies used for the necessary andsufficient condition design maps, with experimental valuesshown for comparison wherever available. Figure 2shows the
unit cells used for the three planar defect calculations.
First, recall that the self-energies of a screw and an edge
dislocation are
66
Es=Gb2
s
4πlnr
r0, (A1)
Ee=Gb2
e
4π(1−ν)lnr
r0, (A2)
where G=c44is the shear modulus; bsandbeare the Burgers
vectors of the screw and edge dislocations, respectively; r0is
the radius of the dislocation core; and ris the cutoff radius
of the dislocation interaction. Note that ris finite because itsstrain field is canceled by the strain field of other dislocations.67
The pure screw-screw and edge-edge interaction energies are66
Ess=Gb2
s
2πlnr
w, (A3)
Eee=Gb2
e
2π(1−ν)lnr
w, (A4)
where wis the separation distance between APBs or SFs.
In Eq. (7a) of the main text, the /angbracketleft111/angbracketrightscrew dislocation
dissociates into perfect dislocations along the cube edges;hence, there are no APBs or SFs. The dissociation doesnot result in any change in the total energy. Thus, withb
a,s=a/angbracketleft111/angbracketrightandb2
a,s=3a2, the total energy of the screw
dislocation is
Ea=Gb2
a,s
4πlnr
r0=3Ga2
4πlnr
r0. (A5)
In Eq. (7c) of the main text, the /angbracketleft111/angbracketrightscrew dislocation
can dissociate into two1
2/angbracketleft111/angbracketrightpartials bounding an APB. The
partials are purely screw, with Burgers vector bc,s=a
2/angbracketleft111/angbracketright
andb2
c,s=3a2/4. Given the separation width wAPBof the
partials, with planar defect energy γAPB, the total energy is
Ec=2Ec,s+Ec,ss+γAPBwAPB. (A6)
We minimize the energy with respect to wAPB, from which
we find that
wAPB=Gb2
c,s
4πγAPB=3Ga2
8πγAPB. (A7)
Then, purely on energy grounds, in order for /angbracketleft001/angbracketright{1¯10}slip
to be more favorable than APB {1¯10}formation, Eain Eq. (A5)
must be less than Ecin Eq. (A6) :
3Ga2
4πlnr
r0<3Ga2
8π/parenleftbigg
lnr
r0+lnr
wAPB+1/parenrightbigg
or
lnwAPB
r0<1. (A8)
The dislocation core is r0=kafor some constant k. Then, by
Eq.(A8) , the condition for /angbracketleft001/angbracketrightslip is
w1¯10
APB
a<k e , (A9)
or, in terms of the planar defect energy, as
γ1¯10
APB
Ga>3
8πke,or (A10a)
lnC> ln/parenleftbigg3
8πke/parenrightbigg
∼−3.126−lnk. (A10b)
The core of the dislocation can be simulated using semiempir-
ical and first-principles calculations, from which the radius ofthe core, and hence k, can be obtained for each B2 material.
In general, r
0has a range68ofbto 5b,s okis between√
3/2
and 5√
3/2 in our case. Eshelby69estimated analytically r0
to be about 1 .5bfor screw dislocations, which, according
to Read,70is an underestimate. A simulation study by Xu
and Moriarty71shows that 2 b, where b=a
2/angbracketleft111/angbracketright, is a good
approximation for r0in bcc Mo. We expect the core radius to
be somewhat larger in B2 systems than in bcc metals, sincebrepresents a partial dislocation in B2 instead of a perfect
104107-10STABILITY MAPS TO PREDICT ANOMALOUS DUCTILITY ... PHYSICAL REVIEW B 87, 104107 (2013)
dislocation in bcc. Thus, we simply take r0to be between
2.5b(k≈2.17) and 5 b(k≈4.33) for all B2 materials. Then
kein Eq. (A9) is between 5.9 and 11.8, which justifies the
criterion imposed by Rachinger and Cottrell. For k=2.17,
we obtained Eq. (8), which gives the vertical line analytically
in Fig. 3at lnC=−3.9 and provides a quantitative measure
for comparison of many systems.
The derivation of the necessary [Eq. (9)] condition, δ>
0.119C−1/4, is more involved. It results from an energy-based
comparison between APB and SF planar defect energies.45
Equation (7b) shows that the a/angbracketleft001/angbracketrightdislocation can further
dissociate intoa
2/angbracketleft001/angbracketrightpartial dislocations, creating a super-
intrinsic stacking fault. The screw and edge components ofthe partial dislocation can be found by projecting
a
2/angbracketleft001/angbracketrightonto
a/angbracketleft111/angbracketright:
a
2/angbracketleft001/angbracketright=a
6/angbracketleft111/angbracketright
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
bb,s+a
6/angbracketleft¯1¯12/angbracketright
/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright
bb,e. (A11)
Thus, with b2
b,s=a2/12 and b2
b,e=a2/6, and using the self-
energies from Eqs. (A1) and(A2) and the interaction energies
from Eqs. (A3) and(A4) , the total energy associated with SF
formation is
Eb=Ga2
24π/parenleftbigg
lnr
r0+lnr
wSF/parenrightbigg
+Ga2
12π(1−ν)/parenleftbigg
lnr
r0+lnr
wSF/parenrightbigg
+γSFwSF.(A12)
Minimizing energy with respect to wSF, the separation width
is found to be
wSF=3−ν
1−νGa2
24πγSF. (A13)
Then Eq. (A12) becomes, with ν=1/3,
Eb=3−ν
1−νGa2
24π/parenleftbigg
lnr
r0+lnr
wSF+1/parenrightbigg
. (A14)
Finally, we compare SF formation [Eq. (7b)] with APB
formation [Eq. (7c)]. Note, we must multiply Ebb y3f o raf a i r
comparison, since each /angbracketleft111/angbracketrightdislocation dissociates into three
families of /angbracketleft001/angbracketrightdislocations, where each of them can create
SFs independently. So, for APBs to be more energeticallyfavorable than SFs, we need E
cin Eq. (A6) to be less than 3 Eb
in Eq. (A14) ; i.e.,
3l nr
wAPB−2ν
1−ν/parenleftbigg
1+lnr
r0/parenrightbigg
<3−ν
1−νlnr
wSF.(A15)
Withr0=ka, and assuming consistently that ν=1/3 and
r=2r0≈4.33a, we get a criterion for APB formation to be
more favorable than SF formation, i.e., Eq. (9).
APPENDIX B: ELASTIC CONSTANTS
FOR BISTABILITY MAP
Here we provide an overview of how compliance elements
Sijare related to standard elastic constants cij. The derivations
can be found by combining information in Refs. 46and47.Bistability is determined by the anisotropic elastic response
of the B2 lattice. Mis a function of cij.46,47LetH=2c44+
c12−c11=(c11−c12)(A−1), where Ais the Zener ratio,
Eq.(1). Rotating the cubic elastic constants to the [111]axis
yields
c/prime
ij=⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎣c
/prime
11c/prime
12c/prime
13 0 c/prime
150
c/prime
12c/prime
11c/prime
13 0−c/prime
150
c/prime
13c/prime
13c/prime
33 000
000 c/prime
44 0−c/prime
15
c/prime
15−c/prime
1500 c/prime
44 0
000 −c/prime
150 c/prime
66⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎦,(B1)
where
c
/prime
11=c11+1
2H, c/prime
12=c12−1
6H,
c/prime
13=c12−1
3H, c/prime
33=c11+2
3H,
c/prime
44=c44−1
3H, c/prime
66=c44−1
6H,
c/prime
15=−√
2
6H.
Note, for an isotropic material, A=1,H=0, and c/prime
ij=cij.
The third row and column are deleted to obtain the inverse,46
yielding
Sij=⎡
⎢⎢⎢⎢⎢⎣S
11S12 0 S15 0
S12S11 0 −S15 0
00 S44 0−2S15
S15−S15 0 S44 0
00 −2S15 0 S66⎤
⎥⎥⎥⎥⎥⎦, (B2)
where
S
11=c/prime
11c/prime
44−c/prime2
15
2(c/prime
11+c/prime
12)/parenleftbig
c/prime
44c/prime
66−c/prime2
15/parenrightbig,
S12=−c/prime
12c/prime
44+c/prime2
15
2(c/prime
11+c/prime
12)/parenleftbig
c/prime
44c/prime
66−c/prime2
15/parenrightbig,
S44=c/prime
66
c/prime
44c/prime
66−c/prime2
15,S 66=c/prime
44
c/prime
44c/prime
66−c/prime2
15,
S15=−c/prime
15
2/parenleftbig
c/prime
44c/prime
66−c/prime2
15/parenrightbig.
Finally, the parameter Mis defined as
M=/radicalBigg
S11S44
S11S44−S2
15. (B3)
As discussed by Sun,48M/greaterorequalslant1, where the equality holds only
for isotropic materials, where A=0 and, hence, H=0.
104107-11RUOSHI SUN AND D. D. JOHNSON PHYSICAL REVIEW B 87, 104107 (2013)
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104107-12 |
PhysRevB.101.054509.pdf | PHYSICAL REVIEW B 101, 054509 (2020)
Robustness of unconventional s-wave superconducting states against disorder
D. C. Cavanagh1,*and P. M. R. Brydon2,†
1Department of Physics, University of Otago, P .O. Box 56, Dunedin 9054, New Zealand
2Department of Physics and MacDiarmid Institute for Advanced Materials and Nanotechnology, University of Otago,
P .O. Box 56, Dunedin 9054, New Zealand
(Received 12 September 2019; revised manuscript received 13 January 2020; accepted 5 February 2020;
published 20 February 2020)
We investigate the robustness against disorder of superconductivity in multiband systems where the fermions
have four internal degrees of freedom. This permits unconventional s-wave pairing states, which may transform
nontrivially under crystal symmetries. Using the self-consistent Born approximation, we develop a general theoryfor the effect of impurities on the critical temperature, and find that the presence of these novel s-wave channels
significantly modifies the conclusions of single-band theories. We apply our theory to two candidate topologicalsuperconductors, YPtBi and Cu
xBi2Se3, and show that the novel s-wave states display an enhanced resilience
against disorder, which extends to momentum-dependent pairing states with the same crystal symmetry. Therobustness of the s-wave states can be quantified in terms of their superconducting fitness, which can be readily
evaluated for model systems.
DOI: 10.1103/PhysRevB.101.054509
I. INTRODUCTION
It is a textbook result that the critical temperature Tcof
a conventional s-wave spin-singlet superconductor is insen-
sitive to nonmagnetic disorder [ 1]. This is a consequence of
Anderson’s theorem [ 2]: Since this state has an isotropic gap
and pairs electrons in time-reversed partner states, there isno depairing effect from the time-reversal-invariant scatteringoff the impurities. On the other hand, the sign-reversing gapsof unconventional superconductors are averaged to zero bythe impurity scattering across the Fermi surface, and thesepairing states are suppressed by weak disorder with normalstate elastic scattering rate (SR) τ
−1∼kBTc.
Recently there has been much interest in s-wave pairing
states which do not pair time-reversed partner states [ 3–10].
This can occur in systems where the electrons have additionaldiscrete degrees of freedom, such as orbital or sublatticeindices. These permit novel ways to satisfy the fermionicantisymmetry of the Cooper pair wave function in a relatives-wave, e.g., a spin-triplet orbital-singlet state. Such pairing
states typically belong to a nontrivial irreducible representa-tion (irrep) of the point group. They have been proposed in avariety of materials [ 3–8], but here we focus on Cu
xBi2Se3[9]
and YPtBi [ 10]. Experiments indicate a fully gapped nematic
superconducting state in Cu xBi2Se3[11–13], which naturally
arises from a time-reversal-invariant combination of the odd-parity s-wave states in the E
uirrep [ 14]. In YPtBi there is
evidence of a nodal superconducting gap [ 15], which could be
explained by a time-reversal symmetry-breaking combinationof even-parity s-wave states which support exotic Bogoliubov
Fermi surfaces [ 10,16].
*david.cavanagh@otago.ac.nz
†philip.brydon@otago.ac.nzSince the novel s-wave states do not pair time-reversed
partners, Anderson’s theorem does not apply and we mayexpect them to be highly sensitive to disorder. Indeed, ex-pressed in a pseudospin band basis, the novel s-wave states
show a sign-changing gap, which averages to zero across theFermi surface [ 10,17]. However, since the impurity potential
in the pseudospin band basis may be anisotropic, the impurity-
averaged gap may not vanish, which can lead to uncon-
ventional impurity effects [ 18,19]. This anisotropy naturally
appears when the states at the Fermi surface have a strongspin-orbital texture. Indeed, it was shown in Ref. [ 20] that the
spin-orbital texture of the electronic states at the Fermi surfacein Cu
xBi2Se3generates such an anisotropy, granting the novel
s-wave A1ustate enhanced robustness against disorder. It is
nevertheless unclear if a general principle underlies this result,or if it applies to other pairing states.
In this paper we use the self-consistent Born approxi-
mation to study the effect of disorder on the critical tem-perature of a superconducting state in a system where thefermions have four degrees of freedom. In Sec. IIwe develop
a general framework which explicitly keeps track of thesedegrees of freedom, naturally generalizing the familiar resultsof single-band theories with disorder [ 1]. Our computation-
ally straightforward approach generalizes and extends earlierworks [ 20–22], and can be readily applied to new materials.
As concrete examples, in Secs. IIIand IVwe apply our
theory to YPtBi and Cu
xBi2Se3, respectively. We find that
nontrivial s-wave states proposed for these systems show a
parametrically enhanced robustness against disorder, whichis shared with other states in the same irrep according totheir similarity to the s-wave states at the Fermi surface. In
the discussion of Sec. V, we show that the robustness of the
s-wave states is quantified in terms of the superconducting
fitness [ 23,24], which can be readily evaluated for model
Hamiltonians. Although robust unconventional states are
2469-9950/2020/101(5)/054509(7) 054509-1 ©2020 American Physical SocietyD. C. CA V ANAGH AND P. M. R. BRYDON PHYSICAL REVIEW B 101, 054509 (2020)
generally possible, systems with a nontrivial inversion oper-
ator are particularly favorable.
II. GENERAL THEORY
Our starting point is a generic model of a fermionic system
with four internal degrees of freedom that is invariant undertime reversal and inversion. The normal-state HamiltonianisH=/summationtext
kc†
kHkck, where ckis a four-component spinor
encoding the internal degrees of freedom, and the matrix Hk
has the general form [ 25],
Hk=/epsilon1k,014+/vector/epsilon1k·/vectorγ, (1)
where 14is the 4 ×4 unit matrix and /vectorγ=
(γ1,γ2,γ3,γ4,γ5) is the vector of the five mutually
anticommuting Euclidean Dirac matrices. The real functions/epsilon1
k,0and/vector/epsilon1k=(/epsilon1k,1,/epsilon1k,2,/epsilon1k,3,/epsilon1k,4,/epsilon1k,5) are the coefficients of
these matrices. The Hamiltonian in Eq. ( 1) has the doubly
degenerate eigenvalues Ek,±=/epsilon1k,0±|/vector/epsilon1k|. The internal
degrees of freedom can either transform trivially ( I=14)
or nontrivially ( I=γ1) under inversion. The time-reversal
operator is T=UTK, where Kis complex conjugation and
the unitary part can be expressed in terms of the EuclideanDirac matrices without loss of generality as U
T=γ3γ5.
The pairing potential for a general superconducting state is
/Delta1k=/Delta10˜/Delta1kwhere /Delta10is the magnitude and
˜/Delta1k=fkγαγβUT. (2)
Here fkis a normalized form factor, chosen such that
fermionic antisymmetry ˜/Delta1k=− ˜/Delta1T
−kis satisfied. Because the
pairing potential ˜/Delta1kis a 4 ×4 matrix, there are six terms
in Eq. ( 2)f o rw h i c ha n s-wave form factor (i.e., fk=1) is
permitted by fermionic antisymmetry. This is always possi-ble for α=β=0 (where γ
0=14), which describes pairing
between electrons in time-reversed partner states, and hencegeneralizes the usual s-wave spin-singlet state. The five other
channels where an s-wave form factor is allowed have a
nontrivial dependence on the internal degrees of freedom,where αandβin Eq. ( 2) are different and not both zero. These
additional s-wave channels typically belong to nontrivial ir-
reps.
The nontrivial s-wave channels do not generally pair elec-
trons in time-reversed partner states, and hence typicallyinvolve both intraband and interband pairing. To quantifythe degree of interband pairing for a pairing state ˜/Delta1
kat
wave vector k,R e f .[ 24] introduced the quantity FC(k)=
1
4Tr{|Hk˜/Delta1k−˜/Delta1kHT
−k|2}, where Hk˜/Delta1k−˜/Delta1kHT
−kis referred
to as the “superconducting fitness” [ 23] and is vanishing if
there is no interband pairing. The superconducting fitnessalso controls the form of the superconducting gap in the low-energy spectrum. Specifically, the s-wave states (i.e., ˜/Delta1
k=˜/Delta1)
open a gap of magnitude [ 26],
/Delta10/radicalBig
1−˜FC(k), (3)
where ˜FC(k)=4FC(k)/|/vector/epsilon1k|2Tr{˜/Delta1˜/Delta1†}is normalized such that
˜FC(k)/lessorequalslant1. If ˜FC(k)=1, there is no intraband pairing, and so
thes-wave states must necessarily display a gap node. Since
the spin-singlet analog state is perfectly fit [i.e., ˜FC(k)=0],
it hence opens a full gap and there is no interband pairing, asanticipated by the fact that it pairs time-reversed partners. In
contrast, the nontrivial matrix structure of the anomalous s-
wave states typically results in a nonzero fitness and possiblythe formation of nodes.
We consider isotropic scattering off potential impurities
distributed randomly at positions r
j, described by the Hamil-
tonian,
Himp=V
/Omega1/summationdisplay
j/summationdisplay
k,k/primeei(k/prime−k)·rjc†
kck/prime, (4)
where Vis the impurity potential and /Omega1is the volume. We
restrict ourselves here to the use of a scattering potentialthat is isotropic in the spin and orbital indices, as is thestandard approach for nonmagnetic impurities [ 1,20,21]. Al-
though more complicated impurity potentials are possible insystems with orbital degrees of freedom [ 8,27], our intention
here is to understand the relationship between the spin-orbitaltexture of the normal-state bands and the robustness of thesuperconducting state to disorder. To this end, we focus onthe simplest possible scattering potential in the spin-orbitalbasis. This simplification does not imply that intra- andinterband scattering processes are equivalent, however, assuch processes depend on matrix elements introduced by thetransformation to the band basis. Within the self-consistentBorn approximation, the Green’s functions of the disorderedsystem are
¯G(k,iω
n)=/summationdisplay
j=±1
i˜ωn,j−Ek,jPk,j, (5)
where Pk,±=1
2(14±ˆ/epsilon1k·/vectorγ) projects into the ±band at
momentum kand ˆ/epsilon1k=/vector/epsilon1k/|/vector/epsilon1k|. The effect of impurities is
accounted for in the renormalized Matsubara frequencies
˜ωn,j=ωn−(2τk,j)−1sgn(ωn), where the SR in band jis
1
τk,j=πnimpV2/summationdisplay
m=±Nm(1+jmˆ/epsilon1k·/angbracketleftˆ/epsilon1k/angbracketrightFS,m). (6)
Here nimpis the concentration of impurities, Nmis the density
of states of band m=± at the Fermi surface, and /angbracketleft.../angbracketrightFS,m
denotes the average over the Fermi surface of this band. The
second term in the parentheses of Eq. ( 6) is an additional
contribution to the scattering rate which arises from a netaverage polarization in the internal degrees of freedom on the(single band) Fermi surface. In the following we will assumea weak momentum dependence of the SR and replace τ
−1
k,jby
its Fermi surface average in Eq. ( 5).
The critical temperature in the presence of disorder can
be determined from the lowest-order terms of the Ginzburg-Landau free energy in the Born approximation, expanded inpowers of the gap [ 27,28],
F
2=|/Delta10|2
gν+1
2β/summationdisplay
iωn/integraldisplayd3k
(2π)3Tr{/Delta1†
k¯G(k,iωn)(/Delta1k+/Delta10/Sigma12)
ׯGh(k,iωn)}, (7)
where ¯Gh(k,iωn)=¯GT(−k,iωn) is the Green’s function for
the holes, gν<0 is the attractive interaction in a particular
superconducting channel ν, and /Sigma12is the anomalous self-
energy due to the impurity scattering. The critical temperature
054509-2ROBUSTNESS OF UNCONVENTIONAL s-WA VE … PHYSICAL REVIEW B 101, 054509 (2020)
FIG. 1. Diagrammatic form of the linearized gap equation, tak-
ing Cooperon ladder diagrams into account. The dotted line repre-
sents the interaction with the impurity, denoted by the star, and the
double line is the Green’s function dressed by interactions with theimpurity via the normal self-energy.
is found by minimizing the free energy with respect to /Delta1∗.
Canceling an overall factor of the gap magnitude /Delta10gives an
expression for the linearized gap equation,
1
gν=1
2β/summationdisplay
iωn/integraldisplayd3k
(2π)3Tr{˜/Delta1†
k¯G(k,iωn)(˜/Delta1k+/Sigma12)¯Gh(k,iωn)},
(8)
which includes the Cooperon ladder diagrams (see Fig. 1)v i a
the anomalous self-energy.
The self-energy obeys the self-consistency equation de-
fined diagrammatically in Fig. 2,
/Sigma12=−nimpV2/integraldisplayd3k
(2π)3¯G(k,iωn)(˜/Delta1k+/Sigma12)¯Gh(k,iωn).(9)
The anomalous self-energy vanishes unless the lowest-order
contribution is nonzero:
/Sigma1(0)
2=−nimpV2/integraldisplayd3k
(2π)3¯G(k,iωn)˜/Delta1k¯Gh(k,iωn)
=πnimpV2/summationdisplay
j=±Nj
|˜ωn,j|/angbracketleftbig
Pk,j˜/Delta1kPT
−k,j/angbracketrightbig
FS,j, (10)
where, in the final line, we have made the assumption that
the bands are well separated and interband contributions tothe self-energy are therefore small and can be neglected.Equation ( 10) is the central result of our analysis. Because
of the nontrivial form of the projection operators, the
FIG. 2. Diagrammatic form of the anomalous self-energy in the
self-consistent Born approximation.Fermi-surface average will not necessarily vanish for an
unconventional state. Although our theory has been developedfor a two-band model, this result readily generalizes toan arbitrary number of bands. For the two-band systemconsidered here, explicitly evaluating Eq. ( 10) for the general
pairing state Eq. ( 2) yields
/Sigma1
(0)
2=πnimpV2/summationdisplay
j=±Nj
4|˜ωn,j|
×⎡
⎣/angbracketleftfk/angbracketrightFS,jγαγβ+j5/summationdisplay
l=1/angbracketleftfkˆ/epsilon1k,l/angbracketrightFS,j{γαγβ,γl}
+5/summationdisplay
l,m=1/angbracketleftfkˆ/epsilon1k,lˆ/epsilon1k,m/angbracketrightFS,jγlγαγβγm⎤
⎦UT. (11)
Due to the Fermi surface averages of the form-factor fkwith
the coefficients of the γmatrices in the Hamiltonian Eq. ( 1),
the self-energy may be nonzero even for nontrivial formfactors. Moreover, we observe that since /Sigma1
(0)
2(and hence
/Sigma12) is independent of momentum, it must belong to one of
thes-wave channels, and can thus be nonzero for any state
in the same irrep. This modifies the solution of Eq. ( 8) such
that these states acquire some protection against the disorder.This represents the crucial difference to the single-band case,where the trivial form of the projection operators impliesthat the anomalous self-energy vanishes for any state with asign-changing gap, and the critical temperature of these statesis suppressed in a universal fashion [ 1].
Although our theory applies to a general two-band system,
the analysis of systems with multiple Fermi surfaces is com-plicated. To more clearly reveal the universal physics due tothe spin-orbital texture, therefore, in the following we studytwo examples of the simpler case where only one of the bandsintersects the Fermi energy.
III. APPLICATION TO YPtBi
YPtBi is a zero-band-gap semimetal, where the states close
to the Fermi energy belong to the /Gamma18band. Ignoring a weak
antisymmetric spin-orbit coupling due to the broken inversionsymmetry [ 10], this is described by the Luttinger-Kohn model
for the j=
3
2states in a cubic material,
H=(α|k|2−μ)14+β1/summationdisplay
ik2
iJ2
i+β2/summationdisplay
i/negationslash=i/primekiki/primeJiJi/prime,(12)
where iand i/primeenumerate the Cartesian coordinates.
The j=3
2internal angular momentum of the electrons
constitutes the four degrees of freedom in ourgeneral model, and the γmatrices in Eq. ( 1) can
be parametrized as /vectorγ=(
1√
3(J2
x−J2
y),1
3(2J2
z−J2
x−
J2
y),1√
3{Jy,Jz},1√
3{Jx,Jy},1√
3{Jx,Jz}) with /vector/epsilon1k=(√
3β1(k2
x−
k2
y)/2,β1(3k2
z−|k|2)/2,√
3β2kykz,√
3β2kxky,√
3β2kxkz).
The j=3
2index transforms trivially under inversion.
Experiments show holelike carriers in YPtBi, and so weset the chemical potential to lie in the lower band.
The six s-wave pairing states in YPtBi are tabulated in
Table I. Apart from the A
1gsinglet state, there are also five
quintet states which pair electrons with total internal angular
054509-3D. C. CA V ANAGH AND P. M. R. BRYDON PHYSICAL REVIEW B 101, 054509 (2020)
TABLE I. The six s-wave pairing states for YPtBi. The first line
gives the irrep of Oh, the second line gives the form of the pairing
potential in terms of the γmatrices defined in the text, the third line
gives the nodal structure, and the fourth and fifth lines give the valuesoflcorresponding to the γmatrices for which λ
l=1a n dλl=−1,
respectively.
irrep A1g Eg T2g
˜/Delta1U†
T 14 γ1γ2γ3γ4γ5
Nodes None Line Line Line Line Line
l,λl=1 A l l 12345
l,λl=−1 None 2,3,4,5 1,3,4,5 1,2,4,5 1,2,3,5 1,2,3,4
momentum J=2, and which belong to the EgandT2girreps.
Evaluating Eq. ( 11), we find that the lowest-order contribution
to the anomalous self-energy for the s-wave gaps in YPtBi is
/Sigma1(0)
2=πnimpV2N
4|˜ωn|/bracketleftBigg
1+5/summationdisplay
l=1λl/angbracketleftbig
ˆ/epsilon12
k,l/angbracketrightbig
FS/bracketrightBigg
˜/Delta1, (13)
where λl=±1 is tabulated for each channel in Table I.
Solving Eq. ( 9) we obtain the full self-energy,
/Sigma12=¯/Sigma1(0)
2
1−¯/Sigma1(0)
2˜/Delta1, (14)
where /Sigma1(0)
2=¯/Sigma1(0)
2˜/Delta1. Inserting this into the linearized gap
equation, we find that the critical temperature Tcof the s-wave
state in channel νis given by the solution of
log/parenleftbiggTc
Tc0/parenrightbigg
=ψ/parenleftbigg1
2/parenrightbigg
−ψ/parenleftbigg1
2+1
4πkBTcτν/parenrightbigg
, (15)
where Tc0is the critical temperature in the absence of disorder,
ψ(z) is the digramma function, and the effective SR is
1
τν=1
2τ0/parenleftBigg
1−5/summationdisplay
l=1λl/angbracketleftbig
ˆ/epsilon12
k,l/angbracketrightbig
FS/parenrightBigg
, (16)
withτ−1
0=2πnimpV2N. We see that λl=+1 decreases the
effective SR, whereas λl=−1 brings it closer to the normal-
state value τ=τ0. Since all λl=1f o rt h e A1gs-wave state,
we find that τ−1
A1g=0 and it is hence insensitive to disorder,
consistent with Anderson’s theorem. The effective SR of the
other s-wave states are reduced relative to the normal state
value, as in each case there is one lfor which λl=+1. This
gives a modest degree of protection against disorder, as shownin Fig. 3.
The enhanced stability of the nontrivial s-wave states ex-
tends to other pairing potentials: the critical temperature foran arbitrary state ˜/Delta1
ksatisfies
log/parenleftbiggTc
Tc0/parenrightbigg
=ψ/parenleftbigg1
2/parenrightbigg
−(1−αν(˜/Delta1k))ψ/parenleftbigg1
2+1
4πkBTcτ0/parenrightbigg
−αν(˜/Delta1k)ψ/parenleftbigg1
2+1
4πkBTcτν/parenrightbigg
, (17)
where
αν(˜/Delta1k)=/angbracketleftTr{˜/Delta1†
kPk˜/Delta1νPk}/angbracketright2
FS
/angbracketleftTr{˜/Delta1†
kPk˜/Delta1kPk}/angbracketrightFS/angbracketleftTr{˜/Delta1†
νPk˜/Delta1νPk}/angbracketrightFS.(18)0.0 0.2 0.4 0.6 0.80.00.20.40.60.81.0
FIG. 3. Critical temperature Tcfor various gaps in the EgandT2g
irreps as a function of the disorder strength nimpπV2Nin YPtBi. The
lineτν=τcorresponds to the case where the effective SR in Eq. ( 15)
is equal to the normal-state SR, which applies to pairing states inall other nontrivial irreps. We use parameters for the normal-state
Hamiltonian Eq. ( 12)f r o mR e f .[ 10].
This parameter measures the similarity of ˜/Delta1kto the s-wave
state ˜/Delta1νat the Fermi surface. The closer ανis to one, the
more similar these states are to one another, and hence theirresponse to disorder is also similar. In this way, a general statein an irrep with a nontrivial s-wave pairing potential can also
acquire some robustness against disorder. Indeed, as shownin Fig. 3, the singlet d-wave E
gstate ˜/Delta1k=(ˆk2
x−ˆk2
y)UTis
almost as stable against disorder as the quintet s-wave Eg
states, reflecting the nearly identical form of these states at
the Fermi surface. It is instructive to examine the lowest-ordercontribution to the anomalous self-energy for this state. Inparticular, the second term inside the brackets of Eq. ( 11)
gives the overlap with the s-wave E
gstateγ1UT:
/Sigma1(0)
2=−πnimpV2N
2|˜ωn|/angbracketleftbig
ˆ/epsilon1k,1/parenleftbigˆk2
x−ˆk2
y/parenrightbig/angbracketrightbig
FSγ1UT. (19)
This is nonzero since /epsilon1k,1=√
3β1(k2
x−k2
y)/2. The full
anomalous self-energy will have the same form as Eq. ( 14),
where ¯/Sigma1(0)
2is the coefficient of γ1UTin the expression above.
IV . APPLICATION TO Cu xBi2Se3
The low-energy electron states in Cu xBi2Se3derive from
pz-like orbitals which are located on opposite sides of each
Bi2Se3quintuple layer, implying a sublattice degree of free-
dom. The k·pHamiltonian for these states to lowest order in
kfor each term is given by [ 29]
H=−μσ0⊗η0+mσ0⊗ηx+vzkzσ0⊗ηy
+v(kxσy−kyσx)⊗ηz+λkx/parenleftbig
k2
x−3k2
y/parenrightbig
σz⊗ηz,(20)
where σνandηνare the Pauli matrices in spin and sublattice
space, respectively. We choose the γmatrices to be /vectorγ=(σ0⊗
ηx,σ0⊗ηy,σx⊗ηz,σy⊗ηz,σz⊗ηz). The copper intercala-
tion in Cu xBi2Se3dopes electrons into the system, giving a
Fermi surface in the upper band.
The six s-wave pairing channels in Cu xBi2Se3are summa-
rized in Table II: In addition to two A1gstates, there are four
odd-parity states, which are permitted due to the swappingof the sublattice index under inversion. The A
1gstates are
insensitive to disorder [ 20], although the analysis is more
054509-4ROBUSTNESS OF UNCONVENTIONAL s-WA VE … PHYSICAL REVIEW B 101, 054509 (2020)
TABLE II. The six s-wave pairing states for Cu xBi2Se3.T h efi r s t
line gives the irrep of D3d, and the second line gives the form of the
pairing potential in terms of the γmatrices defined in the text. The
third line gives the nodal structure, while the fourth and fifth linesgive the values of lcorresponding to the γmatrices for which λ
l=1
andλl=−1, respectively.
irrep A1g A1g A1u A2u Eu
˜/Delta1U†
T 14 γ1iγ1γ5iγ1γ2iγ1γ3iγ1γ4
Nodes None None None Point Point None
l,λl=1 All 1 2,3,4 3,4,5 2,4,5 2,3,5
l,λl=−1 None 2,3,4,5 1,5 1,2 1,3 1,4
involved than for YPtBi since the anomalous self-energy
includes components from both pairing potentials. The criticaltemperatures of the odd-parity channel νis the solution of
Eq. ( 15) where the effective SR is
1
τν=1
4τ0/parenleftBigg
1+2/angbracketleftˆ/epsilon1k,1/angbracketright2
FS−5/summationdisplay
l=1λl/angbracketleftbig
ˆ/epsilon12
k,l/angbracketrightbig
FS/parenrightBigg
, (21)
and the λlare tabulated in Table II. The second term in the
brackets is due to the nonzero sublattice polarization of thenormal-state bands arising from the mass term. Our resultis consistent with the analysis for the A
1uchannel in [ 20].
We plot the critical temperature as a function of disorderstrength for each channel in Fig. 4. Other odd-parity states in
Cu
xBi2Se3also enjoy some degree of protection against dis-
order. In particular, a nontrivial dependence on the sublatticedegrees of freedom is not required. For example, consider thetwop-wave spin-triplet sublattice-trivial pairing states in A
1u:
A(a)
1u:˜/Delta1k=ˆkzσx⊗η0, (22)
A(b)
1u:˜/Delta1k=− ˆkxσz⊗η0+ˆkyiσ0⊗η0. (23)
As shown in Fig. 4, the robustness of these p-wave states is
comparable to the A2uandEus-wave states, because of their
overlap with the significantly more stable A1us-wave state.
0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0
FIG. 4. Critical temperature Tcfor various gaps in the A1u,A2u,
and Euirreps as a function of the disorder strength nimpπV2N
in Cu xBi2Se3. The line τν=τcorresponds to the case where the
effective SR in Eq. ( 15) is equal to the normal-state SR, which applies
to pairing states in all other nontrivial irreps. We use parameters
for the normal-state Hamiltonian Eq. ( 20) from Ref. [ 29]a n ds e t
μ=0.5e V[ 30].V . DISCUSSION
Our analysis reveals a remarkable robustness of the non-
trivial s-wave pairing states against disorder, which is mani-
fested by an effective SR which can be greatly reduced fromthe normal-state SR. The s-wave states play a crucial role, as
their robustness can be shared with, but not exceeded by, anyother state in the same irrep.
Since they do not exclusively pair time-reversed partners,
the nontrivial s-wave states may involve both pairing of elec-
trons in the same (intraband pairing) and different (interbandpairing) bands. Remarkably, the effective SR of the s-wave
state in channel νcan be expressed in terms of the Fermi
surface average of the superconducting fitness function ˜F
C(k),
which measures the degree of interband pairing:
1
τν=1
τ−1
τ0(1−˜FC). (24)
Here ˜FC=/angbracketleft˜FC(k)/angbracketrightFS=1 (0) implies completely interband
(intraband) pairing across the Fermi surface. The effective SRis reduced, and hence the robustness against disorder is en-hanced, according to the degree that the s-wave state involves
intraband pairing. This result follows from the observationthatλ
l=+1(−1) when γl˜/Delta1ν−˜/Delta1νγl,∗=0( 2γl˜/Delta1ν). We
emphasize that Eq. ( 24)only applies to the s-wave states: for
other states, the value of ˜FCdoes not supply any information
about the robustness against disorder.
The extreme limit where a nontrivial s-wave potential ˜/Delta1νis
perfectly fit (i.e., ˜FC=0) is instructive. As shown in [ 9], the
Bogoliubov–de Gennes Hamiltonian can then be mapped tothat for the trivial s-wave state using c
k→exp(iπ
4˜/Delta1νU†
T)ck.
This global transformation leaves the impurity HamiltonianEq. ( 4) invariant, and so the nontrivial s-wave state is in-
sensitive to nonmagnetic disorder, giving a generalization ofAnderson’s theorem [ 22]. The Hamiltonian will generally
contain terms which violate the fitness condition (i.e., ˜F
C>
0), however, which spoils this correspondence. Nevertheless,the nontrivial s-wave state will retain some robustness against
disorder.
This effect is very sensitive to the material parameters. For
example, it is known that the robustness of the odd-paritys-wave states in Cu
xBi2Se3is enhanced by reducing the mass
term min Eq. ( 20)[20,21]. This is immediately evident in
our framework, where the effective scattering rate is alwaysenhanced by a finite mass. For the odd-parity s-wave gaps,
˜/Delta1=iγ
1γjUT, the enhancement is τ0/τν=ˆm2+/angbracketleftˆ/epsilon12
k,j/angbracketrightFS/2,
and the A1ustate is the most stable as /angbracketleftˆ/epsilon12
k,5/angbracketrightFSis the smallest
component of the Hamiltonian. This is a direct consequence ofthe fact that the mass term in the Hamiltonian is proportionalto the nontrivial inversion symmetry operator I=γ
1, and
thus the odd parity gaps must by definition have λ1=−1.
Equation ( 24) gives a simple diagnostic for the existence
of a highly robust nontrivial irrep in a general system: theremust be an s-wave state in this irrep such that ˜F
C/lessmuch1. A
nontrivial inversion operator is highly desirable: In this case,the odd-parity s-wave states involve the product of two γ
matrices (one of which is the inversion operator), and hencecommute with three γmatrices in the general Hamiltonian
Eq. ( 1). In contrast, the even-parity s-wave states commute
with only one γmatrix in the Hamiltonian when inversion is
054509-5D. C. CA V ANAGH AND P. M. R. BRYDON PHYSICAL REVIEW B 101, 054509 (2020)
trivial. Assuming roughly equal values of all the coefficients
/epsilon1k,lat the Fermi surface, ˜FCwill typically be smaller for the
s-wave states in the system with nontrivial inversion. This is
exemplified by the greater robustness of the s-wave states in
CuxBi2Se3compared to YPtBi.
The analysis presented above has focused entirely on un-
derstanding the role of the spin-orbital texture of the normal-state bands. The impurity physics of superconductors is arich field [ 31], and although the self-consistent Born approx-
imation utilized here can successfully account for the pair-breaking physics in the dilute impurity limit, effects beyondthis approximation can be important. For example, it hasrecently been shown that the enhancement of the local densityof states due to the presence of resonant levels at the impuritysites can increase the critical temperature in unconventionalmultiorbital superconductors above the clean-limit result [ 32].
We nevertheless expect our results to remain qualitativelyvalid for more sophisticated treatments, as the spin-orbitaltexture and the superconducting fitness are properties of theclean-limit Bogoliubov–de Gennes equations. Indeed, the roleof the mass term in controlling the robustness against dis-order in Cu
xBi2Se3has been numerically confirmed using a
self-consistent T-matrix theory [ 21]. Extending our theory
beyond the self-consistent Born approximation is a promisingdirection for future work.
During final preparation of our manuscript we became
aware of a similar analysis in Ref. [ 26]. However, our re-
sults for the effective SR disagree: Whereas we find thatthis involves the superconducting fitness with respect to thenormal-state Hamiltonian, in Ref. [ 26] the superconducting
fitness with respect to the impurity Hamiltonian appears. Thisgives a complete insensitivity of the pairing state to disorder,in contrast to the parametric enhancement of the robustnessfound here, and disagrees with previous studies [ 20–22].
VI. CONCLUSIONS
In this manuscript we have shown that unconventional
superconducting states in multiband systems are genericallyless sensitive to the presence of nonmagnetic disorder thanunconventional states in single-band materials, due to thespin-orbital texture of the normal-state bands. The enhancedstability occurs for pairing states in irreps for which there is anontrivial s-wave state. The degree to which an s-wave state is
robust against disorder can be quantified in terms of the Fermisurface average of the superconducting fitness parameter, andprovides an upper bound for the stability of all other statesin the same irrep. Our theory offers a straightforward wayto assess the robustness against disorder of unconventionalpairing states for any multiband system, and can thus guidethe search for novel superconducting states.
ACKNOWLEDGMENTS
This work was supported by the Marsden Fund Council
from Government funding, managed by Royal Society TeAp¯arangi.
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054509-7 |
PhysRevB.72.233103.pdf | Quantum dynamics of spins coupled by electrons in a one-dimensional channel
Dmitry Mozyrsky,1Alexander Dementsov,2and Vladimir Privman2
1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
2Department of Physics, Clarkson University, Potsdam, New York 13699, USA
/H20849Received 19 September 2005; published 9 December 2005 /H20850
We develop a unified theoretical description of the induced interaction and quantum noise in a system of two
spins /H20849qubits /H20850coupled via a quasi-one-dimensional electron gas in the Luttinger liquid regime. Our results
allow evaluation of the degree of coherence in quantum dynamics driven by the induced indirect exchangeinteraction of localized magnetic moments due to conduction electrons, in channel geometries recently experi-mentally studied for qubit control and measurement.
DOI: 10.1103/PhysRevB.72.233103 PACS number /H20849s/H20850: 85.85. /H11001j, 05.30. /H11002d, 05.60.Gg
Recently, there has been much interest in coherent quan-
tum dynamics of coupled two-level systems /H20849qubits /H20850for
quantum information processing. Realizations are soughtsuch that qubit-qubit interactions can be externally controlledover short time scales of quantum “gate functions,” in theparameter regime ensuring that relaxation and decoherenceare negligibly small over a large number of gate cycles.There have been several proposals for qubit systems in semi-conductor heterostructures, with direct coupling,
1typically
via shared electron wave functions, or indirect coupling, spe-cifically via excitations of the conduction electron gas.
2In
the latter approaches, the medium that induces the indirectinteraction, can also act as a “heat bath” resulting in relax-ation and decoherence. Usually, strongly correlated, low-temperature conditions have been assumed
3in order to en-
sure high degree of coherence. In this work we, for the firsttime, develop a unified theoretical derivation of theRuderman-Kittel-Kasuya-Yosida /H20849RKKY /H20850type
4induced in-
teraction incorporating the description of relaxation effectsresulting from the electron gas “bath.”
A recent experiment
5on coupled quantum dots has dem-
onstrated the realizability of indirect interaction for controlof two-qubit dynamics. Several experimental setups
6suggest
that a quasi-one-dimensional /H208491D/H20850channel geometry for the
conduction electron gas is promising for quantum measure-ment required for quantum computing. Furthermore, there isexperimental evidence
7of Luttinger liquid behavior in elec-
tron transport in quasi-1D structures. Therefore, we are goingto consider the 1D-channel qubit-qubit coupling via indirectRKKY interaction mediated by Luttinger liquid of electrons,
and we assume spin-
1
2qubits.
It has been commonly accepted8that the ground state ex-
citations of a 1D interacting electron gas within the Luttingerliquid model can be described by the following Hamiltonian:
H
0=/H20858
i=c,svi
4/H9266/H20885dx/H20851gi/H20849/H11509x/H9278i/H208502+gi−1/H20849/H11509x/H9258i/H208502/H20852. /H208491/H20850
The phase fields, /H9278c/H20849s/H20850/H20849x/H20850and/H9258c/H20849s/H20850/H20849x/H20850, with subscript indices
describing charge and spin degrees of freedom, respectively,
obey commutation relations /H20851/H11509x/H9278c/H20849s/H20850/H20849x/H20850,/H9258c/H20849s/H20850/H20849x/H11032/H20850/H20852=2/H9266i/H9254/H20849x
−x/H11032/H20850. Consequently, /H9258and/H11509x/H9278can be viewed as canonical
variables. Here and in what follows we set /H6036=1 and kB=1.The Hamiltonian has a simple additive structure as a result of
spin-charge separation in 1D systems, with the charge andspin density waves of the liquid having, generally speaking,different velocities,
vc=vF/gcand vs=vF/gs, respectively,
where vFis the Fermi velocity. The constant gc/H110220 accounts
for the electron-electron interaction and is related to the pa-rameters of the Hubbard model
9as follows: gc/H11229/H208491
+U/2EF/H20850−1/2, where EF=vFkF/2 is the Fermi energy, kFis
the Fermi momentum, and Uis the effective interaction be-
tween the electrons, U/H11011e2/a, where ais the short distance
cutoff, a/H11011kF−1. Also, we assume rotational symmetry, SU /H208492/H20850,
in the spin space,9–11which implies that gs=1.
The localized magnetic moments /H20849spins /H20850are coupled to
conduction electrons via the contact interaction,
Hint=/H20858
jJjs/H20849xj/H20850·Sj. /H208492/H20850
Here j=1,2 labels impurity spins Sjpositioned at xj,Jjare
the exchange coupling constants, and s/H20849x/H20850is the local elec-
tron spin density. The spin density can be explicitly ex-
pressed in terms of the Luttinger phase fields, see Refs. 12and 13,
s
z=/H11509x/H9258s
2/H9266+/H9268z
/H9266acos/H208492kFx+/H9258c/H20850cos/H9258s, /H208493/H20850
s±=e±i/H9278s
/H9266a/H20851±i/H9268ycos/H9258s+/H9268xcos/H208492kFx+/H9258c/H20850/H20852, /H208494/H20850
where /H9268x,y,zare the Pauli matrixes. The Luttinger liquid de-
scription of the problem is generally valid in the “hydrody-namic limit” of spin separations x=/H20841x
1−x2/H20841/H11271a.
Our goal is to obtain an effective description of the dy-
namics of the system of two spins, with electronic degrees offreedom integrated out. We first consider the equilibrium par-tition function of the system, defined as Z=Tr /H20851exp/H20849−
/H9252H/H20850/H20852,
where /H9252=1/Tis the inverse temperature and H=H0+Hint.
The partition function, Z, can be expressed in terms of the
spin- and temperature-dependent effective action Seff, see
Ref. 14, Z=/H20848DS1DS2exp/H20851−Seff/H20849S1,S2/H20850/H20852. As usual, the evalu-
ation of the effective action can only be carried out pertur-
batively in the couplings Jjof the spins to the electrons. The
leading nonvanishing contribution is generated byPHYSICAL REVIEW B 72, 233103 /H208492005 /H20850
1098-0121/2005/72 /H2084923/H20850/233103 /H208494/H20850/$23.00 ©2005 The American Physical Society 233103-1/H208491/2/H20850/H208480/H9252d/H92701d/H92702/H20855THint/H20849/H92701/H20850Hint/H20849/H92702/H20850/H20856H0, where Tstands for Mat-
subara time ordering,15and the equilibrium averaging is
taken with respect to the noninteracting Hamiltonian H0. The
perturbative approach is, generally speaking, invalid at suf-ficiently large Matsubara time scales, i.e., low temperatures,as evident from the perturbative RG analysis carried out inRefs. 11 and 12. At sufficiently low temperatures, spin dy-namics results in a nontrivial strong coupling fixed point, i.e.,Kondo effect. For quantum information processing, we are
interested in qubits that retain their localized spin-
1
2inden-
tity. Therefore, we limit our consideration to temperaturesT/H11022T
Kondo.
The resulting Matsubara action is Seff=SBerry+Sself. Here
SBerry is the Berry action term for free spins.16The self-
action for the spins is given by
Sself=J2
2/H9252/H20858
i/H9275n/H20851/H9273/H20849/H9275n,0/H20850S1/H20849i/H9275n/H20850·S1/H20849−i/H9275n/H20850
+/H9273/H20849/H9275n,x/H20850S1/H20849i/H9275n/H20850·S2/H20849−i/H9275n/H20850+/H20849S1↔S2/H20850/H20852. /H208495/H20850
In/H208495/H20850,/H9273/H20849/H9275n,x/H20850is the spin-spin correlation function of the
electron gas /H20849in the imaginary time represenation /H20850, and /H9275n
=2/H9266n//H9252are the Matsubara frequencies. For simplicity, in /H208495/H20850
and in the following we assume that J1=J2=J. The correla-
tion function /H9273/H20849/H9275n,x/H20850, which generally speaking is a tensor,
/H20855si/H20849−i/H9275n,x/H20850sj/H20849i/H9275n,0/H20850/H20856, in the SU /H208492/H20850symmetric case reduces
to a scalar function, /H9273/H20849/H9275n,x/H20850=/H20855sz/H20849−i/H9275n,x/H20850sz/H20849i/H9275n,0/H20850/H20856.I tc a n
be evaluated by using /H208491/H20850and /H208494/H20850. Assuming that the electron
gas is dense to satisfy EF/H11271/H9252−1, one obtains /H20849with g/H11013gc/H20850,
/H9273/H20849/H9275n,x/H20850=−/H20841/H9275n/H20841
4/H9266vF2exp/H20873−/H20841/H9275nx/H20841
vF/H20874
+cos/H208492kFx/H20850
/H208492/H9266/H20850g+1a1−g/H20885d/H9270exp/H20849−i/H9275n/H9270/H20850
/H20849x2+vF2/H92702/H208501/2/H20849x2+vF2/H92702/g2/H20850g/2.
/H208496/H20850
The correlation function in /H208496/H20850contains two contributions,
one due to forward scattering, peaked at wave vector q=0,
and another due to backwards scattering, peaked at q
=±2 kF. As expected, the forward scattering contribution is
g-independent.
In the perturbative regime considered, the dynamics of the
spins is slow and controlled by small parameter J2/vF2.
Therefore, we can use the small-frequency asymptotic formof the correlation function
/H9273/H20849/H9275n,x/H20850for/H20841/H9275nx/H20841/vF/H112701,
/H9273/H20849/H9275n,x/H20850/H11229C1/H20849g/H20850cos/H208492kFx/H20850
vFa1−gxg
−1
4/H9266vF2/H20875/H20841/H9275n/H20841+C2/H20849g/H20850cos/H208492kFx/H20850/H20841/H9275n/H20841g
vFg−1a1−g/H20876,/H208497/H20850
where C1=/H208492/H9266/H20850−g−1gg/H20848dz/H208491+z2/H20850−1/2/H20849g2+z2/H20850−g/2and C2
=4/H208492/H9266/H20850−ggg−1/H9003/H208491−g/H20850sin/H20851/H20849/H9266/2/H20850/H208491−g/H20850/H20852. The first term in /H208497/H20850
corresponds to interaction between the spins. The interaction
is oscillatory and decays as a power law x−g. This result is
consistent with Ref. 13.The second, /H9275ndependent term in /H208497/H20850corresponds to re-
laxation of the spins. To demonstrate this property, we per-form a transition to the real time dynamics of the spins ac-cording to analytical continuation rule, see Ref. 17. Weintroduce a standard Keldysh contour with forward and re-turn branches time-ordered and anti-time-ordered, respec-tively. In the Keldysh representation, the effective action, /H208495/H20850,
reads
J
2
2/H20885d/H9275
2/H9266/H20851S1T/H20849/H9275/H20850/H9273ˆ/H20849/H9275,x/H20850S1/H20849−/H9275/H20850+S1T/H20849/H9275/H20850/H9273ˆ/H20849/H9275,0/H20850S2/H20849−/H9275/H20850
+/H20849S1↔S2/H20850/H20852. /H208498/H20850
Here Si=1,2are two-element column vectors, with spin-vector
operators as elements, such that their transposes are SiT
=/H20849Sic,Siq/H20850. The “classical,” Sic, and “quantum,” Siq, compo-
nents of the spin Siare the following combinations of spin
operators on the forward /H20849f/H20850and return /H20849r/H20850branches of the
Keldysh contour, Sic=/H20849Sif+Sir/H20850/2 and Sq=Sif−Sir. The re-
sponse /H20849correlation /H20850function /H9273ˆ/H20849/H9275,x/H20850is then a 2 /H110032 matrix,
which can be expressed in terms of the retarded and ad-
vanced response functions, /H9273Rand/H9273A,
/H9273ˆ=/H208730/H9273A
/H9273R/H9273K/H20874. /H208499/H20850
The retarded and advanced response functions are related to
the Matsubara response function via the analytic continua-tioni
/H9275→/H9275±i/H9254. In thermal equilibrium /H9273Kcan be expressed
in terms of the retarded and advanced components by usingthe fluctuation-dissipation theorem,
/H9273K=coth /H20849/H9252/H9275/2/H20850/H20849/H9273R
−/H9273A/H20850.
Let us consider first the noninteracting case, g=1, and
later we will extend the results to g/HS110051. For noninteracting
electrons, the response function corresponds to an Ohmicheat bath, with C
1/H208491/H20850=1/ /H208494/H9266/H20850and C2/H208491/H20850=1 in /H208497/H20850. Upon
Fourier transform, /H208498/H20850yields several terms. Those containing
products Sic·Sjqrepresent interaction between spins, while
theS˙ic·SjqandSiq·Sjqterms are responsible for energy dis-
sipation and pure dephasing /H20849decoherence /H20850, respectively. The
dissipative /H20849time-derivative /H20850terms are small and can be ne-
glected here. Indeed, since S˙i/H11011O/H20849J2/vF2/H20850, the S˙-dependent
terms in the action in /H208498/H20850are of order J4/vF4. The resulting
dynamics of the spins is governed by the action SselfK=SintK
+SdecK, where
SintK=Jeff/H20885dt/H20851S1c/H20849t/H20850·S2q/H20849t/H20850+S1q/H20849t/H20850·S2c/H20849t/H20850/H20852, /H2084910/H20850
SdecK=i/H9253
2/H9266/H20885dt1dt2/H20885
0/H11009
d/H9275/H9275cos/H20851/H9275/H20849t2−t1/H20850/H20852coth/H20873/H9252/H9275
2/H20874
/H11003/H208532S1q/H20849t1/H20850·S1q/H20849t2/H20850−/H208511 + cos /H208492kFx/H20850/H20852
/H11003S1q/H20849t1/H20850·S2q/H20849t2/H20850+/H20851S1q↔S2q/H20852/H20854, /H2084911/H20850
withJeff=J2cos/H208492kFx/H20850//H208494/H9266vFx/H20850and/H9253=J2//H208492/H9266vF2/H20850.
Equation /H2084910/H20850corresponds to coherent interaction of the
spins according to the scalar S1·S2coupling. Equation /H2084911/H20850
represents quantum noise resulting from thermal fluctuationsBRIEF REPORTS PHYSICAL REVIEW B 72, 233103 /H208492005 /H20850
233103-2of the Luttinger liquid. The noise, generally speaking, is col-
ored. Moreover, the noises experienced by the two spins arecorrelated.
The action in /H2084911/H20850can be simplified further if we recall
that the dynamics of the spins is slow, i.e., S
iq/H20849t2/H20850=Siq/H20849t1/H20850
+O/H20849J2/vF2/H20850. Then, the self-action can be replaced by an in-
stantaneous action, by setting Siq/H20849t1/H20850=Siq/H20849t2/H20850in/H2084911/H20850, which
now corresponds to the white noise source. The full action of
the spin system can now be written as
SK=SBerryK−/H20885dt HK/H20849S1,S2/H20850, /H2084912/H20850
where the Berry action plays the role of a velocity
/H11003momentum term in the Largangian-Hamiltonian trans-
formation.16The “Keldysh Hamiltonian” in /H2084912/H20850is
HK=−Jeff/H20849S1f·S2f−S1r·S2r/H20850
−4i/H9253T/H20851/H20849sin/H9278S1q− cos/H9278S2q/H208502+/H20849S1q↔S2q/H20850/H20852,
/H2084913/H20850
where 0 /H33355/H9278/H33355/H9266/2 is defined by sin /H208492/H9278/H20850=cos2/H20849kFx/H20850.
The density matrix of the two-spin system evolves ac-
cording to the “Schrödinger equation” /H9267˙=−iTK/H20849HK/H9267/H20850, where
Keldysh time-ordering implies that the “forward” operators,
with subscript f, are positioned to the left of the density
matrix, while the “return” operators, labelled by r, are to the
right of the density matrix,
/H9267˙=iJeff/H20851S1·S2,/H9267/H20852−4/H9253T/H20858
/H9251=x,y,z/H20849/H20851sin/H9278S1/H9251− cos/H9278S2/H9251,/H20851sin/H9278S1/H9251
− cos/H9278S2/H9251,/H9267/H20852/H20852+/H20849S1↔S2/H20850/H20850. /H2084914/H20850
This expression contains both the mediated interaction and
dephasing due to the electron environment.
It is instructive to illustrate the dynamics of the two-spin
system, described by /H2084914/H20850, for a particular initial state of the
system, /H20841↑↓/H20856. Without the quantum noise term, proportional
to/H9253Tin/H2084914/H20850, the S1·S2interaction would split the singlet
and triplet spin states. As a result, the system would oscillatebetween the /H20841↑↓/H20856and /H20841↓↑/H20856states with frequency determined
by the singlet-triplet energy gap. The effects of the noiseinclude damping of these oscillations, as illustrated in Fig. 1.Furthermore, for t/H110220, the system subject to the noise will no
longer remain in a pure quantum state. The departure of theresulting mixed state from a pure state can be measured bydeviation of Tr /H20851
/H92672/H20849t/H20850/H20852from the pure-state value of 1, as illus-
trated in Fig. 2. We point out that effective evolution equa-
tions that involve only commutators, linear in the densitymatrix, on the right-hand side /H20849RHS /H20850, typically fail to repro-
duce thermal equilibrium at large times. Instead, as seen inFigs. 1 and 2—note the assymptotic values—the fully ran-dom mixed state is obtained. Therefore, the present approxi-mation should not be used beyond the relaxation time de-
fined by /H2084914/H20850, namely, it only applies for t/H110211//H20851T/H20849J
2/vF2/H20850/H20852,
and the theory is therefore applicable in the regime of inter-
est for quantum computing applications, for short and inter-mediate times, because both factors in the denominator, TandJ
2/vF2, are small. Similar results have been obtained in
Ref. 18, studying mediated interaction and decoherence dueto noninteracting electron gas.
Finally, we extend our results to the interacting case, g
/HS110051. Modification of the interaction term, /H2084910/H20850, is straightfor-
ward and comes from the first term on the RHS of /H208497/H20850. Equa-
tion /H2084910/H20850applies with
J
eff/H20849g/H20850=C1/H20849g/H20850J2cos/H208492kFx/H20850
vFa1−gxg. /H2084915/H20850
Thus, indirect interaction between spins becomes longer
range as a result of electron-electron interaction.
Modification of the decoherence term, /H2084911/H20850, is less obvi-
ous. Consider for simplicity a single spin situation, when thefirst term on the RHS of /H208497/H20850can be ommited, and in the
second term we can set x=0. The /H20841
/H9275n/H20841term then in the large
FIG. 1. Lower curve, the probability, /H9267/H20841↑↓/H20856,/H20841↑↓/H20856, to find the two-
spin system in the /H20841↑↓/H20856state as a function of time, for a convenient
set of parameter values, Jeff=1,/H9253T=0.0125, kFx=/H9266. Upper curve,
the sum of the probabilities to find the two-spin system in the states/H20841↑↓/H20856or/H20841↓↑/H20856, given by
/H9267/H20841↑↓/H20856,/H20841↑↓/H20856+/H9267/H20841↓↑/H20856,/H20841↓↑/H20856. The large-time limiting
values of these probabilities are1
4and1
2, respectively.
FIG. 2. Solid line, the quantity Tr /H20849/H92672/H20850, with the initial value of 1
corresponding to a pure state, as a function of time for the sameparameter values as in Fig. 1, with positive J
eff=1. Dashed line,
parameter values illustrating the case of negative Jeff=−2, with
/H9253T=0.0125, kFx=/H9266/2. Dotted line, the deviation from a pure state
/H20849the quantum noise effect /H20850is present even when the leading-order
induced interaction is Jeff=0, with /H9253T=0.0125, kFx=/H9266/4. The
curves for Jeff/HS110050 have weak oscillations superimposed on the de-
cay. The large-time limiting values are1
4.BRIEF REPORTS PHYSICAL REVIEW B 72, 233103 /H208492005 /H20850
233103-3parentheses in /H208497/H20850produces the same, g-independent, contri-
bution to the decoherence rate as in the noninteracting case.The /H20841
/H9275n/H20841gterm in the large parentheses requires careful con-
sideration. Indeed, we are primarily interested in the repul-sive, U/H110220, Hubbard-model interaction, i.e., 0 /H11021g/H110211. The
two-spin contribution then yields a divergence: after settingS
iq/H20849t1/H20850=Siq/H20849t2/H20850in/H2084911/H20850and integrating over t1−t2, the fre-
quency integral /H20848d/H9275/H9275g/H9254/H20849/H9275/H20850coth /H20849/H9252/H9275/2/H20850is divergent for g
/H110211. The origin of this divergence is actually not in the in-
stantaneous assumption but instead it can be traced back totoo liberal a use of the large-
/H9252approximation in our evalu-
ation of the response function in /H208496/H20850, and specifically, extend-
ing the limits of the /H9270integration from − /H11009to/H11009, while we
should have integrated from − /H9252to/H9252. The response function
in/H208496/H20850can be readily evaluated for /H9275n→0/H20849and x=0/H20850for
finite/H9252as well. After a straightforward repetition of steps
leading to Eq. /H208496/H20850one obtains the dephasing rate for a para-
magnetic spin imbedded in Luttinger liquid,
/H9270dec−1=4/H9253/H20873T+C3/H20849g/H20850Tg
vFg−1a1−g/H20874, /H2084916/H20850where C3/H20849g/H20850=2ggsin/H20851/H9266/H208491−g/H20850/H20852//H20851/H208492/H9266/H20850g/H208491−g/H20850/H20852. The second
term on the RHS of Eq. /H2084916/H20850/H20849due to backscattering /H20850repre-
sents modification of the Korringa law for spin relaxation inthe interacting 1D electron gas. Thus, electron-electron re-pulsion leads to the reduction of spin relaxation rate.
In summary, our main result, /H2084914/H20850, is the first theoretically
derived dynamical equation that incorporates both the coher-ent RKKY-type induced interaction and the effects of quan-tum noise due to interacting electrons in the 1D conductionchannel. The approximations and assumptions involved,limit our results to temperatures in the range T
K/H11021T/H11270TF.
Furthermore, to have the noise term small, the temperatureshould be actually in T
K/H11021T/H11021vF/x, where xis the qubit
/H20849spin /H20850separation; restoring the constants earlier set to 1, the
upper bound here is /H6036vF/kBx. The relative strength of the
interaction vs noise terms can be also controlled by position-ing of the qubits, owing to the oscillatory dependence on x,
typical for RKKY-coupled systems.
The authors acknowledge helpful discussions with D. F.
James, I. Martin, and A. Shnirman. The work was supportedby US DOE, and by the NSF, Grant No. DMR-0121146.
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233103-4 |
PhysRevB.88.165403.pdf | PHYSICAL REVIEW B 88, 165403 (2013)
Nonequilibrium spin-current detection with a single Kondo impurity
Jong Soo Lim
Institut de F ´ısica Interdisciplin `aria i de Sistemes Complexos IFISC (CSIC-UIB), E-07122 Palma de Mallorca, Spain
and School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
Rosa L ´opez
Institut de F ´ısica Interdisciplin `aria i de Sistemes Complexos IFISC (CSIC-UIB), E-07122 Palma de Mallorca, Spain
and Departament de F ´ısica, Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
Laurent Limot
Institut de Physique et Chimie des Mat ´eriaux de Strasbourg, Universit ´e de Strasbourg, CNRS, 67034 Strasbourg, France
Pascal Simon
Laboratoire de Physique des Solides, CNRS UMR-8502, Universit ´e Paris Sud, 91405 Orsay Cedex, France
(Received 18 July 2013; published 4 October 2013)
We present a theoretical study based on the Anderson model of the transport properties of a Kondo impurity
(atom or quantum dot) connected to ferromagnetic leads, which can sustain a nonequilibrium spin current. Weanalyze the case where the spin current is injected by an external source and when it is generated by the voltagebias. Due to the presence of ferromagnetic contacts, a static exchange field is produced that eventually destroysthe Kondo correlations. We find that such a field can be compensated by an appropriated combination of thespin-dependent chemical potentials leading to the restoration of the Kondo resonance. In this respect, a Kondoimpurity may be regarded as a very sensitive sensor for nonequilibrium spin phenomena.
DOI: 10.1103/PhysRevB.88.165403 PACS number(s): 72 .10.Fk, 72 .15.Qm, 73 .63.−b, 68.37.Ef
I. INTRODUCTION
In the last decades, there has been a revived interest
in Kondo physics. This many-body effect is produced byhigh-order correlated tunneling events consisting of electronicspins hopping inand outof a localized impurity, which
ultimately lead to an efficient screening of the impurity spin.The Kondo effect has been extensively investigated for the
anomalous behavior it produces on the resistivity versus
temperature in bulk metals possessing magnetic impurities.
1
Experimental advances allow nowadays probing the Kondoeffect in single objects through the detection of a zero-biaspeak known as Kondo resonance. It is now possible to tacklenontrivial many-body effects in a controlled environment.The Kondo resonance has been investigated through scanningtunneling microscopy (STM) in single atoms either isolated
2–4
or coupled to other atoms,5–8in single-atom contacts,9–11
and in single molecules.12–16It has also been successfully
evidenced in nanoscale devices,17–21in particular quantum
dots,17–19,22–24carbon nanotubes,25–27and nanowires.28
Of particular interest—especially in the context of spin-
tronics, is the issue of screening in the presence of a magneticenvironment such as spin-polarized electrodes
29–35and spin-
polarized edge states.36A spin-dependent hybridization for
the spin-up and spin-down energy levels of the impurity isthen predicted, resulting in an effective static magnetic field atthe impurity site (this field can eventually be compensatedby an external magnetic field).
29,31,35In the presence of
ferromagnetism, the Kondo resonance therefore splits apartas confirmed experimentally.
32,37
While such a splitting is well understood, the impact of
a nonequilibrium spin current on the Kondo resonance hasso far been little addressed in correlated nanostructures. Thisremains an open question since a decade ago it was shown that
a spin current flowing from a Co wire through a Cu(Fe) wireis able to strongly suppress the resistivity of the Cu(Fe) Kondoalloy near the interface.
38As demonstrated by Johnson39(see
also Refs. 40–42), a spin current induces spin accumulation
yielding spin-dependent chemical potentials μ↑/negationslash=μ↓, which
is equivalent to a spin bias. Using the equation of motionapproach, Qi et al. analyzed the fate of a Kondo resonance in
the presence of spin accumulation.
43They showed in particular
that the Kondo resonance is split into two peaks attached tothe two spin-dependent chemical potentials. Kobayashi et al.
44
recently validated this prediction by studying experimentally
a Kondo quantum dot in contact with a spin accumulatedelectrode and two normal electrodes. A simplified geometryrelated to this experiment is sketched in Fig. 1(a). Besides
demonstrating that the Kondo splitting can be controlledthrough spin accumulation, they also showed that the Kondoresonance may be restored through an external magneticfield.
Single-atom contacts with STM are another appealing way
for investigating the interplay of a spin current with a Kondo
impurity. In STM tunneling spectra, the Kondo resonance is
detected as a Fano line shape due to the interference betweenelectrons tunneling into the conduction band of the substrateand those involved into the Kondo state.
45,46When the tip is
brought into contact with the atom such a picture remains validalthough the Kondo resonance becomes more symmetric andof order the conductance unit 2 e
2/h.11As shown recently,47
it it possible to introduce a spin current in the single-atom
contact by using a ferromagnetic tip coated with a thick normalcopper spacer [see Fig. 1(b) for a sketch of the setup]. As
in macroscopic spintronic devices,
39,41,42the copper spacer
165403-1 1098-0121/2013/88(16)/165403(9) ©2013 American Physical SocietyLIM, L ´OPEZ, LIMOT, AND SIMON PHYSICAL REVIEW B 88, 165403 (2013)
FIG. 1. (Color online) (a) Sketch of a quantum dot connected to
a normal lead (right) and to a spin-accumulated lead (left). The spin
current is injected by external means. The symbols /Gamma1L/Rrepresent
the hybridization between the dot level and the left/right leads. IL,↑
andIL,↓denote the spin currents for electrons with spin ↑and↓.
(b) Schematic diagram of a magnetic atom adsorbed on a surfacein contact with a spin-accumulated tip. (c) Spin accumulation gives
rise to spin-dependent chemical potentials and eventually to a spin
polarization. Both phenomena may lead to a splitting of the impuritylevels. The color code is as follows: blue (resp. purple) denote
electrons with spin ↑(resp. ↓).
aims at minimizing the direct or indirect magnetic exchange
interactions between the cobalt atom and the tip. The Kondosplitting observed can then be assigned to spin accumulationin the copper spacer. In this respect, the Kondo resonance actsas a very sensitive local sensor for spin current. We want toemphasize that in the STM setup the tip is simultaneously thesource of the spin accumulation and also the transport probe.
Therefore the spin current becomes voltage dependent contrary
to Ref. 44where the spin current was supplied by an external
spin-accumulated electrode while the differential conductancewas probed using two different leads.
The purpose of this paper is to provide a microscopic
description of a magnetic Kondo impurity embedded betweena spin-polarized electrode able to carry a spin current and anormal metallic electrode. The impurity can be either artificial,such as a quantum dot [see Fig. 1(a)], or a genuine magnetic
atom adsorbed on a surface [see Fig. 1(b)]. We consider
both cases in which the spin current is either driven by anexternal source (and therefore constant) or driven by the sameelectrode (and therefore voltage dependent). By using theequation of motion techniques
48–51and comparing various
truncation methods to obtain a consistent picture, we showthat the Kondo ground state depends sensitively on the spinpolarization of the electrode and the spin accumulation thatit generates. We investigate both the spin-resolved spectraldensity of the localized spin and the nonlinear conductance.In the case of a constant spin current, we demonstratethat spin accumulation leads to a splitting of the impurityKondo resonance as shown previously
43,44and schematically
summarized in Fig. 1(c). Taking into account both the
static spin polarization and the spin accumulation, we showadditionally that both effects can actually compensate eachother and therefore the Kondo resonance can be restored.
The case of a voltage dependent spin current turns out tobe more subtle. Through a phenomenological approach weshow that a nonlinear dependence of the spin current withvoltage bias is required to split the Kondo resonance. As weshow in this work, our finite- Qapproximation (which amounts
to voltage-independent spin-dependent chemical potentials atlarge bias) turns out to be rather accurate when the impurity iseasy to spin polarize.
The plan of the paper is as follows: In Sec. II,w e
introduce our model consisting of an impurity in contact with aspin-polarized electrode and a nonmagnetic electrode. We alsodiscuss the method and approximation we use to tackle sucha nonequilibrium interacting problem. In Sec. III, we study
both analytically and numerically the case where one electrodehas a finite spin polarization and sustains a constant spinaccumulation. In Sec. IV, we investigate the more subtle case
where the spin accumulation becomes bias dependent. Finally,in Sec. Vwe provide a summary of our main results and discuss
some perspectives. Details of the truncated equation of motionapproach used are presented in Appendix.
II. MODEL HAMILTONIAN AND METHOD
We consider an impurity—a quantum dot or atom—coupled
to left and right electrodes as depicted in Fig. 1.W eh a v ei n
mind situations in which the quantum dot is used as a detectorof a nonequilibrium spin accumulation; therefore we focus onthe asymmetric situation in which one electrode [the left one inFig.1(a)or the STM tip in Fig. 1(b)] may be partially polarized
and able to sustain a spin current.
We assume that the impurity (the quantum dot or the
magnetic adatom) correspond to a spin S=1/2 impurity.
In order to model the magnetic impurity, we consider anAnderson-type Hamiltonian
H=/summationdisplay
α,k,σ(εαkσ−μασ)c†
αkσcαkσ+/summationdisplay
σεσd†
σdσ+Un↑n↓
+/summationdisplay
α,k,σ(Vαkσc†
αkσdσ+H.c.). (1)
Here,c†
αkσ(cαkσ) denotes the creation (annihilation) operator in
contact αandd†
σ(dσ) is the corresponding operator in the dot.
Vαkσdescribes a tunneling matrix element between contacts
and localized levels and can eventually be spin dependent. U
andεσparametrize the on-site Coulomb interaction and the
spin-dependent localized energy level, respectively. Noticethat an initial energy difference between localized levels,/Delta1
Z/negationslash=ε↑−ε↓, may model an external magnetic field. As we
emphasized in the introduction, a nonequilibrium spin accu-mulation entails spin-dependent chemical potentials μ
ασand
polarizations. The spin polarization in the contacts is lumpedinto spin-dependent hybridization functions /Gamma1
ασ. Following
Refs. 29–31we write a spin-polarization parameter as
Pα=/Gamma1α↑−/Gamma1α↓
/Gamma1α↑+/Gamma1α↓, (2)
where /Gamma1ασ=/Gamma1α(1+σPα)with/Gamma1α=π/summationtext
k|Vαkσ|2ρσ.ρσ
is the spin-dependent lead DOS at the Fermi energy, which
165403-2NONEQUILIBRIUM SPIN-CURRENT DETECTION WITH A ... PHYSICAL REVIEW B 88, 165403 (2013)
is assumed flat for both electrodes. Similarly, we introduce a
spin bias parameter which is defined as
Qα=μα↑−μα↓
μα↑+μα↓. (3)
We consider the commonly used wideband limit for the
tunneling rates where the hybridization /Gamma1ασare constant.
Following Ref. 52, the spin-dependent current Iσreads
Iσ=−4e
h/Gamma1Lσ/Gamma1Rσ
/Gamma1Lσ+/Gamma1Rσ/integraldisplay
dω[fLσ(ω)−fRσ(ω)]Im/bracketleftbig
Gr
σ,σ(ω)/bracketrightbig
.
(4)
Here, eis the elementary (positive) unit charge. Although
the spin current expression may look simple, it is worthunderlining that the retarded Green function G
ris the exact
bias-dependent Green function. Computing such quantityremains a tremendous task. Since we are dealing with anonequilibrium interacting problem, we must rely on someapproximate approach able to capture qualitatively the physics.We have employed the equation of motion technique to calcu-late the retarded Green’s function. This theoretical approachuses a truncated system of equations of motion for the retardedGreen’s function. There are several schemes for the truncationin order to obtain a close set of equations. In our case, wefollow Refs. 53and54in order to compute the Keldysh Green’s
functions. This procedure has been demonstrated to be suitableto treat systems with spin-polarized contacts. Details of thetruncation scheme we have used can be found in Appendix.
III. KONDO RESONANCE IN THE PRESENCE OF A
CONSTANT SPIN CURRENT AND POLARIZATION
In this section, we focus on a quantum dot connected to spin-
polarized electrodes that are able to sustain a nonequilibriumspin current. In order to provide a qualitative understanding ofthe physics, it turns out to be useful to first perform a second-order perturbation theory in the tunneling matrix elements,which generates an effective but nonequilibrium local Zeemanterm in the dot Hamiltonian.
A. Effective magnetic fields
Before presenting the results of our numerical calculations,
we would like to present an extension of the heuristic argumentdeveloped in Ref. 55aiming at interpreting the effect of a finite
polarization and/or finite spin current in a lead as an effectivelocal exchange magnetic field viewed/felt by the spin impurity.In order to calculate this effective magnetic field, we proceedas in Ref. 55to investigate the functional dependence of the
effective magnetic field on temperature and gate voltage. Todo so, we derive an effective Hamiltonian H
effusing second-
order perturbation theory. Physically, the split Kondo peak canbe understood in terms of the dot valence instability (virtualcharge fluctuation) and spin-dependent tunneling amplitudes.To deal with this instability, we perform a Schrieffer-Wolff-type transformation of the Hamiltonian given by Eq. (1)andobtain
H
spin=/summationdisplay
α,k/summationdisplay
β,q/bracketleftbiggVαk↑Vβq↑
εd↑−εβq↑cαk↑c†
βq↑−Vαk↓Vβq↓
εd↓−εβq↓cαk↓c†
βq↓
+Vαk↑Vβq↑
U+εd↑−εβq↑c†
αk↑cβq↑
−Vαk↓Vβq↓
U+εd↓−εβq↓c†
αk↓cβq↓/bracketrightbigg
Sz+[···], (5)
where [ ···] includes the usual terms corresponding to the spin-
flip terms and potential scatterings that show up in the KondoHamiltonian. At this point, unlike in the usual Schrieffer-Wolff transformation, we employ a mean-field approximation
for the lead electrons: /angbracketleftc
αkσc†
βqσ/angbracketright=[1−f(εαkσ)]δα,βδk,q
and/angbracketleftc†
βqσcαkσ/angbracketright=f(εαkσ)δα,βδk,q.55The resulting effective
Hamiltonian can be written as Heff=−BeffSz. The effective
magnetic field generated by having spin-accumulation, i.e.,spin-dependent chemical potentials and spin-polarized con-tacts, then reads
B
eff∝/summationdisplay
α/integraldisplay
dω/bracketleftbigg/Gamma1α↑[1−fα↑(ω)]
ω−εd↑−/Gamma1α↓[1−fα↓(ω)]
ω−εd↓
+/Gamma1α↑fα↑(ω)
ω−εd↑−U−/Gamma1α↓fα↓(ω)
ω−εd↓−U/bracketrightbigg
. (6)
As emphasized in the introduction, we will mainly focus on the
asymmetrical situation and assume that the spin accumulationand polarization occurs only in the left contact. The generalcase can be trivially extended. We also assume spin-degeneratelocalized levels ε
d↑=εd↓=εd. Therefore the spin-dependent
chemical potentials are parametrized as follows:
μL↑=μL(1+Q) andμL↓=μL(1−Q),μ R↑=μR↓=0,
(7)
and the lead polarization Pis defined by
/Gamma1L↑=/Gamma1L(1+P) and/Gamma1L↓=/Gamma1L(1−P),/Gamma1 Rσ=/Gamma1R.(8)
With this parametrization, the effective field can be then
written as
Beff∝/Gamma1L/integraldisplay
dω/bracketleftbigg(1+P)[1−fL↑(ω)]
ω−εd
−(1−P)[1−fL↓(ω)]
ω−εd+(1+P)fL↑(ω)
ω−εd−U
−(1−P)fL↓(ω)
ω−εd−U/bracketrightbigg
. (9)
Up to leading order in PandQ, the previous expression
simplifies to
Beff∝−2P/Gamma1LRe/braceleftbigg
/Psi1/parenleftbigg1
2−iβ(εd−μL)
2π/parenrightbigg
−/Psi1/parenleftbigg1
2−iβ(εd+U−μL)
2π/parenrightbigg/bracerightbigg
+2QμL/Gamma1L/integraldisplay
dω f/prime(ω−μL)/bracketleftbigg1
ω−εd−1
ω−εd−U/bracketrightbigg
,
(10)
withβ=1/kBTand/Psi1denotes the digamma function defined
as the logarithmic derivative of the gamma function. Taking
165403-3LIM, L ´OPEZ, LIMOT, AND SIMON PHYSICAL REVIEW B 88, 165403 (2013)
the limit T→0, the effective magnetic field takes the compact
form
Beff∝−2P/Gamma1Lln/vextendsingle/vextendsingle/vextendsingle/vextendsingleεd−μL
εd+U−μL/vextendsingle/vextendsingle/vextendsingle/vextendsingle
+2QμL/Gamma1LU
(εd−μL)(εd+U−μL).(11)
Note that for the particle-hole symmetry point ( |εd−μL|=
U/2) the effective field due to the finite spin polarization
vanishes, contrary to the effective field due to the spin-dependent chemical potentials. This latter field can be canceledby applying an external static magnetic field B
extsuch
thatBext+Beff=0 as has been shown experimentally by
Kobayashi et al.44
However, we want to stress that this is not the only way
to cancel this effective magnetic field. The two contributionsrelated to the spin accumulation and to the static polarizationmay indeed have different signs. The sign of the formerterm is determined by the difference of the spin-dependentchemical potentials while the sign of the latter is fixed by thedifference of the density of states at the Fermi energy betweenup and down electrons. Therefore this heuristic argumentsuggests that we can control the spin current independentlyof the polarization or vice versa and thus restore the Kondoresonance. This could for example be achieved by controllingthe polarization of the left lead. We check that this is indeedthe case in the next subsection.
B. Spectral weights and differential conductance
We now present our results for the total and spin-resolved
spectral weights which have been obtained with the truncatedequation of motion approach.
53,54We work in units of
/Gamma1L+/Gamma1R=/Gamma1=1. We adopt the following set of parameters
/epsilon1d=−3.5 and D=50. For simplicity, the U→∞ limit is
considered, but our results can be generalized and remainqualitatively correct in the finite- Ulimit. Now, we vary P
andQand show the total and spin-resolved spectral density
evolutions in Fig. 2. First, when both contacts are normal
(P=Q=0), the low-energy spectral density shows a single
peak at the Fermi energy corresponding to the Kondo singu-larity [see Fig. 2(a)]. When there is a finite polarization but
no spin-dependent chemical potentials ( P/negationslash=0,Q=0), the
spin-↑spectral density moves towards negative frequencies,
whereas spin- ↓does the opposite, resulting in a split Kondo
resonance. This behavior is shown in Fig. 2(b).N e x t ,w e
consider the situation with some degree of spin-dependentchemical potentials, but no spin polarization ( P=0,Q/negationslash=0).
Such a situation applies when a spin current is injectedfrom an external terminal (see Ref. 44). We observe that the
peaks in the spin-resolved spectral densities are located atω
↑(↓)≈μL↓(↑),μR; this is illustrated in Fig. 2(c) for which
we use Q=0.5, and μL=0.2 leading to μL↑=0.3, and
μL↓=0.1 while μR=μR↑=μR↓=0. The position of the
peaks is determined by the poles of the impurity retardedGreen’s function [see Appendix, particularly Eq. (A21) ]. We
find that the real part of the denominator of G
r
σ,σ(ω) has zeros
atμα¯σwhen the Kondo correlation develops. With a finite
polarization, the renormalized levels become spin-dependentso that the peak positions are no longer at μ
ασbut depend00.050.10.150.2Aσ(a)
P=0
Q=0
μL=0A↑A↓A↑+A↓(b)
P=0.2
Q=0
μL=0
00.050.10.15Aσ(c)
P=0
Q=0.5
μL=0.2(d)
P=0.2
Q=0.5
μL=0.2
00.050.10.15
-1.5 -0.75 0 0.75Aσ
ω/Γ(e)
P=0.2
Q=0.5
μL=0.5
-1.5 -0.75 0 0.75 1.5
ω/Γ(f)
P=-0.18
Q=0.5
μL=0.2
FIG. 2. (Color online) Spectral weights vs PandQ. Parame-
ters:/Gamma1L=/Gamma1R=0.5,εd=−3.5,D=50,μRσ=0,T=TK,a n d
U→∞ .
on the degree of polarization P. In general, when P/negationslash=0
andQ/negationslash=0, the total spectral density shows a four peak
structure. However, if two of the four peaks encountered forA
↑andA↓coincide, the total spectral function displays only
three peaks. This situation is depicted in Fig. 2(e) for which
P=0.2, and Q=0.5 with μL=0.5. The three peaks can
be also designed by considering P=−0.18 and Q=0.5 and
μL=0.2 as shown in Fig. 2(f). Whereas the peak splitting in
the spectral weights due to the static polarization can occurboth under equilibrium and non equilibrium conditions, wewant to stress again that the split spectral weights due tothe spin-dependent chemical potentials can only occur undernonequilibrium conditions.
In addition, we investigate the nonlinear conductance which
is an experimentally accessible quantity. For practical pur-poses, it turns out to be more convenient to configure the biasesin such a way that the spin-dependent chemical potentials ofthe left contact are fixed, while the spin-independent chemicalpotential of the right contact, μ
Rσ=μL+eV, is varied. The
differential conductance reads
dIσ
dV=4e2
h/Gamma1Lσ/Gamma1Rσ
/Gamma1Lσ+/Gamma1Rσ/integraldisplay
dω/braceleftbiggdfR(ω)
d(eV)Im/bracketleftbig
Gr
σ,σ(ω)/bracketrightbig
−[fLσ(ω)−fR(ω)]Im/bracketleftbiggdGr
σ,σ(ω)
d(eV)/bracketrightbigg/bracerightbigg
. (12)
Figure 3(a) illustrates the nonlinear conductance in the
absence of spin-dependent chemical potentials ( Q=0). The
observed splitting is attributed to the effective field generatedby the presence of the spin-polarized contacts. Notice thatthe two peaks in the nonlinear conductance are almostsymmetrically located and the splitting (therefore the peakpositions) grows with P. In the absence of spin polarization
(P=0) but with spin-dependent chemical potentials, the
165403-4NONEQUILIBRIUM SPIN-CURRENT DETECTION WITH A ... PHYSICAL REVIEW B 88, 165403 (2013)
0.050.10.150.2dI/dV(2e2/h)(a)
Q=0P=0.00
=0.25
=0.50
=0.75(b)P=0
Q=0.00
=0.25
=0.50
=0.75
0.050.10.15
-0.5 -0.25 0 0.25dI/dV(2e2/h)
V/Γ(c)P=0.2
Q=0.00
=0.25
=0.50
=0.75
-0.5 -0.25 0 0.25 0.5
V/Γ(d)P=-0.18
Q=0.00
=0.25
=0.50
=0.75
FIG. 3. (Color online) Differential conductance vs Pand
Q. Parameters: /Gamma1L↑=0.3(1+P),/Gamma1L↓=0.3(1−P),/Gamma1Rσ=0.7,
μL=0.2,εd=−3.5,D=50,T=1.5TK,a n dU→∞ .
nonlinear conductance shows similarly two peaks. Contrary
to the P=0 case, we easily identify that the dI/dV con-
ductance shows two peaks at eV=μL↑(/↓)−μL[Fig. 3(b)].
This corresponds to adjusting the spin-independent chemicalpotential of the right lead with the spin-dependent chemicalpotentials for the left lead. This can be understood as follows.The leading part of the differential conductance is given bythe first term in Eq. (12). At zero temperature, this term is
proportional to the spin-dependent local DOS of the impurity(A
σ(ω)=−1
πIm[Gr
σ,σ(ω)]). When we neglect the second term
in Eq. (12), we therefore find that dIσ/dV∝Adσ(μL+eV).
The total differential conductance dI/dV =/summationtext
σdIσ/dV will
be maximum when the condition
eV=±(μL↑−μL↓)
2=±μLQ (13)
is satisfied.47
In the presence of both spin bias and polarization ( Q/negationslash=0,
andP/negationslash=0), the nonlinear conductance for positive Palso
exhibits two peaks and the splitting between two peakswidens compared with the P=0 cases. On the contrary, for
negative P, the splitting observed in the dI/dV is reduced
and eventually vanishes at some particular value of Q.T h i si s
shown in Fig. 3(d) where the nonlinear conductance shows a
single resonance for P=−0.18 and Q=0.5. In this case, the
Kondo effect is restored. From the heuristic argument givenin Sec. III A , we can interpret this restoration of the resonance
as the compensation of the effective fields generated by thespin-dependent polarization and the spin bias. Our numericalcalculation therefore confirm nicely the qualitative results wediscuss in Sec. III A that the accumulation spin current and the
static polarization may have antagonist effect on the Kondoresonance which results in its restoration.
IV . KONDO RESONANCE IN THE PRESENCE OF
A BIAS-DEPENDENT SPIN ACCUMULATION
In the previous section, we studied the case where the spin
current is injected by an external terminal as in Ref. 44.T h i s
was inherently a nonequilibrium situation even at V=0. Letus now consider the two-terminal situation with the left lead
spin polarized. We assume that at equilibrium (for V=0)
no spin current is generated but only a static magnetic spinpolarization P. This implies μ
L↑=μL↓atV=0. A finite V
generates both charge and spin currents. The purpose of thissection is to analyze what will be the effect of such a bias-dependent spin current on the Kondo resonance. As we haveseen, a finite spin current generates spin-dependent chemicalpotentials that are now encoded through the function Q(V)
verifying Q(0)=0.
Let us first discuss the case where a static induced Zeeman
field is not present; i.e., B
eff=0. Note that this can be achieved
by fine-tuning the gate voltage of a quantum dot to the particle-hole symmetric point according to Eq. (11) where B
eff=0.
In the STM setup, this corresponds to the situation whereP=0. This was implemented experimentally in Ref. 47by
coating a magnetic tip with several layers of copper, in otherwords by introducing a normal metallic spacer between thespin-polarized tip and the atom.
One may first try to expand the function Q(V)i np o w e r so f
Vwhich should correctly capture the behavior at low voltage.
Keeping the first linear order in Vsuch that Q(V)≈aV,
we found numerically by solving the equation of motion foreach value of the bias that the Kondo resonance does not split(we take P=0). Indeed, this is consistent with the condition
given in Eq. (13) which implies a=0. Moreover, at small
bias, we expect a small nonequilibrium effective magneticfield according to Eq. (11) which does not split the Kondo
resonance for B
eff/lessorsimilarTK.F r o mE q . (4), we see that the spin
current Ispin=IL↑−IL↓is obviously a function of the bias V
but also of Q(V). However, in the left lead, we expect /Delta1μL=
μL↑−μL↓=2μLQto be proportional to Ispin.40Therefore,
the function Q(V) is highly nonlinear and needs ap r i o r i to
be determined self-consistently. However, this turns out to bea very difficult task (this is a nonlinear and nonequilibriuminteracting problem). We have chosen a different and simplerstrategy by assuming various phenomenological forms forQ(V) taking into account the constraints imposed on the
function Q(V) at small and large bias.
Since all the current must proceed through the single
impurity states, the spectral weight of the impurity necessarilylimits the total amount of current. In other words, a finitespin current entails a finite spin accumulation which cannotgrow infinitely in a nanoscale structure. Therefore, the functionQ(V) is upper bounded and must converge asymptotically to
a constant Q
0at large V.
In order to reconcile both the small and large Vlimits
we discussed, we have tested the following phenomenologicalforms for Q(V) given by
Q
1(V)=Q0tanh (|V|/Vc), (14a)
Q2(V)=Q0[1−exp(−|V|/Vc)]. (14b)
At large |V|,Qi(V)→Q0, with i=1,2. We have used Vc
as a phenomenological energy scale which is related to the
impurity orbital nonequilibrium polarization. The larger Vc,
the less susceptible to be spin polarized the impurity is. WhenV
cis small compared to the voltage bias range explored, one
can neglect the exponential and Q1/2(V)≈Q0. We therefore
recover the constant- Qcase studied in Sec. III, which as
165403-5LIM, L ´OPEZ, LIMOT, AND SIMON PHYSICAL REVIEW B 88, 165403 (2013)
0.050.10.150.2
-0.5 -0.25 0 0.25 0.5dI/dV (2e2/h)
V/ΓP=0
Q0=0.5
μL=0.2Vc/TK=0.1
=1
=10
=100
FIG. 4. (Color online) Differential conductance for P=0a n di n
the presence of bias-dependent spin chemical potentials determined
byQ1(V)=Q0tanh (−|V|/Vc)for different values of Vc. The other
parameters are the same as before.
we have seen leads to a splitting of the Kondo resonance.
Since the main energy scale entering into our problem is thebare Kondo temperature T
K, one has to compare VctoTK.
Note here that the bare Kondo temperature is the Kondo tem-perature the impurity would acquire in absence of polarizationor spin accumulation. We have first computed the differentialconductance for Q
0=0.5 without any static polarization
(P=0) for different values of Vcusing the function Q1(V).
The results are summarized in Fig. 4. We found that for
Vc/lessorsimilar10TK, the constant- Qapproximation provides results
qualitatively similar to the constant- Qapproximation with a
peak splitting. Only for large value of Vc/greatermuch10TK,d ow e
recover a Kondo resonance. Indeed for large Vc/greatermuch10TK,
we can expand the functions Q1/2(V)i nV/V csince we are
interested in bias of order of a few TK. One can check that
keeping the lowest terms of the expansion does not lead to asplitting of the peak using Eq. (13). We have also performed
the same calculations with the function Q
2(V) defined in
Eq.(14b) . The results are similar to the ones in Fig. 4.
We have also considered the case of a finite polarization
P. When PandQhave the same sign, the Kondo resonance
is always split as explained in Sec. III. We have computed
the differential conductance for P=−0.18 and Q0=0.5f o r
different values of VcusingQ2(V). Our results are shown in
Fig. 5(b). We found that for Vc/lessorsimilar10TK, as in the constant-
Qapproximation, the static polarization and the spin-bias-
dependent chemical potential provide opposite effects whichresult in a restoration of the Kondo peak. Only for large valueofV
c/greatermuch10TKare we dominated by the static polarization,
which entails a splitting of the Kondo peak.
V . CONCLUSION
In this paper we have studied a ferromagnetic-impurity-
normal geometry paying attention to the fact that the fer-romagnetic electrode is able to also sustain a spin current.Such a generic geometry can describe an artificial impurity(a quantum dot, a molecule, a carbon nanotube, etc.) or agenuine impurity contacted between a spin-polarized electrode0.050.10.150.2
-0.5 -0.25 0 0.25 0.5dI/dV(2e2/h)
V/ΓP=-0.18
Q0=0.5
μL=0.2Vc/TK=0.1
=1
=10
=100
FIG. 5. (Color online) Differential conductance for P=−0.18
andQ2(V)=Q0[1−exp(|V|/Vc)] for different values of Vc.T h e
other parameters are the same as before.
and a normal one. For STM experiments, this corresponds
to a magnetic adatom adsorbed on a metallic surface and incontact with a spin-polarized tip. We have first considered thecase in which the spin current is injected in one electrodeand therefore maintained constant as realized experimentallyin Ref. 44. By computing the spectral functions and the
differential conductance, we found that this leads to a splittingof the Kondo resonance. In that respect, the Kondo effect turnsout to be a very sensitive phenomenon to detect a microscopicnonequilibrium spin current. Since the nonequilibrium spinaccumulation and the equilibrium polarization have differentorigin, they can have antagonist effects. We have shown bothanalytically and numerically that the Kondo resonance can berestored when this is the case. Since the effect of the staticpolarization on an artificial impurity can be controlled by agate voltage, this offers a knob to observe such a restoration ofthe Kondo resonance. We have also considered the case wherea spin current is generated when a voltage is applied betweenthe two electrodes. Under this hypothesis, the spin-dependentchemical potentials become voltage dependent. Assumingsimple phenomenological functions for this dependence thatmatch the low-voltage and large-voltage case, we were ableto show that the splitting of the Kondo resonance depends ona energy scale V
cwhich can be related to the polarizability
of the impurity. For Vc/TK/greatermuch1, no splitting is found which
corresponds to the case where the generated spin current at afixed voltage Vis too small to split the resonance of width
of order T
K. In the other limit, we recover results similar
to the constant spin accumulation. We think that this semi-phenomenological treatment captures the essential physics.Future work will be necessary to describe this phenomenonmicroscopically without further assumptions.
We have analyzed in the paper an anisotropic situation in
which only one electrode is spin polarized. A natural extensionof the present analysis is the more isotropic case in which themagnetic impurity is coupled to two spin-polarized electrodesthat are able to sustain a spin current. Such a situation mayapply to atomic contacts made from ferromagnetic materials
165403-6NONEQUILIBRIUM SPIN-CURRENT DETECTION WITH A ... PHYSICAL REVIEW B 88, 165403 (2013)
where the observation of Kondo-Fano line shapes in the
conductance have been reported.56,57
ACKNOWLEDGMENTS
We would like to thank M. V . Rastei for fruitful dis-
cussions. J.-S.L. and R.L. were supported by MICINNGrant No. FIS2011-2352. P.S. has benefited from financialsupport from the ANR under Contract No. DYMESYS(ANR 2011-IS04-001-01).
APPENDIX: EXPLICIT EXPRESSIONS FOR THE
RETARDED GREEN’S FUNCTION Gr
σ,σ(ω)
In this appendix we derive the retarded Green’s function to
be used for the calculation of the differential conductance. Forthat purpose we employ the equation of motion technique
48–51
and in particular the truncation schemes proposed in Refs. 53
and 54. These truncation procedures have been demon-
strated to describe properly systems attached to spin-polarized contacts. The retarded impurity Green’s function isdefined as
G
r
σ,σ(ω)≡/angbracketleft /angbracketleftdσ,d†
σ/angbracketright/angbracketrightrω=/integraldisplay
dteiωt/angbracketleft/angbracketleftdσ,d†
σ/angbracketright/angbracketrightrt, (A1)
where
/angbracketleft/angbracketleftdσ,d†
σ/angbracketright/angbracketrightrt=−i/Theta1(t)/angbracketleft[dσ(t),d†
σ(0)]/angbracketright. (A2)
For a generic two-particle operator /angbracketleft/angbracketleftA,B/angbracketright/angbracketrightr
ωwe have for its
equation of motion
ω/angbracketleft/angbracketleftA,B/angbracketright/angbracketrightr
ω+/angbracketleft /angbracketleft[H,A],B/angbracketright/angbracketrightr
ω=/angbracketleft[A,B]+/angbracketright. (A3)When A=dσ,B=d†
σ, the previous expression gives rise the
equation of motion for the impurity Green’s function in thefrequency domain. The imaginary part i0
+going alongside ω
is implicitly assumed. To simplify the notations, hereafter wewrite the retarded Green’s functions /angbracketleft/angbracketleftA,d
†
σ/angbracketright/angbracketrightrωas/angbracketleft/angbracketleftA/angbracketright/angbracketright.T h e
first equations of motion in the hierarchy are
(ω−εσ)/angbracketleft/angbracketleftdσ/angbracketright/angbracketright = 1+U/angbracketleft/angbracketleftdσn¯σ/angbracketright/angbracketright +/summationdisplay
α,kVαkσ/angbracketleft/angbracketleftcαkσ/angbracketright/angbracketright,
(ω−εαkσ)/angbracketleft/angbracketleftcαkσ/angbracketright/angbracketright =Vαkσ/angbracketleft/angbracketleftdσ/angbracketright/angbracketright, (A4)
where we take Vαkσas real. Then, we have
[ω−εσ−/Sigma10σ(ω)]/angbracketleft/angbracketleftdσ/angbracketright/angbracketright = 1+U/angbracketleft/angbracketleftdσn¯σ/angbracketright/angbracketright, (A5)
where we define /Sigma10σ(ω) as the hopping self-energy
/Sigma10σ(ω)=/summationdisplay
α,k|Vαkσ|2
ω−εαkσ. (A6)
To go to the next order we need to calculate /angbracketleft/angbracketleftdσn¯σ/angbracketright/angbracketrightas well:
(ω−εσ−U)/angbracketleft/angbracketleftdσn¯σ/angbracketright/angbracketright =/angbracketleftn¯σ/angbracketright+/summationdisplay
α,kVαkσ/angbracketleft/angbracketleftcαkσn¯σ/angbracketright/angbracketright
−/summationdisplay
α,kVαk¯σ/angbracketleft/angbracketleftc†
αk¯σd¯σdσ/angbracketright/angbracketright
+/summationdisplay
α,kVαk¯σ/angbracketleft/angbracketleftd†
¯σcαk¯σdσ/angbracketright/angbracketright.(A7)
The next step is to consider the equation of motion for
each of the three higher order Green’s functions /angbracketleft/angbracketleftcαkσn¯σ/angbracketright/angbracketright,
/angbracketleft/angbracketleftc†
αk¯σd¯σdσ/angbracketright/angbracketright, and/angbracketleft/angbracketleftd†
¯σcαk¯σdσ/angbracketright/angbracketrightthat appear on the right-hand
side of Eq. (A7) . Below we write these three equation of
motions approximated as
(ω−εαkσ)/angbracketleft/angbracketleftcαkσn¯σ/angbracketright/angbracketright =Vαkσ/angbracketleft/angbracketleftdσn¯σ/angbracketright/angbracketright −/summationdisplay
β,qVβq¯σ/angbracketleft/angbracketleftcαkσc†
βq¯σd¯σ/angbracketright/angbracketright +/summationdisplay
β,qVβq¯σ/angbracketleft/angbracketleftcαkσd†
¯σcβq¯σ/angbracketright/angbracketright, (A8)
(ω−εαk¯σ+ε¯σ−εσ)/angbracketleft/angbracketleftd†
¯σcαk¯σdσ/angbracketright/angbracketright = /angbracketleftd†
¯σcαk¯σ/angbracketright+Vαk¯σ/angbracketleft/angbracketleftdσn¯σ/angbracketright/angbracketright −/summationdisplay
β,qVβq¯σ/angbracketleft/angbracketleftc†
βq¯σcαk¯σdσ/angbracketright/angbracketright +/summationdisplay
β,qVβqσ/angbracketleft/angbracketleftd†
¯σcαk¯σcβqσ/angbracketright/angbracketright, (A9)
(ω+εαk¯σ−εσ−ε¯σ−U)/angbracketleft/angbracketleftc†
αk¯σd¯σdσ/angbracketright/angbracketright = /angbracketleftc†
αk¯σd¯σ/angbracketright−Vαk¯σ/angbracketleft/angbracketleftdσn¯σ/angbracketright/angbracketright +/summationdisplay
β,qVβqσ/angbracketleft/angbracketleftc†
αk¯σd¯σcβqσ/angbracketright/angbracketright +/summationdisplay
β,qVβq¯σ/angbracketleft/angbracketleftc†
αk¯σcβq¯σdσ/angbracketright/angbracketright.
(A10)
In the next step we truncate the system of equations keeping the Kondo correlations. To abbreviate the notation, we introduce
a shorthand Fσ
a;b≡/angbracketleftc†
aσcbσ/angbracketright. We consider the onset of Kondo correlations by approximating Eqs. (A8) ,(A10) , and (A9) in the
following way:
/summationdisplay
α,kVαkσ/angbracketleft/angbracketleftcαkσn¯σ/angbracketright/angbracketright ≈/summationdisplay
α,k|Vαkσ|2
ω−εαkσ/angbracketleft/angbracketleftdσn¯σ/angbracketright/angbracketright =/Sigma10σ(ω)/angbracketleft/angbracketleftdσn¯σ/angbracketright/angbracketright, (A11)
/summationdisplay
α,kVαk¯σ/angbracketleft/angbracketleftd†
¯σcαk¯σdσ/angbracketright/angbracketright ≈/Sigma10¯σ(ω,¯σ;σ,αk ¯σ)/angbracketleft/angbracketleftdσn¯σ/angbracketright/angbracketright +/summationdisplay
α,kVαk¯σ/angbracketleftd†
¯σcαk¯σ/angbracketright
ω−εαk¯σ+ε¯σ−εσ[1+/Sigma10σ(ω)/angbracketleft/angbracketleftdσ/angbracketright/angbracketright]
−/summationdisplay
α,k/summationdisplay
β,qVβq¯σVαk¯σF¯σ
βq;αk
ω−εαk¯σ+ε¯σ−εσ/angbracketleft/angbracketleftdσ/angbracketright/angbracketright, (A12)
/summationdisplay
α,kVαk¯σ/angbracketleft/angbracketleftc†
αk¯σd¯σdσ/angbracketright/angbracketright ≈ − /Sigma10¯σ(ω,αk ¯σ;σ,¯σ,U)/angbracketleft/angbracketleftdσn¯σ/angbracketright/angbracketright +/summationdisplay
α,kVαk¯σ/angbracketleftc†
αk¯σd¯σ/angbracketright
ω+εαk¯σ−εσ−ε¯σ−U[1+/Sigma10σ(ω)/angbracketleft/angbracketleftdσ/angbracketright/angbracketright]
+/summationdisplay
α,k/summationdisplay
β,qVαk¯σVβq¯σF¯σ
αk;βq
ω+εαk¯σ−εσ−ε¯σ−U/angbracketleft/angbracketleftdσ/angbracketright/angbracketright, (A13)
165403-7LIM, L ´OPEZ, LIMOT, AND SIMON PHYSICAL REVIEW B 88, 165403 (2013)
where the self-energies appearing in Eqs. (A11) ,(A12) , and (A13) are defined accordingly
/Sigma10¯σ(ω,¯σ;σ,αk ¯σ)=/summationdisplay
α,k|Vαk¯σ|2
ω−εαk¯σ+ε¯σ−εσ, (A14a)
/Sigma10¯σ(ω,αk ¯σ;σ,¯σ,U)=/summationdisplay
α,k|Vαk¯σ|2
ω+εαk¯σ−εσ−ε¯σ−U. (A14b)
Using the previous truncated high-order propagators we can show that Eq. (A7) takes the expression
/angbracketleft/angbracketleftdσn¯σ/angbracketright/angbracketright =/angbracketleftn¯σ/angbracketright−/Sigma12σ(ω)/angbracketleft/angbracketleftdσ/angbracketright/angbracketright
ω−εσ−U−/Sigma10σ(ω)−/Sigma11σ(ω), (A15)
where we have defined
/angbracketleftn¯σ/angbracketright=/angbracketleftn¯σ/angbracketright+/summationdisplay
α,kVαk¯σ/angbracketleftd†
¯σcαk¯σ/angbracketright
ω−εαk¯σ+ε¯σ−εσ−/summationdisplay
α,kVαk¯σ/angbracketleftc†
αk¯σd¯σ/angbracketright
ω+εαk¯σ−εσ−ε¯σ−U, (A16)
together with the self-energies
/Sigma11σ(ω)=/Sigma10¯σ(ω,¯σ;σ,αk ¯σ)+/Sigma10¯σ(ω,αk ¯σ;σ,¯σ,U), (A17)
/Sigma12σ(ω)=/summationdisplay
α,k/summationdisplay
β,qVβq¯σVαk¯σF¯σ
βq;αk
ω−εαk¯σ+ε¯σ−εσ+/summationdisplay
α,k/summationdisplay
β,qVαk¯σVβq¯σF¯σ
αk;βq
ω+εk¯σ−εσ−ε¯σ−U
−/bracketleftBigg/summationdisplay
α,kVαk¯σ/angbracketleftd†
¯σcαk¯σ/angbracketright
ω−εαk¯σ+ε¯σ−εσ−/summationdisplay
α,kVαk¯σ/angbracketleftc†
αk¯σd¯σ/angbracketright
ω+εk¯σ−εσ−ε¯σ−U/bracketrightBigg
/Sigma10σ(ω). (A18)
Taking back Eq. (A5) into Eq. (A7) the dot retarded Green’s function is finally obtained:
/angbracketleft/angbracketleftdσ/angbracketright/angbracketright =1−/angbracketleftn¯σ/angbracketright
ω−εσ−/Sigma10σ(ω)+U/Sigma1 2σ(ω)
ω−εσ−U−/Sigma10σ(ω)−/Sigma11σ(ω)+/angbracketleftn¯σ/angbracketright
ω−εσ−U−/Sigma10σ(ω)+U[/Sigma12σ(ω)−/Sigma11σ(ω)]
ω−εσ−/Sigma10σ(ω)−/Sigma11σ(ω). (A19)
It is worth considering the limit U→∞ where the impurity Green’s function expression (A19) is greatly simplified as
/angbracketleft/angbracketleftdσ/angbracketright/angbracketright =1−/angbracketleftn¯σ/angbracketright−/summationtext
α,kVαk¯σ/angbracketleftd†
¯σcαk¯σ/angbracketright
ω−εαk¯σ+ε¯σ−εσ
ω−εσ−/Sigma10σ(ω)−/summationtext
α,k/summationtext
β,qVβq¯σVαk¯σF¯σ
βq;αk
ω−εαk¯σ+ε¯σ−εσ+/summationtext
α,kVαk¯σ/angbracketleftd†
¯σcαk¯σ/angbracketright
ω−εαk¯σ+ε¯σ−εσ/Sigma10σ(ω). (A20)
N o ww ef o l l o wR e f . 50to obtain a much simpler truncated impurity Green’s function by setting /angbracketleftd†
σckσ/angbracketright=0 andFσ
qk=δk,qf(εkσ).
By doing this, the impurity Green’s function is given by
/angbracketleft/angbracketleftdσ/angbracketright/angbracketright =1−/angbracketleftn¯σ/angbracketright
ω−εσ−/Sigma10σ(ω)−/Sigma12σ(ω), (A21)
where
/Sigma12σ(ω)≈/summationdisplay
α,k|Vαk¯σ|f(εαk¯σ)
ω−εαk¯σ+ε¯σ−εσ+i0+≈−/summationdisplay
α/Gamma1α¯σ
π/bracketleftbigg
iπfα¯σ(ω+ε¯σ−εσ)+ln/radicalbig
(π/β)2+(ω+ε¯σ−εσ−μα¯σ)2
D/bracketrightbigg
.(A22)
According to Ref. 53,i nE q . (A21) the bare dot level εσin/Sigma11σ(ω) must be replaced by the renormalized one /tildewideεσthat must be
self-consistently found from the relation
/tildewideεσ=εσ+Re[/Sigma10σ(/tildewideεσ)+/Sigma11σ(/tildewideεσ,/tildewideε↑,/tildewideε↓)]. (A23)
By doing this, we include spin-dependent high-order contributions that produce the effective exchange field when spin-dependent
tunneling couplings are considered.
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PhysRevB.98.174201.pdf | PHYSICAL REVIEW B 98, 174201 (2018)
Editors’ Suggestion
Configuration-controlled many-body localization and the mobility emulsion
Michael Schecter, Thomas Iadecola, and Sankar Das Sarma
Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland,
College Park, Maryland 20742, USA
(Received 27 August 2018; revised manuscript received 12 October 2018; published 2 November 2018)
We uncover a new nonergodic phase, distinct from the many-body localized (MBL) phase, in a disordered
two-leg ladder of interacting hardcore bosons. The dynamics of this emergent phase, which has no single-particleanalog and exists only for strong disorder and finite interaction, is determined by the many-body configurationof the initial state. Remarkably, this phase features the coexistence of localized and extended many-body states
at fixed energy density and thus does not exhibit a many-body mobility edge, nor does it reduce to a modelwith a single-particle mobility edge in the noninteracting limit. We show that eigenstates in this phase can bedescribed in terms of interacting emergent Ising spin degrees of freedom (“singlons”) suspended in a mixturewith inert charge degrees of freedom (“doublons” and “holons”) and thus dub it a mobility emulsion (ME).
We argue that grouping eigenstates by their doublon/holon density reveals a transition between localized andextended states that is invisible as a function of energy density. We further demonstrate that the dynamics of thesystem following a quench may exhibit either delocalizing or localized behavior depending on the doublon/holondensity of the initial product state. Intriguingly, the ergodicity of the ME is thus tuned by the initial state of themany-body system. These results establish a new paradigm for using many-body configurations as a tool to studyand control the MBL transition. The ME phase may be observable in suitably prepared cold atom optical lattices.
DOI: 10.1103/PhysRevB.98.174201
I. INTRODUCTION
Nonergodic quantum systems have attracted much atten-
tion in recent years due to rapid progress in a growing numberof experimental systems including Rydberg atoms [ 1–3] and
interacting disordered systems such as cold atoms [ 4–12] and
trapped ions [ 13,14]( s e ea l s oR e f .[ 15]), making possible
a systematic study of their nontrivial dynamical behavior.
They are also of deep conceptual interest in understanding
the applicability of quantum statistical mechanics to isolatedsystems. Nonergodic systems are exceptional in that theyare fundamentally incompatible with the laws of statisticalmechanics and generally do not relax to thermal equilibriumas described by the Gibbs ensemble. It is well-known thatnonergodicity arises in exactly solvable quantum integrable
models (similar to what happens in classical integrable sys-
tems, e.g., the Fermi-Pasta-Ulam model [ 16–18]) due to their
extensive number of integrals of motion (see Ref. [ 19]f o r
a recent discussion). However, it is frequently believed thatquantum integrability in a many-body system is generallyunstable to the addition of weak integrability-breaking per-turbations due to the absence of a quantum analog of the
KAM theorem (see also [ 20]). Thus such explicitly integrable
quantum systems are not generic, although the associatedlack of thermalization has been studied in carefully preparedlaboratory experiments [ 21].
An alternative route to robust nonergodicity has recently
emerged in the context of many-body localization (MBL),which occurs in (presumably generic) interacting quantumsystems subject to quenched disorder [ 22–50]. It is well
established that the MBL phase possesses a robust and emer-
gent integrability associated with an extensive number oflocal integrals of motion (LIOMs) [ 30,31,35,39,42–45]. This
manifestation of nonergodicity represents a spectacular de-parture from the laws of statistical mechanics (i.e., isolatedinteracting quantum systems may not be thermal generically)as it allows the possibility of spontaneous symmetry-breakingand topological phase transitions even at infinite temperature,i.e., for eigenstates with arbitrary energy density [ 51–63].
Despite the richness of the MBL phase itself, most previousstudies have focused on systems that exhibit only two typesof dynamical behavior: the fully MBL phase with completeemergent integrability (i.e., the number of LIOMs matches thenumber of degrees of freedom) or the usual thermal ergodicphase that satisfies the eigenstate thermalization hypothesis(ETH) [ 64–68] and obeys the laws of statistical mechanics.
The present work introduces the possibility of an intriguingintermediate phase between the MBL and ETH phases, whichoccurs at finite interaction and large disorder, and is trulyemergent in the sense that it has no single-particle analog.
The intermediate phase we find is qualitatively different fromother intermediate phases which have recently been discussedin the MBL literature, as it is not rooted in any mobility-edge physics (either single-particle or many-body) or Griffithsphysics of rare regions.
Intermediate phases that possess a single-
particle [ 48,49,69,70] or many-body [ 23,36,50] mobility
edge represent possible nonergodic phases with incompleteintegrability, but the existence of a many-body mobility edgeis still controversial [ 71]. However, studying this physics
experimentally is challenging; for example, probing thecritical energy or energy density of such an intermediate phasevia quench dynamics requires the ability to prepare the initialstate in an energy-resolved manner. Other models whose
2469-9950/2018/98(17)/174201(18) 174201-1 ©2018 American Physical SocietySCHECTER, IADECOLA, AND DAS SARMA PHYSICAL REVIEW B 98, 174201 (2018)
FIG. 1. Schematic depiction of the mobility emulsion. The solid
black bonds represent entanglement between emergent Ising spins
(singlons), cf. Fig. 2. (a) Above the critical doublon/holon density, a
typical eigenstate consists of a sparse network of singlons suspendedin a background of localized doublons and holons. The system sup-
ports clusters of interacting singlons, but these clusters are typically
much farther apart than their typical size, so that the network ofsinglons is many-body localized. Processes admixing these clusters
with doublons and holons are far off resonance at strong disorder,
and the “emulsion” is stable. (b) Below the critical doublon/holondensity, a typical eigenstate consists of a sparse set of doublons
and holons suspended in a delocalizing bath of singlons. Doublons
exchange energy with this bath and undergo variable-range hopping
(dashed bonds), thereby mediating the transport of charge.
disorder respects continuous SU(2) symmetries [ 72–75]
have been argued to exhibit nonergodicity with incompleteintegrability for small system sizes, but it is likely thatsuch systems ultimately undergo thermalization at longtimes in the thermodynamic limit due to the fundamentalincompatibility of MBL and the (continuous) non-Abeliansymmetry [ 58–60,63,74]. Here we focus on a model that has
only a global Z
2symmetry, which is known to be compatible
with MBL (see, e.g., Ref. [ 51]).
In this paper, we report on the existence of a nonergodic
intermediate phase that neither satisfies ETH nor is fullyMBL, but at the same time is not connected with any many-body (or single-particle) mobility edge. Rather, this phase isdefined by the coexistence of localized and extended many-
body states at fixed energy density. We show that eigenstatesof the system can be described in terms of a mobility emulsion
(ME), wherein emergent interacting Ising spin degrees offreedom (denoted “singlons”) become suspended in a mixturewith inert charge degrees of freedom (denoted “doublons”and “holons”), see Fig. 1. The disorder in this case acts as
a “surfactant” that stabilizes the singlon/doublon emulsionprovided that the emergent doublon/holon density, n
DH,i s
sufficiently large. We argue that grouping eigenstates by nDH
reveals a transition, at a critical value n∗
DH, between localized
and extended states that is absent as a function of energydensity. Thus the ME phase, in addition to requiring finiteinteraction and strong disorder, is configurationally controlledthrough the relative singlon and doublon/holon densities in theinitial state. We emphasize that the standard ETH and MBLphases also exist in the system for small and large disorder,respectively.
FIG. 2. Two-leg ladder with identical (mirror-symmetric) disor-
der potentials on the two legs. For strong disorder, mirror-related
sites that share one boson form emergent Ising spin degrees of
freedom (red arrows) suspended in a mixture of doublons (doubly
occupied rungs) and holons (empty rungs) denoted by full and open
circles, respectively.
In eigenstates with sufficiently large doublon/holon den-
sity,nDH>n∗
DH, the sparse singlons interact weakly and are
frozen into paramagnetic configurations, remaining MBL. Ineigenstates with n
DH<n∗
DH, the enhanced density of singlons
strengthens their mutual interactions, allowing the singlonsto delocalize among themselves. The delocalized bath of sin-glons then mediates variable-range hopping of the remainingdilute doublons and holons, which in turn interact as theypropagate.
These two cases can be distinguished sharply in an exper-
imental setting by observing the dynamics of the system fol-lowing quenches from initial product states with different val-ues of n
DH. Initial states with nDH>n∗
DHretain memory of the
initial state at long times, while initial states with nDH<n∗
DH
lose this memory. Remarkably, this distinction can be made
even in a fixed disorder realization and for fixed Hamiltonianparameters; one need only tune the doublon/holon density ofthe initial state. This configuration-controlled localization inan intermediate phase between ETH and MBL leads to a newparadigm in the study and manipulation of nonergodic phasesof matter.
To exemplify this paradigm, we focus on interacting hard-
core bosons in a two-leg ladder whose legs are subject toidentical disorder potentials, see Fig. 2. This system can
equivalently be viewed as a coupled pair of identical random-field XXZ spin chains. Here, the singlon and doublon/holonstates correspond to rungs of the ladder that either host a singleparticle or are full/vacant, respectively (see Fig. 2). Because
the two legs have the same disorder potential, the systempossesses a Z
2mirror symmetry that exchanges the legs of the
ladder. As we show below, in the MBL phase this symmetrycan break spontaneously, giving rise to “mirror-glass” orderin a nonzero fraction of states at nonzero energy density.The mechanism behind this symmetry breaking was studiedrecently [ 76] in a case involving a single chain with mirror-
symmetric disorder. In the current two-leg ladder system, themirror symmetry arises naturally by virtue of the disorderbeing the same in the two individual chains. Remarkably,the long-range mirror-glass order that arises in the two-leg
174201-2CONFIGURATION-CONTROLLED MANY-BODY … PHYSICAL REVIEW B 98, 174201 (2018)
FIG. 3. The dynamical phase diagram of Eq. ( 2.1)a t/Delta1=0.5.
Here and in the remainder of the paper, we work in units such thatthe intraleg hopping amplitude J=1. In addition to the ETH and
MBL phases, we find a new phase—the mobility emulsion (ME)—
that predominates at strong disorder. We determine transitions usingthe mirror-glass order parameter q
n[Eq. ( 2.4)] for the MBL/ME
phase boundary line and the doublon correlator pn[Eq. ( 2.6)] for
the ETH/ME phase boundary line. See Sec. II Bfor representative
examples of the exact-diagonalization data used to obtain the points
on the above phase diagram.
ladder (and in Ref. [ 76]) can occur only in states with a
nonzero energy density above (below) the ground (ceiling)states, which remain symmetric throughout the MBL phase.This “inverse freezing” effect is due to the fact that the groundstate essentially contains only doublons and holons, whichtransform trivially under the mirror symmetry. Singlons, onthe other hand, carry nontrivial representations of the mirrorsymmetry, but only become activated in excited states [ 76].
When a nonzero density of singlons are activated, their mutualinteractions drive the mirror-symmetry-breaking transition.Importantly, however, doublon/holon states are not restrictedto the tails of the spectrum and can be found with any fraction,n
DH>0, at any energy density.
Unlike the case of discrete non-Abelian symmetries
[58–60,63], in the ladder model studied here, there is no
obstruction to having mirror-symmetric MBL eigenstates at fi-nite energy density in which the singlons form a paramagneticstate. Indeed, as one tunes the interleg hopping amplitude J
⊥
at large disorder, we find that the MBL mirror-glass phase
melts directly into the symmetric nonergodic ME phase asshown in the phase diagram of Fig. 3. Thus, the ME phase
requires finite interaction, large disorder, and strong interchainhopping for its existence in the two-leg ladder system. Wemention here that, in addition to not having a single-particleanalog (e.g., no single-particle mobility edge), the ME phasealso has no single-chain analog for spinless particles as it isdriven explicitly by tuning the interchain hopping to producethe appropriate singlon-holon-doublon dynamics necessaryfor its existence.
The remaining part of the paper is organized as follows. In
Sec. II, we introduce the model of the two-leg ladder studied
throughout. We present its infinite-temperature phase diagramand discuss the numerical diagnostics used to construct it.In Sec. III, we derive an effective singlon Hamiltonian using
a Schrieffer-Wolff transformation to systematically eliminatethe longitudinal hopping at large disorder. This allows us tocharacterize and develop intuition for both the MBL and ME
phases, and to estimate the critical point that separates them.In Sec. IV, we present further numerical results characterizing
the ME phase, showing in particular the lack of a mobilityedge as a function of many-body energy density and the cor-relation between the degree of entanglement in an eigenstateand its doublon/holon density. In Sec. V, we consider the
dynamics of local observables following quantum quenchesfrom various local-density product states, which are read-ily preparable experimentally. We find substantial qualitativedifferences in the late-time behavior of local observablesdepending on the initial value of the doublon/holon density.Discussion and conclusions are presented in Sec. VI.
II. MODEL AND PHASE DIAGRAM
We study hardcore bosons hopping on a disordered two-
leg ladder with a Z2leg-permutation (mirror) symmetry. The
Hamiltonian is given by
H=H1+H2+H⊥, (2.1a)
where
Hα=L/summationdisplay
i=1/bracketleftbiggJ
2(b†
α,ibα,i+1+H.c.)+/Delta1/parenleftbigg
nα,i−1
2/parenrightbigg/parenleftbigg
nα,i+1−1
2/parenrightbigg/bracketrightbigg
+2L/summationdisplay
i=1hα,i/parenleftbigg
nα,i−1
2/parenrightbigg
(2.1b)
and
H⊥=J⊥
2L/summationdisplay
i=1(b†
1,ib2,i+H.c.). (2.1c)
Here, b†
α,i/bα,iare boson creation/annihilation operators on
rungiand leg α=1,2,nα,i=b†
α,ibα,iis the local boson
density, and Lis the system length. We assume the bosons
interact strongly onsite and thus satisfy the hardcore constraint
nα,i(nα,i−1)=0 with commutation relations [ bα,i,b†
β,j]=
δijδαβ(1−2nα,i), which allows one to map the problem onto
a pair of coupled XXZ chains. The Hamiltonian Hpossesses
a global U(1) symmetry associated with conservation of totalparticle number and a Z
2mirror symmetry M, which inter-
changes the leg indices, 1 ↔2, and implies that the disorder
in the two legs is identical,
h1,i=h2,i≡hi. (2.1d)
We focus on the case of half-filling (equivalently, on the
zero-magnetization sector of the XXZ ladder) with periodicboundary conditions and consider independent random onsitepotentials h
idrawn from a normal distribution with mean
zero and standard deviation W/2f o ri=1,...,L . The mirror
symmetry can be realized experimentally in a number of ways,but perhaps the simplest is to use a two-dimensional opticallattice subject to an additional longitudinal disorder potentialand a transverse confining potential supporting two minima,see Fig. 2.
We performed an exact diagonalization study of the
model ( 2.1) at small system sizes. The numerically
174201-3SCHECTER, IADECOLA, AND DAS SARMA PHYSICAL REVIEW B 98, 174201 (2018)
determined infinite-temperature phase diagram of Has a
function of WandJ⊥is shown in Fig. 3. We devote the
remainder of this section to understanding this phase diagramand describing how it is obtained.
A. Qualitative understanding of the phase diagram
We begin by considering the limit of decoupled legs, J⊥=
0, where the phase diagram is easiest to understand. In thiscase, the problem reduces to that of two independent copiesof the random-field XXZ chain, where it is numerically estab-lished [ 24,25,27–30,32–36,38,40–42,45–47] that the system
undergoes a phase transition from an ergodic phase to anMBL phase at a critical disorder strength. At finite J
⊥,i ti s
reasonable to expect that the ergodic phase persists so longas the disorder strength Wis sufficiently weak. Eigenstates
at nonzero energy density remain mirror-symmetric in theergodic phase, since long-range order at nonzero temperatureis thermodynamically forbidden in one dimension [ 77–79].
We now turn to the MBL phase at J
⊥=0. In this case,
the eigenstates on each leg can be labeled by suitably dressedoccupation factors, which constitute the eigenvalues 0 or 1of the LIOMs ˜n
α,i. When the intraleg hopping J=0, the
local state on rung ican be written as |n1,i,n2,i/angbracketright≡|n1,n2/angbracketrighti,
withn1,2=0,1; this yields four states per rung. Of these
four states, only two transform nontrivially under the mirrorsymmetry M, namely the “singlon” states
|1,0/angbracketright
i≡|↑ /angbracketright iand|0,1/angbracketrighti≡|↓ /angbracketright i, (2.2a)
which are degenerate because the disorder potential respects
M. Indeed, from Eqs. ( 2.1b) and ( 2.1d), one sees that the sin-
glon states do not couple directly to the disorder potential. Incontrast, the mirror-symmetric “doublon” and “holon” states,
|1,1/angbracketright
i≡| • /angbracketright iand|0,0/angbracketrighti≡| ◦ /angbracketright i, (2.2b)
respectively, are split in energy by an amount of order W
due to the disorder potential. Doublon and/or holon stateson distinct rungs iandjare thus far off-resonance at large
W, whereas nearby singlon states are split comparatively
weakly by interactions /Delta1/lessmuchW. We emphasize that singlons,
doublons, and holons are only well-defined (i.e., long-lived)degrees of freedom at strong disorder. In the ETH phase, thereis no meaningful distinction among these degrees of freedom,as the local boson density on each rung is not approximatelyconserved, as it is at strong disorder.
At strong disorder, the singlon states ( 2.2a) can thus be
viewed as local states of an effective spin-1 /2 chain in which
the mirror symmetry Mbecomes an onsite Z
2symmetry
(the doublons and holons, on the other hand, are essentiallyinert) [ 76]. Crucially, this effective spin-1 /2 chain is disorder-
free atJ=0, since the disorder potential does not couple
directly to the singlons. At small but finite intraleg hoppingJ/lessmuchW, an effective interaction is generated between these
effective spin configurations, due to the repulsion /Delta1between
neighboring particles, and is randomly renormalized by thedressing of the occupation factors, /Delta1n
α,inα,j→˜/Delta1ij˜nα,i˜nα,j.
In Sec. III, we calculate the random corrections to the interac-
tion energy in the limit of strong disorder, where perturbationtheory in J/W is controlled (up to rare-region effects). These
renormalized random interactions among singlons lead tospontaneous breaking of the Z
2symmetry of the effective
singlon spin chain in eigenstates for which a finite fractionof the rungs occupy singlon states [ 76]. (As mentioned earlier,
these eigenstates are necessarily at finite energy density owingto the fact that the ground and ceiling states consist over-whelmingly of doublon and holon states on each rung.) TheMBL phase at J
⊥=0 thus breaks the mirror symmetry M
by default, yielding long-range mirror-glass order in typicaleigenstates at finite energy density.
Next, we consider the fate of the decoupled-chain MBL
phase upon adding a finite interchain coupling J
⊥. In Sec. III,
we show that this coupling induces a “transverse field” in thesinglon spin model, thereby enhancing the quantum fluctua-tions of the singlons. If the transverse field is much weakerthan the random interactions induced by the finite intraleghopping J, then we expect that the system remains MBL and,
furthermore, that Mremains spontaneously broken. However,
if the quantum fluctuations induced by the transverse fielddominate over the effective bond randomness in the inter-actions, localization is no longer guaranteed. Thus, as J
⊥
increases, we expect MBL eigenstates with broken mirror
symmetry to give way to delocalizing eigenstates that neces-sarily preserve the mirror symmetry. It is the delocalization ofsinglons due to the interchain coupling that drives the strong-disorder transition between the MBL mirror-glass phased atweakJ
⊥and the ME phase at larger J⊥. Furthermore, because
the singlon bond randomness is induced by virtual transitionsbetween singlon and doublon states (see Sec. III), it becomes
weaker asymptotically as the direct disorder strength Wis
increased; thus, we expect the critical value of J
⊥at which
the transition takes place to decrease with increasing W,a s
observed in Fig. 3. At strong disorder, the phase diagram thus
becomes dominated by the ME phase at nonzero J⊥.T h e
phase diagram for different finite values of interaction ( /Delta1) and
intraleg hopping ( J) is similar to Fig. 3.
We stress that the ME phase is not simply a reentrance
of the ergodic phase, where ETH holds in alleigenstates
at finite energy density; on the contrary, there is a sharptransition between them, as we show below. Indeed, althoughthe singlons tend to delocalize as Wis increased at finite J
⊥,
the doublons and holons tend to localize more strongly as theybecome further off-resonance with the singlons and with eachother. Thus, while eigenstates in which singlons predominatetend to delocalize, eigenstates in which doublons and holonspredominate tend to localize more strongly. This is perhapssuggestive of a many-body mobility edge that separates thedelocalized states dominated by singlons from the localizedstates dominated by doublons and holons. However, the term“many-body mobility edge” presupposes the existence of acritical many-body energy density that separates localized andextended states; this is not the case here. Eigenstates in whichall rungs of the ladder are occupied by singlons genericallyarise in the middle of the many-body spectrum, but eigenstatesin which all rungs are occupied by doublons or holons canarise at anyenergy density. This suggests that there is no
discernible transition between MBL and ETH eigenstates asa function of energy density. (We will present vivid numericalproof of this fact in Sec. IV.) Rather, we will argue in Sec. III
that there is a finite doublon/holon density n
∗
DHat which
a transition between delocalizing and localizing behavior
174201-4CONFIGURATION-CONTROLLED MANY-BODY … PHYSICAL REVIEW B 98, 174201 (2018)
occurs. Generic eigenstates of the two-leg ladder consist of
a mixture of “hot” singlons and “cold” doublons and holons,and their interconversion is heavily suppressed by disorder.The singlons, doublons, and holons are thus suspended ina mixture with one another, and configurational propertiesof this mixture determine whether or not an eigenstate isdelocalized or MBL. This is the essence of the mobilityemulsion.
B. Quantitative understanding of the phase diagram
We now discuss the quantitative indicators used to calcu-
late the phase diagram shown in Fig. 3. Given the discussion
in the previous section, it is necessary to keep track of thesinglon and doublon degrees of freedom that become stableexcitations at strong disorder. In the MBL mirror-glass phase,both singlons and doublons/holons are localized in space asthe onsite boson density is approximately conserved. More-over, the singlon “spin” degree of freedom is frozen into apattern of “magnetization” that spontaneously breaks the Z
2
mirror symmetry M. The “spin state” ( |↑/angbracketrightior|↓/angbracketrighti) of each
singlon can be measured using the local polarization
σi=n1,i−n2,i, (2.3a)
which gives ±1 when acting on the state |↑/angbracketrightior|↓/angbracketrighti, respec-
tively, and 0 when acting on the doublon ( |•/angbracketrighti) or holon ( |◦/angbracketrighti)
state. Note that the polarization σiis odd under M.I nt h eM E
phase, the singlons delocalize in the manner discussed in theprevious section; however, a finite fraction of all eigenstatescontain localized doublons and holons. One can keep trackof whether rung ihosts a doublon or holon using the local
density
d
i=n1,i+n2,i−1, (2.3b)
which gives ±1 when acting on the doublon and holon states
|•/angbracketrightiand|◦/angbracketrighti, respectively, and 0 when acting on the singlon
states |↑/angbracketrightiand|↓/angbracketrighti. Note that the number of doublons must
equal the number of holons in the system at half-filling.We will use these quantities to define two indicators thatdistinguish the three phases in Fig. 3.
To keep track of the freezing of singlons in the MBL
mirror-glass phase, we make use of the mirror-glass orderparameter defined in Ref. [ 76]. This order parameter is defined
at the level of individual eigenstates |E
n/angbracketrightwith many-body
energy En:
qn=1
L2L/summationdisplay
i,j=1/angbracketleftEn|σiσj|En/angbracketright2. (2.4)
The definition of qnis motivated as follows. At infinite
temperature, a generic eigenstate in the mirror-glass phasehas nonzero local polarization, /angbracketleftσ
i/angbracketright/negationslash=0, but the sign of
this polarization is generically random, so that/summationtext
i/angbracketleftσi/angbracketright=0.
However, the squares of these expectation values add coher-ently to yield q
n>0 for such states. In this sense, qnis a
faithful detector of spontaneous mirror-symmetry breaking ina many-body eigenstate. Such spontaneous symmetry break-ing can only occur at infinite temperature (i.e., q
naveraged
over all eigenstates) if a finite fraction of the eigenstatesare MBL. [However, the converse of this statement does nothold since singlons can form symmetric (paramagnetic) MBLFIG. 4. Representative plots of the mirror-glass order parameter
(2.4) and the doublon correlator ( 2.6) used to determine the horizon-
tal and vertical phase boundary lines, respectively, in Fig. 3. (Top)
The energy- and disorder-averaged mirror-glass order parameter(2.4) in the zero doublon/holon sector, rescaled by the system size L,
atW=4. For all data shown, we set /Delta1=0.5a n d J=1. The inset
shows the scaling collapse of the data near the transition point J
∗
⊥,f o r
a≈1,ν≈2/3, and J∗
⊥≈0.049. The approximate location of the
transition is shown as a grey vertical line in the main plot. (Bottom)
The infinite-temperature disorder-averaged doublon correlator ( 2.6),
rescaled by the system size L,a tJ⊥=0.1. The inset shows the
scaling collapse of the data near the transition point W∗,f o ra≈1,
ν≈2/3, and W∗≈1.4.
states when their typical separation is large and their mutual
interactions are weak.] Thus we also take the presence ofa nonvanishing infinite-temperature expectation value of themirror-glass order parameter as evidence of MBL. Moreover,q
n=0 generically in the ergodic phase, owing to the no-go
theorems mentioned in the previous section.
We plot the infinite temperature average of qnin Fig. 4
across the MBL-ME phase boundary (the horizontal bluecurve in Fig. 3) as a function of the interleg coupling J
⊥for
a representative choice of disorder strength W. To compute it,
we performed shift-invert exact diagonalization to target statesin the middle of the many-body spectrum at system sizesup to L=8 (i.e., for systems containing as many as 2 L=
16 sites). Results are averaged over 50 energy eigenstatesand at least 1250 disorder realizations. In order to obtainclearer finite-size scaling, we calculate q
nfor eigenstates in
the zero-doublon/holon sector. Eigenstates in this sector havethe strongest tendency to delocalize, as we will see in the nextsection. We use the following definition of the doublon/holondensity:
n
DH,n=1
LL/summationdisplay
i=1/angbracketleftEn|di|En/angbracketright2, (2.5)
174201-5SCHECTER, IADECOLA, AND DAS SARMA PHYSICAL REVIEW B 98, 174201 (2018)
which evaluates to zero in an eigenstate containing only sin-
glons, and which evaluates to one in an eigenstate containingonly doublons and holons. At finite W/J/greatermuch1, there is a small
admixture between doublons and singlons, which requires oneto use a threshold. The threshold n
DH<1/Lis sufficient to
define the zero-doublon/holon sector since the doublon/holondensity is approximately quantized in units of 2 /Lat half-
filling for W/J/greatermuch1 and for the system sizes studied. Once
this postselection has been made, a clear finite size scalingcollapse of the quantity Lqis observed near the transition (see
inset of top panel of Fig. 4). The correlation-length exponent ν
obtained from this scaling collapse is approximately 2 /3. The
critical interleg coupling J
∗
⊥is estimated from the crossing
point of the finite-size curves: to the left of the crossing, thequantity Lqscales to a finite value, while to the right of the
crossing, it scales to zero.
To probe the localization of doublons and holons at strong
disorder, which is characteristic of both the MBL and MEphases, we define the doublon/holon correlator,
p
n=1
L2/summationdisplay
ij/angbracketleftEn|didj|En/angbracketright2. (2.6)
This correlator operates on a similar principle to the mirror-
glass order parameter ( 2.4), although it is defined in terms of
the operators di, which are even under the mirror symmetry
M. In particular, when finite, it indicates that the eigenstate
|En/angbracketrightcontains a frozen pattern of doublons and holons. In the
ETH phase, the infinite temperature average of pn, namely/summationtext
npn/D, where Dis the Hilbert space dimension, is zero
because doublons are not generically stable at nonzero energydensities (with or without interactions). We show this behaviorin the bottom panel of Fig. 4, where we show the infinite
temperature average of p
nfor system sizes Lup to 7 (i.e.,
for systems with up to 14 sites) as a function of the disorderstrength Wfor a representative value of J
⊥. We obtain these
data using full diagonalization and average the results over atleast 1250 disorder realizations. We again obtain finite-sizescaling collapse with a correlation-length exponent ν≈2/3
(see inset of bottom panel of Fig. 4). In the ETH phase, we
see that the quantity Lpscales to zero with increasing L,
while it grows with Lin the MBL phase and scales to a
nonzero value. The crossing point of the finite-size curves canagain be used to estimate the critical disorder strength W
∗
for the transition out of the ETH phase. We expect a weak
dependence of the critical parameters J∗
⊥andW∗(see Fig. 4)
on the interaction ( /Delta1) and intraleg coupling ( J) strength. The
exponent νis likely to be universal although its precise value
may necessitate more numerical studies beyond the scope ofthe current work.
Ultimately, we use the mirror-glass order parameter qand
the doublon correlator pto distinguish among the ETH, MBL,
and ME phases in Fig. 3as follows. In the ETH phase, neither
qnorpscales to a finite value in the thermodynamic limit—
the system is thermalizing, so neither singlons, doublons, norholons are stable degrees of freedom. In the MBL mirror-glassphase, both qandpscale to finite values. The singlons freeze
and break Mspontaneously at infinite temperature, so that qis
finite, but the doublons and holons are also localized. Indeed,the singlon polarization σ
iand the doublon density dican beused to reconstruct the local densities nα,ion the legs α=
1,2, which are both approximately conserved quantities and
have finite overlap with the LIOMs in the MBL phase. Finally,the ME phase is characterized by qscaling to zero (due to
the delocalization or paramagnetism of singlons), while p
scales to a finite value due to the localization of doublons andholons. We emphasize that the characterization of the threedynamical phases—ETH, ME, and MBL—using the qandp
parameters as described above and shown in Fig. 4is unique
and computationally tractable. A finite pvalue (“localized
doublons/holons”) along with a vanishing q(“delocalized or
paramagnetic singlons”) uniquely distinguishes the ME phasefrom both the ETH and MBL phases. We provide furthernumerical evidence for the delocalization of the singlons inSecs. III–V.
III. SINGLON DELOCALIZATION AND ITS BREAKDOWN
IN THE MOBILITY EMULSION
In this section, we derive an effective model that takes
explicit advantage of the approximate conservation of thedoublon/holon density ( 2.5) in the limit of strong disorder
W/greatermuchJ,J
⊥,/Delta1. In this limit, all doublons and holons in
the system are strongly confined to their rungs and can beconsidered completely frozen to first approximation. Indeed,any eigenstate in which each rung of the ladder is occupied bya doublon or holon (i.e., n
DH=1) is manifestly fully localized
in this limit, since hopping between legs is forbidden by Pauliexclusion and each leg is assumed to be MBL in the decoupledlimitJ
⊥=0.
However, since the singlons do not couple directly to
the disorder potential (due to the mirror symmetry), theeigenstates in the all-singlon sector (i.e., n
DH=0) are highly
nontrivial. To determine the fate of such states we derivean effective model for the singlon degrees of freedom usinga Schrieffer-Wolff transformation. The resulting model willgive insight into the delocalization of eigenstates in the all-singlon sector in the ME phase. Furthermore, we will showthat this delocalization is stable to the addition of a finitedensity of doublons and holons. However, we will also arguethat this delocalization breaks down when the singlons aresufficiently dilute and occupy a small but finite fraction ofthe rungs of the ladder. This will lead us to hypothesize thatthere is a critical doublon/holon density n
∗
DHabove which
the system remains localized, and below which the systemdelocalizes.
A. Effective singlon model
To derive an effective model for the singlons, we use a
Schrieffer-Wolff transformation [ 80] to eliminate the intraleg
hopping piece of the full Hamiltonian H, which we denote as
ˆJ=J
2/summationdisplay
α,i(b†
α,ibα,i+1+H.c.). (3.1)
This is achieved with the unitary transformation
Heff=eSHe−S=H+[S,H ]+1
2[S,[S,H ]]+..., (3.2a)
174201-6CONFIGURATION-CONTROLLED MANY-BODY … PHYSICAL REVIEW B 98, 174201 (2018)
with the generator Schosen such that
ˆJ=[H0,S], (3.2b)
where
H0=H−ˆJ−H⊥. (3.2c)
Working to second order in J, one then obtains
Heff=H0+H⊥+1
2[S,ˆJ]+1
2[S,[S,H ⊥]], (3.3)
which can now be projected into a sector with a fixed
configuration of doublons and holons. After projection, theHamiltonian for the singlons can be written in terms of theeffective Ising spin operators
σ
x
i=| ↑ /angbracketright i/angbracketleft↓|i+| ↓ /angbracketright i/angbracketleft↑|i (3.4a)
and
σz
i=| ↑ /angbracketright i/angbracketleft↑|i−| ↓ /angbracketright i/angbracketleft↓|i, (3.4b)
which are defined in terms of the local singlon states ( 2.2a).
To zeroth order in the intraleg hopping J, the effective model
is just the projection of H0+H⊥, which gives the clean
transverse field Ising chain,
Heff=J⊥
2/summationdisplay
iσx
i+/Delta1
2/summationdisplay
iσz
iσz
i+1+δHeff. (3.5a)
Higher-order virtual processes are contained in δHeff, which
is obtained by projecting1
2[S,ˆJ]+1
2[S,[S,H ⊥]]+··· into
a sector with a fixed configuration of doublons and holons.For the remainder of this section we will focus on the termsthat arise in the zero-doublon sector; we will consider whatoccurs for singlons in a generic doublon-holon background inSec. III B.
In addition to the clean part of Eq. ( 3.5), there are vir-
tual processes that involve pairs of singlons transitioning todoublon-holon pairs and back. Such processes are generatedby the Schrieffer-Wolff transformation discussed above andare the mechanism by which randomness enters the effec-tive model, as the intermediate doublon-holon states dependexplicitly on the disorder potential. Keeping corrections tosecond order in Jand first order in /Delta1,J
⊥, we obtain
δHeff=/summationdisplay
iJ2/Delta1
32δh2
ii+1/parenleftbig
σz
i−1σz
i+σz
i+1σz
i+2−σz
iσz
i+2
−σz
i−1σz
i+1/parenrightbig
−J2J⊥
16/summationdisplay
i/parenleftbig
δh−2
ii−1+δh−2
ii+1/parenrightbig
σx
i,
(3.5b)
where δhij=hi−hj∼W. We now see that as the disorder
strength Wincreases, the energy scales for the effective ran-
domness in δHeff, namely /Delta1J2/W2andJ⊥J2/W2,decrease
rapidly. This implies that the effective singlon HamiltonianH
effbecomes cleaner the stronger the potential disorder.
In the limit J⊥/lessmuch/Delta1, where the phase diagram in Fig. 3
resides, the effective model ( 3.5) can be understood as a
theory of weakly interacting domain walls in a weak disorderpotential. The transverse field constitutes a kinetic term forthe domain walls, while the nearest-neighbor Ising interactionconstitutes a potential energy term that is disordered on thescale ( J/W )
2. The next-nearest-neighbor Ising interaction
can be viewed as a density-density interaction for the domainwalls; hence the effective singlon model cannot be mappedto one of Anderson-localized free particles. Heuristically,one expects that when the transverse field is much smallerthan the effective domain-wall disorder potential, i.e., J
⊥/lessmuch
J2/Delta1/W2/lessmuch/Delta1, the system remains fully localized since the
domain wall kinetic energy, of order J⊥, is insufficient to over-
come the relatively large effective disorder potential. How-ever, in the range J
2/Delta1/W2/lessmuchJ⊥/lessmuch/Delta1, the kinetic energy of
the domain walls dominates and one expects ergodic behaviorat nonzero energy densities as the domain walls begin topropagate and interact. As a result, we predict the singlons toexhibit a localization phase transition at strong disorder as afunction of the interleg coupling J
⊥at a critical value of order
J2/Delta1/W2. Note that this behavior is reflected qualitatively
in the phase diagram in Fig. 3, which was obtained by simulat-
ing the full model ( 2.1): at sufficiently large disorder strength
W, the critical coupling J∗
⊥at which the mirror-glass order is
lost decreases as Wincreases.
Interestingly, the primary transport that arises due to the
delocalization of singlons is energy , rather than charge, trans-
port. This is because the singlons are actually quasidegeneratestates of a single particle localized to a rung of the ladder.Thus charge transport is still heavily suppressed at short timesdespite the delocalization of the singlons. Ultimately, how-ever, when the singlon density in the initial state is sufficientlylarge we predict slow charge transport to arise, as discussed inSec. III B.
To further substantiate the claim that the all-singlon sector
indeed undergoes a localization transition as a function ofJ
⊥, we have performed an exact diagonalization study of the
effective model ( 3.5). This allows us to reach system sizes
up toL=14, which is roughly twice as large as those used
to construct Fig. 3. We focus on our results for the bipartite
entanglement entropy SA, which is calculated by partitioning
the system into subsystems AandBof size L/2:
SA=−ρAlnρA, (3.6)
where ρA=trBρandρis the density matrix of Heff.I n
Fig. 5, we show the infinite-temperature average of SAover
the all-singlon sector as a function of system size L.O u r
results are averaged over at least 20 disorder realizations, anderror bars representing one standard error are smaller thanthe plot markers. At small interleg coupling, J
⊥=10−4,t h e
entanglement entropy exhibits a very weak dependence onsystem size and appears to have saturated to a small valueof order ln 2 by L=14. This is consistent with the expected
area-law entanglement scaling in the MBL phase. At larger in-terleg coupling, J
⊥=0.4, the entanglement entropy increases
monotonically with system size in an approximately linearfashion, consistent with the volume-law scaling expected fora delocalizing system.
We tested the stability of the area- and volume-law regimes
by adding a weak uniform longitudinal field to Eqs. ( 3.5), such
that
H
eff→Heff+h/summationdisplay
iσz
i (3.7)
174201-7SCHECTER, IADECOLA, AND DAS SARMA PHYSICAL REVIEW B 98, 174201 (2018)
FIG. 5. Bipartite entanglement entropy in the effective model
(3.5) describing states in the all-singlon sector nDH=0. We fix
W=10 and, as in Figs. 3and4,w ew o r ka t /Delta1=0.5a n d J=1.
At small J⊥(blue curves), the entanglement entropy exhibits a very
weak dependence on system size and appears to saturate to a valueof order ln 2. At larger J
⊥(red curves), it increases monotonically
with system size. To obtain the dashed curves, a weak uniform
“longitudinal field” h=J⊥/10, which breaks the mirror symmetry
M, was added. This field does not change the qualitative behavior,
indicating that the ME phase is not protected by mirror symmetry.
withh=J⊥/10, which weakly breaks the Z2symmetry of
Heff. In the full model ( 2.1), this amounts to weakly breaking
the mirror symmetry by adding a uniform bias between thetwo legs of the ladder (up to small random corrections to thisbias that would result from the Schrieffer-Wolff transforma-tion). The results for this case are plotted as dashed lines inFig.5. Evidently, the area- and volume-law regimes are stable
to breaking the underlying mirror symmetry of the problem;in fact, the discrepancy between the two regimes is enhanced,as may be expected due to the mixing of the symmetrysectors. This provides evidence that the ME phase is not asymmetry-protected phase and can exist even in the absence ofmirror symmetry (unlike the mirror glass, by definition). Wethus expect the main results reported here to be qualitativelysimilar in the presence of generic mirror-symmetry breakingperturbations.
B. Stability of the localized and delocalizing limits
So far, we have demonstrated with numerical and ana-
lytical evidence that the ME phase is characterized by fullMBL in the all-doublon/holon sector with n
DH=1, and by
delocalization in the all-singlon sector with nDH=0. We
will now argue that these two extreme limits are stable tothe addition of a finite density of singlons and doublonsor holons, respectively. This will lead us to the hypothesisthat the ME phase is characterized by the existence of acritical doublon/holon density n
∗
DHbelow which eigenstates
delocalize, and above which MBL sets in.
1. Stability of the localized limit
Consider an eigenstate in which the vast majority of rungs
occupy doublon/holon states, but that contains a very sparserandom distribution of singlons [see Fig. 6(a)]. For an isolated
singlon embedded in such a localized background, we gener-ically find a random renormalization of the interleg hoppingthat depends on the local doublon-holon configuration. Inparticular, a special case arises when a singlon is surroundedby only doublons or by only holons. In this case the interleghopping in fact does notget renormalized, because in this
sector there is only one boson (or one hole) in the entire
FIG. 6. Schematic depiction of the localized and delocalizing
limits discussed in Secs. III B 1 andIII B 2 . (a) Eigenstates containing
a sparse distribution of singlons remain localized because the self-generated random field δJ
⊥is parametrically stronger at strong
disorder than the intersinglon interaction Jr, which arises at a higher
order in perturbation theory when the intersinglon distance ris
sufficiently large. (b) Eigenstates containing a sparse distribution of
doublons and holons delocalize because the doublons and holons
undergo variable-range hopping mediated by the delocalizing bathof singlons. A doublon absorbing energy Efrom the bath can hop
a distance that scales as W/E , but the amplitude for this process
is exponentially suppressed by the decay of its wave function. This
leads to an optimal hopping distance and rate that is calculated in the
Appendix.
system and the hardcore constraint becomes irrelevant. As a
result, the Hamiltonian in the single-particle sector commuteswith the interleg hopping term, Eq. ( 2.1c), and leads to the
conservation of H
⊥, whose value is carried and preserved by
the boson as it hops. Since the eigenvalues of H⊥cannot mix
in the single-particle sector, the local transverse hopping J⊥
does not get renormalized.
However, in every sector with two or more particles (or
holes) in the system, the term H⊥is no longer conserved
because the transverse and longitudinal hopping terms do notcommute for hardcore bosons. This is an interaction effectthat leads to a weak renormalization of the interleg tunnelingamplitude that scales as δJ
⊥∼J⊥(J/W )2when a singlon
neighbors another singlon, cf. Eq. ( 3.5b). However, when
a singlon is surrounded by a doublon-holon pair, e.g., asdepicted in Fig. 6, the renormalization is weaker and scales,
to leading order in 1 /W,a s
δJ
⊥∝J⊥/parenleftbiggJ
W/parenrightbigg4
. (3.8)
This renormalization can be calculated explicitly by using
the generator ( 3.2b) to obtain the effective Schrieffer-Wolff
Hamiltonian ( 3.2a) to fourth order in J, and then projecting
into the space of states with singlons on the appropriate sites.All lower-order corrections to J
⊥vanish identically. Intu-
itively, the extra factor of ( J/W )2in Eq. ( 3.8) as compared
to Eq. ( 3.5b) results from the price a doublon-holon pair must
pay to fluctuate into and out of a virtual singlon pair with at
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least one of the virtual singlons neighboring the target singlon,
which then generates a random correction as in Eq. ( 3.5b).
The renormalized interleg hopping acts as a random local
transverse field applied to the singlon [see Fig. 6(a)] that must
compete with the mutual interactions between the singlons.The interaction between the sparse singlons arises at higherorder in perturbation theory and decays exponentially withtheir separation due to the strong localization of the doublonsand holons that mediate the coupling. Two singlons separatedby a distance rexperience a mutual interaction of order
J
r∼/Delta1/parenleftbiggJ
W/parenrightbigg2(r−1)
, (3.9)
see Fig. 6(a). Thus, when the average intersinglon spacing
is sufficiently large, this effective interaction is parametri-cally weaker than the random part of the “transverse field,”Eq. ( 3.8). As a result, singlons in states with a low (but finite)
singlon density (corresponding to n
DHnear, but not equal to,
one) are localized and paramagnetic. This implies that thedynamics of an initial density product state with a sufficientlysmall number of singlons will show neither particle nor energytransport. Such states will also respect mirror-symmetry sincethe effective spins are polarized along the σ
xdirection, i.e.,
there is no mirror-glass order. This manifests itself dynami-cally as spin precession of singlons initialized along the σ
z
direction, which corresponds to uncorrelated single-particle
hopping between the two legs of the ladder. Thus the localizedsector is stable to the addition of a small singlon density.
2. Stability of the delocalizing limit
We now turn to the opposite limit, considering the case of
a single doublon embedded in a sea of delocalizing singlons[see Fig. 6(b)]. We shall see that the coupling between the
doublon and the singlons leads to singlon-mediated hop-ping processes reminiscent of Mott variable-range hopping(VRH) [ 81]. A key difference, however, is that we must con-
sider states for which the singlon bath is at infinite temperatureT. In the limit of strong disorder W/greatermuchJ
⊥,/Delta1, the singlon
single-particle bandwidth is narrow, so that individual exci-tations cannot mediate hopping with large energy transfers oforder∼W. Thus the doublon must either hop to a faraway
site that has small energy transfer E∼J
⊥,/Delta1,o ri tm u s t
absorb an N-particle excitation with higher energy NJ⊥or
N/Delta1∼W. The former (single-particle) process is limited by
the decay of the localized wave function with distance fromthe initial site, while the latter ( N-particle) process is limited
by the smallness of the coupling between the doublon andsinglons, which suppresses the amplitude of the multiparticleresonance. As we show in Appendix, the competition betweenthese processes leads to an optimal hopping distance and ratewhich controls the mobility of doublons. The optimal hoppingrate is of the form
J
∗∼Je−α√W//Delta1, (3.10)
where αdepends only weakly on J,W,J ⊥,and/Delta1.A tl a r g e
W, the optimal VRH rate is parametrically smaller than J⊥
and/Delta1, which set the rate of energy transport by the excitationsof the delocalizing effective singlon spin chain. At sufficiently
low doublon/holon density nDH, such singlon-mediated VRH
leads to the motion and subsequent interaction of doublonsand holons in the chain, thereby ultimately leading to slowdiffusive charge transport. Thus the delocalized sector is sta-ble to the introduction of a small density of holons/doublons.
The fact that singlons carry energy and not charge indicates
that charge transport in the ME phase is heavily suppressedat all timescales when n
DHis zero. However, when the
doublon/holon density is small but finite, charge transportis mediated via the VRH discussed above. This suggests aparametrically large separation of timescales for charge andenergy transport when the system is deep in the ME phaseand the doublon/holon density is low.
3. Critical doublon/holon density for localization
As the density of singlons is increased, the strength of the
interactions among them increases rapidly due to the expo-nential nature of their effective coupling. We focus now on theregime where the clean component of the transverse field J
⊥is
much larger than the induced randomness (in a zero-doublonbackground), /Delta1(J/W )
2, so that the zero-doublon sector is
strongly delocalizing. We want to estimate the critical dou-blon/holon density n
∗
DHwhere the singlon delocalization tran-
sition takes place. Starting from nDH∼1, i.e., from a dilute set
of singlons, one expects that when their interaction becomescomparable to the random component of the “transverse field”(in a zero-singlon background), Eq. ( 3.8), then the singlons
will undergo the delocalization phase transition. InspectingEq. ( 3.9), one sees that third-neighbor singlons separated by
doublon/holon sites interact with a strength /Delta1(J/W )
4, which
is comparable to their random field when J⊥∼/Delta1.I ft h e
singlons are separated by a further neighbor, their interactionis suppressed by an additional power of J/W and becomes
negligible compared to the random field. This suggests thatthe critical singlon spacing r∼3 sites, corresponding to a
critical doublon/holon density n
∗
DH≈2/3 when the system is
deep in the ME phase and J⊥∼/Delta1/lessmuchW.
This simple estimate of n∗
DHdoes not take into account
what happens in eigenstates in which clusters of severalsinglons are separated by intervening regions of doublonsand holons. Any two such clusters interact to leading ordervia a coupling of the form ( 3.9); the clusters effectively
decouple when this interaction is smaller than the minimumlevel spacing of the two clusters. Thus the critical separationbetween clusters depends on the cluster size in a nontrivialway, and this dependence must be taken into account in orderto precisely determine n
∗
DH.
Furthermore, as the system is tuned towards the ETH or
mirror-glass phase transitions, it is possible, if not highlylikely, that n
∗
DHdeviates strongly from its value deep in the
ME phase. Near the mirror-glass phase, for example, it isreasonable to expect n
∗
DH→0 continuously upon entering the
MBL phase where all singlons are localized. This is becausethe delocalizing singlon states near the phase boundary areextremely fragile so that adding even a small fraction ofdoublons and holons would immediately lead to localization.Near the ETH phase transition, however, the localized dou-blon/holon states become fragile so that adding any fraction
174201-9SCHECTER, IADECOLA, AND DAS SARMA PHYSICAL REVIEW B 98, 174201 (2018)
of singlons leads to delocalization. This would imply n∗
DH→
1 upon approaching the ETH phase boundary. We shouldnote that this speculation also does not take into account theeffect of rare configurations at fixed n
DHin which singlons
are anomalously close together. Further studies with largersystems will be required to reach more definitive conclusionsregarding the precise value of the critical density discussedabove, which we have argued is generically finite and nontriv-ial 0<n
∗
DH<1 in the ME phase. A precise determination of
n∗
DHis an important future problem of interest, but our work
establishes that such an n∗
DHexists in the ME phase.
It is interesting to note that the arguments above also
point to the existence of a critical value of nDHthat separates
eigenstates with and without mirror-glass order in the MBLphase, although finite-size restrictions also pose a challenge toconfirming this idea numerically. Indeed, interactions betweensinglons are the crucial ingredient that drives the spontaneousZ
2mirror symmetry breaking that gives rise to mirror-glass
order, and their strength must be compared to that of thelocal transverse field to determine whether a given singlonremains paramagnetic or participates in the long-range order.It is important to stress in this case that the putative criticaldoublon/holon density for mirror-glass order does not implythe presence of delocalized states in the many-body spectrum;all states are localized in the MBL mirror glass phase, and afinite fraction of them participate in the mirror-glass order, asour numerical results in Sec. IIindicate. Any states that do not
participate in the long-range order are simply “paramagnetic”MBL states.
4. Analogy with MBL coupled to a heat bath
Finally, we note that one can make an analogy between
the above stability analysis of the localized and delocalizinglimits and the problem of an MBL system coupled to a heatbath [ 82–86]. In this analogy, one can view the interacting
singlons as the “bath” and the inert doublons and holons asthe “system” of interest. If n
DH/lessmuch1/2, then there are many
more singlons in the system than there are doublons andholons. In this case, the many-body density of states of thesinglons is nearly continuous and their many-body bandwidthis much larger than that of the doublons and holons. In thiscase, the system truly resembles the generic case of an MBLsystem coupled to a heat bath, where it is known on generalgrounds that the initially localized system will delocalize atinfinite time [ 82,83,85]. However, if n
DH/greatermuch1/2, then the
“bath” contains many fewer degrees of freedom than thesystem and can itself become localized due to their coupling[84,86].
The fascinating aspect of this problem in the context of
the ME phase is that the emergent parameter n
DHessentially
tunes the “quality” of the delocalized bath as a function of
the many-body configuration of an eigenstate. When nDH/lessmuch
1/2, the singlon “bath” has a dense many-body spectrum and
can easily mediate transport via the VRH process outlined inSec. III B 2 . Increasing n
DHdegrades the quality of the bath
until ultimately it is incapable of delocalizing the system, andbecomes localized itself via the self-generated random fielddiscussed in Sec. III B 1 .IV . HILBERT SPACE STRUCTURE OF THE MOBILITY
EMULSION: ABSENCE OF A MANY-BODY
MOBILITY EDGE
The discussion in Sec. IIIindicates that it is the dou-
blon/holon density nDH, which is an emergent approximately
conserved quantity at strong disorder, that controls whethereigenstates are localized ( n
DH>n∗
DH) or delocalizing ( nDH<
n∗
DH). This is in sharp contrast to the case of a putative
many-body mobility edge, where localization is controlled bythe many-body energy density /epsilon1, which is not an emergent
quantity. This implies that the ME phase is both sharplydistinct from the ETH phase as we have already argued,and from other possible intermediate phases with putativemany-body mobility edges. (In particular, unlike in several
recently studied incommensurate MBL models in the litera-
ture [ 11,48,49,69,70], in the noninteracting limit our system
manifests no single particle mobility edge.) In this section,we analyze this claim in further detail and present additionalnumerical results on the full model ( 2.1) that substantiate it.
To demonstrate the absence of a many-body mobility edge
in this model, we investigate the distribution of the bipartiteentanglement entropy S
Adefined in Eq. ( 3.6)a saf u n c t i o no f
the many-body energy density /epsilon1. More concretely, we perform
full exact diagonalization of the model ( 2.1)a tL=8 for 100
realizations of the disorder and record SAand/epsilon1for each eigen-
state in each realization. We choose a large disorder strength,W=1000, so that the discrepancy between the degree of
localization of the singlons and the doublons and holons ismore pronounced. We plot the results in a two-dimensionalhistogram for two representative values of J
⊥in Fig. 7. Deep
in the MBL phase, at J⊥=10−8[Fig. 7(a)], the distribution
shows that the entanglement entropy clusters around twocharacteristic values, 0 and ln 2, for any energy density. Thisis to be expected, since at such large disorder strengths stateswhere nearly all rungs are occupied by doublons or holonsare essentially product states, while states with more singlons,which spontaneously break the mirror symmetry, form many-body cat states with entanglement ln 2.
In contrast, for J
⊥=0.4 [Fig. 7(b)], deep in the ME
phase, the distribution of the entanglement entropy is much
broader, encompassing values between 0 and ∼4 ln 2. Indeed,
in going from the MBL to the ME phase, a finite fraction ofeigenstates are redistributed to entanglement entropies largerthan∼ln 2, as one can see by comparing the top panels
in Figs. 7(a) and7(b). Moreover, the largest values of the
entanglement entropy increase with system size, as shown inFig.9. However, these high-entanglement eigenstates coexist
with low-entanglement eigenstates at the same energy density,
with no obvious demarcation between them as a function of /epsilon1.
In Fig. 8, we show the same data as in Fig. 7,b u t
binned as a function of the doublon/holon density n
DHrather
than/epsilon1. At such strong disorder, nDHassumes sharply quan-
tized values, as can be seen in the top panels of Figs. 8(a)
and 8(b). Once the data have been reorganized in this
way, a clear trend emerges. In both the MBL and MEphases, states in the all-doublon/holon sector at n
DH=1
have nearly zero entanglement, and are essentially productstates. In the MBL phase, the average entanglement entropyin each eigenstate increases as n
DHis decreased from 1, but
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J⊥=1 0−8total % ≈98
100101102103104distribution of SA≤1.1l n2 distribution of SA≤1.1l n2
total % ≈81
J⊥=0.4
100101102103
FIG. 7. Histogram of the bipartite entanglement entropy SAand many-body energy density /epsilon1accumulated from every eigenstate of 100
disorder realizations. The color scale indicates the logof the number of counts in each bin in order to make outliers more visible. In both
panels J=1,/Delta1=0.5,W=1000, and L=8 (16 sites). (a) In the MBL mirror-glass phase, the strong peaks of SAnear the values 0 and
ln 2 result from fully-localized paramagnetic states and symmetry-broken cat states, respectively. (b) In the ME phase, the distribution ofentanglement entropy spreads out to much larger values, indicating a stronger tendency towards delocalization. This is particularly evident for
high-entanglement states near /epsilon1≈0.5, which coexist with the band of localized states near S
A≈0. The strong increase of entanglement in the
ME phase is notaccompanied by a many-body mobility edge, as a nonzero fraction of localized states exist and violate ETH for all /epsilon1[compare
top panels of (a) and (b)].
saturates once it reaches a value near ln 2. In contrast, in
the ME phase, the entanglement entropy does not saturate as
nDHdecreases. Instead, the entanglement entropy increases
monotonically until nDHreaches 0 and the all-singlon limit
is achieved. Thus, in the ME phase, there is a clear corre-lation between the density of singlons and the entanglemententropy: eigenstates containing more singlons have substan-tially more entanglement on average than those with fewersinglons.In Fig. 9, we compare the data from Figs. 7and8, obtained
atL=8, to data obtained at L=6 with the same parame-
ters and 2000 disorder realizations. We plot the means andstandard deviations of the entanglement distributions for eachsystem size as functions of n
DHin both the MBL and ME
phases. (Note that if we had instead plotted the standard error ,
indicating convergence of the mean of the distribution as a
function of the number of samples, the error bars would becomparable to or smaller than the size of the plot markers.) In
distribution of nDH
J⊥=1 0−8
100101102103104105distribution of nDH
J⊥=0.4
100101102103104105
FIG. 8. Histogram of the bipartite entanglement and doublon/holon density nDHusing the same data set and color scale as in Fig. 7.T h e
distribution of entanglement organizes into well-separated doublon/holon density bands labeled by quantized values of nDH=2k/L (L=8
shown) with k=0,1,...,L / 2 [see top panels of (a) and (b)]. (a) In the MBL phase, the mean of the distribution of SAin a given density band
(red dots) grows with decreasing nDHbut saturates to ∼ln 2 for nDH/lessorsimilar0.5. (b) In the ME phase, SAgrows monotonically without saturating
asnDHdecreases. This indicates that the degree of localization of a given eigenstate is configuration-controlled .
174201-11SCHECTER, IADECOLA, AND DAS SARMA PHYSICAL REVIEW B 98, 174201 (2018)
J⊥=1 0−8
J⊥=0.4
FIG. 9. Average bipartite entanglement entropy SAin each dou-
blon/holon density sector at L=6a n d8( f o r L=8, we use the
same data sets as in Figs. 7and8). Error bars represent one standard
deviation of the distribution of SA(horizontal error bars are smaller
than the point size). (Top) In the MBL phase, SAsaturates to ln 2
fornDH/lessorsimilar1/2 and is insensitive to the system size. (Bottom) In
the ME phase, SAis nearly independent of nDHfornDH/greaterorequalslant2/3
and increases with decreasing nDHfornDH<2/3. For nDH/lessorsimilar1/2,
SAclearly increases with L, suggesting delocalization below some
critical doublon/holon density.
the MBL phase, the two entanglement versus doublon/holon
density curves are nearly indistinguishable, consistent withthe expected area-law scaling. In the ME phase, the two curvesare indistinguishable for n
DH/greaterorsimilar1/2, and begin to diverge
from one another for nDH/lessorsimilar1/2. The entanglement growth
asnDHdecreases appears to be faster , and the final mean
value of SAatnDH=0 markedly higher, for the larger system.
Furthermore, we observe that SAappears to be independent
of both LandnDHfornDH/greaterorequalslant2/3, which may indicate that
the singlons “freeze-out” above nDH∼2/3, as argued heuris-
tically in Sec. III B 3 . However, in order to test the predictions
of Sec. III B 3 more rigorously, it is necessary to consider
larger system sizes.
The data presented in this section serve as an important
consistency check on the picture of the ME phase developedin Sec. III. However, by no means do they constitute proof
that there is a critical doublon/holon density for localization inthe ME phase. Indeed, finite size scaling at fixed n
DHrequires
access to much larger system sizes than are available to exactdiagonalization. Nevertheless, these data show that there is nomany-body mobility edge in the ME phase, but that instead theemergent doublon/holon density controls the (de)localizationof eigenstates. Thus eigenstate properties in the ME phaseare controlled by the many-body configuration rather than themany-body energy density.
V . DYNAMICAL SIGNATURES OF THE ME PHASE
In this section, we discuss dynamics in the ME phase. In
particular, we consider how the dynamics after a quantumquench depends on the choice of the initial state. We take
the initial states to be local density product states, which aremost relevant to ongoing experiments studying nonergodicdynamics. The defining dynamical feature of the mobilityemulsion is that particle and energy transport depend stronglyon the choice of initial configuration of the system. For initialstates with a subcritical doublon/holon density, n
DH<n∗
DH,
the system will show delocalizing behavior due to the largefraction of interacting singlons. For initial states with n
DH>
n∗
DHthe singlons, doublons, and holons are frozen and remain
localized in the initial configuration. By contrast, in the ETHand MBL phases, a generic product state will instead relaxtowards thermal equilibrium or remain localized, respectively,regardless of the value of n
DHin the initial state.
This strong dependence of the quench dynamics on the
choice of initial product state is striking (and to the best of ourknowledge, never discussed before in the MBL literature) andprovides a useful tool in the study of nonergodic dynamics inthat the initial state can be used to select the dynamical regime
of interest. In this sense, the physics of the ME phase can bequalitatively altered by the appropriate tuning of the initialstate! We demonstrate this ability by studying the behaviorof several spatially averaged autocorrelation functions thatsharply distinguish the ETH, MBL, and ME phases.
We first consider the local density autocorrelation function,
C(t)=1
2L/summationdisplay
α=1,2L/summationdisplay
i=1/angbracketleft2nα,i(t)−1/angbracketright/angbracketleft2nα,i(0)−1/angbracketright, (5.1)
where we have used the fact that the initial state is a local
density product state, so that the connected part of the auto-correlator vanishes. C(t) probes the localization of individual
particles. Defined such that C(0)=1, it remains finite as t→
∞when particles remain confined to their initial positions
and tends to zero as t→∞ when particles are delocalized.
The dynamics of CatL=8 starting from an all-doublon and
an all-singlon state are shown in Fig. 10. In the ETH phase, C
rapidly decays to zero on a timescale of order 1 /Jirrespective
of the initial state. In the MBL phase, Cremains frozen near
its initial value out to arbitrarily late times, again irrespectiveof the initial state. However, in the ME phase, Cremains finite
when the system is initialized in the all-doublon state, while itdecays on a timescale 1 /J
⊥when the system is initialized in
the all-singlon state. This provides evidence that the mobilityof particles in the ME phase is configuration-controlled ,i n
sharp contrast to the ETH and MBL phases, where the late-time dynamics of Cis independent of the choice of initial state.
Next, we consider autocorrelators that allow one to probe
separately the dynamics of the singlons and doublons/holonscontained in the initial state. We first define the singlonautocorrelation function,
C
S(t)=1
nSLL/summationdisplay
i=1/angbracketleftσi(t)/angbracketright/angbracketleftσi(0)/angbracketright, (5.2)
where nS=1−nDHis the density of singlons in the initial
density product state, so that CS(0)=1. We implicitly assume
nS>0 when using CSandnDH>0 when using CDHdefined
below. With this normalization, CS(t) measures the fraction of
singlons that remain in their initial configuration under time
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W=6.0
J⊥=0.14W=6.0
J⊥=1 0−4W=0.3
J⊥=0.14
FIG. 10. Dynamics of the local density autocorrelator ( 5.1)i nt h e
ETH (top), MBL (middle), and ME (bottom) phases at J=1,/Delta1=
0.5,andL=8. In each case, we compare the dynamics starting
from two initial states: the all-doublon state | •◦•◦•◦•◦ /angbracketright and the
all-singlon state |↑↑↓↓↑↑↓↓/angbracketright . In the ETH and MBL phases, the
dynamics of the autocorrelator does not depend on the choice of
initial state; in the former case, it quickly decays to zero, whereasin the latter case it remains frozen near its initial value. In the ME
phase, however, the autocorrelator does not decay when the system is
initialized in the all-doublon state (similar to its behavior in the MBL
phase), while it does when initialized in the all-singlon state (similar
to its behavior in the ETH phase). Note that the data for the MBL andME phases were obtained using the same disorder realization—the
only change in going from the middle to the bottom panel is the
increase in J
⊥.
evolution. If one starts in a configuration with only singlons,
nS=1, and the singlons are localized (i.e., σi=±1), then
CSwill remain nonzero as t→∞ . Such dynamical behavior
sharply distinguishes the MBL phase from both the ETH andME phases, where the all-singlon state delocalizes and C
Sde-
cays to zero at late times. Such delocalization and decay of CS
is qualitatively similar in both the ETH and ME phases, thus
requiring at least one other measure to distinguish the two.
The distinction between the ETH and ME phases can
be observed by studying the doublon/holon autocorrelationfunction, defined as
C
DH(t)=1
nDHLL/summationdisplay
i=1/angbracketleftdi(t)/angbracketright/angbracketleftdi(0)/angbracketright. (5.3)
Here, CDHmeasures the fraction of the doublons and holons
that remain confined to their initial sites. This measure alsodetermines the charge transport of the system when initializedin a mirror-symmetric charge density wave (CDW) state. Inthe MBL phase neither doublons nor holons propagate, and inthis case C
DHwill remain close to its initial value as t→∞ .
In the ETH phase, CDHwill generically decay to zero dueFIG. 11. Dynamics of the moving average of the imbalance CS
as a function of J⊥atJ=1,/Delta1=0.5,andL=7 .T h ed a t as h o w n
were generated using a single disorder realization at W=6. The
initial state is taken to be the all-singlon configuration |↑↑↓↓↑
↑↓ /angbracketright . In the MBL phase J⊥/lessorsimilar0.03 the imbalance has a nonzero
late time average, while in the ME phase, J⊥/greaterorsimilar0.0 3 ,t h el a t et i m e
average vanishes.
to the spreading of charge. However, in the ME phase the
late-time behavior of CDHis instead determined by the initial
configuration and will show either localized or delocalizingbehavior depending on the value of n
DHin the initial state.
For example, in the all-doublon/holon sector, nDH=1, the
system is fully localized in the ME phase and any initial CDWwill survive indefinitely under time evolution (just as in theMBL phase). In the ETH phase, however, a generic CDWwill thermalize rapidly. The doublon/holon autocorrelationfunction C
DHthus sharply distinguishes the ETH phase from
both the ME and MBL phases and, in combination with thesinglon autocorrelation function C
S, allows the three phases to
be uniquely determined.
In Figs. 11and12, we plot the moving time average of
the singlon and doublon/holon autocorrelators, defined for anyquantity O(t)a s
O(t)=/integraldisplayt
0dt/prime
tO(t/prime). (5.4)
In Fig. 11, we show the behavior of CSstarting from a fixed
all-singlon configuration for various values of J⊥at large W
where only the MBL and ME phases exist. At small transversehopping, J
⊥<J∗
⊥, one sees that the late-time value of CS(t)
converges to a nonzero value due to singlon localization thatarises in the MBL phase. Beyond a critical value J
∗
⊥≈0.03
the system enters the ME phase and CSdecays to zero at late-
times, indicating a delocalizing state in the all-singlon sector,n
S=1. This behavior is consistent with the data in Figs. 8
and9that shows the entanglement growing with every singlon
added to the system at nDH=0. Near the transition ( J⊥=
0.01), the timescale on which CS(t) converges becomes much
longer than it is when the system is deep in either phase.
To distinguish the ETH and ME phases, in Fig. 12,w e
plot the doublon-holon autocorrelation function starting froma fixed doublon CDW initial state. The data are taken at afixed transverse hopping strength where only the ETH andME phases exist. We fix a random disorder configuration atW=6.0 (the same one used in Fig. 11) and globally rescale it
to vary its strength W; this drives the system through the ETH
to ME transition. At strong disorder the system is in the MEphase and the CDW is dynamically stable. As one lowers the
174201-13SCHECTER, IADECOLA, AND DAS SARMA PHYSICAL REVIEW B 98, 174201 (2018)
FIG. 12. Dynamics of the moving average of the imbalance CDH
as a function of disorder strength at J=1,/Delta1=0.5,J⊥=0.14,
andL=7. The initial state is taken to be the all-doublon/holon
configuration | •◦•◦•◦• /angbracketright . To investigate the dependence of the
late-time average of CDHon the disorder strength, we fix the disorder
potential to be the one used in Fig. 5, and globally rescale it to change
the effective value of W. In the ME phase, W/greaterorsimilar1.4, the imbalance
has a nonzero late-time average, while in the ETH phase, W/lessorsimilar1.4,
the late time average vanishes. The gray horizontal indicates thevalue 1 /L=0.14..., below which the late-time average should be
viewed as indistinguishable from zero. Note that the parameters and
disorder potential used for the purple curve at W=6 coincides with
those of the purple curve in Fig. 11; the only difference between the
two curves (besides the quantity being measured) is the initial state
used.
disorder strength the late-time average of CDHis reduced until
it vanishes upon reaching the critical point. Below the criticaldisorder strength the CDW melts at late times, indicatingbehavior consistent with the ETH phase. At W=2, near the
transition W
∗≈1.4 shown in Fig. 3, the timescale on which
CS(t) converges becomes longer than it is when the system
is deep in either phase, similar to what is observed near thetransition in Fig. 11.
We emphasize that the dynamical data shown in
Figs. 10–12further exemplify a defining feature of the ME
phase, namely that the dynamics of the system is localizedwhen the system is prepared in an all-doublon/holon initialstate, and delocalizing when the system is prepared in anall-singlon initial state. By strong contradistinction, in theMBL phase both of the two states studied (as well as genericstates) remain localized, while in the ETH phase both statesdelocalize (see Fig. 10). The fact that two distinct diagnostics,
namely, C
SandCDH, are needed to uniquely characterize the
ME phase makes perfect sense since the ME phase sharesproperties of both ETH and MBL phases in a configuration-dependent (tuned by n
DH) manner, necessitating two separate
correlators to distinguish it from the ETH and MBL phases. Infact, any intermediate phase in any situation is likely to requiretwo distinct diagnostics to distinguish it from both ETH andMBL phases whereas ETH and MBL phases themselves canbe distinguished by one diagnostic.
We should note that when the singlons are in a param-
agnetic state, as occurs when singlons are sufficiently dilute,
they localize along the σ
xdirection and in this case CS(t) will
show oscillations associated with singlons initialized in a σz
i
eigenstate precessing about the local σxfield of strength ∼J⊥
(see Sec. III). To see localization in this case, one should first
perform a π/2 rotation about the σyaxis on the singlons inthe initial σz-basis product state to obtain a σx-basis product
state. [Note that here we are talking about the spin axes of theemergent singlon spin states; the hardcore bosons we considerare spinless.] After evolving the state with the Hamiltonian(2.1), one performs an additional π/2 rotation before measur-
ing in the σ
zbasis. In this way, one will effectively measure
the dynamics of σx
istarting from a σx
iproduct state, which
will remain localized under time evolution for a paramagneticsinglon state. It is interesting to note that the necessary π/2
rotation of the singlons around the σ
yaxis can be achieved in
situin a system of hardcore bosons. Starting from a deep lat-
tice, such that hopping is suppressed both within and betweenthe legs, one can first apply the interleg hopping HamiltonianH
⊥for a time π/(2J⊥), which enacts a π/2 rotation about
theσxaxis for the singlons and does nothing to the doublons
and holons. Subsequently, one can turn on a bias of the formh/summationtext
i(n1,i−n2,i) for a time π/(4h), which performs a π/2
rotation about the σzaxis for the singlons and again does
nothing to the doublons and holons.
Our arguments in Sec. IVsuggest that the qualitatively
distinct dynamical behavior of the ME phase, illustrated inFigs. 10–12, persists away from the special limits n
DH=0
andnDH=1. However, due to severe finite size restrictions
imposed by exact diagonalization, we are not able to see asharp transition between these two behaviors upon changingn
DHin dynamical or eigenstate properties. This is primarily
due to the strong quantization of nDHthat arises for small
systems at half-filling (i.e., nDHcannot be tuned continuously
likeJ⊥orW), cf. Fig. 8. It would be very interesting to
implement another method, e.g., the recently developed time-dependent variational principle (TDVP) [ 87–90], that can
study dynamics in larger systems where the sensitive depen-dence of the dynamics on the initial value of n
DHcan be seen
explicitly. This would also potentially allow one to observe thedrastic separation of charge and energy transport discussed inSec. III B 2 , which is not visible in the small systems studied
here. This issue is also coupled to an accurate determinationof the critical n
∗
DHitself, requiring future studies using much
larger system sizes.
VI. DISCUSSIONS AND CONCLUSION
A. Summary
In this paper, we have proposed a new nonergodic phase of
matter—the mobility emulsion. It has the peculiar property ofneither satisfying ETH nor being fully MBL, and as such is anintermediate phase; indeed, in Sec. II, we presented numerical
evidence that it is separated from both phases by a phasetransition, and is thus sharply distinct. It is characterized by(1) the coexistence of localized and delocalizing eigenstates
at fixed energy density (and thus manifests no mobility edge),(2) the emergence of a parameter, the doublon/holon density
n
DHin the case of the model ( 2.1), that can be used to label
eigenstates and that determines whether they exhibit localizedor delocalizing behavior, and (3) the ability to select the dy-
namical behavior following a quantum quench by initializingthe system in many-body configurations with different valuesof this emergent parameter.
174201-14CONFIGURATION-CONTROLLED MANY-BODY … PHYSICAL REVIEW B 98, 174201 (2018)
The ME phase of the model ( 2.1) provides a striking
example of a quantum system with robust and incomplete
emergent integrability, wherein the number of integrals ofmotion (here represented by the emergent dressed doublonsand holons) is a nonzero, but nonunity, fraction of the totalnumber of degrees of freedom. Thus it is neither ETH norMBL, but is a distinct new phase. As discussed in Secs. III
andIV, this feature has a natural explanation and character-
ization in terms of many-body configurations labeled by thedoublon/holon density n
DH. Morever, the strong configuration
dependence of the quench dynamics examined in Sec. V
enables one to selectively access various dynamical regimesby initial state preparation. This suggests a new paradigmin which the choice of initial state can be used as a tool tostudy and manipulate nonergodic disordered phases of matter.This paradigm could be implemented, e.g., in systems ofcold atoms, trapped ions, or Rydberg atoms, as long as asystem with the appropriate hierarchy of energy scales can beengineered.
Quite apart from possible experimental realizations, which
should be possible since the system we study is equivalentto two coupled XXZ spin chains, the ME phase is of con-siderable fundamental interest, explicitly demonstrating thatthe presence of hopping, disorder, and interaction could leadto sufficient dynamical frustration in a quantum system soas to produce only partial emergent integrability so that theresultant phase, depending on its internal configurations, may,even at infinite temperature, be neither an ergodic metal nor anonergodic insulator. The fact that such a phase could arisewithout the physics of mobility edge playing any role isindeed extremely intriguing.
B. Discussion and outlook
Although various aspects of the ME phase have been
studied and characterized numerically and analytically in thiswork, some important properties remain yet to be fully charac-terized. Perhaps the most notable is the critical doublon/holondensity n
∗
DHthat separates localized from delocalizing states.
While we have argued for the existence of a nontrivial criticalpoint, 0 <n
∗
DH<1, its precise location is difficult to extract
using exact diagonalization due to severe finite size restric-tions. We do, however, establish the stability of the localizedand delocalizing limits of the ME phase to small changes inn
DH.
On the other hand, analytical attempts to determine n∗
DHare
challenging due to the presence of rare region effects, the im-portance of which is well-appreciated but that are notoriouslydifficult to address [ 71,91–95]. In our setup, rare regions of
weak disorder can influence the form of the effective singlonHamiltonian ( 3.5) by locally admixing singlon and doublon
states. Such regions are statistically rare for W/greatermuchJ(where
we find the ME phase—see Fig. 3) and are not expected
to qualitatively modify the phase diagram as compared tothe case of a quasiperiodic potential where rare regions areabsent. Even in the latter case, however, rare configurations
with anomalously high local singlon density could play animportant role near the critical doublon/holon density. (Theeffects of similar rare configurations may have been observedexperimentally in Ref. [ 9].) Consequently, it would be highlydesirable to implement a numerical method, such as the TDVP
[87–90], that not only captures rare region effects but also
allows one to access substantially larger system sizes wherethe doublon/holon density quantization is less prominent.Such a future work could also help to better pin down thecritical density n
∗
DH. Moreover, using such a method one
might be able to observe the strong separation of timescalesfor energy and charge transport, the latter being associatedwith the slow variable-range hopping of doublons and holons.This physics should be directly visible in a sufficiently largeexperimental system with on-site density resolution and a longparticle lifetime.
One intriguing aspect of the model studied here is that it
resembles the Fermi-Hubbard model in a transverse field. Inthis analogy, the two legs of the ladder represent the spin-up and spin-down states of spin-1 /2 fermions on a single
chain. The doublons and holons of the resulting model arethose of the usual Fermi-Hubbard model, and the singlonstates on a rung correspond to the spin states of a fermionon a singly occupied site. However, because of the form ofEq. ( 2.1), there is no SU(2) spin-rotation symmetry, even
forJ
⊥=/Delta1=0. This breaking of SU(2) symmetry lifts the
obstruction to localization [ 58,74], ultimately leading to the
mobility emulsion. In the bosonic language, the interactionbetween particles on the same leg (or spin state) is hard-core, i.e., U
↑↑=U↓↓=∞ , while the interaction between
particles on opposite legs vanishes, U↓↑=0. The breaking
ofSU(2) symmetry thus arises not only from the transverse
hopping J⊥, which can be viewed as a homogeneous trans-
verse field in spin space, but also from the spin-anisotropicinteractions that further break conservation of spin U(1)symmetry associated with rotations about the transverse-fieldaxis.
This feature represents a crucial difference from the Fermi-
Hubbard model, which always preserves a spin U(1) sym-metry in the presence of a uniform field. The similarity tothe Fermi-Hubbard model nevertheless provides an additionalstarting point and guiding principle in the search for andengineering of systems that may harbor the ME phase. Thereis particular motivation to use such a broken SU(2) Fermi-Hubbard prescription for the laboratory observation of theME phase since recent experimental work has been able tostudy the Fermi-Hubbard model with single-site resolution inoptical lattices [ 96–98].
A more promising route to experimentally observing the
separation of singlon and doublon dynamics in the ME phasewould be to implement the disordered Bose-Hubbard modelon a two-leg ladder in a quantum gas microscope. This modelshould exhibit qualitatively similar physics when the hardcoreconstraint is softened by a finite on-site repulsion U<∞;a
nearest-neighbor interaction like /Delta1can then be generated via
fluctuations into and out of intermediate states with doubleoccupancies. A recent experiment by Lukin et al. [12] has
shown the feasibility of this setup for studying entanglementin a single-leg disordered Bose-Hubbard model, while Kauf-man et al. [7] studied the clean Bose-Hubbard model on
the two-leg ladder with single-site resolution. By combiningthese techniques, one could prepare and directly image thenonergodic dynamics of singlons and doublons that can becontrolled exclusively by the initial state.
174201-15SCHECTER, IADECOLA, AND DAS SARMA PHYSICAL REVIEW B 98, 174201 (2018)
ACKNOWLEDGMENTS
We acknowledge helpful discussions with Vedika Khe-
mani, Markus Müller, and Ivan Protopopov. This work issupported in part by the Laboratory for Physical Sciences andMicrosoft. T.I. acknowledges a JQI postdoctoral fellowship.
M.S. and T.I. contributed equally to this work.
APPENDIX: DERIV ATION OF THE OPTIMAL DOUBLON
V ARIABLE-RANGE HOPPING RATE
We consider the variable-range hopping of a doublon
embedded in a delocalizing background of singlons. Thecoupling between the doublon and singlons is easiest to derivewhen the interleg hopping J
⊥is either much smaller or much
greater than the nearest-neighbor interaction /Delta1. In both cases,
we work in the limit W/greatermuchJ, J ⊥,/Delta1, so that the doublons and
holons are strongly localized and the effective disorder for thesinglons is very small.
1. Domain-wall limit
We first consider the limit J⊥/lessmuch/Delta1. In this limit, the
domain walls of the singlon “spin” configurations are goodquasiparticles. In this case, the doublon can annihilate adomain wall with amplitude J
⊥because the former acts essen-
tially as a hard boundary at large disorder W/greatermuchJ. Denoting
a domain wall creation/annihilation operator on bond iby
a†
i/ai,w eh a v e
Hint=J⊥
2/summationdisplay
if†
ifi(ai+a†
i), (A1)
where f†
i/fiis a doublon creation/annihilation operator on
sitei. The domain walls form a bath with single-particle
spectrum
Hdw=/summationdisplay
kωka†
kak,ω k=/Delta1+J⊥
2cosk, (A2)
where klabels the domain-wall momentum and we neglected
the weak disorder-dependent terms. In addition to the domainwall Hamiltonian, we have the localized doublon Hamiltonianwritten in the basis of (exponentially) localized states αas
H
dbl=/summationdisplay
α˜hαf†
αfα. (A3)
In the localized basis, Hinttakes the form
Hint=J⊥
2/summationdisplay
i,α,α/primeψ∗
i,αψi,α/primef†
αfα/prime(ai+a†
i), (A4)
where ψi,αis the wave function of a localized state α.T h e
coupling ( A4) shows that domain wall absorption or emission
leads to transitions between distinct localized states whoseamplitude decays exponentially (due to the decay of ψ
i,α)
with the separation from the site of absorption i.A tt h e
same time the typical energy difference between states α,α/prime,
˜hα−˜hα/prime∼W, is much larger than the energy of a single
domain wall, ∼/Delta1. To find the optimal transition rate we first
find the transition amplitude for a process involving an energytransfer E<W .
Starting from some site i, we can estimate the distance
rrequired before finding another site with probability O(1)within the energy window h
i±E/2t ob e r(E)∼W/E [see
Fig.6(b)]. In order for the doublon to gain or lose energy E,
it must absorb or emit N(E)∼E//Delta1domain walls. This re-
quires going to Nth order perturbation theory in Hint, leading
to an effective doublon hopping amplitude scaling as
J(E)∼Je−r(E)/ξ/parenleftbiggJ⊥/2
N(E)/Delta1/parenrightbiggN(E)
, (A5)
where ξis the doublon localization length. At strong disorder
ξ∼1/ln(W/J )/lessmuch1. The factor N(E) in the denominator of
Eq. ( A5) stems from the fact that the energy difference grows
linearly in each order of perturbation theory, and we used theapproximation n!/similarequaln
n. Optimizing J(E) over Eleads to an
optimal hopping rate
J∗∼Je−2r∗/ξ, (A6a)
where
r∗≈/radicalBigg
Wξ
/Delta1ln/parenleftbigg√W/Delta1/ξ
J⊥/2/parenrightbigg
. (A6b)
2. Paramagnetic limit
In the limit J⊥/greatermuch/Delta1, the domain walls are no longer good
quasiparticles, but the local spin projections onto the xaxis,
|→/angbracketright iand|←/angbracketright i, are. Such “spin flips” have a single-particle
spectrum given by
Hspin=/summationdisplay
kεks†
ksk,ε k=J⊥+/Delta1
2cosk, (A7)
where kis the spin-flip momentum and s†
k/skare spin-flip
quasiparticle creation/annihilation operators. The coupling tothe doublon now acquires a scattering form
H
int=/Delta1/summationdisplay
j,k,k/primegk,k/primeeij(k−k/prime)f†
jfjs†
ksk/prime, (A8)
with dimensionless form factor gk,k/primewhose precise structure
we neglect in the following estimate of the effective doublonhopping amplitude. In each scattering process, the doubloncan acquire a maximal energy of /Delta1from the spin excitations
scattered across the Brillouin zone from k=0t ok
/prime=πor
vice versa. The multiparticle scattering process that absorbsenergy E/greatermuch/Delta1acquires a structure similar to Eq. ( A5) and
thus gives rise to an effective doublon hopping amplitudescaling as
J(E)=Je
−r(E)/ξ/parenleftbigg/Delta1
E/parenrightbiggE
/Delta1
. (A9)
Optimizing over energy Eleads to the hopping rate
J∗∼Je−2r∗/ξ, (A10a)
where
r∗≈/radicaltp/radicalvertex/radicalvertex/radicalbtWξ
/Delta1ln/parenleftBigg/radicalBigg
W
ξ/Delta1/parenrightBigg
. (A10b)
The result, Eq. ( A10), can be obtained from Eq. ( A6) upon
the substitution J⊥/2→/Delta1due to the difference between the
interaction Hamiltonians ( A4) and ( A8).
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174201-18 |
PhysRevB.93.245158.pdf | PHYSICAL REVIEW B 93, 245158 (2016)
Phase boundary of spin-polarized-current state of electrons in bilayer graphene
Xin-Zhong Yan,1Yinfeng Ma,1and C. S. Ting2
1Institute of Physics, Chinese Academy of Sciences, P .O. Box 603, Beijing 100190, China
2Texas Center for Superconductivity, University of Houston, Houston, Texas 77204, USA
(Received 13 April 2016; revised manuscript received 27 May 2016; published 29 June 2016)
Using a four-band Hamiltonian, we study the phase boundary of spin-polarized-current state (SPCS) of
interacting electrons in bilayer graphene. The model of spin-polarized-current state has previously been shownto resolve a number of experimental puzzles in bilayer graphene. The phase boundaries of the SPCS with andwithout the external voltage between the two layers are obtained in this work. An unusual phase boundary wherethere are two transition temperatures for a given carrier concentration is found at finite external voltage. Thephysics of this phenomenon is explained.
DOI: 10.1103/PhysRevB.93.245158
I. INTRODUCTION
From a framework of free-electron system in bilayer
graphene (BLG), there can be a tunable gap between theconduction and valence bands under an external electric field.Because of this property, BLG is a promising material with agreat potential for application to new electronic devices [ 1–4].
The experimental observations on high-quality suspendedBLG samples [ 5–8] has revealed that the ground-state of
the electron system at the charge neutrality point (CNP) isinsulating with a gap about 2 meV that can be closed by aperpendicular electric field of either polarity. In an externalmagnetic field, the gap grows greatly with increasing themagnetic field much larger than the Zeeman splitting [ 7].
The observed quantum Hall states at the integer fillingsfromν=0t o±4[9,10] are different from the prediction
of free-electron model by which the ν=0 state should be
eightfold degenerated. These puzzling properties of the systemat low temperature stem from the electron interactions. Anumber of theoretical models for the ground state of theinteracting electron system in BLG has been proposed [ 11–24].
Among these theories, the experimental observations can bereasonably explained only by the model of spin-polarizedcurrent state (SPCS) for the electrons [ 22]. The SPCS is a
symmetry-broken state due to the electron interactions at lowtemperature and at low carrier concentration. For applicationof BLG, it is necessary to know the phase boundary of theSPCS.
In this work, we intend to investigate the phase transition
between the SPCS and the normal state of the interactingelectrons in the BLG with and without external voltagebetween the two layers. Using the four-band model for theelectrons, we derive and solve the equation for the phaseboundary of the SPCS. At finite voltage, the electron systemcan be in a state with the layer-charge polarization (LCP).Above the LCP background, there may exist spin-polarized-current ordering. The phase transition between the SPCS witha LCP background and the state of the pure LCP should beunusual. This study not only is of the scientific interest but alsoprovides the knowledge for real application of the BLG.
II. SPIN-POLARIZED-CURRENT STATE
The lattice of the BLG shown in Fig. 1(left) contains atoms
aandbon top layer, and a/primeandb/primeon bottom layer with latticeconstant a≈2.4˚A and interlayer distance d≈3.34˚A. The
Hamiltonian of the electron system in BLG is
H=−/summationdisplay
ijσtijc†
iσcjσ+U/summationdisplay
jδnj↑δnj↓+1
2/summationdisplay
i/negationslash=jvijδniδnj,
(1)
where c†
iσ(ciσ) creates (annihilates) an electron of spin σ
at site i,tijis the hopping energy between sites iandj,
δni=ni−nis the number deviation of electrons at site
ifrom the average occupation n, and Uandv’s are the
Coulomb interactions between electrons. By the tight-bindingmodel, we consider only the intralayer nearest-neighbor (NN)[between a(a
/prime) andb(b/prime)] electron hopping with t=3 eV and
interlayer NN (between banda/prime) electron hopping with t1=
0.273 eV [ 25,26].
We use the mean-field theory (or the self-consistent Hartree-
Fock approximation) (MFT) to treat the interactions. By theMFT, the interaction part in Eq. ( 1) is approximated as
H
int=U/summationdisplay
jσ/angbracketleftδnj¯σ/angbracketrightδnjσ+/summationdisplay
i/negationslash=jvij/angbracketleftδni/angbracketrightδnj
+/summationdisplay
i/negationslash=j,σvij/angbracketleftciσc†
jσ/angbracketrightc†
iσcjσ, (2)
where the first and second lines in the right-hand side of
Eq. ( 2) are, respectively, the Hartree and Fock factorizations
and ¯σmeans the inverse spin of spin σ. According to the
many-particle theory, while the direct interactions in theHartree term are given by the bare Coulomb interactions,the interactions in the exchange part include the screeningdue to the electronic charge fluctuations. We will adopteffective exchange interactions [ 22,27] that qualitatively take
into account the screening effect. From Eq. ( 2), we extract out
the self-energy of the spin- σelectron,
/Sigma1
σ(i,j)=(U/angbracketleftδnj¯σ/angbracketright+/summationdisplay
j/prime/negationslash=jvj/primej/angbracketleftδnj/prime/angbracketright)δij
+veff
ij(/angbracketleftciσc†
jσ/angbracketright−/angbracketleftc†
jσciσ/angbracketright)/2|i/negationslash=j, (3)
where veffmeans the effective interactions with electron
screenings.
Define the order parameters mj=(/angbracketleftδnj↑/angbracketright−/angbracketleftδnj↓/angbracketright)/2 and
ρj=(/angbracketleftδnj↑/angbracketright+/angbracketleftδnj↓/angbracketright) for the spin and charge orderings,
2469-9950/2016/93(24)/245158(6) 245158-1 ©2016 American Physical SocietyXIN-ZHONG Y AN, YINFENG MA, AND C. S. TING PHYSICAL REVIEW B 93, 245158 (2016)
tt1ab
a'b'
FIG. 1. (Left) Lattice structure of the BLG. The unit cell contains
atoms aandbon top layer and a/primeandb/primeon bottom layer. The intralayer
and interlayer NN electron hoppings are tandt1, respectively. (Right)
First Brillouin zone and the two valleys KandK/primein the momentum
space.
respectively. These parameters depend only on the index of
the sublattice; within a sublattice, they are constants, mj=ml
andρj=ρl, where the position jbelongs to the sublattice
l. Because of the charge neutrality, we have ρa=−ρb/primeand
ρb=−ρa/prime, which comes from the broken layer-inversion
symmetry. In terms of these order parameters, the average/angbracketleftδn
jσ/angbracketrightis given by /angbracketleftδnjσ/angbracketright=σml+ρl/2 where jbelongs to
sublattice landσ=1(−1) for spin up (down). The Hartree
term in Eq. ( 3) can be written as
/Sigma1σH(l,l)=−σUm l+(Vll+U/2)ρl+Vl˜lρ˜l,
where ˜lmeans that ˜a(˜b)=b(a) and ˜a/prime(˜b/prime)=b/prime(a/prime), and
Vaa=−v(rab/prime)+/summationdisplay
/vectorr/negationslash=0[v(r)−v(|/vectorr+/vectorrab/prime|)],
Vab=/summationdisplay
/vectorr[v(|/vectorr+/vectorrab|)−v(|/vectorr+/vectorraa/prime)],
Vbb=−v(d)+/summationdisplay
/vectorr/negationslash=0[v(r)−v(|/vectorr−/vectord|)].
Here, v(r)=vijwithrthe distance between the position i
andj,t h e/vectorrsummations run over the positions on sublattice
a,/vectorrab/prime=(1,1/√
3,−d) and /vectorrab=(1,1/2√
3,0) and /vectorraa/prime=
(1,1/2√
3,−d) are, respectively, the vectors from atom ato
atoms b/prime,b, and a/primein the unit cell, and /vectord=(0,0,d). The
other quantities are given by Va/primea/prime=Vbb,Vb/primeb/prime=Vaa, and
Va/primeb/prime=Vb/primea/prime=Vab=Vba.
In the exchange (XC) part, the average /angbracketleftciσc†
jσ/angbracketrightcan be a
complex containing an imaginary part [ 28],
/angbracketleftciσc†
jσ/angbracketright=Rijσ+iIijσ. (4)
The imaginary part Iijσcorresponds to a current and is
self-consistently determined by the approximation. In a recentwork [ 29], we have shown that within the range of physical
interaction strength only the intrasublattice current orderingsare possible. There is no intersublattice current orderingbecause it breaks the translational invariance; more symmetrybreaking would happen in a stronger interacting system. Theremaining real part R
ijσforiandjin different sublattices
gives rise to the renormalization of the intersublattice electronhoping. We suppose this renormalization has been alreadyincluded in the original hoping terms. Therefore we here
consider only the current orderings (and the self-energies)between the sites of same sublattice.
In momentum space, the exchange part of the self-energy
is given by
/Sigma1
σXC
l(k)=−1
N/summationdisplay
k/primeveff(|/vectork−/vectork/prime|)(/angbracketleftc†
lk/primeσclk/primeσ/angbracketright−1/2),
where Nis the total number of the unit cells of the BLG lattice,
c†
lk/primeσ(clk/primeσ) creates (annihilates) an electron of momentum k/prime
and spin σon sublattice l, andk/primesummation runs over the first
Brillouin zone. Here, the main points are that (1) the quantity
/angbracketleftc†
lkσclkσ/angbracketright−1/2a saf u n c t i o no f kis sizable only when k
is close to the Dirac points KandK/prime[21], (2) for carrier
concentration close to the charge neutral point, we need toconsider only low-energy quasiparticles with kclose to the
Dirac points, and (3) v
eff(q) is a slowly varying function of q
because of the electron screening. Under these considerations,the exchange self-energy /Sigma1
σXC
l(k)f o rkin valley v=KorK/prime
can be approximated as
/Sigma1vσXC
l=−1
N/summationdisplay
v/primek/primeveff(|/vectorv−/vectorv/prime|)(/angbracketleftc†
lv/prime+k/primeσclv/prime+k/primeσ/angbracketright−1/2)
=−vc
N/summationdisplay
v/primek/prime(/angbracketleftc†
lv/prime+k/primeσclv/prime+k/primeσ/angbracketright−1/2)
−svvs
N/summationdisplay
v/primek/primesv/prime/angbracketleftc†
lv/prime+k/primeσclv/prime+k/primeσ/angbracketright
where k/primeis measured from the Dirac point v/primeand the k/prime-
summation runs over a circle k/prime/lessorequalslant1/ain valley v/prime[see Fig. 1
(right)], vc,s=[veff(0)±veff(2K)]/2, and sv=1( - 1 )f o r
v=K(K/prime). The first term in the last equal can be written
as−vc(σml+ρl/2+δ/2) with δas the average electron
doping concentration per atom. The last term corresponds tothe current ordering since the imaginary part in Eq. ( 4)i sg i v e n
by
I
ijσ=1
N/summationdisplay
vksv/angbracketleftc†
lv+kσclv+kσ/angbracketrightsin(/vectorK·/vectorrij). (5)
The “current” (up to a constant factor) Iijσis finite only when
the distributions in the two valleys are unbalanced. Since thesublattice is a triangular lattice, the current flows in threedirections with equal magnitude. However, the current densityat each atom vanishes. Note that the current I
ijσdepends on the
relative vector /vectorrijfrom position itojand does not change the
translational invariance of the system. Therefore the currentcan exist in the uniform triangular lattice.
The total self-energy in momentum space /Sigma1
vσ
l=
/Sigma1σH(l,l)+/Sigma1vσXC
l now can be written as
/Sigma1vσ
l=/epsilon1l−σu 0ml−sv/Delta1lσ−vcδ/2, (6)
where /epsilon1l=ullρl+ul˜lρ˜lwithull=Vll+U/2−vc/2 and
ul˜l=Vl˜l,u0=U+vc, and/Delta1lσis the current order parameter.
The relation /epsilon1l=−/epsilon1¯lwith ¯a(¯b/prime)=b/prime(a) and ¯b(¯a/prime)=a/prime(b)i s
valid because of the charge neutrality condition. Since the term−v
cδ/2 is a constant (independent of the layer, valley, and
spin), we hereafter will discard this term in the self-energy.
245158-2PHASE BOUNDARY OF SPIN-POLARIZED-CURRENT . . . PHYSICAL REVIEW B 93, 245158 (2016)
The order parameters are calculated by
ρl=1
2N/summationdisplay
vkσ(/angbracketleftc†
lv+kσclv+kσ/angbracketright−/angbracketleftc†
¯lv+kσc¯lv+kσ/angbracketright), (7)
ml=1
2N/summationdisplay
vkσσ/angbracketleftc†
lv+kσclv+kσ/angbracketright, (8)
/Delta1lσ=vs
N/summationdisplay
vksv/angbracketleftc†
lv+kσclv+kσ/angbracketright. (9)
The interaction parameters have been determined in the
previous work [ 22] with the results uaa≈ubb=3.3/epsilon10,uab=
6.58/epsilon10,u0=6.38/epsilon10,vc=5.38/epsilon10, andvs=6.372/epsilon10with/epsilon10=√
3t/2.
Define the operator
C†
vkσ=(c†
a,v+k,σ,c†
b,v+k,σ,c†
a/prime,v+k,σ,c†
b/prime,v+k,σ).
The effective Hamiltonian under the MFT is obtained as
H=/summationdisplay
vkσC†
vkσHvkσCvkσ
with
Hvkσ=⎛
⎜⎜⎜⎝/Sigma1vσ
1evk 00
e∗
vk/Sigma1vσ
2−t1 0
0−t1/Sigma1vσ
3evk
00 e∗
vk/Sigma1vσ
4⎞
⎟⎟⎟⎠, (10)
where e
vk=svkx+ikyin units of /epsilon10=1, and the sublattice
index lruns from 1 to 4 for the sublattices a,b,a/prime, andb/prime,
respectively.
In the absence of an external magnetic field, we have shown
that there is no spin ordering, ml=0[22]. Then, the current
ordering parameters satisfy the relations /Delta11σ=−/Delta14σ,/Delta12σ=
−/Delta13σ, and/Delta1l↑=−/Delta1l↓[22]. The charge ordering can appear
only when an external voltage is applied between the twolayers. With such a voltage, the electrons experience differentpotentials −uanduin the top and bottom layers, respectively.
The Hamiltonian matrix H
vkσis then modified by adding to it
a diagonal matrix,
Hex=Diag{−u,−u,u,u},
or/epsilon11and/epsilon12in the self-energy are replaced with /epsilon11−uand
/epsilon12−u, respectively.
To proceed, we start with the Green’s function of the
electrons. The Green’s function Gof the electron system in
the imaginary τspace is defined as
Gvσ(k,τ−τ/prime)=− /angbracketleftTτCvkσ(τ)C†
vkσ(τ/prime)/angbracketright.
In the Matsubara-frequency space, G(a 4×4 matrix) is
expressed as
Gvσ(k,iω /lscript)=(iω/lscript+μ−Hvkσ)−1, (11)
where μis the chemical potential determined by
δ=1
4N/summationdisplay
vkσ/bracketleftBigg
T/summationdisplay
/lscriptTrGvσ(k,iω /lscript)e x p (iω/lscriptη)−2/bracketrightBigg
,(12)
where Tis the temperature, ω/lscript=(2/lscript+1)πTis the Matsubara
frequency, and ηis an infinitesimal small positive constant.Note that the Hamiltonian matrix can be transformed to a
simple form. Denote the angle of the vector ( svkx,ky)a sφv
and define the matrix
M(φv)=Diag{exp(iφv),1,1,exp(−iφv)}.
With M(φv), the transformed Hamiltonian M†(φv)
HvkσM(φv)≡hvkσis independent of the momentum angle.
Similarly, we have M†(φv)Gvσ(k,iω /lscript)M(φv)≡gvσ(k,iω /lscript)
independent of the angle φv. It is then convenient to work in
the space of the transformed Hamiltonian hvkσ. By denoting
theαth component of the λth eigenfunction of hvkσwith
eigenvalue Evσ
λ(k)a sWvσ
αλ(k), theαβth element of the Green’s
function gvσis expressed as
gvσ
αβ(k,iω /lscript)=/summationdisplay
λWvσ
αλ(k)Wvσ
βλ(k)//bracketleftbig
iω/lscript+μ−Evσ
λ(k)/bracketrightbig
.
For our purpose, we write the order parameters ρland
/Delta1l↑≡/Delta1lin terms of the Green’s function. Using the definition
for the Green’s function gvσ(k,iω /lscript), we have
ρl=1
2N/summationdisplay
vkσ/bracketleftbig
gvσ
ll(k,iω /lscript)−gvσ
¯l¯l(k,iω /lscript)/bracketrightbig
, (13)
/Delta11=vsT
N/summationdisplay
vk/lscriptsvgv↑
11(k,iω /lscript), (14)
/Delta12=vsT
N/summationdisplay
vk/lscriptsvgv↑
22(k,iω /lscript). (15)
III. PHASE TRANSITION
The phase boundary of the SPCS is the relation between the
critical temperature Tcand the carrier doping concentration δ.
We will consider the cases for zero and finite external voltages.
A. Zero voltage
For zero voltage, u=0, there is no charge ordering, ρl=0
and/epsilon1l=0[22]. The Hamiltonian matrix hvkσhas the property
hvkσ=Sh−vkσS=Shvk−σS, where S=τ1σ1with the Pauli
matrix τ1implying the exchange of top and bottom layers and
σ1the exchange of ( a,b) and ( a/prime,b/prime)a t o m s .I f Wvσ(k)i sa n
eigenfunction of hvkσwith eigenvalue Evσ, then SWvσ(k)i s
an eigenfunction of h−vkσorhvk−σwith the same eigenvalue.
Therefore the whole eigenstates can be obtained from the oneonly for a given spin in a single valley. Because of this propertyof the effective Hamiltonian, we only need to consider theGreen’s function in the Kvalley for spin-up electrons. We
hereafter drop the valley and spin subscripts vandσin the
Green’s function and g(k,iω
/lscript) is understood to be the Green’s
function in the Kvalley for spin-up electrons.
As we approach the phase boundary from the SPCS side,
/Delta11and/Delta12become vanishingly small. We expand Eqs. ( 14)
and ( 15) to the first order in /Delta11and obtain
1=−vsT
N/summationdisplay
k/lscript[(gDg )11−(gDg )44], (16)
∂/Delta1 2
∂/Delta1 1=−vsT
N/summationdisplay
k/lscript[(gDg )22−(gDg )33], (17)
245158-3XIN-ZHONG Y AN, YINFENG MA, AND C. S. TING PHYSICAL REVIEW B 93, 245158 (2016)
k
kvs D
FIG. 2. Diagrammatic equation for the matrix D(green triangle).
The solid lines are the Green’s functions and the dashed line is theeffective interaction v
s.
where D=−∂hk/∂/Delta1 1is a matrix obtained as
D=Diag/braceleftbigg
1,∂/Delta1 2
∂/Delta1 1,−∂/Delta1 2
∂/Delta1 1,−1/bracerightbigg
.
In deriving Eqs. ( 16) and ( 17), we have used ∂g=
−g(∂g−1)g=g(∂hk)g. The Green’s functions in Eqs. ( 16)
and ( 17) are now calculated in the normal state with /Delta11,2=0.
In the normal state, since the system is symmetric for theexchange of top and bottom layers, the term −(gDg )
44in the
sum of Eq. ( 16) gives rise to the same contribution as ( gDg )11.
Similarly, the term −(gDg )33contributes the same as ( gDg )22
in Eq. ( 17). Therefore the two summations in Eqs. ( 16) and ( 17)
can be simplified. On the other hand, the summations over theMatsubara frequency can be carried out immediately with theresult given as
T/summationdisplay
/lscriptgll/primegl/primel=/summationdisplay
γγ/primeWlγWl/primeγWl/primeγ/primeWlγ/primeF(Eγ,Eγ/prime)
≡fll/prime(k) (18)
and
F(Eγ,Eγ/prime)=f(Eγ)−f(Eγ/prime)
Eγ−Eγ/prime,
where f(Eγ) is the Fermi distribution function and γandγ/prime
run over the indexes of the four energy levels. When Eγ=Eγ/prime,
Fis defined as F=df(Eγ)/dE γ.N o w ,E q s .( 16) and ( 17)
can be rewritten in a compact form:
Dl=−2vs
N/summationdisplay
kl/primefll/prime(k)Dl/prime, (19)
withDlthelth element in the diagonal of the matrix D.
Recalling that /Delta1l’s represent the current orderings, the matrix
Dactually describes the particle-hole propagator in the current
channel. The diagrammatic representation is shown in Fig. 2.
To search the phase boundary, we need to solve the Green’s
function at a series of selected points ( δ,T) in the normal phase.
For a given carrier concentration δ, the transition temperature
Tcis found by gradually lowering temperature Tfrom a value
higher than Tc. At each point ( δ,T), we self-consistently solve
Eq. ( 19)f o rl=2 to determine ∂/Delta1 2/∂/Delta1 1. Then we apply
the result in the right-hand side of Eq. ( 19)f o rl=1 and
denote the calculated value as λ. By inspecting this value λ,t h e
transition temperature Tcis reached when λis unity. Figure 3
(left) shows the value λas a function of temperature Tat chargeT/Δ00.2 0.4 0.6 0.8 1.0λ
0.981.001.02δ = 0
δ (10-6)1.2 1.4 1.6 1.8 2.0Τ /Δ0 = 0.3
FIG. 3. (Left) Quantity λas a function of temperature Tat charge
neutrality point δ=0. (Right) λas function of δatT//Delta1 0=0.3.
neutrality point δ=0. The transition temperature Tcatδ=0
is determined as Tc=0.567/Delta10with/Delta10=1 meV the gap
parameter observed by experiment [ 7]. However, at doping
concentration δ> 1.4×10−6, the transition temperature is
not in a one-to-one correspondence with the doping. In thiscase, we solve the equation with varying doping at a fixedtemperature. In Fig. 3(right), λis presented as a function
ofδatT//Delta1
0=0.3. We have thus determined the phase
boundary of the SPCS. The result is shown in Fig. 4.T h e
highest Tc=0.567/Delta10appears at the CNP δ=0. The largest
carrier concentration for the SPCS is about δ≈1.7×10−6
withTc//Delta1 0≈0.3. Note that the Fermi energy is EF=
8πδ/epsilon12
0/√
3t1.A tδ=1.7×10−6,w eh a v e EF//Delta1 0=0.61.
Therefore the Fermi energy at the largest carrier concentrationfor the SPCS is about the same order of magnitude as thelargest T
cat the CNP. Since the Hamiltonian is symmetric
about the carrier doping, Tcis an even function of δ.
B. Finite voltage
At finite voltage, the system is layer-charge polarized with
ρl/negationslash=0. In Fig. 5, we present the charge order parameters ρ1
δ (10-6)0.0 0.5 1.0 1.5 2.0Tc /Δ0
0.00.20.40.6
FIG. 4. Phase boundary of the spin-polarized current state.
245158-4PHASE BOUNDARY OF SPIN-POLARIZED-CURRENT . . . PHYSICAL REVIEW B 93, 245158 (2016)
u/Δ00.0 0.5 1.0 1.5 2.0ρl (10-4)
01234
δ = 0ρ1
-ρ2
Τ /Δ0 = 0.567
FIG. 5. Charge order parameters ρ1andρ2as functions of the
external potential uatδ=0a n dT//Delta1 0=0.567.
andρ2of electrons as functions of the external potential u
at the charge neutrality point δ=0. The temperature is at
the transition point T=0.567/Delta10foru=0. The potential
difference between the bottom layer and top layer is 2 u.F o r
positive u, the polarized electron number per unit cell at top
(bottom) layer is ρ1+ρ2>0(ρ3+ρ4=−ρ2−ρ1<0). The
polarization increases with increasing u.
At low temperature and low carrier concentration, the
current ordering may coexist with the charge ordering whenu/negationslash=0. To search the boundary of the spin-polarized current
phase, we take the derivative of the order parameters withrespect to /Delta1
1.F r o mE q s .( 14) and ( 15), we have
1=−vsT
N/summationdisplay
vk/lscript[gv↑(k,iω /lscript)Dgv↑(k,iω /lscript)]11, (20)
∂/Delta1 2
∂/Delta1 1=−vsT
N/summationdisplay
vk/lscript[gv↑(k,iω /lscript)Dgv↑(k,iω /lscript)]22.(21)
Since the layer inversion symmetry is now broken, these
equations are different from Eqs. ( 16) and ( 17). Note that
the dependence of the charge ordering ρlon/Delta11is negligibly
small since ρlis mainly determined by the external voltage.
(We have numerically checked this point.) The summationsover the Matsubara frequency in Eqs. ( 20) and ( 21) can be
performed similarly as shown in Eq. ( 18). The phase boundary
of the SPCS is now determined by Eqs. ( 20) and ( 21) with
/Delta1
l=0 in the Green’s function.
The obtained phase boundary of the SPCS at finite uis
shown in Fig. 6. By comparing the case of zero ushown in
Fig. 4, the phase area of the SPCS shrinks with increasing
u. The phase of the SPCS eventually disappears at certain
strength of the potential difference u. As seen from Fig. 6,t h e
unusual feature of the phase diagram for a finite uin a certain
range of strength is that there are two transition temperaturesfor a given carrier concentration. We analyze this result below.
First, there is a gap between the conduction and valence
bands because of the finite potential u. At low temperature
close to zero, for carrier concentration close to the CNP, theδ (10-6)0.0 0.5 1.0 1.5 2.0Tc /Δ0
0.00.20.40.6
u/Δ0 = 0.2
u/Δ0 = 0.15
FIG. 6. Phase boundary of the spin-polarized current state at finite
potential difference ubetween bottom and top layer.
chemical potential μ(approximately the Fermi energy) is close
to the bottom of the conduction band. The current orderinghappens when there exist a valley polarization because of theexchange effect; the energy levels of spin- σelectrons in one
valley are raised with /Delta1
1σwhile they are lowered by - /Delta11σin
another valley, resulting in the spin- σelectrons transferring
from the former to the latter valley. The level change /Delta11σ
and the electron transferring are self-consistently determined
by themselves. Below the first transition temperature, thisprocess cannot happen because there are not enough electronsbelow the level μin the conduction band for transferring.
However, with increasing the temperature, the electrons inthe valence band can be excited to the conduction band.Especially, above the first transition temperature, the excitedelectrons can participate in the transferring process and assistthe current ordering. On the other hand, the thermal excitationsof electrons between two valleys are also allowable and areweakening the exchange effect. At higher temperature abovethe second transition temperature, the exchange effect isquenched by the thermal excitations and there is no currentordering. Therefore there is a second transition temperaturehigher than the first one.
In Fig. 4, we have seen that there are two transition
temperatures for 1 .4×10
−6<δ< 1.7×10−6where the
external voltage is zero. Within this doping range and belowthe first transition temperature, there is no gap between theconduction and valence bands. The SPCS emerges above thefirst transition temperature just because the thermal excitationsof electrons from the low levels in one valley to the levelsabove the chemical potential in another valley assist theelectron transferring from the former to the latter valley. Themechanism for the two transition temperatures is the same asexplained above.
IV . SUMMARY
Using the four-band model, we have studied the phase
boundary of the spin-polarized current state of the interacting
245158-5XIN-ZHONG Y AN, YINFENG MA, AND C. S. TING PHYSICAL REVIEW B 93, 245158 (2016)
electrons in bilayer graphene. In the absence of external
voltage, the highest transition temperature is found as Tc=
0.567/Delta10=0.567 meV appearing at the charge neutrality point
δ=0. The SPCS phase extends to a carrier concentration
about δ≈1.7×10−6withTc≈0.3 meV . At finite voltage
between the two layers, we find there are two transitiontemperatures corresponding to a given carrier concentration.The physics of such an unusual phase boundary is explainedas the two effects of the thermal excitations: (1) the excitedelectrons participate in the process of transferring from onevalley to another valley and assist the current ordering and
(2) excitations between two valleys at higher temperaturequench the current ordering. The result should be useful forreal application of the BLG.
ACKNOWLEDGMENTS
This work was supported by the National Basic Research
973 Program of China under Grant No. 2012CB932302 andthe Robert A. Welch Foundation under Grant No. E-1146.
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245158-6 |
PhysRevB.79.165322.pdf | Density-matrix theory of the optical dynamics and transport in quantum cascade structures:
The role of coherence
C. Weber,1,*A. Wacker,1and A. Knorr2
1Mathematical Physics, Lund University, P .O. Box 118, 22100 Lund, Sweden
2Institut für Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Technische Universität Berlin,
Hardenbergstr. 36, 10623 Berlin, Germany
/H20849Received 23 November 2008; revised manuscript received 3 April 2009; published 30 April 2009 /H20850
The impact of coherence on the nonlinear optical response and stationary transport is studied in quantum
cascade laser structures. Nonequilibrium effects such as the pump-probe signals, the spatiotemporally resolvedelectron density evolution, and the subband population dynamics /H20849Rabi flopping /H20850as well as the stationary
current characteristics are investigated within a microscopic density-matrix approach. Focusing on the station-ary current and the recently observed gain oscillations, it is found that the inclusion of coherence leads toobservable coherent effects in opposite parameter regimes regarding the relation between the level broadeningand the tunnel coupling across the main injection barrier. This shows that coherence plays a complementaryrole in stationary transport and nonlinear optical dynamics in the sense that it leads to measurable effects inopposite regimes. For this reason, a fully coherent consideration of such nonequilibrium structures is necessaryto describe the combined optical and transport properties.
DOI: 10.1103/PhysRevB.79.165322 PACS number /H20849s/H20850: 78.20.Bh, 74.78.Fk, 42.65. /H11002k, 73.63. /H11002b
I. INTRODUCTION
Quantum cascade lasers /H20849QCLs /H20850/H20849Ref.1/H20850are semiconduc-
tor heterostructure lasers employing the transitions betweenquantized intersubband levels in quantum well structures
2–6
and act as a source of radiation in the terahertz /H20849THz /H20850/mid-
infrared regime. The laser consists of multiquantum well pe-riods comprising the electron injector and the optically activeregion. These periods are repeated tens or even hundreds oftimes over the length of the structure.
7To drive the electrons
through the sample, an external bias is applied. Scatteringprocesses and optical recombination between the conductionsubbands in the doped structure within as well as betweenperiods lead to stationary electronic occupations out of equi-librium. While on the technological side, this light sourcecan be used for spectroscopy in the fields of environmentaldetection or medicine,
7,8it offers on a fundamental ground
an interesting model system to study intersubband chargedynamics in a structure where the optical and the transportproperties are closely interrelated.
The first semiconductor heterostructure laser operating in
the intersubband regime was realized by Faist et al.
1in 1994.
Since then, many types of QCLs of different design havebeen built and optimized; see, e.g., Refs. 7and9for an
overview. The QCL has been the subject of extensive theo-retical research. The stationary properties of QCLs werestudied by a rate equation
10,11and a Boltzmann-type
approach12–17as well as a quantum theory employing both
nonequilibrium Green’s functions18–22and density-matrix
theory.23–25Here, the gain, the current-voltage characteris-
tics, and the stationary charge distributions have been estab-lished. First results regarding the nonlinear optical propertiessuch as optically induced subband population dynamics havebeen presented by us within a density-matrix theory.
26
One central result of these studies is that coherence can
play an important role in the determination of the stationarycurrent, requiring a fully microscopic theory of the currentincluding nondiagonal density-matrix elements.
20,27,28Here,
the often applied rate and Boltzmann equation approaches,based on Wannier-Stark hopping /H20849WSH /H20850, fail. Experimen-
tally, indications of coherent charge transport have been ob-served in the oscillatory gain recovery in pump-probe experi-
ments of mid-infrared QCLs /H20849Refs. 29–32/H20850as well as
recently in THz structures.
33This oscillatory behavior has
been attributed to resonant tunneling through the injectionbarrier of the laser. These observations suggest that coherenteffects might also become visible in optics in the time regimebeyond the light-matter interaction. However, differentstudies
34,35reveal a simple relaxation in the gain recovery,
showing that not all samples exhibit this coherent effect.
Motivated by these investigations, this paper is focused
on the role and importance of coherence in the interplaybetween ultrafast optical dynamics and stationary transportin quantum cascade structures. We investigate the regimeswhere coherence is of relevance in the combined optics-transport system. To this end, we present a fully microscopictheory describing the dephasing and tunneling processes in aQCL structure. In order to systematically investigate the op-tical and transport regimes, we focus on two quantities de-scribing the coherent and the incoherent evolution within thesystem: the tunnel coupling 2 /H9024between the two states
across the main injection barrier and the level broadening /H9003
of the states. We systematically vary the width of the maintunneling barrier in the QCL structure, thus establishing arelation between these two central quantities. Using this re-lation, we investigate the importance of coherence in thecalculated signals: In the transport regime, we compare a rateequation /H20849WSH /H20850approach with a fully microscopic current
theory to investigate the regime of validity of both models.In the optical regime, we consider pump-probe calculations,complemented by the spatiotemporally resolved electrondensity evolution and the subband population dynamics dueto strong ultrafast excitation. Using these nonlinear results,aspects of the nature of the optically induced charge dynam-PHYSICAL REVIEW B 79, 165322 /H208492009 /H20850
1098-0121/2009/79 /H2084916/H20850/165322 /H2084914/H20850 ©2009 The American Physical Society 165322-1ics and the importance of coherence can be addressed.23
Comparing the optical and transport results, we find that
the influence of coherence is observable in opposite param-eter regimes: the inclusion of coherence in stationary trans-port calculations becomes important for 2 /H9024/H11351/H9003 , where the
coherence between subbands limits the current flowing
through the structure, while in the nonlinear optical gain dy-namics, the inclusion of coherence drives the observed oscil-
lations in the gain recovery for 2 /H9024/H11407/H9003 . Combining the two
results, we thus find that it is necessary to consider a fullycoherent theory in order to understand the combined optics-transport problem independent of the parameter range of op-eration.
II. THEORY
The QCL heterostructure is modeled as a multiconduction
subband system, with each period comprising an injector andan active region. Within the effective mass and envelopefunction approximations,
36the wave functions, assumed to
be separable and infinitely extended in the quantum wellplane, are given by
/H9023
ik/H6023/H20849x/H6023,z/H20850=1
/H20881Aeik/H6023x/H6023/H9264i/H20849z/H20850uk/H6023/H110150/H20849x/H20850, /H208491/H20850
where Ais the in-plane quantization area, x/H6023andk/H6023are the
in-plane spatial and momentum vector, respectively, /H9264i/H20849z/H20850is
the envelope function in growth direction, and uk/H6023/H110150/H20849x/H20850is the
Bloch function taken at the band edge. There are several
natural possibilities to choose the wave functions /H9264i/H20849z/H20850to
describe the system. Since we consider the optical responseof the system as well as the WSH model in the current cal-culation, we choose the Wannier-Stark /H20849WS/H20850basis which di-
agonalizes the heterostructure potential as well as the exter-nally applied bias. This offers an intuitive physicalinterpretation of the optical transitions and the scattering be-tween single-particle WS states as well as of the electroncounting between approximate system eigenstates. It shouldbe noted that, even though the physical observables are inprinciple independent of the choice of basis, different resultsare expected for different basis choices due to the necessaryapproximations applied in many-particle problems. In thefollowing, the theory is applied to the THz QCL discussed inRef. 37. The band structure of this QCL in the original de-
sign is shown in Fig. 1with the important subbands marked.
This structure is considered throughout this paper, consider-ing different injection barrier widths b. The parameters used
in the calculations are found in Table I.
In order to describe the ultrafast nonlinear optical and
stationary transport properties of the laser, it is preferable toconsider a fully microscopic treatment of the scatteringmechanisms in the structure. In this work, we consider theinteraction of the electronic system with longitudinal optical/H20849LO/H20850phonons via the polar /H20849Fröhlich /H20850coupling as well as
with the ionized doping centers in the laser. Even thoughCoulomb scattering can be treated in the same way,
38,39we
do not consider this interaction here. For the relatively lowlydoped THz laser considered here, it was shown that scatter-ing with impurities typically dominates over Coulomb
scattering.
40Acoustic phonons are excluded since they in-
duce only very long scattering times in quantum wells andact mainly as a low-energy intrasubband thermalizationmechanism.
41Interface roughness scattering is excluded due
to the difficulty of quantifying it microscopically, while alloyscattering is small in structures with a binary material for thewells /H20849GaAs /H20850and therefore neglected.
42The included scat-
tering mechanisms are treated in a density-matrix correlationexpansion approach within a second-order Born-Markovapproximation.
43,44We do not include renormalizations of
the subband energies due to scattering in the form of princi-pal values /H20849see Sec. I IB3 for a discussion of this approxi-
mation /H20850; it is assumed that these are small and approximately
constant for the different transitions, leading only to an ab-solute energy shift without physical consequences /H20849see Ref.
45for a discussion of scattering-induced energy renormaliza-
tions in quantum wells /H20850.
A. Hamiltonian
To derive the dynamical equations, we divide the Hamil-
tonian of the system into three parts,
H=H0+Hel−light +Hscatt. /H208492/H20850
The first part,
H0=HSL+H/H9255+H0,ph=/H20858
ik/H6023/H9255ik/H6023aik/H6023†aik/H6023+/H20858
q/H6036/H9275qbq†bq,/H208493/H20850
describes the kinetics of the electrons in the heterostructure
potential HSLas well as the externally applied bias H/H9255and-100-500
-20 0 20 40energy (meV )
growth position (nm)5
42’
1’b
FIG. 1. /H20849Color online /H20850Band structure of the THz QCL from Ref.
37for the injection barrier width b=5.5 nm under resonance con-
dition. The subbands involved in the laser transition /H208494,1 /H11032/H20850as well
as the injector subbands /H208495,2 /H11032/H20850are marked.
TABLE I. Structural and material parameters used in the
calculations.
material system GaAs /Ga0.15Al0.85As
doping density n2d 2.945/H110031010cm−2
well electron mass m/H115690.067 m 0
LO phonon energy /H6036/H9275LO 36.7 meV
high-frequency permittivity /H9255/H11009 10.9
static permittivity /H9255s 12.9WEBER, WACKER, AND KNORR PHYSICAL REVIEW B 79, 165322 /H208492009 /H20850
165322-2the kinetics of the phonons H0,ph. The electronic part is di-
agonal due to the choice of basis. aik/H6023†/H20849aik/H6023/H20850denotes the cre-
ation /H20849annihilation /H20850operator of an electron in subband iwith
quasimomentum k/H6023and energy /H9255ik/H6023=/H9255i+/H60362k/H60232//H208492mi/H20850andbq†/H20849bq/H20850
the creation /H20849annihilation /H20850operator of an LO phonon with
three-dimensional quasimomentum qand energy /H6036/H9275q
/H11013/H6036/H9275LO. Subband nonparabolicity is neglected, and the sub-
band effective masses are assumed to be constant, mi/H11013m/H11569,
where m/H11569is the effective mass of the well material.
Hel−light describes the interaction of the system with a co-
herent classical light source. The polarization of the lightfield is chosen in the direction of the dipole moment, i.e., inthe growth /H20849z-/H20850direction, and the field is assumed to be a
spatially homogeneous Gaussian pulse,
A/H20849t/H20850=A/H20849t/H20850e
z,A/H20849t/H20850=A0exp/H20853−/H20849t//H9270/H208502/H20854cos/H20849/H9275Lt/H20850, /H208494/H20850
with the laser frequency /H9275Land Gaussian pulse duration /H9270.
The pulse area is defined via the envelope of the pulse.46
Under the assumption of a homogeneous excitation, theHamiltonian is momentum diagonal and reads as
H
el−light =eA/H20849t/H20850/H20858
ij/H20858
k/H6023Mijaik/H6023†ajk/H6023, /H208495/H20850
with the elementary charge e/H110220, the coupling elements
Mij=1
2/H20855i/H20841/H20853pˆz/m/H20849z/H20850+eA/H20849t/H20850//H208512m/H20849z/H20850/H20852/H20854+H.c. /H20841j/H20856, and the momen-
tum operator pˆz=/H20849/H6036/i/H20850/H11509zwith the space-dependent effective
mass m/H20849z/H20850.
Finally, Hscatt describes the scattering processes:
electron-LO phonon interaction Hel−ph as well as scattering
with ionized doping centers Hel−imp .
1. Electron-LO phonon interaction
The electron-LO phonon coupling Hamiltonian is given
by
Hel−ph=/H20858
ij/H20858
k/H6023,qgijqaik/H6023†bqajk/H6023−q/H6023+gijq/H11569ajk/H6023−q/H6023†bq†aik/H6023, /H208496/H20850
where q/H6023is the in-plane projection of q. The coupling matrix
element is given by the Fröhlich coupling,
gijq=−i/H20875e2/H6036/H9275LO
2/H92550V/H208731
/H9255/H11009−1
/H9255s/H20874/H208761/2e/H20849q/H20850·q
q2/H20855/H9264i/H20841eiq/H11036z/H20841/H9264j/H20856./H208497/H20850
Here,/H9255sand/H9255/H11009are the static and high-frequency permittiv-
ity, respectively, Vis the quantization volume, e/H20849q/H20850is the
displacement unit vector, and q/H11036is the projection of the mo-
mentum qin growth direction.
2. Interaction with ionized doping centers
Since the quantum cascade structure is doped, it is neces-
sary to take into account the interaction of the electrons withthe ionized doping centers. Typically, either a barrier or awell in the QCL is doped, making it necessary to distributethe ions in this layer. Here, we treat the interaction followingRef. 19, where the dopant density is distributed on several
/H9254
sheets located at zlin the doped barrier/well. The ionized
doping centers are considered as classical scattering centers.The Hamiltonian for the interaction is given by
Hel−imp =/H20858
ij/H20858
k/H6023,q/H6023/H20858
lVijl/H20849q/H6023/H20850aik/H6023+q/H6023†ajk/H6023, /H208498/H20850
with the screened electron-impurity interaction potential
Vijl/H20849q/H6023/H20850=1
A−e2
2/H92550/H9255s/H20881q/H60232+/H92612/H20855/H9264i/H20841e−iq/H6023x/H6023le−/H20881q/H60232+/H92612/H20841z−zl/H20841/H20841/H9264j/H20856. /H208499/H20850
Here,/H9261is the screening constant and llabels the position of
the randomly distributed individual ion /H9254distributions in the
growth direction. For the screening, we use the static limit ofthe Lindhard formula for a homogeneous electron gas.
47
B. Dynamical equations
1. Equation structure and approximations
We derive the dynamical equations using a correlation
expansion within a density-matrix approach in second-orderBorn-Markov approximation, applying a bath approximationfor the LO phonons.
43,44This leads to equations of motion
for the microscopic polarizations fij,k/H6023=/H20855aik/H6023†ajk/H6023/H20856/H20849i/HS11005j/H20850and the
subband occupations nik/H6023=/H20855aik/H6023†aik/H6023/H20856. On the level of the
phonon- /H20849impurity /H20850-assisted coherences /H20855aik/H6023†bq/H20849†/H20850ajk/H6023/H11032/H20856
/H20849/H20855aik/H6023†Vmnl/H20849q/H6023/H20850ajk/H6023/H11032/H20856/H20850, respectively, we neglect the interaction
with the optical field since these terms are of higher order in
the coupling.48We assume the system to be homogeneous in
the plane perpendicular to the growth direction z; then, we
can restrict to a density-matrix diagonal in the in-plane mo-
mentum: /H20855aik/H6023†ajk/H6023/H11032/H20856=fij,k/H6023/H9254k/H6023k/H6023/H11032.
The general dynamical equations are found in the Appen-
dix. While for the interaction with the doping centers, thecomplete equation structure is considered, only the terms lin-ear in the density matrix are taken along for the electron-phonon interaction due to numerical reasons. This approxi-mation is justified if we assume that we are working in theregime of nondegenerate electron gases, where n
ik/H6023/H112701 which
is typically fulfilled in these QCL structures.
We should stress at this point that, due to the spatial ex-
tension of the Wannier-Stark wave functions, it is essential toconsider the whole set of matrix elements in the calculations.Testbed calculations considered only certain sets of matrixelements, e.g., we restricted to the diagonal/nondiagonal
scattering terms g
ijqgijq/H11569, considered typically in nonbiased
/H20849equilibrium /H20850quantum well systems,5,38,39,49or alternatively
to terms where the overlap of the wave functions is the main
argument, i.e., terms such as giiqgijq/H11569along with the above set
of terms. Both versions lead to nonstable results in someparameter regimes where the coherences are important. It isthus essential to consider the full set of matrix elementswhen considering the properties of the QCL at resonancecondition and for large injection barrier widths, i.e., smalltunnel coupling across the injection barrier.
When considering the WSH model, the relaxation pro-
cesses are restricted to the Boltzmann scattering given byDENSITY-MATRIX THEORY OF THE OPTICAL DYNAMICS … PHYSICAL REVIEW B 79, 165322 /H208492009 /H20850
165322-3n˙ik/H6023/H208490/H20850=−/H9003ik/H6023outnik/H6023+/H9003ik/H6023in/H208491−nik/H6023/H20850. /H2084910/H20850
The in- and out-scattering rates /H9003ik/H6023inand/H9003ik/H6023outare functions of
the subband occupations nlk/H6023/H11032and are given for both the
electron-phonon and electron-impurity interactions in the
Appendix.
The system temperature enters the calculations through
the scattering rates which include the phonon distribution,assumed to be given by an equilibrium Bose-Einstein distri-bution n
q=/H20851exp/H20849/H6036/H9275q/kBT/H20850−1/H20852−1, and the screening param-
eter/H9261for the interaction of the electrons with the doping
centers /H20849see also Ref. 50/H20850. Here, the approximation is made
that the lattice and the electronic temperature are the same.
2. Periodic boundary conditions
In order to describe the extension of the periodically
coupled structure, it is necessary to apply appropriate bound-ary conditions in the growth direction. Since the inclusion ofcoherence between states of different periods is important todescribe the nonlinear optical dynamics due to the spatialextension of the wave functions,
26we consider a nearest-
neighbor approach where the coherence and wave functionoverlap between two adjacent periods /H20849n,n
/H11032/H20850are taken into
account. We checked this approximation by showing that
/H20841finjn /H11032,k/H6023/H20841,ginjn /H11032q,Vinjn /H11032l/H20849q/H6023/H20850/H110150 for /H20841n−n/H11032/H20841/H113502, where
finjn /H11032,k/H6023/H20849ginjn /H11032q,Vinjn /H11032l/H20849q/H6023/H20850/H20850denotes the coherence /H20849coupling ele-
ment /H20850between the states /H20841i/H20856and /H20841j/H20856in the periods nandn/H11032,
respectively. We apply translational invariance of the densitymatrix between periods,
51
finjn /H11032,k/H6023=fi/H20849n+1/H20850j/H20849n/H11032+1/H20850,k/H6023,nin,k/H6023=ni/H20849n+1/H20850,k/H6023. /H2084911/H20850
For the wave functions and the band edge energies /H9255i,w e
apply a coordinate shift and a bias drop, respectively,
/H9264in/H20849z/H20850=/H9264i/H20849n+1/H20850/H20849z+Lper/H20850,/H9255in=/H9255i/H20849n+1/H20850−eEL per,/H2084912/H20850
where Lperis the period length and Ethe applied electric
field. In the following, the index i/H11013/H20849i,n/H20850is taken as a com-
posite index.
3. Shortcomings of the model
Even taking along the whole set of equations within the
approximations discussed above, we partially encounternegative occupations n
ik/H20849see Fig. 2/H20850which is known from
the treatment of Redfield-type equations such as consideredhere.52–54However, this typically does not lead to unphysical
results if we focus on averaged observables. For all dynami-cal calculations, we obtain physical values for the currentdensity, the total electron densities in each subband, and thegain spectrum. However, the broadening in Eq. /H2084913/H20850may
become negative for regimes where coherences play an im-portant role due to the negative occupations. This may bedue to the fact that /H20849i/H20850the scattering rates as such are not
observables and /H20849ii/H20850the simple formula for the broadening
used here, which is typically applied in the literature, doesnot take into account stationary coherence. In order to guar-antee strictly positive values for the averaged scattering ratesto calculate the broadening in Eq. /H2084913/H20850, which we can use as
a measure of the lifetime of the Wannier-Stark states, weperform a Gaussian smoothening of the stationary distribu-tions and use these in the calculations of the broadening /H9003.
Figure 2shows an example of this smoothening for two dif-
ferent barrier widths: while for small barrier widths and low
temperatures the negative occupations are negligible, andthus the corresponding averaged scattering rates are alwayspositive, the smoothening becomes important for large bar-rier widths where the negative occupations can lead to nega-tive values. Thus, the values obtained for the broadening /H9003
should be viewed as approximative values. Since we are con-cerned with qualitative results only, this is justified. For thecases where the negative occupations are negligible, and thusthe averaged scattering rates strictly positive, the smoothen-ing induces changes in the broadening of 2%–3%. We wouldlike to stress again, however, that all dynamical calculationsand all results except for Fig. 5are performed without arti-
ficial modifications of the calculated data, such as any kindof smoothening. In addition, it should be noted that the cal-culations here are performed at the point of “maximal coher-ence” with respect to the tunnel coupling across the injectionbarrier, i.e., at resonance /H20849cf. Sec. III A /H20850. Away from reso-
nance, the influence of coherence and the correspondinglyrelated problems such as nonpositivity are strongly reduced.
The reason for the quite large negative values for the oc-
cupations partially encountered may be explained by the factthat the system under consideration is a very complicatedone: for the nearest-neighbor coupling considered in the cal-culations, we have 35 independent coherences, and thusthere is a very strong interplay between the different frequen-cies of the coherences which can destroy the strict positivityof the occupations. For small barrier widths and low tem-peratures, the general dynamical Eqs. /H20849A1/H20850and /H20849A5/H20850reduce
to the Boltzmann dynamics given in Eq. /H2084910/H20850. Here, any
negative occupations observed in our full calculations arenegligible /H20849n
ik/H11407−10−8/H20850/H20851see, e.g., Fig. 2/H20849a/H20850/H20852. In the case of
the Boltzmann scattering dynamics, i.e., including only thediagonal elements of the density matrix, the stationary occu-pations remain strictly positive for all barrier widths andtemperatures.
The specific oscillation energies of the peaks can be ex-
plained by the energetic structure of the system: the peakscorrespond to different combinations of the bias drop perperiod eFL
perand the LO phonon energy ELO. In Fig. 2/H20849b/H20850,
e.g., the large peaks correspond to the energy eFL per−ELO
=56.40−36.7 meV=19.7 meV, while the smaller peak po-
sitions are given by 2 eFL per−3ELO=2.7 meV. The latter are00.010.020.03
0 10 20 30 40nik
−h2k2/(2 m) (meV)subb. 1
subb. 5
-0.0100.010.02
0 10 20 30 40nik
−h2k2/(2 m) (meV)subb. 1
subb. 5
(b) (a)
FIG. 2. /H20849Color online /H20850Gaussian smoothening of the subband
occupations nikfor the two main subbands /H20849injector 5, upper laser
state 1 /H20850atT=10 K: /H20849a/H20850b=4.5 nm; /H20849b/H20850b=11.0 nm.WEBER, WACKER, AND KNORR PHYSICAL REVIEW B 79, 165322 /H208492009 /H20850
165322-4weaker since they represent a higher order process of itera-
tive transport and scattering. For the two subbands, the peaksare shifted by /H9004E/H110150.1 meV, corresponding to the tunnel
splitting between the two states, E
1/H11032−E5/H110150.1 meV.
As stated above, there is a strong interplay between dif-
ferent coherences of sharp energy. For T=50 K and a certain
barrier width regime, we need to incorporate additionaldamping in order to obtain convergence. In the calculations,we do this by adding a phenomenological, low-energy intra-subband scattering mechanism with an energy which is un-commensurable with the LO phonon energy /H20849E/H110150.5 meV /H20850
and use the LO phonon coupling element with an increasedstrength. We should note that this damping is of a purelyphenomenological character and not a physical scatteringmechanism as implemented here. The concerned barrierwidths are marked in Figs. 5and6. The problem with con-
vergence may be a consequence of the combination of ne-glecting low-energy scattering channels and applying theBorn-Markov approximation within the density-matrixtheory. Another explanation of this convergence problem isthat scattering-induced energy renormalizations are ne-glected at this order of the perturbation expansion. This is anapproximation often applied within the Born-Markovapproximation.
25,38Since the renormalizations, like the scat-
tering rates, are temperature dependent, this could explainwhy the convergence problems only appear for certain tem-peratures. On the other hand, since the resonances depend onthe specific energy structure, and thus on the barrier width b,
problems are expected to appear only for certain barrierwidths, as witnessed in the results.
Including an artificial damping as discussed above, we
find that the value of the stationary current is approximatelyindependent of the scattering strength. This has already beenobserved in earlier studies of the QCL /H20849Refs. 15and41/H20850and
attributed to the fact that a low-energy dissipative scatteringmechanism such as LA phonons leads mainly to a thermali-zation within the subbands but has no strong effect on theactual value of the stationary current. We adopt the validityof this statement here and take the obtained values of thecurrent as physical results. On the other hand, it is clear thatthe inclusion of additional large damping leads to a strongoverestimation of the total broadening as calculated in Eq./H2084913/H20850. Since the relation between the broadening /H9003and the
tunnel coupling 2 /H9024is used to discriminate qualitatively dif-
ferent regimes, the inclusion of additional scattering mecha-nisms for the concerned barrier widths /H20849which typically leads
to a larger broadening /H20850does not lead to qualitatively new
regimes of interest /H20849see Fig. 5/H20850. It is for this reason that one
can safely disregard these values in the discussion of thebroadening and the tunnel coupling as is done in the follow-ing.
III. NUMERICAL RESULTS
In this section, we apply the theory of Sec. IIto the THz
laser from Ref. 37. This structure has been investigated with
respect to its transport in the stationary laser regime as wellas its gain properties.
19In the regime of ultrafast nonlinear
optics, Rabi oscillations were considered recently.26First, the systematic setup is discussed, along with the
calculation of the central quantities used for the further dis-cussion of the transport and optical properties. The stationarynonequilibrium due to the different scattering mechanisms isdetermined. This is necessary for the determination of thestationary value of the current but also as a starting point forthe consideration of the optical response. Then, the WSHapproach is compared with a full current calculation to in-vestigate the importance of the inclusion of coherence in thestationary transport. Finally, the nonlinear optical response ofthe structure is considered, focusing on pump-probe signalsas well as the spatiotemporally resolved electron density andthe optically induced population dynamics, to study the roleof coherence in the optical regime.
A. Systematic setup and stationary state
In order to systematically investigate the importance of
coherence, we vary the width of the main tunneling barrierconnecting the injector and the active region of the laser.When resonant tunneling between the injector and the upperlaser state is small, the WS states are approximately localizedin the injector and in the active region. For a certain appliedbias, the WS states become delocalized across the injectionbarrier and an anticrossing occurs, where the injector and theupper laser level form a pair of binding/antibinding states.The system is assumed to be in resonance at the center of the
anticrossing, i.e., for the bias at which the splitting energybetween the two respective levels has a minimum. Here, thelevel splitting /H9004Eequals twice the tunnel coupling 2 /H9024be-
tween the localized injector and the upper laser state /H20849see
Fig.3/H20850.
The QCL as a multisubband nonequilibrium system is a
very complex structure, and thus a straightforward analyticaldetermination of the stationary population and coherence dis-tributions due to scattering is not possible. We thus numeri-cally determine the initial conditions to be stationary solu-tions for f
ij,k/H6023andnik/H6023without the optical field by solving all
scattering contributions from an arbitrary state of fixed total
population until a steady state is reached. Figure 4shows the
process of determining the initial distributions by startingfrom subband populations n
i=2 /A/H20858k/H6023nik/H6023given by single-
subband Fermi distributions with equal populations in each
subband. An approximate stationarity is reached on a time-150-100-50
-40 -20 0 20 40 60energy (meV)
growth position (nm)b 541’
∆E
00.10.20.3
56 56.5 57 57.5∆E (meV)
bias drop per period (mV)b = 11.0 nm b = 10.0 nm
∆E=2Ω
(b) (a)
FIG. 3. /H20849Color online /H20850/H20849a/H20850Injector /H208495/H20850and upper laser state /H208491/H11032/H20850
showing an anticrossing at the main tunneling barrier at exact reso-nance of the corresponding Wannier states. Lower laser state /H208494/H20850is
also shown. /H20849b/H20850Tunnel splitting energy /H9004E=/H20841E
5−E1/H11032/H20841for varying
bias and different barrier widths bdetermining the resonance bias at
the center of the anticrossing and the tunnel coupling 2 /H9024/H11015/H9004E.DENSITY-MATRIX THEORY OF THE OPTICAL DYNAMICS … PHYSICAL REVIEW B 79, 165322 /H208492009 /H20850
165322-5scale of a few picoseconds, while long time intraband redis-
tributions can last up to several nanoseconds /H20849not shown /H20850.
For small barrier widths, i.e., large tunnel coupling /H9024, the
relaxation is described well by a Boltzmann relaxation n˙ik/H6023/H208490/H20850.
For larger barrier widths, the influence of the coherence on
the stationary populations cannot be neglected any longerand becomes important which is discussed in detail in thenext section.
Having determined the stationary distributions, we can
calculate the scattering-induced broadening of the states. Thelevel broadening of the considered states is approximatelygiven by the mean of the two tunnel-split Wannier-Starkstates, /H9003=/H20849/H9003
1/H11032+/H90035/H20850/2, where the broadening is determined
by the stationary Boltzmann out-scattering rate,
/H9003i=2
A/H20858k/H6023nik/H6023/H208490/H20850/H9003ik/H6023out/H20853nlk/H6023/H208490/H20850/H20854
ni/H208490/H20850, /H2084913/H20850
for the stationary occupations nlk/H6023/H208490/H20850. At these low tempera-
tures, the broadening is typically dominated by the impurity
scattering, which is strongly temperature dependent due tothe screening length
50and thus increases strongly from 10 to
50 K, while the interaction with phonons is essentially givenby spontaneous emission.
Figure 5shows the calculated values of /H9024and/H9003as a
function of the barrier width b. As is expected, the broaden-
ing remains roughly constant for varying barrier widths sincethe scattering rates are not strongly influenced by the width,while the tunnel coupling across the barrier approaches zero
forb→/H11009. For T=10 K, the two energies are roughly the
same for b/H110156.0. This is the regime which is physically in-
teresting, and thus barrier widths slightly larger and less thanthis value are the focus in the following investigations. ForT=50 K, the broadening of the states is much larger then the
tunnel coupling for all considered barrier widths, and thus notransition between different regimes is expected.
B. Stationary results: current calculations
The stationary distributions far from equilibrium deter-
mined in Sec. III A are the starting point for the optical dy-
namics discussed later in this chapter. Here, we consider theaccompanying evolution to the steady state value of the cur-rent flowing through the structure. In order to investigate theinfluence of coherence on the system, we focus on two dif-ferent approaches to the current calculation: the first restrictsto the occupations n
ik/H6023to consider a rate equation approach—
the so-called Wannier-Stark hopping model—and the second
takes the coherences fij,k/H6023into account.
We start with the rate equation approach, where in effect
the electrons are counted as they cross an interface at a fixedpoint in the growth direction of the structure due to scatter-ing between states which are localized at different spatialparts of the structure. The WSH current is just given by theapplication of Fermi’s golden rule to each set of states. Forthe electron-phonon interaction, it is determined via /H20849n
−q
=nq/H20850,
JWSH/H11006/H20849t/H20850=−22/H9266
/H6036AL/H20858
ij/H20858
k/H6023,q/H20841gijq/H208412/H9254/H20849/H9255ik/H6023−/H9255jk/H6023+q/H6023/H11006/H9255 LO/H20850
/H11003/H20873nq+1
2/H110071
2/H20874/H20849dii−djj/H20850nik/H6023/H208491−njk/H6023+q/H6023/H20850, /H2084914/H20850
where J/H11006denotes the absorption/emission of a phonon, dii
=−eziiis the expectation value of zfor the WS state /H20841i/H20856, and
the sum i,jis carried out over all states. The factor 2 arises
due to spin degeneracy and L=NLperis the length of the
structure with Nperiods of length Lper. The current is fully
determined by the diagonal elements of the density matrixn
ik/H6023, considering jumping of electrons from one state to the
next, and reaches its steady state value as determined by the
relaxation in Fig. 4when restricting to Boltzmann dynamics.
An analogous expression is found for the elastic impurityscattering. While Eq. /H2084914/H20850can be motivated by hopping of
electrons between different positions z
iiandzjj, it is in fact
an approximation for the full current driven by coherences/H20851Eq. /H2084915/H20850/H20852; see Ref. 56.
When deriving the current microscopically from the cur-
rent operator Jˆ=−e//H208492m
0/H20850/H9023ˆ†/H20849x,t/H20850/H20851pˆ+eA/H20849t/H20850/H20852/H9023ˆ/H20849x,t/H20850+H.c.,
one arrives at another picture of the current, where it is thecoherence between states which carries the current flowingthrough the structure. Here, the current is given completelyvia the nondiagonal elements of the density matrix f
ij,k/H6023,
which in growth direction yields00.40.81.2
0 20 40 60subband pop. (10-4nm-2)
time (ps)T=1 0K1
2
345
0 1 2 3
00.40.81.2
0 20 40 60subband pop. (10-4nm-2)
time (ps)T=5 0K1
2
345
(b) (a)
FIG. 4. /H20849Color online /H20850Population relaxation dynamics using the
full set of dynamical equations for a barrier width of b=5.5 nm at
T=10 and 50 K, determining the stationary nonequilibrium of the
system. Inset: Short-time evolution of the subband populations.
0246
5 6 7 8 9 10 11 12 13Γ(meV )
2Ω(meV )
barrier width (nm)2Ω
Γ10 K
Γ50 K
FIG. 5. /H20849Color online /H20850Tunnel coupling 2 /H9024and mean broaden-
ing/H9003=/H20849/H90035+/H90031/H11032/H20850/2 of the injector and the upper laser state for
different barrier widths band at T=10 and 50 K /H20849see remark in
Ref. 55/H20850.WEBER, WACKER, AND KNORR PHYSICAL REVIEW B 79, 165322 /H208492009 /H20850
165322-6Jfull/H20849t/H20850=−2e
AL/H20858
ij/H20858
k/H6023vijfij,k/H6023, /H2084915/H20850
where vij=1
2/H20855i/H20841/H20851pˆz/m/H20849z/H20850+eA/H20849t/H20850/m/H20849z/H20850/H20852+H.c. /H20841j/H20856are the veloc-
ity matrix elements and the factor 2 again arises due to spindegeneracy. For the calculation, the polarizations f
ij,k/H6023are ini-
tially set to zero. The finite occupations in the subbands lead
to a stationary current which is fully determined by the scat-tering in the structure.
We now carry out current calculations using both the
WSH and the full approach for varying barrier widths. Theresult is shown in Fig. 6. The WSH current remains approxi-
mately constant for varying barrier widths. This is due to thefact that at resonance, the injector and the upper laser levelform a pair of binding/antibinding states whose nature hardlychanges with b. Thus, the hopping current is not affected by
the barrier width; see also Ref. 27. The slight increase in the
current may be attributed to the impact of the other levelswhich change slightly with decreasing bias, which is neededto keep the two tunnel-coupled states in resonance.
In contrast, the full calculation /H20851Eq. /H2084915/H20850/H20852based on the
coherences shows the expected drop of the current with bar-rier thickness. We first focus on the case T=10 K. For small
barrier widths /H20849where 2 /H9024/H11407/H9003 /H20850, the two stationary current
results are approximately the same and show a qualitativelysimilar behavior, i.e., the slope of the two curves are almostparallel. In this regime, the WSH and the full model bothdescribe the stationary current well. The slightly smallervalue for the full model stems from the fact that only themain injection barrier is investigated systematically here.Resonant tunneling at the other barriers in the structure is notconsidered, so that a similar argumentation as applied in thefollowing can also be applied to them. Thus, it is expectedthat a constant offset can occur, probably due to the extrac-tion barrier where the two delocalized states /H20849lower laser
level, extraction level /H20850are also close to resonance.
For large barrier widths /H20849where 2 /H9024/H11351/H9003 /H20850, the full current
decreases until it almost vanishes in the limit of large barrierwidths. This decrease is the physically expected behavior, as
the growing injection barrier width restricts the current flow-ing across the barrier and thus through the whole structure.For larger barrier widths, the growing localization of thecharge in the injection region leads to the fact that theWannier-Stark states which are delocalized across the injec-tion barrier no longer constitute a “good” basis to describethe states of the system. For this reason, the coherence be-tween these two states becomes important in the stationarystate, whereas it is negligible for smaller barrier widths. Thiscan be seen in the relaxation dynamics as well, where forlarge barrier widths, the influence of the coherences on thepopulations, mainly of the injector and the upper laser state,becomes very important, and thus a pure Boltzmann relax-ation description fails for the stationary populations.
For the case of T=50 K, the situation is different. For
small barrier widths, the drop of the full current is morepronounced than for T=10 K, showing that even for those
barrier widths the WSH model for the current fails. Weshould note that this difference is not as pronounced aswould be expected; however, since we are interested inqualitative results, it suffices to find that the onset of devia-tion between the WSH and the full calculation is shifted tolower barrier widths. It should be noted that for all barrierwidths here, the relation 2 /H9024/H11270/H9003 is fulfilled.
It is thus found that in the stationary current calculations,
the inclusion of coherence becomes important if 2 /H9024/H11351/H9003 un-
der resonance. For 2 /H9024/H11407/H9003 , the WSH approach to the current
calculation is a good approximation, and thus the inclusionof coherence is negligible. For 2 /H9024/H11351/H9003 , only the full model
shows the physically expected decrease in the current as thetunnel coupling decreases since the coherence between theinjector and the upper laser state becomes important to de-scribe the localization of the eigenstates. The decrease in thetunnel coupling across the main tunneling barrier limits thecurrent flowing through the structure. This limitation is notreproduced by the WSH approach. It should be noted thatthis result has already been discussed before in Ref. 27for
the QCL and in Ref. 57for superlattices.
C. Ultrafast optical dynamics
We now turn to the nonlinear optical response due to an
ultrafast external perturbation of the laser structure from sta-tionarity. Here, the situation is not as clear as in the transportcase. Obviously, the light-matter interaction as such is a co-herent process as the vector potential couples to the nondi-agonal elements of the density matrix f
ij,k/H6023. However, it is
more interesting to consider the dynamics after the passage
of the pulse, where the signal is typically dominated by in-coherent scattering.
As discussed in Sec. III B, coherences between Wannier-
Stark states are important in the regime of large barrierwidths and high temperatures, where the stationary states arecoherent superpositions of the WS states; they describe thetunneling through the barriers and thus the return to the sta-tionary state. This is expected to remain important in theoptical dynamics in order to describe the return to the steadystate after optical excitation. However, the comparatively00.20.40.6
5 6 7 8 9 10 11 12 13current density (kA/cm2)
barrier width (nm)T=5 0K
WSH
full00.20.4current density (kA/cm2)
T=1 0K
WSH
full
FIG. 6. /H20849Color online /H20850Stationary current for both the WSH and
the full calculation at resonance for varying barrier widths at T
=10 and 50 K /H20849see remark in Ref. 55/H20850.DENSITY-MATRIX THEORY OF THE OPTICAL DYNAMICS … PHYSICAL REVIEW B 79, 165322 /H208492009 /H20850
165322-7large scattering with respect to the tunneling /H20849see Fig. 5/H20850is
expected to destroy all observable coherent effects in thisregime. The question arises whether coherent optical effectsafter ultrafast optical excitation can be observed in the oppo-site regime, where coherences were shown not to be impor-tant in the stationary transport. Recently, optical pump-probemeasurements in mid-infrared
29–32and THz QCLs /H20849Ref. 33/H20850
have shown an oscillatory behavior attributed to /H20849coherent /H20850
resonant tunneling through the injection barrier. In this sec-tion, this effect is investigated with respect to the relationbetween the tunnel coupling and the level broadening.
We begin by considering the linear response of the system
to characterize the laser. Then, we analyze the nonlinear op-tical response in form of the experimentally studied pump-probe signals to study the regime of importance and the rolecoherence plays in the optical signal. We complement thisstudy by looking at the optically induced charge as well asthe subband population dynamics.
The linear absorption of the structure is calculated via the
complex susceptibility
/H9273/H20849/H9275/H20850=1
/H92550/H9254J/H20849/H9275/H20850
/H92752A/H20849/H9275/H20850. /H2084916/H20850
/H9254J/H20849t/H20850=J/H20849t/H20850−J0denotes the change in the current density in-
duced by the optical field with vector potential A/H20849/H9275/H20850, where
J0is the stationary value of the current density determined in
the previous section and J/H20849t/H20850is given by Eq. /H2084915/H20850. From this,
the absorption /H9251/H20849/H9275/H20850/H11011/H9275Im/H9273/H20849/H9275/H20850is calculated.
In Fig. 7, the absorption spectrum of the gain transition is
shown for different injection barrier widths. Due to the anti-crossing at resonance, the spectrum shows a double-peakstructure which can be resolved for small barrier widthswhere the tunnel splitting is sufficiently large. For larger bar-rier widths, the two peaks merge to form a single resonancefor sufficiently small tunnel splitting energies compared tothe dephasing of the transition. Due to this splitting, it is a
priori unclear which of the transitions is to be considered the
optical gain transition. In the following considerations, thesystem is excited resonantly on the transition yielding thelarger peak gain which here is the lower gain transition fre-quency, and the accompanying optical dynamics is investi-gated. The second peak appearing for b=9.0 nm at /H9004E
/H1101516 meV arises from an enhanced dipole moment between
the two resonant states 1
/H11032,5 and extractor state 3.When increasing the barrier width further, the coherence
between the injector and the upper laser state becomes im-portant so that the stationary state of the system is in a linearsuperposition of the two WS states, localized in the injectorregion. This coherence can lead to a strong decrease in oreven a vanishing of the gain. To illustrate this, it is helpful toconsider a simple three-level system consisting of the tworesonant states /H20849injector 5 and upper laser state 1
/H11032/H20850and lower
laser state 4. We assume that the coherences between thelower laser state and other two states is small compared tothe stationary tunneling coherence at the time of the probepulse t
0, which is taken as a /H9254pulse, i.e., /H20841f41/H11032/H20849t0/H20850/H20841,/H20841f45/H20849t0/H20850/H20841
/H11270/H20841f1/H110325/H20849t0/H20850/H20841. Solving the semiconductor Bloch equations, the
gain at the laser frequency is then given by /H20849under the as-
sumption /H20841E5−E1/H11032/H20841/H11270/H20841E5−E4/H20841/H20850
/H9251/H11011−/H20853d452/H20849n5−n4/H20850+d41/H110322/H20849n1/H11032−n4/H20850+2d45d41/H11032Re/H20851f1/H110325/H20849t0/H20850/H20852/H20854,
/H2084917/H20850
with the constant level densities ni. If the system is now in
the state /H20841/H9023/H20856/H11015/H208415/H20856+/H208411/H11032/H20856/H20849d45/H11015−d41/H11032/H20850which is localized in
the injection region, as is the case for large barrier widths,
we have n5/H11015n1/H11032. The gain is then given by the stationary
inversion of the states n5−n4as well as a further term
−2 Re /H20851f1/H110325/H20849t0/H20850/H20852containing the coherence between the two
resonant states,
/H9251/H11011−d452/H208532/H20849n5−n4/H20850−2R e /H20851f1/H110325/H20849t0/H20850/H20852/H20854. /H2084918/H20850
Typically, the first term containing the inversion determines
the optical spectrum. In the presence of a strong stationarycoherence f
1/H110325/H20849t0/H20850, the gain can be strongly reduced or even
vanish, even in the presence of a strong inversion, which is
the case for all barrier widths considered. The stationary stateas a superposition of the two WS states is fully localized inthe injector, leading to a very small dipole overlap with thelower laser state which is witnessed in the vanishing gain.
For the nonlinear optical dynamics, we focus on the
pump-probe signals which have been experimentally mea-sured recently.
29–35An ultrafast nonlinear pump pulse reso-
nant on the laser transition excites the sample, leading to again saturation at the laser energy, and a subsequent weakprobe pulse tests the laser transition as a function of thedelay time between the two pulses /H20849see Fig. 8/H20850. Correspond-
ing to the experiments, a 170 fs 1
/H9266pump pulse /H20849P/H20850and a /H20849in
comparison to all optical transitions of interest /H20850spectrally
broad probe pulse /H20849Pr, with T=50 fs /H20850are used. In contrast to
a standard differential transmission spectrum where the com-plete energy range is recorded,
47the focus is on a monochro-
matic measurement, yielding the absorption at a fixed energyfor different delay times. The experimental separation ofpump and probe pulses which is done via directional filteringof the response is carried out here in such a way that theresponse to only the pump pulse
/H9254JPis subtracted from the
response to both the pump and probe pulses /H9254PP+Pr, yielding
the purely linear response after the influence of the pumppulse whose direct response is filtered out. Thus, we considerfor the absorption0
8 10 12 14 16 18absorption (arb. units)
ener gy (meV)increasing
barrier width4.5nm
5.5 nm
6.5 nm
7.0 nm
9.0 nm
11.0 nm
13.0 nm
FIG. 7. /H20849Color online /H20850Dependence of the absorption spectrum
on the injection barrier width at T=10 K.WEBER, WACKER, AND KNORR PHYSICAL REVIEW B 79, 165322 /H208492009 /H20850
165322-8/H9251/H20849td,/H9275¯/H20850/H11011Im/H20875/H9254JP+Pr/H20849td,/H9275¯/H20850−/H9254JP/H20849/H9275¯/H20850
/H9275¯APr/H20849td,/H9275¯/H20850/H20876, /H2084919/H20850
with the delay time td, the fixed pump and probe energy /H9275¯
/H20849which is equal to the laser energy here /H20850, and the probe field
APr./H9254Ji/H20849/H9275¯/H20850is the Fourier transform of the corresponding
simulated time signal /H9254Ji/H20849t/H20850taken at the fixed frequency /H9275¯.
In Ref. 58, it is shown that the nonlinear absorption accord-
ing to Eq. /H2084919/H20850describes the signal contribution measured in
the probe direction. While the signal contains all coherenteffects also in the limit of delay times shorter than or com-parable with the dephasing times, additional coherent effectsare included for short delay times which are not measuredexperimentally due to directional filtering; these effects areexpected to be small. Our result reproduces well-knownstudies in the probe direction, for instance, Ref. 59. The dif-
ferential pump-probe response is then given by exp /H20853−/H20851
/H9251/H20849td/H20850
−/H92510/H20849td/H20850/H20852L/H20854−1/H11011−/H9251/H20849td/H20850+/H92510/H20849td/H20850/H11013PP/H20849td/H20850, where /H92510/H20849td/H20850de-
notes the linear response without the pump pulse and it isassumed that
/H9251L/H112701 where Lis the length of the structure.
For reasons of presentation, we plot the negative differentialpump-probe response. Then, positive signals correspond to adecreased gain compared to the stationary reference gainvalue.
The pump-probe response for different barrier widths and
atT=10 and 50 K is shown in Fig. 8/H20849see also Fig. 11for the
corresponding population dynamics caused by the pumppulse for b=4.5 and 7.0 nm at T=10 K /H20850. The ultrafast pump
pulse saturates the gain transition, leading to a strong absorp-tion at this energy /H20849positive pump-probe signal /H20850. The follow-
ing return to the steady state shows two main features: /H20849i/H20850a
decay of the signal on a time scale of a few picoseconds.This feature is due to the incoherent scattering which leads toa return to the steady state, yielding the original gain for thelaser transition. /H20849ii/H20850For smaller barrier widths, an oscillatorymodulation of the signal whose amplitude decreases for
growing barrier widths, while its period of the order of pico-seconds increases correspondingly. In addition, fast oscilla-tions with a period on the order of 0.1 ps are visible whichare addressed briefly below but which are not the focus ofthe paper.
The oscillatory feature is a coherent effect in the dynam-
ics after the passage of the pulse: The pump pulse depopu-lates the upper laser state locally in the excited region. Atresonance condition, such a locality is represented by a co-herent superposition of levels 1
/H11032and 5. This causes coherent
charge oscillations between the two superpositions which arelocalized in the injector and in the active region,
60leading to
a modulation of the coherence between the two laser statesand thus to an oscillation of the gain signal. The oscillationhardly causes any changes in the WS subband populationsand no oscillating inversion as will be discussed later in thissection. Thus, the oscillation is a coherent effect. The charge
oscillations are visible for the case that the tunneling periodT
osc/H110111//H9004Eis less than the scattering period Tscatt/H110111//H9003,
i.e., Tosc/H11351Tscatt. Let us first focus on T=10 K. Here, gain
oscillations are strongly visible for b=4.5 and 5.5 nm where
2/H9024/H11407/H9003 , i.e., Tosc/H11351Tscatt. Already for b=6.5 nm, the oscilla-
tions become hardly visible and fully vanish for d
/H113507.0 nm, where Tosc/H11271Tscatt. For very large barrier widths,
it is not possible to invert the gain transition due to the strongcoherence between the injector and the upper laser state. Thetwo states are now in a superposition state localized in theinjector, and thus a dipole interaction via the electric fieldbetween the lower laser state and the upper state is not pos-sible since the charge is localized outside the active region.Thus, no charge inversion occurs /H20849compare the discussion of
the gain for large barrier widths in Fig. 7/H20850.A t T=50 K,
2/H9024/H11351/H9003 , i.e., T
osc/H11407Tscattis fulfilled for all barrier widths /H20849cf.
Fig.5/H20850, and thus only very weak gain oscillations are found
forb=4.5 nm. Again, for large barrier widths, no inversion
occurs, leading to an almost vanishing pump-probe signal. Itshould be noted that the fast oscillations on a time scale ofaround 300 fs seen in the insets of the pump-probe signalsfor small delay times t
d/H113511 ps can be attributed to tunneling
due to the coherence between the lower laser state and thetwo resonant states which show an energy splitting of /H9004E
/H1101513 meV, corresponding to an oscillation period of T
osc
/H11015300 fs. Due to the changing applied bias and the corre-
spondingly changing laser energy, the oscillation periodchanges slightly between T
osc/H11015200–300 fs.
Experimentally, gain oscillations have been observed in a
mid-infrared laser in Refs. 29–32and in a THz laser in Ref.
33. In the case of the mid-infrared laser, pronounced gain
oscillations were found up to relatively high temperatures,including a gain overshoot when probing close to the reso-nance. This effect depends strongly on the strength of thescattering mechanisms, specifically on the depletion of thelower laser subband and the strong coupling through the in-jection barrier, and has not been observed in our calculations.According to Ref. 31, the lifetime of the superposition of the
two resonant states T
scatt/H110151 ps which is much longer than
the observed oscillation of Tosc/H11015500 fs, and thus 2 /H9024/H11022/H9003
which is the regime where gain oscillations are expected. InRef. 35, time-resolved pump-probe differential transmission
FIG. 8. /H20849Color online /H20850Pump-probe signals after ultrafast strong
pumping and subsequent linear probing at the laser energy for dif-ferent delay times t
dand barrier widths at T=10 and 50 K. Both
pump and probe pulses are chosen to be resonant on the laser tran-sition. Insets: Pump-probe signals for small delay times.DENSITY-MATRIX THEORY OF THE OPTICAL DYNAMICS … PHYSICAL REVIEW B 79, 165322 /H208492009 /H20850
165322-9measurements did not show signs of gain oscillations. As the
authors argue in the paper, this is due to the very short scat-tering rates due to Coulomb scattering /H20849T
scatt/H11021100 fs /H20850which
is much smaller than the tunneling oscillation period /H20849Tosc
=517 fs /H20850, and thus no gain oscillations are expected.
The interpretation of the oscillations in the pump-probe
signals being caused by coherent charge transfer across theinjection barrier is strengthened by considering the spatioen-ergetic structure of the stationary state obtained from ourGreen’s function model.
20In Figs. 9/H20849a/H20850and9/H20849c/H20850, the spectral
function A/H20849E,k=0,z/H20850which describes the density of states
for vanishing in-plane momentum is shown. The peaksroughly correspond to the WS wave functions /H20841
/H9264i/H20849z/H20850/H208412with an
energetic width /H9003i. For b=4.5 nm, the binding and antibind-
ing combinations of the injector and the upper laser level canbe resolved around E/H1101536 meV, while this is not the case
forb=7.0 nm as /H9004E/H11021/H9003. The right figures show the lesser
Green’s function Im /H20851G
/H11021/H20849E,k=0,z/H20850/H20852describing the corre-
sponding carrier density. For b=4.5 nm, the carrier density
qualitatively follows the spectral function corresponding totwo occupied levels in the stationary state. Thus, when per-turbing the stationarity on an ultrashort time scale, a super-position of the two states leads to an oscillatory modulationof the gain. For b=7.0 nm, the carrier density does not fol-
low the density of states; only the superposition of the twostates, localized in the injector, is occupied at stationarity,and thus no superposition can be excited and no oscillationsoccur. The stationary distribution is already a superpositionof the two WS states, which is also the reason why the in-clusion of coherence is important to correctly describe thestationary transport of the system /H20849see Fig. 6/H20850.In addition to the spatio energetic resolution of the station-
ary charge density, it is insightful to consider its spatio tem-
poral evolution during and especially after the excitation
with an ultrafast laser pulse. To do this, we consider thespatiotemporally resolved electron density which is given by
n/H20849z,t/H20850=21
A/H20858
ij/H9264i/H11569/H20849z/H20850/H9264j/H20849z/H20850/H20858
k/H6023fij,k/H6023/H20849t/H20850, /H2084920/H20850
where the time dependence is included in the density matrix
fij,k/H6023/H20849t/H20850. In order to simplify the representation and to focus on
the optically induced dynamics, the difference between
n/H20849z,t/H20850and the stationary electron density n0/H20849z/H20850,/H9004n/H20849z,t/H20850
=n/H20849z,t/H20850−n0/H20849z/H20850, is considered in the following. Similar calcu-
lations have been presented in Ref. 23, where however no
detailed modeling of the pump pulse was done.
In Fig. 10, the electron density evolution is shown for the
same parameters of the pump pulse as used for the calcula-tion of the pump-probe signal discussed above. Shortly afterthe pulse has passed at t/H110150.3 ps, the density is depleted in
the active region due to the fast nonradiative extraction fromthe lower laser level to the extractor state and into the injec-tor.
After this, two qualitatively different behaviors are seen
for the two barrier widths. For b=4.5 nm, the resonant tun-
neling across the injection barrier leads to a fast refilling ofthe upper laser state and thus to an increased density in theactive region at t/H110151 ps. At t/H110152 ps, the density in the ac-
tive region again decreases. Thus, the oscillation of chargewhich occurs on a time scale of 1.8 ps as detected by theprobe pulse /H20851see Fig. 8/H20849left/H20850/H20852can be directly seen in the
(b) (a)
(c) (d)
FIG. 9. /H20849Color online /H20850/H20849a/H20850,/H20849c/H20850Spectral function and /H20849b/H20850,/H20849d/H20850the imaginary part of the lesser Green’s function for two different barrier
widths at T=10 K.WEBER, WACKER, AND KNORR PHYSICAL REVIEW B 79, 165322 /H208492009 /H20850
165322-10electron density evolution. When considering the electron
density without the polarizations, i.e., restricting to the diag-onal elements of the density matrix for the calculation, oreven just without including the coherences between differentperiods /H20849not shown /H20850, this oscillation is not observed at all.
For the case of b=7.0 nm, a roughly monotonous return
of the density depletion to the steady state is found, and thusno gain oscillations are expected, as verified in the pump-probe signals. Due to the weaker tunnel coupling betweenthe injector and the active region, the charge transfer is slowso that the oscillation period induced by the tunnel couplingT
oscis larger than the scattering period Tscatt, and thus charge
oscillations cannot be observed. Additionally, there are fastoscillations in the active region on a time scale of around 300fs which we already addressed in the discussion of the pump-probe signals. The oscillations in the injector region can be
explained by coherences between the injector states where
E
2−E5/H110155 meV= ˆ850 fs.
It has already been mentioned that the charge oscillations
are not seen in the electron density evolution when only thediagonal elements of the density matrix are used in Eq. /H2084920/H20850.
Thus they constitute a coherent effect. This is better exem-plified by considering directly the subband population evo-lution caused by the strong pump pulse excitation. We thusconsider the dynamics of the WS populations n
i/H20849t/H20850
=2 /A/H20858k/H6023nik/H6023/H20849t/H20850under ultrafast optical excitation. It should be
stressed here that the dynamical calculations are carried out
with the full equation structure including both the diagonaland nondiagonal density-matrix elements. Again, the dynam-ics is considered for the same nonlinear excitation used inthe pump-probe calculations. The results are shown in Fig.11.
During the excitation with the pump pulse, the popula-
tions of the laser levels undergo Rabi flopping, while theother levels follow adiabatically.
26,61–63After the excitation,
the populations return to the steady state due to the /H20849incoher-
ent/H20850scattering processes. While for b=4.5 nm, the Rabi
flopping is restricted mainly to the two resonantly excitedsubbands /H208491
/H11032,4/H20850; the subband flopping for b=7.0 nm in-
cludes both the two laser states /H208494,1 /H11032/H20850and injector level /H208495/H20850
since both transitions from the lower laser state are energeti-cally very close due to the very small tunnel splitting. Thisweakens the oscillator strength of each transition, reducingthe effective pulse area compared to the full population in-version seen for a 1
/H9266pump pulse in an ideal two-level sys-tem. Still, in both cases, the gain transition is saturated, lead-
ing to absorption at this energy. An oscillating inversion ofthe upper und lower laser states corresponding to the oscil-lations in the pump-probe signals /H20849Fig. 8/H20850or in the spa-
tiotemporal electron density evolution on a time scale of afew picoseconds /H20849Fig.10/H20850is found neither for b=4.5 nm nor
forb=7.0 nm. As a matter of fact, if the dynamics are con-
sidered without scattering /H20849not shown /H20850, no further population
dynamics are found after the passage of the pulse, while thecorresponding pump-probe signals show an undamped oscil-lation in the case of b=4.5 nm. This illustrates again that the
oscillations in the gain recovery are a coherent effect.
In summary, it is found that gain oscillations are observed
where the tunnel oscillation period is sufficiently less thanthe scattering period, where the relation 2 /H9024/H11407/H9003 is valid. The
tunnel coupling leads to pronounced coherent charge transferbetween the injector and the active region as witnessed in thepump-probe signals and the spatiotemporally revolved elec-tron density evolution. Considerations of quantities which donot directly relate to the coherence, such as the dynamics ofthe WS populations, show that the oscillations observed inthe pump-probe signals constitute a coherent effect whichrequires a fully coherent theory to be observed. It should benoted here that compared to the current calculations, where astationary localization of charge leads to the drop of the
current signal, it is here the dynamical transfer of charge
between different locations in the laser structure which con-stitutes the coherent effect. Thus, the inclusion of coherencebecomes essential in the nonlinear optical dynamics for theregime where 2 /H9024/H11407/H9003 to describe the observed coherent ef-
fects. Here, the coherence in the system drives the gain os-cillations, while for 2 /H9024/H11351/H9003 , the scattering destroys the co-
herent effects.
IV. CONCLUSION
We have established a microscopic quantum theory of
scattering for quantum cascade lasers to treat stationarytransport as well as nonlinear optical dynamics. A key issueis the importance of coherence, i.e., the nondiagonal ele-ments of the density matrix, and the role which it plays indifferent parameter regimes of the tunnel coupling /H9024be-
tween the two states across the main injection barrier and the
FIG. 10. /H20849Color online /H20850Spatiotemporal evolution of the electron
density of the QCL after ultrafast nonlinear optical excitation at T
=10 K. The pump pulse parameters are the same as for the pump-probe results.00.40.81.2
0 1 2 3 4subband pop. (10-4nm-2)
time (ps)12
5
4
3b = 4.5 nm
00.40.81.2
0 1 2 3 4subband pop. (10-4nm-2)
time (ps)12
5
43b = 7.0 nm
(b) (a)
FIG. 11. /H20849Color online /H20850Optically induced population dynamics
ni/H20849t/H20850for the barrier widths b=4.5 and 7.0 nm at T=10 K, showing
coherent Rabi oscillations as well as subsequent scattering-inducedrelaxation back into the steady state. The pump pulse parameters arethe same as for the pump-probe results /H20849the pump pulse is shown by
the dashed line in the figures /H20850.DENSITY-MATRIX THEORY OF THE OPTICAL DYNAMICS … PHYSICAL REVIEW B 79, 165322 /H208492009 /H20850
165322-11mean level broadening /H9003of the states. In the transport re-
gime, we have compared a rate equation /H20849WSH /H20850approach to
the current with our microscopic theory at resonance condi-tion and found that for 2 /H9024/H11407/H9003 , the WSH approach is a good
approximation. For 2 /H9024/H11351/H9003 , only the full approach shows the
physically expected decrease of the current for decreasingtunneling across the main injection barrier, whereby the in-cluded coherence limits the current flowing through thestructure via a stationary localization of charge in the injec-tion region. On the other hand, the consideration of ultrafastnonlinear optical pump-probe signals has shown that the in-clusion of coherence is important to describe the experimen-tally observed coherent effects for 2 /H9024/H11407/H9003 , where it drives
the dynamical charge transfer between the injector and theactive region, resulting in oscillations in the gain recovery.For 2/H9024/H11351/H9003 , only the typical decrease in the pump-probe
signal due to incoherent scattering is observed. It should benoted again that, in general, coherences are necessary to de-scribe the return to the stationary state after optical excitationdue to the strong interplay between transport and optics inthis system. The spatially resolved electron density evolutionin time shows the density oscillating between the injectorand the active region for 2 /H9024/H11407/H9003 . The consideration of the
population dynamics of the levels does not show these oscil-lations, revealing that they are an inherent coherent effectresulting from the nondiagonal elements of the density ma-trix.
Thus, the inclusion of coherence is important in opposite
parameter regimes in stationary transport and nonlinear op-tical dynamics: in the former, it becomes important for2/H9024/H11351/H9003 , while in the latter, it allows the observability of
coherent effects for 2 /H9024/H11407/H9003 . In this sense, the coherence acts
in complementary ways: in the transport regime, it leads to alimitation of the current which is not reproduced by the rate
equation approach. In the optical regime, the coherencedrives the oscillatory modulation of the gain recovery whichis not destroyed by the scattering. Thus, it is necessary toconsider a fully coherent theory in order to describe the com-bined system of optics and transport in this nonequilibriumstructure in the entire parameter range.
ACKNOWLEDGMENTS
We acknowledge fruitful discussions with Rikard
Nelander, Marten Richter, Stefan Butscher, MichaelWoerner, and Klaus Reimann. This work was supported bythe Deutsche Forschungsgemeinschaft /H20849DFG /H20850and the Swed-
ish research council /H20849VR/H20850.
APPENDIX: SCATTERING EQUATIONS
1. Electron-LO phonon interaction
The general form of the scattering equations in Born-
Markov approximation is given by /H20849fornq=n−q/H20850f˙ij,k/H6023=/H9266
/H6036/H20858
lmn/H20858
qNmnli/H20849q/H20850/H20853/H20849/H9254nj−fnj,k/H6023/H20850flm,k/H6023+q/H6023/H20851/H20849nq+1/H20850/H9254/H20849/H9255nk/H6023
−/H9255mk/H6023+q/H6023+/H9255LO/H20850+nq/H9254/H20849/H9255nk/H6023−/H9255mk/H6023+q/H6023−/H9255LO/H20850/H20852−/H20849/H9254lm
−flm,k/H6023+q/H6023/H20850fnj,k/H6023/H20851/H20849nq+1/H20850/H9254/H20849/H9255nk/H6023−/H9255mk/H6023+q/H6023−/H9255LO/H20850+nq/H9254/H20849/H9255nk/H6023
−/H9255mk/H6023+q/H6023+/H9255LO/H20850/H20852/H20854+/H9266
/H6036/H20858
lmn/H20858
qNmnlj/H11569/H20849q/H20850/H11003/H20853/H20849/H9254in
−fin,k/H6023/H20850fml,k/H6023+q/H6023/H20851/H20849nq+1/H20850/H9254/H20849/H9255nk/H6023−/H9255mk/H6023+q/H6023+/H9255LO/H20850+nq/H9254/H20849/H9255nk/H6023
−/H9255mk/H6023+q/H6023−/H9255LO/H20850/H20852−/H20849/H9254ml−fml,k/H6023+q/H6023/H20850fin,k/H6023/H20851/H20849nq+1/H20850/H9254/H20849/H9255nk/H6023
−/H9255mk/H6023+q/H6023−/H9255LO/H20850+nq/H9254/H20849/H9255nk/H6023−/H9255mk/H6023+q/H6023+/H9255LO/H20850/H20852/H20854, /H20849A1/H20850
where Nmnij/H20849q/H20850=gijqgmnq/H11569. In the calculations, the considerations
are restricted to terms linear in the density matrix since in thesystems considered here, /H20841f
ij,k/H6023/H20841/H112701 so that /H20841fij,k/H6023flm,k/H6023/H20841/H11270/H20841fij,k/H6023/H20841.
The Boltzmann relaxation is determined via
n˙ik/H6023/H208490/H20850=−/H9003ik/H6023outnik/H6023+/H9003ik/H6023in/H208491−nik/H6023/H20850, /H20849A2/H20850
with the semiclassical Boltzmann in- and out-scattering rates
given by
/H9003ik/H6023in=2/H9266
/H6036/H20858
q,l/H20841gilq/H208412nlk/H6023+q/H6023/H20851/H20849nq+1/H20850/H9254/H20849/H9255ik/H6023−/H9255lk/H6023+q/H6023+/H9255LO/H20850
+nq/H9254/H20849/H9255ik/H6023−/H9255lk/H6023+q/H6023−/H9255LO/H20850/H20852, /H20849A3/H20850
/H9003ik/H6023out=2/H9266
/H6036/H20858
q,l/H20841gilq/H208412/H208491−nlk/H6023+q/H6023/H20850/H20851nq/H9254/H20849/H9255ik/H6023−/H9255lk/H6023+q/H6023+/H9255LO/H20850
+/H20849nq+1/H20850/H9254/H20849/H9255ik/H6023−/H9255lk/H6023+q/H6023−/H9255LO/H20850/H20852. /H20849A4/H20850
2. Ionized doping centers—impurity scattering
The general form of the scattering equations is given by
f˙ij,k/H6023=−/H9266
/H6036/H20858
mn/H20858
q/H6023/H20851Mnmmi/H20849q/H6023/H20850fnj,k/H6023/H9254/H20849/H9255nk/H6023−/H9255mk/H6023+q/H6023/H20850
−Mjnmi/H20849q/H6023/H20850fmn,k/H6023+q/H6023/H9254/H20849/H9255jk/H6023−/H9255nk/H6023+q/H6023/H20850+Mnmmj/H20849q/H6023/H20850fin,k/H6023/H9254/H20849/H9255nk/H6023
−/H9255mk/H6023+q/H6023/H20850−Minmj/H20849q/H6023/H20850fnm,k/H6023+q/H6023/H9254/H20849/H9255ik/H6023−/H9255nk/H6023+q/H6023/H20850/H20852, /H20849A5/H20850
where the average Mmnij/H20849q/H6023/H20850=/H20855/H20858l¯,l¯/H11032Vijl¯/H20849q/H6023/H20850Vmnl¯/H11032/H20849−q/H6023/H20850/H20856Scover the
statistically distributed doping atoms is introduced. Since the
semiclassical interaction is linear in the density matrix, allterms are taken along for this interaction.
The semiclassical Boltzmann in- and out-scattering rates
are given by
/H9003
ik/H6023in=2/H9266
/H6036/H20858
q/H6023,lMilil/H9254/H20849/H9255ik/H6023−/H9255lk/H6023+q/H6023/H20850nlk/H6023+q/H6023, /H20849A6/H20850
/H9003ik/H6023out=2/H9266
/H6036/H20858
q/H6023,lMilil/H9254/H20849/H9255ik/H6023−/H9255lk/H6023+q/H6023/H20850/H208491−nlk/H6023+q/H6023/H20850. /H20849A7/H20850WEBER, WACKER, AND KNORR PHYSICAL REVIEW B 79, 165322 /H208492009 /H20850
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165322-14 |
PhysRevB.95.125106.pdf | PHYSICAL REVIEW B 95, 125106 (2017)
Bound-state dynamics in one-dimensional multispecies fermionic systems
P. Azaria
Laboratoire de Physique Th ´eorique de la Mati `ere Condens ´ee, Sorbonne Universit ´es,
UPMC Universit ´e de Paris 06 and CNRS, 4 Place Jussieu, 75005 Paris, France
and Physics Department, Technion, 32000 Haifa, Israel
(Received 17 August 2016; revised manuscript received 10 January 2017; published 6 March 2017)
In this work we provide for a description of the low-energy physics of interacting multispecies fermions in
terms of the bound states that are stabilized in these systems when a spin gap opens. We argue that at energiesmuch smaller than the spin gap, these systems are described by a Luttinger liquid of bound states that depends,on top of the charge stiffness νand the charge velocity u, on a “Fermi” momentum P
Fsatisfying qPF=NkF
where qis the charge of the bound state, Nis the number of species, and kFis the Fermi momentum in the
noninteracting limit. We further argue that for generic interactions, generic bound states are likely to be stabilized.They are associated with emergent, in general nonlocal, symmetries and are in the number of five. The first twoconsist of either a charge q=Nlocal SU( N) singlet or a charge q=Nbound state made of two local SU( p)
and SU( N−p) singlets. In this case the Fermi momentum P
F=kFis preserved. The three others have an
enhanced Fermi vector PF. The latter are either charge q=2 bosonic p-wave and s-wave pairs with SO( N)a n d
SP(N) symmetry and PF=NkF/2 or a composite fermion of charge q=1 with PF=NkF. The instabilities
of these Luttinger liquid states towards incompressible phases and their possible topological nature are alsodiscussed.
DOI: 10.1103/PhysRevB.95.125106
I. INTRODUCTION
As is well known the Luttinger liquid constitutes the
universality class of a large number of gapless quantumsystems in one dimension [ 1,2]. Spinless bosons or fermions
on a lattice, spin models such as the XXZ spin chain [ 3,4],
and edge states in the fractional quantum Hall effect [ 5,6]a r e
all well-known examples of one-dimensional systems whichare described by the Luttinger liquid theory. The Luttingerliquid is also expected to describe the low-energy sector of
more involved models with Nspecies of particles: when a
gap is present in the species, or spin, sector the low-energyphysics is expected to be captured by the total charge, ordensity, fluctuations which are described by the Luttingerliquid Hamiltonian. Examples can be found, among others,in electronic ladders [ 7–10] or cold-atoms systems [ 11,12,15]
with hyperfine spin F=(N−1)/2. All these systems have
in common that their low-energy physics depend on two
Luttinger parameters, a stiffness νand a velocity u. These
parameters, which may be eventually taken as phenomenolog-ical input parameters, completely determine the asymptoticsof the correlation functions of physical observables. Does thismean that the low-energy physics of these systems is the same?As we shall argue in this work, though this is certainly true asfar as particle-hole (or plasmon) excitations are involved, the
nature of the elementary excitations in these Luttinger liquids
is different.
Indeed when a gap opens in the spin sector, single-particle
correlation functions fall off exponentially and only certainsinglet combinations remain massless. These combinations arebound states of the elementary fermions and are, at energies E
much smaller than the spin gap /Delta1, the relevant excitations of
these systems. A celebrated example is that of the stabilizationof bosonic BCS pairs by the opening of a spin gap in the S=
1/2 attractive Hubbard model [ 3]. Many other examples with
more than two components were also reported in the literature.In cold-atom problems, s-wave pairs made of hyperfine spinF> 1/2 singlets, as well as trionic or quartet bound states
made of SU( N)(N=3,4) singlets, were also shown to exist
[12–15,17–21].
The purpose of this work will be to present a description
of the dynamics of these bound states within the frameworkof the Luttinger liquid theory and to provide for a commonview of the low-energy physics of multispecies fermionicsystems when a spin gap is present. As we shall see, theLuttinger liquid theory offers a natural framework to describebound states. Indeed, Luttinger liquids may be distinguished bytheir nonzero charge Qand current Jspectrum or zero-mode
spectrum. When a bound state of charge qis stabilized by
the spin gap, the total charge Qis to be quantified in units of
the elementary bound-state charge q∈NwithQ=nq.T h e
fundamental excitation of charge q, which is either a boson
when qis even or a fermion when qis odd, is the minimal
charge that one can add (remove) to (from) the system andplay an analogous role as the electron in a one-species system.Similar considerations yield the quantization of the currentJ=mjfor fermions and J=2mjfor bosons where j∈N
and 2j∈Nare the minimal nonzero currents both systems
can support. Therefore, in order to completely characterizethe bound-state Luttinger liquid state, one needs, on top of theLuttinger parameters uandν, to specify the elementary charge
and current quantum numbers ( q,j). A bound-state Luttinger
liquid can then be viewed as an additional selection rule onthe zero-mode spectrum ( Q,J). The latter selection rules keep
track of the underlying possible orders in the high-energy spinsector.
As a first result, we shall see in Sec. IIthat, independently of
the nature of the high-energy physics involved, the bound-statequantum numbers ( q,j) are not arbitrary. For instance, with
the additional assumption that the bound states are local in
terms of the elementary fermions, we find that they have tobe dual in the sense qj=N. Hence, a bound-state Luttinger
liquid is characterized, on top of the Luttinger parameters ν
2469-9950/2017/95(12)/125106(19) 125106-1 ©2017 American Physical SocietyP. AZARIA PHYSICAL REVIEW B 95, 125106 (2017)
andu, by the charge qand the current jquanta solutions of
the latter constraint. Owing to the relation between current andmomentum one can associate a momentum scale to the boundstates, P
F=jkF, where kFis the Fermi momentum of the
elementary fermions, and consequently rewrite the constraintasqP
F=NkF. Of course, at some point, the specific nature of
the ordering in the spin sector should come into play and selectspecific values of qandj. We shall see in Sec. IIIthat under
the assumption of dynamical symmetry enlargement in thespin sector, some generic bound states, i.e., particular valuesof (q,j), are likely to be stabilized for generic Hamiltonians.
The latter are in the number of five and are associated withemergent duality symmetries. The first two types of boundstates have ( q,j)=(N,1) and are either a SU( N) singlet or
a bound state made of two SU( p) and SU( N−p) singlets
with 1 /lessorequalslantp<N . The three other types are either SP( N)
singlet s-wave and SO( N) singlet p-wave bosonic states with
(q,j)=(2,N/2) or SO( N) singlet composite fermions with
(q,j)=(1,N). In Sec. IVwe shall give explicit forms of the
associated wave functions and give their expressions in termsof the elementary fermionic species. We shall show, using alow-energy approach, that after averaging over the gapped spindegrees of freedom, they have a finite overlap with the single-particle creation operator of the Luttinger liquid. After havingcharacterized these generic bound states, we shall finallyinvestigate the instabilities of the corresponding Luttinger
liquid states toward possible incompressible phases in Sec. V.
As one of the consequences of the bound-state dynamics weshall find that, since it is P
F=jkFand not kFthat controls the
commensurability effects with the lattice, when j> 1 possible
nondegenerate Mott phases with topological order might bestabilized for systems. We finally conclude in Sec. VI, where
we discuss open problems and further directions of works.
In the following we shall consider systems with Nspecies
of fermions on a one-dimensional lattice of length Lwith a
generic Hamiltonian
H=−t/summationdisplay
j,a[c†
a,jca,j+1+c†
a,j+1ca,j]+Vint(c†
a,i,cb,j),(1.1)
where the operators c†
a,jcreate a fermion of species a=
(1,..., N ) at lattice site jand are subject to periodic boundary
conditions: c†
a,j+L=c†
a,j. We assume that the potential Vintis
short range, translationally and parity invariant, and preservesthe total number of particles. We shall further assume that agap/Delta1opens everywhere in the spin sector and that the system
remains massless. The interaction pattern between the species
is supposed to be in such a way that none of them decouple;
if the system decouples into two or more subsets, then weconsider applying the analysis to each one of these separately.For simplicity a balanced incommensurate density per species
¯ρ
a=¯ρ=N/Lis also assumed so that there is only one Fermi
momentum kF=¯ρπ.
II. BOUND-STATE LUTTINGER LIQUIDS
Our approach is a low-energy one in which the electron
operators ca,j,a=(1,..., N ), decompose into right and left
components as
ca,j/√a0∼e−ikFxψa,L(x)+eikFxψa,R(x), (2.1)where x=ja0(a0being the lattice spacing) and kF=π¯ρis
the Fermi momentum associated with each species. The aboveright and left fermions can be in turn expressed in terms of twodual bosonic fields [ 7]θ
aandφasatisfying [ φa(x),θb(y)]=
iδabY(y−x), where Y(u) is the step function ( Y(0)=1/2).
We have
ψa,L(R)=κa√
2πe−i√π[θa±φa], (2.2)
where the κa=1,...,Nare anticommuting Klein factors, {κa,κb}=
2δab, that ensure the anticommutation between fermions of
different species. For each species, the bosonic fields θaand
φaare related to the current densities, ja(x)=∂xθa/√π,
and uniform particle densities (relative to the ground state)ρ
a(x)=∂xφa/√π. The zero modes of the charge and current
densities, Qa=/integraltext
dx∂ xφa/√πandJa=/integraltext
dx∂ xθa/√π,a s -
sociated with each species
Qa=/integraldisplay
dx[ψ†
a,Lψa,L+ψ†
a,Rψa,R],
Ja=/integraldisplay
dx[ψ†
a,Lψa,L−ψ†
a,Rψa,R], (2.3)
are topological quantities which, as befits from charge quan-
tization, are integers. In a system with periodic boundaryconditions they are subjected to the additional constraint [ 3,4]
(Q
a±Ja)∈2Zeven,a=(1,..., N ). (2.4)
When a gap /Delta1is present in spin space, we expect the low-
energy physics to be captured by a Luttinger liquid describingthe fluctuations of the total charge and current densities of the
system described by the bosonic fields
/Phi1
c=1√
NN/summationdisplay
a=1φa,/Theta1 c=1√
NN/summationdisplay
a=1θa, (2.5)
where [ /Phi1c(x),/Theta1c(y)]=iY(y−x). Integrating out the spin
degrees of freedom, the effective Hamiltonian at scales E/lessmuch/Delta1
is therefore expected to be [ 1]
H=u
2/integraldisplay
dx/bracketleftbigg1
K(∂x/Phi1c)2+K(∂x/Theta1c)2/bracketrightbigg
, (2.6)
where uis a velocity and Kis the Luttinger parameter that
measures the interaction between the elementary fermions.
Seemingly, the spin degrees of freedom only affect the
parameters uandKwhich anyhow depend in a nonuniversal
way on the details of the microscopic Hamiltonian and can betaken as phenomenological input parameters. The underlyingspin order, though, has a nontrivial effect on the topologicalexcitations associated with the zero-mode part of the bosonicfields/Phi1
cand/Theta1c,
Q=N/summationdisplay
a=1Qa,J=N/summationdisplay
a=1Ja, (2.7)
orQ=/radicalBig
N
π/integraltext
dx∂ x/Phi1c,J=/radicalBig
N
π/integraltext
dx∂ x/Theta1c. Indeed, as dis-
cussed above, when a spectral gap /Delta1opens in the spin
sector, excitations involving arbitrary nonzero values of thecharge and current operators ( 2.3),Q
aandJa,a r ea l s o
gapped in general. Only certain singlet combinations of the
125106-2BOUND-STATE DYNAMICS IN ONE-DIMENSIONAL . . . PHYSICAL REVIEW B 95, 125106 (2017)
elementary fermions survive at low energies and remain
massless. This restricts, on top of the constraints ( 2.4), the
allowed eigenvalues of both QandJzero-mode operators
(2.7). Taking into account the constraint ( 2.4), we find it
suitable to parametrize the bound states with the help of twointegers ( q,j)a s
qeven Q=nq,J =2mj, (2.8)
qodd Q=nq,J =mj, (2.9)
(nq±mj)e v e n,
where ( n,m) are relative integers. In the latter equation ( 2.8)
and ( 2.9) are bosonic and fermionic solutions, respectively.
The two quantities ( q,j) are in fact not independent. Let us
consider indeed the vertex operator that creates a state withcharge Qand current J,
V
Q,J≡exp/bracketleftbigg
i/radicalbiggπ
N[Q/Theta1c+J/Phi1c]/bracketrightbigg
, (2.10)
and look at the (imaginary time) correlation function,
/angbracketleftVQ1J1(x1,τ1)VQ2J2(x2,τ2)/angbracketright=|z1−z2|/Delta112e−i/Theta112/Gamma112,(2.11)
where z=τ+ix/u and/Theta112=Arg(z1−z2). We find
forqodd/Gamma112=qj(n1m2+m1n2)/2N,/Delta112=(n1n2q2+
m1m2K2j2)/2KN while for qeven /Gamma112=qj(n1m2+
m1n2)/N,/Delta112=(n1n2q2+4m1m2K2j2)/2KN. Analyticity
of the correlation function in the complex plane [ 22] requires
/Gamma112to be an integer which, using the constraints ( 2.9) and ( 2.8),
implies that qj=lNwhere lis an arbitrary integer. As we shall
see below, only the case with l=1 corresponds to local bound
states when expressed in terms of the elementary fermions. Webelieve that these are the states that can be stabilized with aHamiltonian of the kind ( 1.1) and from now on we shall focus
on the sets of the bound-state solutions ( q,j)o f
qj=N. (2.12)
We shall comment briefly later on the l/negationslash=1 states. To get some
physical insight on the meaning of ( 2.12) we notice that in a
Luttinger liquid, the quantum of current jdefines a momentum
scaleP
F=jkFwhich, due to ( 2.12), must satisfy
qPF=NkF. (2.13)
For fermions we may interpret ( 2.13) as an extended Luttinger
theorem [ 23,24] in one dimension when a spin gap is present.
For bosonic bound states 2 PF(for practical purposes we use
the same symbol for fermion and bosons) governs the periodof the oscillations of the charge density wave and is related tothe bound-state density by P
F=πρBS. The constraint ( 2.13)
then yields for the bound-state density
ρBS=N¯ρ/q, (2.14)
which is the one that we would calculate in a limit where the
bosonic bound states are free hard-core bosons.
The constraint ( 2.12) [or equivalently ( 2.13)] is not trivial.
For instance, the solutions with j> 1 are reminiscent of some
degree of confinement of the current which is the signalthat the nonzero momenta components of the bound statesmight involve composites of particle-hole of the elementaryfermions. For the moment let us comment qualitatively onthe solutions of ( 2.12). Given the number Nof species, the
possible bound-state solutions ( q,j) are strongly constrained
byj=N/q∈N. First we find that only bosonic bound states
exist for even Nwhile for odd Nthey are fermions [ 51].
For example, we find that in the simplest case of N=2t h e
only bound-state solution is given by ( q=2,j=1) which
corresponds to spin F=1/2 BCS pairs. In the case of N=3
we get two solutions: ( q=3,j=1) and ( q=1,j=3). The
first solution corresponds to a charge q=3 fermionic trionic
bound state [ 18,19] with a preserved Fermi momentum at
P
F=kF. The second bound-state solution with charge q=1
displays an enhanced Fermi surface, PF=3kF. As we shall
see in the next section, this bound state is a composite fermionmade of two particles and one hole. For N=4t h e r ea r et w o
solutions with ( q=4,j=1) and ( q=2,j=2). These are
bosonic quartet bound states and spin F=3/2 BCS pairs
[17]. More solutions can be found for higher values of Nwith
or without an enhancement of the Fermi momentum.
A. Universal description of the bound-state Luttinger liquids
Though in general the different bound-state solutions
describe different physics, they can be described by the same
effective bosonic theory provided one uses suitable rescaledfields. Introducing new bosonic fields ¯φand¯θwith the help of
the canonical transformation
¯φ=√
N
q/Phi1c,¯θ=√
N
j/Theta1c, (2.15)
the Luttinger liquid Hamiltonian ( 2.6) can be brought into the
universal form
H=u
2/integraldisplay
dx/bracketleftbigg1
ν(∂x¯φ)2+ν(∂x¯θ)2/bracketrightbigg
, (2.16)
where
ν=NK/q2. (2.17)
The latter relation shows that if Kis a measure of the interac-
tion between the species, νmeasures the interaction between
the bound states. In particular νis the Luttinger parameter that
controls the power-law behaviors of the different correlationfunctions of the system.
The charge and current operators ( 2.7),QandJ, express in
terms of the “dimensionless” (i.e., independent of both qand
j) charge and current
Q=q¯Q, J =j¯J, (2.18)
where ¯Q=/integraltext
∂
x¯φ(x)/√πand ¯J=/integraltext
∂x¯θ(x)/√π. With the
use of ( 2.15) and depending on the parity of qthe conditions
(2.8,2.9)a r en o w
qeven ¯Q=n,¯J=2m, (2.19)
qodd ¯Q=n,¯J=m, (2.20)
(n±m)e v e n.
The latter constraints are the ones defining both bosonic
and fermionic Luttinger liquids [ 4] and the Hamiltonian
(2.16) describes the low-energy physics of spinless bosons or
fermions with periodic boundary conditions. Once a bound-state solution ( 2.12) is given in terms of ( q,j) the low-energy
125106-3P. AZARIA PHYSICAL REVIEW B 95, 125106 (2017)
dynamics of the bound-state Luttinger liquid is that of a charge
qboson with density ρBS(2.14) or of a charge qspinless
fermion with Fermi momentum
PF=πρBS. (2.21)
The relations ( 2.15) and ( 2.18) allow us to translate all known
results for the spinless fermionic and bosonic Luttinger liquid[3,4]. In particular, the relevant physical operators can be
expressed in the basis of the vertex operators
V
¯Q,¯J≡expi√π[¯Q¯θ+¯J¯φ], (2.22)
which carry physical charge q¯Q, current j¯J, and momentum
¯JPF. For instance both bosonic and fermionic bound-state
single-particle creation operators have charge ¯Q=1 and are
given by [ 3,25]
/Psi1†
B/F/similarequal/summationdisplay
¯Jeven/oddα¯Jei[¯JPFx+¯J√π¯φ+√π¯θ]. (2.23)
Similarly the density operator (relative to the ground state) is
given by
ρ(x)/similarequalq√π∂x¯φ+/summationdisplay
¯Jevenβ¯Jei¯J[PFx+√π¯φ]. (2.24)
In both the latter expressions, the constants α¯Jandβ¯Jare
nonuniversal and depend on the details of the microscopicHamiltonian. To leading order the boson operator is given byitsP=0 component
/Psi1
†
B/similarequalexp [i√π¯θ], (2.25)
while the fermionic bound-state creation operator has leading
components at P=±PF
/Psi1†
F/similarequal/Psi1†
LeiPFx+/Psi1†
Re−iPFx, (2.26)
where
/Psi1†
F,L(R)/similarequalexp[i√π(¯θ±¯φ)]. (2.27)
In both the bosonic and fermionic cases the phase diagram
of the Luttinger liquid is well known [ 3] and depends on
ν. From the long-distance behavior of the equal-time cor-
relation functions /angbracketleft/Psi1†
B(x)/Psi1B(0)/angbracketright∼x−1/2ν,/angbracketleft/Psi1†
F(x)/Psi1F(0)/angbracketright∼
x−(ν+ν−1)/2cos (PFx), and/angbracketleftρ(x)ρ(0)/angbracketright∼x−2νcos (2PFx), one
deduces that the dominant instability is given by the bound-state/bound-state correlation function for ν>1/2 and ν>
1/√
3 in both bosonic and fermionic cases. For stronger
repulsions and smaller values of νthe dominant instability
becomes eventually of the charge density wave type at thewave vector 2 P
F. We notice though that even in this regime of
couplings, as far as the energy scale is much smaller thanthe spin gap, the bound-state description is still sensible.In particular, the bound states manifest themselves in thenontrivial wave vector 2 P
F. At this point let us stress that due to
(2.17), a repulsive interaction between the bound states ( ν<1)
might result from either attractive or repulsive interactionsbetween the elementary fermionic species ( K> 1o rK< 1).
In particular both ( 2.25) and ( 2.27) are the operators for a free
hard-core boson or a fermion in the limit ν=1 which does
not correspond in general to K=1 (except when q=j).
Let us close this section by commenting on the solutions
withl>1. The constraint on the Fermi vector is then givenbyqP
F=lNkFinstead of ( 2.13). In this case, as we shall
argue, the bound states are nonlocal objects when expressed interms of the elementary fermions. To see this let us consideras an example fermionic bound-state solutions ( lNodd). Then
one may bring the Luttinger Hamiltonian to its universal form(2.16) with, instead of ( 2.15), the canonical transformation
¯φ=√
N/Phi1c/q,¯θ=l√
N/Theta1c/j.As a consequence the values
of the current ¯Jarestillconstrained to be ¯J=lm,m∈Z.I n
particular the dimensionless vertex operator ( 2.22) that creates
a state with charge qand current ±jis given by
V1,±l=expi√π[¯θ∓l¯φ]. (2.28)
This state is a composite fermion and is nonlocal when
expressed in terms of the original fermions. In the simplestcase of a single species N=1 with charge q=1 and current
quantum number j=lwe have V
1,−l(x)∼ψ†
R(ψ†
RψL)(l−1)/2
andV1,l(x)∼ψ†
L(ψ†
LψR)(l−1)/2. As is well known when ν=
1/lthese states identify with the electron operator at the edges
of fractional quantum Hall devices [ 5,26,27]. As we shall see,
whenl=1, a bound state with ( q=1,j=N) may be still a
composite fermion but can be made local thanks to the Nspin
degrees of freedom.
B. Stiffnesses, Luttinger parameters, and transport properties
Thanks to the relation ( 2.12) a bound-state Luttinger
liquid is characterized by the three quantities ( PF,u,ν)o r
equivalently ( q,u,ν ). These independent parameters could be
in principle extracted from the knowledge of the differentstiffnesses of the problem [ 28]. As is well known the two
Luttinger parameters uandνcan be related to different
stiffnesses or rigidities associated with ground-state propertiesof the system [ 28]. The first stiffness is related to the
macroscopic compressibility at zero temperature which canbe related to the second derivative of the ground-state energyE
0with respect to the total number of particles:
κ−1=L∂2E0
∂N2=νq2/πu. (2.29)
The other stiffness is the zero-temperature phase stiffness Dα
which is related to the response of the system to an infinitesimal
twistαin the boundary conditions: c†
a,i+L=eiαc†
a,i.I na
Luttinger liquid the ground-state energy in the presence ofthe twist αis to be found in the reduced space with total zero
charge (and zero particle-hole excitations) described by theprojected Hamiltonian
H(α)=E
0+1
2LDα/parenleftbiggπ
q¯J−α/parenrightbigg2
, (2.30)
where ¯Jis the dimensionless current operator ( 2.18) and
Dα=L∂2E0
∂α2=uνq2/π (2.31)
is the phase stiffness associated with the capability of the
system to sustain a persistent current. From ( 2.29) and ( 2.31)
we see that uandνcan be obtained from κandDαonly
ifqis known. To determine the value of the bound-state
charge qone has to consider the full dependence of the
ground-state energy on the twist α[29,30]. As ¯Q=0, and
125106-4BOUND-STATE DYNAMICS IN ONE-DIMENSIONAL . . . PHYSICAL REVIEW B 95, 125106 (2017)
whatever the parity of qis,¯Jhas to be even. Therefore
the ground-state energy E0(α) is a periodic function with
period 2 π/q and has minima at αm=2πm/q ,m∈Z.T h e
corresponding eigenstates have quantum numbers ¯J=2mand
carry persistent currents J=2mj. We thus find that by varying
αin the interval [0 ,2π[ the ground-state energy E0(α) has
exactly qminima, a result that could allow us, in principle, to
determine q. Both the Luttinger parameter νand the charge q
may be also obtained from transport properties. For instancethe dc limit of the conductance of spinless fermions of chargeeand Luttinger parameter Kis given by [ 31]G
0=Ke2/h,
while for Nchannels it is given by [ 32]
G0=NKe2/h=ν(qe)2
h. (2.32)
We can infer from the latter relation that νis the Luttinger
parameter associated with a single channel consisting of abound state of charge qe. However we also see that the
measurement of G
0alone does not fully characterize the
Luttinger liquid state. To do so one needs an independentmeasurement of the Luttinger parameter ν. This could be
achieved, in principle, by the measurement of the nonlinear-ities in the I-Vcurrent-voltage curve in the presence of an
impurity [ 31,33–35].
III. GENERIC BOUND STATES AND DYNAMICAL
SYMMETRY ENLARGEMENT
The requirement of the analyticity of the correlation
functions and the constraint of locality of the bound statesin the Luttinger liquid framework does not completely fix theallowed charges of the possible bound states: even if theircharges qare severely restricted by the constraint ( 2.12), there
is still room for a fairly large number of possible bound-statesolutions. It is obvious that at some point the knowledgeof the allowed charges qand current jquantum numbers
should ultimately rely on the type of ordering in spin spacestabilized by the opening of the spin gap /Delta1. At first glance
it seems unlikely that more can be said about the possiblebound states that can be stabilized by generic Hamiltoniansof the form ( 1.1). Fortunately it is largely recognized that
forgeneric interactions and fillings the low-energy physics
associated with multispecies interacting systems is capturedby renormalization group (RG) asymptotic trajectories whichdisplay an enlarged symmetry [ 36–39]. This is the so-called
dynamical symmetry enlargement (DSE) phenomenon. Inthe cases we are concerned with in this work, i.e., that ofN-species fermionic systems, the large SU( N) symmetry of
the noninteracting limit is in general broken down to somesubgroup G
/bardblby the interactions. According to the DSE
mechanism, the low-energy properties are expected to bedescribed by an approximate enlarged low-energy symmetryG. In particular, both the ground-state properties as well as the
low-energy excitations are adiabatically connected to thoseof the simpler G-symmetric theories. In the simplest case
the symmetry of the noninteracting theory is fully restoredat low energies and G=SU(N). The resulting effective
theory is then given by the SU( N) Gross-Neveu model and is
integrable. Other high-symmetry effective theories were alsoshown to be stabilized by the DSE mechanism. They display
FIG. 1. RG flow diagram of the XXZ model. On the ray g/bardbl=
g⊥the theory is SU(2) symmetric. For rays sufficiently close to it,
the effective theory at low energies displays an approximate SU(2)
invariance. The ray g/bardbl=−g⊥is dual to the SU(2) ray in the sense of
Eq. ( 3.4). Models described by rays close to the above ray display an
approximate dual/hatwiderSU(2) symmetry generated in the continuum limit
by Eq. ( 3.5).
dual symmetries/hatwiderSU(N) obtained from SU( N) by, in general
nonlocal, duality transformations on the elementary fermions.Remarkably enough, the set of all possible dualities is knownand they fall into a finite number of classes [ 39]. This allows
us in the following to propose a classification of the possiblebound states that can be stabilized by generic interactions andfillings.
A. Warm up: The XXZ model
Before going further let us pause and consider, as a proto-
typical example of the DSE mechanism, a system with N=2
and spin-orbit coupling such that the spin is conserved alongone direction, say the zdirection. The resulting symmetry of
the problem is then reduced from SU(2) down to G
/bardbl=U(1).
The interacting part in ( 1.1) of the effective low-energy
Hamiltonian can be expressed, in the continuum limit, in termsof the SU(2) spin currents ( 3.9)[3]
V
int=/integraldisplay
dx/bracketleftbig
g⊥/parenleftbig
Ix
LIx
R+Iy
LIy
R/parenrightbig
+g/bardblIz
LIz
R/bracketrightbig
. (3.1)
The physics described by the latter model is well known and
the associated RG flow diagram is depicted in the Fig. 1. Apart
from a massless phase there are two distinct phases where aspin gap opens whose low-energy physics are governed by thetwo attractive rays with g
/bardbl=±g⊥>0.
On the first ray g/bardbl=g⊥the effective low-energy Hamil-
tonian ( 3.1) clearly displays an SU(2) symmetry. For initial
conditions of the RG flow not too far from the SU(2)-invariantray the effective low-energy theory display an approximateSU(2) invariance. This is the DSE mechanism. For thesemodels both the ground state and the low-energy spectrum areadiabatically connected to that of the SU(2)-symmetric theory.As is well known this phase is characterized by quasi-long-range spin-singlet superconducting fluctuations with orderparameter [ 3]
O
SS=ψ†
R,↑ψ†
L,↓+ψ†
L,↑ψ†
R,↓. (3.2)
125106-5P. AZARIA PHYSICAL REVIEW B 95, 125106 (2017)
The massless charge excitations are bound states which carry
a charge q=2. A bare bound-state creation operator (or wave
function) with a finite overlap with ( 2.23) is the BCS pair
creation operator
B†
2,i=c†
↑,ic†
↓,i. (3.3)
On the second ray g/bardbl=−g⊥the effective low-energy Hamil-
tonian ( 3.1) can be brought to an SU(2)-invariant form by
means of a duality transformation /Omega1(which is a πrotation of
theRspin currents around the zdirection)
/hatwideIx,y,z
L=Ix,y,z
L,/hatwideIx,y
R=−Ix,y
R,/hatwideIz
R=Iz
R. (3.4)
In terms of the classification that we shall introduce in
Sec. III B the above duality belongs to the AIIIclass ( 3.28).
On this ray, the theory is invariant under a dual /hatwiderSU(2) group
generated in the continuum limit by
/hatwideIx,y,z=/integraldisplay
dx/parenleftbig/hatwideIx,y,z
L+/hatwideIx,y,z
R/parenrightbig
, (3.5)
which thanks to ( 3.4) is a highly nonlocal operator when
expressed in terms of the bare fermion operators. The orderparameter in the dual phase can be obtained from ( 3.2)b y
duality. To this end we need the effect of the duality /Omega1on the
fermions themselves:
/hatwideψ
L,↑(↓)=ψL,↑(↓),/hatwideψR,↑(↓)=(±)ψR,↑(↓). (3.6)
Using the latter duality transformation we find that the
dual phase is characterized by quasi-long-range spin-tripletsuperconducting fluctuations with order parameter /hatwidestO
SS=Oz
ST
where
Oz
ST=ψ†
R,↑ψ†
L,↓−ψ†
L,↑ψ†
R,↓. (3.7)
The massless charge excitations are still bound states which
carry a charge q=2, but a bare bound-state creation operator
(or wave function) with a finite overlap with ( 2.23)i sn o w
given by the spin-triplet pair
(B1)†
2,i=c†
↑,ic†
↓,i+1−c†
↑,ic†
↓,i−1. (3.8)
As we have just described in the simplest case of N=2 there
exist two finite domains of equal size in the ( g/bardbl,g⊥), close to
the two rays g/bardbl=±g⊥, where the low-energy symmetry is
dynamically enlarged. These two phases are dual to each otherand support two different types of bound states.
The question we shall address in the following is how this
picture extends to larger values of N> 2. The strategy we
shall adopt will be first to study the SU( N)-invariant case
and then discuss all possible dual /hatwiderSU(N) phases that can be
obtained with use of duality transformations of the type ( 3.4).
It turns out that all the possible duality transformations thatcan be implemented on the SU( N) spin currents are known
and fall into a finite number of classes [ 39]. These classes
are in one to one correspondence with the symmetry-breakingpatterns SU( N)→G
/bardblwhere the group G/bardblcan be chosen to be
either SO( N),SP(N)o rS U ( p)×SU(N−p) are the maximal
subgroups of SU( N). One may thus think the situation at hand
as qualitatively similar to that of the XXZ model we have
just discussed except for the nature of the subgroup G/bardbl.A s
we shall show, this will enable us to characterize a finite set ofbound-state solutions of ( 2.12) that we shall call generic boundstates. They are generic in the sense that, thanks to the DSE
mechanism, they are the ones which are likely to be stabilizedfor a generic interaction.
B. General case
In the following, will shall use the fact that due to the
presence of a spin gap /Delta1, the low-energy wave functions have
to be singlets of either the SU( N) group or the/hatwiderSU(N) groups.
The latter conditions, when translated in terms of the fermioniccharges and currents Q
aandJa, will yield constraints on the
total charge and current zero-mode operators QandJand
hence on the bound-state charge and current quantum numbers(q,j).
Assuming spin-charge separation and weak enough interac-
tions, the low-energy physics of the generic Hamiltonian ( 1.1)
is to be described by the sum of two commuting charge andspin Hamiltonians H=H+H
swhere His given by ( 2.6)
andHsdescribes the spin fluctuations. In order to discuss
the properties of Hs, and as we shall focus on the symmetry
properties, it is useful to describe the dynamics in the spinsector using non-Abelian bosonization [ 7]. To this end let us
introduce the right-left SU( N) spin currents
I
A
L(R)=N/summationdisplay
(a,b)=1ψ†
a,L(R)TA
abψb,L(R), (3.9)
where TA,A=(1,..., N2−1), are the generators of the
Lie algebra of SU( N) which are normalized as Tr( TATB)=
δAB/2. These currents satisfy the SU( N)1Kac-Moody algebra
given by the operator product expansion (OPE)
IA
L(R)(x)IB
L(R)(y)∼−δAB
8π2(x−y)2±fABC
2π(x−y)IC
L(R)(y).
(3.10)
In terms of these quantities the effective Hamiltonian in the
spin sector may be written as a Wess-Zumino-Witten-Novikov(WZWN) SU( N)
1[7] perturbed by a marginal current-current
interaction
Hs=2πvs
N+1/integraldisplay
dx/summationdisplay
A/bracketleftbig/parenleftbig
IA
L/parenrightbig2+/parenleftbig
IA
R/parenrightbig2/bracketrightbig
+/integraldisplay
dx/summationdisplay
ABgABIA
LIB
R. (3.11)
When gAB=0, the first part of the Hamiltonian describes the
spin dynamics of Nfree fermions with independent SU( N)L
and SU( N)Rsymmetries. With these definitions, the statement
of the DSE phenomenon can be phrased as follows: whenthe interaction is relevant the couplings g
AB(t) grow with the
RG time tand ultimately reach some attractive ray where
the symmetry is dynamically enlarged to some group G.A s
stated above the symmetry can be maximally enlarged in theinfrared to G=SU(N) but as well to duals [ 39]o fS U ( N)i n
which cases G=/hatwiderSU(N). The constraints on the bound-state
quantum numbers ( q,j) will be different. In the following we
shall assume that the symmetry is dynamically enlarged up tosmall symmetry-breaking corrections.
125106-6BOUND-STATE DYNAMICS IN ONE-DIMENSIONAL . . . PHYSICAL REVIEW B 95, 125106 (2017)
C. SU( N) bound states
Let us start by discussing the simplest case of a maximally
enlarged SU( N) symmetry. In this case the RG trajectory has
the asymptotic gAB(t)∼g(t)δABand the interacting part of
(3.11) takes the asymptotic SU( N)-invariant form
Hint=g/integraldisplay
dx/summationdisplay
AIA
LIA
R. (3.12)
When g> 0 a spin gap opens and the ground state of ( 3.12)
displays an (approximate) SU( N) symmetry. More precisely,
the effective low-energy symmetry is given by the diagonalgroup SU( N)=SU(N)
L×SU(N)R|diagwhich is generated
by
IA=/integraldisplay
dx/parenleftbig
IA
L+IA
R/parenrightbig
. (3.13)
Thanks to the gap in the SU( N) sector the low-energy sector
is obtained by projecting into the SU( N) singlet sector
IA≡0,A=1,..., N2−1. (3.14)
The latter equations impose constraints for the eigenvalues Qa
andJaof the charge and current operators ( 2.3). Indeed let
us consider the N−1 conserved charges associated with the
SU(N) symmetry. They are the subset of Cartan generators hα,
α=(1,..., N −1), of the SU( N) generators that are mutually
commuting: [ hα,hβ]=0. They are expressed in terms of the
fermion charges ( 2.3) (see the Appendix) as
hα=N/summationdisplay
a=1ωα
aQa, (3.15)
where the vectors /vectorωasatisfy /vectorωa·/vectorωb=δab−1/N and/summationtextN
a=1/vectorωa=0. Equation ( 3.14) implies in particular hα=0
for all α=(1,..., N −1) which together with the property/summationtextN
a=1/vectorωa=0 yields for the species charges Qa
Qa=n∈Z,a=(1,..., N ). (3.16)
As Eq. ( 3.14) does not yield other constraints on the values of
the current quantum numbers than ( 2.4), we thus find for the
total charge and current eigenvalues
Neven Q=nN,J =2m, (3.17)
Nodd Q=nN,J =m, (3.18)
(n±m)e v e n,
where ( n,m) are relative integers. From ( 2.8) and ( 2.9)w e
immediately find
(q=N,j=1). (3.19)
The above solution satisfies the constraint ( 2.12) and we
identify these bound states as charge- Nbosons for Neven
and charge- Nfermions for Nodd, both with density ρBS=¯ρ
and a preserved Fermi momentum PF=kF.
D. Dual/hatwidestSU(N) bound states
On top of the dynamical enlargement of the SU( N)
symmetry there are three other possibilities of DSE which arerelated to emergent duality symmetries [ 39]. These dualities/Omega1act on one chiral sector of the theory and in particular on
the SU( N) currents as follows:
/Omega1:I
A
L(R)→/hatwideIA
L(R), (3.20)
where
/hatwideIA
L=IA
L,/hatwideIA
R=/summationdisplay
B/Omega1A
BIB
R, (3.21)
with/Omega12=1. These dualities are symmetries of the problem
and preserve the Kac-Moody algebra ( 3.10). Therefore, to any
set of initial conditions of the RG flow gAB(0) that are attracted
by the SU( N)-invariant ray there exists models with couplings
/hatwidegAB(0)=/summationtext
C/Omega1C
AgCB(0) that will flow toward
Hint=/hatwideg/integraldisplay
dx/summationdisplay
A/hatwideIA
L/hatwideIA
R. (3.22)
Similarly to the SU( N) case, a spin gap opens and the ground
state of ( 3.22) displays an approximate dual /hatwiderSU(N) symmetry
generated by
/hatwideIA=/integraldisplay
dx/parenleftBigg
IA
L+/summationdisplay
B/Omega1A
BIB
R/parenrightBigg
. (3.23)
We can now look at the constraints imposed on the total charge
and current quantum numbers QandJwhen a spin gap is
present. The low-energy sector we are interested in is the
/hatwiderSU(N)-singlet sector obtained by the projection
/hatwideIA≡0. (3.24)
The resulting constraints on the charge and current operators
(Q,J) can then be obtained from the knowledge of the duality
/Omega1in (3.20). Remarkably enough, the set of all possible /Omega1
is known [ 39] and fall into a finite number of classes named
AI,AII, andAIII. They act [up to a simultaneous change of
basis in the L(R) chiral sectors] on the left and right fermions
(2.1)a s
/Omega1:ψa,L(R)→/hatwideψa,L(R), (3.25)
where /hatwideψa,L=ψa,Land
AI:/hatwideψa,R=ψ†
a,R, (3.26)
AII:/hatwideψa,R=N/summationdisplay
b=1Jabψ†
b,R(Neven), (3.27)
AIII:/hatwideψa,R=N/summationdisplay
b=1(Ip)abψb,R. (3.28)
In the above equations the matrix Jab=(−iσ2)⊗IN/2is the
SP(N) metric and Ip(0<p<N ) is the diagonal matrix
withN−pentries +1 and pentries −1. In order to obtain
the constraints on Q=/summationtextN
a=1QaandJ=/summationtextN
a=1Jaimposed
by the singlet dual projection ( 3.24) we use the fact that in
the dual/hatwiderSU(N) ground state of ( 3.22) the dual charge and
current eigenvalues /hatwideQand/hatwideJsatisfy ( 3.17) and ( 3.18). With
the knowledge of /Omega1in each class we then find for both AIand
125106-7P. AZARIA PHYSICAL REVIEW B 95, 125106 (2017)
AIIclasses
AI:Neven Q=2n,J=mN, (3.29)
Nodd Q=n, J=mN, (3.30)
(n±m)e v e n,
AII:Neven Q=2n,J=mN. (3.31)
For the class AIIIthe constraints are the same as the ones given
in the SU( N) case by ( 3.17) and ( 3.18) and does not yield to
a new bound-state solution but the ( q=N,j=1) one. As we
shall see in the next section the duality class AIIIprovides
for an internal structure of the bound states. In contrast, bothA
IandAIIduality classes yield to new selection rules and
hence to new types of bound states. The main reason for thisis that they contain the charge conjugation operator
C:ψ
a,R→ψ†
a,R, (3.32)
which induces an electromagnetic duality and exchanges the
charge and current operators in ( 3.17) and ( 3.18). The new
bound-state solutions depend on the parity of N.
N even . In this case the bound states are bosons and for the
two duality classes AIandAIIwe find charge q=2 states
with
(q=2,j=N/2), (3.33)
which satisfy the locality constraint ( 2.12). As we shall see
below, these bound states correspond to genuine antisym-metric and symmetric pairing states for the duality classesA
IandAII. Their density is from ( 2.14)ρBS=N¯ρ/2 and
their Fermi momentum is enlarged to PF=NkF/2.
N odd . In this case the AIclass yields to a fermionic bound
state of charge q=1 with
(q=1,j=N), (3.34)
satisfying ( 2.12). Though this bound state has the same charge
as the elementary fermions, it is of a completely differentnature. The bound-state density in this case is ρ
BS=N¯ρand
the fermionic bound state carries left and right momenta ∓PF
with an enlarged Fermi momentum PF=NkF. As we shall
see, these states are local composite fermions.
Let us end this section by noticing that the two bound-state
solutions ( 3.33) and ( 3.34) (associated with the duality classes
AIandAII) are duals to the SU( N) bound states ( 3.19). The
corresponding duality symmetry of the bound-state Luttingerliquid Hamiltonian, which is also a symmetry of the bound-state equation ( 2.12), is given by
qeven ( q,j)→(2j,q/2), (3.35)
qodd ( q,j)→(j,q), (3.36)
together with /Phi1
c↔/Theta1candK↔1/K. The above duality
transformations preserve the bosonic and fermionic selectionrules ( 2.8) and ( 2.9) and hence map a bosonic (fermionic)
Luttinger liquid to another bosonic (fermionic) Luttingerliquid. The above duality transformations ( 3.35) and ( 3.36)
are equivalent, upon rescaling the fields in the dimensionlessbasis, to the one of a bosonic Luttinger liquid: ¯/Phi1→¯/Theta1/2,
¯/Theta1→2¯/Phi1, andν→1/4ν, and to that of a fermionic Luttinger
liquid: ¯/Phi1↔¯/Theta1andν→1/ν[4].IV . WA VE FUNCTIONS OF THE SU( N)
AND/hatwidestSU(N) BOUND STATES
So far we have determined the charge and current quantum
numbers ( q,j) of the generic bound states and it remains to
characterize them in terms of the elementary fermions. Theidea is to look at bare operators made of the lattice fermions
c
a,i, in either SU( N)o r/hatwiderSU(N) singlets, which after averaging
over the gapped spin degrees of freedom have a finite overlapwith the bound-state creation operators ( 2.23). The choice of
the bare wave function is of course not unique but, as we shallsee, some choices turn out to be physically more transparent.In the case of SU( N) bound states, the natural choice is a local
singlet with wave function made of the Nelementary fermions.
For the/hatwiderSU(N) bound states, the situation is less obvious since
the dual enlarged symmetries generated by ( 3.23)a r e nonlocal
in the original fermion basis ( 2.1). There exists though, for
each class of duality, subgroups G
/bardblof/hatwiderSU(N) that act locally
in the elementary fermion basis. These are the subgroups ofSU(N) that are invariant under the duality transformations
(3.26), (3.27), and ( 3.28). The transformations U∈G
/bardblact
simultaneously on the two chiral sectors
ψa,L(R)→Uabψb,L(R), (4.1)
with
AI:U∈O(N), (4.2)
AII:U∈SP(N), (4.3)
AIII:U∈S(U(N−p)×U(p)). (4.4)
These latter local symmetries G/bardblwill help us to characterize
the bound states stabilized in each duality class as local G/bardbl-
singlet operators made of the elementary fermions. As weshall see below, we shall find fivedifferent types of bound
states which are related to the three duality classes we havejust discussed. They distinguish themselves by their charge q,
their Fermi wave vector, and the isotropy group G
/bardbland hence
can be labeled by the duality classes ( 4.2), (4.3), (4.4). Before
going into more details we summarize our results in Table I.
A. SU( N) bound states: Baryons
Let us start with the simplest case of the SU( N) singlet
bound states ( 3.19)
(q=N,j=1), (4.5)
and hence PF=kF. In terms of the elementary electrons these
excitations are naturally related to a bound state made of N
electrons in an SU( N) singlet state
B†
N,i=c†
1,i...c†
N,i, (4.6)
which is a boson for Neven and a fermion for Nodd. By
analogy with QCD, we may call these bound-state baryons.Using the low-energy expansion ( 2.1) and averaging over the
spin degrees of freedom in the SU( N) ground state of ( 3.12)
we find
/Psi1†
B/F=/angbracketleft(BN)†(x)/angbracketright|SU(N)∼N/summationdisplay
¯J=−Nα¯Jei¯JPFx+i√π(¯/Theta1+¯J¯/Phi1),
(4.7)
125106-8BOUND-STATE DYNAMICS IN ONE-DIMENSIONAL . . . PHYSICAL REVIEW B 95, 125106 (2017)
TABLE I. The five types of bound states that can be stabilized in a system with generic interactions. They differ by their charge and Fermi
momentum PFand are labeled by the duality classes A0,I,II,III. We give the corresponding isotropic groups G/bardbland pictorial wave functions in
terms of the bare fermions.
Class Local Symmetry G/bardbl PF Bound-State Wave Function (Pictorial) Charge Type
A0 SU(N) kF B†
N,i=/Pi1N
a=1c†
a,i N baryon
AI SO(N),Neven NkF/2 /Pi1†
i=/summationtextN
a=1c†
a,ic†
a,i+1 2 p-wave supra
SO(N),Nodd NkF /Xi1†
i=/summationtext
{aj}/epsilon1a1...aNc†
a1,i...c†
aN+1
2,icaN+1
2+1,i...caN,i 1 composite fermion
AII SP(N),Neven NkF/2 P†
i=/summationtext
a,bc†
a,ic†
b,iJab 2 s-wave supra
AIII SU(p)×SU(N−p) kF (Bp
N)†(x)=B†
p(x)B†
N−p(x+x0) N baryon
where B†
N(x)=B†
N,i/(a0)N/2and the sum is over ¯Jeven for
Neven (bosons) and ¯Jodd for Nodd (fermions). Similar
considerations lead for the relative density operator,
ni=N/summationdisplay
a=1c†
a,ica,i−N¯ρ, (4.8)
to
ρ(x)=/angbracketleftn(x)/angbracketright|SU(N)∼q∂x¯/Phi1/√π+β2e2iPFx+2i√π¯/Phi1+H.c.,
(4.9)
withn(x)=ni/a0andq=N. In both expressions ( 4.7) and
(4.9) we have rescaled the charge fields /Phi1cand/Theta1caccording to
(2.15) withq=Nandj=1, i.e., ¯/Phi1=/Phi1c/√
N,¯/Theta1=/Theta1c√
N.
The coefficients α¯Jandβ¯Jare related to the primary operators
of the SU( N)1WZWN model as
α¯J/similarequal(γ∗)(N−¯J)/2/angbracketleftTr(/Phi1(N−¯J)/2))/angbracketright|SU(N),α −¯J=α∗
¯J,
β2/similarequal/angbracketleftTr(/Phi1(1))/angbracketright|SU(N),β −2=β∗
2, (4.10)
where /Phi1(m)is the primary operator of SU( N)1that transforms
according to the representation of SU( N) consisting of Young
tableaus with mboxes and one column. Finally γ=±iis a
cocycle (see the Appendix). The operator /Phi1(m)has the scaling
dimension dm=m(1−m/N ) and therefore the coefficients
(4.10) scale with the spin gap /Delta1asα¯J∼/Delta1(N2−¯J2)/4Nand
β2∼/Delta11−1/N. We thus find that, to this order, the expressions
for both ( 4.7) and ( 4.9) match the expansions ( 2.23) and ( 2.24).
Notice though that higher harmonics in 2 mkFx,m> 1, are
missing in the density operator expansion ( 4.9). In any case,
the coefficients α¯Jandβ¯Jare expected to be renormalized.
Indeed, from the renormalization group point of view, theintegration over the (high-energy) spin degrees of freedomis expected to generate corrections to ( 3.12). Among these,
for instance, are oscillating contributions to the Hamiltoniandensity ( 3.12) such as/summationtext
mHme2imkFx. Though they do not
contribute to the Hamiltonian once the/integraltext
dxis performed,
they do renormalize locally the various Fourier componentsof both the wave function ( 4.7) and of the density ( 4.9)i n
which higher components at 2 mk
Fxare thus expected to be
generated. Another source of renormalization comes from thefact that the SU( N) symmetry of ( 3.12) is expected to be only
approximate. In general, there will be subleading correctionsdue to symmetry-breaking operators which are supposed to besmall for not too large anisotropies.At this point we may compare our findings with existing
results. The baryonic bound states we have just described werefound in the attractive SU(N) Hubbard model, with Coulomb
interaction U< 0, away from half filling [ 15,18,21]. DMRG
results [ 15] for both N=3 andN=4 cases strongly support
the existence of massless charge q=3 fermionic trions and
charge q=4 bosonic quartets bound-state excitations, while
the single-fermion excitations are shown to be gapped. In bothcases, the baryon-baryon correlation functions exhibit power-law behaviors with oscillations at wave vectors ±2k
F.T h e
physics in these cases were found to agree with that of spinlessfermions or hard-core bosons [ 3] in a wide range of densities
¯ρand couplings U< 0. In particular, for a sufficiently large
|U|and density, typically smaller than ¯ ρ∼1/N(for which
ν>1/2o rν>1/√
3 for bosons and fermions, respectively),
the baryon-baryon correlation function was found to bedominant. For larger ¯ ρand smaller |U|,t h e2 k
Fdensity
wave was found to be the dominant instability. All togetherthese results provide strong evidence for the relevance of thebound-state description. Let us add that further investigationsalso show that the effect of various anisotropies [ 17,18], like
small breaking of the SU( N) symmetry, do not modify the
above picture. This shows that these baryonic bound states arerobust and generic and in particular that the DSE hypothesisis sensible.
B. Dual AIbound states
As discussed above these bound states are the duals under
(3.35) and ( 3.36) of the SU( N) baryonic states. They are
nontrivial states as they involve an enhanced Fermi momentumP
F. The bound states associated with the duality AIare of
two types depending on the parity of Nand are either charge
q=2 bosons for Neven ( 3.33) or charge q=1 fermions for
Nodd ( 3.34). As the duality ( 3.26) is nonlocal in terms of
the elementary fermions, we shall look, as discussed above, atwave functions which are from ( 4.2)G
/bardbl=O(N) singlets with
either charge q=2o rq=1.
1. N even: p-wave pairing
The corresponding bound-state solution is given by ( 3.33)
(q=2,j=N/2) (4.11)
and hence PF=NkF/2. Given the O( N) symmetry of the
problem it is natural to look at the p-wave symmetric lattice
125106-9P. AZARIA PHYSICAL REVIEW B 95, 125106 (2017)
pairing operator
/Pi1†
i=N/summationdisplay
a=1c†
a,ic†
a,i+1. (4.12)
Using bosonization we find /Pi1†
i/a0=sin (kFa0)/Pi1†(x) where
/Pi1†(x)/similarequalTr(/hatwide/Phi1(1))ei√4π/N /Theta1 c. (4.13)
In the latter expression we have omitted terms that average to
zero in the/hatwiderSU(N) ground state of ( 3.22). The operator /hatwide/Phi1(1)
entering in ( 4.13)i st h e dual of the SU( N)1primary operator
obtained from /Phi1(1)with the help of the duality transformation
AI(see Appendix). It has the same scaling dimension d1=1−
1/Nand is odd under parity, i.e., P:T r (/hatwide/Phi1(1))→− Tr(/hatwide/Phi1(1)).
As/Theta1c→/Theta1cunderP, we find that ( 4.13) is odd as it should
be. A similar calculation yields for the density operator n(x)
(4.8)
n(x)/similarequal/radicalbig
N/π ∂ x/Phi1c, (4.14)
where here again we have discarded terms that average to zero
in the/hatwiderSU(N) ground state of ( 3.22) .T h en e x ts t e pt ob et a k e ni n
order to obtain both the bound-state wave function /Psi1†
B(x) and
the density ρ(x) is to average over the spin degrees of freedom
in the/hatwiderSU(N) ground state of ( 3.22) with duality class AI.T o
do so we notice that, as the duality transformations AI,II,IIIare
symmetries of the problem, we have for any operator O
/angbracketleft/hatwideO/angbracketright|/hatwiderSU(N)=/angbracketleftO/angbracketright|SU(N), (4.15)
where /hatwideOis the dual of O. Hence we get
/Psi1†
B(x)=/angbracketleft/Pi1†(x)/angbracketright|/hatwiderSU(N)∼α0ei√π¯/Theta1, (4.16)
ρ(x)=/angbracketleftn(x)/angbracketright|/hatwiderSU(N)∼q∂x¯/Phi1/√π, (4.17)
withq=2 and
α0/similarequal/angbracketleftTr(/hatwide/Phi1(1))/angbracketright|/hatwiderSU(N)=/angbracketleftTr(/Phi1(1))/angbracketright|SU(N)
∼γ/Delta11−1/N. (4.18)
In both Eqs. ( 4.16) and ( 4.17) we have rescaled the charge
fields according to ( 2.15) with q=2 andj=N/2:
¯/Phi1=√
N/Phi1c/2,¯/Theta1=2/Theta1c/√
N. (4.19)
Notice that in Eq. ( 4.18) we have singled out the cocycle γ
to keep track of the parity transformation properties of thep-wave wave function, P:γ→γ
∗=−γ. In contrast with
the baryonic bound-state wave function we find that ( 4.16) and
(4.17) match the expansions ( 2.23) and ( 2.24) only to leading
order in the momentum expansion. In particular, the harmonicsat±2mP
FxwithPF=NkF/2 are absent for both the bosonic
wave function and the density. This is not very satisfying asone of the hallmarks of the bound-state solution in the dualA
Iclass is the emergence of oscillations at the enlarged wave
vectors ±2PF=±NkF.
Composite density. As we shall now see, these harmonics
are generated by composite operators. Indeed, in the RGframework, we are at liberty to add to the effective Hamiltonianany term which is compatible with the symmetries of theproblem and that would be generated anyway at energy scalesE/lessmuch/Delta1. In the following we shall accordingly consider adding
to the Hamiltonian ( 3.22) the neutral (i.e., charge Q=0),
/hatwiderSU(N)-singlet and parity-invariant composite density operator
with momentum components at ±Nk
F. To do this, let us first
consider the charge Q=Nand current J=0 singlet operator
under the [SU( N)L×SU(N)R]|diaggroup. It is obtained from
the rank- Ninvariant tensor /epsilon1a1...aNof SU( N):
/summationdisplay
{aj}/epsilon1a1...aNψ†
a1,L...ψ†
aN/2,Lψ†
aN/2+1,R...ψ†
aN,R.(4.20)
The above operator is in fact proportional to the zero-
momentum component of the SU( N) baryon wave function
(4.7) when Nis even. We can now use the duality transfor-
mation ( 3.26) to obtain the NkFcomponent of the composite
density operator, the −NkFcomponent being obtained with
help of the parity transformation. Using ( 3.26) and imposing
parity invariance, we find for the /hatwiderSU(N) composite density
operator in the AIclass ( Neven)
RN(x)=eiNkFxRNkF(x)+e−iNkFxR−NkF(x),(4.21)
where
RNkF(x)/similarequaleiNπ/ 4/summationdisplay
{aj}/epsilon1a1...aNψ†
a1,L
...ψ†
aN/2,LψaN/2+1,R...ψaN,R. (4.22)
The phase factor eiNπ/ 4in (4.22) has been chosen in such a way
that under P,RNkF(x)→R†
NkF(x)=R−NkF(x). This ensures
that ( 4.21) is indeed parity invariant. Using bosonization we
finally find
RN(x)=WN/2(x) cos (√
Nπ/Phi1 c+NkFx), (4.23)
where
WN/2(x)/similarequalγN/2Tr(/hatwide/Phi1(N/2)). (4.24)
In the above equation /hatwide/Phi1(N/2)is the dual, under AI,o ft h e
SU(N)1primary operator /Phi1(N/2)transforming in the self-
conjugate representation of SU( N). As shown in the Appendix,
the operator WN/2(x), which has the scaling dimension N/4,
is parity invariant and real: WN/2(x)=PWN/2(x)=W∗
N/2(x).
We may now write the contribution of RN(x)t ot h e
interacting Hamiltonian ( 3.22)a s
Hint→Hint+λ/integraldisplay
dxR N(x), (4.25)
where λis some nonuniversal coupling. For generic fillings,
kF/negationslash=2π/N , it is oscillating and gives a negligible contribution
to the total Hamiltonian. However, as discussed above, it doeshave an effect on the renormalization of the different vertexoperators. For instance, it generates the ±Nk
Fxcomponents of
both the p-wave bound-state wave function and of the density
operator ( 4.16) and ( 4.17). To leading order in λwe have
/Psi1†
B(x)→/Psi1†
B(x)+λδ/Psi1†
B(x), (4.26)
ρ(x)→ρ(x)+λδ ρ(x), (4.27)
125106-10BOUND-STATE DYNAMICS IN ONE-DIMENSIONAL . . . PHYSICAL REVIEW B 95, 125106 (2017)
where δ/Psi1†
B(x) and δρ(x) are given by the operator product
expansions (OPE)
δ/Psi1†
B(x)∼/angbracketleftRN(z,¯z)·/Pi1†(w,¯w)/angbracketright|/hatwiderSU(N),
δρ(x)∼/angbracketleftRN(z,¯z)·n(w,¯w)/angbracketright|/hatwiderSU(N), (4.28)
where z=τ+i(x+a0) and w=τ+ix. Performing the
necessary OPE and averaging over the spin degrees of freedomwe find, for the bound-state wave function and the densityoperator, the corrections to ( 4.16) and ( 4.17)
/Psi1
†
B(x)/similarequalα0ei√π¯/Theta1+α2ei(√π¯/Theta1+2√π¯/Phi1+2PFx)
+α−2ei(√π¯/Theta1−2√π¯/Phi1−2PFx), (4.29)
ρ(x)/similarequalq∂x¯/Phi1/√π+β2ei(2√π¯/Phi1+2PFx)+β−2e−i(2√π¯/Phi1+2PFx),
(4.30)
where α0is given by ( 4.18) and [ 52]
α2=α−2/similarequalλγ/Delta1(N/4−1/N),
β2=−β−2/similarequalia0
|a0|λ/Delta1N/4. (4.31)
In order to obtain the latter expressions we have made use of
(4.15) and have rescaled the charge fields according to ( 4.19).
The momentum expansions ( 4.29) and ( 4.30) fit the general
expressions ( 2.23) and ( 2.24) with nonvanishing coefficients
up to ±2PF. Higher momenta components can be obtained
similarly by including higher harmonics to the Hamiltoniandensity or going to higher order in λ. The important point
is that these harmonics, being /hatwiderSU(N) symmetric, must carry
multiples of ±Nk
F. In this respect, the composite density
RN(x)(4.23) is the minimal/hatwiderSU(N)-invariant object that one
can build from the bare fermions. As we shall see in thenext section, it plays also a crucial role when discussing theincompressible phases associated with the dual phases. So farwe have obtained the bound-state wave function assuming
the dual symmetry/hatwiderSU(N) is dynamically enlarged. Other
corrections to the coefficients α
¯Jandβ¯Jare also expected
from small symmetry-breaking operators. We expect thesecorrections to be small and the p-wave bound state to be robust.
2. N odd: Composite fermions
We now discuss the bound-state solution ( 3.34)
(q=1,j=N), (4.32)
which is a fermion with an enlarged Fermi momentum PF=
NkF. As discussed above, this is a nontrivial excitation since,
though it has the same charge as the elementary fermions, itcarries an excess of current of ±(N−1) in its left and right
components. The situation is similar to the composite fermionconstruction [ 5,26,27](2.28). In the present case though the
composite fermion can be made a local object thanks to the
Nindependent spin degrees of freedom. Let us consider for
instance the fermionic charge Q=1 and O( N)-symmetric
lattice operator
/Xi1
†
i=/summationdisplay
{aj}/epsilon1a1...aNc†
a1,i...c†
aN+1
2,icaN+1
2+1,i...caN,i.(4.33)Using the low-energy expansion ( 2.1) we find that /Xi1†(x)=
/Xi1†
i/(a0)N/2has left and right components at ±NkF:
/Xi1†(x)=/Xi1†
NkFeiNkFx+/Xi1†
−NkFe−iNkFx, (4.34)
where
/Xi1†
NkF/similarequal/summationdisplay
{aj}/epsilon1a1...aNψ†
a1,L...ψ†
a(N+1)/2,Lψa(N+1)/2+1,R...ψaN,R,
/Xi1†
−NkF/similarequal/summationdisplay
{aj}/epsilon1a1...aNψ†
a1,R...ψ†
a(N+1)/2,Rψa(N+1)/2+1,L...ψaN,L,
(4.35)
which, upon bosonization, can be expressed as
/Xi1†
NkF/similarequalγ(N+1)/2Tr(/hatwide/Phi1(N+1)/2)ei√π(√
N/Phi1c+/Theta1c/√
N),
/Xi1†
−NkF/similarequalγ(N−1)/2Tr(/hatwide/Phi1(N−1)/2)ei√π(−√
N/Phi1c+/Theta1c/√
N).(4.36)
We may now average over the spin degrees of freedom in the
dual/hatwiderSU(N) ground state of ( 3.22) with duality class AIto get
the composite fermion wave function
/Psi1†
F(x)=/angbracketleft/Xi1†(x)/angbracketright|/hatwiderSU(N)=α1eiPFxei√π(¯/Theta1+¯/Phi1)
+α−1e−iPFxei√π(¯/Theta1−¯/Phi1), (4.37)
withPF=NkFandα1=α−1/similarequal/Delta1(N/4−1/4N).I nE q .( 4.37)
we have rescaled the charge fields according to ( 2.15) with
q=1 andj=N:
¯/Phi1=√
N/Phi1c,¯/Theta1=/Theta1c/√
N. (4.38)
The result ( 4.37) shows that the local composite fermion
(4.33) has a finite overlap with the bound-state solution
(3.34). In particular, when ν=1, it can be interpreted as
a free fermion with a sharp extended Fermi surface withFermi momentum P
F. Notice that this limit corresponds to
strong repulsive interaction between the elementary fermions
asK=ν/N=1/N. The expression for the density is the
same as for the even- Ncase ( 4.14), i.e., n(x)/similarequal√N/π ∂ x/Phi1c,
and there too, the ±2mPF=±2mNk F,m> 1, components
are missing.
Composite density . Following the same strategy as in the
even-Ncase, we are led to consider adding to the interacting
Hamiltonian ( 3.22) the neutral,/hatwiderSU(N)-symmetric, and parity-
invariant composite density operator. In contrast with theeven-Ncase, when Nis odd the latter operator must have
momentum components at multiples of ±2Nk
Fsince the
quantum of current is now j=N. The only density operator
with such a property is the self-dual, i.e., both SU( N) and
/hatwiderSU(N) symmetric, operator given by
R2N(x)=e2iNkFxR2NkF(x)+e−2iNkFxR−2NkF(x),(4.39)
where
R2NkF(x)/similarequal/Pi1N
j=aψ†
a,Lψa,R, (4.40)
andR−2NkF=R†
2NkF.A s( 4.40)i sS U ( N) invariant, it is
proportional to the identity operator which gives us
R2NkF(x)/similarequal(γ)Nei√
4πN/Phi1 c. (4.41)
125106-11P. AZARIA PHYSICAL REVIEW B 95, 125106 (2017)
AsNis odd and γ∗=−γwe finally get
R2N(x)/similarequal(iγ)s i n(√
4πN/Phi1 c+2NkFx). (4.42)
At this point, it is worth stressing that the above expression
isPinvariant despite the presence of the sin function. This
is due to the presence of the cocycle γsince under P,/Phi1c→
−/Phi1candγ→−γ. As we shall see below, this will be of
crucial importance when discussing boundary effects in theincompressible phase. Proceeding as with the p-wave function
and performing the necessary OPE we find, using ( 4.28) and
(4.38), the expression for the density
ρ(x)/similarequalq∂
x¯/Phi1/√π+β2ei(2√π¯/Phi1+2PFx)+β−2e−i(2√π¯/Phi1+2PFx)
(4.43)
withq=1,PF=NkF, andβ2=β−2∼(−iγ)λa0/|a0|.The
momentum expansions ( 4.37) and ( 4.43) match the general
expressions ( 2.23) and ( 2.24) to leading nontrivial order with
nonvanishing coefficients up to ±2PF.I naw a ys i m i l a rt o
that for the p-wave bosonic bound state, higher momenta
components may be generated at higher orders in λand
additional renormalizations of the coefficients α¯Jandβ¯J
are to be expected. For the same reasons as for the p-wave
bound states, we expect also, despite the fact that the dual
/hatwiderSU(N)s y m m e t r yo ft h e AIclass is only approximate, that
the composite fermion will also be robust against small
/hatwiderSU(N)-symmetry-breaking operators.
In sharp contrast with SU( N) baryonic bound states,
both the bosonic p-wave ( 4.12) and the composite fermion
(4.33) bound-state wave functions display an enlarged Fermi
momentum at PF=NkF/2 and PF=NkF. In order to
account for this high-momenta physics within the low-energyexpansion (made around the two bare Fermi points ±k
F), we
have seen that composite operators play a crucial role. To
start with, the composite fermion wave function itself is abound state made of an elementary fermion and a compositeof (N−1)/2 particle-hole excitations ( 4.35) that account for
excess of current needed to build up a total current J=±N.
In the bosonic case, we also find that the ±2P
F=±NkF
components of the bosonic wave function are due to the
fusion with the composite density RN(x)(4.23) made of N/2
particle-hole excitations. This is the signature that the groundstate in the spin sector is highly nontrivial. This is particularlytrue for the composite fermion since, as we see from ( 4.33),
there is no simple atomic limit where this fermion can bedefined contrarily with the baryonic SU( N) fermions ( 4.7). To
our knowledge, both the p-wave and composite fermion bound
states with SO( N) symmetry have not yet been predicted or
observed.
C. Dual AIIbound states
These bound states exist for Neven only and correspond
to the same bound-state solution as for the AIclass
(q=2,j=N/2), (4.44)
and here again the Fermi momentum PF=NkF/2 is enlarged.
Though the situation looks similar, in the present case thebound-state wave function and the underlying physics is differ-ent. The main reason for this is that the relevant local symmetryat present is SP( N) and this has important consequences. These
bound states were studied in Refs. [ 12,14,16,17] in the context
of cold fermionic atoms with hyperfine spin F=(N−1)/2.
In the following we shall review some of these previousfindings in the light of the present work.
The SP( N) symmetry has two important consequences.
First is the symmetry of the bound-state wave function whichhas to be a local SP( N) singlet which implies an s-wave pairing
of the BCS type in contrast to the p-wave wave function of the
classA
I. A local lattice operator with this property is given by
P†
i=/summationdisplay
a,bc†
a,ic†
b,iJab, (4.45)
where Jabis the SP( N) metric defined in ( 3.26). To get a
better understanding of the physics behind ( 4.45) we may use
a basis where the Nspin indices correspond to the 2 F+1
spin components of a half-integer spin F:a=(−F,..., F ).
The SP( N) metric is then proportional to the Clebsch-Jordan
coefficient projecting onto the total spin-zero subspace: Jab/similarequal
/angbracketlefta,F;b,F|00/angbracketright. Hence ( 4.45) may be seen as the s-wave BCS
wave function for a half-integer spin F.
The second consequence is the existence, on top of the
SP(N) symmetry, of a discrete local ZN/2symmetry for N>
2:
c†
a,i→e2imπ/Nc†
a,i,m=0,..., N/ 2−1. (4.46)
As discussed in [ 12,16] the latter ZN/2symmetry plays
a crucial role in the low-energy limit and the associatedexcitations are related to that of generalized two-dimensionalZ
N/2Ising models [ 41]. In a way similar to that for the
Ising model, these models display a two-phase structure: anordered phase where the Z
N/2is spontaneously broken, and
a disordered phase where it is not. Accordingly there existN/2−1, mutually nonlocal, order and disorder parameters
σ
kandμk,k=1,..., N/ 2−1, such as in the ordered phase
/angbracketleftσk/angbracketright/negationslash=0 and /angbracketleftμk/angbracketright=0 and in the disordered phase /angbracketleftσk/angbracketright=0
and/angbracketleftμk/angbracketright/negationslash=0. These operators are of scaling dimensions
dk=2k(N−2k)/[N(N+4)]. On top of these spin fields,
theZN/2conformal field theory (CFT) possesses neutral
fields, /epsilon1j(j=1,..., [N/4]), with scaling dimensions dj=
4j(j+1)/(N+4), which are the thermal operators of the
theory.
The important point with which we are concerned here
is that the ZN/2degrees of freedom have their own energy
scale, or gap m, which is independent of the SP( N) oneM.
In the generic situation m/negationslash=Mand a faithful description
of the physics involved in this system requires a detailedunderstanding of the interplay between both Z
N/2and SP( N)
degrees of freedom. This was done in Ref. [ 16]u s i n gt h e
CFT embedding SU( N)1∼SP(N)1×ZN/2where the ZN/2
fluctuations are captured by the parafermionic CFT introduced
in Ref. [ 41]. Without loss of generality, we shall consider here
the case where M/greatermuchmand integrate out the SP( N) degrees
of freedom. The extension to the M∼mcan be done using
the results of Ref. [ 16] and does not change qualitatively our
results.
In the continuum limit the s-wave pairing operator ( 4.45)
expresses in terms of the first order parameter σ1of the ZN/2
125106-12BOUND-STATE DYNAMICS IN ONE-DIMENSIONAL . . . PHYSICAL REVIEW B 95, 125106 (2017)
Ising model
P†(x)∼σ1ei√4π/N/Theta1 c, (4.47)
withP†(x)=P†
i/a0, while the density operator is given only
in terms of the charge field
n(x)∼/radicalbig
N/π ∂ x/Phi1c. (4.48)
Given these results, two remarks are in order. First as ( 4.45)
is parity invariant, ( 4.47) has to be so. Therefore, as /Theta1cis
invariant under P,σ1has to also be parity invariant which
is indeed the case [ 41]. Second, as the s-wave pairing term
(4.45) is not invariant under the ZN/2symmetry ( 4.46), the
mere existence of the bound state ( 4.45) requires the ZN/2
symmetry to be spontaneously broken and hence the ZN/2
Ising model to be in its ordered phase with /angbracketleftσ1/angbracketright/negationslash=0.
Composite density . As with the AIclass of bound states,
higher harmonics at 2 mNk Fare missing to this order and
have to be generated by some composite density operator. Therelevant composite density operator in the present case can beobtained following the strategy of the preceding subsectionby taking the dual under A
IIof the charge Q=NandJ=
0S U (N) singlet operator ( 4.20). Doing so, we find
QN(x)=eiNkFxQNkF(x)+e−iNkFxQ−NkF(x), (4.49)
where Q±NkFexpresses in terms of the elementary fermions
as
QNkF(x)/similarequal/epsilon1a1...aN/2b1...bN/2ψ†
a1,L...ψ†
aN/2,Lψc1,R
...ψcN/2,RJc1b1...JcN/2bN/2, (4.50)
andQ−NkFis obtained with the change L↔R.U s i n gt h e
results of Ref. [ 16] and averaging over the SP( N) degrees of
freedom we obtain in the limit M/greatermuchm
QN(x)∼/epsilon11cos (√
πN/Phi1 c+NkFx), (4.51)
where /epsilon11is the first thermal operator of the ZN/2Ising
models which is even under P. Using the OPE [ 41]/epsilon11(z,¯z)·
σ1(w,¯w)∼σ1(z,¯z) we find, following the steps of the preced-
ing subsection, the same expansion for both the bound-state
wave function /Psi1†
Band for the bound-state density ρas in the
p-wave case ( 4.29), (4.30) with coefficients
α0/similarequal/angbracketleftσ1/angbracketright∼m2(N−2)/N(N+4),α ±2/similarequalλα0,
β2=−β−2∼iλa0
|a0|/angbracketleft/epsilon11/angbracketright∼iλa0
|a0|m8/N(N+4).(4.52)
The above predictions are in agreement with the results,
obtained by extended QMC and DMRG calculations, on a1D lattice model with spin-3 /2 fermions ( N=4) [14,17]: at
quarter filling ( ¯ ρ=1/4) an extended phase with deconfined
s-wave BCS pairs ( 4.45) together with gapped single-particle
excitations was shown to exist. In addition, density fluctuationswith wave vector 2 P
F=πwere clearly observed in a wide
range of parameters, a result which is consistent with abound-state density ρ
BS=1/2 when ¯ ρ=1/4,N=4, and
q=2.
D. Dual AIIIbound states
These are the last types of generic bound states. As
discussed previously, they have the same quantum numbersas the baryonic states with SU( N) symmetry ( 4.5)
(q=N,j=1) (4.53)
andPF=kF. However, the duality ( 3.28) is still nontrivial
and provides for an internal structure of the bound states. Thisis the manifestation of the fact that the local symmetry groupassociated with the duality class A
IIIis not SU( N) but rather,
from ( 4.4),G/bardbl=S(U(p)×U(N−p)). Therefore, one may
naturally anticipate that they are made of a bound state ofboth SU( p)-singlet and SU( N−p)-singlet baryons. In order
to shed light on the physics that hides behind ( 3.28), it is useful
to first consider the density per spin or species n
a(x)=ρa,i/a0
na(x)=∂xφa/√π+(/Phi11)a,ae(2ikFx+i√4π/N/Phi1 c)+H.c.,
(4.54)
where ( /Phi11)a,aare the diagonal components of the SU( N)1pri-
mary operator transforming in the fundamental representationof SU( N). When averaging over the spin degrees of freedom
in the/hatwiderSU(N) ground state of ( 3.22) with duality class A
IIIwe
make use of ( 4.15) with
/angbracketleft(/Phi11)a,a/angbracketright|/hatwiderSU(N)=/angbracketleft(/Phi11Ip)a,a/angbracketright|SU(N), (4.55)
where Ipis the diagonal matrix defining the duality transfor-
mation ( 3.28). Using SU( N) invariance we find
/angbracketleftna(x)/angbracketright|/hatwiderSU(N)=∂x/Phi1c/√
Nπ+(β2e(2ikFx+i√4π/N/Phi1 c)+H.c.),
a=(1,..., p ),
=∂x/Phi1c/√
Nπ−(β2e(2ikFx+i√4π/N/Phi1 c)+H.c.),
a=(p+1,..., N ),(4.56)
where β2/similarequalγ/Delta11−1/N.The latter result shows that the 2 kF
components of the density waves of the pspecies or spins,
labeled a=(1,..., p ), are out of phase from those of the re-
maining N−pones, labeled a=(p+1,..., N ). Therefore,
the two density profiles are shifted by a distance
x0=π/2kF. (4.57)
Considering now the total density
ρ(x)=/summationdisplay
a/angbracketleftna(x)/angbracketright|/hatwiderSU(N), (4.58)
we find, upon rescaling the charge fields, the same expansions
as in the SU( N) baryonic case ( 4.9) with coefficients at ±2kF:
β2/similarequal(2p−N)γ/Delta11−1/N,β −2=−β2. (4.59)
We notice that these coefficients vanish when p=N/2(N
even) due to the πphase shift. This effect is not expected to
survive corrections due to symmetry-breaking operators unlessthe system possesses an additional Z
2symmetry interchanging
the two sets a=(1,..., p ) and a=(p+1,..., N ). In any
case, one may also define a relative density between the twosets, which reads (in an obvious notation) δρ(x)=ρ
p(x)−
ρN−p(x), that exhibits ±2kFoscillations.
From the above discussion we are naturally led to look after
a bound state made of two SU( p) and SU( N−p) singlets
separated by a distance x0. With the notation of ( 4.7) let us
consider now the wave function
/parenleftbig
Bp
N/parenrightbig†(x)=B†
p(x)B†
N−p(x+x0). (4.60)
125106-13P. AZARIA PHYSICAL REVIEW B 95, 125106 (2017)
For not too small densities ¯ ρ, in which case x0is of order the
lattice spacing a0, one may use the low-energy expansion ( 2.1)
and average over the spin degrees of freedom in the /hatwiderSU(N)
ground state of class AIII. As result we find for the bound-state
wave function
(/Psi1p)†
B/F=/angbracketleft(Bp
N)†(x)/angbracketright|/hatwiderSU(N)(4.61)
the same expansion ( 4.7)a sf o rt h eS U ( N) baryons with, up
to a phase, the same coefficients αp
¯J=αN
¯J. We notice at this
point that one could also have defined the bound state ( 4.60)
at another value of the relative distance, y/negationslash=x0=π/2kF,
between the two SU( p) and SU( N−p) baryons in ( 4.60).
In general, the corresponding amplitudes αp
¯J(y) are nonzero
but the |αp
¯J(y)|are maximal at y=±x0, a result which is
consistent with the behavior of the density waves ( 4.56). The
baryonic wave function ( 4.60)o r( 4.61) might be even or odd
under the reflexion x0→−x0as
B†
p(x)B†
N−p(x−x0)=(−1)N−pB†
p(x)B†
N−p(x+x0).(4.62)
With these results at hand one may now draw the following
physical picture: the bound states ( 4.60) may be seen as
symmetric or antisymmetric pairs of baryonic SU( p) and
SU(N−p) singlets. These pairs might be bosons or fermions
depending on the parity of N. In the fermionic case, i.e., when
Nis odd, the pair is made of a boson and a fermion. When N
is even the pair is bosonic and may consist of two charged p
andN−pbosons ( peven) with a symmetric wave function,
or fermions ( podd) with an antisymmetric wave function.
Until now we made the assumption that the density per
spin ¯ρis not too small so that x0∼1/2¯ρis of order of
the lattice spacing. When ¯ ρ/lessmuch1 (which corresponds to the
strong interaction regime) we might expect the two SU( p) and
SU(N−p) singlets to be weakly bounded over a separation δx
such as kFδx/lessmuch1. Although we have no general proof, in this
regime, we expect the pairs to be unstable toward decoupling,for example due to a repulsive interaction between the two
SU(p) and SU( N−p) baryons. This is actually what has
been demonstrated [ 18] in the simplest case of N=3 and
p=2 where, at small enough densities, a trionic bound state
made of a F=1/2 BCS pair and a single fermion was found
to be unstable toward decoupling upon switching on a smallrepulsive interaction between them. It is beyond the scope ofthe present work to elaborate on the general case.
V . INCOMPRESSIBLE PHASES
In the preceding sections we have provided for a description
of the low-energy physics of generic Hamiltonians of the type(1.1) in terms of the bound state that are stabilized by the
opening of a spin gap /Delta1. Once a bound-state solution of
(2.12) is given in terms of ( q,j), the low-energy physics at
energy scales much smaller than the spin gap is captured by aLuttinger liquid Hamiltonian with momentum scale P
F=jkF
(j=N/q ). Equipped with this result, it is natural to look at the
possible instabilities of such a state in the regime E/lessmuch/Delta1.I n
the charge sector of the theory the most important instabilityis due to commensurability effects with the lattice and theopening of a Mott gap stabilizing an incompressible phase. Inthe following we shall relate the nature of the Mott phases to
that of the low-energy bound states we discussed above.
As is well known, the general strategy to investigate the
Mott transition is to look at small umklapp perturbations tothe Luttinger liquid state. In the framework of the bound-state Luttinger liquid we can express things in terms of the“dimensionless” charge fields ¯φand ¯θprovided one uses the
bound-state density ρ
BSas the relevant parameter that controls
the commensuration effects. To this end, we shall considersmall perturbations of the Luttinger liquid Hamiltonian
H→H+V
Mott, (5.1)
where His given in ( 2.16) and VMott is any potential
allowed by the symmetries of the problem which are, ontop of charge conservation, translational and parity invariance.Decomposing V
Mottin the basis of the vertex operators ( 2.22)
and taking into account the global U(1) symmetry associatedwith charge conservation, one finds that the allowed vertexoperators lie in the zero-charge sector ¯Q=0 and hence carry
even currents ¯J=2m. One thus has
V
Mott=/summationdisplay
m/greaterorequalslant0λm/integraldisplay
dxe−2im√π¯φ(x)+H.c. (5.2)
The constraint imposed by translational invariance on the
lattice arises, after noticing that each term in the sum ( 5.2)
carries a momentum Pm=2mPF(PF=πρBS), from the
conservation of momentum up to a lattice reciprocal vector ≡
2nπ. This imposes the commensurability condition 2 mPF=
2nπwhich reads in terms of the bound-state density
ρBS=n
m, (5.3)
or in terms of the bare density
¯ρ=n
m1
j. (5.4)
Keeping the most relevant term in the expansion ( 5.2)
compatible with the commensurability condition ( 5.3), we are
led to write the effective Hamiltonian describing the Motttransition for commensurate bound-state fillings as
H=/integraldisplay
dx/braceleftbiggu
2/bracketleftbigg1
ν(∂x¯φ)2+ν(∂x¯θ)2/bracketrightbigg
+λcos [2m√π¯φ(x)+η]/bracerightbigg
, (5.5)
where λis a nonuniversal coupling and ηis a phase. The
last constraint on ( 5.2) comes from the parity symmetry,
P:VMott→VMott, which should fix the phase η. The latter
depends on how the vertex operators e−2im√π¯φ(x)transform
under parity which we find a nontrivial issue for a generalbound-state Luttinger liquid. We shall come back later tothis problem when focusing on the particular cases of integerbound-state densities where ηplays a crucial role.
Fractional bound-state fillings . Let us first focus on generic
fractional fillings, i.e., when ( n,m) are coprime integers. The
physics behind ( 5.5) is well known [ 3]. When νm
2/lessorequalslant2t h e
cosine term becomes relevant, a gap opens in the charge sector,and the system becomes an insulator. Translational symmetry
125106-14BOUND-STATE DYNAMICS IN ONE-DIMENSIONAL . . . PHYSICAL REVIEW B 95, 125106 (2017)
on the lattice which reads in terms of the bosonic field
¯φ(x)→¯φ(x)+PF/√π (5.6)
is spontaneously broken leading to an m-fold-degenerated
ground state. The gapped elementary excitations are solitonsor kinks that interpolate between two ground states and havea fractional charge
Q
s=q
m. (5.7)
What we just described is similar to what happens in a one-
species problem provided one uses as the relevant physicalquantity the bound-state density ρ
BS=¯ρj/q rather than the
species density ¯ ρ. The fact that it is ρBSand not ¯ ρthat controls
commensurability effects with the lattice has an importantconsequence for integer bound-state densities and leads tonew physics.
Integer bound-state fillings . Let us now consider the case
of integer bound-state densities:
ρ
BS=n, (5.8)
which implies m=1i n( 5.3) and ( 5.5). Since ρBS=j¯ρsuch
a situation can only occur when j> 1 (this is due to Pauli
principle that requires ¯ ρ< 1). This situation can therefore
only happen for bound-state solutions ( q,j) where the Fermi
momentum is enhanced, i.e., PF=jkF>kF. This is only
possible when the number of species N> 2. In these cases,
translation symmetry ( 5.6) remains unbroken in the insulating
or Mott phase and the ground state is not degenerate. Thisopens the interesting possibility that some of these insulatorsmay be topological insulators.
Charge edge states and generic bound states
The topological character of these insulating phases rely
on the possible existence of zero-energy modes (ZEMs), oredge states, in the problem [ 42]. At the level of this work,
where we focus on the instability of the bound-state Luttingerliquid, one may only address the possible existence of ZEMsin the charge sector and can gain no information on whathappens in the spin sector. Even in this case the situation iscomplex since, as we shall see, the existence of charge ZEMsultimately relies on the phase ηin (5.5) and hence on the way
the vertex operators transform under the parity symmetry P.
For a general bound-state Luttinger liquid, as said above, wefind it a difficult problem. However, this issue can be solvedfor the generic bound-state solutions we have discussed inthe previous section. Out of the five types of generic boundstates, the constraint of an integer bound-state density ( 5.8)
can be possibly realized only with the classes A
IandAIIfor
which ( q=2,j=N/2) or (q=1,j=N). The relevant Mott
potentials in these cases are given by the composite densities
RN(x),R2N(x), and QN(x)o fE q s .( 4.23), (4.42), and ( 4.51).
At integer bound-state density these composite fields are notoscillating, charge neutral, and parity invariant. On top of that,as they all carry momentum ±2P
Fthey identify with the
operators with the smallest scaling dimension in ( 5.2). One
then finds for Neven in both AIandAIIclasses
VMott/similarequal/integraldisplay
dx/angbracketleftRN(x)/angbracketright,/integraldisplay
dx/angbracketleftQN(x)/angbracketright, (5.9)and for Nodd in the AIclass
VMott/similarequal/integraldisplay
dx/angbracketleftR2N(x)/angbracketright, (5.10)
where /angbracketleft.../angbracketrightdenotes the average over the spin degrees of
freedom in the corresponding dual ground states. Uponrescaling the charge fields according to the “dimensionless”basis ( 2.15) we finally end up with two different types of
effective Hamiltonians
H
B=H+g/integraldisplay
dxcos (2√π¯φ), (5.11)
HF=H+iγ g/integraldisplay
dxsin (2√π¯φ), (5.12)
where HBis the effective Hamiltonian for bosonic p-wave or
s-wave charged q=2 pairs for the classes AIandAIIand
the Hamiltonian HFdescribes the charge q=1 composite
fermions of class AI. The coupling constant gis, from
Eqs. ( 4.24), (4.51), and ( 4.41), proportional to /angbracketleftWN(x)/angbracketright∼
/Delta1N/4,/angbracketleft/epsilon11/angbracketright∼m8/N(N+4)for the bosonic bound states of class
AIandAIIand for composite-fermion bound state of class
AI,g/similarequalcst. The two Mott potentials, in the bosonic and
fermionic cases, are different as they involve cos and sinfunctions of the charge field ¯φ. They essentially differ in the
way the parity symmetry Pis realized. As under P:¯φ→− ¯φ
andγ→γ
∗=−γboth Mott terms are two independent
P-invariant potentials. As far as bulk properties are concerned,
this difference has no important consequences but wheninvestigating boundary properties , as the existence of possible
edge states, it is crucial.
Let us now consider the system in the semi-infinite geom-
etry [0 ,∞[ with an open boundary condition (OBC) at x=0.
To get some insights let us focus on the Luther-Emery point atwhich the Luttinger parameter ν=1 and both Hamiltonians
(5.11) and ( 5.12) can be expressed in terms of that of free
massive fermions. Indeed introducing the chiral fermionicoperators
/Psi1
R(L)/similarequalexp−i√π(¯θ∓¯φ) (5.13)
one may rewrite ( 5.11) and ( 5.12) as a 1D Dirac Hamiltonian
HB(F)=−/integraldisplay∞
0dx/Psi1†hB(F)/Psi1, (5.14)
hB(F)=iσ3∂x+mσ 2(1), (5.15)
where σa=1,2,3are the Pauli matrices, /Psi1is the two-component
spinor
/Psi1=/parenleftbigg
/Psi1R
/Psi1L/parenrightbigg
, (5.16)
andm=−πgis a mass parameter [ 53]. The fermionic
operators ( 5.13) have different physical origins in both the
bosonic and fermionic cases. While in the composite fermioncase described by H
F,(5.13) are allowed eigenstates of the
Luttinger liquid, in the bosonic case described by HBthey
are not. In this case, the fermions ( 5.13) are rather Laughlin
quasiparticle (at ν=1) states [ 4] that span the zero-charge
sector of the Luttinger liquid spectrum and always occur inparticle-hole pairs.
125106-15P. AZARIA PHYSICAL REVIEW B 95, 125106 (2017)
The 1D Dirac Hamiltonian possesses (for suitable boundary
condition) a zero-energy solution /Psi10(x) localized at the
boundary [ 40]a tx=0. As the mass terms in ( 5.15)d i f f e r
for both bosonic and composite fermionic bound states thelocalized ZEM wave functions are different in both cases. Forthe composite fermion bound states we have
/Psi1
0F(x)=/radicalbig
|m|e−|m|x/parenleftbigg
1
isgn(m)/parenrightbigg
(5.17)
while for bosonic bound states
/Psi10B(x)=/radicalbig
|m|e−|m|x/parenleftbigg
1
−sgn(m)/parenrightbigg
. (5.18)
The question is now whether such states exist for the lattice
model with open boundary conditions. To see this, let us asusual modelize the open boundary condition on the lattice byc
a,i=0=0,a=(1,..., N ),which implies for the continuum
fermions
/Psi1L,a(0)+/Psi1R,a(0)=0 (5.19)
for each species. We immediately find the corresponding
boundary conditions on the rescaled fields
¯φ(0)=N√π
2q, (5.20)
and hence on the Dirac spinors for both the charge q=1
composite fermion and charge q=2 bosonic s-wave or p-
wave bound states
q=1:/Psi1R(0)=−/Psi1L(0), (5.21)
q=2:/Psi1R(0)=e−iNπ/ 2/Psi1L(0). (5.22)
We now arrive at the important conclusion that in the composite
fermion case there is no massless edge state localized at theboundary x=0 since the ZEM solution of the Dirac equation
(5.17) does not match with the OBC ( 5.21). In contrast, for
bosonic bound states, a ZEM solution may exist depending onthe sign of the mass m.F r o m( 5.18) one finds that for N/2e v e n
andm< 0 and for N/2 odd and m> 0 the OBC on the lattice
(5.22) is compatible with ( 5.18). When such a ZEM exists, the
ground state is to be doubly degenerated corresponding to thepresence of a fractional charge [ 43]±Q
edgeat the edge. As the
bound state charge is q=2 the charge at the edge is
Qedge=q/2=1 (5.23)
in units of the elementary fermion charge. In a system with
open boundary conditions at both ends [0 ,L], where Lis
the system size, we may expect, by symmetry, a fourfolddegeneracy (to the e
−|m|Laccuracy). Thus, as far as the
charge degrees of freedom are concerned, we find a linkbetween the nature of the low-energy bound states and thepossible topological nature of the associated insulating phases.Although all the discussion on the existence of the ZEMs wasmade at the Luther-Emery point, we expect our results to holdqualitatively when one departs from ν=1. The reason for
this is that a value of ν/negationslash=1 reflects the interaction between
the fermions which, we expect, only affect the bulk properties[54]. With this said two remarks are in order.
First, the fact that a given model exhibits either p-wave or s-
wave pairings of the type A
IorAIIis not a sufficient conditionfor it to become a topological insulator at integer bound-state
density; at issue is the sign of the mass term in ( 5.15) which
is model dependent. More importantly even when the ZEMexists, i.e., when m(−1)
N/2<0, we have no general proof
that time reversal and either the SO( N)o rS P ( N) symmetries
are enough to protect the charge edge states [ 48]. On the basis
of the results presented so far one may only state that theexistence of either p-wave or s-wave pairings of the type A
I
orAIIat integer bound-state densities are necessary conditions
for the existence of charge edge states.
Second, the above discussion focuses only on edge modes
in the charge sector. It could well be that edge modes existin the spin sector so that the above analysis does notallow
us to conclude about the total degeneracy of the groundstate. In particular, the absence of edge states in the chargesector does not imply that a given system is not a topologicalinsulator. This is particularly true for the composite fermioncase. Though we certainly believe that there are no chargeedge states in these systems, there exists the possibility thatZEMs in the spin sector may be stabilized in the Mott phase.
In this respect, in the simplest case of N=3 at the filling
¯ρ=1/3, preliminary investigations [ 47] on a particular SO(3)-
invariant fermionic model that display composite fermions aslow-energy excitations may exhibit spin-1 /2 ZEMs at each
end of an open chain. We hope to come back soon to this topicin a forthcoming publication [ 47].
The existence of charge edge states was first predicted
in a one-dimensional lattice bosonic system with extendedinteractions [ 44]. In a very nice series of works, Nonne
and co-workers [ 45,46] further demonstrated the existence
of charge edge states in a system of spin F=3/2 fermions
with SP(4) symmetry at half filling, i.e., ¯ ρ=1/2. In these
phases, called Haldane insulators [ 44], the charge degrees of
freedom are described by an effective spin S=1( w h i c ht h r e e
components describe states with zero, one, and two boundstates); the topological order is similar to the spin-1 Haldanechain, with a spin S=1/2 localized at each edge. When trying
to see how these results fit with our prediction, we face theproblem that they were obtained at half filling where there is
no spin-charge separation and therefore our approach does notstrictly apply. If we assume though that it does, thanks to theSP(N) symmetry involved in these studies, the relevant bound
states are s-wave pairs belonging to the A
IIclass with q=2.
At half filling, edge states in the charge sector are predictedwhen the bound-state density is an integer which, from ( 5.4),
implies N/4 to be integer. Even then, the issue depends on the
sign of the mass term min (5.15), so that for a given model
both a trivial and a topological insulator may be stabilizeddepending on the mass parameter. This is exactly what hasbeen first shown to happen in Refs. [ 45,46]. Though we find
this agreement encouraging, it would be more satisfactory tocheck our predictions to inquire whether these charge edgestates exist in these systems for fillings other than one-half.For instance, our analysis opens the possibility of charge edgestates for N=6, or spin 5 /2 fermions, at the filling ¯ ρ=1/3.
VI. CONCLUSIONS AND OPEN QUESTIONS
In this work we have provided for a description of the
low-energy physics of interacting multispecies fermions in
125106-16BOUND-STATE DYNAMICS IN ONE-DIMENSIONAL . . . PHYSICAL REVIEW B 95, 125106 (2017)
terms of the bound states that are stabilized in these systems
by the opening of a spin gap. We focused essentially on themassless charge degrees of freedom and on the associatedbound-state Luttinger liquid states. We have found that aconsistent bound-state Luttinger liquid state requires boththe charge qand current jquantum numbers, defining its
zero-mode spectrum, to satisfy the constraint qj=Nor
equivalently qP
F=NkFfor local bound states. The latter
condition may be viewed as some form of the Luttingertheorem [ 23,24]. Indeed assuming from the outset a gapless
phase, the Luttinger liquid state, the equation qP
F=NkF
shows that there exists a massless mode at 2 PF=2jkF
where j=N/q for local bound-state solutions. Among the
solutions of the latter equation a small finite subset (i.e., fivetypes) of generic bound states were characterized in terms of
the elementary fermions. In terms of the charge and currentquantum numbers they are ( q,j)=(N,1) for both SU( N)
baryons and A
IIIclass, ( q,j)=(2,N/2) for both AIandAII
classes when Nis even, and finally ( q,j)=(1,N)f o rt h e AI
class when Nis odd. We found that these are likely to be
stabilized in systems which display an enlarged symmetryat low energies and are associated with emergent dualitysymmetries in the spin sector. Our results are in agreementwith previous findings for three types of bound states thatwere identified in the attractive SU( N) Hubbard model with
N=3 and N=4[15,18–21] as well as for SP( N) models
[12,14,16,17] relevant to describe general spin F=(N−1)/2
fermionic cold atoms. They are fermionic or bosonic SU( N)
baryonic singlets and bound states of them as well as s-wave
pairing states associated with SP( N) symmetry. An important
output of the previous studies is that these states are stableagainst small symmetry-breaking fields. On top of these boundstates we also predict two new types of generic bound stateswith O( N) symmetry: p-wave bosonic pairs when Nis even
and composite fermions for odd N. To our knowledge, the
latter bound states have not yet been observed.
Apart from these generic bound states we also predict
the possible existence of other types of bound states. WhenN/greaterorequalslant8 for example, out of the two dual solutions ( q,j)=
(8,1) and ( q,j)=(2,4) which are either SU(8) baryons or
s-wave pairing of the A
IIclass with SP(8) symmetry, there is
another solution with ( q,j)=(4,2) which is self-dual. Upon
increasing Nmore solutions can be found (not necessarily
self-dual) that are not generic bound states. These states mightbe stabilized in systems which do not exhibit a dynamical
enlarged symmetry at low energies. One way to think aboutthese bound-state solutions is to regard them as bound statesmade of generic bound states themselves. For instance, in asystem with N=qjspecies, one may build up SU( q)-singlet
baryons made of either SO( j)-singlet charge-1 composite
fermions or charge-2 p-wave pairs. The total symmetry in
such a situation is then SU( q)×SO(j). For odd jthe bound
state has a charge qand the unit of current is j. For even j
the total charge of the bound state would be 2 qand the unit of
current j/2. Other possibilities involving other combinations
of generic bound states are of course possible. The point in theabove construction is that it requires a hierarchy of scale. In theexample we just gave, the gap in the SO( j) sector should be
much greater than the one in the SU( q) one which is consistent
with the fact that the symmetry is not dynamically enlarged.More generally, one may anticipate that the bound states which
are expected to be generic in the sense of the DSE mechanismmight be the building blocks for more general bound states.
Another important result of the present work concerns
the relation between the possible existence of topologicalinsulating phases with the nature of the bound states—inparticular, the fact that it is the bound-state density ρ
BS=jρ/q
that controls commensuration effects with the lattice. Thisopens the possibility of nondegenerate Mott phases whenj>q or, equivalently, when the Fermi momentum P
F=jkF
associated with these bound states is enlarged. The fact that
these phases display topological order is a highly nontrivialproblem. We though gave arguments that, in the particularcase of the generic bound states, zero-energy edge states in thecharge sector may be stabilized for either p-wave or s-wave
bosonic bound states associated with the classes A
IandAII
whenNis even. We finally stress that zero-energy edge states
could also be stabilized in the spin sector. In particular, thisleaves open the question of the topological nature of theMott phase associated with the composite fermions at integerbound-state densities [ 47].
ACKNOWLEDGMENTS
We gratefully acknowledge illuminating discussions with
E. Altman, E. Berg, F. Cr ´epin, B. Doucot, and P. Lecheminant.
We also want to thank E. Boulat, S. Capponi, G. Roux, and A.M. Tsvelik for collaborations related to this work. The authoris grateful to A. Auerbach and the Physics Department of theTechnion for their kind hospitality during the past academicyear while parts of this work were completed.
APPENDIX: BOSONIZATION CONVENTIONS
In this appendix we discuss our bosonization conven-
tions. We recall the bosonized expressions of the elementaryfermions
c
a,i/√a0=κa√
2π[e−i(kFx+2√πφa,L)+ei(kFx+2√πφa,R)],(A1)
where the κa=1,...,N anticommuting Majorana fermions,
{κa,κb}=2δab, ensure the anticommutation between fermions
of different species. The bosonic fields φa,Landφa,Rdo not
commute and their commutators are given by [ φa,L,φb,R]=
∓iδab/4 in order to ensure the anticommutation between L
andRfermions of the same species. The above commutators
emerge in general through the quantity
γ=e−2π[φa,L,φa,R], (A2)
which takes the values γ=±i. We find it important to keep
it explicit in the bosonization expressions in order to discussparity issues as under P:
φ
a,L↔−φa,R,γ→γ∗=−γ. (A3)
In this paper we make use of a basis in which spin and charge
degrees of freedom are described by charge bosonic fields /Phi1c
and/Theta1cas well as N−1 component spin bosonic fields /vector/Phi1and
125106-17P. AZARIA PHYSICAL REVIEW B 95, 125106 (2017)
/vector/Theta1such that
φa=1√
N/Phi1c+/vectorωa·/vector/Phi1,
θa=1√
N/Theta1c+/vectorωa·/vector/Theta1, (A4)
where the Nvectors /vectorωa=1,...,N are not independent and satisfy/summationtextN
a=1/vectorωa=0 together with /vectorωa·/vectorωb=δab−1/N. With these
definitions the parity symmetry acts on the spin fields as
P:/vector/Phi1→−/vector/Phi1,/vector/Theta1→/vector/Theta1, γ→γ∗. (A5)
The SU( N)1currents IA
L(R)of Eqs. ( 3.9) can be expressed
in terms of the spin fields /vector/Phi1and/vector/Theta1. Among them, the N−
1 Cartan generators of SU( N) take a simple form. In each
chirality sector one has
/vectorhL(R)=∂x/vector/Phi1L(R)/√π=N/summationdisplay
a=1/vectorωa∂xφa,L(R)/√π (A6)
from which one deduces that
/vectorh=/integraldisplay
dx(/vectorhL+/vectorhR)=N/summationdisplay
a=1/vectorωaQa (A7)
as given in ( 3.15).
SU(N)1primary operators .
Following Affleck [ 49] we define the SU( N)1primary
operators /Phi1(m)as
ψ†
a1,L...ψ†
am,Lψb1,R...ψbm,R=/Phi1(m)
a,beim√4π/N/Phi1 c.(A8)
Using the bosonization formula ( A1)a sw e l la s( A4) one may
obtain the expression of /Phi1(m)in terms of the spin fields /vector/Phi1and
/vector/Theta1. For instance, the trace of /Phi1(m)is given by
Tr(/Phi1(m))=γm/Gamma12
m
(2π)m/summationdisplay
{/vectorλm}ei√
4π/vectorλm·/vector/Phi1, (A9)where /vectorλm=/summationtextm
j=1/vectorωajand/Gamma1m=/Pi1m
a=1κa.I n( A9)t h es u m
runs over independent permutations of the set {aj}compatible
with the antisymmetry of ( A8). These operators have the
scaling dimension dm=/vectorλ2
m=m(1−m/N ) and transform
under parity as P:T r (/Phi1(m))→Tr(/Phi1(m))∗.
Duals of the SU(N)1primary operators . These objects
appear naturally in discussing both classes AIandAIIIof
dualities. In the case of the class AIIwe find it more convenient
to rely on the CFT embedding SU( N)1∼SP(N)1×ZN/2
which is discussed in detail in Ref. [ 39] to which we refer.
BothAIandAIIIduality transformations have a simple
representation in terms of the spin fields /vector/Phi1and/vector/Theta1.F o rAI
we have
/vector/Phi1↔/vector/Theta1, γ →γ∗, (A10)
while for AIII
/vector/Phi1→/vector/Phi1+√π
2/vectorep,/vector/Theta1→/vector/Theta1−√π
2/vectorep,γ→γ,(A11)
where /vectorep=/summationtextp
j=1/vectorωj. Using the above representations we find
for the duals of Tr( /Phi1(m))
AI:T r (/hatwidest/Phi1(m))=(γ∗)m/Gamma12
m
(2π)m/summationdisplay
{/vectorλm}ei√
4π/vectorλm·/vector/Theta1, (A12)
AIII:T r (/hatwidest/Phi1(m))=γm/Gamma12
m
(2π)m/summationdisplay
{/vectorλm}ei√
4π/vectorλm·/vector/Phi1−iπ/vectorλm·/vectorep.(A13)
In the particular case of the AIduality class we find
that under the parity transformation ( A5)P:T r (/hatwidest/Phi1(m))→
(−1)mTr(/hatwidest/Phi1(m)) and that consequently the combination
γmTr(/hatwidest/Phi1(m))i sPinvariant. When m=N/2 the latter quantity
γN/2Tr(/hatwider/Phi1(N/2)) is also real. The reason is that in ( A12)t h e
sum over {/vectorλN/2}contains both configurations ±/vectorλN/2thanks to
the property/summationtextN
j=1/vectorωj=0.
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[51] Since the parametrization given in ( 2.8)a n d( 2.9)i sr a t h e r
compact this statement might not be obvious. Trivially, due tothe constraint qj=N,w h e n Nis odd both qandjare odd. In
addition if qis even then Nis even. The main point is whether
one can have Neven and qodd. To see this let us suppose N
even and qodd; then, according to the constraint qj=N,jis
necessarily even. Therefore from ( 2.9) one finds that qis even
which leads to a contradiction. Therefore qandNhave the same
parity.
[52] Notice the common factor
γin the coefficients α¯0andα±2which
reflects the odd parity of the p-wave function.
[53] Notice that the mass parameter does not depend on the cocycle
γanymore. In the composite-fermion case it is reabsorbed in the
fermions /Psi1L(R). For bosonic bound states, the cos term yields a
cocycle γupon refermionization which, as shown in Ref. [ 50],
is fixed to γ=−iby the OBC.
[54] There is though a particular value of ν=2 where the Mott
potential displays an SU(2) symmetry in the charge sector. Inthis case, as argued in Ref. [ 45], the number of edge states may
be greater than two.
125106-19 |
PhysRevB.81.115412.pdf | Convergence in quantum transport calculations: Localized atomic orbitals
versus nonlocalized basis sets
J. A. Driscoll and K. Varga
Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA
/H20849Received 14 December 2009; published 9 March 2010 /H20850
The convergence and basis set dependence of quantum transport calculations is studied using localized
atomic orbitals and nonlocalized basis sets. The nonequilibrium Green’s function formalism and ground-statedensity-functional theory are used in the calculations. Numerical examples show that the extended nonlocal-ized basis sets give more accurate results with much lower basis dimension than the atomic orbitals. Examplesalso show that a low-dimensional atomic orbital basis can also be accurate provided that the self-consistentground-state potential is accurately calculated.
DOI: 10.1103/PhysRevB.81.115412 PACS number /H20849s/H20850: 73.40. /H11002c, 85.65. /H11001h, 72.10. /H11002d, 73.63. /H11002b
I. INTRODUCTION
The convergence of energy and other physical properties
in ground-state electronic structure calculations has been in-tensively studied in the past. Benchmark calculations havebeen established and state of the art electronic structure cal-culations are validated against them. Depending on the basisstates used in the calculation there are various ways to con-trol the convergence. In plane-wave calculations
1the conver-
gence of the energy is controlled by the energy cutoff. Inreal-space grid calculations
2–6convergence can be reached
by increasing the number of grid points. Convergence ofcalculations using atomic orbitals are checked against theincrease in the number of orbitals in the basis.
The convergence of the conductance or the transmission
coefficient in quantum transport calculations
7–20is more
complicated. There is no simple minimum principle /H20849such as
the minimum of the variational energy in ground-state calcu-lations /H20850that can be used to judge the quality of the calcula-
tions. Most quantum transport calculations are based on thenonequilibrium Green’s function /H20849NEGF /H20850formalism which,
in its most common implementation, uses localized basissets. The basis set dependence of these calculations is obvi-ously a very important issue. Nevertheless, only a very fewcalculations have investigated the convergence of the trans-mission coefficient as a function of the basis size.
21,22In Ref.
21Gaussian-based atomic orbitals were used and the conver-
gence with respect to the number of Gaussian orbitals wasinvestigated for a model gold-benzeneditholate-gold struc-ture. In Ref. 22several test systems were used and both
atomic orbitals /H20849SIESTA /H20850
23and Wannier function-based ba-
sis functions were tested. The slow convergence of the trans-port properties with localized atomic orbitals is apparent inboth calculations. The main reason behind the slow conver-gence of atomic orbital-based calculations is that it is hard torepresent the rapidly oscillating extended current-carryingstates with localized orbitals centered on the atoms.
In this work the convergence of transport properties is
investigated using different basis sets. Three different basissets, /H208491/H20850localized atomic orbitals /H20849AOs /H20850,/H208492/H20850AOs augmented
with floating Gaussians, and /H208493/H20850box basis functions are
tested. The first set, the AO basis, is very popular in transportcalculations and most transport codes use this representation.These basis functions are centered at atomic positions and
give a very good description of the wave function near theatoms /H20849where they are optimized /H20850, but the AO representation
of the wave function between atoms is less flexible. Theadvantage of these basis sets is that they are localized and
this localization property can be exploited to speed up largescale electronic structure calculations. On the other hand,these basis sets cannot be systematically enlarged in a simpleway and the results are subject to basis set errors. To improvethe description of the electron scattering wave function in thespace between atoms one can augment the atomic orbitals inthe interstitial region with suitably chosen basis functions. Inthis work we add a grid of “floating” Gaussian functions toimprove the representation of the wave function between at-oms. The third set, the “box basis functions” consists of basisfunctions that are obtained by diagonalizing the Hamiltonianin an appropriately chosen region /H20849box /H20850. Unlike the AOs
these basis functions are not tied to atomic positions and areproven to be an efficient representation for electronic struc-ture and transport calculations.
24
To calculate the transport properties the NEGF formalism
is used in the density-functional theory /H20849DFT /H20850
framework.7–19In the NEGF implementation one has to have
basis states that do not connect the left and right leads and itis also advantageous to have basis functions that only con-nect the nearest periodically repeated layers in the lead. Thebasis function sets employed in this work satisfy both ofthese conditions.
The transport calculation has two steps. First the ground-
state density and potential are calculated self-consistentlyand then, using this self-consistent potential, the NEGF for-malism is used to calculate the transmission as a function ofenergy. The accuracy of both of these steps depends on thebasis set chosen. The basis set dependence of the ground-state DFT calculation has been extensively researched in thepast.
25,26The main objective of the present work is to inves-
tigate the basis set dependence of the second step, the trans-mission calculation, which consists of the calculation of theGreen’s function of the system in a suitable basis represen-tation.
The paper is organized as follows. Following the intro-
duction, Sec. IIgives a brief presentation of the transport
formalism and the definition of the basis states. NumericalPHYSICAL REVIEW B 81, 115412 /H208492010 /H20850
1098-0121/2010/81 /H2084911/H20850/115412 /H208497/H20850 ©2010 The American Physical Society 115412-1examples are presented in Sec. IIIand a summary is given in
Sec. IV.
II. FORMALISM
In this section we briefly describe the calculation of the
transmission coefficients and describe the definition of thebasis function sets used in the calculations.
A. Calculation of the Transmission Coefficient
In the NEGF framework the system is divided into left
and right leads and a device part as shown in Fig. 1.
The leads consist of periodically repeated layers /H20849boxes /H20850.
The Hamiltonian is defined as
HKS=−/H60362
2m/H116122+VA/H20849r/H20850+VH/H20851/H9267/H20852/H20849r/H20850+Vxc/H20851/H9267/H20852/H20849r/H20850, /H208491/H20850
where VAis the Coulomb potential of the atomic nuclei, VH
is the Hartree potential, and Vxcis the exchange-correlation
potential.
Each region is represented by a set of basis functions /H9023L,
/H9023C, and/H9023R. Here the index X=L,C,Rrefers to the fact that
these basis functions are situated in the X=L,C,R/H20849left, cen-
ter, and right /H20850regions. As has been discussed, only neighbor-
ing regions overlap, that is
/H20855/H9023iL/H20841/H9023jC/H20856/HS110050 /H20855/H9023iR/H20841/H9023jC/H20856/HS110050, /H208492/H20850
but there is no overlap between the leads’ basis functions
/H20855/H9023R/H20841/H9023L/H20856=0 . /H208493/H20850
In this basis representation the Hamiltonian and the overlap
matrices of the left-lead—device—right-lead system, underthe assumption that there is no interaction between the leads,takes the form
H=
/H20898HLHLC 0
HLC†HCHRC†
0HRC HR/H20899O=/H20898OLOLC 0
OLC†OCORC†
0ORC OR/H20899,
where HL/H20849OL/H20850,HC/H20849OC/H20850, and HR/H20849OR/H20850are the Hamiltonian
/H20849overlap /H20850matrices of the leads and the device. HLC/H20849OLC/H20850and
HRC/H20849ORC/H20850are the coupling matrices between the central re-
gion and the leads defined as
HijXY=/H20855/H9023iX/H20841HKS/H20841/H9023jY/H20856OijXY=/H20855/H9023iX/H20841/H9023jY/H20856. /H208494/H20850
By defining the self energies of the leads /H20849X=L,R/H20850as
/H9003X/H20849E/H20850=i/H20851/H9018X/H20849E/H20850−/H9018X†/H20849E/H20850/H20852, /H208495/H20850
where
/H9018X/H20849E/H20850=/H20849EO XC−HXC/H20850gX/H20849E/H20850/H20849EO XC−HXC/H20850, /H208496/H20850
and where
gX/H20849E/H20850=/H20849EO X−HX/H20850−1, /H208497/H20850
is the Green’s function of the semi-infinite leads and defining
the Green’s function of the central regionGC/H20849E/H20850=/H20851EO C−HC−/H9018L/H20849E/H20850−/H9018R/H20849E/H20850/H20852−1, /H208498/H20850
the transmission probability is given by7
T/H20849E/H20850=Tr/H20851GC/H20849E/H20850/H9003L/H20849E/H20850GC†/H20849E/H20850/H9003R/H20849E/H20850/H20852. /H208499/H20850
In this equation the transmission coefficient, T/H20849E/H20850,i se x -
pressed by the Green’s functions of the device and the semi-infinite leads. The Green’s function of the semi-infinite leads,g
X, is calculated by the decimation27technique.
B. Basis Functions
In the following two subsections we briefly introduce the
basis functions used in the calculations.
1. Localized orbitals
The atomic orbitals are defined as
/H9278k/H9251AO/H20849r−Rk/H20850=/H20858
jcpj/H9272lm/H9263j/H20849r−Rk/H20850/H20849 10/H20850
where Rkis the position of atom kand/H9251is an index for
/H20849p,l,m/H20850. Gaussian functions are used in the expansion of the
atomic orbitals
/H9272lm/H9263/H20849r/H20850=/H208734/H92664l
/H208492l+1/H20850!!/H208741/2
/H20849/H20881/H9263r/H20850l/H208732/H9263
/H9266/H208743/4
e−/H9263r2Ylm/H20849rˆ/H20850.
The linear combination coefficients cpiare determined by
solving the Kohn-Sham equation for a single atom confinedin a sphere of radius R
cutoff,28–31and the AOs vanish beyond
that radius. AOs similar to these are very popular in elec-tronic structure and transport calculations. The AO basis de-pends on the maximum angular momentum l
max, the cut-off
radius Rcutoff, and the maximum number pmaxof radial parts
for each value of the angular momentum l. By increasing lmax
andpmaxthe accuracy of the calculations improves but the
computational time also increases, losing the advantages ofthe localized basis states. The accuracy of the calculationsalso depends on the cutoff radius and shape of the AOs. Forthe light elements used in the present work R
cutoff =5 Å and
lmax=2 give well-converged results.25,30
AOs have proven to be accurate in describing ground-
state properties. The AO representation, however, is lessflexible for extended continuum wave functions that describetunneling and current-carrying states. One way to alleviatethis is to augment the localized AO with a basis set thatflexibly represents the wave functions in the interatomic re-gions. For this purpose we introduce a grid of Gaussians withl=0,
/H9278/H9252grid/H20849r/H20850=/H927200/H9263/H20849r−sk/H20850, /H2084911/H20850
where skis a point on a grid where the Gaussian is centered
and/H9252is an index for /H20849/H9263,sk/H20850. The grid points skand the
Gaussian width parameter /H9263are selected in such a way that
the overlap between neighboring functions is small and thereis no linear dependence in the basis. The accuracy of thecalculations can be increased by increasing the number ofgrid points for appropriately chosen
/H9263parameters. Grids of
Gaussians are often used as a basis; a recent illustration ofJ. A. DRISCOLL AND K. V ARGA PHYSICAL REVIEW B 81, 115412 /H208492010 /H20850
115412-2the applicability of such functions to represent unbound elec-
trons in nanoelectronics can be found in Ref. 32.
Both of these basis sets are localized in a radius around
their centers. In the next subsection we introduce a basis setthat is not tied to the atomic positions and is nonzero in alarger spatial region.
2. Box basis functions
In this subsection we define box basis functions as an
alternative to the AO representation. These basis functionsare defined in each box /H20849see Fig. 1/H20850. The jth basis function in
theith box is expanded in terms of a tensorial product of
Lagrange basis functions
33as
/H9278jbox,i/H20849r/H20850=/H20858
l=1Mxi
/H20858
m=1My
/H20858
n=1Mz
Cj,lmniLli/H20849x/H20850Lm/H20849y/H20850Ln/H20849z/H20850. /H2084912/H20850
In the xdirection, the Lagrange functions are defined on grid
points ai−h/H11021xki/H11021bi+h, where aiis the left and biis the
right boundary of box i,a s
Lni/H20849x/H20850=/H9266n/H20849x/H20850/H20881w/H20849x/H20850/H9266n/H20849x/H20850=/H20863
k=1
k/HS11005nMxx−xki
xni−xki, /H2084913/H20850
where w/H20849x/H20850is the weight function and the index iindicates
that the Lagrange function is defined in the ith box. The
Lagrange functions in neighboring boxes can overlap and theregion of overlap is determined by the parameter h. We use
the same Lagrange basis L
m/H20849y/H20850andLn/H20849z/H20850in the yandzdirections in each box. By this construction there are Mi
=Mxi/H11003My/H11003MzLagrange basis functions in box i. These
functions are used to generate the box basis functions that areused in the transport calculations.
The box basis functions
/H9278jbox,iare selected by solving the
eigenvalue problem
HiCji=EjCji, /H2084914/H20850
where Hiis the matrix of the Hamiltonian HKSin the ith box
in the orthogonal Lagrange function representation. After di-agonalization this Hamiltonian has M
ieigenstates. By using
a cutoff energy Ecutoff the lowest nieigenstates of this Hamil-
tonian are used as box basis states in the transport calcula-tions. The convergence properties and accuracy of thisapproach has been studied in Ref. 24.
III. NUMERICAL RESULTS
Two systems were used to study the convergence of trans-
port calculations. The Au-CO system is a linear chain of Auatoms to which a CO molecule has been adsorbed. This sys-tem has been studied in Ref. 22and we adopted the geom-
etry of the atoms from that source. The other system, Al-C-Al, consists of two bulk Al leads joined by a linear chain ofseven carbon atoms /H20849Fig. 2/H20850. The Al lattice constant isHC hR hLhLhL hRhR
12 3 4 56 7
FIG. 1. /H20849Color online /H20850Organization of the system into left/right
leads /H20849L/R /H20850and a central device /H20849c/H20850. The self-consistent potential is
calculated for the central region between the planes. Only a fewlayers of the leads need to be included to obtain a converged po-tential for the central region.
(a)
(b)
FIG. 2. /H20849Color online /H20850Structure of the /H20849a/H20850Au-CO and /H20849b/H20850Al-
C-Al systems. The division into lead and device regions, displayedgenerally in Fig. 1, is shown.-4 -2 0 2 4
E(eV)00.511.52T(E)
(a)
-4 -2 0 2 4
E (eV)00.511.52T(E)
(b)
FIG. 3. /H20849Color online /H20850Transmission vs relative energy for the
/H20849a/H20850Au-CO and /H20849b/H20850Al-C-Al systems using an atomic orbital basis. In
these plots, both the ground state and the conductance calculationswere performed on the same AO basis /H20849p
max=1 blue dot-dashed
line, pmax=2 red dashed line, and pmax=3 black solid line /H20850.CONVERGENCE IN QUANTUM TRANSPORT … PHYSICAL REVIEW B 81, 115412 /H208492010 /H20850
115412-34.05 Å, the distance of the C atoms from the Al surfaces is
1 Å, and the C-C distance is 1.25 Å
These are relatively simple systems but they allow us to
study the convergence properties using large basis sets. Theincreased computational cost associated with systems thatcontain more atoms would prevent us from a systematic en-largement of the bases.
In the calculations we first determine the self-consistent
potential by diagonalizing the Kohn-Sham Hamiltonian /H20851Eq.
/H208491/H20850/H20852in the region that includes three layers in the left, three
layers in the right, and the central region /H20849see Fig. 2/H20850using
periodic boundary conditions on the computational cell.
Numerical tests show that adding these three layers is
sufficient to obtain the converged self-consistent potential inthe middle region containing layers 2–3, the device and lay-ers 5–6 /H20849between the planes in Fig. 2/H20850. The self-consistent
potential obtained in this way does not change in the middleregion if further layers are included in the computationalcell. That is, this self-consistent potential is the same as it isin the infinite system containing the semi-infinite leads andthe device. Using the self-consistent potential of this middleregion one can calculate the matrix elements needed in thetransmission calculations using the basis functions definedabove.
First we calculated the transmission coefficient using
AOs. In these calculations both the self-consistent groundstate and the transmission coefficient are calculated using thesame AO basis set; that is, the transmission coefficient has atwofold dependence on the basis set.
There are three parameters that influence the convergence
behavior: R
cutoff,lmax, and pmax. Out of the three, pmax, the
number of orbitals per angular momentum, is the most sen-sible to vary and the easiest to change. We have kept l
max
=2 and Rcutoff =5 Å as these are typical choices in AO cal-culations and give good ground-state properties for these
systems. Figure 3shows the transmission as a function of
energy /H20849relative to the Fermi energy /H20850forpmax=1,2,3. Con-
sidering the value of the transmission at the Fermi energy,the calculated conductances are shown in Table I/H20849Case I /H20850.
The transmission curves obtained for different p
maxvalues
are quite different, and the convergence as a function of thenumber of basis states is slow, similar to what has been ob-served in previous calculations.
21,22For clarity Fig. 3only
shows the results up to pmax=3, as the transmission for
pmax=4 and pmax=5 is still changing; that is, convergence
has not been reached. Adding more AOs is difficult becausethe computational time becomes prohibitively large. Theslow convergence is especially noticeable in the Au-CO case.The somewhat better convergence in the Al-C-Al case is dueto the fact that the dl=2 states are included in these calcu-
lations. Without the dstates, a truncation that is often used in
transport calculations, the convergence is much worse. Theinclusion of the dstates is computationally demanding and
by omitting them, the basis dimensions would decrease toless than half of what is shown in Table I. The convergence
of the conductances in the table shows a very similar patternto what has been discussed above concerning the transmis-sion curves.
Next, we will study the effect of varying the basis just in
the transmission calculation, and leaving the ground statebasis fixed. To this end we calculate the self-consistent po-tential without employing AOs, instead using the Lagrangefunction method
33which provides an accurate self-consistent
potential. Any other approach, e.g., plane-wave basis calcu-lations could have been used for this purpose; the main pointis that the self-consistent potential has been calculated inde-pendently of the basis function sets that are to be tested inthe transmission calculations. For all subsequent calcula-TABLE I. Convergence of conductance /H20849in units of G0/H20850. The asterisk appears when the basis size was too
large to complete the calculation. NCandNLdenotes the basis dimension in the central region and in the unit
cell of the lead.
Au-CO Al-C-Al
NC NL GN C NL G
Case I pmax=1 99 27 0.244 423 162 0.868
pmax=2 198 54 0.248 846 324 0.819
pmax=3 297 81 0.208 1269 486 0.788
pmax=4 396 108 0.204 1692 648/H11569
pmax=5 495 135 0.212 2115 810/H11569
Case II pmax=1 99 27 0.185 423 162 0.739
pmax=2 198 54 0.078 846 324 0.693
pmax=3 297 81 0.039 1269 486 0.817
pmax=4 396 108 0.017 1692 648 0.874
pmax=5 495 135 0.001 2115 810 0.902
Case III 1 50 25 0.001 100 50 0.721
2 60 30 0.001 120 60 0.7553 70 35 0.001 140 70 0.7624 80 40 0.001 160 80 0.7635 100 50 0.001 180 90 0.763J. A. DRISCOLL AND K. V ARGA PHYSICAL REVIEW B 81, 115412 /H208492010 /H20850
115412-4tions, we will fix the ground-state potential to that obtained
by the Lagrange function basis. This allows us to see just theinfluence of varying the AO basis in the transmission calcu-lation.
Figure 4shows the basis set dependence of the AO-based
calculation for the fixed self-consistent potential. Comparedto the previous case, the convergence is much faster and amuch smaller basis set is sufficient to calculate accuratetransport properties. The conductance values in Table I/H20849Case
II/H20850nicely converge with the number of basis states. One can
notice that the converged values are different from thoseshown in Case I. This shows that a large part of the basis set
error in the previous case is due to the basis set dependenceof the self-consistent potential. This is mostly due to thedifference of the Fermi energies obtained by the Lagrangebasis grid and by the AO self-consistent potential calcula-tions. Due to the Fermi energy difference the curves are
-4 -2 0 2 4
E(eV)00.511.52T(E)
FIG. 5. /H20849Color online /H20850Same as Fig. 4/H20849a/H20850, except that the atomic
orbital bases used were spatially smaller.-4 -2 0 2 4
E(eV)00.511.52T(E)
FIG. 6. /H20849Color online /H20850Au-CO transmission calculation with the
Gaussian-augmented AO basis. For pmax=2 the transmission is cal-
culated with /H20849dashed red line /H20850and without /H20849solid black line /H20850the
addition of a uniform grid of Gaussian basis functions.
-4 -2 0 2 4
E (eV)00.511.52T(E)
(a)
-4 -2 0 2 4
E (eV)00.511.52T(E)
(b)
FIG. 7. /H20849Color online /H20850Convergence of the transmission using
box basis states for /H20849a/H20850Au-CO and /H20849b/H20850Al-C-Al. The curves corre-
spond to the number of box basis states as defined by sets 1 /H20849blue
dot-dashed line /H20850,2 /H20849red dashed line /H20850, and 3 /H20849black solid line /H20850in
Table I/H20849Case III /H20850. The results obtained by using larger basis sets /H208494
and 5 in Table I/H20850are identical with the black solid line within the
resolution of the figure.-4 -2 0 2 4
E (eV)00.511.52T(E)
(a)
-4 -2 0 2 4
E (eV)00.511.52T(E)
(b)
FIG. 4. /H20849Color online /H20850Transmission vs relative energy for the
/H20849a/H20850Au-CO and /H20849b/H20850Al-C-Al systems using an atomic orbital basis. In
contrast with Fig. 3, here the ground-state potentials for all curves
were the same, calculated with a Lagrange basis for high accuracy.CONVERGENCE IN QUANTUM TRANSPORT … PHYSICAL REVIEW B 81, 115412 /H208492010 /H20850
115412-5slightly shifted leading to this shift in the conductances. It is
interesting to note the strong dependence of the results onR
cutoff. Figure 5shows the same calculation as Fig. 4/H20849a/H20850us-
ing AOs with Rcutoff =3.5 Å. The smaller radius does not
significantly affect the energy of the system, but the calcu-lated transmission has substantially changed, especially inthe higher energy region.
We have also investigated the effect of augmenting the
AO with floating Gaussian states. The calculations show thatthese states do not significantly improve the convergence.For example, adding a grid of Gaussians to p
max=2 /H20849Fig. 6/H20850
does not substantially change the transmission coefficient.Adding a Gaussian grid to AOs which have a smaller cutoffradius changes the results somewhat more /H20849these results are
not shown here /H20850but it is found to be hard to optimize their
positions and widths. Moreover, although adding floatingGaussians have somewhat changed the transmission coeffi-cients, no clear convergence pattern could be found.
Next we show our results using the box basis states. The
convergence of the transmission is shown in Fig. 7and the
conductances can be seen in Table I/H20849Case III /H20850. The most
important result of this case is that the transmission con-verges rapidly and systematically as a function of the numberof basis states. One can also note that the number of basisstates needed for convergence is much less than in the AOcalculations. The converged box basis calculation is com-pared to the best AO results in Fig. 8. The agreement is good;
for a better agreement probably more AOs should be in-cluded, which as we have already emphasized, is computa-
tionally unfeasible.
Finally, we compare our results to the benchmark calcu-
lation published in Ref. 22. In the benchmark paper the
transmission functions are calculated using two differentdensity functional theory methods, an ultrasoft pseudopoten-tial plane-wave code in combination with maximally local-ized Wannier functions and the norm-conserving pseudopo-tential code SIESTA which applies an atomic orbital basisset. Figure 9compares the results of the benchmark calcula-
tion to our box basis results. The agreement is very good.
IV . CONCLUSIONS AND SUMMARY
The development of efficient quantum transport
calculations7–20is an area of very active research. These cal-
culations have yet to reach the accuracy and efficacy ofground state DFT calculations. Using the combination of thenonequilibrium Green’s function formalism and ground-statedensity-functional theory we have studied the convergenceand basis set dependence of quantum transport calculations.
Atomic orbitals are widely used in NEGF implementa-
tions, but the convergence of transport properties with re-spect to the number of basis states is slow. Examples in thispaper show that this is partly due to the basis set dependenceof the self-consistent potential. In other words, if the self-consistent potential is accurately calculated then much fewerAOs are needed in the calculation of the transport properties.The calculation of the transport coefficient is a time consum-ing part of the transport calculations because it involvesmany inversions of large matrices and this has to be repeatedfor many energy values.
The box basis orbitals lead to much faster convergence on
a much smaller basis than the AOs. The typical number ofbasis states per electron orbital is about 2–3. This signifi-cantly reduces the computational cost. Another useful prop-erty of the box basis is that by increasing the number of basisstates the Hilbert space is enlarged and the transport coeffi-cients systematically converge. In the case of AOs the trans-port coefficients always change and due to computationallimitations the same level of accuracy cannot be reached.-4 -2 0 2 4
E (eV)00.511.52T(E)
(a)
-4 -2 0 2 4
E(eV)00.511.52T(E)
(b)
FIG. 8. /H20849Color online /H20850Comparison of the results obtained by the
box basis /H20849solid black line /H20850and the AO /H20849dashed red line /H20850for /H20849a/H20850
Au-CO and /H20849b/H20850Al-C-Al.-4 -2 0 2 4
E(eV)00.511.52T(E)
FIG. 9. /H20849Color online /H20850Comparison of the box basis calculation
/H20849solid black line /H20850to benchmark results. The blue dot-dashed line is
the Wannier, the red dashed line is the AO calculation of Ref. 22.J. A. DRISCOLL AND K. V ARGA PHYSICAL REVIEW B 81, 115412 /H208492010 /H20850
115412-6The accuracy of box basis states is due to the facts that /H208491/H20850
they are spatially extended and can represent scattering statesefficiently and /H208492/H20850the box basis states are optimized for the
converged self-consistent potential.
The present calculations are restricted to zero bias volt-
age. The study of transport calculations in nonequilibriumconditions is more challenging and left for future work.
ACKNOWLEDGMENTS
This work is supported by NSF grants No. ECCS0925422
and No. CMMI0927345.
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Joannopoulos, Rev. Mod. Phys. 64, 1045 /H208491992 /H20850.
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115412-7 |
PhysRevB.98.245137.pdf | PHYSICAL REVIEW B 98, 245137 (2018)
Efficient O( N) divide-conquer method with localized single-particle natural orbitals
Taisuke Ozaki and Masahiro Fukuda
Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan
Gengping Jiang
College of Science, Wuhan University of Science and Technology, Wuhan, 430081, China
(Received 28 August 2018; revised manuscript received 5 November 2018; published 26 December 2018)
An efficient O( N) divide-conquer (DC) method based on localized single-particle natural orbitals (LNOs)
is presented for large-scale density functional theory (DFT) calculations of gapped and metallic systems. TheLNOs are noniteratively calculated by a low-rank approximation via a local eigendecomposition of a projectionoperator for the occupied space. Introducing LNOs to represent the long-range region of a truncated clusterreduces the computational cost of the DC method while keeping computational accuracy. A series of benchmarkcalculations and high parallel efficiency in a multilevel parallelization clearly demonstrate that the O( N) method
enables us to perform large-scale simulations for a wide variety of materials including metals with sufficientaccuracy in accordance with development of massively parallel computers.
DOI: 10.1103/PhysRevB.98.245137
I. INTRODUCTION
First-principles electronic structure calculations based on
the density functional theories (DFT) [ 1,2] have been playing
a versatile role in a wide variety of materials sciences todeeply understand the physical and chemical properties ofexisting materials and even to design novel materials havinga desired property before actual experiments [ 3–6]. In recent
years, more complicated materials with secondary structuressuch as heterointerfaces [ 7,8] and dislocations [ 9] have been
becoming a scope of application by DFT calculations. Sincethese complicated structures cannot be easily modeled by
a small unit cell, development of efficient DFT methods in
accordance with development of massively parallel computersis crucial to realize such large-scale DFT calculations. Amongefficient DFT methods [ 10–34], O(N) methods whose com-
putational cost scales linearly as a function of the numberof atoms have enabled us to extend the applicability of DFTto large-scale systems [ 12–34]. Nevertheless, applications of
the O( N) methods to metallic systems have been still limited
because of the fundamental difficulty of a truncation schemein real space, which is an idea commonly adopted in most ofthe O( N) methods [ 12,13], in realizing the O( N) methods [ 14]
as discussed in the Appendix. A theoretically proper approach
to go beyond the truncation scheme is to take account of the
contribution from the external region beyond the truncatedregion via a self-energy in Green function formalism [ 26–30].
Another straightforward approach is to use a relatively largecutoff radius in the truncation scheme in order to reach asufficient accuracy. The latter approach might be suited tothe divide-conquer (DC) method [ 31,32] among the O( N)
methods proposed, so far, in the following twofold aspects.(i) There is a way to reduce the computational cost in theframework of the DC method by lowering the dimension ofmatrices with an introduction of a Krylov subspace [ 33,34].
(ii) The computational time of the DC method can be reducedby a massive parallelization, since the calculations in the DCmethod are performed nearly independently for each atom
[35,36]. With the two aspects, the improvement of the DC
method can be a promising direction to develop an accurate,efficient, and robust O( N) method applicable to not only
insulators and semiconductors, but also metals. Along thisline, the DC method based on the Krylov subspace has beenproven to be an efficient and accurate O( N) method by a
wide variety of applications such as dynamics of Li ions ina lithium ion battery [ 37–39] and structure optimization of
semicoherent heterointerfaces in steel [ 7,8]. However, there
exist drawbacks in the generation of the Krylov subspace[34]. Since the Krylov subspace is generated at the first self-
consistent field (SCF) step and kept unchanged during the
subsequent SCF calculation, calculated quantities such as the
electron density and total energy depend on the initial guessof electron density or Hamiltonian matrix elements. If theKrylov subspace is regenerated every SCF step to avoid thedependency on the initial guess, the computational efficiencymust be largely degraded. In addition, the iterative calcula-tions in the generation of the Krylov subspace tend to sufferfrom numerical round-off error, leading to an uncontrollablebehavior in the SCF calculation if the Krylov subspace isregenerated every SCF step [ 34]. Therefore a more robust
approach needs to be developed to improve the DC method,which overcomes the drawbacks inherent in the DC method
based on the Krylov subspace.
In this paper, we focus on localized single-particle natural
orbitals (LNOs) calculated by a low-rank approximation toperform a coarse graining of basis functions, and apply theLNOs to the DC method, which will be referred to as theDC-LNO method hereafter, to reduce the dimension of ma-trices without sacrificing the accuracy. The DC-LNO method
overcomes the drawbacks in the DC method based on the
Krylov subspace, while taking account of reduction of the
prefactor in the computational cost and a simple algorithmwith less communication leading to a high parallel efficiency.
2469-9950/2018/98(24)/245137(14) 245137-1 ©2018 American Physical SocietyTAISUKE OZAKI, MASAHIRO FUKUDA, AND GENGPING JIANG PHYSICAL REVIEW B 98, 245137 (2018)
A series of benchmark calculations clearly demonstrates that
the DC-LNO method is an accurate, efficient, and robustO(N) method applicable to not only insulators and semicon-
ductors, but also metals in step with recent development ofmassively parallel computers.
The paper is organized as follows. In Sec. II, we propose a
method to generate LNOs, and present the DC-LNO methodas an extension of the DC method. In Sec. III, the implemen-
tations of the method are discussed in detail. In Sec. IV,a
series of benchmark calculations is presented. In Sec. V,w e
summarize the theory of the DC-LNO method and numericalaspects.
II. THEORY
A. General
We consider an extension of a divide-conquer (DC) ap-
proach [ 31,32] by introducing a coarse graining of basis
functions as shown in Fig. 1. For each atom, the Kohn-Sham
(KS) Hamiltonian and overlap matrices are truncated within agiven cutoff radius, and the resultant truncated cluster problemis solved atom by atom, leading to the O( N) scaling in the
computational cost. As expected the error of truncation canbe systematically reduced in exchange for the increase ofcomputational cost as the cutoff radius increases. The numberof atoms in a truncated cluster exceeds 300 atoms in manycases in order to attain a sufficient accuracy as discussed lateron. To reduce the computational cost we introduce a coarsegraining of basis functions that the original basis functions,pseudo-atomic orbitals (PAOs) in our case [ 40,41], are re-
placed by localized single-particle natural orbitals (LNOs) inthe long-range (yellow) region to represent the Hamiltonianand overlap matrices, while the PAO functions remain un-changed in the short-range (orange) region. In the following
FIG. 1. Truncation of a system in the DC method with LNOs.
The short-range (orange) and long-range (yellow) regions are repre-
sented by PAOs and LNOs, respectively.sections, a method of generating LNOs and a DC method with
LNOs is discussed in detail.
B. Generation of LNOs
We present a method of generating LNOs based on a
low-rank approximation via a local eigendecomposition of aprojection operator. The method might be applicable to anylocal basis functions, and not limited to the application to theDC method we discuss in the paper. Even in the conventionalO(N
3) calculations, the LNOs can be easily obtained by a
noniterative calculation using the density matrix and overlapmatrix elements. Since a smaller number of LNOs well repro-duce dispersion of occupied bands, they can be an alternativebasis set for a compact representation for the Hamiltonianand overlap matrices. Though in this subsection we presenta method of calculating LNOs in a general form starting fromBloch functions to clarify a mathematical basis of LNOs, itshould be clearly noticed in this place that the density matrixis directly calculated in the DC-LNO method without calcu-lating the Bloch functions. The general formulation presentedhere provides a theoretical basis of three step algorithm, whichwill be discussed in the end of the subsection.
Under the Born-von Karman boundary condition, we ex-
pand a Kohn-Sham (KS) orbital φ
kμ, indexed with a k-vector
kin the first Brillouin zone and band index μ,u s i n gP A O s χ
[33,34], being a real function, as
|φkμ/angbracketright=1√NBC/summationdisplay
Reik·R/summationdisplay
iαckμ,iα|χRiα/angbracketright, (1)
where R,N BC, andcare a lattice vector, the number of cells
in the boundary condition, and linear combination of pseu-doatomic orbital (LCPAO) coefficients, respectively. It is alsonoted that /angbracketleftr|φ
(k)
μ/angbracketright≡φ(k)
μ(r) and/angbracketleftr|χRiα/angbracketright≡χiα(r−τi−R),
where iandαare atomic and orbital indices, respectively, and
τiis the position of atom i. We assume that MiPAO functions
are allocated to atom i. Throughout the paper we do not
consider the spin dependency on the formulation for sake ofsimplicity, but the generalization is straightforward. By con-sidering overlap matrix elements S
Riα,R/primejβ≡/angbracketleftχRiα|χR/primejβ/angbracketright,
one can introduce two alternative localized orbitals:
|χRiα/angbracketright=/summationdisplay
R/primejβ|χR/primejβ/angbracketrightS−1/2
R/primejβ,Riα, (2)
|/tildewideχRiα/angbracketright=/summationdisplay
R/primejβ|χR/primejβ/angbracketrightS−1
R/primejβ,Riα, (3)
where S−1/2andS−1are calculated from an overlap matrix
Sfor a supercell consisting of NBCprimitive cells in the
Born-von Karman boundary condition. Hereafter, χand/tildewideχ
will be referred to as Löwdin [ 42] and dual orbitals [ 33],
respectively. It is noted that we have the following relations:
/angbracketleftχRiα|χR/primejβ/angbracketright=δRR/primeδijδαβ, (4)
/angbracketleftχRiα|/tildewideχR/primejβ/angbracketright=/angbracketleft/tildewideχRiα|χR/primejβ/angbracketright=δRR/primeδijδαβ, (5)
and that the identity operator ˆIcan be expressed in Bloch
functions φ, Löwdin orbitals χ,o rP A O s χand dual orbitals
245137-2EFFICIENT O( N) DIVIDE-CONQUER METHOD WITH … PHYSICAL REVIEW B 98, 245137 (2018)
/tildewideχas
ˆI=/summationdisplay
kμ|φkμ/angbracketright/angbracketleftφkμ|
=/summationdisplay
Riα|χRiα/angbracketright/angbracketleftχRiα|
=/summationdisplay
Riα|χRiα/angbracketright/angbracketleft/tildewideχRiα|=/summationdisplay
Riα|/tildewideχRiα/angbracketright/angbracketleftχRiα|. (6)
We now define a projection operator for the occupied space by
ˆP=/summationdisplay
kμ|φkμ/angbracketrightf(εkμ)/angbracketleftφkμ|, (7)
where fandεare the Fermi-Dirac function and an eigenvalue
of the KS equation, respectively. By using the second line ofEq. (6), one obtains an alternative expression of the projection
operator as
ˆP=/summationdisplay
Riα,R/primejβ|χRiα/angbracketrightρRiα,R/primejβ/angbracketleftχR/primejβ| (8)
with
ρRiα,R/primejβ=/summationdisplay
kμ/angbracketleftχRiα|φkμ/angbracketrightf(εkμ)/angbracketleftφkμ|χR/primejβ/angbracketright
=1
NBC/summationdisplay
kμeik·(R−R/prime)f(εkμ)bkμ,iαb∗
kμ,jβ,(9)
where bkμ=S1/2ckμ, and ckμis a column vector whose
elements are LCPAO coefficients {ckμ,iα}. As well, one can
derive another expression of the projection operator by apply-ing the third line of Eq. ( 6)t oE q .( 7)a s
ˆP=/summationdisplay
Riα,R/primejβ|χRiα/angbracketrightρRiα,R/primejβ/angbracketleftχR/primejβ| (10)
with
ρRiα,R/primejβ=/summationdisplay
kμ/angbracketleft/tildewideχRiα|φkμ/angbracketrightf(εkμ)/angbracketleftφkμ|/tildewideχR/primejβ/angbracketright
=1
NBC/summationdisplay
kμeik·(R−R/prime)f(εkμ)ckμ,iαc∗
kμ,jβ.(11)
It is noted that ρis related to ρby the following relation:
ρ=S1/2ρS1/2. (12)
Remembering that the number of electrons Nelein the su-
percell consisting of NBCprimitive cells can be obtained by
the trace of ˆP, and that we have two alternative expressions
Eqs. ( 8) and ( 10)f o r ˆP, one has the following expressions for
Nele:
Nele=2tr[ˆP]
=2/summationdisplay
Riα/angbracketleftχRiα|ˆP|χRiα/angbracketright=2tr[S1/2ρS1/2]
=2tr[ρS]=2/summationdisplay
Riα/angbracketleft/tildewideχRiα|ˆP|χRiα/angbracketright, (13)
where the factor of 2 is due to spin degeneracy. Since each
term in the summation over Requally contributes to Nele,we have Nele=NBCN(0)
ele, where N(0)
ele=2/summationtext
iα/angbracketleft/tildewideχ0iα|ˆP|χ0iα/angbracketright.
Thus it is enough to consider N(0)
eleinstead of Nelefor further
discussion. By introducing a notation for a subset of orbitals{χ}and{/tildewideχ}with
|χ
Ri)=(|χRi1/angbracketright,|χRi2/angbracketright,···,|χRiMi/angbracketright), (14)
|/tildewideχRi)=(|/tildewideχRi1/angbracketright,|/tildewideχRi2/angbracketright,···,|/tildewideχRiMi/angbracketright), (15)
one can write N(0)
eleas
N(0)
ele=2/summationdisplay
itr0i[(/tildewideχ0i|ˆP|χ0i)]=2/summationdisplay
itr0i[/Lambda10i],(16)
where tr 0imeans a partial trace over orbitals associated with
an atom iin the central cell with R=0, and/Lambda10iis defined by
/Lambda10i=/summationdisplay
Rjρ0i,RjSRj,0i (17)
with definition of block elements:
ρRi,R/primej=(/tildewideχRi|ˆP|/tildewideχR/primej), (18)
SRi,R/primej=(χRi|χR/primej). (19)
These block elements ρRi,R/primejandSRi,R/primejareMi×Mjma-
trices, where MiandMjare the number of PAO functions
allocated to atoms iandj, respectively. Therefore /Lambda10idefined
by Eq. ( 17)i sa Mi×Mimatrix. It should be emphasized
that Eq. ( 16)g i v i n g N(0)
eleby the sum of the partial trace is
an important relation in calculating LNOs, since it shows thatalocal similarity transformation on an atomic site idoes not
change N(0)
elebecause of a property of the trace. Noting that /Lambda10i
is nonsymmetric, we consider a general eigendecomposition
of a nonsymmetric matrix for /Lambda10iamong similarity transfor-
mations as
V−1
0i/Lambda10iV0i=λ0i, (20)
where λ0iis a diagonal matrix having eigenvalues {λ0iγ}
of/Lambda10ias diagonal elements. Since an eigenvalue λ0iγof
/Lambda10igives the population for the corresponding eigenstate of
/Lambda10i, one can distinguish LNOs spanning the occupied space
from others among all the eigenstates of /Lambda10iby monitoring
the eigenvalues. To see the idea more clearly, we define anoperator by
ˆ/Lambda1
0i=/summationdisplay
γ|v0iγ/angbracketrightλ0iγ/angbracketleft/tildewidev0iγ|, (21)
where |v0iγ/angbracketrightis theγth column vector of V0i, and/angbracketleft/tildewidev0iγ|is the
γth row vector of V−1
0i. Note that /angbracketleft/tildewidev0iγ|is the dual orbital of
|v0iγ/angbracketright, and/angbracketleft/tildewidev0iγ|v0iη/angbracketright=δγη. It is easy to confirm that /Lambda10iand
λ0iin a matrix form can be obtained by representing ˆ/Lambda10iwith
{χ0iα}and{/tildewideχ0iα}, and with {v0iγ}and{/tildewidev0iγ}, respectively.
Thus we have tr 0i[ˆ/Lambda10i]=tri0[/Lambda10i]. If the eigenvalue λ0iγis
nearly zero, the contribution of the corresponding eigenstate|v
0iγ/angbracketrightis negligible in the summation of Eq. ( 21). Therefore
ˆ/Lambda10ican be approximated by excluding eigenstates whose
eigenvalues are less than a threshold value λththat we will
discuss later on. The treatment can be regarded as a low-rank
245137-3TAISUKE OZAKI, MASAHIRO FUKUDA, AND GENGPING JIANG PHYSICAL REVIEW B 98, 245137 (2018)
approximation [ 43]. Then, using Eqs. ( 16) and ( 21) we can
approximate the projection operator ˆPdefined by Eq. ( 7)a s
ˆP/similarequal/summationdisplay
Riλth/lessorequalslantλRiγ/summationdisplay
γ|vRiγ/angbracketrightλRiγ/angbracketleft/tildewidevRiγ|, (22)
where terms satisfying a condition λth/lessorequalslantλRiγare taken into
account in the summation over γ. If all the terms are included,
Eq. ( 22) becomes equivalent to Eq. ( 7). We now define our
LNOs by {v}whose eigenvalues are larger than or equal to the
threshold value λth. By comparing Eq. ( 20) with Eq. ( 16), one
hasV−1
0i/Lambda10iV0i=V−1
0i(/tildewideχ0i|ˆP|χ0i)V0i. Thus LNOs are defined
by{v}, and it should be noted that {/tildewidev}are the corresponding
dual orbitals. The approximate formula Eq. ( 22)f o r ˆPimplies
that a set of orbitals {v}, LNOs, well spans the occupied
space, while the number of orbitals is reduced comparedto the original PAOs. It is worth noting that LNOs can beindependently calculated for each atom by a noniterativecalculation via Eqs. ( 17) and ( 20). The fact makes the method
of generating LNOs very efficient, and also guarantees that theresultant LNOs associated with an atom iare expressed by a
linear combination of PAOs allocated to only the atom i[44].
The computational procedure to generate LNOs is summa-
rized as follows. (i) Calculation of /Lambda1. For each atom i,t h e
matrix /Lambda1
0iis calculated by Eq. ( 17), where the summation
over Ris limited within a finite range because of the locality
of PAOs in real space. (ii) Diagonalization of /Lambda1. Since the
matrix/Lambda10iis nonsymmetric, the diagonalization in Eq. ( 20)i s
performed by a generalized eigenvalue solver for a nonsym-metric matrix such as
DGEEV inLAPACK [48]. (iii) Selection
ofv. Eigenvectors {v0iγ}whose eigenvalues are larger than or
equal to the threshold value λthare selected as LNOs.
Only the overlap and density matrices are required to
calculate LNOs through the steps (i)–(iii) above. Therefore,either conventional O( N
3) methods or O( N) methods can
be employed as eigenvalue solvers as long as they generatethe density matrix. As for LNOs other than those in thecentral cell with R=0, it is apparent from the derivation
that one can obtain |v
Riγ/angbracketrightby parallel translation of |v0iγ/angbracketright
with the lattice vector R. The method can also be extended to
choose another energy window. In the projection operator ˆP
defined by Eq. ( 7), the Fermi-Dirac function is introduced to
choose the occupied space. However, one can choose a properenergy window in the definition of the projection operator ˆP
depending on what we discuss, which enables us to focus onspecific bands such as localized d-bands near the Fermi level.
In this sense, LNOs can be utilized like Wannier functions(WFs) [ 49,50], while WFs are obtained through a unitary
transformation of Bloch functions rather than the low-rankapproximation, and they are orthonormal each other unlikeLNOs. It is also worth pointing out that our method shares thebasic idea based on the projection with other methods such asthe quasiatomic orbitals scheme [ 51–53], a projection method
[54], and a method via selected columns of the density matrix
[55,56].
It might be possible for the method we present in the
paper to be applied for other localized basis functions such asfinite element methods [ 57,58] and finite difference methods
[59–61]. In those cases one may introduce spatial partitioning
methods such as the Voronoi tessellation to decompose basisfunctions rather than focusing on basis functions on a single
grid. A similar procedure can be applied for a set of parti-tioned basis functions.
C. DC method with LNOs
Here we consider an extension of the DC method [ 31,32]
using LNOs discussed in the previous subsection. Our the-oretical basis to formulate the O( N) DC method is that the
total energy and atomic forces in the KS framework can becalculated by using electron density n(r), density matrix ρ,
and energy density matrix edefined by
n(r)=/summationdisplay
i⎛
⎝2/summationdisplay
α,Rjβρ0iα,Rjβχ0iα(r)χRjβ(r)⎞
⎠
=/summationdisplay
ini(r), (23)
ρ=−1
πIm/integraldisplay∞
−∞G(E+i0+)f(E)dE, (24)
and
e=−1
πIm/integraldisplay∞
−∞G(E+i0+)f(E)EdE, (25)
where Gis the Green function defined by G(Z)≡(ZS−
H)−1with the overlap matrix Sand KS matrix H, and the
factor of 2 in Eq. ( 23) is due to spin degeneracy. It is remarked
that forces on atoms in the DC method are not calculatedvariationally, but evaluated by using the formula derivedby assuming that numerically exact KS wave functions areavailable as discussed in Ref. [ 34]. The DC method calculates
the Green function G(Z) approximately by introducing the
truncation scheme as shown in Fig. 1. The KS matrix of the
truncated cluster for atom iare constructed using PAOs and
LNOs as follows:
H
(i)=/parenleftbigg
PAO−PAO PAO−LNO
LNO−PAO LNO−LNO/parenrightbigg
, (26)
where the top left and bottom right blocks correspond to
the short-range (orange) region represented by PAOs, andthe long-range (yellow) region represented by LNOs, respec-tively, as shown in Fig. 1. The top right and the bottom
left block consist of the hopping matrix elements bridgingthe two regions, and they are represented by both PAOs andLNOs. As well, the same structure is found for the overlapmatrix. Noting that the computational bottleneck is mainlygoverned by the eigenvalue problem for the truncated clusters,and that the matrix size can be reduced by introducing LNOscompared to the conventional DC method, one can expect aconsiderable reduction of the computational cost as the sizeof the long-range region increases. The idea of reducing thematrix dimension by introducing an effective representationof Hamiltonian is similar to that in the O( N) Krylov subspace
method [ 34] and the absolutely localized molecular orbitals
(ALMO) method [ 23]. By solving the eigenvalue problem
H
(i)c(i)
μ=ε(i)
μS(i)c(i)
μfor the truncated cluster of atom i,w e
calculate matrix elements associated with the atom ifor the
245137-4EFFICIENT O( N) DIVIDE-CONQUER METHOD WITH … PHYSICAL REVIEW B 98, 245137 (2018)
Green function as [ 62]
G(i)
0iα,Rjβ(Z)=/summationdisplay
μc(i)
μ,0iα/parenleftbig
c(i)
μ,Rjβ/parenrightbig∗
Z−ε(i)
μ. (27)
Matrix elements ρ0iα,Rjβande0iα,Rjβassociated with the atom
ican be analytically calculated by inserting Eq. ( 27)i n t o
Eqs. ( 24) and ( 25), respectively. We only have to calculate
ρ0iα,Rjβande0iα,Rjβonly if S0iα,Rjβis nonzero, since the other
elements do not contribute to the total energy and forces onatoms in case of semilocal functionals such as local densityapproximations (LDA) [ 63,64] and generalized gradient ap-
proximations (GGA) [ 65]. Then, n
i(r)i nE q .( 23) is easily
computed from ρ0iα,Rjβ. By applying the procedure for all the
atoms in a system, all the necessary information to calculatethe total energy and forces on atoms are obtained.
The overall procedure of the DC method with LNOs is
summarized as follows: (i) calculation of LNOs. LNOs arecalculated by Eq. ( 20) for all atoms in the central cell with
R=0. At every SCF step, LNOs are updated, leading to
self-consistent determination of LNOs. (ii) Construction ofH
(i)andS(i). For each atom ithe KS and overlap matrices for
a truncated cluster associated with the atom iare constructed
by Eq. ( 26). (iii) Diagonalization of H(i)c(i)
μ=ε(i)
μS(i)c(i)
μby
making use of a parallel eigenvalue solver. (iv) Finding acommon chemical potential to conserve the total numberof electrons in the system using Eq. ( 27), as discussed in
Ref. [ 34] in detail. (v) Calculation of density matrix ρand
energy density matrix eusing Eqs. ( 24), (25), and ( 27).
(vi) Calculation of electron density nusing Eq. ( 23).
III. IMPLEMENTATIONS
We have implemented the DC-LNO method into the
OPENMX DFT software package [ 66] which is based on norm-
conserving pseudopotentials (PPs) [ 67,68] and optimized
pseudo-atomic orbitals (PAOs) [ 40,41] as basis set. All the
benchmark calculations were performed with a computationalcondition of a production level. The basis functions usedare C6.0-s2p2d1, Si7.0-s2p2d1, Ti7.0-s2p2d1, O6.0-s2p2d1,Li8.0-s3p2, Al7.0-s2p2d1, and Fe5.5-s3p2d2 for carbon,silicon, titanium, oxygen, lithium, aluminum, and iron, re-spectively, where in the abbreviation of basis functions suchas C6.0-s2p2d1, C stands for the atomic symbol, 6.0 thecutoff radius (Bohr) in the generation by the confinementscheme, and s2p2d1 means the employment of two, two, andone optimized radial functions for the s,p, anddorbitals,
respectively. The radial functions were optimized by a vari-ational optimization method [ 40]. These basis functions we
used can be regarded as double zeta plus polarization basissets if we follow the terminology of Gaussian basis functions.As valence electrons in the PPs we included 2 sand 2p,3s
and 3p,3s,3p,3d, and 4 s,2sand 2p,1s,2s, and 2 p,3s,
and 3p, and 3 s,3p,3d, and 4 sstates for carbon, silicon,
titanium, oxygen, lithium, aluminum, and iron, respectively.All the PPs and PAOs we used in the study were takenfrom the database (2013) in the
OPENMX website [ 66], which
were benchmarked by the delta gauge method [ 69]. Real
space grid techniques are used for the numerical integrationsand the solution of the Poisson equation using FFT withthe energy cutoff of 300 Ryd [ 70]. We used a generalized
gradient approximation (GGA) proposed by Perdew, Burke,and Ernzerhof to the exchange-correlation functional [ 65]. An
electronic temperature of 300 K is used to count the numberof electrons by the Fermi-Dirac function for all the systemswe considered.
The short- and long-range regions depicted in Fig. 1are
determined as follows. (i) We first pick up atoms in a spherewith a given cutoff radius r
L. (ii) Among the atoms selected by
the step (i) we distinguish the first neighboring atoms (FNAs)having nonzero overlap with the central atom in terms of basisfunctions, and remaining atoms other than FNAs are calledthe second neighboring atoms (SNAs), where the numberof FNAs and SNAs are N
FandNS, respectively. (iii) The
short-range region is determined by adjusting a cutoff radiusr
Sso that the number of atoms in a sphere with a radius of rS
can be as close as possible to NF+κNS, where the parameter
κcan vary from 0 to 1, and we will discuss the choice of
κlater on. (iv) The long-range region consists of remaining
atoms other than atoms selected by the step (iii). If we assignFNAs to atoms in the short-range region, the total energy doesnot converge to the numerically exact one calculated by theconventional diagonalization method even if the cutoff radiusr
Lincreases systematically. This is because the error with
the low-rank approximation by Eq. ( 22) keeps increasing as
the cutoff radius increases. To avoid the situation, we adda buffer region consisting of about κN
Satoms as described
by the step (iii) above, which guarantees the convergenceof the total energy and other quantities as a function of thecutoff radius r
L. Throughout the study, we used κof3
10for
all the systems by taking the accuracy into account morethan the efficiency, and did not adjust the parameter, whilea smaller value, which well balances both the accuracy andefficiency, can be employed for some systems. The way ofparallelization for the DC-LNO method on parallel computerswill be discussed together with its benchmark calculationslater on.
IV . NUMERICAL RESULTS
A. Band dispersions by LNOs
In order to investigate to what extent LNOs can span
occupied spaces, we compare band dispersions of gappedand metallic systems calculated with PAOs and LNOs.Figure 2(a)and2(b) show band dispersions of diamond and
silicon calculated by a conventional O( N
3) diagonalization
method with PAOs and LNOs. For both the cases, the SCFcalculations were performed by using PAOs. For the case ofLNOs, the band dispersions were calculated with the LNOsafter the SCF calculations with PAOs. The number of LNOsper atom is 4 for both carbon and silicon atoms. It is foundthat in both the cases the band dispersions of occupied spaceare well reproduced with LNOs compared to those calculatedby PAOs, while a large difference can be seen in conductionbands between PAOs and LNOs as expected. The good agree-ment between PAOs and LNOs in describing the occupiedbands implies that the low-rank approximation by Eq. ( 22)i s
reasonably valid. As shown in Table I, we see that the first four
eigenvalues of the matrix /Lambda1are actually dominant for both
245137-5TAISUKE OZAKI, MASAHIRO FUKUDA, AND GENGPING JIANG PHYSICAL REVIEW B 98, 245137 (2018)
(a) (b)
-22-20-18-16-14-12-10-8-6-4-20246810
W L G XWKEnergy (eV)
-12-10-8-6-4-20246
W L G XWKEnergy (eV)
PAO(13)
LNO(4)PAO(13)LNO(4)
FIG. 2. Band dispersions of (a) diamond and (b) silicon in the
diamond structure with experimental lattice constants (3.567 and
5.430 Å) calculated by PAOs (black) and LNOs (red). A conventionalO(N
3) method was used for the diagonalization, where the number
ofkpoints for the Brillouin zone sampling is 71 ×71×71 for both
the cases. In the case of LNOs, the SCF calculations were performedby using PAOs, and after determining the SC electron density, LNOs
were used to calculate the band dispersion. The numbers of PAOs
and LNOs per atom are shown in the parenthesis.
diamond and silicon, justifying the low-rank approximation.
The largest and the next three eigenvalues correspond to ansorbital and p-like orbitals deformed by contribution of d
orbitals, respectively. It is also noted that the eigenvaluescan be negative, which is related to a negative value ofMulliken populations for delocalized orbitals [ 71,72]. As well
as the gapped systems, similar calculations were performedfor metals, lithium in the body centered cubic (BCC) structureand aluminum in the face centered cubic (FCC) structure as
TABLE I. Eigenvalues λof the matrix /Lambda1for diamond, silicon,
BCC lithium, and FCC aluminum. The corresponding eigenvectors
were used as LNOs to calculate the band dispersions shown in
Figs. 2–4.
Diamond Si Li Al
λ1 0.514 0.639 0.999 0.551
λ2 0.483 0.430 0.277 0.253
λ3 0.483 0.430 0.075 0.253
λ4 0.483 0.430 0.075 0.253
λ5 0.012 0.018 0.074 0.049
λ6 0.012 0.018 0.001 0.049
λ7 0.012 0.018 0.001 0.049
λ8 0.011 0.012 0.001 0.033
λ9 0.011 0.012 −0.002 0.033
λ10 −0.001 0.002 - −0.002
λ11 −0.001 0.002 - −0.002
λ12 −0.001 0.002 - −0.002
λ13 −0.018 −0.011 - −0.015(a) (b)PAO(9)
LNO(2)-4-20246
G H N G PEnergy (eV)
-4-20246
G H N G PEnergy (eV)
PAO(9)LNO(5)
FIG. 3. Band dispersions of BCC lithium with an experimental
lattice constant of 3.491 Å calculated by PAOs (black) and (a) two
and (b) five LNOs (red). The number of kpoints for the Brillouin
zone sampling is 101 ×101×101. The other details are the same as
in the caption of Fig. 2.
shown in Figs. 3and4, respectively. The band dispersions
calculated with the minimal LNOs are reasonably comparedto those by PAOs, while the use of the five and nine LNOsfor Li and Al atoms fully reproduce the band dispersionsincluding conduction bands as shown in Figs. 3(b) and4(b),
respectively. One can confirm again in Table Ithat eigenvalues
for the minimal LNOs are dominant even for metals, whilethe magnitude of the subsequent eigenvalues is relativelylarge compared to those of the gapped systems. Thus weconclude that LNOs can be regarded as a compact basis set
(a) (b)PAO(13)
LNO(4)
-14-12-10-8-6-4-20246810
W L G XWKEnergy (eV)
-14-12-10-8-6-4-20246810
W L G XWKEnergy (eV)
PAO(13)
LNO(9)
FIG. 4. Band dispersions of FCC aluminum with an experimen-
tal lattice constant of 4.050 Å calculated by PAOs (black) and (a) fourand (b) nine LNOs (red). The number of kpoints for the Brillouin
zone sampling is 111 ×111×111. The other details are the same as
in the caption of Fig. 2.
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0 100 200 300 400 500 600 70010−510−410−310−2
0 100 200 300 400 500 600 70010−510−410−310−2
0 100 200 300 400 500 600 70010−510−410−310−2(a) Diamond
(b) Silicon
(c) r−TiO 2DC
DC−LNO
DC
DC−LNO
DC
DC−LNO
Number of atoms in a truncated clusterAbsolute error in total energy (Hartree/atom)0 100 200 300 400 500 600 700 800 90010−510−410−310−2
0 100 200 300 400 500 600 700 80010−510−410−310−2
0 100 200 300 400 500 600 70010−510−410−310−2(d) BCC Li
(e) FCC Al
(f) BCC FeDC
DC−LNO
DC
DC−LNO
DC
DC−LNO
Number of atoms in a truncated clusterAbsolute error in total energy (Hartree/atom)
FIG. 5. Absolute error in the total energy (Hatree/atom) for (a) diamond, (b) silicon in the diamond structure, (c) rutile TiO 2, (d) BCC
lithium, (e) FCC aluminum, and (f) BCC iron as a function of the number of atoms in a truncated cluster calculated by the DC and DC-LNO
methods. The experimental lattice constants were used for all the cases.
spanning well the occupied space for both gapped and metallic
systems.
B. Total energies by DC-LNO
As a first step of validation of the DC-LNO method with
respect to computational accuracy and efficiency, we showin Fig. 5the absolute error in the total energy of gapped
and metallic systems calculated by the DC and the DC-LNOmethods, where the reference energies were calculated bythe conventional O( N
3) method with dense kpoints for the
Brillouin zone sampling as given in the caption of Figs. 2–4.
For all the cases the threshold value λthin Eq. ( 22)w a ss e t
to be 0.1, which gives us the minimal LNOs corresponding toorbitals of valence electrons. In the gapped systems, diamond,silicon, and rutile TiO
2, the absolute error decreases almost
exponentially as the number of atoms in a truncated clusterincreases. The overall behavior of the error between the DCand DC-LNO methods is similar, while the error by the DC-LNO method is accidentally much smaller than that by theDC method in the case of large truncated clusters of diamond.Similar to gapped systems, the absolute error for metalscalculated by the DC-LNO method decreases with increasingnumber of atoms in a truncated cluster in a similar way to theDC method. The relatively large oscillating behavior observedin the metals might be related to long-range characteristics ofthe off-diagonal Green functions as discussed in Appendix.
For all the cases including metals, it is found that a truncatedcluster including about 300 atoms is required to attain themillihartree accuracy corresponding to an error less than a fewmillihartree/atom in the total energy. From the comparisonbetween the DC and DC-LNO methods, we see that the com-putational accuracy does not degrade largely even if the basisfunctions for atoms in the long-range region are approximatedby LNOs, and that thereby the computational accuracy canbe controlled mainly by the size of truncated cluster justlike for the DC method. Since controlling only the singleparameter allows us to balance the computational accuracyand efficiency, it is expected that the feature makes the DC-LNO method easy to use for a wide variety of applications.
C. Computational time
Since the total number of basis functions to represent the
Hamiltonian of the truncated cluster is reduced by introduc-ing LNOs while keeping the accuracy, it is expected thatthe computational time can be substantially reduced. In thewhole procedure of the DC-LNO method, the calculation ofLNOs and construction of Hamiltonian and overlap matricesoccupy a small fraction of the whole computational time,typically less than 10%, and thereby the computational timeis mainly governed by solving of the eigenvalue problem
245137-7TAISUKE OZAKI, MASAHIRO FUKUDA, AND GENGPING JIANG PHYSICAL REVIEW B 98, 245137 (2018)
100 200 300 400 500 600 7000200400600
200 400 600 800 10000200400600
Number of atoms in a truncated clusterComputational time (sec./MD step)DC
DC−LNO
DC
DC−LNO(a) Diamond
(b) BCC Li
FIG. 6. Comparison of computational time of the diagonalization
part per molecular dynamics (MD) step between the DC and DC-LNO methods for (a) diamond and (b) BCC lithium. Both the
calculations were performed for the primitive cell using 14 MPI
processes per atom on Intel Xeon CPUs E5-2690v4 @ 2.60 GHz.
H(i)c(i)
μ=ε(i)
μS(i)c(i)
μfor each atom i. Noting that the com-
putational time to solve the eigenvalue problem scales as thethird power of the dimension of the matrices, the ratio ofcomputational time between the DC-LNO and DC methodsfor elemental systems might be estimated by
t
DC−LNO
tDC=[MP(NF+κNS)+ML(1−κ)NS]3
M3
P(NF+NS)3,(28)
where MPandMLare the number of PAOs and LNOs associ-
ated with each atom, respectively, and κis a factor, which is
fixed to3
10in this study, to control the size of the buffer region
as discussed in the section Implementations. For example, ifthe cutoff radius r
Lis set to be 8.7 Å in the diamond structure
with the experimental lattice constant of 3.567 Å, NFandNS
are found to be 167 and 298, and the resultant number of
atoms in the short-(long-)range region becomes 275 (190).Then, t
DC-LNO /tDCcan be estimated to be 0.37 in the case
that the number of PAOs and LNOs per atom is 13 and 4,which implies that the computational time of the DC-LNOmethod becomes about one-third of that calculated by the DCmethod. Figure 6shows actual timing results of the DC and
DC-LNO methods for (a) diamond and (b) BCC lithium. Wesee that the actual t
DC-LNO /tDCfor the diamond case is 0.40
in the case that the number of atoms is 465 ( =167+298) as
shown in Fig. 6(a), which is well compared to the estimated
value of 0.37. As indicated by Eq. ( 28) and in Figs. 6(a)and
(b), it is concluded that the DC-LNO method becomes muchfaster than the DC method as the size of the truncated clusterincreases.0 1000 2000 3000 4000 5000050010001500
Number of atomsComputational time (sec.)DC−LNOO(N3)
FIG. 7. Computational time (seconds) of the diagonalization part
per 10 SCF steps for Si in the diamond structure. In the DC-LNO
method, the cutoff radius rLof 11.3 Å was used, resulting in the
truncated cluster containing 293 atoms. In the O( N3) method, the
kgrids of 2 ×2×2 were used for the Brillouin zone sampling in
all the cases. All the calculations were performed using 1280 MPI
processes on a cluster machine consisting of Intel Xeon CPUs E5-2680v2 @ 2.80 GHz connected with Infiniband 4X FDR @ 56 Gbps.
In Fig. 7, we further show timing results of the DC-LNO
method as a function of the number of atoms in the unitcell of Si crystal. It is confirmed from the linear increase ofthe computational time that the DC-LNO method is a linearscaling approach in practice. As a comparison the computa-tional time is also shown for the conventional diagonalizationmethod [ 73]. The crossing point between the two methods
in the computational time is located around 100 atoms when1280 CPU cores is used. Over 100 atoms the DC-LNO methodis much faster than the conventional diagonalization method.The reason why the crossing point is located at such a smallnumber of atoms is partly due to a better parallel efficiency ofthe DC-LNO method as discussed in the next section.
D. Parallelization
To minimize the computational time on massively parallel
computers we introduce a multilevel parallelization usingmessage passing interface (MPI). In our implementation thereare three levels for the parallelization, i.e., atom level, spinlevel, and diagonalization level as explained below. (i) Paral-
lelization in the atom level . If the number of MPI processes
is smaller than that of atoms, only the parallelization in theatom level is taken into account. The allocation of atoms toMPI processes is performed by a bisection method, which isbased on a projection of atoms onto a principal axis calculatedfrom an inertia tensor and a modified binary tree of MPIprocesses to minimize memory usage and the amount of MPIcommunications [ 36]. (ii) Parallelization in the spin level .
If the number of MPI processes exceeds that of atoms, anda spin-polarized calculation is performed, the parallelizationin the spin level is introduced on top of the parallelizationin the atom level, where a loop for the spin index is fur-ther parallelized. (iii) Parallelization in the diagonalization
level. If the number of MPI processes is larger than the
245137-8EFFICIENT O( N) DIVIDE-CONQUER METHOD WITH … PHYSICAL REVIEW B 98, 245137 (2018)
0 500 1000 1500 20000500100015002000
The number of MPI processesSpeed−up ratio
Parallel efficiency
70.0%Ideal
FIG. 8. Speed-up ratio in the MPI parallelization of the DC-LNO
method for a diamond supercell containing 64 atoms, where the
cutoff radius rLof 8.0 Å was used, leading to the numbers of
atoms of 239 and 142 in the short and long-range regions, and thedimension of matrices of 3675 for the truncated cluster problem. The
calculations were performed using the same cluster machine as for
Fig.7.
product of the number of atoms and the multiplicity of spin
index, corresponding to 1, 2, and 1 for nonspin polarized,spin-polarized, and noncollinear calculations, respectively, aparallelization in the diagonalization level is further taken intoaccount on top of both the parallelizations in the atom andspin levels. The parallelization in the diagonalization levelis made by employing a parallel eigenvalue solver ELPA[73]. It is noted that the parallelization in the diagonalization
level requires a considerable amount of MPI communications,while the parallelizations in the atom and spin levels haveless MPI communications. So, one would expect a highparallel efficiency in the atom and spin levels, while theparallelization in the diagonalization level might be limitedup to several tens of MPI processes. To achieve a betterscaling for the parallelization in the diagonalization level, itis important to allocate CPU cores in the same computernode as MPI processes to avoid the internode communicationas much as possible. We have implemented the multilevelparallelization so that amount of the internode communicationcan be minimized especially for the parallelization in thediagonalization level. In Fig. 8, the speed-up ratio in the MPI
parallelization of the DC-LNO method is shown for nonspinpolarized calculations of a diamond supercell containing 64atoms. Since the multiplicity of spin index is 1, we see anearly ideal behavior up to 64 MPI processes. Beyond 64MPI processes the parallelization in the diagonalization levelis taken into account on top of the parallelization in the atomlevel. A superlinear speed-up is observed at 128 and 256MPI processes, which might be due to an effective use ofcache by the reduction of memory usage, and a good scalingis achieved up to 1280 MPI processes at which the parallelefficiency is calculated to be 70% using the elapsed time at1 MPI process as reference. Since each computer node has20 CPU cores in this case, it would be reasonable to observe1 5.0 0−0.003−0.002−0.00100.0010.0020.003
1 5.0 0−0.003−0.002−0.00100.0010.0020.003Energy (Hartree/atom)Kinetic
EDFT
Sum(a) Si
Time (ps)Energy (Hartree/atom)Kinetic
EDFT
Sum(b) AlTime (ps)
FIG. 9. The kinetic energy of atomic nuclei (kinetic), the internal
total energy ( EDFT), and the sum of them as a function of time in
NVE molecular dynamics simulations by the DC-LNO method for(a) Si and (b) Al, where the time step of 2 fs was used, and random
velocities corresponding to 400 K were given for atomic nuclei at
the first MD step. The simulation cells with experimental latticeconstants for Si and Al contain 64 and 108 atoms, respectively. For
the DC-LNO method, the cutoff radii r
Lof 11.3 and 10.1 Å were
used for Si and Al, respectively, resulting in the truncated clustercontaining 293 and 249 atoms in the ideal bulk structure. Each energy
curve was shifted by adding a constant.
a good scaling up to 1280 ( =64×20) MPI processes. Thus
we see that the multilevel parallelization is very effectiveto minimize the computational time in accordance with therecent development of massively parallel computers.
E. Molecular dynamics simulations
To verify the accuracy of forces on atoms calculated by
the DC-LNO method, results of NVE molecular dynamics(MD) simulations are shown in Fig. 9. We see that the sum of
the kinetic energy of atomic nuclei (kinetic) and the internaltotal energy ( E
DFT), being a conserved quantity, is reasonably
conserved as a function of time, and the fluctuation is aboutone tenth of the kinetic energy or the internal total energy. Itshould be noted that the approximate conservation of the sumis achieved for not only Si being a semiconductor, but also Albeing a metal. Thus, it can be concluded that the accuracy
245137-9TAISUKE OZAKI, MASAHIRO FUKUDA, AND GENGPING JIANG PHYSICAL REVIEW B 98, 245137 (2018)
2468012
246801232468012
246802468O(N3)
DC−LNO
r (Å)Radial distribution function O(N3)
DC−LNO(a) Si
(b) AlO(N3)
DC−LNO
r (Å)Radial distribution function O(N3)
DC−LNO(c) Li
(d) SiO2
FIG. 10. Total radial distribution function (RDF) of (a) silicon at 3500 K, (b) aluminum at 2500 K, (c) lithium at 800 K, and (d) SiO 2
at 3000 K, calculated by the conventional O( N3) diagonalization and the DC-LNO methods. The MD simulations were performed for cubic
supercells containing 64, 108, 128, and 192 atoms with a fixed lattice constant of 10.86, 12.15, 14.04, 14.25 Å for silicon, aluminum, lithium,
and SiO 2, respectively, for 10 ps with the time step of 2 fs. The temperature was controlled by a velocity scaling scheme by Woodkock [ 74]. The
coordinates for the first 1 ps were excluded to calculate RDF. In the DC-LNO method the cutoff radii rLof 11.3, 10.1, 12.5, and 11.0 were used
for silicon, aluminum, lithium, and SiO 2, respectively, resulting in truncated clusters consisting of 293, 249, 339, and 344 (on average) atoms
in the ideal bulk structures. In the conventional O( N3) diagonalization method, kpoints of 7 ×7×7,8×8×8,7×7×7, and 5 ×5×5
were used for the Brillouin zone sampling in silicon, aluminum, lithium, and SiO 2, respectively,
of forces on atoms calculated by the DC-LNO method is
sufficient for practical purposes, while it was remarked inSec. IIthat the forces on atoms in the DC-LNO method
are not calculated variationally. Sufficient accuracy of thecalculated forces is achieved by the use of large cutoff radiiin constructing the truncation clusters, which is realized byboth the introduction of LNOs and the massive parallelizationwith the multilevel parallelism.
To further demonstrate the applicability of the DC-LNO
method for MD simulations, we show radial distributionfunctions (RDFs) in the liquid phases of silicon, aluminum,lithium, and SiO
2in Fig. 10. Since the electronic structures
exhibit metallic features in the liquid phases of silicon, alu-minum, and lithium, the MD simulations can be consideredas a severe benchmark to validate the applicability of theDC-LNO method to metals. The cutoff radii r
Lwe used
corresponds to truncated clusters consisting of about 300atoms in the ideal bulk structures. It turns out that in all thecases the DC-LNO method reproduces well the results by theconventional O( N
3) diagonalization method, and that the ob-
tained RDFs are well compared to other computational results[75–78]. The considerable agreement between the DC-LNO
and conventional methods strongly implies that a sufficientaccuracy in reproducing at least RDF for MD simulations canbe attainable with a cutoff radius r
Lresulting in truncated
clusters consisting of about 300 atoms for not only insulatorsbut also metals. Thus adjusting the cutoff radius r
Lso that the
number of atoms in a truncated cluster can be ∼300 atoms
would be a compromise to balance the computational accu-racy and efficiency, while the difference between the DC andDC-LNO methods in terms of the computational efficiencymay not be significant for truncated clusters of this size. It is
crucial to minimize the elapsed time for realization of longtime MD simulations. With the computational condition, theelapsed time per SCF step for silicon is 1.5 (sec.) on averageusing 1280 MPI processes on the same machine used for thecalculations shown in Fig. 8.
V . CONCLUSIONS
We have presented an efficient O( N) method based on the
DC approach and a coarse graining of basis functions by lo-calized single-particle natural orbitals (LNOs) for large-scaleDFT calculations. A straightforward way to attain sufficientaccuracy in the DC method is to employ a relatively largecutoff radius for the truncation of a system, which is themost fundamental parameter in most of O( N) methods to
control the computational accuracy and efficiency. We haveadopted the rather brute force approach, and attempted todecrease the computational cost by introducing LNOs as basisfunctions in the long-range region of the truncated cluster,and to minimize the elapsed time in the computation with thehelp of a multilevel parallelization. The method of generatingLNOs is based on a low-rank approximation to the projectionoperator for the occupied space by a local eigendecompositionat each atomic site, and the band structure calculations withPAOs and LNOs clearly show that the resultant LNOs spanwell the occupied space of not only gapped systems but alsometals. It is also worth mentioning that the computational costof generating LNOs is almost negligible thank to the indepen-dent calculation at each atomic site. By replacing PAOs withLNOs in the long-range region of the truncated cluster in the
245137-10EFFICIENT O( N) DIVIDE-CONQUER METHOD WITH … PHYSICAL REVIEW B 98, 245137 (2018)
DC method, the computational cost of the DC method can
be reduced without largely sacrificing the accuracy. Notingthat the DC-LNO method holds the simple algorithm of theoriginal DC method suited for parallel calculations, we haveimplemented a multilevel parallelization using MPI by takingaccount of the atom level, spin level, and diagonalizationlevel. It was demonstrated that the speed-up of the DC-LNOmethod by the multilevel parallelization can be expected up toa specific number of MPI processes which corresponds to theproduct of the number of atoms, the multiplicity of spin index,and the number of CPU cores in a single computer node. Forexample, if a spin-polarized calculation is performed for asystem consisting of 1000 atoms on a parallel computer with20 CPU cores per node, a high parallel efficiency might beexpected up to 40 000 MPI processes. As a validation of theapplicability of the DC-LNO method, we have performed MDsimulations for liquid phases of an insulator, semiconductor,and metals, and confirmed that the RDFs calculated by theDC-LNOs are in good agreement with those by the con-ventional O( N
3) diagonalization method, which may lead to
its various applications to structural determinations of amor-phous and liquid structures of complicated materials [ 79–83].
Considering the simplicity and robustness of the algorithm,we conclude that the DC-LNO method is an efficient andaccurate approach to large-scale DFT calculations for a widevariety of materials including metals.
ACKNOWLEDGMENTS
This work was supported by Priority Issue (Creation of new
functional devices and high-performance materials to supportnext-generation industries) to be tackled by using Post ’K’Computer, MEXT, Japan. Part of the computation in the studywas performed using the computational facility of the JapanAdvanced Institute of Science and Technology.
APPENDIX: ASYMPTOTIC BEHA VIORS
OF THE OFF-DIAGONAL GREEN FUNCTIONS
As an example we show asymptotic behaviors of the off-
diagonal Green functions for an one-dimensional (1D) tight-binding (TB) model with a single sorbital on each site,
and relate the asymptotic behaviors to electronic structures ingapped and metallic systems. The analysis interprets evidentlythe oscillating behavior of the error in the total energy of themetallic systems as shown in Fig. 5, and the rapid convergence
in a high electronic temperature [ 24,33].
Let us consider an orthogonal chain model with the nearest
neighbor interaction tand the on-site energy εas defined by
ˆH=ε/summationdisplay
iˆc†
iˆci+t/summationdisplay
i(ˆc†
iˆci+1+H.c.), (A1)
where tis assumed to be positive. By tridiagonalizing the
Hamiltonian with a Lanczos algorithm starting from a site(i=0), and calculating the diagonal Green function via a
continued fraction using the recursion method [ 84,85], one
obtains a well known result for the diagonal Green functionG
00as follows:
G00(Z)=1/radicalbig
(Z−ε)2−4t2. (A2)The off-diagonal Green functions can be obtained by us-
ing a recurrence relation [ 86] derived from G(L)(Z)(Z−
H(L))=I, where G(L)andH(L)are the Green function and
Hamiltonian matrices represented by the Lanczos vectors, andby performing a back unitary transformation as
G
01(Z)=G00(Z)γ
2−1
2t, (A3)
G02(Z)=G00(Z)/parenleftbiggγ2
2−1/parenrightbigg
−γ
2t, (A4)
G03(Z)=G00(Z)/parenleftbiggγ3
2−3
2γ/parenrightbigg
−γ2−1
2t, (A5)
G04(Z)=G00(Z)/parenleftbiggγ4
2−2γ2+1/parenrightbigg
−γ3−2γ
2t,(A6)
where γ=(Z−ε)/tandG0jis the off-diagonal element of
Green function between the sites 0 and j. It turns out that
the off-diagonal Green functions can be expressed by G00and
γ. To see the asymptotic behavior of the off-diagonal Green
functions, by employing the following formula [ 87]:
1√
a2−x2=1
a∞/summationdisplay
n=0/parenleftbig2n
n/parenrightbig
4na2nx2n(A7)
with the radius of convergence |a|, we Taylor expand G00at
γ−1=0a s
G00(Z)=1
t∞/summationdisplay
n=0/parenleftbig2n
n/parenrightbig
4n2−2nγ−(2n+1)
=1
t/parenleftbigg1
γ+2
γ3+6
γ5+20
γ7+···/parenrightbigg
,(A8)
where the convergence is guaranteed for |γ|>2. By inserting
Eq. ( A8) into Eqs. ( A3)–(A6), and taking the leading terms,
we obtain the following relation:
G0j(Z)∝1
tγj+1. (A9)
Thus, we see that G0japproaches to zero asymptotically for
|γ|>2a sj→∞ . On the other hand the Green functions at
Z=εcorresponding to γ=0a r eg i v e nb y
G0(2k−1)(ε)=(−1)k
2t, (A10)
G0(2k)(ε)=(−1)kG00(ε), (A11)
where G00(ε)=−i
2t. It is found that G0jatγ=0 exhibits an
oscillating behavior as a function of j, and never decays.
We now relate the asymptotic behaviors of Green functions
to the calculation of density matrix, which is defined byEq. ( 24). Introducing the Matsubara expansion of the Fermi-
Dirac function, and changing the integration path with theCauchy theorem, one has [ 88]
ρ
0j=1
2δ0j+Im⎡
⎣2i
β∞/summationdisplay
p=1G0j(αp)⎤
⎦, (A12)
where αpare Matsubara poles located at μ+i(2p−1)π
βwith a
chemical potential of μandβ=1
kBT. The expression allows
245137-11TAISUKE OZAKI, MASAHIRO FUKUDA, AND GENGPING JIANG PHYSICAL REVIEW B 98, 245137 (2018)
2wr
wReIm(a) Metal
w2wr2wr
ReIm(b) Insulator
w
FIG. 11. Relation between the position of Matsubara poles
(black filled circles), the spectrum range on the real axis (blue filled
rectangles), and the convergent region of the off-diagonal Greenfunctions whose boundaries are shown by red circles in the complex
plane for (a) metal and (b) insulator. In the simple TB model, the
bandwidth wis given by 4 t, and the off-diagonal Green function G
0j
decays asymptotically as jincreases in the exterior region of the red
circles.
us to figure out a relation between the Matsubara poles, where
the Green functions are evaluated, and the convergent regionof the off-diagonal Green functions as illustrated in Fig. 11.
Remembering that in the simple TB model the bandwidth w
is given by 4 t, and assuming that the single band is half-
filled in the metallic case, we may have Matsubara polesin the red circle, which is the nonconvergent region of theoff-diagonal Green functions as shown in Fig. 11(a) . Since
the off-diagonal Green functions evaluated at the Matsubarapoles in the red circle do not simply decay in real spaceasjincreases, the truncation scheme commonly adopted in
most of O( N) methods should suffer from the long-rangecharacteristics of the off-diagonal Green functions, while the
effect can appear in a different way depending on underlyingprinciples of each O( N) method. In the DC-LNO method the
truncated eigenvalue problem H
(i)c(i)
μ=ε(i)
μS(i)c(i)
μis solved
for each atom i, and the integration of Eqs. ( 24) and ( 25)
can be easily performed on the real axis since we have theapproximate spectrum representation of Eq. ( 27). The way of
evaluating the density matrix is numerically equivalent to thecomputational method via a generalized formula of Eq. ( A12)
to the nonorthogonal basis set, where the Green functions forthe truncated problem are computed at each Matsubara poleby the inverse calculation, since the Green function computedthrough the spectrum representation is exactly the same asthe one computed by the inverse calculation. Therefore theoscillating behavior of error in the total energy calculation ob-served in Figs. 5(d)–5(f)should be attributed to the long-range
characteristics of the off-diagonal Green functions. It can alsobe understood that the use of a higher electronic temperaturesuppresses the deficiency since all the Matsubara poles canbe placed in the exterior region of the red circles beyond acritical temperature [ 24,33]. On the other hand, we model an
insulator by considering two bands as shown in Fig. 11(b) ,
where each of them is expressed by the 1D TB model and thebands are separated by a finite gap. Unlike the metallic case,all the Matsubara poles are located in the exterior region ofthe red circles. The feature guarantees that ρ
0jdecays as j
increases since all the Green functions in the summation ofEq. ( A12) decay as jincreases, theoretically justifying that
the truncation scheme is valid for gapped systems, althoughour benchmark calculations imply that the use of a large cutoffradius diminishes the effect of the long-range characteristicsof the off-diagonal Green functions even to metals at least forthe calculations of density matrix and energy density matrixin a practical sense.
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245137-14 |
PhysRevB.82.144527.pdf | Microscopic study of the superconducting state of the iron pnictide RbFe 2As2via muon
spin rotation
Z. Shermadini,1,2J. Kanter,3C. Baines,1M. Bendele,1,4Z. Bukowski,3R. Khasanov,1H.-H. Klauss,2H. Luetkens,1
H. Maeter,2G. Pascua,1B. Batlogg,3and A. Amato1
1Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
2Institut für Festkörperphysik, TU Dresden, D-01069 Dresden, Germany
3Laboratory for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland
4Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
/H20849Received 21 May 2010; revised manuscript received 16 August 2010; published 28 October 2010 /H20850
A study of the temperature and field dependence of the penetration depth /H9261of the superconductor
RbFe 2As2/H20849Tc=2.52 K /H20850was carried out by means of muon-spin rotation measurements. In addition to the
zero-temperature value of the penetration depth /H9261/H208490/H20850=267 /H208495/H20850nm, a determination of the upper critical field
Bc2/H208490/H20850=2.6 /H208492/H20850T was obtained. The temperature dependence of the superconducting carrier concentration is
discussed within the framework of a multigap scenario. Compared to the other “122” systems which exhibitmuch higher Fermi level, a strong reduction in the large gap BCS ratio 2 /H9004/k
BTcis observed. This is interpreted
as a consequence of the absence of interband processes. Indications of possible pair-breaking effect are alsodiscussed.
DOI: 10.1103/PhysRevB.82.144527 PACS number /H20849s/H20850: 74.70.Xa, 76.75. /H11001i, 74.25.Ha, 74.20.Mn
I. INTRODUCTION
The iron arsenide AFe2As2systems /H20849where Ais an
alkaline-earth element /H20850crystallize with the tetragonal
ThCr 2Si2-type structure /H20849space group I4/mmm /H20850.1The interest
for these compounds arises from the observation of super-conductivity with transition temperatures T
cup to 38 K upon
alkali-metal substitution for the A element2–4or partial
transition-metal substitution for iron.5A huge number of
studies were already devoted to unravel the properties oftheir superconducting ground state. However, some studiesare hampered by the fact that to date no clear picture couldbe drawn about the bulk character of the superconductivity.For example, in superconducting systems obtained from thesubstitution of the Aelement /H20849like K for Ba /H20850, muon-spin
rotation/relaxation /H20849
/H9262SR/H20850measurements studies clearly indi-
cate the occurrence of phase separation between magneticand superconducting phases.
6–8On the other hand, substitu-
tion performed on the superconducting plane, as cobalt sub-stitution for iron, does not reveal any phase separation asreported also by
/H9262SR.9
The alkali-metal iron arsenide RbFe 2As2was discovered
some years ago10but was only recently found, by Bukowski
et al. ,11to exhibit type-II bulk superconductivity below Tc
/H112292.6 K. The reported studies were hindered by a limited
temperature range of the equipment and the full developmentof the Meissner state could not be recorded. The estimatedvalue of the upper critical field at zero temperature, B
c2
/H112292.5 T, was obtained from magnetization measurements
performed at various field down to 1.5 K in the mixed stateand by assuming a temperature dependence provided by theWerthamer-Helfand-Hohenberg theory.
11
Compared to the better known compound BaFe 2As2,
RbFe 2As2possesses a lower Fermi level and is characterized
by the absence of magnetic instability. Furthermore, the elec-tron deficiency in RbFe
2As2leads also to a change /H20849i.e., a
decrease /H20850in the number of bands contributing to the super-conducting state, compared, for example, to Ba 1−xKxFe2As2.
Hence, one expects a strong decrease in the contribution ofthe electronlike bands at the M point of the Fermi surface.Such a decrease has been observed by angle-resolved photo-emission spectroscopy
12in the analog system KFe 2As2,
which also presents a case of naturally hole- /H20849over /H20850doped sys-
tem when compared to the alkaline-earth “122” iron-basedsuperconductors.
As exemplified by a number of recent studies, the
/H9262SR
technique is very well suited to investigate the superconduct-ing properties of iron-based systems /H20849see, for example, Ref.
13/H20850. In addition, due to its comparatively low upper critical
field B
c2and its reduced Tc, the system RbFe 2As2opens a
unique opportunity to fully study the B-Tphase diagram of
an iron-arsenide compound.
In this paper, we report on a detailed study of the tem-
perature and field dependence of the magnetic penetrationdepth of RbFe
2As2, which is closely related to the supercon-
ducting carrier concentration.
II. EXPERIMENT
Polycrystalline samples of RbFe 2As2were synthesized in
two steps as reported recently.11The/H9262SR measurements
were performed at the /H9266M3 beamline of the Paul Scherrer
Institute /H20849Villigen, Switzerland /H20850, using the general purpose
spectrometer instrument /H20849for temperatures down to 1.6 K and
field up to 0.6 T /H20850as well as the low temperature facility
instrument /H20849for temperatures down to 0.02 K and higher
fields /H20850. Both zero-field /H20849ZF/H20850and transverse-field /H20849TF/H20850/H9262SR
measurements were performed. Additional transport studieswere performed on the very same sample at the ETH-Zürichusing an ac transport option of a Quantum Design 14T-PPMS.
III. RESULTS AND DISCUSSION
To exclude the occurrence of any magnetic contributions
of the Fe ions at low temperature, we performed first ZFPHYSICAL REVIEW B 82, 144527 /H208492010 /H20850
1098-0121/2010/82 /H2084914/H20850/144527 /H208495/H20850 ©2010 The American Physical Society 144527-1measurements above and below Tc. As exemplified by the
data reported in Fig. 1/H20849a/H20850, no sign of static magnetism could
be detected on the ZF response RbFe 2As2. The data are well
described by a standard Kubo-Toyabe depolarizationfunction,
14reflecting the field distribution at the muon site
created by the nuclear moments. The marginal increase in thedepolarization rate, not related to the superconducting tran-sition, possibly points to a slowing down of the magneticfluctuations.
Figure 1/H20849b/H20850exhibits the TF
/H9262SR time spectra measured in
an applied field of 0.01 T, above /H20849T=4 K /H20850and below
/H20849T=0.02 K /H20850the superconducting transition temperature. The
strong muon-spin depolarization at low temperatures reflectsthe formation of the flux-line lattice /H20849FLL /H20850in the supercon-
ducting state. The long-lived component detectable at lowtemperatures is due to a background contribution from thesample holder. In a polycrystalline sample the magnetic pen-etration depth /H9261/H20849and consequently the superconducting car-
rier concentration n
s/H110081//H92612/H20850can be extracted from the
Gaussian muon-spin depolarization rate /H9268s/H20849T/H20850/H20851see also be-
low Eq. /H208492/H20850/H20852, which reflects the second moment /H20849/H9268s2//H9253/H92622/H20850of
the magnetic field distribution due to the FLL in the mixedstate. The TF data were analyzed using the polarization func-tion
A
0P/H20849t/H20850=Asexp/H20875−/H20849/H9268s2+/H9268n2/H20850t2
2/H20876cos /H20849/H9253/H9262Bintt+/H9272/H20850
+Ashexp/H20873−/H9268sh2t2
2/H20874cos /H20849/H9253/H9262Bsht+/H9272/H20850. /H208491/H20850
The first term on the right-hand side of Eq. /H208491/H20850represents
the sample contribution, where Asdenotes the initial asym-
metry connected to the sample signal; /H9268sis the Gaussian
relaxation rate due to the FLL; /H9268nis the contribution to the
field distribution arising from the nuclear moment and whichis found to be temperature independent, in agreement withthe ZF results; B
intis the internal magnetic field, sensed by
the muons; and /H9272is the initial phase of the muon-spin en-
semble. The second term reflects the muons stopping in thesilver sample holder, where Ashdenotes the initial asymmetry
connected to the holder signal; /H9268shis the relaxation rate due
to the nuclear moments /H20849which is very close to zero in this
case /H20850; and Bshis the magnetic field in the sample holder,
which has essentially the value of the external field.
In Fig. 2, we report the temperature dependence of /H9268s
extracted from TF- /H9262SR measurements in four different
fields. We note first that the perfect fits obtained by assuminga Gaussian field distribution of the FLL point to a ratherlarge anisotropy of the magnetic penetration depth in oursystem. This is confirmed by recent
/H9262SR measurements per-
formed on hole- and electron-doped 122 systems.8,9As ex-
pected, /H9268sis zero in the paramagnetic state and starts to
increase below Tc/H20849B/H20850when the FLL is formed. Upon lower-
ing the temperature, /H9268sincreases gradually reflecting the de-
crease in the penetration depth or, alternatively, the increasein the superconducting density. The overall decrease in
/H9268sat
very low temperatures observed upon increasing the appliedfield is a direct consequence of the decrease in the width ofthe internal field distribution when increasing the field to-ward B
c2. In order to quantify such an effect, one can make
use of the numerical Ginzburg-Landau model, developed byBrandt.
15This model allows one to calculate the supercon-
ducting carrier concentration with good approximationwithin the local /H20849London /H20850approximation /H20849/H9261/H11271
/H9264,/H9264is the co-
herence length /H20850. This model predicts the magnetic field de-
pendence of the second moment of the magnetic field distri-bution or, alternatively, of the
/H9262SR depolarization rate,
which can be expressed as
/H9268s/H20849/H9262s−1/H20850= 4.83 /H11003104/H208491−B/Bc2/H20850
/H11003/H208511 + 1.21 /H208491−/H20881B/Bc2/H208503/H20852/H9261−2/H20849nm/H20850. /H208492/H20850
The field dependence of /H9268swas measured down to 0.02 K
and, as illustration, the inset of Fig. 2exhibits the measure-
ments at 1.6 K. For each data point, the sample was fieldcooled from above T
cand the recorded /H9262SR spectra were
analyzed with Eq. /H208491/H20850. At each temperature, the field depen-
dence of /H9268swas analyzed with Eq. /H208492/H20850by leaving the param-FIG. 1. /H20849Color online /H20850Typical /H9262SR spectra recorded above and
below Tc, in: /H20849a/H20850zero field and /H20849b/H20850transverse field.FIG. 2. /H20849Color online /H20850Temperature dependence of the depolar-
ization rate due to the FLL in RbFe 2As2and obtained in fields of
1.5, 0.5, 0.1, and 0.01 T /H20849lines are guides to the eyes /H20850. Inset: field
dependence of /H9268sobtained at 1.6 K and analyzed using the Eq. /H208492/H20850.SHERMADINI et al. PHYSICAL REVIEW B 82, 144527 /H208492010 /H20850
144527-2eters/H9261andBc2free. The corresponding fitted values of the
penetration depth /H20849related to the superconducting carrier con-
centration /H20850and of the upper critical field are reported in Figs.
3/H20849b/H20850and4. As demonstrated by Fig. 3/H20849b/H20850, the values of Bc2
obtained by fitting the field dependence of /H9268s/H20849assuming a
field-independent penetration depth /H20850agree very well with the
values of Bc2obtained from the magnetoresistivity and the
ones deduced directly from the temperature dependence of
/H9268s/H20849see Fig. 2/H20850. This is a strong support that the assumption
of a field-independent penetration depth is indeed valid. Thisrules out the possibility that RbFe
2As2is a nodal supercon-
ductor, since a field should have induced excitations at thegap nodes due to nonlocal and nonlinear effects, thus reduc-ing the superconducting carrier concentration n
sand there-
fore affecting /H9261/H20849see, for example, Ref. 16/H20850.
By looking at the temperature dependence of /H9261−2obtained
using Eq. /H208492/H20850with the values of the parameter Bc2/H20849T/H20850pre-
sented in Fig. 3/H20849b/H20850, the zero-temperature value of the pen-
etration depth /H9261/H208490/H20850=267 /H208495/H20850nm can be deduced. The ob-
tained temperature dependence of /H9261−2was analyzed, in a first
step, within the framework of a BCS single s-wave symme-
try superconducting gap /H9004/H20849see Fig. 4/H20850, using the form17
/H9261−2/H20849T/H20850
/H9261−2/H208490/H20850=1−2
kBT/H20885
/H9004/H11009
f/H20849/H9280,T/H20850/H208511−f/H20849/H9280,T/H20850/H20852d/H9280, /H208493/H20850
where f/H20849/H9280,T/H20850=/H208491+exp /H20851/H20881/H92802+/H9004/H20849T/H208502/kBT/H20852/H20850−1and with a stan-
dard BCS temperature dependence for the gap function. Asevidenced in Fig. 4, this analysis is not satisfactory. We note
also that a d-wave symmetry model does not fit the data,
confirming at posteriori the discussion of a field-independent
penetration depth. These results are actually not unexpected,as there are growing evidences that several disconnectedFermi-surface sheets contribute to the superconductivity, asrevealed by angle-resolved photoemission spectroscopy,
12re-
sulting into two distinct values of superconducting gaps.Hence, in a second step, the experimental /H9261
−2/H20849T/H20850data were
analyzed by assuming two independent contributions withdifferent values /H9004
iofs-wave gaps.8,18,19In Fig. 4the solid
line shows a s+smultigap function which fits to the experi-
mental data rather well. The parameters extracted from the fitare/H9004
1/H208490/H20850=0.15 /H208492/H20850meV for the small gap value /H20849contribut-
ing/H9275=36% to the total amount of ns/H20850and /H90042
=0.49 /H208494/H20850meV for the larger one. However, note that ac-
cording to Eq. /H208493/H20850,/H9261−2is insensitive to the phase of the
superconducting gap /H20849s/H20850. By considering the intrinsic hole
doping in RbFe 2As2compared to the optimally doped 122
iron-based system, it is natural to consider that the gaps val-ues are connected, respectively, to the outer /H20849
/H9252/H20850and inner /H20849/H9251/H20850
holelike bands at the /H9003point of the Fermi surface. In this
frame, RbFe 2As2can be considered as hole overdoped with
electronlike /H9253and/H9254bands at the M point, which shift to the
unoccupied side. Note that in optimally doped 122 systems,one observes the occurrence of
/H9280hole bands /H20849so-called
“blades” /H20850around the M point, which also slightly contribute
to the superconducting carrier concentration.
An additional support for a two-gap superconducting state
could be provided by the observed positive curvature of theB
c2/H20849T/H20850near Tc, in sharp contrast to the usual Bc2BCS tem-
perature dependence /H20851see Fig. 3/H20849b/H20850/H20852. Note first that the values
ofBc2extracted from the fit with Eq. /H208492/H20850are in perfect agree-
ment with: /H20849i/H20850the values corresponding to the complete sup-
pression of the electrical resistivity in field and /H20849ii/H20850to theFIG. 3. /H20849Color online /H20850/H20849a/H20850Field dependence of the electrical
resistivity. /H20849b/H20850Upper critical field for RbFe 2As2. The open circles
are obtained by analyzing the field dependence of /H9268susing Eq. /H208492/H20850,
as explained in the text. The diamonds are the value obtained byanalyzing the temperature dependence of
/H9268s. The stars correspond
to the complete disappearance of the resistivity in field. The line isa guide to the eyes.FIG. 4. /H20849Color online /H20850Magnetic penetration depth as a function
of temperature. Above 0.5 K, the values obtained with Eq. /H208492/H20850co-
incides with the values measured in a field of 0.01 T, and only theselatter are plotted for this temperature range. The red dashed linecorresponds to a BCS s-wave gap symmetry whereas the solid one
to represents a fit using a two-gap s+smodel. The inset exhibits the
penetration depth as a function of /H20849T/T
c/H208502.MICROSCOPIC STUDY OF THE SUPERCONDUCTING … PHYSICAL REVIEW B 82, 144527 /H208492010 /H20850
144527-3values obtained by analyzing the temperature dependence of
/H9268sin different magnetic fields /H20849see Fig. 2/H20850. An additional
indication that bulk superconductivity occurs when the elec-trical resistivity completely vanishes is provided by specific-heat measurements
20performed in zero-applied field for
which the observed Tccorresponds to 2.52 /H208491/H20850K.
Similar positive curvature of the Bc2/H20849T/H20850near Tcwere ob-
served in MgB 2/H20849Refs. 21and22/H20850and in the borocarbides,23
where it was explained within a two-gap model. However,
one should keep in mind that alternative explanations for theobserved positive curvature in B
c2/H20849T/H20850are possible and that
complementary measurements, as here our /H9261−2/H20849T/H20850data, are
necessary to draw conclusions.
If on one hand, the two-gap model scenario appears to
best fit the temperature dependence of the penetration depth,on the other hand one could argue that it does not appearfully consistent with the observation that the field evolutionof the field distribution follows Eq. /H208492/H20850. Hence for a two-gap
model, one expects a deviation from the simple field depen-dence reflecting the occurrence of distinct lengths scales
/H9264i
for both gaps /H20849associated to the coherence length, for a clean
single gap system /H20850. Such behavior is, for example, clearly
observed on the archetypical two-gap superconductorMgB
2.24The experimental observation that Eq. /H208492/H20850repro-
duces our data indicates a small difference between the /H9264i
parameters for both bands. This is also inline with the verygood agreement between the extracted values of B
c2with Eq.
/H208492/H20850and the observed values by resistivity. In this frame, we
also mention that ARPES measurements25on members of the
122 family indicate that the Fermi velocity of the inner/H9003-barrel band /H20849
/H9251band /H20850is substantially higher that the one for
the outer /H9003-barrel band /H20849/H9252band /H20850, which therefore weakens
the difference of the gap values on the /H9264iparameters /H20849as/H9264
/H11008/H20855vF/H20856//H9004/H20850. Finally, we note that the observed depolarization
rate in RbFe 2As2is about 40 times weaker than the one
reported for MgB 2, hampering therefore the determination of
possible distinct /H9264ilength scales.
For completeness, we discuss now the slight deviation
observed at very low temperatures from the s+sfit and the
/H9261−2/H20849T/H20850data. Recently, it was shown that the observation of
universal scalings in the whole iron-pnictides superconduct-
ors, for the specific-heat jump /H20849/H9004C/H11008Tc3/H20850and the slope of
upper critical field at Tc/H20849dBc2/dT/H11008Tc/H20850could be interpreted
as signatures for strong pair-breaking effects,26as, for ex-
ample, magnetic scattering. In the same frame it wasdeduced
27that such an effect should lead to a very low-
temperature dependence of the penetration depth deviatingfrom an usual exponential behavior and transforming into aquadratic one, i.e., /H9261/H11008T
2, which is indeed reported in a num-
ber of studies /H20849see, for example, Refs. 28and29/H20850. In the inset
of Fig. 4we report the extracted penetration depth as a func-
tion of /H20849T/Tc/H208502. The good scaling is inline with the presence
of magnetic scattering in RbFe 2As2, as previously reported
for hole- or electron-doped 122 systems.27
IV. CONCLUSION
To conclude, /H9262SR measurements were performed on a
RbFe 2As2polycrystalline sample. From the temperature and
field dependence of the superconducting response of the
/H9262SR signal, the values of the upper critical field and of the
magnetic penetration depth could be extracted. The zero-temperature values of B
c2/H208490/H20850and/H9261/H208490/H20850were estimated to be
2.6/H208492/H20850T and 267 /H208495/H20850nm, respectively. The temperature de-
pendence of the penetration depth and similarly of the super-conducting carrier concentration are reproduced assuming amultigap model with possibly pair-breaking effects at lowtemperatures. The multigap scenario is supported by the ob-servation of a clear positive curvature on the temperaturedependence of the upper critical field. We attribute thesegaps to the holelike bands around the /H9003point of the Fermi
surface and possibly also to the hole-bands blades around theM point. Assuming that the
/H9253and/H9254electronlike bands
around the M point are in the unoccupied side, one wouldexpect an absence of nesting conditions in RbFe
2As2. The
consequence would be an absence of magnetic order, as con-firmed by our ZF data, and a strong decrease in the interbandprocesses between the
/H9251and/H9253/H20849/H9254/H20850bands. In this frame, it is
remarkable to see that the ratio between the gaps values isdecreased by a factor more than 2 compared to optimallydoped 122 systems. Similarly, we note that the BCS ratio2/H9004/k
BTcfor the small gap that we assign to the /H9252band is
almost identical to the values observed for optimally dopedBa
1−xKxFe2As2, i.e., 2 /H90041/kBTc/H112291.4. On the other side, for
the large gap of the /H9251band, this ratio is strongly reduced,8,25
confirming therefore the possible role played by interband
processes in optimally hole-doped iron-based 122 supercon-ductors.
ACKNOWLEDGMENTS
Part of this work was performed at the Swiss Muon
Source /H20849S/H9262S/H20850, Paul Scherrer Institute /H20849PSI, Switzerland /H20850.
The work of M.B. was supported by the Swiss National Sci-ence Foundation. The work at the IFW Dresden has beensupported by the DFG through FOR 538.
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144527-5 |
PhysRevB.74.195110.pdf | Mott transition of MnO under pressure: A comparison of correlated band theories
Deepa Kasinathan,1J. Kuneš,1,2K. Koepernik,3Cristian V . Diaconu,4Richard L. Martin,4Ionu ţD. Prodan,5
Gustavo E. Scuseria,5Nicola Spaldin,6L. Petit,7T. C. Schulthess,7and W. E. Pickett1
1Department of Physics, University of California Davis, Davis, California 95616, USA
2Institute of Physics, ASCR, Cukrovarnická 10, 162 53 Praha 6, Czech Republic
3IFW Dresden, P .O. Box 270116, D-01171 Dresden, Germany
4Theoretical Division, MSB269, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
5Department of Chemistry, Rice University, Houston, Texas 77005, USA
6Materials Research Laboratory and Materials Department, University of California Santa Barbara, Santa Barbara,
California 93106, USA
7Computer Science and Mathematics Division and Center for Nanophase Materials Science, Oak Ridge National Laboratory,
Oak Ridge, Tennessee 37831-6493, USA
/H20849Received 18 May 2006; revised manuscript received 1 October 2006; published 15 November 2006 /H20850
The electronic structure, magnetic moment, and volume collapse of MnO under pressure are obtained from
four different correlated band theory methods; local density approximation+Hubbard U/H20849LDA+ U/H20850, pseudo-
potential self-interaction correction /H20849pseudo-SIC /H20850, the hybrid functional /H20849combined local exchange plus
Hartree-Fock exchange /H20850, and the local spin density SIC /H20849SIC-LSD /H20850method. Each method treats correlation
among the five Mn 3 dorbitals /H20849per spin /H20850, including their hybridization with three O 2 porbitals in the valence
bands and their changes with pressure. The focus is on comparison of the methods for rocksalt MnO /H20849neglect-
ing the observed transition to the NiAs structure in the 90–100 GPa range /H20850. Each method predicts a first-order
volume collapse, but with variation in the predicted volume and critical pressure. Accompanying the volumecollapse is a moment collapse, which for all methods is from high-spin to low-spin
/H208495
2→1
2/H20850, not to nonmagnetic
as the simplest scenario would have. The specific manner in which the transition occurs varies considerablyamong the methods: pseudo-SIC and SIC-LSD give insulator-to-metal, while LDA+ Ugives insulator-to-
insulator and the hybrid method gives an insulator-to-semimetal transition. Projected densities of statesabove and below the transition are presented for each of the methods and used to analyze the character of eachtransition. In some cases the rhombohedral symmetry of the antiferromagnetically ordered phase clearlyinfluences the character of the transition.
DOI: 10.1103/PhysRevB.74.195110 PACS number /H20849s/H20850: 71.10. /H11002w, 71.27. /H11001a, 71.30. /H11001h, 75.20.Hr
I. INTRODUCTION
For 50 years the metal-insulator transition has been one of
the central themes1of condensed matter physics. The type
we address here does not involve spatial disorder nor changeof the number of charge carriers per cell; the competing ten-dencies arise solely from the kinetic and potential energies inthe Hamiltonian, favoring itineracy and localization, respec-
tively, and the many real-material complexities that arise.The classic categorization is that of the Mott transition,treated in its most basic form with the single-band Hubbardmodel. Much has been learned about this model, butthere are very few physical systems that are modeled faith-fully by such a model. Real materials involve multiorbitalatoms and thus extra internal degrees of freedom, and anenvironment that is often very active and may even react tothe configuration of active sites.
MnO is a transition metal monoxide /H20849TMO /H20850with open 3 d
shell that qualifies as one of the simpler realizations of aprototypical, but real, Mott insulator. It is, certainly, a mul-tiorbital system with the accompanying complexities, but thehalf-filled 3 dbands lead to a spherical, spin-only moment at
ambient pressure. Applying pressure to such a system leadsto a number of possibilities, including insulator-metal transi-tion, moment reduction, volume collapse if a first-order tran-sition /H20849electronic phase change /H20850occurs, and any of these may
be accompanied by a structural phase transition, that is, achange in crystal symmetry. The 3 dbandwidth Wof sucha Mott insulator is very susceptible to applied pressure, and
is one of the main determining factors of the strength ofcorrelation effects.
The “closed subshell” aspect makes MnO an atypical 3 d
monoxide, as shown for example by Saito et al. , who
compiled
2effective parameters for this system from spectro-
scopic information. An effective intra-atomic Coulomb re-pulsion energy as defined by them, for example, is roughlytwice as large as for the other 3 dmonoxides. The complexity
of this compound can be considered in terms of the energyscales that are involved in the electronic structure and mag-netism of these oxides. These include the 3 dbandwidth W,
an intra-atomic Coulomb repulsion strength U, an intra-
atomic d-dexchange energy /H20849Hund’s rule J, or exchange
splitting /H9004
ex/H20850, the crystal field splitting /H9004cf=/H9255eg−/H9255t2g, and the
charge transfer energy /H9004ct/H11013/H9255d−/H9255p/H20849the difference in mean
Mn 3 da n dO2 psite energies /H20850. In the magnetically ordered
antiferromagnetic /H20849AFM /H20850state, there are further symmetry
lowering and ligand field subsplittings involving 3 d-2phy-
bridization. All of these scales change as the volumechanges, making the pressure-driven Mott transition achallenging phenomenon to describe.
Although the objective of the current paper is to compare
methods within the fcc /H20849rocksalt /H20850phase, it is useful first
to recount what is known about the Mott transition at thistime. The current experimental information, mostly at roomtemperature, on the behavior of MnO under pressure is sum-PHYSICAL REVIEW B 74, 195110 /H208492006 /H20850
1098-0121/2006/74 /H2084919/H20850/195110 /H2084912/H20850 ©2006 The American Physical Society 195110-1marized in Fig. 1. Resistance measurements3provided the
first evidence of the Mott transition in MnO near 100 GPa.Recent x-ray diffraction and emission spectroscopy measure-ments of the crystal structure and magnetic moment by Yooet al.
4have clarified the behavior. Around 90 GPa there is a
structural transformation from the distorted B1 /H20849rocksalt /H20850
phase to the B8 /H20849NiAs /H20850structure. This structure change is
followed at 105 GPa by the Mott transition, consisting of asimultaneous volume collapse and moment collapse signify-ing a qualitative change in the electronic structure of thecompound.
On the theoretical side, little is known about how the Mott
transition occurs in a real multiband TMO in spite of theextensive studies of the Mott transition in the single-bandHubbard model, which has a simple spin-half moment atstrong coupling and half-filling. The numerous energy scales
listed above, and the S=
5
2moment on Mn arising from the
five 3 delectrons, allow many possibilities for how the mo-
ment might disintegrate as the effective repulsion decreases.The high pressure limit is clear: a nonmagnetic 3 d-2pband
metal in which kinetic energy overwhelms potential energy.This is the competition studied in the /H20849simplified /H20850Hubbard
model. The multiband nature has attracted little attention un-til recently, when for example the question of possibleorbital-selective Mott transitions
6,7have aroused interest.
One can imagine one scenario of a cascade of moment re-
ductions S=5
2→3
2→1
2before complete destruction of mag-
netism, as electrons use their freedom to flip spins /H20849as some
competing energy overcomes Hund’s rule, for example /H20850.I n
such a scenario there is the question of which orbital flips itsspin at each spin-flip, which involves a question of orbitalselection and ordering. At each flip the system loses ex-change /H20849potential /H20850energy while gaining kinetic energy /H20849or
correlation energy through singlet formation /H20850. The manner in
which kinetic energy changes is difficult to estimate becausesubband involvement means that there is no longer a singlebandwidth Wthat is involved. The increasing hybridization
with O 2 pstates under pressure strongly affects the kinetic
energy, directly and through superexchange /H20849a kinetic energy
effect /H20850.
It is known that conventional band theory /H20851local density
approximation /H20849LDA /H20850/H20852that does so well for so many mate-
rials gives poor results for 3 dmonoxides in many respects,
and some predictions are qualitatively incorrect /H20849viz.no band
gap when there should be a large gap of several eV /H20850. Thus
even at the density functional level /H20849ground state energy,
density, and magnetization /H20850some correlated approach is re-
quired. In the past 15 years several approaches, which werefer to as correlated band theories, have been put forward,and each has had its successes in providing an improveddescription of some aspects of correlated TMOs. Althoughcommonly called mean-field approaches with which theyshare many similarities, they are not mean-field treatments ofany many-body Hamiltonian. Rather, they are energy func-tionals based on the complete many-body Hamiltonian,which must then be approximated due to limited knowledgeof the exchange-correlation functional.
In this paper, we provide a close comparison of certain
results from four such methods: full potential LDA+ U, the
hybrid exchange functional /H20849HSE /H20850approach, the self-
interaction-corrected local spin-density method /H20849SIC-LSD /H20850,
and a nonlocal pseudopotential-like variation of SIC/H20849pseudo-SIC /H20850. Our main focus is to compare the predicted
changes in energy, moment, and volume within the AFMIIrocksalt phase of MnO. To keep the comparison manageablewe confine our attention to the rocksalt phase, since our em-phasis is on comparison of methods and not yet the ultimatebut more daunting task of modeling structural changes thatmay precede, or accompany, the Mott transition.
II. STRUCTURE AND SYMMETRY
Rocksalt MnO has an experimental equilibrium lattice
constant a0=4.45 Å /H20849cubic cell volume V0=88.1 Å3/H20850. Den-
sity functional theory, like Hartree-Fock theory, deals in itsmost straightforward form with ground state properties, i.e.,zero temperature. The ground state is known to be the AFMIIphase in which /H20855111/H20856layers have spins aligned, and succes-
sive layers are antiparallel. The resulting symmetry is rhom-bohedral, with Mn ↑and Mn ↓being distinct sites /H20849although
related through a translation+spin-flip operation /H20850. Thus,
while most of the lore about transition metal monoxides isbased upon cubic symmetry of the Mn /H20849and O /H20850ion, in the
ordered state the electronic symmetry is reduced. It is obvi-ous that individual wave functions will be impacted by thissymmetry, viz.fourfold symmetry around the cubic axes is
lost. It has been emphasized by Massidda et al.
8that zone-
integrated, and even spin-integrated, quantities show theeffects of this symmetry lowering; for example, Born effec-tive charges lose their cubic symmetry. Since this issuearises in the interpretation of our results, we provide somebackground here.
In cubic symmetry the Mn 3 dstates split into the
irreducible representations /H20849irreps /H20850denoted by t
2gand eg.
Rhombohedral site symmetry results in the three irreducible
FIG. 1. /H20849Color online /H20850A schematic /H20849conceptual /H20850P-Tphase dia-
gram for MnO based on recent high pressure work at LawrenceLivermore National Laboratory /H20849Refs. 3and 4/H20850. The thick phase
line signifies the first-order Mott transition which simultaneouslyaccompanies the loss of Mn magnetic moment, a large volume col-lapse, and metalization. This transition should end at a critical point/H20849solid circle /H20850. The gray fan above the critical point signifies a region
of crossover to metallic behavior at high temperature. The star de-notes earlier shock data, /H20849Ref. 5/H20850, and the diamond marks the am-
bient pressure Néel temperature. Only the distorted B1 /H20849dB1 /H20850phase
is magnetically ordered.KASINATHAN et al. PHYSICAL REVIEW B 74, 195110 /H208492006 /H20850
195110-2representations ag,eg,1, and eg,2, the latter two being
twofold degenerate. The coordinate rotation from cubic torhombohedral /H20849superscript candr, respectively /H20850is, with a
specific choice for the orientation of the xandyaxes in the
rhombohedral system,
/H20898xr
yr
zr/H20899=/H208981
/H2088161
/H208816−/H208812
/H208813
−1
/H2088121
/H2088120
1
/H2088131
/H2088131
/H208813/H20899/H20898xc
yc
zc/H20899.
Applying this rotation of coordinates gives the 3 dorbitals in
the rhombohedral frame in terms of those in the cubic frame/H20849d
z2/H11013d3z2−r2/H20850,
dxyr=1
/H208813/H20849dxzc−dyzc−dx2−y2c/H20850, /H208491/H20850
dyzr=1
/H208816/H20849dyzc−dxzc/H20850−/H208812
3dx2−y2c, /H208492/H20850
dxzr=/H208812
3dxyc−1
3/H208812/H20849dxzc+dyzc/H20850−/H208812
/H208813dz2c, /H208493/H20850
dx2−y2r=−1
3/H20873dxzc+dyzc+2
3dxyc/H20874−1
/H208813dz2c, /H208494/H20850
dz2r=1
/H208813/H20849dxyc+dyzc+dxzc/H20850. /H208495/H20850
In rhombohedral coordinates it is useful to categorize the
3dorbitals in terms of their orbital angular momentum pro-
jections along the rhombohedral axis, dz2r↔m/H5129=0;
dxzr,dyzr↔m/H5129=±1 ; dxyr,dx2−y2r↔m/H5129= ±2. It is easy to see that
/H20841m/H5129/H20841specifies groups of states that only transform into
combinations of themselves under trigonal point groupoperations.
Note that the unique a
gsymmetry state in rhombohedral
coordinates is the fully symmetric combination of the cubict
2gstates. The other two irreps are both egdoublets. While
/H20841m/H5129/H20841=1 and /H20841m/H5129/H20841=2 form representations of these irreps, if
there are components of the crystal field that are not diagonalin the L=/H208492,m
/H5129/H20850basis, these states will mix. Then each of
the resulting /H20849orthonormal /H20850irreps eg,1, and eg,2will contain
both /H20841m/H5129/H20841=1 and /H20841m/H5129/H20841=2 components. Such mixing does oc-
cur in MnO and complicates the symmetry characterizationof the 3 dstates.
III. METHODS
A. LDA calculations
For LDA band structure plot /H20849Fig.2/H20850we used version 5.20
of the full-potential local orbital band structure method/H20851FPLO /H20849Ref. 9/H20850/H20852. Relativistic effects were incorporated on a
scalar-relativistic level. We used a single numerical basis setfor the core states /H20849Mn 1 s2s2pa n dO1 s/H20850and a double nu-
merical basis set for the valence sector including two 4 sand
3dradial functions, and one 4 pradial function, for Mn, and
two 2 sand 2 pradial functions, and one 3 dradial function,
for O. The semicore states /H20849Mn 3 s3p/H20850are treated as valence
states with a single numerical radial function per nlshell.
The local density exchange-correlation functional PW92 ofPerdew and Wang
10was used.
B. LDA+ Umethod
The LDA+ Uapproach of including correlation effects is
to/H208491/H20850identify the correlated orbital, 3 din this case, /H208492/H20850aug-
ment the LDA energy functional with a Hubbard-like term/H20849Coulomb repulsion U/H20850and Hund’s /H20849exchange J/H20850energy be-
tween like spins, /H208493/H20850subtract off a spin-dependent average of
this interaction energy to keep from double-counting repul-sions /H20849once in LDA fashion, once in this Uterm /H20850, and /H208494/H20850
include the correlated orbital occupation numbers in the self-consistency procedure, which leads to an orbital-dependentHartree-Fock-like potential acting on the correlated orbitals.The addition to the energy functional has the schematic form
E
U=1
2/H20858
m/H9268/HS11005m/H11032/H9268/H11032/H20849U−J/H9254/H9268/H9268/H11032/H20850/H20849nm/H9268nm/H11032/H9268/H11032−n¯/H9268n¯/H9268/H11032/H20850. /H208496/H20850
We actually use the coordinate-system independent form
of LSDA+ U /H20849Refs. 11–13/H20850implemented in FPLO,9
which leads to four mindices on UandJwhich for simplic-
ity have not been displayed /H20849nor has the full off-diagonal
form of the occupation matrices nmm/H11032/H9268/H20850. This treatment of
the on-site interactions UandJincorporates on-site correla-
tion effects in the Mn 3 dshell. We have used the so-called
atomic-limit /H20849strong local moment /H20850form of the double-
FIG. 2. /H20849Color online /H20850LDA band structure of AFM MnO along
rhombohedral symmetry lines, calculated with the FPLO method/H20849Ref. 9/H20850, with horizontal line /H20849“Fermi level” /H20850placed at the top of the
gap. The /H9003-Tlies along the rhombohedral axis, while /H9003-Llies in the
basal plane. The O 2 pbands lie in the −8 eV to −3.5 eV range,
with the majority Mn 3 dbands just above /H20849−3 eV to −1 eV /H20850. The
five minority 3 dbands are just above the gap. Note the small mass,
free-electron-like band that lies below the unoccupied 3 dbands at
the/H9003point.MOTT TRANSITION OF MnO UNDER PRESSURE: A … PHYSICAL REVIEW B 74, 195110 /H208492006 /H20850
195110-3counting correction, the last term in Eq. /H208496/H20850. This form is
appropriate for the high-spin state, but it is less obviously sofor the low-spin state that is found at reduced volumes. TheSlater parameters were chosen according to U=F
0=5.5 eV,
J=1
14/H20849F2+F4/H20850=1 eV, and F2/F4=8/5.
The shape of the basis orbitals has been optimized yield-
ing a sufficient accuracy of the total energy over the range ofgeometries considered in this work. The kintegrals are per-
formed via the tetrahedron method with an irreducible meshcorresponding to 1728 /H2084912
3/H20850points in the full Brillouin zone.
C. SIC-LSD method
The SIC-LSD method addresses the unphysical self-
interaction in the LDA treatment of localized states. Itinerantstates, being spread over space without finite density in anygiven region, do not experience this self-interaction withinthe LDA treatment. Should there be localized states, confinedto some region and giving a finite density, they will suffer anunphysical self-interaction in the LSD method. This issuethen clearly arises in the itinerant-localized transition inMnO and other correlated systems. The basic premise of theSIC-LSD method is that localized electrons should experi-ence a different potential from that of itinerant electrons,
14–16
analogous to that of an atomic state whose self-interaction
must be removed. Then electrons on the surrounding atomsare allowed to accommodate self-consistently. This distinc-tion of localized versus itinerant state is addressed inSIC-LSD by extending the energy functional in the form
E
SIC-LSD=ELSD−/H20858
/H9251occ.
/H9254/H9251SIC,
/H9254/H9251SIC=U/H20851n/H9251/H20852+ExcLSD/H20849n/H9251/H20850. /H208497/H20850
Here U/H20849n/H20850represents the Hartree /H20849classical Coulomb /H20850energy
of a density n/H20849r/H20850. The self-Coulomb energy U/H20849n/H9251/H20850and self-
exchange-correlation ExcLSD/H20849n/H9251/H20850energies are subtracted off for
each localized state /H9274/H9251with density n/H9251. Whether states are
localized or not /H20849with nonzero, respectively, zero self-
interaction /H20850is determined by minimization of this functional,
allowing localized as well as itinerant states /H9274/H9251. Since the
correction vanishes for itinerant states, the sum finally in-cludes only the self-consistently localized states. The local-ized and itinerant states are expanded in the same basis set,and minimization becomes a process of optimizing the coef-ficients in the expansion of the states /H20849as other band structure
methods do, except that Bloch character is imposed in othermethods /H20850. The implementation of Temmerman and
collaborators
17used here incorporates the atomic-sphere ap-
proximation /H20849ASA /H20850of the linear muffin-tin orbital electronic
structure method18/H20849LMTO /H20850in the tight-binding
representation.19Further details can be found in Ref. 20.
D. Pseudo-SIC method
The large computing requirements /H20849compared to LDA /H20850of
the SIC-LSD method, even for materials with smallunit cells, has led to an alternative approach,
21in which theself-interaction part of the Kohn-Sham potential is approxi-
mated by a nonlocal, atomiclike contribution included withinthe pseudopotential construction. The original implementa-tion of this scheme has given important improvements overLSDA results for nonmagnetic II-VI and III-V semiconduc-tors, but was not applicable to metals or to magnetic andhighly correlated systems where there is a coexistence ofstrongly localized and hybridized electron charges.
The pseudopotential self-interaction corrected calcula-
tions presented here were performed using the recently de-veloped “pseudo-SIC” method of Filippetti and Spaldin.
22
This pseudo-SIC approach represents a compromise betweenthe fully self-consistent implementations of Svane et al.
17,23
and the alternative method of V ogel et al. ,21in that the SIC
calculated for the atom /H20849as in Ref. 16/H20850is scaled by the elec-
tron occupation numbers calculated self-consistently withinthe crystal environment. This allows the SIC coming fromlocalized, hybridized, or completely itinerant electrons to bediscriminated, and permits the treatment of metallic as wellas insulating compounds, with minimal computational over-head beyond the LSDA. In this pseudo-SIC procedure, theorbital SIC potential is taken from the isolated neutral atomand included in the crystal potential in terms of a nonlocalprojector, similar in form to the nonlocal part of the pseudo-potential. The Bloch wave functions are projected onto thebasis of the pseudoatomic orbitals, then, for each projection,the potential acting on the Bloch state is corrected by anamount corresponding to the atomic SIC potential. Note that,within this formalism, a physically meaningful energy func-tional which is related to the Kohn-Sham equations by avariational principle is not available. However, a suitable ex-pression for the total energy functional was formulated inRef. 22and shown to yield structures in good agreement
with experiment. We use this functional here to calculate thebondlength dependence of the total energy. We have usedultrasoft pseudopotentials with an energy cutoff of 35 Ry. An8
3Monkhorst-Pack grid was used for k-point sampling. The
low-spin and high-spin solutions were obtained by settinginitial magnetization to 5
/H9262Bor 1/H9262B, respectively.
E. Hybrid functional method
The hybrid-exchange DFT approximation mixes a frac-
tion of the exact, nonlocal, exchange interaction /H20851which uses
the Hartree-Fock /H20849HF/H20850expression /H20852with the local, or semi-
local, exchange energy of the LDA or the generalized gradi-ent approximation /H20849GGA /H20850. The PBE0 functional takes the
form
E
xc=aExHF+/H208491−a/H20850ExPBE+EcPBE, /H208498/H20850
where ExHFand ExPBEare the exchange and the PBE GGA
functionals.24The mixing parameter a=1/4 was determined
via perturbation theory21and EcPBEis the PBE correlation
energy.
In this work we use the hybrid method recently developed
by Heyd, Scuseria, and Enzerhof /H20849HSE /H20850.25It is based upon
the PBE0 functional, but employs a screened, short-range/H20849SR/H20850Hartree-Fock /H20849HF/H20850exchange instead of the full exact
exchange, which results in a more efficient evaluation forKASINATHAN et al. PHYSICAL REVIEW B 74, 195110 /H208492006 /H20850
195110-4small band gap systems. In this approach, the Coulomb op-
erator is split into short-range /H20849SR/H20850and long-range /H20849LR/H20850
components, respectively,
1
r=1 − erf /H20849/H9275r/H20850
r+erf/H20849/H9275r/H20850
r, /H208499/H20850
where /H9275is a parameter that can be adjusted for numerical or
formal convenience.
The expression for the HSE exchange-correlation energy
is
Exc=aExHF,SR/H20849/H9275/H20850+/H208491−a/H20850ExPBE,SR/H20849/H9275/H20850+ExPBE,LR/H20849/H9275/H20850+EcPBE,
/H2084910/H20850
where ExHF,SR/H20849/H9275/H20850is the SR HF exchange computed for the
SR part of the Coulomb potential, ExPBE,SR/H20849/H9275/H20850andExPBE,LR
/H20849/H9275/H20850are the SR and the LR components of the PBE exchange,
respectively. The cited papers should be consulted for further
details. The HSE functional has been found to yield results ingood agreement with experiment for a wide range of solidsand molecules.
26,27
This functional is implemented in the development ver-
sion of the Gaussian quantum chemistry package.28We use
the Towler basis29–31of Gaussian functions for our basis set.
It consists of a /H2085120s12p5d/5s4p2d/H20852contraction for Mn and a
/H2085114s6p/4s3p/H20852basis for O, optimized for HF studies on
MnO. A pruned grid for numerical integration with 99 radial
shells and 590 angular points per shell was used. The kspace
was sampled with a 163mesh. Low spin and high spin anti-
ferromagnetic initial guesses were obtained using the crystalfield approach by patching the density matrix obtained fromdiagonalization of the Harris functional
32with the density
matrices obtained from calculation on ions in the appropriateligand field.
33,34
IV. PREVIOUS ELECTRONIC STRUCTURE STUDIES
The origin, and the proper description, of the moments
and the band gaps in transition metal monoxides have a longhistory. The earliest question centered on the connection be-tween the antiferromagnetic /H20849AFM /H20850order and the insulating
behavior. Slater’s band picture
35could account in a one-
electron manner for a gap arising from AFM order, whereasMott’s picture of correlation-induced insulating behavior
36
was a many-body viewpoint with insulating behavior notconnected to the magnetic order. The proper general picturein these monoxides arose from studies of transport above theNéel temperature and with introduction of defects, givingthem the designation as Mott insulators.
Much progress on the understanding of MnO and the
other monoxides came from early studies using LDA. Whileunderstanding that LDA does not address the strong correla-tion aspect of the electronic structure, Mattheiss
37and
Terakura et al.38quantified the degree and effects of 3 d-2p
interactions, and pointed out the strong effect of magneticordering on the band structure. More recently, Pask andcollaborators
39have studied the structural properties, and the
rhombohedral distortion, with LDA and GGA approxima-tions. The symmetry lowering and resulting structure isdescribed well, and in addition they found that AFM order-
ing results in significant charge anisotropy. Effects of AFMorder were further probed by Posternak et al. by calculating
and analyzing maximally localized Wannier functions for theoccupied states.
40
The application of correlation corrections in MnO already
has a colorful history. The first work, by Svane andGunnarsson
23and by Szotek et al. ,41was in the application
of the SIC-LSD method. The former pair correctly obtainedthat MnO, FeO, CoO, NiO, and CuO are AFM insulators,while VO is a metal. They calculated a gap of 4 eV for MnO.Szotek et al. used a fairly different implementation of the
SIC-LSD approach but find a similar gap /H208493.6 eV /H20850. Their 3 d
states lay about 6 eV below the center of the 2 pbands, al-
though hybridization was still clearly present. In this sametime frame, Anisimov, Zaanen, and Andersen introduced
42
the LDA+ Umethod with application to the transition metal
monoxides. They obtained a band gap of 3.5 eV but fewother results on MnO were reported.
Kotani implemented
43–45the “DFT exact-exchange”
method of Talman and Shadwick46to crystal calculations.
This method consists of taking the Fock expression for theexchange energy in the DFT functional, then performing aKohn-Sham solution /H20849minimization /H20850, giving a local exchange
potential /H20849“optimized effective potential” /H20850. In Kotani’s re-
sults for MnO, the Mn e
gandt2gbands form very narrow
/H20849almost atomiclike /H20850bands between the occupied O 2 pbands
and the conduction bands. Takahashi and Igarashi47proposed
starting from the Hartree-Fock exchange and adding correla-tion from a local, three-body scattering viewpoint. Theircorrections were built on a parametrized tight-binding repre-sentation, and they obtained small self-energy correctionsfor MnO, much smaller than they obtained for the othertransition metal monoxides.
The effective potential approach used by Kotani was ex-
tended by Solovyev and Terakura
48in an unconventional
way. They obtained an effective potential using the criterionthat it had to reproduce the spin-wave spectrum, i.e., that ithad to describe the magnetic interactions correctly. Theyfound clear differences when comparing to the LDA+ Uand
the optimized effective potential results, and discussedlimitations of the one-electron band method itself.
More recently, Savrasov and Kotliar applied a dynamical
extension
49of the LDA+ Umethod /H20849dynamical mean field
theory /H20850to MnO and NiO. Being a self-energy method, this is
not really a correlated band theory. For the properties theycalculated /H20849band gap, effective charges, dielectric constant,
optic phonon frequencies /H20850the dynamical results are similar
to the LDA+ Uresults and differ considerably from LDA
values.
Even though hybrid-exchange DFT applications to solids
are still in their infancy, there have been two previous studiesof MnO. The first, by Bredow and Gerson,
50utilized the
B3LYP hybrid functional. Unlike the LDA and GGA, theyfound B3LYP provided an excellent band gap for MnO.More recently, Franchini et al. have examined MnO in more
detail using the PBE0 approximation.
51They also found a
gap, lattice constant and density of states in quite goodagreement with experiment. In particular, the distorted dB1rhombohedral structure was determined to be the minimumMOTT TRANSITION OF MnO UNDER PRESSURE: A … PHYSICAL REVIEW B 74, 195110 /H208492006 /H20850
195110-5energy geometry, in agreement with experiment. Neither the
B3LYP nor the PBE0 approximation can be applied to themetallic side of the transition of interest here. For that, wemust turn to the screened hybrid-exchange of HSE.
Therefore, while there has been thorough LDA studies of
MnO and a variety of approaches to treatment of the corre-lation problem, nearly all of these have considered only am-bient pressure or small variations of volume near zero pres-sure. The work described in the following sections focuseson testing the four different correlated band methods fromambient conditions to high pressures, through the volumecollapse regime, to see whether some basic foundation canbe laid for the understanding and theoretical description ofpressure-driven Mott transitions in real materials.
V. RESULTS
Our principal results revolve around the first-order transi-
tion, high →low volume which is also high →low moment in
nature. For convenience we use from here on the specificvolume
v/H11013V/V0, the volume referenced to the experimental
zero-pressure volume. The equations of state have been fittedfor both high volume and low volume phases for each com-putational method, and the resulting constants are presentedand analyzed below.
A. Baseline: LDA bands and equation of state
In Table Iwe illustrate the magnitude of variation of three
properties calculated from the four codes used here, as anindication of what size of differences should be given mean-ing in properties presented below. Before the method-specificbeyond-LDA corrections are applied, there are differences inthe band gap /H208490.95±0.2 eV /H20850due to the algorithms applied in
the codes, to the basis quality, and because different LDA
functionals are used. It is evident that these variations havevery little effect on the calculated equilibrium volume andminor effect on the calculated Mn moment /H208494.47±0.05
/H9262B/H20850.
Such differences will not affect the comparisons of theresults given here, at the level of precision of interest at this
time.
The LDA band structure of AFM MnO is shown in Fig. 2
as the reference point for the following calculations. There isa band gap of /H110110.7 eV. The five bands immediately below
the gap are the majority Mn 3 dbands, those lying below are
t h eO2 pbands. The charge transfer energy mentioned in the
Introduction is /H9004
ct=/H9255d−/H9255p=6 eV, and the exchange splitting
is/H110153.5 eV. It is tempting to interpret the 3+2 separation of
occupied 3 dstates as t2g+eg, but the rhombohedral symme-
try renders such a characterization approximate. The fivebands above the gap are primarily the minority Mn 3 dbands.
However, a free-electron-like band at /H9003lies lower in energy
than the 3 dbands, but disperses upward rapidly, so over
most of the zone the lowest conduction band is Mn 3 dand
the gap is 1 eV. The presence of the non-3 dband does com-
plicate the interpretation of the band gap for some of thecorrelated methods, presented below.
The behavior of MnO under compression within GGA
has been given earlier by Cohen, Mazin, and Isaak.
53They
obtained an equilibrium volume 2% higher, and bulk modu-lus 13% smaller, than measured. Pressure studies includingextensive structural relaxation have also been provided byFang et al.
54Their structural relaxations make their study
more relevant /H20849within the restrictions of GGA /H20850but also
make comparison with our /H20849structurally restricted /H20850results
impossible.
B. Energetics and equation of state
The equation of state /H20849EOS /H20850energy vs volume curves for
the various functionals are collected in Fig. 3. For each cor-
related band method a large volume, high spin state and asmall volume, low spin state are obtained. The analysis toobtain the first-order volume collapse transition was done asfollows. For each high volume and low volume phase sepa-rately, an EOS function E
h,l/H20849V/H20850was determined /H20849h,l=high,
low/H20850by a fit to the Murnaghan equation. Both fits give
minima, with the most relevant one being for the high spinTABLE I. Local density approximation results obtained from the four codes used in this work, for the
equilibrium lattice constant relative to the experimental value /H20849a/a0/H20850, the bulk modulus B, the energy gap,
and Mn moment. The experimental values for Bfall in the range 142–160 GPa /H20849see caption to Table II/H20850.
All calculations are carried out for the antiferromagnetic II phase of MnO in the cubic /H20849NaCl /H20850structure.
Differences are due to /H208491/H20850differing exchange-correlation functions that are used, /H208492/H20850approximations
made in algorithms to solve the Kohn-Sham equations and evaluate the energy, /H208493/H20850basis set quality, and
/H208494/H20850for the moment, the operational definition of the “Mn moment” differs. The notation follows. FPLO:
full potential local orbital method with Perdew-Wang /H208491992 /H20850exchange-correlation functional. Gaussian:
Gaussian local orbital code, Slater exchange /H20849/H9251=2/3 /H20850and V osko-Wilk-Nusair correlation. PW-USSP: plane-
wave basis using ultrasoft pseudopotentials, with the Perdew-Zunger exchange-correlation functional.LMTO-ASA: linear muffin-tin orbital code in the atomic sphere approximation, using the Perdew-Zungerexchange-correlation functional.
FPLO Gaussian PW-USSP LMTO-ASA
a/a
0 0.97 0.97 0.96 0.97
B/H20849GPa /H20850 170 196 169 205
Gap /H20849eV/H20850 0.72 1.13 0.92 1.04
Moment /H20849/H9262B/H20850 4.52 4.53 4.42 4.42KASINATHAN et al. PHYSICAL REVIEW B 74, 195110 /H208492006 /H20850
195110-6phase and being the predicted equilibrium volume V0th. The
pressure is obtained from the volume derivative of the EOS,which is inverted to get V/H20849P/H20850. Equating the enthalpies
E/H20851V/H20849P/H20850/H20852+PV/H20849P/H20850of the two phases gives the critical pressure
P
c. The volumes at this pressure then give the volume
collapse /H9004V=Vh/H20849Pc/H20850−Vl/H20849Pc/H20850.
The various quantities for all four computational schemes
are given in Table II, along with the uncorrelated results of
Cohen et al.53Not surprisingly given the other differences
that will be discussed, there are substantial variations among
the critical pressures and related quantities. Particularly no-ticeable already in the EOS is the result that the energy dif-ference between the low-spin energy minimum and the high-spin one is /H110110.2 eV for the LDA+ Uand pseudo-SIC
methods, while the HSE method gives roughly twice the en-ergy difference /H208490.4 eV /H20850, and the SIC-LSD method gives
roughly 0.6 eV.
We now mention other noteworthy features of the
calculated data in Table II./H208491/H20850The predicted equilibrium volume from the LDA+ U
method is the smallest of the four methods /H20849
v0th=0.93 /H20850, thus
overbinding. The HSE value is almost indistinguishable from
the observed value, while the SIC-LSD and pseudo-SIC
methods give underbinding /H20849v0th=1.04, 1.09, respectively /H20850.
/H208492/H20850The pseudo-SIC method predicts the transition to oc-
cur at a relatively small volume reduction /H20849vh=0.86 /H20850; the
other methods give the onset of transition at v=0.63±0.03.
/H208493/H20850The critical pressure Pc=56 GPa predicted by pseudo-
SIC is smallest of the methods. Pcin LDA+ U/H20849123 GPa /H20850is
comparable to that of LDA; those of SIC-LSD and HSE are
higher /H20849204, 241 GPa, respectively /H20850.
/H208494/H20850The SIC-LSD method predicts a transition to a low
volume phase that is much softer than the high volumephase, a phenomenon that is extremely unusual in practicebut not disallowed. There are two possible sources of thisdifference: /H20849i/H20850in SIC-LSD the system becomes completely
LDA-like in the low volume phase whereas in the othermethods the 3 dstates are still correlated, or /H20849ii/H20850the LMTO-
TABLE II. Quantities obtained from fits to the Murnaghan equation of state for the various functionals,
except for the GGA column, which are taken from Ref. 53.v0is the calculated equilibrium volume, BandB/H11032
are the bulk modulus /H20849in GPa /H20850and its pressure derivative. vh,vlare the calculated volumes of the high and
low pressure phases, respectively, at the critical pressure Pc/H20849in GPa /H20850./H9004vis the amount of volume collapse
that occurs at the transition pressure Pc. All volumes are referred to the experimental equilibrium volume.
The experimental values are B=142–160 GPa, B/H11032/H110154; see Zhang /H20849Ref. 52/H20850and references therein.
GGA LDA+ UHSE
exchange pseudo-SIC SIC-LSD
v0 1.02 0.93 0.99 1.09 1.04
vh 0.70 0.66 0.60 0.86 0.64
vl 0.62 0.61 0.55 0.73 0.52
/H9004v 0.08 0.05 0.05 0.13 0.12
Bh 196 192 187 138 159
Bl 195 224 230 67
Bh/H11032 3.9 3.2 3.3 3.6 3.3
Bl/H11032 3.6 4.0 3.5 4.7
Pc 149 123 241 56 204
FIG. 3. /H20849Color online /H20850The calculated total
energy/MnO versus volumes for the various func-tionals for the AFMII rocksalt phase, referred tozero at their equilibrium volume. The filled sym-bols denote the calculated energies of the largevolume, high spin configuration and the opensymbols denote the calculated energies of thesmall volume, low spin configuration. The con-tinuous and dashed lines are the least square fittedcurves to the Murnaghan equation of state forhigh and low spin configurations, respectively.MOTT TRANSITION OF MnO UNDER PRESSURE: A … PHYSICAL REVIEW B 74, 195110 /H208492006 /H20850
195110-7ASA method involves approximations that pose limitations
in accuracy.55
/H208495/H20850The values of Bin the large volume phase vary al-
though not anomalously so, given the differences discussedjust above.
/H208496/H20850The values of B
/H11032in the high volume phase are reason-
ably similar across the methods: B/H11032=3.4±0.2. The variation
in the collapsed phase is greater.
C. Magnetic moment
The moment collapse behavior of each method is col-
lected in Fig. 4. For comparison, the GGA result presented
by Cohen et al.53was a moment collapse from 3.4 /H9262Bto
1.3/H9262Bat the volume given in Table II. On the broadest level,
the predictions for the Mn moment show remarkable simi-
larity . At low pressure, all methods of course give the high
spin S=5
2configuration of the Mn2+ion, with the local mo-
ment being reduced slightly from 5 /H9262Bby 3d-2pmixing. This
electronic phase persists over a substantial volume reduction,
giving way in all cases to an S=1
2state, notthe nonmagnetic
S=0 result that might naively be anticipated. Three methods
give a stable moment very near 1 /H9262Bover a range of volumes.
The SIC-LSD method /H20849which in the collapsed phase is sim-
ply LDA /H20850is alone in giving a varying moment, one that
reduces from 1.4 /H9262Batv=0.68 down to 0.5 /H9262Batv=0.48.
One difference between the methods lies in how soon the
low spin state becomes metastable, i.e., when it is possible toobtain that state self-consistently, as opposed to when it be-comes the stable solution /H20849which was discussed in the EOS
section /H20850. The state is obtained already at ambient volume in
the HSE method; the pseudo-SIC method obtains the lowspin state just below
v=0.80; for the LDA+ Umethod, it wasnot followed above v=0.68. It should be emphasized
however that no concerted measures were taken to try tofollow all solutions to the limit of their stability.
D. Fundamental band gap
In Fig. 5the calculated band gap of both high spin and
low spin states are shown for all methods. Here the behaviordiffers considerably between the methods, in part because atcertain volumes the gap lies between different bands forsome of the methods. At ambient pressure the pseudo-SICand HSE methods obtain a gap of 3.5–4 eV, while that forSIC-LSD is 2.9 eV, and that of the LDA+ Umethod is even
lower, less than 2 eV. Experimental values lie in the3.8–4.2 eV range. Referring to Fig. 2, it can be observed that
the large volume gap depends on the position of the majority3dstates with respect to the free-electron-like band, i.e., it is
not the 3 d-3dMott gap. Both of the former approaches show
only a slight increase as pressure is applied, reaching a maxi-mum around
v=0.76 where a band crossing results in a de-
creasing gap from that point. For pseudo-SIC, there is analmost immediate collapse to a metallic low spin state.Within HSE, MnO collapses to the low spin state at a volume
v=0.55, coincident with the closing of the gap in the low
spin state. The SIC-LSD and LDA+ Ugaps, smaller initially,
show a much stronger increase with pressure, and also incurthe band crossover that leads to decrease of the gap /H20849within
the high spin state /H20850.
VI. ANALYSIS OF THE TRANSITION
In this section we analyze the character of the states just
above and just below the Mott transition, as predicted by
FIG. 4. /H20849Color online /H20850Calculated values of the moment /H20849shown
as symbols /H20850on each Mn site as a function of volume for the various
functionals. All the methods predict a distinct collapse /H20849first order /H20850
of magnetic moment with decrease in volume. At large volumes, thehigh spin state with S=5/2 /H20849single occupancy of each of the 3 d
orbitals /H20850is realized while the low spin state with S=1/2 is favored
for smaller volumes. Note: the computational methods calculate theMn moment in inequivalent ways, so differences of less than 0.1
/H9262B
may have no significance.
FIG. 5. /H20849Color online /H20850Calculated band gap as a function of
volume; the lines are simply a guide to the eye. For LDA+ U, the
band gap increases with decrease in volume for the high spin state,but decreases with volume in the low spin state. For the HSE cal-culations, the band gap monotonously decreases with volume whilstpseudo-SIC shows a first order jump in going from high to low spinstate. At very low volumes, where the low spin configuration ispreferred LDA+ Ugives a substantial gap and is still an insulator,
while HSE and pseudo-SIC calculations converge to a metallicsolution.KASINATHAN et al. PHYSICAL REVIEW B 74, 195110 /H208492006 /H20850
195110-8each of the methods. Due to the differing capabilities of the
codes, the quantities used for analysis will not be identical inall cases. In the Figs. 6–9we present for uniformity the DOS
in the high volume phase at the equilibrium volume /H20849a
0/H20850and
in the collapsed phase at a=0.85 a0/H20849v=0.6 /H20850. Note /H20849from
Table I/H20850that this specific volume does not correspond to any
specific feature in the phase diagram for any method, al-though it lies in the general neighborhood of the volume atthe collapse. Changes within the collapsed phase are continu-ous, however, so the plots at a=0.85 a
0are representative of
the collapsed phase.
A. LDA+ U
The projected DOSs /H20849PDOSs /H20850in Fig. 6refer to projec-
tions onto Mn 3 dorbitals, with the zaxis being the rhombo-
hedral axis, the ag3z2−r2/H20849/H20841m/H20841=0/H20850state; the eg/H11032pair /H20853xz,yz/H20854
/H20849/H20841m/H20841=1/H20850; and the egpair /H20853x2−y2,xy/H20854/H20849 /H20841m/H20841=2/H20850. Because the
two egrepresentations have the same symmetry, they can
mix and the actual combinations eg,1,eg,2are orthogonal lin-
ear combinations of eg,e/H11032gwhich depend on interactions. For
the LDA+ Uresults, however, there is little mixing of the
eg,e/H11032gpairs. The character of the transition is simple to de-
scribe: the eg/H11032pair /H20849/H20841m/H20841=1/H20850with respect to the rhombohedral
axis /H20850simply flips its spin.
This S=1
2state is unexpected and quite unusual. First,
each 3 dorbital is still singly occupied, verified by plotting
the charge density on the Mn ion and finding it just as spheri-cal as for the high spin state. Second, each 3 dorbital
is essentially fully spin-polarized, with the configuration
being a
g↑eg↑eg/H11032↓. A plot of the spin density56reveals theunanticipated strong anisotropy with nodal character, charac-
teristic of spin-up m=0 and /H20841m/H20841=2 orbitals, and spin-down
/H20841m/H20841=1 orbitals /H20849in the rhombohedral frame /H20850. Third, it makes
FIG. 6. /H20849Color online /H20850Projected DOS onto symmetrized Mn 3 d
orbitals in the rhombohedral AFMII rocksalt phase using theLDA+ Umethod. Top panel: High spin solution at the LDA+ U
equilibrium volume. Bottom panel: Low spin solution at 60% of theLDA+ Uequilibrium volume. The a
gorbital is the 3 z2−r2oriented
along the rhombohedral axis, other symmetries are described in thetext. The overriding feature is the spin-reversal of the m=±1 e
g/H11032
orbitals between the two volumes.
FIG. 7. /H20849Color online /H20850Spin- and orbital-projected DOS from the
HSE /H20849hybrid-exchange /H20850method. Orbitals are expressed in a coordi-
nate system that is specific to the code, neither cubic nor rhombo-hedral. In the collapsed phase two orbitals are /H20849nearly /H20850doubly oc-
cupied, and the net moment arises from the single occupation of theorbital labeled d
4. In a cubic representation the configuration can be
specified as t2g5=t2g↑3+t2g↓2. At the volume v=0.55 of onset of the
collapsed phase /H20849see Table I/H20850, the gap is essentially zero.
FIG. 8. /H20849Color online /H20850Spin- and symmetry-projected DOS of
the high spin /H20849upper panel /H20850and low spin /H20849lower panel /H20850states result-
ing from the pseudo-SIC method. In the low spin state, the agor-
bital is unpolarized due to occupation by both spin directions, andshows little exchange splitting. One of the e
g-symmetry pairs, here
called eg/H11032, is unpolarized due to being unoccupied in both spin di-
rections. It is the one labeled egthat is spin split across the Fermi
level and is responsible for the moment.MOTT TRANSITION OF MnO UNDER PRESSURE: A … PHYSICAL REVIEW B 74, 195110 /H208492006 /H20850
195110-9this transition with essentially zero change in the gap, which
is 3.5 eV. The band structure changes completely, however,so the close similarity of the gaps on either side is accidental.
B. HSE method
In the high volume phase the distribution and overall
width of the occupied 3 dstates, shown in Fig. 7, is similar to
that of the LDA+ Umethod /H20849preceding section /H20850. The gap is
larger, as discussed earlier. The collapsed phase shows newcharacteristics. The gap collapses from 4 eV at a
0to essen-
tially zero at the onset of the collapsed phase at v=0.55,
making this an insulator-to-semimetal transition. The metal-lic phase then evolves continuously as the pressure is in-creased beyond P
c. The 3 dconfiguration can be character-
ized as t2g5=t2g↑3+t2g↓2, resulting in a moment of 1 /H9262B. The
corresponding spin density is strongly anisotropic, althoughin a different manner than is the case for LDA+ U.
C. Pseudo-SIC method
The spin-decomposed spectrum from this method is
shown in Fig. 8, symmetry projected as done above for the
LDA+ Umethod. The PDOSs in the high spin state are quite
similar to those given by the LDA+ Umethod. The transition
could hardly be more different, however. The gap collapsesin an insulator-to-good-metal character, the Fermi level lyingwithin both majority and minority bands. The majority bands
are the e
g/H11032pair /H20849/H20841m/H20841=1/H20850and are only slightly occupied. In the
minority bands both egandeg/H11032are roughly quarter filled. Thereason this solution is /H20849at least locally /H20850stable seems clear: EF
falls in a deep valley in the minority DOS.
In the collapsed, low spin state, the agorbital of both
spins is occupied, and the majority egpair is also fully occu-
pied. This results in a configuration that can be characterized
roughly as ag1↑eg2↑;ag1↓eg0.5↓/H20849eg/H11032/H208500.5↓, giving spin3
2−1
2−1
2
=1
2. Thus the fact that the same moment is found in the low
spin state as was found with the LDA+ Uand pseudo-SIC
methods seems accidental, because in those methods the en-ergy gap required integer moment whereas the pseudo-SICsolution is firmly metallic. It is in fact close to half metallic,which accounts for the near-integer moment. In pseudo-SIC,thea
gorbital is unpolarized /H20849spin-paired /H20850, the egpair /H20853xz,yz/H20854
/H20849/H20841m/H20841=1/H20850is positively polarized, and the eg/H11032pair /H20853x2−y2,xy/H20854
/H20849/H20841m/H20841=2/H20850is negatively polarized but to a smaller degree.
D. SIC-LSD method
This method give much more tightly bound 3 dstate in the
high spin state than the other methods. An associated featureis that the majority-minority splitting, the “effective U”
/H11013U
SICplus the exchange splitting 13 eV, more than twice as
large as used in the LDA+ Umethod /H20849both in this paper and
elsewhere /H20850. Note that in the SIC-LSD method USICis a true
Slater self-Coulomb integral, whereas in the LDA+ U
method the value of Urepresents the /H20849somewhat screened /H20850
Coulomb interaction between a 3 delectron and an additional
3delectron, so agreement between the two is not expected.
Nevertheless the difference is striking. All five majority 3 d
states are localized, leading to the self-interaction potentialthat binds them. The majority 3 dstates lie 6 eV below the
center of the 2 pbands and hybridize very weakly, which
accounts for the very narrow, almost corelike 3 dbands.
In the collapsed phase, there are no localized states and
the usual LSDA results reemerge. All 3 dstates make some
contribution to the moment, but the strongest contributionarises from the a
gorbital. One might argue in the LDA+ U
result that the net moment arises from the agorbital /H20849with the
moments arising from the egandeg/H11032orbital canceling /H20850. How-
ever, the electronic structure in the collapsed phase of thisSIC-LSD method is very different from that of the LDA+Umethod.
VII. DISCUSSION AND SUMMARY
The results of four correlated band methods—LDA+ U,
SIC-LSD, pseudo-SIC, and HSE—have been compared forthe equation of state /H20849for both normal and collapsed phases /H20850,
electronic structure /H20849including 3 dconfiguration /H20850, and the Mn
moment under pressure. In order to make the comparisonas straightforward as possible, the crystal structure waskept cubic /H20849rocksalt /H20850. To compare seriously with experiment,
one must account for the coupling of the AFM order tothe structure, because this results in a substantial rhombohe-dral distortion of the lattice.
54Then structural transitions
/H20849particularly to the B8 phase /H20850must also be considered.
The large volume, high-spin phases are qualitatively the
same for the various functionals: AFM with a fully polarized3d
5configuration. Due to the large charge transfer energy,
FIG. 9. /H20849Color online /H20850Spin- and symmetry-projected DOS of
the high spin /H20849upper panel /H20850and low spin /H20849lower panel /H20850states given
by the SIC-LSD method. Orbital characters are expressed in therhombohedral coordinate system. In the high spin state, the majority3dstates are are centered 6 eV below the center of the occupied 2 p
bands, resulting in little hybridization and very narrow bands. Theexchange splitting of the 3 dstates is about 13 eV, providing an
“effective U” from this method. The collapsed moment phase is
representative of a band /H20849LSD /H20850ferromagnet.KASINATHAN et al. PHYSICAL REVIEW B 74, 195110 /H208492006 /H20850
195110-10the configuration remains d5at all volumes studied here. The
predicted equations of state /H20849which give the equilibrium vol-
ume, bulk modulus, and its pressure derivative /H20850show rather
strong variation, suggesting that the extension of the Mn 3 d
a n dO2 pfunctions, or their hybridization, differ substan-
tially even in this large volume phase. Of course, since the
functionals are different, any given density would lead todifferent energies.
Under pressure, the gap initially increases /H20849all methods
give this behavior /H20850, and the system suffers a first-order tran-
sition /H20849isostructural, by constraint /H20850to a collapsed phase
where hybridization must be correspondingly stronger. Uni-formly among the methods, the moment collapse reflects an
S=
5
2toS=1
2transition, rigorously so for the LDA+ U
method for which the collapsed phase retains a gap, and lessrigorously so for the the other methods where the collapsedphase is metallic. It is remarkable that none of the methodsgives a collapse to a nonmagnetic state, which probablywould be the most common expectation.
This S=
5
2→1
2moment collapse is related in some cases to
the local symmetry of the Mn 3 dorbitals in the AFM phase
/H20849being most obvious for the LDA+ Uresults of Fig. 6/H20850. The
symmetry is ag+eg+e/H11032g, i.e., a singlet and two doublets per
spin direction. Without further symmetry breaking /H20849orbital
ordering /H20850anS=5
2→3
2transition requires a single spin-flip,
which could only be the agspin. However, the agstate is
more tightly bound than at least one of the two doublets bothfor LDA+ U/H20849Fig. 6/H20850and for pseudo-SIC /H20849Fig. 8/H20850.I nL D A
+Uthee
gdoublet flips its spin, while in pseudo-SIC the ag
singlet flips its spin leaving the minority egandeg/H11032doublets
partially occupied and therefore metallic. This symmetry-related behavior depends of course on the magnetic orderingthat gives rise to the /H20849electronic /H20850rhombohedral symmetry.
Above the Néel temperature, the moment collapse atthe Mott transition may proceed differently because theMn moment would lie at a site of cubic symmetry /H20849a dy-
namic treatment could include the effect of short range spincorrelations /H20850.
Our study provides some of the first detailed information
on how magnetic moments in a real material may begin todisintegrate without vanishing identically, at or near a Motttransition, when correlation is taken into account. It is ac-cepted that dynamic processes will be required for a trulyrealistic picture of the Mott /H20849insulator-to-metal /H20850transition.
However, a moment collapse between two insulating phases/H20849as described here by two of the methods /H20850may be described
reasonably by a correlated band /H20849static /H20850approach.
Beyond this similar amount of moment collapse, the four
functionals give substantially different collapsed phases: dif-ferences in the Mn 3 dmagnetic configuration /H20849although allremain d
5/H20850and differences in conducting versus insulating
behavior. It is not surprising therefore that the collapsed-phase equations of state differ considerably between themethods.
The differences in predictions can be traced, in principle,
to the different ways in which exchange and correlation arecorrected with respect to LDA. One clear shortcoming ofLDA is in the local approximation to the exchange energy.The HSE method deals with this problem directly, by using25% Hartree-Fock exchange. The self-interaction of the SIC-LSD method is largely a self-exchange energy correction,subtracting out the spurious self-Coulomb energy that occursin the Hartree functional if an orbital chooses to localize. Aself- /H20849local density /H20850correlation correction is also included in
SIC-LSD. The pseudo-SIC method includes the same correc-tion if applied to an atom, but in a crystal the pseudo-SICenergy correction and change in potential takes a substan-tially different form, as the difference in predictions reflects.The LDA+ Umethod is rather different in this respect: it
specifically does not subtract out any self-interaction /H20849al-
though it is sometimes discussed in this way /H20850. In the form
Eq. /H208496/H20850of LDA+ Ucorrection, the second term is simply an
LDA-like average of the first term. The on-site Coulombrepulsion is treated Hartree-Fock like, leading to an orbital-dependent, occupation-dependent potential. Each method hasits own strengths, and each is only an anticipated improve-ment on LDA toward a better, more general functional. It isexpected that more details of the results may be presentedseparately by the respective practitioners.
ACKNOWLEDGMENTS
Three of the authors /H20849D.K., J.K., and W.E.P. /H20850acknowl-
edge support from Department of Energy Grant No. DE-FG03-01ER45876. Two of the authors /H20849R.L.M. and C.V .D. /H20850
thank the DOE BES heavy element chemistry program andLANL LDRD for support. The work at Rice University wassupported by DOE Grant No. DE-FG02-01ER15232 and theWelch Foundation. The authors acknowledge important in-teractions within, and some financial support from, the De-partment of Energy’s Stewardship Science Academic Alli-ances Program. Two of the authors /H20849L.P. and T.C.S. /H20850were
supported by the Office of Basic Sciences, U.S. Departmentof Energy. One of the authors /H20849N.S. /H20850was supported by the
National Science Foundation’s Division of Materials Re-search Information Technology Research program, Grant No.DMR-0312407, and made use of MRL Central Facilities sup-ported by the National Science Foundation Grant No. DMR-05-20415. This collaboration was stimulated by, and sup-ported by, DOE’s Computational Materials Science Network.
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195110-12 |
PhysRevB.72.024512.pdf | BCS-BEC crossover of collective excitations in two-band superfluids
M. Iskin and C. A. R. Sá de Melo
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
/H20849Received 26 August 2004; revised manuscript received 2 March 2005; published 11 July 2005 /H20850
We use the functional integral approach to study low energy collective excitations in a continuum model of
neutral two-band superfluids at T=0 for all couplings with a separable pairing interaction. In the long wave-
length and low frequency limit, we recover Leggett’s analytical results in weak coupling /H20849BCS /H20850fors-wave
pairing, and further obtain analytical results in strong coupling Bose-Einstein condensation for both two andthree dimensional systems. We also analyze numerically the behavior of the out-of-phase exciton /H20849finite fre-
quency /H20850mode and the in-phase phonon /H20849Goldstone /H20850mode from weak to strong coupling limits, including the
crossover region. In principle, the evolution of Goldstone and finite frequency modes from weak to strongcoupling may be accessible experimentally in the superfluid phase of neutral Fermi atomic gases, and couldserve as a test of the validity of the theoretical analysis and approximations proposed here.
DOI: 10.1103/PhysRevB.72.024512 PACS number /H20849s/H20850: 74.20.Fg, 03.75.Kk, 05.30.Fk, 74.40. /H11001k
I. INTRODUCTION
A two-band model for superfluidity was introduced by
Suhl et al.1in 1959 soon after the BCS theory in order to
allow for the possibility of multiple band crossing throughthe Fermi surface. Suhl et al.
1observed that a larger number
of energy bands crossing the Fermi surface could increasethe overall electron state density and lead to the onset ofadditional interactions. Since then, this model has been usedto describe high temperature superconductivity in copper ox-ides and more recently it has been used in connection toMgB
2.2,3
In the seminal work by Leggett,4the existence of collec-
tive phase modes in two-band superfluids has been predictedin both neutral and charged systems. Within the weak cou-pling /H20849BCS /H20850limit, Leggett showed that if s-wave interactions
are attractive in both bands, an undamped long-wavelengthexciton /H20849finite frequency /H20850mode, as well as an undamped
long-wavelength phonon /H20849Goldstone /H20850mode may exist. These
modes were further studied theoretically
5–7in the BCS limit,
however, there was no experimental evidence of their exis-tence until manifestations of two-gap behavior
8–12were
found in MgB 2.
However, it has not been possible to study the evolution
of the spectrum of collective excitations from weak /H20849BCS /H20850to
strong coupling /H20851Bose-Einstein condensation /H20849BEC /H20850/H20852limit
until very recently. Advances in experiments with neutralFermi gases enabled the tuning and control of two-particleinteractions between atoms in different hyperfine states byusing Feshbach resonances.
13,14This kind of control is not
fully present in standard fermionic condensed matter sys-tems, and has hindered the development of experiments thatcould probe systematically the effects of strong correlationsas a function of coupling or density of fermions. It wasthought theoretically for many years that a weakly coupled/H20849BCS /H20850superfluid could evolve smoothly into the limit of
tightly bound pairs which undergo Bose-Einstein condensa-tion /H20849BEC /H20850.
15–21It was not until recently that experimental
evidence that hyperfine states of40K/H20849Refs. 22 and 23 /H20850and
6Li/H20849Refs. 24–27 /H20850can form weakly and tightly bound atompairs /H20849Cooper pairs /H20850, when the magnetic field is swept
through a s-wave Feshbach resonance.
Considering these recent findings in condensed matter
systems and advances in atomic physics experiments, we ex-pand Leggett’s calculation of collective modes in neutraltwo-band /H20849s-wave /H20850superfluids to all couplings by following
a similar one-band approach.
28These collective modes for
two-band s-wave systems are undamped in the low-
frequency and low-momentum limits, provided that the two-quasiparticle threshold is not reached. We present results ofthe evolution of the finite frequency and Goldstone’s modesfrom weak coupling /H20849BCS /H20850and to strong coupling /H20849BEC /H20850
limits, and discuss briefly the possibility of observing thesemodes in experiments involving multicomponent ultracoldatomic Fermi gases. While it is still a matter of debate thatextentions of Eagles’,
15Leggett’s,16and the Nozieres and
Schmitt-Rink’17/H20849NSR /H20850suggestions are good quantitative de-
scriptions of the crossover phenomena, the evolution of col-lective modes can serve as a test of these ideas.
The rest of the paper is organized as follows. In Sec. II,
we discuss the effective action and the saddle point approxi-mation for a two-band continuum Hamiltonian with attrac-tive interactions in the s-wave channel and a Josephson in-
terband coupling term. We analyze the effects of Gaussianfluctuations in Sec. III, and derive effective amplitude andphase actions, from which the in-phase /H20849Goldstone /H20850andout-
of-phase /H20849finite frequency /H20850collective mode spectra are cal-
culated in Sec. IV . The evolution of these modes is analyzedas a function of interaction strengths from the BCS to theBEC limit both analytically and numerical in Sec. V . In Sec.VI, we summarize our conclusions and propose that ultra-cold Fermi atoms can be used to test our results regarding theevolution of the finite frequency and Goldstone modes intwo-band superfluids. Finally, in Appendix A, we presentdetails of the matrix elements involved in the phase and am-plitude effective actions, while in Appendix B, we show thelong wavelength expansion coefficients needed to evaluate
the phase modes.PHYSICAL REVIEW B 72, 024512 /H208492005 /H20850
1098-0121/2005/72 /H208492/H20850/024512 /H2084913/H20850/$23.00 ©2005 The American Physical Society 024512-1II. EFFECTIVE ACTION METHOD
A Hamiltonian for multiband /H20849or multicomponent /H20850super-
fluids with singlet pairing can be written as
H=/H20858
n,k,/H9268/H9264n/H20849k/H20850an,k,/H9268†an,k,/H9268+/H20858
n,m,qUnm/H20849q/H20850/H9267n,q/H9267m,−q
+/H20858
n,m,k,k/H11032,qVnm/H20849k,k/H11032/H20850bn,k,q†bm,k/H11032,q/H11032, /H208491/H20850
where the indices nandmlabel different bands /H20849or compo-
nents /H20850,/H9268labels the spins /H20849or pseudospins /H20850, and klabels the
momentum. In addition, an,k,/H9268†is the fermion creation opera-
tor,/H9267n,q=/H20858k,/H9268an,k−q,/H9268†an,k,/H9268is the density operator, and bn,k,q†
=an,k+q/2,↑†an,−k+q/2,↓†is the pair creation operator. Here,
/H9264n/H20849k/H20850=/H9255n/H20849k/H20850−/H9262, where
/H9255n/H20849k/H20850=/H9255n,0+k2/2mn /H208492/H20850
/H20849with/H6036=1/H20850is the kinetic energy of fermions. The reference
energies are /H92551,0=0 and/H92552,0=E0/H110220/H20849see Fig. 1 /H20850;mnis the
effective mass for the nth band. Furthermore, the terms Unm
andVnmcorrespond to the Coulomb and pairing interaction
matrix elements, respectively. Since in this paper we are in-terested only in the neutral case, we will ignore Coulombinteractions, and consider only the pairing term. This choiceis more appropriate to ultracold atomic Fermi gases, whilethe inclusion of Coulomb terms is more appropriate in thecase of superconductors. The discussion of the charged caseis postponed to a future manuscript. For the neutral case, wetake
V
nm/H20849k,k/H11032/H20850=Vnm/H9003n*/H20849k/H20850/H9003m/H20849k/H11032/H20850/H20849 3/H20850
to be separable, and we consider in general a two-band sys-
tem with distinct intraband interactions V11andV22, and in-
terband interactions V12andV21. Notice that the off-diagonal
terms V12andV21play the role of Josephson coupling terms.
For the purpose of this paper we will consider all Vnmto be
negative. The /H9003n/H20849k/H20850coefficients are symmetry factors char-
acterizing the chosen angular momentum channel.
In the imaginary-time functional integration formalism
/H20849/H9252=1/T,/H6036=kB=1/H20850, the partition function is written as
Z=/H20885D/H20851a†,a/H20852e−S/H208494/H20850
with an action given byS=/H20885
0/H9252
d/H9270/H20875/H20858
n,k,/H9268an,k,/H9268†/H20849/H9270/H20850/H20849/H11509/H9270/H20850an,k,/H9268/H20849/H9270/H20850+H/H20849/H9270/H20850/H20876. /H208495/H20850
The Hamiltonian from Eq. /H208491/H20850can be rewritten in the form
H/H20849/H9270/H20850=/H20858
n,m,k,/H9268/H9264n/H20849k/H20850an,k,/H9268†/H20849/H9270/H20850an,k,/H9268/H20849/H9270/H20850
+/H20858
n,m,qBn†/H20849q,/H9270/H20850VnmBm/H20849q,/H9270/H20850, /H208496/H20850
with Bn/H20849q,/H9270/H20850=/H20858k/H9003n/H20849k/H20850bn,k,q/H20849/H9270/H20850. We first introduce the
Nambu spinor /H9274n†/H20849p/H20850=/H20849an,p↑†,an,−p↓/H20850, where p=/H20849k,iw/H5129/H20850is
used to denote both momentum and Matsubara frequencies
/H20851w/H5129=/H208492/H5129+1/H20850/H9266//H9252/H20852. Furthermore, we use the Hubbard-
Stratanovich transformation
exp/H20877−/H20858
n,m,qBn†/H20849q/H20850VnmBm/H20849q/H20850/H20878=/H20885D/H20851/H9021†,/H9021/H20852
/H11003exp/H20877/H20858
n,m,q/H9021n†/H20849q/H20850gnm/H9021m/H20849q/H20850
+/H20858
n,q/H20851Bn†/H20849q/H20850/H9021n/H20849q/H20850+ H.c. /H20852/H20878
/H208497/H20850
to decouple the fermionic degrees of freedom at the expense
of introducing the bosonic fields /H9021n/H20849q/H20850, with q=/H20849q,iv/H5129/H20850
where v/H5129=2/H5129/H9266//H9252. The tensor gnmassociated with the
bosonic pairing fields /H9021m/H20849q/H20850can be written as
/H20873g11g12
g21g22/H20874=1
detV/H20873V22 −V12
−V21 V11/H20874, /H208498/H20850
where det V=V11V22−V12V21. Performing an integration over
the fermionic part /H20849D/H20851/H9274†,/H9274/H20852/H20850leads to
S=−/H9252/H20858
n,m,q/H9021n†/H20849q/H20850gnm/H9021m/H20849q/H20850
+/H20858
n,p,p/H11032/H20851/H9252/H9264n/H20849k/H20850/H9254p,p/H11032−T rl n/H9252Gn−1/H20852. /H208499/H20850
Here, the inverse Nambu matrix is
Gn−1=/H9021n/H20849q/H20850/H9003n*/H20873p+p/H11032
2/H20874/H9268++/H9021n†/H20849−q/H20850/H9003n/H20873p+p/H11032
2/H20874/H9268−
+/H20851iw/H5129/H92680−/H9264n/H20849k/H20850/H92683/H20852/H9254p,p/H11032, /H2084910/H20850
where/H9268±=/H20849/H92681±/H92682/H20850/2 and/H9268iare the Pauli spin matrices.
We choose an approximation scheme where the field
/H9021n/H20849q/H20850is written as a sum of a /H9270-independent /H20849stationary /H20850part
and a/H9270-dependent contribution
/H9021n/H20849p−p/H11032/H20850=/H9021n/H20849q/H20850=/H9004n,0/H9254q,0+/H9011n/H20849q/H20850. /H2084911/H20850
We first write the inverse Nambu matrix in terms of two
matrices:
Gn−1=Gn,0−1/H208491+Gn,0Gn,1−1/H20850. /H2084912/H20850
The first one is the saddle point inverse Nambu matrix given
by
FIG. 1. Schematic figure of two bands with reference energies
/H92551,0=0 for the band 1 and /H92552,0=E0for band 2.M. ISKIN AND C. A. R. SÁ DE MELO PHYSICAL REVIEW B 72, 024512 /H208492005 /H20850
024512-2Gn,0−1=/H9254p,p/H11032/H20873iw−/H9264n/H20849k/H20850/H9004n,0/H9003n*/H20849p/H20850
/H9004n,0*/H9003n/H20849p/H20850iw+/H9264n/H20849k/H20850/H20874, /H2084913/H20850
and the second one is the fluctuation matrix
Gn,1−1=/H208980 /H9011n/H20849q/H20850/H9003n/H20873p+p/H11032
2/H20874
/H9011n*/H20849−q/H20850/H9003n*/H20873p+p/H11032
2/H20874 0/H20899,
/H2084914/H20850
used in the expansion of the natural logarithm of
Tr ln Gn−1=T rl n Gn,0−/H20858
j=1/H11009/H20849−1/H20850j
jTr/H20849Gn,0Gn,1−1/H20850j./H2084915/H20850
Within this approximation, we expand the action to sec-
ond order in the fluctuation field /H9011n/H20849q/H20850. This procedure pro-
duces an effective action of the form S=S0+S2, where
S0=−/H9252/H20858
n,mgnm/H9004n,0*/H9004m,0+/H20858
n,p,p/H11032/H20851/H9252/H9264n/H20849k/H20850/H9254p,p/H11032−T rl n/H9252Gn,0−1/H20852
is the effective saddle point action, and
S2=/H20858
n,piGn,0/H2084912/H20850Gn,1−1/H2084923/H20850Gn,0/H2084934/H20850Gn,1−1/H2084941/H20850
is the Gaussian correction to it. The notation /H20849ij/H20850in the G
matrices is understood as the momentum labels /H20849pi,pj/H20850.A n
important comment about S0is in order. Writing /H9004n,0in
terms of its amplitude and phase
/H9004n,0=/H20841/H9004n,0/H20841exp /H20849i/H9272n/H20850, /H2084916/H20850
the first term of S0becomes the sum of two contributions:
the standard band-diagonal terms − /H9252/H20858ngnn/H20841/H9004n,0/H208412, and the
band-off-diagonal −2 /H9252g12/H20841/H90041,0/H20841/H20841/H90042,0/H20841cos /H20849/H92722−/H92721/H20850correspond-
ing to the Josephson coupling between bands. Since
g12=−V12/detV, with det V/H110220 and all Vnm/H110210, the
saddle point thermodynamic potential /H90240=S0//H9252, has its
quadratic term of the form /H20849/H20841V22/H20841/H20841/H90041,0/H208412+/H20841V11/H20841/H20841/H90042,0/H208412
−2/H20841V12/H20841/H20841/H90041,0/H20841/H20841/H90042,0/H20841cos /H20849/H92722−/H92721/H20850/H20850/detV, which shows explicity
the Josephson energy. For the case chosen, where all Vnmare
negative, the in-phase /H92722=/H92721is the only stable solution.
However, if V12were positive, another stable solution would
appear where /H92722=/H92721+/H9266. This is the so-called /H9266-phase solu-
tion, which we will not discuss here.
From the stationary condition /H11509S0//H11509/H9004n*/H20849q/H20850=0 we obtain
the order parameter equations
/H9004n,0=−/H20858
m,kVnm/H9004m,0/H9003m2/H20849k/H20850
2Em/H20849k/H20850tanh/H9252Em/H20849k/H20850
2, /H2084917/H20850
where En/H20849k/H20850=/H20849/H9264n2/H20849k/H20850+/H20841/H9004n/H20849k/H20850/H208412/H208501/2is the quasiparticle energy
spectrum. Note that the order parameter
/H9004n/H20849k/H20850=/H9004n,0/H9003n/H20849k/H20850/H20849 18/H20850
is a separable function of temperature Tand momentum k.
In this manuscript, we focus on s-wave superfluids, and
thus consider only the zero angular momentum channel ofthe interaction Vnm/H20849k,k/H11032/H20850. In addition, instead of taking
/H9003n/H20849k/H20850=1 /H20849independent of k/H20850which would cause ultraviolet
divergences /H20849logarithmic in two dimensions and linear in
three dimensions /H20850in the integrations over momentum for the
order parameter equation, we take
/H9003n/H20849k/H20850=1 / /H208491+k/kn,0/H208501/2/H2084919/H20850
as the corresponding symmetry factor in two dimensions
/H208492D/H20850, and
/H9003n/H20849k/H20850=1 / /H208491+k/kn,0/H20850/H20849 20/H20850
as the corresponding symmetry factor in three dimensions
/H208493D/H20850. Here kn,0/H11011Rn,0−1, where Rn,0plays the role of the inter-
action range in real space, sets the scale at large momenta,and it is necessary to produce the physically correct behaviorof a generic interaction at short wavelengths. In the caseof 2D, V
nn/H20849k,k/H11032/H20850/H110111 for small kandk/H11032, while Vnn/H20849k,k/H11032/H20850
/H110111//H20881kk/H11032for large kandk/H11032, when a generic real potential is
used.29There is no ultraviolet divergence in our theory since
the momentum integrations always produce finite results.This choice of interactions has the advantage of making un-necessary the introduction of the T-matrix approximation to
renormalize the order parameter equation, and redefine theinteraction amplitude in terms of the two-body bindingenergy.
30,31In the case of 3D, Vnn/H20849k,k/H11032/H20850/H110111 for small kand
k/H11032, while Vnn/H20849k,k/H11032/H20850/H110111/kk/H11032for large kandk/H11032, when a ge-
neric real potential is used.17Notice that, our three-
dimensional interaction /H9003n/H20849k/H20850has the same behavior at both
low and high momenta as the one used by NSR /H20849Ref. 17 /H20850
/H9003/H20849k/H20850=1//H208811+/H20849k/k0/H208502. Either choice /H20849ours or NSR’s /H20850pro-
duces qualitatively similar results as it will be later dis-
cussed.
Furthermore, the order parameter equations need to be
solved self-consistently with the number equation
N=/H20858
n,k,/H9268/H208751
2−/H9264n/H20849k/H20850
2En/H20849k/H20850tanhEn/H20849k/H20850
2kbT/H20876, /H2084921/H20850
which is obtained from N=−/H11509/H90240//H11509/H9262, where/H90240is the saddle
point thermodynamic potential. The inclusion of fluctuationsare very important for the number equation near the criticaltemperature of the system, and in this case the saddle pointthermodynamic potential needs to be corrected to /H9024=/H9024
0
+/H9024G, where/H9024Gshould be calculated at least at the Gaussian
level. In this manuscript, however, we limit ourselves to fluc-tuation effects at low temperatures, as discussed next.
III. GAUSSIAN FLUCTUATIONS
We now investigate Gaussian fluctuations in the pairing
field/H9021n/H20849q/H20850about the the static saddle point /H9004n,0. The Gauss-
ian /H20849quadratic /H20850effective action can be written as
SG=S0+/H9252
2/H20858
q/H9011†/H20849−q/H20850M/H20849q/H20850/H9011/H20849q/H20850/H20849 22/H20850
where the fluctuation field is
/H9011†/H20849−q/H20850=/H20849/H90111*/H20849q/H20850,/H90111/H20849−q/H20850,/H90112/H20849−q/H20850,/H90112*/H20849q/H20850/H20850 /H20849 23/H20850
andM/H20849q/H20850is the fluctuation matrix given byBCS-BEC CROSSOVER OF COLLECTIVE EXCITATIONS … PHYSICAL REVIEW B 72, 024512 /H208492005 /H20850
024512-3M/H20849q/H20850=/H20898M111M1210 −g12
M211M221−g12 0
0 −g21M222M212
−g21 0 M122M112/H20899, /H2084924/H20850
where the matrix elements M11n/H20849q/H20850=M22n/H20849−q/H20850, and M12n/H20849q/H20850
=M21n*/H20849q/H20850are given by
M11n=−gnn+/H9252−1/H20858
p/H9003n/H20849p+q/2/H208502/H9277n/H20849p+q/H20850/H9277n/H20849−p/H20850
M12n=/H9252−1/H20858
p/H9003n2/H20849p+q/2/H20850/H9269n/H20849p+q/H20850/H9269n/H20849p/H20850.
The expressions /H9277n/H20849p/H20850=/H9264¯n/H20849−p/H20850/F/H20849p/H20850and/H9269n/H20849p/H20850=/H9004n/H20849k/H20850/F/H20849p/H20850
are the matrix elements /H20849Gn,0/H2085011and /H20849Gn,0/H2085012, respectively.
Here, we use the definitions /H9264¯n/H20849p/H20850=iw/H5129−/H9264n/H20849k/H20850, and F/H20849p/H20850
=/H20841/H9264¯n/H20849p/H20850/H208412+/H20841/H9004n/H20849k/H20850/H208412. Notice that while M12n/H20849q/H20850andM21n/H20849q/H20850are
even under the transformations q→−qand iv/H5129→−iv/H5129;
M11n/H20849q/H20850andM22n/H20849q/H20850are even only under the transformation
q→−q, having no defined parity in w/H5129. In addition, notice
that when g12=g21=0, the fluctuation matrix becomes block
diagonal indicating that the two bands are uncoupled. Thiscorresponds to the case where the Josephson coupling V
12
=V21=0, since g12=−V12/detVas indicated in Eq. /H208498/H20850.
Performing Matsubara summations over w/H5129in the expres-
sions for M11nandM12nleads to
M11n/H20849q/H20850=−gnn+/H9008n,11qp-qh+/H9008n,11qp-qp, /H2084925/H20850
M12n/H20849q/H20850=/H9008n,12qp-qh+/H9008n,12qp-qp, /H2084926/H20850
where the explicit form of the /H9008n,ijfunctions is given in
Appendix A, for the peruse of the reader. We choose to sepa-rate the contributions of the matrix elements M
ijin terms of
quasiparticle-quasiparticle /H20849qp-qp /H20850and quasiparticle-
quasihole /H20849qp-qh /H20850processes in order to isolate the channels
that contribute to Landau damping of the collective modes tobe discussed in the next section.
IV . COLLECTIVE MODES AT T=0
The collective modes are determined by the poles of the
propagator matrix M−1/H20849q/H20850for the pair fluctuation fields /H9011/H20849q/H20850,
which describe the Gaussian deviations about the saddle
point order parameter. The poles of M−1/H20849q/H20850are determined
by the condition det M=0, and lead to a dispersion for the
collective modes w=w/H20849q/H20850, when the usual analytic continu-
ation iv/H5129→w+i0+is performed.
We will focus here only at the zero temperature limit, but
we will analyze phase and amplitude modes. At T=0, only
the qp-qp terms contribute, as the qp-qh terms vanish iden-tically /H20849see Appendix A /H20850. In this limit, we separate the diag-
onal matrix elements of M/H20849q/H20850into even and odd contribu-
tions with respect to w
M
11n,E/H20849q/H20850=−gnn+/H20858
k/H9003/H110322/H20851/H9264/H9264/H11032+EE/H11032/H20852/H20851E+E/H11032/H20852
2EE/H11032/H20851w2−/H20849E+E/H11032/H208502/H20852, /H2084927/H20850M11n,O/H20849q/H20850=−/H20858
k/H9003/H110322/H20851/H9264/H9264/H11032+EE/H11032/H20852w
2EE/H11032/H20851w2−/H20849E+E/H11032/H208502/H20852. /H2084928/H20850
The off-diagonal term is even in w, and it reduces to
M12n/H20849q/H20850=−/H20858
k/H9003/H110322/H9004/H9004 /H11032/H20851E+E/H11032/H20852
2EE/H11032/H20851w2−/H20849E+E/H11032/H208502/H20852. /H2084929/H20850
We used in the previous expressions the following simplified
notation: the kinetic energies /H9264n/H20849k/H20850→/H9264,/H9264n/H20849k+q/H20850→/H9264/H11032; the
quasiparticle energies En/H20849k/H20850→E,En/H20849k+q/H20850→E/H11032; the order
parameters /H9004n/H20849k/H20850→/H9004,/H9004n/H20849k+q/H20850→/H9004/H11032; and the symmetry
factors/H9003n/H20849k/H20850→/H9003,/H9003n/H20849k+q/2/H20850→/H9003/H11032.
In order to obtain the collective mode spectrum, we ex-
press/H9011n/H20849q/H20850=/H9270n/H20849q/H20850ei/H9278n/H20849q/H20850=/H20851/H9261n/H20849q/H20850+i/H9258n/H20849q/H20850/H20852//H208812 where/H9270n/H20849q/H20850,
/H9278n/H20849q/H20850,/H9261n/H20849q/H20850, and/H9258n/H20849q/H20850are all real. Notice that the new fields
/H9261n/H20849q/H20850=/H9270n/H20849q/H20850cos/H9278/H20849q/H20850, and/H9258n/H20849q/H20850=/H9270n/H20849q/H20850sin/H9278/H20849q/H20850can be re-
garded essentially as the amplitude and phase fields, respec-
tively, when /H9278/H20849q/H20850is small. This change of basis can be de-
scribed by the following unitary transformation:
/H9011/H20849q/H20850=1
/H208812/H208981i00
1−i00
00 −i1
00 i1/H20899/H20898/H92611/H20849q/H20850
/H92581/H20849q/H20850
/H92582/H20849q/H20850
/H92612/H20849q/H20850/H20899. /H2084930/H20850
Since we are considering the case where the saddle point
order parameters /H90041/H20849k/H20850and/H90042/H20849k/H20850are in phase the fluctua-
tion matrix in the rotated basis then reads
M˜=/H20898x1 iz1 0 −g12
−iz1y1−g12 0
0 −g21 y2−iz2
−g21 0 iz2 x2/H20899, /H2084931/H20850
where xn=M11n,E+M12n,yn=M11n,E−M12n, and zn=M11n,Owith the
qdependence being implicit. The rotated matrix M˜is as
complex as the un-rotated matrix M, and the real advantage
of working in the rotated basis involving fields /H9261nand/H9258nis
that the interpretation in terms of amplitude and phase fieldsis straightforward. For instance, by inspection, notice that thephase fluctuation fields
/H92581and/H92582corresponding to different
condensates are coupled by interband elements g12=g21. The
same applies to amplitude fluctuation fields /H92611and/H92612. When
the interaction V12=0, the matrix element g12vanishes, the
matrix M˜becomes block diagonal and the two bands are
decoupled. In this case the one-band results previously ob-tained are recovered for each independent band.
28
Next, we focus on phase-phase and amplitude-amplitude
collective modes. The easiest way to get the amplitude-amplitude collective modes is to integrate out the phasefields to obtain an amplitude-only effective action
S
/H92611/H92612=/H9252
2/H20858
q/H9261†/H20849q/H20850Ma/H20849q/H20850/H9261/H20849q/H20850, /H2084932/H20850
where/H9261†/H20849q/H20850=/H20851/H92611/H20849q/H20850,/H92612/H20849q/H20850/H20852. The amplitude-amplitude fluc-
tuation matrix has the formM. ISKIN AND C. A. R. SÁ DE MELO PHYSICAL REVIEW B 72, 024512 /H208492005 /H20850
024512-4Ma=/H20873x1+y2z12/Wa −g12/H208491−z1z2/Wa/H20850
−g12/H208491−z1z2/Wa/H20850 x2+y1z22/Wa/H20874,/H2084933/H20850
where Wa=g122−y1y2. The dispersion relation for the
amplitude-amplitude collective modes is obtained from thecondition det M
a=0. In this paper, however, we are mostly
interested in the phase-phase collective modes. Thus, uponintegration of the amplitude fields we obtain a phase-onlyeffective action
S
/H92581/H92582=/H9252
2/H20858
q/H9258†/H20849q/H20850Mp/H20849q/H20850/H9258/H20849q/H20850, /H2084934/H20850
where/H9258†/H20849q/H20850=/H20851/H92581/H20849q/H20850,/H92582/H20849q/H20850/H20852. The phase-phase fluctuation ma-
trix has the form
Mp=/H20873y1+x2z12/Wp −g12/H208491−z1z2/Wp/H20850
−g12/H208491−z1z2/Wp/H20850 y2+x1z22/Wp/H20874 /H2084935/H20850
with Wp=g122−x1x2. Again, the dispersion relation for the
phase-phase collective modes is obtained from the conditiondetM
p=0 corresponding to poles of the phase-phase corre-
lation matrix Mp−1. In the next section, we discuss both ana-
lytical results for the phase-phase modes in BCS and BEClimits, as well as numerical results and the crossover regime.
We note in passing that at finite temperatures, for con-
tinuum s-wave systems, the qp-qp terms are well behaved in
the long-wavelength and low frequency limit, while theqp-qh terms do not in general allow for a simple expansionin the same limit due to the presence of Landau damping /H20849see
Appendix A /H20850. However, at zero temperature, the qp-qh terms
vanish, and a well-defined expansion is possible at low fre-quencies provided that the collective modes cannot decayinto the two-quasiparticle continuum. Thus, the collectivemode dispersion w/H20849q/H20850must satisfy the following condition:
w/H20849q/H20850/H11270min /H20853E
1/H20849k/H20850+E1/H20849k+q/H20850,E2/H20849k/H20850+E2/H20849k+q/H20850/H20854.
/H2084936/H20850
To obtain the long wavelength dispersions for the collective
modes at T=0, we expand the matrix elements of M˜/H20851Eq.
/H2084931/H20850/H20852in the amplitude-phase representation to second order
in/H20841q/H20841and fourth order in wto get
xn=An+Cn/H20841q/H208412−Dnw2+Enw2/H20841q/H208412+Fnw4, /H2084937/H20850
yn=Pn+Qn/H20841q/H208412−Rnw2+Snw2/H20841q/H208412+Tnw4, /H2084938/H20850
zn=Bnw+Hnw3, /H2084939/H20850
where expansion coefficients are given in Appendix B. As it
will become clear in Sec. V , the /H20841q/H208414order terms in the ex-
pansion are not necessary to calculate the collective modefrequencies w/H20849q/H20850accurately to order /H20841q/H20841
2.
The expressions of the coefficients found in Appendix B
are valid for values of the interaction range parameter kn,0
that satisfy the diluteness condition /H20849kn,0/kF/H112711/H20850. However,
in order to make analytical progress in the calculation of
collective modes to be discussed next, we take the limitk
n,0→/H11009, since all the momentum integrals are convergent.
This is in contrast to the situation encountered with the orderparameter equation, where we used kn,0/H11011104kFin order to
ensure convergence of our numerical calculations for /H20841/H9004n/H20841
and/H9262as will be seen in the next section.
V . ANALYTICAL AND NUMERICAL RESULTS
In this section, we will focus only on the long-wavelength
/H20849small /H20841q/H20841/H20850limit phase-phase modes determined by the con-
dition det Mp=0. We begin our discussion with the trivial
case when g12=0 /H20849where V12=0 but V11andV22can have
any negative value /H20850which corresponds to two uncoupled
bands. In this case, the phase-phase fluctuation matrix Mp
=0 becomes block diagonal, and we find two Goldstone
modes satisfying the relation
wn2=cn2/H20841q/H208412, /H2084940/H20850
where the square of the speed of sound cninnth band is
given by
cn2=AnQn
Bn2+AnRn. /H2084941/H20850
In the limit q→0, the corresponding eigenvectors are given
by
/H9258†/H20849q=0 ,w=0/H20850=/H20849/H92581,/H92582/H20850/H11008/H208491,0 /H20850/H20849 42/H20850
for the first band, and
/H9258†/H20849q=0 ,w=0/H20850=/H20849/H92581,/H92582/H20850/H11008/H208490,1 /H20850/H20849 43/H20850
for the second band, respectively. These results are identical
to the known results in the one-band case.28
However, in the nontrivial case when g12/HS110050, which cor-
responds to a finite Jopephson coupling V12between bands,
we find two modes. In order to determine the collectivemode spectra w/H20849q/H20850accurately to order /H20841q/H20841
2it is sufficient to
rewrite the determinant condition in the form
w4/H20849/H92511+/H92512/H20841q/H208412/H20850+w2/H20849/H92513+/H92514/H20841q/H208412/H20850+/H92515/H20841q/H208412=0 , /H2084944/H20850
where/H9251nare nontrivial and extremely complicated functions
of the expansion coefficients An,Bn,Cn,Dn,..., given in Ap-
pendix B. The exact dependence can be obtained via a sym-bolic manipulation program, but we will not quote these gen-eral results here or in appendices, as they are not particularlyilluminating. Instead, we will present simple limits, wheretheir behavior can be easily understood.
In this nontrivial case of g
12/HS110050, we find two collective
modes. The first mode is the Goldstone mode satisfying therelation
w
2=c2/H20841q/H208412, /H2084945/H20850
where the square of the speed of sound cis given by
c2=−/H92515//H92513/H110220. /H2084946/H20850
In the limit q→0, the eigenvector of the Goldstone mode is
given by
/H9258†/H20849q=0 ,w=0/H20850=/H20849/H92581,/H92582/H20850/H11008/H20849/H20841/H90041/H20841,/H20841/H90042/H20841/H20850, /H2084947/H20850
which is valid for all values of the V11andV22couplings.BCS-BEC CROSSOVER OF COLLECTIVE EXCITATIONS … PHYSICAL REVIEW B 72, 024512 /H208492005 /H20850
024512-5Notice that this mode is associated with the in-phase fluctua-
tions of the phases of the order parameters around theirsaddle point values.
In the particular case, where g
12/min /H20853N1,N2/H20854is the small-
est expansion parameter /H20849small g12limit /H20850we can perform a
Taylor expansion of the /H9251ncoefficients around g12=0, and
obtain
c2=t1t2/H20849P2Q1+P1Q2/H20850//H20849P1t1+P2t2/H20850, /H2084948/H20850
as the square of speed cof the Goldstone mode. Here, we
introduced the coefficient
tn=/H20849An−Pn/H20850//H20851Bn2+Rn/H20849An−Pn/H20850/H20852/H110220, /H2084949/H20850
which is positive definite since An/H11022PnandRn/H110220. The pre-
cise meaning of the smallest expansion parameter
g12/min /H20853N1,N2/H20854will be clear in Secs. V A and V B, where
we discuss analytically the weak and strong coupling limits.
The eigenvector in the small g12limit has the same as in the
case of general g12, since/H9258†/H20849q=0,w=0/H20850=/H20849/H92581,/H92582/H20850is not ex-
plicitly dependent on g12./H20851See Eq. /H2084947/H20850/H20852
The second mode is a finite frequency mode satisfying the
relation
w2=w02+v2/H20841q/H208412, /H2084950/H20850
where the square of a finite frequency w0of the mode is
w02=−/H92513//H92511/H110220 /H2084951/H20850
and the square of the speed vof the mode is
v2=/H92513/H92512//H925112−/H92514//H92511+/H92515//H92513. /H2084952/H20850
The eigenvector for this mode has a complicated expression
/H9258†/H20849q,w/H20850=/H20849/H92581,/H92582/H20850/H11008/H20851g12/H20849Wp−z1z2/H20850,y1Wp+x2z12/H20852,/H2084953/H20850
for a general value of g12.
In the small g12limit, the coefficients simplify to
w02=P1t1+P2t2/H110220, /H2084954/H20850
v2=Q1t1+Q2t2−c2/H110220. /H2084955/H20850
In the limit of w=w0,q→0, and small g12, the eigenvector
expression simplifies to
/H9258†/H20849q=0 ,w=w0/H20850=/H20849/H92581,/H92582/H20850/H11008/H20849/H20841/H90042/H20841t1,−/H20841/H90041/H20841t2/H20850, /H2084956/H20850
and it becomes transparent that this mode is associated with
out-of- phase fluctuations of the phases of the order param-
eters around their saddle point values, since tnis positive
definite.
Before discussing the collective modes in the analytically
tractable weak and strong coupling limits, notice that a finiteqGoldstone mode is only possible when c
2/H110220, and that a
finite qfinite frequency mode is only possible when w02/H110220.
If these conditions are violated the modes are nonexistent.Furthermore, caution should be exercised by recalling thatthe small wapproximation used in the expansion of general
fluctuation matrix /H20849M
˜/H20850elements breaks down when the fre-
quency wof any of the collective modes moves up into the
continuum of two-quasiparticle states. Therefore, our resultsare strictly valid only forw/H11270min /H208532E
1/H20849k/H20850,2E2/H20849k/H20850/H20854, /H2084957/H20850
which corresponds to
w/H11270min /H208532/H20841/H90041/H20841,2/H20841/H90042/H20841/H20854 /H20849 58/H20850
in the weak coupling /H20849BCS /H20850limit, and to
w/H11270min /H208532/H20881/H20841/H9262/H208412+/H20841/H90041/H208412,2/H20881/H20849/H20841/H9262/H20841+E0/H208502+/H20841/H90042/H208412/H20854/H20849 59/H20850
in the strong coupling /H20849BEC /H20850limit. With these conditions in
mind, we discuss next the weak and strong coupling limits.
A. Weak Coupling Limit
The s-wave weak coupling limit is characterized by the
criteria/H9262/H110220,/H9262/H11022E0,/H9262/H11015/H9255F/H11271/H20841/H90041/H20841, and/H9262−E0/H11271/H20841/H90042/H20841. Ana-
lytical calculations are particularly simple in this case sinceall integrals for the coefficients needed to calculate the col-lective mode dispersions are peaked near the Fermi surface/H20849see Appendix B /H20850. In addition, we make use of the nearly
perfect particle-hole symmetry, which forces integrals tovanish, when their integrands are odd under the transforma-tion
/H9264→−/H9264. For instance, the coefficients that couple phase
and amplitude modes within a given band /H20849BnandHn/H20850van-
ish. Thus, in this case, there is no mixing between phase andamplitude fields within nth band, as can be seen by inspec-
tion of the fluctuation matrix M˜/H20851Eq. /H2084931/H20850/H20852.
We would like to focus on the phase-phase collective
modes, as they correspond to the low energy part of thecollective mode spectrum. Notice that, all expansion coeffi-cients appearing in the phase-phase fluctuation matrix M
p
are analytically tractable. The expansion of the matrix ele-
ments to order /H20841q/H208412andw4is performed under the condition
/H20849w,/H20841q/H208412/2mn/H20850/H11270min /H208532/H20841/H90041/H20841,2/H20841/H90042/H20841/H20854. To evaluate xnfor each
band n, we need A1=g12/H20841/H90042/H20841//H20841/H90041/H20841+N1,A2=g21/H20841/H90041/H20841//H20841/H90042/H20841+N2,
which are the coefficients of the /H20849q=0,w=0/H20850term; Cn
=cn,w2Nn/12 /H20841/H9004n/H208412, which are the coefficients of /H20841q/H208412;Dn
=Nn/12 /H20841/H9004n/H208412, which are the coefficients of w2; and Fn
=−Nn/120 /H20841/H9004n/H208414, which are the coefficients of w4. To evaluate
yn, we need P1=g12/H20841/H90042/H20841//H20841/H90041/H20841andP2=g21/H20841/H90041/H20841//H20841/H90042/H20841, which are
the coefficients of the /H20849q=0,w=0/H20850term; Qn=cn,w2Nn/4/H20841/H9004n/H208412,
which are the coefficients of /H20841q/H208412;Rn=Nn/4/H20841/H9004n/H208412, which are
the coefficients of w2; and Tn=−Nn/24 /H20841/H9004n/H208414which are the
coefficients of w4. Here,
cn,w=vn,F//H20881dn /H2084960/H20850
is the velocity of the sound mode in the one-band case,18dn
is the dimension, vn,Fis the Fermi velocity, and Nnis the
density of states at the Fermi energy per spin in the nth band.
While we use Nnas the density of states per spin at the Fermi
energy /H20849Nn=mnL2/2/H9266in 2D and Nn=mnL3kn,F/2/H92662in 3D /H20850in
thenth band, in Ref. 4 the density of states used includes
both spins. The off-diagonal elements are /H20849Mp/H2085012=/H20849Mp/H2085021
=−g12, since B1=B2=0 and H1=H2=0, as discussed above.
Notice that the expressions above are valid for both 2D and3D bands.
In order to bring our results in contact with Leggett’s,
4
and Sharapov et al. ,5,6we make use of the order parameter
saddle point Eq. /H2084917/H20850atT=0M. ISKIN AND C. A. R. SÁ DE MELO PHYSICAL REVIEW B 72, 024512 /H208492005 /H20850
024512-6/H20841/H90041/H20841/H208491+V11F1/H20850=− /H20841/H90042/H20841V21F2, /H2084961/H20850
/H20841/H90042/H20841/H208491+V22F2/H20850=− /H20841/H90041/H20841V12F1, /H2084962/H20850
and consider the small g12limit, where
g12/min /H20853N1,N2/H20854/H11270min /H20853/H20841/H90041/H20841,/H20841/H90042/H20841/H20854/max /H20853/H20841/H90041/H20841,/H20841/H90042/H20841/H20854./H2084963/H20850
A simple evaluation of det Mp=0 leads to a Goldstone mode
w2=c2/H20841q/H208412, characterized by the speed of sound
c2=N1c1,w2+N2c2,w2
N1+N2, /H2084964/H20850
and a finite frequency /H20849Leggett /H20850mode w2=w02+v2/H20841q/H208412, char-
acterized by
w02=N1+N2
2N1N2/H208418V12/H20841/H20841/H90041/H20841/H20841/H90042/H20841
V11V22−V122, /H2084965/H20850
v2=N1c2,w2+N2c1,w2
N1+N2, /H2084966/H20850
where w0is a finite frequency, and vis the speed of propa-
gation of the mode. Here we reintroduced all the couplingconstants of the original Hamiltonian in Eq. /H208491/H20850. These re-
sults are valid only in the weak-coupling limit, with allV
nm/H110210, and det V/H110220. Notice that if V12=0 the Leggett
mode does not exist as the two bands are uncoupled. Further-more, the trivial limit of one-band /H20849say only band 1 exists /H20850is
directly recovered by taking /H20841/H9004
2/H20841=0,N2=0 and c2,w=0 which
leads to c2=c1,w2,w02=0, and v2=0.
It is also very illustrative to analyze the eigenvectors as-
sociated with the two solutions. For Goldstone’s mode, in thelimit of q→0; for any g
12, it is easy to see that
/H9258†/H20849q=0 ,w=0/H20850=/H20849/H92581,/H92582/H20850/H11008/H20849/H20841/H90041/H20841,/H20841/H90042/H20841/H20850, /H2084967/H20850
corresponding to an in-phase mode. In the degenerate case,
where /H20841/H90041/H20841=/H20841/H90042/H20841, and N1=N2, the eigenvector /H9258†/H20849q=0,w
=0/H20850/H11008/H208491,1 /H20850and the phase fields are perfectly in phase. The
eigenvector for the finite frequency mode is /H9258†/H20849q,w/H20850
=/H20849/H92581,/H92582/H20850/H11008/H20849g12,y1/H20850. In the particular limit where g12is small
/H20849Leggett mode /H20850, this leads to an eigenvector
/H9258†/H20849q=0 ,w=w0/H20850=/H20849/H92581,/H92582/H20850/H11008/H20849N2/H20841/H90041/H20841,−N1/H20841/H90042/H20841/H20850, /H2084968/H20850
which corresponds to an out-of-phase mode. In the degener-
ate case, this simplifies further to /H9258†/H20849q=0,w=0/H20850=/H208491,−1 /H20850be-
coming a perfectly out-of-phase mode.
Now, we would like to turn our attention to the analysis of
the phase-phase modes in the strong coupling limit, which isalso analytically tractable.
B. Strong coupling limit
The s-wave strong coupling limit is characterized by the
criteria/H9262/H110210, and /H20841/H9262/H20841/H11271/H20841/H90041/H20841and /H20841/H9262/H20841+E0/H11271/H20841/H90042/H20841. The situation
encountered here is very different from the weak couplinglimit, because one can no longer invoke particle-hole sym-metry to simplify the calculation of many of the coefficients
appearing in the fluctuation matrix M
˜/H20849see Appendix B /H20850.I nparticular, the coefficients Bn,Hn/HS110050 indicate that the ampli-
tude and phase fields within an individual band nare mixed.
We will concentrate here only on phase-phase modes
characterized by the fluctuation matrix Mp. The expansion of
the matrix elements to order /H20841q/H208412andw4is performed under
the condition /H20849w,/H20841q/H208412/2mn/H20850/H112702/H20841/H9262/H20841. All the coefficients
/H20849xn,yn,zn,Wp/H20850appearing in Mpmatrix Eq. /H2084935/H20850are evaluated
to order /H20849/H20841/H90041/H20841//H20841/H9262/H20841/H208502and /H20851/H20841/H90042/H20841//H20849/H20841/H9262/H20841+E0/H20850/H208522in the strong cou-
pling limit for both two and three dimensional systems. To
evaluate xnfor each band n, we need A1=g12/H20841/H90042/H20841//H20841/H90041/H20841
+/H92601/H20841/H90041/H208412/2/H20841/H9262/H20841,A2=g21/H20841/H90041/H20841//H20841/H90042/H20841+/H92602/H20841/H90042/H208412/2/H20849/H20841/H9262/H20841+E0/H20850, which
are the coefficients of the /H20849q=0,w=0/H20850term; Cn=/H9260n/4mn,
which are the coefficients of /H20841q/H208412;D1=/H92601/8/H20841/H9262/H20841,D2
=/H92602/8/H20849/H20841/H9262/H20841+E0/H20850, which are the coefficients of w2, and F1=
−/H92601/H9253/512 /H20841/H9262/H208413,F2=−/H92602/H9253/512 /H20849/H20841/H9262/H20841+E0/H208503, which are the coef-
ficients of w4. To evaluate yn, we need P1=g12/H20841/H90042/H20841//H20841/H90041/H20841and
P2=g21/H20841/H90041/H20841//H20841/H90042/H20841, which are the coefficients of the /H20849q=0,w
=0/H20850term; Qn=/H9260n/4mn, which are the coefficients of /H20841q/H208412;
R1=/H92601/8/H20841/H9262/H20841,R2=/H92602/8/H20849/H20841/H9262/H20841+E0/H20850, which are the coefficients of
w2, and T1=−/H92601/H9253/512 /H20841/H9262/H208413,T2=−/H92602/H9253/512 /H20849/H20841/H9262/H20841+E0/H208503, which
are the coefficients of w4. To evaluate zn, we need Bn=/H9260n,
which are the coefficients of w, and H1=/H92601/H9253¯/96 /H20841/H9262/H208412,H2
=/H92602/H9253¯/96 /H20849/H20841/H9262/H20841+E0/H208502which are the coefficients of w3. The
computation of Wp=g122−x1x2, just relies on the knowledge
ofxnalready obtained above. While the expressions above
are valid for any value of /H9262andE0in the 2D bands, they are
only rigorously valid for /H20841/H9262/H20841/H11271E0in the 3D case through the
use of the dimensionally dependent variables /H9260n, and/H9253and/H9253¯
which in the 3D case become
/H92601=/H9266N1/8/H20881/H20841/H9262/H20841/H9255F, /H2084969/H20850
/H92602=/H9266N2/8/H20881/H20849/H20841/H9262/H20841+E0/H20850/H9255F, /H2084970/H20850
and/H9253=5,/H9253¯=3. To write our expressions in compact notation
and to recover the degenerate limit /H20849E0→0/H20850, we keep E0in
all 3D coefficients. All expressions for the 3D case can be
converted into the 2D case through the following procedure.In all coefficients /H20849x
n,yn,zn,Wp/H20850, the dependence in /H20841/H9262/H20841,
/H20849/H20841/H9262/H20841+E0/H20850is transformed into /H20841/H9262/H20841/2, /H20849/H20841/H9262/H20841+E0/H20850/2, and the vari-
ables/H9260n, and/H9253and/H9253¯need to be redefined as
/H92601=N1/8/H20841/H9262/H20841, /H2084971/H20850
/H92602=N2/8/H20849/H20841/H9262/H20841+E0/H20850, /H2084972/H20850
and/H9253=2,/H9253¯=2.
In the limit q→0, the condition det Mp=0 leads again to
two modes. General expressions for the collective modes arehighly nontrivial and, therefore, we consider analyticallyonly the asymptotic small g
12limit defined as
g12/min /H20853N1,N2/H20854/H11270/H20851/H20841min /H20853/H20841/H90041/H20841,/H20841/H90042/H20841/H20854/H20841//H20849/H20841/H9262/H20841+E0/H20850/H208522/H112701.
/H2084973/H20850
In this case, the Goldstone mode corresponds to w2=c2/H20841q/H208412,
whereBCS-BEC CROSSOVER OF COLLECTIVE EXCITATIONS … PHYSICAL REVIEW B 72, 024512 /H208492005 /H20850
024512-7c2=/H92601/H20841/H9262/H20841c1,s2+/H92602/H20849/H20841/H9262/H20841+E0/H20850c2,s2
/H92601/H20841/H9262/H20841+/H92602/H20849/H20841/H9262/H20841+E0/H20850/H2084974/H20850
is the square of the speed of sound. The finite frequency
mode /H20849which is the extension of Leggett’s mode in the weak
coupling limit /H20850corresponds to w2=w02+v2/H20841q/H208412, where
w02=/H92601/H20841/H9262/H20841+/H92602/H20849/H20841/H9262/H20841+E0/H20850
2/H92601/H20841/H9262/H20841/H92602/H20849/H20841/H9262/H20841+E0/H20850/H20841V12/H90041/H90042/H20841
V11V22−V122, /H2084975/H20850
v2=/H92601/H20841/H9262/H20841c2,s2+/H92602/H20849/H20841/H9262/H20841+E0/H20850c1,s2
/H92601/H20841/H9262/H20841+/H92602/H20849/H20841/H9262/H20841+E0/H20850, /H2084976/H20850
are the finite frequency, and speed of propagation of the
mode, respectively. Here, the quantitites
c1,s=/H20841/H90041/H20841//H208818m1/H20841/H9262/H20841, /H2084977/H20850
c2,s=/H20841/H90042/H20841//H208818m2/H20841/H20849/H9262/H20841+E0/H20850/H20849 78/H20850
are the velocities of the sound mode in the one-band case.28
These results are valid only in the strong-coupling limit, with
allVnm/H110210, and det V/H110220. Notice that if V12=0 the finite
frequency mode does not exist as the two bands are un-coupled. Furthermore, the trivial limit of one band /H20849say only
band 1 exists /H20850is directly recovered by taking
/H92602=0 and
c2,s=0 which leads to c2=c1,s2,w02=0, and v2=0.
The eigenvectors associated with these solutions are as
follows. For Goldstone’s mode, in the limit of q→0 and for
any value of g12,
/H9258†/H20849q=0 ,w=0/H20850=/H20849/H92581,/H92582/H20850/H11008/H20849/H20841/H90041/H20841,/H20841/H90042/H20841/H20850, /H2084979/H20850
corresponding to an in-phase mode. For the finite frequency
mode /H20849in the small g12limit /H20850, the eigenvector becomes
/H9258†/H20849q=0 ,w=w0/H20850/H11008/H20849/H92602/H20849/H20841/H9262/H20841+E0/H20850/H20841/H90041/H20841,−/H92601/H20841/H9262/H20841/H20841/H90042/H20841/H20850,/H2084980/H20850
which corresponds to an out-of-phase mode, as/H9260n/H110220. In the
degenerate case /H20849/H92601=/H92602,/H20841/H90041/H20841=/H20841/H90042/H20841andE0→0/H20850, this simpli-
fies further to /H9258†/H20849q=0,w=0/H20850=/H208491,−1 /H20850becoming a perfectly
out-of-phase mode, similar to the weak coupling case.
Next, we would like to turn our attention to the analysis of
the phase-phase modes in the crossover region which is notanalytically tractable and requires numerical calculations.
C. Numerical results and crossover region
Thus far, we have focused on the analytically tractable
limits corresponding to weak and strong couplings. In orderto gain further insight into the behavior of the phase-phasecollective excitations at T=0, we present numerical results of
the evolution from weak to strong coupling for the Goldstoneand finite frequency modes. We limit ourselves to numericalcalculations of the fully degenerate /H20849identical /H20850bands case,
from which the known one-band results
28for the Goldstone
mode can be easily recovered in the BCS, BEC, and cross-over regimes for an s-wave superfluid. While the limit of
degenerate bands is probably harder to find in nature, it pro-vides us with qualitative and quantitative understanding ofthe evolution of the finite frequency collective modes fromweak to strong coupling. At the same time the degenerate
problem is easier to solve numerically, and serves as a testmodel to our analytical results. Thus, we postpone a detailednumerical calculation of nondenegerate bands case for a fu-ture publication, where the more complex numerical problemwill be attacked.
In this particular case /H20849identical bands /H20850, the band offset is
E
0→0, the density of states at the Fermi energy are N1
=N2=N; the Fermion masses are m1=m2=m; the Fermi ve-
locities are v1,F=v2,F=vF; and the intraband interactions are
V11=V22=V; while the order parameter amplitudes are /H20841/H90041/H20841
=/H20841/H90042/H20841=/H20841/H9004/H20841/H20850. Furthermore, the low-momentum and low-
energy expansion coefficients become identical, i.e., A1=A2
=A;B1=B2=B;D1=D2=D;H1=H2=H;R1=R2=R; and T1
=T2=T, and we define A˜=A−g12=/H20858k/H20841/H9004/H20849k/H20850/H208412/2E/H20849k/H208503. For
any value of g12in the degenerate bands limit, we obtain
c2=v2=QA˜
B2+A˜R/H110220, /H2084981/H20850
for the squares of the speeds c/H20849Goldstone /H20850and v/H20849finite
frequency /H20850. In addition, we obtain
w02=P2/H20849g12/H20850
P2/H11032/H20849g12/H20850/H2084982/H20850
for the square of the finite frequency which is also valid for
any value of g12. The numerator P2/H20849g12/H20850=2/H20849B2+A˜R/H20850/H20849g12A˜
+2g122/H20850is positive definite and the denominator
P2/H11032/H20849g12/H20850=/H20849B2+A˜R/H208502+g12/H20851R/H20849B2+A˜R/H20850+A˜/H208492TA˜−4BH /H20850+8A˜RD
+6B2D/H20852+g122/H208514TA˜+8/H20849RD−BH /H20850+5B2D/A˜/H20852must be also
positive definite to guarantee that w02/H110220. Notice that these
functions are not strictly second-order polynomials in g12,
since all coefficients depend implicitly on g12. In the small
g12limit, expressions for c2,v2, and w02simplify to Eqs. /H2084948/H20850,
/H2084955/H20850, and /H2084954/H20850, with the degenerate case coefficients. Further-
more, since P2/H20849g12/H20850/H110220 is positive definite, Eq. /H2084982/H20850is only
valid strictly for P2/H11032/H20849g12/H20850/H110220.
In our numerical calculations of /H20841/H9004n/H20841and/H9262/H20851via the saddle
point Eqs. /H2084917/H20850and /H2084921/H20850/H20852, we choose a momentum cutoff of
value k1,0=k2,0=k0/H11011104kFto ensure convergenge of all
k-space integrations. The fermion density can be expressed
in dimensionless units as ns=n/nmax, where nmax=kFmax2/2/H9266
in 2D, and nmax=kFmax3/3/H92662in 3D. The maximal value of the
Fermi momentum kFmax/H20849that fixes the maximal density nmax/H20850
is chosen by fixing the ratio kFmax/k0=10−4, which easily
satisfies the diluteness conditions /H20849k0/kFmax/H208503/H112711/H20849ornmaxR03
/H112701/H20850in 3D, and /H20849k0/kFmax/H208502/H112711/H20849ornmaxR02/H112701/H20850in 2D. See
Sec. II to recall how the interactions depend on k0=2/H9266/R0,
where R0plays the role of the interaction range in real space.
For any kF/kFmax/H110211 all conditions are satisfied, thus we
work at fixed density ns=1/2/H9266/H110150.159, corresponding to
kF/kFmax=1/ /H208812/H9266=0.398 in 2D, and ns=3/ /H208498/H9266/H20850/H110150.119, cor-
responding to kF/kFmax=/H208493/8/H9266/H208501/3/H110150.492 in 3D.
We also confine ourselves to the asymptotic small g12
limit, which means g12/N/H112701 in weak coupling, and g12/N
/H11270/H20851/H20841/H9004/H20841//H20841/H9262/H20841/H208522in strong coupling limits. Therefore, in 3D, weM. ISKIN AND C. A. R. SÁ DE MELO PHYSICAL REVIEW B 72, 024512 /H208492005 /H20850
024512-8choose V12=10−7V /H20849since V11=V22=V/H20850which leads to
g12/N/H1101110−4, for a 3D Fermion density of ns=0.119. In the
2D case, we choose V12=10−5V, which leads to g12/N
/H1101110−4, for a 2D Fermion density of ns=0.159. This particu-
lar choice satisfies the small g12condition for the range of
couplings shown in Figs. 2, 3, 4, and 5.
We solve the saddle point equations for the order param-
eter Eq. /H2084917/H20850together with the number equation Eq. /H2084921/H20850
self-consistently for fixed densities. The order parameter am-plitude /H20841/H9004/H20841and chemical potential
/H9262in 2D /H208493D/H20850are pre-
sented in Figs. 2 /H20849a/H20850and 2 /H20849b/H20850/H20851Figs. 4 /H20849a/H20850and 4 /H20849b/H20850/H20852as a func-
tion of the dimensionless intraband interaction parameterV
r=NV//H9266. Notice that the system crosses over from the BCS
/H20849/H9262/H110220/H20850to BEC /H20849/H9262/H110210/H20850regimes at Vr=1.33/H1100310−2in 2D,
and at Vr=1.59/H1100310−4in 3D, where the chemical potential /H9262
crosses zero /H20849the bottom of the degenerate bands /H20850.
For the 2D /H208493D/H20850case, we show in Figs. 3 /H20849a/H20850and 3 /H20849b/H20850
/H20851Figs. 5 /H20849a/H20850and 5 /H20849b/H20850/H20852, numerical plots of the sound velocity c,
normalized by the Fermi velocity vF, and of the ratio of the
finite frequency w0with respect to the minimum quasi-
particle excitation energy min /H208532E/H20849k/H20850/H20854as a function of intra-
band couplings Vr. In addition, notice the very good agree-
ment between the numerical results and the analyticalapproximations in their respective /H20849BCS or BEC /H20850limits.
The analytical value for the weak coupling sound velocity
follows from Eq. /H2084964/H20850for the nondegenerate case with N
1
=N2andc1,w=c2,w, which leads to c2=vF2/2 for 2D and c2
=vF2/3 for 3D bands. Similarly, the analytical value for the
strong coupling sound velocity follows from Eq. /H2084974/H20850with
/H92601=/H92602andc1,s=c2,s, which leads to c2=/H20841/H9004/H208412/4m/H20841/H9262/H20841for 2Dand c2=/H20841/H9004/H208412/8m/H20841/H9262/H20841for 3D bands. Thus, we recover the
Goldstone mode in both BCS and BEC limits as in the caseof the presence of only one band.
20,28The numerical values
/H20851solid circles in Figs. 3 /H20849a/H20850and 5 /H20849a/H20850/H20852for the sound velocity as
a function of the dimensionless coupling Vrare calculated
from Eq. /H2084981/H20850. Notice the very good agreement with the ana-
lytical results in weak and strong coupling /H20849dotted lines in
the same figures /H20850. As a further consistency check, notice that
this agreement is very reasonable since Eq. /H2084981/H20850is identical
to the expression for the sound velocity given in Ref. 28 forthe one-band model, with the correspondence that our coef-
ficient A˜has the same expression as the coefficient Adefined
in their paper.
Notice in Fig. 3 /H20849a/H20850that the sound velocity cis essentially
a constant for all couplings Vrin the k0/kFmax→/H11009/H20849k0/kFmax=104/H20850limit. For smaller values of k0/kFmaxthe sound velocity
decreases as a function of Vr/H20849not shown in figure /H20850. We will
not discuss here the dependence of con the ratio k0/kFmaxsince we are mostly concerned with checking the consistency
of our calculations with the analytically tractable limits. Inthis case, the sound velocity cis not a good indicator that the
BCS-BEC crossover is occuring in 2D. However, in the 3Dcase /H20851see Fig. 5 /H20849a/H20850/H20852, the speed of sound changes very fast in
the neighboorhood of
/H9262=0, thus manifesting itself as an in-
dicator of the crossover regime. Therefore, measurements ofGoldstone mode frequency can offer an indication of theBCS-BEC crossover possibly only in 3D two-band superflu-ids.
With the confidence of recovering the sound mode results
for a one-band model from a two-band model with identical
FIG. 2. Plots of /H20849a/H20850chemical potential /H9262scaled by/H9255Fand /H20849b/H20850
order parameter /H20841/H9004/H20841scaled by/H9255Fversus dimensionless coupling Vr
/H20849see text for definition /H20850for two-dimensional degenerate bands. The
region where /H9262changes sign is shown in the inset. Note that /H9262
=0 when Vr=0.0133.
FIG. 3. Numerical calculation /H20849solid circles /H20850of/H20849a/H20850square of
sound velocity c2scaled by vF2, and of /H20849b/H20850the square of finite fre-
quency w02scaled by the two-quasiparticle threshold min /H208532E/H20849k/H20850/H20854
versus dimensionless coupling Vrfor two-dimensional degenerate
bands. The dotted lines represent analytical results for the weak andstrong coupling limits. Notice that we scaled w
02by 4 /H20841/H9004/H208412for/H9262
/H110220, and by 4 /H20849/H20841/H9262/H208412+/H20841/H9004/H208412/H20850for/H9262/H110210.BCS-BEC CROSSOVER OF COLLECTIVE EXCITATIONS … PHYSICAL REVIEW B 72, 024512 /H208492005 /H20850
024512-9bands, we proceed with the discussion of the finite frequency
mode, which is shown in Figs. 3 /H20849b/H20850and 5 /H20849b/H20850for the 2D and
3D cases, respectively.
The analytical value of w02for the weak coupling finite
frequency mode follows from Eq. /H2084965/H20850for the nondegenerate
case with V11=V22=V,N1=N2=N, and /H20841/H90041/H20841=/H20841/H90042/H20841=/H20841/H9004/H20841, which
leads to w02=8g12/H20841/H9004/H208412/Nfor both 2D and 3D bands. Simi-
larly, the analytical value for strong coupling follows from
Eq. /H2084975/H20850, which leads to w02=8g12/H20841/H9004/H208412/Nfor 2D, and w02
=8g12/H20841/H9004/H208412/H20881/H9255F//H20841/H9262/H20841//H9266Nfor 3D bands, respectively. Numerical
values /H20851solid circles in Figs. 3 /H20849b/H20850and 5 /H20849b/H20850/H20852for the finite
frequency w0as a function of the dimensionless coupling Vr
are calculated from Eq. /H2084982/H20850. Notice the very good agreement
between the numerical results and their analytical counter-parts in both weak and strong coupling limits /H20849dotted lines in
the same figures /H20850. It is important to notice that the scales for
weak and strong coupling used in the finite frequency plotsare not the same. We scaled w
0by min /H208532E/H20849k/H20850/H20854, which corre-
sponds to the two-quasiparticle excitation threshold, thus w02
is scaled by 4 /H20841/H9004/H208412since min /H208532E/H20849k/H20850/H20854=2/H20841/H9004/H20841for/H9262/H110220, and w02
is scaled by 4 /H20849/H20841/H9262/H208412+/H20841/H9004/H208412/H20850since min /H208532E/H20849k/H20850/H20854=2/H20881/H20841/H9262/H208412+/H20841/H9004/H208412for
/H9262/H110210. This choice is natural, because it indicates that the
finite frequency w0always lies below the two-quasiparticle
excitation threshold for the parameters used, meaning thatthe collective mode is undamped.
Notice in Figs. 3 /H20849b/H20850and 5 /H20849b/H20850/H20849k
0/kFmax→/H11009,k0/kFmax=104/H20850that the finite frequency w0changes qualitatively near
the coupling Vrwhere/H9262changes sign for both 2D and 3D
cases. We will not discuss here the dependence of w0on the
ratio k0/kFmaxsince we are mostly concerned with checkingthe consistency of our calculations with the analytically trac-
table limits. However, it is important to emphasize that thefinite frequency mode is a good indicator that the BCS-BECcrossover is occuring in both 2D and 3D. Thus, measure-ments of the finite frequency w
0can reveal BCS-BEC cross-
over behavior in two-band superfluids.
VI. CONCLUSIONS
We studied the evolution of low energy collective excita-
tions from weak /H20849BCS /H20850to strong /H20849BEC /H20850coupling limits in
two-band s-wave superfluids at T=0 for all intraband cou-
pling strengths with ranges satisfying the diluteness condi-tion. We assumed that the two bands were coupled via aninterband Josephson interaction. We focused on the phase-phase collective modes and showed that there can be twoundamped phase-phase modes in the evolution from weak tostrong coupling. In the weak coupling limit, we recoveredLeggett’s results corresponding to an in-phase mode /H20849Gold-
stone /H20850mode and an out-of-phase mode /H20849Leggett’s mode /H20850in
the appropriate asymptotic limits. Furthermore, we general-ized Leggett’s weak coupling results to include the BCS-BEC crossover and the strong coupling regime. In addition,we presented analytical results in the strong coupling limit,in the asymptotic limit of small g
12corresponding to weak
Josephson coupling between the bands. All the analytical re-sults were presented for the cases of two-dimensional andthree-dimensional bands, and the for cases of nondegenerateand degenerate bands.
FIG. 4. Plots of /H20849a/H20850chemical potential /H9262scaled by/H9255Fand /H20849b/H20850
order parameter /H20841/H9004/H20841scaled by/H9255Fversus dimensionless coupling Vr
/H20849see text for definition /H20850for three-dimensional degenerate bands. The
region where /H9262changes sign is shown in the inset. Note that /H9262
=0 when Vr=1.592/H1100310−4.
FIG. 5. Numerical calculation /H20849solid circles /H20850of/H20849a/H20850square of
sound velocity c2scaled by vF2, and of /H20849b/H20850the square of finite fre-
quency w02scaled by the two-quasiparticle threshold min /H208532E/H20849k/H20850/H20854
versus dimensionless coupling Vrfor three-dimensional degenerate
bands. The dotted lines represent analytical results for the weak andstrong coupling limits. Notice that we scaled w
02by 4 /H20841/H9004/H208412for/H9262
/H110220, and by 4 /H20849/H20841/H9262/H208412+/H20841/H9004/H208412/H20850for/H9262/H110210.M. ISKIN AND C. A. R. SÁ DE MELO PHYSICAL REVIEW B 72, 024512 /H208492005 /H20850
024512-10On the numerical side, we analyzed fully the limit of de-
generate bands, from which the one-band results28can be
easily recovered in the BCS, BEC, and crossover regimes fora3 D s-wave superfluid. The limit of degenerate bands, al-
though less likely to be found in nature, provides us with agood basis for the more challenging numerical work for thenon-degenerate case, which will be performed in the future,as they require the self-consistent solutions of three simulta-neous nonlinear integral equations in order to determine theorder parameters /H20841/H9004
1/H20841,/H20841/H90042/H20841, and the chemical potential /H9262.
We have also briefly described in this manuscript the pro-
cedure to compute the amplitude-amplitude collectivemodes, although we have not discussed in detail their ex-plicit form, we would like to mention that amplitude-amplitude collective modes are higher energy modes in com-parison to lowest energy /H20849phase-phase /H20850collective modes
discussed here. The phase-phase and amplitude-amplitudecollective modes are potentially important to the analysisof multi-component ultracold neutral Fermi gases, wherecollective mode frequencies can be measuredspectroscopically.
25,26
The results for collective modes described here /H20849neutral /H20850
is not strictly valid in standard /H20849charged /H20850condensed matter
systems. In particular, in the BCS limit, the effect of theCoulomb interaction is to plasmonize the the Goldstonemode. Furthermore, the Coulomb interaction modifies thevelocity of the Leggett mode, but does not change its finitefrequency offset.
4,5There is some experimental evidence that
the Leggett mode in the BCS limit has been observed12in
MgB 2, but there is presently no experimental charged two-
band condensed matter system where the BEC limit can bereached via the tunning of an experimental parameter. Theinteresting extension to the BEC limit and the crossover re-gion in the charged case is currently underway and will bepublished elsewhere.
To conclude, the main contribution of our manuscript is to
study the evolution of the Goldstone /H20849in-phase /H20850and the finite
frequency /H20849out-of-phase /H20850collective modes from weak /H20849BCS /H20850
to strong /H20849BEC /H20850couplings in neutral two-band superfluids.
Our results are potentially relevant to multicomponent ultra-cold Fermi atoms, and can be used, in principle, to test thevalidity of the BCS-BEC evolution based on extensions
18,19
of Eagles’,15Leggett’s,16and the Nozieres and
Schmitt-Rink17suggestions.
ACKNOWLEDGEMENT
We would like to thank NSF /H20849DMR-0304380 /H20850for finan-
cial support.
APPENDIX A: MATRIX ELEMENTS
In the evaluation of the elements of the fluctuation matrix
M/H20849q/H20850appearing in Sec. III, and defined in Eqs. /H2084925/H20850, we need
to calculate the functions /H9008n,11qp-qh,/H9008n,11qp-qp,/H9008n,12qp-qh, and/H9008n,12qp-qp
given by
/H9008n,11qp-qh=/H20858
kX/H20875/H20841u/H208412/H20841v/H11032/H208412
iv+E−E/H11032−/H20841v/H208412/H20841u/H11032/H208412
iv−E+E/H11032/H20876,/H9008n,11qp-qp=/H20858
kY/H20875/H20841v/H208412/H20841v/H11032/H208412
iv−E−E/H11032−/H20841u/H208412/H20841u/H11032/H208412
iv+E+E/H11032/H20876,
/H9008n,12qp-qh=/H20858
kX/H20875u*vu/H11032*v/H11032
iv+E−E/H11032−u*vu/H11032*v/H11032
iv−E+E/H11032/H20876,
/H9008n,12qp-qp=/H20858
kY/H20875u*vu/H11032*v/H11032
iv+E+E/H11032−u*vu/H11032*v/H11032
iv−E−E/H11032/H20876.
In the previous expressions the indices qp-qh and qp-qp clas-
sify quasiparticle-quasihole and quasiparticle-quasiparticleterms, respectively. Furthermore, we used the following sim-plified notation: the kinetic energies
/H9264n/H20849k/H20850→/H9264,/H9264n/H20849k+q/H20850
→/H9264/H11032; the quasiparticle energies En/H20849k/H20850→E,En/H20849k+q/H20850→E/H11032;
the order parameters /H9004n/H20849k/H20850→/H9004,/H9004n/H20849k+q/H20850→/H9004/H11032; the symme-
try factors/H9003n/H20849k/H20850→/H9003,/H9003n/H20849k+q/2/H20850→/H9003/H11032; and the Fermi func-
tions fn/H20851En/H20849k/H20850/H20852→f,fn/H20851En/H20849k+q/H20850/H20852→f/H11032. We also made use of
the definition of the first coherence factor
/H20841u/H208412=1
2/H208731+/H9264
E/H20874, /H20849A1/H20850
/H20851/H20841u/H11032/H208412=/H208491+/H9264/H11032/E/H11032/H20850/2/H20852, the second coherence factor
/H20841v/H208412=1
2/H208731−/H9264
E/H20874 /H20849A2/H20850
/H20851/H20841v/H11032/H208412=/H208491−/H9264/H11032/E/H11032/H20850/2/H20852, and the phase relation between them
u*v=/H9004
2E/H20849A3/H20850
/H20851u/H11032*v/H11032=/H9004/H11032/2E/H11032/H20852. Finally, we used the notation X=/H20849f
−f/H11032/H20850/H9003/H110322andY=/H208491−f−f/H11032/H20850/H9003/H110322to indicate the combinations of
Fermi functions /H20849f,f/H11032/H20850and symmetry coefficients /H20849/H9003,/H9003/H11032/H20850ap-
pearing in the quasiparticle-quasihole, and quasiparticle-
quasiparticle terms, respectively.
APPENDIX B: EXPANSION COEFFICIENTS
From the rotated fluctuation matrix M˜expressed in the
amplitude-phase basis as defined in Sec. IV , we can obtainthe expansion coefficients necessary to calculate the collec-tive modes at T=0. In the long-wavelength /H20849q→0/H20850, and low
frequency limit /H20849w→0/H20850the matrix M
˜defined in Eq. /H2084931/H20850is
fully determined by the knowledge of the matrix elements xn,
yn, and zn. Note that, this expansion requires
w,/H20841q/H208412/2mn/H11270min /H208532E1/H20849k/H20850,2E2/H20849k/H20850/H20854 /H20849 B1/H20850
and these coefficients are valid for all couplings for s-wave
pairing.
In all the expressions below we use the following simpli-
fying notation /H9003˙=/H11509/H9003//H11509k,/H9264˙=/H11509/H9264//H11509k,/H9003¨=/H115092/H9003//H11509k2,/H9264¨=/H115092/H9264//H11509k2,
andk=/H20841k/H20841.
The coefficients necessary to obtain the matrix element xn
areBCS-BEC CROSSOVER OF COLLECTIVE EXCITATIONS … PHYSICAL REVIEW B 72, 024512 /H208492005 /H20850
024512-11An=−gnn−/H20858
k/H92642
2E3/H90032, /H20849B2/H20850
corresponding to the /H20849q=0,w=0/H20850term,
Cn=/H20858
k1
8E5/H20877/H9264¨/H9264/H20849E2−3/H90042/H20850/H90032+/H92642/H208492/H90042−/H92642/H20850/H9003/H9003¨
−/H9264˙2/H20875E2−1 0/H90042/H208731−/H90042
E2/H20874/H20876/H90032cos2/H9251
+2/H9264˙/H9264/H20875E2+/H90042/H208731−1 0/H92642
E2/H20874/H20876/H9003/H9003˙cos2/H9251
−/H20875/H92642/H20849/H92642−7/H90042/H20850−5/H90044/H208731−2/H92642
E2/H20874/H20876/H9003˙2cos2/H9251/H20878,
/H20849B3/H20850
corresponding to the /H20841q/H208412term with/H9251being the angle be-
tween kandq
Dn=/H20858
k/H92642
8E5/H90032, /H20849B4/H20850
corresponding to the w2term, and
Fn=−/H20858
k/H92642
32E7/H90032, /H20849B5/H20850
corresponding to the w4term.
The coefficients necessary to obtain the matrix element yn
arePn=−gnn−/H20858
k1
2E/H90032, /H20849B6/H20850
corresponding to the /H20849q=0,w=0/H20850term,
Qn=/H20858
k1
8E5/H20853/H9264¨/H9264E2/H90032−/H9264˙2/H20849E2−3/H90042/H20850/H90032−/H92642E2/H9003/H9003¨
+/H20851/H9264˙/H9264/H208492E2−6/H90042/H20850/H9003/H9003˙−/H92642/H20849/H92642−2/H90042/H20850/H9003˙2/H20852cos2/H9251/H20854,/H20849B7/H20850
corresponding to the /H20841q/H208412term
Rn=/H20858
k1
8E3/H90032, /H20849B8/H20850
corresponding to the w2term, and
Tn=−/H20858
k1
32E5/H90032, /H20849B9/H20850
corresponding to the w4term.
The coefficients necessary to obtain the matrix element zn
are
Bn=/H20858
k/H9264
4E3/H90032, /H20849B10 /H20850
corresponding to the wterm, and
Hn=/H20858
k/H9264
16E5/H90032, /H20849B11 /H20850
corresponding to the w3term.
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PhysRevB.95.155417.pdf | PHYSICAL REVIEW B 95, 155417 (2017)
Long-range exchange interaction in triple quantum dots in the Kondo regime
YongXi Cheng,1,2YuanDong Wang,1JianHua Wei,1,*ZhenGang Zhu,3and YiJing Yan4,5
1Department of Physics, Renmin University of China, Beijing 100872, China
2Department of Science, Taiyuan Institute of Technology, Taiyuan 030008, China
3School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China
4Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China,
Hefei, Anhui 230026, China
5Department of Chemistry, Hong Kong University of Science and Technology, Kowloon, Hong Kong
(Received 1 August 2016; revised manuscript received 30 March 2017; published 12 April 2017)
Long-range interactions in triple quantum dots in the Kondo regime are investigated by accurately solving
the three-impurity Anderson model. For the occupation configuration of ( N1,N2,N3)=(1,0,1), a long-range
antiferromagnetic exchange interaction ( JAF) is demonstrated and induces a crossover from the separated Kondo
singlet to the long-range spin singlet state between edge dots. In the long-range spin singlet phase, a long-rangeoverlapping or entanglement of Kondo clouds is discovered, which induces a transition peak in the spectralfunction of the middle dot under equilibrium conditions. Under nonequilibrium conditions, the long-rangeentanglement of the Kondo clouds is characterized by the conductance peak at zero bias, which can be observedin experiments.
DOI: 10.1103/PhysRevB.95.155417
I. INTRODUCTION
Long-range interaction as a high-order interaction origi-
nates from the superpositions of indirectly coupled states. Itplays an important role in many-body physics and quantumcomputing [ 1–4]. For the latter, the long-range interaction
makes it possible to manipulate a distant quantum gate orqubit in one step, which is of higher operating efficiencyand fault-tolerant capability than nearest-neighbor control inexchange-based quantum gates [ 5]. The triple-quantum-dot
(TQD) device provides an ideal platform for investigatingthe quantum manipulation [ 6–11]. The long-range transport
in serially coupled TQDs has been observed in recent ex-periments [ 3,12,13]. For example, Platero et al. measured
a resonant transport line (in the area of the bipolar spinblockade) between the edge dots, which suggests a long-range coherent superposition near the degenerate point of(N
1,N2,N3)=(1,1,1)/(2,1,2) (Niis the number of electrons
inith QD) [ 12]. Shortly afterwards, the same group reported
a long-range spin transfer near another degenerate point of(1,0,1)/(2,0,2), where QD2 remains unoccupied during the
tunneling process [ 13]. Vandersypen et al. demonstrated a
high-order coherent tunneling between QD1 and 3 near thedegenerate point of (0 ,1,0)/(1,1,1) through the observation of
Landau-Zener-Stückelberg interference [ 3].
In order to produce measurable current, all of above
experimental results are achieved in the boundary of Coulombblockade near degenerate points in the stability diagram.However, these regimes are not suitable for theoretical analysisof the long-range interaction (especially the long-range spincorrelation or exchange interaction), since occupation numbersand magnetic moments of QD1 and 3 are not conservedduring the transport under bias in those boundaries. One betterchoice is to push the range of study deeply into the Coulombblockade region far away from the degenerate points, such
*wjh@ruc.edu.cnas the local moment regime of QD1 and 3 where both theoccupation number and spin are well defined. In order toproduce measurable current or other observable features, weinvestigate the long-range exchange interaction and its effectsin the Kondo regime.
The Kondo phenomenon itself is an important and in-
teresting issue in TQDs. It results from the screening of alocalized spin by the delocalized spins from reservoirs (orleads), which presents a pronounced zero-bias conductancepeak at temperatures below the Kondo temperature in QDsystems, with a Kondo singlet (KS) formed [ 14–16]. Recently
considerable theoretical efforts have been made in the topicof serial TQDs, such as the equilibrium and nonequilib-rium Kondo transport properties [ 17], Fermi-liquid versus
non-Fermi-liquid behavior [ 18], and two-channel Kondo
physics [ 19]. In addition, the Kondo phenomenon in other
structures of TQDs has been discussed as well, including themirror symmetry TQDs [ 20–22], triangular TQDs [ 23–26],
and parallel TQDs [ 27]. To the best of our knowledge, none of
these works concern the long-range exchange interaction andits effect on the Kondo phenomenon in TQDs.
In the present work, we study the long-range exchange
interaction between QD1 and 3 in the Kondo regime in serialTQDs, by accurately solving the three-impurity Andersonmodel with the hierarchical equation of motion (HEOM)formalism [ 28,29]. The geometry is depicted in Fig. 1(a).
Two symmetrical edge dots (QD1 and 3) are in the localmagnetic moment regime ( N
1=N3=1), and are coupled to
the source (S) and drain (D) reservoir but decoupled from eachother ( t
13=0). The intermediate one (QD2) symmetrically
couples to the QD1 and 3 ( t12=t23=t) via a variable
singly occupied level ε2modulated by a gate voltage Vg.
In order to highlight the long-range correlation, we focuson the occupation configuration of ( N
1,N2,N3)=(1,0,1) by
pushing ε2high enough, as schematically shown in Fig. 1(b).
In the limit of (1 ,1,1), we have reported a reappearance of the
Kondo phenomenon and worked out an effective ferromagneticexchange interaction between QD1 and 3 [ 30]. In the present
2469-9950/2017/95(15)/155417(6) 155417-1 ©2017 American Physical SocietyCHENG, WANG, WEI, ZHU, AND YAN PHYSICAL REVIEW B 95, 155417 (2017)
FIG. 1. (a) The schematic diagram of the triple-quantum-dot
system. In present work, QD1 and 3 are symmetric and both in
the localized momentum regime with N1=N3=1. QD2 is nearly
unoccupied with a gate-modulated on-site energy ε2=−U/2+eVg.
(b) The schematic diagram is shown for the long-range antiferromag-
netic exchange interaction ( JAF) between QD1 and 3 via high-order
tunneling processes.
work, as schematically shown in Fig. 1(b), we will demonstrate
a long-range antiferromagnetic exchange interaction ( JAF>
0), which can be simply expressed in terms of JAF≈4t4/ξ2U,
where ξ≡ε2−ε1(ε1being the on-site energy of QD1) is
called the detuning energy, and U(Ui=U;i=1,2,3) is
the on-dot Coulomb interaction. The effect of JAFon Kondo
features including spectral characteristics and Kondo currentin TQDs will be discussed in detail.
II. MODEL AND THEORY
The total Hamiltonian for the system is described by the
three-impurity Anderson model
H=Hdots+Hres+Hcoup, (1)
where the isolated TQD part is
Hdots=/summationdisplay
σ,i=1,2,3[/epsilon1iσˆa†
iσˆaiσ+Uniσni¯σ]
+t/summationdisplay
σ(ˆa†
1σˆa2σ+ˆa†
2σˆa3σ+H.c.), (2)
with ˆa†
iσ(ˆaiσ) being the operator that creates (annihilates) a
spin-σelectron with energy /epsilon1iσinith QD. niσ=ˆa†
iσˆaiσis the
operator of the occupation number.
In what follows, the symbol μis adopted to denote the
electron orbital (including spin, space, etc.) in the systemfor brevity, i.e., μ={σ,i,... }. The device reservoirs are
treated as single-particle systems with the Hamiltonian as
H
res=/summationtext
kμα=S,D/epsilon1kαˆd†
kμαˆdkμα, with /epsilon1kαbeing the energy of
an electron with wave vector kin the αreservoir, and
ˆd†
kμα(ˆdkμα) the corresponding creation (annihilation) operator
for an electron with the α-reservoir state |k/angbracketrightof energy /epsilon1kα.
The Hamiltonian of the dot-reservoir coupling is Hcoup=/summationtext
kμαtkμαˆa†
μˆdkμα+H.c. To describe the stochastic nature
of the transfer coupling, it can be written in the reservoirH
res-interaction picture as Hcoup=/summationtext
μ[f†
μ(t)ˆaμ+ˆa†
μfμ(t)],
withf†
μ=eiHrest[/summationtext
kαt∗
kμαˆd†
kμα]e−iHrestbeing the stochasticinteractional operator and satisfying the Gauss statistics.
Here, tkμαdenotes the transfer coupling matrix element.
The influence of electron reservoirs on the dots is takeninto account through the hybridization functions, which as-sume Lorentzian form, /Delta1
α(ω)≡π/summationtext
ktαkμt∗
αkμδ(ω−/epsilon1kα)=
/Delta1W2/[2(ω−μα)2+W2], where /Delta1is the effective quantum
dot–reservoir coupling strength, Wis the bandwidth, and μα
is the chemical potentials of the αreservoir.
In this paper, the three-impurity Anderson model is accu-
rately solved by the HEOM approach, which is establishedbased on the Feynman-Vernon path-integral formalism with ageneral Hamiltonian, in which the system-environment corre-lations are fully taken into consideration [ 28,29]. The HEOM
formalism is in principle accurate and applicable to arbitraryelectronic systems, including Coulomb interactions, under theinfluence of arbitrary applied bias voltage and external fields.The outstanding issue of characterizing both equilibrium andnonequilibrium properties of a general open quantum systemis referred to Refs. [ 28–33]. It has been demonstrated that
the HEOM approach achieves the same level of accuracyas the latest high-level numerical renormalization group andquantum Monte Carlo approaches for the prediction of variousdynamical properties at equilibrium and nonequilibrium [ 29].
The reduced density matrix of the quantum dot system
ρ
(0)(t)≡trres[ρtotal(t)] and a set of auxiliary density matrices
{ρ(n)
j1···jn(t);n=1,..., L }are the basic variables in HEOM. L
denotes the truncated tier level. The equations governing thedynamics of open systems are in the form of [ 28,29]
˙ρ(n)
j1···jn=−/parenleftBigg
iL+n/summationdisplay
r=1γjr/parenrightBigg
ρ(n)
j1···jn−i/summationdisplay
jA¯jρ(n+1)
j1···jnj
−in/summationdisplay
r=1(−)n−rCjrρ(n−1)
j1···jr−1jr+1···jn, (3)
where A¯jandCjrare Grassmannian superoperators which are
illustrated in detail in Refs. [ 28,29].
The dynamical quantities can be acquired via the HEOM-
space linear response theory [ 34]. The spectral function A(ω)
exhibiting prominent Kondo signatures at low temperaturescan be evaluated by a half Fourier transformation of correlationfunctions as
A
μ(ω)=1
πRe/parenleftbigg/integraldisplay∞
0dt{˜Cˆa†
μˆaμ(t)+[˜Cˆaμˆa†
μ(t)]∗}eiωt/parenrightbigg
.(4)
The electric current from the αreservoir to the system is
given by
Iα(t)=i/summationdisplay
μtrs[ρ†
αμ(t)ˆaμ−ˆa†
μρ−
αμ(t)], (5)
where ρ†
αμ=(ρ−
αμ)†is the first-tier auxiliary density operator.
The details of the HEOM formalism and the derivation ofphysical quantities are supplied in Refs. [ 28,29].
III. RESULTS AND DISCUSSION
As shown in Fig. 1, we assume that QD1 and 3 always keep
electron-hole symmetry and their parameters are the same,in which ε
1=/epsilon11=−U/2 and ε3=/epsilon13=−U/2. In order to
figure out whether there may exist a long-range exchange
155417-2LONG-RANGE EXCHANGE INTERACTION IN TRIPLE . . . PHYSICAL REVIEW B 95, 155417 (2017)
FIG. 2. The spin-spin correlation function C13≡/angbracketleft/vectorS1·/vectorS3/angbracketright−
/angbracketleft/vectorS1/angbracketright·/angbracketleft/vectorS3/angbracketrightvaries as a function of the on-site energy of QD2, i.e., ε2,
and the nearest interdot coupling strength t. Two phases are shown: the
Kondo singlet (KS, C13∼0) and long-range spin singlet (LSS, C13<
0). The horizontal dashed line marks the gradual change of phasesfrom KS to LSS, which is further elucidated by the spectral functions
of the scatter points (see Fig. 3). The vertical dashed line marks the
gradual change of phases with ε
2att=0.7 meV . The parameters
are as follows: U=1.2 (in units of meV), /epsilon11=/epsilon13=−0.6, the
bandwidth of reservoirs W=5.0, the temperature KBT=0.03, and
the hybridization width between reservoirs and QDs is /Delta1=0.3.
interaction, we calculate the spin-spin correlation function
between QD1 and 3,
C13≡/angbracketleft/vectorS1·/vectorS3/angbracketright−/angbracketleft/vectorS1/angbracketright·/angbracketleft/vectorS3/angbracketright. (6)
In Fig. 2, we depict C13as a function of the modulated
on-site energy of QD2 ( ε2=/epsilon12+eVg) and the nearest interdot
coupling strength t. The other parameters are as follows: the
on-dot Coulomb correlation U=1.2 (in unit of meV), the
bandwith of reservoirs W=5.0, the temperature KBT=0.03
which is much lower than the Kondo temperature of QD1 or3 derived from the analytical formula in the literature [ 15],
and the hybridization widths between reservoirs and QDs/Delta1=0.3. In present work, we set ε
2>1.0 meV to keep the
configuration of (1 ,0,1). Two phases are shown in the figure.
The first one is the Kondo singlet (KS) which is characterized
by near zero correlation between /vectorS1and/vectorS3(C13∼0) at small
t. The second one is the main finding of the present work,
called the long-range spin singlet (LSS) characterized by finiteC
13(C13<0 in the figure), which proves that a long-range
exchange interaction between /vectorS1and/vectorS3dose exist although
direct coupling between them is absent. From the sign of C13,
we conclude that the long-range exchange is antiferromagneticand thus we suggest an effective interaction term as
H
13=JAF/vectorS1·/vectorS3. (7)
It is expected that H13plays an important role in quan-
tum computing, which can expand the original idea of theexchange-based quantum gates [ 5], by manipulating a distant
quantum gate or qubit in one step. We comment that thesmall value of C
13in the bottom right corner of Fig. 2results
from the competition between the LSS phase and the effectiveferromagnetic phase we have reported in Ref. [ 30].
More detailed information of the phase diagram in Fig. 2
can be illustrated by the spectral functions A
iσ(ω)i nd i f f e r -FIG. 3. The spectral functions Ai(ω) of the TQDs are shown at
ε2=3.0 (in unit of meV , ξ=3.6) for different interdot coupling
strengths along the horizontal line in Fig. 2:( a )t=0.01, (b) t=0.5,
and (c) t=0.7. The top panel: i=1; the middle panel: i=2; and
the bottom panel: i=3. The other parameters are the same as those
in Fig. 2.
ent phases. The spin degeneracy makes Ai↑(ω)=Ai↓(ω)=
Ai(ω) and the symmetry in our model also suggests A1(ω)=
A3(ω). We select three characteristic points at t=0.01, 0.5,
and 0.7 meV along the dashed line in Fig. 2atε2=3.0(ξ=
3.6) meV , and depict their corresponding Ai(ω)i nF i g s . 3(a)to
3(c). By referring to Figs. 2and3, we find in the limit of weak
interdot coupling ( t<0.2 meV) the absence of long-range
correlation ( C13∼0) results in the individual screening of
local momentums by the nearest reservoirs; thus the degenerateKS state is formed. The spectral function A
1(ω)/A3(ω)s h o w s
similar behavior to that in a single QD with one Kondo peakatω=0, while A
2(ω)∼0 around ω=0 due to the empty
occupation, as shown in Fig. 3(a). With the increase of the
interdot coupling, the single peak of A1(ω)/A3(ω)g r o w s
slightly higher due to the “ t-enhanced Kondo phenomenon”
(figure not shown) [ 31].
Further increasing ttot>0.3 meV distinctly changes
the Kondo features. As shown in Fig. 3(b), the central
peak of A1(ω)/A3(ω) becomes much higher and wider at
t=0.5 meV than that at t=0.01 meV; meanwhile a small
peak develops in A2(ω) near ω=0. The latter is unusual,
since QD2 is still empty and the emerging peak impossiblyresults from Kondo screening directly. The most possiblemechanism is the long-range tunneling between QD1 and3 by the aid of J
AF, or equivalently speaking, the electron
wave functions separately localized in QD1 and 3 at t∼0
becomes overlapping within QD2 now. At first glance, it isanalogous to the ordinary double-well model in the textbooks;however, what are localized in QD1 and 3 here are not ordinaryelectrons but Kondo quasiparticles. This means that the Kondoquasiparticle in QD1 can tunnel to QD3 through QD2, viaoverlapping their wave functions which are nothing but thewidely studied “Kondo cloud” [ 35]. Although we cannot
present the spatial distribution of Kondo clouds here, wehave demonstrated their long-range overlapping, or long-rangequantum entanglement [ 36].
155417-3CHENG, WANG, WEI, ZHU, AND YAN PHYSICAL REVIEW B 95, 155417 (2017)
FIG. 4. The I-Vcurves of the TQD system at various detuning
energies ξatt=0.7 (in units of meV) along the vertical line in
Fig.2at (a) low temperature KBT=0.03, and (c) high temperature
KBT=0.3. The corresponding differential conductance dI/dV -V
curves at (b) low temperature KBT=0.03 and (d) high temperature
KBT=0.3. The other parameters are the same as those in Fig. 2.
In the limit of strong interdot couplings, e.g., t=0.7m e V ,
the overlapping of Kondo clouds of QD1 and 3 becomes muchstronger to induce a distinct peak in A
2(ω); meanwhile the
long-range JAFhas induced the crossover from the degenerate
KS state of individual QD to the LSS state (see Fig. 2),
characterized by the splitting of the Kondo peaks of QD1a n d3i nF i g . 3(c). Since Kondo clouds are hard to be observed
in experiments [ 35], we suggest that they can be captured by
their long-range overlapping or entanglement.
In order to relate to experiments and highlight the effect of
long-range entanglement of Kondo clouds under nonequilib-rium conditions, we then calculate the current-voltage ( I-V)
curves and corresponding differential conductances dI/dV at
various detuning energy ξatt=0.7 meV and summarize the
results respectively in Figs. 4(a) and4(b) at low temperature
K
BT=0.03. Interestingly, we find that current can increase
with the increase of ξ, or with the increase of the height of
the potential barrier. For example, the current at V=0.02 mV
increases from 1.3 nA at ξ=1.6 meV to 2.0 nA at ξ=3.6
meV , as shown in Fig. 4(a). Meanwhile, a conductance peak at
zero bias develops with the increase of ξ, as shown in Fig. 4(b).
The anomalous enhancement of transport ability obviouslyresults from the long-range entanglement of Kondo clouds. Att=0.7 andξ=3.6 meV , the long-range entanglement of the
Kondo clouds is strong enough to form an extended conductivechannel (see Fig. 3); thus the electrons can transfer through
QD1 to 3 along this channel no matter how high the barrier
in QD2, which induces the enhancement of current and thedevelopment of conductance peak at zero bias.
The long-range transport with the aid of the entanglement of
Kondo clouds is a many-body effect that is distinctly differentfrom the low-order sequential tunneling and cotunneling. Thatpoint can be verified by checking the transport properties attemperature higher than the Kondo temperature, as shown inFigs. 4(c) and4(d), which depicts the I-VanddI/dV -V
curves with the same parameters as those in Figs. 4(a)and4(b)
except K
BT=0.3 meV . Now, the current decreases normallyFIG. 5. The differential conductance dI/dV -Vcurves of the
TQD system at ξ=3.6 (in units of meV) for different phases along
the horizontal line in Fig. 2:K S(t=0.2), crossover region ( t=0.5),
and LSS ( t=0.7). The other parameters are the same as those
in Fig. 2.
with the increase of ξor with the increase of the height of the
barrier. Of course, no conductance peak can be seen any more[Fig. 4(c)], and all of the dI/dV -Vcurves become flat with
the values of conductance much smaller [Fig. 4(d)] than those
shown in Fig. 4(b).
In order to verify the conclusion that the long-range
entanglement of the Kondo clouds is characterized by the con-ductance peak at zero bias, we calculate the dI/dV -Vcurves
atξ=3.6 meV in different phases along the horizontal line
in Fig. 2:K S(t=0.2 meV), crossover region ( t=0.5m e V ) ,
and LSS ( t=0.7 meV). The results are shown in Fig. 5, where
the scale of the value of dI/dV of the KS phase is expanded
by a factor of 10. As shown in the figure, at small t(KS
phase), although the Kondo peaks in QD1 and 3 contribute apossible conductive channel, the high barrier of QD2 preventsthe transfer of electrons from QD1 to QD3. Increasing tto
0.5 meV drives the TQDs into the crossover region betweenKS and LSS. The weak entanglement of Kondo clouds [seeFig.3(b)] can assist the tunneling of electrons and introduce a
conductance peak ∼0.4e
2/hat zero bias, as shown in Fig. 5.
Further increasing twill broaden and heighten the conductance
peak. In the LSS phase at t=0.7 meV , the conductance
peak at zero bias increases to a much higher value ∼0.9e2/h
(see Fig. 5) due to the strong entanglement of Kondo clouds
[see Fig. 3(c)]. We confirm the maximum value of the
conductance peak only accessible in the LSS phase. If onedrags the phase out of the LSS by changing torε
2from the
top right corner of Fig. 2, the peak will shrink and decrease
(figures not shown).
Finally, we will derive an analytical expression of JAFfor
isolated TQDs, and then prove it is also valid in the open TQD
system over a wide range of parameters. We start from the
Hamiltonian of Eq. ( 2) for isolated TQDs and constrain our
derivation in the subspace with a total occupation number ofN
T≡N1+N2+N3=2. The states of double occupation in
QD2 (with zero occupation in QD1 and 3, i.e., the |0,2,0/angbracketright
states) are excluded because their energy is much higher thanothers. When t=0,E
|1,0,1/angbracketright=2/epsilon11,E|2,0,0/angbracketright=E|0,0,2/angbracketright=2/epsilon11+U,
andE|1,1,0/angbracketright=E|0,1,1/angbracketright=2/epsilon11+ξ. Since only low-energy states
are concerned, we substitute high-energy eigenvalues withtheir unperturbed ones and solve the secular equation to obtainthe singlet and triplet states, where their splitting are defined
155417-4LONG-RANGE EXCHANGE INTERACTION IN TRIPLE . . . PHYSICAL REVIEW B 95, 155417 (2017)
FIG. 6. The dependence of the long-range antiferromagnetic
exchange interaction JAFont4and 1/ξ2is calculated in (a) and
(b), respectively. The other parameters are U=1.2a n d ξ=2.6i n
(a),t=0.55 and U=1.2 in (b). The black lines and red scattered
points are calculated by the analytic formula [Eq. ( 8)] and numerical
HEOM approach, respectively. The other parameters are the same as
those in Fig. 2.
asJAF≡ET−ES. After some algebra and assuming t/lessmuchU,
we get
JAF≈4t4
ξ2U. (8)
Equation ( 8) is similar to the antiferromagnetic exchange
interaction in double QDs (DQDs) if one defines an effectivenext-neighbor interdot coupling between QD1 and 3 as t
/prime=
t2/ξto rewrite Eq. ( 8)a sJAF≈4t/prime2/U. However, a rigorous
proof is required for open TQD systems. By investigating thesplitting of Kondo peaks of QD1 or 3, the features of J
AFcan
be studied, as in DQD systems [ 29]. The distance between the
splitting peaks should equal to 2 JAF.
The results of HEOM calculations on the formula of JAF
are summarized in Fig. 6, where ε2=2.0 meV is chosen. The
other parameters are the same as those in Fig. 2unless specified
otherwise. The numerical JAFas functions of t4and 1/ξ2
are respectively shown in Figs. 6(a) and6(b), together with
the analytical results obtained from Eq. ( 8) for comparison.
One can see that almost all of the numerical data fall in theanalytical lines in the range of parameters explored here. Wethus conclude that J
AF≈4t4/ξ2Uis also valid in open TQD
systems over a wide range of parameters. We emphasize somefeatures shown in Fig. 6as follows: (i) The t
4dependence
shown in Fig. 6(a)indicates that JAFin TQDs is more sensitive
tot; thus a larger tis required to induce the crossover from
the KS to the LSS phase (cf. Fig. 2). (ii) The 1 /ξ2dependence
shown in Fig. 6(b) suggests an easy way to manipulate JAFinTQDs via gate control of the detuning energy. In this sense, the
phase diagram shown in Fig. 2is experimentally accessible.
It is well acknowledged that the nearest-neighbor antiferro-
magnetic exchange can induce a non-Fermi-liquid quantumcritical point in DQDs, which is called the “two-impurityproblem.” Let us make some comments on the relation betweenthe results here and those in DQDs in the literature. In thetwo-impurity problem, Sela and Affleck found a crossoverfrom the critical point to the low-energy Fermi liquid phase atfinite temperature [ 37]. The quantum phase transition has ever
been proved to be very robust against both the asymmetryof the device (parity) and electron-hole asymmetry [ 38].
That crossover behavior has been verified by our HEOMcalculations on parallel-coupled DQDs [ 29], as well as on
serial-coupled ones. For nonequilibrium transport, the HEOMcalculations have found the zero-bias conductance peakexhibits a single peak for the weak interdot coupling t. With
the increase of t, the zero-bias conductance peak also shows a
continuous evolution from single to double peaked behavior.Those nonequilibrium characteristics confirm the conclusionsand predictions in Refs. [ 37,38] and other relevant references.
For the TQDs with an unoccupied middle dot as studiedhere, although the long-range antiferromagnetic exchange issimilar to that of DQDs if an effective next-neighbor interdotcoupling defined, the characteristics in the strong-couplinglimit are quite different. As shown in Fig. 4(b), the zero-bias
conductance peak rather than a splitting dip (see Ref. [ 29])
develops at low temperature. Even at high temperature, thedip character will not appear as shown in Fig. 4(d). Therefore,
the HEOM calculations have revealed new nonequilibriumtransport characteristics in TQDs, which can improve theunderstandings in the above-mentioned literature, includingour work.
IV . SUMMARY
In summary, we have investigated the long-range interac-
tions in triple quantum dots (TQDs) in the Kondo regime, byaccurately solving the three-impurity Anderson model withthe hierarchical equation of motion (HEOM) formalism. Forthe occupation configuration of ( N
1,N2,N3)=(1,0,1), we
demonstrate that there exists a long-range antiferromagneticexchange interaction, J
AF, which can induce a crossover from
the separated Kondo singlet (KS) to the long-range spinsinglet (LSS) state between edge dots. In the LSS phase, along-range overlapping or entanglement of Kondo clouds isdiscovered, which induces a transition peak in the spectralfunction of the middle dot under equilibrium conditions. Undernonequilibrium conditions, the long-range entanglement of
Kondo clouds induces an anomalous enhancement of current
and a conductance peak at zero bias, which can be observed inexperiments. The expression of J
AF≈4t4/ξ2Uis analytically
derived and numerically verified, according to which JAFcan
be conveniently manipulated via gate control of the detuningenergy.
ACKNOWLEDGMENTS
This work was supported by the NSF of China
(No. 11374363) and the Research Funds of Renmin University
155417-5CHENG, WANG, WEI, ZHU, AND YAN PHYSICAL REVIEW B 95, 155417 (2017)
of China (Grant No. 11XNJ026). Computational resources
have been provided by the Physical Laboratory of HighPerformance Computing at Renmin University of China.
Z.G.Z. is supported by the Hundred Talents Program of CAS.
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155417-6 |
PhysRevB.101.035427.pdf | PHYSICAL REVIEW B 101, 035427 (2020)
Editors’ Suggestion
Inelastic light scattering by intrasubband spin-density excitations in GaAs-AlGaAs
quantum wells with balanced Bychkov-Rashba and Dresselhaus spin-orbit interaction:
Quantitative determination of the spin-orbit field
S. Gelfert,1C. Frankerl,1C. Reichl,2D. Schuh,1G. Salis,3W. Wegscheider,2D. Bougeard,1T. Korn,4and C. Schüller1,*
1Institut für Experimentelle und Angewandte Physik, Universität Regensburg, D-93040 Regensburg, Germany
2Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland
3IBM Research-Zurich, 8803 Rüschlikon, Switzerland
4Institut für Physik, Universität Rostock, D-18059 Rostock, Germany
(Received 24 October 2019; revised manuscript received 7 January 2020; published 29 January 2020)
Inelastic light scattering experiments on low-energy intrasubband spin-density excitations (SDEs) are per-
formed in (001)-grown modulation-doped GaAs-AlGaAs single quantum wells in in-plane external magneticfields. The investigated samples possess balanced linear Bychkov-Rashba ( α) and Dresselhaus ( β) spin-orbit
strengths in two different configurations, α=βandα=−β. Both configurations lead to an extreme anisotropy
of the spin splitting of the conduction band, where the in-plane directions of maximum spin splitting for bothconfigurations are perpendicular to each other. The spin splitting asymmetry can be directly detected via theSDE by breaking of the time-reversal symmetry due to transfer of a momentum qin the quantum-well plane.
In addition, the application of an in-plane magnetic field B
ext⊥qallows us to modulate the effective magnetic
field. Via a numerical line-shape analysis of the experimental SDE spectra, we determine the relevant parametersof the samples. We find that the linear spin-orbit strength |α|=βis comparable for both samples, while the
electron gfactors are markedly different. Furthermore, we experimentally quantify the values of the maximum
internal spin-orbit fields, which are as high as B
so∼18 T for both samples.
DOI: 10.1103/PhysRevB.101.035427
I. INTRODUCTION
Resonant inelastic light scattering (RILS) is a very pow-
erful tool for the investigation of electronic excitations insemiconductor nanostructures. In fact, more than two decadesago, the first experimental proof of the interplay of the so-called Bychkov-Rashba [ 1] and Dresselhaus [ 2] spin-orbit
interaction in two-dimensional electron systems (2DES) inGaAs-AlGaAs quantum wells was made by RILS in thepioneering experiments of Jusserand and Richards et al. [3–6].
The collective electronic excitations in such systems are co-herent density oscillations of the 2DES, either oscillationsof the charge density, so-called charge-density excitations(CDE), or oscillations of the spin density, so-called spin-density excitations (SDE). Due to direct Coulomb interac-tion, the CDEs are typically blue-shifted with respect tothe corresponding SDEs [ 7]. The electronic excitations can
either be excitations between different subbands of the 2DES(intersubband excitations), or excitations within a subband(intrasubband excitations). In crystals with zinc blende struc-ture, CDEs and SDEs can be separated in RILS experimentsby polarization selection rules [ 8]: CDEs are visible for
parallel polarizations of the incoming and scattered light,while SDEs require perpendicular linear polarizations. Underconditions of extreme resonance, also excitations which showcharacteristics of single-particle excitations (SPEs) can be
*christian.schueller@ur.deobserved [ 9,10]. These excitations do not obey polarization
selection rules. It is assumed that they are incoherent densityoscillations with no fixed phase relation between individualelectrons, and, hence, their energies are close to the excitationenergies of noninteracting particles, i.e., they can be regardedas SPEs [ 11,12].
Typically, spin-orbit interaction in low-dimensional semi-
conductor systems has been considered as an effective,k-dependent spin-orbit field acting on individual electrons.
Generally, for electrons in quantum wells with zinc blende lat-tice, the spin-orbit field consists of a Dresselhaus contributionwith linear strength β, due to the bulk inversion asymmetry
of the host crystal, and a Bychkov-Rashba term (with linearstrength α), which is present if there is an asymmetry of
the structure, caused by, e.g., external electric fields and /or
space charges due to asymmetric modulation doping. Theeffect of this single-particle-like Bychkov-Rashba and Dres-selhaus spin-orbit interaction on the propagation of intrasub-band CDEs (plasmons) in quantum wells was theoreticallyconsidered by Badalyan et al. [13]. On the other hand, on the
basis of RILS experiments on intersubband and intrasubbandSDEs, it was suggested by Baboux et al. [14,15] that—due to
collective effects—the spin-orbit field in the coherent SDEsmay be enhanced by up to a factor of five [ 14], compared to
the spin-orbit interaction of individual electrons, e.g., insidea spin packet. Furthermore, in Ref. [ 14] it was shown that
the intersubband SDE, which microscopically is a tripletexcitation with spin S=1, splits in the collective spin-orbit
field into three components with magnetic quantum numbers
2469-9950/2020/101(3)/035427(9) 035427-1 ©2020 American Physical SocietyS. GELFERT et al. PHYSICAL REVIEW B 101, 035427 (2020)
mS=0,±1 if a finite wave vector qis transferred to the SDE,
allowing a break of time-reversal symmetry. In this work weshow that for intrasubband SDEs in two comparable GaAs-AlGaAs quantum-well structures, the sample parameters canbe consistently described by the well-known Bychkov-Rashbaand Dresselhaus spin-orbit terms. This may be due to the factthat the intrasubband SDEs are strongly Landau damped, and,hence, are dominantly of single-particle character. However,we observe significantly different gfactors in our two samples,
which may be taken as indicative of weak collective contribu-tions to the SDEs.
Two-dimensional systems with balanced Rashba and Dres-
selhaus spin-orbit strength, i.e., α=±β, have moved into
the focus of spintronic research, motivated by a theoreticalwork of Schliemann et al. [16], who proposed a nonballistic
spin field-effect transistor, and, later, of Bernevig et al. [17],
where the authors proposed a new spin rotational SU(2)symmetry, which should lead to the formation of a persistentspin helix (PSH). The PSH was subsequently experimentallydemonstrated by different groups [ 18–20]. In an earlier work,
some of the authors and others have investigated the spin-orbit spin splitting of a 2DES with balanced Rashba andDresselhaus spin-orbit strengths, α=β, by RILS from the
intrasubband SDE [ 21]. In these experiments, a wave vector q
was either transferred parallel to the [110] in-plane directionof the GaAs-AlGaAs quantum well, where the two contri-butions of spin-orbit interaction add up, or parallel to [1 ¯10],
where they cancel each other. Furthermore, in the presence ofexternal magnetic fields, we could show that a superpositionof the intrinsic spin-orbit field and the external magnetic fieldoccurs [ 22].
In this work, we use RILS measurements to obtain further
understanding of spin-orbit field parameters for samples withbalanced Rashba and Dresselhaus spin-orbit contribution. Inparticular, we compare the two different configurations, α=
βandα=−β. By rotating the samples on a rotary stage, we
precisely map the anisotropic spin splitting, and, dependingon the configuration α=β(sample A) or α=−β(sample
B), we show that the maximum spin splitting occurs parallelto [110] or parallel to [1 ¯10], respectively. Furthermore, by
applying in-plane external magnetic fields, we directly deter-mine the gfactors of the samples and the absolute maximum
strengths of the intrinsic spin-orbit fields. Surprisingly, thegfactors turn out to be significantly different, though the
nominal quantum-well widths and compositions are the samefor both samples. On the other hand, the maximum spin-orbitfield B
so∼18 T is almost identical for both samples. It is
governed by the relation kF/g(|α|+β), with gthe electron g
factor and kFthe Fermi momentum. All extracted parameters
can be consistently described and verified on a single-particlebasis. Thus, we detect no significant influence of many-particle interaction on the strength of the spin-orbit field inSDEs, as was reported in Refs. [ 14,15]. However, we observe
a∼50% difference in the gfactors.
II. EXPERIMENTAL DETAILS
The two investigated samples were grown via molecular-
beam epitaxy and contain (001)-oriented, n-modulation-doped GaAs-Al
xGa1−xAs (x=0.30 for sample A, x=0.33
dominant
dopingα = β
CBE-field
[001](a) (b)
(c)
θ
BLaser
ϕSample
Rotary stageBack
scattered
lightα = −β
doping
layerE-field
CB
A
qsecondary
doping
FIG. 1. (a) Asymmetric conduction-band profile of sample A.
(b) Same for sample B. (c) Sketch of the experimental configuration
for the RILS experiments (see text).
for sample B) single quantum wells with 12-nm well width.
Their parameters were designed for equal strengths of thelinear Rashba and Dresselhaus coefficients, αandβ, respec-
tively. Sample A is from the same wafer which was usedin Refs. [ 19,21,22]. In Ref. [ 19], the PSH in this sample
was imaged for the first time by direct spatial mapping,using time-resolved Kerr microscopy. The wafer was grownwith a so-called inverted doping profile, where the dominantdoping layer is grown before the quantum-well layer. This wasdone in order to induce an electric field across the quantumwell in growth direction ([001] direction). Figure 1(a) is a
schematic of the conduction-band potential profile of thissample. The space charges inside the quantum well and in thedoping regions lead to an electric field in [001] direction [ 23].
The 2DES has an electron carrier density and mobility ofn∼5.0×10
15m−2andμ∼33 m2V−1s−1, respectively, as
determined from transport experiments [ 19]. From the line-
shape analysis of our RILS experiments (see below), wedetermine a reduced carrier density of n∼(2.7±0.3)×
10
15m−2, which is on the one hand due to the redistribution of
electrons from the quantum well to the ionized donors in thebarriers via laser illumination. On the other hand, fluctuationsin the donor distribution may contribute to this quite largedifference since a different piece of the wafer was used. Forour further analyses, we use this value of n, since it follows
directly from the spectroscopic data. The effective mass of theelectrons, m
∗∼0.075m0, was determined from Raman exper-
iments on cyclotron-resonance excitations in a perpendicularmagnetic field (not shown). The measured effective mass islarger than the band-edge mass of GaAs, m
∗=0.067m0,f o r
two reasons. (i) The first is the nonparabolicity of the con-duction band: The electrons that contribute to the intrabandand cyclotron-resonance excitations are close to the Fermienergy, which is about 10–20 meV above the conduction-bandedge. (ii) The second is the penetration of the electron wavefunction into the barrier material (AlGaAs), which has a largereffective mass.
035427-2INELASTIC LIGHT SCATTERING BY INTRASUBBAND … PHYSICAL REVIEW B 101, 035427 (2020)
Sample B is n-modulation-doped only on the surface side
of the quantum-well layer. So the built-in electric field pointsalong the [00 ¯1] direction, which leads to the configuration of
α∼−β, with βbeing positive [ 23]. A sample of the same
wafer was used in Ref. [ 24], where the existence of a PSH
was shown via time-resolved Kerr microscopy. Figure 1(b)
shows a schematic picture of the conduction-band profile ofthe quantum well in this sample. The 2DES of the sam-ple has an electron density of n∼5.9×10
15m−2and mo-
bility of μ∼84 m2V−1s−1, as determined from magneto-
transport measurements. Again, from our spectroscopic data(see below) we extract a slightly lower electron density ofn∼(5.8±0.7)×10
15m−2. The effective electron mass of
m∗∼0.079m0was also determined from cyclotron resonance
Raman experiments. The slightly larger effective mass ofsample B, as compared to sample A, may be due to the largercarrier density in this sample and, hence, larger Fermi energy.
For the RILS experiments, a tunable continuous-wave
Ti:Sapphire laser was used, which was tuned slightly abovethe band gap of the quantum well for near-resonant excitation.For the experiments presented in this manuscript, the sampleswere mounted in a He flow magnetocryostat at a nominaltemperature of T=1.8 K. However, from the line-shape
analysis of our spectra we extract a temperature of the 2DESof about T=12 K, which is due to heating of the 2DES
by the incident laser. All experiments were performed inbackscattering geometry. By tilting the sample normal withrespect to the direction of the incoming and scattered lightby an angle θ, a finite wave vector qcan be transferred to
the 2DES, as shown in Fig. 1(c). For this configuration, the
wave-vector transfer qis given by q∼4π/λ sinθ, where λ
is the wavelength of the laser light. Tilting angles of θ=35
◦
and 40◦were used in our experiments. For the transfer of q
into arbitrary in-plane crystal directions ϕ, the samples were
mounted on an Attocube piezo-driven rotary stage with re-sistive position encoder. Optional in-plane external magneticfields Bwith magnitudes of up to 6 T were generated by a
superconducting split coil magnet. The direction of qwith
respect to Bwas fixed at 90
◦[see Fig. 1(c)]. The scattered light
was analyzed in a triple Raman spectrometer and detected bya liquid-nitrogen-cooled charge-coupled-device camera. Forall experiments, a depolarized scattering geometry was used,i.e., crossed linear polarizations of incident and scattered lightin order to be sensitive for SDEs. The asymmetric line shapesof the experimental spectra were analyzed via a computationalline-shape analysis, based on the Lindhard-Mermin line shapeof intrasubband excitations [ 25,26].
III. THEORETICAL CONSIDERATIONS
The effect of relativistic spin-orbit interaction on an elec-
tron, moving in a periodic crystal potential, can be describedin terms of an effective magnetic field B
so. For a superposition
of the linear Bychkov-Rashba and the linear Dresselhauscontributions with strengths αandβ, respectively, the intrinsic
effective magnetic field in a (001)-grown quantum well isgiven by
B
so=2
gμB{(αky+βkx)ex−(αkx+βky)ey}. (1)[110]
[110]k010
k100ϕ
BSOk100
Δk[110]k010
ϕ(a) (b)
(c) (d)
Δk[110]
[110]k010
k100ϕ
BSOk100k010 [110]A
ϕ
FIG. 2. (a) Schematic picture of the unidirectional effective spin-
orbit field Bsoof a 2DES with balanced Bychkov-Rashba and
Dresselhaus spin-orbit strengths α=β(sample A). The angle ϕ
is measured clockwise with respect to the [ ¯110] in-plane direction.
(b) Fermi contour for spins parallel and antiparallel to Bsofor sample
A. (c) Schematic picture of Bsofor sample B ( α=−β). (d) Fermi
contour for sample B.
Here x,y, and zare parallel to the [100], [010], and [001]
directions, respectively. gis the effective gfactor and μB
the Bohr magneton. For balanced Rashba and Dresselhaus
interactions, α=±β,E q .( 1) reduces to
Bso=|α|+β
gμB(kx±ky)(ex∓ey). (2)
This situation is schematically displayed in Fig. 2(a) forα=
β:Bsois either parallel or antiparallel to the [1 ¯10] in-plane
direction, which leads to a spin splitting for in-plane spins,as schematically shown in Fig. 2(b) for the Fermi contour
of the 2DES. The two energy paraboloids for spins paralleland antiparallel to [1 ¯10] are shifted in kspace by a maximum
value of /Delta1k
110relative to each other. For arbitrary in-plane
directions, denoted by the angle ϕin Figs. 2(a) and2(b),t h e
magnitude of the shift of the two parabolas in this direction isgiven by
/Delta1k
ϕ=/Delta1k110|sinϕ|. (3)
For the situation α=−β, the patterns in Figs. 2(a) and2(b)
are rotated clockwise by 90◦and time reversed, i.e., Bsois
then parallel or antiparallel to the [110] in-plane direction [seeFigs. 2(c)and2(d)].
We turn now to the discussion of intrasubband SDEs,
as investigated in this work via RILS. For simplicity, wediscuss here explicitly the case α=β(sample A) only. The
discussion for α=−βis exactly the same, except that ϕ=0
◦
then has to be replaced by ϕ=90◦, because of the clockwise
rotation of the relevant spin and field patterns. In the follow-ing, the terms “spin up” and “spin down” refer to the twodifferent in-plane spin orientations [ 21]. In the backscattering
035427-3S. GELFERT et al. PHYSICAL REVIEW B 101, 035427 (2020)
qEFEqm = ± 1sq
q2ΔES,ϕ
EFm= + 1s
Δkϕm= - 1s(a) (b)
Wavevector q
Energy02ΔES,ϕ
ΔES,ϕΔkϕϕ ≠ 0°
ϕ = 0°(c)
FIG. 3. (a) Single-particle continua for intrasubband spinflip
excitations for in-plane directions of the wave-vector transfer q
corresponding to ϕ=0◦andϕ/negationslash=0◦, as valid for α=β(sample A).
(b) Cut through the energy paraboloids [from Fig. 2(b)]: Spinflip
transitions (arrows), which correspond to the high-energy cutoffs ofthe single-particle continua for ϕ/negationslash=0
◦. (c) Same as (b), for ϕ=0◦,
where the spin splitting is zero.
geometry, only spinflip intrasubband transitions are allowed
(cf. Ref. [ 21]), i.e., for the intrasubband SDE, spinflip tran-
sitions between states of different in-plane spin orientationshave to be considered only. Consequently, this means thatthe SDE is a triplet excitation with magnetic quantum num-bers m
s=±1 only and ms=0 does not contribute [ 14]. In
Fig. 3(a), the gray-shaded areas are displaying the single-
particle continua for intrasubband spinflip excitations. Forϕ=0
◦, the spin-up and spin-down states are degenerate [see
Fig. 3(c)], i.e., the two excitations with ms=±1 are degen-
erate. The high-energy cutoff of the corresponding single-particle continuum for wave-vector transfer in this direction
is given by E
q=¯h2
m∗kFq, indicated by an orange dashed line
in Fig. 3(a) [central dashed line in Fig. 3(a)]. For ϕ/negationslash=0◦,
the spinflip excitations with ms=+1 and ms=−1a r en o
longer degenerate, i.e., there are two spinflip transitions withdifferent energies for a given wave-vector transfer q[see
Fig.3(b)] with high-energy cutoffs of E
+
q=/Delta1ES,ϕ+¯h2
m∗kFq,
and E−
q=¯h2
m∗kF(q−/Delta1kϕ) [blue and green dashed lines in
Fig. 3(a) and solid arrows in Fig. 3(b), i.e., right and left
dashed lines in Fig. 3(a) and long and short curved arrows
in Fig. 3(b)]. Here /Delta1ES,ϕis the maximum spin splitting at the
Fermi level for in-plane direction ϕ. For a given wave-vector
transfer q, the difference between E+
qandE−
qis just twice the
spin splitting, E+
q−E−
q=2/Delta1ES,ϕ[see Fig. 3(b)], since
/Delta1ES,ϕ=¯h2
m∗kF/Delta1kϕ=2(α+β)kF|sinϕ|. (4)
For sample B, the right-hand side of Eq. ( 4) becomes
/Delta1ES,ϕ=2(|α|+β)kF|cosϕ|. (5)In Fig. 3(a), the Lindhard-Mermin-type line shapes [ 25,26]o f
intrasubband spinflip excitations for a given qare schemati-
cally indicated as solid green, orange, and blue curves. In theunderlying formalism, the excitation spectrum is calculatedon the basis of single-particle transitions by evaluating theimaginary part of the dielectric response function χ
0(ω,q,τ)
of a system of noninteracting particles and taking into accounta finite single-particle scattering time τand the temperature
T. The four parameters, which determine the line shape, are
the magnitude of the wave-vector transfer q, the electron
density n, the electron temperature T, and the single-particle
scattering time τ[26]. Since qis adjusted in the experiment
and therefore known, there are only three free parameters.The maxima of the excitations are around the high-energycutoffs of the corresponding single-particle continua, whichare proportional to the Fermi wave vector k
F. Therefore, via
the relation kF=√
2πn, the electron density nof the 2DES
is determined very accurately by the energetic positions ofthe high-energy cutoffs of the spectra. As mentioned above,for in-plane spin orientations, only spinflip transitions of elec-trons are Raman allowed in depolarized scattering geometry,i.e., excitations with m
s=±1. Therefore, for directions of
the wave-vector transfer qwithϕ/negationslash=0◦(sample A) or ϕ/negationslash=
90◦(sample B), we expect a superposition of the green and
blue spectra of Fig. 3(a) (cf. Ref. [ 21]), corresponding to
two maxima. By contrast for the particular cases of ϕ=0◦
(sample A) or ϕ=90◦(sample B), i.e., q/bardbl[1¯10] or q/bardbl[110],
respectively, there should be a single maximum, only [orangecurve in Fig. 3(a)]. For the two extreme cases, ϕ=0
◦and 90◦,
this was experimentally confirmed in Ref. [ 21] for sample A.
So far, we have discussed the intrasubband SDE in terms
of single-particle spinflip excitations. It is well known thatthe intrasubband SDE is a collective excitation of the 2DESdue to exchange Coulomb interaction. However, since theexchange interaction leads to a redshift of the collectiveSDE [ 9], its energy lies within the single-particle continua,
which are displayed in Fig. 3(a). Hence, the collective SDE
is strongly Landau damped due to the decay into uncorrelatedspinflip excitations of individual electrons. For this reason, thecollective shift of this excitation due to exchange interactionis known to be rather small [ 27]. It can be deduced from
the comparison of RILS spectra in polarized geometry (seeSupplemental Material [ 28]), where intrasubband nonspinflip
single-particle excitations can be observed, with spectra mea-sured in depolarized geometry, where the intrasubband SDEis visible [ 27].
IV . EXPERIMENTS AND RESULTS
With these considerations we are now ready to discuss the
first set of experiments. Figure 4(a) shows a waterfall plot of
depolarized RILS spectra of sample A for a tilt angle of θ=
35◦, i.e., a fixed wave-vector transfer of q=9.1×106m−1.
Then, via the rotary stage, the sample is rotated in stepsof 15
◦between positions ϕ=0◦andϕ=360◦. For each
position, a RILS spectrum of the low-energy SDE is taken.The normalized spectra are displayed with vertical offsets forbetter comparison in Fig. 4(a). The gray shaded area indicates
the cutoff of the triple Raman spectrometer. The evolutionof the spectra from a Lindhard-Mermin spectrum at ϕ=0
◦
035427-4INELASTIC LIGHT SCATTERING BY INTRASUBBAND … PHYSICAL REVIEW B 101, 035427 (2020)
012Intensity (norm., shifted) ϕ = 0°ϕ = 90°2ΔES,ϕ
Raman shift (meV)+Iq−IqSum/2(b)
Simulation(a)Intensity (norm., shifted)
Raman shift (meV) 180 0
90
270
360ϕ(deg)
0 123
FIG. 4. (a) Waterfall plot of depolarized RILS spectra of sample
A at a tilt angle of θ=35◦, corresponding to a wave-vector transfer
ofq=9.1×106m−1for different in-plane directions ϕ(relative to
[¯110]) at zero external magnetic field. The gray shaded area marks
the spectrometer cutoff. The positions of the maxima are traced by
the gray dashed lines. (b) Lindhard-Mermin line-shape analysis of
measured spectra (black solid lines) exemplarily for minimum spin
splitting at ϕ=0◦(bottom spectrum) and maximum spin splitting at
ϕ=90◦(top spectrum).
to a superposition of two spectra with two different maxima
(indicated by gray dashed lines) can clearly be recognized.When rotating ϕby 360
◦, a periodic pattern for the shift
of the maxima can be found, which resembles the expectedspin splitting due to the effective spin-orbit field, shown inFig.2(b). To accurately extract the positions of the maxima in
the spectra, i.e., the cutoff energies of the spinflip transitions,all measured spectra were reproduced via a computationalline-shape analysis implemented in Python, based on theLindhard-Mermin line shape [ 25,26], where χ
0(ω,q,τ)i s
calculated [ 8,29] by numerically integrating over all possible
spinflip single-particle transitions in kspace.
Our line-shape analysis procedure is exemplarily shown
in Fig. 4(b) for the two extreme cases ϕ=0◦andϕ=90◦
for sample A. For ϕ=0◦the measured single-peak spectrum
(black solid line) in the lower part of Fig. 4(b) can nicely
be reproduced by the simulation (orange open dots), usingthe following material parameters: Electron temperature of
T=12 K, single-particle scattering time of τ=3 ps and
electron density of n=(2.7±0.3)×10
15m−2. We assume
that the short lifetime of 3 ps is due to the strong Landau
damping of the intrasubband SDE, since its energy is in-side the single-particle continuum. The smaller electron den-
sity here, as compared to the transport measurements from
Ref. [ 19], may arise from the strong laser illumination in our
experiments. The experimental spectrum for ϕ=90
◦[solid
black line in the upper part of 4(b)] is nicely reproduced by the
superposition of two simulated curves I+
qandI−
q[green and
blue open dots in Fig. 4(b)], which are obtained by varying
/Delta1kϕ[see Eqs. ( 3) and ( 4)] until the simulation matches the
measured spectra [red crosses in Fig. 4(b)]. The extracted peak
splittings are just twice the spin splitting, /Delta1ES,ϕ, for the corre-
sponding in-plane direction, as explained in the considerations
above.0.00.10.20.3 0.10.20.3 0.00.10.20.3 0.10.20.3ΔES,ϕ (meV) (a) (b)
09 0
180 27009 0
180 270ϕ
(deg)
A225 0T 6T Measured
CalculatedΔES,ϕ (meV)
FIG. 5. (a) Polar plot of extracted peak splittings versus angle
ϕ(relative to [ ¯110] in-plane direction) for sample A at zero external
magnetic field (inner open circles) and for external in-plane magneticfields of 6 T (outer solid circles). The gray lines show the calculated
sinϕdependence of the peak splitting. (b) Same as (a) but for sample
B at a tilt angle of θ=40
◦, corresponding to a wave-vector transfer
ofq=10.2×106m−1. The increased peak splitting is due to the
higher electron density of this sample.
The results of this analysis for sample A, i.e., for the
condition α=β, are summarized in Fig. 5(a). The inner open
symbols in Fig. 5(a)are showing the spin splittings /Delta1ES,ϕver-
sus the angle ϕ(relative to [ ¯110] in-plane direction), derived
via the above discussed line-shape analysis procedure for allmeasured spectra at zero external magnetic field. At ϕ=90
◦
and 270◦a maximum splitting of /Delta1ES=(0.19±0.01) meV
can be extracted for sample A, whereas for ϕ=0◦and 180◦
no spin splitting is observable (cf. Ref. [ 21]). The solid gray
line shows the computed values for the spin splitting, basedon the above discussed sin ϕdependence [see Eq. ( 3)]. For
the intrinsic spin-orbit parameters αandβ[after Eq. ( 4)], we
receive for ( α+β)/2=α=β=(3.50±0.25) meVÅ the
best agreement with the experimental data, when using n=
2.7×10
15m−2for the 2DES density, as determined from the
line-shape analysis (see above). The parameters αandβ,a s
deduced from the experiments in Ref. [ 19] on a different piece
of the same wafer, are somewhat smaller. Those experimentswere conducted at a higher temperature of T=40 K. Addi-
tionally, in Ref. [ 19] a transient in /Delta1kdue to a finite excitation
spot size [ 30] has not been included in the analysis. Similarly,
the RILS experiments in Ref. [ 21] delivered slightly smaller
values for αandβ. In these experiments, a different laser
excitation energy was used, leading to different resonanceconditions and, hence, a different electron density in theoptical experiment, corresponding to a different electric fieldacross the quantum well.
Next we discuss the experiments in external magnetic
fields. As indicated in Fig. 1(c), the externally applied field
B
extlies in-plane with the effective spin orbit field Bso
[Figs. 2(a)and2(c)] and, for each position of ϕ, perpendicular
to the transferred wave vector ( Bext⊥q). This generally
leads to a disturbance of the unidirectionality of the effec-
tive magnetic field acting on an electron, since the intrinsic
and the external fields superimpose. The above described
035427-5S. GELFERT et al. PHYSICAL REVIEW B 101, 035427 (2020)
measurement series, i.e., recording depolarized RILS spectra
of the intrasubband SDE, while rotating the sample stepwisewith respect to the fixed direction of q, is repeated for both
samples with a fixed external magnetic field of B
ext=6T ,
andBext⊥q. The blue solid dots in Fig. 5(a) display the
extracted spin splittings for sample A. A global increase of
the spin splittings can be found with maxima /Delta1ES=(0.24±
0.01) meV at ϕ=90◦and 270◦, where the external magnetic
field Bextis oriented parallel to the intrinsic spin-orbit field
Bsoso that they add up. For the orthogonal in-plane axis
alongϕ=0◦and 180◦, a finite splitting of /Delta1ES=0.15 meV
appears. Because of the unidirectional character of Bso,t h i s
splitting can only stem from the Zeeman splitting due to the
applied external magnetic field Bext. The measured values
of/Delta1ES,ϕare in good agreement with the computed values
[gray dashed line in Fig. 5(a)], which were calculated via the
relation /Delta1ES,B(ϕ)=gμB|Btot(ϕ)|, where the superposition of
Bextand Bsoleads to a total effective magnetic field Btot.
Using a |g|factor of 0 .16±0.04, and the same values for
α=βandnas for the Bext=0 measurement analysis above,
the computed values [dashed gray line in Fig. 5(a)] fit best
with the measured spin splittings. This gfactor, which would,
according to Ref. [ 31], correspond to a symmetric GaAs
quantum well with width of about 8 nm, will be discussed
in more detail below.
We now turn to measurements on sample B with negative
Bychkov-Rashba spin-orbit parameter α=−β. The intrin-
sic spin-orbit field Bsois still unidirectional according to
Eq. ( 2), but, compared to sample A, now parallel or an-
tiparallel to the orthogonal in-plane direction. Figure 5(b)
shows the spin splittings /Delta1ES,ϕ, as extracted from the line-
shape analysis of depolarized RILS spectra, for zero external
magnetic field (inner open dots) and for an external magnetic
field of Bext=6 T (green solid dots). The maximum spin
splitting now emerges for ϕ=0◦and 180◦with a magni-
tude of /Delta1ES=(0.20±0.01) meV for zero external magnetic
field and /Delta1ES=(0.34±0.01) meV for Bext=6T .F r o mt h e
Lindhard-Mermin line-shape analysis we obtain an electrontemperature of T=12 K, a single-particle scattering time
ofτ=3 ps and an electron density of n=(5.8±0.7)×
10
15m−2. For the calculation of the spin splittings [gray solid
and dashed lines in Fig. 5(b)], the density of the 2DES is
kept fixed at n=5.8×1015m−2(see above), and the spin-
orbit parameters [see Eq. ( 5)] are chosen to be α=−β=
(−3.25±0.25) meV Å. The measured data points for Bext=
0 T in Fig. 5(b) show a deviation from the calculated values.
This could be due to inhomogeneities of the donor distribution
in the sample, resulting in an inhomogeneity of the electron
density n: By rotating the sample during a ϕseries, the laser
spot may vary locally on the sample surface if the rotationaxis is not perfectly aligned to the laser-spot position. ForB
ext=6 T, the measured data points are much closer fitting
to the calculated values. Compared to sample A, the enlargedspin splitting [solid gray line in Fig. 5(b)] can be explained
by the higher electron density, which, due to the relation k
F=√
2πnand Eq. ( 4), leads to an increase of /Delta1ES,ϕ.F r o mt h i s
measurement, an effective |g|factor of 0 .23±0.07 can be
determined for sample B. Surprisingly, this gfactor is almost
50% larger than the above presented gfactor for sample A.
According to Ref. [ 31], it would correspond to a symmetric
GaAs quantum well with a width of about 10 nm.We note here that in our analysis we have assumed
an isotropic in-plane gfactor. Actually, for GaAs-AlGaAs
quantum wells with an asymmetric quantum-well potential,the in-plane gfactor should be anisotropic [ 32–35], and the
strength of the anisotropy depends on the potential asymme-try. In Ref. [ 35], e.g., a difference of about 20% was detected
for the two in-plane directions [110] and [1 ¯1 0 ]i naG a A s -
AlGaAs heterojunction. However, within our experimentalerror margins, we cannot clearly resolve such an anisotropy(cf. Fig. 5). We will come back to this point in more detail
below, in Sec. V.
Finally, we demonstrate how the intrinsic effective spin-
orbit field B
socan be determined directly by superimposing
an external magnetic field. In the above measurements, a fixedmagnetic field was applied, and the sample was rotated in or-der to change the in-plane crystal directions with respect to themagnetic field. From the variation of the spin splittings, ex-tracted from RILS spectra of intrasubband SDE, first, the spin-orbit parameters are determined from the measurements atB
ext=0, and then the gfactors are derived from experiments
atBext/negationslash=0. As outlined above, this measurement procedure
is prone to be influenced by inhomogeneities of the sample ifthe laser spot is not perfectly aligned with the rotation axis.The carrier densities are determined by the positions of thehigh-energy cutoffs of the spectra. In principle, we could nowcompute the maximum spin-orbit field with these parameters.Forα=±β, the energy paraboloids for spin up and spin
down are shifted against each other in kspace [see Figs. 2(b)
and2(d)] by a momentum /Delta1kwith magnitude
/Delta1k=4m
∗α
¯h2. (6)
Inserting this into Eq. ( 4), and assuming a Zeeman-type
energy splitting /Delta1ES=gμBBso, one gets for the magnitude
of the maximum intrinsic spin-orbit field
Bso=4αkF
gμB. (7)
Inserting the parameters, as extracted above from the mea-
surements of samples A and B, into Eq. ( 7), we receive
for sample A Bso∼(19.9±4.1) T and for sample B Bso∼
(18.6±3.9) T. The uncertainties are quite large, due to the
uncertainties in the contributing individual parameters. In thefollowing we will verify these field values by a direct, moreaccurate measurement. In order to do so, we fix the directionof wave-vector transfer qin the directions of maximum spin
splittings, i.e., the [110] direction for sample A and the [1 ¯10]
direction for sample B. Since B
ext⊥q, the external field is
then parallel or antiparallel to the maximum internal spin-orbitfield B
so[36]. Then we record a series of RILS spectra of the
SDE for different external fields, ranging from −6Tt o +6T .
In doing so, the laser spot is kept fixed on the sample surface,which reduces the possible influence of inhomogeneities toa minimum. Figure 6(a) shows such a series of depolarized
RILS spectra for different external magnetic fields of sampleB, recorded with a wave-vector transfer along ϕ=0
◦(q
||[1¯10]), where the maximum spin splitting appears. A linear
convergence of the peak splittings from positive to negativeexternal magnetic fields is clearly visible.
035427-6INELASTIC LIGHT SCATTERING BY INTRASUBBAND … PHYSICAL REVIEW B 101, 035427 (2020)
0.000.050.100.150.200.250.300.35ΔES (meV)
Magnetic field (T)012345 Intensity (norm., shifted)
Raman shift (meV)-6 T 0 T 6 TBext
ϕ = 0°
ϕ = 90°
ϕ = 0°(a) (b)
-18 -12 -6 0 6 Sample B
Sample A
Linear fit
FIG. 6. (a) Depolarized RILS spectra of sample B for different
external magnetic fields, oriented parallel for Bext>0Tand an-
tiparallel for Bext<0T. The peaks are converging linearly due to a
superposition of BextandBso. (b) Extracted spin splittings for both
investigated samples, recorded with a wave-vector transfer alongmaximum spin splitting crystal direction. The orange shaded area
is the experimentally accessible magnetic field range of the cryostat.
Dashed line: Linear fit and extrapolation of the spin splittings.
We expect the effective spin-orbit field Btotto be enhanced
if the external magnetic field is oriented parallel to Bso,
which is the case for Bext>0TandBtotto be attenuated
for antiparallel orientation, i.e., Bext<0T. The values for the
splitting were extracted via the Lindhard-Mermin line-shapeanalysis and plotted against the external magnetic field inFig. 6(b) for both investigated samples. For both samples a
linear shift of the peak splitting occurs in the orange shadedarea of Fig. 6(b), which marks the accessible magnetic field
range of the cryostat. In particular, this measurement seriesshows a clear difference of the spin splitting of both samplesforB
ext=0. The value of this splitting for sample B nicely
agrees with the calculated value in Fig. 5(b) for 0◦and 180◦
[solid gray line in Fig. 5(b)], confirming the above-derived
values for αandβ. As discussed above, the corresponding
experimental data in this figure [open green dots in Fig. 5(b)]
may have been influenced by sample inhomogeneities. Thestrength of the intrinsic spin-orbit field B
socan be extracted
by extrapolating the spin splittings (dashed lines) until theirintersection with the xaxis. For both samples we extract
a value of ∼(18±1) T, in good agreement with the com-
puted values above. From the slope of the dashed lines weextract an effective |g|factor of 0 .17±0.01 for sample A
and 0.24±0.01 for sample B, in almost perfect agreement
with the values we have received from the angular-resolvedmeasurements with fixed magnetic field [dashed gray lines inFigs. 5(a)and5(b)].
V . DISCUSSION
We have investigated two single quantum-well samples
with nominally identical GaAs wells of 12-nm width but op-posite doping profiles, leading to oppositely oriented electricfields in the quantum well and, hence, linear Bychkov-Rashbaparameters, α, with opposite sign. The electron densities n
of the samples can be relatively accurately determined fromthe high-energy cutoffs of the intrasubband SDE spectra via
a Lindhard-Mermin line-shape analysis. This delivers differ-ent values for the two samples, namely n∼(2.7±0.3)×
10
15m−2for sample A and n∼(5.8±0.7)×1015m−2for
sample B. Furthermore, from the line-shape analysis of thepeak splittings at B
ext=0, comparable magnitudes of the
spin-orbit parameters ( |α|+β)/2=(3.50±0.25) meV Å
for sample A and ( |α|+β)/2=(3.25±0.25) meV Å for
sample B can be determined within the error bars. Thesevalues are consistently confirmed by the high accuracy mea-surements, displayed in Fig. 6, where the laser spot is kept
at a fixed position on the sample surface. We believe thatthe approximate equivalence of the spin-orbit parameters inboth samples is just by coincidence. It seems that the differentdoping profiles in the two samples [see Figs. 1(a) and1(b)]
result in approximately the same magnitude of the Bychkov-Rashba coefficient |α|for both samples. The gfactors, ex-
tracted from the experiments in Fig. 6with B
ext/negationslash=0a r e ,
however, puzzling. It is obvious from Fig. 6that the gfactors
for samples A and B are distinctly different. Puzzling is thefact that sample B, which has the larger electron density n,
shows a gfactor of |g|=0.24±0.01, which is almost 50%
larger than the gfactor of sample A. This is counterintuitive:
Assuming a negative gfactor (as expected for a well width
of 12 nm [ 37,38]),|g|should decrease with increasing Fermi
energy (and same well width). This is further corroborated bythe fact that the effective mass, m
∗, follows the expectation:
It is slightly larger in sample B, which has the larger Fermienergy. This is expected because of the nonparabolicity of theconduction band of GaAs.
At a first glance, one could think that the difference in g
factors for the two samples could be due to the anisotropy ofthe in-plane gfactor [ 32,34,35]: The directions of maximum
spin splitting in samples A and B are the [110] and the[1¯10] directions, respectively (cf. Fig. 5). In Ref. [ 35], these
directions were experimentally found to be the directionsof maximum anisotropy in a GaAs-AlGaAs heterojunction.However, due to the reversed electric fields in the two samples,the in-plane directions of maximum and minimum gfactors
are expected to switch between the two samples. As a result,in both samples the directions of maximum spin splittingcorrespond to the directions of maximum gfactor. This can
be proven as follows: A theoretical treatment yields for thenondiagonal elements of the in-plane gtensor [ 32,34]
g
xy=gyx=2γe
¯h3μB/parenleftbig/angbracketleftbig
p2
z/angbracketrightbig
/angbracketleftz/angbracketright−/angbracketleftbig
p2
zz/angbracketrightbig/parenrightbig
, (8)
with the bulk Dresselhaus coefficient γ[2] and /angbracketleft/angbracketrightdenoting
the expectation value for the electron wave function. The twoterms in the brackets on the right-hand side of Eq. ( 8) exactly
cancel if the quantum-well potential is symmetric, i.e., thenthe in-plane gfactor is isotropic. The larger the asymmetry of
the potential, the larger is the magnitude of the offdiagonalelement g
xy.W i t hE q .( 8) and the electronic wave functions
of samples A and B [cf., Figs. 1(a) and1(b)] it follows that
gxy>0 for sample A and gxy<0 for sample B, provided that
γ> 0. For a small magnetic field Bext, pointing in an arbitrary
in-plane direction ϕ, as defined above, the anisotropic g(ϕ)i s
035427-7S. GELFERT et al. PHYSICAL REVIEW B 101, 035427 (2020)
given by [ 34]
g(ϕ)=−/radicalBig
g2xx+g2xy+2gxxgxycos(2ϕ). (9)
We have to take into account now that in our experiments
we have Bext⊥q. This means that, e.g., in an experiment
on sample A with q/bardbl[110], i.e., in direction of maximum
spin splitting, Bext/bardbl[1¯10], and so on. Altogether, employ-
ing Eq. ( 9), this results in the relations |g110|=| g(180◦)|>
|g1¯10|=| g(−90◦)|for sample A and |g1¯10|=| g(−90◦)|>
|g110|=| g(180◦)|for sample B. This confirms the above
statement that the directions of maximum spin splitting inboth samples ([110] for sample A and [1 ¯10] for sample B)
are the directions of maximum in-plane gfactor. Hence, the
quite significant difference of the measured gfactors cannot
be explained by the gfactor anisotropy.
We speculate that the difference in gfactors is due to
residual collective effects in the Landau-damped SDE, whichmay lead to an enhanced spin-orbit field and, hence, anenlarged splitting. If we rewrite Eq. ( 4) for the situation when
the external field is parallel to the intrinsic spin-orbit field[as in the experiments, displayed in Fig. 6] for a Zeeman-like
energy splitting, then we have for the spin splitting
/Delta1E
S,max=2(|α|+β)kF+gμBBext=gμB[Bso+Bext].
(10)
Equation ( 10) may contain gas an effective gfactor, modeling
the effect of a collective spin-orbit field. However, at themoment it is not clear whether it is just an enhancement ofthegfactor due to many-particle interaction or an enhanced
spin-orbit field. This is so far a naive assumption and furthertheoretical elaboration is needed for an accurate description ofthe relevant effects.Finally, we note that having approximately the same mag-
nitude of the spin-orbit field B
soin both samples, e.g., on
the basis of Eq. ( 7) means that the relation kF/gin both
samples has to be the same. This would mean that a largerk
F(sample B) is compensated for by a larger gfactor and
vice versa for sample A. For our extracted parameters, thisis the case (see above). The conclusion regarding whetherthis is significant or just coincidence for the two investigatedsamples will need more investigations from the experimentalas well as theoretical sides.
VI. CONCLUSION
In conclusion, we compared two samples possessing bal-
anced Bychkov-Rashba and Dresselhaus spin-orbit couplingbut with different sign of the Rashba parameter, i.e., α=±β,
by their spectra of intrasubband SDEs. For each sample, weprecisely mapped the spin splitting of the 2DES and couldshow that the unidirectional spin-orbit field is pointing paralleland antiparallel along the [1 ¯10] in-plane direction for α=β
and along [110] for α=−β. With in-plane external magnetic
fields, we were able to deduce the strengths of the maximumintrinsic spin-orbit fields. We found an effective gfactor,
which is significantly enhanced for the sample with the largerdensity, possibly indicating the influence of collective effectsin the intrasubband SDEs.
ACKNOWLEDGMENTS
Funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) Project-ID 314695032-SFB1277 (subprojects A01 and B06) and project SCHU1171 /7-
1, as well as from the Swiss National Science Foundationthrough NCCR QSIT.
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035427-9 |
PhysRevB.78.144517.pdf | Spin-triplet p-wave pairing in a three-orbital model for iron pnictide superconductors
Patrick A. Lee and Xiao-Gang Wen
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
/H20849Received 12 September 2008; published 28 October 2008 /H20850
We examine the possibility that the superconductivity in the newly discovered FeAs materials may be caused
by the Coulomb interaction between delectrons of the iron atoms. We find that when the Hund’s rule ferro-
magnetic interaction is strong enough, the leading pairing instability is in spin-triplet p-wave channel in the
weak-coupling limit. The resulting superconducting gap has nodal points on the two-dimensional Fermi sur-faces. The kdependent hybridization of several orbitals around a Fermi pocket is the key for the appearance of
the spin-triplet p-wave pairing.
DOI: 10.1103/PhysRevB.78.144517 PACS number /H20849s/H20850: 74.20.Mn, 74.20.Rp, 74.25.Jb, 74.70. /H11002b
I. INTRODUCTION
Recently, a new class of superconductors—iron-based
superconductors—was discovered.1–10The superconducting
transition temperature can be as high as 52 K.8The undoped
samples /H20849for example LaOFeAs /H20850appear to have a spin or-
dered phase below 150 K.5,9,10The electron doped2
LaO 1−xFxFeAs and hole doped4LaO 1−xSrxFeAs samples are
superconducting with Tc/H1101125 K. The magnetic-field depen-
dence of the specific heat in the electron doped material sug-gests the presence of gapless nodal lines on the Fermisurface.
3
It appears that the electron-phonon interaction is not
strong enough to give rise to such high transitiontemperature.
11In this paper, we will examine the possibility
that Coulomb interaction between delectrons on Fe drives
the superconductivity in the electron or hole doped samplesand the spin order in the undoped samples. In this case, wefind that a p-wave spin-triplet superconducting order with
gapless nodal lines is the most likely superconducting orderin the weak-coupling limit. Naively short-range repulsiondoes not have the requisite kdependence to drive p-wave
pairing. It turns out that the kdependent hybridization of
several orbitals around a Fermi pocket makes the p-wave
spin-triplet pairing possible. The same model is also shownto have spin-density wave /H20849SDW /H20850order for undoped
samples.
II. THREE-ORBITAL TIGHT-BINDING MODEL
First, let us examine the Fermi surfaces of the iron-based
superconductor. For concreteness, we will consider theLaOFeAs sample. We will assume that the properties of thesample are mainly determined by the Fe-As planes. The Featoms in a Fe-As plane form a two-dimensional /H208492D/H20850square
lattice /H20849see Fig. 1/H20850. Due to the buckling of the As atoms, the
real unit cell contains two Fe atoms. The real unit cell is alsoa square /H20849see Fig. 1/H20850. According to band-structure calcula-
tions, the Fermi surfaces of the Fe dbands are formed by
two hole pockets at the /H9003point and two electron pockets at
theMpoint /H20851see Fig. 2/H20849a/H20850/H20852.
12–15All those pockets, mainly
formed by the dxz,dyz, and dxyorbitals of Fe,11have similar
size, shape, and Fermi velocity.
To understand the mixing of the orbitals near those Fermi
pockets, we follow Refs. 13and14to unfold the band struc-ture to the extended Brillouin zone /H20849BZ/H20850. Ordinarily, such an
unfolding of the band structure extends the BZ superficiallyand there are certain ambiguities in assigning the location ofeach band. We emphasize that this is not the case here. Theband structure of the extended BZ is uniquely defined due toan additional symmetry, i.e., the Fe-As plane is invariantunder P
zTxandPzTy, where Tx/H20849Ty/H20850is the translation in the x
/H20849y/H20850direction by the Fe-Fe distance and Pzis the reflection
z→−z/H20849see Fig. 1/H20850. Thus, if we combine the translation and
the reflection Pz, then the electron hopping Hamiltonian H
has a symmetry described by a reduced unit cell with onlyone Fe per unit cell /H20849see Fig. 1/H20850. Since /H20851P
zTx,H/H20852=/H20851PzTy,H/H20852
=/H20851PzTx,PzTy/H20852=0, we can use the eigenvalues of PzTxand
PzTyto label the single-body energy eigenstates
PzTx/H20841k˜/H20856=eik˜
x/H20841k˜/H20856,PzTy/H20841k˜/H20856=eik˜
y/H20841k˜/H20856, /H208491/H20850
where k˜=/H20849k˜x,k˜y/H20850plays a role of crystal momentum. We will
use such a pseudocrystal momentum to label states in an
energy band. It is the pseudocrystal momenta k˜that form the
extended Brillouin zone /H20851see Fig. 2/H20849b/H20850/H20852which corresponds to
the reduced unit cell with only one Fe.
Let us work out an explicit example by considering a
tight-binding model involving the dxz,dyz, and dxyorbitals.
First, consider the hopping terms that do not mix the orbitals
y
x(b) (a)xy yzyz xzxzxy
xz
FIG. 1. /H20849Color online /H20850/H20849a/H20850The Fe-As plane and the dorbitals of
Fe. The filled dots are Fe atoms and the empty dots are As atoms.The plus and minus signs in the empty dots indicate if the As atomis above or below the Fe plane. The large square is the 2D unit cell.The dashed square is the reduced unit cell which contains only oneFe. The d
xz,dyz, and dxyorbitals of the Fe atoms are described by
the red, blue, and green curves, and marked by xz,yz, and xy
respectively. /H20849b/H20850The yhopping between the dxzordyzwith dxy
orbitals.PHYSICAL REVIEW B 78, 144517 /H208492008 /H20850
1098-0121/2008/78 /H2084914/H20850/144517 /H208496/H20850 ©2008 The American Physical Society 144517-1H1=−/H20858
/H20855ij/H20856/H20851tijxz/H20849cixz/H20850†cjxz+tijyz/H20849ciyz/H20850†cjyz+tijxy/H20849cixy/H20850†cjxy+ H.c. /H20852,/H208492/H20850
where i,jlabel the positions of the Fe. Since the dxz,dyz, and
dxyorbitals are eigenstates of Pzreflection, the PzTxandPzTy
symmetries require that tijxz,tijyz, and tijxyonly depend on i−j.
The mixing term between the dxzanddyzorbitals is given by
H2=−/H20858
/H20855ij/H20856/H20851tijxz,yz/H20849cixz/H20850†cjyz+ H.c. /H20852, /H208493/H20850
where tijxz,yzalso depends only on i−jas required by the PzTx
and PzTysymmetries. On the other hand, the mixing term
between the dxzanddxyorbitals as well as that between the
dyzanddxyorbitals are given by
H3=/H20858
/H20855ij/H20856/H20849−/H20850ix+iy/H20851tijxz,xy/H20849cixz/H20850†cjxy+tijyz,xy/H20849ciyz/H20850†cjxy+ H.c. /H20852,/H208494/H20850
where tijxz,xyandtijyz,xyonly depend on i−j. We note that the dxz
anddxyorbitals have opposite eigenvalues /H110061 under Pz. The
PzTxandPzTysymmetries require the presence of the factor
/H20849−/H20850ix+iy. Thus, cxzandcyzwith conventional crystal momen-
tum k+Qcan mix with ckxywith crystal momentum kwhere
Q=/H20849/H9266,/H9266/H20850.
However, in the pseudocrystal momentum k˜space, only
operators with the same pseudocrystal momentum can mix.Let
/H9023˜k˜=/H20849c˜k˜xz,c˜k˜yz,c˜k˜xy/H20850T/H208495/H20850
be the operators with pseudocrystal momentum k˜, where
c˜k˜xz/H11011/H20858
ie−i/H20849k˜+Q/H20850·icixz,
c˜k˜yz/H11011/H20858
ie−i/H20849k˜+Q/H20850·iciyz,
c˜k˜xy/H11011/H20858
ie−ik˜·icixy. /H208496/H20850
We see that the pseudocrystal momentum k˜and the conven-
tional crystal momentum kare related by k=k˜for the dxyorbital and k+Q=k˜for the dxzand dyzorbitals. The total
hopping Hamiltonian H=H1+H2+H3can be written as H
=/H20858k˜/H9023˜
k˜†Mk˜/H9023˜k˜with
Mk˜=/H20898/H9280xz/H20849k˜/H20850/H9280xz,yz/H20849k˜/H20850/H9280xz,xy/H20849k˜/H20850
/H9280xz,yz/H20849k˜/H20850/H9280yz/H20849k˜/H20850/H9280yz,xy/H20849k˜/H20850
/H9280xz,xy/H11569/H20849k˜/H20850/H9280yz,xy/H11569/H20849k˜/H20850/H9280xy/H20849k˜/H20850/H20899. /H208497/H20850
It is worth noting that the above Hamiltonian and the result-
ing energy bands are defined on the extended Brillouin zone
/H20849labeled by k˜/H20850of the reduced unit cell /H20851see Fig. 2/H20849b/H20850/H20852.
The nearest-neighbor admixture between dxzanddyzvan-
ishes by symmetry. Keeping the next-nearest-neighbor term
only, we have /H9280xz,yz=−2txz,yz/H11032/H20851cos/H20849k˜x+k˜y/H20850−cos /H20849k˜x−k˜y/H20850/H20852. Note
that this vanishes at /H9003˜,M˜,X˜, and Y˜points /H20849see Fig. 3/H20850, but is
maximal midway between X˜and Y˜. Next, consider the
nearest-neighbor admixture between dxzordyzwith dxyalong
theydirection /H20851see Fig. 1/H20849b/H20850/H20852. These orbitals can admix only
because of the asymmetry introduced by the As ions, whichhasxzsymmetry. This implies that the d
xz−dxyoverlap inte-
gral is odd under x→−xand vanishes. Only the dyz−dxy
matrix element survives. Also note that under the 180°
rotation in the x−yplane about the isite, /H20849ci+yyz/H20850†cixy
→−/H20849ci−yyz/H20850†cixy. Thus only the combination /H20849ci+yyz/H20850†cixy
−/H20849ci−yyz/H20850†cixythat preserves such a symmetry can appear in the
hopping Hamiltonian which has a form
/H20858
i/H20851/H20849−/H20850ix+iytyz,xy/H20851/H20849ci+yyz/H20850†cixy−/H20849ci−yyz/H20850†cixy/H20852+ H.c. /H20852
=/H20858
k˜/H20851−tyz,xy/H20849eik˜
y−e−ik˜
y/H20850/H20849c˜k˜yz/H20850†c˜k˜xy+ H.c. /H20852. /H208498/H20850xx
(π,π)
y
(0,0)y
∼
(a)ΓΜ
(b)Υ
Γ∼Χ∼Μ∼
FIG. 2. /H20849Color online /H20850/H20849a/H20850The Fermi surfaces of the Fe dbands.
The plus sign marks the hole pockets and the minus sign marks theelectron pockets. The square is the Brillouin zone. /H20849b/H20850The extended
Brillouin zone /H20849dashed square /H20850for the reduced unit cell and the
positions of the Fermi pockets in the extended Brillouin zone. The
/H9003˜point has /H20849k˜x,k˜y/H20850=/H208490,0/H20850. The folding of the extended Brillouin
zone in /H20849b/H20850produces /H20849a/H20850. After folding, the /H20849/H9003˜,M˜/H20850and /H20849X˜,Y˜/H20850in the
extended Brillouin-zone map into the /H9003andMin the original Bril-
louin zone, respectively. The yellow shading marks the regionwhere the p-wave pairing order parameter may have the same sign.(0,0)(π,π)
(d)(b)
(c)(a) Χ∼Υ∼
Μ∼
Γ∼
Γ∼Υ∼
Χ∼Μ∼
xz,yzxy
xyyzxz
FIG. 3. /H20849Color online /H20850/H20849a/H20850The zero energy contour of /H9280xz/H20849k˜/H20850
/H20849red/H20850and/H9280yz/H20849k˜/H20850/H20849dashed blue /H20850. The/H11006are signs of /H9280xz/H20849k˜/H20850and/H9280yz/H20849k˜/H20850
in the region. The /H9003˜point has k˜=0. /H20849b/H20850The hybridization of the dxz
anddyzbands. The curves are the zero energy contours of the hy-
bridized bands. /H20849c/H20850The zero energy contour of /H9280xy/H20849k˜/H20850./H20849d/H20850The solid
curves are the Fermi surfaces of the three-band tight-binding modelas a result of the hybridization of /H20849b/H20850and /H20849c/H20850.PATRICK A. LEE AND XIAO-GANG WEN PHYSICAL REVIEW B 78, 144517 /H208492008 /H20850
144517-2This allows us to conclude that the nearest neighbor
dxz−dxyand dyz−dxymixing give rise to /H9280xz,xy/H20849k˜/H20850
=−2i txz,xysin/H20849k˜x/H20850and/H9280yz,xy/H20849k˜/H20850=−2i tyz,xysin/H20849k˜y/H20850.
Now we can see how the three-orbital model can repro-
duce the four-pocket Fermi surface. Let us begin by assum-
ing that /H9280xz/H20849k˜/H20850lies just below the Fermi energy /H20849−0.2 eV /H20850at
k˜=Y˜and disperses rapidly upward toward M˜, reaching 2.5
eV. From /H9003˜, it descends toward X˜where its energy is
−1.4 eV. The dispersion is relatively flat along X˜-/H9003˜-Y˜and a
shallow local maximum appear at /H9003˜. The Fermi surface cor-
responding to this band is shown in Fig. 3/H20849a/H20850. The dyzband is
similar except rotated by 90°. The reason for the choice of
locating the dxz+dxyband at Y˜/H20849as opposed to X˜/H20850will be
explained later.
Next, we turn on the hybridization between dxzanddyz.
The hybridization is maximal at X/H20849midway between X˜and
Y˜/H20850and creates two sheets which touch at /H9003˜and two Fermi
surfaces. We also find two concentric small hole pockets at /H9003˜
which become the well-known hole pockets at /H9003after fold-
ing.
The dxyband/H9280xy/H20849k˜/H20850is assumed to be at −0.5 eV at k˜
=X˜and k˜=Y˜and disperses rapid upwards toward /H9003˜but
rather flat toward M˜. The Fermi surface is sketched in Fig.
3/H20849c/H20850. Now we turn on /H9280xz,xyand/H9280yz,xy. This gives the electron
pockets centered at X˜andY˜which become the two electron
pockets at Mafter the folding. Note that because /H9280xz,yz=0 at
X˜andY˜, the band at Y˜/H20849orX˜/H20850is purely dxz+dxy/H20849ordyz+dxy/H20850.
Furthermore, near Y˜,/H9280xz,xy/H11011sink˜x. Thus along Y˜−/H9003˜, the hy-
bridization is zero and the dxzanddxybands cross as shown
in Fig. 4/H20849a/H20850. Along Y˜−M˜, the upper band is mostly dxzwhile
thedxzamplitude in the lower band increases linearly with
the distance from Y˜. These features are in agreement withband calculations11,14and are strong evidences that our as-
signment of the dxzband is correct.
The final result is an ellipse shaped pocket, with the long
axis of the ellipse pointing in the ydirection at Y˜/H20849toward the
hole pockets at /H9003˜/H20850. The Fermi surface crossing along the long
direction is purely dxyand the short direction is mainly dxz
/H20851see Fig. 3/H20849d/H20850/H20852.
The pocket formed between the dashed lines in Fig. 3/H20849d/H20850
is eliminated for sufficiently strong hybridization. However,within the three-orbital model, it is impossible to remove the
Fermi surface surrounding M˜/H20851the small gold solid loop in
center of Fig. 3/H20849d/H20850/H20852because the hybridization elements are all
zero at M˜. We need a fourth band which crosses and hybrid-
izes with the dxz+dyzband to eliminate the unwanted Fermi
surface. Apart from this, the fourth band plays no role as faras the remaining two electrons and two hole pockets areconcerned. It is in this sense that we maintain that the three-orbital model gives an adequate description of the low en-ergy Hamiltonian.
III. INTERACTION BETWEEN dELECTRONS
Next, let us consider the single-ion interaction between the electrons on the Fe dorbitals which is given by
HI=1
2/H20858
/H9251,/H9251/H11032/H20885d2xd2x/H11032c/H9251†/H20849x/H20850c/H9251/H20849x/H20850V/H20849x−x/H11032/H20850c/H9251/H11032†/H20849x/H11032/H20850c/H9251/H11032/H20849x/H11032/H20850
=1
2/H20858
a1,a2,a3,a4,/H9251,/H9251/H11032c/H9251a1†c/H9251/H11032a2†c/H9251/H11032a3c/H9251a4/H20885d2xd2x/H11032/H9278a1/H20849x/H20850/H9278a2/H20849x/H11032/H20850V/H20849x−x/H11032/H20850/H9278a3/H20849x/H11032/H20850/H9278a4/H20849x/H20850, /H208499/H20850
where a1,¯,a4=xz,yz,xylabel the orbitals, /H9278a/H20849x/H20850is the wave function of the orbitals, and cais the electron operator for the
aorbital. Due to the symmetry of the orbitals, aimust appear in pairs and the above can be rewritten as
HI=1
2U1/H20858
ac/H9251a†c/H9252a†c/H9252ac/H9251a+1
2U2/H20858
a/HS11005bc/H9251a†c/H9252b†c/H9252bc/H9251a+1
2J/H20858
a/HS11005bc/H9251a†c/H9252b†c/H9252ac/H9251b+1
2J/H20858
a/HS11005bc/H9251a†c/H9252a†c/H9252bc/H9251b, /H2084910/H208500~
xz
xy0ε
εY
(Γ) (Μ)eV
0
−0.2
−0.5xz+yzxyxz
Μ (Γ)~Γ~Y~
(a)−0.49−0.25 0.25
0.49
(b)0
FIG. 4. /H20849Color online /H20850/H20849a/H20850The energy bands near Y˜and/H9003˜. The
bands near Y˜come from the dxzanddxyorbitals. The thickness of
the curve indicates the weight in the dxzorbital. The bands near /H9003˜
came from the dxzanddyzorbitals. /H20849b/H20850The contour plot of i ukvk
near Y˜. The dashed loop is the Fermi surface.SPIN-TRIPLET p-WAVE PAIRING IN A THREE- … PHYSICAL REVIEW B 78, 144517 /H208492008 /H20850
144517-3where U1=Uaa=Ubb,U2=Uab=U1−2J,16and
Uab=/H20885d2xd2x/H11032/H9278a/H20849x/H20850/H9278b/H20849x/H11032/H20850V/H20849x−x/H11032/H20850/H9278b/H20849x/H11032/H20850/H9278a/H20849x/H20850,
J=/H20885d2xd2x/H11032/H9278a/H20849x/H20850/H9278b/H20849x/H11032/H20850V/H20849x−x/H11032/H20850/H9278a/H20849x/H11032/H20850/H9278b/H20849x/H20850./H2084911/H20850
Here the U1term represents the Coulomb interaction be-
tween electrons on same orbital. The U2term represents the
interaction between electrons on different orbitals. The first J
term is the exchange effect that favor parallel spins on thesame Fe and is responsible for the Hund’s rule. The second J
term is pair hopping between different orbitals.
IV. PAIRING INSTABILITY
The above d-electron interaction may cause a pairing in-
stability. One possible pairing instability is the spin-triplet
pairing between X˜andY˜electron pockets.17In this paper, we
will consider a different type of pairing—the spin-triplet
p-wave pairing within the same Fermi pocket.
First, let us consider the p-wave pairing on the Fermi
pocket near the Y˜point /H20849see Fig. 3/H20850. Such a Fermi pocket is
a mixture of the dxzanddxyorbitals. Let /H9274xzand/H9274xybe the
electron operators in the continuum limit in the dxzanddxy
orbitals near the Y˜point. The electron operator /H9274near the
pocket at the Y˜point /H20851see Fig. 2/H20849a/H20850/H20852is a mixture of /H9274xzand
/H9274xy.
The Hamiltonian has the PzPx,PzPy,Pxy,PzTx, and PzTy
symmetries, where Px:x→−xandPy:y→−yare reflections
about a Fe atom. These symmetries dictate certain forms ofthe continuum Hamiltonian. Alternatively, we can expand the
tight-binding picture near Y˜to obtain the following form
which respects all the symmetry requirements:
H0=/H20849t1kx2+t2ky2+/H9280xz0/H20850/H9274xz†/H9274xz+/H20849t˜1kx2+t˜2ky2+/H9280xy0/H20850/H9274xy†/H9274xy
+t3kx/H20849i/H9274xy†/H9274xz+ H.c. /H20850. /H2084912/H20850
Here kis the pseudocrystal momentum measured from the Y˜
point. From Fig. 4/H20849a/H20850, we see that the dxzband is flat along
theyaxis, which implies t1/H11271t2. Similarly t˜2/H11271t˜1. In the fol-
lowing, we will ignore t2andt˜1.
The electrons have two bands with energies
E/H11006/H20849k/H20850=/H92800/H11006/H20881/H928022+/H928032, /H2084913/H20850
where
/H92800=1
2/H20849t1kx2+t˜2ky2+/H9280xz0+/H9280xy0/H20850,
/H92803=1
2/H20849t1kx2−t˜2ky2+/H9280xz0−/H9280xy0/H20850,
/H92802=t3kx. /H2084914/H20850
/H9274xzand/H9274xyare related to the electron operator /H9274in the upper
band E+as/H9274xz/H20849k/H20850=uk/H9274/H20849k/H20850,/H9274xy/H20849k/H20850=vk/H9274/H20849k/H20850, /H2084915/H20850
where
uk=i/H92802
/H208812/H928022+2/H928032−2/H92803/H20881/H928022+/H928032,
vk=/H92803−/H20881/H928022+/H928032
/H208812/H928022+2/H928032−2/H92803/H20881/H928022+/H928032. /H2084916/H20850
To obtain the pairing interaction near the Y˜point in the
spin-triplet channel we set /H9251=/H9252=/H9251/H11032=/H9252/H11032=↑in Eq. /H2084910/H20850.W e
find that the first and the fourth terms in Eq. /H2084910/H20850vanish. The
second and the third terms become
/H20858
k1,k2VY˜/H20849k2,k1/H20850/H20851/H9274↑/H20849k2/H20850/H9274↑/H20849−k2/H20850/H20852†/H9274↑/H20849k1/H20850/H9274↑/H20849−k1/H20850/H2084917/H20850
in the spin triplet and /H20849k,−k/H20850pairing channel, where the
effective pairing interaction of /H9274is
VY˜/H20849k2,k1/H20850=− /H20849J−U2/H20850uk2/H11569v−k2/H11569uk1v−k1. /H2084918/H20850
From the effective pairing interaction, we can obtain a di-
mensionless coupling constant18
/H9261=−/H20885d/H9268k
/H208492/H9266/H208502/H20841vk/H20841/H20885d/H9268k/H11032
/H208492/H9266/H208502/H20841vk/H11032/H20841g/H11569/H20849k/H20850V/H20849k,k/H11032/H20850g/H20849k/H11032/H20850
/H20885d/H9268k
/H208492/H9266/H208502/H20841vk/H20841/H20841g/H20849k/H20850/H208412,
/H2084919/H20850
where /H20848d/H9268kis the integration over the Y˜Fermi surface and
vkis the Fermi velocity. The function g/H20849k/H20850is a square har-
monics which describes the shape of the superconductinggap, e.g., g/H20849k/H20850=1 corresponds to a s-wave and g/H20849k/H20850=sin k
xor
sinkyto ap-wave superconductor. The superconducting tran-
sition temperature Tcis given by Tc=/H9024e−1 //H9261, where /H9024is of
order of the Fermi energy of the pocket /H9024/H110110.2 eV.
From Eq. /H2084918/H20850, we find that, when J/H11022U2, the pairing
interaction VY˜/H20849k2,k1/H20850induces a pairing g/H20849k/H20850that have the
same symmetry as ukvk. Because ukvkis odd in kx/H20851see Fig.
4/H20849b/H20850/H20852, the induced pairing is in p-wave channel g/H20849k/H20850=sin kx.
Thus when J/H11022U2, the d-electron interaction will cause a
spin-triplet p-wave pairing. Note that the location of the node
depends on the choice of sin kxversus sin ky, which in turn
hinges on our assignment of the orbital to be xz-like near Y˜.
Triplet pairing has been proposed earlier,19and fully gapped
states such as px+ipywere suggested. In contrast, after in-
cluding the k-dependent orbital mixing, we find the on-site
ferromagnetic interaction to favor a particular nodal pxand
pystates in the Y˜andX˜valleys, respectively.
As long as /H9280F/H11022/H9280xz0, we see from Fig. 4/H20849a/H20850that the Fermi
surface changes its character from pure dxyto mostly dxzas a
function of angles. /H20841uk/H208412and /H20841vk/H208412must cross at some angles
where /H20841ukvk/H20841takes its peak value 1/2 /H20851see Fig. 4/H20849b/H20850/H20852. Thus the
effect of /H20841ukvk/H20841or/H9261is relatively insensitive to doping pro-
vided that /H9280F/H11022/H9280xz0. Since the density of states is also inde-
pendent of doping in 2D, this explains why Tcis somewhatPATRICK A. LEE AND XIAO-GANG WEN PHYSICAL REVIEW B 78, 144517 /H208492008 /H20850
144517-4insensitive to doping and may extend to the hole doped side,4
except near zero doping. There the U1term will drive a SDW
instability due to the nesting between the electron and thehole pockets.
13,23
We would like to mention that since E/H20849k/H20850=E/H20849−k/H20850, our
intrapocket p-wave pairing appears even when the attraction
J−U2is weak. When the attraction J−U2is strong enough to
overcome the interpocket energy splitting /H20849/H110110.05 eV /H20850, our
model also has the instability in the spin-triplet interpocketpairing channel proposed in Ref. 17. Since the intrapocket
effective attraction is reduced by the matrix elements asshown in Eq. /H2084918/H20850, the interpocket pairing may be stronger in
large J−U
2limit while the intrapocket pairing is stronger in
small J−U2limit.
Similarly, we can consider the spin-triplet pairing on the
Fermi pocket near the /H9003˜point /H20851see Fig. 2/H20849b/H20850/H20852. Such a Fermi
pocket is a mixture of the dxzanddyzorbitals. The dispersion
and the dxz-dyzmixing near /H9003can be determined from the
symmetry consideration. The crucial difference is that thehybridization matrix element is now proportional to k
xky.
After a similar calculation, we find that the spin-triplet pair-ing potential V
/H9003˜/H20849k2,k1/H20850satisfies V/H9003˜/H20849k2,k1/H20850=V/H9003˜/H20849k2,−k1/H20850
=V/H9003˜/H20849−k2,k1/H20850.Ap-wave pairing will result in a vanishing
dimensionless coupling /H9261=0. Thus the Coulomb interaction
in the dorbitals does not induce spin-triplet p-wave pairing
on the two pockets near /H9003even when J/H11022U2. We see that the
dxz-dxy/H20849dyz-dxy/H20850mixing at Y˜/H20849X˜/H20850in the three-orbital model is
crucial for the appearance of our p-wave instability. A two-
orbital model has the wrong Fermi-surface topology in that
one hole pocket is located at /H9003˜andM˜in Fig. 2/H20849b/H20850, instead of
both being at /H9003˜.22Although the two-orbital model may allow
certain superconducting states,20,21it does not have the
proper symmetry and the orbital mixing to generate the
p-wave pairing proposed here.
V. CONCLUSION
In this paper, we examine the possibility that Coulomb
interaction between the electrons in the Fe d-orbitals may
induce a superconducting phase. We find that when theHund’s rule ferromagnetic interaction on Fe is strongenough, i.e., when J/H11022U
2, the Coulomb interaction can in-
duce pairing instability. In the weak-coupling limit, the lead-ing instability is found to be a spin-triplet p-wave pairing on
the electron pockets at M/H20849or at X˜and Y˜in the extended
Brillouin zone /H20850. Although the Coulomb interaction can only
directly induce pairing on the electron pockets at M, the
proximity effect will lead to a p-wave pairing on the hole
pockets at /H9003. So the resulting superconducting state has node
lines on the three-dimensional /H208493D/H20850Fermi surfaces.
The spin-triplet pairing order parameter is a complex vec-
tord. The relative phases and relative orientations of the two
order parameters dX˜anddY˜on the two pockets X˜andY˜can
have interesting relations which cannot be determined from
the linear-response calculation adopted here. One possibledistribution of the phases is given in Fig. 2/H20849b/H20850, where p-wave
pairing gap is positive in the shaded regions and negative inunshaded regions.At the atomic level, U/H110153–4 eV and J/H110150.7 eV, so U
2
−Jis positive. However, in a tight-binding model involving
only the Fe dorbitals, the As orbitals have been projected out
and the appropriate UandJare those corresponding to the
Wannier orbitals which are much more extended than theatomic dorbitals. A recent estimate by Anisimov et al.
24
found the average Uto be strongly renormalized down to 0.8
eV while Jremains large at 0.5 eV. These are the more
appropriate bare parameters in a three-band model. Further-more, in a crystal, the strong hopping leads to extended qua-siparticles near the Fermi surface. The on-site U
2andJwill
induce effective interaction U2/H11569andJ/H11569between those quasi-
particles. The term induced by U2will remain short ranged.
However, Jmay induce a long-range couple because parallel
spin configuration favors hopping, i.e., Hund’s rule and hop-ping are compatible. Since the Fermi pockets are small, the
effective interaction is given by U
2/H11569/H20849q/H20850−J/H11569/H20849q/H20850with q/H11011kF
/H112701/a. A long range Jcoupling enhances J/H11569/H20849q=0/H20850and can
potentially lead to a sign change and a net effective attrac-tion.
Since the original submission of this paper the experimen-
tal situation has evolved rapidly. Here we attempt a brief
summary of the relevant experimental data. The issue ofwhether gap nodes exist remains open to debate. Photoemis-sion data on Ba
1−xKxFe2As2indicate an almost isotropic gap
on the hole pocket in this hole doped material.25It was re-
cently found that the magnetic-field dependence of the spe-
cific heat in this material is linear,26in contrast with the /H20881H
behavior taken as evidence for nodes in the electron dopedmaterial.
3At the same time, the nuclear-spin relaxation rate
1
T1fits the T3law over three decades in La /H20849O1−xFex/H20850FeAs.27
Thus at the moment, existing data seem to point to gap nodes
in electron doped materials and their absence in hole dopedmaterials. We note that, because the size of the Fermi pock-ets is small, it is usually advantageous to hide the nodes inthekspace between Fermi pockets. In order to produce a gap
node on the Fermi surface of the small pocket, one needs aneffective pairing potential which varies rapidly on the scaleof the small pocket. Our theory is one of the few that will dothis, and we rely on the rapid kdependence of the hybridiza-
tion matrix element. The message that wave function andmatrix elements may play an important role and must there-fore be handled properly has validity beyond the special
p-wave pairing scenario described here.
What about singlet vs triplet pairing? The Knight shift is
probably the best way to answer this question. In a tripletsuperconductor, the spin contribution to the Knight shiftdrops below T
cto zero if the magnetic field is parallel to the
dvector, but remains unchanged if it is perpendicular. If the
dvector is free to rotate, it will turn perpendicular to Hand
no change in Knight shift is predicted. This is apparently thecase for Sr
2RuO 4. On the other hand, if dis locked to the
lattice, we expect to see in a polycrystalline sample a drop of2/3 of the value compared with singlet pairing. Without ac-curate knowledge of the orbital contribution to the Knightshift, this is hard to distinguish. Thus NMR on single crystalsis needed to settle this question. As of this writing, the onlysingle-crystal data available are from Ning et al.
28on
BaFe 1.8Co0.2As2. The data do not support triplet pairing in
that a drop in the Knight shift is seen for field directions bothSPIN-TRIPLET p-WAVE PAIRING IN A THREE- … PHYSICAL REVIEW B 78, 144517 /H208492008 /H20850
144517-5parallel and perpendicular to the plane. On the other hand, a
recent paper by Nakai et al.29on the FeP system
La0.87Ca0.13FePO shows that the magnetic behavior is quite
different from the FeAs system. The Knight shift increaseswith decreasing temperature, indicative of ferromagnetic
fluctuations above T
cand1
T1Tshows a very unusual increase
below Tc. The authors speculate that magnetic fluctuations
associated with triplet pairing may be responsible for theincrease. The experimental situation remains in flux and itmay be possible that different pairing scenarios may be com-
peting and win out in different materials.
ACKNOWLEDGMENTS
This research is supported by DOE under Grant No. DE-
FG02-03ER46076 /H20849P.A.L. /H20850and by NSF under Grant No.
DMR-0706078 /H20849X.G.W. /H20850.
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144517-6 |
PhysRevB.62.7440.pdf | Three-wave interaction among plasmons in a weakly coupled quasi-two-dimensional Fermi gas:
Down-conversion of high-power terahertz radiation
J. P. Mondt, *Hyun-Tak Kim,†and Kwang-Yong Kang‡
Telecommunications Basic Research Laboratory, ETRI, Taejon 305-350, Korea
~Received 4 August 1999; revised manuscript received 8 May 2000 !
It is shown that, unlike in three dimensions, and as a result of their acoustic character, three plasmons of the
same type in the same subband of a quasi-two-dimensional electron gas ~Q2DEG !can satisfy the frequency
matching conditions among themselves across different regimes of collisionality. The lowest frequency in-volved in the three-wave interaction can be tuned, through the use of segments of different impurity dopinglevels within the two-dimensional layer and through the total carrier density ~gate voltage !. A wide range of
frequencies within the terahertz regime can thus be covered. The present theory is built on the flow equationsbased upon the Bhatnagar-Gross-Krook approximation as an extension of the Euler equations for quasi-two-dimensional electron layers for the low-frequency, collisional regime and the Lindhard theory based on therandom phase approximation for the high-frequency, collisionless regime within the context of kinetic theoryfor an arbitrary Fermi-Dirac distribution. The mode-coupling equations show the possibility of generatingplasmons in the terahertz range through frequency difference generation, yielding nonlinear growth withinabout 1 to 2 ps. The criterion for parametric instability based on one pump plasmon is also given. It is shownthat the quasi-two-dimensional pump plasmon needed for the three-wave interaction within the Q2DEG foundin this paper can be resonant with a three-dimensional plasmon in the bulk with a wave number correspondingto the peak of stimulated Raman scattering against plasmons for some parameters corresponding to a low-gapsemiconductor. The dependence of the terahertz amplitude and rise time on the three-dimensional stimulatedRaman scattering process providing the pump plasmon in the quasi-two-dimensional layer is quantified.
I. INTRODUCTION
Efforts to create a tunable, high-power, cw source of tera-
hertz radiation through the excitation and subsequentgrating-induced radiative decay of plasmons in quasi-two-dimensional electron gases ~Q2DEG’s !have widely been
recognized for their potential.
1–6The most extensively devel-
oped and interesting scenarios are based on the excitation ofcurrent-driven
1,2instabilities in a single layer or pair of coun-
terstreaming layers, on the field-induced instability in a su-perlattice of alternating electron and hole layers,
3or on the
shallow-water wave type electron plasma instability in theshort field effect transistor.
4–6For the current-driven or field-
induced methods it is not clear whether the fields requiredfor acceleration of the carriers inevitably cause a degradationof the plasma or whether nonradiative decay of plasmonsthrough acoustic phonons is too strong a competitiveprocess. The strongly coupled nature of the
Al
xGa12xAs/InxGa12xAs system considered for the short
field effect transistor as well as for GaAs/Ga xAl12xAs sys-
tems aimed for in the work on two-stream instability strictlyspeaking invalidates the application of the random phase ap-proximation, thus also of a fluid-dynamical description. The
strongly coupled nature of the plasma may well be a limitingfactor in efforts for the practical realization of these methodsby limiting the lifetime of plasmonlike structures to micro-dynamical time scales associated with the unscreened many-body system. This theoretical point
7,8is underscored
experimentally9in the case of optically excited plasmons in
AlxGa12xAs/GaAs heterojunctions. Their quick decay was
attributed to strong carrier-carrier interactions. Hence, boththeoretical and experimental arguments seem to favor the use
ofweakly coupled Q2DEG’s for the generation of plasmons,
to which case we restrict ourselves in this work @rs2!1,rs
[ks/(2kF), where kFandksare the Fermi and screening
wave numbers, respectively #.
Particularly, we consider the coherent excitation10–12of
plasmons as this method has several advantages. First, thelevel of excitation can be adjusted more readily by control-ling the driver. Second, parametric excitation does not relyon the drift velocity as the sole source of free energy. Third,by varying the parameters ~carrier density, pump frequency,
temperature, drift velocity, doping concentration !the tera-
hertz frequency could be tuned. To avoid the need for bulkacceleration that is necessary for two-stream instability butkeep the advantage of high power through exploiting thelong-range, collective interactions in the plasma, long-pulse,high-power, tunable THz radiation may be achieved by theapplication of stimulated Raman scattering ~SRS!to nonme-
tallic media. Carrier densities of even intrinsic semiconduc-tors such as InSb are experimentally known
13to be increased
within 1 ps to about 1018cm23through the application of a
CO2laser ( l’10.6 mm) in the 100 2200 MW/cm2power
range and at a laser frequency below the intrinsic gap butabove the expected range of subintrinsic peaks in the opticalconductivity usually attributed to phonon activity. Laser
power levels of several tens of MW/cm
2are sustainable over
much longer periods without damage to the crystal. Stimu-lated Raman scattering in InSb has also been established,
14
while stimulated Raman scattering against the LO phononand coupled LO-phonon–plasmon modes in GaP has longbeen known to produce extremely high gain, exceeding gainsPHYSICAL REVIEW B 15 SEPTEMBER 2000-I VOLUME 62, NUMBER 11
PRB 62 0163-1829/2000/62 ~11!/7440 ~14!/$15.00 7440 ©2000 The American Physical Societyfrom a/o CS 2, LiNbO 3, and potassium dihydrogenphosphate
~KDP!.15It has long been recognized16that SRS at intensi-
ties .10 MWcm22can produce Stokes wave intensities
approaching those of the incident laser beam. In the case of
InSb stimulated Brillouin scattering is known to interferewith SRS after about 1 ns. Numerical simulations
17indicate
that such interference may be due to the local violation of thecondition, necessary for propagation of the plasma wave andidler, that the carrier density be less than one-quarter critical.Therefore, a material with a slightly higher intrinsic gap mayoverall be preferable despite a lower relative power effi-ciency because of a higher down-conversion ratio. It is notthe purpose of the present paper to specify a particular ex-perimental configuration by which SRS in three-dimensionalmedia could be exploited to excite the pump plasmon in theQ2DEG required for the presently discussed three-wave in-teraction. However, a numerical example is worked outbased on a material with an energy gap and other materialproperties similar to InAs because of its low gap ~0.35 eV at
room temperature, 0.4 eV at 100 K without doping, and areduction by less than 0.1 eV for doping concentrations not
exceeding 10
18cm23).18This may enable, under certain cir-
cumstances, the use of a laser wavelength down to as low as
4.2mm, although the experimentally obtainable power
seems to be an open question. The plasmon frequency fromthree-dimensional SRS would only be moderately tunable, asit depends on three-dimensional parameters and moreoverhas an optical character; it also tends to be slightly above thedesired THz range of ~linear !frequencies between 0.2 and
2.0 THz. Therefore, moderate down-conversion and a
method for tuning would be desirable. We stress, however,
that on the one hand, to the best of the authors’ knowledgethe usefulness of InAs for this purpose is not established,while on the other hand, the physics of the currently foundthree-wave interaction among plasmons in Q2DEG’s doesnot depend on the use of stimulated Raman scattering againstplasmons, nor is it the focus of the present communication.
It is the main purpose of this paper to show that condi-
tions for three-plasmon interaction within the electron ~hole!
plasma pertaining to one subband in a Q2DEG can be ful-filled, and that this offers the possibility to achieve down-conversion and tuning of high-power, three-dimensional lon-gitudinal waves. Because of the limited range of the plasmafrequency in three dimensions, and because of the weak andconcave dependence on wave number of the three-dimensional plasma frequency, three-wave interactionamong plasmons in a three-dimensional isotropic plasma isknown to be impossible.
12However, plasmons in Q2DEGs
have an acoustic character, with a Akbehavior of the fre-
quency for small wave number. Furthermore, the dispersioncurve has an inclination point at the screening wave numberbeyond which it is slightly convex, while the group velocity
of the high-frequency plasmons with finite k/k
sis higher
than what would be predicted from hydrodynamic theory.The latter property is caused by a difference in the electron~or hole !sound speed corrections in the collisional and col-
lisionless regimes because of a difference in the rate of ap-proach towards isotropy of the stress tensor relative to theoscillation frequency. If the rate of approach towards isot-ropy is higher than the oscillation frequency the pressureperturbation is isotropic within the Q2DEG, hence two-dimensional, and ~for low temperature, T/T
F!1) the elec-
tron sound velocity is s’vF/A2; if instead the rate of ap-
proach towards pressure isotropy is lower than the oscillationfrequency the dynamics is essentially that of a one-dimensional, collisionless shock wave and the sound speed is
closer to
vF. A third, and, as it turns out, more important
factor contributing to three-wave interaction in the terahertzregime is the role of the linear momentum relaxation rate
(
nc) as a sink of momentum, but not of particle number
density, thus providing an offset to the square of the real
frequency ( v2!v22nc2/4) in addition to a linear damping
(g.2nc/2). That the conditions for three-wave interaction
can be fulfilled is illustrated in Figs. 1 and 2. As a result ofthe offset the real phase velocity of the low-frequency plas-mon varies over a range that can include the high-frequencyplasmon group velocity within the terahertz regime for mod-erate values of the mobility, i.e., typically several thousands
of cm
2/(Vs), while at the same time the frequency is tunable
through the electron mobility ~hence through the neutral im-
purity doping level !and through the total carrier density ~i.e.,
gate voltage !.
In addition to the generic process of three-plasmon inter-
action we also discuss a specific mechanism for producinghigh-frequency pump plasmons in the Q2DEG as an ex-ample of a possible application. When moving at close dis-
tance (d<k
21) past a Q2DEG, a three-dimensional longitu-
dinal wave interacts electrostatically with the Q2DEG chargecarriers as if the wave occurs within it. This can be seen fromthe quasi-two-dimensional Poisson equation,
19of which the
Green function after Fourier transformation in the coordi-nates tangential to the planes is given by
g
~z,z0!5~2k!21exp~2kuz2z0u!, ~1!
where the exponential factor is non-negligible for any two
points at vertical coordinates ~z!differing by not substan-
tially more than k21. Consequently, when its frequency and
wave number match those of a plasmon in the Q2DEG it willbe able to propagate as a quasi-two-dimensional plasmonthere. As will be shown, a plasmon with frequency and wavenumber corresponding to the peak ~backscatter !of SRS pro-
duced by a laser source with a ~vacuum !wavelength of 4.2
mm in a three-dimensional medium with a refractive index
nr,3D53.5 can be resonant with a two-dimensional plasmon
in an InSb layer for wave numbers of the order of the screen-ing wave number, while at the same time conditions aremanifestly fulfilled for three-wave interaction within thelayer and while the grating is sufficiently close by to achieveradiative decay ~cf. Figs. 3 and 4 !.
In passing it is noted here that the use of InSb as both the
Q2DEGandthe 3D medium is complicated for the case of
SRS against plasmons because the peak wave number, beingtwice the wave number of the laser in the medium, is neces-sarily rather much lower than the screening wave number ofQ2DEG’s for moderate densities because of the extremelylow-energy gap of InSb ~0.17 eV at 300 K, 0.23 eV at 77
K!, whereas the carrier density of the 3D medium is close to
one-quarter critical. The collision frequency in the case ofInSb as the 3D medium is just above the terahertz frequency~see Fig. 5 !, casting further doubt on the feasibility to use
InSb. Therefore, although electron-neutral collisions, as-PRB 62 7441 THREE-WAVE INTERACTION AMONG PLASMONS IN A . . .sumed here to limit the mobility, typically make the distri-
bution function isotropic after just one collision, importantkinetic corrections must be expected in the case of InSb. Asshown in Fig. 5, apart from the aforementioned difficulties itis possible to obtain three-plasmon resonance also in thiscase, while the intensities obtained for any given pump waveexceed those in the case of InAs precisely because of thecloseness of terahertz and collision frequencies.
Returning to the InSb-InAs system described above, the
terahertz radiation is shown to be tunable across a wide por-tion of the terahertz frequency range through variation of thecarrier density and, more importantly, impurity concentrationacross segments of the layer, thereby allowing the use of thelinear momentum relaxation rate for tuning. However, theterahertz power can only be quantified in terms of the powerof the three-dimensional pump wave, the former dependingquadratically on the latter. For the main purpose of the paperthe specific nature of the three-dimensional longitudinalpump wave does not matter as long as it resonates. However,for the present purpose we restrict considerations to SRSagainst plasmons.
The organization of the paper is as follows. In Secs. II and
III we derive the dispersion relations for plasmons in thecollisionless, high-frequency regime as described by kinetictheory based on the random phase approximation and in thecollisional, hydrodynamic regime as described by aBhatnagar-Gross-Krook extension of the quasi-two-dimensional Euler equations, respectively. In Sec. IV we ad-dress the possibilities to obtain exact frequency matching. InSec. V mode-coupling equations are derived based on a con-tinuum description of the essential convective nonlinearities.In Sec. VI the nonlinear evolution is discussed, based on thenonlinear equations with full incorporation of linear damp-ing, driving forces, and mode coupling for the case of fre-quency difference generation ~FDG!. The condition for para-
metric instability is also derived. In Sec. VII we present thenumerical analysis on the creation and use of three-dimensional plasmons obtained from SRS as pump plasmonsin the Q2DEG within this dynamical context. We end withconcluding remarks.
II. HIGH-FREQUENCY PLASMON DISPERSION
EQUATION
Within the context of the random phase approximation for
the case of a spin-independent electron distribution the dis-persion relation for a fully collisionless plasmon can be de-rived from the Schro ¨dinger equation for the density
operator
20with the Hamiltonian
Hˆ5\2
2mD2ew~t,rW!, ~2!
subject to linearization about a homogeneous dynamical
equilibrium, r5r01dr, where r0only depends on RW[rW1
2rW2. The density operator r0is related to the equilibrium
electron momentum distribution by
n0~pW!5NeEr0~RW!exp~2ipWRW!d2x, ~3!whereNeis the total number of electrons. As the occupation
number of quantum states of electrons with definite values ofthe momentum and spin component, the number of states in
an element d
2pof momentum space and with either value of
the spin component is 2 d3p/(2p\)2; hence the electron dis-
tribution function is given by f(pW)52n(pW)/(2p\)2. After
Fourier decomposition c˜5cexp(2ivt1ikWxW), wherexWis the
spatial coordinate in the plane and zis the distance to the
plane, the density response to a fluctuation in the electro-
static potential w˜is obtained in the usual manner20as
n˜5ew˜
\Ed2pf~pW1\kW/2!2f~pW2\kW/2!
v2kWpW/m, ~4!
with the Landau prescription for the pole. The Fermi-Dirac
distribution function may be written21as a convolution over
its zero-temperature limit f052/h2u(m82p2/2m), where u
is the Heavyside function and
f5b
4E
0‘
dm8f0~m82p2/2m!
cosh2F1
2b~m2m8!G, ~5!
where b[1/(kBT). The density response and electrostatic
potential fluctuation are also related through the quasi-two-dimensional Poisson equation
19for free oscillations in a
plane under the influence of the ~three-dimensional !electro-
static potential, i.e.,
~]2/]z22k2!ew~kW,z,v!522pek21n˜~ukWu,v!exp~2kuzu!,
~6!
where eisthedielectricfunction.Forthepurposeofderiving
the three-wave interaction among Q2DEG plasmons we may
setz50. Substituting Poisson’s equation into the dynamical
response and performing the integration over momentumprior to the convolution, Eqs. ~4!and~6!lead to the follow-
ing dispersion relation:
0511pe2
2e\kb2
h2E
0‘
dm8~J12J2!
cosh2F1
2b~m2m8!G,~7!
where we defined
Js[Ed2pu~m82p2/2m!
vs2kWpW/m, ~8!
withs51,2andv6[v6\k2/(2m). The integration over
the angle of the momentum can be performed to yield
Js5m
k@A1(s)2A2(s)#, ~9!
where
A6(s)[E
upxu<pMdpxFlnSmvs
kpM~px!D61G. ~10!7442 PRB 62 J. P. MONDT, HYUN-TAK KIM, AND KWANG-YONG KANGHere,pM[(pm22px2)1/2withpm[(2mm8)1/2. Use of the
substitution x(u)5sin(u)o n uP(2p/2,p/2) followed by
the inverse of x(u)5tan(u/2) that is monotonic on (0, p/2)
yields
A1(s)2A2(s)52ppm@v˜f,s22p~v˜f,s221!1/2#, ~11!
where the tilde in v˜f[v/(kvm) denotes normalization of the
phase velocity in units of vmandv6[v6vq, with vq
[\k2/(2m). The resulting dispersion equation is
0511kTF
kH~11e2bm!211b
8vqE
0‘ dm8
cosh2@b~m2m8!/2#
3@~v222k2vm82!1/22~v122k2vm82!1/2#J, ~12!
where vm8is the Fermi velocity corresponding to the chemi-
cal potential m8.kTF[2me2/(\e¯) is the Thomas-Fermi
wave number corresponding to the average dielectric con-
stant e¯of the semiconductor and insulator. Without the de-
pendence on m8of the numerator of the integrand the inte-
gration would just yield the overall multiplier (1 1e2bm)21
for the entire term }kTF/k. Thus finite temperature would
only have an exponentially small effect in the degenerateregime; with it, however, electron plasmons in the sound
wave regime ( k@k
TF) are affected by wave-particle reso-
nance, in particular, Landau damping.
It has been shown22that screening and hence the effective
kTFare modified by nearby conductors on both sides of the
Q2DEG. In the presence of a nearby grating, needed forradiative plasmon decay, the insulator thickness is limited by
the distance d
insof the Q2DEG to the grating. On the other
hand, the thickness dscof the semiconductor would be lim-
ited by enhanced levels of the charge carrier density in it, asin the case of SRS against plasmons. The full expression
for the modification is k
TF!kTF2e¯/@einscoth(kdins)
1esccoth(kdsc)#.
Equation ~12!will be used in the numerical work. Its form
illustrates that finite temperature manifests itself mainlythrough distributing the relevant Fermi velocity over a rangeof the order of the thermal velocity. It is clear that finitetemperature effects are most important when the ~Doppler-
shifted !phase velocities approach the Fermi velocity
vm,
which occurs for k@kTF. As will be shown in the Appendix,
the integral equation can be approximated by a local equa-
tion provided T!TFandthe thermal velocity is small com-
pared with the difference between vmandvf,sfors51,2.
The latter condition is the most restrictive; when not metLandau damping sets in. The local equation in its simplestform tends to agree quantitatively even for rather high
T/T
F,1u pt ok’kTF, but for realistic values of rsit sud-
denly breaks down for wave numbers comparable to theThomas-Fermi wave number ~see Fig. 6 !. The authors do not
know whether Eq. ~12!for arbitrary temperature or its math-
ematical equivalent exists in the literature to date. It is usedhere because, in addition to demonstrating the importance of
Landau damping for k.k
TFand in addition to our interest in
finiteT/TFfor room-temperature applications, the pump
waves in the three-plasmon interaction must have finitek/kTF, while small errors in the difference of pump and idler
frequencies magnify in the expression for the resonance fre-quency. As discussed in the Appendix, its lowest-order ap-proximation reduces to the more familiar forms
23through the
Taylor expansion for small k/kTF.
III. LOW-FREQUENCY PLASMON DISPERSION
RELATION
The description of the low-frequency plasmon in principle
involves the nonlinear integro-differential kinetic equationswith full incorporation of electron-electron and unlike-species collision terms. For simplicity we will adhere to theEuler equations for a quasi-two-dimensional Fermi gas
19of
spin 1/2 augmented by a velocity drag term based on theBhatnagar-Gross-Krook ~BGK !model
24,25for the linear mo-
mentum relaxation processes. The coupling of fluctuations inthe impurity and phonon distributions back onto the elec-trons is ignored. Linear momentum relaxation through colli-sions between different species also has an effect of creatingpressure isotropy in their time scale through the randomiza-tion of the specific velocities. However, because electron-electron collisions conserve total electron momentum, theireffect on momentum is mostly indirect, although it has beenreported that they sometimes do contribute to mobility, spe-cifically in narrow wires, where they cause an increase inelectron collisions with the wall. Two collisional rates haveto be distinguished, at least in principle, i.e., the linear mo-mentum relaxation and the pressure anisotropy relaxation
rate. The latter rate is the sum of the former and the e-e
collision rate, although practically the e-ecollision rate does
not play an important role ~see next section !. In the present
BGK model of the low-frequency plasmon we consider theelectron plasma as a Q2DEG coupled to the outside throughthe~three-dimensional !electrostatic field and through
unlike-particle collisions.
The complete fluid-dynamical equations based on the
BGK approximation are
24
Dtn1nuW50, ~13!
HDtuWs1P
mn2e
mEWJ
k5njk~uWj2uWk!, ~14!
whereuWis the electron fluid velocity ~species k !,Pis the
pressure tensor, on the right-hand side summation over all
speciesjÞkis assumed, and Dt[]t1vW„W. The pressure
gradient term in the case of a two-dimensional Fermi gas is
equal to ( s02/n0)„n, wheres0is the adiabatic sound velocity.
njkis the resistance coefficient24per mass density of species
kagainst the drag exerted by species jand as such only
depends on the collision frequency of species kwith species
j, the effective masses of species jandk, and the density of
target species j. The linear momentum collision terms for the
case of massive scattering targets lead to perturbations in the
momentum balance that can be approximated by 2nduW,a s
only the electron species is perturbed. There is no counter-part in the continuity equation to the drag in momentumbalance, because we only consider particle-conserving colli-sions. Because the low-frequency plasmon is described in theregime where the pressure perturbation is fully isotropic, noPRB 62 7443 THREE-WAVE INTERACTION AMONG PLASMONS IN A . . .viscosity term is included in the force balance. Density and
electrostatic fields are also coupled through Poisson’s equa-tion@Eq.~6!#. Linearization and Fourier decomposition of
the continuity equation and force balance yield in the ab-sence of background mass ~fluid!flow
2i
vn˜1in0kjvj˜50 ~15!
and
2ivvi˜2iek’,iw
m1ikis02
n0n˜52ncvi˜. ~16!
Substituting Poisson’s law into the inner product of force
balance with kW, and eliminating the fluid velocity through the
continuity equation, we find the dispersion equation
v21incv52pe2n0ukWu
erm1k2s02. ~17!
In terms of the screening wave number ks[kTFvm2/(2s02) the
eigenfrequency may be written
v5@~ksk1k2!s022nc2/4#1/22nc/2. ~18!
The sound velocity s0is, at any temperature, given by26
s052P
mn. ~19!
Starting from the grand-canonical thermodynamic potential
for an ideal Fermi gas with spin 1/2 over a surface area A,
FFD52mA
p\2b22E
2bm‘
dxln~11e2x!, ~20!
the pressure, as P52FFD/A(A!0), is seen to have the
following exact temperature dependence:
P
P~T50!52~bm!22E
2bm‘
ln~11e2x!dx, ~21!
where b[1/(kBT). This determines the temperature depen-
dence of the sound velocity through Eq. ~19!and thereby
also that of the screening wave number ks5kTFvm2/(2s02).
The logarithm in the above integral can be analytically
determined ~convergence radii are 1 !without approximations
by expanding the logarithm in ln(1 1e2x)5ln(11ex)2xfor
smallexin the negative domain and expanding ln(1 1e2x) for
smalle2xin the positive domain. This yields
s0251
2vF2H11F1
3p222(
n51‘
~21!n21z2nG~bm!22J,
~22!
where the sum involving the fugacity z[ebmcan be approxi-
mated by its first term even for the case when T’TF.
IV. FREQUENCY MATCHING
While the electron-electron scattering rate with regard to
the energy of a quasiparticle is quadratic in T/TF,27,28ne-e5p
8t2Fln~t!1ln~8!
220.083G, ~23!
where t[T/TF, and the electron-electron momentum an-
isotropy relaxation rate for degenerate systems is reported to
be much slower than ne-eas given above, namely a factor
}t2slower, except within a cone of order At.29In the
present work we will therefore neglect the electron-electroncollisions altogether. The three-wave interaction, then, islimited to nonideal systems. Because the electron momentumrelaxation rate due to all other types of collisions affects themomentum of the electron fluid as a whole, it is necessary tokeep this rate as low as possible in order to prevent strong
damping ( }
nc/2). This prompts consideration of the regime
v,nc,v0. The linear momentum relaxation reduces the
real frequency for a given k,kTF, thereby reducing the
phase velocity of the low-frequency plasmon such that itbecomes equal to the group velocity of the pump plasmon ~s!,
which is roughly the phase matching condition. As can beseen from the position of the participating modes in the lin-ear dispersion curves ~cf. Fig. 2 !, the idler wave is rather
close to the pump, such that a Taylor expansion of the dif-ference of their frequencies is allowed for purposes of illus-tration. However, in the numerical calculations the exact ex-pressions for the pump and idler frequencies are taken intoaccount, as the danger otherwise might be that in relation tothe~much !lower excited frequency the error in the Taylor
expansion is not negligible. Let the group velocity of high-
frequency plasmons near the pump and idler be
vg,h. Then,
within the context of the Taylor expansion, wave number k
and frequency vof the low-frequency mode both match
whenkvg,h’v, i.e.,
k2~vg,h22s02!’kkss022nc2/4. ~24!
Because vg,h.vFas the pump wave has to be in the
purely oscillatory regime while, except for highly nondegen-
erate regimes, s0.vF/A2,vF, there is a solution to the
resonance condition provided the plasma frequency in theabsence of linear momentum relaxation exceeds half the lin-ear momentum relaxation rate. Note that the plasmon fre-quency in the presence of linear momentum relaxation isreduced, such that the required inequality for pressure isot-ropy on the time scale of the excited frequency can be met.From Eqs. ~18!and~61!the wave number is found to be
k5k
s
2@vg,h2/s0221#S16H11F12~vg,h2/s02!nc2
ks2s02GJD1/2
,
~25!
while the threshold linear momentum relaxation rate is
nc<~vg,h2/s0221!21/2kss0. ~26!
Inclusion of linear momentum relaxation in the high-
frequency plasmon dispersion equation does not significantlymodify the frequency matching condition, because in orderof magnitude this modification could be estimated from thesame effect on a mode of higher frequency given by theEuler equations with a BGK term added. From the above-7444 PRB 62 J. P. MONDT, HYUN-TAK KIM, AND KWANG-YONG KANGmentioned threshold condition with vg,h2>vF2’2s02,w eo b -
tainn2/4<ks2s02/4!2ks2s02. Therefore, up to an error of rela-
tive order ( nv/v0)2/8,
v.v0S12nc2
8v02D, ~27!
where v0is the high-frequency eigenfrequency in the ab-
sence of collisions. Applying this rough estimate we see that
the frequency matching condition v02v15vonly acquires
an additional term through BGK effects on the pump and
idler by 2nc2/(8v12) on the left, which is small compared
with the excited frequency v, because, as a prerequisite for
the matching v1@nc. Figure 2 indicates the positions on the
dispersion curves in a typical example of a three-wave reso-nance.
V. MODE-COUPLING EQUATIONS
To analyze the mode coupling we will assume, for sim-
plicity, one-dimensional perturbations in a fully degenerateFermi gas. The high-frequency modes are only needed in anarrow portion of wave numbers, between idler and pumpwaves, in which we can take the dependence on wave num-ber of frequency to be linear, as evidenced from the high-frequency dispersion equation. To avoid a very complicatednonlinear kinetic treatment we will represent the high-frequency modes by a dispersion equation that is linear as faras the dependence of frequency on wave number betweenpump and idler waves is concerned ~although with an offset !
and adopt otherwise the same nonlinear fluid equations as forthe low-frequency modes. Implicitly we thereby approximateitsform ~as opposed to the location of the points on the
dispersion curve !by that of the low-frequency dispersion
equation in the range where we do not need the dispersionequation. Eliminating the electrostatic potential from the
fully linear Poisson equation @Eq.~6!#the normal modes a
L
andaHfor the low- and high-frequency waves are combina-
tions of the dynamical variables n¯/n0;exp(ikx2iwt) and v¯x,
i.e.,a5n¯/n01lv¯x, where in both low- and high-frequency
cases the coefficient lis determined by requiring ato be an
eigenmode in the linear case, which yields
aL,H5n¯
n01k
vL,H*v¯x, ~28!
in which the vL,Hare the eigenfrequencies of the low- and
high-frequency modes, respectively, the latter being given bythe fully kinetic dispersion relation. From the continuityequation the relation between the normal modes and the elec-tron density perturbation is obtained, i.e.,
a
H,L5S11vH,L
vH,L*Dn¯H,L
n0. ~29!
Turning to the nonlinear fluid equations, the nonlinearity
involving the sound velocity, which vanishes when s02.vF2,
will be neglected, and the only nonlinearities that remain are
]x(nvx) in the continuity equation and vx]x(vx) in force
balance. Expressing the real variables ~e.g.,n˜L) in terms ofthe dynamical variables @in this example1
2(n˜(0)1n˜(1)) and
its complex conjugate !and collecting terms with the same
Fourier component, the following set of mode-couplingequations is obtained:
~]t1ivL!aL5cL(0,1)aH(0)aH(1)*, ~30!
~]t1ivH(0)!aH(0)5cH(1,L)aH(1)aL, ~31!
~]t1ivH(1)!aH(1)5cH(0,L)aH(0)aL*, ~32!
where the mode-coupling coefficients are given by
cL(0,1)52ikL
16SvH(0)1vH(1)1vH(0)vH(1)
vL*D, ~33!
cH(1,L)52ikH(0)
16vL*
Re~vL!SvH(1)1vL1vLvH(1)
vH(0)D,~34!
cH(0,L)52ikH(1)
16vL
Re~vL!SvH(0)1vL*1vL*vH(0)
vH(1)D,~35!
where vf,Handvf,Lare the average phase velocity of the
pump and idler waves, and the ~complex !phase velocity of
the low-frequency wave, respectively.
To simplify notation define
a0[aH(0),a1[aH(1),a2[aL, ~36!
c0,1[cL(0,1),c1,2[cH(1,L)*,c0,2[cH(0,L), ~37!
v0[vH(0),v1[vH(1),v2[vL, ~38!
and define moduli and phases for all amplitudes and coupling
coefficients, aj5uje2iRe(vj)t1ifj,cij5vijeiuij, the mode-
coupling equations can be shown to be equivalent to the realequations
]tu05v12u1u2cos~F1u12!, ~39!
]tu15v02u0u2cos~F1u02!, ~40!
]tu2521
2ncu21v01u0u1cos~F1u01!, ~41!
]tF52Fv12u1u2
u0sin~F1u12!1v02u0u2
u1sin~F1u02!
1v01u0u1
u2sin~F1u01!G, ~42!
where F[f02f12f2. From Eqs. ~33!–~35!it follows that
u0,15u0,252p/21uc,u1,25p/21uc, ~43!
where uc[arg@11vHvL/(2uvLu2)#represents the effect of
dissipation on the phases of the coupling coefficients.
VI. NONLINEAR EVOLUTION
Although linear damping is slow compared with the os-
cillation frequency of the high-frequency plasmons and thusPRB 62 7445 THREE-WAVE INTERACTION AMONG PLASMONS IN A . . .does not affect their dispersion relations, it is comparable to
the linear damping of the low-frequency plasmons and thenonlinear time scales of evolution. Therefore, in the descrip-tion of the nonlinear evolution of the three-wave system it isnecessary to include linear damping for all plasmons. Fur-thermore, a model of the SRS driving force is required. Tobe specific, let us assume that in the absence of the modecoupling the overall effect of the driving force and losses on
any driven high-frequency plasmon u
His to bring and main-
tain its amplitude to a certain level uH,mat a characteristic
ratenH,dand proportional to its relative deviation uH,m
2uHfrom it. Defining Fˆ[F1uc2p/2 the dynamical sys-
tem may then be written as
]tu052v12u1u2cosFˆ1n0,d~u0,m2u0!21
2ncu0,
~44!
]tu15v02u0u2cosFˆ1n1,d~u1,m2u1!21
2ncu1,~45!
]tu25u0u1cosFˆ21
2ncu1, ~46!
]tFˆ5sinFˆSv12u1u2
u02v02u0u2
u12v01u0u1
u2D.~47!
As the relevant collisions are between electrons and neu-
tral species within the framework of a BGK model we as-
sume all three plasmons share the same ncwhile for the case
of FDG we will take the rates nd,H[ndand asymptotic lev-
elsuH,m[uHto be the same for both high-frequency plas-
mons. Because the estimate gLandau ’2Im(e)/]vRe(e)
shows that the Landau damping is less than 1/1000 of the
collisional damping for the cases studied here we approxi-
mate their damping rate by the collision rate nc, which when
dominated by electron-neutral collisions as assumed here canbe taken to be the same for all three waves. We consider thecase when both high-frequency modes are driven through the
use of two lasers with different wave numbers k
0,1~hence
different pump wave numbers in the Q2DEG !and the use of
two separate slabs with different carrier density within a dis-
tanced<k0,1from and parallel to the Q2DEG, such
that the three-dimensional plasma frequencies vp,3D
[(4pn3De2)/(m*e3D)1/2are equal to the frequencies v0,1,
respectively. Alternatively, one slab with a gradient in thecarrier density could be considered. No distinction will bemade between the growth time of the three-dimensionalstimulated Raman signals for the high-frequency modes. Af-
ter an initial rise time (
gSRS21) due to the startup of SRS in the
3D medium to a level for the amplitudes of the normal
modes (ui) that would be equal to a certain level umin the
absence of the mode coupling a stationary state sets in. If the
initial phase Fˆis set to zero it remains so during the evolu-
tion; if not it relaxes to zero typically in a fraction of 1 ps@Fig. 7 ~a!#. The time needed for the low-frequency plasmon
to reach its asymptotic value typically is of the order of apicosecond or two @Fig. 7 ~b!#. Figure 8 shows the numerical
solution for the asymptotic values of the ratio of the ~dy-
namical !density perturbation of the low-frequency mode di-vided by that of the high-frequency mode in percentage as a
function of both the strength and growth rate of the 3D SRSprocess as represented by the asymptotic normal mode am-plitude of the high-frequency modes in the Q2DEG in theabsence of mode
FIG. 1. Frequency mismatch and terahertz frequencies ~both lin-
ear rather than angular !versus the ratio of the terahertz wave num-
ber divided by the Thomas-Fermi wave number for ~a!T577 K,
and~b!T5300 K, for the case when the total carrier density
ntotal5231012cm22. The infrared dielectric constants of the in-
sulator and semiconductor are chosen to be eins53 and esc516,
respectively. The distances from the Q2DEG to the grating and tothe illuminated portion of the semiconductor are d
ins52000 Å and
dsc51000 Å, respectively. The pump wave number is chosen to
correspond with the peak of SRS against plasmons ( k052klaser
.1.0472 3105cm21.0.097kTF) for a 3D refractive index nr,3D
53.5 and a vacuum laser wavelength of 4.2 mm. The wave number
k0determines the pump frequency through the high-frequency dis-
persion relation. This frequency should equal the three-dimensionalplasma frequency excited by SRS. Its ~linear !value isf
054.578
THz. The mobility was selected to be m56708 cm2/(Vs). The
semiclassical approximation for the mass dependence for narrow-gap semiconductors is adopted ~Ref. 30 !, such that m
*50.03 for
this total carrier density. The ~linear !electron momentum relaxation
frequency is therefore fc51.542 THz. The excited frequency is f
50.833 THz. The 0-subband density is taken ~Ref. 30 !to ben
50.67ntotal.7446 PRB 62 J. P. MONDT, HYUN-TAK KIM, AND KWANG-YONG KANGcoupling. Additional losses due to the distance between the
3D medium and the Q2DEG and between the Q2DEG andthe grating are not taken into account in Fig. 8 and amount to
a multiplication of the ordinate by e
2kHdsce2kLdins(.0.28
for the parameters of Fig. 8 !.
Finally, we give an analytical criterion for the onset of
parametric instability ~one driver only !. Setting ]t50 in all
dynamical equations we see that the asymptotic value of the
normal mode of the pump wave u0,‘corresponds to the con-
dition for the marginal point of a purely growing instabilityfor the envelope of the pump plasmon, i.e.,
u
0‘5nc/2
Av01v02. ~48!
Then a steady state with nonzero u2,‘transpires if and
only if
u2‘5Sv01
v02D1/4S@nd~um2u0‘!2ncu0‘/2#
v02D1/2
~49!
is real and nonzero, whence if and only ifAv01v02um.~nd1nc/2!~nc/2!
nd. ~50!
This criterion agrees with numerical solutions. However,
the threshold is not met for the presently studied parameterset~see Fig. 3 !chosen to obtain resonance with three-
dimensional SRS in a moderately low-gap semiconductor.
VII. NUMERICAL ANALYSIS ON THE USE OF PUMP
PLASMONS FROM STIMULATED RAMAN SCATTERING
In the numerical analysis we use a fit to the semiclassical
approximation30for the dependence on total carrier density
of both the effective mass and the density of the zeroth sub-band of a Q2DEG consisting of InSb. This approximationagrees closely with the more complete theory.
31,32For the
electro-optical properties of the low-gap semiconductorsInSb and InAs as three-dimensional media we refer to ar-ticles in a recent compendium.
18,33
Our main objective in this section is to demonstrate that
SRS against plasmons in a three-dimensional medium adja-cent to the Q2DEG may in principle be used for the genera-tion of the high-frequency pump plasmons needed for thecurrently proposed three-plasmon interaction in a Q2DEG.This issue depends admittedly on the maximum obtainablepower through SRS in three-dimensional media, which isultimately an experimental issue beyond the scope of thepresent article. However, we show here that the resonanceconditions, first between the three-dimensional plasmon and
a Q2DEG plasmon with finite k/k
TF, and subsequently be-
tween the Q2DEG plasmon thus created and two lower-frequency plasmons, can be met, such that the lowest fre-quency plasmon is in the terahertz range and such that,through variation of total carrier density and impurity level,this frequency can be tuned across the terahertz range. Fur-thermore, the resonance between the three-dimensional plas-
mon and the Q2DEG plasmon of finite k/k
TFcan be accom-
plished at peak SRS power ~backscatter !. Leaving aside
considerations of manufacturability, we also show that thesestatements can be made not only for fairly low temperature
~77 K!, but also for room temperature ( T5300 K !.
The illumination by a laser of a frequency below the ma-
terial’s energy gap but at least a factor well over 2 above theplasma frequency associated with the charge carriers in thethree-dimensional medium, and at a Poynting flux ~S!below
the breakdown voltage but exceeding the stability thresholdfor SRS would yield a spectrum of plasma waves peaking at
the backscatter wave number k
p,3D52klaser. In terms of the
electric field amplitude of the laser light
E05S8pS
cAeD1/2
~51!
and assuming the predominance of collisional damping
(nc,3D) the threshold condition for the onset of SRS is given
in the underdense regime by34
eE0
km*cSn
ncrD1/4
.nvp3D
v1. ~52!
FIG. 2. Fast and slow wave dispersion relations showing the
interacting triplet and the electron momentum relaxation frequencyfor~a!T577 K and ~b!T5300 K for the parameters pertaining to
Fig. 1.PRB 62
7447 THREE-WAVE INTERACTION AMONG PLASMONS IN A . . .Provided a strip of the three-dimensional medium adja-
cent to the quasi-two-dimensional layer is excited such thatthe distance of the strip to the layer does not exceed the wavenumber of the excited three-dimensional plasmon, the two-and three-dimensional media can be considered as interpen-etrating, as discussed before. A two-dimensional plasmon
(
v0,k0) is then excited by the three-dimensional plasmon
when v05vp,3Dandk052klaser. Because the group veloc-
ity of the two-dimensional pump ~and idler !plasmon is low
the phase velocity of the excited low-frequency plasmonshould be considerably lower than the sound velocity of thelow-frequency plasmon. Therefore, any excited low-frequency plasmon must have a significant correction in the
real frequency caused by the linear momentum relaxation
rate. This can only happen if the relaxation rate is not too farremoved from the lowest plasmon frequency. The sharpnessof the resonance and the location of the triplet on the high-and low-frequency plasmon dispersion curves are illustratedin Figs. 1 and 2, respectively. As shown in Fig. 3 the tera-hertz frequency is tunable throughout a wide range by vary-ing the total carrier density and ~more important !the impu-
rity level, such that the momentum relaxation rate itself isvaried through the dependence of the mean free path on thetotal carrier density. Tunability could thus be achieved in
FIG. 3. Dependence of the terahertz frequency on the ~total!carrier density for three different impurity contents including the choice for
Figs. 1 and 2: ~a!T577 K, ~b!T5300 K. Because the mobilities are assumed to be dominated by impurities they depend on carrier density
through the Fermi velocity and the mean free path. Therefore we assume m}1/An. At a reference total carrier density of 1011cm22the
values of the mobility were taken to be 30000 ~solid squares !, 40000 ~solid circles !, and 50000 ~upward triangles !cm2/(Vs). All other
parameters are the same as in Figs. 1 and 2. Also, ~c!its dependence on the mobility for ntotal5331012cm22forT577 K and T5300 K,
and the same parameter setting. Finally, ~d!the dependence on mobility of the required three-dimensional carrier density for the same
parameter setting.7448 PRB 62 J. P. MONDT, HYUN-TAK KIM, AND KWANG-YONG KANGprinciple by varying the doping level in the Q2DEG from
one segment to another, using the total carrier density ~gate
voltage !for fine tuning. That the frequency of the excited
plasmon could be in the terahertz regime and that the relax-
ation rate satisfies the conditions v/vc!1!v1/vc, albeit
for the former inequality only marginally, is illustrated in
Figs. 3 and 5. The parameters t[T/TFandks/kFare small
as required ~Fig. 4 !. Furthermore, these conditions are met
while the illuminated area is within about one inverse pumpwave number from the Q2DEG andwhile the grating is well
within one inverse terahertz plasmon wave number from theQ2DEG @see Figs. 4 ~a!,4~b!, and 5 ~b!#. The pump wave
number in the Q2DEG is limited by the energy gap of thethree-dimensional medium, for which in the present numeri-cal results, we assumed a value not less than that of InAs,which is given by
18
Eg50.415 22.7631024T2
T183eV, ~53!
whereTis in K (0 ,T,300). We assume here that the en-
ergy gap narrowing due to high doping levels is such that a
value for the laser pump wavelength equal to 4.2 mm
(’0.28 eV !is still allowable. Finally, we note that the re-
sults for pure InSb are limited to 77 K because of the laser
wavelength of 8 mm and the low-energy gap of InSb.
VIII. CONCLUDING REMARKS
Three-wave interaction between three plasmons in the
same subband of a weakly coupled, quasi-two-dimensional,Fermi gas of spin 1/2 was shown to be possible in principleacross regimes of different collisionality. In three-dimensional plasmas three-wave interaction among electronplasmons is known to be impossible. The nonlinear evolutionhas been quantified through the derivation of the mode-coupling equations using a fluid approach to the convectivenonlinearities. It was shown that SRS against plasmons atpeak power ~backscatter !through the use of a nearby three-
dimensional semiconducting medium of a moderately nar-row gap and a gradient in the charge carriers, or, alterna-tively, two parallel segments of different carrierconcentration, could in principle provide plasmons with thecorrect wave numbers to generate a Q2DEG plasmon in theterahertz range.
One open question is the output power that is maximally
obtainable in the three-dimensional SRS process, as this out-put power is input into the presently proposed three-waveinteraction within the Q2DEG. The terahertz output powerdepends quadratically on it. In this work only the relationbetween input and output power within the quasi-two-dimensional plasma has been quantified. Therefore, the ob-tainable power levels are an open question depending on theexperimentally achievable three-dimensional density fluctua-tion level of coherent longitudinal excitations. Given such alevel the present theory predicts terahertz output through fre-quency difference generation. For the purpose of generatinga suitable pump plasmon in the Q2DEG it is not essentialthat the SRS in the nearby three-dimensional medium beagainst plasmons, the only requirement being that the excitedlongitudinal mode must correspond to a solution of thequasi-two-dimensional dispersion equation for high-frequency plasmons. However, because power transfer from
the three-dimensional medium to the Q2DEG occurs throughresonance, a sharp peak in the spectrum of the particular SRSprocess used for this purpose would be an advantage.
Another open question is the control over the three-
dimensional plasma frequency. Because the three-dimensional plasma frequency depends on the effective massand infrared refractive index, which in turn depend on theabundance of optically excited carriers ~of effective mass
;1) and on the total carrier density, hence on the laser
power, it is a rather complicated task to theoretically predict
FIG. 4. For the parameter setting of Fig. 3 and for a reference
mobility corresponding to the lowest curve in Fig. 3(50000 cm
2/Vs) ~actualmobility depends on total carrier density
}1/Antotalthe following parameters are plotted: rs[kTF/2kF,t
[T/TF, the ratios of the frequencies of the low-frequency plasmon
divided by the collision frequency ( v/nc), the collision frequency
divided by the frequency of the idler plasmon ( nc/v1), and the
thickness of the insulator between the Q2DEG and the grating interms of the wave number of the low-frequency plasmon ( kd
ins):
~a!forT577 K; ~b!forT5300 K.PRB 62 7449 THREE-WAVE INTERACTION AMONG PLASMONS IN A . . .conditions under which the three-dimensional plasma fre-
quency will equal the quasi-two-dimensional pump fre-quency of a given wave number. The problem is aggravatedby uncertainties about the lifetime of carriers optically ex-cited into the conduction band of low-gap semiconductors,particularly in relation to the plasma period. Without anyenhancement of the product of effective mass and permittiv-ity the carrier density required for the conditions pertaining
to Fig. 3 ~c!is in the 10
16–1017range; see Fig. 3 ~d!.
The essence of the present finding of three-plasmon inter-action depends on the shift in real frequency caused by the
linear momentum relaxation rate ncincorporated through a
BGK model. This shift is due to the existence of a velocitydrag and the absence of a corresponding dissipative processin the equation of continuity. Such a shift is possible, but ofa different physical origin ~ionization recombination !and
hence would always be an independent physical quantity.The above-mentioned shift results in a drastic phase velocityreduction of the low-frequency plasmon enabling a reso-
nance for low wave number ( k!k
s) and with pump and idler
wave numbers down to a finite fraction of the effectivescreening wave number of the low-frequency plasmon,which is the screening wave number calculated with the av-erage of the semiconductor and insulator dielectric constants.Because the phase velocity of the excited low-frequencyplasmon is approximately equal to the group velocity of thehigh-frequency plasmons, retardation effects are negligible,provided this group velocity, typically between one and sev-eral times the Fermi velocity, is small compared with thevelocity of light, which is assumed here and which is true forthe quantified examples. Frequency difference generation ispredicted to result in a saturated amplitude of the low-frequency plasmon after approximately 1 or 2 ps and at apower level quantified by Fig. 8, which information might beused for an experimental test.
ACKNOWLEDGMENT
It is a pleasure to acknowledge Dr. Bun Lee for stimulat-
ing discussions and logistical support.
APPENDIX
We discuss the conditions under which the integral repre-
sentation @Eq.~12!#of the high-frequency plasmon disper-
FIG. 5. InSb as both the Q2DEG and the 3D medium for the
same parameter setting as Fig. 3 except for the 3D refractive indexn
r,3D54.0, and the vacuum laser wavelength llaser,vacuum
58mm:~a!dependence of the ~linear !terahertz frequency on total
carrier density for three values of the reference mobility @m(ntotal
51012cm22)580000, 100000, and 120000 cm2/Vs from top to
bottom #;~b!parameters rs,t[T/TF,v/vc,vc/v1,kdinsversus
the total carrier density for a reference mobility m(ntotal
51011cm22)5100000 cm2/V s. These results are valid for T
577 K but not extendable to 300 K because of the low-energy gap,
given the present relatively short vacuum laser wavelength of8
mm.
FIG. 6. Comparison between the lowest-order analytical ap-
proximation for the high-frequency plasmon dispersion equation@Eqs. ~A8!#and the integral equation @Eq.~12!#, forr
s50.16 and
t[T/TF50.16. All other data have been obtained by computing
therealroot vrtotherealpartofthepermittivity egivenbythefull
integral equation @Eq.~12!#, and approximating the imaginary part
of the frequency by g’2Im(e)/@]vrRe(e)#.7450 PRB 62 J. P. MONDT, HYUN-TAK KIM, AND KWANG-YONG KANGsion equation may be approximated through an expansion for
low electron temperature. The kernel in the integral represen-
tation depends on the integration variable through vm8[(2m8/m)1/2. The relevant nontrivial integrals to be per-
formed are
I6[kTF
k1
4vq~v622k2vm2!1/2J6, ~A1!
with the definition
J6[E
(2bm/2)‘ ~12a6x!1/2
cosh2xdx, ~A2!
where we defined a6[(2T/TF)/(v˜f,6221), an inverse mea-
sure of the closeness of the ~Doppler-shifted !phase veloci-ties and the Fermi velocity in units of the thermal velocity.
The real ( J6,r) and imaginary ( J6,i) parts are equal to
J6,r51
a6E
(2a6bm/2)1 ~12x!1/2
cosh2~x/a6!dx, ~A3!
and
J6,i5a621E
1‘~x21!1/2dx
cosh2~x/a6!. ~A4!
Provided a6!1, cosh22(x/a6)’0 unlessx!1. Only then is
the imaginary part exponentially small. For the full integralequation no a prioriassumption is necessary, since Landau
damping manifests itself through the disappearance of a realroot to the real part of the permittivity, while if the solution
can be obtained
g’2Im(e)/]vRe(e). In the expression for
the real part of J6the square root in the numerator can then
be expanded for small x!1. Keeping only the first three
terms and partially integrating the second and third term wefind
J
r51
2tanh1
a611
2S211
a6221DtanhTF
T
1a6Slncosh1
a62lncoshTF
2TD
11
4a62FE
0‘
djj2
cosh2~x!
2~a6221a62111/2!e22/a6
2S1
4TF2/T211
2TF/T11
2DeTF/TG. ~A5!
FIG. 7. Nonlinear evolution: ~a!Time dependence of the phase
Fˆ, essentially the phase difference between the normal modes, in
Eqs. ~44!–~47!for various initial conditions, demonstrating subpi-
cosecond relaxation to zero. Parameters: v0150.579 31012s21,
v025v1258.5131012s21,vc55.0931012s21,u0m50.03, nd
52.531012s21. The low-frequency plasmon amplitude saturates
to the same value at a much later time ~1–2ps !:~b!evolution of the
normal mode of the low-frequency plasmon for the same case as in~a!.
FIG. 8. Time-asymptotic ratio of the density fluctuations of the
terahertz plasmon and the high-frequency plasmons in percent, as afunction of both the density fluctuation relative to background den-sity of the high-frequency plasmon ~in percent !and the character-
istic rate
gSRS,3Dof the three-dimensional SRS ~in units 1012s21).
Parameters as in Fig. 3 with the actual mobility equal to10047 cm
2/Vs, and a density of 3 31012cm22. Other ~depen-
dent!parameters are v0155.7931012s21;v025v1251012s21;
nc52.8731012s21.PRB 62 7451 THREE-WAVE INTERACTION AMONG PLASMONS IN A . . .Here, the only remaining integral is equal to 0.8225. The
degree of smallness of the imaginary part follows from the
substitution cosh( j)’euju/2 under the same restriction:
Ji5A2pa6exp~22/a6!, ~A6!
which is exponentially small when a6!1, i.e., when the
difference between the phase velocity Doppler-shifted by
6vq/2 and the Fermi velocity is small compared with the
thermal velocity. In conclusion, for a6!1 and to within an
error of order a62/4 the integrals J6may be approximated by
the above expressions. The high-frequency plasmon disper-
sion equation reduces to its zero-temperature limit when in
additionT/TF!1, since then J652. Therelativeerrors in
the integrals J6each are therefore half of the absolute values
given above. The condition a6!1 reflects the importance of
finite temperature effects upon waves with a phase velocitythat differs from the Fermi velocity by about the thermalvelocity or less. With this approximation the permittivity be-comes
e511kTF
kS11Av˜22212Av˜1221
k/kFD, ~A7!
v5A~11d!vsc, ~A8!
where
vsc[~kTFk1k2!vm
~2kTFk1k2!1/2~A9!
is the semi-classical limit ( k/kF#0) of the eigenfrequency
and where
d[Sk2
2kTFkFD2
~112kTF/k!. ~A10!Because the errors are additive the combined relative error in
the numerator involving the square roots in the above expres-
sion for the permittivity equals a2/4, where aby definition is
the maximum of the a6.
The above-derived simplified dispersion relation exhibits
the well-known23first-order Taylor expansion in k/kTF, i.e.,
v~k!kTF!’vsc’S113
4k
kTFDA2kkTFvm.~A11!
Even in the low-temperature regime it is only possible to
satisfy both the real and imaginary parts of the dispersionequation for purely oscillatory waves when both Doppler-shifted phase velocities exceed the Fermi velocity in magni-tude and provided the expression obtained by squaring both
sides of the dispersion equation
e50 as given by Eq. ~A7!
allows for non-negative values of the product of the twosquare roots occurring in it. The latter condition is, in terms
of
v˜f[v/(kvm):
v˜f2>111
2Sk2
kTFkFD2S11kTF
kD2
21
4Sk
kFD2
.~A12!
In terms of wave number the above condition is met if and
only if
k312kTFk2<2kTF2kF. ~A13!
A quantitative comparison between the solutions of the
integral form of the dispersion equation and even the lowest-
order approximation in T/TFgiven above shows agreement
~see Fig. 6 !within a few percent for the frequency and even
better agreement for the ~more relevant !group velocity up to
the wave number at which the integral equation fails to havea solution altogether due to Landau damping. All resultsother than Fig. 6 given here are obtained from the full inte-gral dispersion equation.
*Email: jpmondt@yahoo.com. Corresponding author: 2323B 38
Street, Los Alamos, NM 87544.
†Email: kimht45@hotmail.com
‡Email: kykang@etri.re.kr
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PhysRevB.93.035435.pdf | PHYSICAL REVIEW B 93, 035435 (2016)
Resonant second-harmonic generation in a ballistic graphene transistor with an ac-driven gate
Y . Korniyenko, O. Shevtsov, and T. L ¨ofwander
Department of Microtechnology and Nanoscience, MC2, Chalmers University of Technology, SE-412 96 G ¨oteborg, Sweden
(Received 10 June 2015; revised manuscript received 4 January 2016; published 20 January 2016)
We report a theoretical study of time-dependent transport in a ballistic graphene field effect transistor. We
develop a model based on Floquet theory describing Dirac electron transmission through a harmonically drivenpotential barrier. Photon-assisted tunneling results in excitation of quasibound states at the barrier. Underresonance conditions, the excitation of the quasibound states leads to promotion of higher-order sidebandsand, in particular, an enhanced second harmonic of the source-drain conductance. The resonances in the maintransmission channel are of the Fano form, while they are of the Breit-Wigner form for sidebands. For weakac drive strength Z
1, the dynamic Stark shift scales as Z4
1, while the resonance broadens as Z2
1. We discuss the
possibility of utilizing the resonances in prospective ballistic high-frequency devices, in particular frequencydoublers operating at high frequencies and low temperatures.
DOI: 10.1103/PhysRevB.93.035435
I. INTRODUCTION
Already in the early years of graphene research, analog
high-frequency electronics was recognized as a potential nichefor applications [ 1–4]. Although many devices have probably
been limited by parasitics due to problems with developinggood recipes for making graphene transistors, the current speedrecord [ 5] is already a cutoff frequency of over 400 GHz. At
the same time, we have seen a rapid improvement of graphene
material quality. Mobilities reaching 10
5cm2/V s at room
temperature and larger than 106cm2/V s at low temperature
have been achieved [ 4]. Promising paths towards improved
mobility include encapsulation of graphene between layers ofother two-dimensional (2D) crystals, notably hexagonal boronnitride, or suspension of graphene between contacts. Using the
latter approach, ultrahigh-quality p-njunctions were recently
made [ 6]. Fabry-Perot resonances at zero magnetic field were
measured, and so-called snake states were possible to seeat small magnetic fields of order 20 mT. With such rapidimprovements of device quality, it has become increasinglyimportant to study in detail ballistic high-frequency devices.
One of the key ideas behind using 2D materials for
high-frequency electronics is the favorable scaling towardsshort gate lengths without so-called short-channel effects [ 1].
Thin channels (2D is the extreme) allow for short gates, highspeed, and high-density integration. High speed, reaching THzfrequencies [ 7], is the ultimate goal. Another advantage of
graphene is the possibility of tuning the electron density, forinstance by means of a back gate: the Fermi energy can betuned from the electron to the hole band (through the so-calledDirac point at charge neutrality). Such ambipolarity is veryadvantageous, in that both n-type and p-type devices can in
principle be made at will across a single wafer.
A challenge is to capitalize on the unique properties of
graphene and derive device functionality directly from thefact that electrons in graphene behave like massless Diracparticles with linear spectrum and a pseudospin degree offreedom. Several works in this direction show indeed thatac transport in graphene is a rich subject. Studies includequantum pumping [ 8–12], nonlinear electromagnetic response
[13–18], and photon-assisted tunneling phenomena [ 19–25].
In theoretical investigations for low doping (Fermi energyE
Fclose to the Dirac point) and high frequencies /Omega1, withEFand/Omega1of comparable magnitude (we put /planckover2pi1=1), a
true quantum mechanical description becomes necessary. Fortime-dependent transport in two-dimensional electron gases(2DEGs) in semiconducting heterostructures, displaying aquadratic dispersion relation, photon-assisted tunneling intime-harmonic potentials is described well within a Floquettheory framework and have been investigated for a longtime [ 26–29]. Here, we study theoretically a ballistic field
effect transistor with a harmonic drive applied to the topgate, see Fig. 1(a), within a Floquet theory applicable to
graphene. The harmonic drive of frequency /Omega1supports
inelastic scattering from the Fermi energy E
F, to sideband
energies En=EF+n/Omega1, where nis an integer. Near charge
neutrality, on the scale of the drive frequency, the barrieris close to transparent due to Klein tunneling. At the sametime, a quasibound state on the barrier can be inelasticallyexcited through a resonant process (supported by the harmonicdrive) that interferes with direct elastic transmission. Thisleads to a Fano resonance in direct transmission, as recentlyfound numerically [ 23–25]. Analogous Fano resonances were
earlier studied [ 26] for few-channel waveguides in 2DEGs with
attractive point scatterers (for a review of Fano resonances innanostructures, see Ref. [ 30]). For graphene, the ambipolar
band structure guarantees the existence of a bound stateand resonances for both attractive and repulsive scatterers ofarbitrary strength [see Eq. ( B9) in Appendix B]. In this paper,
we show that higher-order sidebands are resonantly enhancedsimultaneously as the Fano resonance in direct transmissiondevelops. This leads to the possibility of building a frequencydoubler based on a ballistic graphene device that we study indetail.
The paper is organized as follows. In Sec. IIwe specify
the model assumptions and outline the main steps in thecalculation. In Sec. IIIwe present results for the electron
transmission functions for inelastic scattering at the barrierfrom incidence energies Eto outgoing sideband energies E
n.
We also analyze in detail the above mentioned resonances. InSec. IVwe present results for the dc and ac conductances,
with a focus on the second ac harmonic that can be resonantlyenhanced. In Sec. Vwe discuss in detail the dynamic Stark
shift (resonance shift) and how it scales with the drive strength.In Sec. VIwe summarize the paper. Finally, the appendices
2469-9950/2016/93(3)/035435(12) 035435-1 ©2016 American Physical SocietyY . KORNIYENKO, O. SHEVTSOV , AND T. L ¨OFWANDER PHYSICAL REVIEW B 93, 035435 (2016)
BGTG
D S
insulatorgraphene
VBGTGV (t)Vbias(a)
(b)
EFEb
(2)E
(1)(4)
(5) E1
E0E2S D
(3)
FIG. 1. (a) A graphene field effect transistor, where the overall
doping level is controlled by a back gate (BG), and the source
(S)–drain (D) current is controlled by the top gate (TG) dc and ac
signals. (b) The harmonic ac signal of frequency /Omega1leads to inelastic
scattering that under resonance conditions excites an otherwise
unoccupied bound state in the top gate barrier potential at energyE
b. This leads to a Fano resonance in the transmission to E0due
to interference between processes (1) and (2) and a Breit-Wigner
resonance in the transmission to E2[process (4)]. Process (4) leads
to higher-harmonic generation, in particular the 2 /Omega1harmonic.
contain all details of the derivation outlined in Sec. IIand all
formulas used to obtain the results in Secs. IIIandIV.
II. MODEL
We are interested in the intrinsic properties of the graphene
transistor in Fig. 1(a), and neglect parasitics. This allows us to
make a minimal model in terms of a Dirac Hamiltonian
H=−iσx∇x+σyky+[Z0+Z1cos(/Omega1t)]δ(x),(1)
where we have set the Fermi velocity in graphene equal to
unity,vF=1. The Pauli matrices are as usual denoted σxand
σy. The top gate barrier potential is considered smooth on the
atomic scale and cannot induce scattering between the twovalleys in the band structure. In the end all observables willcontain an extra factor of 2 to account for valley degeneracy,in addition to spin degeneracy. At the same time, on the Diraclength scale (given by /planckover2pi1v
F/EFafter reinstating /planckover2pi1andvF),
we consider the potential width Dto be small but its height
Vto be large, such that we can take the limits D→0 and
V→∞ keeping the product VD=Zconstant. The strengths
of the time-independent component Z0and the time-dependent
component Z1can be different. The δfunction in Eq. ( 1)i s
therefore smooth on the atomic scale but sharp on the Diraclength scale. We consider the barrier to be translationallyinvariant along the transverse direction, which guarantees thatthe corresponding wave vector component k
yis conserved.
The spatial dependence then enters through the coordinate x
perpendicular to the barrier. We assume homogeneous dopingof the graphene sheet tuned by the back gate and that source
and drain contacts are sufficiently far away from the barrierthat evanescent waves from the contacts can be neglected; i.e.,we avoid in this paper the so-called pseudodiffusive limit [ 31].
The methodology to solve the problem at hand is to first
solve the scattering problem for the wave functions satisfyingthe Dirac equation
Hψ(x,k
y,t)=i∂tψ(x,ky,t). (2)
The solution can be collected into a unitary scattering matrix
for reflection and transmission coefficients between incomingwaves at energy Eand scattered waves at energies E
n=
E+n/Omega1. For the current, the Landauer-B ¨uttiker approach [ 32]
is used to compute the current operator in terms of creationand annihilation operators for incoming and scattered waves,where the latter are related to the former through the scatteringmatrix. A statistical average is performed to obtain the time-dependent current that depends on the occupation factors of thesource and drain leads, which are given by the Fermi functionwith chemical potentials shifted by the applied source-drainvoltage eV. Below, we shall give results for the conductance
in linear response to the applied source-drain voltage at zerotemperature.
For the scattering problem, since the Hamiltonian is
periodic in time, we use a general Floquet ansatz
ψ(x,k
y,t)=/summationdisplay
nψn(x,ky,E)e x p (−iEnt). (3)
When plugged into Eq. ( 2) it yields a set of coupled differ-
ential equations for the (formally infinitely many) sidebandamplitudes ψ
n(x,ky,E). In the following we do not write
the arguments x,ky, and Ein order to keep the notation
compact. The sideband amplitudes can be arranged into avector /Phi1=[..., ψ
−1,ψ0,ψ1,...]T, which then satisfies
∇x/Phi1=ˇMtd/Phi1, (4)
where
ˇMtd=[kyσz+iEnσx−iZ0σxδ(x)]⊗ˇ1−iZ1
2δ(x)σx⊗ˇ2.
(5)
Here, ˇMtdis a tridiagonal matrix in sideband space with
(ˇ1)nm=δnmand ( ˇ2)nm=δn,m+1+δn,m−1. After integration
overx=0 we obtain a boundary condition
/Phi1(x=0−)=exp/bracketleftbigg
iZ0σx⊗ˇ1+iZ1
2σx⊗ˇ2/bracketrightbigg
/Phi1(x=0+)
≡ˇM/Phi1(x=0+). (6)
This boundary condition can also be derived by solving a
square barrier problem first, and in the end let D→0 andV→
∞keeping the product VD=Zconstant; see also Ref. [ 33].
This boundary condition gives an elegant view of scatteringoff a potential in graphene in terms of pseudospin rotation.For instance, the static barrier leads to a rotation around thepseudospin xaxis by an angle −2Z
0. If the pseudospin is
aligned with σx, i.e., electron propagation along the xaxis
with perpendicular incidence, the rotation has no effect (Kleintunneling [ 34]). For other angles, transmission is nonperfect.
The formulas for the transmission amplitudes t
n(ky,E) derived
from Eq. ( 6) are given in Eq. ( B14).
035435-2RESONANT SECOND-HARMONIC GENERATION IN A . . . PHYSICAL REVIEW B 93, 035435 (2016)
(a)
(b) T (E φ, ) 2T (E φ, ) 0
FIG. 2. Energy and incidence angle dependence of transmission
probabilities for (a) elastic scattering T0(E,ϕ) and (b) inelastic
scattering between energy EandE+2/Omega1,T2(E,ϕ). The black regions
to the left of the white dashed lines in (b) are regions where the
second-sideband wave functions are evanescent waves decaying away
from the barrier. The barrier strengths are Z0=0.4πandZ1=0.45.
Inset: Transmission probabilities for fixed ϕ=π/9.
III. TRANSMISSION AMPLITUDES
In Fig. 2we display the transmission probabilities
Tn(E,ϕ)=|tn(E,ϕ)|2forn=0 and n=2, where Tn(E,ϕ)
denotes incidence on the barrier at energy Eand transmission
at sideband energy En, keeping the parallel momentum ky=
|E|sinϕconserved (the angle ϕis measured relative to the
barrier normal). In the main transmission channel T0(E,ϕ),
Klein tunneling is apparent in that the transmission is veryclose to unity. Deviation from unity transmission is due to thestatic barrier of strength Z
0and finite incidence angle (nonzero
ϕ) and, in addition, scattering to other sidebands with n/negationslash=0.
The transmission probability to the second sideband T2(E,ϕ)
is in general very small. For certain energies there are Fanoresonances [ 23–25] induced by the time-dependent drive and
a bound state at the barrier, which give rise to a peak-dipstructure dispersing with ϕ, one feature at positive energies and
another one at negative energies. The Fano resonances occur ina parameter range where the outgoing waves (from the barrier)at a sideband E
nare evanescent (below we shall concentrate
onn=±1, which are the most pronounced resonances in
Fig. 2). This happens when inelastic scattering from E2=
kx(E)2+k2
ytoE2
n=kx(En)2+k2
y(with conserved ky) causes
kx(En) to become imaginary. Resonant behavior occurs due
to the existence of a bound state on the barrier at energyE
b(Z0,ky)=−sgn(Z0)|ky|cosZ0[cf. Eq. ( B9)] that can be
excited by the ac drive (in which case it becomes quasibound).The Fano resonance at Er=Eb±/Omega1is a quantum mechanical
interference between direct elastic tunneling and a tunnelingprocess involving excitation to the first sideband (for n=±1
atE
r=Eb∓/Omega1) and deexcitation back to energy E; see paths
(1) and (2) in the diagram in Fig. 1(b) for the E< 0 case.
On resonance, inelastic tunneling to the second sideband isenhanced and T
±2(E,ϕ) display Breit-Wigner resonance peaks
atEr=Eb∓/Omega1; see Fig. 2(b). The resonance in T2(E,ϕ) can
be viewed as due to transmission in energy space througha double barrier structure with barrier heights proportionaltoZ
1.
To extract more information about the above numerical
results, we proceed with an analytic approximation. We canexpand the boundary condition in Eq. ( 6) to second order in
the ac drive strength Z
1, assuming Z1/lessmuch1,
ˇM≈eiZ0σx/bracketleftbigg
ˇ1+iZ1
2σx⊗ˇ2−Z2
1
8(2·ˇ1+ˇ3)/bracketrightbigg
, (7)
where ( ˇ3)nm=δn,m+2+δn,m−2in sideband space. To second
order in Z1, the transmissions to the first two sidebands can
be computed by solving a system of equations for t0,t±1, and
t±2; see Appendix C. We separate two cases: (i) off-resonant
transmission and (ii) on-resonant transmission. For case (i)off-resonant transmission, the equation system can be inverteddirectly and we get (for each ϕ; we suppress the argument ϕ
below for brevity)
t
0(E)≈/bracketleftbigg
1+Z2
1
4+Z2
1t(0)(E)A0,1(E)t(0)(E1)A1,0(E)
+t(0)(E)A0,−1(E)t(0)(E−1)A−1,0(E)/bracketrightbigg
t(0)(E),
t±1(E)≈−Z1t(0)(E±1)A±1,0(E)t(0)(E), (8)
t±2(E)≈Z2
1[t(0)(E±2)A±2,±1(E)t(0)(E±1)A±1,0(E)
−t(0)(E±2)A±2,0(E)]t(0)(E),
where t(0)(En) is the transmission amplitude without ac drive
computed at energy En, andAn,m(E) is a transition amplitude
in energy space between energies EmandEn, which can be
related to off-diagonal matrix elements (in sideband space)of the matrix ˇMin Eq. ( 7). The above expressions make the
inelastic tunneling processes at play explicit, see enumeratedprocesses in Fig. 1(b). For instance, the expression for t
1(E),
read from right to left, has a transparent physical meaning. Itconsists of transmission amplitudes at EandE
1, separated by
a transition in energy space A1,0, corresponding to absorption
of one quantum /Omega1. Consequently, the process is of order Z1.
Direct transmission has corrections to the static transmissionamplitude due to excitation and deexcitation to neighboringsidebands (processes of order Z
2
1), while t2(E) consists of a
direct process of absorbing two quanta, 2 /Omega1, and a sequential
process involving the first-sideband energy; both are of orderZ
2
1. This tells us that the sideband amplitudes are in general
very small when Z1is small.
The above picture changes for case (ii) on-resonant trans-
mission, for energies near Er=Eb±/Omega1[we shall concentrate
onEb−/Omega1in the following discussion, as in Fig. 1(b)]. In
this case, the equation determining the function t(0)(En,ky)a t
energy En=Eb(heren=1) has to be reconsidered. There is a
035435-3Y . KORNIYENKO, O. SHEVTSOV , AND T. L ¨OFWANDER PHYSICAL REVIEW B 93, 035435 (2016)
pole in the matrix equation determining the scattering matrix at
this energy for fixed ky, corresponding to formation of a bound
state with evanescent waves decaying away from the barrier.The bound state is unoccupied (decoupled from reservoirs)in the absence of the ac drive. For case (ii) on-resonanttransmission, we get for energies δEaround the resonance
energy E
r
t0(Er+δE)≈δE−Z2
1h2(Er)
δE+iZ2
1h1(Er)t(0)(Eb), (9)
t2(Er+δE)≈Z2
1h3(Er)
δE+iZ2
1h1(Er)t(0)(Eb+2/Omega1),(10)
where hj(Er)≡hj(Er,ky,Z0),j=1,2,3, are complex-
valued functions that can be read off from Eqs. ( C9) and ( C10)
[their explicit forms are not important in the discussion below,except to note that h
1(Er) is purely real]. Note that t1is not well
defined near resonance (it was eliminated in the calculation)because it is related to the excitation of the bound state. Theconductance computed below will not get contributions fromthis sideband energy [crossed process (3) in Fig. 1(b)]. For
the direct transmission probability T
0(Er+δE), neglecting
for a while the second-sideband contribution (setting h2=0
above), there is a characteristic Fano resonance form T0(Er+
δE)∝(q/Gamma1/2+δE)/[δE2+(/Gamma1/2)2], where /Gamma1∝Z2
1/Omega1andq
is of order unity, O[(Z1)0]. This is the blue dotted line
displayed in the inset of Fig. 2(b). Taking into account
tunneling (in energy space) to the second sideband ( h2finite
above) and higher, we obtain the corrected line shape, theblack solid line in the inset of Fig. 2(b). For the probability to
scatter inelastically to the second sideband, we obtain from theabove a Breit-Wigner resonance with the characteristic formT
2(Er+δE)∝(/Gamma1/2)/[δE2+(/Gamma1/2)2], which is displayed as
the black dashed line in the inset of Fig. 2(b). Thus, in a
rangeδE∝Z2
1/Omega1around Er, the response is highly nonlinear
and higher-order harmonics can be resonantly enhanced, inparticular the second harmonic.
IV . DC AND AC CONDUCTANCES
To quantify the resonant generation of the second harmonic,
we present calculations of the linear conductances Gn, both
the time-averaged component ( n=0) in Fig. 3and the first
two harmonics ( n=1, 2) in Fig. 4. Note that in the linear
response (small source-drain voltage), the source-drain accurrent I=/summationtext
nIne−in/Omega1t, with its harmonics In, naturally
defines ac conductance components Gn; see Appendix D.I n
Fig. 3(a) we plot the angle resolved conductance G0(EF,ϕ),
which reflects the sum over transmission functions, includingthe ones displayed in Fig. 2. After angle integration, we obtain
the dc linear conductance; see the solid black line in Fig. 3(b).
Due to the ac drive, the dc conductance contains features notpresent in the static case (included as thin green straight lines).The Fano resonance is clearly visible as a peak-dip featureinG
0(EF). Thus, it is enough to study the time-averaged
conductance to infer influence of the ac drive.
In Fig. 4we present the real and imaginary parts of the
first and second harmonics, G1(EF) andG2(EF). For small
drive amplitude Z1, the harmonics generally scale as Zn
1in
perturbation theory and the second harmonic is expected to
G (E Fφ, ) 0
FF 4Ω(a)
(b)
FIG. 3. Zero-temperature source-drain dc linear conductance in
the presence of ac drive of strength Z1=0.45 on the top gate. Upper
panel: Impact angle resolved dc conductance G0(EF,ϕ). Lower panel:
Angle integrated time-average dc conductance G0(EF) (black solid
line). The green straight line is G0(EF) in the static case, Z1=0, for
comparison. The dip-peak structures are related to the Fano and Breit-Wigner resonances in the elastic and inelastic transmission functions.
The static barrier strength is Z
0=0.4π.
be small. Near resonance, however, it is enhanced to order
unity, O[(Z1)0], within a window of doping ∼Z2
1/Omega1around
EF=Er. This results in the second harmonic becoming of the
same magnitude as the first harmonic already for weak driving(Z
1=0.45 in the figure). This leads us to the conclusion that
the device can operate as a frequency doubler.
Parameter regimes
Parameter regimes available experimentally will affect the
possibility of measuring frequency doubling. First we discusstypical gate lengths and drive amplitudes. These two quantitiesare connected by the dimensionless parameter Z
1=V1D//planckover2pi1vF
(reinstating /planckover2pi1andvF). For graphene, vF∼106m/s. For gate
lengths of order 100 nm, we obtain a drive amplitude of V1∼
6Z1meV . For relatively weak driving as considered above,
Z1∼1, the required drive amplitude is a few meV , which is
reasonable [ 35]. If the source is weaker, the gate length has to
be increased. Typical gate lengths used in experiments todayare of order 1 μm, which with the help of nanowires have been
brought down to 170 nm [ 36].
Until today, graphene devices operating at room tem-
perature display frequency doubling in the frequency range1–10 GHz [ 36]. Their operating principle is classical, only
relying on the bipolar gate voltage dependence of the dcconductance [see thin green line in Fig. 3(b)]. With the
Fermi energy at dc operation aligned to the conductanceminimum, for instance by applying a suitable back gatevoltage V
BG, an additional ac signal with frequency /Omega1on
035435-4RESONANT SECOND-HARMONIC GENERATION IN A . . . PHYSICAL REVIEW B 93, 035435 (2016)
F(a)
1 Re( ) F 4Ω
2 Re( ) F 4Ω(c)
F
1 Im( ) F 4Ω(b)
F
(d)
2 Im( ) F 4Ω
F
FIG. 4. Zero-temperature source-drain linear conductances for sideband currents with n=±1a n dn=±2 in the presence of an ac drive
of strength Z1=0.45 on the top gate. Upper panels: Impact angle resolved average conductances Gn(EF,ϕ). Lower panels: Angle integrated
real and imaginary parts of average conductances Gn(EF). The static barrier strength is Z0=0.4π.
the top gate (locally modulating the Fermi energy) leads
to a source-drain conductance response at frequency 2 /Omega1
simply because G(−EF)=G(EF). For our proposed device,
on the other hand, the operating principle is based on aquantum mechanical resonance effect. For weak driving, lowtemperature is therefore needed. For Z
1/lessmuch1, the resonance
width scales as Z2
1/planckover2pi1/Omega1. To avoid broadening, we must require
small temperature kBT/lessmuchZ2
1/planckover2pi1/Omega1. For a frequency of order
50 GHz we get T∼2Kf o r Z1∼1. For stronger drive, this
constraint is less restrictive. It remains to be experimentallyexplored whether also room temperature operation and higherfrequency, 100 GHz to 1 THz, are possible to achieve forstronger driving in a ballistic device. We note that another
quantum effect, the quantum Hall effect, has been observed atroom temperature in graphene devices [ 37].
V . DYNAMIC STARK SHIFT
As we have seen above, in the presence of the ac drive,
the barrier-induced bound state is excited and appears asa quasibound state, which is broadened as well as shiftedin energy with respect to the original bound state. Thisdynamic Stark shift can be investigated quantitatively bycomputing the determinant of the matrix M
sdefined in
035435-5Y . KORNIYENKO, O. SHEVTSOV , AND T. L ¨OFWANDER PHYSICAL REVIEW B 93, 035435 (2016)
FIG. 5. Dynamic Stark shift as illustrated by plotting det[ Ms]a s
a function of complex energy E, where the matrix Msis defined in
Eq. ( B16). The points Ezeroin the complex energy plane where this
function vanishes coincide with resonances in the Floquet scattering
matrix. (a) Without the ac drive, Z1=0, there is a true bound state
and the determinant vanishes at Re[ E]=Eband Im[ E]=0. (b) and
(c) Including the ac drive, Z1>0, the minimum moves out into the
complex plane. The dashed lines indicate the position of the boundstate in the absence of the ac drive. The static barrier strength is
Z
0=0.4πand the impact angle is ϕ=π/9.
Eq. ( B16). In the absence of driving, the matrix is diagonal
with the nth matrix element given by D(ky,En); cf. Eq. ( B4).
The determinant therefore vanishes, det[ Ms]=0, if one of
the sideband energies equals the bound state energy Eb,
since D(ky,En=Eb)=0 as explained in Appendix B2.
Including drive, Z1>0, the zeros of the determinant move
into the complex energy plane [ 26,38],Eb→Ezero, signaling
a quasibound state. The resonance energy Er=Re(Ezero)i s
then shifted from Eb. The resonance width can be related to
Im(Ezero). This effect is illustrated in Fig. 5,w h e r ew ep l o t
det[Ms] as function of complex energy. In Fig. 6we plot
the resonance energy shift and the broadening as a functionof drive strength. For weak drive, the shift is proportionaltoZ
4
1/Omega1, while the broadening scales as Z2
1/Omega1. For stronger
drive, higher-order contributions lead to a deviation from thisscaling. The resonance shift could be mapped out directly inan experiment, since the Fermi energy of graphene is tunableby the back gate.
VI. SUMMARY
In summary, we have investigated time-dependent transport
in a ballistic graphene field effect transistor with ac drive on itstop gate. We find resonances in inelastic scattering to sidebandenergies related to excitation of a quasibound state in the topgate barrier. This leads to substantial resonant enhancement ofthe second harmonic of the source-drain conductance, thatcould possibly be used in developing a frequency doublerbased on a ballistic device.
ACKNOWLEDGMENTS
It is a pleasure to thank V . S. Shumeiko and J. Stenarson for
valuable discussions. We acknowledge financial support fromthe Swedish Foundation for Strategic Reseach (SSF) and theKnut and Alice Wallenberg Foundation (KAW).FIG. 6. (a) The dynamic Stark shift scales with the drive strength
asZ4
1for small Z1. (b) The broadening scales as Z2
1. The static barrier
strength is Z0=0.4πand the impact angle is ϕ=π/9.
APPENDIX A: WA VE SOLUTIONS IN GRAPHENE
1. General solution
We start by introducing general wave solutions in graphene
without time-dependent perturbation. They are known (see,e.g., Refs. [ 39,40]) and we write them down here to establish a
coherent notation for subsequent sections. As mentioned in themain text, we consider only one valley (one Kpoint) described
by the Hamiltonian
H
0=−iσ·∇,σ=(σx,σy). (A1)
We have to solve the Dirac equation i∂tψ(x,y,t )=
H0ψ(x,y,t ), which is done by the standard ansatz
ψ(x,y,t )∝eikxxeikyye−iEtψ(kx,ky,E). (A2)
We obtain the following eigenvalues and eigensolutions,
Eλ(kx,ky)=λ/radicalBig
k2x+k2y,λ=±1, (A3)
ψλ(kx,ky,E)=1√
2/parenleftbigg1
kx+iky
Eλ/parenrightbigg
. (A4)
2. Scattering basis
Note that once we have found the spectrum, Eq. ( A3), there
are only two independent parameters labeling eigenstates, e.g.,(k
x,ky)o r(ky,E). Since we are going to build a scattering
theory following B ¨uttiker [ 32,41] the latter choice is natural
because we assume translational invariance along the barrier(yaxis); cf. Eq. ( 1). In order to introduce the scattering basis we
have to find the group velocity of states propagating along thexaxis (perpendicular to the barrier). Using standard definitions
035435-6RESONANT SECOND-HARMONIC GENERATION IN A . . . PHYSICAL REVIEW B 93, 035435 (2016)
we have
u(ky,E)=∂E
∂kx=±v(ky,E),
v(ky,E)=κx(ky,E)
E, (A5)
κx(ky,E)=sgn(E)/radicalBig
E2−k2y,
where the upper and lower signs describe particles moving in
the positive and negative directions along x, respectively. Then
we can introduce a scattering basis via
ψ→(x,ky,E)=1/radicalbig2v(ky,E)/parenleftbigg
1
η(ky,E)/parenrightbigg
eiκx(ky,E)x,
ψ←(x,ky,E)=1/radicalbig2v(ky,E)/parenleftbigg
1
¯η(ky,E)/parenrightbigg
e−iκx(ky,E)x,
(A6)
η(ky,E)=κx(ky,E)+iky
E,
¯η(ky,E)=−κx(ky,E)+iky
E,
where arrows indicate the direction of propagation. The
normalization in Eq. ( A6) is chosen such that particles carry
unit probability flux along the xaxis, defined as
jx(x,ky,E)=ψ†(x,ky,E)σxψ(x,ky,E). (A7)
This basis is used to find a scattering matrix and build the
scattering field theory below.
APPENDIX B: FLOQUET SCATTERING MATRIX
IN GRAPHENE WITH AC δPOTENTIAL
Let us now discuss the Floquet scattering matrix for
graphene in the presence of an oscillating line scatterer; i.e.,we consider a system described by [cf. Eq. ( 1)]
H=H
0+[Z0+Z1cos(/Omega1t)]δ(x). (B1)
Before discussing the solution associated with the full time-
dependent Hamiltonian, it is instructive to consider thestationary case, Z
1=0.
1. Static δbarrier
For a static barrier, scattering is elastic, and it is easy to write
down a scattering ansatz, first assuming incoming particlesfrom the left,
ψ(x,k
y,E)=/braceleftBigg
ψ→(x,ky,E)+r(0)ψ←(x,ky,E),x < 0,
t(0)ψ→(x,ky,E),x > 0.
(B2)
The superscript X(0)indicates functions Xcomputed for a
static barrier. The unknown transmission t(0)(ky,E) and reflec-
tionr(0)(ky,E) coefficients are found through the boundary
condition at x=0, which reads [cf. Eq. ( 6)]
ψ(0−,ky,E)=exp[iZ0σx]ψ(0+,ky,E). (B3)
It is straightforward to find a solution to Eq. ( B3), but it is
convenient for what follows to write down an equation satisfiedbyt(0),
D(ky,E)t(0)=1,
D(ky,E)=1
2v(ky,E)/parenleftbig−¯η(ky,E)1/parenrightbig
(B4)
×exp[iZ0σx]/parenleftbigg
1
η(ky,E)/parenrightbigg
.
Using Eq. ( A6) we can easily simplify Eq. ( B4) and obtain
t(0)(ky,E)=D(ky,E)−1=/parenleftbigg
cosZ0+isinZ0
v(ky,E)/parenrightbigg−1
.(B5)
Let us introduce an incidence angle ϕ, measured relative to the
barrier normal, via ky=|E|sinϕ. Equation ( B5) can then be
rewritten as
t(0)(ky,E)=cosϕ
cosϕcosZ0+isinZ0. (B6)
2. Barrier-induced bound state
It is well known [ 42] that poles of the scattering matrix
correspond to bound states. In our case the static δbarrier
induces exactly one bound state as will be shown now. Weequate to zero the denominator of Eq. ( B5), giving
κ
x(ky,E)=−iEtanZ0. (B7)
Equation ( B7) is periodic in Z0and we consider for defi-
niteness −π
2<Z 0<π
2. We then impose a condition that the
bound state solution has to be decaying away from the barrier,which means
κ
x(ky,E)=i/radicalBig
k2y−E2. (B8)
Collecting the above, the energy of the bound state is given by
Eb=−sgn(Z0)|ky|cosZ0. (B9)
It is interesting to note that the bound state plays no role in dc
transport in our model setup since it is disconnected from thecontinuum of propagating waves connecting the contacts. Thiscircumstance changes as soon as we allow inelastic scatteringon the barrier, when Z
1/negationslash=0.
3. Oscillating δbarrier
In the case when Z1/negationslash=0 the Hamiltonian, Eq. ( B1),
is periodic in time, which enables us to use the Floquettheorem[ 27–29] for finding eigenvectors,
ψ(x,k
y,t)=e−iEt+∞/summationdisplay
n=−∞e−in/Omega1tψn(x,ky,E). (B10)
We organize the (formally infinite) many sideband amplitudes
ψn(x,ky,E) into a column vector
/Phi1(x,ky,E)=⎛
⎜⎜⎜⎜⎜⎝...
ψ
−1(x,ky,E)
ψ0(x,ky,E)
ψ1(x,ky,E)
...⎞
⎟⎟⎟⎟⎟⎠, (B11)
035435-7Y . KORNIYENKO, O. SHEVTSOV , AND T. L ¨OFWANDER PHYSICAL REVIEW B 93, 035435 (2016)
and write down the condition to be satisfied at x=0[ c f .
Eq. ( 6)],
/Phi1(0−,ky,E)=ˇM/Phi1(0+,ky,E),
ˇM=exp/bracketleftbigg
iZ0σx⊗ˇ1+iZ1
2σx⊗ˇ2/bracketrightbigg
,(B12)
[ˇ1]n,m=δn,m,[ˇ2]n,m=δn,m+1+δn,m−1.
Since the barrier is active only at x=0, asymptotic solutions
are still given by a linear combination of the static solutionsin Eq. ( A6). The barrier only scatters an incident particle
with quantum numbers ( E,k
y) into a linear combination of
states with quantum numbers ( En,ky). In the end, for our
model, we have to consider only propagating outgoing waves,E
n>|ky|, for calculating transport properties. Therefore we
use the following ansatz:
ψn(x,ky,E)
=/braceleftBigg
δn,0ψ→(x,ky,En)+rnψ←(x,ky,En),x < 0,
tnψ→(x,ky,En),x > 0.
(B13)
We can eliminate reflection coefficients rn(ky,E) and find a
system of equations for tn(ky,E) only, which reads
/summationdisplay
m1
2/radicalbigv(ky,En)v(ky,Em)(−¯η(ky,En)1 )
×[ˇM]nm/parenleftbigg
1
η(ky,Em)/parenrightbigg
tm=δn,0. (B14)Equation ( B14) must be solved numerically, in principle for
infinite number of sidebands. In practice, the number ofsidebands can be limited to n∈[−N
c,Nc], where the cutoff
Nccan be estimated by expanding Eq. ( B12) and studying the
coupling to sidebands. In this way we find a tolerance measure/epsilon1for the cutoff,
(Z
1/2)Nc
[(Nc/2)!]2</epsilon1 . (B15)
With the sideband cutoff, the above system of equations can
be collected into the matrix form
Ms·t=δn,0, (B16)
where Msis a (2 Nc+1)×(2Nc+1) matrix, tis a vector
of length 2 Nc+1 with the transmission functions tn(ky,E)
forn∈[−Nc,Nc], and δn,0is a vector of length 2 Nc+1
with elements δn,0. The transmission functions are found by
inverting the matrix Ms.
APPENDIX C: ANALYSIS OF SIDEBAND TRANSMISSION
COEFFICIENTS: FANO AND BREIT-WIGNER
RESONANCES
ForZ1/lessmuch1, we expand ˇMup to terms of order O(Z2
1),
and consider five outgoing channels with n={0,±1,±2}.
Then we obtain a system of five coupled equations which reads[omitting the arguments ( k
y,E) for brevity]
/parenleftbigg
1−Z2
1
4/parenrightbigg
D2t2+Z1A2,1t1+Z2
1A2,0t0=0,
Z1A1,2t2+/parenleftbigg
1−Z2
1
4/parenrightbigg
D1t1+Z1A1,0t0+Z2
1A1,−1t−1=0,
Z2
1A0,2t2+Z1A0,1t1+/parenleftbigg
1−Z2
1
4/parenrightbigg
D0t0+Z1A0,−1t−1+Z2
1A0,−2t−2=1, (C1)
Z2
1A−1,1t1+Z1A−1,0t0+/parenleftbigg
1−Z2
1
4/parenrightbigg
D−1t−1+Z1A−1,−2t−2=0,
Z2
1A−2,0t0+Z1A−2,−1t−1+/parenleftbigg
1−Z2
1
4/parenrightbigg
D−2t−2=0,
where we have used the following notations:
Dn(ky,E)=D(ky,En),
An,m=(i/2)|n−m|
|n−m|1
2/radicalbigv(ky,En)v(ky,Em)(−¯η(ky,En)1)
×exp[iZ0σx]σ|n−m|
x/parenleftbigg
1
η(ky,Em)/parenrightbigg
,n/negationslash=m.(C2)
Note that from Eq. ( C2) and Eq. ( B4) it is obvious that
D−1
n(ky,E)≡t(0)(ky,En) provided the corresponding wave
is propagating, i.e., En>|ky|. On the other hand the new
functions An,mhave a meaning of transition matrix between the
sidebands. Now we recall that the presence of a (static) δbarrierimplies the existence of a bound state, see Appendix B2, which
now can be coupled to the propagating waves via inelasticscattering. In this case one of the functions D
n(ky,E) vanishes
when En=Eb. This possibility leads to resonances in the
transmission spectrum of the sidebands (see Fig. 7), as will be
discussed in detail below.
1. Off-resonant transmission
We will first consider the rather trivial case of transmission
in different sidebands away from the resonances. In this casewe can straightforwardly estimate orders of magnitude for thesideband transmission coefficients keeping only contributions
035435-8RESONANT SECOND-HARMONIC GENERATION IN A . . . PHYSICAL REVIEW B 93, 035435 (2016)
T (E φ, ) 2
T (E φ, ) 1
T (E φ, ) -1 T (E φ, ) -2(a)
(c) (d)(b)
FIG. 7. Energy and incidence angle dependence of transmission probabilities for inelastic scattering to the sidebands. The static barrier
strength is Z0=0.4πand the ac drive strength is Z1=0.45.
O(Z2
1),
t0=t(0)
0+Z2
1τ0,
t±1=Z1τ±1, (C3)
t±2=Z2
1τ±2,
where we introduced for convenience t(0)
n(ky,E)≡t(0)(ky,En).
Keeping the same order of approximation in Eq. ( C1) we can
easily solve it with the following results,
τ0=/parenleftbigg1
4+t(0)
0A0,1A1,0
D1+t(0)
0A0,−1A−1,0
D−1/parenrightbigg
t(0)
0,
τ±1=−A±1,0
D±1t(0)
0, (C4)
τ±2=/parenleftbiggA±2,±1
D±2A±1,0
D±1−A±2,0
D±2/parenrightbigg
t(0)
0,
which were also collected into Eq. ( 8). The transmission
coefficients, Eq. ( C3), supplemented by Eq. ( C4) have a physi-
cally transparent form if they describe propagating waves, i.e.,waves with all E
n>|ky|. In this case we can identify D−1
n=
t(0)
n[see Eq. ( B4)] and, reading the resulting expressions from
right to left, we can distinguish the transmission processesdepicted in Fig. 1(b) [except that E
1/negationslash=Ebin the process (3),
according to our assumption].
2. Close-to-resonance transmission
Now we will focus on the resonances associated with the
case when the energy of one of the n=±1 sidebands hitsthe bound state, E±1=Eb, and the corresponding channel
is closed. They are observed as zeros in T±1and maxima
inT±2, dispersing with the incidence angle ϕ(see Fig. 2).
For definiteness we will consider the resonance condition forn=1, but this analysis is straightforward to repeat for n=−1.
The resonance condition reads
D
1(ky,Er)=0. (C5)
We expand the D1coefficient in Eq. ( C1) around the resonance
energy assuming
E=Eb−/Omega1+δE=Er+δE, |δE|/lessmuch{/Omega1,|ky|},
D1(ky,E)≈δE
|ky|sin2Z0. (C6)
Evaluating all other functions in Eqs. ( C1)a tE=Er,w es o l v e
the resulting system of equations keeping only terms of orderO(δE,Z
2
1). The solution for t0andt2reads
t0=δED 2−Z2
1A1,2A2,1|ky|sin2Z0
δED 0D2−Z2
1(D2A0,1A1,0+D0A1,2A2,1)|ky|sin2Z0,
(C7)
t2=Z2
1A2,1A1,0|ky|sin2Z0
δED 0D2−Z2
1(D2A0,1A1,0+D0A1,2A2,1)|ky|sin2Z0.
(C8)
We note that for |ky|→ 0,|t0|2will be close to unity due to
Klein tunneling [ 34] and there is no resonance behavior. If
we consider the case when both the main channel n=0 and
035435-9Y . KORNIYENKO, O. SHEVTSOV , AND T. L ¨OFWANDER PHYSICAL REVIEW B 93, 035435 (2016)
-1.3 -1.2 -1.1 -1.0 -0.9
E/Ω0.00.20.40.60.81.0T
T
Eq. (C9)
Eq. (C10)
FIG. 8. Energy dependence of transmission probabilities T0and
T2for incidence angle ϕ=π/9,Z0=0.4π,a n dZ1=0.45. The blue
dotted line is the result for T0when neglecting scattering to the second
sideband at E2. In this case, the Fano resonance is fully developed
(peak at unit transmission and dip at zero transmission).
the second sideband n=2 are propagating, then Eqs. ( C7)
and ( C8) can be rewritten as
t0=δE−Z2
1A1,2t(0)
2A2,1|ky|sin2Z0
δE−Z2
1/parenleftbig
t(0)
0A0,1A1,0+A1,2t(0)
2A2,1/parenrightbig
|ky|sin2Z0t(0)
0,
(C9)
t2=Z2
1A2,1A1,0t(0)
0|ky|sin2Z0
δE−Z2
1/parenleftbig
t(0)
0A0,1A1,0+A1,2t(0)
2A2,1/parenrightbig
|ky|sin2Z0t(0)
2,
(C10)
with the shorthand notation t(0)
0=t(0)(Er) andt(0)
2=t(0)(Er+
2/Omega1). If we analyze these expressions we can see the following:
(1) if we compute the corresponding transmission proba-
bilities, T0=|t0|2, andT2=|t2|2, we clearly see that T0has a
Fano-type resonance shape, while T2is of Breit-Wigner type
with the width of the resonances ∝Z2
1|ky|sin2Z0;
(2) exactly at the resonance, δE→0, both t0andt2have
finite values independent of Z1due to constructive interference
between the first and the second sidebands.
In Fig. 8we compare the approximate solution we have
found with the exact numerical calculation [see inset ofFig. 2(b) in the main text]. We clearly see that Eqs. ( C9)
and ( C10) correctly describe all the essential features of the
transmission probabilities discussed above.
Finally, in a more strict expansion of all functions in
Eq. ( C1) to linear order in δE, more cumbersome expressions
are obtained, but the above conclusions will not change, asalso supported by the good agreement between the black andred lines in Fig. 8.
APPENDIX D: SCATTERING FIELD THEORY
OF AC CURRENT
In this section we briefly describe the method we used
to compute ac electric current. The theory below is validas soon as a single-particle approach is justified, i.e., whenparticle-particle interactions can be neglected. Without loss ofgenerality we assume particles incident on the barrier from thecontact α[e.g., the source contact; see Fig. 1(a) in the main
text]. Using the scattering basis, Eq. ( A6), found above we
construct a field operator
ˆ/Psi1
α(x,y,t )=/integraldisplay+∞
−∞dky√
2πeikyy/integraldisplay
|E|>|ky|dE√
2π
×e−iEt[ˆγα,in(ky,E)ψ→(x,ky,E)
+ˆγα,out(ky,E)ψ←(x,ky,E)] (D1)
in the local coordinate system of the contact, where
ˆγα,in/out(ky,E) are the corresponding annihilation operators for
the incoming/outgoing particles, which satisfy
{ˆγα,in(ky,E),ˆγ†
β,in(k/prime
y,E/prime)}=δα,βδ(ky−k/prime
y)δ(E−E/prime),
{ˆγα,in(ky,E),ˆγβ,in(k/prime
y,E/prime)}=0, (D2)
{ˆγ†
α,in(ky,E),ˆγ†
β,in(k/prime
y,E/prime)}=0.
According to the scattering theory the outgoing operator
ˆγα,out(ky,E) is, via a scattering matrix, related to the incoming
one. For our case of an ac barrier and static contacts thisrelation reads
ˆγ
α,out(ky,E)=/summationdisplay
β/summationdisplay
n,propag .Sαβ(ky;E,E n)ˆγβ,in(ky,En),(D3)
where we restrict the sum over sidebands to propagating
waves only, which is equivalent to setting the scattering matrixelements to zero if an incoming/outgoing wave is evanescent.Then we construct the current operator defined by the standardexpression [ 41]
ˆI
α(x,t)=e/integraldisplay
dyˆ/Psi1†
α(x,y,t )σxˆ/Psi1α(x,y,t ), (D4)
where eis the electron charge. Note that δ(ky−k/prime
y)i nE q .( D2)
must be understood in the sense of a Kronecker symbolmeaning that we use Born–von K ´arm´an periodic boundary
conditions in the ydirection. This means that there are
correspondences
δ(k
y−k/prime
y)=/integraldisplay+∞
−∞dy
2πei(ky−k/prime
y)y(D5)
⇔1
Ly/integraldisplayLy
0dy ei(kn
y−km
y)y=δn,m (D6)
and
2π
Ly/summationdisplay
kny⇔/integraldisplay+∞
−∞dky. (D7)
To obtain an observable quantity Iα(x,t) we compute a
statistical average of Eq. ( D4) with the help of
/angbracketleftˆγ†
α,in(ky,E)ˆγβ,in(ky,E/prime)/angbracketright=δα,βδ(E−E/prime)fα(E),(D8)
where fα(E) is a Fermi-Dirac distribution in the contact α.
The resulting expression has the form
Iα(x,t)=+∞/summationdisplay
n=−∞e−in/Omega1tIα,n(x),Iα,−n(x)=I∗
α,n(x),(D9)
035435-10RESONANT SECOND-HARMONIC GENERATION IN A . . . PHYSICAL REVIEW B 93, 035435 (2016)
where
Iα,n(x)=e/integraldisplay+∞
−∞dky/integraldisplay
|E|>|ky|dE/braceleftbigg
δn,0fα(E)+η∗(ky,E)+¯η(ky,En)
2/radicalbigv(ky,E)v(ky,En)e−i[κx(ky,E)+κx(ky,En)]xSαα(ky;En,E)fα(E)
+¯η∗(ky,E−n)+η(ky,E)
2/radicalbigv(ky,E−n)v(ky,E)ei[κx(ky,E−n)+κx(ky,E)]x[Sαα(ky;E−n,E)]†fα(E)
+/summationdisplay
β+∞/summationdisplay
m=−∞¯η∗(ky,E)+¯η(ky,En)
2/radicalbigv(ky,E)v(ky,En)ei[κx(ky,E)−κx(ky,En)]x[Sαβ(ky;E,E m)]†Sαβ(ky;En,Em)fβ(Em)/bracerightbigg
. (D10)
Using unitarity of the scattering matrix [ 43],
/summationdisplay
α/summationdisplay
n[Sαβ(ky;En,Em)]†Sαγ(ky;En,E)=δβ,γδm,0, (D11)
/summationdisplay
β/summationdisplay
nSγβ(ky;Em,En)[Sαβ(ky;E,E n)]†=δα,γδm,0, (D12)
we can rewrite Eq. ( D10) in the following form:
Iα,n(x)=e/integraldisplay+∞
−∞dky/integraldisplay
|E|>|ky|dE/braceleftbiggη∗(ky,E)+¯η(ky,En)
2/radicalbigv(ky,E)v(ky,En)e−i[κx(ky,E)+κx(ky,En)]xSαα(ky;En,E)fα(E)
+¯η∗(ky,E−n)+η(ky,E)
2/radicalbigv(ky,E−n)v(ky,E)ei[κx(ky,E−n)+κx(ky,E)]x[Sαα(ky;E−n,E)]†fα(E)+/summationdisplay
β+∞/summationdisplay
m=−∞¯η∗(ky,Em)+¯η(ky,En+m)
2/radicalbigv(ky,Em)v(ky,En+m)
×ei[κx(ky,Em)−κx(ky,En+m)]x[Sαβ(ky;Em,E)]†Sαβ(ky;En+m,E)[fβ(E)−fα(Em)]/bracerightbigg
. (D13)
In contrast with the usual B ¨uttiker theory [ 32], one cannot in general neglect the energy dependence of κx(ky,E) andv(ky,Em)
in Eq. ( D13), because the Fermi energy EFin graphene can be tuned to the Dirac point. On the other hand, if we keep the first
two terms on the right-hand side of Eq. ( D13), we see that the ac current components are formally determined by the full Fermi
sea rather than states close to the Fermi surface only.
APPENDIX E: AC DIFFERENTIAL CONDUCTANCE
In this section we present formulas that we use to compute
ac conductance for different sidebands in the main text. Weassume that our system [see Fig. 1(a)] is at low temperature
and compute a linear differential conductance with respect tothe source-drain bias voltage V
S,
fα(E)=f(E−eVα),−∂f(E)
∂E→δ(E−EF),
Gn(EF)=∂ID,n(x=0+,VS)
∂VS/vextendsingle/vextendsingle/vextendsingle/vextendsingle
VS→0. (E1)
Note that in principle the current, Eq. ( D9), is a function
of coordinate and we choose the point x=0+in ourcalculations. If we use the results of the previous section we
obtain
Gn(EF)=e2
h/integraldisplay∞
−∞dky+∞/summationdisplay
m=−∞η∗(ky,Em)+η(ky,En+m)
2/radicalbigv(ky,Em)v(ky,En+m)
×t†
m(ky,E)tn+m(ky,E)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
E=EF, (E2)
where we have restored Planck’s constant hto obtain the well-
known conductance unit. This formula was used in Fig. 3.
Finally, in Fig. 4we present more details of the results obtained
with the help of Eq. ( E2) for sidebands with n=±1,±2.
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035435-12 |
PhysRevB.85.205434.pdf | PHYSICAL REVIEW B 85, 205434 (2012)
Impact of intercalated cobalt on the electronic properties of graphene on Pt(111)
M. Gyamfi, T. Eelbo, M. Wa ´sniowska,*and R. Wiesendanger
Institute of Applied Physics, University of Hamburg, Jungiusstraße 11, D-20355 Hamburg, Germany
(Received 23 March 2012; published 18 May 2012)
We report on the strong interaction between graphene and intercalated cobalt revealed by scanning tunneling
spectroscopy (STS). A significant difference between pristine graphene and Co-intercalated graphene is identifiedat high bias voltages in the regime of field emission. For MLg/Co/Pt a resonance is observed at 3 eV which isnot visible for pristine graphene. Additionally, we observe a change of the phonon energy upon Co intercalation.Our STS results indicate that the Dirac point will not be preserved for graphene on Pt(111) if a single atomiclayer of Co is intercalated.
DOI: 10.1103/PhysRevB.85.205434 PACS number(s): 73 .22.Pr, 68.37.Ef, 73.20.−r
Epitaxially grown graphene has received much attention
because of the possibilities to engineer specific chemical
and physical properties for future nanodevices. Electrons
in monolayer graphene (MLg) are described by the Diracequation revealing the remarkable properties of massless
quasiparticles. Numerous examples can be found in the
literature demonstrating the unique electronic properties of
graphene, e. g., a peculiar integer quantum-Hall effect,
1high
carrier mobility,2or chemical doping.3These special proper-
ties can be influenced by a strong or weak interaction between
graphene and the supporting substrates. For example, it was
demonstrated that graphene prepared on Rh(111),4Co(0001),5
or Ni(111)6single crystals is strongly bonded and does not
preserve the Dirac point. Another example is graphene onRu(0001), demonstrating the possibility of forming a moir ´e
pattern which reflects strongly (valleys) or weakly (hills)
bonded areas of graphene and the substrate.
7,8In contrast,
MLg on Ir(111)9or Pt(111)10can reveal the properties of free-
standing graphene. Moreover, there has been a lot of interestin the interaction between graphene and magnetic substrates,
e. g., Ni(111)
11,12or Co(0001)5surfaces. These studies are
driven by possible applications in spintronic devices. Several
theoretical studies13predicted that it is possible to either induce
magnetism in MLg through hybridization between πand 3d
states (although experimental results are contradictory)5,11
or to produce a spin valve.14In contrast to graphene on
Ni(111) which is intensively studied, only a few experimental
results have been reported concerning the electronic properties
of graphene prepared on Co surfaces.5,15,16It was shown
that graphene strongly hybridizes with the Co 3 dstates
although the magnetic properties of thick Co layers or single
crystals are hardly affected. However, it could not be clarified
whether the intercalation of cobalt results in a strong or weak
interaction with MLg. To this end, we used scanning tunnelingmicroscopy/spectroscopy (STM/STS) to investigate the local
electronic properties of graphene on Pt(111) intercalated by
cobalt. This method has a high spatial resolution which allows
us to directly correlate the local electronic properties with the
topography of the sample.
STM and STS measurements were performed in an ul-
trahigh vacuum system equipped with a home-built STMoperating at 5 K. The d I/dUand d
2I/dU2signals were
measured using a lock-in technique with a modulation voltage
of 10 mV and a frequency of 5 kHz. The Pt(111) single crystal
was cleaned by repeated cycles of Ar+-ion sputtering at 300 K
and subsequent annealing to 1000 K. MLg was grown on thePt(111) by thermal decomposition of C 2H4.17Co is evaporated
from a rod, which is heated by electron bombardment, at a rateof 1 ML per minute at 300 K. In order to avoid any alloyingbetween Co and Pt(111), the samples were immediately cooleddown to low temperatures after Co deposition. Co atoms startto migrate underneath graphene and to form islands already at
room temperature. This is due to the interlayer mass transport
of Co atoms across platinum’s step edges. These nucleationcenters for Co are due to an imperfect matching betweengraphene’s and platinum’s edges, as is indicated by blue arrowsin the white box in Fig. 1(a).
After Co deposition on the partially graphene covered
Pt(111), the sample surface is composed of one atomic layer
high Co islands coexisting with MLg islands [see Fig. 1(a)].
Depending on the exact conditions during the preparation ofMLg on Pt(111) more than five different moir ´e patterns can
be observed due to the lattice mismatch between graphene andPt(111) as shown in Refs. 10,17,18. Two of them, i.e., the moir ´e
pattern with a periodicity of 0.5 nm and 1.8 nm, were mainlyobserved in our measurements. At first glance Co islands on
Pt(111) with a triangular shape and sharp edges are observed
in the topography. In contrast MLg islands exhibit edgeswith irregular shapes and are incorporated into platinum stepssometimes. The observed differences in the morphology of theislands allow the distinction between MLg and metal islandsin STM topographies. One can easily identify whether Co isintercalated below MLg islands or not by the unequal apparentheight. The separation between MLg and the Pt(111) surface
is found to be about 1.9 ±0.4˚Ao r3 . 4 ±0.2˚A for the case in
which Co is intercalated. This large apparent height differenceindicates the incorporation of a Co monolayer between MLgand Pt(111), which was reported for other metals as well; seeRefs. 6,19–21. The change of the height goes hand in hand with
a change of the corrugation of the pattern of MLg intercalatedby Co; see the line profiles in Fig. 1(b). We note that the line
profiles were acquired with the same tunneling conditions.
Interestingly at bias voltages close to the Fermi energy, wefound a change of the relative intensities of the corrugation ofthe moir ´e pattern and the atomic lattice of the graphene for
MLg/Co/Pt(111) compared to pristine MLg. We attribute thiseffect to the interaction between carbon and Co atoms, whichinfluences the bonds between carbon atoms in graphene aswell.
22These changes would cause a shift in phonon energy.
In order to confirm this, inelastic scanning tunneling
spectroscopy (IESTS) was performed on several locations ofthe sample. The IESTS has been successfully used to study
205434-1 1098-0121/2012/85(20)/205434(4) ©2012 American Physical SocietyGY AMFI, EELBO, W A ´SNIOWSKA, AND WIESENDANGER PHYSICAL REVIEW B 85, 205434 (2012)
0123450.00.20.40.60.0 0.5 1.0 1.5 2.0 2.50.00.61.2
MLg/Co/Pt(111)
MLg
distance (nm)corrugation (Å)MLg/Co/Pt(111)
MLg
-50 0 50-505Pt(111)
Mlg/Co/Pt(111)
Co/Pt(111)
MLg/Pt(111)d2I/dU2(arb. units)
bias voltage (mV)
6.23Å
0.00ÅMLg/CoMLg/CoMLg/Co
Co islandCo
5n m
Co
island
(2 MLg×2 )
(a)
(b) (c)
FIG. 1. (Color online) (a) Atomically resolved topography reveal-
ing an overview of the Pt(111) surface coated by a submonolayer ofgraphene coexisting with Co islands and Co underneath MLg islands
(I=2.3 nA and U=− 0.26 V). The STM image is composed of the
STM topograph and its derivative. (b) Line profiles along the linesindicated in (a) comparing the apparent corrugation of the moir ´e
pattern of pristine and Co intercalated graphene for the (2 ×2)
and√
61×√
61R26◦superstructure. The tunneling parameters:
I=2.3 nA and U=0.06 V . (c) d2I/dU2curves taken on different
areas of the sample ( I=2n Aa n d U=− 0.2 V). The spectra for
MLg/Co/Pt(111) and MLg were averaged over an area of the unitcell. We note that the peak’s position for MLg/Co/Pt does not vary
within the unit cell.
vibrational excitations of single molecules23or phonons in
graphene24in the past. In Fig. 1(c) we show a compari-
son between four corresponding IESTS spectra for Pt(111),MLg/Pt(111), Co/Pt(111), and MLg/Co/Pt(111), respectively.The investigations reveal no signature of inelastic tunneling forCo islands; only the spectra for Pt(111) show weak features atabout 12 meV , which can be related to contaminations (e.g.,CO or C
2H4molecules) observed at the surface [e.g., white
dots in Fig. 1(a)]. However, significant differences between
d2I/dU2spectra measured on graphene and MLg/Co/Pt(111)
were observed. For graphene a peak (dip) is observed ataround 50 meV ( −50 meV) and is attributed to K-point
graphene phonons
24with a node perpendicular to the surface
of graphene. Furthermore similar spectra are obtained whenthe tip is positioned above MLg/Co/Pt [see blue curve inFig. 1(c)] but the peak (dip) position is shifted ( /Delta1E=8m e V )
toward the Fermi energy. The fact that features are observedon graphene structures but not on Co/Pt(111) and Pt(111)suggests an excitation of graphene phonons. The shift ofthe peak (dip) position for MLg/Co/Pt with respect to MLg-0.6 -0.3 0.0 0.3 0. 60 .90246dI /dU (arb. units)
bias voltage (V)nanobubble
Co intercalated
MLg/Pt(111)
Pt(111)
Co island
FIG. 2. (Color online) STS spectra recorded on clean Pt(111), Co
islands, pristine and Co intercalated graphene. Tunneling parameters
for STS are I=0.5 nA and U=1 V . The spectrum of the Co island
(the nanobubble) is shifted with an offset of 3.1 (1.9) arb. units withrespect to the one of MLg.
indicates a change of the phonon’s energy. It can be related
either to a strain-induced deformation of the graphene latticeby intercalated cobalt or to a formation of bonds betweengraphene and Co in line with changes of the corrugation in theline profiles [see Fig. 1(b)].
In order to gain a further understanding of the impact
of intercalated Co on the electronic properties of MLg, weperformed STS as is shown in Fig. 2. In the following we
concentrate on the electronic properties of MLg/Co/Pt(111)exhibiting a moir ´e pattern with a periodicity of about 1.8 nm,
because the differential conductance is found to be the same
for all moir ´e patterns. At first glance, a remarkably strong
difference is visible for nanobubbles,
25Co islands, and
Pt(111). The spectra taken on Co islands, Pt(111), and thenanobubbles are used as a reference due to the previouslyidentified surface state of Pt(111) at +0.3 eV
26and a d-like
minority resonance state of Co nanostructures measured onPt(111) at about −0.3 eV .
27In the case of nanobubbles, the
features which are symmetrically located around the Fermi
level are correlated to K-point graphene phonons.25These
features are easier to observe on nanobubbles than on flatMLg/Pt(111) due to a different distance between graphene andPt(111). Almost no differences are visible for d I/dUspectra
recorded on the Pt surface, MLg/Pt(111), and MLg/Co/Pt.We note that it is not possible to unambiguously identify the
Dirac point of MLg,
10due to its overlap with the surface
state of Pt(111). Moreover, the MLg/Co/Pt spectrum revealsa lack of the 3 dCo resonance state for intercalated cobalt at
occupied electronic states, thereby confirming that a Co layerunderneath graphene exhibits different electronic propertiesthan Co islands without graphene on top. This observationindicates a strong bonding between the carbon layer and theCo atoms due to hybridization between metal 3 dstates and p
z
orbitals of MLg, being in line with measurements performed
on graphene prepared on bulk Co(0001) in Refs. 5,15,16.
These experiments revealed that in the energy range aroundthe Fermi level spin-polarized Co bulk states dominate thelocal density of states (LDOS). In the case of a single layer
205434-2IMPACT OF INTERCALATED COBALT ON THE ... PHYSICAL REVIEW B 85, 205434 (2012)
24680369121518 dI/dU(arb. units)
bias voltage (V)valley MLg/Co/Pt(111)hill
MLg/Ru(0001)
hill MLg/Co/Pt(111)hcp valley MLg/Ru(0001)Pt(111)
fcc valley MLg/Ru(0001)Ru(0001)
Co/Pt(111)
MLg/Pt(111)
bias voltage (V)
92 pm 0p m4n m (a) (b)
FIG. 3. (Color online) Comparison of high-voltage STS spectra
taken with closed feedback loop (a) clean Pt(111), Co islands, MLg,and MLg/Co/Pt(111) and (b) clean Ru(0001) and different locations
of the moir ´e pattern within MLg/Ru(0001). The spectra in panels (a)
and (b) are shifted by an offset of 5 arb. units and 4 arb. units to eachother, respectively. The spectra were taken with a tunneling current
of 0.5 nA. The inset in (b) shows the unit cell of MLg/Ru(0001) and
the locations where the spectra were measured ( U=0.25 eV and
I=1 nA). The subunit cell is shown as a red rhombus.
of cobalt intercalated below MLg/Pt(111), only the surface
resonance states contribute to the LDOS. As we do not observeany peak in the MLg/Co/Pt(111) spectrum at around −0.3 eV ,
the interaction between Co 3 dstates and the graphene seems
to be so strong that the dstate is quenched. Previously, a
similar behavior was reported for graphene on Ni(111).
12
To confirm the strong interaction between Co and carbon
atoms, we performed scanning tunneling spectroscopy in a
larger bias range with a closed feedback loop. It was recentlyreported that this kind of measurement allows the successfuldistinction between areas where graphene is weakly or stronglybonded to the substrate.
28Figures 3(a) and 3(b) present
dI/dUspectra of several areas, i.e., Pt(111), MLg/Pt(111),
Co/Pt(111), and MLg/Co/Pt(111) in comparison to MLg
prepared on a Ru(0001) single crystal. The features observed in
thedI/dU spectra originate from field emission resonances
which appear as well-defined peaks. Deeper insight can beobtained by comparing spectra, in particular those acquiredon different regions of monolayer graphene on Pt(111) andRu(0001). We identify the first image-potential state for theclean Pt surface at 5.8 V in agreement with Ref. 14. However,for pristine MLg it is found at 4.3 V , i.e., at a bias voltage which
corresponds well to the work function reported in Ref. 29.I t
is a signature of weakly bonded graphene as found on hills ofMLg/Ru(0001); cf. Fig. 3(b). The low and high regions of the
moir ´e patterns of MLg/Co/Pt exhibit the same spectrum which
indicates a similar chemical interaction in all areas. This obser-vation is in contrast with the moir ´e pattern of MLg/Ru(0001)
where the strength of the coupling changes from hills to valleys
as revealed by Borca et al.
28,30and by our STS measurements;
cf. Fig. 3(b) and Ref. 8. The first resonance measured on
MLg/Co/Pt(111) is found at a lower bias (about 3 eV) thanthose values for Pt, MLg, and Co/Pt(111). According toMLg/Ru(0001) this resonance at 3 eV on MLg/Co/Pt(111)originates from the different coupling of graphene to Pt(111)and graphene to Co/Pt(111), similar to hills and valleys ofMLg/Ru(0001). Since all spectra shown in Fig. 3(a) were
measured in one experiment (one data set), i.e., with the same
tip, we conclude that a strong bonding exists between MLgand intercalated Co atoms. As a consequence the Dirac pointshould no longer be preserved for MLg/Co/Pt(111) contrary topristine MLg/Pt(111). In addition, the spectrum for MLg/Co/Ptreveals a resonance at 4.6 eV with a lower intensity comparedto the neighboring peaks. In the case of MLg/Co/Pt(111) we in-
vestigate a system containing three layers of different chemical
elements. The Co atoms will mainly occupy hollow sites of thePt(111) surface if one assumes an unmodified atomic structureof the intercalated Co with respect to the pristine Co islands onPt(111).
27Consequently, the carbon atoms are located either
on top of Co or Pt atoms depending whether they occupy topor hollow sites. This chemical inhomogeneity does not exist in
the case of MLg/Ru(0001) and can explain the observed differ-
ences. At high bias voltages (when measuring field emissionresonances) the STS curves represent the properties of bothchemical environments and reflect an averaged charge redis-tribution originating from the C-Co and C-Pt dipole moments.
In summary, we have presented a thorough investigation
of Co and MLg on Pt(111) with an emphasis on the Co-intercalated MLg. On Pt(111) the intercalation of Co begins atroom temperature and does not require any high-temperatureannealing. The comparison of STS spectra in the range of fieldemission resonances of pristine and Co-intercalated MLg onPt(111) provides evidence that the cobalt and carbon atomsstrongly hybridize. This strong hybridization is also revealedby IESTS measurements showing a shift of 8 meV for theexcitation of phonons on intercalated graphene in comparisonto pristine MLg.
Financial support from the DFG (SFB 668-A8 and Grant
No. WI 1277/25), the ERC Advanced Grant FURORE, andthe Cluster of Excellence “Nanospintronics” is gratefullyacknowledged.
*Corresponding author: mwasniow@physnet.uni-hamburg.de
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205434-4 |
PhysRevB.80.214516.pdf | Quantum kinetic approach to the calculation of the Nernst effect
Karen Michaeli1and Alexander M. Finkel’stein1,2
1Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
2Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA
/H20849Received 13 August 2009; revised manuscript received 2 November 2009; published 16 December 2009 /H20850
We show that the strong Nernst effect observed recently in amorphous superconducting films far above the
critical temperature is caused by the fluctuations of the superconducting order parameter. We employ thequantum kinetic approach /H20851K. Michaeli and A. M. Finkel’stein, Phys. Rev. B 80, 115111 /H208492009 /H20850/H20852for the
derivation of the Nernst coefficient. We present here the main steps of the calculation and discuss some subtleissues that we encountered while calculating the Nernst coefficient. In particular, we demonstrate that in thelimit T→0 the contribution of the magnetization ensures the vanishing of the Nernst signal in accordance with
the third law of thermodynamics. We obtained a striking agreement between our theoretical calculations andthe experimental data in a broad region of temperatures and magnetic fields.
DOI: 10.1103/PhysRevB.80.214516 PACS number /H20849s/H20850: 74.40./H11001k, 72.15.Jf, 74.78. /H11002w
I. INTRODUCTION
After many years in the shade, the Nernst effect /H20849the
transverse thermoelectric signal /H20850entered the spotlight in
condensed-matter physics as well as other fields of research,such as the theory of gravitation.
1,2The “rediscovery” of the
Nernst effect by the condensed-matter community occurredafter the measurement of the effect in high- T
cmaterials
above the superconducting transition temperatures.3,4Since
then, the Nernst effect was also observed in conventionalamorphous superconducting films far above T
c.5,6The Nernst
effect in high- Tcsuperconductors3,4has been attributed to the
motion of vortices7–9existing even above Tc/H20849the vortex-
liquid regime /H20850. In conventional amorphous superconducting
films the strong Nernst signal observed deep in the normalstate
5,6cannot be explained by the vortexlike fluctuations.
Rather, the authors of Refs. 5and6suggested that the effect
is caused by fluctuations of the superconducting order pa-rameter. Here we present a comprehensive analysis of thismechanism using the quantum kinetic technique and demon-strate a quantitative agreement between the theoretical ex-pressions and the experiment.
6No fitting parameters have
been used; the values of Tcand the diffusion coefficient were
taken from independent measurements /H20849see Refs. 5and6/H20850.I n
particular, we succeeded in reproducing the nontrivial depen-dence of the signal on the magnetic field. Our results implythat in the quest for understanding the thermoelectric phe-nomena in high- T
cmaterials the fluctuations of the order
parameter should not be ignored.
The Nernst effect and its counterpart, the Ettingshausen
effect, are effective tools for studding the superconductingfluctuations because in metallic conductors the contributionof the quasiparticle excitations is negligible. the approxima-tion of a constant density of states at the Fermi energy, whichis a standard approximation for the Fermi-liquid theory, thiscontribution vanishes completely.
10On the other hand, the
collective modes describing all kinds of fluctuations can, ingeneral, generate significant contributions to the Nernst ef-fect. Since the neutral modes are not deflected by the Lorentzforce, they do not contribute to the transverse thermoelectriccurrent. The charged modes, such as fluctuations of super-conducting order parameter, are a possible source for the
giant Nernst effect even far from the superconducting transi-tion. The fact that the main contribution to the Nernst signaloriginates from the superconducting fluctuations is in con-trast to other transport phenomena such as the electric con-ductivity. The contributions to the electric conductivitycaused by the superconducting fluctuations/H20849paraconductivity
11–13/H20850can be observed close enough to the
superconducting transition where the paraconductivity in-creases rapidly and may even overcome the Drude conduc-tivity. Far from the transition the superconducting fluctua-tions produce only one among many corrections to theconductivity and, therefore, can hardly be identified. Owingto the fact that in the absence of fluctuations the Nernst effectis negligible, measurements of the Nernst signal provide aunique opportunity to study the superconducting fluctuationsdeep inside the normal state.
The transport coefficients for the electric and thermal cur-
rents are defined via the standard conductivity tensor,
/H20873je
jh/H20874=/H20873/H9268ˆ/H9251ˆ
/H9251˜ˆ/H9260ˆ/H20874/H20873E
−/H11633T/H20874. /H208491/H20850
When thermomagnetic phenomena are studied in films /H20849or
layered conductors /H20850the magnetic field is conventionally di-
rected perpendicularly to the conducting plane, see Fig. 1.
Then, each element of the conductivity tensor corresponds to
V
T1 T2Hz
xy
T∆
FIG. 1. The setup of the Nernst effect measurement. The sample
is placed between two thermal baths of different temperatures. Thetemperature gradient is in the xdirection, the magnetic field is along
thezdirection and the electric field is induced in the ydirection.PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
1098-0121/2009/80 /H2084921/H20850/214516 /H2084923/H20850 ©2009 The American Physical Society 214516-1a2/H110032 matrix describing the conductivity components in the
x-yplane /H20849see Fig. 1/H20850. The different components of the con-
ductivity tensor are connected through the Onsager relations.
In particular, /H9268ij/H20849H/H20850=/H9268ji/H20849−H/H20850and/H9251˜ij/H20849H/H20850=T/H9251ji/H20849−H/H20850.I na n
open circuit setup, one gets from the condition je=0 that the
Nernst coefficient is
eN=Ey
−/H11633xT=/H9268xx/H9251xy−/H9268xy/H9251xx
/H9268xx2+/H9268xy2. /H208492/H20850
We checked that the second term in the numerator is negli-
gible in comparison to the first one /H20851see the comment below
Eq. /H2084942a /H20850/H20852. This observation has been experimentally verified
as it follows from Fig. 2 /H20849a/H20850in Ref. 5. Therefore, the leading-
order term in the expression for the Nernst coefficient is eN
/H11015/H9251xy//H9268xxand our goal is to find the transverse Peltier coef-
ficient,/H9251xy.
The electric current generated as a response to an external
force, such as the electric field, can be found in the linearregime by the Kubo formula
14which expresses the response
in terms of a corresponding correlation function. Extendingthe Kubo formalism to the calculation of the response to atemperature gradient is not trivial because this gradient is notdirectly connected to any mechanical force. Following thescheme used in the derivation of the Einstein relation,Luttinger
15made a connection between the responses to a
temperature gradient and to an auxiliary gravitational field.As a result, Luttinger succeeded in relating all transport co-efficients with various current-current correlation functions.A main ingredient in the Kubo formula is the quantum-mechanical expressions for the current operators that enterthe correlation function, e.g., the electric and heat currents incase of the thermoelectric transport. When the electron-electron interactions are neglected, the expression for theheat current operator is
j
h/H20849q=0 ,/H9270/H20850=/H20858
p,/H9268/H11509/H9255p
/H11509p/H20849/H9255p−/H9262/H20850cp,/H9268†/H20849/H9270/H20850cp,/H9268/H20849/H9270/H20850
+/H20858
p,p/H11032,/H9268/H11509/H9255p
/H11509pVimp/H20849p,p/H11032/H20850cp,/H9268†/H20849/H9270/H20850cp/H11032,/H9268/H20849/H9270/H20850, /H208493/H20850
where cp,/H9268†/H20849/H9270/H20850/H20851cp,/H9268/H20849/H9270/H20850/H20852is the creation /H20851annihilation /H20852operator
of an electron in a state with energy /H9255p. Here/H9262is the chemi-
cal potential, Vimp/H20849p,p/H11032/H20850is the potential created by the dis-
order and /H9270is the imaginary time. With the help of the equa-
tions of motion and after transforming to the Matsubarafrequencies, the current operator can be written as
16
jh/H20849q=0 ,/H9275n/H20850=/H20858
p,/H9280n,/H9268/H11509/H9255p
/H11509p2i/H9280n−i/H9275n
2cp,/H9268†/H20849/H9280n/H20850cp,/H9268/H20849/H9280n−/H9275n/H20850.
/H208494/H20850
When electron-electron interactions are included, jhis a
more complicated function /H20849it contains terms with four fer-
mion operators /H20850. In general, the resulting expression for the
heatcurrent is not just the frequency multiplied by the velocity as
it is for free electrons. Unfortunately, very often the expres-sion for the heat current of free electrons presented in Eq. /H208494/H20850
is used in the presence of electron-electron interactions,when there is no real justification for it. In Appendix B ofRef. 17we showed that this simplified form of the Kubo
formula fails to reproduce the known result for the thermalconductivity of Fermi liquids. The incorrect result thatemerges from Eq. /H208494/H20850does not imply that the use of the Kubo
formula for the thermal transport coefficients is necessarilywrong. The weak point is in replacing the full expression forthe heat current by the one in Eq. /H208494/H20850. The problem with the
full expression for the heat current is in its complexity.
In addition, the Kubo formalism meets with some diffi-
culties when the thermoelectric currents are considered in thepresence of a magnetic field. Obraztsov
18pointed out that
when a magnetic field is applied, the heat current describingthe change in the entropy must include a contribution from
the magnetization. This is because the thermodynamic ex-pression for the heat contains the magnetization term. Thus,additional problem of the Kubo formula is that the currentcannot be expressed entirely by a correlation function. Inorder to determine the transverse thermoelectric currents oneneeds to combine the quantum mechanical response to theexternal field with the magnetization, which is a thermody-namic quantity.
18–20
In the derivation of the thermoelectric currents we de-
cided, instead of applying the Kubo formula, to employ adifferent approach and to use the quantum kineticequation.
21–23One main advantage of the quantum kinetic
approach is that the problem of the magnetization current issolved straightforwardly. We directly obtained the expressionfor the thermoelectric current which includes the magnetiza-tion current. In this way, the electric current generated by thetemperature gradient can be related to the flow of entropy.Therefore, according to the third law of thermodynamics theNernst signal must vanish at T→0.
24As we will see, this
argument imposes a strict constraint on the magnitude of thePeltier coefficient in a broad range of temperatures. Note thatthe calculation of the thermoelectric transport using the ki-netic equation also allows a direct verification of the Onsagerrelations between the off-diagonal components of the con-ductivity tensor /H20849see Appendix A /H20850.
The paper is organized as follows: in Secs. IIandIIIwe
present the main steps in the derivation of the electric currentas a response to a temperature gradient in the presence offluctuations of the superconducting order parameter using thequantum kinetic equation. Then, in Secs. IVandVwe give
details of the calculation that are specific to the transversecurrent. We devote Sec. VIand Appendix B to the contribu-
tion of the magnetization current to the transverse thermo-electric current. We demonstrate that the magnetization cur-rent ensures the vanishing of the Peltier coefficient in thelimit T→0. This makes the Nernst signal compatible with
the third law of thermodynamics. The result of the calcula-tion of the Nernst effect and comparison with the Nernstsignal measured in amorphous superconducting films
6are
presented in Sec. VII. The content of Sec. VIIhas already
been published25as a separate letter; we include it here for
completeness. In Appendix A we demonstrate that the twoKAREN MICHAELI AND ALEXANDER M. FINKEL’STEIN PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-2off-diagonal coefficients of the conductivity tensor that are
found independently using the quantum kinetic approach sat-
isfy the Onsager relations, /H9251ij/H20849B/H20850=T/H9251˜ji/H20849−B/H20850. In view of the
frequently used argument that the particle-hole symmetrylimits the magnitude of the Nernst effect /H20849see, e.g., Ref. 26/H20850
we discuss this issue in Appendix C. We demonstrate that thevalue of the Nernst coefficient is not constrained by theparticle-hole symmetry. Rather, the contribution from thequasiparticle excitations is zero when their density of statesis taken to be constant, which is often confused with theparticle-hole symmetry.
II. QUANTUM KINETIC EQUATION ABOVE TcIN THE
PRESENCE OF A TEMPERATURE GRADIENT
In this paper we extend the scheme developed in Ref. 17
to the case of electrons interacting with superconductingfluctuations in the presence of a magnetic field. Here wedescribe the system using two fields; one is the quasiparticlefield
/H9274while the other represents the fluctuations of the su-
perconducting order parameter /H9004. The matrix functions
Gˆ/H20849r,t;r/H11032,t/H11032/H20850and Lˆ/H20849r,t;r/H11032,t/H11032/H20850written in the Keldysh
form21–23describe the propagation of these two fields, re-
spectively. Throughout the paper, we work in the basis of theretarded, advanced, and Keldysh propagators,
Gˆ/H20849r,t;r
/H11032,t/H11032/H20850=/H20873GR/H20849r,t;r/H11032,t/H11032/H20850GK/H20849r,t;r/H11032,t/H11032/H20850
0 GA/H20849r,t;r/H11032,t/H11032/H20850/H20874, /H208495/H20850
where a similar expression can be written for Lˆ./H20851Notice that
we use the term propagators when referring to both these
functions while separately we name Gˆ/H20849r,t;r/H11032,t/H11032/H20850the quasi-
particle Green’s function and Lˆ/H20849r,t;r/H11032,t/H11032/H20850the propagator of
the superconducting fluctuations. /H20852The derivation of the
transport coefficients in the quantum kinetic equation isseparated into two steps. First, the propagators are foundusing the quantum kinetic equations. Then, the expressionfor the current in terms of the propagators is derived.
We now derive the quantum kinetic equations for the
propagators GˆandLˆin the presence of a temperature gradi-
ent. Inspired by Luttinger,
15we introduce an auxiliary gravi-
tational field of the form /H9253/H20849r/H20850=T0/T/H20849r/H20850/H20849where T0is the con-
stant part of the temperature /H20850. The purpose of the
gravitational field is to compensate for the nonuniform tem-perature at the initial state. In other words, the temperaturegradient and the gravitational field are applied in such a waythat at t=−/H11009the system is in equilibrium. Then, starting at
t=−/H11009, the gravitational field is adiabatically switched off.
From the response to switching off the gravitational field, wecan learn about the effect of the temperature gradient on thesystem /H20849for more details see Ref. 17/H20850. The general expression
for the action in the presence of a gravitational field
/H9253/H20849r/H20850and
a vector potential A/H20849r/H20850isS=/H20885drdt/H9253/H20849r/H20850/H20877/H20858
/H9268/H20877i
/H9253/H20849r/H20850/H9274/H9268†/H20849r,t/H20850/H11509
/H11509t/H9274/H9268/H20849r,t/H20850
−1
2m/H20879/H20875/H11633−ie
cA/H20849r/H20850/H20876/H9274/H9268/H20849r,t/H20850/H208792
−Vimp/H20849r/H20850
/H11003/H20858
/H9268/H9274/H9268†/H20849r,t/H20850/H9274/H9268/H20849r,t/H20850−/H9268
2/H20851/H9004/H20849r,t/H20850/H9274/H9268†/H20849r,t/H20850/H9274−/H9268†/H20849r,t/H20850+ H.c. /H20852/H20878
−/H20841/H9004/H20849r,t/H20850/H208412
/H9261/H20878. /H208496/H20850
Here/H9261is the coupling constant of the interaction /H20849we are
interested in the case of an s-wave coupling /H20850. The choice of
signs is such that /H9261/H110220 corresponds to an attractive interac-
tion. The spin index /H9268=1/H20849−1/H20850, or equivalently ↑/H20849↓/H20850, indi-
cates the spin direction up /H20849down /H20850. In the above equation and
throughout the paper we set /H6036=1.
The Dyson equation for the Green’s function in the pres-
ence of the gravitational field is
/H20877i/H11509
/H11509t+1
2m/H20875/H11633−ie
cA/H20849r/H20850/H20876/H9253/H20849r/H20850/H20875/H11633−ie
cA/H20849r/H20850/H20876
−/H9253/H20849r/H20850/H20851Vimp/H20849r/H20850−/H9262/H20852/H20878Gˆ/H20849r,t;r/H11032,t/H11032/H20850
=/H9254/H20849r−r/H11032/H20850/H9254/H20849t−t/H11032/H20850
+/H9253/H20849r/H20850/H20885dt1dr1/H9018ˆ/H20849r,t;r1,t1/H20850/H9253/H20849r1/H20850Gˆ/H20849r1,t1;r/H11032,t/H11032/H20850./H208497/H20850
In general, the Green’s function Gˆand self-energy /H9018ˆcontain
spin indices. Since we do not consider scattering mecha-nisms that flip the spins and ignore the Zeeman splitting,here and in the following we do not indicate the spin indiceswhenever it is possible. Next, we introduce the followingtransformation:
Yˆ/H20849r,t;r
/H11032,t/H11032/H20850=/H9253−1 /2/H20849r/H20850Yˆ=/H20849r,t;r/H11032,t/H11032/H20850/H9253−1 /2/H20849r/H11032/H20850, /H208498/H20850
where Yˆcan be either Gˆor/H9018ˆ. For the calculation of the
response to switching off the gravitational field in the linearregime, we set
/H9253/H20849r/H20850=1+r·/H11633T/T0. Then, the quantum kinetic
equation for the Green’s function of the quasiparticles be-comes
/H20877i/H208731−r·/H11633T
T0/H20874/H11509
/H11509t+1
2m/H20875/H11633−ie
cA/H20849r,t/H20850/H208762
−Vimp/H20849r/H20850+/H9262/H20878Gˆ=/H20849r,t;r/H11032,t/H11032/H20850=/H9254/H20849r−r/H11032/H20850/H9254/H20849t−t/H11032/H20850
+/H20885dt1dr1/H9018ˆ=/H20849r,t;r1,t1/H20850Gˆ=/H20849r1,t1;r/H11032,t/H11032/H20850. /H208499/H20850
The dependence of this equation on the temperature gradient
is much simplified by the transformation to Gˆ=and/H9018ˆ=because
/H11633T/T0was eliminated from all the terms in the equation
except the derivative with respect to time. In field theories
which include a nontrivial space-time metric gˆthe transfor-QUANTUM KINETIC APPROACH TO THE CALCULATION … PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-3mation of the kind Y=/H20849x,x/H11032/H20850=/H20881−det gˆ/H20849x/H20850Y/H20849x,x/H11032/H20850/H20881−det gˆ/H20849x/H11032/H20850
/H20849where xis a 4-vector /H20850is standard. The success of this trans-
formation in simplifying the quantum kinetic equation is dueto the relatively simple structure of the metric.
We write Eq. /H208499/H20850in coordinate space because all the
propagators and self-energies that enter the kinetic equationare not translationally invariant. There are three sources forthe lack of translation invariance. The first one is because thepropagators and self-energies depend on the magnetic fieldthrough the vector potential, which is a function of the coor-dinate. The second reason is due to the fact that we did notyet perform the averaging over the disorder. Finally, andmost important, in the presence of a temperature gradient/H20849even in the absence of a magnetic field /H20850the propagators
become functions of the center-of-mass coordinate.
We choose to postpone the averaging over impurities until
the last stage of the derivation of the current. Therefore, theGreen’s function of the quasiparticles contains open impuritylines as illustrated in the two coupled equations presented inFig. 2,
Gˆ/H20849r,t;r
/H11032,t/H11032/H20850=Gˆint/H20849r,t;r/H11032,t/H11032/H20850
+/H20885dr1dt1Gˆint/H20849r,t;r1,t1/H20850Vimp/H20849r1/H20850Gˆ/H20849r1,t1;r/H11032,t/H11032/H20850,
/H2084910a /H20850
Gˆint/H20849r,t;r/H11032,t/H11032/H20850=Gˆb/H20849r,t;r/H11032,t/H11032/H20850+/H20885dr1dt1dr2dt2Gˆb/H20849r,t;r1,t1/H20850
/H11003/H9018ˆ/H20849r1,t1;r2,t2/H20850Gˆint/H20849r2,t2;r/H11032,t/H11032/H20850. /H2084910b /H20850
Here, Gˆint/H20849r,t;r/H11032,t/H11032/H20850is the Green’s function of interacting
electrons while Gˆb/H20849r,t;r/H11032,t/H11032/H20850is free from both the interac-
tions and the scattering by impurities. Note that
Gˆint/H20849r,t;r/H11032,t/H11032/H20850includes partially the scattering by impurities.
Next we write the quantum kinetic equation using the
center-of-mass coordinates for space and time, R=/H20849r
+r/H11032/H20850/2 and T=/H20849t+t/H11032/H20850/2, and the relative space and time co-
ordinates, /H9267=r−r/H11032and/H9270=t−t/H11032. Since the gravitational field
is independent of time and we are interested in the steady-state solution, the Green’s function will be taken to be inde-
pendent of T. On the other hand, the dependence of the
Green’s function on Rremains because the temperature gra-
dient enters the equation as the product r·/H11633T=/H20849R
+/H9267/2/H20850·/H11633T. This dependence on Ris the main difference
between the response to a temperature gradient and the re-sponse to an electric field. The point is that in the presence ofan electric field the quantum kinetic equation can be formu-lated in such a way that the electric field enters only as aproduct with the relative coordinate, /H20849r−r
/H11032/H20850·E. Therefore,
after averaging over the disorder the electric field-dependentGreen’s function becomes translationally invariant.
In order to find the expression for the /H11633T-dependent
Green’s function using the quantum kinetic equation, weseparate the Green’s function into three parts,
Gˆ
==gˆeq+Gˆloc-eq+Gˆ/H11633T. /H2084911/H20850
The first part describes the propagation at equilibrium. The
retarded and advanced components of gˆeqare described by
Eq. /H208499/H20850with/H11633T=0,
/H20877/H9280+1
2m/H20875/H11633−ie
cA/H20849R+/H9267/2,t/H20850/H208762
−Vimp/H20849R+/H9267/2/H20850
+/H9262/H20878geqR,A/H20849/H9267,/H9280;A,imp /H20850
−/H20885dr1/H9268eqR,A/H20849/H9267−r1,/H9280;A,imp /H20850geqR,A/H20849r1,/H9280;A,imp /H20850=/H9254/H20849/H9267/H20850.
/H2084912/H20850
This is the usual Dyson equation for the Green’s function at
equilibrium in which we performed the Fourier transform ofthe relative time
/H9270. In the above equation we introduced the
equilibrium self-energy, /H9268ˆeq. The Green’s function gˆeqde-
pends on the center-of-mass coordinate through the vectorpotential and the potential V
impcreated by the impurities at a
specific realization. Correspondingly, we use the notationg
eq/H20849/H9267,/H9280;A,imp /H20850in which these dependencies on Rare in-
corporated into Aand imp. The gradient, /H11633=1
2/H11633R+/H11633/H9267,i nt h e
equation for geqR,Acontains the derivatives with respect to both
Rand/H9267.
According to the standard rule, the Keldysh component of
the Green’s function at equilibrium can be written in terms ofthe Fermi distribution function n
F/H20849/H9280/H20850and the retarded and
advanced Green’s functions,
geqK/H20849/H9267,/H9280;A,imp /H20850=/H208511–2 nF/H20849/H9280/H20850/H20852/H20851geqR/H20849/H9267,/H9280;A,imp /H20850
−geqA/H20849/H9267,/H9280;A,imp /H20850/H20852. /H2084913/H20850
In the presence of a uniform and constant in time mag-
netic field, the expressions given in Eq. /H2084912/H20850can be rewritten
as a product of the phase= +(a)
(b)= +G Gint
ΣGintG
Gint Gb Gb Gint{G,V}
FIG. 2. /H20849a/H20850Illustration of Eq. /H2084910a /H20850for the full Green’s function
Gˆ./H20849b/H20850The Dyson equation for Gˆint/H20851see Eq. /H2084910b /H20850/H20852. Note that Gˆint
includes scattering by impurities only through /H9018ˆ/H20849G/H20850which is a
function of the full Green’s function Gˆ. The bare Green’s function,
i.e., free from the interactions and the scattering by impurities, is
denoted by Gˆb.KAREN MICHAELI AND ALEXANDER M. FINKEL’STEIN PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-4exp/H20875ie
c/H20885
r/H11032r
A·/H20849r1/H20850dr1/H20876= exp/H20877−ieB
4c·/H20851/H20849r−r/H11032/H20850/H11003/H20849r+r/H11032/H20850/H20852/H20878
/H2084914/H20850
and the gauge-invariant Green’s functions, g˜ˆ. The retarded
and advanced components of g˜ˆsatisfy the equation,
/H20875/H9280+1
2m/H20873/H11633−ieB/H11003/H9267
2c/H208742
−Vimp−/H9268eqR,A/H20876
/H11003g˜eqR,A/H20849/H9267,/H9280;imp /H20850=/H9254/H20849/H9267/H20850, /H2084915/H20850
where the product of the Green’s function and the self-
energy should be understood as a convolution in real space/H20851see Eq. /H2084912/H20850/H20852. In the following, the permeability is taken to
be 1 and correspondingly we will not distinguish between B
/H20849the magnetic-flux density /H20850and the magnetic field H. After
averaging over the disorder, the gauge-invariant Green’sfunctions at equilibrium become translationally invariant,i.e., functions of the relative coordinate
/H9267alone /H20849see Ref. 27
and references therein /H20850,
/H20875/H9280+1
2m/H20873/H115092
/H11509/H92672−e2H2/H92672
4c2/H20874+/H9262/H11006i
2/H9270−/H9268eqR,A/H20876
g˜eqR,A/H20849/H9267,/H9280/H20850=/H9254/H20849/H9267/H20850, /H2084916/H20850
where/H9270is the elastic mean-free time of the electrons.
As we have already discussed, when we turn from the
equilibrium Green’s function to the /H11633T-dependent Green’s
function, an additional dependence on the center-of-mass co-ordinate appears. We wish to isolate this dependence on R
from the others. Similar to gˆ
eq/H20849/H9267,/H9280;A,imp /H20850, we denote the
dependencies of Gˆ=on the center-of-mass coordinate caused
by the impurity potential and the vector potential by imp and
A, respectively. Then, the remaining explicit dependence on
RinGˆ=/H20849R;/H9267,/H9280;A,imp /H20850arises due to the temperature gradi-
ent. Therefore, in the process of linearizing the equation in
/H11633T/T0, we expand Gˆ=and/H9018ˆ=in the collision integral with
respect to this explicit dependence on R. In other words, we
may write
/H20885dr1/H9018ˆ=/H20873R+r1
2;/H9267−r1,/H9280;A,imp/H20874
/H11003Gˆ=/H20873R−/H9267−r1
2;r1,/H9280;A,imp/H20874
/H11015/H20885dr1/H9018ˆ=/H20849R;/H9267−r1,/H9280;A,imp /H20850Gˆ=/H20849R;r1,/H9280;A,imp /H20850
+/H20885dr1r1
2·/H11509/H9018ˆ=/H20849R;/H9267−r1,/H9280;A,imp /H20850
/H11509RGˆ=/H20849R;r1,/H9280;A,imp /H20850
−/H20885dr1/H9018ˆ=/H20849R;/H9267−r1,/H9280;A,imp /H20850/H9267−r1
2
·/H11509Gˆ=/H20849R;r1,/H9280;A,imp /H20850
/H11509R. /H2084917/H20850As we shall see below, the last two terms in the expansion
are actually proportional to /H11633T/T0.
The equation for the local-equilibrium Green’s function is
/H20885dr1gˆeq−1/H20849/H9267−r1,/H9280;A,imp /H20850Gˆloc-eq/H20849R;r1,/H9280;A,imp /H20850
=/H20885dr1/H9018ˆloc-eq/H20849R;/H9267−r1,/H9280;A,imp /H20850gˆeq/H20849r1,/H9280;A,imp /H20850
+R·/H11633T
T0/H9280gˆeq/H20849/H9267,/H9280;A,imp /H20850. /H2084918/H20850
This equation is solved by
Gˆloc-eq/H20849R;/H9267,/H9280;A,imp /H20850=−R·/H11633T
T0/H9280/H11509gˆeq/H20849/H9267,/H9280;A,imp /H20850
/H11509/H9280,
/H2084919/H20850
where the corresponding self-energy should be taken as
/H9018ˆloc-eq/H20849R;/H9267,/H9280;A,imp /H20850=−/H20849R·/H11633T/T0/H20850/H9280/H11509/H9268ˆeq/H20849/H9267,/H9280;A,imp /H20850//H11509/H9280.
We see that the local-equilibrium Green’s function is astraightforward extension of the equilibrium Green’s func-tion for a nonuniform temperature. Since the same holds for
/H9018ˆ
loc-eq, the equation for Gˆloc-eqis a closed equation deter-
mined by the equilibrium properties of the system.
The Green’s function Gˆloc-eqdescribes the readjustment of
quasiparticles to the nonuniform temperature when the sys-tem is trying to maintain a local equilibrium. This responseof the electrons to the temperature gradient tends to inducemodulation of the density. Since for charged particles it isimpossible to have a large-scale charge modulation, the tem-perature gradient transforms into a gradient of the electro-
chemical potential. Therefore, j
e=/H9268ˆ/H20849E−/H11633/H9262/e/H20850=/H9268ˆE/H11569, where
the effective field E/H11569is the one measured in experiments.
The role of the local-equilibrium Green’s function is most
peculiar when the response to the temperature gradient isconsidered in the presence of a magnetic field. these condi-tions, as we show in Secs. IIIandVI,G
loc-eq/H20849R;/H9267,/H9280;A,imp /H20850
is responsible for the nonvanishing contribution to the elec-tric current from the magnetization.
All the remaining terms in the quantum kinetic equation
determine the last term of the Green’s function, Gˆ
/H11633T,/H20885dr1gˆeq−1/H20849/H9267−r1,/H9280;A,imp /H20850Gˆ/H11633T/H20849R;r1,/H9280;A,imp /H20850
−/H9267·/H11633T
2T0/H9280gˆeq/H20849/H9267,/H9280;A,imp /H20850
+1
2m/H20875/H11509
/H11509/H9267−ie
cA/H20849R+/H9267/2/H20850/H20876·/H11509Gˆloc-eq/H20849R;/H9267,/H9280;A,imp /H20850
/H11509R
=/H20885dr1/H9018ˆ/H11633T/H20849R;/H9267−r1,/H9280;A,imp /H20850gˆeq/H20849r1,/H9280;A,imp /H20850
+/H20885dr1r1
2·/H11509/H9018ˆloc-eq/H20849R;/H9267−r1,/H9280;A,imp /H20850
/H11509Rgˆeq/H20849r1,/H9280;A,imp /H20850
−/H20885dr1/H9268ˆeq/H20849/H9267−r1,/H9280;A,imp /H20850/H9267−r1
2QUANTUM KINETIC APPROACH TO THE CALCULATION … PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-5·/H11509Gˆloc-eq/H20849R;r1,/H9280;A,imp /H20850
/H11509R. /H2084920/H20850
In the above equation, the derivatives with respect to the
center-of-mass coordinate act only on the explicit depen-
dence of Gˆloc-eq/H20849R,/H9267,/H9280;A,imp /H20850and/H9018ˆloc-eq/H20849R,/H9267,/H9280;A,imp /H20850
onR/H20849i.e., through the spatially dependent temperature /H20850. Re-
call that the derivatives with respect to the center-of-masscoordinate which act on V
impandAin the local-equilibrium
Green’s function was already included in geq−1that appears in
Eq. /H2084918/H20850.
Once the explicit expressions for Gˆloc-eqand/H9018ˆloc-eqare
inserted, the equation becomes much simpler
Gˆ/H11633T/H20849/H9267,/H9280;A,imp /H20850
=gˆeq/H20849/H9280/H20850/H9018ˆ/H11633T/H20849/H9280/H20850gˆeq/H20849/H9280/H20850
−i/H9280/H11633T
2T0·/H20875/H11509gˆeq/H20849/H9280/H20850
/H11509/H9280vˆeq/H20849/H9280/H20850gˆeq/H20849/H9280/H20850−gˆeq/H20849/H9280/H20850vˆeq/H20849/H9280/H20850/H11509gˆeq/H20849/H9280/H20850
/H11509/H9280/H20876.
/H2084921/H20850
The product of matrices should be understood as a convolu-
tion in real space. The velocity vˆeqis the renormalized ve-
locity at equilibrium,
vˆeq/H20849r,t;r/H11032,t/H11032/H20850=−ilim
r/H11032→r/H11633−/H11633/H11032
2m−i/H20849r−r/H11032/H20850/H9268ˆeq/H20849r,r/H11032,/H9280/H20850.
/H2084922/H20850
Let us point out an important difference between the two
parts of the Green’s function depending on the temperature
gradient, Gˆloc-eqandGˆ/H11633T. As was already mentioned, Gˆloc-eq
and/H9018ˆloc-eqare a straightforward extension of the equilibrium
Green’s function and self-energy for a nonuniform tempera-
ture. On the other hand, the equation for Gˆ/H11633Tcontains the
/H11633T-dependent self-energy which by itself is a function of
Gˆ/H11633T. Thus, this is a self-consistent equation and in order to
find a close expression for Gˆ/H11633T, one has to determine the
structure of the self-energy. Once the form of the self-energyis known, one should take into consideration in the coarse oflinearization with respect to /H11633Tthat all the propagators in
/H9018ˆ
/H11633Tmay depend on the temperature gradient.
To complete the derivation of the electric current as a
response to a temperature gradient, we must also find thedependence of the propagator of the superconducting fluc-
tuations Lˆ/H20849r,t;r
/H11032,t/H11032/H20850on/H11633T. In the regime of linear re-
sponse, the explicit dependence on the temperature gradient
can be eliminated from the kinetic equation for Lˆby trans-
forming to the propagator Lˆ=,
/H9261−1Lˆ=/H20849r,t;r/H11032,t/H11032/H20850=/H9254/H20849r−r/H11032/H20850
−/H20885dr1dt1/H9016ˆ=/H20849r,t;r1,t1/H20850Lˆ=/H20849r1,t1;r/H11032,t/H11032/H20850.
/H2084923/H20850
Thus, the entire dependence of the propagator on the tem-perature gradient is through the self-energy term /H9016ˆ=, which is
a function of the quasiparticle Green’s functions.
Let us separate the solution of Eq. /H2084923/H20850into the equilib-
rium and /H11633T-dependent propagators, Lˆ==Lˆeq+Lˆloc-eq+Lˆ/H11633T.
The propagator at equilibrium satisfies the equation,
Vˆeq/H20849R;/H9267,/H9275/H20850=/H20851/H9261−1+/H9016ˆeq/H20849R;/H9267,/H9275/H20850/H20852−1. /H2084924/H20850
The entire dependence of Lˆeq/H20849/H9275/H20850on the frequency is due to
dressing of the bare propagator by its self-energy /H9016ˆeq/H20849/H9275/H20850.I n
the above equation the propagator of the superconductingfluctuations is a function of the temperature T
0. Similar to
Eq. /H2084922/H20850, we may define the “renormalized velocity” of the
collective mode describing the superconducting fluctuationsat equilibrium to be
Vˆ
eq/H20849r,t;r/H11032,t/H11032/H20850=−i/H20849r−r/H11032/H20850/H9016ˆeq/H20849r,t;r/H11032,t/H11032/H20850. /H2084925/H20850
Note that in fact Vˆdoes not have the dimension of a velocity.
The equations for the /H11633T-dependent propagators remind
the first term in Eq. /H2084921/H20850forGˆ/H11633T,
Lˆloc-eq/H20849R;/H9267,/H9275/H20850=−Lˆeq/H20849/H9275/H20850/H9016ˆloc-eq/H20849/H9275/H20850Lˆeq/H20849/H9275/H20850/H20849 26/H20850
and
Lˆ/H11633T/H20849R;/H9267,/H9275/H20850=−Lˆeq/H20849/H9275/H20850/H9016ˆ/H11633T/H20849/H9275/H20850Lˆeq/H20849/H9275/H20850. /H2084927/H20850
Once again, one should understand the product as a convo-
lution of the spatial coordinate.
III. ELECTRIC CURRENT AS A RESPONSE TO A
TEMPERATURE GRADIENT
For the calculation of the Nernst effect we have to derive
the expression for the electric current as a response to atemperature gradient. In the presence of a magnetic field, theelectric current is a sum of two terms,
j
e=jecon+jemag. /H2084928/H20850
The first one, jecon, is derived using the continuity equation
for the electric charge. The second contribution to the elec-tric current originates from the magnetization current. Sincethe magnetization current is divergenceless it cannot be ob-tained using the continuity equation and it is found sepa-rately.
As follows from the action in Eq. /H208496/H20850, the fields
/H9274and/H9004
carry electric current. Therefore, the charge continuity equa-tion must include both fields,
−e/H20858
/H9268/H11509t/H20841/H9274/H9268/H20849r,t/H20850/H208412+/H11633·jecon/H20849r,t/H20850=−2 ie/H9253/H20849r/H20850
/H11003/H20851/H9004†/H20849r,t/H20850/H9274↓/H20849r,t/H20850/H9274↑/H20849r,t/H20850−/H9004/H20849r,t/H20850/H9274↑†/H20849r,t/H20850/H9274↓†/H20849r,t/H20850/H20852,
/H2084929/H20850
where jecon/H20849r,t/H20850=−ie/H20858/H9268/H9253/H20849r/H20850/H20851/H9274/H9268†/H20849r,t/H20850/H11633/H9274/H9268/H20849r,t/H20850
−/H11633/H9274/H9268†/H20849r,t/H20850/H9274/H9268/H20849r,t/H20850−2ieA/H20841/H9274/H9268/H208412/c/H20852/2m. The terms in the
right-hand side /H20849RHS /H20850describe absorption and emission ofKAREN MICHAELI AND ALEXANDER M. FINKEL’STEIN PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-6quasiparticles by the superconducting fluctuations; the factor
2 reflects the fact that the Copper pairs carry charge of 2 e.
To find the expression for jeconin terms of the Green’s
function, we rewrite the charge density using the lesser com-ponent of the Green’s function
/H9267/H20849r,t/H20850=−elim
r/H11032→r
t/H11032→t+/H20858
/H9268/H20855/H9274/H9268†/H20849r/H11032,t/H11032/H20850/H9274/H9268/H20849r,t/H20850/H20856
=ielim
r/H11032→r
t/H11032→t/H20858
/H9268G/H11021/H20849r,t;r/H11032,t/H11032/H20850. /H2084930/H20850
We use the notation t/H11032→t+to indicate that the limit should
be taken in such a way that tis on the upper branch of the
Keldysh contour while t/H11032is on the lower branch.22The sum-
mation over the spin projection results in a factor of 2. Here/H20855A/H20856denotes the quantum mechanical averaging with the ac-
tion given in Eq. /H208496/H20850. Therefore, the Green’s function is fully
dressed by the interactions and depends on the impurity po-
tential. In addition, Gˆis a function of the temperature gradi-
ent. Since we find the current by extracting it from the con-tinuity equation, we assume that the temperature gradient hassome spatial modulations that will be set to zero at the end ofthe procedure.
Next, we insert the above expression into the continuity
equation given in Eq. /H2084929/H20850and rewrite /H11633·j
econas a sum of
two terms,
/H11633·jecon=I1+I2,
I1=elim
r/H11032→r
t/H11032→t+/H20873/H11509
/H11509t+/H11509
/H11509t/H11032/H20874/H20858
/H9268/H20855/H9274/H9268†/H20849r/H11032,t/H11032/H20850/H9274/H9268/H20849r,t/H20850/H20856,
I2=−2 ielim
r/H11032→r
t/H11032→t+/H20855/H9253/H20849r/H11032/H20850/H9004†/H20849r/H11032,t/H11032/H20850/H9274↓/H20849r,t/H20850/H9274↑/H20849r,t/H20850
−/H9274↑†/H20849r/H11032,t/H11032/H20850/H9274↓†/H20849r/H11032,t/H11032/H20850/H9253/H20849r/H20850/H9004/H20849r,t/H20850/H20856. /H2084931/H20850
To resolve the expression for jeconwe need to find the equa-
tions of motion for the field /H9274. The variational derivative of
the action in Eq. /H208496/H20850with respect to the /H9274†yields the equa-
tion of motion for the field /H9274,
i/H11509/H9274/H9268/H20849r,t/H20850
/H11509t=−1
2m/H20875/H11633−ie
cA/H20849r/H20850/H20876/H9253/H20849r/H20850/H20875/H11633−ie
cA/H20849r/H20850/H20876/H9274/H9268/H20849r,t/H20850
+/H9253/H20849r/H20850Vimp/H20849r/H20850/H9274/H9268/H20849r,t/H20850+/H9268/H9253/H20849r/H20850/H9004/H20849r,t/H20850/H9274−/H9268†/H20849r,t/H20850.
/H2084932/H20850
the average, the equations of motion allow us to rewrite theexpression for I1as
I1=−ielim
r/H11032→r
t/H11032→t+/H20883/H20858
/H9268/H20877−1
2m/H20875/H11633−ie
cA/H20849r/H20850/H20876/H9253/H20849r/H20850/H20875/H11633−ie
cA/H20849r/H20850/H20876
+1
2m/H20875/H11633/H11032−ie
cA/H20849r/H11032/H20850/H20876/H9253/H20849r/H11032/H20850/H20875/H11633/H11032−ie
cA/H20849r/H11032/H20850/H20876
+/H9253/H20849r/H20850Vimp/H20849r/H20850−/H9253/H20849r/H11032/H20850Vimp/H20849r/H11032/H20850/H20878/H9274/H9268†/H20849r/H11032,t/H11032/H20850/H9274/H9268/H20849r,t/H20850
+/H20858
/H9268/H9268/H9274/H9268†/H20849r/H11032,t/H11032/H20850/H9274−/H9268†/H20849r,t/H20850/H9253/H20849r/H20850/H9004/H20849r,t/H20850
−/H20858
/H9268/H9268/H9253/H20849r/H11032/H20850/H9004†/H20849r/H11032,t/H11032/H20850/H9274−/H9268/H20849r/H11032,t/H11032/H20850/H9274/H9268/H20849r,t/H20850/H20884. /H2084933/H20850
Now, we wish to express the electric current in the pres-
ence of a gravitational field in terms of the propagators. The
expression /H20855/H9004/H20849r,t/H20850/H9274/H9268†/H20849r/H11032,t/H11032/H20850/H9274−/H9268†/H20849r,t/H20850/H20856and its counterpart are
averaged with respect to the Hamiltonian that includes boththe interactions and the gravitational field. These expressionscan be written in terms of the self-energy of the quasiparti-
cles. For example, /H20855
/H9268/H9253/H20849r/H20850/H9004/H20849r,t/H20850/H9274/H9268†/H20849r/H11032,t/H11032/H20850/H9274−/H9268†/H20849r,t/H20850/H20856=
−i/H9253/H20849r/H20850/H20848dr1dt1/H9018ˆ/H9268/H20849r,t;r1,t1/H20850/H9253/H20849r1/H20850Gˆ/H9268/H20849r1,t1;r/H11032,t/H11032/H20850./H20849Here, the
factor iappears because in real time the evolution operator is
of the form e−iHtand because of the conventional definition
of the propagators and self-energies.22/H20850. As a result, we ob-
tain,
I1=−2 elim
r/H11032→r
t/H11032→t/H20877−/H11633/H9253/H20849r/H20850/H20851/H11633−ieA/H20849r/H20850/c/H20852
2mGˆ/H20849r,t;r/H11032,t/H11032/H20850
+/H11633/H11032/H9253/H20849r/H11032/H20850/H20851/H11633/H11032+ieA/H20849r/H11032/H20850/c/H20852
2mGˆ/H20849r,t;r/H11032,t/H11032/H20850
+/H9253/H20849r/H20850/H20885dr1dt1/H9018ˆ/H20849r,t;r1,t1/H20850/H9253/H20849r1/H20850Gˆ/H20849r1,tt;r/H11032,t/H11032/H20850
−/H20885dr1dt1Gˆ/H20849r,t;r1,t1/H20850/H9253/H20849r1/H20850/H9018ˆ/H20849r1,t1;r/H11032,t/H11032/H20850/H9253/H20849r/H11032/H20850/H20878/H11021
.
/H2084934/H20850
The factor 2 is a consequence of the sum over the spin index.
Similarly, we can express the averages in the equation for I2
in terms of the self-energy of the superconducting
fluctuations, e.g., /H20855/H9253/H20849r/H11032/H20850/H9004†/H20849r/H11032,t/H11032/H20850/H9274↓/H20849r,t/H20850/H9274↑/H20849r,t/H20850/H20856=
−i/H20848dr1dt1/H9016ˆ/H20849r,t;r1,t1/H20850/H9253/H20849r1/H20850Lˆ/H20849r1,t1;r/H11032,t/H11032/H20850/H9253/H20849r/H11032/H20850.
In the regime of linear response we may eliminate the
explicit dependence of the current on /H9253/H20849r/H20850by expressing the
current in terms of Gˆ=,Lˆ=,/H9018ˆ=, and/H9016ˆ=/H20851as defined in Eq. /H208498/H20850/H20852.
Then, the sum of the two contributions to /H11633·jeconbecomesQUANTUM KINETIC APPROACH TO THE CALCULATION … PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-7/H11633·jecon=−2 elim
r/H11032→r
t/H11032→t/H20877−/H11633/H20851/H11633−ieA/H20849r/H20850/c/H20852
2mGˆ=/H20849r,t;r/H11032,t/H11032/H20850
+/H11633/H11032/H20851/H11633/H11032+ieA/H20849r/H11032/H20850/c/H20852
2mGˆ=/H20849r,t;r/H11032,t/H11032/H20850
+/H20885dr1dt1/H9018ˆ=/H20849r,t;r1,t1/H20850Gˆ=/H20849r1,t1;r/H11032,t/H11032/H20850
−/H20885dr1dt1Gˆ=/H20849r,t;r1,t1/H20850/H9018ˆ=/H20849r1,t1;r/H11032,t/H11032/H20850
+/H20885dr1dt1/H9016ˆ=/H20849r,t;r1,t1/H20850Lˆ=/H20849r1,t1;r/H11032,t/H11032/H20850
−/H20885dr1dt1Lˆ=/H20849r,t;r1,t1/H20850/H9016ˆ=/H20849r1,t1;r/H11032,t/H11032/H20850/H20878/H11021
./H2084935/H20850
Note that the current still depends on the gravitational field
/H20849i.e., on the temperature gradient /H20850through the propagators
and self-energies.
In the final step of the derivation one has to resolve the
expression for the current out of the gradient. In other words,to reformulate the products of propagators and self-energiesas a derivative with respect to the center-of-mass coordinate.As discussed in the previous section, we can isolate the de-pendencies on the center-of-mass coordinate created by thevector potential and by the impurities. After we average thecurrent over the disorder and transform to the gauge-invariant propagators and self-energies, these dependenciesvanish. Pay attention that when the limit r
/H11032→ris taken, one
may rewrite Eq. /H2084935/H20850in terms of the gauge-invariant quanti-
ties alone. Therefore, we expand the products of the propa-gators and the self-energies with respect to the deviationfrom Rexactly in the same way as performed in Eq. /H2084917/H20850.A s
a consequence of the symmetric form of the terms in Eq.
/H2084935/H20850,/H20851/H9018ˆGˆ−Gˆ/H9018ˆ/H20852
/H11021and /H20851/H9016ˆLˆ−Lˆ/H9016ˆ/H20852/H11021, one may check that all
even orders in the expansion vanish. In the regime of linearresponse it is enough to keep only the lowest nonvanishingorder in the expansion. Eventually, the expression for thecurrent becomes
j
econ/H20849r,t/H20850=ie/H20885dr/H11032dt/H11032/H20851vˆ=/H20849r,t;r/H11032,t/H11032/H20850Gˆ=/H20849r/H11032,t/H11032;r,t/H20850/H20852/H11021
+ie/H20885dr/H11032dt/H11032/H20851Vˆ=/H20849r,t;r/H11032,t/H11032/H20850Lˆ=/H20849r/H11032,t/H11032;r,t/H20850/H20852/H11021+ H.c.
/H2084936/H20850
We use the notation /H20851¯/H20852/H11021to remind that the expression
inside the square brackets is a product of matrices and toindicate that the current corresponds to the lesser componentof the resulting matrix. The matrices vˆ
=andVˆ=are the renor-
malized velocities defined in Eqs. /H2084922/H20850and /H2084925/H20850with the
/H11633T-dependent self-energies /H9018ˆ=and/H9016ˆ=replacing the equilib-
rium ones.
The velocity of the quasiparticles vˆis renormalized by the
self-energy, /H9254vˆ/H20849r,t;r/H11032,t/H11032/H20850=−i/H20849r−r/H11032/H20850/H9018ˆ/H20849r,t;r/H11032,t/H11032/H20850. We find it
useful to rewrite this expression as follows: − /H20851i/H20849r/H11032−r/H20850
+2i/H20849r−r/H11032/H20850/H20852/H9018ˆ/H20849r,t;r/H11032,t/H11032/H20850. The idea behind this representation
can be explained using, as an example, the first-order expan-sion of the self-energy with respect to the superconductingfluctuations presented in Fig. 3. In this example, the self-
energy contains a quasiparticle Green’s function propagatingfrom r
/H11032torand a propagator of the superconducting fluc-
tuations that goes from rtor/H11032. Correspondingly, the first
difference of the coordinates in the square brackets acts on Gˆ
while the second /H20849which appears with the factor 2 /H20850acts on Lˆ.
As mentioned in the beginning of this section, there is
another contribution to the electric current arising from themagnetization,
j
emag=−2 ic/H11633/H11003M/H20849r/H20850lim
r/H11032→r
t/H11032→t/H20851Gˆ=/H20849r/H11032,t/H11032;r,t/H20850/H20852/H11021, /H2084937/H20850
where M/H20849r/H20850=er/H11003v/2mcdenotes the magnetization and the
factor of 2 is due to the summation over the spin index. Wewould like to emphasize that since the magnetization is cre-ated by itinerant electrons, the magnetization current canequally contribute to the transverse transport electric current.
IV. DERIVATION OF THE TRANSVERSE
COMPONENT OF jecon
At this stage of the derivation we shall consider only jecon
keeping for a while the magnetization current aside. Inserting
the expressions for the /H11633T-dependent propagators given in
Eqs. /H2084919/H20850,/H2084921/H20850, and /H2084927/H20850into Eq. /H2084936/H20850and extracting the
lesser component, we obtain jeconas a response to the tem-
perature gradient. First of all, one may observe that the con-
tributions of the local-equilibrium functions Gˆloc-eqand
Lˆloc-eqtojeconvanish upon averaging the current over the
sample. Since we are not interested in terms that vanish after
averaging over the volume, the nonzero part of jeconbecomesω−εω
FIG. 3. The self-energy in the first order with respect to the
propagator of superconducting fluctuations before averaging overthe disorder.KAREN MICHAELI AND ALEXANDER M. FINKEL’STEIN PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-8jeicon=−e/H11633jT
2T0/H20885d/H9280
2/H9266/H9280/H11509nF/H20849/H9280/H20850
/H11509/H9280/H20851viR/H20849/H9280/H20850gR/H20849/H9280/H20850vjA/H20849/H9280/H20850gA/H20849/H9280/H20850
+viR/H20849/H9280/H20850gR/H20849/H9280/H20850vjR/H20849/H9280/H20850gA/H20849/H9280/H20850−viR/H20849/H9280/H20850gR/H20849/H9280/H20850vjR/H20849/H9280/H20850gR/H20849/H9280/H20850
−gR/H20849/H9280/H20850vjR/H20849/H9280/H20850gR/H20849/H9280/H20850viA/H20849/H9280/H20850/H20852−e/H11633jT
T0/H20885d/H9280
2/H9266/H9280nF/H20849/H9280/H20850
/H11003/H20875viR/H20849/H9280/H20850/H11509gR/H20849/H9280/H20850
/H11509/H9280vjR/H20849/H9280/H20850gR/H20849/H9280/H20850−viR/H20849/H9280/H20850gR/H20849/H9280/H20850vjR/H20849/H9280/H20850/H11509gR/H20849/H9280/H20850
/H11509/H9280/H20876
−ie/H20885d/H9280
2/H9266viR/H20849/H9280/H20850gR/H20849/H9280/H20850/H20853/H9018/H11633T/H11021/H20849/H9280/H20850/H208511−nF/H20849/H9280/H20850/H20852+/H9018/H11633T/H11022/H20849/H9280/H20850nF/H20849/H9280/H20850/H20854
/H11003/H20851gR/H20849/H9280/H20850−gA/H20849/H9280/H20850/H20852+ie/H20885d/H9275
2/H9266ViR/H20849/H9275/H20850LR/H20849/H9275/H20850/H20853/H9016/H11633T/H11021/H20849/H9275/H20850
/H11003/H208511+nP/H20849/H9275/H20850/H20852−/H9016/H11633T/H11022/H20849/H9275/H20850nP/H20849/H9275/H20850/H20854/H20851LR/H20849/H9275/H20850−LA/H20849/H9275/H20850/H20852+ c.c.
/H2084938/H20850
Here and from now on we omit the notation eqfrom the
equilibrium quantities such as the propagators, self-energies,and velocities.
As we are interested in the Gaussian fluctuations, we re-
place the equilibrium Green’s function by gˆ/H20849r,r
/H11032,/H9280/H20850
=gˆ0/H20849r,r/H11032,/H9280/H20850+/H20848dr1dr2gˆ0/H20849r,r1,/H9280/H20850/H9268ˆ/H20849r1,r2,/H9280/H20850gˆ0/H20849r2,r/H11032,/H9280/H20850. Be-
sides, we keep only the contribution to the self-energy withone propagator of the superconducting fluctuations as illus-trated in Fig. 3,
/H9018
/H11021,/H11022/H20849r,r/H11032,/H9280/H20850=−i/H20885d/H9275
2/H9266G/H11022,/H11021/H20849r/H11032,r,/H9275−/H9280/H20850L/H11021,/H11022/H20849r,r/H11032,/H9275/H20850,
/H9018R,A/H20849r,r/H11032,/H9280/H20850=−i/H20885d/H9275
2/H9266G/H11021/H20849r/H11032,r,/H9275−/H9280/H20850LR,A/H20849r,r/H11032,/H9275/H20850
+GA,R/H20849r/H11032,r,/H9275−/H9280/H20850L/H11021/H20849r,r/H11032,/H9275/H20850. /H2084939/H20850The propagator of the superconducting fluctuations /H20849see the
end of Sec. II/H20850is determined by the standard geometrical
series Lˆ/H20849/H9275/H20850=/H20851−/H9261−1+/H9016ˆ/H20849/H9275/H20850/H20852−1, where /H9016ˆis approximated by
the particle-particle polarization operator as shown in Fig. 4,
/H9016/H11021,/H11022/H20849r,r/H11032,/H9280/H20850=−i/H20885d/H9280
2/H9266G/H11021,/H11022/H20849r,r/H11032,/H9275−/H9280/H20850G/H11021,/H11022/H20849r,r/H11032,/H9280/H20850,
/H9016R,A/H20849r,r/H11032,/H9280/H20850=−i/H20885d/H9280
2/H9266G/H11021/H20849r,r/H11032,/H9275−/H9280/H20850GR,A/H20849r,r/H11032,/H9280/H20850
+GR,A/H20849r,r/H11032,/H9275−/H9280/H20850G/H11022/H20849r,r/H11032,/H9280/H20850. /H2084940/H20850
One may check that at equilibrium /H9016eqK=/H208511+2nP/H20849/H9275/H20850/H20852/H20849/H9016eqR
−/H9016eqA/H20850, where nP/H20849/H9275/H20850is the Bose distribution function. After
averaging over the disorder, /H9016eqRand/H9016eqAare given by the
standard expressions.
We may now obtain the leading-order corrections in the
interaction to the electric current as a response to a tempera-ture gradient in the linear regime. We should consider allpossibilities to linearize the expressions for /H9018
/H11633Tand/H9016/H11633T
with respect to /H11633Tin Eq. /H2084938/H20850. The diagrammatic interpreta-
tion for the different contributions to the transverse electriccurrent obtained in the quantum kinetic equation techniquecorresponds to the three diagrams shown in Fig. 5. After
averaging over the disorder the leading contributions to theNernst signal in the diffusive regime are obtained from thediagrams with three Cooperons
13presented in Figs. 6/H20849a/H20850and
6/H20849b/H20850and the Aslamazov-Larkin diagram11shown in Fig. 6/H20849c/H20850.
/H20851The Cooperon is a singular diffusion propagator which de-
scribes the rescattering on impurities in the particle-particlechannel. /H20852Since we generate these terms using the quantum
kinetic equation, the analytic structure of the diagrams isgiven by the equation.
To get the explicit expression for the current we return to
the gauge-invariant equilibrium Green’s functions g
˜given in
Eq. /H2084915/H20850. Since we restrict our calculation to the limit /H9275c/H9270
/H112701/H20849where/H9275c=eH /m/H11569cis the cyclotron frequency of the
quasiparticles /H20850, we may neglect the dependence of g˜eqon the
magnetic field entering through the Landau quantization ofε
ω−εε
ω−ε+ +...+ =ε
ω−εFIG. 4. The geometrical series
describing the fluctuations propa-gator in the Cooper channel.
(a) (b)( c)
FIG. 5. The diagrammatic contributions to the transverse com-
ponent of jeconbefore averaging over the disorder. /H20851The obvious
counterpart diagram for /H20849a/H20850is not shown. /H20852(a) (b)( c)
FIG. 6. The diagrammatic contributions to the transverse com-
ponent of the jecon. Diagrams /H20849a/H20850and /H20849b/H20850describe the fluctuation of
the superconducting order parameter decorated by three Cooperonsand /H20849c/H20850is the Aslamazov-Larkin diagram. /H20851The obvious counterpart
diagrams for /H20849a/H20850and /H20849b/H20850are not shown. /H20852QUANTUM KINETIC APPROACH TO THE CALCULATION … PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-9the quasiparticles states. Therefore, the entire dependence of
the quasiparticle Green’s functions on the magnetic field isthrough the phase. Unlike the quasiparticles, the Landauquantization of the collective modes /H20849both the Cooperons
and the fluctuations of the superconducting order parameter /H20850
cannot be neglected because the quantization condition forthese modes is /H9024
c/T0/H110221, where in the diffusive regime
/H9024c=4eHD /cis the cyclotron frequency in the Cooper chan-
nel. Note that /H9024c/H11008/H9275c/H20849/H9255F/H9270/H20850/H11271/H9275cbecause the product of the
Fermi energy and the mean-free time is assumed to be a largeparameter. /H20851In/H9024
cthe effective charge is equal to 2 eand the
diffusion coefficient Dreplaces 1 /2mbecause in the Coop-
erons and the fluctuations propagators the term Dq2substi-
tutes the kinetic energy p2/2m./H20852
Similar to the quasiparticle Green’s functions, in the limit
of low magnetic field the Cooperons can be separated into
the phase exp /H208532ie/H20848r/H11032rA/H20849r1/H20850dr1/c/H20854and the gauge-invariant
part at H=0, C˜R,A/H20849/H9267,/H9280,/H9275−/H9280/H20850=/H20851/H11007i/H208492/H9280−/H9275/H20850/H9270−D/H11633/H92672/H20852−1, see
Appendix C in Ref. 27. At a finite magnetic field, one may
express the gauge-invariant part of the Cooperon propagatorusing the Landau-level quantization
C˜
NR,A/H20849/H9280,/H9275−/H9280/H20850=/H20851/H11007i/H208492/H9280−/H9275/H20850/H9270+/H9024c/H9270/H20849N+1 /2/H20850/H20852−1,/H2084941/H20850
where Nis the index of the Landau level. Similarly, the
propagator of the superconducting fluctuations written interms of the Landau levels becomes
L˜
NR,A/H20849/H9275/H20850=−1
/H9263/H20875ln/H20873T
Tc/H20874+/H9274R,A/H20849/H9275,N/H20850−/H9274/H208731
2/H20874+/H9269/H9275/H20876−1
,
/H2084942a /H20850/H9274R,A/H20849/H9275,N/H20850=/H9274/H208731
2/H11007i/H9275
4/H9266T+/H9024c/H20849N+1 /2/H20850
4/H9266T/H20874. /H2084942b /H20850
Here,/H9274/H20849x/H20850is the digamma function. The primary goal of this
calculation is to analyze the measurements of the Nernst ef-fect in superconducting films.
5,6In such films the electron
states are not quantized and therefore /H9263is the density of
states of three-dimensional electrons /H20849as well as D/H20850. The pa-
rameter /H9269/H110081//H20849/H9263/H9261/H9255F/H20850is important for understanding the dif-
ference in magnitude between the longitudinal and transversePeltier coefficients. The longitudinal Peltier coefficient,
/H9251xx,
contains an integral over the frequency that vanishes when/H9269=0 while the integrand determining
/H9251xyremains finite even
in the absence of /H9269. As a result, in the expression for the
Nernst coefficient given in Eq. /H208492/H20850the second term in the
numerator is smaller than the first one by a factor of the orderT//H20849
/H9263/H9261/H9255F/H20850.28
Using the expressions for the quasiparticle Green’s func-
tions, the Cooperons and the propagators of the supercon-
ducting fluctuations in the equilibrium state we may investi-
gate the contributions of Gˆ/H11633TandLˆ/H11633Tto the current. Recall
that we are interested in the transverse current. For illustra-tion, let us show how to find the transverse current for onerepresentative term out of the few contributions to theAslamazov-Larkin diagram,
jexcon/H20849r1/H20850=e/H11633yT
2T0/H20885d/H9280d/H9280/H11032d/H9275
/H208492/H9266/H208503/H20885dr2¯dr12lim
r12→r1/H20873/H116331x
2m+ieHy 1
4mc−/H1163312x
2m+ieHy 12
4mc/H20874
/H11003lim
r6→r7/H20873/H116337y
2m−ieHx 7
4mc−/H116336y
2m−ieHx 6
4mc/H20874g0R/H20849r1,r2,/H9280/H20850g0A/H20849r11,r2,/H9275−/H9280/H20850g0R/H20849r11,r12,/H9280/H20850CR/H20849r2,r3,/H9280,/H9275−/H9280/H20850
/H11003CR/H20849r10,r11,/H9280,/H9275−/H9280/H20850LeqR/H20849r3,r4,/H9275/H20850LeqA/H20849r9,r10,/H9275/H20850g0R/H20849r5,r6,/H9280/H11032/H20850g0A/H20849r5,r8,/H9275−/H9280/H11032/H20850g0R/H20849r7,r8,/H9280/H11032/H20850CR/H20849r4,r5,/H9280/H11032,/H9275−/H9280/H11032/H20850
/H11003CR/H20849r8,r9,/H9280/H11032,/H9275−/H9280/H11032/H20850F/H20849/H9280,/H9280/H11032,/H9275/H20850. /H2084943/H20850
In Fig. 7we indicate the spatial coordinates corresponding to the expression given above. Since in this part of the calculation
we concentrate on the integration over the spatial coordinates, we collect all the frequency-dependent factors into the functionF/H20849
/H9280,/H9280/H11032,/H9275/H20850=/H9280/H20853tanh /H20851/H9280/2T/H20852−tanh /H20851/H20849/H9280−/H9275/H20850/2T/H20852/H20854tanh /H20851/H20849/H9275−/H9280/H11032/H20850/2T/H20852/H11509nP/H20849/H9275/H20850//H11509/H9275and leave them aside for a while.
Next, we rewrite the Cooperons and the propagators of the superconducting fluctuations using the basis of the Landau levels
states,/H9272N,n/H20849r/H20850=RN,n/H20849r/H20850ein/H9278//H208812/H9266/H20851where RN,n/H20849r/H20850are the generalized Laguerre polynomials /H20852. In addition, we separate the qua-
siparticles Green’s functions into the phases and the gauge-invariant Green’s functions. Then, following the flux technique
introduced in Ref. 27, we rearrange Eq. /H2084943/H20850asr1
r12r11r10 r9r8r7r6r5r4r3r2
FIG. 7. The Aslamazov-Larkin diagram.KAREN MICHAELI AND ALEXANDER M. FINKEL’STEIN PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-10jexcon/H20849r1/H20850=e/H11633yT
4/H9266T0/H5129H2/H20885d/H9280d/H9280/H11032d/H9275
/H208492/H9266/H208503/H20858
N,M/H20885dr2¯dr12e−ieH/H20849r11−r1/H20850/H11003/H20849r1−r2/H20850/2ce−ieH/H20849r5−r6/H20850/H20849r6−r8/H20850/2clim
r12→r1/H20875/H116331x
2m+ieH /H20849y1−y2/H20850
4mc−/H1163312x
2m
−ieH /H20849y11−y12/H20850
4mc/H20876g˜0R/H20849r1−r2,/H9280/H20850g˜0A/H20849r11−r2,/H9275−/H9280/H20850g˜0R/H20849r11−r12,/H9280/H20850lim
r6→r7/H20875/H116337y
2m−ieH /H20849x7−x8/H20850
4mc−/H116336y
2m+ieH /H20849x5−x6/H20850
4mc/H20876
/H11003g˜0R/H20849r5−r6,/H9280/H11032/H20850g˜0A/H20849r5−r8,/H9275−/H9280/H11032/H20850g˜0R/H20849r7−r8,/H9280/H11032/H20850e−ieH/H20849r8−r11/H20850/H11003/H20849r11−r2/H20850/c−ieH/H20849r2−r5/H20850/H11003/H20849r5−r8/H20850/c/H9272N,0/H20849r2−r5/H20850/H9272M,0/H20849r8−r11/H20850
/H11003CNR/H20849/H9280,/H9275−/H9280/H20850CM2/H20849/H9280,/H9275−/H9280/H20850LNR/H20849/H9275/H20850LMA/H20849/H9275/H20850CNR/H20849/H9280/H11032,/H9275−/H9280/H11032/H20850CMR/H20849/H9280/H11032,/H9275−/H9280/H11032/H20850F/H20849/H9280,/H9280/H11032,/H9275/H20850, /H2084944/H20850
where /H5129H=/H20881c/2eHis the magnetic length in the Cooper
channel. In the last step we used the orthogonality of thegeneralized Laguerre polynomials /H20849an example for the treat-
ment of the propagators in this basis can be found in Ref.29/H20850. The first two exponents in Eq. /H2084944/H20850contain the magnetic
fluxes accumulated in the triangles /H20849r
1,r2,r11/H20850and
/H20849r5,r6,r8/H20850, respectively. One way to get the transverse cur-
rent is to extract the magnetic field from these two fluxes orfrom the diamagnetic terms. As a result the transverse currentappears with the coefficient
/H9275c/H9270. We neglect these terms; we
will see that when the magnetic field responsible for turningthe current to the transverse direction is extracted from theCooperons or the propagators of the superconducting fluc-tuations one gets a much larger factor of the order /H9024
C/T.
Therefore, the integration over the coordinates of the twotriangles can be done with the quasiparticle Green’s func-tions taken at H=0,
j
excon=−e/H11633yT
8/H92662T0/H5129H2/H92632/H92704/H20885d/H9280d/H9280/H11032d/H9275/H20885dr/H20858
N,M
/H11003/H208752D/H20873/H11509
/H11509x+ieHy
c/H20874/H9272N,0/H20849r/H20850/H20876
/H11003/H208752D/H20873/H11509
/H11509y−ieHx
c/H20874/H9272M,0/H20849r/H20850/H20876
/H11003CNR/H20849/H9280,/H9275−/H9280/H20850CMR/H20849/H9280,/H9275−/H9280/H20850LNR/H20849/H9275/H20850LMA/H20849/H9275/H20850CNR/H20849/H9280/H11032,/H9275−/H9280/H11032/H20850
/H11003CMR/H20849/H9280/H11032,/H9275−/H9280/H11032/H20850F/H20849/H9280,/H9280/H11032,/H9275/H20850. /H2084945/H20850
The integral over the coordinate corresponds to the matrix
element of the velocity operators /H20855N,0/H20841VxVy/H20841M,0/H20856, where
/H20841M,0/H20856=/H9272M,0is the quantum state of a particle with a mass
equal to 1 /2Din the MLandau level and zero angular mo-
mentum in the zdirection. Using the known properties of the
Laguerre polynomials, the matrix element can be written as
/H20855N,0/H20841VxVy/H20841M,0/H20856=2iD2/H20851/H20849N+1/H20850/H9254N,M−1−/H20849M+1/H20850/H9254M,N−1/H20852//H5129H2.
Finally, the contribution to the current becomes
jexcon=−ie/H11633yT
4/H92662T0/H5129H4/H92632D2/H92704/H20885d/H9280d/H9280/H11032d/H9275/H20858
N=0/H11009
/H20849N+1/H20850
/H11003CNR/H20849/H9280,/H9275−/H9280/H20850CN+1R/H20849/H9280,/H9275−/H9280/H20850CNR/H20849/H9280/H11032,/H9275−/H9280/H11032/H20850
/H11003CN+1R/H20849/H9280/H11032,/H9275−/H9280/H11032/H20850/H20851LNR/H20849/H9275/H20850LN+1A/H20849/H9275/H20850
−LN+1R/H20849/H9275/H20850LNA/H20849/H9275/H20850/H20852F/H20849/H9280,/H9280/H11032,/H9275/H20850. /H2084946/H20850In the limit H→0 when the quantization of the collective
modes can be neglected, one may replace the Cooperons andthe propagators of the superconducting fluctuations in Eq./H2084943/H20850by the product of the phase terms /H20849with charge 2 e/H20850and
the corresponding propagators in the absence of a magneticfield. Then, the contribution to the current at vanishinglysmall magnetic field can be found by employing the fluxtechnique of Ref. 27. One may check that the same result is
obtained when the transformation from the discrete sum intoan integral over a continuous variable is performed in Eq./H2084946/H20850.
Let us conclude with a remark regarding the diagram-
matic interpretation of the different contributions to j
econ.A s
already mentioned, the analytical structure and the expres-sions for the vertices of these diagrams were found from the
quantum kinetic equation. In principle, the same diagramscan be calculated using the Kubo formula. However, if forsimplicity one uses in the Kubo formula the heat currentoperator of noninteracting electrons described in Eq. /H208494/H20850, the
resulting expressions for these diagrams differ from thoseobtained in the quantum kinetic approach. Most important,as one can see from Eq. /H2084938/H20850, in the quantum kinetic ap-
proach the frequency accompanies the renormalized velocityso that the expression for the electric current is generally ofthe form eg/H20849
/H9280/H20850vi/H20849/H9280/H20850g/H20849/H9280/H20850/H9280vj/H20849/H9280/H20850/H11633jT/T0. In other words, the fre-
quency appears together with the velocity that was alreadyrenormalized by the interaction. On the other hand, owing tothe fact that the frequency in the simplified version of theKubo formula is attached to the external vertex before therenormalization of the velocity, the expression for the currenthas a totally different structure.
This is also the proper place to explain what is so unique
in the superconducting fluctuations in the diffusive limit thatleads to the giant Nernst effect. As any transverse currentcoefficient,
/H9251xycontains a difference of two almost equal
terms. In addition, like all thermoelectric coefficients the in-tegral over the frequency in
/H9251contains a factor of the qua-
siparticle frequency. Consequently, as discussed in AppendixC the contribution of the quasiparticles to the transversePeltier coefficient includes two small parameters. The first isthe usual
/H9275c/H9270that appears in all transverse currents. The
second is a reminiscence of the fact that the frequency factor/H20849that in the Boltzmann equation is converted into the energy /H20850
is responsible for the vanishing of the Peltier coefficient un-der the approximation of a constant density of states. When anonconstant density of states is considered, the integrationover the energy yields another small parameter proportionalQUANTUM KINETIC APPROACH TO THE CALCULATION … PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-11toT0//H9255F. Now we turn to the contribution of the supercon-
ducting fluctuations to the transverse component of jecon, and
consider Eq. /H2084946/H20850as a representative example. For the mo-
ment we ignore the factor /H9280inF/H20849/H9280,/H9280/H11032,/H9275/H20850associated with the
thermoelectric current. Then, the difference between the twoalmost identical terms results in an odd integrand with re-spect to the frequency of the superconducting fluctuations,
/H9275,
which potentially may lead to the vanishing of /H9251xy. So, how
can the superconducting fluctuations induce a strong Nernstsignal? The explanation lies in the fact that the Cooperonsaccompanying the superconducting fluctuation depend on thefrequency of the incoming/outgoing quasiparticles and notonly on the frequency
/H9275carried by the fluctuations /H20851Eq.
/H2084941/H20850/H20852. The dependence of the Cooperons on /H9280combined with
the frequency factor /H9280inF/H20849/H9280,/H9280,/H9275/H20850save the situation. This is
because the integration over /H9280results in an integrand, that is,
an even function of /H9275and hence, there is no longer danger
that the transverse Peltier coefficient vanishes. We shall seethat instead of the two small parameters obtained for thequasiparticles, the contribution of the superconducting fluc-tuations includes only one. Because of the extra sensitivity ofthese fluctuations to the magnetic field this parameter is/H9024
c/T0.
V. FINAL EXPRESSIONS FOR THE TRANSVERSE
COMPONENT OF jecon
If one examines Eq. /H2084938/H20850which presents the general ex-
pression for the contributions to the electric current from
Gˆ/H11633TandLˆ/H11633T, one may notice that not all the terms contain
the derivative of a Fermi distribution function. As one mayexpect, the terms in which the Fermi distribution function isnot differentiated contribute only to the transverse compo-
nent of j
econand not to the longitudinal one. After integration
over the Fermion degrees of freedom /H20849the frequency /H9280and
the coordinates of the quasiparticles Green’s functions /H20850, the
terms proportional to /H11509nF/H20849/H9280/H20850//H11509/H9280give two nonvanishing con-
tributions. The first one corresponds to the Aslamazov-Larkin diagram presented in Fig. 6/H20849c/H20850,
j
eicon1=/H9255ije/H11633jT
16/H92662T0/H92632/H20885d/H9275/H20858
N=0/H11009
/H20849N+1/H20850/H11509nP/H20849/H9275/H20850
/H11509/H9275/H20851LNR/H20849/H9275/H20850LN+1A/H20849/H9275/H20850
−LN+1R/H20849/H9275/H20850LNA/H20849/H9275/H20850/H20852/H20851/H9274R/H20849/H9275,N/H20850−/H9274R/H20849/H9275,N+1/H20850+/H9274A/H20849/H9275,N/H20850
−/H9274A/H20849/H9275,N+1/H20850/H20852/H20853/H9024c/H20849N+1 /2/H20850/H20851/H9274R/H20849/H9275,N/H20850−/H9274A/H20849/H9275,N/H20850/H20852
−/H9024c/H20849N+3 /2/H20850/H20851/H9274R/H20849/H9275,N+1/H20850−/H9274A/H20849/H9275,N+1/H20850/H20852/H20854, /H2084947/H20850
where the upper index in jcon1enumerates the contribution to
the current and /H9255ijis the antisymmetric tensor. The second
contribution generated by terms with the derivative
/H11509nF/H20849/H9280/H20850//H11509/H9280corresponds to the diagram with three Cooperons
shown in Fig. 6/H20849a/H20850,
jeicon2=−/H9255ije/H11633jT
4/H92662T0/H9263/H20885d/H9275/H20858
N=0/H11009
/H9024c/H20849N+1/H20850
/H11003/H208771
4LNR/H20849/H9275/H20850/H11509nP/H20849/H9275/H20850
/H11509/H9275/H20877i/H9275+/H9024c/H20849N+1 /2/H20850
4/H9266T0/H20851/H9274A/H11032/H20849/H9275,N/H20850
−/H9274R/H11032/H20849/H9275,N/H20850/H20852+i/H9275+/H9024c/H20849N+3 /2/H20850
/H9024c/H20851/H9274A/H20849/H9275,N/H20850−/H9274R/H20849/H9275,N/H20850−/H9274A/H20849/H9275,N+1/H20850+/H9274R/H20849/H9275,N+1/H20850/H20852/H20878+i
2LNAnP/H20849/H9275/H20850
/H11003/H20877/H11003i/H9275+/H9024c/H20849N+1 /2/H20850
/H208494/H9266T0/H208502/H9274A/H11033/H20849/H9275,N/H20850
+i/H9275+/H9024c/H20849N+3 /2/H20850
4/H9266T0/H9024c/H20851/H9274A/H11032/H20849/H9275,N/H20850−/H9274A/H11032/H20849/H9275,N+1/H20850/H20852/H20878
+N↔N+1/H20878+ c.c. /H2084948/H20850
Here/H9274R,A/H11032and/H9274R,A/H11033correspond to the first and second deriva-
tives of the digamma function defined in Eq. /H2084942b /H20850. The no-
tation N↔N+1 means that Nis replaced by N+1 and the
other way around in all the terms inside the curly brackets.Notice that there are no contributions proportional to thederivative of the distribution function which can be attrib-uted to the diagram shown in Fig. 6/H20849b/H20850.
Next, we discuss the group of terms that are proportional
ton
F/H20849/H9280/H20850. The diagrammatic interpretation of these terms,
which are generated by Eq. /H2084938/H20850, includes all three diagrams
presented in Fig. 6. However, one may check that the contri-
butions from the diagrams shown in Figs. 6/H20849b/H20850and6/H20849c/H20850are
canceled by a part of the contribution from the diagramgiven in Fig. 6/H20849a/H20850. The remaining contribution is
j
eicon3=−i/H9255ije/H11633jT
4/H92662T0/H9263/H20885d/H9275/H20858
N=0/H11009
/H20849N+1/H20850nP/H20849/H9275/H20850/H20877/H20851LNR/H20849/H9275/H20850
+LN+1R/H20849/H9275/H20850/H20852/H20851/H9274R/H20849/H9275,N/H20850−/H9274R/H20849/H9275,N+1/H20850/H20852
+/H9024c
4/H9266TLNR/H20849/H9275/H20850/H9274R/H11032/H20849/H9275,N/H20850+/H9024c
4/H9266T0LN+1R/H20849/H9275/H20850/H9274R/H11032/H20849/H9275,N+1/H20850/H20878.
/H2084949/H20850
In the derivation of the different contributions to jeconwe used
the following identities for products of the distribution func-tions:
n
F/H20849/H9280/H20850nF/H20849/H9275−/H9280/H20850=nP/H20849/H9275/H20850/H20851nF/H20849/H9280−/H9275/H20850−nF/H20849/H9280/H20850/H20852,
/H11509nF/H20849/H9275−/H9280/H20850
/H11509/H9275nF/H20849/H9280/H20850=/H11509nP/H20849/H9275/H20850
/H11509/H9275/H20851nF/H20849/H9280−/H9275/H20850−nF/H20849/H9280/H20850/H20852
−/H11509nF/H20849/H9275−/H9280/H20850
/H11509/H9275nP/H20849/H9275/H20850. /H2084950/H20850
Further analysis of jeconat arbitrary temperatures and mag-
netic fields shows that in jeicon1andjeicon2the integration over
the frequency accumulates at /H9275/H11011T/H112701//H9270. As a consequence
of the narrow range of the integration, the final expressionsfor these two contributions vanish in the limit T→0. In con-
trast, in j
eicon3the integration over the frequency is not limited
to small frequencies and, hence, the outcome of the integra-
tion depends logarithmically on the scattering rate 1 //H9270which
acts as an ultraviolet cutoff. In addition, as the temperaturegoes to zero there is even a more serious problem with thisterm because its prefactor is proportional to /H9024
c/T0. Such a
dependence on the temperature violates the third law of ther-modynamics. /H20851The connection between the third law of therKAREN MICHAELI AND ALEXANDER M. FINKEL’STEIN PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-12-l modynamics and the Nernst effect was discussed in Sec. I./H20852
We shall see that the dangerous parts in jeicon3are canceled out
by the magnetization current that up to now we have not yet
considered.
VI. MAGNETIZATION CURRENT AND THE THIRD LAW
OF THERMODYNAMICS
In this section we examine the magnetization current
given in Eq. /H2084937/H20850. In general, we need to insert the
/H11633T-dependent part of the Green’s function, Gˆloc-eq+Gˆ/H11633T,
into Eq. /H2084937/H20850. Since after the averaging over the disorder
Gˆ/H11633T/H20849r→r/H11032,/H9280/H20850is translationally invariant, it is clear that this
part of the Green’s function does not contribute to the mag-netization current. On the other hand, the explicit depen-dence of the local-equilibrium Green’s function on thecenter-of-mass coordinate leads to a nonzero contribution tothe magnetization current,
j
emag=2ic/H11633R/H11003M/H20849R/H20850/H20885d/H9280
2/H9266lim
/H9267→0/H9280R/H11612T
T0/H11509geq/H11021/H20849/H9267,/H9280;A,imp /H20850
/H11509/H9280.
/H2084951/H20850
Thus, Gˆ/H11633TandGˆloc-eqare complementary to each other while
the first contributes only to jecon, the other one fully deter-
mines jemag. One should recall that we are looking for a cur-
rent that does not vanish after spatial averaging, i.e., afterintegration with respect to the center-of-mass coordinate R.
Since in the process of averaging over Rwe may integrate
by parts, the magnetization current can be written as
j
eimag=2i/H9255ijcM zlim
/H9267→0/H20885d/H9280
2/H9266/H11633jT
T0geq/H11021/H20849/H9267,/H9280;A,imp /H20850. /H2084952/H20850
Here we integrated by parts over the frequency as well. One
may recognize that jemagis directly related to the magnetiza-
tion density at equilibrium,
jeimag=−/H9255ijc/H20855Mz/H20856/H11633jT
T0. /H2084953/H20850
The result demonstrates the strength of the quantum kinetic
approach. This method provides a way to derive both com-ponents of the current without engaging any thermodynami-cal arguments.
Actually, at this point one may employ in Eq. /H2084953/H20850the
known expression for the magnetization in the presence ofsuperconducting fluctuation. Still, since we are interested inthe interplay between the quasiparticle excitations and thefluctuations of the superconducting order parameter, let usderive the expression for the first-order correction to themagnetization induced by the fluctuations starting with Eq./H2084952/H20850. Using the standard identities for the Keldysh Green’s
function at equilibrium, one getsj
eimag=ic/H9255ij/H11633jT
T0lim
r/H11032→r/H20885d/H9280
2/H9266dr1dr2/H20851Mz/H20849r/H20850+Mz/H20849r/H11032/H20850/H20852
/H11003nF/H20849/H9280/H20850/H20851g0A/H20849r−r1,/H9280;A,imp /H20850/H9268A/H20849r1−r2,/H9280;A,imp /H20850
/H11003g0A/H20849r2−r/H11032,/H9280;A,imp /H20850−g0R/H20849r−r1,/H9280;A,imp /H20850
/H11003/H9268R/H20849r1−r2,/H9280;A,imp /H20850g0R/H20849r2−r/H11032,/H9280;A,imp /H20850/H20852./H2084954/H20850
Here, for convenience, we returned to the initial coordinates.
Next, we use the fact that the equilibrium Green’s function inthe absence of fluctuations satisfies the following identity
−
/H11509g0R,A//H11509H=g0R,AMg0R,A. Therefore, the expression for the
magnetization current can be rewritten as
jeimag=−2 ic/H9255ij/H11633jT
T0lim
r/H11032→r/H20885d/H9280
2/H9266dr1nF/H20849/H9280/H20850
/H11003/H20875/H11509g0A/H20849r−r1,/H9280;A,imp /H20850
/H11509Hz/H9268A/H20849r1−r/H11032,/H9280;A,imp /H20850
−/H11509g0R/H20849r−r1,/H9280;A,imp /H20850
/H11509Hz/H9268R/H20849r1−r2,/H9280;A,imp /H20850/H20876.
/H2084955/H20850
Finally, using the explicit expression for the self-energy and
rearranging all the terms we reformulate the expression forthe magnetization current in terms of the propagator of thesuperconducting fluctuations,
j
eimag=−ic/H9255ij/H11633jT
T0lim
r/H11032→r/H20885d/H9275
2/H9266dr1nP/H20849/H9275/H20850
/H11003/H20875/H11509/H9016R/H20849r−r1,/H9275;A,imp /H20850
/H11509HzŁR/H20849r1−r/H11032,/H9275;A,imp /H20850
−/H11509/H9016A/H20849r−r1,/H9275;A,imp /H20850
/H11509HzLA/H20849r1−r2,/H9275;A,imp /H20850/H20876
=−ic/H9255ij/H11633jT
T0/H11509
/H11509Hzlim
/H9267→0/H20885d/H9275
2/H9266nP/H20849/H9275/H20850/H20851lnLR−1/H20849/H9267,/H9275;A,imp /H20850
−l nLA−1/H20849/H9267,/H9275;A,imp /H20850/H20852. /H2084956/H20850
The transition from Eq. /H2084954/H20850to the last line in Eq. /H2084956/H20850is
illustrated in Fig. 8. Averaging over the disorder and trans-
forming from the expression for the propagator as a functionof the coordinates to the basis of Landau levels, one obtainsthe known expression for the correction to the magnetizationin the lowest order with respect to the fluctuations,
13,30
jeimag=/H9255ij/H11633jT
T0/H11509
/H11509HeH
/H9266/H20885d/H9275
2/H9266i/H20858
N=0/H11009
nP/H20849/H9275/H20850/H20853ln/H20851LNR/H20849/H9275/H20850/H20852−1
−l n /H20851LNA/H20849/H9275/H20850/H20852−1/H20854. /H2084957/H20850
The discussion of higher order corrections to the magnetiza-
tion is given in Ref. 30.
Similar to jecon3in Eq. /H2084949/H20850, the integration over the fre-
quency in the magnetization current is restricted by the scat-
tering rate and at low temperature jemagalso diverges as
/H9024c/T. The opposite sign of the magnetization current relative
tojecon3suggests that these dangerous parts may cancel eachQUANTUM KINETIC APPROACH TO THE CALCULATION … PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-13other making the Nernst signal compatible with the third law
of thermodynamics. Another hint for this cancellation is the
similar analytical structure of jemagandjecon3./H20851All the terms in
Eqs. /H2084949/H20850and /H2084957/H20850are a product of either retarded or ad-
vanced functions only. /H20852In Appendix B we show that the
diverging parts of the magnetization current indeed identi-
cally cancel out the diverging parts of jecon3. We demonstrate
that the total current is independent of /H9270in the whole tem-
perature range T/H112701//H9270. As a result, the Nernst signal is regu-
lar at T→0. Moreover, the contributions which are constant
with respect to the temperature also vanish and the remainingterms are linear in T.
VII. PHASE DIAGRAM FOR THE NERNST EFFECT—
COMPARISON BETWEEN THE THEORETICAL RESULTS
AND THE EXPERIMENT
In the following part we present the theoretical expres-
sions for the transverse Peltier coefficient for a superconduct-ing film in the normal state for various regions of the tem-perature and the magnetic field. The phase diagram for thePeltier coefficient is plotted in Fig. 9. In the area below the
line ln /H20851T/T
c/H20849H/H20850/H20852=/H9024c/4/H9266Tthe Landau-level quantization of
the superconducting fluctuations becomes essential. The lineln/H20851H/H
c2/H20849T/H20850/H20852=4/H9266T//H9024cseparates the regions of classical and
quantum fluctuations. From now on T0is replaced by T
which represents the spatially averaged temperature.
For a small magnetic field, /H9024c/H11270T, close to the transition
temperature /H20849T/H11015Tc/H20850the leading contribution to /H9251xyis given
by the Aslamazov-Larkin term /H20851see Fig. 6/H20849c/H20850/H20852and the mag-
netization current,
/H9251xy/H11015e/H9024c
192TlnT/Tc/H20849H/H20850. /H2084958/H20850
In the previous section we discussed in details the impor-
tance of the magnetization current in canceling the quantumcontributions to the Nernst signal. In the vicinity of T
cone
can interpret the expression in Eq. /H2084958/H20850in terms of the clas-
sical picture in which the Cooper pairs with a finite lifetimeare responsible for the thermoelectric current. The magneti-zation current is just equal to −2 /3 of the leading-order con-
tribution from the Aslamazov-Larkin term. Note that unlikethe electric conductivity,
/H9268xx, for which the anomalous
Maki-Thompson12and the Aslamazov-Larkin terms yield
comparable corrections, the contribution from the anomalousMaki-Thompson term to the Nernst signal is /H11011/H20849T//H9255
F/H208502/H112701
smaller than the one given by Eq. /H2084958/H20850. Therefore, it is natu-
ral that in the vicinity of Tcour result coincides with the
expression13,31obtained phenomenologically from the time-dependent Ginzburg-Landau /H20849TDGL /H20850equation.
When temperature is increased further away from the
critical temperature, the sum of the contributions to the trans-verse Peltier coefficient from all the diagrams and the mag-netization current yields
/H9251xy/H11015e/H9024c
24/H92662TlnT/Tc. /H2084959/H20850
A comparison between the transverse Peltier coefficient in
the vicinity of Tc/H20851Eq. /H2084958/H20850/H20852and far from the transition /H20851Eq.
/H2084959/H20850/H20852reveals that the two expressions differ only by a nu-
merical coefficient. The similarity between the expressionsfor
/H9251xyin the two different limits is not seen in paraconduc-
tivity. This is a consequence of the cancellation of the quan-
tum contributions to jeconby the magnetization current, which
is specific for the Nernst effect. Away from the critical regionT/H11271T
c, the quantum nature of the fluctuations reveals itself in
contributions to jecon3andjemagthat contain an integration
over a wide interval of frequencies between Tand 1 //H9270.A sa
result, these terms become of the order ln /H20849ln 1 /T/H9270/H20850
−ln /H20849lnT/Tc/H20850. However, as we show in Appendix B these
/H9270-dependent terms in jecon3andjemagcancel each other.32The
Peltier coefficient far from Tcdemonstrates how the third law
of thermodynamics constrains the magnitude of the Nernstsignal not only at T→0 but also at high temperatures, T
/H11271T
c.
The comparison of our result with the experimental ob-
servation of Ref. 6for two Nb 0.15Si0.85films of thicknesses
35 and 12.5 nm /H20849with critical temperatures Tc=380 mK and
Tc=165 mK, respectively /H20850is given in Fig. 10. The Peltier
coefficient depends on the mean-field temperature of the su-
perconducting transition, TcMF, and on the diffusion coeffi-
cient through /H9024c. Throughout the paper we fit the data using
the same diffusion coefficient D=0.187 cm2/s which is
within the measurement accuracy of the value that is ex-tracted from experiment as described in Ref. 6.Mz ln −ddH-1
FIG. 8. An illustration of the relation between the magnetization
current term that is obtained from the local-equilibrium Green’sfunction and the thermodynamic diagram for the magnetization.
Eq. 58Eq. 59
Eq. 60 Eq. 61
FIG. 9. The phase diagram for the Peltier coefficient /H9251xy.W e
indicate the equations in the text which give the corresponding ex-pressions for
/H9251xyin the different limits. /H9024c=4eHD /cis the cyclo-
tron frequency for the fluctuations of the superconducting orderparameter in the diffusive regime.KAREN MICHAELI AND ALEXANDER M. FINKEL’STEIN PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-14The cancellation of the terms proportional to /H9024c/Tin the
limit T/H11270/H9024 cwas described in the previous section. /H20849Without
this cancellation we would get a nonzero Nernst effect in thelimit T→0 and the third law of thermodynamics would be
violated. /H20850After the cancellation, the remaining contributions
to
/H9251xyin the limit T→0 are linear in the temperature,
/H9251xy/H11015−eTln 3
3/H9024c/H20851lnH/Hc2/H20849T/H20850/H208522forH/H11015Hc2/H2084960/H20850
and
/H9251xy/H110152eT
3/H9024clnH/Hc2forH/H11271Hc2. /H2084961/H20850
Similar to the limit /H9024c/H11021Tthe integrals determining the final
expression for /H9251accumulate at low frequency. This situation
is not typical for fluctuations induced by a quantum phasetransition. Notice that
/H9251xychanges its sign in this region.
Since the transverse signal is nondissipative the sign of theeffect is not fixed. As mentioned before, in the vicinity of T
c
for/H9024c/H11270T, the main contribution to the Peltier coefficient is
from the Aslamazov-Larkin term and the magnetization cur-rent. The magnetization current is opposite in sign to the
Aslamazov-Larkin terms and equals 2/3 of it. When crossingto the region ln /H20851T/T
c/H20849H/H20850/H20852/H11021/H9024 c/T/H20849see the phase diagram in
Fig. 9/H20850the contribution from the magnetization current
grows. To the first order in /H9024c/Tthe magnetization current
cancels the Aslamazov-Larkin term and the Peltier coeffi-cient turns out to be proportional to O/H20851/H20849/H9024
c/T/H208502/H20852. Lowering
further the temperature and increasing the magnetic field onereaches the region /H9024
c/H11022Tand ln /H20851H/Hc2/H20849T/H20850/H20852/H11021T//H9024c. In this
region the magnetization current becomes dominant. Since
the magnetization current gives a contribution that is oppo-site in sign to the Aslamazov-Larkin term, we obtain that thePeltier coefficient is negative.
In Fig. 11we plot the Peltier coefficient for the 35 nm
film as a function of the magnetic field at a temperature
higher than T
c. We take TcMF=385 mK to be slightly higher
than the measured critical temperature anticipating a smallsuppression of the temperature of the transition by fluctua-tions. /H20851The data in Fig. 11, unlike the data in Fig. 10/H20849a/H20850,i s
presented in linear rather than a logarithmic scale. Therefore,this fit is much more sensitive to the input parameters com-pared to the one in Fig. 10/H20849a/H20850. As a result, the small deviation
ofT
cMFfrom the measured Tccan be noticed. For consis-
tency, we use the same value of TcMFalso in Fig. 10/H20849a/H20850./H20852
Figure 11demonstrates the agreement between the theoreti-
cal expressions and the experimental observation for a broadrange of magnetic fields. In addition, we show that the ex-perimental data is well described by Eq. /H2084958/H20850in the limit of
vanishing magnetic field; see inset of Fig. 11. Since Eq. /H2084958/H20850
is valid in the limit /H9024
c/H11270T, it can describe only the first few
point in the measurement. In order to fit the entire range ofthe magnetic field we had to include higher order terms in/H9024
c/T. For that we needed to sum the contributions from all
diagrams and the magnetization current. We performed thecalculation assuming that ln /H20851T/T
c/H20849H/H20850/H20852/H112701; therefore the the-
oretical curve starts to deviate from the measured data whenln/H20851T/T
c/H20849H/H20850/H20852is no longer small /H20849H/H110151T /H20850.0.02 0.05 0.1 0.5 1 210−510−410−310−2
ln(T/Tc )αxy/H(µA/KT)
ln(T/Tc)αxy/H(µA/KT)
0.02 0.1 110−2
10−3
10−4
(a)
ln(T/Tc )αxy/H(µA/KT)
1 2 0.5 3 0.210−410−2
10−3
10−510−5ln(T/Tc)αxy/H(µA/KT)
0.2 0.5 1110−410−310−3
(b)
FIG. 10. /H20849Color online /H20850The transverse Peltier coefficient /H9251xy
divided by the magnetic field Has a function of ln T/Tcin the limit
H→0 for films of thicknesses /H20849a/H2085035 nm and /H20849b/H2085012.5 nm. The
experimental data of Ref. 6is presented by the black squares and
the solid line corresponds to the theoretical curve given by Eq. /H2084959/H20850.
The inset presents the fitting of the data in the vicinity of Tcwith
Eq. /H2084958/H20850.H(Tesla )αxy(µA/K)
0.5 1 1.5 201234x1 0−4
0 0.1 0.2 0.30123x1 0−4
H (Tesla)αxy(µA/K)
ln(T/Tc(B)=
Ωc/T
FIG. 11. /H20849Color online /H20850The transverse Peltier coefficient /H9251xyas
a function of the magnetic field measured at T=410 mK. The black
squares correspond to the experimental data of Ref. 8while the
solid line describes the theoretical result. The arrow on the phasediagram illustrates the direction of the measurement. In the inset thelow magnetic field data is fitted with the theoretical curve given byEq. /H2084958/H20850.QUANTUM KINETIC APPROACH TO THE CALCULATION … PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-15VIII. SUMMARY
We demonstrated that the contribution from the fluctua-
tions of the superconducting order parameter to the Nernsteffect in disordered films is dominant and can be observedfar away from the transition. We showed that the importantrole of the magnetization current is in canceling the quantumcontributions, thus making the Nernst signal compatible withthe third law of thermodynamics. The third law of thermo-dynamics constrains the magnitude of the Nernst signal notonly at low temperatures but also far from T
c. As a conse-
quence of this constraint the phase diagram is less rich anddiverse than one expects in the vicinity of a quantum phasetransition.
The Nernst effect provides an excellent opportunity to test
the use of the quantum kinetic approach in the description ofthermoelectric transport phenomena. We showed that in thisscheme we get automatically all contributions to the Nernstcoefficient as response to the temperature gradient, in par-ticular, the one from the magnetization current. This is anadvantage of the quantum kinetic approach but it is not theonly one. This method also allows us to verify the Onsagerrelations; a comprehensive discussion of this issue is pre-sented in Appendix A. The fact that the we were able to findindependently the two off-diagonal components of the con-
ductivity tensor,
/H9251ijand/H9251˜ij, and verify that they are con-
nected through the Onsager relation assures that the quantumkinetic approach developed in this paper and in Ref. 17gives
the correct expressions for the electric and heat currents.
Finally, we should remark that our results for the Peltier
coefficient differ in few aspects from those obtained recentlyin Ref. 33using the Kubo formula. As we already discussed
in the end of Sec. IV, the simplified Kubo formula cannot
give the correct electric current as a response to a tempera-ture gradient. Therefore, the claim of the authors of Ref. 33
that the difference between the Nernst signal calculated usingthe simplified Kubo formula and the quantum kinetic ap-proach is only in the numerical coefficients is unacceptable.The expression given in Ref. 33for
/H9251xyin the vicinity of Tc
cannot fit the experimental data and it also contradicts the
phenomenological result of the TDGL.31The only fit of the
experimental data presented in Ref. 33is a logarithmic plot
of the Nernst signal as a function of temperature using theformula for temperatures not too close to T
c. Such a logarith-
mic plot is not very sensitive to the numerical coefficients.The striking agreement between our results and the experi-mental data, in particular, our ability to obtain the nontrivialdependence of the Nernst signal on the magnetic field andthe fact that we reproduced the phenomenological result
31
reinforces us in the correctness of our method.
ACKNOWLEDGMENTS
We thank O. Entin-Wohlman and the condensed-matter
theory group in TAMU for their interest in this work and forthe extended discussions. The research was supported by theU.S.-Israel BSF and the German-Israeli Foundation for Sci-entific Research and Development.APPENDIX A: ONSAGER’S RELATIONS
In this appendix we compare the electric current arising as
a response to a temperature gradient and the heat currentgenerated by an electric field. We verify that for the Gaussianfluctuations of the superconducting order parameter, the twoexpressions are connected through the Onsager relations,
34
/H9251˜ij/H20849B/H20850=T0/H9251ji/H20849−B/H20850./H20851In a similar way, in Ref. 17we demon-
strated the Onsager relations for the longitudinal current inthe presence of the Coulomb interaction. /H20852
The derivation of the electric current induced by the tem-
perature gradient was presented in Sec. II, where we dis-
cussed the two contributions to the current. The first one,
j
econ, was found using the continuity equation and its expres-
sion before the expansion in the superconducting fluctuationsis given in Eq. /H2084938/H20850. The second contribution, analyzed in
Sec. VI, is from the magnetization current. This term con-
tributes only to the transverse current and it can be written as
j
eimag=c/H9255i,jMz/H20849−/H11633jT/T0/H20850.
Next, we sketch the derivation of the heat current as a
response to a uniform electric field. For this purpose we use
the heat continuity equation, Q˙/H20849r,t/H20850+/H11633jhcon/H20849r,t/H20850=jeconE. The
product jeconEdescribes the work performed by the electric
field on the current. Since the electric field cannot perform
any work on the magnetization current, only jeconenters to the
RHS of the continuity equation.35The heat density in the
presence of an electromagnetic field is a function of the mag-netization and the electrochemical potential,
dQ/H20849r,t/H20850=dh/H20849r,t/H20850−/H20851
/H9262−e/H9272/H20849r/H20850/H20852dn/H20849r,t/H20850+MdH/H20849r,t/H20850,
/H20849A1/H20850
where h/H20849r,t/H20850is the Hamiltonian density. To find the time
derivative of the magnetic field, we turn to the Maxwell
equation H˙=−c/H11633/H11003E. Thus, the heat current is described by
the following equation:
/H11633jh=−dh/H20849r,t/H20850+/H20849/H9262−e/H9272/H20850dn/H20849r,t/H20850−jecon/H11633/H9272+cM/H11633/H11003E.
/H20849A2/H20850
In Ref. 17we showed that for H=0 the expression for the
heat current found from the continuity equation is
jhcon/H20849H=0/H20850= lim
r/H11032→r
t/H11032→t/H20851/H11509t+ie/H9272/H20849r/H20850−/H11509t/H11032+ie/H9272/H20849r/H11032/H20850/H20852
/H11003/H20885dr1dt1/H20851vˆ=/H20849r,t;r1,t1/H20850Gˆ=/H20849r1,t1;r/H11032,t/H11032/H20850/H20852/H11021+ H.c.
/H20849A3/H20850
This result was obtained for the Coulomb interaction but it
also holds for the superconducting fluctuations because, un-like the charge, there is no principle difference between theway the fluctuations in the density and Cooper channelscarry heat. Here, we use the fact that in the presence of aninteraction field, such as /H9004, which does not have its own
dynamics, the heat current can be formulated in terms of thequasiparticle Green’s function alone. This is compatible withthe observation that according to the kinetic equation given
in Eq. /H2084923/H20850, the temperature gradient is coupled to Lˆ
=onlyKAREN MICHAELI AND ALEXANDER M. FINKEL’STEIN PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-16through the quasiparticle Green’s functions inside /H9016ˆ=./H20851When
interactions with dynamic fields such as phonons are studied
the heat current acquires additional terms. /H20852
Although we restrict our derivation to the regime of linear
response with respect to the electric field, the source term
jeconEis still important. This is because the source term
makes the heat current to be gauge invariant as Eq. /H20849A3/H20850
reveals. In principle, there may be an additional contributionto the heat current from the charge current carried by thesuperconducting fluctuations /H20851corresponding to the RHS of
Eq. /H2084929/H20850/H20852. This issue is not addressed here because such con-
tributions are beyond the linear response.
When we consider the effect of applying a magnetic field
the expression for the heat current given in Eq. /H20849A3/H20850has to
be modified. The first change is simply to include the vectorpotential in the velocity as it is shown in Eq. /H2084922/H20850. We denote
the contribution from the heat current given in Eq. /H20849A3/H20850with
the modified velocity as j
hcon1. Besides, there is an additional
contribution to the heat current, jhcon2, from the last term in
Eq. /H20849A1/H20850that contains the magnetization,
/H11633jhcon2=cM/H20849/H11633/H11003E/H20850=c/H11633/H20849E/H11003M/H20850. /H20849A4/H20850
Here we used the fact that under the condition of a constant
magnetic field /H11633/H11003M=0. For a setup similar to the Nernst
measurement /H20849see Fig. 1/H20850in which the magnetic field is
aligned along the zdirections, the contribution of the mag-
netization to the heat current is
jhicon2=c/H9255ijEjMz. /H20849A5/H20850
At this stage, one may wonder whether there is a contribu-
tion to the transverse current that cannot be found from thecontinuity equation, i.e., a term of the form /H11633/H11003W.I nt h e
case of the electric current generated by the temperature gra-dient, we saw the term of this kind is the magnetizationcurrent. This term does not vanish and contributes to thetransport electric current because the nonuniform tempera-ture induces a coordinate dependent magnetization /H20849see Sec.
VI/H20850. However, in the presence of a constant electric field, the
system remains uniform. Therefore, the quantity that we de-noted by Wshould be independent of the spatial coordinate
and, hence, /H11633/H11003W=0.
Let us compare between the electric current as a response
to a temperature gradient and the heat current generated byan electric field when a magnetic field is applied. One may
immediately notice that j
hcon2andjemag/H20851given in Eqs. /H20849A5/H20850
and /H2084953/H20850, respectively /H20852satisfy the Onsager relations,
jhicon2/H20849H/H20850
Ej=T0jejmag/H20849−H/H20850
−/H11633iT. /H20849A6/H20850
It is interesting that these two terms coincide although they
seem to have a different origin. The contribution of the mag-netization, /H11633/H11003M, to the electric current cannot be found
from the continuity equation and it is nonzero due to the
dependence of Gˆ
loc-eqon the center-of-mass coordinate.
When the response to a uniform electric field is considered,the Green’s functions are independent of the center-of-masscoordinate. Still, there is an equivalent contribution to theheat current arising from the continuity equation.
Now, we show that j
econandjhcon1also satisfy the Onsager
relations. In order to find jhcon1we need to know the expres-
sion for the electric field-dependent propagators. The electricfield-dependent Green’s function can be written in the fol-lowing form:
17
GˆE/H20849/H9267,/H9280;A,imp /H20850=gˆeq/H20849/H9280/H20850/H9018ˆE/H20849/H9280/H20850gˆeq/H20849/H9280/H20850
−ieE
2/H20875/H11509gˆeq/H20849/H9280/H20850
/H11509/H9280vˆeq/H20849/H9280/H20850gˆeq/H20849/H9280/H20850−gˆeq/H20849/H9280/H20850vˆeq/H20849/H9280/H20850/H11509gˆeq/H20849/H9280/H20850
/H11509/H9280/H20876.
/H20849A7/H20850
The above equation is similar to Eq. /H2084921/H20850forGˆ/H11633T. Since/H9018ˆE
contains also the electric field-dependent propagator of the
superconducting fluctuations, we have to find the equation
forLˆE. Owing to the fact that the superconducting fluctua-
tions carry charge, their coupling to the electric field is morecomplicated than their dependence on the temperature gradi-ent described in Eqs. /H2084926/H20850and /H2084927/H20850,
Lˆ
E/H20849/H9267,/H9275;A,imp /H20850=−Lˆeq/H20849/H9275/H20850/H9016ˆE/H20849/H9275/H20850Lˆeq/H20849/H9275/H20850
+ieE/H20875/H11509Lˆeq/H20849/H9275/H20850
/H11509/H9275Vˆeq/H20849/H9275/H20850Lˆeq/H20849/H9275/H20850
−Lˆeq/H20849/H9275/H20850Vˆeq/H20849/H9275/H20850/H11509Lˆeq/H20849/H9275/H20850
/H11509/H9275/H20876. /H20849A8/H20850
Inserting the expression for GˆEinto Eq. /H20849A3/H20850and extracting
the lesser component, we get
jhicon1=eEj
2/H20885d/H9280
2/H9266/H9280/H11509nF/H20849/H9280/H20850
/H11509/H9280/H20851viR/H20849/H9280/H20850geqR/H20849/H9280/H20850vjA/H20849/H9280/H20850geqA/H20849/H9280/H20850
+viR/H20849/H9280/H20850geqR/H20849/H9280/H20850vjR/H20849/H9280/H20850geqA/H20849/H9280/H20850−viR/H20849/H9280/H20850geqR/H20849/H9280/H20850vjR/H20849/H9280/H20850geqR/H20849/H9280/H20850
−geqR/H20849/H9280/H20850vjR/H20849/H9280/H20850geqR/H20849/H9280/H20850viA/H20849/H9280/H20850/H20852+eEj/H20885d/H9280
2/H9266/H9280nF/H20849/H9280/H20850
/H11003/H20875viR/H20849/H9280/H20850/H11509geqR/H20849/H9280/H20850
/H11509/H9280vjR/H20849/H9280/H20850geqR/H20849/H9280/H20850
−viR/H20849/H9280/H20850geqR/H20849/H9280/H20850vjR/H20849/H9280/H20850/H11509geqR/H20849/H9280/H20850
/H11509/H9280/H20876+i/H20885d/H9280
2/H9266/H9280viR/H20849/H9280/H20850geqR/H20849/H9280/H20850
/H11003/H20853/H9018E/H11021/H20849/H9280/H20850/H208511−nF/H20849/H9280/H20850/H20852+/H9018E/H11022/H20849/H9280/H20850nF/H20849/H9280/H20850/H20854/H20851geqR/H20849/H9280/H20850−geqA/H20849/H9280/H20850/H20852
+ c.c. /H20849A9/H20850
The fulfillment of Onsager relation demands microscopic re-
versibility, which in our case implies that Gˆ/H20849r,r/H11032,/H9280;H/H20850
=Gˆ/H20849r/H11032,r,/H9280;−H/H20850andLˆ/H20849r,r/H11032,/H9280;H/H20850=Lˆ/H20849r/H11032,r,/H9280;−H/H20850. Since the
currents contain a trace over the coordinates, it is obviousQUANTUM KINETIC APPROACH TO THE CALCULATION … PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-17that the first two terms in Eq. /H20849A9/H20850and the first two terms in
Eq. /H2084938/H20850are connected through the Onsager relations.
Let us examine the remaining terms in jeconandjhcon1. The
electric current contains not only the contribution of /H9018ˆ/H11633Tbut
also terms with /H9016ˆ/H11633Twhile the remaining part of the heat
current contains only /H9018ˆE. Actually, when we treat /H9018ˆEwe
must consider the possibility that Eenters also through Lˆ.
Then, as we show later, since the equation for LˆEincludes
more terms than the equation for Lˆ/H11633T/H20851see Eqs. /H20849A8/H20850and
/H2084927/H20850/H20852, the corresponding contributions to the electric and heat
currents coincide.
Let us start with the contributions to the currents in which
/H9018ˆdepends on the electric filed/temperature gradient through
Gˆrather than Lˆ. Since we are interested in the effect of
Gaussian fluctuations, we can use the expressions for the GE
andG/H11633Tin the absence of interactions,
GF/H11021/H20849/H9280/H20850=i
2Fj/H20849/H9280/H20850nF/H20849/H9280/H20850/H20875/H11509g0R/H20849/H9280/H20850
/H11509/H9280v0jg0R/H20849/H9280/H20850−g0R/H20849/H9280/H20850v0j/H11509g0R/H20849/H9280/H20850
/H11509/H9280/H20876
+i
2Fj/H20849/H9280/H20850/H11509nF/H20849/H9280/H20850
/H11509/H9280g0R/H20849/H9280/H20850v0j/H20851g0A/H20849/H9280/H20850−g0R/H20849/H9280/H20850/H20852− c.c.,
/H20849A10 /H20850
where Fj/H20849/H9280/H20850is equal to eEjand/H9280/H11633jT/T0, respectively. The
only difference between the above equation for the lesser
component of GˆFand the one for the greater component is
that in the latter the distribution function, nF/H20849/H9280/H20850should be
replaced by nF/H20849/H9280/H20850−1. /H20851In Eq. /H20849A10 /H20850and below we start to
place the spatial direction indices also as superscripts. /H20852
Using the identities given in Eq. /H2084950/H20850, one may notice that
for the discussed contributions to the currents /H20851denoted as
je,h/H20849GˆF,H/H20850/H20852only the terms proportional to the derivative of
the distribution function in Eq. /H20849A10 /H20850do not vanish. As a
result we get
je/hi/H20849GˆF,H/H20850=−i
2/H20885d/H9280d/H9275
/H208492/H9266/H208502dr2¯dr6fe/h/H20849/H9280/H20850Fj/H20849/H9275−/H9280/H20850v0i/H20849r6,r1/H20850
/H11003/H20851g0R/H20849r1,r2,/H9280/H20850−g0A/H20849r1,r2,/H9280/H20850/H20852/H20851LeqR/H20849r2,r5,/H9275/H20850
−LeqA/H20849r2,r5,/H9275/H20850/H20852v0j/H20849r4,r3/H20850/H20851g0R/H20849r5,r4,/H9275−/H9280/H20850
−g0A/H20849r5,r4,/H9275−/H9280/H20850/H20852/H20851g0R/H20849r3,r2,/H9275−/H9280/H20850
−g0A/H20849r3,r2,/H9275−/H9280/H20850/H20852/H20851g0R/H20849r6,r1,/H9280/H20850
−g0A/H20849r6,r1,/H9280/H20850/H20852/H11509nP/H20849/H9275/H20850
/H11509/H9275/H20851nF/H20849/H9280−/H9275/H20850−nF/H20849/H9280/H20850/H20852.
/H20849A11 /H20850
Here, fe/H20849/H9280/H20850=−eand fh/H20849/H9280/H20850=/H9280. The diagrammatic representa-
tion of the above expression is presented in Fig. 12. Recall
that the bare velocity v0/H20849r,r/H11032/H20850/H11008/H9254/H20849r−r/H11032/H20850. Finally, we shall
change the frequency as follows /H9280→/H9275−/H9280,je/hi/H20849GˆF,H/H20850=−i
2/H20885d/H9280d/H9275
/H208492/H9266/H208502dr2¯dr6fe/h/H20849/H9275−/H9280/H20850Fj/H20849/H9280/H20850v0i/H20849r6,r1/H20850
/H11003/H20851g0R/H20849r1,r2,/H9275−/H9280/H20850−g0A/H20849r1,r2,/H9275−/H9280/H20850/H20852
/H11003/H20851LeqR/H20849r2,r5,/H9275/H20850−LeqA/H20849r2,r5,/H9275/H20850/H20852v0j/H20849r4,r3/H20850
/H11003/H20851g0R/H20849r5,r4,/H9280/H20850−g0A/H20849r5,r4,/H9280/H20850/H20852/H20851g0R/H20849r3,r2,/H9280/H20850
−g0A/H20849r3,r2,/H9280/H20850/H20852/H20851g0R/H20849r6,r1,/H9275−/H9280/H20850
−g0A/H20849r6,r1,/H9275−/H9280/H20850/H20852/H11509nP/H20849/H9275/H20850
/H11509/H9275/H20851nF/H20849/H9280−/H9275/H20850−nF/H20849/H9280/H20850/H20852.
/H20849A12 /H20850
the condition of microscopic reversibility mentioned previ-
ously, we see that the electric current created by a tempera-ture gradient as given in Eq. /H20849A11 /H20850/H20851with f
e/H20849/H9280/H20850=−eand
Fj/H20849/H9275−/H9280/H20850=/H20849/H9275−/H9280/H20850/H11633jT/T0/H20852and the heat current generated by
an electric field as described in Eq. /H20849A12 /H20850/H20851with fh/H20849/H9275−/H9280/H20850
=/H9275−/H9280andFj/H20849/H9280/H20850=Ej/H20852satisfy the Onsager relations. Thus, the
microscopic reversibility and the Onsager relations emergingfrom it correspond to reading the diagram in Fig. 12from
right to left instead of left to right /H20849i.e., reading it in Hebrew
instead of English /H20850.
Next, we shall examine the contribution to the currents in
which the propagator of the superconducting fluctuations /H20849or
the polarization operator /H20850depends on the electric field/
temperature gradient. We start with the corresponding contri-bution to the heat current as a response to an electric field,j
h/H20849LE,H/H20850. Using the identities for the distribution functions
in Eq. /H2084950/H20850, this term can be written as
jhi/H20849LE,H/H20850=/H20885d/H9280d/H9275
/H208492/H9266/H208502dr2¯dr4/H9280v0i/H20849r4,r1/H20850/H20851nF/H20849/H9280/H20850−nF/H20849/H9280−/H9275/H20850/H20852
/H11003/H20851geqA/H20849r1,r2,/H9280/H20850−geqR/H20849r1,r2,/H9280/H20850/H20852/H20851geqA/H20849r3,r4,/H9280/H20850
−geqR/H20849r3,r4,/H9280/H20850/H20852/H20851geqA/H20849r3,r2,/H9275/H9280/H20850−geqR/H20849r3,r2,/H9275−/H9280/H20850/H20852
/H11003/H20853/H208511+nP/H20849/H9275/H20850/H20852LE/H11021/H20849r2,r3,/H9275/H20850−nP/H20849/H9275/H20850LE/H11022/H20849r2,r3,/H9275/H20850/H20854.
/H20849A13 /H20850
Using the definition of /H9016ˆ/H11633T, one may notice that the expres-
sion for the heat current can be rewritten in the formεω−ε
εω−εv(, )i 1r6r2r
5rv( , )j 3r4r
FIG. 12. A diagrammatic representation of the contribution to
the current written in Eq. /H20849A11 /H20850. For simplicity the scattering by
impurities is not indicated.KAREN MICHAELI AND ALEXANDER M. FINKEL’STEIN PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-18jhi/H20849LE,H/H20850=T0
/H11633iT/H20885d/H9275
2/H9266dr/H11032/H20875/H11509nP/H20849/H9275/H20850
/H11509/H9275/H20876−1
/H20853/H9016/H11633iT/H11021/H20849r/H11032,r,/H9275/H20850
/H11003/H208511+nP/H20849/H9275/H20850/H20852−/H9016/H11633iT/H11022/H20849r/H11032,r,/H9275/H20850nP/H20849/H9275/H20850/H20854
/H11003/H20853/H208511+nP/H20849/H9275/H20850/H20852LEj/H11021/H20849r,r/H11032,/H9275/H20850−nP/H20849/H9275/H20850LEj/H11022/H20849r,r/H11032,/H9275/H20850/H20854.
/H20849A14 /H20850
Let us now turn to the equivalent contributions to the
electric current created by a temperature gradient, je/H20849L/H11633T,H/H20850.
Performing manipulations similar to those in the expressionfor the heat current we get
j
ej/H20849L/H11633T,H/H20850=−ie/H20885d/H9275
2/H9266dr2¯dr4/H20853/H20851LeqR/H20849r3,r4,/H9275/H20850
−LeqA/H20849r3,r4,/H9275/H20850/H20852VjR/H20849r4,r1,/H9275/H20850Leq/H20849r1,r2,/H9275/H20850
−LeqA/H20849r3,r4,/H9275/H20850VjA/H20849r4,r1,/H9275/H20850/H20851LeqR/H20849r1,r2,/H9275/H20850
−LeqA/H20849r1,r2,/H9275/H20850/H20852/H20854/H20853/H208511+nP/H20849/H9275/H20850/H20852/H9016/H11633iT/H11021/H20849r2,r3,/H9275/H20850
−nP/H20849/H9275/H20850/H9016/H11633iT/H11022/H20849r2,r3,/H9275/H20850/H20854
+e/H20885d/H9280d/H9275
/H208492/H9266/H208502dr2¯dr4v0j/H20849r4,r1/H20850/H20851g0A/H20849r1,r2,/H9280/H20850
−g0R/H20849r1,r2,/H9280/H20850/H20852/H20851g0A/H20849r3,r2,/H9275−/H9280/H20850
−g0R/H20849r3,r2,/H9275−/H9280/H20850/H20852/H20853/H208511+nP/H20849/H9275/H20850/H20852L/H11633iT/H11021/H20849r2,r3,/H9275/H20850
−nP/H20849/H9275/H20850L/H11633iT/H11022/H20849r2,r3,/H9275/H20850/H20854/H20851nF/H20849/H9280/H20850−nF/H20849/H9280−/H9275/H20850/H20852
/H11003/H20851g0A/H20849r3,r4,/H9280/H20850−g0R/H20849r3,r4,/H9280/H20850/H20852. /H20849A15 /H20850
Keeping in mind the definition of the polarization operator,
we may rewrite the second integral in terms of /H9016ˆE/H20849/H9275/H20850. Then
we may collect the two contributions into a more compactexpressionj
ej/H20849L/H11633T,H/H20850=−1
Ej/H20885d/H9275
2/H9266dr2¯dr4/H20853/H208511+nP/H20849/H9275/H20850/H20852/H9016/H11633iT/H11021/H20849r3,r4,/H9275/H20850
−nP/H20849/H9275/H20850/H9016/H11633iT/H11022/H20849r3,r4,/H9275/H20850/H20854/H20875/H11509nP/H20849/H9275/H20850
/H11509/H9275/H20876−1
/H11003/H20877−ieE j/H11509nP/H20849/H9275/H20850
/H11509/H9275/H20853LeqA/H20849r4,r1,/H9275/H20850ViA/H20849r1,r2,/H9275/H20850
/H11003/H20851LeqR/H20849r2,r3,/H9275/H20850−LeqA/H20849r2,r3,/H9275/H20850/H20852−/H20851LeqR/H20849r4,r1,/H9275/H20850
−LeqA/H20849r4,r1,/H9275/H20850/H20852ViR/H20849r1,r2,/H9275/H20850LeqR/H20849r2,r3,/H9275/H20850/H20854
+LeqA/H20849r4,r1,/H9275/H20850/H20853/H208511+nP/H20849/H9275/H20850/H20852/H9016Ej/H11021/H20849r1,r2,/H9275/H20850
−nP/H20849/H9275/H20850/H9016Ej/H11022/H20849r1,r2,/H9275/H20850/H20854LeqR/H20849r2,r3,/H9275/H20850/H20878. /H20849A16 /H20850
The expression inside the curly brackets in the above equa-
tion can be written as /H208511+nP/H20849/H9275/H20850/H20852LE/H11021/H20849/H9275/H20850−nP/H20849/H9275/H20850LE/H11022/H20849/H9275/H20850. To ob-
tain this identity one should find the lesser and greater com-
ponents of Lˆfrom Eq. /H20849A8/H20850. A simple calculation reveals that
in this combination of LE/H11021andLE/H11022only the terms proportional
to the derivative of the Bose distribution function and those
including /H9016E/H11021,/H11022/H20851which also contain /H11509nP/H20849/H9275/H20850//H11509/H9275/H20852give a non-
zero contribution. Once again, if we invert the direction ofthe propagation of all the ingredients in Eq. /H20849A16 /H20850and also
the direction of the magnetic field we get that the expressionfor this last contribution to the electric current becomes
j
ej/H20849L/H11633T,−H/H20850=−1
Ej/H20885d/H9275
2/H9266dr/H11032/H20875/H11509nP/H20849/H9275/H20850
/H11509/H9275/H20876−1
/H20853/H9016/H11633iT/H11021/H20849r/H11032,r,/H9275/H20850
/H11003/H208511+nP/H20849/H9275/H20850/H20852−/H9016/H11633iT/H11022/H20849r/H11032,r,/H9275/H20850nP/H20849/H9275/H20850/H20854
/H11003/H20853/H208511+nP/H20849/H9275/H20850/H20852LEj/H11021/H20849r,r/H11032,/H9275/H20850
−nP/H20849/H9275/H20850LEj/H11022/H20849r,r/H11032,/H9275/H20850/H20854. /H20849A17 /H20850
Comparing jej/H20849L/H11633iT,−B/H20850given above and jhi/H20849Ej,B/H20850presented
in Eq. /H20849A14 /H20850, one immediately sees that they are indeed con-
nected by the Onsager relations.
In conclusion, we demonstrated the Onsager relation for
the Gaussian fluctuations of the superconducting order pa-rameter /H20851i.e., to the leading order in /H20849/H9255
F/H9270/H20850−1/H20852. The structure of
the expressions for the electric and heat currents, Eqs. /H2084938/H20850
and /H20849A9/H20850, indicates that the same is true for any order. /H20851An
example for a general proof of the Onsager relations is givenin Sec. VI of Ref. 17./H20852
APPENDIX B: CANCELLATION OF THE
THERMOELECTRIC CURRENT IN THE LIMIT T\0
In this appendix we demonstrate how the contribution to
the transverse component of jeconthat does not vanish when
T→0 is canceled by the magnetization current given in Eq.
/H2084953/H20850. From the three terms constituting jecon, which are de-
scribed in Sec. IV, the one that remains at low temperatures,
jecon3, is presented in Eq. /H2084949/H20850. Let us restore the general
expression from which jecon3originatesr1
r8
r7 r6r5r4r3r2
r1
r12r11r10 r9r8r7r6r5r4r3r2vβvµrαA( r )µ4 4
vβvµrαA( r )µ4 4(a)
(b)
FIG. 13. /H20849a/H20850The contribution to jerem/H20849a/H20850before and /H20849b/H20850after
averaging over the disorder.QUANTUM KINETIC APPROACH TO THE CALCULATION … PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-19jeicon3=e/H11633jT
T0/H20885d/H9280
2/H9266dr2¯dr6nF/H20849/H9280/H20850/H20851v0i/H20849r6,r1/H20850g0R/H20849r1,r2,/H9280/H20850/H9268eqR/H20849r2,r3,/H9280/H20850g0R/H20849r3,r4,/H9280/H20850v0j/H20849r4,r5/H20850g0R/H20849r5,r6,/H9280/H20850
−v0i/H20849r6,r1/H20850g0R/H20849r1,r2,/H9280/H20850v0j/H20849r2,r3/H20850g0R/H20849r3,r4,/H9280/H20850/H9268eqR/H20849r4,r5,/H9280/H20850g0R/H20849r5,r6,/H9280/H20850/H20852+ c.c. /H20849B1/H20850
Here we use the notation v0/H20849r,r/H11032/H20850for the bare velocity /H20851see Eq. /H2084922/H20850with/H9268ˆeqtaken to be zero /H20850. Following Refs. 19and36and
recalling that we are interested in the current averaged over the space, we may replace the convolution g0R/H20849/H9280/H20850v0g0R/H20849/H9280/H20850in the
above equation with − i/H20849r−r/H11032/H20850g0R/H20849r,r/H11032,/H9280/H20850. Then, the contribution to the current from jecon3is
jeicon3=−ie/H11633jT
T0/H20885d/H9280
2/H9266dr2¯dr4nF/H20849/H9280/H20850/H20851v0j/H20849r4,r1/H20850
/H11003/H20849r1−r2/H20850ig0R/H20849r1,r2,/H9280/H20850/H9268eqR/H20849r2,r3,/H9280/H20850g0R/H20849r3,r4,/H9280/H20850
−v0i/H20849r4,r1/H20850
/H11003/H20849r1−r2/H20850jg0R/H20849r1,r2,/H9280/H20850/H9268eqR/H20849r2,r3,/H9280/H20850g0R/H20849r3,r4,/H9280/H20850/H20852+ c.c.
/H20849B2/H20850
The above expression can be rewritten using the magnetiza-
tionMdefined below Eq. /H2084937/H20850,
jeicon3=−2 i/H9255ij/H11633jT
T0cM zlim
/H9267→0/H20885d/H9280
2/H9266geq/H11021/H20849/H9267,/H9280;A,imp /H20850
+i/H9280ije/H11633jT
2T0/H20885d/H9280
2/H9266dr2¯dr4nF/H20849/H9280/H20850/H9255z/H9251/H9252
/H11003/H20849r2+r3/H20850/H9251v0/H9252/H20849r4,r1/H20850
/H11003/H20851g0R/H20849r1,r2,/H9280/H20850/H9268eqR/H20849r2,r3,/H9280/H20850g0R/H20849r3,r4,/H9280/H20850
−g0A/H20849r1,r2,/H9280/H20850/H9268eqA/H20849r2,r3,/H9280/H20850g0A/H20849r3,r4,/H9280/H20850/H20852. /H20849B3/H20850
Obviously, when we add the magnetization current presented
in Eq. /H2084952/H20850tojecon3, only the second term in Eq. /H20849B3/H20850remains.
We shall denote this remaining term as jerem.Next, we replace /H9268eqwith its explicit expression given in
Eq. /H2084939/H20850,
jeirem=/H9255ije/H11633jT
2T0/H20885d/H9280d/H9275
/H208492/H9266/H208502dr2¯dr4nP/H20849/H9275/H20850/H9255z/H9251/H9252/H20849r2+r3/H20850/H9251
/H11003v0/H9252/H20849r4,r1/H20850g0R/H20849r1,r2,/H9280/H20850g0A/H20849r3,r2,/H9275/H9280/H20850g0R/H20849r3,r4,/H9280/H20850
/H11003/H20851LR/H20849r2,r3,/H9275/H20850nF/H20849/H9280−/H9275/H20850−LA/H20849r2,r3,/H9275/H20850nF/H20849/H9280/H20850/H20852+ c.c.
/H20849B4/H20850
Here, we dropped terms with three retarded /H20849advanced /H20850qua-
siparticles Green’s functions and we used the identities forthe products of distribution functions presented in Eq. /H2084950/H20850.
We show now that j
eremcontributes to the current only at
finite temperature, jerem→0 in the limit T→0. We demon-
strate that jeremis closely related to the contribution of the
Aslamazov-Larkin diagram to the magnetic susceptibility.37
To calculate jerem, we allow the magnetic field to depend on
the coordinate; in the end of the derivation we shall take thelimit of a uniform magnetic field.
Let us first concentrate on the product /H20849r
+r
/H11032/H20850LR,A/H20849r,r/H11032,/H9275/H20850. Since the magnetic field varies in space,
the dependence of the propagator Lon the center-of-mass
coordinate is through the magnetic field. The result of actingwith the operator R
/H9251onL/H20849R;/H9267,/H9275/H20850is equal to the derivative
with respect to the vector potential A/H9252/H20849r/H20850multiplied by
r/H9251A/H9252/H20849r/H20850,
/H20849r+r/H11032/H20850/H9251LR/H20849r,r/H11032,/H9275/H20850=−4 ie
c/H20885d/H9280/H11032
2/H9266dr1dr2dr3tanh/H20873/H9275−/H9280/H11032
2T/H20874LR/H20849r,r1,/H9275/H20850v0/H9252/H20849r2,r3/H20850r2/H9251A/H9252/H20849r2/H20850g0R/H20849r1,r2,/H9280/H11032/H20850g0R/H20849r3,r4,/H9280/H11032/H20850
/H11003g0A/H20849r1,r4,/H9275−/H9280/H11032/H20850LR/H20849r4,r/H11032,/H9275/H20850. /H20849B5/H20850
Now, we may return to the limit of a constant magnetic field, H/H20849r/H20850=Hzˆ, and set the vector potential to be A/H20849r/H20850=H/H20849r/H20850
/H11003r/2. Inserting the expression given in Eq. /H20849B5/H20850into jeirempresented in Eq. /H20849B4/H20850, we obtain
jeirem=−i/H9255ije2/H11633jT
T0c/H20885d/H9280d/H9280/H11032d/H9275
/H208492/H9266/H208503dr2¯dr8nP/H20849/H9275/H20850tanh/H20873/H9275−/H9280/H11032
2T/H20874/H9255z/H9251/H9252v0/H9252/H20849r8,r1/H20850v0/H9262/H20849r4,r5/H20850r4/H9251/H9255z/H9263/H9262H/H20849r4/H20850r4/H9263g0R/H20849r1,r2,/H9280/H20850
/H11003g0A/H20849r3,r2,/H9275−/H9280/H20850g0R/H20849r3,r4,/H9280/H20850/H20853LR/H20849r2,r3,/H9275/H20850g0R/H20849r3,r4,/H9280/H11032/H20850g0R/H20849r5,r6,/H9280/H11032/H20850g0A/H20849r3,r6,/H9275−/H9280/H11032/H20850LR/H20849r6,r7,/H9275/H20850nF/H20849/H9280−/H9275/H20850
+LA/H20849r3,r2,/H9275/H20850g0A/H20849r3,r4,/H9280/H11032/H20850g0A/H20849r5,r6,/H9280/H11032/H20850g0R/H20849r3,r6,/H9275−/H9280/H11032/H20850LA/H20849r6,r7,/H9275/H20850nF/H20849/H9280/H20850/H20854+ c.c. /H20849B6/H20850
The diagrammatic interpretation of the above expression is
presented in Fig. 13/H20849a/H20850. Unlike the superconducting fluctua-
tions, the electrons are considered to be three dimensional.Then, for an isotropic system we may rewrite the product ofthe antisymmetric tensors as: /H9255
z/H9251/H9252/H9255z/H9263/H9262=/H20849/H9254/H9263,/H9251/H9254/H9262,/H9252
−/H9254/H9263,/H9252/H9254/H9262,/H9251/H20850/3. Now we average over the disorder /H20851see Fig.13/H20849b/H20850/H20852and represent all the propagators /H20849Green’s functions,
propagators of the superconducting fluctuations and Cooper-ons /H20850as a product of the phases and the gauge-invariant
terms. All the phases are collected into the function e
i/H9021. After
performing the Fourier transform, the equation for jeirembe-
comesKAREN MICHAELI AND ALEXANDER M. FINKEL’STEIN PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-20jeirem=−i/H9280ije2/H11633jT
3T0c/H20849/H9254/H9263,/H9251/H9254/H9262,/H9252−/H9254/H9263,/H9252/H9254/H9262,/H9251/H20850/H20885d/H9280d/H9280/H11032d/H9275
/H208492/H9266/H208503dkdk/H11032dqdQ
/H208492/H9266/H208504dnP/H20849/H9275/H20850tanh/H20873/H9275−/H9280/H11032
2T/H20874ei/H9021/H115092H/H20849Q/H20850
/H11509Q/H9251/H11509Q/H9263g˜0R/H20849k,/H9280/H20850v˜0/H9252/H20849k,k+Q/H20850
/H11003g˜0R/H20849k+Q,/H9280/H20850g˜0A/H20849q−k,/H9275−/H9280/H20850/H20853C˜R/H20849q+Q,/H9280,/H9275−/H9280/H20850L˜R/H20849q+Q,/H9275/H20850C˜R/H20849q+Q,/H9280/H11032,/H9275−/H9280/H11032/H20850g˜0R/H20849k/H11032+Q,/H9280/H11032/H20850v˜0/H9262/H20849k/H11032+Q,k/H11032/H20850
/H11003g˜0R/H20849k/H11032,/H9280/H11032/H20850g˜A/H20849q−k/H11032,/H9275−/H9280/H11032/H20850C˜R/H20849q,/H9280/H11032,/H9275−/H9280/H11032/H20850L˜R/H20849q,/H9275/H20850C˜R/H20849q,/H9280,/H9275−/H9280/H20850nF/H20849/H9280−/H9275/H20850+C˜R/H20849q+Q,/H9280,/H9275−/H9280/H20850L˜A/H20849q+Q,/H9275/H20850
/H11003C˜A/H20849q+Q,/H9280/H11032,/H9275−/H9280/H11032/H20850g˜0A/H20849k/H11032+Q,/H9280/H11032/H20850v˜0/H9262/H20849k/H11032+Q,k/H11032/H20850g˜0A/H20849k/H11032,/H9280/H11032/H20850g˜0R/H20849q−k/H11032,/H9275−/H9280/H11032/H20850C˜A/H20849q,/H9280/H11032,/H9275−/H9280/H11032/H20850L˜A/H20849q,/H9275/H20850
/H11003C˜R/H20849q,/H9280,/H9275−/H9280/H20850nF/H20849/H9280/H20850/H20854+ c.c. /H20849B7/H20850
Here, H/H20849Q/H20850=/H208492/H9266/H20850dH/H9254/H20849Q/H20850zˆand v0/H20849k/H11032,k/H20850=/H20849k
2m−ieH
c/H11003/H11509/H6023
/H11509k
+k/H11032
2m+ieH
c/H11003/H11509/H6024
/H11509k/H11032/H20850is the Fourier transform of the velocity; the
arrows above the derivatives indicate on which of the
Green’s functions the derivative acts.
Let us transfer the derivatives with respect to the momen-
tumQfrom the magnetic field to the rest of the expression
using integration by parts. Since /H9021which collects all the
phases contains only derivatives with respect to the mo-menta, it will not be differentiated in the course of this op-eration. In the diffusive limit, the main contribution to the
current is obtained when the derivatives with respect to Qact
on the propagators of the collective modes, either LorC
/H20851here we rely on the arguments given below Eq. /H2084944/H20850/H20852. More-
over, it follows from the tensor structure of Eq. /H20849B7/H20850that
only terms in which
/H115092//H11509Q/H9251/H11509Q/H9263=2/H9254/H9251,/H9263/H11509//H11509Q2survive. /H20851One
should keep in mind that the gauge-invariant propagators ofthe superconducting fluctuations and Cooperons depend on
the square of the momentum. /H20852Then, j
eiremcan be written as
jeirem=−2 i/H9255ije2/H11633jT
3T0cH/H20885d/H9280d/H9280/H11032d/H9275
/H208492/H9266/H208503dkdk/H11032dq
/H208492/H9266/H208503dnP/H20849/H9275/H20850tanh/H20873/H9275−/H9280/H11032
2T/H20874ei/H9021g˜0R/H20849k,/H9280/H20850v˜0/H9252/H20849k,k/H20850g˜0R/H20849k,/H9280/H20850g˜0A/H20849q−k,/H9275−/H9280/H20850/H20877/H11509
/H11509q2
/H11003/H20851C˜R/H20849q,/H9280,/H9275−/H9280/H20850L˜R/H20849q,/H9275/H20850C˜R/H20849q,/H9280/H11032,/H9275−/H9280/H11032/H20850/H20852g˜0R/H20849k/H11032,/H9280/H11032/H20850v˜0/H9262/H20849k/H11032,k/H11032/H20850g˜0R/H20849k/H11032,/H9280/H11032/H20850g˜0A/H20849q−k/H11032,/H9275−/H9280/H11032/H20850C˜R/H20849q,/H9280/H11032,/H9275−/H9280/H11032/H20850L˜R/H20849q,/H9275/H20850
/H11003C˜
0R/H20849q,/H9280,/H9275−/H9280/H20850nF/H20849/H9280−/H9275/H20850+/H11509
/H11509q2/H20851C˜R/H20849q,/H9280,/H9275−/H9280/H20850L˜A/H20849q,/H9275/H20850C˜A/H20849q,/H9280/H11032,/H9275−/H9280/H11032/H20850/H20852g˜0A/H20849k/H11032,/H9280/H11032/H20850v˜0/H9262/H20849k/H11032,k/H11032/H20850g˜0A/H20849k/H11032,/H9280/H11032/H20850g˜0R/H20849q−k/H11032,/H9275−/H9280/H11032/H20850
/H11003C˜A/H20849q,/H9280/H11032,/H9275−/H9280/H11032/H20850L˜A/H20849q,/H9275/H20850C˜R/H20849q,/H9280,/H9275−/H9280/H20850nF/H20849/H9280/H20850/H20878+ c.c. /H20849B8/H20850
The next step is to integrate over the electronic degrees of freedom and to transform to the basis of the Landau levels. This
can be performed following the explanation presented in Sec. IV. The only difference is in the matrix elements for the Landau
levels. While in the calculation presented in the main text the matrix element is /H20855N,0/H20841VxVy/H20841M,0/H20856, here we have /H20855N,0/H20841Vx2
+Vy2/H20841M,0/H20856=4D2/H20851/H20849N+1/H20850/H9254N,M−1+/H20849M+1/H20850/H9254M,N−1/H20852//H5129H2. Finally, after replacing the derivative with respect to q2with a derivative
with respect to the index of the Landau levels, the expression for jeiremacquires the form
jeirem=−4 i/H9266/H92632D2
3/H5129H4/H92704/H9255ije/H11633jT
T0/H20885d/H9280d/H9275d/H9280/H11032
/H208492/H9266/H208503/H20858
N/H20849N+1/H20850tanh/H20873/H9275−/H9280/H11032
2T/H20874np/H20849/H9275/H20850/H11509
/H11509N/H20851nF/H20849/H9280−/H9275/H20850
/H11003CNR/H20849/H9280,/H9275−/H9280/H20850LNR/H20849/H9275/H20850CNR/H20849/H9280/H11032,/H9275−/H9280/H11032/H20850CN+1R/H20849/H9280/H11032,/H9275−/H9280/H11032/H20850LN+1R/H20849/H9275/H20850CN+1R/H20849/H9280,/H9275−/H9280/H20850+nF/H20849/H9280/H20850
/H11003CNR/H20849/H9280,/H9275−/H9280/H20850LNA/H20849/H9275/H20850CNA/H20849/H9280/H11032,/H9275−/H9280/H11032/H20850CN+1R/H20849/H9280/H11032,/H9275−/H9280/H11032/H20850LN+1A/H20849/H9275/H20850CN+1A/H20849/H9255,/H9275−/H9255/H20850/H20852+ c.c. /H20849B9/H20850
To recognize that the above expression goes to zero in the limit T→0, we have to integrate over the fermionic frequencies /H9280
and/H9280/H11032,
jeirem=i
24/H92662/H92632/H9255ije/H11633jT
T0/H20885d/H9275/H20858
N/H20849N+1/H20850np/H20849/H9275/H20850
/H11003/H11509
/H11509NLNR/H20849/H9275/H20850LN+1R/H20849/H9275/H20850/H20877/H9274NR/H208731
2−i/H9275
4/H9266T0+/H9024C/H20849N+1 /2/H20850
4/H9266T0/H20874−/H9274NR/H208731
2−i/H9275
4/H9266T0+/H9024C/H20849N+3 /2/H20850
4/H9266T0/H20874/H208782
+ c.c. /H20849B10 /H20850QUANTUM KINETIC APPROACH TO THE CALCULATION … PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
214516-21now, we can exploit the fact that in the expression differen-
tiated with respect to Nthe argument Nstands together with
the frequency, − i/H9275+/H9024cN. Replacing the derivative /H11509//H11509Nby
a derivative with respect to the frequency and integrating byparts, one immediately gets that the above integral includesthe factor
/H11509nP/H20849/H9275/H20850//H11509/H9275and, hence, vanishes at zero tempera-
ture.
To summaries, we showed that in accordance with the
third law of thermodynamics, the transverse thermoelectric
transport coefficients /H9251xyand/H9251˜xygo to zero in the limit T
→0./H20851As to the longitudinal coefficients, both jecon3andjemag
do not appear while the remaining contributions vanish inde-
pendently at T→0./H20852It follows from this result that at finite
T, one can obtain the temperature dependence of the thermo-
electric current by substituting nP/H20849/H9275/H20850with nP/H20849/H9275/H20850+/H9008/H20849−/H9275/H20850in
all the expressions determining jeconandjemag.
APPENDIX C: THE QUASIPARTICLES CONTRIBUTION
TO THE THERMOELECTRIC TRANSPORT
COEFFICIENTS
In this appendix we discuss the role of the particle-hole
asymmetry and the constant density-of-states approximationin determining the quasiparticles contribution to the thermo-electric transport coefficients. In metallic conductors the qua-siparticle excitations yield a negligible contribution to theNernst effect and to its counterpart, the Ettingshausen effect.Let us consider a system with an electron and hole conduct-ing bands that have a particle-hole symmetry, i.e., the bandsare identical and the two species of particles differ only intheir charge. We shall describe the deviation of the distribu-tion functions of the two species /H20849
/H9254feand/H9254fh/H20850from their
equilibrium value in the linear response to a temperature gra-dient. For that we write the classical Boltzmann equation inthe relaxation-time approximation,
/H9254fe,h/H20849/H9280k/H20850
/H9270=/H11509f0/H20849/H9280k/H20850
/H11509Tvk/H11633T/H11007evk/H11003H
c/H11509f0/H20849/H9280k/H20850
/H11509k. /H20849C1/H20850
Here the equilibrium /H20849Fermi-Dirac /H20850distribution function is
denoted by f0/H20849/H9280k/H20850andvkis the velocity of the particles.
The electric current is the sum of the electric currents due
to the electrons and the holes,
jetotal=−2 e/H20885dk
/H208492/H9266/H20850dvk/H9254fe/H20849/H9280k/H20850+2e/H20885dk
/H208492/H9266/H20850dvk/H9254fh/H20849/H9280k/H20850.
/H20849C2/H20850
Notice that the factor-of-2 results from the sum over the two
spin directions. For simplicity we only examine the limit ofvanishingly small magnetic field. In order to determinewhether a current vanishes in the particle-hole symmetricsystem we just need to count the powers of the electriccharge; an odd power means cancellation of the two contri-butions to the current.We start from the longitudinal electric current induced by
the temperature gradient. In the limit H→0, the longitudinal
current is independent of the magnetic field,
j
ex=2e/H20885dk
/H208492/H9266/H20850d/H11509f0/H20849/H9280k/H20850
/H11509/H9280k/H20851/H9280kDe−/H9280kDh/H20852/H11612xT
T0=0 , /H20849C3/H20850
where De=Dh/H11013D=vk2/H9270/dwith dthe dimension of the sys-
tem. Since the expression includes only one power of thecharge, there is no longitudinal electric current unlessparticle-hole asymmetry is introduced.
The transverse current is obtained when the Lorentz force
in the Boltzmann equation acts on the distribution function.Therefore, the expression for the transverse current containsan additional power of the charge,
j
ey=2e/H20885dk
/H208492/H9266/H20850d/H11509f0/H20849/H9280k/H20850
/H11509/H9280k/H9280kD/H20851/H9275c/H9270−/H20849−/H9275c/H9270/H20850/H20852/H11633xT
T0/HS110050.
/H20849C4/H20850
The additional charge enters through the cyclotron frequency
/H9275c=eH /m/H11569c. The even power of the charge means that the
particle-hole symmetry does not constrain the Nernst effect.
Now, we look at the contribution for the transverse elec-
tric current in a metal with only one conducting band. Weuse the approximate of a constant density of states which isstandard for Fermi-liquid systems. this approximation the ex-pression for the transverse current is
j
ey=2e/H92630D/H20849/H9275c/H9270/H20850/H11633T
T0/H20885d/H9280k/H11509f0/H20849/H9280k/H20850
/H11509/H9280k/H9280k. /H20849C5/H20850
Since near the Fermi energy the integrand is an odd function
of the energy, the resulting current is zero. Therefore, underthe approximation of a constant density of states at the Fermienergy this contribution vanishes.
10One may conclude that
in metallic systems with high Fermi energy the contributionof the quasiparticles to the Nernst signal includes a smallfactor related to the deviation from the constant density ofstates which is of the order T//H9255
F. In semimetals such as Bi
where the constant density-of-states approximation cannot beused, a large Nernst signal was measured.
38
Let us compare the magnitudes of the transverse Peltier
coefficient generated by the quasiparticles and by the super-conducting fluctuations in a film of thickness a. The first is
of the order /H11011/H20849
/H9275c/H9270/H20850e/H9263DaT //H9255Ffor/H9275c/H9270/H112701 while the second
one is of the order /H11011e/H9024c/Tfor/H9024c/T/H112701 and /H11011eT //H9024cfor
the opposite limit. Thus in the limit of vanishing small mag-netic field the ratio between the contribution of the quasipar-
ticles and the fluctuations is
/H9251xyqp//H9251xyfl/H11011/H20849kFa/H20850T2/H9270//H9255F.A t
higher magnetic fields /H20849but still in the limit /H9275c/H9270/H112701/H20850this ratio
becomes /H9251xyqp//H9251xyfl/H11011/H20849kFa/H20850/H9255F/H9270/H20849/H9275c/H9270/H208502/H112701. the condition of the
experiment,5,6the ratio /H9251xyqp//H9251xyfl/H112701u pt o T/H11351100TcandH
/H11351100Hc2. The reason why the Nernst signal generated by the
superconducting fluctuations dominates the one produced by
the quasiparticles was explained in the end of Sec. IV.KAREN MICHAELI AND ALEXANDER M. FINKEL’STEIN PHYSICAL REVIEW B 80, 214516 /H208492009 /H20850
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214516-23 |
PhysRevB.85.075418.pdf | PHYSICAL REVIEW B 85, 075418 (2012)
Optimized confinement of fermions in two dimensions
J. D. Cone,1S. Chiesa,2V . R. Rousseau,3G. G. Batrouni,4and R. T. Scalettar1
1Physics Department, University of California, Davis, California 95616, USA
2Department of Physics, College of William and Mary, Williamsburg, Virginia 23185
3Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803
4INLN, Universit ´e de Nice-Sophia and Institut Universitaire de France, CNRS; 1361 route des Lucioles, F-06560 Valbonne, France
(Received 16 December 2011; revised manuscript received 27 January 2012; published 21 February 2012)
One of the challenging features of studying model Hamiltonians with cold atoms in optical lattices is the
presence of spatial inhomogeneities induced by the confining potential, which results in the coexistence ofdifferent phases. This paper presents quantum Monte Carlo results comparing methods for confining fermions intwo dimensions, including conventional diagonal confinement, a recently proposed “off-diagonal confinement”,as well as a trap which produces uniform density in the lattice. At constant entropy and for currently accessibletemperatures, we show that (1) diagonal confinement results in the strongest signature of magnetic order, primarilybecause of its judicious use of entropy sinks at the trap edge and that (2) for d-wave pairing, a trap with uniformdensity is optimal and can be effectively implemented via off-diagonal confinement. This feature is important toany prospective search for superconductivity in optical lattices.
DOI: 10.1103/PhysRevB.85.075418 PACS number(s): 67 .85.−d, 05.30.Fk, 71 .10.Fd, 71 .27.+a
I. INTRODUCTION
Optical lattice emulators (OLE) control ultracold atomic
gases with lasers and magnetic fields to create experimentalrealizations of quantum lattice models of bosonic or fermionicparticles. For bosons, classic signatures of low temperature
correlated states—superfluidity and the Mott transition—have
been explored now for a decade.
1For fermions, quantum
degeneracy has been established through the observation ofa Fermi surface,
2as has the Mott transition.3,4The observa-
tion of magnetic order is the next immediate experimentalobjective.
5–8One ultimate goal is resolving the long-standing
question of whether the doped two-dimensional (2D) fermion
Hubbard Hamiltonian has long-range d-wave superconducting
order.9
Optical lattice experiments face at least two major obstacles
in simulating the fermion Hubbard model. The first is achiev-ing low enough temperature to pass through phase transitionsand into reduced entropy ordered phases. Present limits inexperiments are to temperatures T∼t(the near-neighbor
hopping energy), and to local entropies per atom ∼0.77k
B,10
values which are at the border for observing short-range
magnetic order.
The other obstacle, which we will be addressing in this
paper, is inhomogeneity arising from the confining potential.11
The external field conventionally used to trap cold atoms inthe lattice, a spatially dependent chemical potential which we
refer to as “diagonal confinement” (DC), causes variations in
the density per site ρ
i, with more atoms, on average, in the
center of the lattice and fewer at the edges. Density plays akey role in determining which correlations are dominant ininteracting quantum systems, but this is especially true of thefermion Hubbard Hamiltonian in two dimensions where themagnetic response is very sharply peaked
12near half-filling
(ρ=1). Various analytic and numerical calculations suggest
that pairing order also has a fairly sharp optimal filling, ρ≈
0.80−0.85.As a consequence of the inhomogeneous density arising
from DC, a trapped gas in an optical lattice will exhibitcoexistence of different phases, complicating the analysis and,potentially, significantly weakening and blurring the signal ofany phase transitions. To some extent, this loss of signal isreduced for antiferromagnetism (AFM), since the Mott gapin a DC trap can produce a fairly broad region of half-filling,where AFM is dominant. But the problem of observing pairingorder with DC seems especially acute since there is no suchprotection of the optimal density for superconductivity.
A recent proposal
13to use a reduction to zero of the hopping
at the lattice edge to confine the atoms allows the realization ofsystems with more uniform density. Such control of hoppingparameters is experimentally possible through holographicmasks
14and is referred to as “off diagonal confinement”
(ODC). Since ODC preserves the particle-hole symmetry ofthe unconfined Hubbard Hamiltonian, this trapping geometrycan lead to a uniform ρ=1 density and, at low entropy, can
produce a pure antiferromagnetic phase.
What is unclear is whether ODC is an effectively superior
way to confine fermions in OLE. This is a nontrivial questionsince OLE experiments do not have direct control over thetemperature—instead, the lattice and trapping potentials areintroduced adibatically, so optical confinement methods mustbe compared at fixed entropy. Here we present determinantquantum Monte Carlo (DQMC)
15calculations which compare
systems with ODC and DC traps, as well as a proposedmelding of these confinement methods to create a constantdensity (CD) trap. We evaluate the effects of these traps onmagnetic order and d-wave pairing correlations across the
lattice.
Our key results are (1) that the conventional DC trap
yields larger spin correlations than an ODC trap at the same,currently accessible entropy, (2) that, however, under the sameconditions, a constant density trap leads to larger pairingcorrelations than those achievable with DC traps, and (3) thatODC can implement a near constant density profile.
075418-1 1098-0121/2012/85(7)/075418(5) ©2012 American Physical SocietyCONE, CHIESA, ROUSSEAU, BATROUNI, AND SCALETTAR PHYSICAL REVIEW B 85, 075418 (2012)
II. TRAPPED HUBBARD MODEL, COMPUTATIONAL
METHODS
The Hubbard Hamiltonian in the presence of spatially
varying hopping and chemical potential is
H=−/summationdisplay
/angbracketleftij/angbracketright,σtij(c†
jσciσ+c†
iσcjσ)
+U/summationdisplay
i/parenleftbigg
ni↑−1
2/parenrightbigg/parenleftbigg
ni↓−1
2/parenrightbigg
−/summationdisplay
iμi(ni↑+ni↓).
(1)
Herec†
iσ(ciσ) are creation (destruction) operators for two
fermionic species σon site i, andniσare the corresponding
number operators. We will study a two-dimensional (2D)
square lattice with hopping between pairs of near neighborsites/angbracketleftij/angbracketright. For a DC trap, t
ijis constant and the chemical po-
tential μi=μ0−Vt(i2
x+i2
y) decreases quadratically toward
the lattice edge. For an ODC trap, instead, μiis constant and
the hopping term varies. Here we choose a parabolic formt
ij=t0−αr2
bond, where rbondis the distance of the center of
bond/angbracketleftij/angbracketrightto the lattice center.13Particle-hole symmetry for
this geometry implies the density ρi=/angbracketleft/summationtext
σc†
iσciσ/angbracketright=1 for all
lattice sites when μi=0.
For ODC, we fix energy units by setting t0=1 at the lattice
center. The parameter α, which controls the hopping decay,
is chosen so that tij→0 at the edge. A similar convention is
used for the CD trap, with the addition that now Vt,a sw e l la s
α, is nonzero.
We characterize and compare traps using the nearest-
neighbor (n.n.) spin-spin correlation, m, and the next-nearest-
neighbor (n.n.n.) d-wave pairing correlation, p. These are
defined as:
S+
i=c†
i↑ci↓/Delta1†
i=c†
i↑(c†
i+ˆx↓−c†
i+ˆy↓+c†
i−ˆx↓−c†
i−ˆy↓)
m(i)=/angbracketleftS−
i+ˆxS+
i/angbracketrightp(i)=/angbracketleft/Delta1i+ˆx+ˆy/Delta1†
i/angbracketright. (2)
III. USE OF LDA TO SIMULATE TRAPS
To simulate different traps, we first compute observables
and entropy values for homogeneous 8 ×8 lattices using
DQMC15which provides exact results for operator expectation
values of the fermion Hubbard Hamiltonian.16The entropy is
obtained via energy integration from T=∞ and values ob-
tained are consistent with other recently published results.17,18
We then use the local density approximation to simulate the
effects of each trapping method for a much larger lattice. Withthe LDA, observables for any position in a trap are determinedby the density ( ρ) of the equivalent homogeneous system, that
is, a system with the same values for U/t,mu/t , andT/t
as the local point in the trapped lattice. So, for each trap, wecompute these values at each position (as a function of r) and
determine the spin and d-wave pairing correlation, m(r) and
p(r), from the equivalent homogeneous result. The accuracy
of the LDA for short-range correlation functions has beendemonstrated for 2D
19and 3D20,21lattices in the regime of
temperature presently considered.
For a trap at a given temperature, the number of fermions
(N) and total entropy ( S) are obtained by integrating thesite density ρ(r) or entropy per site s(r) across the lat-
tice:N=2π/integraltext∞
0rρ(r)drandS=2π/integraltext∞
0rs(r)dr. Discrete
lattice sums are not used, since the simulated lattice sizes(approximately 11 000 sites for r=60) are large enough that
the functions ρ(r) ands(r) can be considered continuous.
IV . COMPARING DIFFERENT TYPES OF TRAPS
Figures 1and2(a) show the variation of the density ( ρ) and
entropy per site with U/t andμ/t for the uniform Hubbard
model at T/t=0.5. Different trapping geometries correspond
to the different paths in the ( μ,U ) plane. Figures 2(b) and
2(c) show, respectively, the variations in n.n. spin and n.n.n.
d-wave correlations for the 2D Hubbard model. The arrows in
Figs. 2(b) and2(c) indicate the trajectories of sample trap paths
projected onto the U/t−μ/t plane. Note that these figures
are at a specific temperature ( T/t=0.5), while for an actual
trap path (ODC, for example), T/twill vary across the lattice;
in what follows, we compare traps with the same entropy, not
at the same temperature.
The four parameters U/t 0,μ0/t0,Vt/t0, andα/t 0determine
the shape and physics for each trap type with the followingconstraints: DC, α=0; ODC, V
t=0; and CD, Vtandα
chosen to approximately follow a constant density path.To estimate the pairing and magnetic order for a certainconfinement type, we use the average, per particle, of thequantities in Figs. 2(b) and 2(c): the average n.n.n. d-wave
pairing correlation for superfluidity /angbracketleftp/angbracketright=
2π
N/integraltext∞
0rp(r)dr,
and the average n.n. spin correlation for magnetism /angbracketleftm/angbracketright=
2π
N/integraltext∞
0rm(r)dr.
We first determined the optimal U,μ0,Vt, andαfor each
trap type (DC, ODC, CD) by selecting the parameter valuesyielding traps that maximize either /angbracketleftp/angbracketrightor/angbracketleftm/angbracketright. Once the
optimal parameters were identified by trap type, we proceed to
FIG. 1. (Color online) The density as a function of U/t andμ/t
for the homogeneous fermion Hubbard model. Data were obtained
on 8×8 lattices with T/t=0.5. The dotted line is a constant density
path (ρ=.80) used in the trap comparisons. Using the LDA, density
profiles of inhomogeneous models can be determined by following an
appropriate ( U/t,μ/t ) path of local parameters as the lattice position
is changed.
075418-2OPTIMIZED CONFINEMENT OF FERMIONS IN TWO ... PHYSICAL REVIEW B 85, 075418 (2012)
FIG. 2. (Color online) (a) Entropy per site, (b) n.n. spin correlation, and (c) n.n.n. d-wave pairing correlation shown as functions of U/t and
μ/t for the homogeneous Hubbard Hamiltonian. Data were obtained on 8 ×8 lattices with T/t=0.5. Paths for optimal DC and ODC traps
are shown as solid lines, with the constant density trap (CD) as a dashed line. The ridge of prominent d-wave pairing (c) occurs at ρ∼0.80
and is nearly linear, so that an ODC path is close to the constant density one.
compare traps of different types. In all comparisons, the total
number of fermions and total entropy are the same for eachtrap.
V . MAGNETIC ORDER
Figure 3compares an optimal DC trap ( U=10.0,μ0=
2.5,Vt=0.0039) with an optimal ODC trap ( U=3.0,μ0=
0.0,α=0.0004). Both parameter sets were selected by max-
imizing /angbracketleftm/angbracketrightunder the common constraints S/N=0.75 and
N=6600. Note that the figure panels show only two trap
types since, when μ0=0.0, the ODC trap is equivalent to the
ρ=1 CD trap.
One might expect ODC to lead to large antiferromagnetic
correlations, since this confinement method allows for auniform half-filled Mott phase where magnetic correlations
FIG. 3. (Color online) (a) n.n. spin correlation, (c) density ( ρ),
and (d) entropy per site ( s) profiles are shown as a function of the
distance ( r) from the trap center for two different trap types: DC and
ODC, using optimal trap parameters. ODC trap with μ=0.0 is also
a constant density trap ( ρ=1). The average n.n. spin correlation is
larger for DC trap (0.14) than ODC (0.08) at the same entropy (0.75)and number of fermions (6600). Panel (b) shows average n.n. spin
correlation as a function of entropy per fermion ( S/N) for optimal
DC and ODC traps.are strongest. However, as Fig. 3(a) shows, DC has a
significantly larger average spin correlation (0.14 vs 0.08)than ODC when the two are compared at the same entropy.
This results because the low density wings in the DC
trap can store entropy that would otherwise accumulate inregion nearer the trap center. The central area in the DCtrap is effectively at lower temperature and has higher spincorrelations than in an ODC trap where there is no entropysink in the wings. Consequences of nonuniform entropydistribution have been emphasized previously in Refs. 17
and18.
VI. PAIRING A WAY FROM HALF-FILLING
We now turn to the question of pairing order. In Fig. 4,w e
show results comparing average next-near-neighbor22d-wave
pairing correlation for optimal DC, ODC, and constant density
FIG. 4. (Color online) (a) average n.n.n. d-wave pairing correla-
tion, (c) density ( ρ), and (d) entropy per site ( s) profiles are shown as
a function of the distance ( r) from the trap center for three different
trap types: DC, ODC, and constant density (CD) using optimal trap
parameters for each type. The average pairing values for CD and ODC
traps are 0.0052 and 0.0051, with the DC trap at 0.0046. Entropyper fermion (0.95) and number of fermions (6600) is the same for
each trap. Panel (b) shows average d-wave response as a function of
entropy per fermion ( S/N) for optimal DC, ODC, and CD traps.
075418-3CONE, CHIESA, ROUSSEAU, BATROUNI, AND SCALETTAR PHYSICAL REVIEW B 85, 075418 (2012)
(CD) traps at the same entropy per fermion (0.95) and number
of fermions (6600). Optimal parameters obtained for each trapare as follows: DC ( U=4.5,μ
0=1.5,Vt=0.00235), ODC
(U=3.0,μ0=− 1.1,α=0.00032), and CD ( U=3.0,α=
0.00032 ,ρ=0.80).
Looking at the trap profiles, we can see that peak pairing
for the DC trap occurs at lattice distances (measured from thecenter) which correspond to densities between 0.9 and 1.1,but pairing dips outside of this region and falls off rapidlytoward the trap edge. The constant density and ODC trapsare characterized by larger average pairing values of 0.0052and 0.0051 (vs 0.0046 for DC). This result can be clearly un-derstood from Fig. 2(c) by observing how the constant density
line (ρ=0.80) and ODC path follow the ridge of high d-wave
pairing response, while the DC trap path cuts through this ridgefor only a portion of the trap area. While the pairing differenceis not large at this high entropy level, the advantage is expectedto grow at lower entropies (see discussion in next section).
Figure 2(c) also emphasizes the narrowness of the optimal
d-wave response region compared to the wider area of
magnetic response seen in Fig. 2(b). This suggests that
OLE trap parameters tuned to follow this ridge of highd-wave response will increase the potential for observing
superconducting order.
VII. SUMMARY OF TRAP COMPARISONS
Figures 3(b) and 4(b) summarize the results of our trap
comparisons by plotting the optimal /angbracketleftm/angbracketrightand/angbracketleftp/angbracketrightfor the
different confining schemes against entropy per fermion ( S/N).
In the range of entropy shown, the DC trap produces a largerantiferromagnetic signal than the corresponding ODC trapat the same entropy. On the other hand, while the averaged-wave pairing correlation in a DC trap flattens at low entropy
(S/N∼0.6−0.9), that of a CD or an ODC trap continues to
increase as entropy is lowered. Due to the sign problem, wewere unable to reach entropy per fermion levels lower than 0.9for ODC and CD traps, but it is evident from Fig. 4(b) that the
d-wave response continues to rise as S/N decreases.VIII. CONCLUSIONS
We have evaluated several trapping geometries for fermions
in a 2D optical lattice. For magnetic properties, the DC trap,which is the common experimental technique used in OLE,continues to be the most promising confinement approachbecause excess entropy can be stored in its low densitywings leaving a low entropy Mott region with large AFMcorrelations. We have also shown that a more robust signalofd-wave pairing is produced with a constant density trap
with optimal ρ∼0.80, where local pairing correlations extend
over a significantly greater fraction of sites. We find that ODCclosely follows the constant density line shown in Fig. 2.
An important conclusion of our work is that while the search
for antiferromagnetic correlations in optical lattices is aidedby the inhomogeneous entropy distribution, this is not the casefor pairing. The local entropy is not reduced in the vicinityofρ=0.80, which is best for d-wave superconductivity.
Thus the same inhomogeneous s(r) which helps the magnetic
signal will weaken the pairing signal. This is a furtherargument for construction of a trap which has constantdensity. By providing an optimal confinement template forfermions in two dimensions, we anticipate that the results willaid experimenters in determining the physics of the dopedHubbard model.
ACKNOWLEDGMENTS
We acknowledge financial support from Army Research
Office Award No. W911NF0710576 and the Defense Ad-vanced Research Projects Agency Optical Lattice EmulatorProgram, Army Research Office Grant 56693-PH, NationalScience Foundation Grant No. OCI-0904972, CNRS-UCDavis EPOCAL LIA joint research grant, and the Depart-ment of Energy SciDAC program (Grant No. DOE-DE-FC0206ER25793). This research was supported in part byan allocation of advanced computing resources provided bythe National Science Foundation. The computations benefittedfrom runs on Kraken at the National Institute for Com-putational Sciences (http://www.nics.tennessee.edu/). We aregrateful for inspiration from G. Thorogood.
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075418-5 |
PhysRevB.57.412.pdf | Quantum and thermal spin fluctuations in itinerant-electron magnets
Fusayoshi J. Ohkawa
Department of Physics, Hokkaido University, Sapporo 060, Japan
~Received 19 May 1997; revised manuscript received 20 August 1997 !
Itinerant-electron magnetism is investigated using the Hubbard model. Local spin fluctuations are taken into
account through mapping to the Anderson model. Intersite spin fluctuations are taken into account by the1/d-expansion method, with dbeing the spatial dimensionality. The free energy as a function of magnetization,
which is similar to Landau’s phenomenological one, is derived at the microscopic level. There are twomechanisms of Curie-Weiss behavior of the susceptibility. One is due to specific structures of the dispersionrelation of quasiparticles and is of leading in 1/ dorO
@(1/d)0#, and the other is due to the mode-mode coupling
between intersite thermal spin fluctuations and is of higher order in 1/ d. The formation of light quasiparticles
can occur in the vicinity of the Mott transition, caused by momentum-dependent self-energy corrections due tointersite quantum spin fluctuations. It can explain the so called spin-bag or spin-gap behavior of high- T
c
cuprate oxides. @S0163-1829 ~98!04501-9 #
I. INTRODUCTION
Electron magnetism has been an important area of inves-
tigation for more than half a century since the advent ofquantum mechanics. Certain magnets are insulators at lowtemperatures, where local-moment magnetism occurs. Wehave a good understanding of local-moment magnetism onthe basis of the concept of the Mott insulator. On the otherhand, other magnets are metallic at low temperatures, whereitinerant-electron magnetism occurs. In spite of manyinvestigations,
1–3quite a few critical issues have not yet been
resolved for itinerant-electron magnetism or magnetism inthe crossover region between the two types of magnetism.
The Hubbard model is one of the simplest effective
Hamiltonians for itinerant-electron magnetism. The strong U
condition of U
r(0).1 is satisfied in typical magnets such as
3d-transition-metal, 5 f-actinide, and 4 f-lanthanide magnets,
withUbeing the on-site repulsion and r(0) the density of
states at the chemical potential for unrenormalized electrons.The Stoner model, the mean-field approximation ~MFA !or
the random-phase approximation ~RPA!predicts that ferro-
magnetic instability occurs at U
r(0)>1, and that antiferro-
magnetic instability or spin-density-wave ~SDW !instability
occurs even at Ur(0),1 when there is substantial nesting of
the Fermi surface. However, the MFA or the RPA is too
naive to predict magnetic instability in the region of Ur(0)
*1; it is valid only for the SDW instability occurring at
Ur(0)!1.
Three papers published in 1963 are distinguished. Ac-
cording to one of these papers, the ferromagnetic instability
predicted by the MFA is suppressed in the T-matrix
approximation.4The other two papers gave different results
that apparently contradict each other; one showed that theGutzwiller heavy quasi-particle band is present in the vicin-ity of the chemical potential for almost half filling,
5whereas
the other showed that the band splits into the lower andupper Hubbard bands.
6According to a recent paper,7the
Gutzwiller band lies between the lower and upper Hubbardbands. Different energy states, the ground and the excitedstates, were discussed in the two different papers.
5,6The superexchange interaction8can also be derived within
the conventional theoretical framework of the virtual ex-change of bosons or bosonic excitations.
9Because the
bosonic excitations are high-energy spin excitations, pair ex-citations of electrons through the spin channels between thelower and upper Hubbard bands, it is obvious that the super-exchange interaction is present in not only insulating but alsometallic states. The exchange interaction responsible for
magnetic instability in the region of U
r(0)*1 is not the
on-site repulsion Uitself but the superexchange interaction.
Local quantum spin fluctuations suppress magnetic insta-
bility. They can be totally taken into account in the single-site approximation ~SSA!that is rigorous in the large limit of
the coordination number of the lattices
10or the spatial
dimensionality11except for few physical properties relating
to magnetic instability.12The SSA or the renormalized
single-site approximation ~RSSA !for the Hubbard model or
the periodic Anderson model is reduced to solving theAnderson model.
13–16The Anderson model is an effective
Hamiltonian for the Kondo effect. The essence of the Kondoeffect is that local quantum spin fluctuations quench mag-netic moments; the ground state is singlet in dilute magneticalloys.
17
Intersite thermal spin fluctuations also suppress magnetic
instability. According to Murata and Doniach’sinvestigation
18and Moriya and Kawabata’s ones,3,19–21the
suppression by the mode-mode coupling between thermalspin fluctuations is responsible for Curie-Weiss behavior ofthe susceptibility. However, Murata and Doniach treated themode-mode coupling phenomenologically. Because Moriyaand Kawabata considered local quantum spin fluctuations inthe RPA at the microscopic level, their formulations cannot
apply to the strong Uregion at least at the microscopic level.
It is desirable to treat both the two types of spin fluctuationsproperly at the microscopic level.
The 1/d-expansion method can easily treat both the two
types of spin fluctuations, with dbeing the spatial dimen-
sionality. Its starting or unperturbed state is constructed inthe RSSA; one of the greatest advantages of the RSSA or the
1/d-expansion method is that many established results for thePHYSICAL REVIEW B 1 JANUARY 1998-I VOLUME 57, NUMBER 1
57 0163-1829/98/57 ~1!/412~14!/$15.00 412 © 1998 The American Physical SocietyKondo effect can be used to treat local quantum spin fluc-
tuations, which are mainly responsible for quenching mag-
netic moments. The 1/ d-expansion method is physically a
perturbative method to consider intersite effects, startingfrom the RSSA . In a previous paper,
22the formation of heavy
quasiparticles and a magnetic Cooper-pair interaction in thevicinity of, but a little away from, the Mott transition have
been investigated by the 1/ d-expansion method in order to
explain high temperature ~high-T
c) superconductivity in cu-
prate oxides. One of the purposes of this paper is to constructa relevant microscopic theory for itinerant-electron magne-tism, which takes into account properly all types of spinfluctuations.
The structure of this paper is as follows. A formulation
developed in the previous paper
22is reviewed in Sec. II. The
free energy as a function of magnetization, similar to Land-au’s phenomenological free energy, is derived in Sec. III.The mode-mode coupling between spin fluctuations is inves-tigated in Sec. IV. The renormalization of quasiparticles byintersite quantum spin fluctuations is investigated in Sec. V.Results obtained in this paper are compared with those ob-tained by other previous formulations in Sec. VI. A conclu-sion is presented in Sec. VII.
II. PRELIMINARIES
We consider the Hubbard model in ddimensions:
H5(
ijstijdis†djs11
2(
isUdis†disdi2s†di2s,~2.1!
where dimensional factors are included in the transfer matri-
cestij.11The irreducible two-point polarization function is
denoted by pst(ivl,q), and the up and the down spins by
s561. The function for the charge channel is given by
pc(ivl,q)51
2(stpst(ivl,q), and the function for the
longitudinal spin channel by ps(ivl,q)
51
2(st(st)pst(ivl,q). Every physical property is divided
into single-site and multisite terms, and every single-siteterm is identical to its corresponding term for the mappedAnderson model ~MAM !determined in the RSSA.
16For ex-
ample, it follows that ps(ivl,q)5p˜s(ivl)1Dps(ivl,q),
withp˜s(ivl) being the single-site term and Dps(ivl,q) the
multi-site term. When only localized electrons, or delec-
trons, are placed in the presence of a magnetic field, thesusceptibility of the MAM that excludes the polarization ofconduction electrons is given by
x˜s~ivl!52p˜s~ivl!/@12Up˜s~ivl!#. ~2.2!
The susceptibility of the Hubbard model is given by
xs~ivl,q!52ps~ivl,q!/@12Ups~ivl,q!#
5x˜s~ivl!F11Is~ivl,q!
2UG
121
4Is~ivl,q!x˜s~ivl!~2.3!
per unit cell, with Is(ivl,q)52@U/p˜s(ivl)#Dps(ivl,q).
The local Kondo temperature TKis defined by @x˜s(0)#T50K[1/(kBTK).23In this paper, strongly correlated electron
systems are characterized by U@kBTKrather than Ur(0)
.1. Equation ~2.3!is described as xs(ivl,q)5x˜s(ivl)/@1
21
4Is(ivl,q)x˜s(ivl)#, with
Is~ivl,q!52U2Dps~ivl,q! ~2.4!
to leading order in kBTK/U.
According to the well-known crossover for the Kondo
effect,24local-moment magnetism occurs at T@TKoruvlu
@kBTK, whereas itinerant-electron magnetism occurs at T
!TKanduvlu!kBTK. In this paper, we confine ourselves to
investigating itinerant-electron magnetism in the region of
U@kBTK.
The exchange interaction Is(ivl,q) is divided into two
terms so that Is(ivl,q)5Js(ivl,q)1JQ(ivl,q). The first
termJs(ivl,q) is due to the virtual exchange of high-energy
spin excitations, whose excitation energies are much larger
thankBTK. Within the Hubbard model, it is nothing but the
superexchange interaction.8Its energy dependence can be ig-
nored for low-energy processes uvlu!U: It is almost instan-
taneous for quasiparticles and will be simply denoted by
Js(q) in the following part. The superexchange interaction is
definitely antiferromagnetic within the Hubbard model. On
the other hand, whether Js(q) is ferromagnetic or antiferro-
magnetic depends on the whole band structure in an ex-tended model.
25The existence of other bands responsible for
high-energy processes is implicitly assumed in this paper.
We regard Js(q) as being phenomenologically given; when
ferromagnetism is investigated we assume that Js(q) has a
maximum at q50, whereas when antiferromagnetism is in-
vestigated we assume that Js(q) has a maximum or maxima
atqÞ0. The second term JQ(ivl,q) is due to the virtual
exchange of low-energy spin excitations within quasiparticlebands and can be properly treated within the Hubbard model.
The intrasite repulsion is present only between electrons
with antiparallel spins so that U
ss851
2U(12ss8). An in-
teraction between electrons is given by
Vss8~ivl,q!5Uss82(
t1t2Ust1pt1t2~ivl,q!Vt2s8~ivl,q!
5Vc~ivl,q!1ss8Vs~ivl,q!, ~2.5!
whereVc(ivl,q)51
2U/@11Upc(ivl,q)#is an interaction
through the charge channel and
Vs~ivl,q!521
2U21
4U2xs~ivl,q! ~2.6a!
521
2U21
4U2x˜s~ivl!21
4U2Fs~ivl,q!,
~2.6b!
with
Fs~ivl,q!5F1
4Is~ivl,q!x˜s~ivl!Gxs~ivl,q!~2.7!
to leading order in kBTK/U, is an interaction through the
longitudinal spin channel. When not only the longitudinal57 413 QUANTUM AND THERMAL SPIN FLUCTUATIONS IN . . .but also the transverse spin channels are considered, an in-
teraction due to spin fluctuations is given by
Vs~ivl,q!52~sxsx81sysy81szsz8!1
4U2xs~ivl,q!,
~2.8!
where snandsn8, with n5x,y,o rz, express the changes of
spin indices of interacting two electrons; for example,
sx5S01
10D,sy5S02i
i0D, ~2.9a!
and
sz5S10
021D. ~2.9b!
When Eq. ~2.8!is derived, the first term 21
2Uin Eq. ~2.6a!
is ignored to leading order in kBTK/U. When the single-site
portion in Eq. ~2.6b!is discarded, an interaction due to in-
tersite spin fluctuations is given by
Vs8~ivl,q!52~sxsx81sysy81szsz8!1
4U2Fs~ivl,q!.
~2.10!
When vertex corrections are considered, the charge channel
interaction Vc(ivl,q) can be ignored in the region of
kBTK/U!1.
Because low-energy and low-temperature properties are
considered in this paper, various physical properties are ex-panded in the following way. The self-energy is divided into
single-site and multisite terms so that S
s(i«n,k)5S˜
s(i«n)
1DS s(i«n,k). The single-site term is expanded so that
S˜
s(i«n)5S˜
01@12f˜m#i«n1forT!TK, and the multi-
site term is expanded so that26DS s(i«n,k)5DS(k)
2Dfm(k)i«n1. The total expansion coefficient is given
byfm(k)5f˜m1Dfm(k). The self-energy of the MAM is
similarly expanded so that27S˜
s(A)(i«n)5S˜
01@12f˜m#i«n
1@12w˜s#sH*1in the presence of an infinitesimally
small Zeeman energy H*51
2gmBH, withgbeing an effec-
tivegfactor, mBthe Bohr magneton, and Ha homogeneous
magnetic field. The single-particle Green function is given
byGs(i«n,k)51/@i«n1m2E(k)2Ss(i«n,k)#, with mbe-
ing the chemical potential and E(k)5(jtijeik(Ri2Rj). The
dispersion relation of quasiparticles is given by
j~k!5@E~k!1S˜
01DS~k!2m#/fm~k!.~2.11!
The coherent part of the single-particle Green function is
given by
Gs~c!~i«n,k!51/$fm~k!@i«n2j~k!#%. ~2.12!
According to Gutzwiller’s5or Brinkman and Rice’s
investigation,28it follows that f˜m@1 and fm(k)@1 in the
vicinity of the Mott transition.
The susceptibility of the MAM that includes the polariza-
tion of conduction electrons is given by x˜s(0)1x˜s(0)sc
52w˜sr(0),where x˜s(0)scis due to the polarization ofconduction electrons, r(«)5(1/N)(krk(«) with rk(«)
52(1/p)ImGs(«1i0,k) is the density of states for the
Hubbard model and is equal to that for the MAM. We as-sume in this paper that Anderson’s compensation theorem
29
is approximately satisfied so that uscu!1.30Following Shi-
ba’s argument,31we obtain
1/x˜s~v1i0!51/x˜s~0!2i~pcP/2!v1 ~2.13!
forT!TKanduvu!kBTK, with
cP5@2w˜sr~0!/x˜s~0!#25~11sc!2.1. ~2.14!
The specific heat coefficient of the Hubbard model is given
by
g52
3p2kB2r*~0! ~2.15!
per unit cell, with r*(«)5(1/N)(kd@«2j(k)#being the
density of states for quasiparticles. It is divided into two
terms so that g5g˜1Dg, where g˜52
3p2kB2f˜mr(0) and
Dg51
3p2kB21
N(
ksDfm~k!rk~«! ~2.16!
are the contributions from local and intersite spin fluctua-
tions, respectively. An energy scale kBTQfor quasiparticles
is defined by
r*~0!51/~4kBTQ!. ~2.17!
The density of states for electrons is given by
r~0!5~11sc!/~2w˜skBTK!51/~4fmkBTQ!,~2.18!
where the kdependence of fm(k) is ignored.
The 1/d-expansion method is a perturbative method to
consider multisite terms, staring from the RSSA. In theRSSA, the dispersion relation of quasi-particles is given by
j0~k!5@E~k!1S˜
02m#/f˜m, ~2.19!
and the coherent part of the single-particle Green function is
given by
gs~c!~i«n,k!51/$f˜m@i«n2j0~k!#%. ~2.20!
The irreducible single-site three-point vertex function for the
spin channels is denoted by l˜s(i«n,i«n1ivl). According
to the Ward-Takahashi relation,32it follows that w˜s
5l˜s(0,0)/ @12Up˜s(0)#and
Ul˜s~0,0!52w˜s/x˜s~0!5~11sc!/r~0!.~2.21!
When this vertex correction is properly considered, multisite
terms can be perturbatively calculated in terms of Eqs. ~2.4!,
~2.8!,o r~2.10!. In principle, single-site and multisite terms
should be self-consistently calculated with each other, be-cause they depend on each other.
III. FREE-ENERGY EXPANSION
As was discussed by Murata and Doniach,18the mode-
mode coupling for the susceptibility is related to the expan-sion coefficients of the free energy as a function of magne-414 57 FUSAYOSHI J. OHKAWAtization. In this section, we consider the free-energy
expansion as preliminaries for the next section, where thesusceptibility will be discussed.
When electrons are placed in the presence of the Zeeman
energyH
in*51
2gmBHin, withHinbeing the ncomponent of
a magnetic field introduced at the ith site, the Zeeman term is
given byHZ52(inabHin*(dia†snabdib), with snabdefined by
Eqs.~2.9a!and~2.9b!. Magnetic moments are given by
min5(
ab^dia†snabdib&, ~3.1!
where ^&stands for the statistical average. Equation ~3.1!
is nothing but the self-consistency condition to determine the
set of the magnetic moments $m%.
We tentatively consider the thermodynamic potential for
the model defined by HV5H1HZ1E01H8($m%)
2H8($p%), withE05(i1
4U(mi22pi2) and
H8~$m%!52(
inab1
2Umin~dia†snabdib!, ~3.2!
instead of that for H1HZ. It should be noted that when
$p%5$m%this model is reduced to H1HZ. As will be
shown below, a minimizing process of this thermodynamic
potential gives nonzero $m%5$p%that satisfy Eq. ~3.1!.
The normal Hartree term is given by1
2Uni8withni8
5(s^ais†ais&. The normal Fock term does not appear in this
model. Anomalous Hartree and Fock terms are given by
21
2Upin8withpin85(ab^aia†snabaib&forn5x,y, andz.W e
require the following condition for $p%:
pix5pix8,piy5piy8, ~3.3a!
and
piz5piz8, ~3.3b!
which define $p%as functions of m,$m%, and $H*%. When
the normalization of diagrams is carried out, terms from
2H8($p%) cancel the anomalous Hartree and the anomalous
Fock terms because of Eqs. ~3.3a!and~3.3b!. The chemical
potential mis adjusted so that (is^ais†ais&might give the
correct number of total electrons.
The thermodynamic potential is calculated perturbatively
in terms ofH8($m%),2H8($p%),HZ, andU. It is a function
of$m%and$H*%asV($m%,$H*%;$p%,m), where $p%andm
are functions of $m%and$H*%. We consider the derivative
of the thermodynamic potential with mbeing fixed:
FdV
dminG
m5]V
]min1(
jn8]V
]pjn8]pjn8
]min. ~3.4!
Because a single vertex which stands for ]/(]min)o r
]/(]pjn8) is fixed in diagrams, the normalization of diagrams
can be carried out. It is straightforward to show that
]V/]pin51
2U(pin82pin)50 and ]V/]min521
2U(pin8
2min). When the thermodynamic potential is minimized in
such a way that @dV/dmin#m50, Eq. ~3.1!and$m%5$p%are
satisfied.
It is easier to calculate the derivative of the thermody-
namic potential than to calculate the thermodynamic poten-tial itself. The derivative is calculated perturbatively in terms
ofH8($m%),HZandU; we do not have to consider
2H8($p%), the anomalous Hartree and the anomalous Fock
terms. Unrenormalized diagrams for the derivative have un-
renormalized electron lines, interaction points of H8($m%)
andHZ, and purely many-body Ulines. The unrenormalized
electron lines are renormalized, when many-body effects areconsidered. Because no magnetic contributions should betaken into account in this renormalization, renormalizedelectron lines stand for the renormalized Green functions inthe absence of magnetization.
BecauseH
8($m%) andHZappear in exactly the same
way, only the combination of m¯i5mi12Hi*/Uappears in
the magnetic terms of the perturbation series. Integrating the
derivative, we obtain the thermodynamic potential itself:
V~m;m!5Vpara~m!2(
im¯iHi*11
4U(
im¯i2
21
21
2U2(
i1i2pn1n2~i1,i2;m!m¯i1n1m¯i2n2
21
41
23U4(
i1i2i3i4pn1n2n3n4~i1,i2,i3,i4;m!
3m¯i1n1m¯i2n2m¯i3n3m¯i4n42, ~3.5!
where Vpara(m) is the thermodynamic potential in the ab-
sence of magnetization, and pn1n2(i1,i2;m) and pn1n2n3n4
(i1,i2,i3,i4;m) are the static irreducible multipoint polariza-
tion functions for the spin channels in the absence of mag-
netization. In Eq. ~3.5!, the relation of Umi2/45(Hi*)2/U
2(m¯iHi*)1Um¯i2/4 is made use of, and (i(Hi*)2/Uis ig-
nored.
In general, the multipoint polarization functions vanish
when they are odd with respect to at least one of the spin
fluctuation channels x,y, andz. For example, it follows that
px50,pxy50,px3[pxxx50,py2x[pyyx50,pxyz50,
pxy3[pxyyy50, and so on. They are the same as each other
for any even permutation among different ni’s. For example,
it follows that px2y25pxy2x5pyx2y5py2x2andpxyxy
5pyxyx.
It is obvious that umiu@2uHi*u/Ufor magnetic states. It is
also obvious that min5O@Hin*/(kBTK)#and uminu
@2uHin*u/Ueven for paramagnetic states. Then, it follows
thatm¯in5minto leading order in kBTK/U. Equation ~3.5!is
similar to Landau’s free energy with the Zeeman energy as afunction of magnetization. An only difference is that the
chemical potential
mshould be determined self-consistently
in this formulation. In the following part of this paper, thechemical potential will not be explicitly shown as an argu-ment.
Unrenormalized diagrams for the derivative of the ther-
modynamic potential are classified into single-site and mul-
tisite diagrams. When all the site indices of
]/]min,min
fromH8($m%),Hin*fromHZ, andUlines are restricted to
single sites, they are single-site diagrams. All the other dia-57 415 QUANTUM AND THERMAL SPIN FLUCTUATIONS IN . . .grams are multisite diagrams. Thus, the thermodynamic po-
tential is divided into single-site and multisite terms.
The single-site term is identical to the corresponding term
for the MAM. When the n-point polarization functions for
the MAM are denoted by p˜n1n2nn, the thermodynamic po-
tential for the MAM is expanded as a function of magneti-
zation in such a way that
V˜~m!5V˜
para11
21
2U@12Up˜s~0!#@mx21my21mz2#
21
41
23U4@p˜x4~mx41my41mz4!12p˜@x2y2#
3~mx2my21my2mz21mz2mx2!#2, ~3.6!
with p˜@x2y2#52p˜x2y21p˜xyxy. The Zeeman term is not in-
cluded in Eq. ~3.6!.
The thermodynamic potential for the Anderson model
with constant hybridization as a function of energies can beused to approximate the multipoint polarization functions forthe MAM.
30According to a previous paper,33this is given by
V˜(H*)5V˜
para2A(kBTK)21(H*)21kBTK. Magnetization
is given by m52]V˜(H*)/]H*5H*/A(kBTK)21(H*)2.
Then, we find34
V˜~m!5V˜~H*!1~mH*!
5V˜
para11
x˜s~0!F1
2m21c4
8m41G,~3.7!
with 1/ x˜s(0)5kBTKandc451. Comparing Eqs. ~3.6!and
~3.7!, we obtain
U@12p˜s~0!#/251/x˜s~0! ~3.8!
and
p˜x45p˜@x2y2#524c4/U4x˜s~0!. ~3.9!
The Kondo temperature TKdepends on the energy depen-
dence of hybridization for the MAM, and low-energy andlow-temperature properties of the MAM are characterized by
only the single energy scale T
K. Therefore, it is expected
that Eq. ~3.7!for constant hybridization is valid when TKfor
actual hybridization is used. In actual, the fact of Eq. ~3.8!
being consistent with Eq. ~2.2!to leading order in kBTK/U
means that the coefficient of the second order magnetization
term in Eq. ~3.7!,1
2, is rigorous to leading order in kBTK/U.
The coefficient c4may be slightly different from unity, when
the MAM is rigorously and self-consistently solved. In this
paper, we approximately use Eqs. ~3.9!withc451.
In lattice systems, magnetic moments can be expanded in
Fourier series so that mi5(qm(q)eiqRi. Because mi’s are
real, it follows that m(q)5m*(2q). The thermodynamic
potential is expanded so that
V~$m%!5Vpara1(
n511`
V2n~$m%!. ~3.10!
When we usepnn8~i,j!5dnn81
N(
qe2iq~Ri2Rj!ps~0,q!,~3.11!
the second-order magnetization term is described as
V2~$m%!51
22NU(
nq@12Ups~0,q!#umn~q!u2
5N(
nq1
2xs~0,q!umn~q!u2~3.12!
to leading order in kBTK/U. The susceptibility xs(0,q) in-
cluding the mode-mode coupling will be discussed in Sec.IV. The fourth-order term is described as
V
4~$m%!521
4323NU4(
$n%$q%pn1n2n3n4~q1,q2,q3,q4!
3mn1~q1!mn2~q2!mn3~q3!mn4~q4!,~3.13!
where $n%and$q%stand for the sets of ni’s andqi’s, respec-
tively, and
pn1n2n3n4~i1,i2,i3,i4!51
N3(
$q%expF2i(
i514
~qiRi!G
3pn1n2n3n4~q1,q2,q3,q4!
~3.14!
is made use of. Higher-order terms than fourth order for the
thermodynamic potential are similarly described.
The four-point polarization functions are divided into
single-site and multisite terms so that
pn1n2n3n4~q1,q2,q3,q4!5dq11q21q31q4@p˜n1n2n3n4
1Dpn1n2n3n4~q1,q2,q3!#.
~3.15!
The single-site terms are approximately given by Eq. ~3.9!.
When the contribution from a diagram as is shown in Fig. 1
is considered, the multisite terms satisfy the relation of Dpx4
5Dpy45Dpz45Dpx2y25Dpy2z25Dpz2x25
2Dpxyxy52Dpyzyz52Dpzxzx, where the arguments of
q1,q2, andq3are implicitly assumed. When the mode-mode
coupling is ignored,35Dpx4(q1,q2,q3) is approximately
given by
Dpx4~q1,q2,q3!52l˜s4~0,0!FkBT
N(
«nkgs~c!~i«n,k!
3gs~c!~i«n,k1q2!gs~c!~i«n,k1q21q3!
3gs~c!~i«n,k2q1!
2kBT(
«n@rs~c!~i«n!#4G, ~3.16!
withgs(c)(i«n,k) given by Eq. ~2.20!andrs(c)(i«n)
5(1/N)(kgs(c)(i«n,k). The single-site portion is subtracted
in Eq. ~3.16!.416 57 FUSAYOSHI J. OHKAWAEquation ~3.16!can be large only for specific qi’s, if the
band structure of quasiparticles has singularities. First, con-
sider Dpx4(0,0,0), which is calculated so that
Dpx4~0,0,0 !524
U4x˜s~0!~CF2C˜L!, ~3.17!
with
CF54w˜s4
x˜s3~0!kBT
N(
«nk@gs~c!~i«n,k!#4
52
3x˜s3~0!Fw˜s
f˜mG4EdxF2df~x!
dxGF2d2r0*~x!
dx2G
~3.18!
and
C˜L54w˜s4
x˜s3~0!kBT(
«n@rs~c!~i«n!#4. ~3.19!
Here, Eq. ~2.21!is made use of, and r0*(x) is the density of
states defined by r0*(x)5(1/N)(kd@x2j0(k)#. When
r0*(«) has a sharp peak in a broad spectrum, Dpx4(q1
.0,q2.0,q3.0) is significant.
Next, consider a case when the Fermi surface shows sub-
stantial nesting for wave vector Q. There are two possible
cases for the structure of the Fermi surface: an ordinary nest-
ing case, where j0(k).2j0(k1Q)Þj0(k2Q) is satisfied
for certain k’s on the Fermi surface, and a specific nesting
case, where j0(k).2j0(k1Q).j0(k2Q) is satisfied for
certaink’s on the Fermi surface. The ordinary case usually
occurs for incommensurate Q’s. The specific case never oc-
curs for incommensurate Q’s except for specific band struc-
tures, but it is likely to occur for commensurate Q’s. For
example, nesting of the Fermi surface with Q5(6p/a,
6p/a) on the simple square lattice is one of the specific
cases, with abeing the lattice constant.In both the ordinary and the specific nesting cases, Eq.
~3.16!is significant for q1.Q,q2.2Q, andq3.Q(q4.
2Q):
Dpx4~Q,2Q,Q!524
U4x˜s~0!~CQ2C˜L!,~3.20!
with
CQ54
x˜s3~0!Fw˜s
f˜mG4
kBT(
«nEdxrn*~0!/2
~i«n2x!2~i«n1x!2
5rn*~0!
x˜s3~0!Fw˜s
f˜mG47z~3!
8p2~kBT!2, ~3.21!
withrn*(0) being the density of states at the chemical poten-
tial for the nested area and 7 z(3)58.415 . Equation
~3.16!is also significant for q1.0,q2.0, andq3.Q(q4
.2Q):
Dpx4~0,0,Q!524
U4x˜s~0!~CQF2C˜L!, ~3.22!
with
CQF54
x˜s3~0!Fw˜s
f˜mG4
kBT(
«nEdxrn*~0!/2
~i«n2x!3~i«n1x!
51
2CQ. ~3.23!
There is a significant coupling between the q.0 and the q
.6Qcomponents, when nesting of the Fermi surface is
substantial. When the specific nesting is substantial, Eq.
~3.16!is also significant for q1.Q,q2.Qandq3.2Q
(q4.2Q):
Dpx4~Q,Q,2Q!524
U4x˜s~0!~CS2C˜L!,~3.24!
withCS.CQ. The multisite polarization function
Dpx4(q1,q2,q3) is also significant for other combinations of
q1,q2, andq3(q452q12q22q3) that are equivalent to
those considered here. Because the perfect nesting of the
Fermi surface is assumed, the coupling constants of CQ,
CQF, andCSdiverge at T50 K. If the Fermi surface of
actual magnets is more precisely treated than it is here, they
do not diverge at T50K .
In another paper,36competition between sinusoidal and
helical magnetic structures has been examined using thistype of the free energy expansion. When there is substantialnesting of the Fermi surface, a sinusoidal structure stabilizes.Otherwise, a helical structure stabilizes.
IV. SUSCEPTIBILITY
A. Curie-Weiss law of leading order in 1/ d
Because the mode-mode coupling between intersite spin
fluctuations is of higher order in 1/ d, it can never cause
Curie-Weiss behavior of the susceptibility in the large d
FIG. 1. Multisite thermodynamic potential of fourth order in
magnetization. A closed circle stands for magnetization, an opencircle for the three-point vertex function, a dotted line for U, and a
solid line for the renormalized Green function in the absence ofmagnetization.57
417 QUANTUM AND THERMAL SPIN FLUCTUATIONS IN . . .limit. The purpose of this subsection is to show that there is
another mechanism of Curie-Weiss behavior, which works
even in the large dlimit.
The intersite exchange interaction due to the virtual ex-
change low-energy spin excitations is divided into two parts
so that1
4JQ(ivl,q)5S(ivl,q)2L(ivl,q), whereS(ivl,q)
is the contribution of the so called two-line diagram22as is
shown in Fig. 2, and L(ivl,q) includes all other terms. One
of the mode-mode coupling effects is the renormalization ofquasiparticles by intersite spin fluctuations; two lowest-orderdiagrams are shown in Fig. 3 ~a!. Although this effect can be
considered as a portion of the two-line diagram when thefully renormalized quasiparticles are used, this effect will beperturbatively considered in the following parts; the renor-malization at elevated temperatures will be considered in
Sec. IV B and the renormalization at T50K in Sec. V.
As in our previous paper,
22the contribution from the two-
line diagram, which does not include mode-mode coupling,is calculated so that
S
~ivl,q!521
2U2kBT(
«nl˜s2~0,0!F1
N(
kgs~c!~i«n,k!gs~c!
3~i«n1ivl,k1q!2rs~c!~i«n!rs~c!~i«n1ivl!G
54cP~kBTK!2@P~ivl,q!2P0~ivl!#, ~4.1!
withcPgiven by Eq. ~2.14!andP~ivl,q!51
N(
ksf@j0~k1q!#2f@j0~k!#
j0~k!2j0~k1q!2ivl,~4.2!
withf(«)51/@e«/kBT11#. The single-site portion P0(ivl)
5(1/N)(qP(ivl,q) is subtracted in Eq. ~4.1!.
Equation ~4.2!is described as
P~ivl,q!5E
2`1`
d«S2df~«!
d«DD~«,q;ivl!,~4.3!
with
D~«,q;ivl!52
N(
ku@«2j0~k1q!#2u@«2j0~k!#
j0~k!2j0~k1q!2ivl.
~4.4!
Here, u(«) is the step function defined by u(«>0)51 and
u(«,0)50. Equation ~4.4!is singular for q50 when the
density of states has a sharp peak around the chemical po-
tential, and it is for qÞ0 when there is substantial nesting of
the Fermi surface. In such specific cases, P(0,Q) shows an
approximate T-linear dependence. For example, assume that
D(«,Q;0) is divided into two terms so that D(«,Q;0)
5D01DD(«,Q;0) for u«u!kBTK, whereD0is a nonsingu-
lar part and
DD~«,Q;0!5Hh
2kBTQF12u«u
4kBu*G,u«u<4kBu*
0,u«u.4kBu*,
~4.5!
with handu*being constants, is a singular part. It follows
that
P~0,Q!5h
4kBTQF22T
u*lnS2
11e24u*/TDG.~4.6!
TheTdependence of P(0,Q) is almost linear for T,u*.
WhenD(«,q;0) is singular for only q.Q, theTdependence
ofP0(0) is weak. The susceptibility shows Curie-Weiss be-
havior for only specific q.Q.
In the large dlimit, the susceptibility shows Pauli para-
magnetism for almost all q’s so that xs(0,q)5x˜s(0) for al-
most allq’s, but shows Curie-Weiss behavior for only spe-
cificq5Q. Localization of intersite spin fluctuations in
momentum space is characteristic of itinerant-electron mag-netism. On the other hand, local spin fluctuations, which arelocalized in real space, are totally taken into account in theRSSA. Both the localization in momentum space and thelocalization in real space can be easily taken into account by
the 1/d-expansion method.
B. Mode-mode coupling
In this subsection, we investigate L(ivl50,q) to first or-
der explicitly in Fs(ivl,q). It is written as
L~0,q!5LL~0,q!1Ls~0,q!1Lv~0,q!. ~4.7!
The first term LL(0,q) is a local term due to interactions
between local and intersite spin fluctuations; it is the sum ofthe contributions shown in Fig. 4. The second and the third
FIG. 2. Two-line diagram.
FIG. 3. Three types of mode-mode coupling diagrams, which
correspond to those considered by Kawabata ~Ref. 21 !:~a!self-
energy type, ~b!vertex-type, and ~c!coupling between charge and
spin fluctuations. A wavy line stands for the spin or the chargefluctuation function. In ~c!, a triangle stands for a three-point irre-
ducible polarization function; both the fluctuation lines cannot bethe spin fluctuation functions because the three-point polarizationfunction for the spin channels vanishes.418 57
FUSAYOSHI J. OHKAWAterms are intersite terms including interactions between in-
tersite spin fluctuations themselves; Ls(0,q) is the self-
energy-type contribution as is shown in Fig. 3 ~a!, and
Lv(0,q) is the vertex-type contribution as is shown in Fig.
3~b!. For the local term and the self-energy type contribution,
the single-site portion of spin fluctuations should not be con-sidered, because it is considered in the RSSA.
The local term is calculated so that
L
L~0,q!521
2U2@3p˜x412p˜@x2y2##kBT
N(
vlp1
4U2Fs~ivl,p!
55
2x˜s~0!kBT
N(
vlpFs~ivl,p!. ~4.8!
Equation ~4.8!includes a contribution of the self-energy
type, which is shown in Fig. 5 and is given by
23
2U2~kBT!2
N(
«nvlp@12Up˜s~0!#2]2S˜s~0!
]~H*!2l˜s2~0,0!
3@R˜s~i«n!#2R˜s~i«n2ivl!1
4U2Fs~ivl,p!.~4.9!
Because this is considered in the RSSA, it should be sub-
tracted from Eq. ~4.8!. This subtraction can be ignored be-
cause ]2S˜
s(0)/](H*)2is small for strongly correlated sys-
tems.
The self-energy type contribution is calculated so that37
Ls~0,q!56
23U4l˜s4~0,0!FkBT
NG2
(
«nvl(
kpgs~c!~i«n,k2q!
3@gs~c!~i«n,k!#2gs~c!~i«n1ivl,k1p!Fs~ivl,p!
2~local term !
53
x˜s~0!kBT
N(
vlpBs~ivl,p;q!Fs~ivl,p!,~4.10!
with
Bs~ivl,p;q!522w˜s4
x˜s3~0!kBT(
«nF1
N(
kgs~c!~i«n,k2q!
3@gs~c!~i«n,k!#2gs~c!~i«n1ivl,k1p!
2@rs~c!~i«n!#3rs~c!~i«n1ivl!G. ~4.11!In Eq. ~4.10!,2~local term !stands for the subtraction of
terms considered as a portion of LL(0,q). The vertex-type
contribution is similarly calculated so that
Lv~0,q!521
2x˜s~0!kBT
N(
vlpBv~ivl,p;q!xs~ivl,p!
2DS8~0,q!, ~4.12!
whereBv(ivl,p;q) is given by
Bv~ivl,p;q!522w˜s4
x˜s3~0!kBT(
«nF1
N(
kgs~c!~i«n,k1q!
3gs~c!~i«n,k!gs~c!~i«n1ivl,k1q1p!
3gs~c!~i«n1ivl,k1p!2@rs~c!~i«n!#2
3@rs~c!~i«n1ivl!#2G, ~4.13!
andDS8(0,q)b y
DS8~0,q!51
2U2kBT(
«n2l˜s~0,0!Dls8~i«n,i«n!F1
N(
kgs~c!
3~i«n,k!gs~c!~i«n,k1q!2@rs~c!~i«n!#2G~4.14!
with
Dls8~i«n,i«n!52kBT
N(
vlk@rs~c!~i«n1ivl!#2
31
4U2l˜s2~0,0!x˜s~ivl!. ~4.15!
In Eq. ~4.12!, terms considered as a portion of LL(0,q) and
S(0,q) are subtracted. It should be noted that when l˜s(0,0)
is substituted with l˜s(0,0) 2Dls8(i«n,i«n) in Eq. ~4.1!the
subtraction term DS8(0,q) can be approximately taken into
account.
The intersite mode-mode coupling terms are significant
only in specific cases as Dpx4(q1,q2,q3) are. For example,
consider L(0,0). When ferromagnetic spin fluctuations are
substantial, it is calculated so that
FIG. 5. Single-site diagram to be excluded. A rectangle stands
for@12Up˜s(0)#2@]2S˜
s(0)/](H*)2#.
FIG. 4. Three diagrams contributing to LL(q). A wavy line
stands for the spin fluctuation function, and p˜s4stands for a four-
point polarization function of the MAM for the spin channels.57 419 QUANTUM AND THERMAL SPIN FLUCTUATIONS IN . . .L~0,0!5516@CF2C˜L#
2x˜s~0!kBT
N(
vlpFs~ivl,p!
2CF2C˜L
2x˜s~0!kBT
N(
vlpxs~ivl,p!2DS8~0,0!,
~4.16!
whereBs(0,p;0)5Bv(0,p;0)5CF2C˜Lis approximately
used forp.0 under the assumption that intersite spin fluc-
tuations are localized in momentum space. On the otherhand, when antiferromagnetic spin fluctuations are substan-tial, it is calculated so that
L
~0,0!5516@CQF2C˜L#
2x˜s~0!kBT
N(
vlpFs~ivl,p!
2CQ2C˜L
2x˜s~0!kBT
N(
vlpxs~ivl,p!2DS8~0,0!,
~4.17!
whereBs(0,p;0)5CQF2C˜LandBv(0,p;0)5CQ2C˜Lare
approximately used for p.6Qand equivalent p’s. In Eqs.
~4.16!and~4.17!, the summation over pis restricted to the
region where Bs(0,p;0) orBv(0,p;0) is large. There is a
significant effect of antiferromagnetic spin fluctuations onthe homogeneous susceptibility, when nesting of the Fermisurface is substantial: Eq. ~4.17!together with the divergence
ofC
QFatT50 K discussed in Sec. III implies that the
homogeneous susceptibility is suppressed at not only el-evated temperatures but also low temperatures. This suppres-sion effect at low temperatures has already been investigatedby Miyake and Narikiyo
38with a semiphenomenological
model called a duality model39–41and will be investigated
from a little different theoretical point of view in Sec. V ofthis paper.
It is straightforward to calculate the mode-mode coupling
term for the susceptibility for nonzero q’s. In general, the
mode-mode coupling term is related to the fourth-order mag-netization term for the thermodynamic potential as was dis-cussed by Murata and Doniach
18and as will be discussed in
Sec. VI of this paper.
When the fully renormalized Green functions are used in
the two-line diagram, all the self-energy type mode-couplingterms can be taken into account as has been discussed in Sec.
IV A. It is easy to see that the
v-linear imaginary part of
1/x˜s(v1i0) given by Eq. ~2.13!is cancelled by the subtrac-
tion of the single-site portion of such a renormalized two-linediagram. When this cancellation is considered, the inverse ofthe dynamical susceptibility is approximately described as
1/
xs~v1i0,q!51/x˜s~0!2Js~q!/41L~0,q!
24cP~kBTK!2@P~v1i0,q!2P0~0!#,
~4.18!
with 1/ x˜s(0)5kBTK. This susceptibility has to be calculated
in a self-consistent way, because the mode-mode coupling
term L(0,q) depends on the susceptibility. When the tem-
perature dependence of coupling constants such as CF,CQF,CQ,CS, andC˜Lis not substantial, this self-
consistency equation is similar to those of the previouspapers
18–21as will be discussed in Sec. VI. Then, it is easy to
see that when the mode-mode coupling is strong enough
L(0,q) also shows an approximate T-linear dependence.
This Curie-Weiss law is of higher order in 1/ d. When the
dispersion relation of quasiparticles has specific structures,both the two mechanisms studied in this section are signifi-cant in finite dimensions. It is interesting to examine which isthe primary cause of the Curie-Weiss behavior of actualitinerant-electron magnets.
V. INTERSITE QUANTUM SPIN FLUCTUATIONS
A. Model of high- Tccuprates
According to the prediction by Gutzwiller5or by Brink-
man and Rice,28the effective mass of quasiparticles diverges
at the Mott transition. It has already been implied in Sec.IV B that this divergence is suppressed by antiferromagnetic
quantum spin fluctuations. However, the divergence of C
QF
orCQatT50 K implies that the renormalization of quasi-
particles should be treated more properly than it has been inSec. IV B. The main purpose of this section is to confirm thissuppression. In this section, we consider only the self-energy
correction at T50 K for the sake of simplicity.
We consider the Hubbard model in the large Uregion of
U
r(0).1o rU@kBTKon the simple square lattice ( d52)
as an example; it is one of the simplest effective Hamilto-
nians for high- Tccuprate oxides. When the transfer integral
between the nearest neighbors 2t1and that between the
next-nearest neighbors 2t2are only taken into account, the
dispersion relation of unrenormalized electrons is given by
E~k!522t1@cos~kxa!1cos~kya!#24t2cos~kxa!cos~kya!.
(5.1)
The superexchange interaction is given by
Js~q!522J1@cos~qxa!1cos~qya!#
24J2cos~qxa!cos~qya!, ~5.2!
withJ154t12/U.0 andJ254t22/U.0. In ordinary situa-
tions, it follows that ut2u,ut1uandJ2!J1.
WhenIs(1i0,q) has maxima at Q.(6p/a,6p/a), the
intersite exchange interaction is expanded as in our previouspaper
22so that
1
4Is~v1i0,Q2q!5kBTKFa21
4~p1xqx21p1yqy2!a2
1i~p22cP!pv
2kBTK1G,~5.3!
with a,p1x,p1yandp22cPbeing expansion coefficients.
Here, 2cPdefined by Eq. ~2.14!is included because of the
subtraction term of the single-site portion. When Is(0,q)i s
isotropic around Q, it follows that
xs~v1i0,Q2q!5xs~Q!k2
q21k22iv/G, ~5.4!420 57 FUSAYOSHI J. OHKAWAwith xs(Q)51/@(12a)kBTK#,k254(12a)/(p1a2), and
G5p1kBTKa2/(2pp2), withp1[p1x5p1y. According to
our previous paper,22the specific heat coefficient from inter-
site spin fluctuations is given by
Dg5~kB/T0!lnA~qc/k!211, ~5.5!
withqcbeing the cutoff wave number and kBT0
52G/a25(p1/pp2)kBTK. It follows from Eqs. ~2.16!and
~5.5!that
Dfm53Dg
2p2r~0!kB25fm6p2TQ
p1pTKlnA~qc/k!211.
~5.6!
Here, the kdependence of fm(k) and Dfm(k) is ignored. It
is likely that 0 ,p2&1 andp1.O(1) in many cases. When
there is substantial nesting of the Fermi surface, p1is much
larger than unity. When p1@1 is satisfied, intersite spin fluc-
tuations are localized in momentum space so that T0@TK
andfm@Dfm.
B. Absence of the divergence of the effective mass
The self-energy correction to first order explicitly in
Fs(ivl,q) is given by
DS s~i«n,k!53kBT
N(
vlq1
4U2Fs~ivl,q!l˜s2~0,0!
3Gs~c!~i«n1ivl,k1q!, ~5.7!
where the factor 3 is due to the longitudinal and the trans-
verse spin channels. Because Gs(c)(i«n,k) given by Eq.
~2.12!depends on the multisite self-energy, Eq. ~5.7!should
be self-consistently treated.
Equation ~2.7!can be divided into two parts so that
Fs(ivl,q)5Fs(1)(ivl,q)1Fs(2)(ivl,q), with
Fs~1!~ivl,q!5x˜s2~ivl!1
4Is~ivl,q!.x˜s2~0!1
4Is~0,q!
~5.8!
and
Fs~2!~ivl,q!5x˜s2~ivl!F1
4Is~ivl,q!G2
xs~ivl,q!
.x˜s2~0!I¯2xs~ivl,q!, ~5.9!
withI¯51
4Is(0,Q)5akBTK. Because high-energy spin exci-
tations with uvu@kBTKare mainly responsible for
Fs(1)(ivl,q), its vldependence is small and can be ignored
foruvlu,kBTK: The main part of Is(0,q) comes from the
superexchange interaction so that
Is~0,q!522I1@cos~qxa!1cos~qya!#
24I2cos~qxa!cos~qya!1, ~5.10!
withI1.J1anduI2u!I1. On the other hand, low-energy spin
excitations with uvu,kBTKare mainly responsible for
Fs(2)(ivl,q). When low-energy spin fluctuations are local-
ized around Qin momentum space, the vland theqdepen-dences of Is(ivl,q) can be ignored in Eq. ~5.9!and only the
vland theqdependences of xs(ivl,q) are significant. Ac-
cording to the division of Fs(ivl,q), the multisite self-
energy is also divided into two parts so that DS s(i«n,k)
5DSs(1)(i«n,k)1DSs(2)(i«n,k), with
DSs~1!~i«n,k!53w˜s2
4kBT
N(
vlqIs~0,q!Gs~c!~i«n1ivl,k1q!
~5.11!
and
DSs~2!~i«n,k!53w˜s2I¯2kBT
N(
vlqxs~ivl,q!
3Gs~c!~i«n1ivl,k1q!. ~5.12!
Here, Eq. ~2.21!is made use of.
Equation ~5.11!is calculated so that
DSs~1!~«1i0,k!523w˜s2
4fm@I1h1@cos~kxa!1cos~kya!#
12I2h2cos~kxa!cos~kya!#, ~5.13!
with
h151
N(
kf@j~k!#@cos~kxa!1cos~kya!#~5.14!
and
h251
N(
kf@j~k!#@2cos~kxa!cos~kya!#.~5.15!
Equations ~5.14!and~5.15!depend on the shape and the
volume of the Fermi surface. When the kdependence of
fm(k) is ignored, they are calculated so that h1
5(4/p2)(t1/ut1u).0.4t1/ut1uandh250 forT50K ,t250
and just the half filling. Although h2is nonzero for t2Þ0,it
follows that uh2u!uh1ufor almost half filling. Equation
~5.13!is approximately given by
DSs~1!~«1i0,k!.23w˜s2
4fmI1h1@cos~kxa!1cos~kya!#
~5.16!
for nonzero t2and almost half filling.
Equation ~5.12!is calculated so that
DSs~2!~«1i0,k!523w˜s2I¯2
fm1
N(
qFf~jq!xs~jq2«2i0,q!
11
pE
2`1`
dvn~v!Im@xs~v1i0,q!#
«1v1i02jqG,
~5.17!
with jqstanding for j(k1q). The real part of Eq. ~5.17!is
calculated for T50 K so that2657 421 QUANTUM AND THERMAL SPIN FLUCTUATIONS IN . . .ReDSs~2!~«1i0,k!5Y21
N(
qH@f~jq8!21/2#pG~q21k2!
@«2jq8#21@G~q21k2!#2
2@«2jq8#lnuG~q21k2!/~«2jq8!u
@«2jq8#21@G~q21k2!#2J,
~5.18!
with jq8standing for j(k2Q1q) here and
Y253w˜s2I¯2xs~Q!Gk2
pfm56w˜s2a2~kBTK!2
p2p2fm.~5.19!
The derivatives of the self-energy are given by
ReF]
]«DSs~2!~«1i0,k!G
«505B~k! ~5.20!
and
ReF]
]kDSs~2!~1i0,k!G.]j~k!
]k@A~k!1B~k!#,
~5.21!
whereA(k) andB(k) are given by
A~k!5pY2
G1
N(
qd@j~k2Q1q!#1
q21k2~5.22!
and
B~k!5Y21
N(
qb@j~k2Q1q!#, ~5.23!
with
b~j!52pG~q21k2!uju
$@G~q21k2!#21j2%2
12@G~q21k2!#21j2
$@G~q21k2!#21j2%2lnUG~q21k2!
jU
11
@G~q21k2!#21j2. ~5.24!
In Eq. ~5.21!, an approximate relation of ]j(k2Q)/]k.
2]j(k)/]kfork’s on the Fermi surface is made use of. In
the first term of Eq. ~5.24!, the relation of @2f(j)21#j5
2ujuforT50 K is made use of.
When two variables, qi5qn(k) andq'5uq2qin(k)u
withn(k)5@]j(k2Q)/]k#/u]j(k2Q)/]ku, are introduced,
the integration ~5.22!is carried out so that
A~k!.3p3~k!w˜s2a2
pfmp1aq¯~k!, ~5.25!
withq¯(k)5Ak21q02(k) andp3(k)5akBTQ/u]j(k2Q)/
]ku.1. Here, q0(k) is defined by j@k2Q1q0(k)n(k)#
50; it is approximately given by q0(k).2j(k2Q)/u]j(k
2Q)/]ku. It should be noted that A(k)/fm5O(1) for
p1aq¯(k)5O(1).Because a straightforward integration of Eq. ~5.23!is dif-
ficult, we use one averaged over kinstead of B(k) itself:
B¯5Y21
N(
qr*~0!E
2`1`
djb~j!
52fm3a2w˜s2TK
2pp1fm2TQlnA11~qc/k!2, ~5.26!
where the integrations of the second and the third terms in
Eq.~5.24!over jcancel each other because of
E
2`1`
dyF211y2
~11y2!2ln1
uyu11
11y2G50. ~5.27!
It follows that B¯/Dfm52a2w˜s2/(4p2fm2).21a si se x -
pected.
The multisite self-energy is given by
DS~«1i0,k!.DSs~2!~0,kF!1B¯«1DSs~1!~0,k!
1~A1B¯!j~k!1, ~5.28!
where a relation ( k2kF)@]j(k)/]k#kF.j(k)2j(kF) for
k.kFwithkFbeing a wave number vector on the Fermi
surface, is made use of, and the kdependence of A(k)i s
ignored. According to the Fermi-surface sum rule, it follows
that uE(kF)1S˜
01DSs(2)(0,kF)2mu/fm!kBTQfor almost
half filling.42Then, we obtain a self-consistency equation for
j(k)o rTQthrough Eq. ~2.11!; it is solved in such a way that
kBTQ.1
fmHU2t113w˜s2
4fmh1I1U1~A1B¯!kBTQJ
.1
12~A1B¯!/fmU2
fmt113w˜s2
4fm2h1I1U,~5.29!
withfm5f˜m2B¯. Here, ut1u@ut2uis assumed.
In the region of fm@ut1u/uh1I1u, the renormalization ef-
fect due to the superexchange interaction is more significantfor the transfer effect of electrons than the original transfer
integralt
1; quasiparticles are dressed in high-energy intersite
spin fluctuations. As long as the superexchange interaction is
nonzero so that h1I1Þ0,TQnever becomes zero even in the
large limit of fm. The divergence of the quasiparticle mass
never occurs even in the vicinity of the Mott transition. It is
interesting that unless ut2uis much smaller than ut1uthe shape
of the Fermi surface in the region of fm@ut2u/uh1I1uis
slightly different from that in the region of fm!ut2u/uh1I1u.
C. Light quasiparticles
Equation ~5.29!also shows that TQdiverges at ( A
1B¯)/fm51; the effective mass of quasiparticles can be-
come very light. This anomaly is due to the momentum de-pendent self-energy correction caused by low-energy inter-site quantum spin fluctuations and must be a precursor effectof antiferromagnetic instability. The purpose of this subsec-tion is to show that the formation of this type of light quasi-422 57 FUSAYOSHI J. OHKAWAparticles dressed in low-energy intersite spin fluctuations can
explain the so-called spin-bag or spin-gap behavior of high-
Tccuprate oxides.43
First of all, assume that fm@1 in the vicinity of the Mott
transition. When intersite spin fluctuations are localized inmomentum space so that the electronic specific heat mainly
comes from local spin fluctuations, it follows that
uB¯u/fm
!1,w˜s/fm.2 andaq¯(k)!1. It is likely that p1aq¯(k)i s
not much larger than unity. Then, it is possible that A/fm
5O(1) in the spin-gap region of actual high- Tccuprate ox-
ides.
One of the most typical properties characterized as the
spin-gap behavior is the suppression of the spin susceptibil-ity at low temperatures.
44Because quasiparticles themselves
are renormalized in this mechanism, the spin-gap behavior isexpected for not only spin excitations but also charge exci-
tations, that is, for any physical property. When T
Qincreases
in the spin-gap region, it is quite trivial from Eqs. ~2.15!and
~2.17!that not only the susceptibility but also the specific-
heat coefficient decrease.
The peak value of the density of states is given by
r~0!.F12~A1B¯!
fmGYU8t113w˜s2
fmh1I1U~5.30!
through Eqs. ~2.18!and~5.29!. Because the terms due to
intersite spin fluctuations such as A,B¯, and h1I1vanish in
the large dlimit, the density of states at the chemical poten-
tialr(0) is not renormalized to leading order in 1/ deven for
fm!1`. However, it is reduced by the higher-order effects
in 1/d. The reduction of the peak value of the density of
states must be observed in the spin-gap region. It should benoted that the spectral weight around the chemical potential
remains almost constant and is as small as 1/
fmper unit cell.
Many physical properties decrease with increasing TQ.
One of the exceptional properties is the conductivity along
CuO2planes, because it increases as the effective mass of
quasiparticles becomes lighter. On the other hand, the effec-
tive mass perpendicular to CuO 2planes must be quite heavy;
it is renormalized by only the frequency dependence of the
self-energy and is about fmtimes as large as that according
to band calculation. Therefore, the anisotropy of the effectivemass must become larger as the spin-gap behavior develops.When this anisotropy is large enough, it is likely that the
conductivity along CuO
2planes is metallic whereas the con-
ductivity perpendicular to CuO 2planes is nonmetallic.
Another exceptional property is the intersite exchange in-
teraction JQ(0,q) due to the virtual exchange of low-energy
spin excitations within the quasiparticle bands. It becomes
large with increasing TQ. When an anisotropic intersite ex-
change interaction between nuclear spins is mediated by
JQ(0,q),45the transverse NMR rate is enhanced rather than
suppressed.
Our investigation has been so far restricted to T50Ki n
this section. It is necessary to calculate self-consistently the
temperature dependence of A/fmtogether with all the physi-
cal properties appearing in this formulation in order to givemore quantitative analysis.VI. COMPARISON WITH OTHER FORMULATIONS
The free energy of this paper includes the mode-mode
coupling. If this free energy were treated in the same way asMurata and Doniach’s,
18the mode-mode coupling would be
doubly counted. The free energy used by Murata and Doni-ach includes no essential part of the mode-mode coupling.When the isotropy of the spin space is considered, it is given
byV
MD($m%)5Vpara1V28($m%)1V4($m%)1, where
the second term is given by V28($m%)
51
2N(nqumn(q)u2/xs8(q) with 1/ xs8(q)51/x˜s(0)21
4Js(q)
2S(0,q)2DS8(0,q), and V4($m%) is the one discussed in
Sec. III. When only ferromagnetic spin fluctuations are sub-
stantial, the fourth order term is decoupled so that V4(m)
.const 11
2NLMD(0)(nqumn(q)u2, with
LMD~0!52U4
24F4!
2!2!p˜x412~2p˜@x2y2#!
15~2Dpx4!~0,0,0 !G(
q^umn~q!u2&
55~11CF2C˜L!
2x˜s~0!(
q^umn~q!u2&. ~6.1!
The eventual susceptibility is given by 1/ xMD(0)51/xs8(0)
1LMD(0).Because intersite spin fluctuations are significant
for only such q’s that satisfy1
4Is(0,q)x˜s(0).1, it follows
thatFs(ivl,q).xs(ivl,q) for such q’s. When ^umn(q)u2&
5(kBT/N)(vlqxs(ivl,q) is used, it follows that xMD(0) is
approximately equal to xs(0,0) given by Eq. ~4.18!.I nt h e
static approximation, it follows that ^umn(q)u2&5(kBT/
N)(qxs(0,q). The formulation of this paper is consistent
with Murata and Doniach’s formulation that takes into ac-count the isotropy of the spin space.
The formulation of this paper is also consistent with
Moriya and Kawabata’s formulations.
3,19–21However, there
is a crucial difference; Moriya and Kawabata considered lo-cal quantum spin fluctuations in the RPA at the microscopiclevel.
46In principle, therefore, the formulations by Moriya
and Kawabata are only applicable to the small Uregion. An
apparent great success of their formulations relies on thecontinuation hypothesis: When general relations are con-
structed for the small Uregion and parameters appearing in
the relations are regarded as phenomenological, the relations
can apply even to the large Uregion as long as there is no
discontinuity as a function of U. This paper has confirmed
the validity of this phenomenological treatment.
Kuramoto and Miyake
39–41proposed a semiphenomeno-
logical model called the duality model, where electronicproperties of strongly correlated systems are described interms of two degrees of freedom: localized spins and itiner-ant fermions. The results of the duality model are apparentlysimilar to but physically different from the results of thispaper. According to Ref. 38, for example, the dynamicalsusceptibility of the duality model consists of two parts: aterm due to localized spins and a term due to itinerant fer-mions. This division is different from that of Eq. ~2.3!: the
division into the term of leading order in k
BTK/Uand the
term of higher order in it. As long as T!TKand uvlu57 423 QUANTUM AND THERMAL SPIN FLUCTUATIONS IN . . .!kBTK, all the physical properties of this paper are of itin-
erant fermions except for the exchange interaction from the
virtual exchange of high-energy spin excitations Js(q).
When the notations of this paper are used, the term due tolocalized spins of the duality model can be described as
1/
xs~v1i0,q!51/x0~v1i0!2Js~q!/41L~0,q!
24cP~kBTK!2P~v1i0,q!. ~6.2!
This is different from Eq. ~4.18!of this paper. In Eq. ~6.2!,
x0(v1i0) is the susceptibility of localized spins. It was as-
sumed that x0(v1i0) has no low-energy excitations in or-
der to make the susceptibility ~6.2!physical.38On the other
hand, x˜s(v1i0) of the MAM has low-energy excitations. In
Eq.~4.18!, itsv-linear imaginary part is cancelled by the
subtraction of the single-site portion.
VII. CONCLUSION
It has been demonstrated that the 1/ d-expansion method is
useful in investigating itinerant-electron but almostlocalized-electron magnetism or magnetism in the crossoverregion between local-moment magnetism and itinerant-electron magnetism, which is characterized by the coexist-ence of the localization of spin fluctuations in momentumspace and the localization of spin fluctuations in real space.All the types of spin fluctuations have been properly takeninto account. Local spin fluctuations have been taken intoaccount through mapping to the Anderson model: TheKondo effect plays an essential role in quenching magnetic
moments even in lattice systems. Intersite spin fluctuations
have been taken into account by the 1/ d-expansion method,
withdbeing the spatial dimensionality, or the mode-mode
coupling between spin fluctuations has been taken into ac-count in essentially the same approximation as the one usedby Murata and Doniach and the ones by Moriya and Kawa-bata. The free energy as a function of magnetization has beenderived at the microscopic level.
There are two mechanisms of Curie-Weiss behavior of the
susceptibility. One is of leading order in 1/ dorO
@(1/d)0#
and is due to specific structures of the dispersion relation of
quasiparticles, and the other is of higher order in 1/ dand is
due to the mode-mode coupling between thermal spin fluc-tuations. In contrast to the prediction by Gutzwiller or byBrinkman and Rice, the effective mass of quasiparticles doesnot diverge in the vicinity of the Mott transition because ofmomentum-dependent self-energy corrections due to antifer-romagnetic quantum spin fluctuations. The formation of lightquasiparticles is possible in the vicinity of the Mott transi-tion. This can explain the so-called spin-bag or spin-gap be-
havior of high- T
ccuprate oxides.
ACKNOWLEDGMENTS
This work was partly supported by a Grant-in-Aid for
Scientific Research ~C!No. 08640434 from the Ministry of
Education, Science, Sports and Culture of Japan.
1See, for example, Proceedings of the International Conference on
Itinerant-Electron Magnetism , edited by R. D. Lowde and E. P.
Wohlfarth, @Physica B&C 91~1977!#.
2See, for example, Electron Correlation and Magnetism in
Narrow-Band Systems , Vol. 29 of Springer Series in Solid-State
Science, edited by T. Moriya ~Springer-Verlag, Berlin, 1981 !.
3See, for example, T. Moriya, Spin Fluctuations in Itinerant Elec-
tron Magnetism , Vol. 56 of Springer Series in Solid-State Sci-
ence, edited by T. Moriya ~Springer-Verlag, Berlin, 1985 !.
4J. Kanamori, Prog. Theor. Phys. 30, 275 ~1963!.
5M. C. Gutzwiller, Phys. Rev. 134, A293 ~1963!.
6J. Hubbard, Proc. R. Soc. London, Ser. A 276, 238 ~1963!.
7F. J. Ohkawa, J. Phys. Soc. Jpn. 58, 4156 ~1989!.
8See, for example, P. W. Anderson, Solid State Physics , edited by
F. Seitz and D. Turnbull ~Academic, New York, 1963 !, Vol. 14,
p. 99.
9F. J. Ohkawa and N. Matsumoto, J. Phys. Soc. Jpn. 63, 602
~1994!.
10Y. Kuramoto and T. Watanabe, Physica B 148,8 0~1987!.
11W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 ~1989!.
12F. J. Ohkawa, Phys. Rev. B 46,1 19 6 5 ~1992!. As was discussed
in this reference, the response function for specific qand the
mean fields, which relate to magnetic instability in repulsivesystems or to charge-density-wave instability or the Bardeen-Cooper-Schrieffer type superconductivity in attractive systems,are of leading order in 1/ dorO
@(1/d)0#. Unless these properties
are discussed, the SSA or the RSSA is rigorous in the large d
limit because no fluctuations for these types of instability exist.13F. J. Ohkawa, Phys. Rev. B 44, 6812 ~1991!.
14F. J. Ohkawa, J. Phys. Soc. Jpn. 60, 3218 ~1991!.
15F. J. Ohkawa, J. Phys. Soc. Jpn. 61, 1615 ~1992!.
16F. J. Ohkawa, J. Phys. Soc. Jpn. 61, 4490 ~1992!. The RSSA is
reduced to the SSA in the large dlimit.
17K. Yosida, Phys. Rev. 147, 223 ~1966!.
18K. K. Murata and S. Doniach, Phys. Rev. Lett. 29, 285 ~1972!.
19T. Moriya and A. Kawabata, J. Phys. Soc. Jpn. 34, 639 ~1973!.
20T. Moriya and A. Kawabata, J. Phys. Soc. Jpn. 34, 669 ~1973!.
21A. Kawabata, J. Phys. F 4, 1477 ~1974!.
22F. J. Ohkawa, Phys. Rev. B 54,1 53 8 8 ~1996!.
23The mapping condition depends on temperature T. Then,TKde-
pends on Tthrough the Tdependence of the MAM itself.
24K. G. Wilson, Rev. Mod. Phys. 47, 773 ~1975!.
25F. J. Ohkawa ~unpublished !.
26The life time of quasiparticles or the imaginary part of the self-
energy can be ignored for u«nu!kBTKandT!TKin the RSSA.
When certain types of the spin-gap behavior of the cuprate ox-ides are discussed, it has to be considered beyond the RSSA.
27The MAM is determined in the absence of magnetic fields in this
formulation. In general, therefore, the expansion coefficient fortheH
*linear term of the MAM is different from that for the
single-site term of the Hubbard model.
28W. F. Brinkman and T. M. Rice, Phys. Rev. B 2, 4302 ~1970!.
29P. W. Anderson, Phys. Rev. 124,4 1~1961!.
30A. Georges and G. Kotliar, Phys. Rev. B 45, 6479 ~1992!. When
the density of states for unrenormalized electrons, r0(«)
5(1/N)(kd@«2E(k)#, is of a Lorentzian shape in the large d424 57 FUSAYOSHI J. OHKAWAlimit, the hybridization energy of the MAM is constant as a
function of energies as was discussed in this reference. It fol-lows that s
c50 for such a model in the large dlimit.
31H. Shiba, Prog. Theor. Phys. 54, 967 ~1975!.
32J. C. Ward, Phys. Rev. 78, 182 ~1950!.
33T. Yamamoto and F. J. Ohkawa, J. Phys. Soc. Jpn. 57, 3562
~1988!.
34Only irreducible diagrams are included in the thermodynamic po-
tential as a function of magnetization, whereas not only irreduc-ible diagrams but also reducible diagrams are included in that asa function of magnetic fields.
35WhenGs(c)(i«n,k) is used instead of gs(c)(i«n,k) in Eq. ~3.16!,
the self-energy-type mode-mode coupling can be taken into ac-count.
36F. J. Ohkawa, K. Onoue, and H. Satoh ~unpublished !.
37A portion of the self-energy-type contribution is cancelled by the
chemical potential shift. ~Ref. 21 !. In general, however, this ef-
fect is small for strongly correlated systems.
38K. Miyake and O. Narikiyo, J. Phys. Soc. Jpn. 63, 3821 ~1994!.39Y. Kuramoto, Physica B 156-157, 789 ~1989!.
40Y. Kuramoto and K. Miyake, J. Phys. Soc. Jpn. 59, 2831 ~1990!.
41K. Miyake and Y. Kuramoto, Physica B 171,2 0~1991!.
42The cancellation of a portion of the self-energy correction by the
chemical potential shift ~Ref. 37 !is taken into account in this
treatment based on the Fermi-surface sum rule.
43Superconducting fluctuations also renormalize quasiparticles.
Even when this renormalization is significant, the formation oflight quasiparticles occurs in a similar way.
44See, for example, T. Shimizu, H. Aoki, H. Yasuoka, T. Tsuda, Y.
Ueda, K. Yoshimura, and K. Kosuge, J. Phys. Soc. Jpn. 62, 3710
~1993!.
45C. H. Pennington and C. P. Slichter, Phys. Rev. Lett. 66, 381
~1991!.
46Because they treated weakly correlated systems, they had to con-
sider a mode-mode coupling between charge and spin fluctua-tions as is shown in Fig. 3 ~c!of this paper. The coupling has
been ignored in this paper, because charge fluctuations play nocritical role in strongly correlated systems./57
425 QUANTUM AND THERMAL SPIN FLUCTUATIONS IN . . . |
PhysRevB.89.161404.pdf | RAPID COMMUNICATIONS
PHYSICAL REVIEW B 89, 161404(R) (2014)
Control of the ionization state of three single donor atoms in silicon
B. V oisin,1M. Cobian,2X. Jehl,1M. Vinet,3Y .-M. Niquet,2C. Delerue,4S. de Franceschi,1and M. Sanquer1,*
1SPSMS, UMR-E CEA/UJF-Grenoble 1, INAC, 17 rue des Martyrs, 38054 Grenoble, France
2SPMM, UMR-E CEA/UJF-Grenoble 1, INAC, 17 rue des Martyrs, 38054 Grenoble, France
3CEA, LETI, MINATEC Campus, 17 rue des Martyrs, 38054 Grenoble, France
4IEMN, 41 Boulevard Vauban, 59046 Lille, France
(Received 6 December 2013; published 16 April 2014)
By varying the front-gate and the substrate voltages in a short silicon-on-insulator trigate field-effect transistor,
we control the ionization state of three arsenic donors. We obtain good quantitative agreement between three-dimensional electrostatic simulations and experiment for the control voltages at which the ionization takes place.It allows us to observe the three doubly occupied states As
−at strong electric field in the presence of nearby
source-drain electrodes.
DOI: 10.1103/PhysRevB.89.161404 PACS number(s): 73 .22.Dj,73.23.Hk,73.63.−b
Semiconducting devices have entered a new era where
single dopants can be used for new (quantum) functionali-ties [ 1,2]. Thin silicon-on-insulator (SOI) devices are partic-
ularly attractive in this perspective, offering good control ofthe transverse electric field in the channel. This electrostaticproperty, at the core of the metal-oxide-semiconductor field-effect transistors (MOSFETs), is crucial to address dopantsindividually and to control their electronic wave functionsand couplings. These are prerequisites for dopant-basedapplications.
In this work we study, both experimentally and with
simulations, how three arsenic donors are charged in ananoscopic MOSFET. The ionization state of each donor isseparately read out by detecting its corresponding resonancein the source-drain (S-D) current. The ionization state of eachdonor—As
+,A s0or As−—is individually controlled at low
temperature by tuning the transverse electric field with a frontand a back gate. The scalability of a compact system of a fewtunable shallow donors beyond the previously studied cases ofone [ 3–7] and two donors [ 4,8] is then shown.
In our small MOSFETs—in the 10 nm size range—dopants
are not isolated in the channel but see a complex electrostaticenvironment which should be considered cautiously. It in-cludes other donors in the channel and in the heavily doped S-Dleads, as well as offset charges in the gate stack. First the inter-action among donors, which may result in a Coulomb glass [ 9],
is screened by the S-D leads [see the inset of Fig. 1(c) which
shows a weak interaction between two donors]. Therefore wecan independently address the gate-induced charge transitionsof a given donor while assuming the other donors to be in aconstant ionization state. Secondly, the ionization of donors atthe graded edges of the S-D leads—which we shall refer to aslead extensions—is explicitly considered in our simulation ina mean-field approach, neglecting Kondo [ 10] and Fermi edge
singularity [ 11] effects, which are not observed in our devices
at 4.2 K. The simulation of a realistic electrostatic environmentexplains the evolution of the donor ionization lines (DIL) as afunction of the control voltages.
The samples, fabricated on 200 mm SOI wafers, are similar
to those described in Ref. [ 3]. A 200-nm-long, 17-nm-thick,
*marc.sanquer@cea.frand 50-nm-wide silicon nanowire was etched from the SOI
film and covered at its center by a 30-nm-long polysilicongate isolated by a 4-nm-thick SiO
2layer, called the front
oxide (FOX) [see Figs. 1(a),1(b)]. This front gate covers three
sides of the silicon channel in a so-called trigate geometry.A 400-nm-thick buried oxide (BOX) separates the channelfrom the silicon substrate, which can be biased using theprocedure described in Ref. [ 12]. The central part of the
channel contains a few arsenic donors as estimated by processsimulations including the rapid thermal annealing step fordonors activation [ 3]. The device differential conductance ( G)
was measured at T=4.2 K using standard lock-in detection.
The extensions of the S-D are located below the gate becausethere are no spacers. Therefore the channel length (between10 and 20 nm) is significantly smaller than the nominalgate length (30 nm) [ 3], resulting in the donors centered in
the channel to have sufficient tunnel couplings to S-D andtherefore to be detected by resonant tunneling.
Figure 1(c) shows a color plot of Gas a function of
substrate voltage V
band front gate voltage Vgat a dc S-D
biasVd=0. Above a certain threshold voltage the S-D
current has contributions coming from the continuum of
delocalized conduction-band states. The donors in the body of
the SOI contribute below this threshold, giving rise to resonanttunneling conduction paths whenever their energy levels liein the bias window [ 3]. AtV
b=0, the channel conduction
starts at Vg/similarequal0–0.1 V , which is the expected threshold voltage
for the gate stack used [ 13].Vb/similarequalVg/similarequal0 corresponds to the
flat-band regime without evidence for donor states. At Vb>0,
conduction starts at negative Vg, and a large transverse electric
field is present in the channel. Carriers are accumulated atthe BOX interface and the threshold shows a kink in the(V
b,Vg) plane [see the blue dashed line in Fig. 1(c)]. In the
case of P-doped, macroscopic SOI films, this kink has beenattributed to the ionization of donors in the body of the channelwhen a vertical electric field is applied [ 14]. In the middle
of a long undoped channel the kink is due to the shift of
the two-dimensional electron gas (2DEG) from the top to the
back-gate interface [ 15]. It is different in our short nanoscale
field-effect transistor where the S-D leads end up with a stronggradient of As atoms near the channel. In these S-D extensionsthe carrier density and the ionization state of the donors canchange with V
gandVb. This affects the potential landscape in
1098-0121/2014/89(16)/161404(5) 161404-1 ©2014 American Physical SocietyRAPID COMMUNICATIONS
B. VOISIN et al. PHYSICAL REVIEW B 89, 161404(R) (2014)
10-1210-1010-810-610-410-2100
AA'BB'CC'
simulated threshold
simulation x=0 y=15 z=3 00 . 51G (µS)
-20 -18G (S)
(1,0,0)(2,0,0)(2,1,0)(0,0,1)(0,0,2)
(1,0,1)(2,0,1)(2,1,1)BOX (SiO 2)FOX (SiO 2)
17 nm
400 nmfront gate (poly-Si)
4 nm
7 nm50 nm
Back-gate (Si)z
yOchannel (Si)
15 nm30 nm
y
xOgate
gateSource
DrainExtension
Extensiontransverse view top view (a)
(c)(b)
FIG. 1. (Color online) (a),(b) Schematic cross-sectional view of
the device. Vertical cut along the transverse z-yplane (a) and
horizontal cut along the x-yplane (b). (c) Color plot of the source-
drain linear conductance vs back- and front-gate voltages ( Vband
Vg, respectively) at T=4.2 K. The blue dashed line is the simulated
threshold voltage assailing no donors in the channel. The three pairs of
DILs, denoted by A-A/prime,B-B/prime,a n dC-C/prime, are found in the subthreshold
regime. They correspond to the As+/As0and As0/As−transitions of
three As donors. The ionization states ( a,b,c ) for the three donors are
indicated between the DILs ( a,b,a n dccan be 0, 1, or 2 depending on
whether the corresponding donor is empty, singly, or doubly occupied,
respectively). States with more than four electrons on the three donors
are barely visible because of the strong coupling with the conductionband. According to our simulation the donor corresponding to the
C-C
/primepair of DILs is closer to the front gate, while the other two
donors are closer to the BOX. The black line is the simulated DILfor a donor located at ( x,y,z )=(0,15,and 3 nm). Inset: Zoom-in of
the crossing between the AandCDIL at T=1 K showing a weak
Coulomb repulsion.
the whole channel and the curvature of the threshold line [ 16].
As we shall show below, the response of the donors andconduction electrons in the S-D extensions affects the slopeand position of both the threshold voltage line and the DILs.
We have simulated the DIL of a few donors in the channel
of our trigate transistor. For that purpose, we have treated thefew donors in the channel as interacting point charges, andthe donors in the highly doped source and drain extensions asa continuum. We have first computed the potential landscapeV(/vectorr) in the nanowire channel at V
d=0[16]. To this aim,
we have solved Poisson’s equation self-consistently using the
Fermi integral F1/2(Ec−eV(/vectorr)−μ
kBT) as an approximation for the
local density of electrons. The density of ionized donors inthe S-D extensions is approximated as in Ref. [ 17]. Here E
candμare the silicon conduction-band edge and the device
chemical potential, respectively. The ∼15-nm-long channel
was left undoped, and the simulations were run at 30 K forcomputational reasons. This simple model shall give a fairaccount of the screening by the quasimetallic S-D extensions.It provides, admittedly, a coarse description of the channel,but our interest here is the physics of individual donors, thusbelow the channel threshold. Once the potential landscape hasbeen computed as a function of V
bandVg, we have added a
few bulklike donors in the channel at positions /vectorri, and have
tracked their bound-state energy levels E1s(Vb,Vg,/vectorri)=Ec−
Eb−eV(/vectorri) (where Ebis the binding energy of these donors,
53 meV). We have also computed the Coulomb interactions Uij
between these donors as the screened Coulomb interactions
between point charges. We have finally used these data asinput for a Coulomb-blockade-like model of the donor systemin order to determine the DIL of each donor.
First of all, we have computed the threshold voltage as the
voltage where the electron concentration, integrated over theSOI thickness, exceeds 10
11cm−2[in the ( Vb,Vg) plane of
Fig. 1(c), this threshold is denoted by a blue dashed line]. The
kink near Vb/similarequalVg/similarequal0 and the absolute values for VbandVg
are in excellent agreement with the experimental data. We start
with one single donor located at ( x=0,y=15,z=3) nm
[see Figs. 1(a),1(b) for a definition of the spatial coordinates;
in our reference, ( x=0,y=0,z=8.5) corresponds to the
center of the channel, |x|/greaterorequalslant7.5 nm are the S-D extensions].
The (x=0,y=15,z=3) nm position is chosen such that
it approximately corresponds to the experimental DIL fordopant A (Fig. 1(c), black solid line). The DIL is curved in
the (V
b,Vg) plane, which cannot be captured with a model
assuming perfectly metallic S-D leads and constant capacitivecouplings between the donor and all the surrounding electrodes(such would yield only straight DILs). The curvature indicatesthat the couplings to the gate and substrate evolve with V
band
Vg, as a result of the ionization of donors in the extensions
and of the accumulation of surface carriers. Taking thiscomplex electrostatic environment into account is necessaryto reproduce the DILs in the channel. In particular the DILbecome less dependent on V
g, when Vbis decreased. The
donor’s ionization thus occurs at higher Vgvalues where
the conduction channel is set in the extensions of S-D. Thelatter screens the gate potential at the bottom of the channel,therefore on the donor site.
In Fig. 2we introduced two more donors whose positions
differ either in x,y,o rz. A change in x[with constant y,z,s e e
Fig. 2(a)] produces three almost parallel DIL. This is because
the lever arm parameters change ( α
g=δφ
δVg,αb=δφ
δVb, where
φis the electrostatic potential at the donor position). A donor
centered in the channel has larger lever arm parameters (whichmeans a better electrostatic control by V
bandVg) than a donor
located closer to the S-D extensions. In other words, there isa significant electric field along xin our structure when finite
V
gandVbare applied. The parallelism between the 3 DIL
suggests that the ratio ofαg
αbis barely affected. As a result
donors close to the S-D ( |x|/similarequal5) are charged (As+to As0)
at the largest negative Vgvalues. Donors more centered in the
channel ( |x|/similarequal0 or 3) are still ionized at this value of Vgthanks
to the potential gradient along x.
161404-2RAPID COMMUNICATIONS
CONTROL OF THE IONIZATION STATE OF THREE . . . PHYSICAL REVIEW B 89, 161404(R) (2014)
-2-10Vg(V)
-100 -50 0
Vb(V)
y
xSource
Drain
-2-10Vg(V)
-100 -50 0
Vb(V)(a)
(b)
DrainBOXfront gate
z
y Ochannel
FIG. 2. (Color online) Calculated DILs in the ( Vb,Vg)p l a n e
for a three-donor configuration. (a) The donors differ by theirxposition ( x,y=15 nm, z=3 nm) as sketched in the in-
set:x=0 (respectively, 3, −5) for the donor represented in
black (respectively, blue, red), keeping a constant distance to thegates. (b) The donors differ by their z[(x=0,y=15,z=
{3 (black) ,7.5 (blue) ,12 (red) }), full lines], as sketched in the inset,
ory[(x=0,y={15 (black) ,20 (blue) ,23 (red) },z=3), dashed
lines] position. Both cases correspond to varying the distance to the
front gate. For all simulations, the black donor is at the same position
as the simulated one in Fig. 1(c).
On the contrary a change in yor inz[see Fig. 2(b)] modifies
the distance between the donor and the gates, which very muchinfluences the DIL’s curvature, i.e.,αg
αb. Donors located at the
bottom center of the channel (near the BOX) are charged firstatV
b/greatermuch0 and Vg/lessmuch0. Donors closer to the front gate (large
yor large z) are charged first at positive Vgand they are less
sensitive to Vb, like the DIL C-C/primein Fig. 1(c). Therefore, a
measurement of the DILs as function of VbandVg[i.e., the
A-A/prime,B-B/prime, andC-C/primeDILs in Fig. 1(c)] allows one to deduce
the position of the corresponding donors [ 18] (i.e., donors
A,B, andC, respectively) relative to the FOX and the BOX
[see Fig. 1(a)].
Remarkably, A-A/prime,B-B/prime, andC-C/primein Fig. 1(c) form three
pairs of approximately parallel DILs, each pair being associ-ated with a different donor (i.e., A,B, andC, respectively).
We attribute the upper line of each pair to the loading of thedoubly occupied donor state (i.e., the As
0/As−transition). Thedistance between this upper line and the corresponding lower
line is set by the intradonor charging energy. The fact thatthe DILs of a given pair are parallel to each other indicatesthat the intradot charging energy does not depend on theelectric field in the channel. The positions and the slopes of theA-A
/primeandB-B/primeDILs indicate two donors near the BOX and
approximately centered in the channel (i.e., small yandz). By
contrast the C-C/primeDILs must originate from a donor close to
the front gate [large yorz, like the red lines in Fig. 2(b)]. We
cannot take into account the double occupation problem in thesimulation yet, as it would deserve to include electron-electroninteractions beyond the mean-field treatment used here. Thestrong electric field in the channel, combined with the smallnumber of donors, also favors the population of the As
−state
of a donor (e.g., A/prime) rather than the loading the As0state of
another donor (e.g., B).
Studies at finite bias Vd(see, e.g., Fig. 3) provide a direct
measurement of the charging energy Ecand the lever-arm
parameter αgof a given donor (donor Ain the case of Fig. 3).
We find Ec/similarequal30 and 20 meV for donors AandB, respectively.
In the case of donor C, we can only estimate a lower bound
Ec/greaterorequalslant30 meV due to the lack of contrast of the second
resonance (As0/As−transition), which occurs too close to
the threshold (not shown). The lever-arm parameter is smallerforA
/prime(αA/prime
g=0.08) than for A(αA
G=0.12), i.e., As−is more
strongly coupled to S-D than As0. Two physical mechanisms
account for this observation. First, the As−electronic orbital
is less localized on the donor. Secondly, the ionization of As−
occurs at higher Vgwhere the S-D are more extended towards
the donor. These two mechanisms increase the capacitivecouplings to the S-D leads with respect to the capacitivecouplings to the gates, resulting in a lower α
g. In addition,
the As−state has a stronger tunnel coupling to the S-D
leads. The Glines parallel to Coulomb-diamond edges (clearly
visible in Fig. 3) are due to local density-of-states fluctuations
in the S-D extensions [ 3]. The similar pattern for AandA/prime
FIG. 3. (Color online) Color plot of the S-D differential con-
ductance at T=4.2Ka n d Vb=10 V . The observed resonances
correspond to DILs AandA/prime. The Coulomb-blockade regimes
associated with the As+,A s0,a n dA s−charge states are indicated.
As expected, the lever arm factor is smaller for A/primethan for A(see
text). The lines of differential conductance appearing at finite Vdand
parallel to the diamond are due to fluctuations in the local density
of states of the S-D extensions [ 3]. They present approximately the
same pattern for AandA/prime, but they are more blurred for A/primedue to the
larger tunnel coupling.
161404-3RAPID COMMUNICATIONS
B. VOISIN et al. PHYSICAL REVIEW B 89, 161404(R) (2014)
(only smoothed for A/primedue to higher tunneling rates) supports
the assumption that AandA/primeare different ionization states
associated with the same donor, feeling the same localenvironment in the S-D.
Several conclusions can be drawn from our observations.
The doubly charged state exists for the three donors. Then,if the charging energy depends on the actual donor positionand consequently on its mesoscopic environment [ 19,20], the
measured E
care much smaller than the ionization energy
for As donors in bulk ( /similarequal53 meV) and the double occupied
states are well separated from the conduction-band threshold.Hence the double occupied state of a donor is more stablein our nanostructure than in the bulk case. Finally, E
cdoes
not depend significantly on the electric field in the channel. Inprevious experiments [ 20] the electric field could not be varied
on demand.
The stability of the doubly occupied state of shallow
donors in the presence of an interface has been the subjectof intense research [ 21–23]. The reduction of E
ccan be due
to the screening by a metallic gate electrode separated fromthe silicon by a very thin dielectric barrier [ 15,22,23], or to the
electric-field-induced hybridization of the donor state with aconduction-band state at Si/SiO
2interface [ 20]. In particular,
our calculations give Ec/similarequal20–30 meV for donors located 3–5
nm from the interface in the presence of a strong transverseelectric field of ∼30 mV /nm, which is in good quantitative
agreement with our experimental results. The fact that E
cdoes
not depend on the electric field indicates that the electric fieldis always large in our device (which is in agreement withour simulation), such that donors lying close to the BOX orto the FOX are strongly hybridized with an interfacial state.As opposed to the previous works mentioned above, here wehave shown that the S-D leads have an important screeningrole leading to a reduced E
cin our devices [ 24]. Moreover,
the predicted binding energy for the doubly occupied stateremains small with or without the gate.
The charging sequence of the hybrid state could be the
following. An As ion located a few nanometers away from aninterface in the bulk is always ionized at large electric fields. Itcreates a local positive potential which forms a donor-inducedpotential dip at the interface. This dip attracts an electron
and forms the singly occupied state As
0. It is important to
note that the first electron does not fully screen the As+
ion but rather forms a dielectric dipole with it (transversetox). This dipole produces a local field which is larger than
the screened central potential which would result from thesingly occupied neutral state in the absence of electric field.The doubly occupied state As
−could be stabilized in that
situation even if it is hard to conclude definitely on thispoint as it involves complex correlation effects between thetwo electrons and their image charges at the interfaces. Thisscenario may explain why the lines AandA
/primerun parallel
to the conduction-band edge at large positive Vb, because the
interface 2DEG and the Coulomb island induced by the ionizeddonor potential have exactly the same coupling to the substrateand to the front gate. Both effects—the shift of the electronfrom the donor under large transverse electric field (along y,z)
and the screening by the S-D electrodes (along x)—can
help to explain the reduction of the charging energy and thestabilization of the As
−state. However, a full simulation of
the two-electron problem is lacking for a more quantitativeanalysis.
In summary we have tuned independently the ionization
state of three randomly implanted As donors in a nanoscalesilicon MOSFET channel at low temperature, by applyingboth a front-gate and a back-gate voltage. We have shown thedominant screening role of the S-D leads, which combinedwith the existence of hybridized states due to the largetransverse electric fields, results in a stabilization of thedonor doubly occupied state with a decreased charging energycompared to the bulk case. This understanding of the complexenvironment around a dopant is essential for the perspectivesof donor-based quantum applications.
We thank B. Sklenard and O. Cueto for extensive process
simulation. The authors acknowledge financial support fromthe EU, through the EC FP7 MINECC initiative under ProjectTOLOP No. 318397 and through the ERC Grant AgreementNo. 280043, and from the French ANR, through the projectSIMPSSON No. 2010-Blan-1015.
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161404-5 |
PhysRevB.84.045126.pdf | PHYSICAL REVIEW B 84, 045126 (2011)
Correlation energy functional from jellium surface analysis
Lucian A. Constantin,1Letizia Chiodo,1Eduardo Fabiano,2Igor Bodrenko,2and Fabio Della Sala2,1
1Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti, 73010 Arnesano (LE)
2National Nanotechnology Laboratory (NNL), Istituto Nanoscienze-CNR, Via per Arnesano 16, I-73100 Lecce, Italy
(Received 12 April 2011; revised manuscript received 17 May 2011; published 18 July 2011)
Using the wave-vector analysis of the jellium exchange-correlation surface energy, we show that the PBEint
generalized gradient approximation (GGA) of Fabiano et al. [Phys. Rev. B 82, 113104 (2010) ] is one of the
most accurate density functionals for jellium surfaces, being able to describe both exchange and correlationparts of the surface energy, without error compensations. We show that the stabilized jellium model allows usto achieve a realistic description of the correlation surface energy of simple metals at any wave vector k.T h e
PBEint correlation is then used to construct a meta-GGA correlation functional, modifying the one-electronself-correlation-free Tao-Perdew-Staroverov-Scuseria (TPSS) one. We find that this new functional (named JS)performs in agreement with fixed-node diffusion Monte Carlo estimates of the jellium surfaces, and is accuratefor spherical atoms and ions of different spin-polarization and for Hooke’s atom for any value of the springconstant.
DOI: 10.1103/PhysRevB.84.045126 PACS number(s): 71 .10.Ca, 71 .15.Mb, 71 .45.Gm
I. INTRODUCTION
Kohn-Sham ground-state density-functional theory
(DFT),1,2the most used method in first-principles
electronic calculations of quantum chemistry andcondensed-matter physics, is based on the approximationsof the exchange-correlation (XC) energy ( E
xc) and potential
(Vxc,σ=δExc/δnσ). Here nσis the spin density, σ=↑and↓.
In recent years, many XC functionals have been built, beingclassified on the so-called “Jacob’s ladder.”
3The ground
on which the ladder lies is the Hartree approximation (XCenergy and potential are zero), and the first rung is the localspin-density approximation (LSDA) constructed entirely fromthe uniform electron-gas model:
1
ELSDA
xc=/integraldisplay
drn/epsilon1unif
xc(n↑,n↓), (1)
where /epsilon1unif
xc(n↑,n↓) is the exchange-correlation energy per
particle of an electron gas with uniform spin densities n↑and
n↓.4,5
The next rungs of the Jacob’s ladder are the generalized
gradient approximation (GGA) and the meta-GGA that usemore ingredients in order to satisfy exact constraints of theXC energy:
E
GGA
xc=/integraldisplay
drn/epsilon1GGA
xc(n↑,n↓,∇n↑,∇n↓), (2)
and
EMGGA
xc =/integraldisplay
drn/epsilon1MGGA
xc (n↑,n↓,∇n↑,∇n↓,τ↑,τ↓),(3)
where τσ(r) are the positive Kohn-Sham kinetic energy
densities,
τσ(r)=occ./summationdisplay
i1
2|∇ψiσ(r)|2, (4)
andψiσ(r) are the occupied Kohn-Sham orbitals of spin σ.
(Unless otherwise stated, atomic units are used throughout,i.e.,e
2=¯h=me=1.)These approximations are nowadays the most used because
of their computational efficiency and accuracy. Due to its
simplicity, no GGA functional can be accurate for both atomsand solids.
6In spite of this severe limitation, there are accurate
GGAs for molecules (e.g., BLYP,7revPBE,8APBE,9PBE10),
for solids (e.g., PBEsol,11AM0512), for surfaces (e.g., PBEsol,
AM05, PBEint13), and for hybrid interfaces (e.g., PBEint). On
the other hand, meta-GGA14,15can achieve good accuracy for
solids, atoms, and molecules, and might overcome difficult
problems in condensed-matter physics, as interaction of CO
molecule with Pt surface.16
Higher rungs of the ladder, such as optimized effective
potentials,17–19hyper-GGA, and random-phase approximation
(RPA) methods have a nonlocal dependence of the Kohn-
Sham orbitals, showing a prohibitively high computational
cost. Moreover, the hyper-GGA, which is a fully nonlocal
correlation functional compatible with exact exchange, has an
important degree of empiricism, and even if it satisfies manyexact constraints, it is not as accurate as it was expected.
20The
random-phase approximation (RPA) method is very expensive
for a full self-consistent calculation, employing the occupied
and unoccupied Kohn-Sham orbitals. Non-self-consistent RPA
evaluations severely fail for Be 2binding energy,21and unfortu-
nately show an important dependence on the chosen orbitals22
for molecules. This is not the case for bulk solids, for whichRPA is remarkably accurate.
23RPA-related methods, such
as the inhomogeneous Singwi-Tosi-Land-Sj ¨olander (ISTLS)
approach,24,25or the ones derived from the linear response
of time-dependent density-functional theory in the context
of adiabatic-connection-fluctuation-dissipation theorem, have
been barely applied to finite systems.
Jellium electron gas has become, along the years, a DFT
paradigm used to construct and to test XC approximations,as well as to derive important exact properties of XC energy.Jellium is a simple model of a simple metal, in which theion cores are replaced by a uniform positive background ofdensity ¯n=3/4πr
3
s=k3
F/3π2, and the valence electrons in
the spin-unpolarized bulk neutralize this background. Here rs
is the bulk density parameter ( rs=2.07 for Al and 3.93 for
Na), and kFis the bulk Fermi wave vector. The jellium surface
045126-1 1098-0121/2011/84(4)/045126(10) ©2011 American Physical SocietyCONSTANTIN, CHIODO, FABIANO, BODRENKO, AND DELLA SALA PHYSICAL REVIEW B 84, 045126 (2011)
energy ( σ) is the energy cost per unit area to create a planar
surface by cutting the bulk. Lang and Kohn26reported for the
first time jellium surface LSDA self-consistent calculationsthat showed early evidence that density functionals may workfor surface science. However, jellium XC surface energy ( σ
xc)
was a long-standing puzzle: density functionals, RPA, andtime-dependent DFT methods
27agree well with each other but
disagree strongly with high-level correlated methods such asFermi hypernetted chain (FHNC//0)
28and old diffusion Monte
Carlo (DMC)29calculations for jellium slabs. This puzzle was
resolved by new DMC calculations,30by the inhomogeneous
Singwi-Tosi-Land-Sj ¨olander approach,25and by the jellium
surface wave-vector analysis at the RPA level.31Indeed, all
the semilocal rungs of the Jacob’s ladder (LSDA, GGA, meta-GGA) are reliable for jellium surfaces.
The wave-vector analysis of jellium XC surface energy
is an important tool for understanding how well an XCapproximation works, because it separates the long-range andshort-range XC effects. Langreth and Perdew
32showed that
the exact XC energy of any inhomogeneous system can beobtained from a three-dimensional (3D) Fourier transform(wave-vector analysis) of the spherical averaged XC holedensity, that is, a function of a 3D wave vector k. For a jellium
surface, exact constraints of this wave-vector-dependent XChole are known: at long wavelengths ( k→0) the surface
plasmons dominate, whereas at short wavelengths ( k→∞ ),
LSDA becomes accurate.
32,33These known limits have been
used to carry out a wave-vector interpolation correction toLSDA,
32PBE-GGA,34and TPSS-metaGGA.35
Moreover, recent 2D and 3D wave-vector analyses of the ex-
act RPA,31,36ISTLS,25and time-dependent DFT (TDDFT)27,37
calculations have been used to solve and understand the
jellium surface problem, and as accurate benchmarks fordensity functionals.
13,31,37Thus such calculations become a
common test for density functionals, because they can revealthe accuracy of the approximation for the surface energy atany wave vector.
Recently, the PBEint GGA functional
13for hybrid inter-
faces has been constructed, with the aim to preserve, as muchas possible at the GGA level, the good properties of bothPBE and PBEsol, i.e., a good description of molecular andsolid-state properties, respectively. Its exchange enhancementfactor is
F
x(s)=1+κ−κ
1+μ(s)s2/κ, (5)
where s=| ∇n|/[2(3π2)1/3n4/3] is the reduced gradient, κ=
0.804 is fixed from the Lieb-Oxford bound,10and
μPBEint(s)=μGE+(μPBE−μGE)αs2
1+αs2, (6)
with α=(μGE)2/[κ(μPBE−μGE)]=0.197. Here μGE=
10/81 is the coefficient of the second-order gradient expansion
(GE) of exchange energy, and μPBE=0.219 51 was derived
from the linear response of bulk jellium, but is also reasonablyaccurate for heavy atoms.
6,9The value of αwas fixed from
the constraint d2FPBEint
x (s)/d(s2)2|s=0=0, which ensures a
smooth functional derivative δEPBEint
x/δn. Equation ( 6) leads
then to the recovery of the second-order gradient expansion ofthe exchange energy over a large range of the reduced gradient(fors/lessorsimilar1 PBEint exchange is close to PBEsol exchange).
13
In the rapidly varying density regime ( s/greaterorequalslant2.5),μPBEint→
μPBE, PBEint exchange behaves as PBE exchange. For the
correlation functional, PBEint has a PBE-like expression withβ=0.052, fitted to jellium surfaces. A very similar value of β
has been also found for the RGE2 functional,
38which uses a
different exchange functional. The PBEint is accurate for bulksolids and jellium surfaces because it recovers the second-ordergradient expansion in the slowly varying limit, as well as formetal-molecule interfaces because it also recovers the PBEbehavior at medium and large values of the reduced gradients. Recently, good performance of the PBEint functional for
the description of electronic and structural properties of goldnanostructures was also reported.
39
PBEint GGA gives similar wave-vector analysis of the
jellium XC surface energy as PBEsol.13However, in Sec. II,
we show that, unlike PBEsol, PBEint does not rely on an errorcancellation between the exchange and correlation parts of thejellium surface energy, being able to accurately account notonly for σ
xc, but also for σxandσcseparately, at every 3D
wave vector k. By employing a reliable model for the simple
metal surfaces (stabilized jellium model), we achieve a realisticpicture of the correlation surface energy σ
c(k) of Al (111), and
we show significant differences from the jellium model. Ourcalculations give indication for the behavior of exact σ
xcand
σcof real metal surfaces.
Because the PBEint correlation parameter β=0.052 can
capture the right physics of the exact correlation for jelliumsurfaces, and from this point of view is a nonempiricalparameter, it can be used further in construction of moreaccurate approximations for the correlation hole and energy.Thus, in Sec. III, we modified the TPSS meta-GGA correlation
functional,
14which is self-correlation free for one-electron
systems, to recover the PBEint correlation energy at slowlyvarying density regime, and at jellium surfaces. This newjellium-surface (JS) correlation functional is remarkably ac-curate for jellium surfaces, atoms, ions, and Hooke’s atom.Finally in Sec. IV, we summarize our conclusions.
II. WA VE-VECTOR ANALYSIS OF JELLIUM AND
STABILIZED JELLIUM XC SURFACE ENERGIES
In our calculations we use self-consistent LSDA orbitals
and densities, as in Refs. 13,31,34,35, and 37.I ti sf a i r
and reasonable to compare the energies predicted by differentfunctionals for the same density. Self-consistency effects on
the density from corrections to LSDA are small for jellium,where the self-interaction errors are negligible, as shown inFigs. 1and 2of Ref. 40and Fig. 1of Ref. 41. Moreover,
because LSDA orbitals give very good results in describingjellium surfaces,
37we also use them for the stabilized jellium
calculations (see Sec. II B below), similar to Ref. 42.
A. Jellium
Let us consider a jellium surface at z=0. This system is
translationally invariant in the plane perpendicular to the zaxis.
045126-2CORRELATION ENERGY FUNCTIONAL FROM JELLIUM ... PHYSICAL REVIEW B 84, 045126 (2011)
234 5 6rs0.50.60.70.80.911.1σxapprox/σxexact
Exact
PBE
PBEint
PBEsol
TPSS
FIG. 1. (Color online) Comparison of the PBE, TPSS, PBEsol,
and PBEint exchange-only jellium surface energies with the exactresults. σ
exact
x were calculated in Ref. 36, using the adiabatic-
connection-fluctuation-dissipation theorem.
This symmetry greatly simplifies the Kohn-Sham equations
because veff(r)=veff(z), and the normalized orbitals become
/Phi1k||,l(r)=1
A1/2eik||r||φl(z), (7)
where Ais the cross-sectional area, k||andr||are the wave
vector and the position in the plane perpendicular to the zaxis,
andφl(z) are solutions of the one-dimensional Kohn-Sham
equations,
/bracketleftbigg
−1
2d2
dz2+veff(z)/bracketrightbigg
φl(z)=/epsilon1lφl(z), (8)
withl=1,2,..., l Mis the subband quantum number ( lMis the
highest occupied level) for a jellium slab, and lis a continuous
quantum index in the case of an infinite jellium surface.26
0 0.2 0.4 0.6 0.8 1 1.2
k/2kF-600-30003006009001200γc(k)TDDFT
PBE
PBEint
PBEsol
LSDA
FIG. 2. (Color online) LSDA, PBE, PBEsol, PBEint, and TDDFT
wave-vector-resolved correlation surface energies γc(k), versus
k/2kF, for a jellium slab of thickness a=2.23λFandrs=2.07.
The area under each curve represents the corresponding correlation
surface energy; see Table I.TABLE I. PBE, PBEint, PBEsol, TDDFT, and DMC jellium
surface correlation energies (erg /cm2)o ft h es l a b ss h o w ni nF i g s . 2
and 3. The DMC values are extrapolated for every slab, taking
into account the quantum size effects of the slabs. The valueswhich best agree with DMC ones are indicated with bold font.
(1 hartree /bohr
2=1.557×106erg/cm2.)
PBE PBEint PBEsol TDDFT DMC
rs=2.07 720 661 604 742 674 ±45
rs=3 262 233 220 275 230 ±10
Because the sums over k||can be done analytically in many
calculations36due to
1
A/summationdisplay
k||−→/integraldisplayd2k||
(2π)2, (9)
the jellium surface becomes in principle a one-dimensional
problem.
The surface exchange-correlation energy is26,34
σxc=/integraldisplay∞
−∞dz n (z){/epsilon1xc([n];z)−/epsilon1unif
xc(¯n)}
=/integraldisplay∞
0d/parenleftbiggk
2kF/parenrightbigg
γxc(k), (10)
where γxc(k) is the 3D wave-vector analysis, that is,
γxc(k)=/integraldisplay∞
0du8kFu2bxc(u)sin(ku)/(ku), (11)
where
bxc(u)=/integraldisplay∞
−∞dz n (z){nxc([n];z,u)−nunif
xc(¯n;u)},(12)
where nxc([n];z,u) is the spherical average of the coupling-
constant averaged XC hole density.31,37Here the 3D wave
vector kis defined as k=√
k||2+k2
z.
The exact low-wave-vector limit of γxcis32
γxc(k→0)=kF
4π/parenleftbigg
ωs−1
2ωp/parenrightbigg
k, (13)
where ωp=(4π¯n)1/2andωs=ωp/√
2 are the bulk- and
surface-plasmon energies, and ¯nis the bulk density. The exact
high-wave-vector limit of γxcis not known, but LSDA is very
accurate in this limit32,33of short-wavelength oscillations.
In Fig. 1we show a comparison of σxat semi-infinite jellium
surfaces, for various density functionals, in the range 2 /lessorequalslantrs/lessorequalslant
6 where most of the metals lie. In these calculations, we usethe physically motivated Eq. ( 15)o fR e f . 41to interpolate
or extrapolate to any r
s(solving four linear equations fitted at
rs=2, 3, 4, and 6). The exact values σexact
x forrs=2, 3, 4, and
6 are taken from Table IIof Ref. 36. Similar calculations were
already reported (see Ref. 35and Fig. S3 of the supplementary
material of Ref. 11).
Both PBEsol and PBEint are very accurate for any rs,b u t
PBEint is almost exact in the range 2 /lessorequalslantrs/lessorequalslant4, outperforming
all other functionals. We recall that σPBEint
xc≈σPBEsol
xc ≈
σTPSS
xc≈σDMC
xc , where σDMC
xc are the fixed-node diffusion
Monte Carlo (DMC) calculations of Ref. 30. (See also Table II
of Ref. 25.) Thus while TPSS meta-GGA and PBEsol GGA
045126-3CONSTANTIN, CHIODO, FABIANO, BODRENKO, AND DELLA SALA PHYSICAL REVIEW B 84, 045126 (2011)
0 0.2 0.4 0.6 0.8 1 1.2
k/2kF-200-1000100200300400500γc(k)PBE
PBEint
PBEsol
LSDA
FIG. 3. (Color online) Same as Fig. 2, but for a jellium slab
of thickness a=2.23λFandrs=3. The area under each curve
represents the corresponding correlation surface energy; see Table I.
rely on an error compensation between σxandσc, PBEint gives
very accurate results for both σxandσc.
In order to understand better the PBEint behavior at jellium
surfaces, we perform the wave-vector analysis of Eq. ( 11). The
LSDA, PBE, PBEsol, and PBEint exchange-correlation holefunctions are accurately known. See Refs. 4,43, and 44for the
LSDA exchange-correlation hole, Ref. 43for the smoothed
PBE exchange hole model and Ref. 45for the PBE correlation
hole, Ref. 37for PBEsol XC hole density, and Ref. 13for
PBEint XC hole density.
In Ref. 13, it was shown that γ
PBEint
xc (k) agrees very well
withγPBEsol
xc (k), being between the most accurate GGAs for
surface energies. A detailed comparison of PBE, PBEsol, andexact jellium exchange surface energies was reported in Fig. 6
of Ref. 37. However, the nonoscillatory model of the GGA
exchange hole is inaccurate near k=2k
F. Thus we focus only
on the correlation part. We choose two jellium slabs of the samethicknesses a=2.23λ
Fandrs=2.07 (which corresponds to
Al) and rs=3, respectively. These slabs have been used also
in Refs. 31and37.
In Fig. 2, we show a comparison of γc(k) for several density
functionals and for a sophisticated TDDFT calculation37that
uses the adiabatic-connection-fluctuation-dissipation theorem.The TDDFT curve is exact at low wave vectors and highwave vectors, but gives surface correlation energies similarwith the RPA ones,
27e.g., considerably higher than the
DMC benchmarks results (see Table I). Thus we use TDDFT
calculation for comparison only in the low- and high-wave-vector regimes.
γ
c(k)PBEintis remarkably close to γc(k)TDDFTin both
low- and high-wave-vector regions, and its integrated surfacecorrelation energy agrees best with the DMC data (Table I).
These facts indicate that γ
c(k)PBEintis the most accurate curve
of Fig. 2, showing that the parameter β=0.052 used in the
construction of PBEint correlation captures the behavior oftheexact correlation energy for simple metals. The same
PBEint good behavior is shown in Fig. 3, for the jellium slab
withr
s=3.
Note that even if the PBEint correlation parameter
(β=0.052) is between the PBE ( β=0.0667) and PBEsol-0.50 -0.25 0.00 0.25 0.50
z/λF00.20.40.60.81n(z) / njellium
stabilized jellium
real Al
FIG. 4. (Color online) Normalized valence electrons densities of
the jellium and stabilized jellium models, at a semi-infinite surface of
rs=2.07 versus z/λF. Also shown is the density of a thick (11-layer)
real Al slab. The surface plane is at z=0, the bulk is at z/lessorequalslant0, and
the vacuum is at z/greaterorequalslant0.
(β=0.046) values, inspection of Figs. 2and 3shows
that PBEint has the steepest curve in the plasmonic region(k/2k
F→0). Moreover, γPBEint
c (k) correctly recovers the right
γTDDFT
c (k)a ts m a l l k,γLSDA
c (k) at large k, and/integraltext∞
0dkγPBEint
c (k)
is the most accurate. Thus β=0.052 can be considered an
(almost) exact correlation hole constraint for jellium surfaces.
B. Stabilized jellium
In the last subsection we have found that PBEint is very
accurate for the ideal, jellium surfaces. However, the jelliummodel has serious deficiencies in describing simple metals: forr
s≈2 the jellium total surface energy σis negative, and for
rs≈6, the jellium bulk modulus is negative. These limitations
were solved in the stabilized jellium model,46by taking into
account the interaction between the ions and electrons througha simple empty-core pseudopotential as
w(r)=/braceleftBigg
−z/r, r/greaterorequalslantr
c
0,r < r c,(14)
and by eliminating the spurious self-repulsion energy of
the positive background in each cell.46Herezis the ion
charge, and rcis the core radius, which is defined by
Eq. ( 26)o fR e f . 46. (The corresponding parameters for the
simple metal Al are rs=2.07,rc=1.11, and z=3.) In the
practical implementation of the model, to avoid the needto exactly define the bulk structure, the stabilized jelliummodel is turned into a simple structureless pseudopotentialmodel. This is achieved by taking the difference between thepseudopotential and the electrostatic potential of the jelliumpositive background to be constant in the bulk (averagingover a Wigner-Seitz cell) and zero outside (see Sec. IIof
Ref. 46). The stabilized jellium model gives accurate surface
energies of simple metals (see Fig. 2of Ref. 46) and, moreover,
the remaining small discrepancy with real surfaces can becorrected by including geometric effects using a “corrugationfactor.”
46,47
045126-4CORRELATION ENERGY FUNCTIONAL FROM JELLIUM ... PHYSICAL REVIEW B 84, 045126 (2011)
0 0.2 0.4 0.6 0.8 1 1.2 1.4
k/2kF010002000300040005000γxc(k)jellium
stabilized jellium
FIG. 5. (Color online) PBEint wave-vector-resolved XC surface
energies γxc(k)v e r s u s k/2kF, for semi-infinite jellium and stabilized
jellium surfaces. The bulk parameter is rs=2.07. The exact slope
i sg i v e nb yE q .( 13). The area under each curve represents the
corresponding XC surface energy; see Table II.
In Fig. 4we show the normalized densities n(z)/¯nat
the semi-infinite jellium, semi-infinite stabilized jellium, and11-layer-slab real Al surfaces. The calculation for real Alwas done at the PBEint self-consistent level (LSDA gives asimilar density), and computational details are given in thenext subsection. Figure 4confirms that the stabilized jellium
density is steeper
46and agrees in detail with the valence density
of real metal, showing that the electrons near the surface aremore tightly bounded than in the jellium model case. For aslab, the same behavior can be seen in Fig. 1of Ref. 42.
Because the stabilized-jellium model is an appropriate
model for real surfaces, it is thus important to understand howthe wave-vector analysis is modified in this more realistic case.We thus report a comparison of the γ
xc(k) andγc(k) for jellium
and stabilized jellium of semi-infinite surfaces of rs=2.07,
at the PBEint level. Stabilized jellium model calculations usethe regular bulk density ¯n.
In Fig. 5we show that γ
xc(k) of the stabilized jellium is
always smaller than the jellium one. Moreover, both curvesrecover the exact behavior at k→0[ s e eE q .( 13)], and the
accurate LSDA at k→∞ . Thus a steeper density decreases
considerably σ
x, whereas σcremains almost constant, as shown
in Table II.
Finally, we show in Fig. 6the surface correlation behavior.
Even if the integrated σcagree well for both jellium and
stabilized jellium (see Table II), their wave-vector analyses
show qualitative differences: (i) the plasmonic region (smallvalues of k) is more important for the stabilized jellium (and
consequently in real simple metals); (ii) the short-wavelength
TABLE II. PBEint semi-infinite jellium and stabilized jellium
correlation and XC surface energies (erg /cm2), for bulk parameter
rs=2.07.
Jellium Stabilized jellium
σc 678 667
σxc 2984 2481(k→∞ ) region is less important for real surfaces, and (iii) at
intermediate wave vectors, the variation of σcdecreases in the
stabilized jellium model.
C. Simple metal: Aluminium
In this last subsection we present results for the real Al(111)
surface, studied in the full self-consistent Kohn-Sham scheme,with PBE, PBEsol, and PBEint functionals. The calculationsare performed within DFT, in a plane-wave pseudopotentialapproach.
48Symmetric slabs of 11 layers thickness have been
used, allowing all layers to relax in the direction perpendicularto the surface. Ultrasoft pseudopotentials with a cutoff of35 Ry, a 10 ×10×1k-point grid, and a vacuum region of 14 ˚A
thick have been employed. Surface energies were computedfollowing Ref. 49.
In Table IIIwe show the equilibrium lattice constant, bulk
modulus, and surface energy of Al(111). The experimentalvalues of the bulk modulus and lattice constants have beencorrected for finite temperature and phonon zero-point effects,according to the careful analysis of Refs. 50and51. Note that
these corrections are more than 7% for bulk moduli.
For the equilibrium lattice constant PBE is in very good
agreement with experiments. This fact shows that in the Albulk, there are important regions with relatively high values ofthe reduced gradient s. We also recall that the maximum value
ofsin Al (111) bulk is s
max=1.4, considerably higher than in
other bulk solids. (See Table IIIof Ref. 52). For bulk modulus,
both PBEint and PBEsol are accurate, being very close to thecorrected experimental value.
For the real surface, PBEint and PBEsol give similar results,
much better than the PBE one. This trend is in accord with ourjellium surface calculations. We recall that for Al(111), thejellium surface is completely wrong ( σ=− 642 erg /cm
2), the
stabilized jellium gives σ=801 erg /cm2, and the stabilized
jellium together with a corrugation factor [see Eq. ( 56)o f
Ref. 46]g i v e s σ=921 erg /cm2,46very close to our full DFT
calculations and to the experimental value.
Clearly the good performance of the PBEint functional is
not only related to the βcoefficient, as the results in Table III
take also into account exchange and self-consistent effects.Nevertheless, these results show that the PBEint functionalyields a very accurate description of simple metals, being moreaccurate than PBEsol for lattice constant and more accuratethan PBE for surface energies and bulk modulus.
TABLE III. Equilibrium lattice constant, bulk modulus, and sur-
face energy of Al(111) from PBE, PBEint, and PBEsol self-consistent
calculations. The experimental data were corrected for thermal
and zero-point effects (see Table Iof Ref. 51). The uncorrected
experimental data are presented in brackets. The best results for each
row are in bold font.
PBE PBEint PBEsol Expt.
a0(˚A) 4.025 4.009 3.998 4.022 (4.05)
B(GPa) 77.26 81.78 81.90 81.3 (76)
σ(erg/cm2) 888 1027 1035 1140
045126-5CONSTANTIN, CHIODO, FABIANO, BODRENKO, AND DELLA SALA PHYSICAL REVIEW B 84, 045126 (2011)
0 0.2 0.4 0.6 0.8 1 1.2 1.4
k/2kF0200400600800100012001400γxc(k)jellium
stabilized jellium
FIG. 6. (Color online) PBEint wave-vector-resolved correlation
surface energies γc(k)v e r s u s k/2kF, for semi-infinite jellium and
stabilized jellium surfaces. The bulk parameter is rs=2.07. The area
under each curve represents the corresponding correlation surface
energy; see Table II.
III. META-GGA CORRELATION FUNCTIONAL
In the previous section we showed that the PBEint param-
eterβ=0.052 correctly describes the correlation in simple
metals. In this sense, β=0.052 is a nonempirical parameter
that can be used further in developing approximations on thehigher rungs of the Jacob’s ladder for both XC energy and hole.
In this section we propose the jellium surface (JS) meta-
GGA correlation functional that is in fact a self-correlationcorrection to the PBEint GGA correlation, and is constructedin a similar way as TPSS and revTPSS meta-GGA correlationfunctionals.
14,15,53
A. Construction of JS correlation functional
The meta-GGA correlation energy is14,53
EMGGA
c =/integraldisplay
drn/epsilon1rev
c/bracketleftbig
1+d/epsilon1rev
czq/bracketrightbig
, (15)
where
z=τW
τ, (16)
withτW=| ∇n|2/(8n) being the von Weizs ¨acker kinetic-
energy density2andτis the total Kohn-Sham positive kinetic-
energy density [see Eq. ( 4)]./epsilon1rev
cis a revision of PKZB
meta-GGA correlation functional,54and has the general form
/epsilon1rev
c=/epsilon1GGA
c[1+C(ζ,ξ)zm]−[1+C(ζ,ξ)]zm/summationdisplaynσ
n˜/epsilon1σ
c,
(17)
with
˜/epsilon1σ
c=max/bracketleftbig
/epsilon1GGA
c(nσ,0,∇nσ,0),/epsilon1GGA
c(n↑,n↓,∇n↑,∇n↓/bracketrightbig
.
(18)
Here
ζ=n↑−n↓
n,ξ=|∇ζ|
2(3π2n)1/3(19)
are the relative spin polarization and its reduced gradient.For any C(ζ,ξ),d, and positive integers qandm,EMGGA
c =
0f o r any fully spin-polarized one-electron density (defined by
ζ=1 andz=1). In the TPSS case, GGA =PBE,q=3, and
m=2 (which represent minimum values for these parameters
in order to satisfy exact constraints, such as recovery ofgradient expansion). revTPSS also uses q=3 andm=2, but
has a GGA expression similar to PBE but βisr
sdependent
[see Eq. ( 3)o fR e f . 15] with, e.g., β(rs=2)=0.059,
β(rs=4)=0.0546, and β(rs=6)=0.0517. The revTPSS
parametrization of βis a fit to the exact β(rs) parameter of
the second-order gradient expansion of the correlation energy,derived by Hu and Langreth.
55
For JS we chose GGA =PBEint, and q=m=4, such
that in the slowly varying density regions, JS recovers fasterthe PBEint behavior. This choice was motivated by ourobservation that for jellium surfaces the PBEint correlationis a remarkably good choice.
For the uniform density scaling
n
γ(r)=γ3n(γr),γ > 0, (20)
in the low-density limit ( γ→0), the JS meta-GGA scales
correctly as EJS
c=γWPBEint, where WPBEintis a negative
constant given by using the value β=0.052 in Eq. (39) of
Ref. 53. In the high-density limit ( γ→∞ ),EJS
ccorrectly
scales to a negative, n-dependent constant. In the large gradient
limit ( s→∞ ), JS correctly vanishes. In the slowly varying
limit (small s), JS recovers PBEint correlation.
We find C(ζ,ξ) anddsimilar as in TPSS construction:53
C(0,0) and dare fixed by the requirement that for jellium
surfaces, where the self-interaction error is negligible andPBEint works very well, JS recovers PBEint for any r
s.
The spin-dependent behavior of C(ζ,ξ) is instead fixed by
requiring that for a Wigner crystal jellium the correlation isindependent of ζfor 0<ζ< 0.7. This is obtained in practice
by considering one-electron Gaussian densities of differentspin polarization 0 <ζ< 1. Thus we propose the following:
d=3.7, (21)
and
C(ζ,ξ)=0.353+0.87ζ
2+0.5ζ4+2.26ζ6
/braceleftbig
1+1
2√ξ[(1+ζ)−4/3+(1−ζ)−4/3]/bracerightbig4.(22)
The denominator of Eq. ( 22) is a dumping factor dependent
on√ξ. In the case of an electron gas with slowly varying
spin densities, the gradient expansion of the correlation energyis known to contain terms depending on |∇ζ|
2and∇n·∇ζ
(which, however, negligibly contribute to the correlationenergy
56), but not on√ξ. Nevertheless, we can state that our
construction is correct, because /epsilon1JS
c→/epsilon1PBEint
c in the slowly
varying limit due to the choice q=m=4. The denominator
of Eq. ( 22) is thus only required to account for self-correlation
in atoms.
In Fig. 7, we show the correlation energy per particle /epsilon1c
of TPSS, JS, and PBEint for the Li atom and the Ar15+ion,
versus the radial distance from the nucleus. For both atoms,/epsilon1
TPSS
c and/epsilon1JS
care correctly zero in the monovalent region,
where the self-correlation correction removes the PBEint badbehavior.
045126-6CORRELATION ENERGY FUNCTIONAL FROM JELLIUM ... PHYSICAL REVIEW B 84, 045126 (2011)
01234 5
Distance (bohr)-0.06-0.04-0.020.00εcJS
TPSS
PBEint
0 0.1 0.2 0.3 0.4 0.5
Distance (bohr)-0.08-0.06-0.04-0.020.00εcJS
TPSS
PBEintLi
Ar15+
FIG. 7. (Color online) Correlation energy per electron (hartree)
versus the radial distance r(bohr) from the nucleus, for Li atom
(upper panel), and for Ar15+(lower panel).
In Table IV, we report correlation energies of the one-
electron Gaussian densities with different spin polarizationζ. All meta-GGAs are correct for ζ=1, and all of them
give correlation energies practically independent on ζin the
range 0 /lessorequalslantζ/lessorequalslant0.7. These results reveal that JS can accurately
describe correlation energies of low-density Wigner crystals,where the electrons are localized near the lattice sites.
B. Results
1. Jellium surfaces
In Table Vwe show that JS is very accurate for jellium
surfaces, all the correlation values being in the range ofDMC estimations. Moreover, JS and PBEint have similarperformance, showing that our self-interaction correctioncorrectly does not have any effect for jellium surfaces, wherethe electrons are delocalized. TPSS and PBE overestimate thejellium surface correlation energies, and PBEsol underesti-mates them for 2 /lessorequalslantr
s/lessorequalslant3. Thus the most accurate values in
Table V, all in the range of DMC calculations, are given by
revTPSS, JS, and PBEint.
TABLE IV . Correlation energies (hartree) of the one-electron
Gaussian densities with relative spin-polarization ζ.
ζ TPSS revTPSS JS
0.0 −0.021 −0.022 −0.019
0.1 −0.021 −0.022 −0.019
0.2 −0.020 −0.022 −0.019
0.3 −0.020 −0.022 −0.019
0.4 −0.020 −0.021 −0.019
0.5 −0.020 −0.021 −0.019
0.6 −0.019 −0.021 −0.019
0.7 −0.018 −0.020 −0.018
0.8 −0.017 −0.018 −0.017
0.9 −0.012 −0.013 −0.013
1.0 0.000 0.000 0.000TABLE V . Semi-infinite jellium surface correlation energies
(erg/cm2) for PBE, PBEint, and PBEsol GGA functionals and TPSS,
revTPSS, and JS meta-GGA functionals. Results within the DMC
error bar are denoted with boldface.
rsPBE PBEint PBEsol TPSS revTPSS JS DMC
2 829 745 708 827 771 749 768±50
3 276 246 234 274 251 248 242±10
4 124 111 105 125 111 112 104±8
6 40.2 35.4 33.3 39.6 34.8 35.9 31 ±...
2. Hooke’s atom
To further assess the JS meta-GGA correlation functional
we consider in this subsection Hooke’s atom. This modelsystem represents two interacting electrons in an isotropic har-monic potential of frequency ω. The XC wave-vector analysis
of the Hooke’s atom is different from the jellium surfaces, be-cause in the large klimit (k→∞ ), the LSDA is not accurate.
33
Thus this system is a challenging test for the JS meta-GGA.
Moreover, at small values of ω, the electrons are strongly
correlated, and at large values of ω, they are tightly bound,
two important cases in many condensed-matter applications.
The exact ground-state solutions of the Hooke’s atom
correlated wave function are known:57Introducing the center
of mass R=(r1+r2)/2, and the relative coordinate r=
r1−r2, the two-particle Hamiltonian is separable, and the
Schr ¨odinger equation decouples in two separate equations
depending only on Randr, respectively [see Eqs. ( 6) and
(7)o fR e f . 57].
The center of mass behaves as a 3D harmonic oscillator
of frequency ωR=2ω,57whose solution is well known. The
eigenvalue problem for the relative motion is57
/bracketleftbigg
−1
2∇2
r+1
2ω2
rr2+1
2r/bracketrightbigg
φ(r)=/epsilon1φ(r), (23)
withωr=ω/2. Exact analytical ground-state solutions of
Eq. ( 23) are known for special values of ω.57They have the
form
φ(r)=φ(r)∼u(r)
r, (24)
where
u(r)=e−ρ2/2ρp−1/summationdisplay
ν=0aνρν, (25)
withρ=√ωrr, andpan integer ( p/greaterorequalslant2). For any p/greaterorequalslant2,
Eqs. ( 24) and ( 25) provide an exact ground-state solution,
whose corresponding ωis obtained from a nonlinear equation
[see Eq. ( 25)o fR e f . 57]. Thus there is a one-to-one
correspondence between the special values of ωfor which
analytical ground state [Eqs. ( 24) and ( 25)] exists, and the
power of the polynomial pthat defines this ground state.
Let us consider the classical electron distance57
r0=/parenleftbig
2ω2
r/parenrightbig−1/3, (26)
which plays a similar role as λFin the case of jellium
surfaces. In Fig. 8we show the normalized exact ground-state
densities n(r)r3
0for different frequencies. The solutions with
045126-7CONSTANTIN, CHIODO, FABIANO, BODRENKO, AND DELLA SALA PHYSICAL REVIEW B 84, 045126 (2011)
0 0.2 0.4 0.6 0.8 1 1.2 1.4
r/r000.30.60.91.21.5n(r)r03p=2
p=4
p=6
p=8
p=11
FIG. 8. (Color online) Exact normalized ground-state electron
densities n(r)r3
0of the Hooke’s atom with different frequencies,
versus the radial distance r/r 0. Here r0is the classical electron
distance defined in Eq. ( 26).prepresents the order of the polynomial
of Eq. ( 25). For p=2,r0=2; for p=4,r0=11.4425; for
p=6,r0=27.9349; for p=8,r0=51.5285; and for p=11,
r0=100.2476. In all cases,/integraltext∞
0dr4πr2n(r)=2.
2/lessorequalslantp/lessorequalslant4[ s e eE q .( 25)] represent the low-correlation regimes
(or tightly bounded regimes), and the solutions with p/greaterorequalslant8
are the strongly correlated regimes. In the strong-correlatedcase, the region near the “nucleus” of the Hooke’s atom hasa deep “hole” due to the strongly correlated electrons, and,moreover, the density becomes more localized over a r
0length.
Asp→∞ , the electrons are perfectly localized around r0
(note that in this case r0→∞ ).
We calculate the exact correlation energies for the first ten
ground-state exact solutions of the Hooke’s atom (2 /lessorequalslantp/lessorequalslant
11), using the same procedure described in Ref. 58. Our exact
correlation results are reported in the lower panel of Fig. 9.
In the upper panel of Fig. 9, we show the accuracy of several
correlation functionals. We use the exact densities and orbitals.As we mentioned at the beginning of Sec. II, it is a common,
fair practice to test different functionals for the same density.
-0.008-0.006-0.004-0.0020Ec-EcexactJS
PBE
PBEint
PBEsol
TPSS
revTPSS
0 2 04 06 08 0 1 0 0r0-0.04-0.03-0.02-0.01Ecexact
FIG. 9. (Color online) Upper panel: Errors of several correlation
functionals Eapprox
c−Eexact
c (hartree) versus the classical electron
distance r0. Lower panel: Exact correlation energy Eexact
c (hartree)
versus the classical electron distance r0(bohr).For these unpolarized systems, τ=τWsoz=1, and ζ=0,
and thus the construction of C(0,0) and dbecomes essential for
describing this two-electron system. The JS meta-GGA is veryaccurate over the whole frequency regimes, including tightlybounded and strongly correlated ones,
58,59outperforming all
the other approximations.
These results, which are not related to jellium surfaces,
show that JS meta-GGA may be one of the most accuratecorrelation functionals for many-electron systems.
3. Atoms and ions
Finally, we consider briefly an application to real systems,
employing the JS correlation functional to compute thecorrelation energy of several atoms. A more comprehensivestudy of real systems (e.g., molecules) is beyond the scope ofthis work and will be the subject of a future investigation.
In Table VIwe report the correlation energy per electron
E
c/Nof 15 spin-unpolarized spherical atoms and ions (Ar6+,
Ar, Kr, Xe, Zn, two-electron systems: He, Li+,B e2+; four-
electron systems: Be, B+,C2+,N3+,O4+; and ten-electron
systems: Ne, Ar8+) , and 11 spin-polarized spherical atoms and
ions (H, N, O+, and three-electron systems: Be+,L i ,A r15+,
C3+,N4+,B2+,O5+,N e7+). We use analytic Hartree-Fock
densities and orbitals.60
For spin-unpolarized systems, all functionals give rather
similar results with differences of few mHa per electron.None of the investigated functionals shows a clearly superiorperformance, nevertheless, the JS and PBEint functionals giveoverall the best mean absolute error (MAE) and mean absoluterelative error (MARE). This shows that PBEint and JS betterdescribe the correlation.
For spin-polarized systems, on the other hand, the JS
functional clearly outperforms the other functionals yieldingalways the best agreement with the reference values, exceptfor the O
+atom, and the best overall performance with a MAE
of 1.1 mHa per electron. This result provides an indication forthe accuracy of Eq. ( 22).
IV . CONCLUSIONS AND FUTURE PERSPECTIVES
In this paper we have shown that the PBEint correlation
functional accurately describes jellium surfaces at any wavevector k, giving a very accurate surface wave-vector analysis
not only for total XC energy, but also for exchange andcorrelation parts. Thus at the simple but accurate PBEint GGAlevel, we have performed a wave-vector analysis of stabilizedjellium, which is a realistic model of simple metals, andwe have obtained a good description of surface energies ofsimple metals at any wave vectors. We have found qualitativedifferences between jellium and stabilized jellium correlationsurface energies, which are reported in Figs. 5and 6. These
findings should be a starting point for the investigation ofother differences between jellium and real metals, e.g., theasymptotic behaviors of XC energy per particle, and XCpotential far outside the surface.
63,64
We have shown that the PBEint correlation parameter
β=0.052 captures the exact physics of jellium surfaces, and
is thus an exact hole constraint. Further, we have performeda self-interaction correction to the PBEint correlation, by
045126-8CORRELATION ENERGY FUNCTIONAL FROM JELLIUM ... PHYSICAL REVIEW B 84, 045126 (2011)
TABLE VI. Correlation energies per electron ( Ec/N, mHa) of spherical atoms and ions with Hartree-Fock analytic orbitals and densities
(Ref. 60). Best results are indicated in boldface.
Atom PBE PBEint PBEsol TPSS revTPSS JS Ref.
Spin-unpolarized atoms and ions
He −21.0 −24.5 −26.3 −21.5 −23.1 −21.1 −21.0a
Li+−22.4 −26.4 −28.3 −22.8 −24.3 −22.5 −21.7a
Be2+−23.1 −27.2 −29.3 −23.5 −24.7 −23.2 −22.2a
Be −21.4 −24.6 −26.1 −21.7 −23.1 −21.7 −23.6a
B+−23.0 −26.5 −28.2 −23.4 −24.7 −23.6 −27.8a
Ne −35.1 −39.2 −41.2 −35.4 −36.5 −38.0 −39.1a
Ar −39.3 −43.5 −45.5 −39.5 −40.5 −42.6 −40.1a
Kr −49.1 −53.8 −56.0 −49.2 −49.9 −53.3 −57.4b
Xe −54.0 −58.8 −61.0 −54.1 −54.7 −58.5 −63.5b
Zn −46.9 −51.5 −53.7 −47.0 −47.8 −51.0 −56.2b
Ar8+−41.0 −46.1 −48.5 −41.4 −42.1 −44.9 −39.9a
Ar6+−38.3 −43.2 −45.6 −38.7 −39.6 −42.2 −41.3a
C2+−24.0 −27.7 −29.6 −24.5 −25.7 −24.9 −35.1a
N3+−24.7 −28.6 −30.5 −25.2 −26.4 −25.8 −35.1a
O4+−25.3 −29.2 −31.2 −25.7 −26.9 −26.4 −38.5a
MAE 5.3 4.2 4.4 5.2 4.8 4.2
MARE 0.13 0.12 0.14 0.13 0.12 0.11
Spin–polarized atoms and ions
H −6.0 −7.2 −7.9 0.0 0.0 0.0 0.0
Ne7+−19.4 −23.2 −25.2 −18.8 −19.5 −18.5 −17.0a
Be+−18.0 −21.3 −23.0 −17.4 −18.3 −16.9 −15.8a
Li −17.2 −20.2 −21.7 −16.5 −17.5 −16.0 −15.1a
Ar15+−19.6 −23.7 −25.8 −19.0 −19.7 −18.8 −17.4a
C3+−18.9 −22.5 −24.3 −18.2 −19.1 −17.8 −16.5a
N4+−19.0 −22.8 −24.7 −18.4 −19.3 −18.0 −16.7a
B2+−18.6 −22.0 −23.8 −17.9 −18.8 −17.4 −16.2a
O+−27.1 −30.8 −32.6 −27.9 −29.0 −28.6 −27.7a
O5+−19.2 −23.0 −24.9 −18.6 −19.4 −18.2 −16.8a
N −25.9 −29.3 −31.0 −26.5 −27.7 −27.0 −26.9a
MAEc2.0 5.3 7.1 1.4 2.2 1.1
MAREc0.12 0.31 0.41 0.08 0.13 0.07
aReference 61.
bReference 62.
cDoes not include H atom.
constructing the jellium-surface-meta-GGA. This new JS
functional is almost exact for jellium surfaces, accurate foratoms and ions, and performs remarkably well for the Hooke’satom of any frequency, including the tightly bounded andstrongly correlated regimes.
The JS meta-GGA correlation hole model can be
constructed using the reverse engineering method proposed inRef. 35for the TPSS case. Because of its good accuracy, our
JS meta-GGA correlation energy (and hole) functional can beemployed in further research for developing new hyper-GGAs.
Finally, we note that in this work we presented non-self-
consistent results, starting from accurate densities, i.e., LSDAfor jellium surfaces, Hartree-Fock for atoms and ions, or exact
density for the Hooke’s atom. This is common procedure for afirst assessment of the correlation functional. In a forthcomingpaper we will investigate self-consistency effects as wellas an appropriate exchange functional to be used togetherwith JS.
ACKNOWLEDGMENTS
This work was partially funded by the European Research
Council (ERC) Starting Grant FP7 Project DEDOM, GrantAgreement No. 207441.
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PhysRevB.85.214431.pdf | PHYSICAL REVIEW B 85, 214431 (2012)
Ab initio study of the factors affecting the ground state of rare-earth nickelates
Sergey Prosandeev,1L. Bellaiche,1and Jorge ´I˜niguez2
1Physics Department and Institute for Nanoscience and Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA
2Institut de Ci `encia de Materials de Barcelona (ICMAB-CSIC), Campus UAB, 08193 Bellaterra, Spain
(Received 16 March 2012; revised manuscript received 7 June 2012; published 26 June 2012)
We have used first-principles methods to investigate the factors that control the ground state of rare-earth
nickelates, studying in detail the case of NdNiO 3. Our results suggest a complex phase diagram, with the bulk
compounds standing on the edge of various instabilities that can be triggered by both electronic (e.g., changesin the Coulomb repulsion) and structural (e.g., epitaxial mismatch) means. In particular, we reveal that severalphase transitions can be induced by epitaxial strain in thin films and predict that a continuous transformationbetween insulating spin-density-wave- and metallic spin-spiral-like solutions occurs at moderate values of thein-plane mismatch. Our results provide a coherent picture of structural and electronic effects in nickelates andhave implications for current experimental and theoretical work on these compounds.
DOI: 10.1103/PhysRevB.85.214431 PACS number(s): 71 .30.+h, 71.15.Mb, 75 .30.−m
I. INTRODUCTION
Rare-earth nickelates RNiO 3(where R=Y , Nd, Pr, La, etc.)
display metal-insulator and complex magnetic andcharge-ordering transitions that can be controlled byproperly choosing the Rcation.
1Thanks to the development
of deposition techniques, nickelates can now be grown ashigh-quality ultrathin films or in complex heterostructures.This is leading to a torrent of outstanding findings—rangingfrom size-
2and dimensionality-driven3metal-insulator
transitions (MITs) to exotic interfacial effects4—and
nickelates are quickly becoming a major field of research inthe functional-oxides community. The physical origin of suchstriking effects is still debated even for the simplest (bulk)cases, and different scenarios have been proposed invokingvarious driving forces for the transformations.
5,6
Here we report on a first-principles investigation of a
relatively simple compound, pure NdNiO 3(NNO), that we
find to be in many ways representative of this family of nickeloxides. Our results render an intricate phase diagram with thebulk material standing on the edge of several instabilities andshow how electronic and structural factors, which we simulateby a varying Hubbard- Uand thin-film epitaxial conditions,
respectively, can induce profound changes in its ground state.In particular, our calculations reveal a complex transitionsequence as a function of epitaxial strain and set the basisfor a detailed understanding of these materials.
II. METHODS
For the simulations we used the generalized gradient
approximation to density functional theory (DFT)—moreprecisely, the so-called PBE scheme proposed by Perdewet al.
7—as implemented in the V ASP package.8We used the
“projector augmented wave” method to represent the ioniccores,
9solving for the following electrons: Ni’s 3 p,3d, and
4s;N d ’ s5 s,5p, and 6 s(Nd’s potential was generated by
assuming a +3 ionization state and leaving three 4 felectrons
frozen in the core); and O’s 2 sand 2p. Wave functions were
represented in a plane-wave basis truncated at 500 eV , and a4×8×2/Gamma1-centered k-point grid was used for integrations
within the Brillouin zone (BZ) corresponding to the 80-atomcell of Fig. 1. The calculation conditions were checked to
render converged results. (We used the software
VESTA10to
prepare some figures.)
It is worth noting a couple of additional points regarding
our simulations. First, we used a “Hubbard- U” correction to
DFT11to evaluate the effect that a varying Coulomb repulsion
has on the ground state of the bulk materials. Note thatchoosing a sound value of Uto treat nickelates is not trivial,
and arguments for using both large
12and small5corrections
have been given. Thus, to investigate the effect of the epitaxialstrain in nickelate films, we performed all our calculations fortwo choices of U: 7 eV (i.e., the relatively large value most
frequently adopted in the literature) and 2 eV .
Second, we worked with the 80-atom cell sketched in
Fig. 1, which is essentially given by the lattice vectors
a=2a
p−2bp,b=ap+bp, and c=4cp, where ap,bp, and
cpdefine the elementary 5-atom perovskite cell. Our 80-atom
cell is compatible with the monoclinic ( P21/c) phase observed
in the bulk materials at low temperatures,13,14which can be
viewed as a distortion of an orthorhombic ( Pnma ) structure
caused by the splitting of the Ni cations in two rocksalt-orderedsublattices. Our cell is also compatible with the various
magnetic orders proposed to occur in these materials, as, e.g.,
the spin-spiral (SS) depicted in Fig. 1(a) or the spin-density
wave (SDW) in Fig. 1(b). As a function of epitaxial strain, we
relaxed the atomic structure by (1) using a distorted Pnma
configuration (with all symmetries broken) as starting point
and (2) imposing the constraint that the in-plane lattice vectors
aandbmatch those of a (001)-oriented cubic substrate with
lattice constant a
sub. By proceeding in this way, we effectively
restricted ourselves to the consideration of phases that can be
obtained as a relatively small distortion of the bulk structures;we did not attempt a more careful search for alternative
phases that might be stabilized at large enough values of the
epitaxial mismatch. Finally, we considered several magneticconfigurations with essentially zero net magnetization; here we
only report the lowest-energy states obtained.
15Most of our
simulations were done within the usual approximation of scalarmagnetism; in selected cases, noncollinear spin arrangements
and spin-orbit effects were considered to better characterize
the ground state.
214431-1 1098-0121/2012/85(21)/214431(7) ©2012 American Physical SocietyPROSANDEEV , BELLAICHE, AND ´I˜NIGUEZ PHYSICAL REVIEW B 85, 214431 (2012)
(a) Spin Spiral (b) Spin Density Waveµ2
µ1
-µ2
-µ1NiO
R
FIG. 1. (Color online) Eighty-atom supercell used in the simula-
tions. The spin arrangements that we call spin-spiral [SS, panel (a)]and spin-density-wave [SDW, panel (b)] are sketched. In panel (a)
we indicate the Ni magnetic moments ( μ
1andμ2) discussed in the
text and from which all others can be obtained. These SS and SDW
arrangements are ideal ones, the SS being characterized by μ1≈μ2
and the SDW by μ2=0. The spin order in panel (a) is of the type
usually denoted by ↑↑↓↓ , alluding to the sequence that occurs along
any of the principal directions of the perovskite structure.
III. RESULTS AND DISCUSSION
A.Udependence of bulk ground state
Figures 2–4show the Udependence of the various physical
parameters characterizing the ground state of bulk NNO.Three regions are readily identified: (1) For U> 8e V ,w e
obtain metallic solutions in which all Ni’s present localizedmagnetic moments of about 1.5 Bohr magneton (i.e., μ
1≈
μ2≈1.5μB), and the spins arrange themselves in a SS as
the one in Fig. 1(a).( 2 )F o r U< 6 eV , we obtain insulating
solutions whose band gap ( Egap) closes continuously as U
decreases, becoming zero at U≈1 eV . As regards the magnetic
configuration, its most significant feature is that we have twodistinct Ni sublattices: one with nonzero magnetic moments(μ
1/negationslash=0) whose magnitude varies continuously with Uand a
second one in which the Ni ions present no localized moment(μ
2=0). We can describe the resulting configuration as a
SDW [Fig. 1(b)]. The splitting in two Ni sublattices is also
evident in the charge disproportionation /Delta1Q/negationslash=0 [Fig. 2(c),
see discussion below] and in the relatively large difference(of as much as 1 eV) in the average electrostatic potentialexperienced by the two types of Ni cations [Fig. 2(d)]. Note
that this large potential modulation will have a considerableimpact in NNO’s electronic conductivity, and should be takeninto account when developing models for this compound.(3) Finally for intermediate values of U(6 eV <U< 8e V ) ,
we obtain solutions that we call SS +SDW and which can be
viewed as an interpolation between the previous two cases, i.e.,asUincreases the magnetic structure transforms continuously
into a perfect SS and the band gap closes.
For all cases investigated, we find that NNO’s structure
is basically characterized by the well-known tilting of theO
6octahedra and antipolar displacements of the Nd cations,
as shown in Figs. 3(a) and3(b), respectively. However, the
structural distortions found to be most directly linked tothe observed electronic transformations are the following:00.20.40.60.8Egap (eV)
00.511.5Mag. Mom. ( μΒ)
-0.1-0.0500.05ΔQ = Q1-Q2 (e)
d electrons
all electrons
01234 56 7 89 1 0
U (eV)00.250.50.751ΔV=V1-V2 (eV)μ2μ1(a)
(b)
(c)
(d)
FIG. 2. Udependence of key electronic parameters for bulk
NNO. (a) Band gap, where filled circles indicate that the zero-gap
result is a marginal one (i.e., the density of states has a minimum atthe Fermi energy). (b) Magnetic moments as defined in Fig. 1(a).
(c) Difference /Delta1Q in the number of electrons around the two
types of Ni cations. We show both the difference in the totalnumber of electrons as well as the difference in electrons with 3 d
character. Essentially, the number of electrons associated to an atom
is estimated by projecting the occupied electronic manifold ontolocalized atom-centered orbitals with specific angular momentum.
(d) Difference /Delta1V in the electrostatic potentials experienced by the
two types of Ni cations.
(1) The breathing -like distortions of the oxygen octahedra
(ubr) depicted in the inset of Fig. 3(c). These modes result in
two rocksalt-ordered Ni sublattices, with the correspondingO
6groups being, respectively, expanded and contracted. As
regards the average Ni–O distances we have ¯dNiO
1>¯dNiO
2,
with ¯dNiO
1≈2.01 ˚A and ¯dNiO
2≈1.90 ˚A obtained for U=
7 eV . (2) The Jahn-Teller (JT) stretching distortion of the O 6
octahedra ( uJT) depicted in the inset of Fig. 3(d).16Note that
the magnitude of these modes correlates strongly with thecomputed magnetic moments and band gaps; more precisely,the breathing distortions are important for U/lessorequalslant7 eV and
almost disappear for U/greaterorequalslant8 eV as the JT modes become
significant.
Our results for U/lessorsimilar8 eV are consistent with the P2
1/c
experimental ground state of these nickelates.17More pre-
cisely, Garc ´ıa-Mu ˜noz et al. reported ¯dNiO
1=1.984 ˚A and
¯dNiO
2=1.910 ˚A for NNO,14in good agreement with our
results mentioned above. Additionally, values of μ1≈1.5μB
andμ2≈0.65μBhave been reported for related compounds
YNiO 318and HoNiO 3,19which are consistent with our results
for bulk NNO obtained using U=7 eV [see Fig. 2(b)].
We also performed simulations allowing for noncollinearmagnetism—which lifts the P2
1/csymmetry—and confirmed
that our lowest-energy solution for U=7 eV is the multiferroic
“N state” discussed by Giovannetti et al.12
214431-2Ab INITIO STUDY OF THE FACTORS AFFECTING ... PHYSICAL REVIEW B 85, 214431 (2012)
xyz81012O6rot. (degree)
00.10.2Nd anti-pol. (C/m2)
00.020.040.06ubr(a.u.)
01234 56 7 89 1 0
U(eV)00.020.040.06uJT(a.u.)X-point (x,y) R-point (x,y)
x,yx,y
z(a)
(b)
(c)
(d)R-point (x,y) M-point (z)
FIG. 3. (Color online) Udependence of key structural parameters
for bulk NNO: (a) Antiphase and in-phase rotations of the O 6
octahedra. The antiphase rotations are about the xandypseudocubic
directions of the perovskite structure [given, respectively, by the ap
andbpdescribed in the text, and sketched in the inset of panel (d)] and
are associated with the Rq-point of the Brillouin zone corresponding
to the 5-atom elemental perovskite cell; the in-phase rotations are
about the zpseudocubic axis (given by the cpvector described in the
text) and associated with the Mq-point. (b) Antipolar displacements
of the Nd sublattice, along the xandypseudocubic directions
and associated to the RandXq-points. To give the distortion in
units of polarization, we multiply its amplitude by the nominal
+3 charge of the Nd cations and divide by the cell volume; we
thus obtain a rough estimate of the magnitude of the local dipolesinvolved in the antipolar pattern. (c) Isotropic-breathing distortion
u
brcorresponding to the atomic displacement pattern shown in the
inset. (d) Jahn-Teller-like distortion uJTsketched in the inset. Note
that the x,y,a n dzcomponents of the various distortions shown are
related in some cases.
The above-mentioned experimental results are the basis
from which a charge order of the Ni atoms is usuallyinferred.
13,14More precisely, the charge disproportionation δ
is estimated to be about 0.25 ein NNO,14where eis the
elemental charge; this would correspond to a difference /Delta1Q=
Q1−Q2=2δ≈0.50ebetween the two Ni sublattices
defined in Fig. 1. This interpretation of the experiments relies
on the so-called bond-valence model ,20the underlying physical
picture being essentially as follows: The larger NiO 6octahedra
would be associated with a larger number of 3 delectrons at the
Ni cation, whose oxidation state would be +3−δ(ideally, +2)
instead of the nominal +3; such ions would be in a high-spin
configuration and display a large magnetic moment (ideally,μ
1=2μB). Conversely, the small NiO 6octahedra would
contain Ni+3+δcations (ideally, Ni4+); these ions would be in-50-2502550pDOS (states/eV)Ni#1
Ni#2
total
-50-2502550pDOS (states/eV)
-6 -4 -2 0 2 4
E - EFermi (eV)-50-2502550pDOS (states/eV)U = 10 eVU = 7 eVU = 2 eV
Ni#1 = Ni#2
(all Ni pDOS x20)(a)
(b)
(c)
FIG. 4. (Color online) Representative results for the electronic
partial density of states (DOS) close to the Fermi energy for bulk
NdNiO 3and various Uvalues. Note that the DOS associated with
individual Ni atoms is scaled for clarity.
a low-spin configuration and display small magnetic moments
(ideally, μ2=0).
Interestingly, while the computed magnetic moments and
interatomic distances are consistent with experiments, ourestimated atomic charges do not fully support the charge orderinferred in the experimental works: As shown in Fig. 2(c),f o r
U< 8 eV the computed atomic charges clearly correspond to
two Ni sublattices (i.e., we have /Delta1Q/negationslash=0), but the magnitude
of the splitting ( |/Delta1Q|<0.1e) is about 6 times smaller
than the values deduced from the bond-valence analysisof the experimental data.
21Further, the sign of the charge
disproportionation obtained from the total number of electronsaround the Ni atoms is opposite to what we would expectfrom the above-described picture; more precisely, for the totalcharge we obtain /Delta1Q=Q
1−Q2<0, implying that the Ni
atoms located at the center of the larger O 6groups have a
relatively small number of electrons bound to them. On theother hand, if we compute /Delta1Q by restricting ourselves to
the electrons with 3 dcharacter, the obtained splitting has the
expected sign (and continues to be small). These results clearlysuggest that a simple ionic picture is not appropriate to describeand understand NNO in detail, as the strong hybridization ofthe nickel and oxygen orbitals may lead to counterintuitiveresults. (Such a strong hybridization probably explains whywecount fewer electrons for the Ni atoms located at the bigger
O
6octahedra: the longer Ni–O distances probably imply that
the electrons shared by Ni and O will be relatively far from theNi center, which will result in a smaller number of electronsassigned to such a Ni cation.) Additionally, the difficultiesto reconcile our results with the usual charge-order picturemay indicate a shift to a Ni-Ni bond-centered charge order, as
suggested in Ref. 12; further investigation is needed to clarify
these issues and their practical implications. In any case, inthe following we will refer to the solutions with μ
1/negationslash=μ2
and/Delta1Q/negationslash=0 as being “charge-ordered” to comply with the
generally adopted terminology.
214431-3PROSANDEEV , BELLAICHE, AND ´I˜NIGUEZ PHYSICAL REVIEW B 85, 214431 (2012)
00.20.40.60.8Egap (eV)
00.511.5Mag. Mom. ( μΒ)
-0.1-0.0500.05ΔQ = Q1-Q2 (e)
3.5 3.6 3.7 3.8 3.9 4 4.1
in-plance lattice constant (Angstrom)00.250.50.751ΔV=V1-V2 (eV)μ2μ1U = 7 eV
U = 2 eV
d
all(a)
(b)
(c)
(d)
FIG. 5. (Color online) Key electronic parameters as a function
of epitaxial strain, as obtained from simulations with U=2e V
(red squares) and U=7 eV (black circles), respectively. Details as
in Fig. 2.
Our solutions for U/greaterorequalslant8 eV qualitatively differ. In this case,
all the Ni atoms present large magnetic moments indicative ofa high-spin state.
22From inspection of the projected density
of states (Fig. 4), we may infer an electronic configuration
somewhere between t3
2g↑e2
g↑t2
2g↓andt3
2g↑e2
g↑t1
2g↓; note that the
hybridization between Ni-3 dand O-2 porbitals is very large,
which complicates a precise assignment. Such Ni species seemcompatible with the observed JT distortion. Additionally, let usnote that these structures display small inversion-symmetry-breaking distortions that result in a polar space group ( Pc, with
polar axes along the aandclattice vectors).
23Nevertheless, our
solutions for U/greaterorequalslant8 eV are metallic and, thus, not ferroelectric.
B. Effect of epitaxial strain
Figures 5–9summarize our results for the effect of epitaxial
strain in NNO films. Once again, three regions are identi-fied: (1) Within a considerable range around a
sub≈3.8˚A,
and irrespective of the value of U, the ground state is
essentially that of the bulk material ( P21/cspace group).24
The simulations with U=7 eV render a SS +SDW solution
that can be tuned continuously toward the ideal SS (SDW)configuration by increasing (decreasing) a
sub. Such a tuning
is most clearly reflected in μ2’s strong dependence on the
epitaxial mismatch. In contrast, the simulations with U=
2 eV render a SDW state whose magnetic structure variesweakly with a
sub. The most significant structural distortions
are the breathing modes that characterize bulk NNO andcorrelate with the μ
1/negationslash=μ2splitting. Naturally, in this case,
there is a difference between the in-plane and out-of-planebreathings; the in-plane distortion is favored as a
subdecreases,
and the out-of-plane component presents a complicated andstrongly U-dependent behavior. Finally, for both U=2 eV and
81012O6 rot. (degree)
00.10.2Nd anti-pol. (C/m2)
00.020.040.06ubr (a.u.)
3.5 3.6 3.7 3.8 3.9 4 4.1
in-plance lattice constant (Angstrom )00.020.040.06uJT (a.u.)R-point (x,y) M-point (z)U = 7 eV
U = 2 eV
x,y
z
x,y(a)
(b)
(c)(d)R-point (x,y) X-point (x,y)
FIG. 6. (Color online) Key structural parameters as a function
of epitaxial strain, as obtained from simulations with U=2e V
(red squares) and U=7 eV (black circles), respectively. Details as
in Fig. 3.
U=7 eV , the obtained solutions are insulating throughout
most of this region; the most significant changes in Egap
correspond to the U=7 eV results and clearly correlate with
the magnitude of the splitting in two Ni sublattices.
(2) For large tensile strains (i.e., asub/greaterorsimilar4.0˚A), we get
a solution in which all Ni atoms are essentially equivalent22
and their localized magnetic moments are arranged in a SSconfiguration. The disappearance of the two Ni sublatticesis reflected in the dominant structural distortions, which arenow of the JT type. Hence, this solution is similar to the oneobtained for bulk NNO in the limit of large U; indeed, the
3.5 3.6 3.7 3.8 3.9 4 4.1
in-plance lattice constant (Angstrom )0.80.911.11.21.3cp/apU = 2 eV
U = 7 eV
FIG. 7. (Color online) Evolution of the aspect ratio of the cell
of NdNiO 3films as a function of epitaxial strain, as obtained from
simulations with U=2 eV (red squares) and U=7 eV (black circles),
respectively. The reported aspect ratio ( cp/ap) corresponds to the cp
andappseudocubic lattice constants (see text) as deduced from our
relaxed structures.
214431-4Ab INITIO STUDY OF THE FACTORS AFFECTING ... PHYSICAL REVIEW B 85, 214431 (2012)
-50-2502550pDOS (states/eV)Ni
total
-6 -4 -2 0 2 4
E - EFermi (eV)-50-2502550pDOS (states/eV)asubs = 4.15 Åasubs = 3.55 Å
(all Ni pDOS x20)(a)
(b)
FIG. 8. (Color online) Representative results for the electronic
partial density of states (DOS) close to the Fermi energy, for NdNiO 3
films simulated at selected asubvalues and with U=2 eV . Note that
the DOS associated with individual Ni atoms is scaled for clarity.
results for U=7 eV and asub/greaterorsimilar4.0˚A—i.e., a metallic phase
withμ1≈μ2≈1.5μB—are essentially identical to our bulk
results for U/greaterorequalslant8e V .T h er e s u l t sf o r U=2 eV slightly differ, as
an insulating solution with μ1≈μ2≈0.8μBand small Egap
is obtained in the limit of large strains. The computed partial
density of states (Figs. 8and9) suggests that this differentiated
behavior is related with a subtle Udependence of the Ni-3 d–
O-2phybridization, which determines the nature of the levels
at the Fermi energy and the existence of a gap.
(3) In the limit of strong in-plane compressions (i.e.,
asub/lessorsimilar3.65 ˚A) we get solutions that somewhat resemble those
obtained for large tensile strains. All Ni atoms present largemagnetic moments of about 1.25 μ
BforU=7 eV and about
0.75μBforU=2 eV . The solutions are metallic and a JT
distortion appears. Note also that, as shown in Fig. 7,i nt h i s
limit the unit cell of our simulated NNO displays a very largeaspect ratio, clearly resembling the so-called supertetragonal
phases that can be strain-engineered in multiferroics likeBiFeO
3.25
Let us remark that our results indicate that the rotations of
the oxygen octahedra, around both in-plane and out-of-planeaxes, are very robust in NNO. Indeed, as shown in Fig. 6(a),
we do not appreciate any cancellation of specific rotationcomponents in the considered range of epitaxial strains. Sucha result may seem surprising in view of the behavior reportedfor other materials (e.g., LaNiO
326and LaAlO 327), namely
that in-plane compression favors the O 6rotations around the
-50-2502550pDOS (states/eV)Ni
total
-6 -4 -2 0 2 4
E - EFermi (eV)-50-2502550pDOS (states/eV)asubs = 4.15 Åasubs = 3.55 Å
(all Ni pDOS x20)(a)
(b)
FIG. 9. (Color online) Same as Fig. 8but for U=7e V .out-of-plane axis and penalizes tilts around in-plane axes.
Yet, let us note that the materials best investigated thusfar have a rhombohedral symmetry (with a rotation patterndenoted as a
−a−a−in Glazer’s notation28), while here we
are considering an orthorhombic compound ( a−a−c+tilting
system). Further studies will be needed to assess the generalityof such a differentiated behavior between rhombohedral andorthorhombic perovskites.
Most experimental works of NNO thin films—especially
for substrates in the range between LaAlO
3(asub=3.79 ˚A) and
SrTiO 3(asub=3.91 ˚A)—indicate that in-plane compression
favors a lower MIT temperature,1,29–31although the resistivity
in the limit of low temperatures is not always observedto decrease accordingly.
31In our case, for small epitaxial
compressions we obtain a complex structural relaxation thatleads (most clearly for U=7 eV) to an enhanced charge
order and increased E
gap; thus, our simulations suggest that
in-plane compression should result in higher low-temperatureresistance and, presumably, a higher MIT temperature. Hence,while it is not totally clear how to connect our results withexperiments, the comparison suggests that the experimentallyobserved effects may not be caused by the mere epitaxialstrain, which is captured exactly in our simulations. Instead,additional structural constraints that may be relevant for verythin films (e.g., O
6rotations/distortions clamped by some
substrates, effect of possible substrate twins, etc.) or extrinsicmechanisms (e.g., defects) may play an important role.
Finally, note that for U=2 eV we obtain the P2
1/c
paraelectric space group23for all asubvalues considered; thus,
the observed transitions are isosymmetric. In contrast, thecalculations with U=7 eV render the P2
1/cphase only in the
3.70 ˚A/lessorequalslantasub/lessorequalslant3.90 ˚A range, as small inversion-symmetry-
breaking distortions occur in all other cases. More precisely,we get the following polar solutions: Pna2
1forasub/lessorequalslant
3.65 ˚A (polar axis along clattice vector), Pcfor 3.95 ˚A/lessorequalslant
asub/lessorequalslant4.00 ˚A (polar axes along aand c), and Pmn 21for
asub/greaterorequalslant4.05 ˚A (polar axis along a). Nevertheless, according
to our simulations all such phases are metallic and, thus, notferroelectric.
C. Further implications of our results
Our simulations suggest that we can realistically simulate
NNO by employing a Hubbard- Ucorrection of about 7 eV , as
in such conditions we get reasonable agreement with existingexperimental results for the bulk material.
15Our calculations
also show that in NNO there is a close relationship betweenspecific structural distortions, the spin state of the Ni cations,and the existence of a band gap. Such a relationship, whichis consistent with the experimental observations, constitutesone of the key effects discussed in theories for the MIT innickelates.
In such an entangled situation, an often-posed question
is: What is the driving force behind the MIT in NNO andsimilar materials? Various electronic mechanisms have beeninvoked to explain the MIT, accounting for the experimentallyproposed charge order
5and even the exotic spin order that
develops in these materials.6Leeet al.6have also addressed
the relationship between charge and spin orders. Let us brieflydiscuss what our simulations suggest regarding these issues.
214431-5PROSANDEEV , BELLAICHE, AND ´I˜NIGUEZ PHYSICAL REVIEW B 85, 214431 (2012)
First, we have found that bulk NNO simulated with
U=7 eV lies midway between two solutions—i.e., the ones
we call pure SS and pure SDW, depicted in Fig. 1—with very
distinct features. In fact, NNO seems to stand on the edge ofstructural and electronic instabilities, and small variations ineither Uor the epitaxial mismatch result in profound changes
in the ground state of the material. This large sensitivitysuggests it may be difficult to identify a unique driving forceresponsible for NNO’s transformations.
We have found that increasing Uresults in bulk states in
which all Ni cations are essentially equivalent; such solutionsprobably correspond to the the strong-coupling limit—i.e.,U/greatermucht, where tis the relevant hopping parameter—in which
allthe Ni cations tend to adopt the same electronic state
resembling that of the isolated ion.
32Interestingly, this U-
driven (and obviously electronic-driven ) transition between
theμ1/negationslash=μ2andμ1≈μ2states carries with it a significant
structural transformation. In particular, the breathing mode—which we identify with the charge order—disappears.
Our results indicate that similar transitions can be induced
by compressing or expanding the lattice in-plane. In this case,the strong epitaxial mismatch seems to penalize the occurrenceof breathing distortions of the O
6octahedra, something that
appears reasonable on geometric and steric grounds. (Thebreathing distortion creates relatively large and relatively smallO
6octahedra, which seems inadequate to accommodate either
the compression or expansion of the lattice.) Then, as a result ofthe small- u
brconstraint, all Ni ions present the same electronic
configuration. Since the epitaxial mismatch is the controlparameter for these transformations, we can interpret themas being structurally driven. Hence, our simulations showthatboth electronic and structural mechanisms can trigger the
manifold changes (structural, electronic, magnetic, band-gap)associated with transitions in NNO and related materials.
Our results show that the charge order in NNO, and the
associated μ
1/negationslash=μ2splitting, strongly relies on the existence
of an O 6-breathing distortion that lowers the symmetry of the
material from orthorhombic ( Pnma ) to monoclinic ( P21/c).
Note that such a requirement goes beyond the one discussedby Lee et al. ,
6who concluded that the orthorhombicity of the
perovskite lattice is necessary for the charge order to occur. Inaddition, we found that the exotic ↑↑↓↓ arrangement of the
spins is still the preferred configuration among the investigatedones,
15even in absence of charge order [i.e., for solutions
withμ1≈μ2andubr≈0]; we checked that this is the case
for the bulk solutions obtained with U> 7 eV , as well as
for the films subject to large epitaxial strains simulated withU=7 eV . This finding supports the claim
6that such a spinorder can exist independently from the charge order. Finally,
our results indicate that small values of Utend to give a very
strong charge order (with μ1/negationslash=0 and μ2=0), while a large
Uof about 7 eV is needed to obtain solutions that reproduce
better the experimental situation (i.e., μ1>μ 2>0). Hence,
we may not fully rule out the possibility that NNO is in theweak-coupling regime,
33as using small Uvalues allows us
to reproduce qualitatively many of the observed effects; at thesame time, we obtain the best agreement with experiment forUvalues around 7 eV .
Finally, let us note that our preliminary results for other
nickelates (YNiO
3, SmNiO 3, and PrNiO 3) indicate that their
qualitative behavior is similar to that of NNO.34This suggests
that the trends discussed here are essentially common to theRNiO
3family, ranging from small (Y) to large (Pr) rare-earth
cations.
IV . SUMMARY
In conclusion, our first-principles investigation of NdNiO 3
and similar materials shows that both electronic and structuralfactors profoundly affect their ground state, evidencing theconnections among the many (electronic, magnetic, structural)effects at play in these compounds. In particular, we predict thatcomplex transformations—e.g., metal-insulator transitions,both drastic and continuous changes of the magnetic structure,etc.—can be induced by means of epitaxial strain in thin films.We thus hope our findings will stimulate new experimentalstudies of these materials, aid in the interpretation of existingand new results, and contribute to the development of realisticmodel theories of nickelates.
ACKNOWLEDGMENTS
J.I. thanks funding from MINECO-Spain (Grants
No. MAT2010-18113, No. MAT2010-10093-E, and No.CSD2007-00041). S.P. and L.B. thank the financial supportof ONR Grants No. N00014-11-1-0384 and No. N00014-08-1-0915 and ARO Grant No. W911NF-12-1-0085. They alsoacknowledge the NSF Grants No. DMR-1066158 and No.DMR-0701558, Department of Energy, Office of Basic EnergySciences, under Contract No. ER-46612 for discussions withscientists sponsored by these grants. Some computations weremade possible thanks to the MRI Grant No. 0722625 fromNSF, the ONR Grant No. N00014-07-1-0825 (DURIP), and aChallenge grant from the Department of Defense. Discussionswith E. Canadell, G. Catal ´an, J. L. Garc ´ıa-Mu ˜noz, and
R. Scherwitzl are gratefully acknowledged.
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large net magnetization, is the predicted ground state of bulk NNOwhen using Uvalues above 3 eV . Thus, for our chosen value of
U=7 eV , the calculations actually predict a ground state that is
in clear disagreement with the experimental observations, and weshould deem the present work as restricted to spin configurationsthat resemble the experimentally observed phases, which haveessentially zero remnant magnetization.
16The breathing mode corresponds to the Rpoint of the Brillouin
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21The scheme employed to estimate atomic charges tends to render
values that are too small, as evidenced by the fact that the sumof the electronic charges assigned to all the atoms does not reachthe total number of electrons. Further, the hybridization betweenNi and O orbitals is clearly very large in these nickelates. As a
result of these two factors, the estimated ionization states deviatesignificantly from the nominal ones that would correspond to theionic limit. Thus, here we only report charge differences, whichare of greater physical significance than the values obtained forindividual atomic charges.
22Having μ1≈μ2does not imply that all Ni atoms are crystallo-
graphically equivalent. In fact, for all Ni’s to be equivalent, wewould need to have a perfect Pnma symmetry, which we never
observed.
23To determine the symmetry of our relaxed structures, we usedthe program
FINDSYM [H. T. Stokes and D. M. Hatch (2004),
http://stokes.byu.edu/isotropy.html ] employing a value of 0.01 ˚A
for the accuracy within which atomic positions and lattice vectorsare known.
24For bulk NNO and U=7 eV , we obtain a pseudocubic in-plane
lattice constant of about 3.87 ˚A. Experimentally, the pseudocubic
in-plane lattice constant of NNO is about 5.38/√
2˚A=3.80 ˚A, as
derived from the results of Ref. 14at 50 K.
25H. Bea, B. Dupe, S. Fusil, R. Mattana, E. Jacquet, B. Warot-Fonrose,
F. Wilhelm, A. Rogalev, S. Petit, V . Cros, A. Anane, F. Petroff,K. Bouzehouane, G. Geneste, B. Dkhil, S. Lisenkov, I. Ponomareva,L. Bellaiche, M. Bibes, and A. Barthelemy, Phys. Rev. Lett. 102,
217603 (2009).
26S. J. May, J. W. Kim, J. M. Rondinelli, E. Karapetrova, N. A.Spaldin, A. Bhattacharya, and P. J. Ryan, Phys. Rev. B 82, 014110
(2010).
27A. J. Hatt and N. A. Spaldin, Phys. Rev. B 82, 195402 (2010).
28A. M. Glazer, Acta Crystallogr. Sect. A 31, 756 (1975).
29G. Catalan, R. M. Bowman, and J. M. Gregg, P h y s .R e v .B 62, 7892
(2000).
30J. Liu, M. Kareev, B. Gray, J. W. Kim, P. Ryan, B. Dabrowski, J. W.Freeland, and J. Chakhalian, Appl. Phys. Lett. 96, 233110 (2010).
31R. Scherwitzl, P. Zubko, I. G. Lezama, S. Ono, A. F. Morpurgo,
G. Catalan, and J. M. Triscone, Adv. Mater. 22, 5517 (2010).
32This would correspond to the strong-coupling limit discussed in
S. B. Lee, R. Chen, and L. Balents, Phys. Rev. B 84, 165119 (2011).
33Mazin et al.5studied the pressure dependence of EgapofRNiO 3
compounds, concluding that small- Ucorrections give better agree-
ment with experiment. Lee et al.6explained the magnetic order in
NdNiO 3as relying on a type of Fermi-nesting mechanism that is
usually associated with weak correlations.
34S. Prosandeev, L. Bellaiche, and J. ´I˜niguez (unpublished).
214431-7 |
PhysRevB.82.075116.pdf | Projector augmented-wave method: Application to relativistic spin-density functional theory
Andrea Dal Corso
International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy
and DEMOCRITOS IOM, CNR, Trieste, Italy
/H20849Received 3 May 2010; revised manuscript received 12 July 2010; published 11 August 2010 /H20850
Applying the projector augmented-wave /H20849PAW /H20850method to relativistic spin-density functional theory /H20849RS-
DFT /H20850we derive PAW Dirac-Kohn-Sham equations for four-component spinor pseudo-wave-functions. The
PAW freedom to add a vanishing operator inside the PAW spheres allows us to transform these PAW Dirac-typeequations into PAW Pauli-type equations for two-component spinor pseudo-wave-functions. With these wavefunctions, we get the frozen-core energy as well as the charge and magnetization densities of RSDFT, witherrors comparable to the largest between 1 /c
2and the transferability error of the PAW data sets. Presently, the
latter limits the accuracy of the calculations, not the use of the Pauli-type equations. The theory is validated byapplications to isolated atoms of Fe, Pt, and Au, and to the band structure of fcc-Pt, fcc-Au, and ferromagneticbcc-Fe.
DOI: 10.1103/PhysRevB.82.075116 PACS number /H20849s/H20850: 71.15.Dx, 71.15.Rf, 75.20.En
I. INTRODUCTION
After a few years from the proposal of density functional
theory /H20849DFT /H20850,1a relativistic extension able to deal with both
nonmagnetic and magnetic systems was presented.2Al-
though not free from subtle conceptual issues—a problemthat we do not address in this paper—the relativistic theory isat the foundations of most DFT studies of materials. Pseudo-potentials /H20849PPs /H20850are constructed starting from the atomic so-
lutions of relativistic DFT /H20849RDFT /H20850equations or of their sca-
lar relativistic /H20849SR/H20850approximation,
3and all-electron methods
deal relativistically with core electrons and often also withvalence electrons.
4,5Formally, the RDFT Kohn-Sham /H20849KS/H20850
equations are rather similar to their nonrelativistic counter-parts: the kinetic-energy operator is replaced with the Dirackinetic energy
6and the wave functions are four-component
spinors. Magnetism is treated in the theory assuming that theexchange and correlation energies are functionals of thecharge and magnetization densities. The resulting relativisticspin-density functional theory /H20849RSDFT /H20850equations have been
solved by many authors in isolated atoms and molecules
7–10
and several codes are available also for solids.11–13
Practical calculations of the DFT total energy require a
basis set and the plane-wave basis together with the projectoraugmented-wave /H20849PAW /H20850/H20849Refs. 14–17/H20850method is becoming
increasingly popular. The PAW method makes a mapping,exact in principle, between pseudo-wave-functions, de-scribed by plane waves, and all-electron wave functionswhose rapid oscillations close to the nuclei are treated byintroducing spheres about each atom and radial grids insidethese spheres. The mapping is carried out with the help of aset of partial waves and projectors calculated in the isolatedatoms. This mapping is exact in the limit of a large numberof partial waves but in practice a compromise must be madebetween the partial-wave completeness and the computa-tional efficiency. So far, the PAW method has been usedwithin nonrelativistic DFT,
14–17although SR effects are in-
cluded in the PAW data sets and sometimes also spin-orbiteffects are calculated.
4,18
In this paper, we introduce RSDFT within the PAW
scheme. We give two formulations of the theory. A fullyrelativistic /H20849FR/H20850Dirac-type version and a FR Pauli-type ver-
sion obtained from the Dirac-type version through approxi-
mations that do not worsen the overall accuracy. At the FRlevel, the theory is quite similar to its nonrelativistic coun-terpart. Nevertheless, we give here some details of its deri-vation because, as far as we know, it is absent in the litera-ture. In the relativistic theory, partial waves and projectorsare four-component spinors and the PAW RSDFT equationsare Dirac-type equations for four-component spinor pseudo-wave-functions with a local effective self-consistent potentialand a nonlocal PP. The coefficients of the nonlocal PP arerecalculated at each self-consistent iteration with the instan-taneous electronic partial occupations, leading to an efficientand accurate picture of the interaction of the valence elec-trons with the nuclei and core electrons.
After the introduction of the exact relativistic PAW theory,
we proceed by showing that often the PAW Dirac-Kohn-Sham equations can be simplified. We show that it would beworthwhile to solve these equations if the transferability er-rors /H20849TEs /H20850of the PAW data sets and the other numerical
errors were kept below 1 /c
2/H20849here cis the speed of light /H20850,a
task that, although feasible, is quite hard. The TEs of modernPAW data sets are larger than 1 /c
2so the PAW Dirac-type
equations can be simplified without losing accuracy by keep-ing the approximation errors below the TEs. The result ofthese simplifications are PAW Pauli-type equations whosesolutions are two-component spinors which are sufficient toevaluate the FR frozen-core energy, as well as the charge andmagnetization densities, with an error either of order 1 /c
2or
comparable to the TE, depending on which is the largest.This is possible because, outside the PAW spheres, the smallcomponents of the pseudo-wave-functions are of order 1 /c
with respect to the large components /H20849independently from the
nuclear charge Z
I/H20850and we can use Pauli-type equations with
an error of order 1 /c2while, inside the PAW spheres, we can
remove the small components of the pseudo-wave-functionsexploiting the peculiarities of the PAW method and makingerrors comparable to the TE. As long as the TE is larger than1/c
2, we can make these approximations without worsening
the overall accuracy.PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
1098-0121/2010/82 /H208497/H20850/075116 /H2084918/H20850 ©2010 The American Physical Society 075116-1The problem of simplifying the four-component Dirac
equations transforming them into two-component Pauliequations has a long history and many approaches have beenproposed.
19,20Usually these methods neglect terms of order
/H20849v/c/H20850k, where vis an estimate of the electron velocity. The
most common expansion neglects terms with k=3 and in-
cludes mass-velocity, Darwin, and spin-orbit terms.20Other
expansions might retain also higher order terms but whenapplied to the calculation of the electronic structure theseexpansions face the problem that close to the nucleus theall-electron potential diverges and the velocity
vbecomes of
order ZI. In heavy atoms ZI/cis quite large. In the zero-order
regular approximation21the problems due to the potential
divergence are avoided and the method has been used exten-sively in molecules and solids.
22,23In our approach, approxi-
mations are used only outside the PAW spheres where v
/H110151 so that, even using Pauli equations with errors of order
/H20849v/c/H208502, the final error is of order 1 /c2, does not increase with
the nuclear charge ZI, and is usually much smaller than
/H20849ZI/c/H20850k, even for large k.
Norm conserving /H20849NC /H20850and ultrasoft /H20849US/H20850/H20849Ref. 24/H20850PPs
are well-defined approximations of the PAW method.15FR
NC-PPs have been known for a long time25,26and several
applications of the FR US-PPs are already available in theliterature as well.
27–32For NC-PPs, the present work proves
the Kleinman’s observation that solutions of the Dirac-typeequations can be mimicked, with errors of order 1 /c
2,b y
solving Pauli-type equations with a PP tailored on the largecomponents of the solutions of the atomic radial Dirac-typeequations.
25,26In the US case, we obtain the FR US-PPs
introduced in Ref. 27and, in addition, we give a few hints
for their construction. Moreover, we explain analytically whyFR US-PPs electronic band structures match all-electron RS-DFT band structures, a result that we were able to prove onlynumerically in Ref. 27.
In this paper, we implement the FR PAW Pauli-type for-
malism and validate it by a few applications. We start withthe atomic frozen-core energy and electron energy levels ofFe, Pt, and Au and show transferability tests of the FR PAWdata sets. Then the band structures of the face-centered-cubic/H20849fcc/H20850Pt and Au in a few high-symmetry points of the Bril-
louin zone are compared with the FR all-electron LAPWapproach
11and with the FR US-PPs.27Finally, the FR PAW
electronic band structures of ferromagnetic body-centered-cubic /H20849bcc /H20850Fe close to the Fermi level are compared with
Refs. 33and34where a NC-PP and an all-electron method
were employed. We find that the FR PAW method yieldstotal energies and electronic levels that match the all-electronresults based on the Dirac-type equations of RSDFT.
This paper is organized as follows. In Sec. I, we summa-
rize the RSDFT and discuss its nonrelativistic limit in anall-electron frozen-core framework and in Sec. II, we sum-
marize the nonrelativistic PAW approach. Section IIIcon-
tains the derivation of the PAW RSDFT equations in theirDirac-type form. In Sec. IV, we discuss how to simplify
these equations and transform them into PAW Pauli-typeequations for two-component spinor wave functions. Finally,Sec. Vcontains a few applications of the FR PAW Pauli-type
theory. More technical questions are treated in the appendi-ces. Appendix A deals with the generation of the FR PAWdata sets while Appendix B discusses how to perform the
summations over the four-component spinor indexes.
II. RELATIVISTIC SPIN-DENSITY FUNCTIONAL THEORY
The basic variables of relativistic DFT are the charge and
the vector current densities, however here we do not use thegeneral theory, but a simplified version /H20849RSDFT /H20850/H20849Refs. 2
and6/H20850in which the dependence of the total energy on the
orbital part of the vector current is neglected and the spindensity is the basic variable. Within RSDFT the total energyof a gas of Ninteracting electrons in the external potential of
fixed nuclei at positions R
Ican be written as a functional of
the four-component spinor one-electron orbitals /H9023i,/H9257/H20849r/H20850,
Etot,ae=/H20858
i,/H92571,/H92572/H20855/H9023i,/H92571/H20841TD/H92571,/H92572/H20841/H9023i,/H92572/H20856+Exc/H20851/H9267e,m/H20852+EH/H20851/H9267e+/H9267Z/H20852,
/H208491/H20850
where iindicates the occupied states, the index /H9257runs on the
four spinor components, and TD/H92571,/H92572are the components of the
Dirac kinetic-energy operator that can be written in terms ofthe momentum operator p=−i/H11612and of the 4 /H110034 matrices
/H9251k
/H20849k=x,y,z/H20850and/H9252. In Hartree atomic units, and subtracting
the electron rest energy, we have20
TD=c/H9251·p+/H20849/H9252−14/H110034/H20850c2, /H208492/H20850
where 14/H110034is the 4 /H110034 identity matrix and cis the speed of
light, about 137 in atomic unit.35EH/H20851/H9267e+/H9267Z/H20852is the Hartree
energy of the electron /H20849/H9267e/H20850and of the nuclear /H20849/H9267Z/H20850charge
densities whereas Exc/H20851/H9267e,m/H20852are the exchange and correla-
tion energies that depend on the electron /H20851/H9267e/H20849r/H20850
=/H20858i,/H9257/H20841/H9023i,/H9257/H20849r/H20850/H208412/H20852and on the magnetization densities /H20851mk/H20849r/H20850
=/H9262B/H20858i,/H92571,/H92572/H9023i,/H92571/H11569/H20849r/H20850/H20849/H9252/H9018k/H20850/H92571,/H92572/H9023i,/H92572/H20849r/H20850, where /H9018k/2 is the
spin angular momentum operator and /H9262Bis the Bohr magne-
ton/H20852. Notice that the spin density /H9267/H92571,/H92572/H20849r/H20850
=/H20858i/H9023i,/H92571/H11569/H20849r/H20850/H9023i,/H92572/H20849r/H20850,o r/H9267e/H20849r/H20850andmk/H20849r/H20850can be considered as
equivalent variables because /H9267e/H20849r/H20850=/H20858/H9257/H9267/H9257,/H9257/H20849r/H20850andmk/H20849r/H20850
=/H9262B/H20858/H92571,/H92572/H9267/H92571,/H92572/H20849r/H20850/H20849/H9252/H9018k/H20850/H92571,/H92572.
The valence frozen-core total energy Etot, calculated by
subtracting from Etot,aethe kinetic and the Hartree energies of
the core electrons, is given by
Etot=/H20858
i,/H92571,/H92572/H20855/H9023i,/H92571/H20841TD/H92571,/H92572/H20841/H9023i,/H92572/H20856+Exc/H20851/H9267+/H9267c,m/H20852+EH/H20851/H9267/H20852
+/H20885d3rVloc/H20849r/H20850/H9267/H20849r/H20850+UI,I, /H208493/H20850
where now the index iruns on the valence states only, /H9267and
/H9267cindicate the valence and core charge densities, Vloc
=VH/H20851/H9267Z+/H9267c/H20852is the Coulomb potential of the core and nuclear
charges, and UI,Iis the long-range ion-ion interaction energy.
The minimization of this functional leads to the Dirac-
type relativistic KS equations,6
/H20858
/H92572/H20853TD/H92571,/H92572+/H20851Veff/H20849r/H20850−/H9255i/H20852/H9254/H92571,/H92572−/H9262BBxc/H20849r/H20850·/H20849/H9252/H9018/H20850/H92571,/H92572/H20854/H20841/H9023i,/H92572/H20856
=0 , /H208494/H20850
where the effective potential Veff/H20849r/H20850=Vloc/H20849r/H20850+VH/H20849r/H20850+Vxc/H20849r/H20850ANDREA DAL CORSO PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-2is the sum of the local, Hartree and exchange and correlation
potentials, and Bxc/H20849r/H20850=−/H9254Exc//H9254mis the exchange and corre-
lation magnetic fields. VH/H20849r/H20850is the Hartree potential of the
valence electrons only whereas Vxc/H20849r/H20850andBxc/H20849r/H20850are calcu-
lated with the total electron charge and magnetization densi-
ties. For latter convenience, we define VLOC/H92571,/H92572/H20849r/H20850
=Veff/H20849r/H20850/H9254/H92571,/H92572−/H9262BBxc/H20849r/H20850·/H20849/H9252/H9018/H20850/H92571,/H92572.
The Dirac-type KS equations /H20851Eq. /H208494/H20850/H20852can be rewritten
introducing the large and the small components of the all-electron wave functions,
/H20841/H9023
i,/H9257/H20856=/H20873/H20841/H9023i,/H9268A/H20856
/H20841/H9023i,/H9268B/H20856/H20874, /H208495/H20850
where /H20841/H9023i,/H9268A/H20856and /H20841/H9023i,/H9268B/H20856are two-component spinors and the
index /H9268runs on the two components. With the help of the
Pauli matrices /H20849/H9268/H20850representation of the /H9251and/H9252matrices,
Eq. /H208494/H20850becomes
/H20858
/H92682/H20853c/H9268/H92681,/H92682·p/H9023i,/H92682B/H20849r/H20850+/H20851VLOC/H92681,/H92682/H20849r/H20850−/H9255i/H9254/H92681,/H92682/H20852/H9023i,/H92682A/H20849r/H20850/H20854=0 ,
/H208496/H20850
/H20858
/H92682/H20853c/H9268/H92681,/H92682·p/H9023i,/H92682A/H20849r/H20850+/H20851V¯
LOC/H92681,/H92682/H20849r/H20850
−/H20849/H9255i+2c2/H20850/H9254/H92681,/H92682/H20852/H9023i,/H92682B/H20849r/H20850/H20854=0 , /H208497/H20850
where now VLOC/H92681,/H92682/H20849r/H20850=Veff/H20849r/H20850/H9254/H92681,/H92682−/H9262BBxc/H20849r/H20850·/H9268/H92681,/H92682and
V¯
LOC/H92681,/H92682/H20849r/H20850=Veff/H20849r/H20850/H9254/H92681,/H92682+/H9262BBxc/H20849r/H20850·/H9268/H92681,/H92682.
Nonrelativistic limit: The Pauli equations
When the electron speed is much smaller than c, these
equations can be approximated by the Pauli equations. FromEq. /H208497/H20850we get
/H9023
i,/H92681B/H20849r/H20850=−/H20858
/H92682/H20851V¯LOC /H20849r/H20850−/H20849/H9255i+2c2/H2085012/H110032/H20852/H92681,/H92682−1
/H11003/H20858
/H92683c/H9268/H92682,/H92683·p/H9023i,/H92683A/H20849r/H20850. /H208498/H20850
The diagonal elements of the matrix /H9255i12/H110032−V¯LOC /H20849r/H20850are
comparable to the electron kinetic energy and the off-diagonal elements are of the same order or smaller. In theregions where these elements are small with respect to 2 c
2,
we can approximate
/H9023i,/H92681B/H20849r/H20850/H11015/H20858
/H92682/H9268/H92681,/H92682·p
2c/H9023i,/H92682A/H20849r/H20850/H20849 9/H20850
with a position-dependent error of order /H20851v/H20849r/H20850/c/H208523, where v
is a parameter of magnitude comparable with the speed of
the electron in that region. By inserting Eq. /H208499/H20850in Eq. /H208496/H20850we
get the Pauli equations,
/H20858
/H92682/H20875p2
2/H9254/H92681,/H92682+VLOC/H92681,/H92682/H20849r/H20850−/H9255i/H9254/H92681,/H92682/H20876/H9023i,/H92682A/H20849r/H20850=0 , /H2084910/H20850
that have an error of order /H20849v/c/H208502. Notice that the solutions of
the Dirac equations /H9023i,/H9257/H20849r/H20850are normalized. This means that/H20858/H9268/H20848Vd3r/H20849/H20841/H9023i,/H9268A/H20849r/H20850/H208412+/H20841/H9023i,/H9268B/H20849r/H20850/H208412/H20850=1. Therefore /H9023i,/H9268A/H20849r/H20850is not
normalized. However the missing term is of order /H20849v/c/H208502, the
same order of the error of the Pauli equations and can beconsistently neglected. For an electron close to an heavynucleus of charge Z
I, a good estimate is v/H11015ZIand /H20849ZI/c/H208502
might be sizable. For the valence states, the nuclear potential
is screened so vis significantly lower than ZI. Nevertheless,
the use of Eq. /H2084910/H20850in the regions close to the nuclei leads to
significant errors. Actually the mass-velocity, Darwin, andspin-orbit terms are neglected in Eq. /H2084910/H20850so spin-orbit split-
tings are completely missing. On the contrary, in the regionsfar from the nuclei or for light elements, Eq. /H2084910/H20850is a quite
good approximation of Eqs. /H208496/H20850and /H208497/H20850. The Pauli-type KS
equations can be obtained directly from the minimization ofthe nonrelativistic DFT total energy written for two-component spinor wave functions and used to deal with non-collinear magnetic structures
36
Etot,nr=/H20858
i,/H92681/H20855/H9023i,/H92681A/H20841p2
2/H20841/H9023i,/H92681A/H20856+Exc/H20851/H9267+/H9267c,m/H20852+EH/H20851/H9267/H20852
+/H20885d3rVloc/H20849r/H20850/H9267/H20849r/H20850+UI,I, /H2084911/H20850
where the charge density is /H9267/H20849r/H20850=/H20858i,/H9268/H20841/H9023i,/H9268A/H20849r/H20850/H208412and
the magnetization density is mk/H20849r/H20850
=/H9262B/H20858i,/H92681,/H92682/H9023i,/H92681A,/H11569/H20849r/H20850/H20849/H9268k/H20850/H92681,/H92682/H9023i,/H92682A/H20849r/H20850.
III. NONRELATIVISTIC PAW METHOD
In the nonrelativistic PAW approach14–16to the electronic
structure problem, the PAW method is used to calculate theenergy in Eq. /H2084911/H20850. We summarize in this section the main
features of the nonrelativistic PAW method following thegeneral scheme presented in Ref. 16to deal with noncol-
linear magnetic structures. The approaches to noncollinearmagnetism implemented in Ref. 37using US-PPs, or in Ref.
38using NC-PPs, are approximations of this PAW formula-
tion. In the PAW approach, a linear mapping transforms the
pseudo-wave-functions /H20841/H9023
˜
i,/H9268A/H20856, which are the variational vari-
ables, into all-electron wave functions /H20841/H9023i,/H9268A/H20856,
/H20841/H9023i,/H9268A/H20856=/H20841/H9023˜
i,/H9268A/H20856+/H20858
I,m/H20851/H20841/H9021mI,AE/H20856−/H20841/H9021mI,PS/H20856/H20852/H20855/H9252mI/H20841/H9023˜
i,/H9268A/H20856./H2084912/H20850
The mapping requires three sets of functions: the all-electron
partial waves /H20841/H9021mI,AE/H20856, the pseudo-partial-waves /H20841/H9021mI,PS/H20856and
the projector functions /H20841/H9252mI/H20856. The all-electron partial waves
are calculated in an isolated nonmagnetic atom by solvingthe nonrelativistic Kohn and Sham equations in sphericalgeometry at N
/H9280values of the energy, for a number of orbital
angular momenta l. Therefore the index Iin/H20841/H9021mI,AE/H20856indicates
the atom and means that the function is centered about theatom at R
Iandmis a composite index that indicates /H9270,l,ml,
where 1 /H11021/H9270/H11021N/H9280identifies the energy, 0 /H11021l/H11021lmaxindicates
the orbital angular momentum and − l/H11021ml/H11021lindicates the
projection of the orbital angular momentum on a quantiza-tion axis. Each pseudo-partial-wave coincides with the cor-responding all-electron partial wave outside a given cutoffradius and is smoothly continued by a pseudization recipePROJECTOR AUGMENTED-WA VE METHOD: APPLICATION … PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-3inside the cutoff radius. For each atom, the maximum cutoff
radius defines the size of the PAW sphere. The projectorfunctions are orthogonal to the pseudo-partial-waves
/H20855
/H9252nI/H20841/H9021mI,PS/H20856=/H9254nmand are localized inside the PAW sphere.
The accuracy of this mapping can be increased systemati-cally by increasing N
/H9280and lmaxand when these are suffi-
ciently large the mapping is virtually exact, meaning thatinside the PAW sphere
/H20858
m/H20841/H9021mI,PS/H20856/H20855/H9252mI/H20841=1 /H2084913/H20850
when applied to any pseudo-wave-function.
Using the above mapping, and the completeness relation-
ship /H20851Eq. /H2084913/H20850/H20852, one can prove that the expectation value of a
local operator Athat acts on all-electron wave functions can
be calculated as the expectation value of a new operator A˜
that acts on pseudo-wave-functions and is given by14
A˜=A+/H20858
I,mn/H20841/H9252mI/H20856/H20851/H20855/H9021mI,AE/H20841A/H20841/H9021nI,AE/H20856−/H20855/H9021mI,PS/H20841A/H20841/H9021nI,PS/H20856/H20852/H20855/H9252nI/H20841.
/H2084914/H20850
The operator A˜is not completely determined by the method.
There is still the additional freedom to add a term of the form
/H9004=B−/H20858
I,mn/H20841/H9252mI/H20856/H20855/H9021mI,PS/H20841B/H20841/H9021nI,PS/H20856/H20855/H9252nI/H20841, /H2084915/H20850
where Bis an arbitrary operator localized inside the PAW
spheres. This is due to the fact that, using Eq. /H2084913/H20850, inside the
PAW spheres, we have /H20858n/H20841/H9021nI,PS/H20856/H20855/H9252nI/H20841/H9023˜
i,/H9268A/H20856=/H20841/H9023˜
i,/H9268A/H20856and there-
fore /H20858/H9268/H20855/H9023˜
i,/H9268A/H20841/H9004/H20841/H9023˜
j,/H9268A/H20856=0.
Using Eq. /H2084914/H20850with the operator A=/H20841r/H20856/H20855r/H20841, one obtains the
PAW expression of the spin-density matrix that is composed
by three terms: /H9267/H92681,/H92682/H20849r/H20850=/H9267˜/H92681,/H92682/H20849r/H20850+/H20858I/H9267/H92681,/H926821,I/H20849r/H20850−/H20858I/H9267˜/H92681,/H926821,I/H20849r/H20850,
where the first term is calculated in real space while the other
two terms are calculated inside the PAW spheres on radialgrids. We have
16
/H9267˜/H92681,/H92682/H20849r/H20850=/H20858
i/H20855/H9023˜
i,/H92681A/H20841r/H20856/H20855r/H20841/H9023˜
i,/H92682A/H20856, /H2084916/H20850
/H9267/H92681,/H926821,I/H20849r/H20850=/H20858
mn/H9267mnI,/H92681,/H92682/H20855/H9021mI,AE/H20841r/H20856/H20855r/H20841/H9021nI,AE/H20856, /H2084917/H20850
/H9267˜/H92681,/H926821,I/H20849r/H20850=/H20858
mn/H9267mnI,/H92681,/H92682/H20855/H9021mI,PS/H20841r/H20856/H20855r/H20841/H9021nI,PS/H20856, /H2084918/H20850
where the partial occupations are defined as
/H9267mnI,/H92681,/H92682=/H20858
i/H20855/H9023˜
i,/H92681A/H20841/H9252mI/H20856/H20855/H9252nI/H20841/H9023˜
i,/H92682A/H20856. /H2084919/H20850
From the spin-density matrix, the charge and magnetization
densities are obtained readily as /H9267/H20849r/H20850=/H20858/H9268/H9267/H9268,/H9268/H20849r/H20850andmk/H20849r/H20850
=/H9262B/H20858/H92681,/H92682/H9267/H92681,/H92682/H20849r/H20850/H20849/H9268k/H20850/H92681,/H92682. With obvious generalizations
we can define also /H9267˜/H20849r/H20850andm˜k/H20849r/H20850on the real-space mesh,
and/H92671,I/H20849r/H20850,mk1,I/H20849r/H20850,/H9267˜1,I/H20849r/H20850, and m˜k1,I/H20849r/H20850inside the spheres.
As written, the charge calculated in the real-space mesh
integrating /H9267˜/H20849r/H20850on all space is not equal to the number ofelectrons. Its actual value depends on the pseudization pro-
cedure of the pseudo-partial-waves. In order to simplify thecalculation of the electrostatic energy, it is convenient
to introduce compensation charges equal to
/H9267ˆI/H20849r/H20850
=/H20858mn/H9267mnIQˆ
mnI/H20849r/H20850inside the PAW spheres /H20849/H9267mnI=/H20858/H9268/H9267mnI,/H9268,/H9268/H20850and
to/H9267ˆ/H20849r/H20850=/H20858I/H9267ˆI/H20849r−RI/H20850in the real-space mesh, such that inside
each sphere /H9267ˆI/H20849r/H20850has not only the same charge but also the
same multipole moments as /H92671,I/H20849r/H20850−/H9267˜1,I/H20849r/H20850. The augmenta-
tion functions Qˆ
mnI/H20849r/H20850are determined in such a way to satisfy
this constraint as explained for instance in Ref. 15. Using the
same augmentation functions we can define the compensa-
tion spin density: /H9267ˆ/H92681,/H926821,I/H20849r/H20850=/H20858mn/H9267mnI,/H92681,/H92682Qˆ
mnI/H20849r/H20850, and hence the
compensation magnetization density mˆk1,I/H20849r/H20850inside the
spheres and mˆk/H20849r/H20850=/H20858Imˆk1,I/H20849r−RI/H20850in the real-space mesh.
Using the relationship between all-electron and pseudo-
wave-functions, the definition of the operators in Eq. /H2084914/H20850,
and the decomposition of the electrostatic energy discussedin Ref. 15, the frozen-core energy /H20851Eq. /H2084911/H20850/H20852can be written in
terms of pseudo-wave-functions as a sum of three terms, thefirst calculated in real space and the other two calculated on
the radial grids inside the spheres: E
tot=E˜+E1−E˜1with16
E˜=/H20858
i,/H9268/H20855/H9023˜
i,/H9268A/H20841p2
2/H20841/H9274˜
i,/H9268A/H20856+Exc/H20851/H9267˜+/H9267ˆ+/H9267˜c,m˜+mˆ/H20852+EH/H20851/H9267˜+/H9267ˆ/H20852
+/H20885d3rV˜loc/H20849r/H20850/H20851/H9267˜/H20849r/H20850+/H9267ˆ/H20849r/H20850/H20852+UI,I, /H2084920/H20850
E˜1=/H20858
I,mn/H9267mnI/H20855/H9021mI,PS/H20841p2
2/H20841/H9021nI,PS/H20856
+/H20858
IExc/H20851/H9267˜1,I+/H9267ˆI+/H9267˜cI,m˜1,I+mˆ1,I/H20852+/H20858
IEH/H20851/H9267˜1,I+/H9267ˆI/H20852
+/H20858
I/H20885
/H9024Id3rv˜locI/H20849r/H20850/H20851/H9267˜1,I/H20849r/H20850+/H9267ˆI/H20849r/H20850/H20852, /H2084921/H20850
E1=/H20858
I,mn/H9267mnI/H20855/H9021mI,AE/H20841p2
2/H20841/H9021nI,AE/H20856+/H20858
IExc/H20851/H92671,I+/H9267cI,m1,I/H20852
+/H20858
IEH/H20851/H92671,I/H20852+/H20858
I/H20885
/H9024Id3rvlocI/H20849r/H20850/H92671,I/H20849r/H20850, /H2084922/H20850
where the core charges /H9267˜cand/H9267˜cIare defined as in Ref. 15
while we used the notation vlocI/H20849r/H20850forvH/H20849/H9267ZcI/H20850andv˜locI/H20849r/H20850for
vH/H20849/H9267˜ZcI/H20850;V˜loc/H20849r/H20850is equal to the sum /H20858Iv˜locI/H20849r−RI/H20850. The mini-
mization of this energy with respect to pseudo-wave-functions that obey to the orthogonality constraint
/H20858
/H9268/H20855/H9023˜i,/H9268/H20841S/H20841/H9023˜j,/H9268/H20856=/H9254i,j, where the overlap matrix Sis
S=1+ /H20858
I,mnqmnI/H20841/H9252mI/H20856/H20855/H9252nI/H20841/H20849 23/H20850
with qmnI=/H20855/H9021mI,AE/H20841/H9021nI,AE/H20856−/H20855/H9021mI,PS/H20841/H9021nI,PS/H20856, yields the nonrelativ-
istic PAW KS equations,ANDREA DAL CORSO PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-4/H20858
/H92682/H20875p2
2/H9254/H92681,/H92682+/H20885d3rV˜
LOC/H92681,/H92682/H20849r/H20850K˜/H20849r/H20850−/H9255iS/H9254/H92681,/H92682
+/H20858
I,mn/H20849DI,mn1,/H92681,/H92682−D˜
I,mn1,/H92681,/H92682/H20850/H20841/H9252mI/H20856/H20855/H9252nI/H20841/H20876/H20841/H9023˜i,/H92682/H20856=0 ,
/H2084924/H20850
where
DI,mn1,/H92681,/H92682=/H20855/H9021mI,AE/H20841p2
2+VLOCI,/H92681,/H92682/H20841/H9021nI,AE/H20856, /H2084925/H20850
D˜
I,mn1,/H92681,/H92682=/H20855/H9021mI,PS/H20841p2
2+V˜
LOCI,/H92681,/H92682/H20841/H9021nI,PS/H20856
+/H20885
/H9024Id3rQˆ
mnI/H20849r/H20850V˜
LOCI,/H92681,/H92682/H20849r/H20850. /H2084926/H20850
Notice that the coefficients of the nonlocal PP are spin de-
pendent in this PAW formulation because they are calculatedwith the spin-dependent partial occupations. In the US-PPscase these coefficients are calculated in the nonmagnetic iso-
lated atom and are spin independent. The function K
˜/H20849r/H20850is
defined in terms of the augmentation functions as
K˜/H20849r/H20850=/H20841r/H20856/H20855r/H20841+/H20858
I,mnQˆ
mnI/H20849r−RI/H20850/H20841/H9252mI/H20856/H20855/H9252nI/H20841, /H2084927/H20850
the potential VLOCI,/H92681,/H92682/H20849r/H20850is calculated with the local potential
vlocI/H20849r/H20850and the charge and magnetization densities /H92671,I/H20849r/H20850and
m1,I/H20849r/H20850,V˜
LOCI,/H92681,/H92682/H20849r/H20850with v˜locI/H20849r/H20850,/H9267˜1,I/H20849r/H20850+/H9267ˆI/H20849r/H20850andm˜1,I/H20849r/H20850
+mˆI/H20849r/H20850, and V˜
LOC/H92681,/H92682/H20849r/H20850with V˜loc/H20849r/H20850,/H9267˜/H20849r/H20850+/H9267ˆ/H20849r/H20850, and m˜/H20849r/H20850
+mˆ/H20849r/H20850.
Notice that, in practical applications, the present scheme
is improved upon by generating the PAW data sets startingfrom the scalar relativistic equations in the isolated atom andtherefore partially including relativistic effects.
IV. RSDFT WITHIN THE PAW METHOD
In the PAW approach to RSDFT, we want to rewrite the
frozen-core total energy of a gas of Ninteracting electrons in
the field of fixed ions at positions RI/H20851Eq. /H208493/H20850/H20852as a functional
of the four-component pseudo-wave-functions /H20841/H9023˜i,/H9257/H20856.I nt h e
FR case, the partial waves and projectors are four-componentspinors and the PAW mapping from pseudo to all-electronwave functions is
/H20841/H9023
i,/H9257/H20856=/H20841/H9023˜i,/H9257/H20856+/H20858
I,m/H20858
/H92571/H20851/H20841/H9021m,/H9257I,AE/H20856−/H20841/H9021m,/H9257I,PS/H20856/H20852/H20855/H9252m,/H92571I/H20841/H9023˜i,/H92571/H20856.
/H2084928/H20850
The all-electron partial waves are a product of radial func-
tions, solutions of the atomic radial Dirac-type KS equations,and spin-angle functions dependent on the angular and spinvariables /H20849see Appendix B /H20850. Therefore, the index mis a
shorthand notation for m=/H20853
/H9270,l,j,mj/H20854that, in addition to /H9270
andldefined as in the nonrelativistic case /H20849lis the orbital
angular momentum of the large component /H20850, contains also jthe total angular momentum and mjits projection on a quan-
tization axis. As in the nonrelativistic case, the pseudo- andall-electron partial waves coincide outside a given radiuswhile inside the PAW spheres, the pseudo-partial-waves areobtained by a pseudization procedure starting from the all-electron partial waves /H20849see Appendix A /H20850. The projectors are
constructed after the pseudo-partial-waves with one of theusual recipes.
14,24The difficulties encountered in the FR case
have been discussed only for NC-PPs in Ref. 39. The PAW
case is simpler to deal with because there is no NC con-straint. There are several different options and a few of themare compared in Appendix A. When the partial waves andprojectors are constructed as described in Appendix A, thefollowing completeness relationship holds inside the spheresfor sufficiently large N
/H9280andlmax:
/H20858
m/H20841/H9021m,/H92571I,PS/H20856/H20855/H9252m,/H92572I/H20841=/H9254/H92571,/H92572/H2084929/H20850
when applied to any pseudo-wave-function. Consequently
also the PAW mapping of the all-electron operators intopseudooperators Eq. /H2084914/H20850can be generalized in an obvious
manner.
For instance, applying this PAW mapping to set the den-
sity matrix, we find its expression in terms of the pseudo-wave-functions.
/H9267/H92571,/H92572/H20849r/H20850is the sum of three terms:
/H9267/H92571,/H92572/H20849r/H20850=/H9267˜/H92571,/H92572/H20849r/H20850+/H20858I/H9267/H92571,/H925721,I/H20849r/H20850−/H20858I/H9267˜/H92571,/H925721,I/H20849r/H20850, where
/H9267˜/H92571,/H92572/H20849r/H20850=/H20858
i/H20855/H9023˜i,/H92571/H20841r/H20856/H20855r/H20841/H9023˜i,/H92572/H20856, /H2084930/H20850
/H9267/H92571,/H925721,I/H20849r/H20850=/H20858
mn/H9267mnI/H20855/H9021m,/H92571I,AE/H20841r/H20856/H20855r/H20841/H9021n,/H92572I,AE/H20856, /H2084931/H20850
/H9267/H92571,/H925721,I/H20849r/H20850=/H20858
mn/H9267mnI/H20855/H9021m,/H92571I,PS/H20841r/H20856/H20855r/H20841/H9021n,/H92572I,PS/H20856, /H2084932/H20850
where the partial occupations are
/H9267mnI=/H20858
i,/H92571,/H92572/H20855/H9023˜i,/H92571/H20841/H9252m,/H92571I/H20856/H20855/H9252n,/H92572I/H20841/H9023˜i,/H92572/H20856. /H2084933/H20850
From the density matrix, the charge and magnetization den-
sities are obtained readily as /H9267/H20849r/H20850=/H20858/H9257/H9267/H9257,/H9257/H20849r/H20850andmk/H20849r/H20850
=/H9262B/H20858/H92571,/H92572/H9267/H92571,/H92572/H20849r/H20850/H20849/H9252/H9018k/H20850/H92571,/H92572.
As in the nonrelativistic case, we can add to the spin
density calculated in real space a compensation spin density
and write: /H9267˜/H92571,/H92572/H20849r/H20850+/H9267ˆ/H92571,/H92572/H20849r/H20850=/H20858i,/H92573,/H92574/H20855/H9023˜i,/H92573/H20841K˜
/H92573,/H92574/H92571,/H92572/H20849r/H20850/H20841/H9023˜i,/H92574/H20856,
where the functions K˜
/H92573,/H92574/H92571,/H92572/H20849r/H20850and the augmentation functions
Qˆ
mn,/H92571,/H92572I/H20849r/H20850which define the compensation spin density in-
side the spheres, /H20851/H9267ˆ/H92571,/H925721,I/H20849r/H20850=/H20858mn/H9267mnIQˆ
mn,/H92571,/H92572I/H20849r/H20850/H20852are related
by
K˜
/H92573,/H92574/H92571,/H92572/H20849r/H20850=/H20841r/H20856/H20855r/H20841/H9254/H92571,/H92573/H9254/H92572,/H92574+/H20858
I,mnQˆ
mn,/H92571,/H92572I/H20849r−RI/H20850/H20841/H9252m,/H92573I/H20856
/H11003/H20855/H9252n,/H92574I/H20841. /H2084934/H20850
The form of Qˆ
mn,/H92571,/H92572I/H20849r/H20850is, to a certain extent, arbitrary. As
long as the compensation charges /H9267ˆI/H20849r/H20850PROJECTOR AUGMENTED-WA VE METHOD: APPLICATION … PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-5=/H20858mn/H20858/H9257/H9267mnIQˆ
mn,/H9257,/H9257I/H20849r/H20850have the same multipole moments of
/H92671,I/H20849r/H20850−/H9267˜1,I/H20849r/H20850,Qˆ
mn,/H92571,/H92572I/H20849r/H20850can be chosen so as to make
K˜
/H92573,/H92574/H92571,/H92572/H20849r/H20850as smooth as possible. The simplest choice
Qmn,/H92571,/H92572I/H20849r/H20850=/H20855/H9021m,/H92571I,AE/H20841r/H20856/H20855r/H20841/H9021n,/H92572I,AE/H20856−/H20855/H9021m,/H92571I,PS/H20841r/H20856/H20855r/H20841/H9021n,/H92572I,PS/H20856leads
to functions K˜
/H92573,/H92574/H92571,/H92572/H20849r/H20850that are too hard to expand in-plane
waves and a pseudization method, like the one adopted in
Ref. 15or in Ref. 40, is necessary. In order not to slow down
to much the exposition, we discuss the details of our ap-proach in Appendix B. Here we just anticipate that the func-
tions Q
mn,/H92571,/H92572I/H20849r/H20850are replaced with smooth functions
Qˆ
mn,/H92571,/H92572I/H20849r/H20850whose Fourier expansion converges rapidly in-
plane waves.
At this point, the charge density is separated as in the
nonrelativistic case. Hence the Hartree and exchange andcorrelation energies can be calculated in the same way withterms evaluated on the real-space mesh and terms calculatedinside the spheres.
15Skipping the derivation that from this
point onward becomes identical to that reported in Ref. 15,
we write the FR frozen-core total energy as Etot=E˜+E1−E˜1
with
E˜=/H20858
i,/H92571,/H92572/H20855/H9023˜i,/H92571/H20841TD/H92571,/H92572/H20841/H9023˜i,/H92572/H20856+Exc/H20851/H9267˜+/H9267ˆ+/H9267˜c,m˜+mˆ/H20852
+EH/H20851/H9267˜+/H9267ˆ/H20852+/H20885d3rV˜loc/H20849r/H20850/H20851/H9267˜/H20849r/H20850+/H9267ˆ/H20849r/H20850/H20852+UI,I,/H2084935/H20850
E˜1=/H20858
I,mn,/H92571,/H92572/H9267mnI/H20855/H9021m,/H92571I,PS/H20841TD/H92571,/H92572/H20841/H9021n,/H92572I,PS/H20856+/H20858
IExc/H20851/H9267˜1,I+/H9267ˆI
+/H9267˜cI,m˜1,I+mˆ1,I/H20852+/H20858
IEH/H20851/H9267˜1,I+/H9267ˆI/H20852+/H20858
I/H20885
/H9024Id3rv˜locI/H20849r/H20850
/H11003/H20851/H9267˜1,I/H20849r/H20850+/H9267ˆI/H20849r/H20850/H20852, /H2084936/H20850
E1=/H20858
I,mn,/H92571,/H92572/H9267mnI/H20855/H9021m,/H92571I,AE/H20841TD/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856+/H20858
IExc/H20851/H92671,I+/H9267cI,m1,I/H20852
+/H20858
IEH/H20851/H92671,I/H20852+/H20858
I/H20885
/H9024Id3rvlocI/H20849r/H20850/H92671,I/H20849r/H20850, /H2084937/H20850
where the core charges /H9267˜cand/H9267˜cIand the local potentials
vlocI/H20849r/H20850,v˜locI/H20849r/H20850, and V˜loc/H20849r/H20850are defined as in the nonrelativis-
tic case.
The minimization of this energy with respect to pseudo-
wave-functions that obey to the orthogonality constraint
/H20858/H92571,/H92572/H20855/H9023˜i,/H92571/H20841S/H92571,/H92572/H20841/H9023˜j,/H92572/H20856=/H9254i,j, where the overlap matrix Sis
S/H92571,/H92572=/H9254/H92571,/H92572+/H20858
I,mnqmnI/H20841/H9252m,/H92571I/H20856/H20855/H9252n,/H92572I/H20841/H20849 38/H20850
with qmnI=/H20858/H9257/H20849/H20855/H9021m,/H9257I,AE/H20841/H9021n,/H9257I,AE/H20856−/H20855/H9021m,/H9257I,PS/H20841/H9021n,/H9257I,PS/H20856/H20850, yields the PAW
Dirac-type KS equations,/H20858
/H92572/H20875TD/H92571,/H92572+/H20858
/H92573,/H92574/H20885d3rV˜
LOC/H92573,/H92574/H20849r/H20850K˜
/H92571,/H92572/H92573,/H92574/H20849r/H20850−/H9255iS/H92571,/H92572
+/H20858
I,mn/H20849DI,mn1−D˜
I,mn1/H20850/H20841/H9252m,/H92571I/H20856/H20855/H9252n,/H92572I/H20841/H20876/H20841/H9023˜i,/H92572/H20856=0 ,
/H2084939/H20850
where
DI,mn1=/H20858
/H92571,/H92572/H20855/H9021m,/H92571I,AE/H20841TD/H92571,/H92572+VLOCI,/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856, /H2084940/H20850
D˜
I,mn1=/H20858
/H92571,/H92572/H20855/H9021m,/H92571I,PS/H20841TD/H92571,/H92572+V˜
LOCI,/H92571,/H92572/H20841/H9021n,/H92572I,PS/H20856
+/H20858
/H92571,/H92572/H20885
/H9024Id3rQˆ
mn,/H92571,/H92572I/H20849r/H20850V˜
LOCI,/H92571,/H92572/H20849r/H20850. /H2084941/H20850
The potential VLOCI,/H92571,/H92572/H20849r/H20850is calculated with the local potential
vlocI/H20849r/H20850and the charge and magnetization densities /H92671,I/H20849r/H20850and
m1,I/H20849r/H20850,V˜
LOCI,/H92571,/H92572/H20849r/H20850with v˜locI/H20849r/H20850,/H9267˜1,I/H20849r/H20850+/H9267ˆI/H20849r/H20850andm˜1,I/H20849r/H20850
+mˆI/H20849r/H20850, and V˜
LOC/H92571,/H92572/H20849r/H20850with V˜loc/H20849r/H20850,/H9267˜/H20849r/H20850+/H9267ˆ/H20849r/H20850, and m˜/H20849r/H20850
+mˆ/H20849r/H20850.
Solving these equations, we get the FR PAW pseudo-
wave-functions. From them the all-electron wave functions/H20851Eq. /H2084928/H20850/H20852, the relativistic total energy /H20851Eqs. /H2084935/H20850–/H2084937/H20850/H20852and
the RSDFT charge and magnetization densities /H20851Eqs.
/H2084930/H20850–/H2084933/H20850/H20852. Ideally, these quantities are as accurate as the
all-electron frozen-core results but, in practice, they are af-fected by the PAW data sets TEs. For instance, in atoms,PAW data sets can recover the frozen-core energies and theenergy levels of atomic configurations close to the referenceconfiguration with an accuracy of a few millirydberg. Ashighlighted in the next section, Eqs. /H2084939/H20850–/H2084941/H20850can be trans-
formed into Pauli-type equations making errors comparableto the TEs or of order 1 /c
2. The TEs can be reduced by
increasing the number of partial waves and projectors, butthe intrinsic error of the Pauli-type equations cannot besmaller than 1 /c
2, so in the future the availability of increas-
ingly larger computational resources might make the imple-mentation and the solution of Eqs. /H2084939/H20850–/H2084941/H20850attractive but
for our present purposes the Pauli-type equations are muchless demanding computationally and, at the same time, ofsimilar accuracy.
V. FULLY RELATIVISTIC PAW VIA PAULI-TYPE
EQUATIONS
A. Kinetic energy
In this section we derive PAW Pauli-type KS equations
that have an accuracy comparable to the PAW Dirac-type KSequations. We start by observing that, outside the PAWspheres, Pauli-type equations like those presented in Sec. II
are a good approximation because in that region relativisticeffects are small. Pauli-type equations introduce an error inthe kinetic energy of order /H20849
v/H11032/c/H208502and, outside the spheres,
v/H11032is on the order of 1 a.u., often even lower about 0.5 a.u.,
so that /H20849v/H11032/c/H208502/H110155–1/H1100310−5, and this error is independentANDREA DAL CORSO PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-6from the nuclear charge ZI. This observation allows us to
write the relativistic kinetic energy in the form
Ekin=/H20858
i,/H92571,/H92572/H20855/H9023i,/H92571/H20841TD/H92571,/H92572/H20841/H9023i,/H92572/H20856/H9024+/H20858
i,/H92681/H20855/H9023i,/H92681A/H20841p2
2/H20841/H9023i,/H92681A/H20856/H9024¯,
/H2084942/H20850
where the symbols /H9024and/H9024¯indicate that the expectation
values are to be calculated inside and outside the PAWspheres, respectively. Inside the spheres, we apply the PAWtransformation of the operators while, outside the spheres,we simply replace the all-electron with pseudo-wave-functions because in that region they coincide. We obtain
E
kin=/H20858
i,/H92571,/H92572/H20855/H9023˜i,/H92571/H20841TD/H92571,/H92572/H20841/H9023˜i,/H92572/H20856/H9024+/H20858
i,/H92681/H20855/H9023˜
i,/H92681A/H20841p2
2/H20841/H9023˜
i,/H92681A/H20856/H9024¯
+/H20858
I,mn,/H92571,/H92572/H9267mnI/H20851/H20855/H9021m,/H92571I,AE/H20841TD/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856/H9024
−/H20855/H9021m,/H92571I,PS/H20841TD/H92571,/H92572/H20841/H9021n,/H92572I,PS/H20856/H9024/H20852. /H2084943/H20850
Now inside the spheres we can add a vanishing operator. The
PAW method has the freedom to add a term written as14
B/H92571,/H92572−/H20858
I,mn,/H92573,/H92574/H20841/H9252m,/H92571I/H20856/H20855/H9021m,/H92573I,PS/H20841B/H92573,/H92574/H20841/H9021n,/H92574I,PS/H20856/H9024/H20855/H9252n,/H92572I/H20841.
/H2084944/H20850
For a complete set of pseudo-partial-waves and projectors,
the expectation value of this term between pseudo-wave-functions vanishes identically whereas in practice it is com-parable to the TE. Specifically, we take B=A−T
D, where Ais
a diagonal 4 /H110034 matrix that has the operator p2/2 in the first
two diagonal elements and is zero elsewhere. With thischoice, Eq. /H2084943/H20850becomes
E
kin=/H20858
i,/H92681/H20855/H9023˜
i,/H92681A/H20841p2
2/H20841/H9023˜
i,/H92681A/H20856
+/H20858
I,mn/H9267mnI/H20875/H20858
/H92571/H92572/H20855/H9021m,/H92571I,AE/H20841TD/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856/H9024
−/H20858
/H92681/H20855/H9021m,/H92681I,PS,A/H20841p2
2/H20841/H9021n,/H92681I,PS,A/H20856/H9024/H20876. /H2084945/H20850
Equation /H2084945/H20850shows that we can get the FR kinetic energy,
applyingp2
2to the pseudo-wave-functions. Outside the
spheres because the Pauli-type equations hold /H20851with an error
of order /H20849v/H11032/c/H208502/H20852, inside the spheres because the difference
between the Dirac kinetic energy and the nonrelativistic ki-netic energy is compensated by the radial grid terms, whereonly the all-electron kinetic energy is evaluated using theDirac operator. Using Pauli-type equations of higher orderoutside the spheres, for instance including mass-velocity,Darwin, and spin-orbit corrections,
20we could reduce the
errors in that region because we would have terms correct upto order /H20849
v/H11032/c/H208502, but in the other regions the accuracy is
lower, because of the TEs and of the errors that we make inthe following, so several other modifications are necessary tobring the overall accuracy to errors of order /H20849
v/H11032/c/H208503.Notice that the kinetic energy in Eq. /H2084945/H20850is now depen-
dent on the size of the PAW spheres /H9024, whereas the FR
expression was independent from /H9024. Specifically, the terms
/H20858/H92571/H92572/H20855/H9021m,/H92571I,AE/H20841TD/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856diverge for unbound partial waves,
a divergence that, in the FR case, is compensated by the
terms /H20858/H92571/H92572/H20855/H9021m,/H92571I,PS/H20841TD/H92571,/H92572/H20841/H9021n,/H92572I,PS/H20856.I nE q . /H2084945/H20850the divergence is
compensated only for the large components not for the small
ones. Now the integrals are finite because they are limited tothe PAW spheres but at the same time they vary with thevolume of the PAW spheres. This is unavoidable because thespheres introduce a separation between the region where theDirac kinetic energy is dealt with exactly /H20849modulo the TEs /H20850
and the region where there is an error of order /H20849
v/H11032/c/H208502.I ft h e
volumes of these two regions change, the kinetic energychanges as well but still with differences of order /H20849
v/H11032/c/H208502.
Notice also that to obtain Eq. /H2084945/H20850, we have not changed the
FR PAW mapping between all-electron and pseudo-wave-functions, or changed the completeness relationship. Wehave only used the freedom of the PAW method to add anoperator localized inside the spheres in the real-space meshand to subtract the same operator inside the spheres.
B. Removal of the small components of the
pseudo-wave-functions
The small components of the pseudo-wave-functions,
which are quantities of order 1 /c, cannot be obtained any
longer through the minimization of the total energy that hasnow errors of order 1 /c
2. These small components are still
present in the partial occupations /H20851Eq. /H2084933/H20850/H20852, in the density
matrix, and in the orthogonality constraints. To remove themeverywhere, we have to make approximations that introduceerrors comparable to the TEs or of order 1 /c
2, depending on
which is the largest. The method illustrated in Appendix A
introduces errors of order v14/c2, where v14is never larger
than about 10 a.u.. This error can be reduced or even can-celed /H20849see A /H20850if it will become larger that the TEs but so far
it is a small error and we have kept it in the formalism.
In general, the shapes of the pseudo-wave-functions in-
side the PAW spheres are arbitrary. They must match con-tinuously the all-electron wave functions at the border of thePAW spheres but they have not the oscillations of the all-electron wave functions. At the border of the spheres, thesmall components of the all-electron wave functions are oforder /H20849
v/H11032/c/H20850independently from the nuclear charge ZIand
the small components of the pseudo-wave-functions can bekept everywhere of order
v1/c/H20849here v1varies with the state
and is also position dependent but usually is smaller than 1or 2 a.u. /H20850. In Appendix A, we show that the small compo-
nents of the projectors can be defined so as to be of order
v13/chence the product /H20858/H9257/H20855/H9023˜i,/H9257/H20841/H9252m,/H9257I/H20856in Eq. /H2084933/H20850is equal to
/H20858/H9268/H20855/H9023˜
i,/H9268A/H20841/H9252m,/H9268I,A/H20856up to a term of order v14/c2that can be ne-
glected thus removing from the partial occupations the smallcomponents of the pseudo-wave-functions.
Let us now consider the density matrix /H20851Eqs. /H2084930/H20850–/H2084933/H20850/H20852.
To remove the small components of the pseudo-wave-functions from its expression, we use again the possibilityoffered by the PAW method to add a vanishing term insidethe spheres /H20851see Eq. /H2084944/H20850/H20852and take as B
/H92571,/H92572a diagonal op-PROJECTOR AUGMENTED-WA VE METHOD: APPLICATION … PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-7erator with − /H20841r/H20856/H20855r/H20841in the third and fourth diagonal elements
and zero elsewhere. After a few steps similar to those illus-trated for the kinetic energy we get
/H9267/H92681,/H92682/H20849r/H20850=/H20858
i/H20855/H9023˜
i,/H92681A/H20841r/H20856/H20855r/H20841/H9023˜
i,/H92682A/H20856+/H20858
I,mn/H9267mnI/H20851/H20855/H9021m,/H92681I,AE/H20841r/H20856
/H11003/H20855r/H20841/H9021n,/H92682I,AE/H20856/H9024−/H20855/H9021m,/H92681I,PS,A/H20841r/H20856/H20855r/H20841/H9021n,/H92682I,PS,A/H20856/H9024/H20852/H20849 46/H20850
for the first upper 2 /H110032 block and
/H9267/H92571,/H92572/H20849r/H20850=/H20858
I,mn/H9267mnI/H20855/H9021m,/H92571I,AE/H20841r/H20856/H20855r/H20841/H9021n,/H92572I,AE/H20856/H9024 /H2084947/H20850
for the other blocks. Equation /H2084946/H20850tells us that the large
components of the pseudo-wave-functions suffice to calcu-late the density matrix on the real-space mesh as far as onlythe large components of the pseudo-partial-waves are used
inside the spheres to get
/H9267˜/H92571,/H925721,I. Only the all-electron-density
matrix inside the spheres /H20849/H9267/H92571,/H925721,I/H20850needs both the large and the
small components of the all-electron partial waves. In Eq.
/H2084946/H20850, we have neglected the small components of the pseudo-
wave-functions outside the spheres with an error of order/H20849
v/H11032/c/H208502. For example, in a Pt atom the charge density due to
these terms is about 10−5electrons, at least two orders of
magnitude smaller that the charge due to the small compo-nents of the all-electron valence wave functions.
The pseudo-wave-functions obey to orthogonality con-
straints /H20851Eq. /H2084938/H20850/H20852, which depend on the both the large and
the small components. We can now transform this constraintby adding, inside the spheres a vanishing term as in Eq. /H2084944/H20850
with B
/H92571,/H92572equal to a diagonal operator with −1 in the third
and fourth diagonal elements and zero elsewhere. By thismethod, we obtain the first upper 2 /H110032 block of the overlap
matrix,
S
/H92681,/H92682=/H9254/H92681,/H92682+/H20858
I,mn/H20841/H9252m,/H92681I,A/H20856/H20875/H20858
/H92571/H20855/H9021m,/H92571I,AE/H20841/H9021n,/H92571I,AE/H20856/H9024
−/H20858
/H92683/H20855/H9021m,/H92683I,PS,A/H20841/H9021n,/H92683I,PS,A/H20856/H9024/H20876/H20855/H9252n,/H92682I,A/H20841. /H2084948/H20850
The other blocks contribute to the orthogonality constraint
with terms of order v14/c2or smaller and can be neglected.
Equation /H2084948/H20850can be obtained also starting from the spin-
density, Eq. /H2084946/H20850, and requiring the integral of the charge
density to be equal to the number of electrons.
C. Total energy and PAW Pauli-type KS equations
We can now define the charge densities inside the spheres,
/H92671,I/H20849r/H20850as in the previous section, and /H9267˜1,I/H20849r/H20850
=/H20858mn/H20858/H9268/H9267mnI/H20855/H9021m,/H9268I,PS,A/H20841r/H20856/H20855r/H20841/H9021n,/H9268I,PS,A/H20856. Introducing a compensa-
tion charge /H9267ˆI/H20849r/H20850with the same multipole moments of
/H92671,I/H20849r/H20850−/H9267˜1,I/H20849r/H20850as in the FR case, we can proceed to the sepa-
ration of the Hartree and exchange and correlation energiesinto terms calculated in the real-space mesh and terms cal-culated inside the spheres. We get the frozen-core total en-
ergy in the form E
tot=E˜+E1−E˜1, where E1is given by Eq.
/H2084937/H20850whileE˜=/H20858
i,/H9268/H20855/H9023˜
i,/H9268A/H20841p2
2/H20841/H9023˜
i,/H9268A/H20856+Exc/H20851/H9267˜+/H9267ˆ+/H9267˜c,m˜+mˆ/H20852+EH/H20851/H9267˜+/H9267ˆ/H20852
+/H20885d3rV˜loc/H20849r/H20850/H20851/H9267˜/H20849r/H20850+/H9267ˆ/H20849r/H20850/H20852+UI,I, /H2084949/H20850
E˜1=/H20858
I,mn,/H9268/H9267mnI/H20855/H9021m,/H9268I,PS,A/H20841p2
2/H20841/H9021n,/H9268I,PS,A/H20856+/H20858
IExc/H20851/H9267˜1,I+/H9267ˆI+/H9267˜cI,m˜I
+mˆI/H20852+/H20858
IEH/H20851/H9267˜1,I+/H9267ˆI/H20852+/H20858
I/H20885
/H9024Id3rv˜locI/H20849r/H20850/H20851/H9267˜1,I/H20849r/H20850
+/H9267ˆI/H20849r/H20850/H20852. /H2084950/H20850
The minimization of the total energy with respect to wave
functions that obey the orthogonality constraints /H20851Eq. /H2084948/H20850/H20852
leads to the following expression for the Pauli-type PAW KSequation:
/H20858
/H92682/H20875p2
2/H9254/H92681,/H92682+/H20858
/H92571,/H92572/H20885d3rV˜
LOC/H92571,/H92572/H20849r/H20850K˜/H20849r/H20850/H92681,/H92682/H92571,/H92572−/H9255iS/H92681,/H92682
+/H20858
I,mn/H20849DI,mn1−D˜
I,mn1/H20850/H20841/H9252m,/H92681I,A/H20856/H20855/H9252n,/H92682I,A/H20841/H20876/H20841/H9023˜
i,/H92682A/H20856=0 , /H2084951/H20850
where
DI,mn1=/H20858
/H92571,/H92572/H20855/H9021m,/H92571I,AE/H20841TD/H92571,/H92572+VLOCI,/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856, /H2084952/H20850
D˜
I,mn1=/H20858
/H92681,/H92682/H20855/H9021m,/H92681I,PS,A/H20841p2
2/H9254/H92681,/H92682+V˜
LOCI,/H92681,/H92682/H20841/H9021n,/H92682I,PS,A/H20856
+/H20858
/H92571,/H92572/H20885
/H9024Id3rQˆ
mn,/H92571,/H92572I/H20849r/H20850V˜
LOCI,/H92571,/H92572/H20849r/H20850. /H2084953/H20850
These Pauli-type KS equations resemble the FR US-PPs
equations and we can exploit this similarity to write them asthe nonrelativistic PAW equations, following the approach ofRefs. 27and28. The projectors depend on spin through the
spin-angle functions; they can be written in terms of spheri-cal harmonics, transferring the spin index to the nonlocal PPcoefficients as shown in Ref. 27for FR US-PPs. The sums
over
/H9257that appear in Eqs. /H2084951/H20850–/H2084953/H20850can be carried out ana-
lytically /H20849see Appendix B /H20850so that the angular and spin de-
pendence can be dealt with as in the US-PPs case with minor
changes in the calculation of DI,mn1. After these transforma-
tions, the Hamiltonian and the total energy become formallyidentical to the nonrelativistic ones, and the Hellmann-Feynman forces as well as density functional perturbationtheory can be written as in Refs. 17and28.
Kleinman’s
25observation that a NC-PP tailored on the
solutions of a Dirac-type equation could yield the FR resultswith errors of order 1 /c
2is implicit in our method. However,
in the NC case there is no well established procedure to dealwith the charge density of the small components of the va-lence all-electron wave functions. If the small componentsare neglected, the large components are not normalized. Forinstance, in a Pt atom, the valence charge of the small com-ponents of the 5 d
3/2and 5 d5/2states, which are occupied byANDREA DAL CORSO PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-8ten electrons, is about 2 /H1100310−3electrons so the missing norm
of each state is about 2 /H1100310−4. This error is small but has to
be corrected in some way. In Ref. 27, we normalized the
large components of the bound all-electron partial waves be-fore applying the pseudization procedure obtaining FR US-PPs with good transferability properties. Alternatively, theradial Dirac-type equation can be inverted accounting for thepresence of the small components, as suggested in Ref. 39.
Within the PAW formalism, the correct charge is recoveredfrom the all-electron partial waves that are calculated ex-actly. In the Pauli-type case, we still have an error of order1/c
2due the missing norm of the small components of the
pseudo-wave-functions outside the PAW spheres. In the cal-culations presented below this error has been compensatednormalizing both the large and the small components of thebound all-electron partial waves. The missing norm is how-ever small. For instance, in Pt, it turns out to be about 10
−6
per state.
VI. APPLICATIONS
The PAW Pauli-type equations /H20851Eq. /H2084951/H20850/H20852have been
implemented in the QUANTUM ESPRESSO package,41and here
we present a few results obtained with this feature. Thepresent method, specialized to FR US-PPs, was applied inRefs. 27–32. That implementation was the starting point of
the present PAW extension /H20849see Appendix B /H20850. As applica-
tions, we compare the PAW electronic band structures offcc-Pt and fcc-Au and of the ferromagnetic bcc-Fe with pub-lished results. The local-density approximation /H20849LDA /H20850/H20849Ref.
42/H20850is used for the exchange and correlation energies infcc-Pt and fcc-Au and the spin-polarized generalized gradi-
ent approximation of Perdew-Burke and Ernzerhof /H20849PBE /H20850
/H20849Ref. 43/H20850is used for bcc-Fe. No relativistic correction has
been included in the exchange and correlation functionals.The PAW data sets of Fe, Pt, and Au are described in thenote.
44For each element, we considered two PAW data sets,
one including semicore states /H208493sand 3 pfor Fe, 5 sand 5 p
for Pt and Au /H20850and one without semicore states. The kinetic-
energy cutoffs for the wave functions of Fe /H20849Pt/H20850/H20851Au/H20852are 45
Ry /H2084940 Ry /H20850/H2085135 Ry /H20852for the data sets without semicore states
and 75 Ry /H2084960 Ry /H20850/H2085180 Ry /H20852for data sets with semicore states.
The cufoffs for the charge density are always 300 Ry exceptfor Fe with semicore states for which we used 400 Ry. Inbcc-Fe /H20849fcc-Pt and fcc-Au /H20850, we sampled the BZ by a 24
/H1100324/H1100324 /H2084916/H1100316/H1100316/H20850uniform k-point grid and a smear-
ing parameter
/H9268=0.02 Ry.
We start by considering the TE of the FR PAW data sets.
We report in Tables I–IIIthe energy levels of a few atomic
configurations and the difference between their total energyand the energy of the reference configuration.
44The magni-
tude of the TEs, indicated in parenthesis, is of a few milliry-dbergs for the data sets without semicore states and of frac-tions of millirydberg for those with semicore states. Thelarger TEs found in the former case can be attributed to thefrozen-core approximation because they are similar to theerrors reported in square brackets that are obtained as thedifference between the all-electron eigenvalues and the ei-genvalues of an all-electron calculation in which the corestates are kept frozen in the reference configuration. Finally,notice that the values of the TEs found in Tables I–IIIare
similar to the typical TEs of SR data sets and are larger than1/c
2.TABLE I. Comparison of the PBE energy eigenvalues and of the total energies calculated by FR PAW
data sets and by solving an all-electron atomic Dirac-type equation for a few atomic configurations of Fe. Thereported eigenvalues are the all-electron ones /H20849in Ry and without the minus sign /H20850. In parenthesis the differ-
ence /H20849in mRy /H20850between the all-electron and the PP results. The latter have been obtained by the FR PAW
Pauli-type method. The PP eigenvalue is the algebraic sum of the all-electron eigenvalue and the number inparenthesis. The total-energy difference /H20849/H9004Ein Ry /H20850is given with respect to the reference configuration. In
square bracket we report the difference of the eigenvalues found by solving the all-electron Dirac-Kohn-Sham equations and the frozen-core all-electron Dirac-Kohn-Sham equations with the same sign convention.
Fe/H208493s
23p6/H20850 3d3/243d5/234s1/2 3d3/243d5/224s1/24p1/2 3d3/243d5/224s1/2
3d3/2 0.2848 /H20849−0.4 /H20850 0.6774 /H20849−0.2 /H20850 1.1772 /H20849−0.3 /H20850
3d5/2 0.2748 /H20849−0.4 /H20850 0.6662 /H20849−0.2 /H20850 1.1659 /H20849−0.3 /H20850
4s1/2 0.3125 /H208490.4/H20850 0.4655 /H208490.0/H20850 0.9057 /H208490.1/H20850
4p1/2 0.0652 /H208490.1/H20850 0.1571 /H208490.0/H20850 0.5431 /H208490.0/H20850
4p3/2 0.0630 /H208490.1/H20850 0.1533 /H208490.0/H20850 0.5372 /H208490.0/H20850
/H9004E −0.0523 /H20849−0.6 /H20850 0.2990 /H208490.0/H20850 0.6413 /H208490.0/H20850
Fe 3 d3/243d5/234s1/2 3d3/243d5/224s1/24p1/2 3d3/243d5/224s1/2
3d3/2 0.2848 /H20849−3.6 /H20850/H20851−3.4 /H20852 0.6774 /H208494.4/H20850/H208514.6/H20852 1.1772 /H208494.0/H20850/H208514.1/H20852
3d5/2 0.2748 /H20849−3.5 /H20850/H20851−3.4 /H20852 0.6662 /H208494.4/H20850/H208514.5/H20852 1.1659 /H208494.0/H20850/H208514.1/H20852
4s1/2 0.3125 /H208491.9/H20850/H208511.7/H20852 0.4655 /H20849−0.2 /H20850/H20851−0.1 /H20852 0.9057 /H208490.0/H20850/H208510.0/H20852
4p1/2 0.0652 /H208490.9/H20850/H208510.8/H20852 0.1571 /H20849−0.2 /H20850/H20851−0.2 /H20852 0.5431 /H208490.0/H20850/H208510.0/H20852
4p3/2 0.0630 /H208490.9/H20850/H208510.8/H20852 0.1533 /H20849−0.2 /H20850/H20851−0.2 /H20852 0.5372 /H208490.0/H20850/H20851−0.1 /H20852
/H9004E 0.0156 /H20849−2.3 /H20850/H20851−2.1 /H20852 0.3356 /H20849−0.6 /H20850/H20851−0.6 /H20852 0.6779 /H208490.5/H20850/H208510.5/H20852PROJECTOR AUGMENTED-WA VE METHOD: APPLICATION … PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-9The theoretical lattice constants and bulk moduli of the
three metals considered in this work are reported in Table IV
and compared with experiment and with previous FRcalculations.
11,27,45In Ref. 11the all-electron Dirac-type
equations are solved, in Ref. 45the all-electron Pauli-type
equations are solved and spin orbit is included while in Ref.27FR US-PPs are used. In fcc-Pt and fcc-Au, PAW data sets
with semicore states give theoretical lattice constants thatagree to better than 0.01 a.u. with the all-electron results ofRef. 11. Slightly larger lattice constants result instead ne-
glecting semicore states. The influence of semicore states onthe bulk moduli is weaker, and there is also a negligibleeffect on the structural and magnetic properties of bcc-Fe.For comparison, in Table IVwe report the structural proper-
ties obtained with SR PAW data sets constructed with thesame parameters. As in Ref. 27, we still find that the inclu-
sion of spin-orbit coupling has only a marginal effect on thelattice constants, smaller than that of semicore states, but has
a detectable effect on the bulk moduli. Surprisingly, the sameapplies also to bcc-Fe, where spin-orbit effects are expectedto be small. The reason for this is presently unclear and willrequire further investigations. We notice only that the sign ofthe spin-orbit effects on the bulk modulus is material depen-dent, with a decrease in bcc-Fe and fcc-Pt and an increase infcc-Au.
In Table V, we report the electronic band structures of
fcc-Pt and fcc-Au in a few high-symmetry points of the Bril-louin zone and compare the results with those obtained bythe FR US-PPs in Ref. 27and by the solution of the Dirac-
type KS equations in Ref. 11. All results have been obtained
at the same lattice constant used in Ref. 11. In Ref. 27we
used a 8 /H110038/H110038k-point mesh as in Ref. 11, while here we
are using a 16 /H1100316/H1100316 mesh, to obtain more converged
results. Although the US-PPs of Ref. 27and the present PAWTABLE II. As in Table Ifor Pt with the LDA functional.
Pt/H208495s25p6/H20850 5d3/245d5/266s1/205d3/245d5/246s1/225d3/245d5/246s1/2
5d3/2 0.4066 /H208490.0/H20850 0.6639 /H208490.2/H20850 1.2896 /H208490.2/H20850
5d5/2 0.3143 /H208490.0/H20850 0.5596 /H208490.1/H20850 1.1823 /H208490.2/H20850
6s1/2 0.3853 /H208490.1/H20850 0.4970 /H208490.0/H20850 1.0508 /H208490.0/H20850
6p1/2 0.0734 /H208490.0/H20850 0.1322 /H208490.0/H20850 0.6135 /H208490.0/H20850
6p3/2 0.0407 /H208490.0/H20850 0.0835 /H208490.0/H20850 0.5301 /H208490.0/H20850
/H9004E 0.0415 /H208490.0/H20850 0.0254 /H20849−0.1 /H20850 0.7944 /H208490.1/H20850
Pt 5 d3/245d5/266s1/205d3/245d5/246s1/225d3/245d5/246s1/2
5d3/2 0.4066 /H20849−1.1 /H20850/H20851−1.1 /H20852 0.6639 /H208493.1/H20850/H208513.2/H20852 1.2896 /H208494.0/H20850/H208513.9/H20852
5d5/2 0.3143 /H20849−0.8 /H20850/H20851−0.8 /H20852 0.5596 /H208492.7/H20850/H208512.8/H20852 1.1823 /H208493.6/H20850/H208513.6/H20852
6s1/2 0.3853 /H208490.9/H20850/H208510.8/H20852 0.4970 /H20849−0.2 /H20850/H20851−0.2 /H20852 1.0508 /H208490.3/H20850/H208510.4/H20852
6p1/2 0.0734 /H208490.7/H20850/H208510.6/H20852 0.1322 /H20849−0.3 /H20850/H20851−0.3 /H20852 0.6135 /H208490.1/H20850/H208510.1/H20852
6p3/2 0.0407 /H208490.6/H20850/H208510.6/H20852 0.0835 /H20849−0.2 /H20850/H20851−0.3 /H20852 0.5301 /H208490.3/H20850/H208510.5/H20852
/H9004E 0.0415 /H20849−0.9 /H20850/H20851−0.9 /H20852 0.0254 /H20849−1.4 /H20850/H20851−1.3 /H20852 0.7944 /H20849−1.4 /H20850/H20851−1.4 /H20852
TABLE III. As in Table Ifor Au with the LDA functional.
Au /H208495s25p6/H20850 5d3/245d5/256s1/225d3/245d5/256s1/2 5d3/245d5/246s1/22
5d3/2 0.7434 /H208490.2/H20850 1.3887 /H208490.1/H20850 1.5674 /H208490.5/H20850
5d5/2 0.6238 /H208490.1/H20850 1.2663 /H208490.1/H20850 1.4384 /H208490.4/H20850
6s1/2 0.5114 /H208490.1/H20850 1.0814 /H208490.0/H20850 1.1625 /H208490.0/H20850
6p1/2 0.1307 /H20849−0.1 /H20850 0.6249 /H20849−0.1 /H20850 0.6805 /H208490.4/H20850
6p3/2 0.0793 /H20849−0.1 /H20850 0.5358 /H20849−0.1 /H20850 0.5816 /H208490.2/H20850
/H9004E 0.0715 /H208490.1/H20850 0.8628 /H208490.0/H20850 1.0924 /H208490.3/H20850
Au 5 d3/245d5/256s1/225d3/245d5/256s1/2 5d3/245d5/266s1/20
5d3/2 0.7434 /H208493.2/H20850/H208513.2/H20852 1.3887 /H208494.0/H20850/H208514.0/H20852 1.2199 /H20849−0.3 /H20850/H20851−0.3 /H20852
5d5/2 0.6238 /H208492.8/H20850/H208512.8/H20852 1.2663 /H208493.5/H20850/H208513.5/H20852 1.1038 /H20849−0.3 /H20850/H20851−0.2 /H20852
6s1/2 0.5114 /H20849−0.2 /H20850/H20851−0.2 /H20852 1.0814 /H208490.3/H20850/H208510.3/H20852 1.0004 /H208490.2/H20850/H208510.2/H20852
6p1/2 0.1307 /H20849−0.3 /H20850/H20851−0.3 /H20852 0.6249 /H208490.1/H20850/H208510.1/H20852 0.5668 /H208490.2/H20850/H208510.2/H20852
6p3/2 0.0793 /H20849−0.2 /H20850/H20851−0.2 /H20852 0.5358 /H208490.3/H20850/H208510.1/H20852 0.4863 /H208490.2/H20850/H208510.1/H20852
/H9004E 0.0715 /H20849−1.4 /H20850/H20851−1.3 /H20852 0.8628 /H20849−1.5 /H20850/H20851−1.4 /H20852 0.7196 /H20849−0.1 /H20850/H20851−0.1 /H20852ANDREA DAL CORSO PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-10data sets have been generated with quite different recipes, the
agreement between the two band structures is good with er-rors mostly below 0.03 eV . The average difference in thefcc-Pt case is of about 0.01 eV . In fcc-Au, the average dif-ference appears larger /H20849about 0.07 eV /H20850but this is due to the
different k-point sampling. With a 8 /H110038/H110038 mesh, the differ-
ence is below 0.03 eV . When compared to the all-electronresults of Ref. 11, the PAW data sets are nearly equivalent to
the US-PPs, both giving average absolute differences ofabout 0.06 eV for fcc-Pt. For fcc-Au, the average absolutedifferences are 0.04 eV and 0.06 eV in PAW and US-PPscase, respectively, but the smaller PAW error is only due tothe different k-point samplings. Since the numerical values
of Ref. 11have been extracted from a figure and have some
uncertainty, and have been obtained with a slightly differentparametrization of the exchange and correlation energies, wecannot give too much meaning to the tiny differences be-tween the two methods and both appear reasonably good.Finally, we notice that although the PAW data sets withoutsemicore states have a worse transferability in the atomictests, they are acceptable to calculate the electronic bandstructures. The energy differences between the data sets withand without semicore states are mostly within 0.01 eV .
The electronic bands of ferromagnetic bcc-Fe close to the
Fermi energy, at the experimental lattice constant /H20849a
0
=5.42 a.u. /H20850, are reported in Fig. 1.48We show the bands
obtained with data set without semicore states but the resultsare almost independent from the data set used, the largestdifferences being of about 0.02 eV . In Fig. 1, we show the
bands along /H9003-H/H20851H=/H208491,0,0 /H20850in units of
2/H9266
a0/H20852and/H9003-H/H11032
/H20851H/H11032=/H208490,0,1 /H20850/H20852. The magnetization vector is, in all points,
parallel to the zaxis, so in the first line, the kvectors are
perpendicular to the magnetization, whereas in the secondline they are parallel to the magnetization. In Fig. 1, we have
indicated with different line types and colors bands of differ-ent symmetry with respect to the operations that do not con-tain time reversal.
49Along the /H9003-Hdirection, the relevant
symmetry group is Cs, the mirror plane containing the mag-netization vector and the kpoints, and the states are divided
among the /H90033and/H90034representations. Along /H9003-H/H11032the rel-
evant symmetry group is C4with the rotation axis parallel to
the magnetization. There are states of four different symme-tries indicated with /H9003
5,/H90036,/H90037, and /H90038. At the points /H9003,H,
and H/H11032the symmetry group becomes C4h. All the bands
shown in Fig. 1are even with respect to inversion and the
representations becomes /H90035+,/H90036+,/H90037+, and /H90038+. Similar bands
calculated using a NC-PP and an all-electron code were pre-sented in Refs. 33and34. Both calculations agree quite well
with each other and with the present result and also the all-electron magnetic moment reported in Ref. 34 /H208492.226
/H9262B/H20850
agrees with our value.
In conclusion, we have written the PAW Dirac-type equa-
tions of RSDFT and we have transformed these equationsinto Pauli-type equations making errors of order 1 /c
2or
comparable to the TEs. In heavy atoms, these errors aremuch smaller than the errors of order /H20849Z
I/c/H208503of the second-
order Taylor expansions of the Dirac-type equation. Wesolved the Pauli-type equations for fcc-Pt, fcc-Au, and ferro-magnetic bcc-Fe and we showed that the PAW band struc-tures match the band structures of the all-electron Dirac-typeequations. The FR US-PPs /H20849Ref. 27/H20850are a further approxi-
mation of the present scheme in which, inside the PAWspheres, the partial occupations are linearized about the
atomic partial occupations so that the bare coefficients of thenonlocal PP can be calculated in the isolated atoms. The FRUS-PPs turn out to give results that compare well with thepresent FR PAW results.
ACKNOWLEDGMENTS
I thank G. Sclauzero for useful discussions. This work
was sponsored by INFM/CNR “Iniziativa transversale cal-colo parallelo.” The calculations have been done on theSISSA Linux cluster and at CINECA in Bologna.
APPENDIX A: THE RELATIVISTIC PAW DATA SET
Partial waves and projectors are calculated in a nonmag-
netic, isolated atom. The magnitude of the small componentsTABLE IV . Theoretical lattice constants /H20849a0/H20850and bulk moduli /H20849B0/H20850of the systems studied in this work
compared with previous calculations and with experiments. For ferromagnetic bcc-Fe, we report also themagnetic moment
/H9262at the equilibrium lattice constant.
Fe/H20849PBE /H20850 Pt/H20849LDA /H20850 Au /H20849LDA /H20850
a0
/H20849a.u. /H20850B0
/H20849GPa /H20850/H9262
/H20849/H9262B/H20850a0
/H20849a.u. /H20850B0
/H20849GPa /H20850a0
/H20849a.u. /H20850B0
/H20849GPa /H20850
No semicore /H20849FR/H20850 5.358 189 2.17 7.403 301 7.666 199
Semicore /H20849FR/H20850 5.360 189 2.16 7.372 300 7.633 200
No semicore /H20849SR/H20850 5.355 196 2.17 7.396 306 7.681 192
Semicore /H20849SR/H20850 5.357 196 2.17 7.365 305 7.648 194
Reference 11/H20849FR/H20850 7.370 297 7.637 195
Reference 45/H20849FR/H20850 7.386 7.648
Reference 27/H20849FR US-PPs /H20850 7.40 292 7.640 198
Expt. 5.42a168a2.22a7.40b, 7.394c283b7.67b, 7.676c173b
aReference 46.
bExtrapolated at T=0. Reference 47.
cExtrapolated at T=0. Reference 45.PROJECTOR AUGMENTED-WA VE METHOD: APPLICATION … PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-11of the pseudo-partial-waves and of the projectors varies with
the construction recipe. /H9021mI,AE/H20849r/H20850depends on the angular and
spin variables through spin-angle functions /H20849see below /H20850,
whereas the radial components are the solutions of the radialDirac-type equations. For a given
/H9270and/H9260, from the solutions
of the coupled equations,
−c/H20873d
dr+/H9260
r/H20874Q/H9270,/H9260/H20849r/H20850+/H20851Veff/H20849r/H20850−/H9255/H9270,/H9260/H20852P/H9270,/H9260/H20849r/H20850=0 , /H20849A1/H20850
c/H20873d
dr−/H9260
r/H20874P/H9270,/H9260/H20849r/H20850+/H20851Veff/H20849r/H20850−/H9255/H9270,/H9260−2c2/H20852Q/H9270,/H9260/H20849r/H20850=0 ,
/H20849A2/H20850
the large and small radial components of /H9021mI,AE/H20849r/H20850are
P/H9270,/H9260/H20849r/H20850/randiQ/H9270,/H9260/H20849r/H20850/r, respectively /H20849here iis the imaginary
unit /H20850./H9260depends on both jandl. When j=l+1 /2,/H9260=l+1 and
when j=l−1 /2,/H9260=−l. The radial components of the pseudo-
partial-waves can be found by applying a pseudization
recipe50–52toP/H9270,/H9260/H20849r/H20850andQ/H9270,/H9260/H20849r/H20850. We call P˜/H9270,/H9260/H20849r/H20850andQ˜/H9270,/H9260/H20849r/H20850these pseudoradial components. By choosing a smooth local
effective potential V˜eff/H20849r/H20850that matches Veff/H20849r/H20850outside the
PAW spheres, the relativistic generalization of the /H9273/H20849r/H20850
functions24is obtained inverting the radial Dirac equation,
/H9273/H9270,/H9260A/H20849r/H20850=c/H20873d
dr+/H9260
r/H20874Q˜/H9270,/H9260/H20849r/H20850+/H20851/H9255/H9270,/H9260−V˜eff/H20849r/H20850/H20852P˜/H9270,/H9260/H20849r/H20850,
/H20849A3/H20850
/H9273/H9270,/H9260B/H20849r/H20850=−c/H20873d
dr−/H9260
r/H20874P˜/H9270,/H9260/H20849r/H20850+/H20851/H9255/H9270,/H9260+2c2−V˜eff/H20849r/H20850/H20852Q˜/H9270,/H9260/H20849r/H20850,
/H20849A4/H20850
where we called /H9273/H9270,/H9260A/H20849r/H20850and/H9273/H9270,/H9260B/H20849r/H20850the radial parts of the
large and small components of /H9273m/H20849r/H20850. The projectors /H9252/H9270,/H9260/H20849r/H20850
are calculated from /H9273/H9270,/H9260/H20849r/H20850by defining the matrix B/H9270,/H9270/H11032
=/H20855P˜/H9270,/H9260/H20841/H9273/H9270/H11032,/H9260A/H20856+/H20855Q˜/H9270,/H9260/H20841/H9273/H9270/H11032,/H9260B/H20856and using the relationship:
/H9252/H9270,/H9260/H20849r/H20850=/H20858/H9270/H11032/H20849B−1/H20850/H9270/H11032,/H9270/H9273/H9270/H11032,/H9260/H20849r/H20850. These projectors are orthogonal
to the pseudo-wave-functions because /H20855/H9252/H9270,/H9260/H20841/H9278/H9270/H11032,/H9260/H20856TABLE V . The energy eigenvalues /H20849in eV and at the lattice constant indicated in the first row /H20850of fcc-Pt
and fcc-Au at high-symmetry points are compared with results obtained by four-component relativisticDirac-type equations /H20849Ref. 11/H20850or by FR US-PPs /H20849Ref. 27/H20850. The Fermi energy is at zero. The reported values
refer to PAW data sets without semicore states while in parenthesis we report the difference, on the last digit,obtained with semicore states.
Pt/H20849this work /H20850Pt/H20849Ref. 27/H20850Pt/H20849Ref. 11/H20850Au /H20849this work /H20850Au /H20849Ref. 27/H20850Au /H20849Ref. 11/H20850
a
0/H20849a.u. /H20850 7.414 7.414 7.414 7.707 7.707 7.707
/H9003 −10.38 /H20849−1/H20850 −10.39 −10.35 −10.07 /H208491/H20850 −10.01 −9.95
−4.35 /H208490/H20850 −4.35 −4.24 −5.41 /H20849−1/H20850 −5.35 −5.32
−3.36 /H208490/H20850 −3.35 −3.28 −4.21 /H20849−1/H20850 −4.14 −4.14
−1.51 /H208490/H20850 −1.52 −1.48 −2.97 /H20849−1/H20850 −2.92 −2.96
X −7.23 /H208490/H20850 −7.22 −7.10 −7.30 /H208490/H20850 −7.23 −7.14
−6.79 /H208490/H20850 −6.78 −6.70 −7.01 /H208490/H20850 −6.94 −6.95
−0.24 /H208490/H20850 −0.25 −0.20 −2.27 /H20849−1/H20850 −2.22 −2.27
0.06 /H208491/H20850 0.05 0.04 −2.06 /H20849−1/H20850 −2.01 −2.07
−1.03 /H20849−1/H20850 −0.98 −1.03
W −5.84 /H208490/H20850 −5.84 −5.72 −6.27 /H208490/H20850 −6.21 −6.16
−4.88 /H208490/H20850 −4.89 −4.83 −5.69 /H208490/H20850 −5.63 −5.67
−4.63 /H208490/H20850 −4.62 −4.58 −5.12 /H208490/H20850 −5.05 −5.08
−1.95 /H208492/H20850 −1.93 −1.87 −3.21 /H208491/H20850 −3.15 −3.20
−1.61 /H20849−1/H20850 −1.56 −1.62
L −7.50 /H20849−1/H20850 −7.48 −7.44 −7.56 /H208490/H20850 −7.48 −7.53
−4.50 /H208490/H20850 −4.49 −4.43 −5.52 /H20849−1/H20850 −5.46 −5.52
−3.49 /H208490/H20850 −3.49 −3.45 −4.33 /H208490/H20850 −4.28 −4.33
−0.72 /H20849−1/H20850 −0.74 −0.79 −2.32 /H208490/H20850 −2.27 −2.37
−0.32 /H20849−1/H20850 −0.33 −0.30 −1.60 /H20849−1/H20850 −1.55 −1.62
−1.23 /H208491/H20850 −1.10 −1.23
K −6.43 /H208491/H20850 −6.42 −6.40 −6.72 /H208490/H20850 −6.66 −6.70
−5.66 /H20849−1/H20850 −5.66 −5.57 −6.14 /H20849−1/H20850 −6.07 −6.11
−3.07 /H208490/H20850 −3.08 −3.00 −4.03 /H208490/H20850 −3.95 −4.04
−1.39 /H208490/H20850 −1.38 −1.38 −2.97 /H208490/H20850 −2.92 −3.00
−0.10 /H208490/H20850 −0.10 −0.10 −1.92 /H208490/H20850 −1.86 −1.97ANDREA DAL CORSO PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-12=/H20858/H9270/H11033/H20849B−1/H20850/H9270/H11033,/H9270/H20855/H9273/H9270/H11033,/H9260/H20841/H9278/H9270/H11032,/H9260/H20856=/H20858/H9270/H11033/H20849B−1/H20850/H9270/H11033,/H9270B/H9270/H11032,/H9270/H11033=/H9254/H9270,/H9270/H11032./H20851To sim-
plify the notation we have indicated with /H9278/H9270/H11032,/H9260/H20849r/H20850the two
functions P˜/H9270,/H9260/H20849r/H20850andQ˜/H9270,/H9260/H20849r/H20850/H20852.
When P˜/H9270,/H9260/H20849r/H20850andQ˜/H9270,/H9260/H20849r/H20850are chosen independently,
/H9273/H9270,/H9260B/H20849r/H20850becomes large, of order c, and the PAW data set can
be used only together with the Dirac-type PAW equations.
Alternatively, as proposed in Ref. 39,P˜/H9270,/H9260/H20849r/H20850can be calcu-
lated by a pseudization method while Q˜/H9270,/H9260/H20849r/H20850can be taken as
Q˜/H9270,/H9260/H20849r/H20850=c
/H20851/H9255/H9270,/H9260+2c2−V˜eff/H20849r/H20850/H20852/H20873d
dr−/H9260
r/H20874P˜/H9270,/H9260/H20849r/H20850./H20849A5/H20850
At variance with Ref. 39,w eu s e V˜eff/H20849r/H20850instead of the total
potential of the given channel /H9260. In the PAW case there are
many partial waves for each /H9260and no unique total potential.
However, as long as Q˜/H9270,/H9260/H20849r/H20850is continuous, the quality of the
data set should be independent from how we define Q˜/H9270,/H9260/H20849r/H20850
so our choice should be completely equivalent to that of Ref.39. We prefer Eq. /H20849A5/H20850because it has two advantages: first
/H9273/H9270,/H9260B/H20849r/H20850vanishes identically together with the corresponding
small component of the projector so that only the large com-ponents of the pseudo-wave-functions are necessary toevaluate the partial occupations /H20851Eq. /H2084933/H20850/H20852and second no
approximation is needed for the calculation of
/H9273/H9270,/H9260A/H20849r/H20850.
With this choice of the pseudo-partial-waves the projec-
tors vanish exactly outside the PAW spheres, Q˜/H9270,/H9260/H20849r/H20850is con-
tinuous and the PAW data set can be used both in the Dirac-type and in the Pauli-type formalism. The above recipemakes no approximation and therefore does not introduceadditional errors, however we have not applied it because,limiting the use of the PAW data set to the Pauli-type equa-
tions, it is simpler to calculate
/H9273/H9270,/H9260A/H20849r/H20850with the nonrelativistic
recipe, a procedure that introduces some errors but does notworsen the overall accuracy. If we take Q˜/H9270,/H9260/H20849r/H20850
=1
2c/H20849d
dr−/H9260
r/H20850P˜/H9270,/H9260/H20849r/H20850we have
/H9273/H9270,/H9260A/H20849r/H20850=/H20875/H9255/H9270,/H9260−V˜eff/H20849r/H20850+1
2d2
dr2−/H9260/H20849/H9260−1/H20850
2r2/H20876P˜/H9270,/H9260/H20849r/H20850,
/H20849A6/H20850
/H9273/H9270,/H9260B/H20849r/H20850=/H20851/H9255/H9270,/H9260−V˜eff/H20849r/H20850/H20852Q˜/H9270,/H9260/H20849r/H20850. /H20849A7/H20850
Obviously, with this choice the function /H9273/H9270,/H9260A/H20849r/H20850does not van-
ish outside the PAW spheres and, at the border, is of order
/H20849v/H11032/c/H208502. This is because Q˜/H9270,/H9260/H20849r/H20850has a jump at the border with
a discontinuity of order /H20849v/H11032/c/H208503. In the PAW Dirac-type ap-
proach these errors are unacceptable and this method has tobe avoided but in the PAW Pauli-type approach they are ofthe same order of the errors made outside the spheres andpresently still much smaller than the TEs. Using Eq. /H20849A7/H20850,
the small components
/H9273/H9270,/H9260B/H20849r/H20850are quantities of order v13/cand
can be neglected when applied to the small components ofthe pseudo-wave-functions that are also of order
v1/c. The
small components of the projectors /H9252/H9270,/H9260B/H20849r/H20850are of the same
order of /H9273/H9270,/H9260B/H20849r/H20850because the elements of the matrix B/H9270,/H9270/H11032as
well as of its inverse are of order one as the first term in the
definition of B/H9270,/H9270/H11032. The second term is of order v14/c2and can
be neglected. Defining the projectors /H9252/H9270,/H9260A/H20849r/H20850
=/H20858/H9270/H11032/H20849B¯−1/H20850/H9270/H11032,/H9270/H9273/H9270/H11032,/H9260A/H20849r/H20850, where B¯/H9270,/H9270/H11032is the matrix B/H9270,/H9270/H11032in which
the terms of order v14/c2are neglected, we obtain projectors
that are exactly orthogonal to the large component of the
pseudo-wave-functions because /H20855/H9252/H9270,/H9260A/H20841P/H9270/H11032,/H9260/H20856
=/H20858/H9270/H11033/H20849B¯−1/H20850/H9270/H11033,/H9270/H20855/H9273/H9270/H11033,/H9260A/H20841P/H9270/H11032,/H9260/H20856=/H20858/H9270/H11033/H20849B¯−1/H20850/H9270/H11033,/H9270B¯/H9270/H11032,/H9270/H11033=/H9254/H9270,/H9270/H11032.
APPENDIX B: SUMS OVER THE FOUR-COMPONENT
SPINOR INDEXES
In this appendix, we discuss how to perform the sums
over the large and small components. These sums show upboth in the PAW Dirac-type equations /H20851Eqs. /H2084939/H20850–/H2084941/H20850/H20852as
well as in the PAW Pauli-type equations /H20851Eqs. /H2084951/H20850–/H2084953/H20850/H20852.
Sums over the
/H9257indexes appear in the expression of the
all-electron charge and magnetization densities /H92671,I/H20849r/H20850,
m1,I/H20849r/H20850, in the coefficients of the nonlocal PP, DI,mn1and
D˜
I,mn1, and in the integral of V˜
LOC/H92571,/H92572/H20849r/H20850andK˜
/H92681,/H92682/H92571,/H92572/H20849r/H20850. As shown
below, we can replace Qmn,/H92571,/H92572I/H20849r/H20850with pseudized functions
Qˆ
mn,/H92681,/H92682I/H20849r/H20850that vanish everywhere except in the first upper
2/H110032 block. As a consequence, sums over the small compo-
nents appear only for quantities calculated on the radial gridsinside the spheres.
We start by considering the form of the all-electron partial
waves. The partial wave /H9021
m,/H9257I,AE/H20849r/H20850can be written as
/H9021m,/H9257I,AE/H20849r/H20850=1
r/H20873P/H9270,l,jI/H20849r/H20850Y˜
l,j,mjI,/H9268/H20849/H9024r/H20850
iQ/H9270,l,jI/H20849r/H20850Y˜
2j−l,j,mjI,/H9268/H20849/H9024r/H20850/H20874, /H20849B1/H20850
where Y˜
l,j,mjI,/H9268/H20849/H9024r/H20850are spin-angle functions whose expression
is reported, for instance, in Ref. 20. The spin-angle functions−0.4−0.200.20.4Energy (eV)
ΓΓ (1,0,0) H (0,0,1)M//(0,0,1)D4h[C4h]D4h[C4h]D4h[C4h] C2v[Cs]D4[C4]
Γ4 Γ3 Γ6 Γ7 Γ8 Γ5
FIG. 1. /H20849Color online /H20850Calculated relativistic PBE-PAW band
structure of ferromagnetic bcc-Fe close to the Fermi level. Thebands along the /H9003-Hdirection are shown for both kperpendicular
and parallel to the magnetization direction. The zero of the energy istaken at the Fermi energy. Different colors and line types indicatebands of different symmetry with respect to the operations that donot contain time reversal /H20849Ref. 49/H20850. The relevant group is indicated,
for each line and symmetry point, in the square brackets above thefigure.PROJECTOR AUGMENTED-WA VE METHOD: APPLICATION … PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-13of the small and large components have different l. They are
related by the relationship: Y˜
2j−l,j,mjI/H20849/H9024r/H20850=−/H9268·rˆY˜
l,j,mjI/H20849/H9024r/H20850,
where rˆ=/H20849r−RI/H20850//H20841r−RI/H20841. This fact, together with the equa-
tion /H20849/H9268·rˆ/H208502=12/H110032, can be used to write /H92671,I/H20849r/H20850as
/H92671,I/H20849r/H20850= /H20858
/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032/H20858
/H9268/H9267/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032I 1
r2
/H11003/H20851P/H9270,l,jI/H20849r/H20850P/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850
+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850/H20852Y˜
l,j,mj/H11569,I,/H9268/H20849/H9024r/H20850Y˜
l/H11032,j/H11032,mj/H11032I,/H9268/H20849/H9024r/H20850,
/H20849B2/H20850
where we have expanded the indexes mand nintroduced
above with the four indexes /H9270,l,j,mjand/H9270/H11032,l/H11032,j/H11032,mj/H11032, re-
spectively.
For the magnetization density, we can employ the rela-
tionship /H20849/H9268·rˆ/H20850/H9268i/H20849/H9268·rˆ/H20850=2rˆi/H20849/H9268·rˆ/H20850−/H9268ito obtain
mi1,I/H20849r/H20850=/H9262B /H20858
/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032/H20858
/H92681,/H92682/H9267/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032I
/H110031
r2/H20853/H20851P/H9270,l,jI/H20849r/H20850P/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850
+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850/H20852/H9268i/H92681,/H92682
−Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H208502rˆi/H20849/H9268·rˆ/H20850/H92681,/H92682/H20854
/H11003Y˜
l,j,mj/H11569,I,/H92681/H20849/H9024r/H20850Y˜
l/H11032,j/H11032,mj/H11032I,/H92682/H20849/H9024r/H20850. /H20849B3/H20850
For practical calculations, we apply the same approach of
Ref. 27to both the charge and the magnetization densities.
Here we discuss only the latter because the charge densitycan be treated in a similar way. We start by recalling Eq. /H208496/H20850
of Ref. 27and writing the spin-angle functions in terms of
spherical harmonics. This equation is
Y˜
l,j,mjI,/H9268/H20849/H9024r/H20850=/H20858
ml=−ll
cmj,ml/H9268,l,jYl,ml/H11032I/H20849/H9024r/H20850, /H20849B4/H20850
where, with the notations of Ref. 27,cmj,ml/H9268,l,j=/H9251mj/H9268,l,jUmj,ml/H9268,l,j,/H9251mj/H9268,l,j
are the Clebsch-Gordan coefficients and Umj,ml/H9268,l,jis the unitary
matrix that selects the spherical harmonics Yl,ml/H11032Ifor each l,j,
mj, and/H9268. Using this relationship, we obtain
mi1,I/H20849r/H20850=/H9262B /H20858
/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032/H20858
/H92681,/H92682/H20858
k/H20858
ml,ml/H11032
/H11003/H9268k/H92681,/H92682cmj,ml/H11569,/H92681,l,jcmj/H11032,ml/H11032/H92682,l/H11032,j/H11032/H9267/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032I
/H110031
r2/H20853/H20851P/H9270,l,jI/H20849r/H20850P/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850/H20852/H9254i,k
−Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H208502rˆirˆk/H20854Yl,ml/H11032/H11569,I/H20849/H9024r/H20850Yl/H11032,ml/H11032/H11032I/H20849/H9024r/H20850.
/H20849B5/H20850
The partial occupations are written in terms of spin-angle
functions too and can be rewritten as/H9267/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032I=/H20858
/H92681,/H92682/H20858
m1l,m1l/H11032cmj,m1l/H92681,l,jcmj/H11032,m1l/H11032/H11569,/H92682,l/H11032,j/H11032/H9267˜/H9270,l,j,m1l;/H9270/H11032,l/H11032,j/H11032,m1l/H11032I,/H92681,/H92682,
/H20849B6/H20850
where /H9267˜/H9270,l,j,m1l;/H9270/H11032,l/H11032,j/H11032,m1l/H11032I,/H92681,/H92682are partial occupations calculated as
in the SR case, with projectors defined by spherical
harmonics27
/H9267˜/H9270,l,j,m1l;/H9270/H11032,l/H11032,j/H11032,m1l/H11032I,/H92681,/H92682=/H20858
i/H20855/H9023˜
i,/H92681A/H20841/H9252/H9270,l,jI,AYl,m1l/H11032I/H20856/H20855/H9252/H9270/H11032,l/H11032,j/H11032I,AYl/H11032,m1l/H11032/H11032I/H20841/H9023˜
i,/H92682A/H20856.
/H20849B7/H20850
We assumed that the small components of the projectors are
negligible /H20849Pauli-type case /H20850or vanish /H20849Dirac-type case /H20850.
When this does not happen, Eqs. /H20849B6/H20850and /H20849B7/H20850are more
complicated but we do not need them. Following now Ref.
27and introducing the functions fl,j,ml;l,j,m1l/H92681,/H92682 =/H20858mjcmj,ml/H92681,l,jcmj,m1l/H11569,/H92682,l,j
and the partial occupations,
/H9267/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032I,/H92681,/H92682=/H20858
/H92683,/H92684/H20858
m1l,m1l/H11032fl,j,m1l;l,j,ml/H92683,/H92681
/H11003fl/H11032,j/H11032,ml/H11032;l/H11032,j/H11032,m1l/H11032/H92682,/H92684/H9267˜/H9270,l,j,m1l;/H9270/H11032,l/H11032,j/H11032,m1l/H11032I,/H92683,/H92684,
/H20849B8/H20850
we can calculate the magnetization density making sums
over the mlandml/H11032/H20849−l/H11349ml/H11349land − l/H11032/H11349ml/H11032/H11349l/H11032/H20850instead of
mjandmj/H11032/H20849−j/H11349mj/H11349jand − j/H11032/H11349mj/H11349j/H20850. We have
mi1,I/H20849r/H20850=/H9262B /H20858
/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032/H20858
k/H20858
/H92681,/H92682/H9268k/H92681,/H92682/H9267/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032I,/H92681,/H92682
/H110031
r2/H20853/H20851P/H9270,l,jI/H20849r/H20850P/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850/H20852/H9254i,k
−Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H208502rˆirˆk/H20854Yl,ml/H11032/H11569,I/H20849/H9024r/H20850Yl/H11032,ml/H11032/H11032I/H20849/H9024r/H20850.
/H20849B9/H20850
This expression and the equivalent expression for m˜i1,I/H20849r/H20850
+mˆi1,I/H20849r/H20850have been implemented in QE.41
Let us now discuss how to define the augmentation func-
tions. These functions have to satisfy two constraints. On onehand, they have to be as smooth as possible to be describedin the real-space mesh and on the other hand they have togive compensation charges with the same multipole mo-
ments as
/H92671,I/H20849r/H20850−/H9267˜1,I/H20849r/H20850. These charge-density differences are
equal to
/H92671,I/H20849r/H20850−/H9267˜1,I/H20849r/H20850= /H20858
/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032/H20858
/H9268/H9267/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032I
/H11003A/H9270,l,j;/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850Y˜
l,j,mj/H11569,I,/H9268/H20849/H9024r/H20850Y˜
l/H11032,j/H11032,mj/H11032I,/H9268/H20849/H9024r/H20850,
/H20849B10 /H20850
where we used the notation A/H9270,l,j;/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850
=1
r2/H20851P/H9270,l,jI/H20849r/H20850P/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850−P˜
/H9270,l,jI/H20849r/H20850P˜
/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850
−Q˜
/H9270,l,jI/H20849r/H20850Q˜
/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850/H20852that is suitable to the PAW Dirac-type
case. As suggested by Eq. /H2084946/H20850, in the PAW Pauli-type caseANDREA DAL CORSO PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-14the term Q˜
/H9270,l,jI/H20849r/H20850Q˜
/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850is omitted whereas the small
components of the all-electron partial waves are included
only inside the PAW spheres.
Equation /H20849B10 /H20850can be rewritten introducing the spherical
harmonics. Following stepwise the derivation of Eq. /H20849B9/H20850,
we have
/H92671,I/H20849r/H20850−/H9267˜1,I/H20849r/H20850= /H20858
/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032/H20858
/H9268/H9267/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032I,/H9268,/H9268
/H11003A/H9270,l,j;/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850Yl,ml/H11032/H11569,I/H20849/H9024r/H20850Yl/H11032,ml/H11032/H11032I/H20849/H9024r/H20850.
/H20849B11 /H20850
Expanding the product of two spherical harmonics into
spherical harmonics,
Yl,ml/H11032/H11569,I/H20849/H9024r/H20850Yl/H11032,ml/H11032/H11032I/H20849/H9024r/H20850
=/H20858
L=/H20841l−l/H11032/H20841l+l/H11032
/H20858
M=−LL
a/H20849l,ml;l/H11032,ml/H11032;L,M/H20850YL,M/H11032I/H20849/H9024r/H20850,/H20849B12 /H20850
we can proceed as in the SR case,40replacing each function
A/H9270,l,j;/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850with many functions, one for each Lso as to
conserve the Lmultipole moment. We call A˜
/H9270,l,j;/H9270/H11032,l/H11032,j/H11032I,L/H20849r/H20850
these functions that can be constructed as described in Ref.
15or with any equivalent method. The final expression of
the compensation charge is/H9267ˆI/H20849r/H20850= /H20858
/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032/H20858
/H9268/H9267/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032I,/H9268,/H9268Qˆ
/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032I/H20849r/H20850,
/H20849B13 /H20850
where the augmentation functions are
Qˆ
/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032I/H20849r/H20850
=/H20858
L,MA˜
/H9270,l,j;/H9270/H11032,l/H11032,j/H11032I,L/H20849r/H20850a/H20849l,ml;l/H11032,ml/H11032;L,M/H20850YL,M/H11032I/H20849/H9024r/H20850
/H20849B14 /H20850
in close analogy with the SR case. The augmentation func-
tions Qˆ
mn,/H92571,/H92572I/H20849r/H20850used in Eqs. /H2084941/H20850and /H2084951/H20850and in the defi-
nition of K˜
/H92573,/H92574/H92571,/H92572/H20849r/H20850can be constructed starting from
Qˆ
/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032I/H20849r/H20850. The way in which we separate the angu-
lar and spin components of Qˆ
mn,/H92571,/H92572I/H20849r/H20850is to a certain extent
arbitrary, provided that the compensation charges /H9267ˆI/H20849r/H20850have
the same multipole moments as /H92671,I/H20849r/H20850−/H9267˜1,I/H20849r/H20850. For compu-
tational convenience, we defined augmentation functions thatvanish everywhere except in the first upper 2 /H110032 block. We
take
Qˆ
/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032;/H92681,/H92682I/H20849r/H20850
=/H20858
ml,ml/H11032cmj,ml/H11569,/H92681,l,jcmj/H11032,ml/H11032/H92682,l/H11032,j/H11032Qˆ
/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032I/H20849r/H20850/H20849 B15 /H20850
and zero elsewhere. With this definition, /H9267ˆ/H92571,/H92572I/H20849r/H20850becomes
/H9267ˆ/H92571,/H92572I/H20849r/H20850=/H20873/H20858/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032/H9267/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032IQˆ
/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H11032;/H92681,/H92682I/H20849r/H208500
00 /H20874 /H20849B16 /H20850
inside the spheres, and /H9267ˆ/H92571,/H92572/H20849r/H20850=/H20858I/H9267ˆ/H92571,/H92572I/H20849r−RI/H20850in the real-
space mesh. Although the lower and the off-diagonal 2 /H110032
blocks of the compensation density matrices are forced tovanish, the compensation charges have the correct multipolemoments and this is a sufficient condition to have the correctelectrostatics so with this choice we are not introducing anyadditional error.
Finally, we discuss how to calculate D
I,mn1and the nonlo-
cal PP terms /H20858ImnDI,mn1/H20841/H9252m,/H92681I,A/H20856/H20855/H9252n,/H92682I,A/H20841.DI,mn1has two parts, one
due to the kinetic-energy operator and one due to the term
VLOCI,/H92571,/H92572. They are quite different and are dealt with separately.
The kinetic-energy part presents no difficulty because thespherically symmetric Dirac operator can be calculated in theisolated atom. The all-electron partial waves solve the atomicDirac-type equations so the sum over the spin componentsand the angular integral can be carried out analytically. Wehave /H20858
/H92571,/H92572/H20855/H9021m,/H92571I,AE/H20841TD/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856/H9024=/H9254l,l/H11032/H9254j,j/H11032/H9254mj,mj/H11032TD,/H9270,/H9270/H11032I,l,jwhere
TD,/H9270,/H9270/H11032I,l,j=/H20885
0rs
dr/H20851P/H9270,l,jI/H20849r/H20850P/H9270/H11032,l,jI/H20849r/H20850+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l,jI/H20849r/H20850/H20852
/H11003/H20851/H9255/H9270/H11032,l,j−Veff,atI/H20849r/H20850/H20852. /H20849B17 /H20850
rsis the radius of the PAW sphere, /H9255/H9270/H11032,l,jis the energy at
which the partial waves and projectors have been con-
structed, and Veff,atI/H20849r/H20850is the atomic effective all-electron po-
tential. TD,/H9270,/H9270/H11032I,l,jare calculated together with the partial waves
and projectors during the PAW data set generation. TheyPROJECTOR AUGMENTED-WA VE METHOD: APPLICATION … PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-15have the same structure of the unscreened coefficients of the
nonlocal FR US-PPs. Following Ref. 27, we can write
/H20858
I,mn/H20858
/H92571,/H92572/H20855/H9021m,/H92571I,AE/H20841TD/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856/H9024/H20841/H9252m,/H92681I,A/H20856/H20855/H9252n,/H92682I,A/H20841
=/H20858
I/H20858
/H9270,l,j,ml/H20858
/H9270/H11032,l/H11032,j/H11032,ml/H11032TD,/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032I,/H92681,/H92682/H20841/H9252/H9270,l,jI,AYl,ml/H11032I/H20856
/H11003/H20855/H9252/H9270/H11032,l/H11032,j/H11032I,AYl/H11032,ml/H11032/H11032I/H20841, /H20849B18 /H20850
where
TD,/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032I,/H92681,/H92682=TD,/H9270,/H9270/H11032I,l,jfl,j,ml;l,j,ml/H11032/H92681,/H92682/H9254l,l/H11032/H9254j,j/H11032, /H20849B19 /H20850
so this term has the same expression as the nonlocal part of a
SR US-PP but with spin-dependent nonlocal PP coefficients.
In general VLOCI,/H92571,/H92572/H20849r/H20850is not spherically symmetric but it is
block diagonal and does not couple the large and small com-ponents of the partial waves. With the same techniques in-troduced in this appendix, we can write
/H20858
/H92571,/H92572/H20855/H9021m,/H92571I,AE/H20841VLOCI,/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856/H9024
=/H20858
/H92681,/H92682/H20855P/H9270,l,jIY˜
l,j,mjI,/H92681/H20841VLOCI,/H92681,/H92682/H20841P/H9270/H11032,l/H11032,j/H11032IY˜
l/H11032,j/H11032,mj/H11032I,/H92682/H20856/H9024
+/H20858
/H92681,/H92682/H20855Q/H9270,l,jIY˜
l,j,mjI,/H92681/H20841VLOCI,/H92681,/H92682/H20841Q/H9270/H11032,l/H11032,j/H11032IY˜
l/H11032,j/H11032,mj/H11032I,/H92682/H20856/H9024
+/H20858
/H92681,/H92682/H20855Q/H9270,l,jIY˜
l,j,mjI,/H92681/H208412/H9262BBxcI·rˆ/H20849/H9268·rˆ/H20850/H92681,/H92682/H20841Q/H9270/H11032,l/H11032,j/H11032IY˜
l/H11032,j/H11032,mj/H11032I,/H92682/H20856/H9024.
/H20849B20 /H20850
This complete expression, which is needed in a magnetic
solid, can be simplified by carrying out the angular integralsanalytically. We first write the angular dependence of theintegrand function by introducing the spherical harmonics,
/H20858
/H92571,/H92572/H20855/H9021m,/H92571I,AE/H20841VLOCI,/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856
=/H20858
/H92681,/H92682/H20858
ml,ml/H11032cmj,ml/H11569,/H92681,l,jcmj/H11032,ml/H11032/H92682,l/H11032,j/H11032/H20885
/H9024Id3r
r2/H20853/H20851P/H9270,l,jI/H20849r/H20850P/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850
+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850/H20852VLOCI,/H92681,/H92682/H20849r/H20850
+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H208502/H9262BBxcI·rˆ/H20849/H9268·rˆ/H20850/H92681,/H92682/H20854
/H11003Yl,ml/H11032/H11569,I/H20849/H9024r/H20850Yl/H11032,ml/H11032/H11032I/H20849/H9024r/H20850, /H20849B21 /H20850
then we expand VLOCI,/H92681,/H92682/H20849r/H20850 and G/H20849r/H20850I,/H92681,/H92682
=2/H9262BBxcI·rˆ/H20849/H9268·rˆ/H20850/H92681,/H92682in spherical harmonics,
VLOCI,/H92681,/H92682/H20849r/H20850=/H20858
L,MVLOC, L,MI,/H92681,/H92682/H20849r/H20850YL,M/H11032I/H20849/H9024r/H20850, /H20849B22 /H20850
GI,/H92681,/H92682/H20849r/H20850=/H20858
L,MGL,MI,/H92681,/H92682/H20849r/H20850YL,M/H11032I/H20849/H9024r/H20850. /H20849B23 /H20850
Using the expansion /H20851Eq. /H20849B12 /H20850/H20852, we obtain/H20858
/H92571,/H92572/H20855/H9021m,/H92571I,AE/H20841VLOCI,/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856
=/H20858
/H92683,/H92684/H20858
m1l,m1l/H11032cmj,m1l/H11569,/H92683,l,jcmj/H11032,m1l/H11032/H92684,l/H11032,j/H11032D¯
LOC,/H9270,l,j,m1l;/H9270/H11032,l/H11032,j/H11032,m1l/H11032I,/H92683,/H92684,
/H20849B24 /H20850
where
D¯
LOC,/H9270,l,j,m1l;/H9270/H11032,l/H11032,j/H11032,m1l/H11032I,/H92683,/H92684=/H20858
L,M/H20885
0rs
dr/H20853/H20851P/H9270,l,jI/H20849r/H20850P/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850
+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850/H20852VLOC, L,MI,/H92683,/H92684/H20849r/H20850
+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850
/H11003GL,MI,/H92683,/H92684/H20849r/H20850/H20854a/H20849l,m1l;l/H11032,m1l/H11032;L,M/H20850.
/H20849B25 /H20850
In these expressions, for simplicity, we used real spherical
harmonics but a straightforward generalization would allowthe use of complex spherical harmonics. We can now writethe contribution of this term to the nonlocal PP. We have
/H20858
I,mn/H20858
/H92571,/H92572/H20855/H9021m,/H92571I,AE/H20841VLOC/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856/H9024/H20841/H9252m,/H92681I,A/H20856/H20855/H9252n,/H92682I,A/H20841
=/H20858
I/H20858
/H9270,l,j,ml/H20858
/H9270/H11032,l/H11032,j/H11032,ml/H11032DLOC,/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032I,/H92681,/H92682/H20841/H9252/H9270,l,jI,AYl,ml/H11032I/H20856
/H11003/H20855/H9252/H9270/H11032,l/H11032,j/H11032I,AYl/H11032,ml/H11032/H11032I/H20841, /H20849B26 /H20850
where
DLOC,/H9270,l,j,ml;/H9270/H11032,l/H11032,j/H11032,ml/H11032I,/H92681,/H92682=/H20858
/H92683,/H92684/H20858
m1l,m1l/H11032fl,j,ml;l,j,m1l/H92681,/H92683 fl/H11032,j/H11032,m1l/H11032;l/H11032,j/H11032,ml/H11032/H92684,/H92682
/H11003D¯
LOC,/H9270,l,j,m1l;/H9270/H11032,l/H11032,j/H11032,m1l/H11032I,/H92683,/H92684, /H20849B27 /H20850
so also the term due to VLOC/H92571,/H92572has the same form of a nonlo-
cal SR US-PP with spin-dependent coefficients and do notintroduce any complication into existing electronic structurecodes.
In nonmagnetic solids several simplifications occur. The
last term in the square bracket of Eq. /H20849B25 /H20850vanishes and
V
LOC/H92681,/H92682/H20849r/H20850is diagonal in the spin indexes, so we obtain
D¯
LOC,/H9270,l,j,m1l;/H9270/H11032,l/H11032,j/H11032,m1l/H11032I,/H92683,/H92684=/H9254/H92683,/H92684/H20858
L,M/H20885
0rs
dr/H20853/H20851P/H9270,l,jI/H20849r/H20850P/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850
+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850/H20852Veff,L,MI/H20849r/H20850/H20854
/H11003a/H20849l,m1l;l/H11032,m1l/H11032;L,M/H20850. /H20849B28 /H20850
In the US-PPs case this term is calculated in the nonmagnetic
isolated atom, so VeffI/H20849r/H20850is spherically symmetric. InANDREA DAL CORSO PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-16Eq. /H20849B20 /H20850the expectation values between spin-angle func-
tions are nonzero only when l=l/H11032,j=j/H11032, and mj=mj/H11032and we
have
/H20858
/H92571,/H92572/H20855/H9021m,/H92571I,AE/H20841VLOCI,/H92571,/H92572/H20841/H9021n,/H92572I,AE/H20856
=/H9254l,l/H11032/H9254j,j/H11032/H9254mj,mj/H11032/H20885
0rs
dr/H20851P/H9270,l,jI/H20849r/H20850P/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850
+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l/H11032,j/H11032I/H20849r/H20850/H20852Veff,atI/H20849r/H20850. /H20849B29 /H20850This term can be combined with the kinetic-energy term to
give the US expression of DI,mn1,
DI;/H9270,l,j,mj;/H9270/H11032,l/H11032,j/H11032,mj/H110321=/H9254l,l/H11032/H9254j,j/H11032/H9254mj,mj/H11032/H9255/H9270/H11032,l,j/H20885
0rs
dr/H20851P/H9270,l,jI/H20849r/H20850P/H9270/H11032,l,jI/H20849r/H20850
+Q/H9270,l,jI/H20849r/H20850Q/H9270/H11032,l,jI/H20849r/H20850/H20852. /H20849B30 /H20850
Similar considerations as the ones reported for DI,mn1, ap-
ply also to D˜
I,mn1that, having no sum over the four compo-
nents indexes, is simpler to deal with.
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44PAW data sets without semicore states have been generated with
the valence electronic configurations: 4 s1/21.74p1/204p3/203d3/243d5/22.3
/H20849Fe/H20850,6 s1/216p1/206p3/205d3/245d5/25/H20849Pt/H20850, and
6s1/216p1/206p3/205d3/245d5/26/H20849Au/H20850. The core radii /H20849in bohr atomic
unit /H20850are Fe rs=2.2, rp=2.2, rd=1.8, Pt rs=2.4, rp=2.5, rd=2.2,
and Au rs=2.3, rp=2.4, rd=2.2. The nonlocal channels are fittedPROJECTOR AUGMENTED-WA VE METHOD: APPLICATION … PHYSICAL REVIEW B 82, 075116 /H208492010 /H20850
075116-17at the eigenvalue and at /H9280s=2.3 Ry, /H9280p=2.5 Ry, /H9280d=−0.3 Ry in
Fe, at /H9280s=2.5 Ry, /H9280p=3.5 Ry, /H9280d=0.8 Ry in Pt, and at /H9280s
=2.3 Ry, /H9280p=3.6 Ry, /H9280d=−0.3 Ry in Au. NLCC is used in Fe,
Pt, and Au /H20849rc=1.2,1.6,1.6 /H20850. The local potential is the all-
electron potential smoothed before rloc=2.1,2.4,2.4 in Fe, Au,
and Pt, respectively. PAW data sets with semicorestates have been generated with the valence electronic configu-rations: 3 s
1/223p1/223p3/244s1/224p1/204p3/203d3/243d5/22/H20849Fe/H20850,
5s1/225p1/225p3/246s1/216p1/206p3/205d3/245d5/25/H20849Pt/H20850, and
5s1/225p1/225p3/246s1/216p1/206p3/205d3/245d5/26/H20849Au/H20850. The core radii are
Fer3s=1.3, r4s=1.4, r3p=1.3, r4p=1.6, rd=2.0, Pt rs=1.6, r5p
=1.6, r6p=1.8, rd=2.3, and Au r5s=1.5, r6s=1.6, r5p=1.4, r5p
=1.9, rd=2.3. The nonlocal channels are fitted at the eigenvalues
and at /H9280d=−0.4 Ry in Fe, at /H9280d=−0.3 Ry in Pt, and at /H9280d=
−0.3 Ry in Au. NLCC is used in Fe, Pt, and Au /H20849rc
=0.6,1.0,1.0 /H20850. The local potential is the all-electron potentialsmoothed before rloc=1.8, 1.8, and 1.6 in Fe, Au, and Pt, respec-
tively. PAW data sets are generated by the ld1 atomic code gen-eralized to relativistic PAW as explained in Appendix A.
45P. Haas, F. Tran, and P. Blaha, Phys. Rev. B 79, 085104 /H208492009 /H20850.
46E. G. Moroni, G. Kresse, J. Hafner, and J. Furthmüller, Phys.
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/H208491995 /H20850.
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Media /H20849Pergamon Press, Oxford, 1960 /H20850.
50N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 /H208491991 /H20850.
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075116-18 |
PhysRevB.100.235413.pdf | PHYSICAL REVIEW B 100, 235413 (2019)
Kondo effect in a Aharonov-Casher interferometer
A. V . Parafilo,1,*L. Y . Gorelik,2M. N. Kiselev ,3H. C. Park ,1,†and R. I. Shekhter4
1Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Expo-ro, 55, Yuseong-gu, Daejeon 34126, Republic of Korea
2Department of Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
3The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34151 Trieste, Italy
4Department of Physics, University of Gothenburg, SE-412 96 Göteborg, Sweden
(Received 15 September 2019; revised manuscript received 20 November 2019; published 9 December 2019)
We consider a model describing a spin field-effect transistor based on a quantum nanowire with a tunable spin-
orbit interaction embedded between two ferromagnetic leads with anticollinear magnetization. We investigate aregime of a strong interplay between resonance Kondo scattering and interference associated with the Aharonov-Casher effect. Using the Keldysh technique at a weak-coupling regime we calculate perturbatively the chargecurrent. It is predicted that the effects of the spin-orbit interaction result in a nonvanishing current for any spinpolarization of the leads including the case of fully polarized anticollinear contacts. We analyze the influence ofthe Aharonov-Casher phase and degree of spin polarization in the leads onto a Kondo temperature.
DOI: 10.1103/PhysRevB.100.235413
I. INTRODUCTION
The Kondo effect is known to play a very important
role for charge transport through nanostructures, facilitatingthe maximal conductance of a nanodevice at zero bias [ 1].
Having a spin nature, the Kondo effect is associated witha resonance scattering accompanied by a spin flip throughthe multiple cotunneling processes in Coulomb blockadednanodevices [ 2]. The Kondo effect in GaAs-based semicon-
ductor nanostructures (quantum dots, quantum point contacts,quantum wires, etc.) attracted enormous attention in boththe experimental and theoretical communities during the lasttwo decades [ 2–5]. Recently, semiconductor quantum wires
fabricated on InAs and InSb heterostructures started to bewidely used in new quantum technological devices [ 6,7]. One
of the most important properties of these materials is related tothe effects of a strong spin-orbit interaction (SOI) which doesnot conserve spin in the resonance scattering processes (see,e.g., Ref. [ 8]). The high tunability of the interplay between
a SOI and the resonance Kondo effect and its influence onthe charge and spin transport through nanostructures pavesa way for practical applications of these materials in spin-tronics devices. It is known that, in contrast to the effectsof an external magnetic field, the effect of SOI on electronscattering is in preserving time-reversal symmetry. While themagnetic field is destructive for the Kondo effect due to thesuppression of spin-flip processes, the influence of SOI onresonance scattering is more delicate.
One of the most remarkable manifestations of SOI in
quantum devices (e.g., a Datta-Das spin field-effect transis-tor) is associated with an accumulation of a spin-dependentphase difference in the electron (spinor) wave function [ 9,10].
This phase accumulation being controlled by an external
*aparafil@ibs.re.kr
†hc2725@gmail.comelectric field applied to a nanodevice is known as theAharonov-Casher effect [ 9]. The electric field manipulation
of Aharonov-Casher interference provides a big advantagecompared to an external magnetic field control. In particular,no magnetization currents are generated both in the nan-odevice and in the leads and there is no extra decoherenceassociated with extra heating. An additional degree of controlassociated with the use of ferromagnetic leads allows one toopen (enhance) and close (suppress) a charge current throughthe nanostructure [ 10,11], similar to the effects of a spin
valve [ 12–14].
In this paper we present an example when a strong
Coulomb blockade and an established quantum coherenceof electrons are simultaneously present and controlled ina quantum nanowire. We consider the Kondo tunneling ofelectrons through the nanowire in the presence of a strongSOI. We show that quantum interference originating fromthe Aharonov-Casher phase accumulated during the tunnelingprocess affects qualitatively the many-body Kondo transmis-sion and results in a strong renormalization of the Kondo tem-perature and a significant enhancement of the charge current.
The paper is organized as follows: In Sec. IIwe introduce
a model Hamiltonian of a spin-orbit active one-dimensionalnanowire placed between spin-polarized electrodes and derivean effective Kondo model. In Sec. IIIwe analyze the charge
current through the nanowire calculated in the lowest orderof perturbation theory. In Sec. IVwe obtain the contribution
to the charge current in the second order of perturbationtheory and evaluate a Kondo temperature as a function ofthe Aharonov-Casher phase and degree of spin polarizationin the leads. In Sec. Vwe analyze the Kondo temperature in a
particular limit of fully polarized anticollinear contacts.
II. MODEL
We investigate a Datta-Das spin field-effect transistor [ 10]
with a spin-orbit active weak link in the Kondo regime.
2469-9950/2019/100(23)/235413(7) 235413-1 ©2019 American Physical SocietyA. V . PARAFILO et al. PHYSICAL REVIEW B 100, 235413 (2019)
FIG. 1. Scheme of a nanodevice. A 1D nanowire of length Lis
placed between two massive magnetically polarized electrodes. Po-larization is chosen to be collinear antiparallel (AP). Spin-dependent
density of states in the leads is defined through ν
L↑=νR↓=ν(1+p)
andνL↓=νR↑=ν(1−p). A back-gate electrode situated near the
nanowire creates an electric field in the zdirection, inducing a
spin-orbit interaction (SOI) in the nanowire. The short nanowire is
treated as a quantum dot (QD) (see the main text for a discussion).We assume that the QD is in a strong Coulomb blockade regime. Odd
Coulomb valleys provide access to the Kondo physics. SOI leads
to an accumulation of an Aharonov-Casher phase in the electronwave function which is equivalent semiclassically to an electron spin
precession.
We consider a one-dimensional (1D) nanowire embedded
between two magnetically polarized electrodes in an antipar-allel configuration (AP). The back gate controls a spin-orbitinteraction (SOI) in the nanowire (see Fig. 1). The one-
dimensional nanowire can be treated as a quantum dot (QD)in a regime when the temperature and bias voltage are smallercompared to a mean level spacing in the QD, δε∼¯hv
FL−1.
We assume an odd number of electrons in the QD to accessthe Kondo regime. The model is described by the HamiltonianH=H
0+Htun, where
H0=/summationdisplay
k,α,σ/parenleftbig
εk−μσ
α/parenrightbig
c†
kασckασ+/summationdisplay
λ(ε0d†
λdλ+UCˆnλˆn¯λ)
(1)
characterizes a 1D nanowire and the magnetically polarized
left (right) electrodes with chemical potentials μσ
L(R). Here,
ε0stands for the energy of the first half-filled level of the
dot counted from the Fermi level of the leads, and UCis the
charging energy in the nanowire. The annihilation (creation)operators of the conduction electrons are denoted by c
kασ
(c†
kασ), where α=L,R. The electron states in the leads are
characterized by a spin quantum number σ=(↑,↓). The
twofold degenerate quantum level in the dot represented bya linear superposition of states with σ=↑ andσ=↓ is
described by the pseudospin quantum number with two eigen-values λ,¯λ. We use notations d
λ(d†
λ) for the electrons in the
QD, ˆ nλ=d†
λdλ(see Appendix A). To describe the partial spin
polarization of the electrodes we introduce a spin-dependentdensity of states at Fermi energies ν
L↑=νR↓=ν(1+p) and
νL↓=νR↑=ν(1−p), where parameter pdefines a degree
of polarization (see Fig. 1). If magnetization in the leads
is collinear and oriented antiparallel, the net magnetic fieldproduced by the leads at the position of the nanowire iszero. Therefore, the net magnetic field does not lift a twofolddegeneracy of the pseudospin state in the QD. If the ori-
entation of the magnetization is parallel, the net magneticfield at the position of the nanowire is nonzero, resulting intime-reversal symmetry-breaking effects.
While the spin is a good quantum number in the leads,
it cannot be used for the characterization of the state in thenanowire due to the presence of SOI. Thus, the tunnel matrixelement computed using wave functions of electrons in theleads and in QD (see, e.g., Ref. [ 15]) is characterized by two
indices σ(spin) and λ(pseudospin). The most general form
of the tunneling Hamiltonian is given by
H
tun=/summationdisplay
k,α;σλ/parenleftbig
Vσλ
kαc†
kασdλ+H.c./parenrightbig
. (2)
We solve the 1D Schrödinger equation for an electron in the
nanowire in the presence of SOI (see details in Appendix A),
and express the tunnel matrix amplitudes in terms of the SOI
parameters. The two-component electron wave functions /vectorψλ
at different points are connected through the operator ˆUin
such a way, /vectorψλ(x2)=ˆU(x2,x1)/vectorψλ(x1), where
ˆU=exp/bracketleftbiggiˆσyϑ(x1,x2)
2/bracketrightbigg
,ϑ=2αpFl
¯hvF(3)
characterizes the accumulation of the Aharonov-Casher
phase, l=|x2−x1|(see Fig. 1). In Eq. ( 3),α∝Ezis the SOI
coupling constant, Ezis the electric field in the zdirection
produced by the back-gate electrode, and ˆ σyis the y-Pauli
matrix. Using Eq. ( 3) and assuming that the tunneling occurs
at the points x1,x2, we parametrize the tunnel matrix elements
as follows,
Vττ/prime
kα=Vkα/parenleftbig
δττ/primecos(ϑ/4)∓iˆσy
ττ/primesin(ϑ/4)/parenrightbig
. (4)
Here, −/+stands for the L/Rlead correspondingly. The
effect of the SOI on the tunneling processes is characterizedby the parameter ϑ. The SOI vanishes for ϑ=0 and reaches
its maximal value at ϑ=π, when both tunneling processes,
diagonal and off diagonal in spin (pseudospin) indices, con-tribute equally [see Eq. ( 4)]. We assume full symmetry in the
tunneling junction V
kL=VkR=Vtun.
The mapping of the Anderson impurity model Eq. ( 1) onto
a Kondo-like model is done using the standard Schrieffer-Wolff transformation [ 16] (see Appendix B). We assume
a single occupied twofold degenerate level in the QD andconsider the energy level width to be smaller compared to thecharging energy /Gamma1=2πν|V
tun|2/lessmuchUC.
The effective Hamiltonian Heff=Hdir+Hexcontains Hdir
describing a direct (potential) electron scattering between the
leads and
Hex=/summationdisplay
αJ/bracketleftbig/parenleftbig
sz
ααSz+sx
ααSx/parenrightbig
cos(ϑ/2)+sy
ααSy/bracketrightbig
+/summationdisplay
α/negationslash=α/primeJ/bracketleftbig
sz
αα/primeSz+sx
αα/primeSx+sy
αα/primeSycos(ϑ/2)/bracketrightbig
−Jsin(ϑ/2)/bracketleftbig/parenleftbig
sz
LL−sz
RR/parenrightbig
Sx−/parenleftbig
sx
LL−sx
RR/parenrightbig
Sz/bracketrightbig
−J
2sin(ϑ/2)jLRSy(5)
235413-2KONDO EFFECT IN A AHARONOV-CASHER … PHYSICAL REVIEW B 100, 235413 (2019)
constituting the effective exchange interaction between
pseudospin-1 /2 in the QD, /vectorS, and spin /vectorsαα/prime=/summationtext
kk/prime
(1/2)c†
kασˆσσσ/primeck/primeα/primeσ/prime of the conduction electrons
in the L/R leads and charge transfer jLR=
i/summationtext
kk/prime(c†
kL↑ck/primeR↑+c†
kL↓ck/primeR↓−H.c.). We used the following
notations for the exchange interaction constant in Eq. ( 5),
J=2UC|Vtun|2
|ε0|(UC−|ε0|). (6)
We concentrate below on the case of electron-hole symmetry,
ε0→(−UC/2), and ignore the irrelevant processes of poten-
tial scattering [ 17].
The influence of SOI effects onto the Kondo scattering
has several facets. First, SOI is responsible for the differenttypes of spin anisotropies in the terms diagonal and offdiagonal in the lead indices [first and second lines in Eq. ( 5)].
Second, SOI produces an additional coupling between thepseudospin in QD and spin density of the conduction elec-trons, also known as the Dzyaloshinskii-Moriya (DM) in-teraction ( ∝/vectore
y·[/vectorsαα×/vectorS]) [18]. Third, the SOI mediates the
interaction between the pseudospin in the QD and the chargetransfer.
III. COTUNNELING CURRENT
Assuming a high-temperature (compared to some emerg-
ing energy scale to be defined below) regime we cal-culate the current through the nanowire perturbatively inνJ/lessmuch1. The first nonvanishing contribution to the charge
current is ∝(νJ)
2. The cotunneling current given by
Eq. ( 7) can be straightforwardly derived either through an
equation of motion method or using the nonequilibriumKeldysh Green’s function technique (we adopt below theunits k
B=1),
I(2)=eπ2
4π¯h(νJ)2eV/braceleftbigg
(1−p2)(2−q2)
+(1+q2)(1+p2)−8pqcoth/parenleftbiggeV
2T/parenrightbigg
/angbracketleftSz/angbracketright/bracerightbigg
, (7)
where we use shorthand notations q=cos(ϑ/2) for the
parametrization of the accumulated Aharonov-Casher phase(q=1 for the case of the absence of SOI, ϑ=0, and q=0
is when the SOI is maximal, ϑ=π). In Eq. ( 7),/angbracketleftS
z/angbracketrightdenotes
an out-of-equilibrium QD (nanowire) magnetization [ 19,20].
The QD magnetization that appears because of an applied biasvoltage in the presence of a finite polarization pis nonvanish-
ing even without an external magnetic field (see Ref. [ 21]).
The temperature Tin Eq. ( 7) stands for the temperature in
the contacts which are assumed to be in equilibrium. Theexpression for QD magnetization /angbracketleftS
z/angbracketrightis obtained from the
steady-state solution of the quantum Langevin equation ofmotion [ 21,22] for the QD spin-1 /2 in the lowest order of
perturbation theory in νJ,
/angbracketleftS
z/angbracketright=pq(eV/T)
2(1−q2p2)+ϕ/parenleftbigeV
T/parenrightbig
(p2+q2), (8)
where ϕ(x)=xcoth( x/2). Nonequilibrium QD magnetiza-
tion described by Eq. ( 8) is limited by /angbracketleftSz/angbracketright=± pq/(p2+q2)
achieved at a large bias voltage eV/greatermuchT.
FIG. 2. Differential conductance in units of g∗=
3e2π2(νJ)2/(4π¯h) as a function of both polarization in
the leads pand effects associated with the accumulation
of the Aharonov-Casher phase ϑparametrized by q=cos(ϑ/2)for
the case of the bias voltage eV/T/greatermuch1.
The appearance of nonequilibrium QD magnetization
/angbracketleftSz/angbracketrightin Eq. ( 7) influences the shape of the peak in the
differential conductance, G(2)=dI(2)/dV|V→0 (see
Refs. [ 21,23]). At large bias voltages eV/greaterorequalslantTthe effects
of saturation of the QD magnetization result in a suppressionof the charge current. At low bias voltages eV/lessmuchT
the contribution to the current proportional to the QDmagnetization is vanishing and the current reaches itsmaximum value. The height of the conductance peakdepends on the QD magnetization slope ∂/angbracketleftS
z/angbracketright/∂(eV)=
pq/[2T(1+p2+q2−p2q2)].
The pandqdependence of the differential conductance
given by Eq. ( 7) is illustrated in Fig. 2ateV/greatermuchT.T h e
leading contribution to the differential conductance calcu-lated in the lowest nonvanishing order of the perturbationtheory for the case of nonmagnetic leads ( p=0) in the
absence of the SOI ( q=1) saturates at G
(2)=g∗, where g∗=
(e2/π¯h)(3/4)(πνJ)2. The conductance peak at zero-bias volt-
age for the case of partial polarization of the leads ( p/negationslash=1) is
G(2)=g∗(1−p2). The linear response (voltage-independent)
part of the differential conductance at a large bias voltage(eV/greatermuchT) is asymptotically given by G
(2)=g∗(3−4p2+
p4)/[3(1+p2)] (see Ref. [ 23]). Spin-dependent tunneling in-
duced by SOI at q<1 enhances the charge transport through
QD at any collinear AP of the reservoirs ( p/negationslash=0). The zero-
bias conductance in the case of fully polarized leads p=
1 is given by G(2)|V→0=g∗(2/3)(1−q2), while the linear
response conductance at large bias voltages is G(2)|V→∞=
g∗(2/3)(1−q2)2/(1+q2). The effect of SOI on the charge
current is maximal at q=0. The polarization dependence of
the differential conductance in this case is given by G(2)=
g∗(1−p2/3).
IV. KONDO CONTRIBUTION TO THE CHARGE CURRENT
The next nonvanishing contribution to the charge current
∝(νJ)3depends on the spin-flip processes and is therefore
described by the Kondo physics. We apply the nonequilib-rium Keldysh Green’s function technique and Abrikosov’spseudofermion representation [ 24] (see details in Ref. [ 20]t o
proceed with the calculations). The current I
(3)=I(3)
K+I(3)
an
235413-3A. V . PARAFILO et al. PHYSICAL REVIEW B 100, 235413 (2019)
consists of two parts: (i) I(3)
Kis originating from the anisotropic
Kondo model [first two lines in Eq. ( 5)],
I(3)
K
g∗4νJ=1
3{(1−p2)[(1+q2)V−2pqSz]
+q2(1+3p2)V−pq(1+q2)(3+p2)Sz}
×log/parenleftbiggD
T∗/parenrightbigg
, (9)
and (ii) I(3)
anis accounting for both the Dzyaloshinskii-Moriya
and charge transfer processes in Eq. ( 5),
I(3)
an
g∗4νJ=(1−q2)
3(1−p2){2V−3pqSz}log/parenleftbiggD
T∗/parenrightbigg
.(10)
Here, Dis the bandwidth of the leads. We use the shorthand
notations Sz=Vcoth( eV/2T)/angbracketleftSz/angbracketrightand T∗=max[|eV|,T].
The third order in the ( νJ)3correction to the charge current
logarithmically grows with a decrease of both the temperatureand the applied bias voltage, revealing a Kondo anomaly .
The validity of the perturbation theory approximation (weak-coupling regime) of Eqs. ( 9) and ( 10) determines the energy
scale T
K, the Kondo temperature. Perturbation theory breaks
down at T/lessorsimilarTK. The effective coupling constants in this
(strong-coupling) regime flow towards the strong-couplingfixed point.
The dependence of the Kondo temperature on the parame-
terspandqis in general determined by the solution of a sys-
tem of coupled renormalization group (RG) equations [ 25,26].
Without losing generality, we parametrize T
Kby the function
f(p,q)[27,28],
TK=Dexp/parenleftbigg
−f(p,q)
2νJ/parenrightbigg
. (11)
The form of f(p,q) is known for several limiting
cases [ 29–31]. In particular, f(p,1)=1 for all |p|/lessorequalslant
1[29,30]. Besides, in the case of nonmagnetic leads ( p=0)
we get f(0,q)=1. The form of f(p,q) on a line p=1f o r
r=1−q/lessmuch1 is found perturbatively from the condition of
breaking down perturbation theory for the differential conduc-tance, f(1,r)=1+r(here, r=ϑ
2/4). With the same logic
we found f(p/lessmuch1,0)=1+2p2/3. A perturbative analysis
leads to a conclusion that f(p,q)/greaterorequalslant1. Moreover, the absolute
minimum of f(p,q) is reached at the symmetry lines (points),
where original Hamiltonian ( 5) is mapped onto an isotropic
Kondo model. We conclude that the Kondo temperature in theDatta-Das transistor with AP polarized electrodes becomes afunction of the Aharonov-Casher phase and degree of spinpolarization in the leads.
The most striking effect of the influence of the Aharonov-
Casher interference onto Kondo scattering is manifested inthe case of full AP polarization, p=1. In particular, the
nonvanishing charge current is controlled by the Aharonov-Casher phase.
The expression for the differential conductance G=
G
(2)+G(3)derived in the limit of small bias voltage eV/lessmuchT
is given by
G=e2
2π¯h(πνJ)2(1−q2)/parenleftbigg
1+q24νJlogD
T/parenrightbigg
. (12)
FIG. 3. Dependence of zero-bias differential conductance [in
units of g∗=(3/4)(e2/π¯h)(πνJ)2] on the Aharonov-Casher phase
for the case of full AP polarization of the leads p=1. The dotted-
dashed line denotes differential conductance G(2)obtained from
Eq. ( 7), and the dashed line is the third-order ∝(νJ)3perturbative
correction to the conductance G(3)determined by Eqs. ( 9)a n d( 10).
The solid line represents the sum of G(2)andG(3). The figure is
plotted for the following values of the model parameters: D=1e V ,
T=1K ,a n d νJ=0.1.
The conductance dependence on the Aharonov-Casher phase
ϑi ss h o w ni nF i g . 3. Charge current at p=1 and q=1
is blocked by the Pauli principle. However, the absence ofcharge current is not in contradiction to the presence ofresonant Kondo scattering. The Kondo effects (and T
K)a r e
fully determined by multiple pseudospin-flip processes on aQD and a single ( LorR) contact [ 29]. As it is seen from
Eq. ( 12), the logarithmic corrections to the conductance are
positive for J>0 when q/negationslash=1 and increase with decreasing
T. While G
(2)is monotonously increased with ϑ, the behavior
ofG(3)is nonmonotonous due to the interplay between the
Kondo effect and the Aharonov-Casher interferometer. As aresult, the maximal current is reached at some critical value
q
cr=√
[1−(4νJ)−1log(T/D)]/2 which depends on Tand
the initial parameters of the model.
V. SOI INFLUENCE ON KONDO TEMPERATURE
The Hamiltonian Eq. ( 5) casts a simple form for p=1,
˜Hex=J{(szSz+sySy) cos(ϑ/2)+sxSx}
−1
2Jsin(ϑ/2)/summationdisplay
kk/primec†
kγck/primeγSx. (13)
The Hamiltonian ( 13) is derived from Eq. ( 5) by retaining the
operators ckγ(c†
kγ) with γ=1f o r( L,↑),γ=− 1f o r( R,↓),
and/vectors=/summationtext
kk/prime(1/2)c†
kγˆσγγ/primeck/primeγ/prime. We omit all other terms in
Eq. ( 5) with zero expectation values. Equation ( 13) describes
ananisotropic Kondo-like model with an additional term,
which couples the spin in the QD with the charge densityin the leads. The last term in ( 13) can be viewed as an
extra potential scattering and is therefore disregarded for theparticle-hole symmetric limit.
As it is known both from the RG and exact Bethe anzatz
solution of the Kondo model, the maximal Kondo temperatureis achieved in the isotropic case [ 27]. The Kondo temperature
in the anisotropic case is defined through the Bethe anzatz
235413-4KONDO EFFECT IN A AHARONOV-CASHER … PHYSICAL REVIEW B 100, 235413 (2019)
solution [ 27] by an equation equivalent to ( 11) with the
dimensionless function f(1,q) dependent on the anisotropy
parameter q=cos(ϑ/2),
f(1,q)=−1/radicalbig
1−q2log/parenleftBigg
1−/radicalbig
1−q2
q/parenrightBigg
. (14)
The anisotropy controlled by SOI suppresses the Kondo tem-
perature. The asymptotic behavior of Eq. ( 14) in the limit
of weak ( ϑ→0) SOI is given by f(1,ϑ)−1≈ϑ2/12.
Similar behavior is also obtained from the renormalizationgroup treatment under condition νJ/lessmuch1. While the cases
of small anisotropy can be accessed perturbatively [see thediscussion after Eq. ( 11)], or by the Bethe anzatz solution
(p=1) [see Eq. ( 14)], the solution for the function f(p,q)
for arbitrary values of its arguments −1<(p,q)<1 remains
an interesting and unsolved problem [ 32].
VI. SUMMARY AND OUTLOOK
The interplay between resonance Kondo scattering in the
quantum wire, effects of SOI in the tunnel barriers, andpartial spin polarization in the leads provides a high levelof controllability for charge transport through a nanodevice.In particular, the fine tuning of the Kondo temperature isachieved by control of three independent tunable parametersof the system. First, the Aharonov-Casher phase is tuned bythe electric field applied to the area of the nanowire. Second,the degree of spin polarization in the leads is manipulated byt h es p i nv a l v e[ 12–14]. Third, the local out-of-equilibrium QD
magnetization of the nanodevice is controlled by the source-drain voltage. The central result of the paper is a predictionof a finite charge current through the nanowire even at fullAP polarization of the leads in the presence of a nonzerospin-orbit interaction. Besides, perturbative (weak-coupling)calculations demonstrate pronounced (logarithmic) effects ofenhancement of the current by SOI at any given partialpolarization of the leads. Competition between the resonancescattering resulting in a maximal Kondo temperature in theabsence of SOI at ϑ=0 and quantum interference due to
the Aharonov-Casher effect that is maximal at ϑ=πallows
one to find an optimal strength of SOI at ϑ
cr(qcr) under the
condition of maximizing the electric current.
ACKNOWLEDGMENTS
This work was financially supported by IBS-R024-D1. A.P.
thanks the Condensed Matter and Statistical Physics section atThe Abdus Salam International Centre for Theoretical Physicsfor hospitality. The work of M.K. was performed in part atthe Aspen Center for Physics, which is supported by NationalScience Foundation Grant No. PHY-1607611 and partiallysupported by a grant from the Simons Foundation.
APPENDIX A: SPIN-ORBIT INTERACTION AND
SPIN-DEPENDENT TUNNEL MATRIX ELEMENTS
In this Appendix we calculate the Aharonov-Casher phase
ϑ[see Eq. ( 3)]. We start from a Schrödinger equation for
the electron in a 1D nanowire in the presence of an externalhomogeneous electric field E
zproduced by the back gate anddirected perpendicular to nanowire,
−¯h2
2m∂2/vectorψ
∂x2−αˆσyi¯h∂/vectorψ
∂x=(EF−ε)/vectorψ, (A1)
where /vectorψis a two-component electron wave function (spinor),
andα∝Ezis a spin-orbit interaction coupling constant. Since
ε/lessorequalslant¯hvFL−1/lessmuchEF, we can present the electron’s wave func-
tion in terms of right- and left-moving parts,
/vectorψ=eipFx/¯h/vectorψ+(x)+e−ipFx/¯h/vectorψ−(x), (A2)
where pF=√2m(EF−ε) is a Fermi momentum. Substitut-
ing the wave function Eq. ( A2) into Eq. ( A1) and neglecting
the second derivative, we get
−ivF¯h∂/vectorψ±
∂x=αpFˆσy/vectorψ±. (A3)
From this equation one can see that the spinor function /vectorψ±
satisfies the relation
/vectorψ±(x)=ˆU(x)/vectorψ±(0),ˆU(x)=eiˆσyϑ(x)/2, (A4)
where ϑ(x)=2αxpF/(¯hvF) is an Aharonov-Casher phase.
Despite the fact that in the presence of SOI the electronic
spin is an unsuitable quantum number for the classificationof the electronic states, the energy levels continue to be
doubly degenerate. If /vectorψ
λis the eigenstate with energy ε, then
/vectorψ¯λ=iˆσy/vectorψ∗
λis also an eigenstate with the same energy. Here,
we used notations /vectorψλ(¯λ)for the normalized electron wave
function in the nanowire. As this takes place one can classifythe states by the spin structure of the wave functions at afixed point, for example, at x=0 the middle point of the
nanowire between the left and right electrodes. Assuming that
eigenstates /vectorψ
λ(¯λ)(0) in the middle of the nanowire correspond
to a state with spin up and spin down, we define the value ofthe wave function in the point x
1,x2where tunneling into the
leads occurs ( x1(2)=±l/2),
/vectorψλ(x1)=ˆU(x1,0)/parenleftbigg
1
0/parenrightbigg
=/parenleftbigg
cos(ϑ/4)
sin(ϑ/4)/parenrightbigg
,
(A5)
/vectorψλ(x2)=ˆU(x2,0)/parenleftbigg
1
0/parenrightbigg
=/parenleftbigg
cos(ϑ/4)
−sin(ϑ/4)/parenrightbigg
,
/vectorψ¯λ(x1)=ˆU(x1,0)/parenleftbigg
0
1/parenrightbigg
=/parenleftbigg
−sin(ϑ/4)
cos(ϑ/4)/parenrightbigg
,
/vectorψ¯λ(x2)=ˆU(x2,0)/parenleftbigg
0
1/parenrightbigg
=/parenleftbigg
sin(ϑ/4)
cos(ϑ/4)/parenrightbigg
. (A6)
The amplitude of electron tunneling from the left lead to the
nanowire in state λ,¯λcan be found as follows,
V↑λ
kL=/summationdisplay
kVkL/angbracketleft(↑,0)|/vectorψλ(x1)/angbracketright,
(A7)
V↓λ
kL=/summationdisplay
kVkL/angbracketleft(0,↓)|/vectorψλ(x1)/angbracketright,
V↑¯λ
kL=/summationdisplay
kVkL/angbracketleft(↑,0)|/vectorψ¯λ(x1)/angbracketright,
(A8)
V↓¯λ
kL=/summationdisplay
kVkL/angbracketleft(0,↓)|/vectorψ¯λ(x1)/angbracketright,
235413-5A. V . PARAFILO et al. PHYSICAL REVIEW B 100, 235413 (2019)
where VkLis the transition amplitude. The tunnel ma-
trix element for tunneling processes from the right leadcan be defined in a similar way by using wave functions
/vectorψ
λ(¯λ)(x2).
APPENDIX B: SCHRIEFFER-WOLFF TRANSFORMATION
In this Appendix we derive the effective Kondo Hamil-
tonian for the general case of spin-dependent tunnel matrixelements. The mapping of the Anderson-like impurity modelEqs. ( 1) and ( 2) onto a Kondo-like model is done using a
standard Schrieffer-Wolff transformation,
H
eff=eSHe−S≡H+[S,H]+1
2[S,[S,H]]+··· .(B1)The first step is to eliminate the first order in the tunneling
amplitude terms using the following condition,
[S,H0]=−Htun. (B2)
As a result, the effective Hamiltonian is transformed to
Heff=H0+1
2[S,Htun]+··· . (B3)
We choose operator Sin the following form,
S=/bracketleftbigg/summationdisplay
(Aασλ+Bασλˆn¯λ)c†
kασdλ−H.c./bracketrightbigg
. (B4)
Using the condition given by Eq. ( B2) one can determine
the constants Aασλ,Bασλ. After straightforward calculations
using Eqs. ( B3) and ( B4), we obtain the effective Hamiltonian
Heff=Hdir+Hex. The first term responsible for the electron
potential scattering between leads is given by
Hdir=/summationdisplayKαα/prime
4/bracketleftbiggnαα/prime
2[V↑↑
kα(V↑↑
k/primeα/prime)∗+V↓↓
kα(V↓↓
k/primeα/prime)∗+V↓↑
kα(V↑↓
k/primeα/prime)∗+V↑↓
kα(V↓↑
k/primeα/prime)∗]
+sz
αα/prime[V↑↑
kα(V↑↑
k/primeα/prime)∗−V↓↓
kα(V↓↓
k/primeα/prime)∗−V↓↑
kα(V↑↓
k/primeα/prime)∗+V↑↓
kα(V↓↑
k/primeα/prime)∗]
+sx
αα/prime[V↑↑
kα(V↑↓
k/primeα/prime)∗+V↑↓
kα(V↓↓
k/primeα/prime)∗+V↓↓
kα(V↓↑
k/primeα/prime)∗+V↓↑
kα(V↑↑
k/primeα/prime)∗]
+isy
αα/prime[V↑↑
kα(V↑↓
k/primeα/prime)∗+V↑↓
kα(V↓↓
k/primeα/prime)∗−V↓↓
kα(V↓↑
k/primeα/prime)∗−V↓↑
kα(V↑↑
k/primeα/prime)∗]/bracketrightbigg
. (B5)
The matrix elements used in Eq. ( B5)a r eg i v e nb y
Kαα/prime=1
εkα−ε0+1
εk/primeα/prime−ε0−1
UC+ε0−εkα−1
UC+ε0−εk/primeα/prime. (B6)
The next step is to find the Hamiltonian responsible for the exchange processes,
Hex=H1+H2+H3, (B7)
where the Hamiltonian H1,
H1=/summationdisplayJαα/prime
2/bracketleftbigg
sz
αα/primeSz[V↑↑
kα(V↑↑
k/primeα/prime)∗+V↓↓
kα(V↓↓
k/primeα/prime)∗−V↓↑
kα(V↑↓
k/primeα/prime)∗−V↑↓
kα(V↓↑
k/primeα/prime)∗]
+sx
αα/primeSx[V↑↑
kα(V↓↓
k/primeα/prime)∗+V↓↓
kα(V↑↑
k/primeα/prime)∗+V↓↑
kα(V↓↑
k/primeα/prime)∗+V↑↓
kα(V↑↓
k/primeα/prime)∗]
+sy
αα/primeSy[V↑↑
kα(V↓↓
k/primeα/prime)∗+V↓↓
kα(V↑↑
k/primeα/prime)∗−V↓↑
kα(V↓↑
k/primeα/prime)∗−V↑↓
kα(V↑↓
k/primeα/prime)∗/bracketrightbigg
, (B8)
is equivalent to an anisotropic Kondo model. The Hamiltonian H2,
H2=/summationdisplayJαα/prime
4/bracketleftbigg
nαα/primeSz[V↑↑
kα(V↑↑
k/primeα/prime)∗−V↓↓
kα(V↓↓
k/primeα/prime)∗+V↓↑
kα(V↑↓
k/primeα/prime)∗−V↑↓
kα(V↓↑
k/primeα/prime)∗]
+nαα/primeSx[V↑↑
kα(V↓↑
k/primeα/prime)∗+V↑↓
kα(V↑↑
k/primeα/prime)∗+V↓↓
kα(V↑↓
k/primeα/prime)∗+V↓↑
kα(V↓↓
k/primeα/prime)∗]
+inαα/primeSy[−V↑↑
kα(V↓↑
k/primeα/prime)∗+V↑↓
kα(V↑↑
k/primeα/prime)∗+V↓↓
kα(V↑↓
k/primeα/prime)∗−V↓↑
kα(V↓↓
k/primeα/prime)∗]/bracketrightbigg
, (B9)
describes the coupling between the spin-1 /2 on the dot and the “charge” density in the leads nαα/prime.
The Hamiltonian H3,
H3=/summationdisplayJαα/prime
2/bracketleftbigg
isy
αα/primeSx[V↑↑
kα(V↓↓
k/primeα/prime)∗−V↓↓
kα(V↑↑
k/primeα/prime)∗−V↓↑
kα(V↓↑
k/primeα/prime)∗+V↑↓
kα(V↑↓
k/primeα/prime)∗]
+isx
αα/primeSy[−V↑↑
kα(V↓↓
k/primeα/prime)∗+V↓↓
kα(V↑↑
k/primeα/prime)∗−V↓↑
kα(V↓↑
k/primeα/prime)∗+V↑↓
kα(V↑↓
k/primeα/prime)∗]
+sx
αα/primeSz[V↑↑
kα(V↑↓
k/primeα/prime)∗−V↑↓
kα(V↓↓
k/primeα/prime)∗−V↓↓
kα(V↓↑
k/primeα/prime)∗+V↓↑
kα(V↑↑
k/primeα/prime)∗]
+sz
αα/primeSx[V↑↑
kα(V↓↑
k/primeα/prime)∗+V↑↓
kα(V↑↑
k/primeα/prime)∗−V↓↓
kα(V↑↓
k/primeα/prime)∗−V↓↑
kα(V↓↓
k/primeα/prime)∗]
235413-6KONDO EFFECT IN A AHARONOV-CASHER … PHYSICAL REVIEW B 100, 235413 (2019)
+isy
αα/primeSz[V↑↑
kα(V↑↓
k/primeα/prime)∗−V↑↓
kα(V↓↓
k/primeα/prime)∗+V↓↓
kα(V↓↑
k/primeα/prime)∗−V↓↑
kα(V↑↑
k/primeα/prime)∗]
+isz
αα/primeSy[−V↑↑
kα(V↓↑
k/primeα/prime)∗+V↑↓
kα(V↑↑
k/primeα/prime)∗−V↓↓
kα(V↑↓
k/primeα/prime)∗+V↓↑
kα(V↓↓
k/primeα/prime)∗]/bracketrightbigg
, (B10)
accounts for the Dzyaloshinskii-Moriya-like interaction.
The exchange coupling constant reads as follows,
Jαα/prime=1
εkα−ε0+1
UC+ε0−εkα+1
εk/primeα/prime−ε0+1
UC+ε0−εk/primeα/prime. (B11)
Substituting parametrization Eq. ( 4) into Eqs. ( B6) and ( B7) and considering exchange coupling constants of conduction
electrons at the Fermi energy, εkα=εk−μσ
α≈0, we can obtain Hamiltonian Eq. ( 5).
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235413-7 |
PhysRevB.73.024109.pdf | Phase transition and electronic properties of fluorene: A joint experimental and theoretical
high-pressure study
Georg Heimel *
School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400, USA
and Institute of Solid State Physics, Graz University of Technology, Petersgasse 16, A-8010 Graz, Austria
Kerstin Hummer and Claudia Ambrosch-Draxl
Institute of Physics—Division of Theoretical Physics, University of Graz, Universitätsplatz 5, A-8010 Graz, Austria
Withoon Chunwachirasiri and Michael J. Winokur
Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA
Michael Hanfland
European Synchrotron Radiation Facility, BP220, F-38043 Grenoble, France
Martin Oehzelt, Andreas Aichholzer, and Roland Resel
Institute of Solid State Physics, Graz University of Technology, Petersgasse 16, A-8010 Graz, Austria
/H20849Received 3 May 2005; revised manuscript received 10 November 2005; published 19 January 2006 /H20850
We report a structural phase transition of fluorene and the resulting changes of its electronic and optical
properties investigated by a combination of experimental and theoretical methods. Fluorene is a /H9266-conjugated
organic compound that crystallizes in the orthorhombic space group Pnma with four molecules per unit cell.
We probe the stability of the molecular arrangement by x-ray powder diffraction experiments under hydrostaticpressure up to 14 GPa. Our measurements reveal a fully reversible crystallographic phase transition at3.6±0.3 GPa indicated by abrupt changes in the lattice constants, which are accompanied by a re-arrangementof the molecules. The orientation of the molecules relative to each other evolves from the familiar herringbonepattern towards
/H9266-stacking. This results in dramatic modifications of the electronic structure and thus the
optical response as revealed by density functional calculations. In particular, the effective hole masses in thehigh-pressure phase become comparable to those of conventional semiconductors.
DOI: 10.1103/PhysRevB.73.024109 PACS number /H20849s/H20850: 61.10.Nz, 61.66.Hq, 61.50.Ks, 71.20.Rv
I. INTRODUCTION
Organic /H9266-conjugated molecules and polymers have at-
tracted great attention over the last three decades. The com-bination of their semiconducting properties with their poten-tial for low-cost and large-area solution processing as well asmechanical flexibility make them promising candidates for/H20849opto- /H20850electronic applications. Indeed, light-emitting
diodes,
1,2field-effect transistors,3–8and photovoltaic devices
/H20849solar cells /H208509–11have been realized on the basis of such
/H9266-conjugated organic polymers and small molecules.
Many material properties of organic molecular crystals
and polymers can be derived from that of an isolated mol-ecule or polymer chain. However, intermolecular interac-tions, inherently present in the solid state, crucially influenceimportant device parameters. Examples of these includecharge transport and optical properties.
12–14The strength and
the nature of these intermolecular interactions is determinedby the distances between individual molecules and their ori-entation relative to each other. A clean way to study the
impact of these attributes is to apply high hydrostatic pres-sure on crystalline samples. This allows for a systematic re-duction in the intermolecular distances and, additionally,changes in their relative orientation without altering thechemical structure of the molecules.In the present study, we investigate crystalline fluorene
/H20849C
13H10/H20850. We performed x-ray powder diffraction experi-
ments on fluorene under hydrostatic pressure of up to
14 GPa. A crystallographic structural phase transition is
found at 3.6±0.3 GPa.15,16The full crystal structure of this
additional high-pressure /H20849HP/H20850phase could be solved from
the powder diffraction data. We report the compressibility aswell as the pressure evolution of the lattice parameters andthe molecular orientation in the unit cell for both the ambientand the HP phase. This discontinuous change in the molecu-lar packing and thus in the intermolecular interactions repre-sents a unique possibility to study the influence of the mo-lecular arrangement on relevant solid-state properties.Complementing the experiments, we performed first-
principles calculations within the framework of density func-
tional theory /H20849DFT /H20850. The pressure dependence of the band
structures and the dielectric tensors in crystalline fluorene isreported for both, the low-pressure /H20849LP/H20850and the HP phase.
This allows for direct comparisons of the optical and elec-tronic properties of the same molecule in vastly differentcrystalline environments.
II. MATERIAL PROPERTIES
In Fig. 1, the chemical structure of fluorene /H20849C13H10/H20850is
shown. At ambient conditions, this compound crystallizes inPHYSICAL REVIEW B 73, 024109 /H208492006 /H20850
1098-0121/2006/73 /H208492/H20850/024109 /H2084913/H20850/$23.00 ©2006 The American Physical Society 024109-1the orthorhombic space group Pnma /H20849no. 62 /H20850with Z
=4 molecules per unit cell /H20849two at y=1
4, two at y=3
4/H20850.17,18
Within one unit cell, the molecules form a double-layered
structure with all their long molecular axes being parallel tothe long unit cell axis b. The methylenic bridges of the mol-
ecules in the top layer point in the direction opposite to thoseof the bottom layer. A side view of the unit cell in theambient-pressure phase is shown in the left panel of Fig. 2.Within one such layer, the two translationally inequivalentmolecules are arranged in the herringbone pattern, typical
for rigid-rod-like organic molecules. The angle between theplanes of two molecules is the herringbone angle
/H9273. The
position of the molecule in the acplane is defined by xand
z/H20849see Fig. 2, right panel /H20850.
III. METHODOLOGY
A. Experimental setup
Two independent sets of experiments were performed. In
both cases, the polycrystalline powder of fluorene was firstfinely ground in order to provide good statistics even from asmall sample volume. The first series of data was recorded atthe 6ID-B beamline of the Advanced Photon Source /H20849APS /H20850at
the Argonne National Laboratory. High pressure was appliedusing a Merryl-Basset type diamond anvil cell
19,20/H20849DAC /H20850from High Pressure Diamond Optics, Inc. /H20849Tucson, AZ /H20850with
16 faceted, brilliant cut diamonds with a culet size of0.6 mm. As the gasket material, 0.2 mm stainless spring steel/H20849pre-indented to an initial thickness of /H110110.1 mm /H20850was uti-
lized with a 0.3 mm sample hole. Cryogenically loaded liq-uid nitrogen was employed as the pressure-transmitting me-dium and the ruby fluorescence method
21,22was applied to
determine the pressure in the cell. A MAR345 image-platedetector was used to collect the data in Debye-Scherrer ge-ometry with an average integration time of /H110113 min. The
x-ray wavelength was determined to be /H9261
APS=0.414 131 Å
via a silicon powder measurement. Additionally, CeO 2pow-
der was measured as a calibration standard for the MAR345.Several pressure runs up to /H1101110 GPa /H20849and back down /H20850were
conducted. The second series of data was collected at theID09 beamline at the European Synchrotron Radiation Facil-
ity/H20849ESRF /H20850in Grenoble, France. High pressure was applied
using a membrane driven DAC with diamonds of a culet sizeof 0.6 mm. As the gasket material, stainless steel /H20849pre-
indented to an initial thickness of /H110110.09 mm /H20850was utilized
with a 0.25 mm sample hole. This time, helium was em-ployed as the pressure-transmitting medium and loaded at apressure of 1200 bar with a gas loading system and the rubyfluorescence method
21,22was used for pressure calibration. A
MAR345 image-plate detector was used to collect the data inDebye-Scherrer geometry with integration times of30–120 s. To improve powder averaging, the DAC wasrocked by ±3° during exposure. Silicon powder was mea-sured as a calibration standard for the MAR345. A singlepressure run of up to /H1101114 GPa /H20849and back down /H20850was con-
ducted.
The two-dimensional image-plate data was further pro-
cessed using the
FIT2D software.23–25For the APS data, the
CeO 2standard was used to calibrate the sample-to-detector
FIG. 1. Chemical structure of fluorene /H20849C13H10/H20850.
FIG. 2. Side /H20849left panel /H20850and top /H20849right panel /H20850view of the fluorene crystal structure at ambient pressure.HEIMEL et al. PHYSICAL REVIEW B 73, 024109 /H208492006 /H20850
024109-2distance /H20849/H1101160 cm /H20850and the detector misalignment. In the
case of the ESRF data, a Silicon standard was used to deter-
mine the wavelength /H20849/H9261ESRF=0.411 592 Å /H20850, the sample-to-
detector distance /H20849/H1101150 cm /H20850, and the detector misalignment.
The beam center was calibrated individually for each Debye-
Scherrer ring pattern. Spikes due to granularity of the samplepowder were masked prior to the subsequent circular integra-tion of the two-dimensional diffraction patterns in order toobtain one-dimensional two-theta scans.
The one-dimensional two-theta scans were refined with
the
GSAS program package.26,27A manually determined back-
ground was subtracted from the raw patterns. In order todecrease the degrees of freedom in the Rietveld refinement,therigid-body approximation
28,29was applied. There are es-
sentially two different kinds of atom-atom interactions inmolecular crystals. On one hand, there are strong covalentintramolecular bonds and, on the other hand, there are weakforces acting between adjacent molecules. Results on com-parable compounds, where this refinement strategy has beensuccessfully applied,
30,31suggest that, upon applying hydro-
static pressure, only the latter are significantly reduced. Thechanges in the intramolecular bondlengths remained below0.015 Å up to a pressure of 10 GPa.
32,33Also in our case,
results from theoretical calculations /H20849presented later in this
work /H20850justify to keep the internal molecular geometry fixed
upon compression of the crystal.
The asymmetric unit in the ambient-pressure space group
Pnma of fluorene is one phenyl ring plus the methylenic
bridge. A mirror plane at y=1
4generates the second half of
the molecule. Leaving open all translational and rotationaldegrees of freedom of the asymmetric unit during the refine-ment procedure could result in chemically and physicallyimplausible conformations of the molecule. Consequently,the refinement was carried out in space group Pna2
1/H20849no.
33/H20850, a nonisomorphic subgroup of Pnma , where the asym-
metric unit contains one whole molecule. The conformationof this molecule was taken from the single-crystal solution.
17
In order to formally conserve full Pnma symmetry, only the
rotation of the molecule around the long unit cell vector b
and the translations along aandcwere refined. The molecule
was held fixed with its center of mass at y=1
4and its long
axis was held parallel to the long unit cell bdirection.
Rigid-rod-like organic molecules that crystallize in a lay-
ered herringbone structure tend to form slab-shaped crystal-lites with the faces of the slabs perpendicular to the /H20851010/H20852
direction /H20849provided bis the long unit cell axis /H20850. Due to the
sample preparation and the loading procedure of the DAC, aslightly preferred orientation of the slabs parallel to the culetsurfaces and thus perpendicular to the incident beam direc-tion /H20849in Debye-Scherrer geometry /H20850is likely to occur. In order
to compensate for this effect in the refinement procedure, theMarch-Dollase function
34,35for preferentially oriented pow-
ders /H20849in the present case /H20851010/H20852/H20850was applied. This approach is
recommended in the GSAS manual26for high-pressure work.
Other than the abovementioned three parameters describ-
ing the molecular arrangement and the March-Dollase pre-ferred /H20851010/H20852orientation, the lattice parameters a,b, and c,
zero, and scaling were refined within the Rietveld
procedure.
36For the peak profile function, GSAS type 4 waschosen26,27,37–41because it includes model parameters for mi-
crostrain broadening42of the diffraction peaks, which seems
appropriate when dealing with high-pressure data. Of all pro-file parameters, only GV,GW, and LXwere refined, as well
asS
400andS220, if applicable. All other parameters were held
at their default values.26,27
B. Theoretical approach
For the ground state we apply the full-potential aug-
mented plane wave plus local orbitals formalism43as imple-
mented in the WIEN2K code44utilizing the local density ap-
proximation /H20849LDA /H2085045for treating exchange and correlation
effects. The performance of LDA versus the generalized gra-dient approximation /H20849GGA /H20850for comparable compounds has
been discussed in prior studies.
32,33The crystal unit cells as a
function of pressure, which have served as input parametersfor the ab initio calculation of the ground-state energies,
band structures, and dielectric functions, were obtained fromthe x-ray data as follows. After interpolating the experimen-tally observed lattice parameters a,b, and cas well as the
molecular positions and orientation x,z, and
/H9273/H20849see Fig. 2 /H20850by
a fit function, the crystal unit cells were accordingly con-structed on an equidistant pressure grid. While feasible inprinciple, the full optimization of the lattice parameters hasbeen omitted for the following reasons: /H20849i/H20850Prior studies have
shown that, while not quite as severely as in purely van derWaals bound systems, LDA still considerably underestimatesthe equilibrium volume of aromatic molecular crystals,whereas it is overestimated by GGA.
32,33Since it is one goal
of this study to investigate the evolution of the electronic andoptical properties of crystalline fluorene with pressure, it ap-pears more relevant for the present work to start from theexperimentally determined lattice parameters. /H20849ii/H20850Ab initio
calculations of systems of this size /H2084992 atoms per unit cell,
large lattice constants, low symmetry /H20850are at the limit of
state-of-the-art computational resources. A full optimizationof all three lattice constants would, therefore, not be possiblewithin a reasonable time frame.
In the ground-state calculation, the internal geometry of
the molecules was optimized; i.e., the atomic positions wererelaxed by minimizing the forces acting on each atom to beless than 2 mRy/a.u. in magnitude. In each self-consistentcycle, the atomic forces were converged better than1 mRy/a.u. resulting in an energy convergence of less than0.1 meV. Based on these ground-state configurations, theband structures were computed on a discrete k-mesh /H20849242
k-points in total /H20850along high-symmetry directions in the irre-
ducible wedge of the Brillouin zone /H20849IBZ/H20850as illustrated in
Fig. 3. The internal coordinates of the high-symmetry points/H9003,X,S,Y,Z,T,R, and Uin units of /H208492
/H9266/a,2/H9266/b,2/H9266/c/H20850are
/H208490,0,0 /H20850,/H208490.5,0,0 /H20850,/H208490.5,0.5,0 /H20850,/H208490,0.5,0 /H20850,/H208490,0,0.5 /H20850,/H208490,0.5,0.5 /H20850,
/H208490.5,0.5,0.5 /H20850, and /H208490.5,0,0.5 /H20850, respectively. Note that /H9003X/H20849/H9003Y/H20850
is parallel to the crystalline a/H20849b/H20850axis, whereas /H9003Zcorre-
sponds to the cdirection.
The imaginary part of the dielectric tensor /H92552i/H20849/H9275/H20850was cal-
culated within the random phase approximation /H20849RPA /H20850:46
/H92552i/H20849/H9275/H20850=8/H92662
/H9024/H92752/H20858
vck/H20841/H20855vk/H20841pi/H20841ck/H20856/H208412/H9254/H20849/H9280ck−/H9280vk−/H9275/H20850, /H208491/H20850
where the indices v/H20849c/H20850andkrepresent the valence /H20849conduc-
tion/H20850band and a vector kin the IBZ. /H9024and/H9275stand for thePHASE TRANSITION AND ELECTRONIC PROPERTIES … PHYSICAL REVIEW B 73, 024109 /H208492006 /H20850
024109-3crystal volume and the frequency of the incoming light, re-
spectively. /H20855vk/H20841pi/H20841ck/H20856are the optical dipole matrix elements,
where pidenotes the ith Cartesian component of the momen-
tum operator. The /H20841nk/H20856are approximated by the Kohn-Sham
orbitals and their corresponding energies /H9280nk. The dipole ma-
trix elements that provide the selection rules for the /H92552i/H20849/H9275/H20850
were computed on a dense grid of /H2084913/H110036/H1100320/H20850k-points in
an energy window from plus to minus 2.0 Ry with respect to
the Fermi level. Regarding the optical properties, we notethat the RPA does not allow for the description of excitoniceffects. The inclusion of the same by solving the Bethe-Salpeter equation, however, did not change the qualitativeconclusions that could be drawn from the pressure depen-dence of the RPA spectra for comparable compounds.
47
IV . RESULTS AND DISCUSSION
A. Experimental results
1. Structure solution
During the pressure runs at the APS synchrotron source,
clear evidence for a phase transition could be found between2.5 and 3.0 GPa. However, the new phase appeared veryinhomogeneously and anisotropically, leading to isolated,very bright spots on the newly appearing, otherwise homo-geneous Debye-Scherrer rings. Even excessive masking forthe circular integration of the two-dimensional patternsyielded one-dimensional two-theta scans of poor quality andbad statistics that rendered further processing impossible.Moreover, the utilized pressure medium, liquid nitrogen,freezes at 2.7 GPa.
19Due to the weak intermolecular bind-
ing, most organic crystals are highly sensitive to anisotropicshearing stresses.
48Nonhydrostatic pressure above 2.7 GPa
could have led to a distorted, poorly ordered new phase. In asecond attempt, gaseous helium, covering a wide pressurerange of guaranteed hydrostaticity,
19was used as the pressure
transmitting medium at the ESRF. This approach yielded dif-fraction patterns of superior quality up to 14 GPa, clearlyexhibiting a structural phase transition between 3.2 and3.9 GPa /H20849see Fig. 4 /H20850. The peaks of the new HP phase were
fitted individually with a pseudo-Voigt line shape in order toextract their two-theta values. This was achieved by employ-
ing the
XFIT program.49The two-theta values of the 20 most
clearly resolved peaks were indexed with DICVOL .50
The outermost left peak in the diffraction patterns shown
in Fig. 4 is the 020 reflection in the LP phase, correspondingto half the unit cell baxis, half the unit cell height /H20849see Fig.
2/H20850. This peak exhibits minimal changes during the phase
transition and thus we conclude that the layered structure ofstanding molecules is conserved in the HP phase. In addition,fluorene is a rather bulky molecule and so would not beexpected to undergo massive rearrangements within onelayer. Furthermore, this phase transition proved to be fullyreversible /H20849judging from the diffraction patterns in the re-
laxed DAC after pressure runs using either nitrogen or he-lium as the pressure-transfer fluid /H20850. Finally, we note that
many other representative rigid-rod-like conjugated mol-ecules having a crystalline layered herringbone structure ex-hibit polymorphism wherein the respective polymorphs arevery similar to each other.
51–56Consequently, the HP unit cell
needs to be large enough to accommodate four molecules ina layered structure /H20849as is the case in the LP unit cell /H20850. The
structure with the highest figure of merit produced by
DICVOL within the orthorhombic crystal class fulfills these
requirements.
In order to determine the exact spacegroup among all
orthorhombic lattices, the software CHECKCELL57was ap-
plied. Among the manifold of all 59 orthorhombic spacegroups, only two left the four fluorene molecules per unit cellchemically intact /H20849noncrosslinked, reasonable inter- versus
intramolecular bond lengths, etc. /H20850:Pnma , the space group of
the LP phase, or a subgroup thereof /H20849Pna2
1/H20850, the space group
in which the further refinements were performed /H20849vide su-
pra/H20850. Both are fully in line with the abovementioned argu-
ments and, at the same time, follow general observations forthe packing of rigid, polyaromatic molecules.
58Within the
space group Pna21, the structure factors were extracted from
the peak intensities by performing a LeBail fit59on the ex-
perimental diffraction pattern that was used for indexing. Theprogram package
RIETICA60was used to accomplish this step.
The unit cell parameters, zero, and the peak profile param-
eters were refined in the course of the LeBail extraction. The
FIG. 3. The k-path in the Brillouin zone used for the computa-
tion of the electronic band structures of fluorene.
FIG. 4. Powder diffraction pattern /H20849ESRF data /H20850of fluorene at a
series of increasing pressure. Between 3.2 and 3.9 GPa, new fea-tures appear /H20849solid ellipse /H20850. The 020 peak /H20849dashed ellipse /H20850and thus
the layered structure is conserved in the additional HP phase.HEIMEL et al. PHYSICAL REVIEW B 73, 024109 /H208492006 /H20850
024109-4space group, unit cell parameters, and experimental structure
factors then entered the Monte-Carlo program suite ESPOIR .61
The molecular shape was fixed to that of the ambient-conditions single-crystal solution /H20849rigid body /H20850.
17This proce-
dure yielded a final structure for the HP phase, where the/H20849calculated /H20850diffraction pattern perfectly matched the mea-
sured one.
In a last step, all the obtained information of the newly
solved HP phase was entered into
GSAS , where a full rigid-
body Rietveld refinement revealed details about peak pro-files, preferred orientation, and exact structural changes withthe greatest accuracy. Top and side views of the HP phase areshown in Fig. 5 /H20849compare Fig. 2 /H20850. As mentioned above, the
molecules retain their double-layer arrangement with theirlong molecular axis parallel to the crystal baxis. In contrast
to the ambient-pressure phase, the two layers are somewhatshifted along awith respect to each other. Within one layer,
the molecular packing changes more drastically. The latticeparameter ais elongated, while cis dramatically shortened.
As a consequence, the molecular arrangement transformsfrom herringbone to
/H9266-stacking , and this new packing is ex-
pected to significantly alter the electronic and optical prop-erties of the crystal.
The refinements for the LP phase /H208491.0 GPa /H20850, both phases
coexistent /H208493.9 GPa /H20850, and for the pure HP phase /H208497.3 GPa /H20850
are shown in Fig. 6. The overall quality of the refinement
further supports the proposed structure of the new HP phase.
2. Lattice parameters
The pressure-dependent lattice parameters a,b, and care
presented in Fig. 7 for both, the LP and the HP phase. For theLP phase, the data obtained at APS is included as well. Theoverall good agreement between the APS and the ESRF datasets emphasizes the reliability of the measurements and thesubsequent treatment of the data. For the equilibrium latticeparameters, we obtain a=8.509 Å, b=18.910 Å, and c=5.726 Å, which is in good agreement with the single-
crystal solution of the fluorene structure at ambient condi-tions /H20849a=8.475 Å, b=18.917 Å, and c=5.717 Å /H20850.
17The con-
sistency of the data points taken while releasing the pressure
after the upwards pressure run demonstrates the full revers-ibility of the phase transition and all pressure-induced struc-tural changes.
The relative changes in the unit cell lattice spacings with
pressure reveal both the direction and magnitude of thedominant intermolecular packing forces. In both the LP andHP phases the unit cell is hardest to compress along the b
direction. In the LP phase, the unit cell length adecreases at
a rate greater than that of the clattice parameter. Similar
behavior is seen in related compounds having this distinctiveherringbone structure.
30,31,62For the HP phase, the opposite
behavior is seen and, in this instance, the unit cell is harderto compress along athan in the
/H9266-stacking direction /H20849along
the crystal axis c/H20850. As a consequence, one should expect
changes in the electronic structure reflective of this crossoverin the relative magnitudes of the isothermal compressibility.
3. Equation of state
The Murnaghan equation of state /H20849EOS /H20850was used to fit
the pressure dependence of the unit cell volume in bothphases.
63,64The evolution of the unit cell volume is shown in
Fig. 8. The values obtained for the bulk modulus B0and its
pressure derivative B0/H11032are listed in Table I. The results for the
LP phase are within the error bars of the values reported by
Bridgman /H20849B0=6.2±0.1 GPa, B0/H11032=7.9±0.2 /H20850.16The unit cell
volume at ambient pressure /H20849LP phase /H20850is in good agreement
with the single crystal solution where a value of V0LP
=916.6 Å3is reported.17As expected, both the density and
the bulk modulus are higher in the new HP phase.
4. Molecular orientation
Rietveld refinement within the rigid-body approximation
allows for the determination of the position and orientation
FIG. 5. Side /H20849left panel /H20850and top /H20849right panel /H20850view of the fluorene crystal structure in the additional high-pressure phase.PHASE TRANSITION AND ELECTRONIC PROPERTIES … PHYSICAL REVIEW B 73, 024109 /H208492006 /H20850
024109-5of the molecules within the unit cell /H20849as indicated in Fig. 2 /H20850.
Symmetry constraints locate the molecular center mass, interms of fractional coordinates, at x,y=1/4 and zalong a,b,
andc, respectively. The orientation of the molecule is given
by the herringbone angle
/H9273/H20849see Fig. 2 /H20850. Clearly seen in Fig.
9 are the large and discontinuous variations in the fluoreneposition and orientation on going from the LP to the HPphase. Increasing pressure increases the herringbone angle
/H9273
of the LP phase and this effect is similar to that of structural
analogs.30–33In contrast, /H9273slowly decreases in the HP phase,
and this difference in the intermolecular packing forcespresages a major change in the electronic structure.
31–33
B.Ab initio results
In order to theoretically investigate the phase transition in
fluorene we calculated the LDA ground-state energies Eofthe fluorene LP and HP phase as a function of the /H20849experi-
mental /H20850pressure p. Since the volume Vchanges significantly
due to the application of pressure, we further calculated theenthalpy H=E+p·V. In Fig. 10 we plot the results of this
evaluation, i.e., Has a function of p. The phase transition
occurs at the intersection point of the two straight lines.These lines are obtained from linear fits to the LP and HPmodel results. The DFT-LDA calculation suggests the phasetransition to occur at a pressure of 3.8 GPa. This is in sur-prisingly good agreement with experiment /H208493.6 GPa /H20850given
the well known tendency of LDA to produce overbinding for
extended solids.
65We argue that, since the nature of intermo-
lecular interactions can be assumed to be similar in the LPand the HP, their respective equilibrium volumes are off by acomparable amount. This situation corresponds to a rigidlyshifted pressure scale. The resulting cancellation of /H20849system-
atic/H20850errors then leads to the transition pressure of 3.8 GPa
that compares so well to the experimentally determinedvalue of 3.6 GPa. In previous studies on comparable com-pounds, we found that both LDA and GGA reproduce theexperimentally determined bulk modulus and the pressureinduced reorientation of the molecules within the unit cellequally well.
32,33Based on this observation that the intermo-
lecular forces in the repulsive regime are adequately de-scribed by both approaches, one can expect GGA to yieldresults similar to LDA for the observed phase transition. Ofcourse, in this case, the equilibrium volumes of the HP andthe LP would both be systematically overestimated and onehas again to invoke the above argument of error cancellation.
A thorough analysis of the LDA optimized molecular ge-
ometries shows that the changes in the C-C bond lengths andangles are smaller than /H110111% /H20849/H110150.01 Å and /H110151°, respec-
tively /H20850over the whole pressure range /H208490–14 GPa /H20850. X-ray
powder diffraction patterns simulated for the experimental
/H20849rigid-body refinement /H20850and theoretical /H20849relaxed LDA geom-
etry/H20850structures at 14 GPa proved to be identical to a level
where no further relevant structural details could be ex-tracted. This a posteriori justifies regarding individual mol-
ecules as rigid bodies in the refinement procedure.
1. Band structures and densities of states
The electronic band structures E/H20849k/H20850and densities of states
/H20849DOS /H20850of the LP phase at ambient pressure and 4 GPa are
shown in Fig. 11, respectively, whereas the HP phase datacorresponding to 4 and 14 GPa are depicted in Fig. 12. Be-fore starting our discussion, we point out that each band inthe band structure splits into a quartet /H20849due to the presence of
four inequivalent molecules per unit cell /H20850unless the bands
are degenerate. In the following, we abbreviate the highestvalence band quartet with “VB” and, analogously, the lowestfour conduction bands with “CB” and these are highlightedin gray in Figs. 11 and 12. The band widths W, which were
evaluated as energy difference between the minimum of thelowest band and maximum of the highest band within the VBand CB, are summarized in Table II and additionally plottedin Fig. 13. Note that the band crossings were taken intoaccount.
We first focus on the LP phase band structure. The four
uppermost VBs exhibit a sequence of a rather flat quartet,
FIG. 6. Rietveld refinements of the LP phase at 1.0 GPa /H20849a/H20850,o f
both coexistent phases at 3.9 GPa /H20849b/H20850, and of the pure HP phase at
7.3 GPa /H20849c/H20850. The thin vertical lines mark the positions of the peaks
in the respective phase.HEIMEL et al. PHYSICAL REVIEW B 73, 024109 /H208492006 /H20850
024109-6followed by two dispersed ones and a flat VB again, when
going downwards in energy from the Fermi level. Similarband structure features, in particular the alternating appear-ance of flat and dispersed bands, have also been reported forthe oligophenylenes.
33,66As is generally found in such mate-
rials, both the band dispersion and the band splitting arehighly anisotropic. The total band splitting of both VB andCB at ambient pressure is in the range of 0.3–0.4 eV and isalso comparable to the total band widths. When examined inmore detail, along
/H9003Sand in the direction /H9003X/H20849parallel to thecrystalline aaxis/H20850, the VB appears as two interpenetrating
doublets. Each doublet exhibits a splitting of 0.2 eV,whereas the two doublets are split only by 0.1 eV. This be-havior can be traced back to the structural features of fluo-rene. While the larger splitting in this particular direction isdue to the herringbone stacking, the latter reflects the double-layer formation. In contrast, two separated doublets are
present along
/H9003Y, which is parallel to the bdirection /H20849the
long molecular axis /H20850, as well as along the caxis /H20849/H9003Z/H20850. In this
case, the splitting, which is originating in the molecular lay-
ers is more strongly pronounced than that produced by theherringbone stacking. As a matter of symmetry, two degen-erate doublets are present in all the other directions, whichare located at the BZ boundaries /H20849see Fig. 3 /H20850.
Applied pressure results in an increase of band dispersion
and splitting. This is a natural consequence of the reductionin intermolecular distances and broader features in the DOS.For example, the band widths of VB and CB at 4 GPa are
FIG. 7. Pressure dependence of the lattice constants a,b, and cfor the LP and the HP phase /H20849top left, top right, bottom left /H20850. In the bottom
right panel, the relative changes of the unit cell parameters with respect to the lowest pressure values available /H208490.0 GPa for the LP phase and
2.0 GPa for the HP phase /H20850are shown. For the sake of clarity, interpolated lines are drawn.
FIG. 8. Unit cell volume of fluorene as a function of hydrostatic
pressure for both the ambient and the HP phases. The solid lines area least-squares fit to the Murnaghan EOS.TABLE I. Bulk modulus B0/H20851GPa /H20852, its pressure derivative B0/H11032
and the unit cell volume at ambient pressure V0/H20851Å3/H20852for the LP
phase and HP phase of fluorene.
LP phase HP phase
B0/H20851GPa /H20852 5.9±0.4 11.3±1.1
B0/H11032 7.5±0.4 5.4±0.2
V0/H20851Å3/H20852 922.0±2.4 862.5±8.1PHASE TRANSITION AND ELECTRONIC PROPERTIES … PHYSICAL REVIEW B 73, 024109 /H208492006 /H20850
024109-70.57 and 0.68 eV, respectively. However, the crossover to
/H9266-stacking produces a very distinctive change to the band
structure and DOS of the HP phase. The intermolecular bandsplitting of both, the uppermost VB and lowest CB, isstrongly reduced. The effect is rigorous in case of the VB,where the total band splitting is less than 0.1 eV along alldirections at 4 GPa. Nevertheless the band dispersion of theVB drastically increases along
/H9003ZandUX, whereas the va-
lence bands are significantly flatter along the other direc-tions, i.e.,
/H9003X,/H9003Y, and /H9003S. In contrast, the dispersion of the
lowest CB decreases and results in a band width of 0.41 eVat 4 GPa. This value is roughly 40% less than the LP phasevalue /H208490.68 eV /H20850at this pressure. Closer inspection of the four
VBs below the Fermi energy show that the band character-
istics have become reversed: The first and fourth VBs, whichwere rather flat in the LP phase, now exhibit a huge disper-sion of approximately 1 eV /H20849along
/H9003ZandUX/H20850, while the
FIG. 9. xandzpositions of the molecule in fractional coordi-
nates along the aandcaxes, respectively, as well as the herring-
bone angle for the LP and HP phases. The solid lines are linearleast-squares fits. The point in brackets in the center panel wasmeasured outside the pressure cell and was thus excluded from thefit.
FIG. 10. Enthalpy Hof the fluorene LP /H20849in black /H20850and HP /H20849in
gray /H20850phase as a function of pressure. Inset: An enlargement of the
range over which the phase transition occurs. PT indicates the phasetransition between LP and HP phase at 3.8 GPa.
FIG. 11. Band structures E/H20849k/H20850/H20849left/H20850and DOS /H20849right /H20850of the
fluorene LP phase at ambient pressure /H20849top/H20850and 4 GPa /H20849bottom /H20850,
respectively. The Fermi level is indicated by a dashed line. Thesolid /H20849dashed /H20850gray arrow marks the lowest direct /H20849fundamental /H20850
gap.HEIMEL et al. PHYSICAL REVIEW B 73, 024109 /H208492006 /H20850
024109-8originally more dispersive second and third VBs are now
slightly less wide and cross the uppermost VB. Upon in-creasing pressure, the band dispersion is significantly en-hanced in these directions for both the VB and CB, resultingin values of 2.13 and 0.67 eV at 14 GPa, respectively.Athough the effect of pressure on the band splitting is rela-tively minor in the case of VB, the CB splitting is stronglyenhanced by increasing pressure. Additional values for thetotal band widths Wat several pressures are given in Table II.
2. Band gaps
We now focus on the fundamental band gap /H20849Eg/H20850and the
lowest direct band gap /H20849/H9004g/H20850given in Table II. Due to the
LDA band gap problem ,67the band gap is underestimated
when compared to experiment. One possibility to overcomethis problem of standard DFT and go beyond is the GWapproximation
68to the self-energy. This approach is compu-
tationally highly demanding and therefore has not been ap-plied to fluorene. On the other hand, a more simple method/H20849the scissors approximation /H20850is widely used to correct the
band gap, which is a k-independent shift of the conduction
states
69by the self-energy /H9004c./H9004ccan be estimated by com-
paring the experimental gap to the LDA gap. However, it isknown that /H9004
citself can be pressure dependent.47For this
reason, the experimental band gaps as a function of pressureare required in order to consistently perform the gap correc-tion, which have not been measured so far. Since this workdoes not focus on absolute values but rather on changescaused by pressure, in particular by the structural rearrange-ments due to the phase transition, this E
gcorrection was
omitted.
All of the LP phase structures exhibit an indirect funda-
mental band gap Eg. This occurs between /H9003and a low-
FIG. 12. Analogous to Fig. 11, the fluorene HP phase at 4 GPa
/H20849top/H20850and 14 GPa /H20849bottom /H20850, respectively.
TABLE II. Band widths of the uppermost valence band /H20849WVB/H20850
and the lowest conduction band /H20849WCB/H20850quartet as well as the fun-
damental band gap /H20849Eg/H20850and the lowest direct band gap /H20849/H9004g/H20850as a
function of pressure p.
p/H20851GPa /H20852 WVB/H20851eV/H20852 WCB/H20851eV/H20852 Eg/H20851eV/H20852 /H9004g/H20851eV/H20852
LP phase:
0 0.30 0.40 3.24 3.411 0.41 0.53 3.09 3.262 0.48 0.60 3.01 3.213 0.54 0.63 2.94 3.164 0.57 0.68 2.91 3.145 0.60 0.71 2.84 3.09
HP phase:
2 1.04 0.28 2.83 2.834 1.34 0.41 2.60 2.606 1.55 0.47 2.42 2.428 1.74 0.53 2.27 2.2710 1.89 0.59 2.16 2.1612 2.03 0.64 2.06 2.0814 2.13 0.67 1.94 1.94
FIG. 13. Graphical analysis of the data given in Table II. The
black /H20849gray /H20850color indicates LP /H20849HP/H20850phase data.PHASE TRANSITION AND ELECTRONIC PROPERTIES … PHYSICAL REVIEW B 73, 024109 /H208492006 /H20850
024109-9symmetry k-point close to the Z-point, as indicated in the
band structure plots /H20849Fig. 11 /H20850by dashed gray arrows. The
smallest direct gap /H9004g/H20849denoted by the solid gray arrows /H20850
generally appears at a certain point along /H9003Zin all calculated
LP phase band diagrams. The only exception is that of am-bient pressure in which it lies at /H9003./H9004
gis roughly 0.2 eV
larger than Egin either case. The data in Table II are shown
graphically in Fig. 13 and both the fundamental and directband gaps strongly decrease with increased pressure. A com-bination of increasing band widths and band splittings/H20849within each VB or CB quartet /H20850contribute to this result.
Microscopically, it relates to the reduction of the intermo-lecular distances and, correspondingly, an increase in thewave function overlap.
The structural reorganization in the HP phase not only
decreases the fundamental gap size but also yields a directgap /H20849i.e., E
gand/H9004gcoincide /H20850.Egis further reduced by
roughly 30% /H208490.9 eV /H20850on going from 2 GPa to 14 GPa. This
drop is much more pronounced in comparison with the
oligoacenes,32especially in the high-pressure region.
In summary, besides the general effect of pressure on the
electronic properties of organic molecular crystals,31–33we
find a very distinct change of the band dispersion along /H9003Z
that is a direct consequence of the structural changes due tothe phase transition: the VB band widths are increased byone order of magnitude and the band gap is strongly reduced.At pressures above 6 GPa, the variation of the band widthsand gap follows a nearly linear trend.
3. Effective electron and hole masses
The nature of charge carrier transport in organic crystals
remains controversial.70–72The unusually large band disper-
sion of the VB along /H9003ZandUX, however, indicates that
coherent, band-like transport along the crystal caxis
/H20849/H9266-stacks /H20850is the predominant mechanism in the HP phase, at
least for holes. To estimate the charge carrier mobility, weemploy the simple approach for parabolic bands in inorganicsemiconductors in which this parameter is, neglecting scat-tering, inversely proportional to the effective mass m
*of the
charge carriers:
1
m*=1
/H60362d2E/H20849k/H20850
dk2. /H208492/H20850
For comparison purposes both the effective hole mass mh*
and the effective electron mass me*were evaluated in both
phases by the fitting the highest valence and the lowest con-duction band /H20849of the quartet /H20850at/H9003as well as at the point Zto
Eq. /H208492/H20850. The corresponding values as a function of pressure
are summarized in Table III. For pentacene, which exhibitsthe highest hole carrier mobilities among the organic
semiconductors,
73anmh*of 1.55±0.2 m0has been reported,74
where m0denotes the electron rest mass. In the fluorene LP
phase, mh*is large compared to mh*in pentacene, but me*turns
out to be even smaller than the latter. In particular, at ambient
pressure me*evaluated at /H9003/H20849Z/H20850is roughly half /H20849one-third /H20850of
mh*, suggesting much better electron transport properties.
If pressure is applied, the effective masses are reduced.
For the HP phase hole transport clearly becomes favored. At4 GPa we obtain 0.4–0.7 m0and 0.9–1.5 m0formh*andme*
along /H9003Z, respectively. Thus, the mh*values of the fluorene
HP phase presented here are as low as in conventional semi-conductors.
4. Linear optical response
The optical response to a-,b-, and c-polarized incident
light described in terms of the dielectric tensor /H92552i/H20849/H9275/H20850formu-
lated within the RPA /H20849Eq. /H208491/H20850/H20850is shown in Fig. 14 for both
phases. A lifetime broadening of 0.05 eV has been includedin order to smoothen the absorption features. For the reasonsoutlined in Sec. IV B 2, the scissors correction /H9004
cto the
conduction bands was omitted. Therefore, the optical absorp-tion features are lower in energy than expected from experi-ment /H20849we are interested only in the changes due to pressure /H20850.
The onset of the optical response was extracted from theunbroadened optical data and it is summarized in Table IVdepending on the polarization iof the exciting light as well
as pressure.
We first focus on the LP phase under pressure. The lowest
optical absorption is generated by light polarized parallel tothe crystalline baxis, which points along the long molecular
axis. Similar behavior is found in oligophenylenes
33but, in
contrast to the oligoacenes’ optical properties,75the lowest
b-polarized absorption process is strongly allowed. The onset
of this transition at 3.4 eV matches /H9004ggiven in Table II
indicating that the very lowest optical absorption process isan allowed interband transition. As a consequence of theband gap reduction and the increase of the band widths, weclearly observe a redshift and a broadening of the opticalabsorption features. This is the general pressure effect on thedielectric function, which has been previously reported foranthracene
47and biphenyl.33The redshift of the absorption
onset is more strongly pronounced for the c-polarized re-
sponse compared to the aandbcomponents; i.e., roughly 0.4
and 0.2 eV, respectively. In addition, a new peak evolvesfrom the lowest b-polarized absorption feature with increas-
ing pressure, which is indicative of the phase transition.TABLE III. Effective electron /H20849me*/H20850and hole /H20849mh*/H20850mass as a
function of pressure pestimated at the /H9003andYpoints. The plus
/H20849minus /H20850sign indicates the positive /H20849negative /H20850curvature of the re-
garding band. m0denotes the electron rest mass.
p/H20851GPa /H20852/H9003 Z
mh*·m0 me*·m0 mh*·m0 me*·m0
LP phase:
0 −2.83 −1.46 3.22 0.962 −2.02 −0.99 2.04 0.734 −1.57 −0.96 2.14 0.57
HP phase:
2 −0.49 1.92 0.73 2.444 −0.39 0.93 0.66 1.5314 −0.30 0.54 0.66 1.28HEIMEL et al. PHYSICAL REVIEW B 73, 024109 /H208492006 /H20850
024109-10The optical absorption features are also strongly altered
by the structural evolution in transforming to the HP phase.A decreased band gap further redshifts the absorption onset.The lowest transition is still generated by b-polarized light,
but the oscillator strength is redistributed. This is particularly
evident in the acomponent of the dielectric tensor /H9255
2a/H20849/H9275/H20850,
which exhibits a massive gain in oscillator strength. In con-
trast to the LP phase, the redshift of the b-polarized absorp-tion onset is twice that of the aandccomponents. Thus, the
former has a value of 0.6 eV while the latter two are 0.3 eV.
V . SUMMARY AND OUTLOOK
In this work, we have presented a thorough investigation
of crystalline fluorene under high pressure by combining ex-perimental and theoretical methods. A reversible structuralphase transition at 3.6±0.3 GPa has been experimentally ob-served in fluorene. This high-pressure fluorene structure hasbeen determined from x-ray powder diffraction measure-ments as orthorhombic space group /H20849Pnma /H20850with signifi-
cantly altered lattice parameters and molecular arrangement.
In particular, the unit cell parameter cis drastically short-
ened, whereas ais lengthened. The molecular arrangement
changes from the herringbone pattern towards
/H9266-stacking. As
a consequence of the phase transition, the density as well asthe bulk modulus increase.
The expected effects on the electronic and optical proper-
ties resulting from the phase transition are clearly evident inthe band structures and dielectric tensor components calcu-lated within the density functional framework. In addition tothe general effects of pressure, i.e., band gap reduction andan increase in band widths, we observe dramatic changes inthe valence bands of the high-pressure phase. Especially no-table is the large curvature at the band edge and, along thecrystal axis c, dispersion on the order of 1 to 2 eV. The ef-
fective hole masses estimated for these bands are therefore assmall as in conventional semiconductors. These findings fur-ther underline the importance of controlling the molecularpacking in the solid state in addition to tailoring the proper-ties of individual molecules for their application in organic/H20849opto- /H20850electronic devices.
76,77
Finally, we would like to encourage measurements of the
optical absorption as well as the effective masses of crystal-line fluorene as a function of pressure as these would behighly desirable for further comparison between theory andexperiment.
ACKNOWLEDGMENTS
The authors want to thank Felix Porsch for technical
assistance in performing preliminary experiments and espe-cially Peter Puschnig for performing these experiments andrefining the respective data. G.H. would like to acknowledgethe financial support of the SFB Elektroaktive Stoffe , the
Austrian Science Fonds FWF /H20849project P14237 and Erwin-
Schrödinger Grant J2419–N20 /H20850, and the Center for Organic
Photonics and Electronics at Georgia Tech. K.H. and
C.A.-D. are grateful for the fundings from the FWF /H20849project
P16227 /H20850and the EU RT network EXCITING /H20849Contract No.
HPRN-CT-2002-00317 /H20850. W.C. and M.J.W. gratefully ac-
knowledge support by NSF DMR-0350383. Operation of theMUCAT Sector 6 beamlines at the APS was supported by theU.S. DOE. The authors thank Doug Robinson for technicalassistance at the APS. Use of the APS was supported by theU.S. DOE-BES, Office of Energy Research /H20849Contract No.
W-31-109-ENG-38 /H20850.
FIG. 14. Linear optical response to a-,b-, and c-polarized inci-
dent light described in terms of the dielectric tensor /H92552i/H20849/H9275/H20850formu-
lated within the RPA as a function of pressure. A lifetime broaden-ing of 0.05 eV has been included. The black /H20849gray /H20850color indicates
the LP /H20849HP/H20850phase.
TABLE IV. /H92552i/H20849/H9275/H20850peak onsets of the lowest optical transitions as
a function of pand polarization iof the incident light.
p/H20851GPa /H20852 /H92552a/H20849/H9275/H20850/H20851eV/H20852 /H92552b/H20849/H9275/H20850/H20851eV/H20852 /H92552c/H20849/H9275/H20850/H20851eV/H20852
LP phase:
0 3.85 3.39 3.854 3.65 3.17 3.40
HP phase:
4 3.60 2.63 3.6314 3.28 2.03 3.33PHASE TRANSITION AND ELECTRONIC PROPERTIES … PHYSICAL REVIEW B 73, 024109 /H208492006 /H20850
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024109-13 |
PhysRevB.93.195138.pdf | PHYSICAL REVIEW B 93, 195138 (2016)
Emergent spinless Weyl semimetals between the topological crystalline insulator
and normal insulator phases with glide symmetry
Heejae Kim1and Shuichi Murakami1,2
1Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan
2TIES, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan
(Received 7 March 2016; revised manuscript received 24 April 2016; published 19 May 2016)
We construct a theory describing phase transitions between the spinless topological crystalline insulator phase
with glide symmetry and a normal insulator phase. We show that a spinless Weyl semimetal phase shouldintervene between these two phases. Here, because all the bands are free from degeneracy in general, a gapclosing between a single conduction band and a single valence band at phase transition generally gives rise toa pair creation of Weyl nodes; hence the Weyl semimetal phase naturally appears. We show the relationshipbetween the change of the Z
2topological number when the system goes through the Weyl semimetal phase, and
the trajectory of the Weyl nodes.
DOI: 10.1103/PhysRevB.93.195138
I. INTRODUCTION
A three-dimensional (3D) topological insulator (TI) [ 1,2]
is one of the most famous symmetry-protected topologicalphases, and this phase is protected by time-reversal symmetry.In 3D TIs, the bulk bands have a gap like an ordinary insulator,while their surface states are metallic. The surface statesare protected by time-reversal symmetry, and robust againstperturbations unless time-reversal symmetry is broken.
On the other hand, a topological crystalline insulator (TCI)
have recently been studied in condensed matter physics [ 3–13].
The TCI phases are protected by crystal symmetries suchas rotational symmetry, mirror symmetry, etc. The surfacestates of the TCI phases are protected by crystal symmetries.Time-reversal symmetry is not necessarily required and theirtopological class is different from that of the TIs.
In recent works, a new class of 3D TCIs with a nonsym-
morphic glide plane symmetry is theoretically predicted bothfor spinless and spinfull systems [ 14,15]. Nonsymmorphic
symmetry groups are defined by combination of a point-groupoperation and a nonprimitive translation operation. It hasbeen known that nonsymmorphic symmetries yield extradegeneracies in band structures because their periodicities
different from the usual symmetries. For these features, the
nonsymmorphic symmetries are recently attracting attentionin condensed matter physics [ 16–19]. This new TCI phase
with nonsymmorphic glide symmetries is characterized by aZ
2topological number [ 14,15]. For 3D spinless TCIs with a
glide symmetry, the surface states typically form a single Diraccone with its Dirac point not pinned to a high-symmetry point,unlike TIs mentioned above.
In the present paper, we study phase transitions between
the 3D TCI and the NI phases, upon changing a parameter inthe system. We assume that the glide symmetry is preserved inthe phase transition. First, we construct a theory describingsuch a phase transition based on an effective model. Wefind that in general the TCI-NI phase transition is alwaysintervened by a spinless Weyl semimetal (WSM) phase, whichis a semimetal phase with a nondegenerate Dirac cones at oraround the Fermi energy [ 20–27]. In this case, all the bands
in spinless systems are free from degeneracy in general, anda gap closing between a single conduction band and a singlevalence band generally gives rise to a pair creation of Weyl
nodes; hence the WSM phase naturally appear. Since the Weylnodes (the apices of the Dirac cones) are topological, i.e.,either monopoles or antimonopoles in momentum space, theWeyl nodes survive within a finite range of the parameter m,
until they are pairwise annihilated. We relate the trajectory ofthe Weyl nodes within the WSM phase, and the change of theZ
2topological number. Then we support this conclusion by
a calculation on a model by Fang and Fu [ 14], but with an
additional term to restore generality of the system. The resultstotally agree with our theory of topological phase transitionin spinless glide-symmetric systems. This paper is organizedas follows. Section IIis devoted to a theory for the TCI-NI
phase transition with glide symmetry. In Sec. III, we show the
calculation on the Fang-Fu model with an additional term. Weconclude the paper in Sec. IV.
II. PHASE TRANSITION FOR THE SYSTEMS
WITH GLIDE SYMMETRY
In this section, we review basic properties of the glide-
symmetric spinless TCI, and then we develop a theory forthe TCI-NI phase transition to show that the WSM phasenecessarily intervenes between the two topologically distinctphases.
A. Previous works: glide-symmetric spinless
TCI and Z2topological number
In the present section, we consider a general spinless system
with glide symmetry, with a parameter mwhich drives the
topological phase transition. In changing, the parameter m,
the glide symmetry is assumed to be preserved. Physically,this parameter mcan be any parameter in the system, such as
pressure or atomic compositions. To study the TCI-NI phasetransition, we first begin with the system with a bulk gap andassume that the gap is closed by changing m. We then discuss
in which cases the system will open a gap again by the changeofm. This eventually shows that a spinless WSM phase should
appear between the TCI and NI phases with a glide symmetry.We then discuss how the evolution of the Weyl nodes arerelated with a Z
2topological number.
2469-9950/2016/93(19)/195138(8) 195138-1 ©2016 American Physical SocietyHEEJAE KIM AND SHUICHI MURAKAMI PHYSICAL REVIEW B 93, 195138 (2016)
We consider systems with the glide symmetry which is a
combination of a reflection with respect to the xyplane and a
half-translation along the xaxis
G:(x,y,z )→/parenleftbigg
x+a
2,y,−z/parenrightbigg
, (1)
where ais the length of the primitive translation vector along
thexaxis. We take the unit of a=1 for the following
calculation for simplicity. Then, the Hamiltonian satisfies
G(kx)H(kx,ky,kz)G−1(kx)=H(kx,ky,−kz), (2)
where G(kx) is the operator representing G. On the glide
symmetric plane kz=/Lambda1(/Lambda1=0,π), the Hamiltonian H(k,m)
commutes with the glide operator G(kx), and the eigenstates
of the Hamiltonian are also eigenstates of the glide operator.Note here that G(k
x)2≡e−ikxrepresents a translation by a
along the xaxis in spinless systems. It then follows that the
eigenvalues of the glide operator is equal to g±(kx)=±e−ikx/2
and therefore it has 4 πperiodicity for kx. Because of
g±(kx+2π)=g±(kx−2π)=g∓(kx), (3)
a “branch cut” has to be introduced, for example, at kx=0.
Because of this 4 πperiodicity, the minimum number of bands
in the conduction band is two, and that for the valence band isalso two. Therefore the minimal number of bands to describethe TCI-NI phase transition is four, and we consider 4 ×4
Hamiltonian H(k,m), where mis the parameter to control
TCI-NI phase transition.
For 3D spinless gapped systems with glide symmetry G,
theZ
2topological number νis defined in Refs. [ 14,15]. For the
definition of ν, we first define the Berry connection in terms
of the periodic part of the Bloch wave function, un(k):
A(k)≡occ./summationdisplay
ni/angbracketleftun(k)|∇k|un(k)/angbracketright, (4)
A±(k)≡±,occ./summationdisplay
ni/angbracketleftun(k)|∇k|un(k)/angbracketright, (5)
where nis the band index. Here the sum/summationtextocc.
nis over
the occupied states, and/summationtext±,occ.
n is over the occupied states
belonging to the glide sector with the glide eigenvalue g±=
±e−ikx/2. The quantities A±(k) are defined only for the wave
vector kon the glide planes, which are kz=0,πin the present
case. We next define the Berry curvatures
Bx=∂kyAz−∂kzAy, (6)
B±
z=∂kxA±
y−∂kyA±
x, (7)
and the Berry phase
γ±(l)=/integraldisplay
ldk·A±(k), (8)
where lis a closed path. Then the Z2topological number is
defined as
ν=1
2π/bracketleftbigg/integraldisplay
ABxdkydkz+/parenleftbigg/integraldisplay
B−/integraldisplay
C/parenrightbigg
B−
zdkxdky/bracketrightbigg
−1
π[γ+(a)+γ+(b)](mod 2) , (9)k =0zk =πz
k =0xk xk yk zk =πy
aAb
k =-πy
k =2π xd
cBC
FIG. 1. 3D Brillouin zone. In Eq. ( 9), the Berry curvatures are
calculated in the regions A, B, and C, and the Berry phases arecalculated along the arrows a and b.
where the Berry phase is calculated along the paths l=a,bin
Fig.1, and the surface integrals are taken over the regions A,
B, and C in Fig. 1. The TCI and NI correspond to ν=1 and
ν=0, respectively.
B. TCI-NI phase transition and Weyl semimetal
We next consider the main problem of this paper, how
the TCI-NI phase transition occurs. Here we show that theremust appear a spinless WSM phase in between. Before goinginto detailed discussions, let us review basic properties ofthe WSM phase. The Weyl semimetals have nondegenerateDirac cones at or near the Fermi level, and the vertex of thenondegenerate Dirac cone is called a Weyl node. The Weylnodes have topological nature in kspace [ 20,28,29]; the 3D
Weyl nodes are either a monopole or an antimonopole for theBerry curvature field B(k) defined as
B
n(k)=∇k×An(k),An(k)=i/angbracketleftun(k)|∇k|un(k)/angbracketright.(10)
The monopole density is defined as ρn(k)=1
2π∇k·Bn(k).
ρn(k) vanishes identically unless the nth band touches another
band, at which value of kthe monopole density has a
δ-function singularity, with an integer coefficient called a
monopole charge. The 3D Weyl nodes are either a monopole(charge +1) or an antimonopole (charge −1). Because of
the monopole charge quantization, under some change of aparameter contained in the Hamiltonian, the monopole chargecannot change except for pair creation or pair annihilation.
We consider the case where the TCI-NI phase transition
occurs by changing a parameter mcontrolling the phase tran-
sition. As the NI-TCI phase transition necessarily accompaniesclosing of the gap, suppose we begin with the NI phase, andchange the parameter mto close the gap. Since all the states
are free from degeneracies in general, we need to consider gapclosing between a single valence band and a single conductionband. Such a closing of the band gap should be generally apair creation of the Weyl nodes [ 20]. After the Weyl nodes
are created the Weyl nodes migrate in kspace because of
their topological stability. This phase is in general a Weylsemimetal phase. If all the Weyl nodes are related with eachother by symmetry operations, they are at the same energy;
195138-2EMERGENT SPINLESS WEYL SEMIMETALS BETWEEN THE . . . PHYSICAL REVIEW B 93, 195138 (2016)
Glide Plane+
-
Glide Plane+
-+
-
+
-(a)
(c)(b)
(d)
Glide Plane+
-
Glide Plane+
-(e)
(f)
(g)
Glide Plane+
-branch cutH+
H-H+
H+ H+
H+
H+ H+H-
H-
FIG. 2. (a)–(d) Possible cases of pair creation/annihilation of
Weyl nodes. Motions of the Weyl nodes with the change of the
parameter mare illustrated by the direction of the arrows. H+(H−)
indicates the subspace sector with the glide eigenvalue g±(kx)=
±e−ikx/2. The signs +(−) in the circles denote the monopole
(antimonopole). (e)–(g) Three examples of the trajectories of the Weyl
nodes within the Weyl semimetal phase, in the transition between twobulk-insulating phases. Due to the glide symmetry, their trajectory
are symmetric with respect to the glide plane. The trajectory of Weyl
nodes are topologically trivial in (e), and nontrivial in (f) and (g); it
means that (f) and (g) are accompanied by the TCI-NI phase transition
whereas (e) is not.
otherwise, not all the Weyl nodes are at the same energy. After
a further change of m, if all the Weyl nodes disappear by pair
annihilation, the system becomes a bulk insulator again. Thisbulk insulating phase may be the spinless TCI phase with glidesymmetry, or the NI phase. Which phase is realized dependson the trajectory of the Weyl nodes within the WSM phase.
The glide symmetry restricts the trajectory of the Weyl
nodes. We note that the eigenstates are classified into the twoglide sectors only on the glide plane . Thus a pair creation
on the glide plane is classified into two cases Figs. 2(a)
and2(c) in terms of the glide sector. Here, H
±means the
pair creation/annihilation on the glide sector with the glideeigenvalue of g
±(kx)=±e−ikx/2. Similarly, a pair annihilation
on the glide plane is classified into two cases, Figs. 2(b)
and2(d). By combining (a)/(c) with (b)/(d), one can consider
several patterns for annihilating all the Weyl nodes; therebythe system turns into a bulk-insulating phase again. As anexample, suppose Weyl nodes are created in the H
+sector.
After the change of the parameter m, we can consider two
cases of pair annihilation: Weyl nodes are annihilated at theH
+sector [Fig. 2(e)]o rH−sector [Fig. 2(f)]. From the second
term of the Z2topological number, Eq. ( 9), the latter case
[Fig. 2(f)] is accompanied by the change of νby one, i.e., the
NI-TCI phase transition, whereas the former case [Fig. 2(e)]i s
not. One can intuitively understand this because the trajectoryof these Weyl nodes in the former case can be continuouslydeformed to null, whereas that in the latter case is topologicallynontrivial. We exclude the cases with Weyl nodes comingexactly on the branch cut, because it does not happen unlessother parameters are finely tuned.
Furthermore, we should mention the existence of branch
cut, since the glide sectors are switched when we go acrossthe branch cut, from the property of the glide eigenval-uesg
±(kx+2π)=g±(kx−2π)=g∓(kx). Because of this
switching, when the pair creation and pair annihilation occurat the same H
+sector, but there is a branch cut between the
positions of the pair creation and pair annihilation [Fig. 2(g)],
theZ2topological number is changed between the two
bulk-insulating phases, since it is equivalent to Fig. 2(f).I n
the formula of the Z2topological number Eq. ( 9), the first
term is changed by one.
Apart from these cases, we can consider numerous patterns
for the evolution of Weyl nodes. From Eq. ( 9), the change of
theZ2topological number /Delta1νis obtained from the trajectory
of the Weyl nodes in the following way. First, as shownin Ref. [ 30], we flip the orientation of the trajectory of the
antimonopole, while that of the monopole is left unchanged;this always leads us to a single oriented loop [ 30]. Then the
change of the Z
2topological number /Delta1νis generally written
as
/Delta1ν=NA+N−
B+N−
C(mod 2) , (11)
where NAis the number of crossings between the loop and the
region A, and N−
B(N−
C) is the number of crossings between
the loop and the region B (C) within the H−sector [ 31]. We
encounter several cases in the following section using a simpletight-binding model.
Here we comment on time-reversal symmetry in such glide-
symmetric spinless systems. As mentioned in Ref. [ 14], theZ
2
topological number is trivial when the time-reversal symmetry
is preserved. It can be seen from our viewpoint of trajectories ofWeyl nodes. Since the time-reversal operator /Theta1≡K(complex
conjugation) satisfies /Theta1G=G
−1/Theta1, when a pair creation of
Weyl nodes occurs within the glide sector H+atk0, there
simultaneously occurs a pair creation within the same glidesector H
+at−k0. It is also the case for the H−sector. Thus
the numbers N−
B, andN−
Care always even, implying that the
change of the Z2topological number is zero. Namely, one
cannot have a NI-WSM-TCI phase transition in time-reversalinvariant spinless systems with glide symmetry, in accordancewith the argument in Ref. [ 14].
C. Effective model for the pair creation/annihilation
of Weyl nodes
We can construct an effective model describing a pair
annihilation/creation of Weyl nodes on the glide plane. On
195138-3HEEJAE KIM AND SHUICHI MURAKAMI PHYSICAL REVIEW B 93, 195138 (2016)
the glide plane the Hamiltonian H(k,m) satisfies
GH(kx,ky,kz,m)G−1=H(kx,ky,−kz,m). (12)
As an example, we focus on the pair creation/annihilation on
the glide plane kz=0, i.e. at k0=(k0,x,k0,y,0),m=m0.F o r
simplicity, we assume there are no symmetries except for theglide symmetry. Then all the states are free from degeneracies,and the gap closes between a single valence band and a singleconduction band. Thus we need to consider a 2 ×2 effective
Hamiltonian:
H(k,m)=/epsilon1(k,m)+/parenleftbigg
a(k,m)b(k,m)
b
∗(k,m)−a(k,m)/parenrightbigg
, (13)
where /epsilon1(k,m) and a(k,m) are real functions and b(k,m)i s
a complex function. Then we expand H(k,m) in terms of
δk≡k−k0andδm≡m−m0.A tk0,m0, the Hamiltonian
has degenerate eigenstates, and so the entities of the matrixhas no zeroth order terms. We note that at the gap closing at
k
0,m0, the two states have the same glide eigenvalue. Hence
the matrix representation of Gis proportional to the identity
matrix. Then from Eq. ( 12),aandbare even functions of
kz=δkzand we can write
a(k,m)=a1δkx+a2δky+a3δk2
z+a0δm, (14)
b(k,m)=b1δkx+b2δky+b3δk2
z+b0δm, (15)
up to the linear order in δkx,δky, andmand the quadratic order
inδkz. The gap closes when a=0, Reb=0, and Im b=0a r e
satisfied simultaneously. Namely,
⎛
⎝a1 a2 a0
Reb1Reb2Reb0
Imb1Imb2Imb0⎞
⎠⎛
⎝δkx
δky
δm⎞
⎠=⎛
⎝−a3
−Reb3
−Imb3⎞
⎠δk2
z.(16)
This equation yields a solution with the form δkx=Axδk2
z,
δky=Ayδk2
z,δm=A0δk2
z, where Ax,Ay, andA0are real
constants. As mis increased, this effective model describes a
pair creation ( A0>0) or pair annihilation ( A0<0) of Weyl
nodes on the glide plane. For example, when A0>0, a pair
of Weyl nodes exists at (Ax
A0δm,Ay
A0δm,±/radicalBig
δm
A0) only when
δm > 0. From the relations δkx=Axδk2
zandδky=Ayδk2
z,
the trajectory of the Weyl nodes is parabolic around k0,m0
and is symmetric with respect to the glide plane.
III. EXAMPLE: 3D FANG-FU LATTICE MODEL
WITH AN ADDITIONAL TERM
In this section, we consider the three-dimensional model
proposed by Fang and Fu. In order to see general properties ofthe NI-WSM-TCI phase transition, we add a term to the modelto remove spurious double degeneracy of the original model.The results confirm the general arguments in the previoussection.
A. Model
To illustrate our scenario, we use the tight-binding model
proposed by Fang and Fu [ 14] with an additional term. Our
model is a four-band spinless system on a 3D orthorhombiclattice with two sublattices, and two orbitals, written as the
following Hamiltonian:
H(k)=(m−t0coskx−t/prime
0cosky−t/prime/prime
0coskz)/Sigma103
+tsinkx−φ
2/parenleftbigg
coskx
2/Sigma111+sinkx
2/Sigma121/parenrightbigg
+t/primesinky/Sigma102+t/prime/primesinkz/Sigma131+p/Sigma1 01, (17)
where /Sigma1ij=σi⊗σjwithσiandσjare Pauli matrices describ-
ing the lattice and orbital degrees of freedom, respectively, andpis a constant. It is invariant under the glide operator given
by
G(k
x)=e−ikx/2/parenleftbigg
coskx
2/Sigma110+sinkx
2/Sigma120/parenrightbigg
, (18)
which satisfies
G(kx)H(kx,ky,kz)G−1(kx)=H(kx,ky,−kz),
G2(kx)=e−ikx. (19)
For simplicity, we assume t0,t/prime
0,t/prime/prime
0,t,t/prime, andt/prime/primeto be positive.
This model without the last term ( p=0) is the model
proposed by Fang and Fu [ 14], in which all the states are doubly
degenerate. Nevertheless, this double degeneracy for every
kdoes not originate from crystallographic symmetries (see
Appendix), and therefore we cannot expect this degeneracy ingeneral spinless systems with glide symmetry. Therefore weadded the term p/Sigma1
01to lift the degeneracy. Our question is
how the phase transition between the NI and the TCI phasesoccurs in systems with glide symmetry.
The energy eigenvalues of the Hamiltonian ( 17)a r e
E(k)=±/radicalBig
f2+t/prime2sin2ky+g2, (20)
where f=m−t0coskx−t/prime
0cosky−t/prime/prime
0coskzandg=/radicalBig
t2sin2kx−φ
2+t/prime/prime2sin2kz±p. We note that the energy spec-
trum is symmetric with respect to E=0. We set EF=0 and
focus on the bulk band gap at E=0. In particular, when
p=0[14], all the bands are doubly degenerate and the gap
closes at
kx=φ,(ky,kz)=(0,0),(0,π),(π,0),(π,π), (21)
m=t0cosφ+t/prime
0cosky+t/prime/prime
0coskz, (22)
as described in Ref. [ 14]. In this case, the bulk gap closes
at four values of m,m=t0cosφ±t/prime
0±t/prime/prime
0, accompanied by
NI-TCI phase transitions. Among them, we focus on the gapclosing at m=t
0cosφ+t/prime
0+t/prime/prime
0,k=(φ,0,0), as an example,
and add the pterm to see emergence of the WSM phase.
Addition of the pterm eliminates artificial double degen-
eracy in the Fang-Fu model. Below, we show that a spinlessWSM phase appears between the TCI and the NI phases. Whenp/negationslash=0, the band gap closes when
t
2sin2kx−φ
2+t/prime/prime2sin2kz=p2,ky=0,π, (23)
m=t0coskx+t/prime
0cosky+t/prime/prime
0coskz. (24)
Thus the gap closing at k=(φ,0,0) and m=t0cosφ+t/prime
0+
t/prime/prime
0in the case of p=0 is split due to the lifting of the
195138-4EMERGENT SPINLESS WEYL SEMIMETALS BETWEEN THE . . . PHYSICAL REVIEW B 93, 195138 (2016)
kx2 0 -1 1(h) p=0.4, m=2.9
kx2 0 -1 1(j) p=0.4, m=2.2kx0.8 00 . 4(g) p=0.1, m=2.8
kx0.8 0 0.4(e) p=0.1, m=2.9
kx0.8 00 . 4E0.4
0
-0.4
kz0.8
0
-0.8
kx2 0 -1 1(b) (c) (d)
(f)
(i)kx0.8 0 0.4
kx0.8 00 . 4
kx0.8 00 . 4
kz0.5
0
-0.5p=0.1, m=2.85
p=0.4, m=2.4p=0.1, m=2.9 p=0.1, m=2.82mp00.5
-0.51.0
-1.03.0 2.8 2.6 2.4 2.2 2 .3 0.2NITCIWSM
WSM(a)
(d)(c)(b)
(g)(f)(e)(j) (i)(h)(d) (c)
(g) (f)(e)(b)
NITCIWSM
WSM(A )
(A )(B)
(B)
p=0.1, m=2.98 +
-
FIG. 3. (a) Phase diagram of our model in the m-pplane with t0=t/prime
0=t/prime/prime
0=t=t/prime=t/prime/prime=1a n d φ=0.4. The bulk energy dispersions
become gapless on the lines (A ±) and (B). On the line (A ±), the bulk bands become gapless on the glide plane within the H±s e c t o r .O nt h e
line (B), the bulk band gap closes outside of the glide plane. (b)–(d) Bulk band structures on the ky=kz=0 line with p=0.1 as we decrease
m.A t( b ) m=2.98, Weyl nodes are created pairwise at k+=(k+
x,0,0) inH+sector. When mbecomes smaller, the Weyl nodes move in the
momentum space [see Fig. 4(b)]. At (c) m=2.9, the Weyl nodes exists but they are away from the kz=0 plane. Eventually, at (d) m=2.82,
Weyl nodes are annihilated in pair at k−=(k−
x,0,0). (e)–(j) Surface Fermi surface at E=0 around the origin for the (010) surface. In (e)–(g),
we fix p=0.1, and change the parameter mas (e)m=2.9 (spinless WSM), (f) m=2.85 (spinless WSM), and (g) m=2.8 (TCI). Similarly,
in (h)–(j) we fix p=0.4, and change the parameter mas (h)m=2.9 (spinless WSM), (i) m=2.4 (spinless WSM), and (j) m=2.2 (TCI). A
surface Dirac cone in the TCI phase appears in (g) and (j), and the surface Fermi arcs in the spinless WSM phase appear in (e), (f), (h), and (i).
degeneracy by the pterm. Now the gap is closed within a finite
range of the value of m. Within this region, the gap-closing
points are Weyl nodes, as can be shown explicitly, and thereforein this region of m, the system is in the WSM phase. As an
example, we show the phase diagram near m=3,p=0i n
them-pplane in Fig. 3(a)fort
0=t/prime
0=t/prime/prime
0=t=t/prime=t/prime/prime=1.
We can see that unless p=0, the WSM phase extends overa finite range of m. In the following, we show how this phase
diagram results.
B. Trajectories of Weyl nodes
We now describe behaviors of the Weyl nodes in the WSM
phase of the present model. We first note that the curve
195138-5HEEJAE KIM AND SHUICHI MURAKAMI PHYSICAL REVIEW B 93, 195138 (2016)
represented by Eq. ( 23) describes a trajectory of the Weyl nodes
inkspace. The trajectory of the Weyl nodes is symmetric with
respect to the glide plane, and monopoles and antimonopolesare switched by glide operation. This curve crosses theglide plane k
z=0 at two points k±=(φ∓2 arcsinp
t,0,0),
when m±=tcosk±
x+t/prime
0+t/prime/prime
0.H e r ew ea s s u m e |p/t|<1,
for simplicity. At these parameters, the Hamiltonian is writtenas
H(k
±,m±)=∓p/parenleftbigg
cosk±
x
2/Sigma111+sink±
x
2/Sigma121/parenrightbigg
+p/Sigma1 01(25)
and the gap closes at E=0 with the eigenstates
/Psi1(1)
±=1
2t(1,1,±eik±
x/2,±eik±
x/2),/Psi1(2)
±=1
2t(1,−1,±eik±
x/2,
∓eik±
x/2), both belonging to the H±sectors (i.e.,
G=±e−ikx/2). These closings of the bulk gap at k±
form=m±in the H±sectors on the glide plane correspond
to the pair creation or pair annihilation of Weyl nodes on theglide plane.
We calculate the trajectory of the Weyl nodes by changing
the parameter mfor several cases for the NI-WSM-TCI phase
transitions. Henceforth we set the parameters t
0=t/prime
0=t/prime/prime
0=
t=t/prime=t/prime/prime=1. First, we set φ=0.4,p=0.1 and decrease
mfrom the NI phase to the TCI phase. The overall trajectory
of the Weyl nodes is shown in Fig. 4(a), and the corresponding
band structures along the ky=kz=0 line are shown in
Figs. 3(b)–3(d). Two Weyl nodes are created pairwise at k+,
and they move in kspace. Eventually, the two Weyl nodes are
annihilated at k−pairwise, and the system becomes the TCI.
Thus these values of m±gives phase boundaries between the
WSM and the TCI/NI phases shown as the curves (A ±)i nt h e
phase diagram Fig. 3, given by m±,k±.
A different type of trajectories appears when we take φ=
0.4,p=0.4. For these values of the parameters, the trajectory
of the Weyl nodes by decreasing mis shown in Fig. 4(b).
Pair creations of Weyl nodes occur at two points ˜k±=
(˜kx,0,˜k±
z),m=m0outside of the glide plane simultaneously,
where t/prime/prime
0t2sin(˜kx−φ)=4t0t/prime/prime2sin˜kxcos˜k±
z. Then the four
Weyl nodes move in kspace. Firstly, a pair of Weyl nodes is
annihilated at k+,m=m+. The remaining pair of Weyl nodes
is then annihilated at k−,m=m−, and the system runs into
the TCI phase. The pair creation of Weyl nodes at ˜k±outside
of the glide plane leads to a phase boundary (B) in Fig. 3(a).
In these two cases shown in Figs. 4(a)and4(b), the two events
of the pair annihilation/creation on the glide plane occur in thedifferent glide sectors. Thus it is consistent with our scenariofor the switching of the Z
2topological number in Sec. II.
C. Evolution of the surface states
To show evolution of the surface states in the above
cases, we numerically perform band structure calculation fora slab geometry with (010) surfaces, which preserves the glidesymmetry G(k
x). Figures 3(e)–3(j)show the Fermi surfaces of
as l a ba t E=0 for several values of the parameter mwith
p=0.1,φ=0.4 [(e)–(g)], and p=0.4,φ=0.4 [(h)–(j)].
The system with p=0.1 is in the spinless WSM for (e)
m=2.9 and (f) m=2.85, with two Weyl nodes, and one
Fermi arc connects the projections of these Weyl nodes. Asmdecreases, the two Weyl nodes are annihilated pairwise,(b)
0.4 1.0 -0.60.5
-0.50Glide Plane kxkz
H+ H-kx-kx+kz+~
kz-~
(c)
0.5 -0.5 1.0 -1.00.5
-0.50Glide Plane kxkz
H+ H-kx-kx+kz+~
kz-~(a)
0.4 0.80.2
-0.2Glide Plane
0kxkz
H+ H-kx-kx+
FIG. 4. Numerical results for the trajectories of the Weyl nodes
for our model ( 17)f o rt0=t/prime
0=t/prime/prime
0=t=t/prime=t/prime/prime=1 with decreasing
m.H±on the glide plane ( kz=0) represent glide sectors G=
±e−ikx/2, respectively. (a) For φ=0.4a n d p=0.1 ,ap a i ro fW e y l
nodes are created on the glide plane in the glide sector H+and
eventually annihilated on the glide plane in the glide sector H−.( b )
Forφ=0.4a n d p=0.4, two pair creations occur simultaneously
outside of the glide plane, and then the four Weyl nodes are annihilated
in the glide sectors H+andH−.( c )F o r φ=0a n d p=0.4, due to
theC2ysymmetry, the trajectory is symmetrical with respect to the
origin. In (a)–(c), the trajectories are symmetric with respect to the
glide plane, with the sign change of the monopole charges.
leading to the TCI phase. In (g) m=2.8, the surface states of
this TCI phase form a Dirac cone.
On the other hand, the system with p=0.4 is in the spinless
WSM with four Weyl nodes for (h) m=2.9, and there are two
Fermi arcs. As we decrease m, one pair of Weyl nodes is
annihilated at k+first, and there are two Weyl nodes, resulting
in a single Fermi arc as shown in (i) for m=2.4. Later, the
other pair is also annihilated at k−and the system becomes
the TCI. Figure 3(j)shows the surface Dirac cone for the TCI
phase at m=2.2. Thus by going from the WSM phase to the
TCI phase, all the Weyl nodes are annihilated, and the twoFermi arcs evolve into the surface Dirac cone. Thus we haveseen that the surface Fermi arcs in the spinless WSM evolveinto the surface Dirac cone in the TCI.
D.φ=0 case: additional C2ysymmetry
In particular, when φ=0, the trajectory becomes symmet-
ric with respect to the origin [see Fig. 4(c)], because of the
emergent C2ysymmetry, in addition to the glide symmetry.
The two pair creations occur simultaneously, and so do the
195138-6EMERGENT SPINLESS WEYL SEMIMETALS BETWEEN THE . . . PHYSICAL REVIEW B 93, 195138 (2016)
(b)
kx1.5 0 -1.5p=0.4, m=2.8
kx1.5 0 -1.5(d) p=0.4, m=2.65
kz0.8
0
-0.8mp(a)
(c)
kx1.5 0 -1.500.5
-0.51.0
-1.03.0 2.8 2.6 2.4 2.2 2 .3 0.2NITCI
WSMWSM
(d) (c) (b)
p=0.4, m=2.7(A ) +
(A ) +(B)
(B)
FIG. 5. (a) Phase diagram in the m-pplane for the Fang-Fu
model with t0=t/prime
0=t/prime/prime
0=t=t/prime=t/prime/prime=1,p=0.4, and φ=0. In
this case, the system possesses C2ysymmetry. (b)–(d) Surface Fermi
surface at E=0 for the (010) surface with the values of mtaken as
(b)m=2.8 (spinless WSM), (c) m=2.7 (spinless WSM), and (d)
m=2.65 (TCI). A surface Dirac cone in the TCI phase appears in
(d), and the surface Fermi arcs in the spinless WSM phase appear in(b) and (c).
pair annihilations. In this case, the number of Weyl nodes is
always four within the WSM phase, and the phase boundaries(A
+) and (A −) become identical. Figure 5(a)shows the phase
diagram for φ=0i nt h e m-pplane. Figures 5(b)–5(d) are the
surface Fermi surfaces at E=0 on the (010) surface, where
we fix p=0.4 and change the value of m. The system is in
the spinless WSM for (b) m=2.8 and (c) m=2.7, and the
surface Fermi arcs are symmetric with respect to the origin. At(d)m=2.65, the system is in the TCI phase, and the surface
states form a Dirac cone. Thus the two Fermi arcs evolve intothe surface Dirac cone as we go from the WSM phase to theTCI phase, and the states are always C
2ysymmetric and glide
symmetric.
IV . SUMMARY AND DISCUSSION
To summarize, in the present paper, we showed that the
spinless Weyl semimetal (WSM) phase emerges betweentopological crystalline insulator (TCI) and normal insulator(NI) phases with glide symmetric systems. We first constructa theory for the TCI-NI phase transition with glide symmetry,and show a generic phase diagram involving the spinless WSMphase using the effective model. The trajectory of the Weylnodes within the Weyl semimetal phase governs the changeof the Z
2topological number. In particular, when the Weyl
nodes are created and annihilated pairwise in the sectors withopposite signs of glide eigenvalues, the Z
2topological number
changes between the two sides of the WSM phase. Thesescenarios are confirmed by our spinless tight-binding model ona three-dimensional orthorhombic lattice with glide symmetry.In this model, we also show that surface Fermi arcs in thespinless WSM phase evolve into a surface Dirac cone in theTCI phase.Let us compare the glide-symmetric spinless TCI and the
topological insulators (TIs). In the topological phase transitionbetween a TI and a NI, the WSM phase appears only whenthe spatial inversion symmetry is broken [ 20,32], provided the
time-reversal symmetry is preserved. On the other hand, wehave shown in the present paper that between the spinlessglide-symmetric TCI and the NI, the WSM phase alwaysappears.
Materials search of the glide-symmetric spinless TCI
would be an interesting and promising topic for first-principlecalculations, since it does not suffer from the constraintof strong spin-orbit coupling, as has been the case for theTIs. For candidate spinless systems, it is necessary to breaktime-reversal symmetry [ 14,33] but to preserve the glide
symmetry. Such candidate materials might be found amongspinless insulators coupled with localized spin systems orspinless insulators in a magnetic field, and search for suchsystems is left as a future work. We note that as is similarto the spinless TCI for electrons, bosonic systems with glidesymmetry can also support nontrivial Z
2topological number,
giving rise to a surface mode within a certain band gap. Fromthe results in this paper, it is also promising to discover newspinless WSM materials from such kind of glide-symmetricTCI materials.
ACKNOWLEDGMENTS
This work was supported by Grant-in-Aid for Scientific
Research (No. 26287062, 26600012) by MEXT, Japan, andby MEXT Elements Strategy Initiative to Form Core ResearchCenter (TIES).
APPENDIX: DOUBLE DEGENERACY AND SYMMETRY
OF THE FANG-FU MODEL
We here briefly mention the symmetry of the Fang-Fu
model, causing the double degeneracy. We define the followingoperators:
O=c/parenleftbigg
t
/prime/primecoskx
2σ1+t/prime/primesinkx
2σ2−tσ3/parenrightbigg
⊗(t/primeσ2+Mσ 3),
(A1)
O/prime=c/parenleftbigg
tcoskx
2σ1+tsinkx
2σ2+t/prime/primeσ3/parenrightbigg
⊗(t/primeσ2+Mσ 3),
(A2)
where c=1√
t/prime/prime2+t21√
t/prime2+M2.I ts a t i s fi e s
[O,H]=0,[O/prime,H]=0,OO/prime=−O/primeO,O2=1=O/prime2.
(A3)
Therefore the eigenstates of the Hamiltonian can be chosen aseigenstates of O. The eigenvalues of Oare either +1o r−1,
andO
/primeswitches the signs of the eigenvalues of Owithout
changing the energy. Thus all the eigenstates are doublydegenerate, having the opposite signs of the eigenvalues ofO. This symmetry described by O
/primeandOcannot come from
any crystallographic symmetries, and one cannot expect suchdouble degeneracy in crystals.
195138-7HEEJAE KIM AND SHUICHI MURAKAMI PHYSICAL REVIEW B 93, 195138 (2016)
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195138-8 |
PhysRevB.19.6456.pdf | PHYSICAL REVIEW 8 VOLUME 19, NUMBER 12 15JUNE1979
Bands,bonds,andcharge-density wavesintheNbsesfamilyofcompounds
J.A.Wilson
BellLaboratories, Murray Hill,¹uJersey07974
(Received 30October 1978)
Through detailed examination ofthecrystalstructure andbonding conditions inthetrichalcogenides, ithas
.beenpossibletogainadeeperunderstanding ofthebandstructure ofthesecomplex materials, andwiththis
oftheirunusual electronic behavior. Thecharacter oftheobserved semimetallicity inTaSe,andNbSe,is
muchclarified. Eachmaterial inthegroup-V familyisseentoachieve averyindividually tailored band
structure, enfolding thevarious structural andenergy-level adjustments. Thenonuniformity andthe
molecularization ofthesestructures contrasts strongly withthedichalcogenide behavior. Thestructural, if
nottheelectronic, dimensionality ofthegroup-V trichalcogenides remains closertotwothantoone.Itis
foundthattheperiodic structural distortions developed byNbSe,areprobably moresuitedtoametal-metal
bonding description thantothetraditional Fermisurface determined instability ofacharge-density wave.
Thefield-induced slidingofthesedistortions canthenbedescribed intermsofcooperative bondflipping
through adiscommensurate superlattice. Weexamine incrystalchemical termswhythismotion isso
unusually weakly pinnedtodefectsandimpurities inNbSe,.Infact,theinvestigation showsthatitwillbe
veryhardtofindamaterial bettersuitedtotherealization ofthisphenomenon.
I.INTRODUCTION
Manypapers'"haveappeared recently incon-
nection withthephysical properties ofthetransi-
tion-metal trichalcogenides, inparticular those
ofNbSe,.Thespecialinterestliesinthefactthat
thetwo"charge-density waves" (CDW)forming
inNbSe,(onsettemperatures T„,=144K,T„=59
K)can,itseems, beinduced toslidethrough the
material undertheapplication ofaverymodest
electric fieldgradient (a0.1Vcm').'"Weare
giventounderstand thattheincommensurate
CDW'sinNbSe,areunusually weakly pinnedto
thelattice.""Indeedalready bymicrowave ex-
citationfrequencies of10GHz(0.05meV)theyhave
becomefreetocarrycurrent.'Thisrestores the
levelofconductivity inthematerial toavalue
closetowhatwouldhaveobtained hadtheFermi
surfaceneverbeenpartially gappedundertheac-'
tionoftheseCDW's. Relatively littleattention
hasbeengivenastowhy,outofallthevarious
CDW-bearing materials, thisphenomenon should
showuponlyinNbSe,.Indeedlittlecrystalchem-
icaldiscussion hasbeengivenconcerning whythe
fourmaterials NbS,NbSe„TaS3, andTaSe,pres-
entveryindividualistic behaviors, whilethecor-
responding dichalcogenides" behaveasaunit.
Thiswetrytoremedy.
Thecrystalstructures ofthegroup-V trichal-
cogenides aremorecomplicated thanthoseofthe
layer-structured dichalcogenides. Inthelatter
thereisonlyonetypeofmetalatompercell(nor-
mally),whereas inNbSe,therearethree.More-
over,despiteallfourmaterials holdingtoacom-
monstructural element (thetrigonal prismatic
column), thestructure ofeachmaterial differssignificantly indetail. Therelated dgroup-IV
trichalcogenides bycontrast areisostructural
fromTiS,(Ref.1)toThTe,(Ref.36).Thestruc-
turaldifferences ingroupVinvolve theaccom-
paniment ofmajorqualitative changes inelectronic
bandstructure, andindeedeventheelectronic con-
figuration ofthecations. Thus,usingthecustom-
aryloosedesignations, NbS,isa4+d'compound,
withproperties andformmuchasexpected given
thoseofd'ZrS3$ whileTaSe,isinessence an
emptyd-band compound (thoughdisplaying a,slight
semimetallic p-doverlap).~36'7Thelackoffull
structural continuity forgroup-V trichalcogenides
onetothenext,andfurthermore tothegroup-IV
family, hasrestricted attempts togrowmixedcry-
stals;aprocedure thatinthedichalcogenides
helpedgreatly toresolve thenatureoftheobserved
latticeinstabilities."
Beforelookingatthevarious crystal-structur-es
andelectronic statesindetail, twopointsshould
beemphasized here.First,thegroup-V trichal-
cogenides arenothalf-filled bandmetals. Inthis
theyarenotlikethedichalcogenides. Indeedthe
latterarealwaysparamagnetic aboveT„and even
belowT,inthe3gcase,whilethetrichalcogenides
areeachdiamagnetic atalltemperatures.
ForthelattertheHallconstants areappropriately
severalorders ofmagnitude thelarger."The
secondpointisthatthetriehalcogenides arenot
one-dimensional compounds inthesensethat
K,Pt(CN)~Bro, nH,O(Ref.39)is,orNbi~,~oor
V(S2),.'Verysubstantial structural cross-linking.
occursbetween thechains, theM-Xcoordination
inthetrichalcogenides movingtowards 8[asfor
PuBr,orThl~(Ref.36)]ratherthan6(asinthe
dichalcogenides, orNbI,).Forexample, inZrSe„
19 6456 1979TheAmerican Physical Society
BANDS, BONDS, ANDCHARGE-DENSITY %AVES INTHE...
besides thesix"intrachain" bonds-2.73A,there
aretwointerchain bonds-2.87A.'Asaresult,
thematerials areasclosetobeinglayerstruc-
turesastochainstructures inthetruesense
[distinct fromwhathaspassedforchainsin,say,
NbO,(Ref.42)orNb,Sn(Ref.43)).'Eventhere-
centlymeasured anisotropy -20withintheslabs
forNbSe,(Ref.29)(i.e.,containing axesbandc)
is,weshallseelater,likelystilltosuggesta
greaterdegreeofonedimensionality forthema-
terialthanreallyexists.Itwouldbeinteresting
tohavethisanisotropy measured againinthesim-
plerstructural andelectronic condition supplied
byZrSe,.
II.COMPARATIVE DESCRIPTION OFTHECRYSTAL
STRUCTURES
Twofactorsaffecttheevolution ofthestructures
ofthegroup-V trichalcogenides fromthatofthe
group-IV prototype ZrSe,.'First,thereismetal-
metalbonding, whichcanleadtouniaxial cation-
cationpairing alongthechaindirection. Thisoc-
curswithNbS„' andputsthematerial inlinewith
manyotherd'compounds liken-Nbl~(Ref.40)and
Nb(S,)Cl,(Ref.44)orV(S,),(Ref.41)andVO,."
TheNb-Nbdistance withinthe5-axispairinNbS,
is3.04Acompared with3.36Aiftherewereno
pairing. Suchpairingsucceeds inmanycasesin
dropping afilledsubband offfromthebottomof
thed-bandmanifold toleaveasmallgap(-1eV)
semiconductor. TheSeebeck coefficient forNbS,
is-500p,V/'C.38Thesecondfactor, actually more
prevalent inthepresentmaterials, istoadjust
thestructure sothatsome(inNbSe,)orall(in
TaSe,)ofthecationsbecome pentavalent."Itis
quiteeasyforthepresentformofstructure toac-
comodate thenecessary augmentation totheelec-
troncontent ofthechalcogen-based valence band,
aswewillsee.Itiswellknown,ascanbeap-
preciated fromFig.1,thatZrSe,iseffectively
Zr~(Se,)'Se'.The(Se,)'pairingleadstos,
tightlyboundvalence subband, veryevident inthe
x-rayphotoemission (XPS)results,'andthereis
correspondingly theejection ofanantibonding
stateupoutofthePband.Thisleavesa16-
ratherthanan18-electron pvalence band.The
0Se-Se,pairinZrSe,isasstrong(2.35A)asin
pyrite-structured MnSe„where itwasevidentthat
theantibondingP stateisejectedasmuchas5eV
abovethetopoftheoccupied pband."
Inthecaseofgroup-V triselenides someofthe
-Se~become tovarying degreeunpaired, andcer-
tainoftheantibonding pstatesfallbackbelowthe
Fermilevelin.thebottomofthedband.Further
electrons passoverthenfromoneormoreofthe
cationsublattices totheanionsublattice. BecauseZrSC3t
p.y8&.---~
4
/is'
FIG.1.Monoclinic celI.ofZrSe3showingstaggered pris-
maticcolumns andtheircrosslinking. Lengths inang-
strems. Notehowtightlyback wallofprismisdrawninby
Se2pairings. Pri.smsshouldbe slightlytallerthandrawn.
Allatomsaty=+~.Centers ofinversion hereandin
remaining figuresmarked bycross.
Zr5e3
AP
A
Igem~'gw~~gcfA
~~~~
~~ ~~~~
~Qta~
2-343-07
FIG.2.PlanofFig.1.Eachunitcellformspartof
slabing-5planetwocolumns thick.Zr-Serangein
prismsfrom2.72-2.74A.Heavyatomsat4bo,light
3a't&50~oftheinterchain cross-linking shownbythestruc-
tures,thevarious cationsublattices (chaintypes)
arenotindependent, andpartialvalencies (be-
tween4and5)maydevelopforthecations onall
sublattices. However, beforeonecanproperly
appreciate theelectronic structures, itfirstis
necessary tostudyingreaterdetailthespecific
structures actually acquired.
(i)ZrSe,.Figures 1and2showhowtheformul-
ationZr(Se,)Seandanemptydbandarises. The
p-tod-bandgapsinthetichalcogenides'~'~are
almostidentical withthoseinthecorresponding
dichalcogenides."InZrSe,thegapis1.1eV,in
ZrS3itis1.9eV.Typically, inthedichalcogenides
6458 J.A.WILSON 19
=CPUB
x-—
l
l
I
t
0fj~
~~
~~&41ji:&"
ZrSe,"~ 4 ~
I..~~''Sy
I
It
I lX X
ZrSe3'
FIG.3.(a)Orthorhombic PuBr3structure. Prisma-
ticcolumns nowfullyinterpenetrate slabs.Space
groupD2I,,withoriginat2/m.Heavyatomsatx=0;
lightatomsatx=2;(b)twovariants ofZrSe3structure,
emphasizing change inintercolumn pathways.between groupsIVandV,thedbandislowered
by1to1—,'eVsothatslightp-doverlapispresent
inIT(2H)-TaSe„etc. ,butthedegreeofd-bandfill-
ingisgreatenoughtherefortheFermileveltolieonly
inthedband.Suchasituationis notlikelytopersist
inthegroup-Vtrichalcogenides duetomodification
fromM-MandX-Xpairings, asweshallsee.
InFig.2theinterchain couplings aredottedin.
Theseformacontinuous M-X-M-X pathwayper-
pendicular tothecolumns, andmustbetakeninto
account whenthebandstructure comestobecal-
culated. InthecloselyrelatedPuBr3structure,'
wherenoanion-anion pairingoccurs andthecation
istrivalent, thetrigonal prismatic columns pack
intonarrower slabs,withtheinterchain linkages
nowimmediately terminating onthe"back"edge
oftheprisms[seeFig.3(a)].Similar packing to
thisrecurs, asweshallsee,inthestructures of
NbSe,andTaSe,atthosecolumns whichshowno
anion-anion pairing.
Further drawing attention tothecrosslinkages
istheexistence ofasecondvariantBoftheZrSe,-typestructure,'~'~measured forTiS„HfSe„and
ZrTe,[Fig.3(b)].Herethetwointerchain M-X
distances, whicharealmostidentical inZrSe",at
at2.870and2.868A,become highlydifferentiated,
beinginHfSe3forexample3.100and2.624A.The
shorterdistance-is nowindeedlessthantheaver-
ageM-Xbondlengthwithintheprism.
(ii)NbS,.Thedistortion oftheZrSe,structure de-
veloped byNbS„'beingintheformofintrachain ca-
tionpairing, becomes ofprimeinterest tothepres-
entwork.Suchpairingisendemic tot,',(ort,',)
chains,whethermetallic initiallyasinVo„'orin-
sulating" asinlow-spind' RuBr,.(Asatisfactory
structural analysis hasnotyetbeen madeford'P-
TiCl„etc. )Accordingly, theformofthedistortion
isnotgoverned bytheFermi-surface geometry, but
byoptimization ofintercellularmatrixelements.
InNbS,thepairingrepresents a10&oreduction in
cationspacing underb-axisshiftsof0.16A.The
bridging triangle ofsulphur atomsispushedout-
wardsinanattempt toconserve M-Xbondlengths,
theprincipal factordetermining theenergetics
ofthemainp-valence band.(SeeRef.48forcase
ofdichalcogenide periodic structural distortions. )
Thetriangleattheotherendoftheprismiscor-
respondingly drawnin.Inneithertriangle, how-
ever,doesthedimension oftheS,'-pairingde-
viatefromwhatistobeexpected, giventhatin
ZrS,(seeTableI).
Thebandgapgenerated withthecationpairing
inNbS,hasnotbeenascertained yet,butislikely
tobenomorethan —,'eV,andsoisd-d(asin
VO,)rather thanp-d.Ifthetriselenide had
adopted thisstructure suchwouldprobably nolong-
erbethecase.
(iii)TaSe,.Thismaterial showsaverydiffer-
entevolution oftheZrSe,structure, asistobe
seenfromFig.4andTableI.Theslabsbreakup
intocantedblocksoffourcolumns each.'~These
blocksarecomposed ofaninnerduoofcolumns
thatarestillintheZrSe,setting, anddisplayfair-
lysubstantial X-Xpairing(2.58A).Theflanking
duo,however, havealmostcompletely relaxed
theirX-Xpairing(2.90Atobecompared with
3.43AinTaSe,).Moreover, theintercolumn link-
agesnowterminate ontheseexpanded columns in-
steadofforming extended pathways. Theseter-
minations hereoccuratboththebackedgeand
frontcorneroftheprism, unlikethesituation in
PuBr,(backedgeonly).
Forconvenience ofreference hereandlater,we
shalldesignate thecolumns withlittleornoX-X
pairing"red,"columns withintermediate X-Xpair-
ing"orange,"andthecolumns withtightX-X pairing
(asinZrSe,)"yellow."InNbSe, weshallfindall
threetypes.
Thefirstsetofcolumns inTaSe3is"orange,"
19 BANDS BONDS,ANDCHARGE-DENSITY WAVES INTHE ~~~ 6459
TABLEI.Crystallographic datafortrichaleogenides.
ap
(A)bp
(A)Cp0
(A) SpacegroupMol.ecular
volume
(A')Anion
pairlengths (A)
Shorter
Intrachain intercha i.nReference
2P2(/m (C2~)
6P2(/m (C~~)2.344 ZrSe, 5.411
NbSe310.00994.97
85.553.7499.44497.48'
3.480 15.62910947'1(b) 3,067
2.374
2.485
2.909
2.58
2.90
2.0902~73
2.9321
4P2(/m(C)I,) TaSe310.4023.4959.829106.26' 85.7637(b) 2.65
2P2)/m (C2~)
4Pl(C/)3(a) 3,035 97.28'
97,17'5.1243.6248.980 82,84
75.75
77.71NbS3 4.9636.7309.144 2,91 20 2.05
TaS3 36,80415,1733.34090' 24Notknown Notknown 13
prallel.tobaxes I.neachcaseexceptTaSwherearal ILNoteprismatic chainsarea
ntherefinement ofthestructure ofNbS(Rf)
apointhalfway betweenthese.ef.20thecentersofinversios'onhavenotbeenchosenasorigin, but
'INbSeInNbSe21.948x10cell.s/cm; inTaSe32.915x10.Multi 1these
particular band.x.ultiplytheseby2fordensityofelectrons neededtofilla
itsX-Xpairingat2.58Abeingconsiderabl ak-
erthywe
erantnZrSea(2.34A)orinthepyrites MnSea
(2.38A),andOsSe,(2.44A)orthemarcasite
FeSe &2.53,(.A,.(N.B.,thelattertwocompounds
aret',semiconductors. )Thepairing inTaSe
bowev~3P
wever,is,asstrongasthatinIrSe„forwhich
aconstitution Ir",(Se'),(Se,),'witht~-e,'semi-
conductivity isstillsecured(seeRef.47F'1).
The
tocepairing evidently remains sufficientl str
ocasttheantibonding pstatethe1or2eVabove
thetopoftheoccupied pbandthatisneeded.
Because oftheaboveratherweakSe-Sebonding
inthecolumns ofTaSe„an intexcolumn Se-Se
separation ofonly2.65Aisincurred
0~urre,ascompared
with3.07AinZrSe,.Thismaywellbeshort
enoughsignificantly toperturb thetopofthep-band
structure, aswillbeseenpresently.
(iv)NbSe,.Thestructure ofNbSe,(Ref.21)in-
cludessimilarblocksoffourcolumns tothosein
TaSe,(though withtheinner"orange" columns
showing somewhat stronger Se-Sepairing) but
theseblocksarenowinterconnected through anad-
ditional duoofstrongly cantedZrSe-like"yellow"3
columns instead ofbeingdirectly linked(Fig.5).
Theinterchain M-Xpathways onceagainterminate
attheredcolumns. Theslabcleavage plane,
thoughnotkinkedasinTaSe„cuts through some
rathershortred-to-yellow spacings of3.30A.
Theyellowcolumns defineinterslab slicesinthe
a-bplane,andinclined yellow-orange sheetsmay
alsobetraced. Withinthetrueslabplanes(b-c)
thepathbetween theyellow andorangecolumns
isofcourseinterrupted bytheredcolumns.
v&a,.Thecrystalstructure forthelastmem-TaSe,q,,g
2Tjgg
2.58165
FIG.4.PlanofTaSe3structure. Heavyfeatures
aty=z lihtaigaty=g;orangecolumn,Q;redcolumn,
()=93.38',Ta-Serangeinprisms2,60-3.66A[Rberofthefamily (nogroup-V tritellurides are
known)isnotyetfullydetermined. Ithasbeen
reported thatthecellcontains 24molecular units
andisface-centered orthorhombic "'&The
originally ascribed spacegroupD,'hadnocenters
ofinversion, andthiswasdifficult toreconcile
withthepresentclassofstructures. Thevery
recentworkofTsutsumi" suggestD"Th'S2~.1Sis
actually thespacegroupforPuBr„" butitstill
seemstoofferdifficulties forTaS,withZ=24.
TaSe„as aglanceatFig.4willshow,almost
presents aface-centered-orthorhombic cellof
eight-molecule content, spanning twoslabs.The
actualangletheretothefour-column-broad unit
was93.38'.Celldimensions quotedforTaS„
6460 J.A.WIISON 19
7'3 S.ss
'~O~~&.~~0~~'
il
2.9t 249
7X
2.7g.~FIG.5.PlanofNbSe3
withdetailsasdetermined
inRef.21.baxisinto
paper.Heavier features
aty=—;lightaty=—'.
Nb-Serangeinprisms
2.63-2.67A.5yellow
{fullypaired),Qorange,Qred{depaired). Yellow
columns inrowsparallel
toa0,orange inrowspar-
allelto(101).
seemtobefora2slab~12columnunit.
Thematerial clearlydoesnotshowM-Ipairing
inthecolumns (unlikeNbS,)sinceitsc-axisparam-
etercorresponds totheheightofoneprismonly(at300
K).TaS,should,however, showastronger adherence
toX-Xpairing thanTaSe,andtherefore bemore
4+like.Nonetheless itisquitestrongly diamag-
netic,evenatroomtemperature,"'sothat,as
withallthesematerials, thenumber offreecar-
riersthatisfinallypresented israthersmall.
Wenowturntolookinmoredetailattheelec-
tronicproperties, theirform,andtheirorigin.:R+~to010
0
:D
~Q :P
III.GENERAL FORMTOTHEBANDSTRUCTURES
ANDTHERESULTING ELECTRONIC CONDITIONS
Whenthenumberofmolecules perunitcellbecomes
largeandof"differentiated disposition,"andparticu-
larlyasthespace-groupsymmetry becomes rather
limited, thebandstructure israpidly brought to
astateofhorizontal stratification. Themany
bands,oftennotdegenerate evenatthespecial
pointsinthezone,areheldfromcrossing each
otherunderthe.veryrestricted setofsymmetry
representations prevailing. Thisisparticularly
sooncethedouble-group representations are
considered.
Afeelingforthetypeofsituation tobeexpected
inZrSe,orTaSe,canbegainedfromthecalculat-
edbandstructure forVO,initsroom-temperature
monoclinic state(Ref.45,Fig.2).Onlytherather
isolated M-Mbonding dbandreaches 1eVin
width. Thespacegroupformonoclinic VO,(Z=4)
isC,'„.AsinthepresentcasewithC,'„,veryfew
degenerate representations arise, thoughthereis~Y
Bg
001a,
Y
FIG.6.Brillouin zoneforNbSe3toscale.Planin
Fig.7(b).Fermisurface shownproduces a1%zone
filling—seetext.
19 BANDS, BONDS, ANDCHARGE-DENSITY WAVES INTHE... 6461
101
00&
00
//slabs
fororbitt15
'A
'~
//c,Q01
10101A''
100
101
~0~
QQT~
ga.A~~~0
101
101
FIG.7.(a)Brillouin zoneforTaSe3inplan,inorienta-
tioncorresponding toFig.4.Theorbitsindicated are
intheplaneoftheslabs(i.e.,containing 5*);(b)Bril-
louinzoneforNbSe3inplan,inorientation correspond-
ingtoFig.5(i.e.,baxisintopaper). Fermisurface
shownisthatwhichappears inFig.6for1%zonefilling.somedegeneracy associated withtimereversal
symmetry. Awayfromthepointsofspecialsym-
metrytherearenevermorethantw.orepresenta-
tions.TheFpointwithlittlegroupP2jmandele-
mentsE,I,6,'~,andpt"'~hasfoursinglydegen-
eraterepresentations. SomepointslikeZhave
justalonedoublydegenerate representation.
Themonoclinic zone,givenproportions ap-
propriate toNbSe,(andZak'slabeling ofthesym-
metrypoints)isshowninFig.8.Thecrosssec-
tionsofthezoneperpendicular to5*(i.e.,tob,
thechaindirection) areshownforTaSe,and
NbSe,inFig.7.Thesecrosssections have
beenshownintheorientation appropriate to
Figs.4and5forthecrystalstructures, mean-
ingthedirection corresponding totheslab-plane
runs"east-west."[NoteforNbSe„ inordertoob-
tainaright-hand setofaxes,b,andb*aredirect-
edintothepaperinFigs.5andV(b).]
Theseemingly highersymmetry totheNbSe,
zoneiscoincidential andisduetotheapproximate-
lyequaldisposition of[100]and[101]about[201](the
originofwhichisillustrated inRef.21).Evenwith
TaSe„points F,B,andAappearmidleadingly
closetothemidpoints ofthezonefaces.Thereis,
ofcourse, nomirror planeparallel tob*.The
dimensions ofthezonesaregiveninTableII.
Thegeneralsortofshapetobeanticipated fora
smallpieceofFermisurface inthesematerials
isportrayed inFig.6.Thereisstrongelonga-
tionperpendicular totheslabdirection because
oflowbanddispersion inthisdirection acrossthe
VanderWaalsgaps.Withintheplaneoftheslabs,
particularly forap-bandsurface, weexpectel-
lipsoidal sections ofrelatively smalleccentricity.
ItshouldbenotedfromFig.5thatthetypicalSe-Se
distance intheslabsJ.b,(-3.30A)isactuallyless
thantheSe-Sedistance alongthecolumns(3.48A).
Sincemuchofthewidthofadbandinmaterials like
thepresentcomesfromP-d mixing,"initial"d-band"
curvatures intheslabarelikelytobecomparable,
despitemetal-metal spacings between columns being
at-4.3Aconsiderably greater thanthoseinthecol-
umns(3.49A).Itshouldbenotedthatintheverygood
pyritemetalP-RhSe,(e')theshortest metal-metal
distance alsois4.3A.4'The"d-band" widthdoesnot
dependprimarily onthisdirectM-M overlap.
Inaccordwiththeabove,atentative measure of
theeffective massesindicates theretobelittle
anisotropy intheslabsfortheverysmallnumber
ofcarriers present inNbSe,atheliu&tempera-
tures,whileperpendicular totheslabsananiso-
tropyof3inmassisreported."Theroom-temp-
erature conductivity anisotropy o,/o,inNbSe,is
-500,'whereas the"inslab"&z,/~,isonly-15."
Thesevaluesprobably aresomewhat overstated
because oftheveryeasycrystalfracture parallel
6462 J.A.WILSON
TABLEII.Heciprocal. -spacedetailsforTaSe3andNbSe3.
b'
(A')P'(deg)Volume
of
zone
(A3)Areaofzone
basalplane
(A-')Megagauss
equivalence
(MG)Areaofb*
containing
section
parallel
toslabs
(A-')Megagauss
equivalence
(MG)
TaSe3
NbSe30.62920.66591.7978
0,66580.42641.805573.74
70.530.7231
0.48320.4022
0.267642
281.38
0.77145
80
to5.
Asevaluated inTableI,alg~filledbandorzone
inNbSe,wouldamount to3.9x10"e/cm'. Theel-
lipsoid drawninFig.6isofapproximately this
filling,metbysemiaxes —,'a,*,—,'cf,+b,*(i.e.
„
0.266,0.085,0.051A').Theexperimental room-
temperature Hallcoefficient" of—lx10'cm'/cb,
wereitattributable simplyton-typecarriers, is
equivalent toonly1.5x10"e/em',sothateven
priortotheCDWtransitions thematerial isvery
muchthesemimetal with=,'/0bandfilling.[The
susceptibility, whilequitestrongly negative
(-0.21x10'emu/g),27isappreciably larger than
inZrSe,(-0.33x10-'), '(~'buttherearegoingtobe
several additional paramagnetic termsbesides the
Pauliterm,itselfenhanced inNbSe,.]TheCDW
transitions inNbSe,seemtoremove about75%of
theFermisurface,'takingthecarriers countdown
toonlys4x10"/em',or0.10%zonefilling. The
aboveconcentrations areevenlessthaninTiSe,
(Ref.52)aboveandbelowitselectron-hole driven
condensation, andthedetailsareintrinsically
evenmoredifficult toanalyze fully,.asweshall
see.
Weturnnowtoconsider insomedetailtheelec-
tronicconditions established inthethreematerials
TaSe„NbSe„and TaS,.
(i)TaSe,.TaSe,showsnolow-temperature distor-
tionsandlossoffreecarriercontent. Fromitssome-
whatlowerdiamagnetism andhigherconductivity at
300Kthecarrier content wouldappearalittlegreat-
erthaninNbSe,.SomeShubnikov-de Haasdatahas
beencollected forTaSe,fororbitsintheplaneof
theslabs, andalsofortheirvariation in"area"
uponinclining thefieldtoproduce tiltedinterslab
orbits. Asmighthavebeenexpected, theobserved
angularresponse functionisoftheformtochar-
acterize theFermi-surface crosssections sampled
ascomingfromeffectively cylindrical elements,
thesituation inalayercompound (seeRef.16,
Fig.12,wherethecommon cylindrical axisfalls
closelyperpendicular totheslabs;compare Fig.
7inRef.56dealing withtheFermisurface of
2H-TaSe,).Mostoftheorbitsarequitesmall,evenorbit15amounting toonlyabout ~~pthearea
oftherelevant bp*-containing section ofthezone
[closely parallel totheslabs;seeFig.7(a)].
IntheSdHdatatheareaoforbit15wouldbe
satisfied byanellipse withsemiaxes 0.20x(~b~)
by0.66x(2d,*),(i.e.,0.18x0.24A').Suchacylin-
derisincluded inFig7(a).,andamounts toazone
fillingofapproximately 10%%uz.FromTableI,we
seethistakesanelectron density ofabout5or
6x10"/cm', whichisverycomparable tothatin
TiSe,atroomtemperature."Orbit4,whichis
—,'thesizeoforbit15,wouldbesatisfied byan
elongated ellipsoid thatjustreaches pointA[see
Fig.7(a)],thecrosssectional semiaxes being
0.08(~ho*)x0. 33(~do*)or0.07x0.12A~.The
volume ofthisellipsoid amounts to.2.5/oofthe
Brillouin zone.
The2a„2epperiodic latticedistortion whichde-
velops inTiSe,comesasaresultofelectron-hole
coupling betweencarrierpockets atI.andI'ona
particularly simplyFermisurface. Theabsence
ofanysuchdistortion inTaSe,mightbeascribed
totheclearlygreater complexity ofitsFermi
surface. Whenwecomebelowtoexamine theform
expected forthebandstructure ofTaSe,itwillbe
seenthatbothp-andd-bandcarriers areindeed
anticipated. Wedonotbelieveeitherthatthe2$p
repeatfoundinNbS,couldbeduetoI'Zelectron-
holeinteraction, because thatdistortion amounts
nottoanion-cation sublattice coupling, butrather
tocation-cation, andthelatterisofmoregeneral
.natureaswasemphasized earlier.
Wehavealready notedinthebeginning ofSec.
IIthemanner inwhichTaSe,isnearlyrendered
semiconducting. Theanionsublattice oftheorange
columntakesfourelectrons perTa(Sea)Se unit,
theredcolumnsixelectrons perTaSe,unit,so
that,with10electrons givenoverperpairofTa,
thejointtantalum-based Pbandisdrained ofelec-
trons.Thatthisprocessisnotinpractice quite
carried tocompletion isduetoslightoverlapbe-
tweenthetopofthepbandandthebottomofthe
dband.Suchoverlap indeedexistsin2H-TaSe,
(asdetected byphotoelectron spectroscopy), but
BARDS, BONDS, ANDCHARGE-DENSITY %AVES IÃTHE...
isnotforthatd'compound abletoaffect(directly)
theFermisurface. Because inTaSe,thetantalum
valenceislargerthaninTaSe„ thedbandwill
actually bewithdrawn —,'eVtohigherenergies.
However, thistendency forthep-dbandoverlap
tobelostisamplycountered bythedestabilization
ofthetopofthepbandinTaSe,relative toTaSe„
forweakly antibonding pstatesnowarepresent
intheredcolumn (andpossibly between redand
orangecolumns; Se-Se2.65A).(Theantibonding
pstatesfromtheorangecolumnareunoccupied
andprobably lie2-3eVabovetheFermilevel.)
AbovewehaveseenthatbothtypesofTaare
rendered d',i.e.,5+inionicterms. However,
sincethebalancing coordination charge inthe
orangeprismsis"4-"andinthered'prisms"6-",
thed-bandstateassociated withthelatterwilllie
tohigherenergy. Itwouldseemthen,thattheob-
servedelectron contentcarried bythelowestd
bandistobeassociated inthemainwiththeorange
columns. Itshouldberemembered, however, that
theanionsublattices ofthetwocolumns arenot
independent, andthatp-dhybridization issubstan-
tial.Theholes, thoughtoastilllesserdegree,
onemightassociated withtheredcolumns. Hole
anisotropy intheslabsshouldbeappreciably less
thanelectron anisotropy.
Theratherlargenumberoforbitsdetected inthe
SdHworkfollowsfromtherebeingtwoatomsofeach
oftheeightkinds perunitcell.Thissecures adual
presence atagreatmanypoints in-thezoneforthe
bandsthatarebasedonanygivenatomicfunction,
say,orangetantalumdp.TheFermisurface
probably incorporates bitsfromthelattertwo
bandsandfromatleasttwosetsofpbands,so
withneckandbellyorbitspossible wesoonreach
alargenumber.
Thereisnoevidence fromitsvariousproper-
tiesofanytransition occurring inTaSe,between
4and'700K'&~'&'~Thesituation withregardto
superconductivity isnotclear.'~'~
(ii)NbSe,.IntheSdHworkitwasfoundthat
NbSe„contrary toTaSe„showed onlyonepri-
mary(i.e.,nonharmonic) orbit,andthatvery
small."""Theorbitalsotakesanunusualro-
tatedangularaspectrelative tothoseforTaSe,.
Thesizeisonlyabout~~o ofthesizeoforbit4in
TaSe,.Theoccupied volumeofkspaceisthus
goingtobedownbyafactor 30.Thisamounts
toa0.1/~fillingoftheNbSe,zone,orclosetothe
samenumber ofcarriers aswassuggested bythe
Hallconstant, namely4x10"/cm'. Theseare
thecarriers leftfollowing theperiodic-structural-
distortion (PSD)interactions thatsetinat144
and59'K,fromwhich-75/poftheoriginalcarrier
contentislost.
Theelectronic condition actually prevailing inNbSe,at300'Kisalsonecessarily significantly dif-
ferentfromthatinTaSe,.Theaddedpresence in
thestructure oftheyellowcolumns meansthere
isnowasmaller percentage demand uponthed
levelsforelectrons tocomplete thepbandofthe
depaired, red-column anionsublattice. Thereck-
oninggoesasfollows: Thesixcationspercell
introduce 30outerelectrons. Ofthese 4&&4elec-
tronsarerequired bythetwopairsoforange and
yellowSe,'containing units.Thepairofred
NbSe,unitsthenrequireafurther 2&&6electrons.
Thisleavesoverforthedbandtwoelectrons.
Thelowestreaching dbandwillbeoneofthetwo
d,2-based bandsassociated withtheyellowcol-
umns.Theobserved noninsulating character of
NbSe,canreadilybeaccepted onthefollowing
counts:First,thetwoyellowbandsaregoingto
betiedtogether bysymmetry atcertain pointsin
thezone.Second, thelowermost d&bandas-
sociated withtheorangecolumnislikelyalsoto
passbelowtheFermilevel.Third,thetopofthe
pbandmaycontinue tooverlapE~,despite thelat-
ternowbeingraisedsignificantly intothedband
(unlike inTaSe,).Thisisbecause thegeneral
d-pseparation inthe4dmaterial willbereduced
by—',eVrelative tothe5d(seeRef.47forthecase
ofNbSe,vsTaSe„etc. ).
.Itshouldbenotedthatintheabsence ofd-dand
p-doverlap, anelectron content ofexactly 1elec-
tronperyellowcationcouldpossibly havebrought
fullcation-cation pairing totheyellowcolumns,
asinNbS,.Lossoftheintegral electron number
peryellowcationcounters this,asalsomustthe
crystalstrainenergysinceonlytwoofthesix
columns wouldshowpairing. Thedistortions act-
uallyfoundinNbSe,seeminfact,asweshallsee
below,tobeseatedbothintheyellowandthe
orangecolumns.
Itispossible thatthegreaterdispersion ofthe
yellow-based dband(fromthecontinuous yellow
connectivity inthea,direction) allowstheorange
bandtobecome significantly filledtoo,despiteits
lowermost pointbeinghigherinenergy. Onemay
infactinferfromthediffraction resultsthatthe
extents ofkspaceoverwhichthetwobandsliebe-
lowtheFermilevelarerathersimilar. Inthose
circumstances bothtypesofchainwillcontain only
about —,'electron perniobium.
%hatthediffraction experiments~~'2'2' revealis
thatthe144-'Ktransition leadstoadistortion with
wavevector(0,0.243,0)whilethesecondtransi-
tionat59'Kleadstoanindependent distortion with
wavevector(0.5,0.263,0.5).Theformer PSDone
mayassociate withtheyellowcolumns. Those
columns definenear-neighbor sheetsparallel to
theaaxis.ThesecondPSDcanbeassociated with
theorangecolumns, whichdefinemuchmore
6464 J.A.WILSON 19
+)0
NbSe,by
~t
~~
r
FIG.8.Typeofbandstructure envisaged forNbSe3,
lightbandsbeingassociated withd»oftheyellowcol-
urnns, heavybandswithd»ofthe'orangecolumns,
anddottedbandswithoccupied slightly antibonding pstates. Inthelowerpartofthefigureisindicated how
themajority ofelectrons andholesmaybeeliminated
fromthissemimetallic situation undernesting inthe
yellowcolumn byq&andintheorangecolumn byq2.The
sizeofthepocketsismagnified forclarity.looselyrelatedsheetshavingthe101orientation
(seeFig.5).Thephasing between neighboring
centers ofinversion withinthea-cplaneforthe
yellowcolumn waveisuniformly0,butforthe
orange-column waveinthe101direction (i.e.,
acrosstheyellowsheets)itisn.(Remember with-
inaduoofcolumns thetwoarestaggered byone-
halfaprismheightabouttheintervening centerof
inversion. )
Alongthecolumns bothwavesareseentobe
incommensurate. Thewavelengths invertoutas
4.115and3.802Qprespectively. Thecloseness
oftheseperiodicities toeachotherandto4&pof-
ferssuggestion forthedriving mechanism behind
thesedistortions.
Thesortofband-structure situation thatwehave
envisaged aboveisportrayed inFig.8.Ahybrid-
izationgap,following bandcrossing ofthelower-
mostyellow andorangebandsnear —,'I'Zisshown
leading totheappearance ofholesinthatregion.
Compensating electron pocketsareintroduced at
IandZ.Fromsuchasemimetallic situation 75~&
ofthecarriers aresubsequently eliminated under
thedistortions. Sincethedistortions eachgrow
fromzeroinaBCS-like fashion,"ithasbeenpre-
sumedthatthelossofcarriers arisesfromFer-
mi-surface nesting andgapping through develop-
mentoftwoindependent charge-density waveswithwavevectors q,andq,.Figure8indicates
howthegapping canfirstsetinwiththeyellow-
columncarriers, andthenextendtotheorange.
Wehavemuchreservation, though, aboutusing
anexcitonic insulator description ofthistypefor
thepresentcase.Itisunfortunate thatsubstitu-
tionaldopinginanisostructural, rigid-band
framework isnotanavailable probehere(un-
like1T-TaSe,).Thefact,however, thata4to
periodicity occurs againinTaS,(Refs.13,15,
21,and25)wherethestructure isdifferent, and
thatoneseestheendemic2t,periodicity fromcation
pairing inthecolumns ofNbS„opens analternative
avenuetoinvestigation—metal-metal bonding.
Undernesting andgapping onlybandstatesfairly
closetotheFermilevelarestabilized. Bycontrast,
thecationpairingprocess leadstostabilization of
sta,testhroughout thebonding band.Normally this
stabilization inalinear-chain compound like
V(S,)„NbI4, orRuBr,issogreatthatthepairing
cannotberemoved thermally. Thislikewise is
trueforNbS,.IgsomecaseslikeVO,andNbO„
whicharenotcrystallographically onedimension-
al,thepairing mayberemoved thermally, norm-
allyinvolving afirst-order transition. Thecation
displacement associated withpairing inV(S,),is
0.20A~andinNbS,0.16A,"butforVO,itisonly
0.12A."
InNbSe3thenumber ofd-electrons intheyellow
andorangecolumns issufficient thereonlytopair
upeverysecondcoupleofcations. This,plusthe
factthattheelectron countper(orange andyellow)
cationisnotquite —,',andthattheredcolumns are
notparticipating, wouldexplain whyforsuchbe-
haviorinNbSe,thedistortions growinsecond-
orderfashion. Onecanalsoaccommodate thefact
thattheincommensurate distortions whicharise
here,although onlymarginally removed fromq,=—,',donotlock-in duringcooling. Indeed, unlike
closelycommensurate 2H-TaSe,or1T-TaS„the
wavevectors inNbSe,arefoundtobeinvariant
withtemperature."Onemaysuspect thattheob-
servedwavevectorsaresimplyapproximations
tothevulgarfractions ~»(i.e.,0.2432,withA,
=4~9b,)and+,(i.e.,0.2632,withX,=3,5,),im-—
plyingthatinNbSe,wearedealing witha"dis-
commensurate" situation" whichisstrongly im-
posed. Theimplied situation intheyellowchains
(es—,')isthatpairing between everysecondcouple
ofniobium atomsproceeds undisturbed fornine
periodsbefore anextraunpaired atomordiscom-
mensuration appears, asinFig.9.Intheorange
columns theelectron count(ea—,')isconversely
suchthatonlyoneunpaired niobiumispresentat
theendoffiveundisturbed periods. Withthese
defectsrepeating regularly every37and19atoms
thediffraction isasfromtherecorded incommen-
BANDS, BONDS, ANDCHAB,GE-DENSITY WAVES INTHE... 6465
suratewaves. However, innonsinusoidal situa-
tionslikethis,higherharmonics mustshowup
(seeRef.53forthecaseof2H-TaSe,).InNbSe,
asecondharmonic isinfactreported" withI,
=0.01I,.
Theredcolumns should, through theabovepro-
cesses, shownocationpairing. Theonlymea-
surements sofarindicate thatthelateraldimen-
sionsinaredcolumnincrease2' (thoughitmust
beremembered thatthelow-temperature crystal-
lographic refinement isofanaveraged structure).
(iii)TaS,.TaS,haslongbeenknowntoundergo
asemimetal tosemiconductor transition around
200'K.'Untilthestructure ofTaS,isresolved,
itisnotpossible tointerpret indetailwhatishap-
peningelectronically. Theherecommensurate
4c,periodalongthechainscouldevenbeduetoa
complex packing offullypairedchains. (N.B.a'
=2ao;b'=8b,:Ref.25.)Thedistortion develops
through alongprecursor rangedownfromroom
temperature, beforelong-range orderappears
around220'K."""InTiSe,(Ref.52)similarly
strongscattering preceeds theadoption ofacom-
mensurate distortion. Inneithercaseisthediffuse
scattering entirelylostatthetransition. The
transition inTaS,appears towipeouttheentire
Fermisurface, producing anelectrical activation
energy of0.15eV."Theresidual conductivity
-3(II—cm)'probably comesfrompoorstoichio-
metry.
IV.SLIDING WAVES INNbSe3: AUNIQUE SITUATION
t
WhathasprovedsofaruniqueaboutNbSe3
amongCDW-bearing materials isthatitsCDW-
PSD'scanbeinduced toslideandtherebycarry
current underaverysmallappliedelectric
field.'"Easeofslidingofcourserequires that
theCDWbeincommensurate (i.e.,notlocked-in
totheparentlattice), andalsothatitspinning by
impurities anddefectsbelow.Itisinthislater
area,thatNbSe,isexceptional.
Thetrichalcogenides appeartohaveaverynarrow
homogeneity range,quiteunlikethedichalcogenides.
Thisisprobably because thecrystalstructures are
socomplex andare"molecular ized"inafashionspe-
cifically determined bytheatomsandstoichiometry
ofthecompound. Itislikelythatsubstitutional de-
fectsarethereby heldtoaminimum [compare
"PS,"withP(S,),j.BeyondthisinNbSe,wehaveanin-
ternalsinkforimpurities, namely, thered
column. Byfarthemostcommon impurity cation
likelytobeincorporated isTa,andthiswouldcer-
tainlypreferthelatter's highervalency situation.
Anionimpurities wouldalsofavortheredcolumns,
notbeingreadilyabletoparticipate intheanion
pairbonding oftheothercolumns. Byresidingyellow
g(1/2orange
8)
I
1
I
II
'~
I
Q.
tt
I
Wt
~
tc0
~~
L0
lA
~~
'U
0
E0
cfC4
g,l
U
0
t58
CL
E0
0
CJ)
FIG.9.Modelofalternate pairingofNbatomsin
yel.lowandorangecolumns ofNbSe3.Deviation ofthe
average electron contentpercationtherefrom 2is
takenupinthediscommensurations (d).Thelatterwill
slidethrough thecolumns bybondflipping. Thecrys-
tal.lographic distortions growinsecond-order fashion.
InthismodelthedetailsoftheFermi-surface geometry
arenotimportant, electronic counting beingsufficient.ontheredcolumns, ananionimpurity canalso
avoidhavingtoparticipate intheintercolumn path-
waysoftheeightfold Nbcoordination. Thedrive
toestablish thatcoordination alsoprobablyre-
stricts theinclusion ofcationvacancies ontheyel-
lowandorangecolumns. Indeedtheanionarray
ofNbSe,isnotatallbasedonclosepacking. Ca-
tionvacancies arefoundinthehcpanion-arrayed
chainstructure ofHfI,"andhelptoinhibitthe
expected metallic conductivity ("broken strands").
Itshouldberemembered, too,thatthe(0andI')
chainsinNbSe,occurininterconnected "duos,"
sothatanycationvacancy whichisincorporated
intooneofthesechainsmaybemorereadilyb'y-
6466 J.A.WILSON 19
passed.
Asaresultofallthesefactors, NbSe,endsup
bypresenting veryhighsinglecarrier mobilit-
ies"forasemimetallic transition metalcom-
pound. Thismirrors theeasewithwhichthe
collective modeisdepinned. Whatisenvisaged Bs
happening inthemotionofthelatteristhatthedis-
commensurations propagate assolitons througha
flipping overofeachsuccessive electron-pair bond
alongthecolumn(seeFig.9).The"sliding" is
foundtoleavetheamplitude andwavevectorofthe
PSDunchanged" (inaddition totheonsettempera-
ture),implying thattheprocessiscooperative
between chains, probably beingstraincoupled
across themanycenters ofinversion oftheparent
structure. Thefactthatthediscommensurations
intheorange andyellowchainshavetoappearat
different intervals restrains thestructure from
excessive distortion, asmustthepresence ofthe
inactive redcolumns.
Itisthisgeneral conglomeration ofunusualfea-
turesinthestructure andchemistry ofNbSe,which
makeitdifficult toenvisage anothermaterial more
suitedtotheappearance ofsliding"charge-den-
sitywaves."Onecertainly wouldnotexpectitin
Zr-substituted NbS„where thenonintegral elec-
troncountisproduced onlyattheexpense ofin-
troducing astrongpinningcenterifthecolumn.
Onepossibility maybetoboostthed-bandelec-
-troncontent inTaSe,bypressure.
Underpressure NbSe,rapidlylosesitsPSD's;
theamplitudes falling inaddition totherebeinga
depression ofbothonsettemperatures attherate
of4K/kB.'Thisisnotsurprising sincepressure
inthesecompounds morethanmostwillmodify
thebandstructure. Actually NbSe,ratherquickly
becomes superconducting. 'Thegreatest sensitiv-
ityisseenascomingfromthetopofthepband.
Acoupleofthenonbonding intercolumn Se-Sedis-
tancesareparticularly short(2.73and2.9A),and
these,together withthestatesfromthered-
columnSe-Sepairthatwehaveregarding asweak-
lybonded, shoulddevelopstronger bonding-anti-
bondingcharacteristics. Littlechangeisneeded
tocarrythelatteroccupied statesabovetheFer-
milevel.Sothereshouldarisearatherrapid
transfer ofelectrons fromtheaniontothecation
sublattice, destroying theelectron countand/or
geometry oftheFermisurface whichoriginally
drovethedistortions (seeFig.8).Itwould, during
thisprocess, beofgreatinterest toknowwhat
happens tothedistortion wavevectors. Doesthe
yellowcolumn waveeverlock-into450?Ithas
beenobserved thattheuppertransition, whichwe
haveassociated withtheyellowcolumn, initially
hasitsamplitude reduced moreslowlythandoes
thelower.'lfthereareindeeddiscommensura-tionsofthetypesuggested bythepair-bonding pic-
ture'thenthewavevectors shouldchangebysmall
butfinitejumps, goingfrom~~x~»(or0.2432)to
~~x~~(or0.2439),etc,.andfrom~~x~»(or0.2632)
to—,'x—,",(or0.2666),etc.aselectrons passfrom
thePintothedband.
V.SUMMARY
Byexamining thestructure andbonding ofthe
transition-metal trichalcogenides inmoredetail
thanpreviously, ithasprovedpossible toobtain
abetterunderstanding oftheelectronic proper-
tiesofthesematerials. Itisseenhowthoseprop-
ertiescometobesignificantly different fromma-
terialtomaterial. TaSe,isasemimetal byp-d
overlap, NbSe,by4-doverlap. Thenumber of
freecarriers issignificantly greater inTaSe„but
thenumber ofelectrons actually inthedbandis
greater inNbSe,.Thelatternumber amounts to
verycloseto—,'anelectron percationinthe
columns showingSe-Sepairs.Theseelectrons
areresponsible forthetwoperiodic structural
distortions whichdevelop inNbSe,.Theupper
transition weassociate withthecolumns having
thetightestSe-Sepairing thatformsheetsparal-
leltoao,andhaveabroader, lower-reaching d
band.Itisconcluded thatthesedistortions are
betterdescribed byadiscommensurate arrayof
cationpairbondsthanbyatraditional CDW,its
geometry strictly determined bythatoftheFer-
mi-surface Kohnanomalies. WhiletheFermi
surface ofthesematerials isnotfullydetermined,
itisclearthatthesematerials shouldnotbere-
gardedaslinear-chain metalsafterthefashion of
KCPandTTF-TCNQ. Thetrichalcogenides clear-
lyprovideachallenge tobandtheorists usingnew
methods whichmaybeabletoyieldmeaningful
resultsforunitcellsofthissizeandcomplexity.
Theeffectoftheseveralvariable anion-anion pair
interactions shouldbeinteresting tofollow. Itis
notgoingtobepossible toneglect manyofthe
intercolumn Se-Seinteractions, inaddition tothe
M-Xcoupling oftheeightfold cationcoordination.
Withthisbackground wearefinallyabletosee
whythedistortions inNbSe,aresoweakly pinned
tothelattice byimpurities ordefects. Thesliding
ofthesedistortions wemodelintermsofbond
flipping intothediscommensurations.
Whatthebondmodeldoes,ofcourse,isto
selectfortheperiodic structural distortion a
specific phasingrelative tothecrystallatticewith-
inthecommensurate segments between discom-
mensurations. Figure10(a)identifies thatphasing
as——,'v.Figure10(b)showsthesituation forQ=0,
appropriate toadiscommensurate charge-density
wavewithchargemaxima andminimacentered
aboutcationsitesratherthanbetween them.
19 BANDS, BONDS, ANDCHARGE-DENSITY WAVES INTHE... 6467
(a)
ctrons 1~I
displnt
~~~~~~
dtsplocernent -down up
electron—deficitexcessdown up
deficitexcess
FIG.10.(a)Phasing ofacosinewave(fortheoriginchosen) thatisappropriate tothebonding modelis-4g.Al-
ternatepairsofatomsapproach andseparate. Thecorresponding charge wavebringselectrons intothebondregion.
Inpractice thewaveswiQdeviateprogressively fromsinusoidal formastheamplitude ofthedistortion increases; (b)
Thisshowsthephasing$=0,whichproduces electron excess oneveryfourthcation.Herethetwochainsoftheduoare
givenarelative phasing mostsuitedtothedominance ofelectrostatic stacking. InFig.10(a)theduowereshownwith
therelative phasing moresuitedtothedominance ofstraincoupling.
Itmustfinallyberecalled thattheactiveyellow
andorangecolumns occurinduos,thesebeing
separated bytheredcolumns. Thetwocolumns
ineachduoarestaggered relative toeachother
byhalf;the heightofaprismatic unit.Figure10
portrays oursuspicion thatthewavephasing within
aduomill,because oftheapproach toeightfold
coordination, bedetermined morebylatticestrain
thanbyelectrostatic considerations. Aswithina
singlecolumn, thestrongcationmotions bringintrainanionmotions thatareprincipally determined
bytheneedtomaintain theM.-Xbondlengths and
thep-bandenergetics. Theworking outofthis
principle wasveryevident intherelative anion-
cationsublattice phasing anddistortion pattern
forthecrystallographically muchsimplercaseof
20-TaSe,.'Inthepresentcasethecomplex parent
crystallography denieseffective modeling and
analysis oftheavailable diffraction dataforthe
lom-temperature state.
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|
PhysRevB.92.041101.pdf | RAPID COMMUNICATIONS
PHYSICAL REVIEW B 92, 041101(R) (2015)
Spin-orbit-induced longitudinal spin-polarized currents in nonmagnetic solids
S. Wimmer,*M. Seemann, K. Chadova, D. K ¨odderitzsch,†and H. Ebert
Department Chemie/Physikalische Chemie, Ludwig-Maximilians-Universit ¨at M ¨unchen, Butenandtstrasse 11, 81377 M ¨unchen, Germany
(Received 26 November 2014; revised manuscript received 9 June 2015; published 2 July 2015)
For certain nonmagnetic solids with low symmetry the occurrence of spin-polarized longitudinal currents is
predicted. These arise due to an interplay of spin-orbit interaction and the particular crystal symmetry. Thisresult is derived using a group-theoretical scheme that allows investigating the symmetry properties of anylinear response tensor relevant to the field of spintronics. For the spin conductivity tensor it is shown that only themagnetic Laue group has to be considered in this context. Within the introduced general scheme also the spin Halland additional related transverse effects emerge without making reference to the two-current model. Numericalstudies confirm these findings and demonstrate for (Au
1−xPtx)4Sc that the longitudinal spin conductivity may be
on the same order of magnitude as the conventional transverse one. The presented formalism only relies on themagnetic space group and therefore is universally applicable to any type of magnetic order.
DOI: 10.1103/PhysRevB.92.041101 PACS number(s): 72 .25.Ba,61.50.Ah,71.15.Rf,72.15.Qm
The discovery of the spin Hall effect [ 1–3] (SHE) with its
particular feature of converting a longitudinal charge currentinto a transverse spin current has sparked numerous studiesthat finally led to a deep understanding of many effects that arespin-orbit induced. Among them are the enigmatic anomalousHall effect (AHE) that shares the same origin as the SHEand many new phenomena emerging from a coupling of spin,charge, and orbital degrees of freedom in electric fields as wellas temperature gradients. Examples of these are the Edelsteineffect (EE [ 4,5]) and the spincaloritronic pendants to the SHE
and AHE, namely the spin and anomalous Nernst effects(SNE [ 6,7], ANE [ 8,9]), respectively. Many models have
been formulated that aim to capture particular contributionsto theses effects. For instance, the concept of the semiclassicalBerry phase that can be determined on the basis of the bandstructure of perfect crystalline systems is connected to so calledintrinsic contributions [ 10–12]. Extrinsic contributions arising
from scattering at impurities in nonperfect systems can, forexample, be obtained from diagrammatic methods [ 13]o r
Boltzmann transport theory [ 14].
The aforementioned transport phenomena and their differ-
ent contributions being linear in the driving fields should, inprinciple, be described using the fundamental Kubo formulafor the corresponding response function [ 15],
τ
ij(ω)=/integraldisplay∞
0dte−iωt/integraldisplayβ
0dλTr[ρˆAjˆBi(t+i/planckover2pi1λ)]. (1)
The effects then emerge from the characteristics of the
underlying Hamiltonian, the pair of chosen operators forperturbation ( ˆA
j) and observable ( ˆBi), and the symmetry
of the system. Due to the intractability of the problem toexactly solve the Kubo formula for a realistic system inpractice one has to resort to approximations and/or models.However, irrespective of this problem one can still analyze thetransformation properties of response tensors τ
determined by
the Kubo formula to make statements about which effects arein principle allowed, i.e., which nonvanishing tensor elements
*sebastian.wimmer@cup.uni-muenchen.de
†diemo.koedderitzsch@cup.uni-muenchen.demay occur given a particular transformation property of theoperators appearing in Eq. ( 1). This route has been followed
by Kleiner [ 15,16], who demonstrated that the occurrence of
the AHE is predicted by such a space-time symmetry analysis.Furthermore, considering in addition heat currents he derivedgeneral Onsager reciprocity relations.
Here, by extending this approach and applying it in the
context of spin current operators [ 17] we demonstrate that in
certain nonmagnetic low-symmetry systems an electric field
can induce a longitudinal spin-polarized current [ 18] that has
hitherto evaded perception, and complements the transversespin Hall effect. Furthermore two additional transverse effectsare found which differ from the SHE by the direction ofpolarization. The results of the group-theoretical analysis areindependently verified for an alloy bulk system performing rel-ativistic first-principles Kubo-type transport calculations. The
presented formalism is furthermore very general, because (i) it
allows identifying nontrivial response phenomena as nonzeroelements in respective response tensors, as, e.g., the AHE, (ii)it applies to both magnetic and nonmagnetic systems, and (iii)it is free of the notion of a two-current model often used as anapproximation in discussing spintronic phenomena; instead itis based on the concept of spin (polarization) current densities.
The material-specific features of any transport property
may be discussed on the basis of the corresponding response
function tensor τ
. Concerning this, the shape of the tensor
τ, i.e., the occurrence and degeneracy of nonzero elements,
reflecting the symmetry of the investigated solid, is obviouslyof central importance. To find, in particular, the shape ofthe spin conductivity tensor, Kleiner’s scheme [ 15] to deal
with the symmetry properties of ordinary transport tensors
has been extended to the case when the response observable
is represented by an arbitrary operator product of the form(ˆB
iˆCj) while an operator ˆAkrepresents the perturbation and the
operators ˆAk,ˆBj, and ˆCiare seen as the Cartesian components
of vector operators. Within Kubo’s linear response formalismthe corresponding frequency-( ω)-dependent response function
is then given by
τ
(ˆBiˆCj)ˆAk(ω,H)=/integraldisplay∞
0dt e−iωt/integraldisplayβ
0dλTr[ρ(H)ˆAk
׈Bi(t+i/planckover2pi1λ;H)ˆCj(t+i/planckover2pi1λ;H)],(2)
1098-0121/2015/92(4)/041101(5) 041101-1 ©2015 American Physical SocietyRAPID COMMUNICATIONS
S. WIMMER et al. PHYSICAL REVIEW B 92, 041101(R) (2015)
TABLE I. Electrical ( σ) and spin ( σk) conductivity tensor forms for the magnetic Laue groups discussed in the text [ 18,19]. Below each
group symbol an example for a material is given in parentheses.
Magnetic Laue Group σ σxσyσz
m¯3m1/prime
(fcc-Pt)⎛
⎝σxx00
0σxx0
00 σxx⎞
⎠⎛
⎝00 0
00 σx
yz
0−σx
yz0⎞
⎠⎛
⎝00−σx
yz
00 0
σx
yz00⎞
⎠⎛
⎝0σx
yz0
−σx
yz00
00 0⎞
⎠
4/mm/primem/prime
(fcc-Fe xNi1−x)⎛
⎝σxxσxy0
−σxyσxx0
00 σzz⎞
⎠⎛
⎝00 σx
xz
00 σx
yz
σx
zxσx
zy0⎞
⎠⎛
⎝00 −σx
yz
00 σx
xz
−σx
zyσx
zx0⎞
⎠⎛
⎝σz
xxσz
xy0
−σz
xyσz
xx0
00 σz
zz⎞
⎠
4/m1/prime
(Au 4Sc)⎛
⎝σxx00
0σxx0
00 σzz⎞
⎠⎛
⎝00 σx
xz
00 σx
yz
σx
zxσx
zy0⎞
⎠⎛
⎝00 −σx
yz
00 σx
xz
−σx
zyσx
zx0⎞
⎠⎛
⎝σz
xxσz
xy0
−σz
xyσz
xx0
00 σz
zz⎞
⎠
2/m1/prime
(Pt3Ge)⎛
⎝σxxσxy0
σxyσyy0
00 σzz⎞
⎠⎛
⎝00 σx
xz
00 σx
yz
σx
zxσx
zy0⎞
⎠⎛
⎝00 σy
xz
00 σy
yz
σy
zxσy
zy0⎞
⎠⎛
⎝σz
xxσz
xy0
σz
yxσz
yy0
00 σz
zz⎞
⎠
where as usual [ 15]ρstands for the density operator, β=
1/kBTwithkBthe Boltzmann constant, Tis the temperature,
andHis a magnetic field that might be present.
The shape of τcan be found by considering the impact
of a symmetry operation of the space group of the solid onEq. ( 2), as this will lead to equations connecting elements of τ
.
Collecting the restrictions imposed by all symmetry operationsthe shape of τ
is obtained. In this context it is important to note
that the relevant space group of the considered system maycontain not only unitary pure spatial ( u) but also antiunitary
symmetry operations ( a) that involve time reversal.
The transformation properties of the operators X=A
i,Bi,
orCiin Eq. ( 2) under symmetry operations can be expressed
in terms of the corresponding Wigner D-matrices [ 15]D(ˆX)(u)
andD(ˆX)(a) belonging to the operator ˆXand the operation uor
a, respectively. Starting from Eq. ( 2) and making use of these
transformation relations one gets the transformation behaviorofτ
under a unitary ( u) or antiunitary ( a) symmetry operation,
respectively [ 19]:
τ(ˆBiˆCj)ˆAk(ω,H)=/summationdisplay
lmnτ(ˆBmˆCn)ˆAl(ω,Hu)
×D(ˆA)(u)lkD(ˆB)(u)miD(ˆC)(u)nj,(3)
τ(ˆBiˆCj)ˆAk(ω,H)=/summationdisplay
lmnτˆA†
l(ˆB†
mˆC†
n)(ω,Ha)
×D(ˆA)(a)∗
lkD(ˆB)(a)∗
miD(ˆC)(a)∗
nj.(4)
It should be noted that in general the tensors τ(ˆBiˆCj)ˆAkand
τˆA†
k(ˆB†
iˆC†
j)are different objects representing different response
functions which are only interrelated by Eq. ( 4). It nevertheless
imposes restrictions on the shape of τ(ˆBiˆCj)ˆAkgiving rise to
(generalized) Onsager relations.
Assuming ˆCi=1 and ˆBi=ˆAi=ˆjiwith ˆjithe current
density operator τcorresponds to the ordinary electrical
conductivity tensor σ. Using the behavior of ˆjiunder sym-
metry operations [ 15], it turns out that only the magnetic
Laue group of the system has to be considered, that isgenerated by adding the (space) inversion operation Ito
the crystallographic magnetic point group [ 20]. The resultingshape of the conductivity tensor σ
is given in Table Ifor four
different magnetic Laue groups [ 19].
When considering the spin conductivity tensor its elements
σk
ijgive the current density along direction ifor the spin
polarization with respect to the kaxis induced by an electrical
field along the jaxis. In this case the perturbing electric
field is still represented by ˆAi=ˆjiwhile the induced spin
current density is represented by the corresponding operator
ˆJk
i=(ˆBiˆCk). As the explicit definition of ˆJk
iis not relevant for
the following, but only its symmetry properties, the frequently
used nonrelativistic definition ˆJk
i=1
2{ˆvi,σk}m a yb eu s e dt h a t
consists of a combination of the Pauli spin matrix σkand
the conventional velocity operator ˆ vi[21]. Alternatively, one
may use the relativistic definition of the spin current operator
ˆJk
i=ˆTkˆjias suggested by Vernes et al. [22] that involves the
spatial part ˆTkof the spin polarization operator [ 23].
Expressing the transformation behavior of ˆJk
iin terms
of the Wigner matrices allows deducing the shape of thecorresponding spin conductivity tensor on the basis of Eqs. ( 3)
and ( 4). As for the electrical conductivity it turns out again that
one has to consider only the magnetic Laue group; i.e., thereare only 37 different cases. Table Igives for the four cases
considered here the shape of the various subtensors σ
k, where
kspecifies the component of the spin polarization.
Considering a nonmagnetic metal with fcc or bcc structure
(m¯3m1/prime) Kleiner’s scheme naturally leads to an isotropic
electrical conductivity tensor σ. The extension to deal with
the spin conductivity tensor sketched above gives in thiscase only a few nonvanishing elements that are associatedwith the SHE and are symmetry related according to σ
x
yz=
σy
zx=σz
xy=−σx
zy=−σy
xz=−σz
yx; i.e., cyclic permutation of
the indices gives no change while anticyclic permutationchanges the sign. In contrast to other derivations, there isobviously no need to artificially introduce a spin-projectedconductivity or to make reference to the conductivity tensorof a spin-polarized solid. For a ferromagnetic metal with fccor bcc structure (4 /mm
/primem/prime) with the magnetization along
the z direction, the well-known shape of the conductivitytensor σ
is obtained that reflects the anomalous Hall effect
(σxy) as well as the magnetoresistance anisotropy ( σxx/negationslash=σzz)
with the symmetry relations σxy=−σyxandσxx=σyy.T h e
041101-2RAPID COMMUNICATIONS
SPIN-ORBIT-INDUCED LONGITUDINAL SPIN- . . . PHYSICAL REVIEW B 92, 041101(R) (2015)
spin conductivity tensor σzshows as for the nonmagnetic
case antisymmetric off-diagonal elements that represent thetransverse spin conductivity. This implies the occurrence of thespin Hall effect in ferromagnets that was investigated recentlyfor diluted alloys [ 24]. For polarization along the x and y
axes, however, different although still interrelated elementsappear as compared to the nonmagnetic case since fewersymmetry relations survive in the presence of a spontaneousmagnetization. Additionally, in contrast to the nonmagneticcase also a longitudinal spin-polarized conductivity ( σ
z
ii)
occurs in a ferromagnet, that for example gives rise to thespin-dependent Seebeck effect [ 25]. A simple explanation for
the corresponding longitudinal spin transport would be basedon Mott’s two-current model assuming different conductivitiesfor the two spin channels. However, it is well known thatspin-orbit interaction leads to a hybridization of the spinchannels and influences even the longitudinal conductivity of aferromagnet this way [ 26]. Accordingly, it cannot be ruled out
that the longitudinal tensor elements σ
z
iiare not only reflecting
the spontaneous spin magnetization of the material but are tosome extent due to spin-orbit coupling.
Indeed the scheme presented above leads for nonmagnetic
systems having low symmetry not only to off-diagonal ele-
ments reflecting transverse spin conductivity, i.e., the SHE, butalso to diagonal elements reflecting longitudinal spin transport,that was not observed so far. For the two magnetic Laue groups4/m1
/primeand 2/m1/primefor nonmagnetic solids considered in Table I,
a 4- and 2-fold, resp., rotation axis is present. As a consequencelongitudinal spin currents show up only with spin polarizationalong this principal axis of rotation.
To verify the results of our group-theoretical approach
independently we calculated the full spin conductivity tensorfor solids having different structures corresponding to differentmagnetic Laue groups. This work employs a computationalscheme that has been used before for numerical studies on theSHE in nonmagnetic transition metal alloys [ 27]. Performing
these calculations without making use of symmetry led
numerically to a spin conductivity tensor that was always fullyin line with the analytical group-theoretical results concerningthe shape and degeneracies of the tensor.
To get a first estimate of the order of magnitude of the
longitudinal spin-polarized conductivity in nonmagnets, cal-culations have been done for the system (Au
1−xPtx)4Sc having
the magnetic Laue group 4 /m1/primefor varying Pt concentration x.
Figure 1(top) shows the corresponding electrical conductivity
that is, in agreement with Table I, diagonal and slightly
anisotropic; i.e., σxx=σyy≈σzz.
Furthermore, the conductivities σiiare strongly asymmetric
with respect to the concentration xwhen replacing Au with
prominent spcharacter at the Fermi level by Pt with dominant
dcharacter. Furthermore, one notes a relatively strong impact
of the vertex corrections on the Au-rich side of the system ( x≈
0) while these are much less important on the Au-poor side(x≈1). This observation is well known from binary transition
metal alloys, such as Cu
1−xPtx[28]o rA g1−xPdx[29], where
the dominance of spcharacter changes to dcharacter when x
is varied from 0 to 1.
The transverse spin conductivity σx
ijis shown in the middle
panel of Fig. 1for x polarization of the spin. As Table I
shows going from m¯3m1/primeto 4/m1/primesymmetry the relation00.511.5
σ (μΩ-1 cm-1)σxx (NV)
σxx (VC)
σzz (NV)
σzz (VC)
0 0.2 0.4 0.6 0.8 1
x in (Au1- xPtx)4Sc-1-0.500.5
σz
xx (mΩ-1 cm-1)
σz
xx (NV)
σz
xx (VC)
0 0.2 0.4 0.6 0.8 1-10-8-6-4-20
σz
xy (mΩ-1 cm-1)
σz
xy (NV)
σz
xy (VC)-3-2-10123
σxij (mΩ-1 cm-1)σx
xz (NV)
σxxz (VC)
-σxzx (NV)
-σxzx (VC)
-3-2-10123
σx
yz (NV)
σx
yz (VC)
-σx
zy (NV)
-σx
zy (VC)
FIG. 1. (Color online) Top: Longitudinal conductivity σiifor
(Au 1−xPtx)4Sc as a function of the concentration xcalculated without
(NV) and with (VC) the vertex corrections. Middle: Transversespin conductivities σ
x
ij. Bottom: Transverse and longitudinal spin
conductivity σz
xyandσz
xx, respectively.
σx
yz=−σx
zydisappears; i.e., the corresponding subtensor is
not antisymmetric anymore. A symmetric component, whichis by definition not present in the ordinary SHE, indeed canbe seen in Fig. 1(middle) although the deviations are not
very pronounced. In line one finds (except for x→0) for
the additional nonzero tensor elements σ
x
xz≈−σx
zx. The first
coefficient relates a spin current jx
xpolarized in the direction
of motion to an electric field Ez, whereas σx
zxdescribes a
spin current jx
ztransverse, but with the spin polarization
parallel to the driving electric field Ex. To our knowledge
the corresponding effects have not been considered so far.Interestingly, both elements occur simultaneously for a givenmagnetic Laue group or both are absent. However, comparedto the spin-Hall-like elements σ
x
yzandσx
zythey are smaller. For
041101-3RAPID COMMUNICATIONS
S. WIMMER et al. PHYSICAL REVIEW B 92, 041101(R) (2015)
-10 -5 0 5
E - EF (eV)00.511.52nα(E ) (sts./Ry)tot (x 1/5)
Au
Pt
Sc
0 0.2 0.4 0.6 0.8 1
x in (Au1- xPtx)4Sc020nα(EF) (sts./Ry)tot (x 1/5)
Au
Pt
Sc
FIG. 2. (Color online) Top: Energy-dependent component-( α)-
resolved DOS nα(E)f o r( A u 0.5Pt0.5)4Sc. Bottom: Component-
resolved DOS nα(EF) at the Fermi energy EFfor (Au 1−xPtx)4Sc
as a function of the concentration x.
y polarization of the spin the corresponding tensor elements
are uniquely related to those for x polarization accordingto Table Iand for this reason not given here. The tensor
elements σ
z
ijfor z polarization are given in the lower panel
of Fig. 1. In line with Table Ithey obey the symmetry relation
σz
xy=−σz
yx(i.e., describing the pure SHE) and differ from
σx
yz. This difference however is, except again for x→0, not
very pronounced. In particular, σx
yzandσz
xyshow a similar
variation with concentration xthat differs clearly from that of
the longitudinal spin conductivity σz
xxshown as well in Fig. 1
(bottom). Although this new type of tensor element is overallsomewhat smaller in magnitude than the dominating transverseelements it has nevertheless the same order of magnitude,especially in the Au-rich regime, and for that reason it shouldbe possible to determine it experimentally.
As can be seen in Fig. 1the curves for the spin conductivity
tensor elements σ
k
ijas function of the concentration xare
much more structured than the electrical conductivity σii; i.e.,
they are much more strongly affected by the variation of theelectronic structure with composition. In particular the spinconductivities σ
k
ijshow pronounced peaks or dips for x≈0.8.
This behavior can be related to the variation of the density ofstates (DOS) with xas can be seen from Fig. 2. The figure
shows the component-resolved DOS n
α(E) as a function of
the energy Efor (Au 0.5Pt0.5)4Sc (top) and at the Fermi energy
EFfor (Au 1−xPtx)4Sc as a function of the concentration x
(bottom). As mentioned above, at the Fermi energy the partialDOSn
Au(EF) of Au is dominated by spstates while that of Pt
has dominant dcharacter. The pronounced dip of the Pt DOS
nPt(EF)a tx≈0.8 is apparently responsible for the prominent
features in the spin conductivity curves shown in Fig. 1
(middle and bottom panels). As mentioned before, for thelongitudinal conductivity σ
iiinclusion of the vertex corrections
has primarily an impact at the Au-rich side of the system. Thesame behavior is found for the transverse ( σ
k
ij)a sw e l la st h e
longitudinal ( σz
ii) spin conductivity components. For the trans-
verse spin Hall conductivity it could be demonstrated that thecontribution connected with the vertex corrections correspondsto the so-called extrinsic contribution that is primarily causedby the skew scattering mechanism [ 24,27]. The very similar
dependence of σ
k
ijandσz
iion the vertex corrections suggests
that this applies also for the longitudinal spin conductivity.
In summary, a group-theoretical scheme has been presented
that allows determining the shape of response tensors relevantfor the field of spintronics. Application to the spin conductivitytensor gave a sound and model-independent explanation for theoccurrence of the transverse tensor elements responsible forthe spin Hall effect and two additional, closely related effects.In addition it was found that for low symmetry longitudinalelements show up in addition even for nonmagnetic solidsthat were not considered before. Independent numericalinvestigations confirmed these results and demonstrated for(Au
1−xPtx)4Sc that the longitudinal spin conductivity may be
on the same order of magnitude as the transverse one. It shouldbe noted that the discussion of the spin conductivity tensorwas referring to the dc limit ω=0. However, the tensor forms
given in Table Ialso hold for finite frequencies, implying
the occurrence of the ac counterparts to the discussed effects.In addition, the formalism is applicable to numerous otherlinear response phenomena as, e.g., the AHE, anisotropicmagnetoresistance (AMR), the Edelstein effect [ 4,5], Gilbert
damping [ 30], spin-orbit torques [ 31], etc. Furthermore, using
the fact that the operators for electrical and heat currents sharethe same transformation properties the presented formalismcan be applied to spincaloritronic phenomena as well.
This work was supported financially by the Deutsche
Forschungsgemeinschaft (DFG) under the priority programSPP 1538 (Spin Caloric Transport) and the SFB 689(Spinph ¨anomene in reduzierten Dimensionen). Discussions
with Ch. Back and H. H ¨ubl are gratefully acknowledged.
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041101-5 |
PhysRevB.77.115344.pdf | Asymmetry of acceptor wave functions caused by surface-related strain and electric field in InAs
S. Loth, M. Wenderoth, *and R. G. Ulbrich
IV . Physikalisches Institut der Universität Göttingen, Friedrich-Hund-Platz. 1, 37077 Göttingen, Germany
/H20849Received 22 November 2007; revised manuscript received 16 January 2008; published 24 March 2008 /H20850
The spatial distribution of the local density of states at Mn acceptors near the /H20849110 /H20850surface of p-doped InAs
is investigated by scanning tunneling microscopy. The shapes of the acceptor contrasts for different dopantdepths under the surface are analyzed. Acceptors located within the first ten subsurface layers of the semicon-ductor show a lower symmetry than expected from theoretical predictions for the bulk acceptor wave function.They exhibit a /H20849001 /H20850mirror asymmetry. The degree of asymmetry depends on the acceptor atoms’ depths. The
measured contrasts for acceptors buried below the tenth subsurface layer closely match the theoreticallyderived shape. Two effects are able to cause the observed symmetry reduction, i.e., the strain field of thesurface relaxation and the tip-induced electric field. While both effects induce similar asymmetries, a compari-son of their relative strengths indicates that surface-related strain is the dominant effect for Mn in InAs.
DOI: 10.1103/PhysRevB.77.115344 PACS number /H20849s/H20850: 71.55.Eq, 73.20. /H11002r, 72.10.Fk, 75.30.Hx
I. INTRODUCTION
Scanning tunneling microscope /H20849STM /H20850studies of the local
electronic contrasts induced by shallow and deep acceptorsin group III–V semiconductors are subject to intensediscussions.
1–8The anisotropic contrasts of magnetic accep-
tors such as Mn are of particular interest because their mi-croscopic coupling to holes and other Mn acceptors influ-ences the macroscopic magnetic properties of the dopedsemiconductor.
9–11For substitutional acceptors in zinc
blende semiconductors, e.g., the group III–V compounds,one expects that the observed contrasts reflect the cubic sym-metry of the host crystal’s band structure /H20849hence, c
2v/H20850.12,13
However, shallow acceptors show up in STM topographies
as triangular contrasts with the dopant atom located in thetriangle’s tip, clearly breaking the c
2vsymmetry.1,2,14Deep
acceptors show an asymmetric bow-tie-like shape reminis-cent of the bulk symmetry, but nevertheless, asymmetric withreference to the /H20849001 /H20850mirror plane.
3,15Tight-binding calcu-
lations were performed to describe the acceptor state in thebulk crystal.
3,16Until now, the semiconductor surface was
not fully included into such calculations because the neces-sary large slab would exceed today’s computing capabilities.The probability density at the sample surface that originatesfrom the wave function of a subsurface acceptor was ex-tracted from the existing bulk calculation by cutting the cal-culated three-dimensional probability density at a certain dis-tance from the acceptor atom and artificially adding thedecay into the vacuum.
13,16However, an acceptor in the vi-
cinity of the surface will differ not only in electronic prop-erties such as binding energy from a bulk acceptor but also inspatial extension of its wave function. The cleavage surfacethat is needed in the STM experiment to access buried dop-ants introduces a symmetry reduction into the system that isnot included in the bulk calculations. Thus, these calculationsdo not completely reproduce the observed asymmetric shape,and especially not the recently reported depth dependentchanges of this asymmetry.
8In this paper, we quantitatively
study the evolution of the acceptor wave function with re-spect to its dependence on the interaction strength withsurface-related and tip-induced effects by comparative im-ages of acceptors in different depths under the surface. The
degree of /H20849001 /H20850mirror asymmetry serves as a measure. Sur-
face strain fields and tip-induced electric fields are discussedon the basis of band structure calculations. Both effects canexplain the symmetry reduction. A comparison of their rela-tive strengths indicates that surface-related strain is the domi-nant effect in this system.
II. EXPERIMENT
The experiments are performed in a low temperature
STM operating in UHV at a base pressure better than2/H1100310
−11mbar. Details of the experimental setup are given
in Ref. 17. The InAs samples are cleaved in situ at room
temperature and they are transferred to the precooled micro-scope where they reach the equilibrium temperature of 5.6 Kwithin less than an hour after cleavage.
To quantitatively analyze the symmetry properties of in-
dividual acceptors, we employ highly diluted samples. Infact, the symmetry analysis requires a dopant–dopant dis-tance larger than the average Mn contrast extension of/H1101110 nm /H20849see Sec. III B /H20850. On the other hand the doping has
to be high enough to ensure stable STM operation at 5.6 K.A doping level of 2 /H1100310
17cm−3is chosen and checked by
large scale topographic STM images. The average distanceof two Mn atoms in the present sample is 17 nm and is thussufficiently large for the following analysis. Additionally, theMn acceptor concentration establishes an impurity band witha few meV spectral widths centered at about 23 meV abovethe valence band edge.
18,19The samples are conducting even
at 4.2 K, which was checked by macroscopic resistance mea-surements.
The asymmetric contrasts discussed here are linked to the
host lattice.
4,14Knowledge of the sample’s exact crystallo-
graphic orientation is crucial for the discussion. The samplesare cut out of a /H20849001 /H20850oriented wafer and the orientation of
each sample piece is documented throughout the preparationprocess. As InAs possesses two groups of nonequivalentcleavage planes, the documented orientation of the cleavageplane is cross-checked in the STM experiment with the rela-tive shift between empty and filled surface resonances inPHYSICAL REVIEW B 77, 115344 /H208492008 /H20850
1098-0121/2008/77 /H2084911/H20850/115344 /H208498/H20850 ©2008 The American Physical Society 115344-1spatially resolved I/H20849V/H20850spectroscopy.20The measurements
presented in this work are carried out on the /H20849110 /H20850surface,
i.e., the In atoms of the surface zigzag row point toward/H20851001 /H20852/H20849compared to the ball-and-stick model in Fig. 2/H20850.
III. RESULTS
A. Acceptor state identification
The first step of the analysis is the identification of the
sample bias voltage at which the acceptor bound hole is im-aged. The tip-induced band bending /H20849TIBB /H20850present at the
/H20853110 /H20854surfaces of InAs has to be considered. It causes a non-
trivial relation of sample energy scale and applied samplebias.
21,22The constant current topography in Fig. 1presents
an 18/H1100318 nm2image of an atomically flat InAs /H20849110 /H20850sur-
face recorded at +1.0 V sample bias. The anisotropic bow-tie-like contrasts of three subsurface Mn acceptors are vis-ible. The differential conductivity /H20849dI/dV/H20850is recorded on two
spots in this region /H20849upper graph in Fig. 1/H20850: The blue curve
/H20849denoted as u/H20850is acquired in the white rectangle in the upper
left corner of the topography and shows the dI/dVsignal of
the undisturbed surface. The red spectrum /H20849c/H20850is recordedabove the Mn contrast labeled with /H20849c/H20850. The lower graph in
Fig. 1presents the numerically derived TIBB /H20849V/H20850depen-
dence, which has been validated with spatially resolved I/H20849V/H20850
spectroscopy in the same manner as described earlier forGaAs.
17The actual TIBB /H20849V/H20850dependence is strongly affected
by parameters that vary for different tips and thus needs to bechecked for each STM measurement. In order to identify theacceptor state in the I/H20849V/H20850spectra, knowledge of the flatband
bias voltage /H20849TIBB=0 meV /H20850is crucial. Thus, the tip work
function, which determines the flatband bias, is experimen-tally evaluated. The TIBB /H20849V/H20850is calculated using this value
/H208494.25 eV for the presented scanning tunneling spectroscopy
measurement /H20850, a typical tip geometry /H2084915 nm tip apex radius,
90° shank slope /H20850, and an estimated vacuum gap of 8 Å. The
numerical model introduced by Feenstra
22is used. The cal-
culated flatband bias is at 1.05 V sample bias. Delocalizedcharge density oscillations appear as a conductivity step inboth dI/dVcurves at 1088 mV . This observation fixes the
flatband bias to a slightly lower value than 1.09 V , which isin good agreement with the calculation. The detection of theacceptor state is expected below the flatband bias. Theprominent conductivity peak at +914 mV that is solely ob-served above the position of the acceptor is therefore identi-fied as the additional tunnel channel into the acceptor groundstate. At positive bias, it becomes accessible when the accep-tor state is lifted above the Fermi energy. Thus, the spatialdistribution of this conductivity peak is associated with thedirect detection of the wave function of the acceptor boundhole. The comparison of the dI/dVcurves above and below
this peak adds further confidence to this: For bias voltageslower than +0.91 V, the acceptor state is below the Fermienergy, i.e., the acceptor is in its ionized state. No hole isbound to it and the dopant core’s negative charge locallyshifts the conduction and valence band states upward. ThedI/dVcurve near the acceptor largely differs from the one
recorded above the undisturbed surface. For bias voltagesexceeding +0.91 V, the dopant core’s negative charge iscompensated by the bound hole. The surrounding area is nolonger electrostatically distorted and both dI/dVcurves
match. To conclude this section, the STM maps the probabil-ity density distribution of the Mn acceptor wave function for/H11022+0.91 V bias voltage.
B. Symmetry analysis
Further analysis is done by topographic measurements. A
total of 29 acceptor contrasts is acquired in a single atomi-cally resolved 210 /H11003210 nm
2multibias measurement. Each
line of the image is subsequently scanned with two differentbias voltages before the scanner moves to the next line. Be-cause two topographies are quasisimultaneously recorded,
thermal drift and piezocreep between them is negligible,
23
and the absolute positions in both images match to an accu-racy better than one surface lattice constant. This was veri-fied by comparing the positions of uncharged surface pointdefects between both images. Figure 2presents the zoomed
images in eight different acceptor contrasts. Taking allzoomed images out of the same multibias measurement en-sures that the different contrast shapes are not caused by
FIG. 1. /H20849Color online /H20850Inset: 18 /H1100318 nm2constant current to-
pography of three subsurface Mn acceptors /H20849a, b, and c /H20850under the
InAs /H20849110 /H20850surface. The topography is recorded at +1.0 V sample
bias and 100 pA tunnel current. The Mn acceptors appear as asym-metric bow-tie-like protrusions. Upper graph: local dI/dVcharac-
teristics acquired in the inset topography. The blue curve /H20849labeled
with u /H20850corresponds to the undisturbed surface and the red one
/H20849labeled with c /H20850was recorded directly above the lower Mn acceptor
/H20849c/H20850. The topographic set point for the I/H20849V/H20850measurement is at 2.0 V
and 0.3 nA. At this set point, the Mn acceptors have no impact onthe topography. Lower graph: numerically derived TIBB /H20849V/H20850depen-
dence adjusted to the presented I/H20849V/H20850spectroscopy. The characteris-
tic bias voltages, i.e., inversion limit and flatband bias, are marked.LOTH, WENDEROTH, AND ULBRICH PHYSICAL REVIEW B 77, 115344 /H208492008 /H20850
115344-2modifications of the tip’s imaging properties. The two biases
are chosen such that the acceptor state is imaged in one to-pography, while the acceptor core position can be determined
in the other. The first topography is recorded at +1.0 V, i.e.,directly above the acceptor state’s conductivity peak. Therespective zoomed images are the blue-red colored images/H20849upper image for each acceptor /H20850. According to the dI/dV
curves they are an image of the acceptor bound hole’s spatialdistribution. The second topography is recorded at −1.0 V.At this bias, the tunnel current is dominated by the valence
band states and the acceptors exhibit circular symmetric pro-trusions. The negative acceptor charge has a circular sym-metric Coulomb potential that influences the band states.
1,2
The center of mass and maximum of this contrast resembles
the projected lateral position of the acceptor core under thesurface. It is indicated by white circles in Fig. 2.
The depth of each acceptor atom under the surface is de-
termined as follows: All visible acceptors are ordered to in-creasing depth under the assumption that the circular protru-sion in the filled state image at −1.0 V is the strongest forthe acceptor nearest the surface and becomes fainter fordeeper acceptors. The acceptor contrasts in Fig. 2are ordered
to increasing depth from top left to bottom right. The black-yellow colored images /H20849lower image for each acceptor /H20850show
the evolution of the circular contrast. To pinpoint not onlythe monotonous depth ordering but also the precise dopantdepth, additional information is used: The symmetry center
of the acceptor contrasts has to follow a certain ordering withrespect to the host lattice.
24,25Mn is a substitutional acceptor
on the In site. The dominant empty state resonance at+1.0 V has its corrugation maxima above the In sites of thesurface zigzag row.
20Therefore, an acceptor contrast in the
first surface layer is centered directly on the corrugationmaximum. If the acceptor is positioned in the second mono-layer, the acceptor atom is located between the corrugationmaxima. The acceptor contrasts in Fig. 2follow the alternat-
ing on-maximum and off-maximum orderings. Recent re-ports suggest that acceptors located in the two monolayersthat form the surface have a different appearance.
8,11Thus,
the label “layer 1” in Fig. 2refers to the first subsurface
layer. The acceptor core positions are determined for accep-tors down to the tenth subsurface monolayer. The analysisshows that no acceptor was found in the fourth and seventhlayer under the surface. About four to five additional depthswere detected but the exact position of the respective accep-tors could not be accurately determined anymore due to thevanishing feature height in the filled state image. Besides, itis worth noting that the STM could resolve acceptors thatwere up to 3 nm /H20849/H1147015th subsurface layer /H20850below the sample
surface. The ordered image sequence of the anisotropic ac-ceptor contrasts /H20849blue-red colored images in Fig. 2/H20850shows a
gradual shift from nearly triangular to rectangular shapes.The acceptor in the first subsurface layer has a pronouncedtriangular shape. The contrast maximum is shifted to the
/H20851001
¯/H20852side of the acceptor atom’s location and the half-plane
to the /H20851001 /H20852side consists only of faint lobes. The acceptor in
the tenth subsurface layer appears as a nearly rectangularfeature centered on the dopant site. Acceptors in intermediatedepths exhibit intermediate contrasts.
The degree of asymmetry with respect to the /H20851001 /H20852direc-
tion is quantified by image analysis, as shown in Fig. 3. The
topography of each acceptor is decomposed into symmetricand asymmetric parts with reference to a /H20849001 /H20850mirror plane
through the exact location of the acceptor core /H20849blue line in
the images of Fig. 3/H20850. The following conditions have to be
met for this analysis.
/H208491/H20850The average dopant-dopant distance has to be larger
than the chosen image size. If the acceptor contrast of inter-
FIG. 2. /H20849Color online /H20850Zoomed images of 13 /H1100313 nm2into the
multibias topography. Each image doublet shows one acceptor. Theblue-red colored image is recorded at +1.0 V and the black-yellowcolored images are acquired at −1.0 V. The images of one rowhave the same color scaling. The adjacent color bar indicates theheight scale for each row in Å. The white circles show the locationof the dopant atom under the surface, as determined by the center ofmass and contrast maximum of the circular contrast at −1.0 V bias.The ball-and-stick model sketches the InAs lattice at the /H20849110 /H20850sur-
face with the orientation of the STM images. The large circles cor-respond to atoms of the first surface layer and the small circles arethe second surface layer.ASYMMETRY OF ACCEPTOR WA VE FUNCTIONS CAUSED … PHYSICAL REVIEW B 77, 115344 /H208492008 /H20850
115344-3est was notably affected by a nearby acceptor, the symmetry
analysis would return symmetry properties of the dopant ar-rangement rather than the symmetry properties of an indi-vidual dopant atom. Here, 13 /H1100313 nm
2topographies are
considered; hence, the average Mn-Mn distance of 17 nm/H20849doping level of 2 /H1100310
17cm−3/H20850is sufficient.
/H208492/H20850The contrast asymmetry is evaluated with reference to
a/H20849001 /H20850mirror plane running through the dopant atom. Thus,
the position of the Mn atom needs to be known in the topog-raphy containing the acceptor state image, i.e., at +1.0 V.
However, the projected dopant position has to be determinedat a different sample bias /H20849here, at −1.0 V /H20850. Due to the em-
ployed multibias acquisition, both topographies at −1.0 Vand +1.0 V precisely match and the acceptor atom positionsthat are determined in the −1.0 V topography are thereforealso known in the +1.0 V topography.
The atomic corrugation of the surface states is suppressed
in the images by fast Fourier transform filtering to minimizethe background signal /H20849upper image of Fig. 3/H20850. The symmet-
ric part z
s/H20849x,y/H20850is deduced from the upper image /H20849middle
image of Fig. 3/H20850. The symmetric part is subtracted from the
topography, which results in an image of the asymmetric partz
a/H20849x,y/H20850/H20849lower image of Fig. 3/H20850. After decomposition of the
filtered image, the degree of asymmetry of each acceptorcontrast is given by the relative weight of symmetric andasymmetric parts. The quotient
/H9257is a quotient of the inte-
grals of the height information of symmetric and asymmetricimages/H9257=/H20885za/H20849x,y/H20850ds
/H20885za/H20849x,y/H20850ds+/H20885zs/H20849x,y/H20850ds.
It describes the ratio of the asymmetric to the symmetric
components of the topography. The graph in Fig. 3plots/H9257
for all acceptors of Fig. 2against the acceptor depth. The
degree of asymmetry is 27% for the acceptor in the secondlayer. With increasing depth, the asymmetry monotonouslydecreases. The slope of a linear fit gives a decrease of 0.024per monolayer depth. The detection limit for this analysis isestimated by performing the same analysis with the mirror
plane /H2084911
¯0/H20850. The Mn acceptor is mirror symmetric with that
plane, but the value of /H9257varies between 0 and 0.05 for this
direction because of residual noise /H20849this limit is indicated by
a dashed line in Fig. 3/H20850. The uncertainty of the symmetry
analysis due to deviations in the mirror plane position deter-mination is lower. It is /H9004
/H9257=0.03 and is plotted as an error
bar. Acceptors in the tenth subsurface layer are fully sym-metric within the accuracy of this analysis. According to thelinear fit, the 12th layer acceptor would be completely sym-metric. As a result, acceptors buried below the 10th to the12th layer under the surface appear as rectangular contrasts
that are mirror symmetric with respect to both the /H2084911
¯0/H20850
plane and the /H20849001 /H20850plane. Acceptors located within the first
ten subsurface layers have a /H20849001 /H20850mirror asymmetry. The
/H20851001¯/H20852side of the acceptor contrast is more pronounced than
the /H20851001 /H20852side.
IV . DISCUSSION
On the basis of the local I/H20849V/H20850spectroscopy /H20849Fig. 1/H20850,w e
conclude that the asymmetric contrasts at +1.0 V are an im-age of the acceptor bound hole, i.e., they resemble the prob-ability distribution of the acceptor wave function at the sur-face. Indeed, the observed probability density distribution ofdeeply buried acceptors has a nearly rectangular shape. Thisfits well with the theoretical expectation for a bulk acceptoras calculated, for example, by effective mass
13or tight-
binding methods.8,16Obviously, the spatial extension of the
wave function for dopants nearer to the surface is reduced bya vertical confinement of the surface. The half-space geom-etry /H208491/2 semiconductor and 1/2 vacuum /H20850will affect other
properties, such as binding energy, as well /H20849see, e.g., Refs. 26
and27/H20850.
However, above all, the surface induces a symmetry re-
duction. The depth dependent measurements demonstratethat the probability density distribution for acceptors near thesurface is deformed compared to deeply buried acceptors.Unfortunately, a quantitative description of surface and dop-
ant in tight-binding calculations or ab initio density func-
tional theory exceeds today’s computational abilities. How-ever, the symmetry properties of the acceptor wave functionmay be qualitatively elucidated by considerations based on
the bulk band structure: The acceptor state is a hybrid of thehighest valence band states. The energy window of the va-lence band needed to form the localized state approximately
FIG. 3. /H20849Color online /H20850Asymmetry factor /H9257of the Mn acceptor
contrasts plotted against dopant depth. These are the results of thesymmetry analysis. If the acceptor in the first monolayer is ex-cluded, the asymmetry decreases linearly to 0 with increasingdepth. The lower limit to which the asymmetry can be detected is0.05, which is reached for an acceptor in the tenth subsurface layer.The images at the right hand side demonstrate the symmetry analy-sis for an acceptor in the fifth layer /H20849indicated by an arrow in the
graph /H20850.LOTH, WENDEROTH, AND ULBRICH PHYSICAL REVIEW B 77, 115344 /H208492008 /H20850
115344-4equals the binding energy of this state.28The bulk Mn accep-
tor in InAs is 23 meV above the valence band maximum,19
so about 10% of the Brillouin zone participates in the hybrid-ization. The symmetry of its wave function is determined bythe host crystal’s band structure. If the band structure is sym-metric along a certain direction, the ground state wave func-tion will be symmetric as well. Anisotropies in the groundstate wave function can only develop when the band struc-ture exhibits this asymmetry. To good approximation, thebulk bands are cubic in this range.
29In particular, they are
symmetric with respect to the /H20851001 /H20852direction. Effects that
break this symmetry are known but usually considered to besmall in the bulk, e.g., the so-called k-linear terms cause a
splitting of less than 1 meV at the valence band edge in thebulk.
30Figure 4/H20849a/H20850presents a band structure calculation for
bulk InAs. Empirical pseudopotentials31,32were used and the
spin-orbit interaction /H20849SOI /H20850was explicitly included.33,34
Since the surface is not explicitly included one primitive
InAs unit cell is modeled and a basis of 65 plane waves isemployed.
35The band structure is evaluated for a cut defined
by a plane consisting of the /H20851001 /H20852and /H20851110 /H20852directions. This
cut visualizes the symmetry properties of the InAs bandstructure that will affect the shape of the acceptor contrasts atthe /H20849110 /H20850surface relative to the /H20851001 /H20852direction. The graph in
Fig. 4shows the energy contour lines of the highest valence
band. The plotted section has an extension of about 10% ofthe Brillouin zone.
For the bulk system without any symmetry reducing field
/H20851Fig. 4/H20849a/H20850/H20852, the well-known shape /H20849c
2vsymmetry /H20850is repro-
duced. The band is symmetric with reference to the /H20849001 /H20850
mirror plane, i.e., the /H20851001¯/H20852and /H20851001 /H20852parts of the graph are
identical. The resulting acceptor wave function will inheritthis symmetry and the acceptor contrast at the surface issymmetric, as depicted in the sketch /H20851Fig. 4/H20849a/H20850, left /H20852. This
agrees well with the measured contrast of the “layer 10”acceptor /H20849see Fig. 2/H20850. As a first result, an acceptor in the tenth
subsurface layer has the bulk symmetry properties. The sur-face has no measurable impact on it. In fact, the tenth layeracceptor closely matches the previously reported theoreticalpredictions for the Mn acceptor wave function.
3,13In contrast
to that, our STM analysis shows that acceptors closer to thesurface exhibit a strong asymmetry along /H20851001 /H20852/H20849refer to
Figs. 2and3/H20850, while they remain symmetric with reference
to the /H208511¯10/H20852mirror plane. This asymmetry cannot be de-
scribed with the bulk band structure only. Obviously, thecleavage surface induces a symmetry breaking that lifts thecubic symmetry with reference to the /H20849001 /H20850mirror plane but
preserves it along the perpendicular /H208491
¯10/H20850mirror plane. In
the following, two effects that introduce strong changes tothe band structure will be discussed: i.e., local strain fieldsand strong electric fields. Both are present under the STM tipat the relaxed surface.
A. Surface effect: Strain
The atoms in the first few layers of the cleaved InAs /H20849110 /H20850
surface relax. The relaxation is usually treated in self-consistent calculations
32,36,37and experimentally investigated
by low energy electron diffraction.38,39In terms of strain, thisrelaxation decomposes into a hydrostatic component, a
uniaxial component, and shear components. The uniaxial
strain is along /H20851110 /H20852for the relaxation, but neither the hydro-
static nor the uniaxial components induce symmetry break-ing with respect to /H20851001 /H20852. However, in zinc blende crystals
FIG. 4. /H20849Color online /H20850Band structure of InAs in a plane defined
by/H20851001 /H20852and /H20851110 /H20852. The colored lines are the isoenergy lines of the
uppermost valence band. The plots show about 10% of the Brillouinzone. The calculated situations are sketched to the left of each plot./H20849a/H20850No additional symmetry reducing field is applied, i.e., bare InAs
band structure. This results in a symmetric contrast with referenceto/H20851001 /H20852./H20849b/H20850A shear strain is applied, which accounts for the strain
induced by the surface relaxation. The In atoms are displaced by0.05% of the /H20849110 /H20850monolayer distance with respect to the As sub-
lattice. /H20849c/H20850An electric field is applied along the /H20851110 /H20852direction with
a strength of 0.1 V/nm. Both /H20849b/H20850and /H20849c/H20850induce a /H20849001 /H20850mirror
asymmetry in the contrasts at the /H20851110 /H20852surface.ASYMMETRY OF ACCEPTOR WA VE FUNCTIONS CAUSED … PHYSICAL REVIEW B 77, 115344 /H208492008 /H20850
115344-5uniaxial strain along the /H20855110 /H20856directions results in shear
components in the strain tensor.40They correspond to the
off-diagonal components /H9255ijof the strain tensor, while the
uniaxial strain is part of the diagonal components /H9255ii.I nr e -
cent experiments, it was shown that even small /H20851110 /H20852
uniaxial strain /H20849that can be applied by an external vice /H20850leads
to large anisotropies in the electron propagation properties inGaAs.
41This gives rise to the idea that shear strain of the
surface relaxation induces the observed symmetry reductionalong /H20851001 /H20852.
To give an estimate of whether such shear strain could
lead to the observed asymmetries, its impact on the bandstructure is approximated within the bulk band structure cal-culation. It is modeled by a slight displacement of the In andAs sublattices against each other. The In atom within themodeled unit cell is displaced along /H20851110 /H20852from its equilib-
rium position, and under the condition of small displace-ments, we assume that the configuration of the unit cell is notchanged. The influence of the symmetry reducing strain fieldon the highest valence band is shown in Fig. 4/H20849b/H20850. The results
show a prominent symmetry reduction with reference to the
/H20849001 /H20850mirror plane. Already very small distortions induce a
considerable asymmetry. The graph in Fig. 4/H20849b/H20850shows the
valence band for an In displacement of 0.05% of the /H20851110 /H20852
monolayer distance, i.e., 0.000 25 a
0//H208812. The valence band
becomes elongated along /H20851111/H20852and compressed along /H20851111¯/H20852.
This asymmetry is highest for the highest valence bandstates. The seven inner /H20849red and yellow /H20850contour lines in Fig.
4/H20849b/H20850correspond to an energy window of about 10 meV start-
ing at the valence band maximum. The degree of asymmetryfor these states exceeds 47%. States further away from thevalence band maximum are less asymmetric, but the outer/H20849blue /H20850contour line is still 26% more elongated along /H20851111/H20852
than along /H20851111
¯/H20852.
Recent reports of the relaxed InAs /H20853110 /H20854surface predict
that even the second subsurface layer, which is the fourthlayer of InAs counted from vacuum, exhibits a /H20851110 /H20852dis-
placement of In and As as high as 0.5% of the monolayerdistance.
42The shear strain employed here is one order of
magnitude smaller and is a reasonable choice for the residualshear of deeper layers.
This simplified model demonstrates that a small shear
strain, which is present at the relaxed surface, already in-duces considerable asymmetry in the valence bands. An ac-ceptor wave function in this environment will extend further
along /H20851111
¯/H20852than along /H20851111/H20852. Thus, the probability density
on the /H20851110 /H20852surface will extend further along /H20851001¯/H20852than it
does along /H20851001 /H20852. The resulting contrast properties are de-
picted in Fig. 4/H20849b/H20850. This matches with the measurement: The
asymmetric bow-tie-like contrasts are more pronounced on
the /H20851001¯/H20852side of the dopant atom. These findings are cor-
roborated by a recent report on Mn acceptors located in thestrain field of a quantum dot. The in-plane strain has a stronginfluence on the wave function shape of a dopant and distortsthe acceptor contrast into the direction of the quantum dot.
7
B. Surface effect: Electric field
The second effect that is capable of producing the /H20851001 /H20852
asymmetry of the Mn contrast is the tip-induced electricfield. The tip exhibits an electric field penetrating into the
semiconductor. It is directed along the surface normal of thecleavage plane /H20851110 /H20852due to the STM geometry. Typical field
strengths are on the order of 10
5–106V/cm.22The STM
images show that the relative weight of the acceptor wavefunction shifts perpendicular to this electric field. An electro-static distortion of the wave function due to such an electricfield, e.g., the Stark effect, would only produce changes thatare symmetric with reference to the /H20849001 /H20850mirror plane. An
elongation or compression of the wave function along /H20851110 /H20852
would not explain the observed asymmetry. An effect is
needed, which acts differently for the /H20851001 /H20852and /H20851001
¯/H20852wave
vector components. The SOI provides this kind of symmetryreduction in the band structure.
30The above calculation is
extended to implement a homogenous electric field in the/H20851110 /H20852direction. It is introduced to the Hamiltonian via
an additional SOI term in the form of the RashbaHamiltonian.
31,34Thus, it represents a homogenous electric
field that acts solely by spin-orbit interaction. For simplicity,other effects of the external field are neglected. As illustratedby the sketch in Fig. 4/H20849c/H20850, the electric field resembles a struc-
ture inversion asymmetry /H20849SIA /H20850.
34Its effect on the highest
valence band is shown in the graph of Fig. 4/H20849c/H20850for an elec-
tric field of 0.1 V/nm. The valence band is elongated along
the /H20851111/H20852direction and compressed along the /H20851111¯/H20852direction.
The calculated deformation of the valence band is caused bythe combination of the bulk inversion asymmetry /H20849BIA /H20850of
the zinc blende crystal and the external field induced SIA.The BIA preserves the cubic shape of the valence bands, i.e.,their elongation in all /H20855111 /H20856directions and compression in
the /H20855100 /H20856directions. The spin splitting of the bands due to
SIA has a different dependence. The sign of the spin splittingdue to SIA and BIA in the /H20851111/H20852direction is the same. Both
effects add up. In the perpendicular /H20851111
¯/H20852direction, SIA and
BIA have opposite signs. The sum of both effects induces thesymmetry reduction over /H20851001 /H20852. Analogous to the previously
discussed strain field, this distortion is inherited by the ac-
ceptor wave function. The /H20851001¯/H20852side of the bow-tie-like con-
trast will be more pronounced than the opposite side. Thefield strength of 0.1 eV/nm is chosen because the inducedvalence band distortion is similar to the distortion due to the0.05% strain field /H20851compare Figs. 4/H20849b/H20850and4/H20849c/H20850/H20852. Thus, the
impact on the acceptor contrast at the /H20851110 /H20852surface will be
comparable if the tip-induced electric field is of this order ofmagnitude.
According to the TIBB /H20849V/H20850dependence, the acceptor state
is detected shortly before the flatband condition is reached.When the Mn acceptor state becomes accessible for tunnel-ing, the TIBB is comparable to the Mn binding energy. Thecorresponding depletion layer involves an electric field of 36kV/cm, i.e., 3.6 mV/nm, which is about a factor of 20smaller than the field needed to induce enough asymmetry inthe valence bands. It should be considered that the tip-induced band bending model employed here is a continuummodel. In particular, the doping is assumed to be homog-enous, but on the level of the STM experiment, the inherentgranularity of doping becomes apparent. Thus, the electricfields present in the STM experiment will be larger than theestimates from continuum models. However, even though theLOTH, WENDEROTH, AND ULBRICH PHYSICAL REVIEW B 77, 115344 /H208492008 /H20850
115344-6present sample is quite diluted, it is unlikely that the neces-
sary level of 0.1 eV/nm is reached.
C. Summary
In summary, the symmetry considerations of the band
structure demonstrate that the strain field of the surface re-laxation and the tip-induced electric field reduce the symme-try of the bulk band structure. Both effects act similarly onthe host crystal’s band structure and will thus introduce simi-lar asymmetries in the acceptor state’s wave function. A/H20849001 /H20850mirror asymmetry is gradually developed for both
cases. This is the same symmetry reduction, as observed inthe experiment. The calculations indicate that the acceptor
contrast should be more pronounced on the /H20851001
¯/H20852side for the
/H20851110 /H20852cleavage surface, which is supported by the measure-
ment /H20851compare sketches in Figs. 4/H20849b/H20850and 4/H20849c/H20850with STM
topographies in Fig. 2/H20852. Thus, both effects, i.e., strain and
electric field, are capable of producing the observed /H20849001 /H20850
mirror asymmetry in the shapes of the Mn acceptors and theyarea priori not distinguishable. Furthermore, the surface-
related strain cannot be influenced in the present experiment.The tip-induced electric field is adjustable via the appliedbias, but because the Mn acceptor state is only visible intopographies ranging from +0.95 to +1.1 V, the possiblevariation of the TIBB is only /H1101135 meV at the surface /H20849see
the lower graph in Fig. 1/H20850and the concomitant variation of
the electric field in the semiconductor is small. Thus, inten-tionally changing the strength of the electric field whilemonitoring the acceptor contrast asymmetry yields no sig-nificant results for this sample system.
Nevertheless, the comparison of Figs. 4/H20849b/H20850and4/H20849c/H20850yields
the estimated relative strengths of both effects and allows adifferentiation: a small shear strain of 0.05% /H20849/H20851110 /H20852mono-
layer distance /H20850originating from the surface relaxation is
capable of producing a considerable mirror asymmetrywith reference to /H20849001 /H20850. On the other hand, a relatively
strong electric field in the /H20851110 /H20852direction of 0.1 V/nm
/H20849=10
6V/cm/H20850is needed to introduce a similar effect solely
by electric fields. Here, the Mn acceptor state is imaged neara flatband condition and the electric field is much smaller.This indicates that the surface-induced strain is the dominanteffect. Additionally, the tip-induced electric field still pen-etrates more than 10 nm into the sample when the acceptorstate is imaged. In contrast to that, the surface-induced strainrapidly decays into the crystal on a length scale of a fewmonolayers. Together with the rapid decrease of asymmetrywith increasing depth /H20849see Fig. 3,/H9004
/H9257=0.024 /monolayer /H20850
and the observation that a Mn acceptor located in the tenthsubsurface layer /H208492.5 nm below the surface /H20850is almost sym-
metric, this lays further weight on the conclusion that the/H20849001 /H20850mirror asymmetry is driven by a surface-induced strain
field.
However, tip-induced fields of 0.1 eV/nm are easily
achievable in the STM experiment. Even though the electricfield is a minor effect for measurements of Mn acceptors inInAs, it should be considered in different systems, e.g., shal-low acceptors in GaAs where the highly anisotropic featuresare imaged well within the depletion bias window when thetip-induced field is large.
17
V . CONCLUSION
Mn acceptors in InAs are analyzed with high resolution
multibias topographic measurements. The anisotropic bow-tie-like features at subsurface acceptors are identified as animage of the probability density distribution of the acceptorground state wave function. This is validated by local I/H20849V/H20850
spectroscopy. Simultaneous acquisition of the circular Cou-lomb contrast of the charged acceptor and the bow-tie-likeneutral acceptor contrast allows high precision registry of thewave function image with respect to the Mn atom position.The contrast shape evolves nearly linearly from almost trian-
gular to rectangular with increasing distance of the dopantatom from the /H20849110 /H20850cleavage surface. Acceptors located
within the first ten subsurface layers of the semiconductorhave a pronounced asymmetry with reference to the /H20849001 /H20850
mirror plane that can be as high as
/H9257=0.27 /H2084927% /H20850. The mea-
sured contrasts for acceptors buried below the tenth subsur-face layer are in good agreement with theoretical predictionsfor the bulk acceptor’s probability density distribution. Inconclusion, the long-known /H20849001 /H20850mirror asymmetry of the
Mn acceptor contrasts is not only influenced by a surface-related effect, it is rather generated by it. Symmetry reduc-tion effects at the surface, such as strain originating from thesurface relaxation and electric fields induced by the STM tipare discussed as sources of the observed asymmetry. Whileboth effects induce similar symmetry reduction, a compari-son of their relative strengths indicates that surface-relatedstrain is the dominant source in this case. These findingsdemonstrate that impurities in different depths under the sur-face give access to the evolution of wave functions in envi-ronments with varying anisotropy and/or reduced dimension-ality.
ACKNOWLEDGMENTS
We thank J. Wiebe, F. Marczinowski, and P. M. Koenraad
for valuable discussions. This work was supported by DFGGrants No. SFB 602 TP A7 and No. SPP 1285, and by theGerman National Academic Foundation.
*wendero@ph4.physik.uni-goettingen.de
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115344-8 |
PhysRevB.91.094519.pdf | PHYSICAL REVIEW B 91, 094519 (2015)
Low-temperature thermal transport at the interface of a topological insulator
and a d-wave superconductor
Adam C. Durst
Department o f Physics and Astronomy, Hofstra University, Hempstead, New York 11549-0151, USA
(Received 25 January 2015; published 31 March 2015)
We consider the low-temperature thermal transport properties of the two-dimensional (2D) proximity-induced
superconducting state formed at the interface between a three-dimensional strong topological insulator (TI) and
a d-wave superconductor (dSC). This system is a playground for studying massless Dirac fermions, because
they enter both as quasiparticles of the dSC and as surface states of the TI. For TI surface states with a single
Dirac point, the four nodes in the interface-state quasiparticle excitation spectrum coalesce into a single node
as the chemical potential p is tuned from above the impurity scattering rate (\p\ F0) to below (|/x| < 5 C T0).
We calculate, via Kubo formula, the universal-limit (T — > 0) thermal conductivity k q as a function of p as it
is tuned through this transition. In the large- and small-|/u.| limits, we obtain disorder-independent, closed-form
expressions for k0 /T. The large-|/x| expression is exactly half the value expected for a d-wave superconductor, a
demonstration of the sense in which the TI surface topological metal is half of an ordinary 2D electron gas. Our
numerical results for intermediate \p\ illustrate the nature of the transition between these limits, which is shown
to depend on disorder in a well-defined manner.
DOI: 10.1103/PhysRevB.91.094519 PACS number(s): 74.25.fc, 73.20.-r, 74.20.Rp, 74.45.+c
I. INTRODUCTION
Topological insulators [1-3] (TIs) represent a novel state of
quantum matter that comes about due to the com bined effects
of spin-orbit interactions and time-reversal symmetry [4-8],
Although characterized by a bulk band gap, they are adiabat-
ically distinct from ordinary insulators and support protected
gapless surface states. In the case of a strong three-dimensional
(3D) TI, these surface states form a novel two-dimensional
(2D) topological metal with a spin-polarized massless Dirac
energy spectrum. The theoretical prediction and subsequent
experimental discovery of TI states in 2D materials [9,10]
(HgTe/CdTe quantum wells), 3D materials [11,12] (B ivSb]_x),
and the cleaner, simpler, second-generation 3D materials
[13-15] (Bi 2Se 3, Bi2Te 3, and Sb2Te 3) has led to great interest
in this area, exploring both the fundamental physics as well
as the potential for applications to fault-tolerant topological
quantum com putation [16,17].
The proximity of either magnetic materials or supercon
ductors to the TI surface can induce an energy gap in the
topological metal, resulting in even more exotic interface states
[1]. Early on, Fu and Kane [16] considered the proximity
effect at the interface between a TI and a conventional s-wave
superconductor, analyzing the proxim ity-induced supercon
ducting interface state and finding that it should support
M ajorana bound states [18-23] at vortices. Subsequent work
has expanded this analysis in many directions and has
included the case of TIs proximity coupled to unconventional
superconductors of different pairing symmetries [24-29]. Such
Tl-interface-state superconductivity has been demonstrated,
experimentally, both for the 5-wave case [30,31] and for the
case o f TIs coupled to high-7) cuprate d-wave superconductors
[32],
This last case, that of the proxim ity-induced superconduct
ing state at the interface of a strong 3D topological insulator
(TI) and a d-wave superconductor (dSC), is our focus here. For
simplicity, we consider a TI with a surface state characterized
by a single Dirac point at the origin o f k space, as is seenin the Bi2Se 3 family of materials [13,14], We are particularly
interested in the low-energy quasiparticle excitations of this
interface state; a system in which massless Dirac fermions
enter in two different ways; as both the surface states of the
TI and the quasiparticles of the dSC. For the former, the TI
surface states, the massless Dirac fermions are isotropic, a
consequence o f band structure, and not pinned to the Fermi
surface, such that one can tune through the Dirac point by
varying the chemical potential. They are described by a Dirac
equation where the gam ma matrices live in 2 x 2 spin space.
For the latter, the dSC quasiparticle states, the massless Dirac
fermions are anisotropic, their energy spectrum squeezed in
k space, and they are pinned to the Fermi surface at four
nodal points. They are described by a Dirac equation where
the gamma matrices live in 2 x 2 particle-hole (Nambu) space.
The Tl-dSC interface state that we consider mixes both spin
and particle-hole space and will have quasiparticle excitations
of its own, with features inherited from both o f the above.
A useful probe for studying massless Dirac quasiparticles
in d-wave superconductors has been low-tem perature thermal
transport [33^12], measurem ents of which can be extrapolated
to the particularly sim ple and interesting regim e where
temperature T is small com pared to the impurity scattering rate
To- This is known as the universal lim it because thermal con
ductivity due to massless Dirac quasiparticles has been shown
to be insensitive to disorder in this very-low-temperature
regime [43^49]. In this paper, we examine the nature of the
low-energy quasiparticle excitations of the Tl-dSC interface
state by calculating the universal-limit thermal conductivity,
Ko/T, as a function of chemical potential p. Although the
Ham iltonian for this interface state couples particle to hole and
spin up to spin down, its quasiparticles carry a well-defined
heat. Thus, thermal transport tracks quasiparticle transport
and is therefore well suited to probing the excitations o f this
system. For \p\ « Tq, it probes the single isotropic Dirac
node inherited from the TI surface. For \p\ T0, it probes the
four anisotropic Dirac nodes resulting from proxim ity-induced
d-wave superconductivity. We study both of these regimes
1098-0121 /2015/91 (9)/094519(11) 094519-1 ©2015 American Physical Society
ADAM C. DURST PHYSICAL REVIEW B 91, 094519 (2015)
and the transition between them as four nodes coalesce into
one.
We begin in Sec. II by writing down the 4 x 4 Hamiltonian
for the proximity-induced interface state, which mixes the
spin-space Dirac equation of the TI surface with the particle-
hole-space Dirac equation of the dSC, and then solve for the
quasiparticle excitation spectrum. In Sec. Ill, we calculate the
matrix spectral function, derive the thermal current operator,
and then use both of these to calculate the universal-limit
thermal conductivity tensor via diagrammatic Kubo formula.
Closed-form analytical expressions for k0/T are obtained
in both the large-|/u| and small-|/r| limits, both of which
are discussed in Sec. IV. Numerical results charting the
disorder-dependent transition between these two limits are
presented in Sec. V. Conclusions are discussed in Sec. VI.
II. PROXIMITY-INDUCED INTERFACE STATE
A. Hamiltonian
We consider the proximity-induced superconducting state
at the interface of a 3D strong topological insulator (TI) and
a d-wave superconductor (dSC). For a TI like those in the
Bi2Se3 family, characterized by surface states with a single
Dirac point at the F point of the Brillouin zone, the TI surface
state is described by the Hamiltonian [16]
Ho = •k - ( i )
t
where irk = (ck^,ck±)T are electron annihilation operators, v
is the slope of the Dirac cone, /z is the chemical potential,
cr = (<7i,er2) are Pauli spin matrices, and we have adopted
units where H = 1. Proximity to a dSC induces d-wave
superconductivity and results in an interface-state Hamiltonian
[16,24,25] that is most compactly expressed in the following
4 x 4 Nambu notation:
H = ( 2)
k
Hk = (va ■ k - /r) t3 + A ^r,, (3)
where the r are particle-hole Pauli matrices that mix the i kk
and ii\k blocks of and the factor of ~ compensates for
particle-hole double counting. Here, the proximity-induced
superconducting order parameter Ak is of dxi_yi symmetry and
is taken to be real. [Note that, in addition to this spin-singlet
d-wave term, the form of the TI surface Hamiltonian allows
for an additional, subdominant spin-triplet (B2u)p-wave term
to also be induced via proximity to a dSC [26]. However, as
shown by Linder et al. [24,25], a spin-triplet p-wave pairing
amplitude in a TI only renormalizes the chemical potential
and never gaps the surface energy spectrum. Thus, while its
inclusion here would likely result in a quantitative correction
to the effect of the singlet term, it is not expected to change
the essential physics. Thus, for simplicity, we shall defer
consideration of the triplet term to future work.] Expanding
Eq. (3) by evaluating the outer products of the Pauli matricesyields the 4 x 4 Hamiltonian
vk~ At 0
Hk =+opi ^
00 At
-vk~
_ 0 At -v k +
where = kx ± iky.
B. Quasiparticle excitation spectrum
The quasiparticle excitation spectrum of the interface state
is obtained by solving for the (positive) eigenvalues of Hk. As
shown in Ref. [16] for the 5-wave case, the resulting spectrum
is
Ek = y (± u |k | - pj1 + A2 k. (6)
Although the precise functional form of Ak is material
dependent, we can proceed, quite generally, as long as At
satisfies two criteria: (1) It has dxi_yi symmetry and therefore
changes sign along the lines ky = ±kx. (2) It vanishes faster
than linearly with k as k — > 0. If these criteria are met, the
quasiparticle spectrum will have the following properties.
For large |/z|, there will be four nodal points in k space,
located at ±kx — dtky = i±/\flv, where one of the two
branches in Eq. (6) goes to zero and quasiparticles can be
excited for zero energy cost. In the vicinity of each of these
nodes,
Ek « yjv2 k] + v2 Akj, (7)
where vA is the slope of Ak at the node and k\ and k2 define
a local coordinate system, centered at each node, with the k\
axis perpendicular to the local Fermi surface (pointing away
from the origin of k space) and the k2 axis parallel to the local
Fermi surface (pointing in the direction of increasing Ak).
For energies small compared to fi, the surfaces of constant
energy are ellipses, elongated parallel to the local Fermi
surface for v > vA (as is typical in cuprate superconductors).
The presence of disorder smears out the nodes, exciting
quasiparticles of energy less than or on the order of the impurity
scattering rate To. For T « To, quasiparticle transport is
dominated by these disorder-induced quasiparticles which
reside within ellipses of semimajor axis ro /u A and semiminor
axis r0 /u about each of the four nodes.
The nodes are distinct for \p\ » T0 but, as \p \ decreases,
the intemode separation decreases, and for | p , | <£ r 0 the nodes
coalesce at the origin of k space. As long as Ak vanishes fast
enough with decreasing k, as per condition (2) above, this
transition reveals the underlying massless Dirac spectrum of
the TI surface state. Thus, for |/z| < $ C r0 ,
Ek « v|k|, (8)
and the system thereby trades the four anisotropic nodes at
nonzero k for a single isotropic node at the origin. Note that
this single node is, however, doubly degenerate, because it
derives from both branches in Eq. (6). For \p ,\ and T small
compared to T0, quasiparticle transport is dominated by the
disorder-induced quasiparticles that reside within the circle of
radius Tq/v about this isotropic node.
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LOW-TEMPERATURE THERMAL TRANSPORT AT THE ... PHYSICAL REVIEW B 91, 094519 (2015)
III. TRANSPORT CALCULATION
Following the approach employed in Refs. [49] and [50],
we now proceed to calculate the universal-limit quasiparticle
thermal conductivity for this system as a function of chemical
potential. Key inputs to this calculation are the spectral func
tion and thermal current operator, which we will calculate first
and then utilize in our calculation of the thermal conductivity.and
A o = r 0(r0 2 + M 2 + U 2* 2 + a* ),
A i = 2F0 fivkx,
A2 = ITotivky,
A den = jr[ro2 + (vk - \± )2 + A2][r 2 + (-vk - n)2 + A2].
(17)
A. Spectral function
To obtain the spectral function, we begin by calculating the
Matsubara Green’s function. Working in our four-component
Nambu basis, the 4x4 bare Green’s function is obtained by
inverting the Hamiltonian
G0(k,ico) = [icol- Hk]~\ (9)
where Hk is the 4x4 Hamiltonian from Eq. (5). The dressed
Green’s function is then found via Dyson’s equation
G(k,ico)-' = G°(kjco)-1 - E (ico), (10)
such that
G(k,ico) = [ioA - E (ico) - Hk]~\ (11)
where E is the Matsubara self-energy matrix. The retarded
Green’s function is then obtained by continuing ico -* c o + f < 5 :
GA (k,w) = [col — E s (cu) — Hk]~l, (12)
where E ft(a>) = E (ico c o + iS) is the retarded self-energy
matrix. We define the matrix spectral function A(k,oj) via
such thatG(k,ico) =,A(k,co')aco---------- ,ic o — 00'(13)
A(k,co) = — [GR(k,oj) - G'4(k,o>)], (14)2n
where GA = GS1 is the advanced Green’s function. Since
our calcul
calculate
Gr ( k,0) =on will only require A(k,a> -> 0), we need only
i To + ix<1
1
10 --1
— vk+ i r 0 + /tt 0 — A *
- A ,
0
-j0
1 vk~ ’
0 — A* vk+”3
1OC - H
(15)
where = kx ± iky. Here we have taken a simple form for
the zero-frequency self-energy matrix, Y.R (to — ► 0) = — 1T0I,
where To is a scalar constant, the impurity scattering rate. In
general, the full 4 x4 self-energy matrix can be calculated for
a particular disorder model, but this simple model captures the
essential physics and establishes an energy scale for disorder.
Performing the inversion in Eq. (15) yields the zero-frequency,
matrix spectral function
A(k,0) =(Ao1c t + A iC q + A 2 C T 2 )lr
A den(16)B. Thermal current operator
To derive an expression for the thermal current density
operator in this system, we generalize the approach developed
for the 5 -wave superconductor case by Ambegaokar and
Griffin [51] and adapted for the d-wave superconductor case
in Ref. [49], We begin by expressing the Hamiltonian in terms
of the coordinate-space field operators, ^ ( x ) and t/u(x), such
that
H = H o + Ht,
H0 = I d2 x(i;\,f\)(-iva ■ V — n) , (18)
H\ = \ J d2xJ yWyp'I'xc',
where a and /3 are spin indices over which summation is
implied, V(x — y) is the effective potential that gives rise to
the proximity-induced superconductivity, and we have adopted
a compact notation whereby 1jr a = % l s xa = ^ra(x) and x f r y p =
yqg(y). Performing the matrix multiplications, H o takes the
form
H0 = J d2 x[-iv(i//\d~ifi + t/'jfi+V't) - di'l'l'c]
= j d2 x[iv{d~f\xlfi r + 3+Vr]t/rt ) - lltylta], (19)
where 3± = ^ ± ( j- and the second equality is the result of
integration by parts. Equations of motion for the field operators
are obtained by noting that
iifa = Wa,H], = (20)
and applying fermion anticommutation relations. Doing so,
we find that
= - vd~xlfX + i < p xx l/t ,
ifl - -n3+^t + icpxfi,
if\ = - v d +^ \ - if\cpx,
ir\ = - i^r\(px, (21)
where we have defined
< P X = < P (x ) = fi — J d 2rV{r - x ) ^ f rY ■ (22)
The thermal current density operator, j^(x), is obtained via
continuity with the thermal density operator, h(x):
h(x) = -V - f (x). ( 23)
where t a is the intrablock (spin) 2x2 identity matrix, Since we have written our Hamiltonian such that all energies
1T is the interblock (particle-hole) 2x2 identity matrix, are measured with respect to the chemical potential, h (x) is
094519-3
ADAM C. DURST PHYSICAL REVIEW B 91, 094519 (2015)
equal to the Hamiltonian density operator and therefore is
defined via
H = J d2xh(x ) (24)
and expressed as
h{\) = -'-^-(f\d~fi - 9~i/rjiA{3 + Vh “ d+ f\f\)
- I f a + ^ J d2yflf\^V{ y - x)fy l i f a, (25)
where we have taken H o to be the average of the first and second
lines of Eq. (19). Taking the time derivative and breaking the
result into two pieces, we write
h (x ) = F A + F B , (26)
where
Fa = - l -^-(f\d~fi ~ d~f\fi + f\d+ ff ~ d + f \ f \
+ f\d~fi - d~f\fi + f\d+ f t - d + f\f^)
- fxflfa - tiflfa, (27)
Fb = ^ / d2yV{ y - ^(.flflpfypfa + flflpfypfa
+ fl f\p fy fi fa + fl flu fy ff fa)- (28)
The first piece F A can be reorganized by using the equations of
motion (20) to sub in for the dotted field operators, regrouping
terms, and then applying the equations of motion again. Doing
so, we find that
Fa = j[d-(f\fi) + d + ( f l f t)
- d~(f\f±) ~ d+ (f\f\)]
- J d2y V ( y - x ) ( f l f l fif yp f a + f l f l p f yp f a ).
(29)
Combining this with F B and applying the continuity equation
(23), we see that it is natural to write the thermal current density
operator as the sum of two terms,
f = ui + u2, (30)
where
V . Ul = - y [9 “ (iA{i/t) + d + (f\fr )
- d - ( f \ f i) - d + (flfO l (3D
v • U 2 = ^ 1 d 2yV(y — x)[{flf\^fy pfa + f l f l pf y pfa )
~ (flflufytifa + flflpfypfa)]- (32)
Expansion of the 9± operators reveals that the right-hand-
side of Eq. (31) is easily expressed as a divergence. Doing so,we extract
ui = ~ y [[(f\fi + f\f\)x - i(f\fi - f\f^)y]
- [(f\fl + f\f\)x - i(f\fi - f\ft)y]}, (33)
which, in 4x4 Nambu notation, becomes
Ui(x,t) = — — — fiLo-rsW], (34)
where 4>1 =qD(x,r) = [f^ f^ f^ , — f^] and a = o \x + < 72 $ .
Fourier transforming in space and time yields
U|(q,£2) = i ^ 4 / A t fw + ^ J i> 5 r3^ +?, (35)
kco ' '
where we have used the shorthand = 'l'(k.ft)) and 'Fk+q =
W (k + q.cu + £2).
To obtain u2, we take the spacetime Fourier transform of
Eq. (32). Doing so yields
iq ■ u2(q,£2) = ^ J d2x d2ydtV{y - x)(e_!qx - e_iqy)
X (flaflpfypfxa + flaflpfypfxa)
= Xi+X2 - Y i - Y 2 , (36)
where we have labeled each of the four resulting terms: X \, X2 ,
Y \ , and T2. Inserting a Fourier representation for the potential
and each of the field operators, the X \ term takes the form
* 1 = \ J2 ^ vh c\iac\ipchpcU a ^ - k \ - h - q )
k\,...,k$ C O
x < $ (& 3 — k-2 H " k ^ )8( ^ c o \ - \ - ( O 2 — (02 — (04 -t- £2). (37)
Making a mean-field approximation, retaining only the terms
for which the average values are over (k f , — k |) pairs
(reduced approximation), and noting that (c L c t^ ) is an even
function of a), this becomes
X^iJ^(cv-Q)A*k c l q ^clki, (38)
kco
where
(39)
k'co'
is the superconducting order parameter for the interface state.
Repeating this calculation for X2 , Y \, and T2, and taking A*
to be real, we find that
q • u2(q,f2) = ]P (A k + q - Ak )
kco
x W c\^c] _(k+q)i + (ft) + Q)c-kirck+q^]. (40)
In the q — ^ 0 limit,
9 A k
A k+ q - A* = q • — - = q • \A k , (41)9k
where vA j. is the slope of the order parameter (the gap velocity)
at k . Plugging into Eq. (40) and taking the £ 2 — » 0 limit, we
094519-4
LOW-TEMPERATURE THERMAL TRANSPORT AT THE ... PHYSICAL REVIEW B 91, 094519 (2015)
find that
“2(0.0) = J 2 ('w + ? )v A * (< W f* H + c -n c k+qt),
k c o ' '
(42)
which, in the 4 x 4 Nambu notation, becomes
“ 2(0,0) = l - J 2 * l ( u + y W n * t+?. (43)
k c o ^ '
Thus, in the q,Q — » 0 limit (which is the limit where we will
need it), the thermal current density operator is
m 0 ) = i £ * l ( r n + f W , (44)
k c o ^ '
where
xM = v a r3 + vAkti (45)
is a vector in coordinate space and a matrix in our 4 x 4 Nambu
space. Here, the first term derives from the massless Dirac
spectrum of the TI surface and has inherited the interesting
spin structure thereof, while the second term derives from
the the d-wave order parameter of the proximity-induced
superconductivity.
C. Thermal conductivity
With the spectral function and thermal current density
operator in hand, we can proceed to calculate the thermal
conductivity in the zero-temperature, zero-frequency limit. For
d-wave superconductors, this limit is known as the universal
limit because thermal conductivity has been shown to be
insensitive to disorder in this regime [43— 49]. We can calculate
the thermal conductivity tensor k(T) for the case at hand by
appealing to the fluctuation-dissipation theorem as expressed
in the Kubo formula [52]:
k(T) imn,(fi)------- = — hm ------- ^-----
T T2 Q(46)
We obtain the retarded current-current correlation function via
analytic continuation from the Matsubara function
n?(£2) = rLo'fi -* g + is), (47)
n*(i£2) = - drein T (T T jK (z)iK m , (48)
J o
where = \/kFT , T x is the r-ordering operator, and the
brackets denote the thermodynamic average. For simplicity,
we proceed by calculating the bare-bubble Feynman diagram
shown in Fig. 1, noting that vertex corrections have been
shown to be small for the 4-wave-superconductor case [49] and
deferring to future work their calculation for the case at hand.
Doing so, the Matsubara thermal current-current correlation
function takes the form
n*(i£ 2)1 1
2H H ( i c o +2
x 7r[G(k,ico)xMG(k,ico + /£2)v«], (49)k,co + Q
k,co
FIG. 1. Feynman diagram representing the bare bubble thermal
current-current correlation function n K (f^2). On each vertex sits a
thermal current density operator j*. Each propagator line denotes
a Green’s function dressed with disorder self-energy, G(k,ia>) and
G (k,i'c<; + i'S 2).
where the u > sum is over fermionic Matsubara frequencies, the
k sum is over the Brillouin zone, the trace is over Nambu space,
and the factor of \ out front compensates for the particle-hole
double counting that is inherent in our 4 x 4 Nambu formalism.
Inserting a matrix spectral representation, as defined in Eqs.
(13) and (14), for each of the Green’s functions, this becomes
n«(i£2)
- £ /dQ}idu>2S(iQ)
xTr[A (k,m ,)vMA(k,m2)vA/],
where
S(iQ)= - p Y l\iM JriQ.
~21
ico — a > \ ico + i£l — a > 2(50)
(51)
Evaluating the Matsubara sum via contour integration (see
Refs. [51] and [49] for a discussion of the technical points)
and continuing / Q -» Q + iS, we obtain the retarded function
s Rm -(«1 + G /2)2H f( «i) — (a) 2 — G /2)2«/r(&>2)
co i — ( x> 2 T £2 ~F / 8(52)
where n F((o) = l/(e^w + \) is the Fermi function. Since
the retarded and advanced Green’s functions are Hermitian
conjugates, the spectral function defined in Eq. (14) must be
Hermitian:
A t = -iGr t - G A t
2nGr - G ai-------------
2n- A .
(53)
And since \M is also Hermitian, the trace in Eq. (50) must be
real:
Tr[A,vM A 2vM ]* =Tr[(A,vM A 2vM )rr
= Tr[(A,vwA2vM)t] = TiW MA \\'MA\}
= Tr[vMA2vMAi] = Tr[A!VMA2vM].
(54)
Therefore,
**R 1Im n^(G ) = - dco\da>2lmSR(S2)
wherex Tr[A(k,cui)vwA(k,<u2)vw], (55)
Im5«(G) = t cQ
22
[nF(co\ + G) - « f(o>i)]
x 8(coi + G — cu2). (56)
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ADAM C. DURSTPHYSICAL REVIEW B 91, 094519 (2015)
Plugging into Eq. (46) and taking the Q , -+ 0 limit yields an
expression for the thermal conductivity tensor:
x ^T r[A (k,a>)vMA(k,a))vM ]. (57)
k
In the zero-temperature limit, (co/T)2 (-dnF /d co ) is sharply
peaked at to = 0. Thus, evaluating the integral
we find that the universal-limit thermal conductivity tensor
takes the form
n3 lc i
= - ^ 2 ] T r [ A ( k ,0 ) v MA(k,0)vM ], (59)
k
where A(k,0) is the spectral function that we evaluated in
Eqs. (16) and (17).
Introducing the shorthand TrA v A v for the trace in the above
expression and plugging in for A(k,0) via Eq. (16) and for \ M
via Eq. (45) yields*o = k(T)
T ~ T
TrAuAu = -2 “ Tr[(A,l T (uC Tr3 + vA j fc lffTi))2 ]
Ad e n
= -^ -T r[u 2(A 5)2l T + va*vmIV21t ], (60)
"den
where N = A0 ta + Aioq + A2 a2 and we have made use of the
multiplicative properties of the particle-hole (r) Pauli matrices.
Noting that a = c s\x + a2 y, making use of the multiplicative
properties of the spin (a) Pauli matrices, evaluating the trace,
and plugging back into Eq. (59), we find that
K0
Tv2 (xx + yy) ^
k
+ v2 (xx — yy) ^
k
+ v2 (xy + yx) ^2
k2A]A2
~a2^den
+^ V A k V A k
kAq + Ay + A22(61)
Since Ak is of d x i_y i symmetry, it must be an even function of
both kx and ky. Therefore, as defined in Eq. (17), A0 and Ad e n
are even functions of kx and ky while Aj is odd in kx but even
in ky and A2 is even in kx but odd in ky. As a result
v— ' 2A] A2 f dkx f dky 2A]A2=/^i^vir= a < 6 2 >
And since exchange of k x for k y sends Ak to - A k , it leaves
Ad e n invariant but exchanges A, for A2. Therefore,
k den k "den(63)Thus, only the first and fourth terms in Eq. (61) survive. Noting
that xx + yy is just the identity tensor 1, plugging in for
Ao, M, A2 , and Ad e n from Eq. (17), and restoring h in the
prefactor, we obtain the following expression for the thermal
conductivity in the zero-temperature limit:
^0 = 4 2jr3
T 3 hVAkVAk(Pk + Q k) , (64)
where P k = A2/A 2 en and Qk = (A2
P k = -Po/tr' ^2) /^den ta^e the form
Po/tt I 2
r2 + (vk - ii)2 + L1 0 r o + (~vk — ii)2 +A2k J
(65)
______ Pq/t t________________ Pq/7 T_______ "l2
+ (vk - /x)2+ A 2 r 2 + (-vk - ix )2 + A2.
(66)
Note that this result depends on integrals of the squares of sums
and differences of Lorentzians centered about the zeros of the
two branches of the quasiparticle excitation spectrum, Eq. (6),
of width given by the impurity scattering rate. For f x y > > T0, k 0
is dominated by impurity-induced quasiparticles in the vicinity
of the zeros of the (+) branch. For /r « - T 0, the ( - ) branch
dominates. For \ i x \ < < C F q, both branches contribute.
IV. ANALYTICAL RESULTS
In both the large- \ f x \ and small-|/x| limits ( \/x \ > T0 and
[M l ^ Pq) the quasiparticle excitation spectrum simplifies, as
described in Sec. IIB, and can be linearized about nodal points
in k space. As a result, in these limits, we can obtain simple,
closed-form expressions for the zero-temperature thermal
conductivity. This is shown in the following sections.
A. Large-1/t | limit
For / X » T0, the Lorentzians in Eqs. (65) and (66) are
sharply peaked about four nodal points, located at ±kx —
±ky = ix/V2v, and are well separated from each other. We
can therefore replace the k sum in Eq. (64) by the sum of four
integrals over local scaled coordinates, p\ and p2 , defined
about the nodal points
4 r
£ - £ /
k j = 1 Jd2 k
V n fy- f d2 p
(2t t)2uua '(67)
where the integrals can be extended to infinity because the inte
grands are so sharply peaked about each node. Here, p\ = vk\
and p2 = v& k 2 , and at each node, k \ and k2 point, respectively,
perpendicular to and parallel to the local Fermi surface, with
k 2 in the direction of increasing A*. In terms of these scaled
coordinates, Ak & p2 and vk = i x + p\, so vk — \ x = p\ and
— vk — p . = — (2 /x + p\) & — 2i x . Therefore, since /r ~ S > T0,
the second Lorentzian can be neglected with respect to the
first in both Eqs. (65) and (66), and we find that
If T o / n
Q k 4 V r2(68)
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where p = yjp2 + p\. Evaluating the integral
[d 2 p ( V _J (2.71 )2 \ E q + P2 ) 4 JT3 ’(69)
and noting that the sum over nodes of the outer product of \ A k
with itself at each node is
E W = 2«il,
7=1
we find that
x , 1 1 1E^ = 4—i
v va 4 4tt2 4 tt3vva(70)
(71)
y^VAkVAkiPk+Qk) = 2v2 Al — 2 ^ - E = ^“ vvA 4 4n3 47r3uuA. (72)
Therefore, the thermal conductivity tensor reduces to a scalar,
< ■ > * */c o = /c q 1, with the simple form
*0 _
T ~ 2 3 h \v A + v )'(73)
The same result is obtained for p < < c — T0, where it is the first
Lorentzian in Eqs. (65) and (66) that can be neglected with
respect to the second. Note that this expression is independent
of disorder and is only a function of the velocity anisotropy,
v/vA, which depends on both p and material parameters. Note
also that this is exactly half the value obtained (per layer) for
the case of an ordinary d-wave superconductor [49], This is
because, unlike the d-wave superconductor case where the
electron dispersion is spin degenerate, here the TI surface
state is nondegenerate and only one of the two branches of
the quasiparticle excitation spectrum [Eq. (6)] contributes to
the thermal conductivity. For p » T0, the (+ ) branch (first
Lorentzian) contributes. For p < S ( — r 0, the (— ) branch (second
Lorentzian) contributes. This factor of two is a clear and
measurable demonstration of the sense in which the TI surface
topological metal is “half” of an ordinary 2D electron gas [16].
B. Small-1/r | limit
For 1 /7 . | < 3 C To, the four anisotropic nodes of the prior section
have coalesced into a single isotropic node at the origin of k
space. The first and second Lorentzians in Eqs. (65) and (66)
are approximately equal and peaked at the origin. The k sum
in Eq. (64) can be replaced by a single integral about scaled
coordinates, p\ — vkx and p2 = vky, and extended to infinity:
/d2 p
(2tc) 2v2(74)
In these scaled coordinates, vk = p = (p2 + p |) 1/2, and since
\p\ <£ T0, vk — p % p and — vk - p — p. As long as A*
vanishes fast enough with decreasing k, as per condition (2) of
Sec. II B, A 1 and \ A kV A k can be neglected in Eqs. (64)-(66)
compared to larger terms. As a result, the two Lorentzians addin P k and cancel out in Q k '.
P k =r 0/7r
, Q k & 0. (75)
J l + P2,
Once again making use of the integral in Eq. (69), we find that
v p = — / d 2 p ( r °^ V =
V * 1,2 J (2 r r)2\ r 2 + p2J1
(2jr)2 VTq + p2 J 4n3v2
X ^ a*vm(Pa . + Qk) « 0,, (76)
(77)
Therefore, the thermal conductivity tensor again reduces to a
scalar, now with an even simpler form:
ko _ kll
T ~ 3H2'(78)
Here, both branches of the quasiparticle spectrum have
contributed to the thermal conductivity, and one obtains
precisely the result one would expect for a single isotropic
massless Dirac node. This expression is clearly independent
of disorder and is just the standard < 7 -wave-superconductor
result [49] for an anisotropy ratio of one divided by a
factor of four since there is only one node here rather than
four.
V . NUMERICAL RESULTS
We would now like to look beyond the large- \p\ and
small-1/71 limits and consider the transition between them
by numerically evaluating Eqs. (64)-(66) as a function of p.
This is easily done, but unlike the large- and small- \p\-limit
calculations which were model independent (aside from the
two conditions in Sec. II B), this calculation requires a
model for A*, the proximity-induced superconducting order
parameter of the Tl-dSC interface state, and its results will
necessarily depend (in the details) on that choice of model.
Since we are primarily interested in understanding the essential
physics of this transition, without delving too deeply into the
material-dependent details, we proceed by considering the
following simple and rather standard expression for the order
parameter of a generic d-wave superconductor:
A* -(cos kxa — cos kya), (79)
which yields a gap velocity at k of the form
_ 3 A k Ao c i
V A k = — = - r - ( “ sin kxa\ + sin kya y). (80)3k 2
Here, we have introduced two new model parameters, the
gap maximum A0 and the lattice constant a. Expressing
all lengths in units of a and all energies in units of v/a,
we define dimensionless parameters jl = pa/v, To = Foa/i>,
and Ao = Ao a/v, as well as a dimensionless wave vector
with components zi = kxa and z2 = kya. Doing so, plugging
Eqs. (79) and (80) into Eqs. (64)-(66), and noting that all terms
not proportional to the identity tensor integrate to zero, we find
that the universal-limit thermal conductivity tensor reduces to
094519-7
ADAM C. DURST PHYSICAL REVIEW B 91, 094519 (2015)
FIG. 2. Calculated universal-limit thermal conductivity, t c 0 /T,
as a function of chemical potential /i. Solid curve denotes numer
ical solution of Eqs. (81)— (83) for parameter values A0 = 0.1 v/a
and r 0 = 0.01 v/a. The solution matches our large-|//| expression
(dashed) for |//| r 0, then reaches a maximum at an intermediate
value of |/z| before decreasing toward the value of our small-|/x|
expression (dotted) for |/x| « F 0.
a scalar and takes the convenient form
a r ^2, r
+ ^ sin2 zi(L(z )2 + L (-z)2)
where
andL( z) =f0
fo + (z - A)2 + A(Z),2’
- AoA(z) = — (coszi - cosz2).(81)
(82)
(83)
The k-space integral is easily computed to obtain k q/ T as a
function of //. Results for Ao = 0.1 v/a and To = 0.01 v/a
are plotted in Fig. 2 alongside the large- and small-|/x| limits.
[For the large- |/x] plot, we have used the model introduced in
Eqs. (79) and (80) to obtain the nodal anisotropy ratio as a
function of //, v/vA = [(Ao/V2)sin(A/V2)]_1, and used that
as input to Eq. (73).] Our numerical result matches the large-//
expression for |/z| To, peaking with decreasing /x |, before
plunging down toward the small-|/z[ value for |/z| < S C To.
This behavior is best understood by considering the evo
lution of the k-space structure of the integrand of Eq. (81) as
a function of //, as shown for a series of // values in Figs. 3
and 4. The upper panel of Fig. 3 illustrates the structure of the
large-// limit. Here, for A = jt/ \ /2, the integrand is peaked
within fo of four well-separated anisotropic nodal points.
Equal-intensity contours are (nearly) elliptical, squeezed in
the direction parallel to the local Fermi surface. In the middle
panel, // is reduced by a factor of two, which draws the nodesFIG. 3. (Color online) Evolution of the It-space structure of the
k0 /T integrand, (a) [// = ^=^]: Large-|//| limit.Four, well-separated,
elliptical peaks within T0 of the nodal points, (b) [// = j ^ ^]: Nodal
peaks closer to the origin, more anisotropic, and curving around the
Fermi circle, (c) [// = Nodal peaks have merged into an
annular peak of width r0 .
094519-8
LOW-TEMPERATURE THERMAL TRANSPORT AT THE ... PHYSICAL REVIEW B 91, 094519 (2015)
FIG. 4. (Color online) Further evolution of the k-space structure
of the k0/ T integrand, zoomed in by a factor of 20. (a) = 4. .*=£];
Closeup view of the same annular peak shown in Fig. 3(c). (b) [fi =
2jo -j i j ] ; Width and radius of the annular peak are now nearly equal,
(c) [/r = 0] Small-1 fx | limit. Annular peak is blurred into single,
isotropic peak within T0 of the origin.closer to the origin. The peaks are still well separated, but less
so than before, since the radius of the Fermi circle has de
creased and the nodal anisotropy ratio has increased. Thus, the
peaks have begun to curve around the Fermi circle, toward each
other, and the independent-node approximation used to derive
the large-1/u. | expression of Eq. (73) has begun to break down.
In the lower panel, g is reduced by an additional factor of ten.
Now the independent-node approximation has completely bro
ken down, and the four anisotropic peaks have curved into each
other, forming an annulus of width To about the Fermi circle.
With decreasing /i, the radius of this annular peak decreases,
resulting in the decrease of k q/ T seen in Fig. 2. We reproduce
this image, zoomed in about the annulus, in the upper panel
of Fig. 4. In the middle panel of that figure, \x is reduced by
another factor of ten such that it is nearly equal to r0 . Now the
width and radius of the annular peak are nearly equal. As / x
decreases further, the system is tuned toward the Dirac point
inherited from the TI surface state and the isotropic node at the
origin is revealed. For |g| T0, the annular peak blurs into a
single isotropic peak at the origin, of width f 0. This is shown
in the lower panel where f x = 0. The integral over this single
isotropic peak recovers the small-1 /x | value of Eq. (78). As //
becomes negative, the process reverses, dominated now by the
(— ) branch of the quasiparticle excitation spectrum instead of
the (+) branch. All else is the same, so k q/T is even in /x.
Results for five different values of the impurity scattering
rate T0 are shown in Fig. 5. Note that, in both the large-1 /x( and
small-1 /x | limits, k q/T is disorder independent. The transition
between these limits does, however, depend on disorder,
with the peaks of the k q/T vs /x curves smoothed out for
greater disorder. This effect can be understood in terms of our
integrand analysis (above). As \ / x \ decreases from its largest
FIG. 5. (Color online) Disorder dependence of calculated
universal-limit thermal conductivity, k0/ T , as a function of chemical
potential /z. We plot numerical solutions of Eqs. (81)— (83) for
A0 = 0.1u/a and five values of the impurity scattering rate T0.
Results are disorder independent in both the large-1/z| and small-|/z|
limits. The transition between limits depends on disorder, with the
peaks more prominent for smaller T0, smoothing out with increasing
disorder.
094519-9
ADAM C. DURST PHYSICAL REVIEW B 91, 094519 (2015)
values, increasing anisotropy ratio yields increasing k q/T via
our large-|/x| expression, Eq. (73). But for greater disorder,
the independent-node approximation that defines the large-|/x|
limit breaks down sooner, as the four anisotropic peaks broaden
with growing disorder and merge together earlier, limiting the
enhancement of k q/T with increasing anisotropy ratio. The
resulting annular peak is of greater width for greater disorder
and therefore blurs into a single peak sooner, ushering in the
small-1/^ | limit as its radius becomes smaller than its width.
VI. CONCLUSIONS
In this paper, we have calculated the universal-limit thennal
conductivity k0 as a function of chemical potential /x due to
quasiparticle excitations of the proximity-induced supercon
ducting state at the 2D interface of a topological insulator and
a d-wave superconductor. In both the Iarge-1/x and small-|/x|
limits, we have obtained simple closed-form expressions for
k q/T, combined here from Eqs. (73) and (78):
^ = + v) for M » r o
T 3h\ ] 2 for |m| «
where v is the slope of the isotropic Dirac cone inherited
from the TI surface state, v/v& is the /x-dependent anisotropy
ratio of the four anisotropic Dirac cones of the proximity-
induced < 7 -wave superconducting state, and r 0 is the impurity
scattering rate, the energy scale characterizing disorder in
the system. Note that the large-|/x| expression is exactly half
the value obtained [49] (per layer) for an ordinary d-wave
superconductor: KqS C /T = (kg/3h)(vp/vA + UA/n^).Thisis
an overt demonstration of the sense in which the underlying
topological metal is “half” of an ordinary metal [ 16] and comes
about because, for large |/z|, only one of the two branches
(positive or negative) of the isotropic Dirac cone contributes
at a time. For |/x| < $ ( To, both branches contribute, but the four
nodes have coalesced into one isotropic node at the origin of k
space. Thus, the small-|/x| expression is equal to the standard
dSC value (with anisotropy ratio equal to one), divided by
four (since there is only one node instead of the usual four):
(1 + l)/4 = 1/2. While kq/ T is disorder independent in both
of these limits, the transition between them, as a function of
/x, depends on disorder. And furthermore, it depends in thedetails on the functional form of the proximity-induced order
parameter A*. Adopting a simple model for Ak [Eq. (79)], we
calculated kq/ T across the full range of /x for different levels of
disorder, as shown in Fig. 5. As /i decreases from its maximum
value, the four nodal peaks of the integrand in Eq. (81) become
more anisotropic, resulting in an increase in Kq/T, as per
our large-1/x| expression. But they also move closer together,
eventually merging into an annular peak about the Fermi
circle. Along the way, the independent-node approximation
that defined the large- |/U.| limit breaks down, and k q/T reaches
its maximum value, decreasing as /x decreases further and
the Fermi circle shrinks. Finally, as /x gets smaller than To,
the annular peak blurs into an isotropic nodal peak at the
origin, and k q/ T reaches its minimum at the value given by
our small-1 ju . | expression. As shown in Fig. 5, the peaks in
the k q/T vs i a curve are more pronounced for smaller T o ,
smoothing out with increasing disorder.
Note that we have assumed herein that the bulk band gap
of the topological insulator extends well above and below
the Dirac point of the surface state, such that /x could be
varied over a wide range of energies without accessing the bulk
valence or conduction bands. In real materials, the available
energy windows may be more restricted. We have also assumed
that the chemical potential can be accurately controlled, via
gating, doping, or other means, and that proper contact can be
made to the Tl-dSC interface. Both may present experimental
challenges.
Our focus in this work has been on the evolution with
changing chemical potential of the massless Dirac quasipar
ticle excitations of the Tl-dSC interface state. The results
shed light on the essential features of low-temperature thermal
transport due to these quasiparticles. Further theoretical devel
opment, including incorporation of a subdominant spin-triplet
order parameter, a more realistic disorder model, and vertex
corrections to our diagrammatic calculation, are left for future
work.
ACKNOWLEDGMENTS
I am grateful to B. Burrington and G. C. Levine for
very helpful discussions. This work was supported by faculty
startup funds provided by Hofstra University.
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PhysRevB.97.235157.pdf | PHYSICAL REVIEW B 97, 235157 (2018)
Suppression and revival of long-range ferromagnetic order in the multiorbital
Fermi-Hubbard model
Andrii Sotnikov,1,2,*Agnieszka Cichy,3,4and Jan Kuneš1
1Institute of Solid State Physics, TU Wien, Wiedner Hauptstrasse 8, 1020 Vienna, Austria
2Akhiezer Institute for Theoretical Physics, NSC KIPT, Akademichna 1, 61108 Kharkiv, Ukraine
3Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland
4Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7, D-55099 Mainz, Germany
(Received 14 February 2018; revised manuscript received 19 June 2018; published 29 June 2018)
By means of dynamical mean-field theory allowing for complete account of SU(2) rotational symmetry of
interactions between spin-1 /2 particles, we observe a strong effect of suppression of ferromagnetic order in the
multiorbital Fermi-Hubbard model in comparison with a widely used restriction to density-density interactions. Inthe case of orbital degeneracy, we show that the suppression effect is the strongest in the two-orbital model (witheffective spin S
eff=1) and significantly decreases when considering three orbitals ( Seff=3/2), thus magnetic
ordering can effectively revive for the same range of parameters, in agreement with arguments based on vanishingof quantum fluctuations in the limit of classical spins ( S
eff→∞ ). We analyze a connection to the double-exchange
model and observe high importance of spin-flip processes there as well.
DOI: 10.1103/PhysRevB.97.235157
I. INTRODUCTION
Symmetry, its discrete or continuous nature, and its explicit
or spontaneous breaking play a crucial role in physics. Incondensed-matter theory, Heisenberg and Ising models aredistinctive examples of systems possessing continuous anddiscrete spin symmetry, respectively. While the spontaneousbreaking of spin symmetry plays a central role in observationsof phase transitions and emerging gapless collective modes,its explicit analog is less “charming” and usually originatesfrom the external source fields, sample imperfections, orsimplifications required to proceed with the correspondingtheoretical description.
Long-range ferromagnetic (FM) order is prominent realiza-
tion of spontaneously broken symmetry responsible for manyimportant physical phenomena, e.g., the colossal magnetore-sistance effect in manganites [ 1]. In a search of a prototypical
lattice system of itinerant interacting particles supportingthe FM ground state, a single-band Hubbard model can besuggested as the simplest one. However, as emphasized andstudied in detail in Refs. [ 2,3], this is not a generic model of fer-
romagnetism, since without lattice loop structures or additionalnearest-neighbor “biasing” interactions the corresponding FMground state can only emerge in the Nagaoka’s limit ( U=∞ ).
The next by simplicity, the two-orbital Hubbard model, pro-vides a minimal number of necessary ingredients (in particular,a local nonzero Hund’s coupling) to support FM ordering.
Nowadays, dynamical mean-field theory (DMFT) [ 4]i s
a powerful nonperturbative theoretical approach to describephysics of strongly correlated materials including transitionsbetween different thermodynamic phases. In a number of pre-vious DMFT studies analyzing FM instability in the two-band
*sotnikov@ifp.tuwien.ac.atHubbard model the computational procedure was restricted tothe density-density interactions [ 5–9]. Although this simplifi-
cation substantially reduces the computational cost, it affectsthe physics of the model, especially close to the critical regime(see, e.g., Ref. [ 10] for a recent material-oriented analysis).
Therefore, a significant effort has been made to account for thefull rotational symmetry of two-particle interactions [ 11–21].
In parallel to the progress of DMFT and other theoretical ap-
proaches, ultracold atoms in optical lattices [ 22] have become
a universal and very accurate experimental tool for gainingnew insights in a rich family of the Hubbard-type models.In these systems, a degree of freedom associated with theelectron spin can be attributed to atoms in different internalstates (i.e., the pseudospin concept is widely applied). This
results in a large tunability—at will, the spin symmetry can beexplicitly broken or restored with a high accuracy by a properexperimental setting [ 23,24]. In particular, recent experiments
with
173Yb and87Sr atoms [ 25–27] show the capability to
realize two-orbital Hubbard models with the FM-type Hund’scoupling and SU( N)-symmetric interactions of pseudospin
flavors.
The purpose of the present paper is to analyze the effect
of interaction parametrization—with the SU(2) spin-rotationalsymmetry or lower—on FM ordering in the multiorbital Hub-bard models. Starting with degenerate orbitals we introducea crystal field in order to link the metallic Stoner-type FMphase of the Hubbard model with overlapping bands to theFM phase of the double-exchange (DE, or the Kondo-lattice)model. The latter has a long history [ 28] and it is widely applied
to describe FM ordering and related effects in manganites. Themodel studies by means of DMFT have already uncoveredimportant effects of electronic correlations [ 29,30] and spin
fluctuations [ 31] on FM ordering in the DE regime.
The paper is organized as follows. In Sec. IIwe introduce
the theoretical model and specify relevant details of the applied
2469-9950/2018/97(23)/235157(6) 235157-1 ©2018 American Physical SocietyANDRII SOTNIKOV , AGNIESZKA CICHY , AND JAN KUNE Š PHYSICAL REVIEW B 97, 235157 (2018)
numerical approach. We analyze the FM instability starting
from the case of degenerate orbitals in Sec. III A and proceed
further with introducing the chemical potential imbalance (i.e.,a nonzero crystal field) in Sec. III B. The main results are
summarized in Sec. IV.
II. MODEL AND METHOD
We consider the Fermi-Hubbard model with multiple
(m=2,3) orbitals described by the Hamiltonian
H=/summationdisplay
kσm/summationdisplay
γ=1(/epsilon1kγ+μγ)c†
kγσckγσ+U/summationdisplay
iγniγ↑niγ↓
+U/prime/summationdisplay
iσ,γ<γ/primeniγσniγ/prime¯σ+(U/prime−J)/summationdisplay
iσ,γ<γ/primeniγσniγ/primeσ
+αJ/summationdisplay
i,γ<γ/prime(c†
iγ↑c†
iγ/prime↓ciγ↓ciγ/prime↑+H.c.)
+α/primeJ/summationdisplay
i,γ<γ/prime(c†
iγ↑c†
iγ↓ciγ/prime↓ciγ/prime↑+H.c.). (1)
The first term includes free-particle energies /epsilon1kγ, chemical
potentials μγ, and fermionic creation (annihilation) operators
c†
kγσ(ckγσ) of electrons on the orbital γwith the spin σ=
↑,↓(and its opposite ¯ σ=↓,↑) and quasimomentum k.U
is the intraorbital interaction amplitude and Jcharacterizes
local ( iis the lattice site index) ferromagnetic ( J> 0) Hund’s
coupling. Here and below, we use the parametrization U/prime=
U−2J, which is generally valid for all electron-electron
interactions that are rotationally invariant in real space. Thecoefficients α,α
/prime∈[0,1] can be set, in principle, independently
of each other. At α=α/prime, the two limiting cases with α=0 and
α=1 correspond to the so-called density-density (Ising-type)
and Slater-Kanamori (SK) parametrization of interactions. Wealso restrict ourselves to all positive interaction amplitudes thatresults in limitations U> 0 andJ/lessorequalslantU/3.
The Hamiltonian ( 1) allows for a separate analysis of spin
and orbital sectors. Since the symmetry of the latter is almostirrelevant for the current study, we omit its discussion [ 32]
for simplicity. Therefore, it is sufficient to point out briefly theinfluence of the Ising-anisotropy parameter αon the symmetry
of the spin sector. In particular, at α=1 the model becomes
SU(2) symmetric with respect to rotations in spin space.A n yo t h e rv a l u eo f α(0/lessorequalslantα< 1) lowers the symmetry of
the spin part to the Z
2×U(1) group, where Z2corresponds
to reflections of the type cγ↑→cγ↓and U(1) suggests the
invariance of the Hamiltonian under rotations around the z
quantization axis, ( cγ↑,cγ↓)→(eiφcγ↑,e−iφcγ↓).
In the DMFT analysis of magnetic ordering in the system
under study, we employ two types of solvers for the auxiliaryAnderson impurity problem. We use the continuous-timehybridization expansion quantum Monte Carlo (CT-HYB) im-purity solver provided via the
W2DYNAMICS software package
[33], which includes necessary generalizations introduced in
Refs. [ 12,15,20]. The second option, the exact diagonalization
(ED) solver is based on extensions discussed in Refs. [ 21,34].
The maximal number of effective bath orbitals per each orbitaland spin flavor in ED is limited in the present study to n
s=4
(form=2) and ns=3 (for m=3) [35]. Since the usedCT-HYB solver does not suffer from a sign problem [ 12], we
use its output for the resulting diagrams and graphical depen-dencies and the output from the ED solver as a supplementarysource supporting main observations.
For simplicity and general analysis purpose, it is sufficient
to restrict below to a semicircular density of states D(/epsilon1)=
(1/2πt
2)√
4t2−/epsilon12and to set the hopping amplitude tas
the scaling unit ( t≡tm=1) in all energy-related quantities
(e.g., the bandwidth for the noninteracting system thus be-comes W=4). Note that estimates for more realistic lattice
geometries can usually be performed by a proper rescaling ofquantities with respect to the coordination number zof the
actual lattice ( t→√
zt)[36]. For example, in simple cubic
(sc) lattice geometry with z=6, the energy-related parameter
P(e.g., critical temperature or interaction strength) becomes
Psc≈√
6PBethe. At weak coupling the rescaling holds only if
there are no Van Hove singularities close to the Fermi level,otherwise, the magnetic ordering tendency can be enhanced.At strong coupling ( U/greatermucht), the shape of the noninteracting
density of states is less relevant.
The FM instability is analyzed in two ways: by direct
measurements of uniform magnetizations (both with the CT-HYB and ED impurity solvers) and by analyzing uniform sus-ceptibilities in the symmetric phase with an external magneticfield (with the ED solver, see, e.g., Ref. [ 8]). Due to restricting
to the hybridization functions that are diagonal in orbital andspin indices, we do not have direct access in measurementsof other potentially competing (e.g., canted antiferromagnetic,Ruderman-Kittel-Kasuya-Yosida coupling, spin-orbit pairing,and excitonic) instabilities.
III. RESULTS
A. Degenerate orbitals
In Fig. 1we show the FM phase boundary as a function of
temperature and filling for several values of the parameter α.
The band filling and interactions parameters, U=12 and J=
U/4[37], fall to the Hund’s metal regime [ 38,39] with a sizable
dynamical mass enhancement. The most striking feature isthat in the case of a two-orbital system FM order completelyvanishes in the SU(2)-symmetric ( α=1) limit. This is in
contrast to previous expectations (based on DMFT analysiswith the density-density interactions only; see Ref. [ 5]) that
the spin-flip term has no strong effect on critical temperaturesin this parameter range. In fact, it does and, as we see fromcomparison with the three-band model, FM ordering is sensi-tive to the number of active orbitals. We attribute this effect tosuppression of quantum fluctuations with increasing orbitaldegeneracy, i.e., effective moments become more classical(see, e.g., Ref. [ 40]).
In Fig. 2, we compare the critical interaction parameters
at fixed temperature ( T=0.025) for the limiting cases α=0
andα=1. For the two-orbital model, we observe a strong
suppression of the FM phase by the spin-flip term. Thiseffect is significantly reduced in the three-orbital model. Thepronounced impact of band multiplicity agrees with the earlystudies on real materials [ 2]. Note that, similarly to arguments
for the single-band SU(2)-symmetric Hubbard model [ 3], the
results confirm absence of the FM instability at U<∞in
235157-2SUPPRESSION AND REVIV AL OF LONG-RANGE … PHYSICAL REVIEW B 97, 235157 (2018)
FIG. 1. Phase diagrams indicating evolution of FM phases under
the change of the Ising anisotropy parameter α(α=0,0.6,0.8,0.9,1
from back to front) at U=12 and J=3.
higher-symmetric (single-band) SU(2 m)-symmetric models,
corresponding to the J→0 limit in Fig. 2.
The form of the interaction, parameter α, affects not only
the FM phase boundaries, but also single-particle observ-ables in the paramagnetic (PM) regime such as the effectivemass and quasiparticle lifetime, relevant in Hund’s metalphysics [ 38,39,41]. Following Ref. [ 42], we use the poly-
nomial fit to the imaginary part of the self-energy /Sigma1(iω
n)
at the six lowest Matsubara frequencies to get the quasipar-ticle mass enhancement Z
−1=1−dIm[/Sigma1(iωn)]/dωn|ωn→0
and the quasiparticle scattering rate /Gamma1(the inverse lifetime)
FIG. 2. U-Jphase diagrams of the degenerate two-orbital (a) and
three-orbital (b) models in the low-temperature region ( T=0.025)
at the lattice filling n=1.5.FIG. 3. Dependencies of the mass enhancement and the quasi-
particle scattering rate (inverse lifetime) on the interaction strength
Ufor the m=3 model in the PM regime at n=2,J=U/4, and
three different temperatures, T=0.0125,0.025,0.05. The vertical
bars indicate the corresponding FM phase boundaries.
/Gamma1Z−1=− Im[/Sigma1(iωn)]|ωn→0. As shown in Fig. 3, the differ-
ence between α=0 and α=1 is maximized in the vicinity
of the FM( α=0) phase boundary. The results for m=3,
n=2, and α=1 agree well with the available quasiparticle
weights Zof Ref. [ 41]. The FM phase boundary in this
case matches well the one obtained for calcium ruthenateCaRuO
3[42].
B. Split orbitals
Next, we analyze the FM instability when the degeneracy
is lifted by a crystal field /Delta1=μm−μγ(γ=1,..., m −1),
which splits one orbital from the rest. The population n=
m−1+δis fixed so that for large Uand/Delta1>t γ(DE regime)
the lower (degenerate) band becomes half filled, while settingthe population of the split-off band to δ=0.2. First, we keep
all bandwidths equal, W
γ=Wm=4, and vary /Delta1. Later, we
will vary the bandwidth of the lower (degenerate) band for thefixed/Delta1.
In Fig. 4we show the FM phase boundaries as functions
of/Delta1andJ. Similar to Sec. III A , the full spin-rotational
symmetry ( α=1) favors the PM phase and the effect is less
pronounced with increasing number of bands. Two regionswith weak dependencies of J
con/Delta1are distinguishable with a
stepwise change between them. The “step” coincides with theonset of integer filling of the lower band. This suggests thatthe DE regime (at higher /Delta1) ends abruptly at this point. At low
/Delta1, the AFM phase is not stable for any J. The pair-hopping
term (α
/prime=1) has only a minor effect on phase boundaries (see
Fig. 4). The likely explanation is provided by a small number
of doubly occupied orbitals in the preset Hund’s couplingregime.
In the DE regime ( n
γ=1f o rγ<m andnm=δ), which
corresponds to the region /Delta1/greaterorsimilar1 andJ/greaterorsimilar1i nF i g . 4, the system
can be described by the ferromagnetic Kondo-lattice model
235157-3ANDRII SOTNIKOV , AGNIESZKA CICHY , AND JAN KUNE Š PHYSICAL REVIEW B 97, 235157 (2018)
FIG. 4. Left: Sketch of the energy states and orbital occupancies of the system with the depicted noninteracting densities of states and
the splitting /Delta1.C e n t e r : J-/Delta1phase diagrams of the two-orbital ( m=2, upper row) and three-orbital ( m=3, lower row) models at the lattice
fillings n=1.2a n dn=2.2, respectively. Right: the orbital occupancies nγ(obtained at α=1) along the vertical lines indicated by arrows on
the corresponding diagrams at J=2.25 (A, dashed lines) and J=5.5 (B, solid lines). The U/J ratio is kept fixed, U=4J,a n dT=0.025
everywhere.
with an additional AFM coupling between local moments
[43–47],
Heff=−m/summationdisplay
γ=1tγ/summationdisplay
/angbracketleftij/angbracketrightσc†
iγσcjγσ+JA/summationdisplay
/angbracketleftij/angbracketrightSi·Sj
−J/summationdisplay
ic†
imστσσ/primecimσ/prime·Si, (2)
where Siare the spin operators for the “localized” spins of
the size S=(m−1)/2,Si=1
2/summationtextm−1
γ=1c†
iγστσσ/primeciγσ/prime,τare
the 2×2 Pauli matrices, and /angbracketleftij/angbracketrightindicates summation over
nearest-neighbor lattice sites. In the strong-coupling limitfor lower orbitals ( U/greatermucht
γ) and nγ≈1, according to the
Schrieffer-Wolff transformations the AFM coupling amplitudescales as J
A∝t2
γ/U(γ<m )[48].
To show the importance of the AFM exchange processes
that are explicitly included in the present extension of the DEmodel ( 2), we perform an additional analysis, where the crystal
field is kept fixed ( /Delta1=1.2), while the bandwidths W
γ(γ<
m) are changed. Due to kinetic exchange mechanism, this is
sufficient to suppress the AFM coupling JA. This is supported
by the DMFT results in Fig. 5, which exhibit an enhancement
of the FM instability due to suppression of its main competitor,the AFM coupling, with the band asymmetry. As before, fromdirect comparison of α=0 and α=1 parametrizations we
observe a more pronounced effect of SU(2) spin symmetry inthe two-orbital model ( m=2) that decreases with inclusion
of an additional orbital flavor into the lower effective band(m=3).The behavior of the critical coupling U
con the band
asymmetry in Fig. 5can be understood as follows. Keeping
the itinerant band fixed, the AFM-FM transition at zero tem-perature and under restriction of collinear magnetic orderingis expected to happen at a particular value of J
A/J. Given that
J∝U(the ratio U/J is kept fixed) and JA∝t2
γ/U, we obtain
the result Uc∝tγ. The corresponding behavior is indicated by
FIG. 5. Dependencies of the critical interaction strengths Ucfor
FM ordering on the bandwidth asymmetry in two-orbital (left) and
three-orbital (right) Hubbard models at /Delta1=1.2,J=U/4,n=m−
1+0.2, and T=0.025. The star-shaped points at equal bandwidths
correspond to the same points indicated in the phase diagrams in
Fig.4. The linear fits (dashed lines) are based on the balance between
the effective FM and AFM couplings in Eq. ( 2); see text. The hopping
amplitude in the upper band is fixed to tm=1.
235157-4SUPPRESSION AND REVIV AL OF LONG-RANGE … PHYSICAL REVIEW B 97, 235157 (2018)
fits in Fig. 5that shows relatively good agreement with the
DMFT data that are obtained by considering the full model ( 1)
in the DE regime at nonzero temperature.
Figure 4shows that the DE description is not valid at low /Delta1.
The stepwise changes in positions of the FM phase boundaries,according to the panels shown on the right-hand side of Fig. 4,
are directly related to the change from the regime of partiallyoverlapping bands (with the metallic behavior of all orbitalflavors) to the DE limit (with the metallic behavior of onlyγ=mflavor). Note also that for the SU(2)-symmetric case and
m=3 (in contrast to m=2) it is possible to drive the system
directly from the FM to the AFM ordered state by changingonly the magnitude of the splitting /Delta1.
The regime of intermediate crystal fields is interesting
in several aspects. First, the ferromagnetism with partiallyoverlapping bands is realized in high-valence transition-metaloxides, such as SrCoO
3[49,50]. Second, this can be experi-
mentally studied with ultracold gases of alkaline-earth atomsin optical lattices (see Refs. [ 25–27]), where the occupations
of the split bands can be tuned independently, therefore, themagnitude of the splitting /Delta1can be changed in a wide range.
To keep the study consistent, we do not extend our current
description to larger degeneracies of orbitals, but the observedbehavior allows us to make a useful extrapolation from thepoint of view of solid-state realizations. For systems charac-terized by the electronic dorbitals in cubic perovskite crystal
structures (and in manganites, in particular), the typical bandasymmetry can be roughly estimated as W
eg/Wt2g≈2. Assum-
ing triple degeneracy of the lower ( t2g) and double degeneracy
of the higher ( eg) states and n=3+δ, the corrections to char-
acteristic critical values due to spin-flip processes are expectedto be still noticeable but, presumably, will not exceed 30%(this agrees, in particular, with the recent studies [ 10]). The
analyzed dependencies also suggest that the antiferromagneticcorrelations within the t
2gorbitals play an important role in
physics of manganites and related compounds, thus must beproperly accounted for.
IV . SUMMARY AND OUTLOOK
We studied the influence of spin symmetry (in particular,
presence of the spin-flip term) in the interaction part of theHamiltonian on the FM instability in the multiorbital Hubbardmodel by means of DMFT. We observe strong effects of sup-pression of FM phases when accounting for full spin-rotationalsymmetry in the two-orbital systems (in contrast to weakereffects for AFM ordering [ 19,21]). By considering the three-
orbital model, it is shown that these effects become weaker(i.e., FM ordering effectively revives) with an increase of thenumber of active orbitals that agrees well with arguments basedon suppression of quantum fluctuations due to approachingthe limit of classical spins. The analysis was extended to thecase of split orbitals, where the corresponding transition fromthe Stoner to the double-exchange regime of FM ordering isobserved, but the suppression effect originating from inclusionof the spin-flip processes remains significant.
The applied approach was restricted to measurements of
observables diagonal in spin and orbital indices. Therefore,a number of instabilities with more complex structure (e.g.,possible canted AFM ordering in the DE regime) were notFIG. 6. Magnetization as a function of the interaction strength U
for the two-orbital Hubbard model at n=1.5,J=U/4, and T=
0.025. In the DMFT results obtained with the ED solver, nsdenotes
the number of effective bath orbitals per each orbital and spin flavor.
studied directly. In view of recent developments in DMFT
schemes with corresponding extensions [ 7,51,52], research
directions aiming to obtain more detailed phase diagrams inthe regimes under study look realistic.
The results for the two-band model are also important from
the viewpoint of the ultracold-atom experiments focused onapproaching the ferromagnetic Kondo-lattice regime in opticallattices. Preliminary estimates for different parametrizationsof interactions, U/lessmuchJ/lessorsimilarU
/prime, i.e., different from the cur-
rent study but closer to the experimentally accessible values[25,27], show that the influence of spin-flip terms remains
crucial for a determination of critical parameters for FMordering. It is also interesting to study the influence of quantumfluctuations in two-orbital models with SU( N)-symmetric
interactions, where magnetic ordering is expected to be sup-pressed with an increase of N(number of pseudospin flavors),
in contrast to the studied direction of SU(2)-symmetric inter-actions and increasing m(number of active orbitals), where
FM phases become stabilized with m.
ACKNOWLEDGMENTS
The authors thank A. Golubeva, A. Hariki, and P. van
Dongen for fruitful discussions. We highly appreciate technicalassistance by A. Hausoel, P. Gunacker, and G. Sangiovanniwith the
W2DYNAMICS package. A.S. and J.K. acknowledge
funding of this work from the European Research Council(ERC) under the European Union’s Horizon 2020 researchand innovation program (Grant Agreement No. 646807-EXMAG). A.C. acknowledges funding of this work by theNational Science Centre (NCN, Poland) under Grant No.UMO-2017/24/C/ST3/00357. Access to computing and stor-age facilities provided by the Vienna Scientific Cluster (VSC)is greatly appreciated. The calculations were also performed atthe Poznan Supercomputing and Networking Center (EAGLEcluster).
APPENDIX: MAGNETIZATION BEHA VIOR IN THE
LOW-TEMPERATURE REGIME
In Fig. 6we provide an explicit comparison of the magneti-
zations M=/summationtext
γ(nγ↑−nγ↓) for two parametrizations ( α=0
andα=1) obtained by DMFT with different (CT-HYB and
ED) impurity solvers. Both parametrizations of interactions
235157-5ANDRII SOTNIKOV , AGNIESZKA CICHY , AND JAN KUNE Š PHYSICAL REVIEW B 97, 235157 (2018)
result in a fully polarized FM state in the limit of large inter-
actions. Compared to more accurate data from the CT-HYBsolver, the depicted dependencies also show the limitations ofthe ED solver due to the finite number of effective bath orbitalsin the Anderson impurity model and indicate relatively goodagreement at n
s=4.At higher temperatures, the magnetization curves become
smoothed out, i.e., the magnitude of the ordered momentsdecreases in the FM phases at the given values of UandJ.
Qualitatively similar magnetization behavior is observed in thepresence of the energy splitting /Delta1between orbitals and in the
three-orbital Hubbard model.
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235157-6 |
PhysRevB.82.024515.pdf | Triplet contribution to the Josephson current in the nonequilibrium
superconductor/ferromagnet/superconductor junction
I. V. Bobkova and A. M. Bobkov
Institute of Solid State Physics, Chernogolovka, Moscow reg., 142432 Russia
/H20849Received 15 February 2010; revised manuscript received 28 May 2010; published 21 July 2010 /H20850
The Josephson current through a long s-wave superconductor/weak ferromagnet/ s-wave superconductor
weak link is studied theoretically in the regime of nonequilibrium spin-dependent occupation of electron statesin the ferromagnetic interlayer. While under the considered nonequilibrium conditions, the standard supercur-rent, carried by the singlet part of current-carrying density of states, practically is not modified, the additionalsupercurrent flowing via the triplet part of the current-carrying density of states appears. Depending on voltage,controlling the particular form of spin-dependent nonequilibrium in the interlayer, this additional current canenhance or reduce the usual current of the singlet component and also switch the junction between 0 and
/H9266
states.
DOI: 10.1103/PhysRevB.82.024515 PACS number /H20849s/H20850: 74.45. /H11001c, 74.50. /H11001r
New states have been predicted and observed in Joseph-
son weak links in the last years. One of them has commonorigin with the famous Larkin-Ovchinnikov-Fulde-Ferrel/H20849LOFF /H20850state.
1,2This mesoscopic LOFF state, which is in-
duced in superconductor/weak ferromagnet/superconductor
/H20849SFS /H20850Josephson junction, was predicted theoretically3,4and
observed experimentally.5–8In this state, Cooper pair ac-
quires the total momentum 2 Qor −2 Qinside the ferromag-
net as a response to the energy difference between the twospin directions. Here Q/H11008h/
vf, where his an exchange en-
ergy and vfis the Fermi velocity. Combination of the two
possibilities results in the spatial oscillations of the conden-sate wave function /H9023/H20849x/H20850in the ferromagnet along the direc-
tion normal to the SF interface. /H9023
s/H20849x/H20850/H11008cos/H208492Qx/H20850for the sin-
glet Cooper pair.9The same picture is also valid in the
diffusive limit. The only thing we need to add is an extradecay of the condensate wave function due to scattering. Inthe regime h/H11271/H20841/H9004/H20841, where /H9004is a superconducting order pa-
rameter in the leads, the decay length is equal to the mag-
netic coherence length
/H9264F=/H20881D/hwhile the oscillation period
is given by 2 /H9266/H9264F. Here Dis the diffusion constant in the
ferromagnet, /H6036=1 throughout the paper.
The presence of an exchange field also leads to the for-
mation of the triplet component of the condensate wavefunction in the interlayer. In the case of a homogeneous ex-change field, only the component with zero-spin projectionon the field direction S
z=0 is induced. Combining the two
pairs with the total momenta 2 Qand −2 Qinto the triplet
combination, we see that the in-plane /H20849Sz=0/H20850triplet conden-
sate wave-function component /H9023t/H20849x/H20850/H11008sin/H208492Qx/H20850, that is, os-
cillates in space with the same period as the singlet one but isshifted by
/H9266/2 with respect to it. Here we do not discuss the
other triplet components with Sz=/H110061, which are typically
induced in case of inhomogeneous magnetization.10
The energy spectrum of the superconducting correlations
is expressed in a so-called supercurrent-carrying density ofstates /H20849SCDOS /H20850.
11–14This quantity represents the density of
states weighted by a factor proportional to the current thateach state carries in a certain direction. Under equilibriumconditions, the supercurrent can be expressed via the SCDOSas
13j/H11008/H20885d/H9255Nj/H20849/H9255/H20850tanh/H9255/2T, /H208491/H20850
where /H9255stands for the quasiparticle energy, /H9272/H20849/H9255/H20850
=tanh /H9255/2Tis the equilibrium distribution function, and
Nj/H20849/H9255/H20850is SCDOS. In the presence of spin effects, SCDOS
becomes a matrix 2 /H110032 in spin space and can be represented
asNˆj=Nj,s+Nj,t/H9268, where /H9268iare Pauli matrices in spin space.
The scalar in spin space part of SCDOS Nj,sis referred to as
the singlet part of SCDOS in the paper and the vector partN
j,tis referred to as the triplet part. Nj,tis directly propor-
tional to the triplet part of the condensate wave function. It iswell known that the spin supercurrent cannot flow throughthe singlet superconducting leads. Therefore, N
j,tdoes not
contribute to the supercurrent in equilibrium. Having in mindthat the triplet part of SCDOS is even function of quasipar-ticle energy, one can directly see that this is indeed the case.
Under nonequilibrium conditions, one can change the
value of the critical Josephson current through the junctionand even realize the
/H9266state by manipulating the quasiparticle
distribution in an interlayer region. This effect was predictedtheoretically
11,12and observed experimentally15–17for a dif-
fusive superconductor/normal metal/superconductor /H20849SNS /H20850
junction. The point is that positive and negative parts of SC-DOS give energy-dependent contributions to the supercur-rent in the positive and negative direction. The size and di-rection of the total supercurrent depends therefore on theoccupied fraction of these states, which is analogous to theoccupation of the discrete Andreev bound states in a ballisticsystem. That is, one can obtain negative Josephson currentresponse to small phase differences and, hence, switch thesystem into
/H9266state by creating an appropriate nonequilib-
rium quasiparticle distribution in the weak link region. Thecombination of the exchange field h/H11270/H20841/H9004/H20841and the spin-
independent nonequilibrium distribution function has beenconsidered as well.
18Under these conditions, the influence of
the nonequilibrium distribution function is also consists ofthe redistribution of the quasiparticles between the energylevels. However, it was shown that in this limit of smallexchange fields, the combined effect of the exchange fieldPHYSICAL REVIEW B 82, 024515 /H208492010 /H20850
1098-0121/2010/82 /H208492/H20850/024515 /H208496/H20850 ©2010 The American Physical Society 024515-1and the nonequilibrium distribution function is nontrivial.
For instance, part of the field-suppressed supercurrent can berecovered by adjusting a voltage between additional elec-trodes, which controls the distribution function.
In the present paper, we investigate the effect of nonequi-
librium occupation of the supercurrent-carrying states on theJosephson current in SFS junction in the parameter range/H20841/H9004/H20841/H11270h/H11270/H9255
f, where /H9255fis the Fermi energy of the ferromagnet.
This regime is relevant to weak ferromagnetic alloys, which
were used for the experimental realization of magnetic /H9266
junctions. It is shown that if the distribution function be-comes nonequilibrium and spin dependent, the supercurrentcarried by the SCDOS triplet component N
j,tin the ferro-
magnet is nonzero. The magnitude of this current contribu-tion j
tcan be of the same order or even larger than the
current contribution jscarried by the singlet component Nj,s.
Due to the fact that the singlet and triplet components of theanomalous Green’s function have the same oscillation periodbut shifted in phase by
/H9266/2,jtcan increase the usual super-
current, carried by the singlet component of SCDOS, orweaken it, or even reverse the sign of the total supercurrent,thus switching between 0 and
/H9266states. Experimentally, the
most probable way to realize the spin-dependent nonequilib-rium in the interlayer is to apply a voltage to /H20849or to pass the
dissipative current through /H20850a spin-active material. Then un-
der appropriate conditions, even quite small voltages shouldbe enough to switch the system from 0 to
/H9266state and vice
versa.
That is, we consider another mechanism of supercurrent
manipulation by creating a nonequilibrium quasiparticle dis-tribution in the interlayer, which cannot be reduced to theredistribution of the quasiparticles between the energy levels.In principle, the both mechanisms can be realized in a junc-tion simultaneously. However, in this particular study we as-sume h/H11271/H9004 and the Thouless energy /H9255
Th=D/d2/H11270h, that is
the interlayer length d/H11271/H9264F. As it is shown below, in this
regime /H20849and for the case of low-transparency SF interfaces /H20850a
spin-independent nonequilibrium distribution of quasiparti-cles in the ferromagnet practically does not affect the Joseph-son current, that is the described above mechanism of thecritical current reversal by the spin-independent redistribu-tion of supercurrent-carrying states population is irrelevant inthis case. For high-transparency SF interfaces, the bothmechanisms contribute to supercurrent.
For a quantitative analysis, we use the formalism of qua-
siclassical Green-Keldysh functions in the diffusive limit.
19
The fundamental quantity for diffusive transport is the mo-mentum average of the quasiclassical Green’s function
gˇ/H20849x,/H9255/H20850=/H20855gˇ/H20849p
f,x,/H9255/H20850/H20856pf.I ti sa8 /H110038 matrix form in the product
space of Keldysh, particle-hole, and spin variables. Here xis
the coordinate measured along the normal to the junction.
The electric current should be calculated via Keldysh part
of the quasiclassical Green’s function. For the plane diffusivejunction, the corresponding expression reads as follows:
j=−d
eRF/H20885
−/H11009+/H11009d/H9255
8/H92662Tr4/H20877/H92700+/H92703
2/H20851gˇ/H20849x,/H9255/H20850/H11509xgˇ/H20849x,/H9255/H20850/H20852K/H20878, /H208492/H20850
where eis the electron charge and RFstands for the resis-
tance of the ferromagnetic region. /H20851gˇ/H20849x,/H9255/H20850/H11509xgˇ/H20849x,/H9255/H20850/H20852Kis 4/H110034 Keldysh part of the corresponding combination of full
Green’s functions. /H9270iare Pauli matrices in particle-hole
space.
It is convenient to express Keldysh part of the full Green’s
function via the retarded and advanced components and the
distribution function: gˇK=gˇR/H9272ˇ−/H9272ˇgˇA. Here the argument
/H20849x,/H9255/H20850of all the functions is omitted for brevity. The distribu-
tion function is diagonal in particle-hole space: /H9272ˇ=/H9272ˆ/H20849/H92700
+/H92703/H20850/2+/H92682/H9272˜ˆ/H92682/H20849/H92700−/H92703/H20850/2. All the matrices denoted byˆare
2/H110032 matrices in spin space throughout the paper. In terms of
the distribution function, current /H208492/H20850takes the form
j=−d
eRF/H20885
−/H11009+/H11009d/H9255
8/H92662Tr2/H20851−/H92662/H11509x/H9272ˆ−gˆR/H11509x/H9272ˆgˆA−fˆR/H11509x/H9272˜ˆf˜ˆA
+/H20849gˆR/H11509xgˆR+fˆR/H11509xf˜ˆR/H20850/H9272ˆ−/H9272ˆ/H20849gˆA/H11509xgˆA+fˆA/H11509xf˜ˆA/H20850/H20852. /H208493/H20850
We assume that the direction of the exchange field his spa-
tially homogeneous and choose the quantization axis alongthe field. In this case, the distribution function and the nor-
mal part gˆ
R,Aof the Green’s function are diagonal matrices in
spin space. The anomalous Green’s functions can be repre-
sented as fˆR,A=fˆ
dR,Ai/H92682andf˜ˆR,A=−i/H92682f˜ˆ
dR,A, where fˆ
dR,Aandf˜ˆ
dR,A
are diagonal in spin space.
The retarded and advanced Green’s functions are obtained
by solving the Usadel equations19supplemented with
Kupriyanov-Lukichev boundary conditions at SFinterfaces.
20It is worth to note that we can safely apply these
boundary conditions to the problem of plane diffusive junc-tion even for high enough dimensionless conductance gof
the SF interface. This can be done in spite of the fact thatthey are the linear in transparency approximation of moregeneral Nazarov boundary conditions.
21The reason is that
the effective number of interface channels N/H11011dy/lis large
and a separate channel transparency T/H11011g/H20849l/dy/H20850is usually
considerably less than unity. Here dyis the junction width
andlis the mean-free path.
Further, the condition d/H11271/H9264Fallows us to find the solution
analytically even for an arbitrary SF-interface transparencyand low temperature, that is in the parameter region, wherethe equations cannot be linearized with respect to the anoma-lous Green’s function. We start from the completely incoher-ent junction /H20849that is, consider the left and right SF interfaces
separately /H20850and then calculate the corrections up to the first
order of the small parameter exp /H20851−d/
/H9264F/H20852to the Green’s func-
tions. Within this accuracy, the anomalous Green’s functionsin the vicinity of left and right SF interfaces /H20849atx=/H11007d/2/H20850
take the form
f
d/H9268R,A=/H9260i/H9266/H20851sinh/H9008/H9268R,Ae−i/H9251/H9273/2+4/H9018/H9268R,A/H20849x/H20850ei/H9251/H9273/2/H20852,
f˜
d/H9268R,A=−fd/H9268R,A/H20849/H9273→−/H9273/H20850. /H208494/H20850
Here/H9268=↑,↓/H20851or +1 /H20849−1/H20850within equations /H20852is the electron-
spin projection on the quantization axis, /H9260=+1 /H20849−1/H20850corre-
sponds to the retarded /H20849advanced /H20850functions, /H9251=+1 /H20849−1/H20850in
the vicinity of the left /H20849right /H20850SF interface, and /H9273is the order-
parameter phase difference between the superconductingleads. The first term represents the anomalous Green’s func-I. V. BOBKOVA AND A. M. BOBKOV PHYSICAL REVIEW B 82, 024515 /H208492010 /H20850
024515-2tion at the ferromagnetic side of the isolated SF boundary
and does not enter the following results. So, we do not giveit explicitly. The second term is the first-order correction,originated from the anomalous Green’s function extendedfrom the other SF interface,
/H9018
/H9268R,A/H20849x/H20850=K/H9268R,Ae−/H20849d/2−/H9251x/H20850/H208491+i/H9260/H9268/H20850//H9264F, /H208495/H20850
where K/H9268R,Ais determined by the equation
/H208491+i/H9260/H9268/H20850K/H9268R,A/H208491−K/H9268R,A2/H20850
=RF/H9264F
4Rgd/H20851sinh/H9008sR,A/H208491+K/H9268R,A2+K/H9268R,A4/H20850
− cosh /H9008sR,A4K/H9268R,A/H208491+K/H9268R,A2/H20850/H20852. /H208496/H20850
Rgstands for the resistance of each SF interface, which are
supposed to be identical. sinh /H9008sR,Aand cosh /H9008sR,Aoriginate
from the Green’s functions at the superconducting side of SFinterfaces. We assume that the parameter
/H20849R
F/H9264s/Rgd/H20850/H20849/H9268F//H9268s/H20850/H112701, where /H9264s=/H20881D//H9004is the supercon-
ducting coherence length in the leads, /H9268Fand/H9268sstand for
conductivities of ferromagnetic and superconducting materi-als, respectively. It allows us to neglect the suppression ofthe superconducting order parameter in the S leads near theinterface and take the Green’s functions at the superconduct-ing side of the boundaries to be equal to their bulk values. Inthis case,
cosh/H9008
sR,A=−/H9260i/H9255
/H20881/H20841/H9004/H208412−/H20849/H9255+/H9260i/H9254/H208502,
sinh/H9008sR,A=−/H9260i/H20841/H9004/H20841
/H20881/H20841/H9004/H208412−/H20849/H9255+/H9260i/H9254/H208502, /H208497/H20850
where /H9254is a positive infinitesimal.
The SCDOS, entering the current /H208493/H20850takes the form
Nj/H9268/H20849/H9255/H20850=/H20849gˆR/H11509xgˆR+fˆR/H11509xf˜ˆR−gˆA/H11509xgˆA−fˆA/H11509xf˜ˆA/H20850/H9268
=8/H92662sin/H9273
/H9268FRgIm/H20875/H9018/H9268R/H20873x=−/H9251d
2/H20874sinh/H9008sR/H20876. /H208498/H20850
The nonequilibrium distribution function in the interlayer is
proposed to be created by applying a voltage along the y
direction between two additional electrodes NbandNt, which
are attached to the central part of the interlayer. It is sup-posed that the conductances of N
bF and NtF interfaces are
spin dependent and equal to gb/H9268andgt/H9268, respectively. The
voltage Vt/H9268/H20849Vb/H9268/H20850between the superconducting leads and
Nt/H20849Nb/H20850electrode can also be spin dependent. It can be real-
ized, for example, by attaching one /H20849or both /H20850of the elec-
trodes NborNtto a strong ferromagnet and applying a volt-
age between the other one and the ferromagnet.
In order to obtain the distribution function up to the first
order of the parameter exp /H20851−d//H9264F/H20852, we solve the kinetic
equation for the distribution function, which is derived fromthe Keldysh part of the Usadel equation. The boundary con-ditions to the kinetic equation are also obtained from theKeldysh part of the general Kypriyanov-Lukichev boundaryconditions. Further it is assumed that /H20841eV
t,b↑,↓/H20841/H11021/H20841/H9004/H20841. Underthis condition, the part of the current associated with the first
three terms in Eq. /H208493/H20850is only determined by the first-order
correction /H9018/H9268R,Ato the anomalous Green’s function and the
distribution function /H9272/H9268/H208490/H20850, calculated up to the zero order of
the parameter exp /H20851−d//H9264F/H20852,
/H20849−/H92662/H11509x/H9272ˆ−gˆR/H11509x/H9272ˆgˆA−fˆR/H11509x/H9272˜ˆf˜ˆA/H20850/H9268=Nj/H9268/H9272˜/H9268/H208490/H20850−/H9272/H9268/H208490/H20850
2, /H208499/H20850
where Nj/H9268/H20849/H9255/H20850is expressed by Eq. /H208498/H20850.
The distribution function /H9272/H208490/H20850does not depend on x. So, it
is convenient to calculate it in the middle of the interlayer,where disregarding the parameter exp /H20851−d/
/H9264F/H20852means disre-
garding the superconducting proximity effect. Then, by con-sidering N
b/F/Ntjunction and applying Kypriyanov-
Lukichev boundary conditions Nb/FandF/Ntinterfaces, we
come to the following expression for /H9272/H208490/H20850/H20849inelastic-
scattering processes are not taken into account /H20850:
/H9272/H9268/H208490/H20850=tanh/H9255−eVt/H9268
2Tgt/H9268/H20849/H9268F+dygb/H9268/H20850
/H9268F/H20849gt/H9268+gb/H9268/H20850+2dygt/H9268gb/H9268
+tanh/H9255−eVb/H9268
2Tgb/H9268/H20849/H9268F+dygt/H9268/H20850
/H9268F/H20849gt/H9268+gb/H9268/H20850+2dygt/H9268gb/H9268, /H2084910/H20850
/H9272˜/H9268/H208490/H20850is connected to /H9272/H9268/H208490/H20850by the symmetry relation /H9272˜↑,↓/H208490/H20850/H20849/H9255/H20850=
−/H9272↓,↑/H208490/H20850/H20849−/H9255/H20850. We focus on the case gt/H9268/H11270/H9268F/dyorgb/H9268/H11270/H9268F/dy,
when the ydependence of the distribution function /H9272/H208490/H20850can
be disregarded.
Substituting Eqs. /H208498/H20850and /H208499/H20850into Eq. /H208493/H20850, we find that the
Josephson current takes the form
j=−d
2eRF/H20885d/H9255
8/H92662/H20858
/H9268/H20851/H20849/H9272/H9268/H208490/H20850+/H9272˜/H9268/H208490/H20850/H20850Nj/H9268/H20852, /H2084911/H20850
where Nj/H9268/H20849/H9255/H20850is expressed by Eq. /H208498/H20850and/H9272/H9268/H208490/H20850should be
taken from Eq. /H2084910/H20850. Electrical current /H2084911/H20850can be divided
into two parts,
j=js+jt,
js=js,csin/H9273=−d
eRF/H20885d/H9255
8/H92662/H20851/H20849/H92720/H208490/H20850+/H9272˜0/H208490/H20850/H20850Nj,s/H20852,
jt=jt,csin/H9273=−d
eRF/H20885d/H9255
8/H92662/H20851/H20849/H9272z/H208490/H20850+/H9272˜z/H208490/H20850/H20850Nj,t/H20852, /H2084912/H20850
where /H92720=/H20849/H9272↑+/H9272↓/H20850/2,/H9272z=/H20849/H9272↑−/H9272↓/H20850/2,/H9272˜0,z=/H20849/H9272˜↑/H11006/H9272˜↓/H20850/2
=/H11007/H92720,z/H20849−/H9255/H20850,Nj,s=/H20849Nj↑+Nj↓/H20850/2 is the singlet part of SC-
DOS, and Nj,t=/H20849Nj↑−Nj↓/H20850/2 is the zcomponent of the SC-
DOS triplet part /H20849the other components equal to zero for the
considered case of homogeneous magnetization /H20850. It is seen
from Eq. /H2084912/H20850thatNj,tgives rise to the additional contribu-
tion to the spinless electrical current if the quasiparticle dis-tribution is spin dependent.
While Eqs. /H2084911/H20850and /H2084912/H20850are valid for arbitrary SF-
interface transparency, at first we concentrate on the discus-
sion of the tunnel limit g
˜/H11013RF/H9264F/Rgd/H112701, where Eq. /H208496/H20850canTRIPLET CONTRIBUTION TO THE JOSEPHSON CURRENT … PHYSICAL REVIEW B 82, 024515 /H208492010 /H20850
024515-3be easily solved and the integral over energy can be calcu-
lated analytically. For the analytical analysis, we choose themost simple form for the distribution function Eq. /H2084910/H20850by
setting g
t/H9268→0 and Vb↓=0. Then /H9272↑/H208490/H20850=tanh /H20851/H20849/H9255−eVb↑/H20850/2T/H20852
while /H9272↓/H208490/H20850=tanh /H20851/H9255/2T/H20852. As it is demonstrated below, the re-
sults corresponding to another set of parameters, which de-termines the particular form of the distribution function sat-isfying Eq. /H2084910/H20850, qualitatively do not differ from that ones
represented here. At low temperature T/H11270/H20841eV
b↑/H20841, we obtain
for the Josephson current the following result:
j=RF/H9264Fsin/H9273
4eRg2d/H20841/H9004/H20841e−d//H9264F/H20875/H208812/H9266cos/H20873d
/H9264F+/H9266
4/H20874
+1
/H208812log/H20879/H20841/H9004/H20841+eVb↑
/H20841/H9004/H20841−eVb↑/H20879sin/H20873d
/H9264F+/H9266
4/H20874/H20876. /H2084913/H20850
The first term in Eq. /H2084913/H20850represents the part of the supercur-
rent jscarried by the singlet component of SCDOS. Under
the conditions T/H11270/H20841eVt,b/H20841and /H20841eVt,b↑,↓/H20841/H11021/H20841/H9004/H20841, it is not affected
by the fact that the distribution function is nonequilibrium, as
can be seen from Eq. /H2084913/H20850. The reason is that for a long
junction and h/H11271/H20841/H9004/H20841in the tunnel limit, the singlet part of
SCDOS is concentrated in the narrow energy intervalsaround /H9255=/H11006/H20841/H9004/H20841, as it is illustrated in panel /H20849a/H20850of Fig. 1. This
is opposed to the cases of diffusive SNS junction with h=0
/H20849Refs. 11–14/H20850and SFS junction with h/H11270/H9004,
18where the sin-
glet part of SCDOS is finite and exhibits nontrivial energydependence in the subgap energy region /H20841/H9255/H20841/H11021/H20841/H9004/H20841. Under the
conditions T/H11270/H20841eV
t,b/H20841and /H20841eVt,b↑,↓/H20841/H11021/H20841/H9004/H20841, the distribution func-
tion /H92720/H208490/H20850+/H9272˜0/H208490/H20850/H11015/H20849sgn/H20851/H9255−eVb↑/H20852+sgn /H20851/H9255+eVb↑/H20852+2sgn /H20851/H9255/H20852/H20850/2,
which enters the expression for js/H20851Eq. /H2084912/H20850/H20852, practically does
not differ from its equilibrium value for energy intervalsaround /H9255=/H11006/H20841/H9004/H20841. That is, under the considered conditions,
the widely discussed in the literature mechanism of supercur-rent manipulation by nonequilibrium redistribution of quasi-particles between energy levels
11,12,15–18does not contribute.
The second term jtis caused by the triplet component of
SCDOS and vanishes in the equilibrium Vb↑=0. As it is seen
in panel /H20849b/H20850of Fig. 1, the triplet part of SCDOS Nj,tis an
even function of energy and has finite value in the subgapenergy region. So, multiplying it by the distribution function,/H9272z/H208490/H20850+/H9272˜z/H208490/H20850/H11015/H20849sgn/H20851/H9255−eVb↑/H20852− sgn /H20851/H9255+eVb↑/H20852/H20850/2, /H2084914/H20850
one obtains current contribution jt. The absolute value of this
contribution is roughly proportional to Vb↑for small enough
values of this parameter and increases sharply when Vb↑ap-
proaches /H20841/H9004/H20841. This behavior is a consequence of two facts: /H20849i/H20850
distribution function /H2084914/H20850is a constant within energy interval
/H20851−eVb↑,eVb↑/H20852and vanishes outside it, /H20849ii/H20850the triplet part of
SCDOS has a particular shape shown in panel /H20849b/H20850of Fig. 1.
To calculate the Josephson current for the case of arbitrary
transparency of SF interfaces, one needs to solve Eq. /H208496/H20850
numerically and make use of Eq. /H2084911/H20850. The resulting curves
as functions of eVb↑are plotted in Fig. 2. Panel /H20849a/H20850shows the
current for low enough dimensionless conductance of SF in-
terface g˜=0.1 while panel /H20849b/H20850represents the case of highly
transparent interface g˜=3.0. Different curves correspond to
different lengths dof the junction. In dependence on Vb↑, the
current can be enhanced or reduced with respect to its valueatV
b↑=0. If the length of the equilibrium junction is not far
from the 0- /H9266transition, then small enough voltage can-2 -1 0 1 2-0.0100.010.02-2 -1 0 1 2-0.03-0.02-0.0100.010.020.030.04
-2 -1 0 1 2-0.0200.020.040.060.08-2 -1 0 1 2-0.1-0.0500.050.1 (a)
(b)(c)
(d)Nj,s(ε)
Nj,s(ε)Nj,t(ε)
Nj,t(ε)˜g=0.1
d=3.4ξF
˜g=0.1
d=3.4ξF˜g=3.0
d=3.6ξF
˜g=3.0
d=3.6ξFε/∆
ε/∆ε/∆
ε/∆
FIG. 1. The singlet and triplet parts of SCDOS in dependence on the quasiparticle energy. The left column corresponds to the low-
transparency limit g˜=0.1 while the right column represents the high-transparency case g˜=3.0.
-1 -0.5 0 0.5 1-0.00100.0010.0020.0030.004
eVb↑/∆jc|e|Rg/∆
˜g=0.1
5.04.33.963.73.4(a)
-1 -0.5 0 0.5 100.010.020.03
eVb↑/∆˜g=3.0jc|e|Rg/∆
5.24.84.464.13.6 (b)
FIG. 2. The critical Josephson current as a function of eVb↑//H9004.
The different curves correspond to different lengths dof the junc-
tion, which are measured in units of /H9264F.I. V. BOBKOVA AND A. M. BOBKOV PHYSICAL REVIEW B 82, 024515 /H208492010 /H20850
024515-4switch between the states. Separate plots of the current con-
tributions jsandjtare represented in panel /H20849a/H20850of Fig. 3for
the low-transparency junction with g˜=0.1 and in panel /H20849b/H20850
for the high-transparency junction with g˜=3.0. It is seen, that
for low-transparency junction, the current and its separatecontributions j
sandjtbehave just as described by the tunnel
limit discussed above. While the tunnel limit qualitativelycaptures the essential physics for the high-transparency junc-tion as well, there are some new features, which are dis-cussed below.
The singlet part of SCDOS for the case of high-
transparency junction is plotted in panel /H20849c/H20850of Fig. 1.I ti sclearly seen that as distinct from the limit of tunnel junction,
it is not only concentrated around superconducting gap edgesbut is finite in the whole subgap region. It results in thesensitivity of j
sto the fact that the distribution function
/H92720/H208490/H20850+/H9272˜0/H208490/H20850is nonequilibrium. In other words, upon increasing
of the SF-interface conductance the mechanism of the cur-rent controlling based on the redistribution of the quasiparti-cles between energy levels starts to contribute. It is seenfrom panel /H20849b/H20850of Fig. 3that the difference j
s,c/H20849Vb↑/H20850
−js,c/H20849Vb↑=0/H20850considerably grows when /H20841eVb↑/H20841approaches
/H20841/H9004/H20841. It is worth to note that the absolute value of js,cis always
reduced by this mechanism. This fact leads to the reductionin the absolute value of the high-transparency total supercur-rent /H20851see Fig. 2/H20849b/H20850/H20852ateV
b↑→/H9004. This behavior should be
compared to the tunnel limit, where the dependence of js,con
Vb↑is negligible and absolute value of the total supercurrent
only increases for eVb↑→/H9004due to growing contribution of
jt,c.
Now we discuss the dependence of the obtained results on
the particular shape of the distribution function. It is illus-trated in Fig. 4. Four different examples of the distribution
function zcomponent
/H9272z/H208490/H20850+/H9272˜z/H208490/H20850, satisfying Eq. /H2084910/H20850are
shown in the right panel. Due to symmetry relations, thiscombination is always an even function of energy. The scalarpart of the distribution function is not plotted in the figurebecause it does not influence the supercurrent in the tunnellimit. The corresponding plots of the critical current in thelow-transparency limit are represented in the left panel. Itcan be concluded that, while the quantitative value of thecritical current depends on the particular choice of the distri-bution function, this choice has no qualitative effect on thesupercurrent behavior, as it was already pointed before.
In conclusion, we have studied the effects of nonequilib-
rium spin-dependent electron distribution in a weakly ferro-magnetic interlayer on the Josephson current through SFSjunction. It is shown that the nonequilibrium spin-dependentelectron distribution gives rise to the supercurrent carried by
-1 -0.5 0 0.5 1-0.00100.0010.0020.0030.0040.005
-2 -1 0 1 211
223
3 4
4jc|e|Rg/∆
ϕz+˜ϕz ˜g=0.1
d=3.4ξF
eVb,↑/∆ε/∆
FIG. 4. Dependence of the critical current on the particular shape of the distribution function. The right panel represents four different
examples of the distribution function vector part. The curves marked by the numbers from 1 to 4 correspond to the following sets ofparameters in Eq. /H2084910/H20850:1 .g
t/H9268→0 and Vb↓=0; 2. gt↑,gb↓→0 and eVt↓=0.8/H9004;3 .gt↑,gb↓→0 and eVt↓=−0.8/H9004, and 4. gt↑=0.1/H9268F/dy,gt↓
=4/H9268F/dy,gb↑=3/H9268F/dy,gb↓=0.2/H9268F/dy,Vb↓=0.5 Vb↑,eVt↑=0.35/H9004, and eVt↓=0.7/H9004. The respective critical current plots in dependence on
eVb↑, calculated for the same sets 1–4 of the distribution function parameters are shown in the left panel. The dashed vertical line marks the
position eVb↑=0.2/H9004, for which the distribution functions are calculated.-1 -0.5 0 0.5 1-0.002-0.00100.0010.0020.0030.004
(a)jc|e|Rg/∆ ˜g=0.1
d=3.4ξF
eVb,↑/∆
-1 -0.5 0 0.5 1-0.0100.010.020.03(b)jc|e|Rg/∆
˜g=3.0
d=3.6ξF
eVb,↑/∆
FIG. 3. Separate plots of the current contributions js,c/H20849dashed
line /H20850and jt,c/H20849dotted line /H20850together with the total critical current
js,c+jt,c/H20849solid line /H20850as a function of eVb↑//H9004. Panel /H20849a/H20850represents the
low-transparency case and panel /H20849b/H20850corresponds to high-
transparency case.TRIPLET CONTRIBUTION TO THE JOSEPHSON CURRENT … PHYSICAL REVIEW B 82, 024515 /H208492010 /H20850
024515-5the triplet component of SCDOS. Depending on voltage,
controlling the particular form of spin-dependent nonequilib-rium in the interlayer, this additional current can enhance orreduce the usual current of the singlet component and alsoswitch the junction between 0 and
/H9266states.We gratefully acknowledge discussions with V. V. Ryaza-
nov, A. S. Melnikov, and M. A. Silaev. The support by RFBRunder Grant 09-02-00779 and the programs of Physical Sci-ence Division of RAS is acknowledged. A.M.B. was alsosupported by the Russian Science Support Foundation.
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024515-6 |
PhysRevB.86.075455.pdf | PHYSICAL REVIEW B 86, 075455 (2012)
Catalytic growth of N-doped MgO on Mo(001)
Martin Grob,*Marco Pratzer, and Markus Morgenstern
II. Institute of Physics B and JARA-FIT, RWTH Aachen University, D-52074 Aachen, Germany
Marjana Le ˇzai´c
Peter-Gr ¨unberg Institute, Forschungszentrum J ¨ulich and JARA, D-52425 J ¨ulich, Germany
(Received 21 February 2012; revised manuscript received 18 July 2012; published 24 August 2012)
A simple pathway to grow thin films of N-doped MgO (MgO:N), which has been found experimentally to be
a ferromagnetic d0insulator, is presented. It relies on the catalytic properties of a Mo(001) substrate using the
growth of Mg in a mixed atmosphere of O 2and N 2. Scanning tunneling spectroscopy reveals that the films are
insulating and exhibit an N-induced state slightly below the conduction band minimum.
DOI: 10.1103/PhysRevB.86.075455 PACS number(s): 68 .55.aj, 68.37.Ef, 81.15.Hi
I. INTRODUCTION
Recently, it has been found that MgO:N-films grown by
molecular beam epitaxy (MBE) exhibit ferromagnetism afterbeing annealed at 1020 K.
1The optimized N concentration
was 2.2% exhibiting coercive fields as large as 60 mT atT=10 K and magnetic moments per N atom of 0.3 μ
B
barely reducing up to room temperature. A Curie temperature
TC/similarequal550 K has been extrapolated and indications that N is
incorporated substitutionally on the O site have been deducedfrom core level spectroscopy. Moreover, independent studiesof N implantation (80 keV) into MgO has led to a hysteresiswith a coercive field of 30 mT at 300 K.
2This raises hope that
reliable d0ferromagnetism avoiding dmetals can be realized
in MgO:N at T=300 K. Such magnetism without dorbitals
has previously been found in thin films of undoped oxides3
including MgO (Ref. 4) or defective carbon systems,5however,
with limited control since relying on defects. ZnO with sp-type
dopants such as C, N, B, Li, Na, Mg, Al, and Ga showsferromagnetic signals, too,
6,7but, likely, Zn or O vacancies
and Zn dorbitals are involved in the magnetic coupling.6,8
Thed0ferromagnetism has been proposed theoretically
relying on the double exchange mechanism in narrow im-purity bands.
9But the high TCproposed originally has been
challenged by going beyond the mean-field approximation10or
by considering correlation effects.11–13Partly, even the absence
of ferromagnetism has been found.11This renders the high TC
observed experimentally in obvious disagreement with current
theory and suggests more detailed studies are needed.
MgO:N films, in addition, exhibit bipolar resistive switch-
ing behavior14prior to annealing. Resistance contrasts as
large as 4 orders of magnitude, switching currents as low as100 nA, and switching times into both states below 10 nshave been obtained.
1This makes MgO:N also interesting for
nonvolatile memories. However, the incorporation of N intoMgO is difficult due to the strongly endothermal incorporationof N atoms with respect to N
2(energy cost per N atom:
10 eV).15It requires, for example, atomic beams of N and
O produced by a high-frequency ion plasma source1or N+
implantation.2Here, we demonstrate a simplified pathway
using the catalytic abilities of a Mo(001) substrate. Thus,we establish a model system of MgO:N for surface science.Mo(001) is chosen since thin MgO films of high qualitycan be grown epitaxially due to the relatively small latticemismatch of 6%.
16–18Moreover, catalytic properties of Mo
with respect to N 2are known, for example, nitrogenase within
bacteria using molybdenum enzymes as catalyst.19Catalytic
N2dissociation on surfaces has been induced successfully for
the electronically similar W(001), whereas growth propertiesof MgO are, however, unknown.
20–22We have grown thin films
of MgO:N on Mo(001) with thicknesses up to 10 monolayers(ML) at optimal doping. Scanning tunneling spectroscopy(STS) has revealed that the Fermi level is well within the bandgap, indicating an insulating behavior of the film. Moreover,an unoccupied state close to the conduction band has beenfound, which is not present in pure MgO films.
II. RESULTS AND DISCUSSION
The experiments are performed in an ultrahigh vacuum at
a base pressure of p=5×10−11mbar. First, the Mo(100)
crystal was cleaned by cyclically annealing within O 2at
an initial pressure of pO2=5×10−7mbar and 1400 K,
followed by flashing to 2300 K.23After every cycle, pO2is
slightly reduced. The MgO:N films are prepared by molecularbeam epitaxy of magnesium at p
O2=1×10−7mbar and N 2
pressure pN2=5×10−6mbar. The deposition temperature
TDis 300 K, if not given explicitly. The deposition rate of
Mg controlled by a quartz microbalance is 0 .5M L/min. After
MgO:N deposition, the samples are annealed at 1100 K for10 min. Figure 1(a) shows a scanning tunneling microscopy
(STM) image of 7 ML MgO
0.973N0.027. It exhibits islands
on top of a MgO:N wetting layer. From comparison of thecoverage determined by the quartz balance and the volume ofthe MgO:N islands, we estimate the wetting layer thicknessto be 1–2 ML. Such relatively rough films are occasionallyalso observed in pure MgO films. So far, we have not beenable to prepare films of similar smoothness as, for example,in Fig. 1 of Ref. 18for MgO:N. More studies are required
to attribute the different morphologies to intrinsic propertiesof MgO and MgO:N. Thicker samples exhibit a plain surfacewith corrugations below 0.5 nm as measured for a film witha thickness of 40 ML by atomic force microscopy (AFM)and shown in Fig. 1(c). We check the crystalline quality
and chemical purity of Mo(100) and the MgO:N films bylow-energy electron diffraction (LEED) and Auger electronspectroscopy (AES). A complete AES spectrum at a primary
075455-1 1098-0121/2012/86(7)/075455(5) ©2012 American Physical SocietyGROB, PRATZER, MORGENSTERN, AND LE ˇZAI´C PHYSICAL REVIEW B 86, 075455 (2012)
FIG. 1. (Color online) (a) STM image of 7 ML MgO 0.973N0.027
film on Mo(001) with thicker islands (bright areas) on top of a
wetting layer of 1–2 ML; (50 ×50) nm2,U=3V ,I=0.5n A .
(b) Line profile along the red line marked in panel (a) with the heightof the island above the wetting layer marked. (c) Tapping mode
AFM image of 40 ML MgO
0.993N0.007film on Mo(001) after transfer
into an ambient environment; (100 ×100) nm2,f=281 kHz, free
amplitude A=30 nm, setpoint 60%. (d) Line profile along the red
line marked in panel (c).
electron energy of 1 keV sensitive to the upper 2 nm (9 to
10 ML)24is shown in Fig. 2(a). Next to the O peak and the N
peak there is a small amount of Mo which remains constantindependent of the number of MgO:N layers. The nominalatomic concentration x
iof element iwithin homogeneous
films is calculated from the AES peak height Yishown in
Fig.2(b) according to25
xi=Yi/Si/summationtext
aYa/Sa, (1)
where Sidenotes the normalized sensitivity factor for element
iandasums over all relevant elements. For Mo this results
inx=4.5% relating to O. We believe that this thickness-
independent amount originates from the Mo sample holderdue to an imperfect focusing of the electron beam onto thesample.
The N concentration xdepends on the deposition tempera-
tureT
Das shown in Fig. 2(c) for a 7-ML MgO:N film. Up to
x=6% is achieved at TD=850 K, indicating a more effective
dissociation of N 2at higher TD. Notice that x=6% at 7 ML
is still far below the amount of N expected from a full Ncoverage of Mo(001). Next, we prepare MgO:N films witha thickness of up to 100 ML at T
D=900 K. The nitrogen
amount within the MgO:N decreases with film thickness asshown in Fig. 3. A 60-ML-thick MgO:N film contains only
0.3% nitrogen in comparison to 3.2% at a thickness of 7 ML.Assuming homogeneous distribution of N, this implies thatthe total amount of N in both films is roughly the same,
FIG. 2. (Color online) (a) Differential AES spectra of 7 ML
MgO 0.979N0.031with assignment of the peaks different elements
recorded at a primary energy of E=1 keV . Above the O peak, there is
no other peak related to another element. (b) Higher resolution spectra
from panel (a) of N and O with the corresponding peak-to-peak height
Yimarked. (c) Nitrogen concentration xof 7-ML-thick MgO:N films
as a function of the deposition temperature TD.
evidencing that N 2is dissociated on the Mo(100) surface only.
We assume that, during MgO growth or annealing, the atomicN from the surface is incorporated into the MgO film. To
FIG. 3. (Color online) (a) Nitrogen concentration xas a function
of monolayers d. Different symbols mark different preparation
methods, the red fit curve assumes a constant amount of N atoms
homogeneously distributed within the MgO film, and the black
curve assumes the same amount of N at the Mo/MgO interface only(see text). (b) Sketch of N
2incorporation. Left: N 2dissociation on
Mo(001). Right: Incorporation of N into MgO leaving the amount of
N independent of the MgO thickness.
075455-2CATALYTIC GROWTH OF N-DOPED MgO ON Mo(001) PHYSICAL REVIEW B 86, 075455 (2012)
support this scenario, we fit the data assuming a constant areal
concentration nof N atoms:
x=n
d, (2)
where ddenotes the number of MgO ML and nrepresents
the concentration of N atoms with respect to the sum of Nand O atoms if only one ML is prepared. If we assume thatall N atoms are substitutionally incorporated into the firstmonolayer of MgO:N, the red fit curve plotted in Fig. 3(a)
shows excellent agreement with the measured data points usingn=21.6%. Assuming interstitial impurities, i.e., dumbbells
of NO,
15Eq.(2)has to be slightly modified and results in
n=18.2%.
The resulting Auger current Iof the N peak at 379 eV can be
calculated, assuming a homogeneous distribution of N in thesample resulting in a N concentration per layer proportional to1/z, with zbeing the film thickness. The amount of incoming
electrons with E=1 keV decays exponentially, as well as the
amount of Auger electrons leaving the sample. This results in
I=I
0
z/integraldisplayz
0e−z/prime/λ1e−z/prime/λ2dz/prime=I0
az(1−e−az),
a=λ1+λ2
λ1λ2, (3)
withI0being a constant depending on the details of the
experiment. One finds an exponential part multiplied by thedominating 1 /zpart. Using the experimentally verified mean
free path of λ
1(1 keV) =2 nm and λ2(379 eV) =1n m ,24
we obtain a decrease in the signal by a factor of 5 from 7
to 40 ML, which agrees well with our measurements. Thisis shown by the red curve in Fig. 3(a), which has been
adapted to the experimental point at x=0.031 to get rid of the
unknown I
0.
Nearly the same ncan be achieved by another preparation
method: Prior to the deposition of Mg in a pure O 2environ-
ment, we expose the Mo crystal to N 2(p=5×10−6mbar)
at 300 K for 10 min. Afterwards, we grow MgO without N 2
atT=900 K, leading to very similar N concentrations as
shown by the blue triangles in Fig. 3(a). Thus, obviously,
the dissociation of N 2takes place at Mo(001) only. Finally,
a third preparation is performed: 10 ML of pristine MgO aregrown first at p
N2<10−11mbar. Subsequently, 10 ML MgO:N
are deposited at TD=300 K and pN2=5×10−6mbar. No
nitrogen is found in the sample, i.e., x< 0.2 %, which has to be
compared to the x=3.1% obtained for 7-ML MgO:N grown
under identical conditions directly on the substrate. Thus, ifMo is covered by MgO, the catalytic effect of the substrateis inhibited. In order to show that the N is incorporated intothe MgO film and does not reside on the substrate, we alsocalculate the expected AES signal of N for this case assuming amaximum concentration of n=28% at the Mo/MgO interface,
i.e., adapting the curve to the 3.1% data point in Fig. 3(a).T h i s
results in
I=I
0e−z/λ 1e−z/λ 2. (4)
The results are plotted in Fig. 3(a), too, revealing a strong
discrepancy with the experimental data. In particular, thestrength of xatd=40 ML is 10
−6, while the experimental
value is 0.69%. Figures 1(c) and 1(d), which have been
FIG. 4. (Color online) (a) dI/dU (U) spectra measured by STS
on an 11-ML-high MgO 0.96N0.04island ( Ustab=3V ,Istab=0.5n A ,
Umod=40 mV) at several positions (straight lines) and on an 11-ML-
thick pristine MgO island (dashed line). The average film thicknessin both cases is 7 ML. (b) Calculated density of states (DOS) for
substitutional and interstitial impurity as well as a N-N dimer at MgO
surface. HOS: Highest occupied state. All states between −1.2 eV
and 3 eV are N induced; CBM is marked by an arrow. The inset shows
the calculated charge density of the unoccupied N-induced states of
a N-N dimer at the surface. The large green and small white spheresshow Mg and O, respectively.
recorded on a 40-ML MgO film exhibiting an AES signal
ofx=0.7%, confirm that the MgO thickness is rather
homogeneous (roughness σ=150 pm), making it impossible
that N at the interface could have been detected by AES.Thus, we conclude that N dissociated at the Mo(001) isincorporated into the MgO film, but we do not know theincorporation site. To tackle this question, we perform in
situ STS, which probes the local density of states at the
surface. Figure 4compares the STS curves of 11-ML films
of MgO
0.96N0.04and undoped MgO.18Doping by N leads to
a shift of the Fermi level EFtowards the valence band by
about 1 eV . The fact that this shift is smaller than expectedforp-type doping by N is possibly due to donation of
075455-3GROB, PRATZER, MORGENSTERN, AND LE ˇZAI´C PHYSICAL REVIEW B 86, 075455 (2012)
electrons, accomplished by the Mo substrate.26An additional
peak with a maximum at 0 .3 eV below the conduction band
minimum (CBM) appears after N doping. This is observed forthree different N concentrations, i.e., MgO thicknesses. Thepeak energy varies laterally by ±0.3 eV . Density functional
theory (DFT) calculations of bulk MgO with substitutional Npredict occupied plevels close to the valence band maximum
(VBM) and an unoccupied, spin polarized level within themiddle of the band gap, if self interaction correction isincluded.
11,12
This disagrees with our experiment. Since surfaces were not
included in these calculations, we performed first-principlesDFT calculations including the surface within the spin-polarized generalized gradient approximation
27using projec-
tor augmented-wave potentials as implemented in the ViennaAb initio Simulation Package (
V ASP ).28Correlation effects on
thepshells of N dopants were accounted for by the DFT +U
scheme in Dudarev et al. ’s approach29with an on-site effective
Coulomb parameter, Ueff=3.4 eV . A kinetic energy cutoff of
500 eV and a 6 ×6×1/Gamma1-centered k-point mesh was used.
The supercell consists of nine atomic layers MgO(001) usingthe experimental lattice parameter and a 16- ˚A-thick layer of
vacuum. All atomic positions as well as the thickness of theMgO slab were fully relaxed.
Three different configurations with N atoms in the surface
layer were calculated: (i) one N-atom substituting an oxygen,(ii) one N-atom at the interstitial site, and (iii) an N-N dimerwith one N atom being substitutional and the other at thenearest interstitial site. In case (ii), the calculation was initiatedwith N at the interstitial site, but the system relaxed to a
configuration where N and O exchanged their places, i.e., Nended up substitutionally and O interstitially. However, theN-derived pstates for cases (i) and (ii) were found within
2.2 eV above the VBM, very similar to the MgO bulk
11,12as
shown in Fig. 4(b) in the first two panels exhibiting no peaks
close to the CBM. We also tried a more complex N structureas, e.g., the N-N dimer at the surface shown in Fig. 4(b),l o w e s t
panel, and exhibiting unoccupied N-type pstates close to the
CBM. However, there is another peak of similar geometricintensity distribution in the center of the band gap not observedin the experiment. Thus, even more complex structures mightbe responsible for the observed peak.
III. CONCLUSION
In conclusion, we prepared thin films of N-doped MgO with
an N concentration up to 6% by using the catalytic effect ofMo(001) for N
2dissociation. Compared with pristine MgO,
an additional state close to the conduction band minimum wasobserved by STS which could not be attributed to simple Nimpurity configurations.
ACKNOWLEDGMENTS
Helpful discussions with P. Mavropoulos, S. Bl ¨ugel and
S. Parkin as well as financial support from the DFG underGrant No. SFB 917-A3 and the HGF-YIG under Grant No.VH-NG-409 are gratefully acknowledged.
*Corresponding author: grob@physik.rwth-aachen.de
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075455-5 |
PhysRevB.75.113407.pdf | Pseudodiffusive magnetotransport in graphene
Elsa Prada,1Pablo San-Jose,1Bernhard Wunsch,2,3and Francisco Guinea3
1Institut für Theoretische Festköperphysik and DFG, Center for Functional Nanostructures (CFN), Universität Karlsruhe,
D-76128 Karlsruhe, Germany
2Departamento de Física de Materiales, Universidad Complutense de Madrid, E-28040 Madrid, Spain
3Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E-28049 Madrid, Spain
/H20849Received 1 February 2007; published 27 March 2007 /H20850
Transport properties through wide and short ballistic graphene junctions are studied in the presence of
arbitrary dopings and magnetic fields. No dependence on the magnetic field is observed at the Dirac point forany current cumulant, just as in a classical diffusive system, both in normal-graphene-normal and normal-graphene-superconductor junctions. This pseudodiffusive regime is, however, extremely fragile with respect todoping at finite fields. We identify the crossovers to a field-suppressed and a normal ballistic transport regimein the magnetic-field-doping parameter space, and provide a physical interpretation of the phase diagram.Remarkably, pseudodiffusive transport is recovered away from the Dirac point in resonance with Landau levelsat high magnetic fields.
DOI: 10.1103/PhysRevB.75.113407 PACS number /H20849s/H20850: 75.47.Jn, 72.80.Rj, 74.45. /H11001c, 75.70.Ak
Low-energy excitations in a monolayer of carbon atoms
arranged in a honeycomb lattice, known as a graphene sheet,have the remarkable peculiarity of being governed by thetwo dimensional massless Dirac equation, which is respon-sible for a variety of exotic transport properties as comparedto ordinary metals. Particularly striking is that for clean, un-doped graphene the density of states is zero, but not so theconductivity, which remains of the order of the quantum unite
2/h.1,2Another intriguing fact is that a wide and short strip
of undoped graphene exhibits “pseudodiffusive” transportproperties in the absence of electron-electron interactionsand impurity scattering.
3By ‘‘pseudodiffusive’’ it is meant
that transport properties are indistinguishable from those of aclassical diffusive system. These include the full transportstatistics /H20849in particular, the Fano factor F=1/3 and the con-
ductance G/H11008W/L,
3where Wis the width and Lis the length
of the graphene strip /H20850, the critical current,4and I-V
characteristics5in Josephson structures, as well as the rela-
tion of the normal-metal–superconductor conductance to thenormal transmissions.
6The same behavior can be expected
in bilayer graphene.7In fact, all of the above similarities can
be explained by noting that at the Dirac point /H20849i.e., for un-
doped graphene /H20850transport occurs entirely via evanescent
modes with a transmission that is equal to the diffusive trans-port theory result /H20849evaluated at k
Fl=1,l/H11013mean free path24/H20850
without quantum corrections,3,6,8
Tky=1
cosh2kyL. /H208491/H20850
Here, kyis the transverse momentum of the channel. In dif-
fusive systems, the above relation holds independently of anexternally applied magnetic field in the limit of many chan-nels /H20849classical limit /H20850, for which any quantum weak localiza-
tion correction is negligible.
8
The question we raise here is as follows: Does the diffu-
sive behavior of ballistic graphene persist in the presence ofa magnetic field? We will show that for zero doping theequivalence is preserved for any magnetic field. Remarkablyfor a ballistic system, the applied magnetic field does not
affect the transport statistics /H20849for any current cumulant /H20850at the
Dirac point. For graphene with disorder, this has recentlyalso been demonstrated at the conductivity level.
9–11At suf-
ficiently strong magnetic fields, an exponentially smallchemical potential is enough to enter a field-suppressed
transport regime. However, at resonance with the Landaulevels /H20849LLs /H20850the pseudodiffusive behavior is recovered for all
current cumulants. For even higher dopings, one observes afinal crossover to the ballistic magnetotransport regime,since clean graphene then resembles a ballistic normal metal.
The magnetic field introduces a fundamental quantum-
mechanical length scale known as the magnetic length l
B
=/H20849/H6036//H20841eB/H6023/H20841/H208501/2. In the complete absence of scattering, localized
LLs are well formed and ballistic transport is suppressed.
The only contribution to transport in this regime comes fromresonant tunneling exactly at Landau energies. In usual met-
als, this happens when the cyclotron diameter 2 r
c=2lB2kFis
smaller than the relevant scattering length /H20849set by system
size, disorder, or temperature /H20850.I f2 rcis of the order of or
larger than the scattering length, then delocalized states con-tribute to transport, leading to Shubnikov–de Haas oscilla-tions and the quantum Hall effect. For wide and short ballis-tic strips, the relevant scattering scale is the strip length L,
and scattering on lateral boundaries can be ignored. Ingraphene, the lowest LL lies precisely at the Dirac point.Besides, at this point k
F=0 and thus rc=0 independent of the
magnetic field. Therefore, in contrast to the normal-metalstrip and to the high doping limit, at the Dirac point nodelocalized bulk transport should take place for any mag-netic field and resonant tunneling should be field indepen-dent.
To confirm this hand-waving picture, we analyze theoreti-
cally magnetotransport effects through normal-graphene-normal /H20849N/H20850and normal-graphene-superconductor /H20849NS/H20850wide
ballistic junctions at arbitrary dopings and magnetic fields.From an experimental point of view, transport properties oflightly doped graphene in contact with superconductors arecurrently being investigated.
12Moreover, the properties ofPHYSICAL REVIEW B 75, 113407 /H208492007 /H20850
1098-0121/2007/75 /H2084911/H20850/113407 /H208494/H20850 ©2007 The American Physical Society 113407-1graphene in strong magnetic fields are also a subject of great
interest,13–15in particular, in relation to weak
/H20849anti/H20850localization.16–20Here, we consider a clean graphene
sheet of width W/H20849assumed to be the largest length scale in
the system /H20850in the ydirection through which transport occurs
in the xdirection. For x/H11021−L/2, it is covered by a normal
contact, and for x/H11022L/2, it is covered either by a supercon-
ducting contact or by a normal one. The central region islightly doped, leading to a finite Fermi energy
/H9262measured
relative to the Dirac point, which can be varied by an exter-nal gate voltage. The contact regions are modeled as heavilydoped graphene, with Fermi energy
/H9262cconveniently fixed to
infinity with respect to both /H9262and the superconducting gap
/H9004. The boundary conditions in the ydirection are irrelevant3
for large aspect ratios W/L/H112711. We choose periodic bound-
aries for simplicity. A constant external magnetic field Bis
applied perpendicular to the graphene sheet. We assume theelectrodes to be magnetically shielded, e.g., by coveringthem with materials with high magnetic permeability. Inthe Landau gauge, we can write the vector potential as
A
/H6023=/H208490,Bx,0/H20850for /H20841x/H20841/H11021L/2, and constant in the contact re-
gions. This gauge is convenient since the motions in the
xand ydirections are uncoupled and kyremains a good
quantum number. We neglect Zeeman splitting, so that theelectron spin only enters as a degeneracy factor of 2 in thefollowing calculation. Finally, we note that edge currentsgenerally give a negligible contribution to transport in theW/H11271Llimit.
We will compute the N and the NS /H20849Andreev /H20850inverse
longitudinal resistivities,
/H9267xx−1=G/H20849L/W/H20850, expressed in terms of
the conductances3,6
GN=4e2
h/H20858
kyTky,GNS=8e2
h/H20858
kyTky2
/H208492−Tky/H208502, /H208492/H20850
and the shot noise using corresponding expressions in terms
of the transmission for normal conducting contacts Tky.21
Note that the above expressions for the NS case are valid
only if Tkyis left-right symmetric, which is not in general
true in the presence of a magnetic field. In our particular
setup, it does indeed turn out to be symmetric. The transmis-sion through the central region is obtained by imposing cur-rent conservation at the interfaces, which translates into con-tinuity of the wave function. In the chosen gauge, thescattering problem is effectively one dimensional, the trans-verse mode profile e
ikyybeing the same in all regions, so we
will only discuss the xdependence of the wave functions
from now on.
The contact region eigenstates at energy /H9280=/H6036vF/H20881kx2+ky2
−/H9262c+/H9262with respect to the central strip Dirac point are given
by
/H9023kxkysN/H20849x/H20850=/H20873szk−s
1/H20874eikxx. /H208493/H20850
The spinor lives in the space of the two triangular sublattices
that conform the graphene hexagonal lattice, s= ±1 is the
“valley” quantum number /H20849for the degenerate KandK/H11032points /H20850, and zk/H11013exp/H20851iarg/H20849kx+iky/H20850/H20852, which tends to
zk/H11015sgn/H20849kx/H20850when/H9262c→/H11009.
The spinor /H9023/H9280kysG/H20849x/H20850=/H20851/H9278/H9280kysA/H20849x/H20850,/H9278/H9280kysB/H20849x/H20850/H20852Tfor the central
region is determined by the one-dimensional Dirac equation
/H208730−iaˆ
iaˆ+0/H20874/H20873/H9278/H9280kysA/H20849x/H20850
/H9278/H9280kysB/H20849x/H20850/H20874=/H9261/H20881n/H9280/H20873/H9278/H9280kysA/H20849x/H20850
/H9278/H9280kysB/H20849x/H20850/H20874, /H208494/H20850
with aˆ/H11013/H20849x˜+/H11509x˜/H20850//H208812,aˆ+/H11013/H20849x˜−/H11509x˜/H20850//H208812,x˜/H11013x/lB+kylB,/H9261
=sgn/H9280, and n/H9280=/H20849lB/H20841/H9280/H20841//H6036vF/H208502/2. Canonical relations
/H20851aˆ,aˆ+/H20852=1 are satisfied. The above equation corresponds to
theKvalley /H20849s=1/H20850, while the s=−1 equation is obtained by
swapping aˆandaˆ+. Since the central region is bounded, no
integrability condition must be met and the eigenspectrum ofEq. /H208494/H20850is continuous. The usual LL solutions, which corre-
spond to integer n
/H9280, are thus complemented by a larger fam-
ily of divergent wave functions, typically localized aroundthe interfaces x=±L/2, with arbitrary n
/H9280/H333560.22At the
Dirac point /H20849n/H9280=0/H20850, the components of the spinor are un-
coupled, and the two eigenstates for s=1 are /H90230ky1G,1/H20849x/H20850
=/H208510,exp /H20849−x˜2/2/H20850/H20852Tand/H90230ky1G,2/H20849x/H20850=/H20851exp/H20849+x˜2/2/H20850,0/H20852T. The
s=−1 solution has interchanged spinor components. At finite
energy /H20849n/H9280/H110220/H20850, the solutions to Eq. /H208494/H20850become
/H9023/H9280ky1G,1/H208492/H20850/H20849x/H20850=/H20873/H9261hn/H9280−1e/H20849o/H20850/H20849x˜/H20850
ihn/H9280o/H20849e/H20850/H20849x˜/H20850/H20874,/H9023/H9280ky−1G,1/H208492/H20850/H20849x/H20850=/H20873/H9261hn/H9280o/H20849e/H20850/H20849x˜/H20850
ihn/H9280−1e/H20849o/H20850/H20849x˜/H20850/H20874.
They have been expressed in terms of the even and odd /H20849inx˜/H20850
solutions hne,o/H20849x˜/H20850of the Klein-Gordon equation a+ahne/H20849o/H20850/H20849x˜/H20850
=nhne/H20849o/H20850/H20849x˜/H20850/H20851the square of Eq. /H208494/H20850/H20852, normalized so that
aˆ+hne/H20849o/H20850=/H20881n+1hn+1o/H20849e/H20850,aˆhne/H20849o/H20850=/H20881nhn−1o/H20849e/H20850. As a function of the con-
fluent hypergeometric function1F1/H20849a,b,z/H20850=1+a
bz
1!+a/H20849a+1/H20850
b/H20849b+1/H20850z2
2!
+¯andSn=sgn /H20853sin/H20851/H9266/H20849n+1/2 /H20850/2/H20852/H20854, these are
hne/H20849x˜/H20850=/H20881/H20849n−1/H20850!!
/H20881/H9266n!!Sn+1e−x˜2/2
1F1/H20873−n
2,1
2,x˜2/H20874,
hno/H20849x˜/H20850=/H208812n!!
/H20881/H9266/H20849n−1/H20850!!Snx˜e−x˜2/2
1F1/H20873−n−1
2,3
2,x˜2/H20874.
Imposing continuity for each /H20853ky,/H9280/H20854at the interfaces re-
sults in the following s-independent transmission probability:
Tky,/H9280=/H20841tky,/H9280/H208412=/H208792gn/H9280N
gn/H9280R−ign/H9280I/H208792
, /H208495/H20850
where
gnR=hne+hn−1e−+hn−1e+hne−−hno+hn−1o−−hn−1o+hno−,
gnI=hno+hne−−hne+hno−+hn−1e+hn−1o−−hn−1o+hn−1e−,
gnN=hn−1e+hne+−hn−1o+hno+=S2n+1/2
/H20881/H9266n
are expressed in terms of the wave functions at the bound-
aries hne/H20849o/H20850±=hne/H20849o/H20850/H20849±L/2
lB+kylB/H20850. This gives the general trans-BRIEF REPORTS PHYSICAL REVIEW B 75, 113407 /H208492007 /H20850
113407-2missions Tkyfor arbitrary doping and magnetic field. It re-
produces previously known results at B=0 and nonzero
doping4as well as Eq. /H208491/H20850for/H9262=0.
In Fig. 1/H20849a/H20850, we plot /H9267xx−1as a function of the Fermi energy
for increasing values of the ratio L/lB/H11008/H20881B. We recover the
results obtained without magnetic field, namely, that /H9268Nand
/H9268NS /H20851where /H9268=/H9267xx−1/H20849B=0/H20850/H20852tend to the known quantum-
limited minimal conductivity value 4 e2//H9266hat zero doping,
whereas for /H20841/H9262/H20841L//H6036vF/H112711 the slope of the asymptotes tends
to 0.38 /H9266and 0.25 /H9266for the NS and N junctions,
respectively.6Remarkably, as we increase B, all the /H9267xx−1
curves remain unchanged at the Dirac point /H9262=0. The Dirac-
point Fano factor in Fig. 1/H20849b/H20850is also unaffected by magnetic
fields and takes the classical diffusive value /H208491/3 for the N
and 2/3 for the NS junction /H20850. This happens for any current
cumulant, since at /H9262=0 the transmission given in Eq. /H208495/H20850
reduces to Eq. /H208491/H20850independent of B.
However, for /H9262/H110220 the resistivities and the Fano factors
do depend on the magnetic field. In particular, for 2 rc/H11021L
/H20849and above a certain critical value of L/lB/H20850transport can take
place only at resonance with the LLs /H20849/H9262L//H6036vF=/H208812nL/lB/H20850,
while for other dopings /H9267xx−1is suppressed as e−/H20849L/lB/H208502/2for the
N junction and as e−/H20849L/lB/H208502for the NS one. The width of the
resonances at the LL energies vanishes for 2 rc/H11270L,a sw e
consider no disorder.15Remarkably, /H9267xx−1at these resonances
for large fields coincides with the one at the Dirac point4e
2//H9266h, a theoretical value that is usually associated strictly
with zero doping and that is interestingly at odds with someexperimental findings.
1In fact, it can be analyticallydemonstrated that not only the conductance but the whole
pseudodiffusive transport statistics is recovered at the reso-nances for high magnetic fields. Under this perspective, thefield-independent resistivity at
/H9262=0 can be understood as
due to resonant transport through the zeroth LL that remainspinned at the Dirac point. The field-suppressed regime is
apparent for small but finite doping in Fig. 1/H20849a/H20850, where
/H9267xx−1
strongly decreases with increasing value of L/lB. Corre-
spondingly, the bulk Fano factor reaches the tunneling limitvalue /H208491 for the N and 2 for the NS junction /H20850as transport gets
suppressed /H20851see Fig. 1/H20849b/H20850/H20852, in which limit the noise of the
edge currents not considered here could be visible. Increas-ing the Fermi energy further, one enters the regime 2 r
c/H11022L,
where /H9267xx−1is composed of two parts. The first part is linear in
/H9262L//H6036vF, in agreement with the scaling with Lbehavior of a
ballistic conductor subject to a magnetic field /H20849L-independent
conductance /H20850. In particular, for sufficiently high dopings, all
curves in Fig. 1/H20849a/H20850become parallel and tend to the same
/H20849average /H20850slope as the zero-field conductivity. The second
contribution to /H9267xx−1is an oscillating part, which for 2 rc/H11022L
can no longer be explained by the resonance with LLs, sincein that regime the effect of the boundaries is dominating thelevel structure in the central region. In fact, for 2 r
c/H11271Lthe
oscillations become equally spaced and are explained ratherby a Fabry-Pérot-type effect, connected to resonant tunnelingthrough the structure.
In the inset of Fig. 1/H20849a/H20850, the ratio G
NSbulk/GNbulkis plotted as
a function of /H9262for the same values of L/lBas in the main
panel. At the Dirac point, the ratio goes to 1. At /H9262/H110220, the
suppressed magnetotransport manifests itself as a decaying
GNSbulk/GNbulk/H11008e−/H20849L/lB/H208502/2, until doping reaches the ballistic
threshold and the ratio starts growing again, finally reachingits asymptotic value 0.38/0.25=1.52. As explained in Ref. 6,
this value is expected in normal ballistic systems with Fermiwavelength mismatch. Note again here that, for sufficientlysuppressed G
bulk, the edge contribution23neglected here will
dominate transport.
All the previous behaviors can be condensed in a quanti-
tative way in the phase diagram shown in Fig. 2. It contains
three regions corresponding to the three different transport
FIG. 1. /H20849Color online /H20850/H20849a/H20850Inverse longitudinal resistivity in units
of 4e2//H9266hand /H20849b/H20850bulk Fano factor for the N /H20849left/H20850and NS /H20849right /H20850
junctions as a function of the /H20849absolute value of the /H20850Fermi energy
/H20849in units of /H6036vF/L/H20850. Different curves correspond to different values
ofL/lB/H20849where lB/H11013/H20881/H6036/eB/H20850, which ranges from zero for red curve
to 4 for the dark blue one in steps of 1. The same for the ratioG
NSbulk/GNbulkin the inset.
FIG. 2. /H20849Color online /H20850Phase diagram representing the cross-
overs from localized, pseudodiffusive, and ballistic transportregimes in the field-doping parameter plane. The solid /H20849dashed /H20850
lines represent the boundaries for a N /H20849NS/H20850junction. LLs are
labeled by n.BRIEF REPORTS PHYSICAL REVIEW B 75, 113407 /H208492007 /H20850
113407-3regimes, namely, pseudodiffusive /H20849red/H20850, field suppressed
/H20849blue /H20850, and ballistic /H20849green /H20850,i nt h e L/lBand/H9262L//H6036vF=kFL
parameter space. The corresponding crossover lines betweenregions are solid /H20849dashed /H20850for the N /H20849NS/H20850junction /H20849note that
the background colors correspond to the boundaries of the Ncase /H20850. The boundaries for the pseudodiffusive region have
been calculated assuming a maximum deviation of ±10%with respect to the Dirac-point conductivity 4 e
2//H9266h. At low
fields, the width of the pseudodiffusive window that bracketsthe Dirac point is roughly field independent, whereas forL/l
B/H110221.8 for N /H208491.35 for NS /H20850the window closes down as
exp/H20851−/H20849L/lB/H208502/4/H20852. Physically, this means that at these higher
fields the quasidiffusive transport regime is extremely fragile
with respect to doping, and an exponentially fast crossover tothe field-suppressed /H20849localized /H20850regime takes place. The
boundaries of the latter /H20849blue region /H20850were set by a crossover
criterion
/H9267xx−1/H110210.1/H208494e2//H9266h/H20850. Its spiked shape is due to the
peaked contributions to the field-suppressed /H9267xx−1discussed in
the analysis of Fig. 1, which are produced by resonant tun-
neling through LLs. When the magnetic field is increased,the positions of these peaks shift to higher dopings, converg-
ing on radial lines with slope 1//H208812n, while their width de-
creases exponentially. Above a certain value of L/lB, the
pseudodiffusive regime is recovered and the resonances arethus colored in red. The third region /H20849green /H20850is characterized
by
/H9267xx−1/H11008Lat fixed field and doping /H20849which would correspond
to radial lines in the phase diagram /H20850, and is therefore a bal-
listic transport regime. As expected from the arguments inthe Introduction, the boundary of the field-suppressed regionclosely follows the ballistic threshold 2 r
c=L. Finally, inter-
mediate regions /H20849white /H20850are characterized by strongly oscil-
lating conductivities.In conclusion, by computing the general transmission
probabilities through short and wide graphene junctions, wehave found that the transport properties at the Dirac pointexactly match those of a classical diffusive system even inthe presence of a magnetic field, which actually does notaffect transport at all at zero doping. This behavior, which isassociated with the existence of a zeroth LL pinned at theDirac point, is, however, found to be exponentially fragilewith respect to doping for high fields. By analyzing inverselongitudinal resistivity and higher current cumulants, wehave identified and interpreted the three distinct regimes thatappear at finite magnetic fields and dopings, correspondingto pseudo-diffusive, field-suppressed, and ballistic transport,and computed the phase diagram for the N and NS junctions
in the relevant field-doping parameter space. Transport reso-nances at the LL energies are found in the field-suppressed
regime, with
/H9267xx−1and all higher bulk current cumulants satu-
rating to the pseudodiffusive Dirac-point values at highfields. The width of these resonances decreases exponentiallywith magnetic field, although broadening due to disorder inreal samples is expected, thus facilitating experimental ob-servation. The reappearance of pseudodiffusive transport atfinite doping could shed light on the 1/
/H9266discrepancy be-
tween experiments and theoretical results for the conductiv-ity at the Dirac point.
We thank G. Schön for support in promoting this collabo-
ration. This work benefited from the financial support of theEuropean Community under the Marie Curie Research Train-ing Networks and the ESR program. F.G. acknowledgesfunding from MEC /H20849Spain /H20850through Grant No. FIS2005-
05478-C02-01 and the European Union Contract No. 12881/H20849NEST /H20850.
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5J. C. Cuevas and A. L. Yeyati, Phys. Rev. B 74, 180501 /H208492006 /H20850.
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Vandersypen, and A. F. Morpurgo, Nature /H20849London /H20850446,5 6
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A. Ponomarenko, D. Jiang, and A. K. Geim, Phys. Rev. Lett. 97,
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24It is noteworthy that the diffusive transport theory actually as-
sumes weak disorder kFl/H112711. Hence, Eq. /H208491/H20850is an extrapolation
of the theory to a dirty metal limit.BRIEF REPORTS PHYSICAL REVIEW B 75, 113407 /H208492007 /H20850
113407-4 |
PhysRevB.96.041109.pdf | RAPID COMMUNICATIONS
PHYSICAL REVIEW B 96, 041109(R) (2017)
Realistic quantum critical point in one-dimensional two-impurity models
Benedikt Lechtenberg,1,2,*Fabian Eickhoff,1and Frithjof B. Anders1
1Lehrstuhl für Theoretische Physik II, Technische Universität Dortmund, 44221 Dortmund, Germany
2Department of Physics, Kyoto University, Kyoto 606-8502, Japan
(Received 30 September 2016; revised manuscript received 31 May 2017; published 7 July 2017)
We show that the two-impurity Anderson model exhibits an additional quantum critical point at infinitely
many specific distances between both impurities for an inversion symmetric one-dimensional dispersion. Unlikethe quantum critical point previously established, it is robust against particle-hole or parity symmetry breaking.The quantum critical point separates a spin doublet from a spin singlet ground state and is, therefore, protected.A finite single-particle tunneling tor an applied uniform gate voltage will drive the system across the quantum
critical point. The discriminative magnetic properties of the different phases cause a jump in the spectral functionsat low temperature, which might be useful for future spintronics devices. A local parity conservation will preventthe spin-spin correlation function from decaying to its equilibrium value after spin manipulations.
DOI: 10.1103/PhysRevB.96.041109
Introduction. The promising perspective of combining
traditional electronics with novel spintronics devices leads tointense research into controlling and switching the magneticproperties of such nanodevices. Experimentally magneticproperties of adatoms on surfaces [ 1–6] or magnetic molecules
[7–14] might serve as the smallest building blocks for
spintronic devices.
From a theoretical perspective, the two-impurity Anderson
model (TIAM) [ 15–17] constitutes an important but simple
system which embodies the competition of interactions be-tween two localized magnetic moments with those betweenthe impurities and the conduction band. The TIAM hasbeen viewed as a paradigm model for the formation of twodifferent singlet phases separated by a quantum critical point(QCP): a Ruderman-Kittel-Kasuya-Yosida (RKKY)-inducedsinglet and a Kondo singlet [ 15]. This QCP investigated by
Jones and Varma [ 17–19], however, turned out to be unstable
against particle-hole (PH) symmetry breaking [ 20] and the two
different singlet phases are adiabatically connected. This led tothe conclusion that for finite distances between the impurities,no QCP exists and the original finding is just a consequence ofunphysical approximations [ 21] which is generically replaced
by a crossover regime.
In this Rapid Communication, we establish that the model
exhibits another realistic QCP for any inversion symmetricone-dimensional (1D) dispersion, depending only on theabsolute value of the wave vector. The existence of thisdifferent QCP relies only on the fact that for specific distancesRbetween both impurities, either the even- or odd-parity
contributions to the conduction band decouple from the impu-rities at low-energy scales, leading to an underscreened Kondoeffect. This underscreened Kondo fixed point (USK FP) [ 22]
has a doublet ground state which is different from the singletground state for large antiferromagnetic interactions betweenboth impurities, excluding a smooth crossover between bothphases. This QCP trivially also exists for the limit R→0
in all dimensions [ 23,24]: For this special case, the QCP
has been recently observed in molecular dimers [ 7], where
*benedikt.lechtenberg@tu-dortmund.dethe different phases can clearly be detected in the scanning
tunneling spectra.
Here, we present the generalization to finite distances, its
robustness against particle-hole symmetry as well as paritybreaking, and demonstrate that the quantum phase transition(QPT) can also be evoked by applying a gate voltage tothe impurities. Since the entanglement between the impurityspins is protected by a dynamical symmetry in the parity-symmetric case, the spin-spin correlation function cannotcompletely decay to its equilibrium value, and, therefore,might be useful for future qubit implementations. Possibleexperimental realizations for finite distances could be inpseudo-1D nanostructures [ 25–29] or optical lattices [ 30–32].
Model. We consider the two-impurity Anderson model
(TIAM) whose Hamiltonian can be separated into the partsH
TIAM=Hc+HD+HI.Hccontains the conduction band
Hc=/summationtext
/vectork,σ/epsilon1(/vectork)c†
/vectork,σc/vectork,σandHDandHIcomprise the impurity
contribution and the interaction between the conduction band
and impurities, respectively,
HD=/summationdisplay
j,σEjd†
j,σdj,σ+U/summationdisplay
jnj,↑nj,↓+/vectorh/summationdisplay
j/vectorSj
+t
2/summationdisplay
σ(d†
1,σd2,σ+d†
2,σd1,σ), (1)
HI=V√
N/summationdisplay
j∈{1,2}k,σc†
k,σeikRjdj,σ+H.c., (2)
withd†
j,σcreating an electron with spin σand energy Ejon
impurity jlocated at position R1/2=±R/2,nj,σ=d†
j,σdj,σ,
a local magnetic field /vectorhapplied to the spin /vectorSj=1
2d†
j,σ/vectorσσ,σ/primedj,σ
of impurity j, and c†
/vectork,σcreating a conduction electron. At
low temperatures, the tunneling tleads to an effective anti-
ferromagnetic exchange interaction K/vectorS1/vectorS2, with K=t2/U
between the impurity spins. Throughout this work, unlessstated otherwise, we will consider the case E
1=E2=E=
−U/2 for simplicity such that both impurities are occupied
with one electron. Below, we will show that the QCP is whollyrobust to a departure from parity and particle-hole symmetries.
For the numerical renormalization-group (NRG) approach
[33–36], it is useful to introduce a parity eigenbasis
2469-9950/2017/96(4)/041109(5) 041109-1 ©2017 American Physical SocietyRAPID COMMUNICATIONS
LECHTENBERG, EICKHOFF, AND ANDERS PHYSICAL REVIEW B 96, 041109(R) (2017)
de/o,σ=1√
2(d1,σ±d2,σ) for the impurity degrees of freedom
[7,18–20,37–40]. In this basis, the orbitals with even/odd
parity couple to corresponding even/odd parity conductionbands via the energy- and distance-dependent hybridizationfunctions (see Supplemental Material [ 41]),
/Gamma1
e(/epsilon1,/vectorR)=2πV2
N/summationdisplay
/vectorkδ(/epsilon1−/epsilon1(/vectork)) cos2/parenleftbigg/vectork/vectorR
2/parenrightbigg
, (3a)
/Gamma1o(/epsilon1,/vectorR)=2πV2
N/summationdisplay
/vectorkδ(/epsilon1−/epsilon1(/vectork)) sin2/parenleftbigg/vectork/vectorR
2/parenrightbigg
. (3b)
A proper consideration of the energy dependence of these
functions generally breaks particle-hole symmetry [ 20,21] and
hence destroys the well-known QCP predicted by Jones andVarma [ 18,19,38].
Hybridization functions. Examining the definitions of the
hybridization functions /Gamma1
e/o(/epsilon1,/vectorR) reveals an important funda-
mental property: If all wave vectors /vectork/primefulfilling /epsilon1(/vectork/prime)=0a l s o
satisfy the condition /vectork/prime/vectorRn=πn, with nbeing an integer, one
of the two hybridization functions exhibits a pseudogap ∝|/epsilon1|2
because either the sine or the cosine in Eqs. ( 3) vanishes for
/epsilon1→0. While for a general dispersion this requirement is not
fulfilled, infinitely many equidistant Rn=|/vectorRn|obeying this
requirement are found for a 1D inversion symmetric dispersionwith/epsilon1(k)=/epsilon1(|k|). Note that the presented results are valid
for the case that the mean free path of the electrons in theconduction channel is larger than the distance R.
Since the Kondo screening breaks down for a pseudogap
hybridization function vanishing as |/epsilon1|
r, withr>1/2[42–45],
the Kondo effect of the even or odd conduction band willdisappear for the specific distances k
/primeRn=πn, leading to an
underscreened spin-1 Kondo fixed point (USK FP) with aneffective free spin-1 /2 remaining.
The odd-hybridization function completely vanishes for
any dispersion and R→0 on all energy scales, leading to
a single-channel model and, thus, trivially to an USK FP. For a1D linear dispersion /epsilon1(k)=v
F(|k|−kF), Eqs. ( 3) yield [ 39,40]
/Gamma11D
e/o(/epsilon1,R)=/Gamma10/braceleftbigg
1±cos/bracketleftbigg
kFR/parenleftbigg
1+/epsilon1
D/parenrightbigg/bracketrightbigg/bracerightbigg
, (4)
with/Gamma10=πρ0V2, the half bandwidth D, the constant density
of states of the original conduction band ρ0=1/2D,k F=
π/2a, andathe lattice constant. The hybridization function
of the even conduction band exhibits a gap for distancesk
FR=(2n+1)πand the one of the odd band for kFR=2nπ.
Note that with increasing distance R, the frequency of the
oscillations in /Gamma11D
e/o(/epsilon1,R) increases and, consequently, the
width of the gap becomes smaller so that the stable low-energyFP is reached at increasingly lower temperatures.
Doublet ground state. Generically, a singlet ground state
is found in the TIAM since either the two impurity spinsare bound in a local singlet for strong antiferromagneticcorrelations between the impurities or the impurity spinsare screened by the surrounding conduction band electronsto spatially extended Kondo singlets [ 18,19,38]. A different
situation arises for the specific distances k
FRn=nπ, where
one conduction band decouples at low energies. This is00.10.20.30.40.5
00.511.522.5
00.511.522.5
10−1010−810−610−410−2100(a)
(b)
(c)μ2
effkFR=1 00π
kFR=1 01π
kFR=1 02π
kFR=1 03πkFR=1 04π
kFR=1 05π
kFR=1 06πSimp/ln(2) Simp/ln(2)
T/Γ0kFR=0.08π
kFR=0.09π
kFR=0.10πkFR=1.03π
kFR=1.04π
kFR=1.05π
FIG. 1. (a) The effective local magnetic moment μ2
effand
(b) entropy of the impurities for the TIAM plotted against the
temperature Tfor different distances. (c) The temperature-dependent
entropy for distances that slightly deviate from Rn=0 (solid
lines) and Rn=1 (dashed lines). Model parameters are t=0,E=
−5/Gamma10,U=10/Gamma10,a n dD=10/Gamma10.
demonstrated in Fig. 1where the effective impurity magnetic
moment μ2
effand the entropy Simp[46] are plotted for different
Rn. The USK FP with a free unquenched spin-1 /2 remaining
is the only stable fixed point for vanishing spin-spin interactionK=0(t=0) characterized by μ
2
eff=0.25 and the impurity
entropy Simp=ln(2).
At very large distances Rn, the gap in one of the
hybridization functions becomes very narrow so that thecrossover to the USK FP only occurs at very low temperatures.For such distances, at first both impurities are screened bythe two conduction bands, leading to an almost vanishingmagnetic moment μ
2
eff≈0 and entropy S≈0. However,
the renormalization of the effective Kondo coupling andconsequently the screening of one local spin always stopsat a finite temperature due to the pseudogap hybridizationfunction and, therefore, the screening is never complete. Sincethe hybridization to one conduction band vanishes at the Fermienergy, the coupling to that band subsequently decreases untilfinally the USK FP emerges at very low temperatures.
In between these two FPs, the model exhibits another
unstable FP with μ
2
eff=0.125 and entropy SImp=2l n ( 2 ) .
The values for μ2
effandSare a feature of the the gapped
Wilson chain [ 47] and are not related to the impurity physics.
While μ2
eff(T) starts to increase until it reaches the value
μ2
eff=0.125 in the regime of the unstable FP, the impurity
spins remain screened so that the local moment of theimpurities μ
2
loc(T)=Tlimhz→0/angbracketleftSz
j/angbracketright/hz[43,48] continues to
decrease linearly with decreasing T. Since the impurity spins
are only completely screened at T=0 in the conventional
Kondo problem, the screening of the impurity spins progressesuntil the USK FP is reached at low temperatures where thelocal moment μ
2
loc(T) and remains constant for T→0a si t
is expected for a free but strongly reduced magnetic momentin the Curie-Weiss law [ 49].
The low-temperature crossover scale from the unstable FP
to the stable USK FP depends on the degree of screening:
041109-2RAPID COMMUNICATIONS
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the smaller μ2
loc(TGap) at the energy scale TGapat which the
pseudogap develops, the smaller the crossover temperaturescale. Such a vigorous screening can be achieved in two ways:either the distance R
nis increased so that the screening stops at
lower temperatures (shown in Fig. 1) or the coupling Vto the
bands is increased so that the impurities are already stronglyscreened at a higher T.
Although a small departure from the specific distances R
n
theoretically always leads to a singlet ground state at very low
temperatures, the system will stay in the now unstable doubletfixed point for all experimentally relevant temperatures if thedeparture is not too large, which can be seen in Fig. 1(c).A s
a result, using k
F=π/(2a) of a linear dispersion yields that
in experiments at finite temperatures, variances of Rnby up to
20% of the lattice constant are still sufficient to detect a sharpchange in the magnetic properties of the system.
Quantum critical point. While for a vanishing spin-spin
interaction between the impurities the ground state is alwaysa doublet at R
n, both impurity spins form a local spin
singlet for sufficiently strong antiferromagnetic interactionsK. Therefore, these two phases must be separated by a QCP.
Unlike the unstable Jones-Varma QCP [ 18–20] separating
two singlet ground states, this QCP is protected by the spinas a conserved quantum number. While the Jones-VarmaQPT is continuous [ 19,20], we found in the parity-symmetric
case a linear decreasing energy scale with decreasing |t−t
c|
typical for a parity-protected level-crossing QPT, while inthe parity-broken case we observed a exponentially van-ishing energy scale indicating a Kosterlitz-Thouless QPT[23,50–52].
The different nature of the QPTs is also revealed in the
local correlation function /angbracketleft/vectorS
1/vectorS2/angbracketright. While in the absence of K
(t=0) a local triplet screened by the Kondo effect at low
Tto a doublet is part of the ground state, a local singlet
forms and suppresses the Kondo effect [ 18,19]f o rl a r g e
antiferromagnetic K. This leads to /angbracketleft/vectorS1/vectorS2/angbracketright>0 in the former
regime, while in the latter, one finds /angbracketleft/vectorS1/vectorS2/angbracketright<0. For the Jones-
Varma QCP, /angbracketleft/vectorS1/vectorS2/angbracketrightvaries continuously across the QPT and
only its derivative diverges at the QCP. This is due to a mixingterm in the Hamiltonian [ 17–19] exchanging even and odd
conduction electrons via impurity scattering processes. Globalparity remains conserved, but local parity on the impuritysubsystem is broken. This is contrasted by the behavior of
/angbracketleft/vectorS
1/vectorS2/angbracketrightatRn, as shown in Fig. 2. Since one band decouples at
low-energy scales, the band mixing term is suppressed and adynamical local parity conservation ensures the conservationof/angbracketleft/vectorS
1/vectorS2/angbracketrightat low temperatures and prohibits the decay to its
equilibrium value after a spin manipulation. Consequently, thecorrelation function has to change discontinuously at the QCPfor a parity-symmetric model [ 53].
Furthermore, the QPT is even robust against parity break-
ing: We have added a small /Delta1Eto one of the two single-particle
levels, i.e., E
1=E+/Delta1E, which is one of several ways of
breaking the parity. Although the spin-correlation functionvaries now continuously in the parity-broken case, as depictedin Fig. 2, other quantities such as the magnetic moment μ
2
eff,
the entropy Simp(shown in the inset of Fig. 2), or the spectral
functions still show a discontinuity at the renormalized critical
tunneling tc(/Delta1E), marked on the xaxis in Fig. 2.−0.6−0.4−0.200.2
00 .25 0 .50 .75 1 1 .25 1 .51 .75 2 tc00.20.40.60.81
0.25 0 .50.75 1S1S2
t/Γ0p. sym.
p. brokenSImp/ln(2)
FIG. 2. The correlation function /angbracketleft/vectorS1/vectorS2/angbracketrightplotted against the tunnel-
ingtfor the distance kFR=π. While in the parity-symmetric cases
(solid lines) /angbracketleft/vectorS1/vectorS2/angbracketrightmust change discontinuously, in the parity-broken
case the correlation function is continuous. The inset shows the
entropy for the parity-broken case and the new critical tc(/Delta1E)i s
marked on the xaxis.
For the parity-conserving case, the spectra of the odd and
even orbital [ 54,55] are shown in Fig. 3for the two different
phases and the distance kFR=π, at which the even orbital
decouples from the conduction band at low-energy scales.The spectral functions exhibit the same features as in theR=0 case [ 7,24], but with the role of even and odd spectra
interchanged.
The spectrum for the odd orbital develops an underscreened
Kondo peak [ 56]a tω=0f o rt<t
c, which collapses once the
tunneling exceeds t>t c. In this phase, both impurity spins are
bound into a local singlet.
In contrast, ρevenalways develops a gap around the Fermi
energy for all t/negationslash=tc: the pseudogap in the even-hybridization
function suppresses the Kondo screening of the spin in theeven orbital. Furthermore, at low frequencies, the orbitaldecouples from the hybridization processes. Injecting/ejectingan electron into/from the even orbital changes the local particlenumber, which cannot relax but induces a suddenly changedCoulomb potential for the odd orbital. The only way the system
0.000.050.100.150.200.25
−10−50 51 0
0.000.501.001.502.002.50
−0.4 −0.20 0 .20 .4−10−50 51 0(a)
(b)ρodd(ω)t/Γ0=1.1
t/Γ0=1.2
t/Γ0=1.3
t/Γ0=1.4
t/Γ0=1.5ρeven(ω)
ω/Γ0
FIG. 3. Spectral function of the (a) odd and (b) even orbital for
tunnelings t<t c(solid lines) and t>t c(dashed lines) and /Delta1E=0.
For the distance kFR=π, the even orbital decouples for T→0 from
the conduction band.
041109-3RAPID COMMUNICATIONS
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FIG. 4. /angbracketleft/vectorS1/vectorS2/angbracketrightphase diagram plotted against EandtforkFR=π
andU=10/Gamma10.
can respond at T=0 is by changing the many-body ground
state. This leads to the well-understood x-ray edge physics [ 57]
also found in the Falicov-Kimball model [ 58]. The excitations
around the Fermi energy thus indicate transitions from thedoublet to the singlet phase for t<t
c, and vice versa for t>t c.
Consequently, the width of the gap in the spectrum is givenby the energy difference between the doublet and singlet stateand vanishes for t→t
c. Note that for distances kFR=2nπ,
the spectral functions of the even and odd are interchanged.
In the general parity-broken case, features of the even
orbital are weakly mixed into the spectral function of theodd orbital, and vice versa, since in this case both orbitalsare coupled to both conduction bands [ 7]. Experimentally, the
QPT can be detected by measuring the differential conductancethough an impurity which is proportional to a superposition ofthe even and odd spectral functions. We predict that for t<t
c,
a clear Kondo peak at the Fermi energy is visible below theKondo temperature T
K. This Kondo peak disappears for t>t c,
and only the finite frequency excitations stemming from thex-ray edge physics of the weakly coupled orbital are mixed inas recently detected in a molecular dimer system [ 7].
Since the tunneling tis generated by the overlap of orbital
wave functions of the adatoms or molecules in experiment[7], variation of the tunneling tis experimentally difficult. The
case of a fixed Ebut different discrete tchanged via molecule
geometry has been recently realized [ 7] for the extreme
case of R≈0, but is not suitable for electronic switching
of the local spin configuration.
However, it is also possible to evoke the QPT for a fixed
tunneling tvia a gate voltage shifting both orbital level energiesE. Figure 4depicts a phase diagram of the correlation function
/angbracketleft/vectorS
1/vectorS2/angbracketrightas a function of Eandt. Ferromagnetic correlations
(red and yellow), indicating that the system is in the doubletphase, are developing inside a tube. If either the tunneling tor
the energy level Eis sufficiently increased or decreased, the
system is found in the singlet phase (blue and green). Insidethe tube, the local magnetic moment and the impurity entropytake the fixed values μ
2
eff=0.25 and Simp=ln(2), while
outside both vanish [ 59]. For very large positive or negative
level energies, |/angbracketleft/vectorS1/vectorS2/angbracketright| → 0 decreases continuously since the
orbitals become either doubly occupied or empty. Note that inthis case, the Kondo effect will also break down in the doubletphase since there is no local moment in the coupled orbital thatcan be screened. To understand the asymmetry with respect toEandt, it is useful to monitor the single-particle energies in
the even-/odd-parity basis where both energies are split by thetunneling E
e/o=E±t/2 so that the even/odd level energy
is increased/decreased with increasing t. In order to evoke a
transition from the singlet to the doublet phase, the decoupledorbital has to be shifted towards half filling such that it becomessingly occupied, again which can only happen discontinuously.Consequently, if the distance is changed from an odd distancek
FR/π=2n+1 ,s h o w ni nF i g . 4, to an even distance, the
roles of the even/odd orbital as the uncoupled/coupled orbitalare interchanged and the phase diagram is hence mirrored atthe line t=0.
Summary. We have shown that the TIAM exhibit a QCP for
a 1D dispersion /epsilon1(k)=/epsilon1(|k|) in the cases that the impurities
are separated by specific distances R
n. In contrast to the
unstable QCP [ 18,19] usually discussed in the context of
the two-impurity models, the QCP presented in this RapidCommunication is stable to departure from particle-hole andparity symmetry.
We believe that this system may be of great relevance
for spintronic devices since it is possible by applying gatevoltages to turn on and off a free magnetic moment which isnot screened at low temperatures. Along with the magneticmoment, one can switch on and off a Kondo effect with itssharp conductance peak at the Fermi energy. Furthermore, inthe parity-symmetric case, the spin-spin correlation betweenboth impurity spins is protected by the parity as a conservedquantity, making this system promising for spin-qubit realiza-tions.
Acknowledgments. We acknowledge useful discussions
with S. F. Tautz and R. Bulla. B.L. thanks the Japan Societyfor the Promotion of Science (JSPS) and the Alexander vonHumboldt Foundation.
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041109-5 |
PhysRevB.102.024206.pdf | PHYSICAL REVIEW B 102, 024206 (2020)
Spin-glass phase transition revealed in transport measurements
Guillaume Forestier , Mathias Solana, and Cécile Naud
Institut Néel, Centre National de la Recherche Scientifique, B.P . 166, 38042 Grenoble Cedex 09, France
Andreas D. Wieck
Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität, Universitätstraße 150, 44780 Bochum, Germany
François Lefloch
Institut Nanosciences et Cryogénie, Commissariat à l’Énergie Atomique, 17 avenue des Martyrs, 38054 Grenoble Cedex 09, France
Robert Whitney
Laboratoire de Physique et Modélisation des Milieux Condensés, B.P . 166, 38042 Grenoble Cedex 09, France
David Carpentier
Laboratoire de Physique, École Normale Supérieure de Lyon, 47 allée d’Italie, 69007 Lyon, France
Laurent P. Lévy and Laurent Saminadayar*
Institut Néel, Centre National de la Recherche Scientifique, B.P . 166, 38042 Grenoble Cedex 09, France
and Université Grenoble-Alpes, B.P . 53, 38041 Grenoble Cedex 09, France
(Received 4 May 2018; revised 5 May 2020; accepted 14 May 2020; published 31 July 2020)
We have measured the resistivity of magnetically doped Ag:Mn mesoscopic wires as a function of temperature
and magnetic field. The doping has been made using ion implantation, allowing a distribution of the dopants in the
middle of the sample . Comparison with an undoped sample, used as a reference sample, shows that the resistivity
of the doped sample exhibits nonmonotonic behavior as a function of both magnetic field and temperature,revealing the competition between the Kondo effect and the RKKY interactions between spins. This proves thattransport measurements are still a reliable probe of the spin-glass transition in nanoscopic metallic wire dopedusing implantation.
DOI: 10.1103/PhysRevB.102.024206
Spin glasses are one of the most fascinating states of mat-
ter. They have attracted the interest of a large community forseveral decades, as it is one of the most fundamental problemsin condensed matter physics [ 1]. A spin glass appears when
magnetic atoms are randomly diluted in a nonmagnetic metal-
lic host. As the spatial distribution of the spins is random,the Ruderman-Kittel-Kasuya-Yoshida (RKKY) interactionsbetween the spins [ 2], which depend on the distance between
them, are also random: This leads to frustration between themagnetic moments. It is this interplay between disorder andfrustration which is at the basis of the formation of a spin glass
below a transition temperature T
sg. It has been shown recently,
using a very tricky experiment, that this subtle scenario forthe formation of a spin glass is actually the real one [ 3]. The
very nature of the ground state is still heavily debated and mayconsist of an unconventional state of matter with remarkablebehaviors [ 4]. Let us mention, however, that spin glasses
may also appear in insulating systems [ 5,6]: This peculiar
situation will not be discussed in this paper as it is focused on
*saminadayar@neel.cnrs.frtransport measurements. Finally, it is worth mentioning that
spin glass models have also been applied to sociology, biology[7], economy [ 8,9], and games [ 10] and are deeply connected
to mathematics (topology) [ 11].
The spin-glass transition itself is quite subtle to detect. In
contrast with other phase transitions, there is, for example, nodivergence of the specific heat at the transition temperature,but rather a cusp in the (low-frequency) magnetic suscepti-bility [ 6]. This has been the most commonly used technique
to detect the transition down to very low concentrations ofmagnetic impurities [ 12]. More surprising is the onset of
irreversibility in the glassy phase: magnetization measured
during a cooldown under magnetic field [field cooled (FC)procedure] is completely different from the magnetizationobtained when the magnetic field is applied after cooling
the sample [zero field cooled (ZFC) procedure]. This dis-crepancy between these two measurements appears exactlyat the spin-glass transition temperature [ 13,14]. This onset of
irreversibility is one of the most characteristic signatures of
the spin-glass transition.
Another way to probe the spin glass state and the spin
glass transition consists of measuring transport properties.
2469-9950/2020/102(2)/024206(7) 024206-1 ©2020 American Physical SocietyGUILLAUME FORESTIER et al. PHYSICAL REVIEW B 102, 024206 (2020)
Historically, people were interested in measuring Kondo ef-
fect as a function of the concentration of magnetic impurities.For high doping levels, they observed deviations from thewell known logarithmic Kondo-like increase of the resistivity:Below a certain temperature, the resistivity exhibits a broadmaximum followed by a subsequent decrease [ 15–18]. Such
a behavior was interpreted as a transition to a spin-glass stateas spin-spin interactions become predominant. This interpre-tation was supported by theoretical work [ 19] which relates
the maximum in the resistivity at a temperature T
gto the
spin-glass transition. Another natural way to detect changesin the spin configuration of a metal consists of measuring theanomalous Hall effect [ 20,21]. As interactions between spins
become predominant, anomalous Hall resistance exhibits abroad maximum followed by a rapid decrease as the numberof free (unfrozen) spins becomes smaller and smaller [ 22]. A
more subtle and difficult way to probe the spin-glass phaseconsists of measuring the resistance noise of a metallic spin-glass sample [ 23]. Finally, it should be stressed that the most
striking feature of the Kondo physics has also been observedin transport measurements: The onset of irreversibility hasbeen observed in field-cooled (FC) /zero-field cooled (ZFC)
resistivity measurements [ 24].
The question of the existence of a spin-glass transition
for small systems has been largely addressed theoreticallyusing numerical simulations. Using state-of-the-art computingfacilities [ 25], it has been shown that samples as small as
30×30×30 spins already exhibit a spin-glass transition
[26–28]. Experimentally, the situation is much more complex:
In a mesoscopic sample (i.e., containing few spins), magneticsignals become too small to be detected, even using themost sensitive techniques like
SQUID measurements. It is thus
tempting to use transport measurements to detect the spin-
glass transition, as resistivity measurement can easily be per-formed on nanometer-size samples. This technique has beenused in the past to measure the temperature dependance of theresistivity of Kondo systems [ 29]: The idea was to detect the
effect of the finite size of the sample on the development ofthe Kondo cloud and thus on the screening of the magneticimpurities by the conduction electrons.
Metallic spin glasses are usually obtained by dilution of
magnetic impurities in the host metal. Below a critical con-centration at which an alloy is formed, impurities will endup in interstitial positions and the crystal structure of thehost metal is preserved. The main problem of this technique,however, is that impurities may move during the evaporationand form clusters. Determination of the actual concentrationand actual spin of the impurities are then sometimes delicate.An alternative technique consists of implanting ions in themetallic host [ 30]. This technique allows us to control per-
fectly the dose implanted [ 31] and to avoid any clustering
as energy barriers forbid any displacement of ions at roomtemperature. Two points, however, differ between these twotechniques: During an implantation, the concentration of theions through the thickness of the sample presents a gaus-sian distribution. Very few ions are present at the surfaceand spin glass physics should not be modified by surfaceeffects as it has been sometimes alleged. Moreover, ions areimplanted at high energy: Damages are thus induced in thecrystalline structure of the host metal. As RKKY interactions
FIG. 1. Electron micrograph of the sample. The wire is 18 μm
long, 200 nm wide, and 50 nm thick. Several voltage probes are set
along the wire distant by 2 μm.
are mediated by conduction electrons, spin glass transition is a
many impurities problem: The equivalence between these twoimplantation techniques, from a physical point of view, is aninteresting and sensible question.
In this paper, we report on transport measurements on
nanometer-size mesoscopic Ag:Mn spin glasses at very lowdoping level. Doping has been obtained by ion implantationtechnique. Measurements have been performed down to verylow temperature (50 mK) and up to high magnetic field (8 T).We show that the behavior of the resistivity can be perfectlyexplained by an interplay between the Kondo effect and aspin-glass phase transition. This proves that our mesoscopicsamples exhibit a phase transition similar to what is observedfor macroscopic samples and that the implantation process,although producing crystalline defects, does not change thespin glass physics in such confined geometries. The observedtransition temperature is identical to the one detected pre-viously on macroscopic samples doped by standard dilutiontechniques and with the same doping level using magneticmeasurements.
Samples have been fabricated on a silicon /silicon ox-
ide wafer using standard electron-beam lithography onpolymethyl-methacrylate resist. Geometry of the sample con-sists of a long (length L≈18μm) and thin (width w≈
200 nm, thickness t ≈50 nm) wire (see Fig. 1). Several con-
tacts have been put along the wire, in order to measure theresistance over different lengths and to thermalize properlythe electrons along the wire [ 32]. Silver has been evaporated
using a dedicated electron gun evaporator and a 99 .9999%
purity source with no adhesion layer. Samples have then beenimplanted with Mn
2+ions of energy 70 keV. This energy has
been chosen after numerical simulations based on calculationsusing the
SRIM software [ 31] in order to ensure that ions will
end up in the sample following a gaussian distribution whosemaximum lies in the middle of the sample thickness. Thistechnique of implantation allows us to avoid clustering ormigration of the Mn
2+ions as no further annealing has been
performed on the samples. The ion dose, measured viathe
current of the implanter, has been chosen in order to givea final ion concentration in the wire of 500 ppm. For this
024206-2SPIN-GLASS PHASE TRANSITION REVEALED IN … PHYSICAL REVIEW B 102, 024206 (2020)
FIG. 2. Resistivity as a function of temperature for the pure Ag
sample. Inset: low temperature part ( T/lessorequalslant1 K) plotted as a function
of 1/√
T. The red line is a fit of the data using Eq. ( 1).
concentration of magnetic impurities, the number of spins in
a section of the wire is approximately 450. This number isquite small (typically a 15 ×30 network of spins) and can be
reasonably compared with the dimensions of state-of-the-artnumerical simulations. Finally, note that some samples havebeen left unimplanted (pure) in order to be used as referencesamples.
Samples have been cooled down in a dilution fridge whose
base temperature Tis≈50 mK and equipped with a two-axis
superconducting coil of maximum magnetic fields B
z=8T
(out-of-plane field) and Bxy=1.5 T (in-plane field). Trans-
port measurements have been carried out using an ac lock-intechnique at a frequency of 11 Hz in a bridge configurationand a very low ac current in order to avoid any overheating ofthe sample. In particular, we have kept eV
ds/lessorequalslantkBTwith Vds
the drain-source voltage, ethe charge of the electron, and kB
the Boltzmann constant. We have seen no dependance of the
resistivity on the frequency of the excitation up to our experi-mentally accessible possibility ( ≈100 kHz). The signal is am-
plified using an ultra-low noise homemade voltage amplifier(voltage noise S
v=500 pV /√
Hz) at room temperature. All
the measuring lines connecting the sample to the experimentalsetup consist of lossy coaxes which ensure a very efficientradio-frequency filtering and thus a good thermalization of theelectrons in the sample [ 33–35]. At 4 .2 K, the resistance of
our samples is R≈10/Omega1for the pure samples and R≈30/Omega1
for the doped ones.
The resistivity of the pure sample as a function of temper-
ature is depicted in Fig. 2(a small magnetic field of ≈1000 G
is applied in order to cancel the weak localization correction[36]). At high temperature, the resistivity is dominated by
electron-phonon interaction and thus decreases rapidly withdecreasing temperature. At low temperature, one observes aslight increase of the resistivity; this is typical of mesoscopicsamples, in which electron-electron interactions modify thedensity of states at the Fermi energy [ 36]. The resistance Rof
the sample as a function of the temperature Tis then given by:
R(T)=0.782λ
σR2
RKLT
L=α√
T, (1)
where LT=√¯hD/kBTis the thermal length, Dthe diffusion
coefficient, Rk=h/e2the quantum of resistance with hthe
FIG. 3. Resistivity as a function of temperature under zero field
of the Ag:Mn spin glass sample doped at 500 ppm. Inset: The black
curve is the same data on a logarithmic scale, while the green curve
has had the electron-electron interaction contribution subtracted.
Planck constant and ethe charge of the electron, and λσa
constant related to the screening parameter of the Fermi liquidtheory [ 36]. For silver, λ
σ≈3.1. The inset of Fig. 2shows
the resistance of the pure sample as a function of 1 /√
T.A s
expected, we observe a nice linear behavior down to 50 mK,proving that the electrons are indeed cooled down to thelowest temperature [ 37].
The same measurements on the doped wire are depicted in
Fig. 3. The behavior is clearly different: At low temperature,
one observes a pronounced minimum, followed by an increaseof the resistivity and an abrupt decrease at very low temper-ature. The temperature dependence of the resistivity can beseparated into four terms:
δρ(T)=δρ
e-ph(T)+δρe-e(T)+δρe-mag(T)+δρwl(T)( 2 )
corresponding, respectively, to the electron-phonon interac-
tion, electron-electron interaction, and electron-magnetic im-purity contribution to the resistivity (which is absent for thepure sample). The term δρ
wl(T) corresponds to the quantum
correction to the conductivity (weak localization).
This magnetic contribution arises from two distinct phe-
nomena. The first one is the well-known Kondo effect whilethe second corresponds to the formation of a spin-glass state.It has to be noted that those two effects do not emerge fromthe same processes. The Kondo effect is due to the scatteringof conduction electrons off up-down degenerated magneticimpurities in the single impurity limit . The typical energy for
this process is the Kondo temperature T
K. Such a behavior has
been extensively studied and leads to a logarithmic tempera-ture dependence of the resistivity [ 38] as the temperature is
reduced.
In order to fit the data, we use Hamann’s law [ 39] which is
ah i g h( T/greatermuchT
K) temperature expansion for the resistivity in
the Kondo regime:
ρKondo (T)=ρ0
Kondo
2/parenleftBigg
1−ln(T/TK)/radicalbig
ln2(T/TK)+π2S(S+1)/parenrightBigg
(3)
024206-3GUILLAUME FORESTIER et al. PHYSICAL REVIEW B 102, 024206 (2020)
FIG. 4. Resistivity of the spin-glass sample as a function of
temperature and after subtraction of the electron-electron interactions
contribution to the resistivity. Left panel: orange line is a fit usingHamann’s law [Eq. ( 3)]. Black line is the same fit but adding a
power law to take into account the phonon scattering at “high” T.
Right panel: same data but restricted to the 0 .05–5.5 K temperature
range. Green line is a fit using Vavilov’s equation [Eq. ( 5)] which
describes the emergence of the spin-glass phase and the correspond-
ing decrease of the resistivity. The different arrows indicate T
Kas the
Kondo temperature, Tsgas the spin-glass transition temperature, and
Tmas the temperature of the maximum in the ρ(T).
with Sthe spin of the magnetic impurities and ρ0
Kondo a
constant given by:
ρ0
Kondo=4π¯hcimp
ne2kF(4)
with nthe electronic density, cimpthe concentration of mag-
netic impurities, ethe charge of the electron, and kFthe
Fermi wave vector. Resistivity of the sample as a function oftemperature is depicted in the left panel of Fig. 4. The orange
line is a fit for T>4 K using equation ( 3), with the fitting
giving S=5/2 and T
K=40 mK. These values are in good
agreement with those found in the literature [ 40] for diluted
manganese ions in silver.
At “high” temperature (from 4 K to 15 K) the main con-
tribution to the resistivity is due to the electron-phononscattering. To take into account for this scattering, we haveadded to Hamann’s law a power law αT
n. Remarkably, this
combination of Kondo effect and electron-phonon scatteringdescribes the experimental data very well between 4 K and15 K.
At “low” temperature (below 4 K), the resistivity deviates
from this simple Kondo description as the spin-glass behaviorstarts to be prominent. Indeed, at those temperatures, RKKYinteractions between magnetic impurities are no more negligi-ble, leading to a progressive lift of the up-down spin degener-acy. The density of spins c
impinvolved in the Kondo processes
is thus lowered, leading to a decrease of the resistivity. In thisregime, the diffusion mechanisms are completely differentand much more complex to calculate. This is related to theappearance of the spin-glass state which is fundamentally verycomplicated to apprehend. Recent theoretical works have beenable, however, to obtain an analytical expression for ρ(T)
by simplifying the RKKY interactions to interactions withinimpurity pairs [ 41,42]. In the limit where T
sg/greatermuchTK, as in ourcase, this leads to:
ρ(T)=A
ln2(T/TK)/parenleftbigg
1−αSTsg
T/parenrightbigg
, (5)
where Ais a constant and αSa constant which depends on
the spin S(forS=5/2,αS=2.33). The temperature of the
maximum in the ρ(T) curve, corresponding to the transition
between the high temperature phase and the spin-glass one, isthen given by:
T
m/similarequalαS
2TsglnTsg
TK. (6)
In the right panel of Fig. 4, we have plotted the low tem-
perature part of the resistivity of the spin-glass sample. Thegreen line is a fit using equation ( 5) with T
K=40 mK. The
fit is in rather good agreement, keeping in mind that equation(5) is valid only in the proximity of T
sg. From this we obtain
that two parameters are A=0.124 and Tsg=500 mK. This
value of Tsgis in perfect agreement with those obtained using
magnetization measurements on macroscopic samples at thesame doping level [ 12] (see especially Fig. 3in this reference).
This suggests that even for such small systems, a spin-glasstransition appears at the same temperature as for macroscopicsystems. It must be stressed, however, that only a completestudy including measurements of thermodynamic quantities(specific heat, magnetization) could unambiguously lead tothis conclusion. Such measurements are, unfortunately, verydifficult on such small samples. The two arrows on the rightpanel of the Fig. 4indicate the spin-glass transition tempera-
tureT
sg(left arrow) and the temperature corresponding to the
maximum of the ρ(T) curve Tm(right arrow). Using equation
(6) with the parameters determined above, we obtain the
theoretical temperature of the maximum Tm≈1.5 K, which
is precisely what is observed experimentally. ConsideringFig. 4, one can see that these resistivity measurements can
be very well described considering the system as a metallicspin glass at low temperature and as a Kondo system athigh temperature. It should be noticed than between 2.5 and3.5 K, experimental data deviate from both fits. This region
corresponds to the range of temperature where Kondo effectand RKKY interactions compete with almost equal strengthand no theoretical approach is able to completely describe thismixed behavior.
Finally, in order to characterize the effect of the magnetic
field on the resistivity of the sample, we have performed highfield magnetoresistance measurements at different tempera-tures. For this purpose, we use the zero field cooled (ZFC)protocol: The sample is cooled down under zero field and themagnetic field is applied at low temperature. The relative vari-ation of the resistivity, /Delta1ρ/ρ
0=(ρ(B)−ρ(B=0))/ρ(B=
0), has been plotted in Fig. 5for temperatures down to
200 mK (i.e., T/lessmuchTsg) and up to 20 K (i.e., T/greatermuchTsg). At
high temperature ( T/greatermuch10 K), curves are perfectly superim-
posed and the magnetoresistivity is perfectly quadratic in B,
as can be seen in Fig. 6. In this regime, the resistivity is
dominated by electron-phonon scattering, and one recoversthe classical quadratic magnetoresistance of metals. In orderto extract the magnetic contribution to the magnetoresistivity,we have subtracted the high temperature contribution in Fig. 6
from the data in Fig. 5. The result, plotted as a function of
024206-4SPIN-GLASS PHASE TRANSITION REVEALED IN … PHYSICAL REVIEW B 102, 024206 (2020)
FIG. 5. Relative magnetoresistivity of a 500 ppm spin glass sam-
ple for different temperatures.
the normalized magnetic field μBB/kBT, with μBthe Bohr
magneton, is depicted in Fig. 7.
For temperatures larger than 1 .5 K, all the curves are rather
well superimposed. This means that, in this range of tem-perature, the resistivity depends only on the ratio μ
BB/kBT,
which is proportional to the polarization, i.e., the number offree spins [ 19,43] (Curie’s law). As mentioned above such
behavior is typical of the Kondo effect as it is a single impurityprocess.
Below 1 K, however, we observe strong deviations to this
behavior, showing that Kondo physics is not the good descrip-tion anymore. Note that this is the temperature at which theρ(T) exhibits a maximum (see Fig. 4). It is quite surprising
that deviations from Kondo physics appear at such high tem-perature as compared to the spin glass transition temperature.We would like to stress, however, that recent measurements ofuniversal conductance fluctuations on mesoscopic spin glasswires suggests that spin-spin interactions already play a rolewell above T
sg[30]. Whether this is due to the nature of
the transition itself or to the reduce dimensionality of thesemesoscopic samples is still an open and intriguing question.
Similar behavior can be observed on the temperature de-
pendance of the resistivity under magnetic field. This mea-surement is depicted in Fig. 8. As we have seen previously,
FIG. 6. Relative magnetoresistivity of a 500 ppm spin glass sam-
ple for temperatures larger than 10 K plotted as a function of B
(left panel) and B2(right panel) (same data as Fig. 5). Blue curve
represents data measured at 10 .1 K when the relative magnetoresis-
tance starts to be positive (see Fig. 5). Green curve represents data
measured well above 10, i.e., far in the saturated regime (it has been
shifted for a sake of clarity).
FIG. 7. Relative magnetoresistivity of a 500 ppm spin glass sam-
ple as a function of the normalized magnetic field for differenttemperatures. The classical B
2contribution has been subtracted.
under zero field, the competition between Kondo physics
and spin-spin interactions leads to a broad maximum in theρ(T) curve. Under magnetic field, the maximum observed
in the ρ(T) curve is progressively suppressed and shifted
towards higher temperatures. This is due to the fact that thefield progressively suppresses the up-down degeneracy andthus the Kondo effect. The slope of the logarithmic increaseinρ(T) becomes weaker: The maximum amplitude in the
resistivity thus decreases [ 43]. Moreover, the shift of the ρ(T)
maximum is explained by the fact that, since Kondo effectis reduced, spin-spin interactions become dominant at highertemperatures; note that this still applies even when magnetic
field becomes larger than the typical spin-spin interactionenergy μ
BB/greatermuchkBTsg.
As a final point, we would like to stress that such a
characterization of nanoscopic spin glass wires fits naturallyinto recent works on mesoscopic spin glasses. In this domain,the goal is to probe magnetic configuration of the spins viathe
dephasing induced on the coherent conduction electrons [ 36].
The most striking result, reproduced in several experiments[30,44–47], of this exploration of mesoscopic systems is that
the configuration of spin glasses are unexpectedly robustagainst the application of high magnetic fields B(μ
BB/greatermuch
kBTsg): Upon cycling samples under magnetic field of several
Tesla, universal conductance fluctuations are perfectly repro-
ducible , meaning that the spins are in the exact same configu-
ration before and after the application of the field. Our work,
FIG. 8. Resistivity of a 500 ppm spin glass sample as a function
of temperature under different magnetic fields.
024206-5GUILLAUME FORESTIER et al. PHYSICAL REVIEW B 102, 024206 (2020)
if it does not shed new light on this strange and remarkable
experimental fact, gives some hints for its interpretation as itexcludes the hypothesis that it may be due to an absence ofspin glass transition in nanoscopic samples.
As a conclusion, we have measured the magnetoresistiv-
ity of a 500 ppm Ag:Mn mesoscopic spin glass wire as afunction of temperature and magnetic field. We have ex-plored temperatures ranging from 50 mK ( T/similarequalT
K/lessmuchTsg)
up to 10 K ( T/greatermuchTsg/greatermuchTK) and magnetic fields up to 8 T
(μBB/greatermuchkBTsg/greatermuchkBTK). Despite the small number of spins
in the section of the wire (typically 15 ×30 spins), we ob-
serve a signature of the spin glass phase transition revealedby a maximum in the ρ(T) curve. Around this maximum,
data can be fitted using Vavilov-Glazman’s law for the lowtemperature (spin-glass) part of the curve and Hamann’slaw for the high temperature (Kondo) part. Moreover, theT
sgandTKextracted from these fits are in agreement with
those obtained by magnetic measurements on macroscopicsamples. These results are validated by magnetoresistancemeasurements in which both the dominance of Kondo physicsand spin-glass behavior are observed in different range oftemperature. This, combined with the recent observation ofirreversibility in the resistivity of mesoscopic samples andnumerical simulations, proves that, even in such small sam-ples, a spin glass phase transition appears when the tem-perature is lower than the typical strength of the spin-spininteraction.
We are indebted to H. Pothier and H. Bouchiat for the use
of their Joule evaporators. We thank T. Costi, L. Glazman, E.Orignac, H. Bouchiat, É. Vincent, A. Rosch, and H. Alloul forfruitful discussions.
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024206-7 |
PhysRevB.54.5097.pdf | Electronic structure, screening, and charging effects at a metal/organic tunneling junction:
A first-principles study
D. Lamoen, P. Ballone, and M. Parrinello
Max-Planck Institut fu ¨r Festko¨rperforschung, Heisenbergstrasse 1, 70569 Stuttgart, Germany
~Received 12 February 1996 !
Byab initio simulation in the density-functional–local-density approximation, we study the structural and
electronic properties of a monolayer of Pd-porphyrin and perylene on a Au ~111!slab, and we investigate the
response of this interface to an external electrostatic field. Our computation closely mimics the metal/organicjunction investigated experimentally by Fischer et al. @Europhys. Lett. 28, 129 ~1994!#that has been shown to
display rectifying behavior and charging effects associated with molecular conduction via single-electrontunneling. The ab initiomodel allows us to determine the conditions for molecular tunneling and to investigate
their dependence on structural and chemical parameters. Moreover, it provides a direct view of screening at themolecular level. @S0163-1829 ~96!04328-7 #
INTRODUCTION
The quest for ever increasing performance of electronic
equipment has focused the attention of researchers towardmolecular electronics that could provide the ultimate devicesin terms of miniaturization and speed.
1The concerted effort
of several experimental groups has recently resulted in ex-amples of molecular systems performing elementary tasks~like conduction, rectification, and switching !of electronic
devices.
2,3With components in the nanometer range, the ex-
perimental control of the active elements is far less directthan in conventional technologies, and theoretical models arelikely to play an increasing role in order to determine thestructure and understand the functionality of these systems.
We apply the ab initio molecular-dynamics ~MD!
technique,
4based on density-functional theory ~DFT!,t o
study the interplay of structural and electronic properties ofordered monolayers of Pd-doped porphyrin ~PdP!and
perylene on the ~111!surface of a gold slab. These organic
molecules are both characterized by a central ring of aro-matic bonds, whose relatively high energy and delocalizationare crucial for applications.
Our computation is directly inspired by the experimental
work of Ref. 2, in which few layers ~from 5 to ;10) of
Pd-substituted phthalocyanine ~PdPc!and a perylene deriva-
tive~PTCDI !are deposited on the ~111!gold surface by the
Langmuir-Blodgett technique. Small-angle x-ray diffractionshows that the molecules ~which are planar !are standing on
the surface
5in a nearly vertical configuration, and are ar-
ranged in one-dimensional stacks whose axes are perpen-dicular to the molecular plane ~see Fig. 1 !. Thermal evapo-
ration of a topmost gold contact creates an organic junctionin which single-electron tunneling has been observed. Thesymmetric Au/PdPc/Au junction displays conduction for
voltage exceeding 0.27 V, while for Au/PTCDI/Au thethreshold for conduction is 0.55 V. The asymmetric sand-
wich Au/PdPc/PTCDI/Au displays rectification of current.The complexity of the interface prevents a direct interpreta-tion of these observations. In particular, recent systematicmeasurements on films with different areas and different filmthickness show that it is still difficult to separate the molecu-
lar contribution to conduction from other effects. Disregard-ing these experimental uncertainties, the qualitative explana-tion of the system behavior is outlined in several papers:
6
Electrons in the molecule occupy discrete energy levels, andresonant tunneling occurs when an external electric fieldaligns the highest occupied molecular orbital ~HOMO !with
the metal Fermi level ( E
F), thus allowing the flow of elec-
trons from the molecule to the metal ~see Fig. 2 !. The con-
duction thresholds therefore measure the offset between thehighest occupied state of the molecule and of the metal. Ex-cited molecular levels are much higher in energy ~at least for
the systems under study here !and do not play a direct role. A
quantitative description with the accuracy and reliability re-quired to have an impact on the development of real devices,however, is much more challenging, and poses to the com-putational investigation several difficult questions concern-ing the following: ~i!the structure and stability of the or-
dered monolayer of organic molecules on the surface; ~ii!the
electronic structure of the Au/PdPc and Au/PTCDI interface,
FIG. 1. Perspective view of a PdP molecule on the ~111!surface
of gold. Molecules in the monolayer form one-dimensional stackswhose axes are perpendicular to the molecular plane.PHYSICAL REVIEW B 15 AUGUST 1996-I VOLUME 54, NUMBER 7
54
0163-1829/96/54 ~7!/5097 ~9!/$10.00 5097 © 1996 The American Physical Societyand its relation with the observed transport properties; ~iii!
the origin and strength of the single-electron phenomena ob-
served in the experiment ~charging effects !;~iv!the influence
of side chains attached to the central molecular body in de-termining the properties of the heterostructure; and ~v!the
effect of temperature on the system.
In the past these questions have been addressed either by
simplified models, or by quantum chemistry computationsfor the isolated molecules. To our knowledge, the direct ab
initiocomputation of structure, electronic properties, and re-
sponse to an external field has never been carried out for asystem of the size and complexity of this heterostructure.
7
The results discussed in the present paper show that the ab
initioMD in conjunction with the density-functional theory–
local-density approximation ~DFT LDA !provides a reliable
model for the system, and highlights the power of themethod for these kinds of problems: the computed propertiesare in good agreement with experiment whenever the com-parison is possible, and the ab initiosimulation allows us to
have a direct and microscopic view of the system behavior.In particular, it offers a convenient way to determine thedependence of the relevant properties on structural param-eters ~like the molecule-surface distance !or chemical substi-
tutions in the organic molecules. Finally, the computation forthe interface in the presence of a static electric field providesuseful insight into the screening of external perturbations bythe organic molecules on the surface and reproduces all the
major features observed in the experiment, including acharging effect discontinuity as a function of the appliedelectric fields.
Our approach is complementary to that of recent papers
on electron tunneling through molecular films.
8These stud-
ies, based on the Landauer formula,9express the conductiv-
ity in terms of a few phenomenological parameters that, inprinciple, can be obtained by ab initio computations like
ours.
THE SYSTEM AND THE AB INITIO MODEL
In our computation we consider mainly monolayers of
Pd-doped porphyrins and perylene ~see Fig. 3 !. Less exten-sive computations on Pd-doped porphyrazine ~PdPz!mono-
layers have also been performed. These organic moleculesare simplified versions of those used in the experiment ofRef. 2 that are characterized by benzene, pentyl, and tetracarboxyl side groups. To gain at least some insight into theeffect of these groups we perform additional computationswith peripheral H atoms substituted by methyl groups, andwe analyze the dependence of the electronic properties of theinterface on the molecule-surface separation, as describedbelow. Our simple substitutions, of course, may only mimicthe effect of saturated groups ~like pentyl !, while still being
inadequate to study in detail the more complex, unsaturatedside groups.
The gold surface is represented by a slab of three hexago-
nal compact planes stacked according to the fcc ABCse-
quence. The slab is perpendicular to the zaxis, and periodi-
cally repeated in the other two directions. The periodicity
along the xdirection ~5.77 Å for PdP and for perylene !
matches the periodicity along the molecular stacks measured
in the experiment. The periodicity along yandzis such to
prevent the lateral interaction among the molecules
(L
y515.0 Å,Lz521.52 Å for PdP, Ly520.0 Å,Lz518.25
Å for perylene !. In the case of PdP/Au ~111!, the slab con-
tains 36 Au atoms, while for perylene/Au ~111!, having a
wider contact area, it contains 48 Au atoms. Our supercell
contains a single organic molecule.
The computation is performed with DFT in the LDA
aproximation, whose standard formulation is described inseveral review papers and books.
11To define quantities that
are discussed later, we summarize here a few basic relations.
For a given atomic configuration $RI%, the ground-state
electronic energy ~defining one point of the Born-
Oppenheimer surface !is the minimum of the functional
FIG. 2. Schematic diagram of the one-electron density of states
at the Au/organic layer/Au junction.
FIG. 3. Atomic structure of ~a!Pd-doped porphyrin, ~b!Pd-
doped porphyrazine, and ~c!perylene.5098 54 D. LAMOEN, P. BALLONE, AND M. PARRINELLOE@RI#5min(
ifi^Ciu21
2D1Vˆ11
2w1eXCuCi&
with respect to variations of the ~Kohn-Sham !independent
electron orbitals $Ci%whose occupation numbers are speci-
fied by $fi%.
The extremum condition is equivalent to a set of
Schro¨dinger-like equations for $Ci%:
$21
2D1VˆKS%Ci5eiCi,
whereVˆKS, the so-called Kohn-Sham potential, includes
contributions from the ion-electron Vˆps(r), Hartree electron-
electron w(r), and the exchange-correlation interaction
mXC@r(r)#,
VˆKS~r!5Vˆps~r!1w~r!1mXC@r~r!#,
and the $ei%, the Kohn-Sham eigenvalues, are introduced as
Lagrange multipliers to ensure the orthonormality of $Ci%.
The optimization of the electronic states and the atomic
structure is achieved by the computational scheme of Carand Parrinello.
12Kohn-Sham orbitals are expanded in a basis
of plane waves with a kinetic energy cutoff of 40 Ry. Onlyvalence electrons are included, and pseudopotentials are usedto describe the electron-ion interaction: hydrogen and thefirst row atoms C and N are described by ultrasoftpseudopotentials,
13while the metal atoms Pd and Au are
modeled by soft scalar-relativistic pseudopotentials of the
Martins-Troullier type.14The 4delectrons of Pd and the
5dof gold are included in the valence charge. Our smallest
gold slab ~36 Au atoms !, therefore, provides a pool of 396
electrons to simulate the metallic environment. The relativeinsensitivity of the computed electronic properties of the slabin contact with the different molecules shows that the size ofthe electron reservoir is sufficient to provide a reasonabledescription of the metallic surface. Of course, the finite sizeof the system ~together with the limitation of our computa-
tion to the Gpoint of the Brillouin zone !implies a discreti-
zation of the Au density of states, which, however, is not too
drastic in the immediate vicinity of the Fermi energy E
F.
For instance, the highest occupied electronic level for theslab~our approximation for the gold Fermi level !is threefold
degenerate, and is separated by less than 0.01 eV from thelowest energy unoccupied state, and by 0.05 eV from thesuccessive pair of unoccupied states. The same gap of 0.05
eV separates E
Ffrom the first underlying occupied state. To
check further the importance of discretization we resort to asimplified model, in which gold is assumed to be monova-
lent, having the 6 selectron as the only valence charge. This
simple scheme is not accurate enough for the quantitativecomputation of the system properties, as we verified by com-parison with the full computation described above. Neverthe-less, it retains some similarity with the full model, especiallyin the description of the eigenstates close to the Fermi level,
which have a strong component derived from the 6 satomic
states. More importantly, the drastic decrease in the numberof electrons allows us to perform computations for severaldifferent system sizes and shapes. The results show that, al-though the metal density of states depends substantially onthe size and shape of the slab, the relative position of themolecular HOMO and the metal Fermi level is much lesssensitive to the slab size. We point out that the G-point ap-
proximation with a unit cell that contains just one organicmolecule implies the neglect of the dispersion for the mo-lecular bands along the axis of the molecular stacking, whichis known to be non-negligible. We comment in the next sec-tion on the effect of this approximation on our results.
For what concerns the description of bonding in the or-
ganic monolayers, recent computations for porphyrins andporphyrazine by two of us
10have shown that the DFT LDA
provides reliable results for geometry and electronic proper-ties of these molecules. In particular, it describes accuratelythe central ring of aromatic bonds, that in all these moleculesprovides the highest occupied molecular orbital.
The final aim of the computation being to gain insight into
the transport properties of the interface, that, in turn, aredependent on the electron eigenvalues, it is necessary to dis-cuss whether our description based on the DFT-LDA is ap-propriate, and expected to be accurate.
As apparent from the schematic diagram of Fig. 2, we are
mainly concerned with the relative position of occupied
states, and, in particular, we are interested in the offset be-tween the gold Fermi energy and the highest occupied mo-lecular orbital. Our computation, therefore, is not affected bythe well known DFT LDA underestimation of the energy gapbetween occupied and unoccupied states, and, in general, ofthe energy of the excited states. Moreover, as discussed indetail below, the interface is made by two weakly interactingparts, i.e., the metal surface and the organic monolayer. Eachof the two eigenstates we compare, therefore, can be thoughtas the highest occupied state of one of the two subsystems,and, as such, they both enjoy the special status of the highestoccupied state: as discussed in Ref. 17, the Kohn-Sham ei-genvalue of the highest occupied state is the only one havinga physical meaning, representing the lowest electron removalenergy. Of course, all the eigenvalues we compute are af-fected by the inaccuracies of LDA, but we may expect thatthese inaccuracies are partially compensated in the eigen-value differences we discuss.
THE STRUCTURE AND ELECTRONIC PROPERTIES
OF THE INTERFACE
The first part of the computation concerns the optimiza-
tion of the molecular position on the surface. Since the slabis very small and the ~111!surface is compact, we do not
relax the Au atomic positions.
15Moreover, the periodicity
alongxis fixed by the experiment, and, therefore, the major
parameter to be determined is the vertical distance of themolecules from the surface. As discussed below, this quan-tity is crucial in determining the transport properties of theheterostructure. Also important, although less crucial, is thelocal relaxation in the region of close contact between themolecule and the surface. These structural properties arecarefully optimized by ab initio MD.
16
Here we describe in detail the results for PdP, mainly to
illustrate the power of the method in the structural investiga-tion of complex systems, while we only summarize the mainfeatures for the other monolayers. At the end of the structuralrelaxation for PdP, the two H atoms closest to the Au surfacesit on the lines joining a central Au atom to two of its neigh-bors. The Au-H distance is 1.78 Å and that is typical of the54 5099 ELECTRONIC STRUCTURE, SCREENING, AND . . .H-metal adsorption distance for several transition and post-
transition-metal ~111!surfaces.18As could be expected, the
main body of the molecules appears to be compressed by theadsorption potential. Vertical distances are contracted withrespect to the free molecule’s values, and the degree of‘‘compression’’ decreases rapidly with increasing distance
from the surface. The C-C bonds closest to the surface, for
instance, are reduced by ;3%; the central ring is already
much less affected, the variation of interatomic distances be-
ing;1.5% in the lower half of the ring, and less than
0.5% in the upper half. The Pd atom is somewhat more af-
fected, but remains tightly at the center of the four inner N
atoms, with bonding angles always very close to 90°. Theadsorption energy is 10.0 eV per molecule. This energy, that
may seem large at first, appears to be rather small if we take
into account the size of the adsorbed molecule: 10 eV is in
fact comparable to the binding energy of two H atoms on thegold surface,
18suggesting that only the atoms closest to the
contact point participate to the metal-molecule bonding.
The results for the perylene/Au interface are very similar:
also in this case H tends to be adsorbed in the bridge con-
figuration, with the shortest H-Au distance of 1.75 Å. The
portion of the molecule closest to the surface is again com-pressed by the adsorption potential, with interatomic dis-
tances changing by 3% at most with respect to the free mol-
ecule values. The adsorption energy is 13.2 eV per molecule.
Once the atomic structure of the interface has been deter-
mined, we turn our attention to the spectrum of single elec-tron eigenvalues. We first analyze the system in the absenceof an external electric field. Since the close contact of mol-ecules and surface occurs at the saturated end of the organicspecies, the hybridization of electron states from the metaland the molecules is not large. This is true even for thehighest energy molecular levels, that, being located at thecentral aromatic region, are only weakly interacting with themetal levels, despite the near coincidence of energies. As aresult, it is always possible to identify each state as belong-ing to the slab or to the molecule.
As mentioned in the Introduction, the quantity we are
interested in is the offset Dbetween the metal Fermi energy
and the highest occupied molecular state: D5E
F2eHOMO.
Our interface has obvious analogies with the classicalSchottky barrier:
19in both cases the fact that one side of the
junction is metallic has simple but important implications onthe relative position of the electron states belonging to themetal and to the overlayer. In principle, the highest occupiedmolecular level may be higher or lower than the Fermi en-ergy of a given metal. However, when the molecule and themetal are brought into contact, the molecular HOMO cannot
be higher than E
F. In such a case, a charge transfer from the
molecule to the empty metal states would decrease the sys-
tem energy and decrease eHOMOdown toEFby a combina-
tion of electrostatic and chemical interactions. In our DFTapproach, that neglects the electron dynamics, the result
eHOMO <EFis valid at any distance. In reality, it will be
relevant only for distances such that the electrons can actu-ally tunnel from the molecule to the surface and equalize thechemical potential of the two subsystems. At short distancesthe direct chemical interaction of the molecule with the metal~a chemically open shell system !will bend the highest mo-
lecular state toward lower energy, while the metal Fermienergy remains constant.
20These simple considerations al-
low us to sketch the behavior of Das a function of the
molecule-surface distance and to distinguish two different
cases, according to whether the eHOMOof the isolated mol-
ecule is higher or lower than EF. In the first case @Fig. 4 ~a!#
a~small !electrostatic interaction will pin the molecular level
toEFfor large separations, while in the second case @Fig.
4~b!#the molecular level is free to attain its intrinsic value
for large separations. At all distances for which
eHOMO ,EF, the molecular levels are filled, and no charge
transfer occurs at the interface.
For the organic overlayers we consider in our study, the
highest occupied states are nonbonding or antibonding orbit-als localized at the central molecular region and character-ized by fairly high energy. As a consequence, the highestoccupied level of the free molecules are close to, and often
higher than E
Fof gold. For large distances, therefore, Dis
either zero or positive and small. Close to the equilibrium
distance the crucial effect determining Dis the bending of
the molecular HOMO due to the interaction between themetal and the central aromatic bonds. By addition of side
chains, which increase the molecule-surface separation, D
will rapidly decrease, although
eHOMOwill always remain
belowEF.
This qualitative analysis of the electronic properties of the
interface greatly helps in organizing the simulation results inan ordered picture that otherwise would be difficult to extractfrom the details of the computation that we describe below.We underline that the qualitative picture discussed here in
FIG. 4. Asymptotic behavior of the offset Das a function of
metal-molecule separation. ~a!TheeHOMOof the free molecule is
higher than the metal Fermi energy EF.~b!TheeHOMOof the free
molecule is lower than EF.5100 54 D. LAMOEN, P. BALLONE, AND M. PARRINELLOterms of the metal EFand the molecular eHOMOremains
valid with only minor modifications in the case in which the
molecular levels give rise to a band with sizable dispersion,as it is known to occur for stacks of porphyrins and phtha-
locyanine. In such a case, the role of
eHOMOis played by the
highest occupied level in the molecular band eHB, that always
satisfies the relation eHB<EF. In the following, however,
we continue to discuss the electronic properties of the inter-
face in terms of eHOMO, which reflects more closely our
model with one molecule per unit cell and the sampling of
the Brillouin zone reduced to the Gpoint.
TheeHOMOof the gas phase PdP molecule is 0.60 eV
higher than EFof gold, and, therefore, D(z) follows the
behavior sketched in Fig. 4 ~a!: it is zero at large distances,
and decreases with a nearly exponential law close to thesurface ~Fig. 5 !. At the equilibrium distance we compute
D50.63 eV, with a characteristic length of 0.75 Å for the
exponential decay. This steep decrease of Dwith distance
suggests that the direct interaction of the HOMO with themetal becomes vanishingly small as soon as the central aro-matic ring is displaced from the surface by the addition ofeven a short side chain. In reality, the effect of chemicalsubstitutions is rather difficult to predict quantitatively with-out a specific computation, since the molecule-surface inter-
action, and therefore D, depends on a variety of factors,
including the energy, symmetry, and degree of localizationof the molecular levels. As an example, substituting the ter-
minal H of PdP with CH
3, equivalent to the insertion of a
CH2group between the central ring of PdP and the H atom
adsorbed on the gold ~PdPCH 3/Au!, increases the molecule-
surface separation by 1 Å. Judging from Fig. 5, this displace-ment would correspond to a Dof 0.07 eV. However, the
chemical addition also increases the energy of the free mol-
ecule HOMO by 0.30 eV and reduces its localization at the
center of the molecule, thus enhancing its bending downwardwhen it is brought into contact with the surface. As a result,
for PdPCH
3at the equilibrium distance we compute
D50.39 eV, which is much closer to the 0.63 eV of PdP/Au.
If we now assume that successive additions of CH 2groups
will reduce Dby the same ratio as the first ~corresponding
again to an exponential dependence on distance !, we esti-
mate D50.06 eV already in the case in which a pentyl group
separates the metal from the molecular central ring.The Pd-doped porphyrazine molecule differs from PdP
only in the central ring, having four nitrogen atoms substi-tuting an equal number of CH groups ~see Fig. 3 !. The two
molecules, therefore, are isoelectronic. The chemical substi-tution stabilizes the molecular HOMO which in the free
PdPz molecule is 0.35 eV lower than E
Fof gold. Interaction
with the surface lowers further eHOMO, resulting in a Dof
1.36 eV at the equilibrium distance. Also, in this case, theaddition of side chains between the molecule and the surface
reduces rapidly D. For instance, the addition of a single
CH
2group reduces Dto 0.94 eV. Extrapolating this behavior
by an exponential law gives D50.2 eV as an estimate for the
interface with a pentyl group between the metal and the por-phyrazine ring. This last interface is indeed rather similar tothe one investigated experimentally in Ref. 2. The Pd-phthalocyanine ~PdPc!molecule used in the experiment dif-
fers from the PdP molecule of our computation mainly byadditional benzene rings interposed between the porphyra-zine and the pentyl chains, which increase further themolecule-surface separation. While PdPc/Au is marginallytoo big for a direct computation, the experience accumulated
with the other molecules allows us to conclude that D50.2
estimated for PdPz/Au is an upper bound for the Dof the
experimental interface. The similarity of this value with theone estimated in the experiment increases our confidence inthe computational scheme.
The
eHOMOof perylene is 0.25 eV lower than EFand the
Dat the equilibrium distance is 1.2 eV. With increasing
molecule-surface distance, Ddecreases exponentially with a
decay length of 2.2 Å. The side chains added to perylene inthe experiment of Ref. 2 will increase the molecule-surfaceseparation, and decrease the interaction of the HOMO with
the metal. Also in this case, therefore, the D51.2 eV com-
puted for the simplified interface ~perylene/Au !has to be
considered an upper bound for the Dof the experimental
interface.
Besides determining D, the interaction with the surface
also modifies the intramolecular electronic structure. Thismodification, however, is rather small, and often does notaffect the shape and relative position of the molecular states.The most apparent modification is a reduction of symmetrythat implies a breaking of molecular degeneracies. For in-
stance, the free PdP molecule has D
4hsymmetry, and few
states close to the highest occupied molecular orbital aretwofold degenerate.
10Adsorption reduces the symmetry and
lifts the degeneracy: at the equilibrium distance for PdP/Au
the highest energy occupied egdoublet is split by 0.1 eV.
Also in this case the effect is very sensitive to the molecule-surface separation.
The results of this section relevant for comparison with
the experimental data can be summarized as follows: explicit
computations give a value of D50.63 eV for the PdP/Au
interface and D51.20 eV for perylene/Au. The analysis of
the dependence of Don the molecule-surface distance and on
simple chemical substitutions allows us to estimate D50.2
eV as an upper bound for the interface PdPc/Au used in theexperiment. Because of size and complexity, it is more dif-ficult to set a precise upper bound for PTCDI/Au. The rapid
decrease of Dwith distance, however, suggests that in
PTCDI/Au the offset will be close to the asymptotic value of
FIG. 5. Offset Das a function of the vertical displacement dof
Pd-doped porphyrin from its equilibrium position on the surface.Dots are the computed values, the continuous line is an exponentialfit. The zero of
dcorresponds to the equilibrium separation.54 5101 ELECTRONIC STRUCTURE, SCREENING, AND . . .perylene/Au for large separations, i.e., D50.25 eV. Both
values are lower than the conduction thresholds measured inthe experiment, and the explicit inclusion of screening isrequired to describe the transport properties of the interfaceat a quantitative level.
TheDestimated here appears to be determined mainly by
the relation
eHOMO <EF, and by the ~weak !interaction of the
electrons in the central molecular ring with the metal surface,while it does not depend much from the intrinsic value of
eHOMOfor the isolated molecule. This result suggests that
our picture is not affected by the fact that the molecularlevels give rise to a band: in such a case the relation
eHB<EFand the interaction with the surface will produce a
similar positive and small offset between EFand the highest
occupied level of the molecular band.
RESPONSE TO AN EXTERNAL STATIC ELECTRIC
FIELD
The energy ordering of the electron eigenvalues and their
identification as slab or molecule states provides the simplestscheme to understand the transport properties of the inter-face. In the case of PdP, for instance, by assuming that themolecular levels are rigidly shifted by an external electro-
static potential V
ext(z), the first electron transfer from the
molecule to the surface is predicted to occur upon applying a
potential difference of 0.63 eV. With further increasing Vext
the conductivity would have a plateau until when, at a volt-
age of 0.72 V, a second molecular level is aligned with the
gold Fermi level, opening a new channel to conduction. Thissimple picture, however, neglects the rearrangement of themolecular levels in the presence of the external electric field,which can be important given the delocalized nature of the
plevels close to the HOMO and the large microscopic field
required to transfer an electron.
In order to account for this effect, we perform computa-
tions for the interface in the presence of an external electro-
static field Vext(z). Due to the use of a plane-wave basis set,
we are forced to periodically repeat our cell in all directions,and, thus, we cannot use a linearly varying potential. Thisdifficulty is solved by applying a sawtooth external potentialwhich varies linearly across the region of contact betweenthe molecule and the metal. We measure the nominal voltage
V
mmapplied to the interface as the difference of Vext(z) be-
tween the center of the slab and the center of the molecule.
As expected, we observe two different regimes in the sys-
tem response: a low-field range, in which the external poten-tial is not sufficient to displace an electron, and a high-fieldregime, in which we observe discontinuous changes in theelectrostatic response due to quantized transfer of chargefrom the molecule to the slab.
21
We first concentrate on the low-field range, extending up
toVmm;2 V, and we describe in detail the results for
PdP/Au as representative of all the systems we analyzed.With increasing amplitude of the external field we observe a
decrease of the offset D~see Table I !. The relation between
the potential-energy difference V
mmandDis linear, but the
slope of DversusVmmis significantly smaller than unity,
thus pointing to an effective screening at the interface. Toidentify the origin of this screening, we analyze the total KSpotential acting on the electrons as a function of externalfield. In Fig. 6 we report the planar average of V
KSas a
function of zin the absence of external field. VKSis modu-
lated by peaks and deep valleys in correspondence with theatomic positions, more pronounced in the slab because of theregular and close packing of the gold atoms. For valence
electrons the scale of the potential modulation is ;15 eV.
Upon application of a weak external field, the total potentialchanges slightly, the change being the superposition of theexternal applied field and the internal field due to the charge
polarization that partially compensates V
ext. This decompo-
sition is illustrated in Fig. 7, displaying the bare
potential Vext(z) and the difference DVKS(z)5VKS(zuVmm)
2VKS(zu0) forVmm51.2 V. The difference of Vext(z) andTABLE I. Offset Das a function of applied potential Vmmfor
the PdP/Au interface.
Vmm~eV!D ~ eV!
0.0 0.63
0.10 0.590.37 0.500.62 0.430.99 0.321.24 0.251.49 0.181.74 0.121.99 0.092.24 0.032.61 0.013.10 0.023.48 0.093.73 0.114.10 0.084.35 0.07
FIG. 6. Kohn-Sham potential across the interface PdP/Au aver-
aged along the xyplane of the surface.5102 54 D. LAMOEN, P. BALLONE, AND M. PARRINELLODVKS(z) is the screening potential Vscr(z). At variance from
VKS,DVKS(z) is smooth. As expected, it is a constant within
the slab, and because of the almost perfect screening of themetal, it raises in the region of close contact between the slaband the molecule, and it is again flat in the central molecularregion, highlighting the metalliclike screening properties ofthe central aromatic rings. Finally, it raises to the bare fieldin the empty space beyond the molecule. The smoothness of
DV
KS(z) allows us to compute an induced potential drop
Vindbetween the metal and the molecule, that is plotted in
Fig. 8 as a function of the external potential difference
Vmm. From the slope of the low-field portion of this curve
we estimate a dielectric constant at the molecular scale
e;3.2. This value of the dielectric constant is instrumental
to bring the values for Destimated in the absence of anexternal field in agreement with the conduction thresholds
measured in the experiment.
The importance of the central ring in the dielectric re-
sponse is underlined by the plot of the polarization charge
@Dr(z)5r(zu0)2r(zuVmm)#forVmm51.2 V, again aver-
aged along the plane parallel to the surface ~Fig. 9 !. The
metallic screening is apparent from the large dipoles inducedat the two metal surfaces, but also apparent is the dipoleinduced on the molecule, whose main contributions comefrom the aromatic molecular backbone.
We measure the charge transfer by computing the total
Mulliken charge on the gold slab and on the molecule, thatis, we compute the projection of the Kohn-Sham orbitals ona basis of localized atomic orbitals. The Mulliken analysis issubject to well-known ambiguities, however, trends shouldbe well reproduced by this approach. Despite the charge po-
larization, and up to V
mm52 V, we do not observe any trans-
fer of charge between the molecule and the surface. The
picture changes almost discontinuously at Vmm52V ,a s
shown in Fig. 10, reporting the variation DQof the total
molecular charge as a function of applied potential. At this
field, with the vanishing of the offset D, we observe the
transfer of one state from the molecule to the slab. The total
charge transferred is ;2e. This is an artifact of the LDA
which does not allow the transfer of a single, spin-unpairedelectron, that is in reality the basic conduction step. Thecharge transfer, in turn, changes discontinuously the proper-ties of the interface as it is apparent from the drop in the
slope of V
indas a function of Vmm~Fig. 8 !, pointing to a
drastic change in the screening by the molecule, and from the
nonmonotonic variation of Dwith increasing Vmmbeyond
2V~Table I !. With a further increase of the applied field, the
process of polarization of the junction and discontinuouschange repeats itself. As can be seen in Fig. 10, we may
FIG. 7. Variation DVKSof the Kohn-Sham potential ~full line !
upon application of the external potential Vext~dash line !. The dif-
ference of the two curves gives the screening potential due to thepolarization charge. Lis the periodicity of the simulation cell in the
zdirection ( L521.52 Å !.
FIG. 8. Induced potential-energy difference Vindbetween the
metal and the molecule as a function of the applied potentialV
ext. The slope of the curve at low field is related to the dielectric
constant of the interface.
FIG. 9. Polarization charge induced by an external electrostatic
field with Vmm51.2 V. The shaded area represents the gold slab.54 5103 ELECTRONIC STRUCTURE, SCREENING, AND . . .observe two of these steps in the molecular charge, although
the behavior of the system is less and less linear, possiblybecause of the small size of the sample we simulate.
If, in analogy with what is done in the experiment, we
interpret these steps in the electron transfer in terms
of a molecular capacitance and apply the relation C
5DQ/DV
ext, we estimate a capacitance of 0.8 310219F. If
we assume that our flat molecule can be represented as adielectric two-dimensional disk, the computed capacitance
corresponds to a radius of 11 Å. Although qualitative, this
estimate is in agreement with the experimental capacitance,and corresponds to the size of the molecular aromatic ring,thus highlighting once more the importance of this region inthe molecular behavior.
CONCLUSIONS
We have presented computations for the structure, elec-
tronic, and transport properties of PdP and perylene mono-layers on the Au ~111!surface with and without an external
electrostatic field. The aim of our computation was to modelthe metal/organic junction studied in a recent experiment~Ref. 2 !, and shown to display rectifying characteristics po-
tentially useful for molecular electronics applications.
The focus on the organic molecules considered in the ex-
periment and our computational study is motivated by theircharacteristic structure, in which a ring of delocalized aro-matic electron levels ~possibly modified by a metal !sits at
the center of a shell of more inert groups. The thermal sta-bility of these molecules, and the ease of their manipulationby the LB technique are also of major relevance for the ap-plication point of view.We have discussed in detail the conditions for molecular
conduction across the interface and their dependence onstructural parameters and chemical composition. In a rigidband model, the potential-energy difference required totransfer one electron from the molecule to the surface isequal to the offset between the metal Fermi energy and themolecular HOMO in the absence of an external field. Al-though qualitatively correct, this picture is quantitatively in-accurate. Direct computations for the interface in the pres-ence of an electrostatic field highlight the importance ofscreening at the molecular level, enhanced by the central ringof aromatic bonds. As a result, conduction occurs with an
applied potential difference ;3.2 times larger than the one
estimated from the unperturbed bands.
A series of computations with external fields of increasing
amplitude reproduces the discontinuous transfer of charge
from the molecule to the surface observed in the experiment,
and related to the matching of the gold E
Fwith successive
molecular levels. The electrostatic potential spacing of thediscontinuities is close to the measured ones, and the capaci-tance associated with these jumps corresponds well to themolecular dimensions.
Although gratifying, the similarity of the computational
and experimental results should be considered together witha strong note of caution. First of all, the computation in-volves many approximations ~such as the LDA, pseudopo-
tentials, finite basis set, small slab to represent the surface !
that could affect the precise value of the computed quanti-ties. More importantly, the complexity of the experimentalsystem still exceeds by far that of the computational model,and further analysis of the behavior of the model and theexperimental system under a variety of conditions is requiredto assess the reliability of the comparison.
In conclusion, we have shown that ab initio simulation
offers a unique tool to study the structure, electronic proper-ties, and response to an external perturbation of these com-plex interfaces, taking into account the interactions of theirdifferent parts in a natural and transparent way. Althoughheavy, the computations described in the present paper, in-
volving ;100 atoms and ;500 electrons, are not among the
largest simulations at the ab initiolevel presently performed
by several groups. The still ongoing progress of computersand algorithms will soon bring our computational modelscloser to reality, and will allow a fully reliable and quantita-tive prediction of the electronic properties of complex inter-faces.
ACKNOWLEDGMENTS
We thank C. Fischer, J. Hutter, and K. von Klitzing for
several useful discussions and a careful reading of the manu-script.
1Nanostructures Based on Molecular Materials , edited by W.
Go¨pel and Ch. Ziegler ~VCH Verlagsgesellschaft, Weinheim,
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4R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 ~1985!.
5M. Burghard et al., Chem. Mater. 7, 2104 ~1995!.
FIG. 10. Variation DQof the total molecular charge
@DQ5Q(Vmm50)2Q(Vmm)#as a function of the applied potential
Vmm. Dots: simulation results from a Mulliken population analysis.
Dashed line: fit with three arctan contributions DQ
5(i2arctan @(Vmm2Vi)/wi#.5104 54 D. LAMOEN, P. BALLONE, AND M. PARRINELLO6A. Aviram and M. A. Ratner, Chem. Phys. Lett. 29, 277 ~1974!.
7We point out, however, that similar computations on simpler sys-
tems have already been reported in the literature. See, for in-stance, A. Aviram, J. Am. Chem. Soc. 110, 5687 ~1988!.
8M. P. Samanta et al., Phys. Rev. B 53, R7626 ~1996!;Y .A .
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~1996!.
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683~1970!.
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~1996!.
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R. G. Parr and W. Yang, Density Functional Theory of Atoms
and Molecules ~Oxford University, Oxford, 1989 !.
12G. Galli and M. Parrinello, in Computer Simulation in Materials
Science, edited by V. Pontikis and M. Meyer ~Kluwer, Dor-
drecht, 1991 !.
13D. Vanderbilt, Phys. Rev. B 41, 7892 ~1990!.
14N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 ~1991!.
15We are not concerned here with the small changes due to the
possible (23 3A3) reconstruction observed for the clean
Au~111!surface. See U. Harten, A. M. Lahee, J. P. Toennies,
and Ch. Wo ¨ll, Phys. Rev. Lett. 54, 2619 ~1985!.16Computations have been performed with CPMDversion 3.0 writ-
ten by J. Hutter, Max-Plank-Institut fu ¨r Festko¨rperforschung,
Stuttgart, 1995 ~unpublished !, with the help of the group for
numerical intensive computations of IBM Research LaboratoryZu¨rich and the Abteilung Parrinello of MPI Stuttgart.
17J. P. Perdew and M. R. Norman, Phys. Rev. B 26, 5445 ~1982!;
M. Levy, J. P. Perdew, and V. Sahni, Phys. Rev. A 30, 2745
~1984!; C.-O. Almbladh and U. von Barth, Phys. Rev. B 31,
3231 ~1985!.
18K. Christmann, Surf. Sci. Rep. 9,1~1988!.
19Metal-Semiconductor Schottky Barrier Junctions and their Appli-
cations, edited by B. L. Sharma ~Plenum, New York, 1984 !.
20Because of the finite size of the slab used to simulate the Au
surface, the computed Fermi energy does change upon adsorp-tion of the organic monolayer. The change, however, is small(;0.05 eV !and, in any case, smaller than the eigenvalue dif-
ferences we are interested in.
21In this context the ‘‘low-field’’ and ‘‘high-field’’ regimes simply
distinguish the different behaviors we observe for external po-tentials below and above 2 V. In fact, a potential drop of 2 V atthe molecular scale corresponds to a very high macroscopicfield.54
5105 ELECTRONIC STRUCTURE, SCREENING, AND . . . |
PhysRevB.93.205404.pdf | PHYSICAL REVIEW B 93, 205404 (2016)
Electron and phonon drag in thermoelectric transport through coherent molecular conductors
Jing-Tao L ¨u,1,*Jian-Sheng Wang,2Per Hedeg ˚ard,3and Mads Brandbyge4
1School of Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, 430074 Wuhan, China
2Department of Physics, National University of Singapore, 117551 Singapore, Republic of Singapore
3Niels Bohr Institute and Nano-Science Center, University of Copenhagen, 2100 Copenhagen Ø, Denmark
4Center for Nanostructured Graphene (CNG), Department of Micro- and Nanotechnology, Technical University of Denmark,
DK-2800 Kongens Lyngby, Denmark
(Received 24 January 2016; revised manuscript received 12 April 2016; published 4 May 2016)
We study thermoelectric transport through a coherent molecular conductor connected to two electron and two
phonon baths using the nonequilibrium Green’s function method. We focus on the mutual drag between electronand phonon transport as a result of ‘momentum’ transfer, which happens only when there are at least two phonondegrees of freedom. After deriving expressions for the linear drag coefficients, obeying the Onsager relation,we further investigate their effect on nonequilibrium transport. We show that the drag effect is closely relatedto two other phenomena: (1) adiabatic charge pumping through a coherent conductor; (2) the current-inducednonconservative and effective magnetic forces on phonons.
DOI: 10.1103/PhysRevB.93.205404
I. INTRODUCTION
The possibility to engineer electron and phonon transport
independently in nanostructures makes them an ideal candidatefor thermoelectric applications, the conversion of heat toelectricity, and vice versa [ 1–11]. Thermoelectric transport in
quantum wells, wires, and dots has been the focus of intensestudy in the past decades. Recently, it has become possibleto measure the thermopower of molecular junctions, theextreme minimization of electronics [ 6,7,9,10]. Although still
in its infancy from an application point of view, academicallythermopower has proven useful as a complementary tool toexplore the transport properties of molecular devices. Forexample, the sign of the thermopower gives information aboutthe relative position of the electrode Fermi level within theHOMO-LUMO gap of the molecule [ 6,7,9,12]; the quan-
tum interference effect [ 13–16] and many-body interactions
[17–20] also show their signatures in the thermopower.
The interaction between electrons and vibrations within
the molecule couples charge and phonon heat transport. Thesignature of this coupling in electrical current has been used asa spectroscopy tool to unambiguously identify the molecule.However, much of the early theoretical work on thermoelectrictransport in molecular conductors treats electron and phonontransport separately, within the linear regime. Recently, thereare more attempts trying to include the electron-phonon(e-ph)interaction, extend the analysis to the nonlinear regime[12,21–37], and consider multiterminal transport [ 38–44].
The e-ph interaction modifies the electronic transmission andconsequently the thermopower. Extending to the nonlinearregime also helps to make connection with the current-inducedheating and heat transport in molecular devices.
In this paper, we study the nonequilibrium thermoelectric
transport through a model device, connected to two electronand two phonon baths, including the e-ph interaction within thedevice. We use the nonequilibrium Green’s function (NEGF)method to take into account the effect of e-ph interaction within
*jtlu@hust.edu.cnthe lowest order perturbation [ 45], assuming the interaction
is weak. Thus, our approach does not apply to molecularjunctions that couple weakly to the electrodes [ 24,28,34].
In the linear regime, we derive the thermoelectric transportcoefficients including the e-ph interaction. We pay specialattention to the drag coefficients, whereby a temperaturedifference between the phonon baths drives an electricalcurrent between the electron baths, and vice versa. Thedrag effect has been well studied in translational invariantsystems but less considered in a nanoconductor. We makeconnections between electron/phonon drag and other relatedeffects, e.g., the current-induced nonconservative, effectivemagnetic force [ 46–48], and adiabatic pumping in a coherent
conductor [ 49,50]. These effects can only emerge in a
nanoconductor with at least two phonon degrees of freedom.This makes our study different and complementary to mostother works [ 24,28,34,38,51]. Furthermore, we extend the
analysis to the nonlinear regime, look at the drag effect onenergy transfer between electrons and phonons, and discussthe possibility of driving heat flow using an electric device orcharge transport using heat [ 38,51].
The paper is organized as follows. In Sec. II, we introduce
our model setup and present analytical results for the chargeand heat currents focusing on the electron/phonon drag effect.In Sec. IIIwe analyze a simple one-dimensional (1D) model
system to illustrate that the drag effect shares the same originas that in a translational invariant lattice and can be understoodas a result of the momentum transfer between electrons andphonons. We also provide numerical results for the modelsystem. Section IVgives concluding remarks. Finally, the
details of the derivation are given in Appendices A–C.
II. THERMOELECTRIC TRANSPORT
A. System setup and Hamiltonian
We consider a model device containing an electronic ( He)
and a phononic (vibrational) part ( Hp), with interactions
between them Hep. The electronic part is linearly coupled
to two separate electron baths ( L,R), so does the vibrational
2469-9950/2016/93(20)/205404(11) 205404-1 ©2016 American Physical SocietyL¨U, WANG, HEDEG ˚ARD, AND BRANDBYGE PHYSICAL REVIEW B 93, 205404 (2016)
FIG. 1. Schematic representation of the model system considered
in this paper and its different possible situations: (a) By applying a
voltage bias, heat can be extracted from one of the phonon baths,
although they remain at the same temperature. (b) By applying atemperature difference between the two phonon baths, an electrical
current can be generated between the two electron baths. This
contributes with a phonon-drag part to the thermopower.
part (Fig. 1). The coupling matrix is denoted by Vα
σ.T h e
Hamiltonian of the entire system is written as
H=/summationdisplay
σ=e,pHσ+Hep+/summationdisplay
α=L,R;σ=e,p/parenleftbig
Hα
σ+Vα
σ/parenrightbig
. (1)
The electron and phonon subsystem (device plus left and
right baths) are noninteracting. For example, the phononsare described within the harmonic approximation, and theelectrons within a single particle picture. We assume no directcoupling between these baths. The only many-body interactionisH
epwithin the device. In a tight-binding description of the
electronic Hamiltonian, it can be written as
Hep=/summationdisplay
i,j,kMk
ijc†
icjuk, (2)
where c†
i(cj) is the electron creation (annihilation) operator
for the i(j)th electronic site, and ukis the mass-normalized
displacement away from the equilibrium position of the kth
degrees of freedom, i.e., uk=√mkrk, with mkthe mass of
thekth degree of freedom, and rkits displacement away
from equilibrium position. Mk
ijis the e-ph interaction matrix
element. We consider spinless electrons throughout this paper.To calculate the electrical and heat current, we assume
the e-ph interaction is weak, and keep only the lowest orderself-energies [ 52]. We perform an expansion of the Green’s
functions and current up to the second order in M, following
the idea of Ref. [ 45]. For example, the lesser Green’s function
is expanded as
G
<≈G<
0+Gr
0/Sigma1<
epGa0+Gr
0/Sigma1r
epGr0/Sigma1<
LRGa0
+Gr
0/Sigma1<
LRGa0/Sigma1a
epGa0, (3)
with/Sigma1LR=/Sigma1L+/Sigma1Rthe self-energy due to coupling to
electrodes, /Sigma1epself-energy due to e-ph interaction, and G0the
noninteracting electron Green’s function. Similar expressionholds for G
>andD>,<. The details of the method can be
found in Appendix Aand Refs. [ 21,45,52].
B. Linear transport coefficients
In the linear response regime, we introduce an infinitesimal
change of the chemical potential or temperature at one of thebaths, α, e.g.,μ
α=μ+δμ,Tα
σ=T+δTσ, withμandTthe
corresponding equilibrium values. We look at the response ofthe charge and heat current due to this small perturbation. Upto the second order in M, the result is summarized as follows:
⎛
⎜⎝Iα
e
Jα
e
Jα
p⎞
⎟⎠=⎛
⎜⎝L0L1˜Q0
L1L2˜Q1
Q0Q1Kp⎞
⎟⎠⎛
⎜⎝δμ
δTe
T
δTp
T⎞
⎟⎠. (4)
We define the positive current direction as that elec-
trons/phonons go from the bath to the device, and Iα,Jα
e,
Jα
pare the electrical current and the heat current carried by
electrons and phonons, respectively. The expressions for thecoefficients LandK
pare given in Appendix A. Both include
three contributions. The first term is the elastic Landauerresult. The second term is the (quasi)elastic correction dueto change of the electron spectral function. The last one is theinelastic term. The effect of each part in Lon the conductance
and Seebeck coefficient have been analyzed in Ref. [ 52]f o r
a single level model. Q
nand ˜Qnare the drag coefficients.
We use the following convention for the drag effect: Theelectron drag effect corresponds to generating phonon flowdue to electron flow, while the phonon drag corresponds to theopposite process. We can write Q
nand ˜Qnas
˜Qn=−/summationdisplay
β/integraldisplaydω
2π/planckover2pi1ωTr/bracketleftbig
/Lambda1(n)
˜αβ(ω)Aα(ω)/bracketrightbig
∂/planckover2pi1ωnB(/planckover2pi1ω,T),(5)
Qn=−/summationdisplay
β/integraldisplaydω
2π/planckover2pi1ωTr/bracketleftbig
/Lambda1(n)
αβ(ω)˜Aα(ω)/bracketrightbig
∂/planckover2pi1ωnB(/planckover2pi1ω,T),(6)
where nB(/planckover2pi1ω,T)=[exp(/planckover2pi1ω
kBT)−1]−1is the Bose-Einstein
distribution function. Throughout the paper, we use Tr[ ·]f o r
trace over phonon indices, tr[ ·] for trace over electronic degrees
of freedom, ˜Aα/Aαis the (time-reversed) phonon spectral
function [Eq. ( A8)], and /Lambda1(n)
˜αβ(ω) is defined as
/Lambda1(n)
˜αβ(ω)=/integraldisplaydε
2π(ε−μα)nX˜αβ(ε,ε−)
×/bracketleftbig
f/parenleftbig
ε,μα,Tα
e/parenrightbig
−f/parenleftbig
ε−,μβ,Tβ
e/parenrightbig/bracketrightbig
, (7)
205404-2ELECTRON AND PHONON DRAG IN THERMOELECTRIC . . . PHYSICAL REVIEW B 93, 205404 (2016)
X˜αβ(ε,ε−)=tr[M˜Aα(ε)MAβ(ε−)], (8)
withε−=ε−/planckover2pi1ω. In the definition of X˜αβand/Lambda1(n)
˜αβ,˜α
means we need to use the time-reversed electron spectralfunction ˜A
α[Eq. ( A7)]./Lambda1(0)(ω) is the coupling-weighted
electron-hole pair density of states (DOS), introduced in ourprevious work [ 47,53,54]. In the linear regime, the Fermi
distribution f(ε,μ
α,Tα
e)=f(ε,μ,T )i st h es a m ef o rb o t h
electrodes, with f(ε,μ,T )=[exp(ε−μ
kBT)+1]−1. Hereafter, the
summation of βis over LandR, and the integration is from
−∞ to+∞ if not specified explicitly. In the linear regime,
without magnetic field, we have ( Dr)T=Drand (Gr)T=Gr.
This leads to Qn=˜Qn, which ensures the Onsager symmetry
(Appendix B).
For one electronic level coupled to one phonon mode, we
can check that our result for Q0/Q1is equivalent to that of
Ref. [ 38] (Appendix C). In order for Q0/Q1to be nonzero,
we need some special design of the system, e.g., asymmetriccoupling to the left and right electron bath [ 38].
Here, we focus on the case where there are two or more
phonon modes. For ˜Q
0, we can do an expansion over the energy
dependence of the electron spectral function. The zeroth ordercontribution is
˜Q(0)
0=/summationdisplay
β/integraldisplaydω
4π2(/planckover2pi1ω)2∂/planckover2pi1ωnB(/planckover2pi1ω,T)
×Tr[ImX˜αβ(μ,μ)ImAα(ω)], (9)
with Re and Im meaning real and imaginary part, respectively.
In order for ˜Q(0)
0to be nonzero, the device needs to have at
least two vibrational modes. This follows from the fact thatA
αis Hermitian.
Relation with adiabatic pumping and current-induced forces
Now we write ˜Q(0)
0in terms of the unperturbed retarded
(advanced) electron scattering states, coming from (leavingto) the left |ψ
L/angbracketright(|˜ψL/angbracketright)o rr i g h t |ψR/angbracketright(|˜ψR/angbracketright) electrode
Xkl
˜αβ(ε,ε−)=/summationdisplay
m,n/angbracketleftbig
ψn
β(ε−)/vextendsingle/vextendsingleMk/vextendsingle/vextendsingle˜ψm
α(ε)/angbracketrightbig/angbracketleftbig˜ψm
α(ε)/vextendsingle/vextendsingleMl/vextendsingle/vextendsingleψn
β(ε−)/angbracketrightbig
.
(10)
Here,mandnare channel indices. The retarded and advanced
scattering states including e-ph interaction are generated from
|/Psi1α(ε)/angbracketright=|ψα(ε)/angbracketright+GrHep|ψα(ε)/angbracketright, (11)
|˜/Psi1α(ε)/angbracketright=|ψα(ε)/angbracketright+GaHep|ψα(ε)/angbracketright. (12)
They are normalized as
/angbracketleft/Psi1α(ε)|/Psi1β(ε/prime)/angbracketright=2πδα,βδ(ε−ε/prime), (13)
and so is |ψα/angbracketright. From the definition of the scattering matrix
2πδ(ε−ε/prime)Smn
αβ=/angbracketleftbig˜/Psi1m
α(ε/prime)/vextendsingle/vextendsingle/Psi1n
β(ε)/angbracketrightbig
, (14)
and Eqs. ( 11) and ( 12), we get
Smn
αβ=δα,βδm,n−i/angbracketleftbig
ψm
α/vextendsingle/vextendsingleHep/vextendsingle/vextendsingle/Psi1n
β/angbracketrightbig
. (15)
Here, Smn
αβis the matrix element connecting the incoming
wave from the nth channel in electrode βto the mth outgoingchannel in electrode α. Taking the derivative over the phonon
displacement yields
∂kSmn
αβ=−i/angbracketleftbig˜/Psi1m
α/vextendsingle/vextendsingleMk/vextendsingle/vextendsingle/Psi1n
β/angbracketrightbig
. (16)
Substituting Eq. ( 16) into Eq. ( 10), taking the ω→0 limit, we
obtain
Im/summationdisplay
βtr[∂lSαβ∂kS†
βα]=Im/summationdisplay
βXkl
˜αβ+··· . (17)
The trace is over the channel indices. We have kept only the
second order terms.
Equation ( 17) makes connection with the Brouwer for-
mula for adiabatic pumping [ 50,55,56]. Here, a temperature
difference between the left and right electrode breaks thepopulation balance between the phonon scattering states fromthese two baths, e.g., there are more phonon waves travelingin one direction, determined by the temperature bias. When
the phonon wave goes through the device, it produces phase-
shifted oscillating potential felt by the electrons. In the spaceof the atomic coordinates, the trajectory may form a closedloop, generating pumped electrical current.
The opposite of this effect is that an electrical current
generates a directed phonon heat current. The term governingthis effect is X
LR. The same term appears in the expressions for
the current-induced nonconservative and effective magneticforces [ 46–48,53,57], e.g., Eqs. (56)– (61) in Ref. [ 53]. This
shows that the electron drag effect is closely related to thesenovel current-induced forces.
C. Nonlinear regime
When the applied temperature or voltage bias is large, addi-
tional energy transfer between the electron and phonon subsys-tem takes place. We consider two situations: electrical-current-driven heat flow in the isothermal case and temperature-drivenelectrical current at zero voltage bias.
1. Electrical-current-driven heat flow (T e=Tp=T, eV /negationslash=0)
In the first setup, all the baths are at the same temperature
(T), but the electron baths are subject to a nonzero voltage
bias (eV=μL−μR). This is the most common situation in a
working electronic device [Fig. 1(a)]. For large bias, there will
be energy transfer from the electron to the phonon subsystem(see Appendix Afor the derivation)
Q=/summationdisplay
α,β/integraldisplaydω
2π/integraldisplaydε
2π/planckover2pi1ωTr[tr[MAα(ε)MAβ(ε+/planckover2pi1ω)]A(ω)]
×fβ(ε+/planckover2pi1ω)(1−fα(ε))(nB(/planckover2pi1ω,T)+1). (18)
Since all the baths are at the same temperature, we have
omitted it in this subsection. Equation ( 18) is a result of
balance between phonon emission and adsorption processes(Fig. 2). For ω> 0, it represents process where an electron
in electrode βcombines with a hole at lower energy in α,
accompanied by a phonon emission process. For ω< 0, it
represents the opposite process, where an electron-hole pairis created between αandβby adsorbing one phonon. While
the Fermi distributions f
β(1−fα) ensures that the phonon
emission process happens only when the applied bias eVis
larger than the phonon energy /planckover2pi1ω, the Bose function nB+1
205404-3L¨U, WANG, HEDEG ˚ARD, AND BRANDBYGE PHYSICAL REVIEW B 93, 205404 (2016)
(1)
(2)(3)
(4)
FIG. 2. Electron-hole pair excitation processes. (1) and (2) are
intraelectrode processes. (3) and (4) are interelectrode ones. At T=0,
(1)–(3) are not possible, and (4) is possible only when the applied
bias is larger than the phonon energy.
prohibits phonon adsorption process at T=0. This equation
can also be written in a compact form as
Q=/integraldisplaydω
2π/planckover2pi1ωTr/bracketleftbig
/Lambda1(0)
LR(ω)A(ω)/bracketrightbig
/Delta1nB(/planckover2pi1ω,T;/planckover2pi1ω−eV,T ),
(19)
with
/Delta1nB(/planckover2pi1ω1,T1;/planckover2pi1ω2,T2)=nB(/planckover2pi1ω1,T1)−nB(/planckover2pi1ω2,T2).(20)
Energy transfer within the device breaks the balance between
the device and bath phonons. As a result, the extra energy isfurther transferred to the two phonon baths. The heat currentflowing out of the phonon bath αis given by the minus of
Eq. ( 19) with ˜Areplaced by ˜A
α, such that
JL
p+JR
p+Q=0, (21)
as required by the energy conservation.
In fact, we can split Jα
pinto two parts according to their
symmetry upon bias reversal Jα
p=Jα,h
p+Jα,p
p, where Jα,h
p
andJα,p
pare even and odd functions of eV. We call them the
Joule heating and Peltier drag current, respectively. Assumingconstant electron DOS, we get
J
α,h
p≈−/integraldisplay+∞
0dω
4π2h(/planckover2pi1ω)Tr[Re XLR(μ,μ)Re˜Aα(ω)],(22)
Jα,p
p≈/integraldisplay+∞
0dω
4π2p(/planckover2pi1ω)Tr[Im XLR(μ,μ)Im˜Aα(ω)].(23)
The two coefficients are
h(/planckover2pi1ω)≡/summationdisplay
s=±1/planckover2pi1ω(/planckover2pi1ω+se V)/Delta1nB(/planckover2pi1ω+seV;/planckover2pi1ω),(24)
p(/planckover2pi1ω)≡/summationdisplay
s=±1s/planckover2pi1ω(/planckover2pi1ω+se V)/Delta1nB(/planckover2pi1ω+seV;/planckover2pi1ω).(25)
The Joule current corresponds to the energy transfer from the
electrons to the phonons in Eq. ( 19), i.e., JL,h
p+JR,h
p+Q=
0. But the drag current is related to the Q0coefficient in
Sec. II B and depends on the direction of current flow, i.e.,
JL,p
p+JR,p
p=0. This relation follows from the fact that
ImA=Im˜A=Im˜AL+Im˜AR=0. We will see later that it
is due to momentum transfer between electrons and phonons.In the limit of high temperature ( k
BT/greatermucheV±/planckover2pi1ω), wehaveh(/planckover2pi1ω)→0, and p(/planckover2pi1ω)→2eVkBT. The drag part will
dominate over the Joule heating part. In this case, it is possibleto extract heat from one of the phonon baths by applying avoltage bias, similar to a refrigerator, as shown in Fig. 1(b).
We note that in Ref. [ 54], we have studied the same
problem using the semiclassical generalized Langevin equa-tion approach. Similar equations were derived there, andthe asymmetric heat flow was attributed to the asymmetriccurrent-induced forces. These two complementary analysesshows that the two effects are closely related.
2. Temperature-driven electric current ( μL=μR,Te/negationslash=Tp)
In the second setup, we apply a temperature difference
between the electron and phonon subsystem at zero voltagebias. This drives an electrical current within the device [see
Eq. ( A19)]
I
α=e/summationdisplay
β,γ/integraldisplaydω
2πTr/bracketleftbig
/Lambda1(0)
˜α˜β(ω)Aγ(ω)/bracketrightbig
/Delta1nB/parenleftbig
/planckover2pi1ω,Tγ
p;/planckover2pi1ω,T e/parenrightbig
.
(26)
Here, ¯αmeans the lead different from α. There are two possible
situations here. The first one is that the phonon baths are at thesame temperature ( T
p) but different from that of electron baths
(Te). We can consider the two phonon baths as an effective
single bath. The four-terminal setup reduces to a three-terminalone, and equation ( 26) simplifies to
I
α=e/summationdisplay
β/integraldisplaydω
2πTr/bracketleftbig
/Lambda1(0)
˜α˜β(ω)A(ω)/bracketrightbig
/Delta1nB(/planckover2pi1ω,T p;/planckover2pi1ω,T e).
(27)
The three-terminal setup has been considered in Ref. [ 38].
For a single electronic level coupling to one phonon mode,equation ( 27) agrees with result therein. Due to the temperature
difference between the electron and phonon systems, therewill be energy flow between them. It has been analyzed inSec. III C of Ref. [ 52]. A similar problem has been considered
in Refs. [ 58–61]. Here we focus on the other situation, where
we apply a temperature difference between the two phononbaths [Fig. 1(b)]. This generates a phonon-drag electrical
current. For constant electronic DOS, we get
I
α≈e/integraldisplaydω
4π2/planckover2pi1ωTr[ImX˜α˜¯α(μ,μ)ImA¯α(ω)]
×/Delta1nB/parenleftbig
/planckover2pi1ω,Tα
p;/planckover2pi1ω,T¯α
p/parenrightbig
, (28)
extending the result in Sec. II Bto the nonlinear regime.
III. MODEL CALCULATION
A. 1D model and qualitative analysis
For the ease of understanding the general results in Sec. II,
we now study a simple 1D atomic chain. The electronicHamiltonian takes the tight-binding form, with the hoppingmatrix element −t,
H
e=−t/summationdisplay
|i−j|=1c†
icj. (29)
The electron dispersion relation is εk=−2tcosk, where kis
the 1D wave vector [Fig. 3(d)]. We have set the lattice distance
a=1. For this 1D lattice, due to translational invariance, the
205404-4ELECTRON AND PHONON DRAG IN THERMOELECTRIC . . . PHYSICAL REVIEW B 93, 205404 (2016)
t( ) t( ) t (u) t (u) t( ) -t (u)
(a)-t (u) -t (u) -t (u) -t (u)
-t0
(b)-t (un)-t (δu) -t0 -t (un+1)
(c) (d)n n+1K0
kL kR
k
FIG. 3. (a) A model 1D lattice, where the electron nearest-
neighbor hopping amplitude t(u) depends on the atomic displacement
u. (b) Localized e-ph interaction at sites nandn+1,δu=un+1−un.
(c) Phonon dispersion relation. (d) Electron dispersion relation.
Blue and red lines depict phonon emission processes due to e-ph
interaction. An electron at state kLis scattered by phonons to an state
kRand emit one phonon, whose wave vector is denoted by the vertical
lines in (c). Depending on the value of kLandkR, the emitted phonon
may travel to the right (blue line, normal process) or the left (red
line, Umklapp process). Normally, the electron energy is much larger
than the phonon energy, so in this figure we have ignored the electronenergy change.
electron Green’s function in real space only depends on the
distance between different sites jandl,n=l−j,
Gr
0,jl(ε)=ei|kL(ε+)n|
2it|sinkL(ε+)|. (30)
Hereafter, we take kL(ε+)>0 andkR(ε+)=−kL(ε+)<0a s
the wave vectors corresponding to the scattering state withenergy ε
+=ε+i0+, coming from the left and the right,
respectively. The corresponding spectral functions, definedwithin the electron energy band, are
A
L,jl(ε)=˜A∗
L,jl(ε)=eikL(ε+)n
2t|sinkL(ε+)|,A R(ε)=A∗
L(ε).
(31)
The ions are connected by 1D springs with spring constant
K0
Hp=/summationdisplay
j/parenleftbigg1
2˙u2
j+K0u2
j/parenrightbigg
−1
2K0/summationdisplay
|i−j|=1uiuj. (32)
The phonon retarded Green’s function is
Dr
0,jl(ω)=ei|q(ω+)n|
2iK0|sinq(ω+)|. (33)
q(ω+) is the phonon wave vector, corresponding to frequency
ω+=ω+i0+. The phonon dispersion relation is ωq=
2√K0|sinq
2|[Fig. 3(c)]. The phonon spectral function, definedwithin the phonon band, is
AL,jl(ω)=˜A∗
L,jl(ω)=eiqL(ω+)n
2K0|sinqL(ω+)|,AR(ω)=A∗
L(ω).
(34)
Here,qL(ω+)>0 is the phonon wave vector corresponding to
scattering wave coming from the left.
To consider the e-ph interaction, we assume the atomic
motion modifies the hopping matrix element linearly, i.e.,
Hep=−m/summationdisplay
juj(c†
jcj+1−c†
jcj−1+H.c.). (35)
For a phonon emission process through electronic transi-
tion from the initial left scattering states |ψL(kL)/angbracketrightto the
final right scattering state |ψR(kR)/angbracketright, only phonon mode
that fulfills the energy and crystal-momentum conservationcan be excited, e.g., /angbracketleftψ
L(kL)|Mq|ψR(kR)/angbracketright∼δ(kL−kR−q+
G)δ(ε(kL)−ε(kR)−/planckover2pi1ω(q)). Here, Gis a reciprocal lattice
vector. It is this selection rule that gives the electron/phonondrag effect in a translational invariant lattice [Figs. 3(c)–3(d)].
To show that similar mechanism works in a coherent
nanoconductor, we artificially switch off the e-ph interaction,except at two sites, e.g., we only consider coupling to u
n
andun+1[Fig. 3(b)]. That is, in Eq. ( 35), the sum over j
only applies to these two sites. Then, only nearest hoppingbetween four sites, {n−1,n,n+1,n+2}, are modified by
atomic motion. We set these four sites as our device, and allother sites as electron and phonon baths LandR.I nt h i s
case, the condition of energy conservation is still valid, butthe conservation of crystal momentum is not, since the locale-ph interaction breaks the translation invariance. The matrixelement is
/angbracketleftψ
L(kL)|Mq|ψR(kR)/angbracketright
=−m
/planckover2pi1√|vLvR|e−ikL[1+ei(q−kL+kR)][1+ei(kL+kR)]
×[1−e−i(kL−kR)]. (36)
Here, vL/R is the group velocity of the L/R scatter-
ing state with wave vector kL/R. We can see that the
squared scattering matrix element |/angbracketleftψL(qL)|Mq|ψR(qR)/angbracketright|2/negationslash=
|/angbracketleftψL(qL)|M−q|ψR(qR)/angbracketright|2, their difference
/Delta1MLR∝sinφsinq. (37)
We have defined φ=kL−kR. This means the electrons have a
different probability of exciting left and right traveling phononwaves. The difference depends on the electron and phononwave vectors. For example, similar to the 1D lattice, electronswithε<0 (below half filling) preferentially emit phonons
traveling to the right. From another point of view, the holesdominate the inelastic transport [ 54]. This breaks the left-
right symmetry, and generates drag effect in a nanoconductor,although the crystal-momentum selection rule is not valid.
To make connection with the NEGF approach in Sec. II,
we can calculate the real space e-ph interaction matrix atsitesnandn+1, and find M
n+1
LR≡/angbracketleftψL(kL)|Mn+1|ψR(kR)/angbracketright=
e−iφ/angbracketleftψL(kL)|Mn|ψR(kR)/angbracketright. So,
Xn,n+1
LR (ε(kL),ε(kR))=/vextendsingle/vextendsingleMn
LR/vextendsingle/vextendsingle2/parenleftbigg
1e−iφ
eiφ1/parenrightbigg
. (38)
205404-5L¨U, WANG, HEDEG ˚ARD, AND BRANDBYGE PHYSICAL REVIEW B 93, 205404 (2016)
Making use of Eq. ( 34), we get, for given electron ( kL,kR) and
phonon wave vectors ( q), that satisfy the requirement of energy
conservation, ε(kL)=ε(kR)+/planckover2pi1ω(q), the electron ‘drag’ term
becomes
Tr[ImXLR(ω)Im˜AL(ω)]∝sinφsinqL(ω+). (39)
This is consistent with our scattering analysis and shows that
the drag effect we discuss here shares the same origin as thatin a lattice system.
B. Numerical results
Now we present our numerical results for the 1D model
with localized e-ph interaction using the formulas developedin Sec. II.I nF i g . 4, we show the calculated phonon drag
contribution to the Seebeck coefficient. The parameters, givenin the figure caption, are chosen to closely resemble thatof a single atom gold chain [ 62–65]. The single electron
contribution to the Seebeck coefficient vanishes since we havea perfect electron transport channel. The drag coefficient iszero at E
F=0 due to electron-hole symmetry [ 54]. But once
moving to EF=−1 eV , the symmetry is broken, and we get
a nonzero value. Positive Smeans the holes dominate over
the electrons in the inelastic scattering process. The saturationof the SwithTcan be understood from Eq. ( 9), i.e., at high
temperature, ˜Q
0∝T, while S∝˜Q0/T.
Figures 5and6show the results for the setup in Fig. 1(a).
The temperatures of all the baths are the same, while there is avoltage bias applied between the two electron baths. We definethe power as Q in Eq. ( 19). Figure 5shows its dependence on
the voltage bias at different temperatures and Fermi levels. Theonset of power flow at the phonon frequency is clear at 4 .2K ,
but smoothed out at 300 K. The inset shows the correspondingconductance drop at the phonon threshold for 4 .2 K. Although
the magnitude of the power changes slightly, the generalbehavior does not depend on the position of the Fermi level.All these results agree with previous studies [ 21,66,67].
0 0.1 0.2 0.3 0.4 0.5 0.6
0 50 100 150 200 250 300S (µV/K)
T (K)Ef=-1 eV
Ef=0
FIG. 4. Phonon drag contribution to the Seebeck coefficient at
different chemical potentials. The parameters used are as follows:
t=1e V ,K0=0.02 eV/(˚A2u),K/prime
0=0.5K,m=0.15 eV/(˚A√u).
Here,K/prime
0is the spring constant connecting the device to the left and
right phonon baths. 0 0.5 1 1.5 2 2.5 3
0 0.05 0.1 0.15 0.2Power (nW)
Bias (V)0 eV, 300 K
0 eV, 4.2 K
-1 eV, 300 K
-1 eV, 4.2 K1.0
-0.03 0 0.03dI/dV (G0)
Bias (eV)
FIG. 5. Energy current going from electron to phonons (power)
as a function of voltage bias at different temperatures and Fermi
l e v e l sf r o mE q s .( 19)–(21). All the electron and phonon baths are
kept at the same temperature, see also Fig. 1(b). The inset shows the
differential conductance ( dI/dV )a tEF=0 (red, solid) and −1e V
(green, dashed) at 4 .2K .
Next, we show in Figs. 6(a) and6(b) that the heat flows
into the left and right phonon baths are drastically different atthe two Fermi levels. For E
F=0, the heat flow into the two
phonon baths are symmetric [Figs. 6(a) and6(b)]. But when
we move to EF=−1.0 eV , they show strong asymmetry, due
to the drag part of the heat current [Figs. 6(c) and6(d)]. It
-1 0 1 2 3 4
0 0.05 0.1 0.15 0.2
Bias (V)(e) 300K, GLE
-1 0 1 2 3 4
0 0.05 0.1 0.15 0.2
Bias (V)(f) 300K, SCBA-1 0 1 2 3 4
0 0.05 0.1 0.15 0.2Heat Current (nW)(c) 300 K
-1 0 1 2 3 4
0 0.05 0.1 0.15 0.2(d) 4.2 K-0.5 0 0.5 1 1.5 2
0 0.05 0.1 0.15 0.2(a) 300 K
-0.5 0 0.5 1 1.5 2
0 0.05 0.1 0.15 0.2(b) 4.2 K
FIG. 6. Heat current carried by phonons, going into the left (red,
solid) and right (black, dashed) phonon bath. (a),(b) EF=0, (c)–(f)
EF=−1 eV . (a)–(d) Calculated from Eqs. ( 19)–(21). We also show
the results calculated from the generalized Langevin equation (GLE)
approach (Ref. [ 53]) (e) and the self-consistent Born approximation
(SCBA) (Ref. [ 21]) (f).
205404-6ELECTRON AND PHONON DRAG IN THERMOELECTRIC . . . PHYSICAL REVIEW B 93, 205404 (2016)
depends on the phase of the electron wave function, which in
our case could be tuned by changing the chemical potential.AtE
F=−1 eV , the probability of emitting right traveling
phonons is larger, resulting in larger heat current into bath R.
This is the same with the lattice system [Figs. 3(c) and3(d)].
Comparing results at T=4.2 and 300 K, we find that the
asymmetry increases with temperature. Interestingly, at 300 K,we can extract heat from the right phonon bath by applyingthe voltage bias. This is a prototype atomic ‘refrigerator.’ Theresults calculated using the generalized Langevin equation(GLE) approach [ 53] [Fig. 6(e)] and self-consistent Born
approximation (SCBA) [ 21] [Fig. 6(f)] show that, although
the magnitude of the heat current changes, the qualitativeconclusions remain no matter which approximation we use.
IV . CONCLUSIONS
In conclusion, by assuming linear coupling between elec-
trons and phonons in a four-terminal nanodevice, we haveshown that in the linear transport regime, in addition tomodifying the normal thermoelectric transport coefficients,
e-ph interaction also introduces new drag type coefficients.The drag effect can be traced back to the momentum transferbetween the electrons and phonons. We have shown thatit is closely related to the adiabatic pumping and current-induced forces in a coherent conductor. So in principlephonon-drag thermopower behaves as an alternative way ofprobing these current-induced forces. The expressions derivedin this paper can be readily applied to the realistic structuresby combining these with first-principles electronic structurecalculation [ 68–70]. Finally, we note that one could also
study similar drag effect in Coulomb coupled all-electronicdevices [ 42–44].
ACKNOWLEDGMENTS
J.T.L. is supported by the National Natural Science Foun-
dation of China (Grant Nos. 11304107 and 61371015). J.-S.W
acknowledges support of an FRC Grant No. R-144-000-343-112. M.B. acknowledges support from the Center forNano-structured Graphene (Project DNRF103).
APPENDIX A: DETAILS OF THE DERIV ATION
Our starting point is the Meir-Wingreen formula for charge and heat current
Iα=e
/planckover2pi1/integraldisplaydε
2πTr[G>(ε)/Sigma1<
α(ε)−G<(ε)/Sigma1>
α(ε)], (A1)
Jα
e=1
/planckover2pi1/integraldisplaydε
2π(ε−μα)Tr[G>(ε)/Sigma1<
α(ε)−G<(ε)/Sigma1>
α(ε)], (A2)
Jα
p=−1
/planckover2pi1/integraldisplaydω
4π/planckover2pi1ωTr[D>(ω)/Pi1<
α(ω)−D<(ω)/Pi1>
α(ω)]. (A3)
The expansion is performed by substituting Eq. ( 3) and similar expression for D>,<into Eqs. ( A1)–(A3). The electrical current
injected from αcan be written as
Iα=e
/planckover2pi1/integraldisplaydε
2π/parenleftbig
tr/bracketleftbig
Gr
0/Sigma1>
LRGa0/Sigma1<
α−Gr
0/Sigma1<
LRGa0/Sigma1>
α/bracketrightbig
+tr/bracketleftbig
Gr
0/Sigma1r
epGr0/Sigma1>
LRGa0/Sigma1<
α−Gr
0/Sigma1r
epGr0/Sigma1<
LRGa0/Sigma1>
α/bracketrightbig
+tr/bracketleftbig
Gr
0/Sigma1>
LRGa0/Sigma1a
epGa0/Sigma1<
α−Gr
0/Sigma1<
LRGa0/Sigma1a
epGa0/Sigma1>
α/bracketrightbig
+tr/bracketleftbig
Gr
0/Sigma1>
epGa0/Sigma1<
α−Gr
0/Sigma1<
epGa0/Sigma1>
α/bracketrightbig/parenrightbig
. (A4)
Similarly, the heat current from phonons injected from α:
Jα
p=−/integraldisplaydω
4π/planckover2pi1ω/parenleftbig
Tr/bracketleftbig
Dr
0/Pi1>LRDa
0/Pi1<α−Dr
0/Pi1<LRDa
0/Pi1>α/bracketrightbig
+Tr/bracketleftbig
Dr
0/Pi1reDr
0/Pi1>LRDa
0/Pi1<α−Dr
0/Pi1reDr
0/Pi1<LRDa
0/Pi1>α/bracketrightbig
+Tr/bracketleftbig
Dr
0/Pi1>LRDa
0/Pi1aeDa
0/Pi1<α−Dr
0/Pi1<LRDa
0/Pi1aeDa
0/Pi1>α/bracketrightbig
+Tr/bracketleftbig
Dr
0/Pi1>eDa
0/Pi1<α−Dr
0/Pi1<eDa
0/Pi1>α/bracketrightbig/parenrightbig
, (A5)
with/Pi1(ω) being the phonon self-energy. We have omitted the argument εorωin the Green’s functions or self-energies.
1. Linear coefficient: L
We group the current into three terms, giving three coefficients,
Ln=3/summationdisplay
i=1L(i)
n. (A6)
Using the expressions for the DOS and time-reversed DOS of electrons and phonons originating from lead α,
Aα(ε)=Gr
0(ε)/Gamma1e
α(ε)Ga
0(ε),˜Aα(ε)=Ga
0(ε)/Gamma1e
α(ε)Gr
0(ε),A (ω)=AL(ω)+AR(ω), (A7)
Aα(ω)=Dr
0(ω)/Gamma1p
α(ω)Da
0(ω),˜Aα(ω)=Da
0(ω)/Gamma1p
α(ω)Dr
0(ω),A(ω)=AL(ω)+AR(ω), (A8)
205404-7L¨U, WANG, HEDEG ˚ARD, AND BRANDBYGE PHYSICAL REVIEW B 93, 205404 (2016)
they can be written as
L(1)
n=1
/planckover2pi1/integraldisplaydε
2π(ε−μ)ntr[A¯α(ε)/Gamma1α(ε)]f/prime(ε),L(2)
n=1
/planckover2pi1/integraldisplaydε
2π(ε−μ)ntr[/Delta1A ¯α(ε)/Gamma1α(ε)]f/prime(ε), (A9)
L(3)
n=i
/planckover2pi1/integraldisplaydε
2π(ε−μ)ntr[(/Sigma1>
ep(ε)−/Sigma1<
ep(ε))˜Aα(ε)]f/prime(ε). (A10)
We have defined f/prime(ε)=−∂f(ε)
∂ε.L(1)
nis the single electron Landauer result. L(2)
nis due to corrections to the electron DOS
/Delta1A ¯α(ε)=Gr
0(ε)/Sigma1r
ep(ε)A¯α(ε)+A¯α(ε)/Sigma1a
ep(ε)Ga
0(ε). (A11)
L(3)is the inelastic term. The effect of e-ph interaction on the Lnin a single level model has been studied in Ref. [ 52].
2. Linear coefficient: K
The phonon thermal conductance has similar form
Kp=3/summationdisplay
i=1K(i)
p (A12)
with
K(1)
p=/integraldisplaydω
4π/planckover2pi1ωTr/bracketleftbig
A¯α(ω)/Gamma1α
p(ω)/bracketrightbig∂nB
∂TpTp,K(2)
p=/integraldisplaydω
4π/planckover2pi1ωTr/bracketleftbig
/Delta1A¯α(ω)/Gamma1α
p(ω)/bracketrightbig∂nB
∂TpTp, (A13)
K(3)
p=i/integraldisplaydω
4π/planckover2pi1ωTr[(/Pi1>
ep(ω)−/Pi1<
ep(ω))˜Aα(ω)]∂nB
∂TpTp. (A14)
Here,K(2)
nis due to corrections to the phonon DOS. /Delta1A¯αis defined similar to Eq. ( A11).
3. Drag coefficients: Qnand ˜Qn
The coefficients Qnand ˜Qncome only from the fourth term in Eqs. ( A4) and ( A5). They are related to the phonon/electron
drag effect, e.g., a phonon temperature difference between the two leads gives rise to an electric current, and vice versa. To deriveit, we write the contribution from the fourth term as
I
α,(4)=e
/planckover2pi1/integraldisplaydε
2πtr/bracketleftbig
Gr
0/Sigma1>
epGa0/Sigma1<
α−Gr
0/Sigma1<
epGa0/Sigma1>
α/bracketrightbig
. (A15)
Using the expression for /Sigma1>,<
ep,
/Sigma1>,<
ep(ε)=i/planckover2pi1/summationdisplay
kl/integraldisplay
(MkG>,<
0(ε−)Ml)D>,<
0,kl(ω)dω
2π, (A16)
and Eqs. ( A7) and ( A8), we get
Iα,(4)=e/summationdisplay
klβγ/integraldisplaydε
2π/integraldisplaydω
2πtr[MkAβ(ε−)Ml˜Aα(ε)]Aγ,kl(ω)(fα(ε)(1−fβ(ε−))(1+nγ(ω))−(1−fα(ε))fβ(ε−)nγ(ω)),(A17)
We have used the abbreviation fα(ε)=f(ε,μα,Tα
e),nγ(ω)=nB(ω,Tγ
p). Using the mathematical relation
[f(x)−/Theta1(t)][f(y)−/Theta1(−t)]=[/Theta1(t)+n(x−y)][f(x)−f(y)], (A18)
with/Theta1(t) the Heaviside step function, f(x)=1/(ex+1),n(x)=1/(ex−1), we have
Iα,(4)=e/summationdisplay
βγ/integraldisplaydω
2πTr[/Lambda1˜αβ(ω)Aγ(ω)](nγ(ω)−nB(/planckover2pi1ω+μβ−μα)). (A19)
It’s easy to see that if μL=μR, andTL=TR,I(4)
α=0. IfTα
p=T¯α
p+δT, we get
Iα,(4)=−e/summationdisplay
β/integraldisplaydω
2π/planckover2pi1ωTr[/Lambda1˜αβ(ω)Aα(ω)]∂nα
∂/planckover2pi1ωδT
T. (A20)
In the same way,
Jα,(4)
e=−e/summationdisplay
β/integraldisplaydω
2π/planckover2pi1ω(ε−μα)Tr[/Lambda1˜αβ(ω)Aα(ω)]∂nα
∂/planckover2pi1ωδT
T. (A21)
From this we get Qnas in Eq. ( 6).
205404-8ELECTRON AND PHONON DRAG IN THERMOELECTRIC . . . PHYSICAL REVIEW B 93, 205404 (2016)
Substituting the phonon self-energy
/Pi1>,<
kl(ω)=−i/integraldisplaydε
2πtr[MkG>,<
0(ε)MlG<,>
0(ε−)] (A22)
into
Jα,(4)
p=−/integraldisplaydω
4π/planckover2pi1ωTr/bracketleftbig
Dr
0/Pi1>eDa
0/Pi1<α−Dr
0/Pi1<eDa
0/Pi1>α/bracketrightbig
. (A23)
Similar manipulation gives ˜Qn.
4. Power
For a finite eV, Joule heating becomes important. The power defined as energy current from the electrons to the phonons
is [21]:
Q=−i/integraldisplaydε
2π/integraldisplaydω
2π/planckover2pi1ωTr[tr[MG>
0(ε)MG<
0(ε−)]D<
0(ω)]. (A24)
Using Eqs. ( A7), (A8), and ( A18), we can show that it is equivalent to
Q=/summationdisplay
αβ/integraldisplaydε
2π/integraldisplaydω
2π/planckover2pi1ωTr[Xαβ(ε,ε−)A(ω)]nB(/planckover2pi1ω)(nB(/planckover2pi1ω+μβ−μα)+1)(fα(ε)−fβ(ε−)). (A25)
It is nonzero only when α/negationslash=β,s o
Q=/integraldisplaydω
2π/planckover2pi1ωTr[/Lambda1LR(ω)A(ω)]/Delta1nB(/planckover2pi1ω,T;/planckover2pi1ω−eV,T ). (A26)
APPENDIX B: ONSAGER SYMMETRY
Here, we show that Qn=˜Qnin the presence of time-reversal symmetry, e.g., ( Gr)T=Gr. In that case, from Eq. ( A8), we
have
˜Aα=A∗
α,˜A=A,˜Aα=A∗
α,˜A=A. (B1)
Furthermore, AandAare both real. Consequently, in the linear response regime, we can set f(ε,μα,Tα)=f(ε,μβ,Tβ)i n
Eq. ( 7), and get
/summationdisplay
β/Lambda1(n)
˜αβ=/summationdisplay
β/Lambda1(n)
˜α˜β=⎡
⎣/summationdisplay
β/Lambda1(n)
αβ⎤
⎦∗
. (B2)
Using the above two equations, we see that Q∗
n=˜Qn.
On the other hand, the coefficients Qnare real Qn=Q∗
n. This can be seen from the fact that: (1) the matrices M,A,Aare
real, and Aα/Aαis Hermitian; (2) the trace of their products is real. Putting them together, we get the desired result Qn=˜Qn.
APPENDIX C: EQUIV ALENCE TO THE RESULTS OF REF. [38]
Here, we show that results of Ref. [ 38][ E q s .( 35) and ( 36)] are a special case of our results. There, the authors considered one
electronic level at ( ε0) coupled to one vibrational mode with angular frequency ω0. The electronic level couples to two electrodes
and the vibrational mode couples to one phonon bath, characterized by γph. In this special case, all the matrices become numbers.
We get Gr
0(ε)=[ε+iγ(ε)/2−ε0]−1, with γ(ε)=γR(ε)+γL(ε), and Dr
0(ω)=[(ω+iγph)2−ω2
0]−1.N o ww eh a v e Q0=0,
since the phonon mode couples only to one bath and Aαis real. Substituting these two equations to Eq. ( 6), and assuming γphis
small, we get
Qn=m2/integraldisplaydε
4π(ε+/planckover2pi1ω0/2−μ)n/vextendsingle/vextendsingleGr
0(ε+/planckover2pi1ω0/2)/vextendsingle/vextendsingle2/vextendsingle/vextendsingleGr
0(ε−/planckover2pi1ω0/2)/vextendsingle/vextendsingle2∂/planckover2pi1ωnB(/planckover2pi1ω0)[γα(ε−/planckover2pi1ω0/2)γ¯α(ε+/planckover2pi1ω0/2)
−γα(ε+/planckover2pi1ω0/2)γ¯α(ε−/planckover2pi1ω0/2)](f(ε+/planckover2pi1ω0/2)−f(ε−/planckover2pi1ω0/2)). (C1)
This is consistent with Eqs. ( 35) and ( 36)o fR e f .[ 38]. The extra factor 1 /2i nE q .( C1) comes from different definition of the
e-ph interaction m. Note that we need γα(ε) to be energy dependent, and γα(ε)/negationslash=γ¯α(ε), in order for Qn/negationslash=0.
205404-9L¨U, WANG, HEDEG ˚ARD, AND BRANDBYGE PHYSICAL REVIEW B 93, 205404 (2016)
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205404-11 |
PhysRevB.88.115209.pdf | PHYSICAL REVIEW B 88, 115209 (2013)
Epitaxial Si 1−xGe xalloys studied by spin-polarized photoemission
A. Ferrari,*F. Bottegoni, G. Isella, S. Cecchi, and F. Ciccacci
LNESS-Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
(Received 10 June 2013; revised manuscript received 3 September 2013; published 20 September 2013)
Spin-polarized photoemission is used to study Si 1−xGexalloys epitaxially grown on a Si substrate. Spin-oriented
electrons are generated in the conduction band upon excitation with circularly polarized light. We prove thatin these structures it is possible to lower the vacuum level of the system below the conduction band minimumat the /Gamma1point of the Brillouin zone and show that electron spin polarization values P=40±3% can be
achieved when promoting electrons with direct transitions at /Gamma1. Such values can be greatly increased, up to
P=72±3%, in compressively strained thin epitaxial films, thus obtaining performances very close to those of
III-V semiconductor heterostructures.
DOI: 10.1103/PhysRevB.88.115209 PACS number(s): 72 .25.Dc, 71 .70.Ej, 73.40.Gk
I. INTRODUCTION
The possibility to generate a spin-polarized electron popula-
tion in the conduction band (CB) of semiconductors by meansof circularly polarized light, known as optical orientation ,
paved the way to the understanding of spin dynamics andtransport in nonmagnetically ordered systems relevant forspintronics applications.
1,2Moreover, the activation proce-
dure, consisting of the alternate deposition in ultrahigh vacuum(UHV) of Cs and O
2on the clean semiconductor surface, may
lower the material vacuum level Evbelow the CB minimum,3
leading to negative electron affinity (NEA) conditions and thus
allowing the emission of photoexcited electrons. The combina-tion of these two effects leads to spin-polarized photoemission,a technique where the electron spin polarization, defined asP=(n
↑−n↓)/(n↑+n↓), with n↑andn↓representing the
densities of electrons with spin parallel or antiparallel to thequantization axis given by the direction of the incident light,is measured as a function of the photon energy.
4Very efficient
spin-polarized electrons sources based on this process weredesigned
5and are nowadays routinely used in high energy6
and solid state physics.5,7–9
Although the first experimental evidence of optical ori-
entation was obtained in Si,10mainly III-V semiconductors
attracted great attention in the past decades, first with bulkGaAs-based structures
4,5,11and then with quantum-confined
and strained heterostructures.12–16However, technological
issues concerning the integration of spintronics deviceswith nowadays Si-based electronics renewed the interest ingroup-IV semiconductors. As a matter of fact, no spin-polarized photoemission measurements from Si have beenreported to date, while a few studies exist for bulk Ge.
17,18
In Ge NEA conditions cannot be achieved,19so that electrons
excited into the CB minima (both to the direct and the indirect
ones, at /Gamma1andL, respectively) cannot escape into vacuum.
Nevertheless, spin-polarized photoelectrons excited at higherenergies into the /Gamma1valley have been observed.
17,18
Recently, the optical generation of spin-polarized carriers
was extensively studied by our group in compressivelystrained
20–22and quantum confined23Ge-based heterostruc-
tures. Spin-polarized electrons created into the /Gamma1valley were
detected also in these structures and a clear indication thatsome degree of polarization survives the /Gamma1toLscattering was
also given.
23,24Such findings, when combined with the highspin diffusion lengths in Ge25–30and Si,31–35make group-IV
semiconductors very appealing for spintronics applications.Moreover, the possibility to exploit a versatile depositiontechnique, namely low-energy plasma-enhanced chemicalvapor deposition (LEPECVD),
36allows the epitaxial growth
of Si 1−xGexalloys within the full range of composition from
pure Si to pure Ge, making band gap engineering, successfullyexploited in III-V compounds, possible also for group-IVsemiconductors. Within this framework, following a similarstudy on the AlGaAs family,
37we present spin-polarized
photoemission data from Si 1−xGexepitaxial layers. We also
analyze the effect on the photoelectron polarization of theapplication of a compressive strain to thin alloy epilayers,as already reported for pure Ge strained thin films.
20The
paper is organized as follows. After the experimental Sec. II,
where the sample structure and the experimental setup arebriefly described, Sec. IIIis devoted to a discussion of the
photoemission process, pointing out the differences with thecase of III-V semiconductors. Original results on bulklike andstrained Si
1−xGexepilayers are presented and discussed in
Secs. IVand V, respectively, while a summary is given in
Sec. VI.
II. EXPERIMENT
The sample structure is schematically shown in Fig. 1.
A high quality relaxed virtual substrate (VS) graded frompure Si to Si
1−xGexis deposited with a rate of 0.07 /μm.
When the desired Ge content in the alloy is reached, aconstant composition 1 μm thick Si
1−xGexlayer is grown
[bulklike alloy, Fig. 1(a)]. In compressively strained samples
[see Fig. 1(b)], a further 10 nm thick Si 1−yGeyepilayer with
a higher Ge content yis deposited on the Si 1−xGexsubstrate.
Since the lattice parameter in Ge is larger than in Si, thesubstrate in-plane lattice parameter a
xis smaller than the one
of the thin Si 1−yGey(ay), resulting in biaxial compressive
strain /epsilon1||=(ax−ay)/ayin the latter. The strained epilayer
thickness is chosen to be smaller than the critical thicknessabove which relaxation occurs.
38
All samples are p-doped (B, 1017cm−3) to induce a down-
wards surface band bending, as needed for effective electronaffinity reduction.
3Because of this high p-doping level, the
Fermi level inside the semiconductor can be assumed to be
115209-1 1098-0121/2013/88(11)/115209(6) ©2013 American Physical SocietyFERRARI, BOTTEGONI, ISELLA, CECCHI, AND CICCACCI PHYSICAL REVIEW B 88, 115209 (2013)
FIG. 1. (Color online) Scheme of the samples structure: (a) fully
relaxed 1 μmt h i c kS i 1−xGexalloy and (b) compressively strained
10 nm thick Si 1−yGey/Si1−xGex.
pinned to the valence band maximum (VBM). Spin-polarized
photoemission measurements were performed in a UHVsystem equipped with a collecting electron optics composedb ya9 0
◦rotator8and a 20 keV mini-Mott polarimeter.21
The samples were annealed at 600◦C in UHV and then
exposed to Cs and O 2until a stable photocurrent condition was
achieved, following standard procedures.3–8Photoexcitation
was performed using circularly polarized light emitted by amonochromatized (0 .8<h ν< 2.8 eV) halogen lamp with an
energy resolution /Delta1E≈5 meV . The electron spin polarization
P(hν) and quantum yield Y(hν), defined as the number of
photoemitted electrons per incident photon, were measured asa function of the photon energy, without any energy filteringon the emitted electrons. All measurements reported herehave been performed at T=120 K. Data taken at room
temperature show polarization values some 30% smaller, inline with what found in III-V photocathodes.
4,5In the SiGe
case, however, the photoemission yield at room temperature isat least one order of magnitude smaller than at low temperature,which makes polarization measurements somehow unreliable.For this reason, in the following we will discuss onlylow temperature data. Since during the annealing procedurestrained samples could in principle undergo plastic relaxation,the strain level /epsilon1
||was determined by high resolution x-ray
diffraction21only after the spin polarization measurements.
III. PHOTOEMISSION PROCESS: GROUP-IV VERSUS
III-V SEMICONDUCTORS
Spin-polarized photoemission data from GaAs and similar
III-V semiconductors are interpreted in terms of electronphotoexcitation across the direct band gap at the /Gamma1point.
Dipole selection rules for circularly polarized light absorptionin systems with cubic symmetry give rise to an electronpolarization line shape with a maximum value P=50% right
at the gap.
2,4,5On the other hand, the Y(hν) spectrum from
NEA samples reflects the absorption coefficient, with a sharpphotothreshold at the energy gap,
3–5as dictated by the critical
point of the joint density of states at /Gamma1. At variance with
GaAs and similar III-V compounds, Si, Ge, and their alloysare indirect gap semiconductors, which makes the processmuch more involved. In Ge, at T=120 K the CB minimum
at/Gamma1is only 160 meV above the absolute CB minimum located
at the Lpoint, and for photon energies larger than the direct
gap (hν > 0.9 eV) direct transitions are by far predominant.
Thus the photogeneration process in Ge is very similar to theIII-V case and indeed optical orientation takes place in Ge
in the same way as in III-V semiconductors.
17,18,20–24,26In
Si the situation is different: The absolute CB minimum lies1.2 eV above the VBM (from now on all the energies arereferred to the VBM, i.e., to the Fermi level position insidethe semiconductor) at low temperature,
39being located along
the [100] direction of the Brillouin zone close to the Xpoint,
while the direct one at /Gamma1is at 4.1 eV . Below this energy (well
above our photon energy range), optical excitation in Si occursvia indirect transitions and the simple scheme mentioned abovefor optical orientation is not applicable. A recent theoreticalstudy,
40based on the analysis of selection rules in phonon
assisted transitions across the indirect band gap in Si, howevershowed that optical orientation is possible also in this case.A spin polarization around 15% of electrons photoexcitedin the CB minimum close to the Xpoint was predicted for
T=10 K.
40
Once excited in the CB with a certain degree of spin polar-
ization, electrons need to be emitted to vacuum to be analyzedby photoelectron spectroscopy: In GaAs photocathodes this isensured by the NEA conditions. However, it is known that inoptimally activated Ge and Si samples the vacuum level canbe lowered down to 1 .1±0.1e V ,
19,21,41which is above the
CB minima (both at /Gamma1and at L) in Ge and slightly below or
coincident with the CB minimum close to Xin Si. Thus, in
Ge, electrons relaxing to the CB minima cannot be emitted invacuum and only ballistic electrons excited at higher energiesinto the /Gamma1valley contribute to the photoemission yield. In Si,
instead, the yield reflects the low optical absorption for indirecttransitions ( α≈10 cm
−1).41As a result, the photoemission
yield from both Ge and Si photocathodes is at least one order ofmagnitude smaller than the one from GaAs.
3,19,41In Si 1−xGex
alloys the Land/Gamma1energy gaps linearly depend on the Ge
content, with a quite large slope for the direct gap, while asubstantial bowing is observed for the Xband gap.
42Thus,
even for a Si content as small as 10% ( x< 0.9), the /Gamma1direct
CB minimum is expected to lie higher in energy than thevacuum level (i.e., >1.1 eV) and thus to be accessible to our
photoemission technique. Moreover, for x< 0.5 the direct gap
goes out of our photon energy range: We thus disregard theseSi-like alloys and focus instead on samples with a larger Gecontent. For these samples the indirect CB minimum close toXlies at an energy lower than the vacuum level and electrons
excited or relaxed to this minimum do not contribute to thephotoemission yield. This is not always the case for the otherindirect minimum at L, thus making electrons promoted or
relaxed there detectable.
IV . BULKLIKE EPILAYERS
Our results for pure Ge (not shown here) have been
discussed elsewhere:20The photoemission yield is due to
ballistic electrons excited above the vacuum level into the/Gamma1valley and emitted with a polarization value at threshold
around 40%, in very good agreement with the reportedliterature.
17,18Original data for activated Si are presented
in Fig. 2, which shows quantum yield and spin-polarization
spectra as a function of the photon energy. A photothresholdat 1.2 eV , corresponding to the band gap, can be seen, as inNEA photocathodes.
41However, the Y(hν) curve increases
115209-2EPITAXIAL Si 1−xGexALLOYS STUDIED BY ... PHYSICAL REVIEW B 88, 115209 (2013)
FIG. 2. (Color online) Electron spin polarization (dots) and
quantum yield spectrum (line) at T=120 K from p-doped bulk Si.
The typical statistical error relative to the polarization measurement
is indicated by the vertical bar.
quite smoothly with photon energy, without the abrupt onset
typical of GaAs in NEA conditions.3–5As noted in the previous
section, this reflects the absorption coefficient behavior that inSi is not due to direct transitions but to phonon-assisted indirectones. More interesting are the results for the spin polarization,which is negligibly small: P≈0 within the experimental
accuracy for all photon energies in the investigated range.This is in striking disagreement with the mentioned theoreticalresults predicting P≈15% for electrons excited close to
theXpoint by indirect phonon-assisted transitions.
40As a
possible explanation of this discrepancy, we note that ourmeasurements are performed at a much higher temperature(T=120 K) than the one used in calculations ( T=10 K) and
temperature is expected to play an important role in processesinvolving phonons. Moreover, because of the very low opticalabsorption ( α≈8c m
−1at 1.16 eV) and long carrier lifetimes
(τ≈10−6s for our p-doping level)43of Si, photogenerated
carriers spend quite a long time inside the material beforereaching the surface and being emitted. The electron spin willbe therefore detectable only for spin lifetimes τ
scomparable
withτ≈10−6s. Experimental43and theoretical40values of τs
for intrinsic Si at T=60 K are in the hundreds of nanosecond
range making the spin signal undetectable in our experiments.5
Figure 3(a) presents Y(hν) spectra for a set of Si 1−xGexalloys
with different Ge content, namely x=0.63, 0.69, and 0 .8.
In Si 0.2Ge0.8, an abrupt photoemission onset is seen, which is
attributed to the direct transitions across the /Gamma1point as in the
case of NEA GaAs: This allows us to locate the /Gamma1CB minimum
atE/Gamma1=1.55 eV . Electrons excited to the Si-like indirect CB
minimum near Xand/or to the Ge-like CB minimum at L,
which according to the deformation potential calculations42
are located at EX=0.93 eV and EL=0.98 eV , respectivelyFIG. 3. (Color online) (a) Quantum yield spectra at T=120 K
from a set of bulklike alloys. Arrows indicate the onset of the direct
transitions from the VBM to the CB at /Gamma1, providing the direct band
gap energy E/Gamma1(energies are referred to the Fermi level, very close to
the VBM). The slowly increasing background seen in Si 0.31Ge0.69and
Si0.37Ge0.63before this main threshold is given by the LCB minimum
lying in proximity of the vacuum level. The inset shows the fitting
procedure used to extrapolate the energy position of EL(see text).
(b) Experimentally determined direct and indirect CB minimaenergies (i.e., band gap energies) (dots) compared with deformation
potential calculations (after Ref. 42) (lines) for Si
1−xGexalloys at
T=120 K.
[cf. Fig. 3(b)], i.e., below the vacuum energy level, cannot be
detected.
TheY(hν) curves for lower Ge content are significantly
different: After a slowly increasing signal similar to thephotothreshold in Si, a change of slope is clearly seen in thespectra. We attribute the latter feature to the onset of directtransitions at /Gamma1: This allows us to locate E
/Gamma1at 1.8 and 1.9 eV
in Si 0.31Ge0.69and Si 0.37Ge0.63, respectively. The behavior of
Y(hν) at low energies is instead ascribed to indirect transitions
towards the LCB minimum, which are expected to rise at
1.11 and 1.2 eV for x=0.69 and x=0.63, respectively42
[Fig. 3(b)], i.e., slightly above the vacuum level position in
activated samples ( EXis not detectable, being always below
1.1 eV). The energy values of the LCB minimum can be
extrapolated from our data44by performing an interpolation
of the Y2
5versus hνthreshold behavior, as expected for
optical absorption at indirect transitions45[Fig. 3(a), inset].
This procedure yields EL≈1.08 eV and EL≈1.27 eV
for Si 0.31Ge0.69and Si 0.37Ge0.63, respectively. The results are
summarized in Fig. 3(b), reporting the extrapolated E/Gamma1and
ELvalues (dots) as a function of the Ge content of the alloy
x, along with the expected behavior based on deformation
potential calculations.42The good agreement between such
calculations and our data supports our interpretation.
Spin polarization data for Si 1−xGexalloys are shown in
Fig. 4. As a general trend, a shift towards higher photon
energies of the P(hν) spectra for decreasing Ge content x
is observed, which is related to the corresponding increase ofthe alloy direct band gap for larger Si concentrations. Somefeatures are more specific to each alloy. In Si
0.2Ge0.8(Fig. 4,
115209-3FERRARI, BOTTEGONI, ISELLA, CECCHI, AND CICCACCI PHYSICAL REVIEW B 88, 115209 (2013)
FIG. 4. (Color online) Electron spin polarization spectra at T=
120 K from Si 1−xGexalloys with x=0.8,0.69, and 0 .63 (bottom,
central, and top panel, respectively). The typical statistical error is
indicated by the vertical bar.
bottom panel), apart from the mentioned blue shift, the P(hν)
curve strongly resembles the line shape of pure Ge17,18,20:T h e
polarization has a maximum ( ≈43%) at threshold and then
decreases approaching zero for increasing photon energies.Despite this similarity, here the situation is radically different.In pure Ge, where the /Gamma1CB minimum lies below the vacuum
level, the photoemission signal is due to ballistic electronsexcited at higher energies into the /Gamma1valley and the measured
polarization value is interpreted in terms of VB mixing.
17,22
In the present case, instead, the band gap increase associated
to the presence of 20% Si atoms in the alloy raises the /Gamma1
CB minimum above the vacuum level, allowing electronsexcited to the /Gamma1point to be emitted. The situation then is
very similar to the well-known case of NEA GaAs,
2–5with
the difference that in Si 0.2Ge0.8also indirect CB minima (at
Land close to X) are present. They are however located
at an energy below the vacuum level and do not contributeto the photoemission signal, giving rise to a polarizationspectrum completely analogous to the one of GaAs. On theother hand, the presence of these indirect minima provides anew energy relaxation channel which strongly influences thephotoemission yield. In fact, electrons photoexcited by directtransitions at /Gamma1may scatter to an indirect CB minimum, from
where they cannot escape into vacuum. Since this process isvery efficient (the /Gamma1toL-Xscattering in Ge occurs within few
hundreds of femtoseconds),
46a very relevant fraction of the
excited electrons is removed from the photoemission channel,strongly reducing the quantum yield.
In samples with a larger Si content the situation changes,
since, as noted above, the LCB minimum energy increasesand becomes accessible by photoemission. This is reflected by
a line shape variation of the polarization spectra with respect tothe previous case. The P(hν) curves from Si
0.31Ge0.69(Fig. 4,
central panel) and Si 0.37Ge0.63(Fig. 4, top panel) are quite
similar: In both cases the polarization has a very small valueat threshold, reaches a maximum, and then decreases to zero.The main difference (apart from the blue shift) is the reductionof the absolute value of the maximum polarization, reaching33% and 27% in Si
0.31Ge0.69and Si 0.37Ge0.63, respectively.
Interestingly enough, the polarization maximum is obtainedfor a photon energy corresponding to the direct gap, i.e., theE
/Gamma1value discussed above. Such a photon energy dependence
of the polarization spectra can be described as follows. Atlow photon energies, below the direct gap, photoexcitationoccurs via indirect transitions, which are theoretically expectedto give rise to maximum polarization values much smallerthan the ones achievable for direct transitions (15%
47versus
50%). When the photon energy increases, opening the directtransition channel, the polarization reaches the maximumvalue, corresponding to direct band gap excitation, and thendecreases as usual because of the contribution arising fromthe spin-orbit mixing in the split VB.
2,4,5The decrease of
the polarization maximum value can hence be ascribed to thecontribution of poorly polarized electrons photoemitted fromthe indirect CB minimum, which, as shown above, increaseswhen lowering the Ge content in the alloy.
V . THIN STRAINED EPILAYERS
Low-symmetry heterostructures play a key role in the
photogeneration of spin oriented carriers: Indeed, the loweris the symmetry of the Brillouin zone of such systems,the higher is the spin polarization of the electrons excitedto the CB.
2,14–16Generally, a symmetry reduction can be
achieved by quantum confinement and/or strain. We discusshere the case of a 10 nm Si
0.18Ge0.82thin film deposited on a
Si0.4Ge0.6substrate, which, as discussed in the experimental
Sec. II, undergoes a compressive in-plane biaxial strain. The
effective strain degree has been determined by high resolutionx-ray diffraction, resulting in /epsilon1
/bardbl=− 0.94%. The presence
of compressive biaxial strain has two main consequences:(i) an increase of the energy band gap and (ii) the removalof the energy degeneracy between the heavy-holes (HH) andlight-holes (LH) valence bands at the /Gamma1point. By using
reported values for the deformation potentials,
48for our sample
we find E/Gamma1=1.50 eV with a HH-LH energy splitting of
71 meV . The experimental data are shown in Fig. 5.T h eY(hν)
spectrum shows the same features as the Si 0.2Ge0.8unstrained
sample, which has a very similar Ge content: In both casesthe photoemission signal is entirely due to the direct band gapexcitation.
49Note that the quantum yields are very similar
also from a quantitative point of view. This indicates thatthe sample region interested by the photoemission process,i.e., the depth from which emitted electrons are originated,does not sizably change when going from thick bulklikesamples to thin films. This can be ascribed to the fact thatin these materials photoexcited electrons can be scattered tothe indirect CB minimum which lies below the vacuum level,thus not contributing anymore to the photoemission signal. Aremarkable difference is instead seen in the P(hν) spectrum:
115209-4EPITAXIAL Si 1−xGexALLOYS STUDIED BY ... PHYSICAL REVIEW B 88, 115209 (2013)
FIG. 5. (Color online) Electron spin polarization (dots) and quan-
tum yield spectrum (line) at T=120 K from compressively strained
Si0.18Ge0.82/Si0.4Ge0.6. The direct band gap energy is indicated by the
arrow.
At threshold the electron spin polarization is P=72±3%,
well above the 50% maximum theoretical value for bulksystems.
2–5The situation is very similar to the one encountered
in low symmetry III-V systems and can be discussed in thesame way.
14–16The large electron spin polarization stems
from the HH-LH degeneracy removal brought about by strain,which in turns allows to selectively excite at threshold onlyelectrons from the HH band, giving rise to a theoreticalvalue of P=100%. Increasing the photon energy, electrons
from the LH band are also excited, bringing the maximumPvalue back to 50%. The present results demonstrate that
performances close to those of III-V heterostructures canbe obtained also with group-IV semiconductors. However,this holds true only as far as the polarization values areconcerned, while the quantum efficiency results one/two ordersof magnitude smaller than in optimized III-V photocathodes.As discussed above, this is due to the intrinsic indirect gapnature of group-IV semiconductors.
VI. SUMMARY
We characterized by means of spin-polarized photoemis-
sion a set of Si 1−xGexepitaxial layers with different Ge
content, pointing out the role of the vacuum level positionon the detection of the photoemitted electrons. The resultswere discussed in terms of similarities and differences withthe case of III-V semiconductors. The experimental data showthat with an appropriate choice of the Ge content xit is
possible to lower the sample vacuum level below the CBbottom at the /Gamma1point, with a maximum spin polarization of the
photoemitted electrons P≈40%. In compressively strained
thin Si
1−xGexepilayers, this value can be further enhanced
up toP≈72%, comparable to those obtained in optimized
photocathodes based on III-V semiconductors. The possibilityto optically generate spin-oriented carriers with high efficiencyin engineered group-IV heterostructures paves the way to theuse of such systems for spintronics applications.
ACKNOWLEDGMENTS
We thank M. Finazzi for the many fruitful discussions and
the critical reading of the manuscript. This work has beensupported by the Fondazione Cariplo through the EIDOS2011-0382 project.
*alberto.ferrari@mail.polimi.it
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115209-6 |
PhysRevB.75.241409.pdf | Gold nanostructures created by highly charged ions
J. M. Pomeroy, *A. C. Perrella,†H. Grube, and J. D. Gillaspy
National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
/H20849Received 14 May 2007; published 28 June 2007 /H20850
Nanometer-sized structures produced by individual highly charged ion /H20849HCI /H20850impacts are now reported on a
high-conductivity surface, and examined by scanning tunneling microscopy /H20849STM /H20850. Highly charged ions, e.g.,
Bi81+, represent an exotic form of terrestrial matter with neutralization energies that can exceed 375 keV per
ion, and velocities in excess of 1000 km/s from only moderate electrostatic potentials /H2084915 kV /H20850. In the experi-
ment presented here, a single-crystal Au /H20849111/H20850sample was irradiated with Xe25+and Xe44+, which are vastly
different in their neutralization energies. They have moderate velocities /H20849slow compared to Bohr velocity /H20850to
maximize the likelihood of observing features and similar nuclear stopping powers. STM analysis indicatesthat the neutralization energy is less significant in forming features on gold than reported in low-free-electron-density systems. These results support the hypothesis that gold’s high free-electron density enables efficientdissipation of the HCI’s potential energy.
DOI: 10.1103/PhysRevB.75.241409 PACS number /H20849s/H20850: 68.65. /H11002k, 61.80.Jh, 68.37.Ef, 81.40.Wx
Precisely controlled laboratory experiments with highly
charged ions /H20849HCIs /H20850are only a couple of decades old, coin-
ciding with an astrophysical demand for improved atomicdata and an industrial need for HCI erosion rates. For astro-physics, spectroscopic studies of trapped HCIs providebenchmark data for comparison with observations ofsupernovae
1and cometary or planetary x rays,2,3due to HCIs
capturing electrons from gas and dust, respectively. On theother hand, studies of HCI-surface collisions have provenimportant for industrial fusion and reliability estimates ofextreme ultraviolet light lithography critical components, buthave also led to new approaches for lithography, by eitherprecisely controlling HCIs,
4,5or utilizing stochastic distribu-
tions of HCIs for the synthesis of nanodevices.6,7
HCI-surface collisions include the well-studied kinetic en-
ergy dissipation relevant to all atomic projectiles, but HCIsalso carry many keV of neutralization /H20849potential /H20850energy /H20849PE/H20850
that is delivered into only a few cubic nanometers of thesurface, massively intensifying the stimulation of secondarymechanisms. Collectively, HCI-target scientific studies haveidentified many different mechanisms that are significantlydependent on the potential energy dissipation, e.g., second-ary electron yields,
8secondary ion yields,9atomic sput-
tering,10and target core-electron excitation.11Taken alto-
gether, these results suggest variations in the relativestrengths of different dissipation mechanisms between differ-ent materials, which we define as the energy distribution pro-file/H20849EDP /H20850./H20849We suggest the EDP as a material specific set of
prefactors for each of the different energy dissipation mecha-nisms, e.g., 30% PE secondary electrons, 20% PE x rays,25% PE sputtered material, etc., for material X./H20850
The widely believed, but essentially unestablished, view
is that the target material’s free-electron density is the singlemost important parameter in determining the EDP, and,therefore, how the potential energy is dissipated. In particu-lar, materials with low free-electron densities /H20849long hole life-
times /H20850are seen to have sputter yields with strong charge state
dependence,
12and are observed to form “hillocks”13–15due
to single ion collisions. Previous measurements of sputteryields on gold did not report a measurable increase in sputteryield,
16and hillock formation on gold was assumed to benegligible /H20849previous attempts by multiple groups failed /H20850.
However, the scanning tunneling microscope /H20849STM /H20850images
here /H20849e.g., Fig. 1/H20850represent clear evidence of nanofeatures
formed by highly charged ions on gold, a free-electron ma-terial. Initially, this observation appears to challenge the as-
sumed dependence on the free-electron density by implyingstrong coupling of potential energy to ionic motion, but de-tailed analysis presented here reveals several indicators thatsuggest the nuclear stopping of the ion is primarily respon-sible for the deformation under these conditions. This analy-sis of the gold nanofeatures formed by HCIs at two extremesin neutralization energy establishes /H208491/H20850that the EDP for gold
is different from that for insulators, and, therefore, /H208492/H20850that
the EDP is not constant for all materials, and, finally, /H208493/H20850that
free-electron density is, indeed, important in determining theEDP of a material.
In many HCI-surface studies, a common strategy is to
correlate the mean object size /H20849or sputter yield /H20850with the po-
tential energy per ion. In this case, the analysis of the meanobject size is less meaningful due to the breadth and varietyof features. Shown at the top in Fig. 2is a STM image of the
Au/H20849111/H20850surface after irradiation with Xe
25+ions, where
many features are observed at a density approximately equal
to the ion dose, i.e., one nanofeature per HCI impact. After
20 nm
FIG. 1. /H20849Color online /H20850STM image of a nanofeature created by a
single Xe44+ion’s impact on a clean Au /H20849111/H20850surface with 8 keV/ q
of kinetic energy—each distinct level is one atomic height.PHYSICAL REVIEW B 75, 241409 /H20849R/H20850/H208492007 /H20850RAPID COMMUNICATIONS
1098-0121/2007/75 /H2084924/H20850/241409 /H208494/H20850 ©2007 The American Physical Society 241409-1an ion collides, the impact zone thermally equilibrates in a
few picoseconds,17i.e., it forms hexagons. One can easily
find at least three different classes of unique object types,e.g., isolated hexagons, hexagonal rings with craters in thecenter, and hexagonal islands with pits /H20849atomic vacancy clus-
ters/H20850immediately adjacent. The hexagonal shapes form dur-
ing the relaxation after the ion’s impact to align with thecrystal’s low-index directions. For comparison of the objectsize distributions, all feature types are considered equally,and the average displaced volume per feature is calculated;data for Xe
25+and Xe44+extracted at 8 kV are shown in
Table I. These two conditions are a factor of 6.5 different in
potential energy, but differ by only 3% in nuclear stoppingpower
25/H20849a factor of 1.5 difference in kinetic energy that goes
into electronic stopping power /H20850. The mean object sizes scale
similar to nuclear stopping powers. The weak dependence onkinetic energy /H20849due to the peak in nuclear stopping power /H20850isalso consistent with kinetic energy data from Parks et al. ,
18
while, in contrast, no saturation in the potential energy scal-
ing for insulators has been observed in this energy range.14
Surface images from the molecular dynamics simulations byBringa et al.
19and other experimental data of singly charged
xenon in the same kinetic energy range find features of a sizeand structure very similar
20to/H20849although slightly smaller
than /H20850those reported here. The “mean crater area” in that
work at 400 keV of kinetic energy with no potential energyis listed as 34 nm
2, and the crater size in Fig. 1is/H1101540 nm2.
Since the /H20849111/H20850face used in this work does not provide open
crystal channels, it is expected that more collisions will oc-cur close to the surface /H20851compared to the /H20849100/H20850face in the
simulation /H20852, and result in a somewhat larger average size for
surface features. In addition, the distribution of feature sizesand types suggests that the ion’s impact parameter with re-spect to the atomic lattice is important, whereas potentialenergies of this magnitude result in classical electron flow atnanometers from the surface, effectively washing out micro-scopic variations.
21
Another more subtle piece of evidence is shown in the
lower frame of Fig. 2, where the 22 /H11003/H208813 “herringbone” re-
construction is observed both around and on top of a hexa-gon formed by the impact of a single Xe
44+ion, which im-
plies that the island’s atoms came from deep in the bulk. Theherringbone reconstruction spontaneously forms on theatomically flat Au /H20849111/H20850face to relax compressive stresses by
allowing atoms to occupy both hcp and fcc lattice
positions.
22The reconstruction is easily disrupted by atomic
vacancies,23since the vacancies change the compressive
strain field. The island shown is composed of more than3000 surface atoms, and the same number of correspondingvacancies must be present somewhere in the crystal /H20849by con-
servation of mass /H20850, yet the local vacancy density is appar-
ently low enough that the reconstruction is not disrupted.Since the potential energy during neutralization transfers intoonly a few cubic nanometers of the surface around the im-pact site, the absence of these defects would require the long-range migration of 3000 vacancies with little adatom annihi-lation, a very low-probability event. On the other hand,defects produced by kinetic energy stopping will be distrib-uted along the ion channel, and interstitial atoms can migratepreferentially along the track to the surface, while the corre-sponding vacancies can either annihilate at surface steps orform metastable clusters in the bulk.
24
Further evidence that the vacancies are being produced
deep within the lattice /H20849due to the kinetic energy /H20850rather thanTABLE I. Mean feature size and standard deviation for nanofea-
tures formed by Xe25+and Xe44+extracted at 8 kV along with the
corresponding kinetic /H20849KE/H20850and potential /H20849PE/H20850energies per ion and
the nuclear stopping power.
HCIPE
/H20849keV /H20850KE
/H20849keV /H20850Nuclear
stopping
/H20849keV/nm /H20850Mean size
/H20849nm3/H20850Standard
deviation
/H20849nm3/H20850
Xe25+8 200 5.7 23.6 17.2
Xe44+51 350 5.9 26.9 18.9
50 nm
10 nm"Herringbone"
reconstruction(a)
(b)
FIG. 2. Top: STM image of the Au /H20849111/H20850surface after exposure
to Xe25+ions, demonstrating a diversity of nanofeatures /H20849contrast
levels are one atomic height /H20850. Bottom: STM image of a single fea-
ture showing the 22 /H11003/H208813 reconstruction around and on top of the
hexagon formed by the impact of a single Xe25+ion; the island is
one atomic layer high /H20849a discontinuous color map is used to empha-
size the reconstruction on the two atomic levels /H20850.POMEROY et al. PHYSICAL REVIEW B 75, 241409 /H20849R/H20850/H208492007 /H20850RAPID COMMUNICATIONS
241409-2on the surface is shown in the sequence of STM images in
Fig. 3. The central feature is a large island that is three
atomic layers high /H20849the color scale is adjusted in most of the
images to emphasize the changes in the surrounding terrace /H20850.
In the first two images, only one pit a single atomic layerdeep is observed to the upper right of the island, althoughsome other lattice deformation is evident.
26Between the sec-
ond and the fourth images, several additional pits appear, andsome have coalesced. In the final two images, the existingpits grow in breadth and depth, but otherwise remain essen-tially stable. This sequence captures atomic vacancies “bub-bling” up from deep in the lattice.
This evidence indicates the features formed primarily due
to the nuclear stopping of the HCI. The feature size datapresented in Table Iare correlated with the nuclear stopping
component /H20849little change /H20850, but are uncorrelated with the po-
tential energy, even though it changes by more than a factorof 6. Further, the similarity in size and structure to featuresobserved in molecular dynamics simulations and experimen-tal data of singly charged xenon in this kinetic energy rangesupports this conclusion. The broad distributions of featuresizes and types imply that the details of the ion’s impactparameter with the surface’s atomic lattice are important, fur-ther implicating kinetic energy as dominant. Finally, preciseanalysis of the nanofeatures provides indications that the at-
oms observed that compose the nano-structures originatedfrom deep within the crystal: the reconstruction observed onsome islands and the percolation of defects to the surface
over the course of hours both support this hypothesis. Takenall together, we conclude that, in this energy regime, mecha-nisms deriving from kinetic energy dissipation dominate thesurface deformation, while little dependence is seen on po-tential energy.
This conclusion, while consistent with previously re-
ported sputter data
16on gold, does not eliminate the the pos-
sibility that potential energy could play a role in surface de-formation of free-electron metals, but does show that it isalmost certainly much weaker than the nuclear stopping ef-fect. This result stands in stark contrast to results for insulat-ing materials and even graphite, where potential energy de-formation was found to be much stronger than kinetic energymechanisms. Therefore, the EDP for gold is clearly differentfrom that for those materials.
Broadly speaking, this starkly different EDP provides
strong evidence that the target free-electron density is impor-tant. By comparison to insulators, the relatively high densityof free electrons in metals, only a few eV below the vacuumenergy, translates to fast electron transfer occurring while theion is farther from the surface. This provides the surface withmore time to relax with less depletion of the electron densityat those shallow levels /H20849near the Fermi energy /H20850. Earlier elec-
tron transfer also provides the HCI with more time to reduceits energy by secondary electron shedding and relaxation tothe core, via x rays or ultraviolet radiation. The potentialenergy remaining when the ion reaches the surface can alsobe more easily absorbed by the fast response of the metal’selectron sea. Conversely, the depletion of shallow electronlevels in semimetals and insulating materials results in morepotential energy transferred to the lattice, i.e., more latticedeformation and more internal stress.
With this analysis, we have presented evidence that irra-
diation of a gold surface by HCIs can produce nanofeaturessimilar to those previously reported on surfaces with lowfree-electron densities. Careful analysis of these features re-veals that the primary formation mechanisms are likelydriven by the HCI’s kinetic energy /H20849nuclear stopping /H20850, not its
potential energy, demonstrating that the free-electron densityis an important parameter in determining any material’s re-sponse to HCI neutralization. When this result is consideredalong with other published HCI-surface data, it is clear thatdifferent materials must have different EDPs, i.e., dissipatethe eletronic energy more efficiently through mechanismsthat depend on the target material. So, whether one’s interestin HCI-matter interactions derives from observations of cos-mological events, basic atomic physics, or the desire tomanufacture nanofeatures, accurate quantitative interpreta-tion of data requires deconvolving the EDP from the result.
+115 min+90 min
+140 min +145 min
+70 min
+90 min20 nm
+135 min
FIG. 3. STM images of the same nanofeature at various times
after ion irradiation. While the feature is initially observed withonly one adjacent pit, additional pits form and grow in laterimages—the island is three atomic layers high, and some pits aremultiple layers deep.GOLD NANOSTRUCTURES CREATED BY HIGHLY CHARGED … PHYSICAL REVIEW B 75, 241409 /H20849R/H20850/H208492007 /H20850RAPID COMMUNICATIONS
241409-3*joshua.pomeroy@nist.gov
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25Stopping powers calculated with SRIM2006 /H20851J. F. Ziegler and J. P.
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26Images were collected at 0.2 V bias with 2 nA tunnel current.
The first image was taken as part of a survey at much lowerresolution. The vertical compression and diagonal lines in thethird image are due to a turbo pump decelerating during imageacquisition.POMEROY et al. PHYSICAL REVIEW B 75, 241409 /H20849R/H20850/H208492007 /H20850
RAPID COMMUNICATIONS
241409-4 |
PhysRevB.70.224433.pdf | O2phole-assisted electronic processes in the Pr 1−xSrxMnO3(x=0.0, 0.3) system
K. Ibrahim,1,3H. J. Qian,1X. Wu,1M. I. Abbas,1J. O. Wang,1C. H. Hong,1R. Su,1J. Zhong,1Y. H. Dong,1Z. Y. Wu,1
L. Wei,1D. C. Xian,1Y. X. Li,2G. J. Lapeyre,3N. Mannella,4,5C. S. Fadley,4,5and Y. Baba6
1Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100039, People’s Republic of China
2Center of Magnetic Materials and Magnetic Technology, Hebei University of Technology, Tianjin 300130, People’s Republic of China
3Department of Physics, Montana State University, Bozeman, Montana 59717, USA
4Department of Physics, University of California—Davis, Davis, California 95616, USA
5Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
6Synchrotron Radiation Research Center, Japan Atomic Energy Research Institute, Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan
(Received 18 March 2004; revised manuscript received 16 July 2004; published 28 December 2004 )
Experimental results, by x-ray absorption (XAS )at the oxygen K-edge and photon-energy dependence of the
O1s2p2pAuger line at the O Kthreshold, below Mn L2,3as well as well above the Mn L2,3edge of colossal
magnetoresistance (CMR )manganites Pr 1−xSrxMnO3(PSMO )withx=0.0 and x=0.3 compositions, demon-
strate the existence of an oxygen 2 phole state and show its importance in the electronic processes. Both XAS
and Auger spectra self-consistently manifest that the oxygen 2 pholes density of state (DOS )increases with
hole doping in the PSMO system, hinting at a hole state transfer from egsymmetry orbitals of Mn 3 dvalence
bands to oxygen 2 pw i t haM n4+ion increase through Sr2+doping.These are discussed in terms of the possible
interatomic hybridization of Mn 3 dwith O 2 porbitals and a different O 2 pvalence band DOS for different
PSMO compositions in the frame of a covalent picture.
DOI: 10.1103/PhysRevB.70.224433 PACS number (s): 71.27. 1a, 79.60. 2i
I. INTRODUCTION
Understanding the microscopic origin of the CMR effect,
which manifests in an almost all perovskite-likeLn
1−xAxMnO3manganites systems, where Ln stands for the
rare earth La, Pr, Nd, Sm atoms with the s1−xdportion equal
to the molar amount of the Mn3+ion with 3 d4, i.e.,t2g3andeg1
configurations, and the Ais for the alkaline earth like Ca, Sr,
Ba atoms with the xportion equal to a molar amount of the
Mn4+site with a 3 d3,t2g3configuration, which challenges our
current knowledge of strongly correlated electron systems.Almost all manganites have the highest ferromagnetic toparamagnetic transition Curie temperature T
cat a hole-doped
side around x=0.33 while progressively regulating the Mn4+
atom number by doping the A atom through x=0tox=1.1,2
Theoretically, the CMR effect of the hole-doped manganites,
i.e., 0 ,x,0.5, at the metal-insulator transition Curie tem-
peratureTcand below Tcin the ferromagnetic phase state has
been interpreted on the basis of the double exchange model.This model was first proposed by Zener
3and visualized in
detail by Anderson and Hasegawa,4and this theoretical base
has further been enriched by adding structural lattice distor-tion, phonon, polaron, and other effects to meet the needs forflattening the energetically order-scale gap exhibiting withexperimental observations.
Although these compounds, Ln
1−xAxMnO3manganites,
have been known for about fifty years, the metal-insulatorphase transition observed both under, increasing of dopingamountxand with increasing temperature T, as well as the
CMR phenomenon itself, are not yet fully understood. In thehole doped compositions, i.e., for Mn
3+rich manganites sx
,0.50d, the double exchange Mn3+-O-Mn4+ferromagnetic
interactions predominate.3–5Consequently, for the hole-
doped manganites, the CMR effect manifests itself as a resultof magnetic field induced transition, from a paramagnetic
insulator to a ferromagnetic metal.6,7The state above Tcbe-
comes an insulator, which is an unexpected property of aparamagnetic state that transforms into a metal upon reduc-ing the temperature. This curious effect is present in all themanganites, and it is a key property of this family of com-
pounds.
The theoretical interpretation based on the double ex-
change model, in which the ferromagnetic state of the man-ganites are considered on the base of a prejudgment fact,dictates that the Ln
1−xAxMnO3systems in which the Mn va-
lence can have and should retain any of two Mn3+-O2−-
Mn4+neighboring integral values, i.e., so called mixed-
valent systems.1,8All the other effects are treated on the basis
of this confinement condition by neglecting the possibility ofoxygen 2 pelectrons enter into effect. Thus in the region of
the metal-insulator transition, the paramagnetic state of thesesystems are also assumed to keep a natural extension of thisconfinement condition with shifting Tor changing the hole
dopingxcontents. On the other side, theoretical fitting of
experimental observations by configuration-interactionmodel,
9,10that takes the Mn3+-O2−-Mn4+chain in the
Ln1−xAxMnO3manganites as a strongly correlated electron
system, shows that strong p-dexchange interaction between
Mn 3dand oxygen 2 p, and gets a best fit to the experimental
results by taking the oxygen 2 pinto account. Systematic
x-ray absorption measurements at the O 1 sedge9and Mn
K-edge XAS11indicate that the ground state of the hole-
doped Ln 1−xAxMnO3systems are preferably of a highly co-
valent character with a p-dcharge transfer band-gap, i.e.,
with an apparent increase of hole density residing on the O2pside with hole doping quantity xstarting from the
LnMnO
3end-point compound and even p-dcharge transfer
in the LnMnO 33d4electron system, instead of the d-dband-PHYSICAL REVIEW B 70, 224433 (2004 )
1098-0121/2004/70 (22)/224433 (9)/$22.50 ©2004 The American Physical Society 224433-1gap character of the Mott insulator that should be retained
under the double exchange model.
In addition to the evidence for oxygen 2 phole density
associated with hole doping observed by XAS (Refs. 9 and
11)measurements, conductivity measurements12and valence
band DOS variation13with hole doping also show evidence
for the existence of O 2 phole state and its nonlinear re-
sponse to the hole doping.11,13In a recent experimental
investigation14of the local electronic and atomic structure of
single crystal La 1−xSrxMnO3(x=0.3,0.4 )by means of core
and valence level PES, XAS, and EXAFS, focusing in par-ticular on the temperature region below and significantlyaboveT
c, give direct evidence for a dramatic change in the
electronic structure not observed previously in these CMRoxides. This change involves charge transfer to and/or local-ization on the Mn atom and an increase of the Mn spin mo-ment by <1
mB. These effects, which are reversible but with
a<200-K-wide hysteresis centered around at Tc. These pro-
vide far more information about the oxygen 2 pinteraction
with Mn 3 dby magnifying them both as main contributor to
the physics that is beyond the theoretical base designed forCMR effects.
In the present work we focus on two kinds of electronic
processes in order to see if the oxygen 2 pelectrons stay
stonily in their fully occupied state, as a pure ionic picturedepicted, or they are movable around. The first kind elec-tronic process we investigate here is through the XAS at theoxygen K edge. The second one is investigated through theAuger decay process of the oxygen 2 pelectrons following
oxygen 1 selectron ionization. These two kinds of experi-
ments, XAS and Auger, result in direct or indirect informa-tion on the oxygen 2 pdensity of states. The first one directly
relates to the DOS of the hole state associated with oxygen2pnear the Fermi edge; the second one provides direct in-
formation on the DOS of electron associated with oxygen 2 p
below the Fermi edge.
The experimental measurements, XAS at the oxygen K
edge and Auger spectra for the oxygen 1 s2p2pprocess, are
carried for manganites Pr
1−xSrxMnO3withx=0.0 and x
=0.3 compositions. In addition, XAS at the oxygen K edge isalso measured for MnO
2for the purpose of comparing the
results of a pure Mn4+valence state with that of Mn3+in
PrMnO 3. The O 1 s2p2pAuger line measurements were per-
formed at the O Kthreshold, below Mn L2,3and well above
the MnL2,3edge to see its excitation energy dependence.
The XAS result shows a decrease of unoccupied egsym-
metry orbitals’ (dx2
−y2,d3z2
−r2)DOS in Pr 0.7Sr0.3MnO3with
the hole doping of PMnO 3end-point system, contrary to the
expectation that the available unoccupied egDOS would in-
crease with an increase of a Mn4+ion in the system. This
hints that the oxygen 2 phole DOS increases with holes dop-
ing.The O 1 s2p2pAuger peak splits. It is consistent with the
XAS spectra that the unfilled oxygen 2 pvalence band state
gives rise to the splitting of core level ionization spectra as aresult of the final state effects. The split peak can be bestfitted to three component sub-peaks, which are ascribed,from the higher kinetic energy peak to the lower one, to thecore-hole over-screened, normal-screened, and poor-screened states upon O 1 sionization, respectively. The aver-
age central kinetic energy weight position of the three com-ponent peaks and full width at half maximum (FWHM ), with
excitation energy through the O Kthreshold to above the Mn
L
2,3edge, change from about 509 eV to 507 eV and 4 eV to
7 eV, for both x=0.0 and 0.3 systems, respectively, showing
an overall decrease of hole screening upon photon energyincrease. The relative intensity of the three peaks featureretains essentially similar values, in an experiment allowederror range of several percent, for both compositions(x=0.0 and 0.3 )PSMO at the O Kthreshold and below Mn
L
2,3excitation, but the total intensity fluctuation that reflects
the final state affects the characteristic features. This situa-tion further changes drastically at above Mn L
2,3excitation
energy by increasing the total intensity of the x=0.3 system
by a factor of 4 to that of x=0.0 and alternating the relative
intensity between the over-screened state and the poor-screened one.
These are discussed in terms of the possible interatomic
response and different valence band DOS for differentPSMO compositions. These final state effect dependent fea-tures point toward similar origins that differ only on the holeportion appearing in the oxygen 2 porbitals with hole-doping
regulation of s1−xdMn
3+/sxdMn4+portions. These results
hint at the essential fact that O 2 porbitals enter into an
interatomic interaction through 3 dorbitals already in their
ground state with its surrounding Mn ions, and this interac-tions should make non-negligible contributions to the man-ganites CMR materials and enable them to have copiouselectronic, magnetic as well as thermodynamic properties.
II. EXPERIMENT
The measurements were performed at the photoemission
station at Beijing Synchrotron Radiation Facility of Instituteof High Energy Physics, Chinese Academy of Sciences. Theexperimental setup used for the photoemission measure-ments has been described in detail elsewhere.
15The ultrahigh
vacuum chamber background pressure was at ,8
310−10Torr and it was at ,1310−9Torr during the mea-
surement.
The Pr 1−xSrxMnO3polycrystalline samples were prepared
using the conventional solid solution method by mixing theappropriate molar amounts of Pr
2O3,M n2O3, and SrCO 3
powders. The mixtures were ground and calcined severaltimes and were shaped into rods with Sr substitution at x
=0 and 0.3. The x-ray diffraction measurements showed thatthey were in a single-phase state and the samples prepared inthe same procedure were subject to use as target for pulsedlaser deposition and a thin film properties investigation.
16
The sample surfaces were scraped before transfer into the
fast sample entrance chamber and they are scraped again inthe sample preparation chamber before transfer into the ana-lytical chamber. The Auger measurements is carried out us-ing both synchrotron photons at the O 1 sthreshold and
above it, and the photons from a conventional XPS source ofnonmonochromatized Al K
aline (1486.6 eV )are used to
measure the Auger peaks at far above the Mn 2 pedge.
In order to have a common base for the experimental data
and make it possible to compare between them, the raw Au-ger spectra got by photon energies at the O 1 sthresholdIBRAHIM et al. PHYSICAL REVIEW B 70, 224433 (2004 )
224433-2through above the Mn 2 pedges are treated against the data
acquisition conditions like spectrum sweeping times, elec-tron storage ring-current, and spectrum background, etc., ex-cept to normalize them to ionization cross sections due to theexcitation at the O 1 sthreshold exceeds the frame of simple
ionization process itself. The data are fitted using an XPSPeak
17by taking the Tougaard background,18and the back-
ground optimization factors B were taken to be the same forthe spectra obtained at the same excitation energy for differ-ent composition manganites.
The XAS spectra were measured in a way equivalent to
the total electron yield measurement by recording the photo-current of the samples. The latter one is believed to havemore bulk sensitive character relative to the total electronyield due to the large difference between the photon penetra-tion length and the mean path of the free electron escapelength. The three excitation photon energies, 530 eV, 580 eV,and 1486.6 eV, are chosen for the purpose of fulfilling such acondition that they cover sequential regions starting from thepoint that we are able to excite only one O 1 selectron to its
threshold through well above its ionization region and thento far above both the O 1 sand Mn 2 pionization thresholds.
The 530 eV photon corresponds to the oxygen 1 sionization
threshold in the PSMO system.
19
III. RESULTS AND DISCUSSION
The XAS at the O 1 sedges of PrMnO 3and
Pr0.7Sr0.3MnO3plus that of MnO 2were measured. The ac-
companying XAS spectrum of MnO 2is shown for a com-
parison of the Mn4+valence state DOS in rutile structure
MnO2with that of Mn3+in PrMnO 3or mixed valence Mn3+/
Mn4+in Pr0.7Sr0.3MnO2. Though the MnO 2and PSMO be-
long to two different crystalline structures, the local atomicstructure of central transition metal ion Mn has the samenearest neighbor O atom surroundings with an octahedralarrangement. In PSMO the crystal structures develop bysharing all six O vertexes between the octahedral, but inMnO
2the crystal structure forms by sharing between the
edges. The 3 dorbitals of Mn ions in both MnO 2and PSMO
are subject to tetragonal crystal field splitting, so the XASspectra of them are worth comparing by taking the differentvalence states of the Mn ions into account. The arrowpointed peak ain Fig. 1 shows the corresponding threshold
photon energy position used to excitation in the O 1 sAuger
measurements.
Aschematic of 3 d
4orbital splitting of transition metal ion
Mn under an octahedral crystal field is given in Fig. 2. Lift-ing oft
2gandegdegeneracy from the highest symmetry cu-
bic crystal field to a lower symmetry tetragonal crystal fieldis mainly due to distortion introduced by the Jahn-Teller ef-fect. The latter effect is closely related to the electron occu-pancy changes of e
gfrom zero to one that invokes a struc-
tural distortion of the perovskite structure from the CaMnO 3
cubicPm3mspace group to the LaMnO 3orthorhombic
Pnmaspace group. At the present ongoing case the average
formal occupancy of egin Pr0.7Sr0.3MnO3is 0.7 that is be-
tween 1 and 0 of PrMnO 3and MnO 2or SrMnO 3.This means
that the crystal field symmetry augments with hole dopingfrom the highly distorted PrMnO 3end point to the highly
relaxed SrMnO 3end point.
At the oxygen Kedge, the PSMO systems show quite
clearly an electronic structure more complicated by combi-nations of various effects such as electron correlation, ioniccontribution to ligand field splitting, and, most importantly,hybridization of the oxygen and transition-metal atoms intheir ground states. The XAS spectra show obvious differ-ences at <530 eV between the three samples, MnO
2giving a
full and intense peak, PrMnO 3with a shape of a clear shoul-
der, and Pr 0.7Sr0.3MnO3with a long flat slope without an
observable peak.
It is believed that at this excitation energy one oxygen 1 s
electron is promoted to egmajority spin character oxygen 2 p
orbitals.21Although the transition involves oxygen orbitals,
the threshold structure observed at the oxygen Kedge is
determined by the electronic structure of the 3 d-transition-
FIG. 1. XAS measured by the photocurrent recording mode for
MnO2, PrMnO 3, and Pr 0.7Sr0.3MnO3systems at the O 1 sedge,
respectively. The photon energy position of peak ashown by an
arrow was used as the threshold excitation energy to excite O 1 s
electrons and consequently for the acquisition of the O 1 s2p2p
Auger spectra.
FIG. 2. A schematic of field splitting of the (a)five-fold degen-
erate atomic 3 d4levels into (b)respective degenerate lower t2gand
highereglevels under cubic crystal field symmetry, and (c)the
particular Jahn-Teller distortion further lifts each degeneracy of t2g
andegunder tetragonal crystal field symmetry. The figure is taken
from Ref. 20.O2pHOLE-ASSISTED ELECTRONIC PROCESSES IN PHYSICAL REVIEW B 70, 224433 (2004 )
224433-3metal ion. Thus the XAS intensity at the oxygen Kedge
threshold region directly relates to both the hybridization ofoxygen 2 pwith Mn 3 dorbitals on the one hand and the
availability of e
gcharacter oxygen 2 porbitals on the other
hand. The 3 dtransition metal is important because the oxy-
gen 2pshell is full in an ionic picture. Empty oxygen 2 p
orbitals are created through ground-state hybridization be-tween 3d-transition-metal orbitals and oxygen 2 porbitals.
In terms of XAS, for the nø5 systems of 3 d
ntransition
metals, the lowest-energy electron addition state is theground state for the n+1 system. The lowest-energy electron
addition state formed by adding an electron is determined
by Hund’s rule. The
5Egsymmetry for d3and6A1gsymmetry
ford4are reached both with the addition of a majority spin
egelectron. On the theoretical side,21for thed4Mn3+andd3
Mn4+systems, the lowest-energy electron addition states of
6A1gand5Egsymmetries are the only possible way of adding
anegmajority spin electron. The first minority-spin egelec-
tron addition state is at 4.5 eV above the majority spin and it
has lost intensity as compared to the6A1gand5Egstates,
because their charge-transfer energies Dare much larger. For
thed3andd4systems, the threshold region of the oxygen
K-edge XAS spectrum is almost fully made up of final eg
electron addition final states. Experimentally, as shown in
Fig. 1, when compared to the configuration interaction cal-culationresults,
21theXASattheoxygen Kedgedemonstrate
an evidently alternated intensity picture of the egelectron
addition between majority spin and minority spin in thethreshold region 526–538 eV range. They mimic the resultsshown for Mn
2+and Fe3+systems in Ref. 21.
Following the above considerations, and the property of
XAS spectra that provide prestigious information on theDOS of the empty state under the dipole selection rule, onecan draw an important picture from the spectra in Fig. 1 thatin the PrMnO
3system still exhibits considerable oxygen 2 p
characteristic empty egmajority spin orbitals which admit
the electrons excited from the O 1 sorbital. Contrary to
PrMnO 3,i nP r 0.7Sr0.3MnO3there exists no or little such
empty orbitals available to allocate electrons excited fromthe O 1sshell. This indicates that actually there is no avail-
able oxygen 2 pcharacteristic empty e
gorbitals in the hole-
doped Pr 0.7Sr0.3MnO3system to match the nominal portion
of 0.7Mn3+/0.3Mn4+.
These show that the oxygen 2 pcharacter empty egorbit-
als are already occupied or shared by electrons from O 2 p
orbitals in its ground state. This picture is consistent with thetrends in Fig. 1, if one takes the two end-point compoundsPrMnO
3and MnO 2as representative of two extreme valence
states for Mn3+and Mn4+, respectively. The XAS threshold
DOS of Pr 0.7Sr0.3MnO3does not close not only to the DOS
of MnO 2, but it shows even a deviated DOS far beyond that
of PrMnO 3. An almost disappearing peak aDOS in
Pr0.7Sr0.3MnO3indicates that hole doping has largely en-
hanced the hybridization of oxygen 2 pwith Mn 3 d, and the
occupation of egsymmetry orbitals by oxygen 2 pelectrons
making the system have more egelectrons than that in
PrMnO 3.
Core ionization, by means of x-ray radiation, can serve
the purpose of element identification, since the energyneeded for the removal of a core electron is characteristic ofthe involved atomic species. Furthermore, in strongly bound
molecules and solids there is a measurable effect of the mo-lecular environment on the energy of the core hole. A corehole can undergo electronic decay, so-called Auger decay,and the spectroscopy of the emitted Auger electron yieldsfurther information of interest. Atoms in the vicinity of thecore hole may induce not only chemical shifts in the kineticenergy spectrum of theAuger electron.They may even affectthe Auger decay rate in an observable manner.
22The O
1s2p2pAuger decay, with kinetic energy of about 508 eV, is
the main secondary electron process in all oxygen containingmaterials following O 1 sionization.
Figure 3 shows the O 1 s2p2pAuger spectra for PrMnO
3
systemathroughcand for Pr 0.7Sr0.3MnO3systemdthrough
fmeasured at three, i.e., 530 eV, 580 eV and 1486.6 eV,
excitation photon energies. The raw data shown with solidlines are given with fitted three component sub-peaks andtheir sum is drawn by dots, and the chi square of residuals as
a baseline beneath the peaks. The fitted three sub-peaks P
1
throughP3are thought to result as a direct response of the
valence band electronic structure to the O 1 score hole for-
mation. Taking the core hole effect on the valence bands andthe covalency between the p-dorbitals into account, P
1is
ascribed as a core hole over-screened state, P2as a core hole
normal-screened state, and the P3as a core hole poor-
screened state.
By a first glance over these spectra, some obvious trends
or differences can be seen from them. The prominent varia-tion in peak shape, and then the FWHM, for both systemsthat broadens with excitation energy are obvious. The rela-tive intensity of three-component sub-peaks P
1–P3essen-
tially maintain similar weights at 530 eV and 580 eV photonenergies for both samples, but this breakdown at 1486 eV,making an alternation in intensity distribution between P
1
andP3. The center of the peak positions shifts toward lower
kinetic energy with an increase of excitation energy in bothsystems. All these main spectral parameters extracted fromthe fitted spectra shown in Fig. 3 are listed in Table I. Theparameters are arranged in such way that the numbers in thefirst three columns are excitation photon energies, total in-tensities of Auger peaks, and relative intensities of compo-nent sub-peaks.The experimental data accumulation was sta-tistically enough to ensure maximum errors within 1%. Theaccuracy of the peak positions and FWHM in the last twocolumns is within 60.16 eV and the photon energy resolu-
tion was better than 0.9 eV.
The final state core hole provokes different effects on the
valence band electronic structure, and these effects mightdemonstrate themselves in different ways by consequentelectron decay processes at threshold excitation and abovethe threshold region. Excitation at threshold creates a highlyexcitedN
*electron state that proceeds mainly through auto-
ionization decay, while excitation at the above threshold re-gion creates a core shell ionized N−1 electron system which
decays through the normal Auger process.
23When the exci-
tation energy moves away from the O 1 sthreshold edge,
ionization of an oxygen 1 selectron is the only possible way
at 580 eV photon excitation. While the excitation energycovers both O 1 sand Mn 2 pedges, it exhibits the ionization
possibility of both O 1 sand Mn 2 pelectrons at 1486.6 eV.IBRAHIM et al. PHYSICAL REVIEW B 70, 224433 (2004 )
224433-4The total intensity of the spectra, which has been ex-
tracted by normalizing the raw data relative to most impor-tant affecting factors, except for the ionization cross section,such as spectra sweeping times and storage ring current inorder to put the experimental data on a comparable base, isprepared to be comparable between two compositions at the
same excitation energy. The total intensity variation with ex-citation energy, as well as the other parameters listed inTableI, is evidence again of the essential fact that O 2 pand Mn 3 d
hybridize in their ground states and evoke different conse-
FIG. 3. (Color online )O1s2p2pAuger decay spectra of (a)–(b)PrMnO 3and(d)–(f)Pr0.7Sr0.3MnO3measured at 530 eV, 580 eV, and
1486.6 eVphoton energies.The overall spectra can be best fitted by three component peaks P1–P3, the solid lines are experimental raw data,
the dotted lines are sums of fitted three components, and the base lines are the chi square of residuals. The right hand ordinate scales werereduced by a factor of 1000.O2pHOLE-ASSISTED ELECTRONIC PROCESSES IN PHYSICAL REVIEW B 70, 224433 (2004 )
224433-5quences of electron DOS by transferring from O 2 pto Mn
3din the PSMO system with a different hole-doping amount.
That is there is higher oxygen 2 pelectron DOS appearing in
the Mn 3 dorbital for the higher hole-doped Pr 0.7Sr0.3MnO3
than that for without hole-doped PrMnO 3. This means that
there is exhibited more oxygen in the 2 phole state in
Pr0.7Sr0.3MnO3than that in PrMnO 3already in their ground
states.
The differences in the total intensity number between
Pr0.7Sr0.3MnO3and PrMnO 3at 530 eV excitation directly
hints information on the differences in the hole number ap-pearing on the oxygen atom with the hole-doping amount, aswell as reflects the probability of creating an excited oxygenN
*state by promoting an oxygen 1 selectron into 2 pcharac-
teregorbitals. A rough estimation from the total intensity
number indicates about a 34% increase of autoionizationAu-ger intensity for Pr
0.7Sr0.3MnO3relative to that of PrMnO 3.
This is consistent with the XAS feature at the oxygen Kedge
that the higher probability of the N*state results in a higher
probability of Auger decay.
It is accepted that a one-particle analysis is not sufficient
to describe the complex x-ray photoelectron spectra for themetal core levels that are often found for transition-metal
ionic materials.
24,25Excitation at the threshold indicates an
important fact that the O-Mn hybridized valence bands reactas a whole against the core hole effect and that the electronsinvolved in covalent state has high movability between thetwo atoms.This high movability of covalent electrons is verylikely induced by a combination effect of both the core holeformed in oxygen 1 sand the addition of the excited electron
onto valence bands, and the latter one introducing an effec-tive environment for the high movability of electrons by per-turbing the overlapped state formed through the contributionfrom both oxygen and manganese valence orbitals. This ex-citedN
*state may decay through two ways by the excited
electron staying as a spectator or involving it in the decayprocess as a participant and both routes eject an Auger-likeelectron that one measures as Auger peaks. Although theexcitation of oxygen 1 sat the threshold region in the initial
state is strictly a dipole selection rule restricted local process,the final state effect induced by an addition of the promotedelectron into the valence bands is very likely to play the roleof easing the decay process through forward or backwardflow of the delocalized covalently shared electrons betweenthe oxygen and manganese atoms.The PrMnO
3’s total intensity at 580 eV photon energy
accounts is with about 2 times larger than that ofPr
0.7Sr0.3MnO3’s showing the same origin consistent with the
results obtained at threshold excitation, again proving alarger amount of oxygen 2 pelectrons sharing with Mn 3 d
orbitals in Pr
0.7Sr0.3MnO3than that in PrMnO 3. The Auger
spectra measured at 580 eV result in an only oxygen 1 s
ionizedN−1 system and proceeds as a normal Auger decay
process. Unlike the complicated localized initial excitationplus the delocalized final state effects at threshold excitation,ionization of the oxygen 1 selectron at 580 eV shows the
system proceeding as only a localized final state effect,which confines the Auger decay process to occur in an oxy-gen atom without affecting its surrounding manganese at-oms’valence band. At this energy the Auger decays directlyproportional to the oxygen valence band DOS by countingthe response of the valence band electronic structure to thecore hole Coulomb potential. From the alternation of totalintensities between Pr
0.7Sr0.3MnO3and PrMnO 3at threshold
through oxygen 1 sionization at 580 eV one finds a sensitive
role of Mn 3 delectron DOS, which is mainly formed by
dragging oxygen 2 pelectrons, i.e., forming a covalent state
between oxygen 2 pelectrons and manganese egcharacter 3 d
orbitals, in the oxygen 1 s2p2pAuger decay. That an en-
hanced decay rate is achieved at oxygen threshold excitationfor a heavily covalent Pr
0.7Sr0.3MnO3system, i.e., an oxygen
2phole rich state, due to the excitation, is sufficient to pro-
voke perturbation on the covalent electronic state, by exert-ing an extended perturbation on both oxygen and manganeseatoms. While an enhanced decay rate is achieved at 580 eVin the slightly covalent PrMnO
3system, i.e., at an oxygen 2 p
electron rich state, where the excitation is only possible toinduce an atomically localized core hole Coulomb interac-tion with the valence band state.
The total density alternation between PrMnO
3and
Pr0.7Sr0.3MnO3at 1486.6 eV relative to that at 580 eV, and a
further enhancement of the decay rate of Pr 0.7Sr0.3MnO3as
large as 4 times that of PrMnO 3, is qualitatively consistent
with the proposed origin for the cases studied, that the twosamples starting with the initial states, that having a higheroxygen 2 phole density in Pr
0.7Sr0.3MnO3than that in
PrMnO 3, responding in different ways to the different exci-
tation energies. At this energy the reason for giving such aresult is similar to that at oxygen threshold excitation with aTABLE I. The main parameters obtained from the spectra on an O 1 s2p2pAuger process shown in Fig. 3. The first column is the
excitation photon energy utilized to excite the underlying systems; the second column is the total Auger peak intensity for both (a)–(c)
PrMnO 3and(d)–(f)Pr0.7Sr0.3MnO3systems obtained after normalizing the raw data to several factors mentioned in Sec. II; the third row is
the relative peak intensity % of three P1,P2,P3component peaks; the fourth and fifth columns show the averaged central weight of kinetic
energy positions and averaged central weight of FWHM of the peaks, respectively.
Total intensityRelative intensity of
three component peaks (%)Central weight of
peaks’ position (eV)Central weight of
FWHM (eV)
PrMnO 3 PrMnO 3 Pr0.7Sr0.3MnO3PrMnO 3 PrMnO 3
hv(eV) Pr0.7Sr0.3MnO3P1P2P3P1P2P3 Pr0.7Sr0.3MnO3 Pr0.7Sr0.3MnO3
530 212940 285572 14.2 42.4 43.4 10.5 43.3 46.2 509.38 509.17 4.61 4.20
580 277053 142416 10.0 47.7 42.3 9.5 48.9 41.6 508.62 508.38 5.89 5.31
1486.6 113503 455666 11.7 53.5 34.8 27.8 62.5 9.7 507.01 507.35 7.05 7.53IBRAHIM et al. PHYSICAL REVIEW B 70, 224433 (2004 )
224433-6resemblance of the final state effect, but with obviously dif-
ferent final states achieved at the two different excitationenergies. The similarity of the final state effect is that at bothenergies, at oxygen threshold excitation and at 1486.6 eVionization, the Mn 3 dorbitals enter to be affected by direct
or indirect perturbation. As it has been described above, atoxygen threshold the Mn 3 delectrons are disturbed indi-
rectly by adding an electron into valence bands, while at farabove the ionization thresholds of both oxygen 1 sand man-
ganese 2pthe final state might be the ionized core holes of
both oxygen and manganese, respectively.
If the inner shell ionization at 1486.6 eV was simply a
process of only the formation of the oxygen 1 score hole as
that below the Mn 2 pthreshold, the Auger decay process
would mimic the oxygen 1 sat 580 eV. Instead the experi-
mental results at 1486.6 eV indicate that there exists a director indirect perturbation effect on the covalent state analogousto the indirect effect at oxygen threshold. The results hint at
the importance of ionization edges of different elements cov-ered by excitation photon energy in a strongly correlatedsystem. Although this energy might lie well beyond the ab-sorption threshold edges region, where the photon energy isnormally considered to cause a direct ionization of all corestate electrons, it can be reached.
Here are some possible reasoning points for the Auger
decay intensity alternation at 1486 eV relative to that at 580eV. The first point is that a perturbation might be introducedto the valence band state by outgoing electrons with differentkinetic energies ionized from the O 1 sstate at two photon
energies. At 580 eV ionization, the outgoing photoelectronwith about 50 eV kinetic energy has a smaller wavevectork=3.62 Å
−1relative to that with 956 eV of 15.85 Å−1at
1486.6 eV photon energy. The electron with a smallerwavevector has smaller momentum p="k, and then with a
smaller velocity taking a longer time to arrive at the Fermisurface. This difference of high and low kinetic energy elec-trons in the timescale might allow a redistribution of valenceband DOS with a sufficient relaxation time for a low kineticenergy electron, breaking down the frozen-orbitalassumption.
26The high photon energy ionization process
might be well described by the frozen-orbital approximationmodel. A further self-consistent proof for this assumptioncomes from the variation of both the central weight of thepeaks’positions and the FWHM of the peaks listed in TableI. A higher kinetic energy position for 580 eV excitationindicates an overall effective screening of the O 1 score hole
relative to that for 1486.6 eV excitation. The widening of theaverage FWHM of the peaks at 1486.6 eV hints of an in-crease of decaying rates of valence band state electrons forsudden frozen orbitals that have “no time” to take place atfurther redistribution like that at 580 eV excitation. A multi-atom interatomic resonant
27decay of the O 1 score hole
following Mn 2 pionization at 1486.6 eV might provide an-
other possibility for disturbing the covalent bonding state.This kind of perturbation is not possible to reach at 580 eVphoton energy. The last possibility is a double photon doublecore hole formation process on both the nearest neighbor O1sand Mn 2 pstates simultaneously. Actually the last two
possibilities should make little contribution to the intensitiesdue to very low probability of a multi-atom resonance effectin such a process
28and the low possibility of creating two
nearest neighbor core holes simultaneously with a very lowintensity of the incident photon compared to the dense atomnumbers in the system.
Solving the problem that if the manganites system keep
its disproportionation state with hole doping, that forms socalled mixed-valence state or lost its disproportionation hadbeen one of the most important key points in understandingthe origin of copious physical properties in the systems. Fol-lowing crystallographic structure refinement, Daoud-Aladineet al.
29introduced a so-called Zener polaron model, involv-
ing both a local double exchange and a polaronic-like distor-tion and the two mechanisms act together to form vibroniclocalized electronic states, proposing no significant chargedisproportionation on the PCMO system. Under this picture,the disproportionation of Mn ions vanishes through couplingof the local double exchange and polaronic-like distortion.More recently, Grenier et al.
30reports resonant x-ray diffrac-
tion results on the same system that supports a charge dis-
proportionation and proposes the partial occupancy of bothorbitals 3 x
2−r2or 3y2−r2o faM n3+ion andx2−y2of a
Mn4+ion, with indirect evidence. Both conclusions that rule
out ion charge disproportionation or keeping disproportion-ation through partial occupation are based on indirect andstatic methods that are difficult to fulfill the requirements fordrawing on the behavior of the electrons that determine vari-ous physical and chemical properties. Extracting electronicstructural information through atomic structure refinementmight turn out to be especially delicate due to the energyscale change of the property-decisive electrons is countedwithin several meV or tens of K while the systems undergophase transitions.
Relative to obtaining the electronic structural information
through crystallographic structure refinement, a combinativeelectronic structure inspection through photoemission mightbe a straight and dynamic way of directly gathering informa-tion on the density of states of the electrons that have deci-sive roles in the systems’various properties. In this work thecombinational investigation of XAS at O 1 sedges andAuger
spectra of O 1 s2p2pprovides a direct picture on the origin of
almost vanishing or indetectable lowering of the dispropor-tionation of Mn
3+/Mn4+ion pairs in the PSMO system. On
the basis of the above discussions of both XAS and Augerresults, an integrated and self-consistent picture on the roleof oxygen 2 pelectrons and orbitals in the hole-doped PSMO
system can be drawn.That is, the holes created on oxygen 2 p
orbitals through electron transfer to Mn 3 dlead to the disap-
pearance or extremely lowering of disproportionation of theMn
3+/Mn4+ion pairs from the formal ratio s1−xd/xin the
system. On the hole-doped side, the Mn ions in the PSMO
system tend to retain their end point proportionality throughtransferring holes to oxygen 2porbitals. In the present ongo-
ing case no traceable evidence manifests for the exhibition ofaM n
4+ion in the Pr 0.7Sr0.3MnO3system.
It is appealing to consider the possible electron transfer
number from the oxygen 2 pto the Mn egstate. As an ex-
tended picture of the above discussed XAS and O 1 s2p2p
Auger spectra of the PSMO systems, experimental results31
on two series of PSMO systems, that were prepared indepen-dently, support the following conclusions as essential trendsO2pHOLE-ASSISTED ELECTRONIC PROCESSES IN PHYSICAL REVIEW B 70, 224433 (2004 )
224433-7of these compounds. In PSMO systems, that from the end
point compound PrMnO 3with Mn3+ion to another end point
compound SrMnO 3with Mn4+, the electron transfer from
oxygen 2 ptoegorbitals reaches its maximum amount at
aboutx=0.33 for Sr substitution of Pr. At about this point,
the maximum total electron number transferred to Mn 3 d
orbitals might reach 100% relative to the formal electron
numbers counted for the Mn3+ion in their stoichiometric
chemical formula under the mixed valence or charge dispro-portionation consideration. This may provide clear hints onthe origin for the experimental results of why almost allmanganites systems at around the x=0.33 hole doping point
have the highest paramagnetic to ferromagnetic transitiontemperatures T
cin their respective R 1−xAxMnO3series.
Here are some speculative consequences of the vanishing
or lowering of the charge disproportionation in the mangan-ites systems through hole creation on the oxygen 2 porbitals.
In addition to its ubiquity in all of the CMR materials, thehighest phase transition temperatures differ largely betweendifferent rare-earth alkaline-earth R
0.67A0.33ion pair series.
These might be thought to result in the different amount ofoxygen 2 pto Mn 3 delectron transitions due to nearest
neighbor local environment variations. Under these pictures,the phase transition temperature Tcs’ change in a vast rangefor different R
0.67A0.33ion pairs, e.g., at the same hole doping
pointx=0.33, could be due to different charge disproportion-
ation of Mn3+/Mn4+ion pairs in those series. For example,
the La0.67Sr0.33MnO3has the highest Tc32among all manga-
nites ofR0.67A0.33ion pairs, and others have lower or lowest
Tc values.According to the above discussion, the La 0.67Sr0.33
ion pair with the highest Tc should largely or even thor-oughly lose its ion charge disproportionation, on the con-trary, the R
0.67A0.33ion pairs with low Tc has a higher possi-
bility of retaining partial or whole ion chargedisproportionation. Thus the higher the Tc values, the lowerthe possibility for double exchange to enter into effect.IV. CONCLUSION
In the present work we have carried both XAS measure-
ments at the O K edge and O 1 s2p2pAuger experiments at
three selected photon energies for PrMnO 3and
Pr0.7Sr0.3MnO3for the purpose of investigating the role of
the oxygen 2 pvalence bands in the case of the PSMO sys-
tem. For the first time we have explicitly demonstrated ex-perimental evidence for the existence of the oxygen 2 phole
state in the CMR materials such as PSMO systems in theirhole-doped sides, as well as showed the roles played by thehole states in consequent electron decay processes followinginner shell excitation/ionization. The experimental resultshint that the mixed-valence concept does not retain its valid-ity in the paramagnetic state of PSMO system.
The excitation photon energies were chosen for the pur-
pose of promoting the inner shell electrons to reach at char-acteristic excited states so that we could distinguish the ori-gins of the Auger peak intensity variation with photonenergy. By the analysis ofAuger peak intensity we concludethat the PrMnO
3and Pr 0.7Sr0.3MnO3two systems possess
different valence band electronic DOS structures already intheir ground states. The PrMnO
3having a lower covalent
DOS formed between O 2 pand Mn 3 d, contrary to the
highly hole-doped Pr 0.7Sr0.3MnO3with higher covalency.
These different initial states of PrMnO 3and Pr 0.7Sr0.3MnO3
systems show different final state effects against various ex-citation energies, which have been conducted to extract thesame conclusions on their origins, respectively, indicatingself-consistency of theAuger decay intensities’variation cor-responding to their initial states.
ACKNOWLEDGMENTS
We are grateful for financial support from the National
Natural Science Foundation of China (NSFC )under Grants
No. 10074063 and No. 10274084.
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224433-9 |
PhysRevB.103.195308.pdf | PHYSICAL REVIEW B 103, 195308 (2021)
Realization of the Chern-insulator and axion-insulator phases in antiferromagnetic
MnTe /Bi2(Se, Te) 3/MnTe heterostructures
N. Pournaghavi ,1M. F. Islam ,1Rajibul Islam,2Carmine Autieri ,2Tomasz Dietl,2,3and C. M. Canali1
1Department of Physics and Electrical Engineering, Linnaeus University, 392 31 Kalmar, Sweden
2International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32 /46, PL-02668 Warsaw, Poland
3WPI-Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
(Received 15 January 2021; accepted 3 May 2021; published 11 May 2021; corrected 14 May 2021)
Breaking time-reversal symmetry in three-dimensional topological insulator thin films can lead to different
topological quantum phases, such as the Chern insulator (CI) phase and the axion insulator (AI) phase. Usingfirst-principles density functional theory methods, we investigate the onset of these two topological phases in atrilayer heterostructure consisting of a Bi
2Se3(Bi 2Te3) TI thin film sandwiched between two antiferromagnetic
MnTe layers. We find that an orthogonal exchange field from the MnTe layers, stabilized by a small anisotropybarrier, opens an energy gap of the order of 10 meV at the Dirac point of the TI film. A topological analysisdemonstrates that, depending on the relative orientation of the exchange field at the two interfaces, the totalChern number of the system is either C=1o r C=0, characteristic of the CI and AI phases, respectively.
Nontopological surface states inside the energy-gap region, caused by the interface potential, complicate thisidentification. Remarkably though, the calculation of the anomalous Hall conductivity shows that such nontopo-logical surface states do not affect the topology-induced transport properties. Given the size of the exchange gap,we estimate that gapless chiral edge states, leading to the quantum anomalous Hall effect, should emerge onthe sidewalls of these heterostructures in the CI phase for widths /greaterorequalslant200 nm. We also discuss the possibility of
inducing transitions between the CI and the AI phases by means of the spin-orbit torque caused by the spin Halleffect in an adjacent conducting layer.
DOI: 10.1103/PhysRevB.103.195308
I. INTRODUCTION
The discovery of three-dimensional (3D) topological
insulators (TIs), characterized by a bulk band gap and dis-sipationless helical surface states protected by time-reversalsymmetry (TRS), has led to intense research in topological
materials over the past decade [ 1–5]. Introducing magnetic
order in TIs breaks TRS, an exchange energy gap opensup at the Dirac point (DP) of the topological surface states,and different topological phases emerge when the chemicalpotential is tuned inside the gap [ 6,7]. Magnetism in TIs
can be achieved either by doping with magnetic impurities[7–12] or through proximity with magnetic layers [ 13–17].
Recently, intrinsic magnetic TIs such as the van der Waals
layered MnBi
2Te4family [ 18–20] and Mn 4Bi2Te7[21]h a v e
also been discovered [ 22]. Magnetic TIs host a wide range
of quantum phenomena, of which the most important andmost investigated ones are the quantum anomalous Hall effect(QAHE) and the topological magneto-electric effect (TME)[4]. The common origin of these phenomena is the so-called
topological θ-axion term [ 23], which in TIs has to be added to
the ordinary Maxwell electrodynamics [ 24,25]. In 3D TIs, the
θterm is directly related to the topological index Z
2, and can
only affect the 2D Dirac surface states. When TRS is brokenat the surface of a TI, a half-integer quantum anomalous Hallconductivity (AHC) ±e
2/2h(his the Planck’s constant and e
is the electron charge) arises at that surface [ 24–27].In a TI thin film, the resulting topological state depends on
how TRS is broken at the two surfaces, see Fig. 1. Specifically,
when the magnetization at the top and the bottom surfacespoints in the same direction (P configuration) and the chemicalpotential is inside the exchange gap, the system is in the Cherninsulator (CI) state. This phase is characterized by a nonzerointeger Chern number C, and may display the QAHE in which
the Hall conductance in a Hall bar geometry is quantized σ
H=
Ce2/h, due to the emergence of chiral edge states on the film
sidewalls [ 28–31]. On the other hand, when the magnetization
at the top and bottom surfaces points in opposite directions(AP configuration), the system is in the axion insulator (AI)state where C=0. In this phase, all the surface states are
gapped, and transport in a Hall bar geometry is characterizedby a large longitudinal and zero Hall resistance. A quantizedversion of the magnetoelectric coupling (the TME) with auniversal coefficient equal to e
2/2his predicted by theory
[24,25,27,32–34], where an applied magnetic field causes a
response in the electrical polarization and vice versa.
Recent experimental work shows that both the CI and
AI phases can be realized in magnetic modulation-doped TImultilayer films [ 35–37]. However, the magnetic gaps in these
doped-TI thin films are small, and the presence of impuritystates [ 11,38] as well as the dissipative nonchiral edge states
inside these gaps [ 39] substantially limit the maximum tem-
peratures and magnetic field intervals where the CI and AIphases are stable. These are also some of the reasons that have
2469-9950/2021/103(19)/195308(12) 195308-1 ©2021 American Physical SocietyN. POURNAGHA VI et al. PHYSICAL REVIEW B 103, 195308 (2021)
so far precluded the realization of the much more elusive TME
in the AI phase.
An alternative approach to generate uniform 2D magnetism
at the surfaces of a TI thin film is to exploit the interfa-cial proximity with an adjacent film of a magnetic insulatoror semiconductor [ 13–15]. The crucial issues here are the
selection of the best magnetic materials and the nature oftheir coupling with the TI film. The magnetic layer shouldbe able to generate sizable exchange gaps; at the same time,the interface hybridization should not be too strong to avoiddamaging the Dirac surface states or shifting them away fromthe Fermi level. For this purpose, over the past few years,several magnetic insulator-TI heterostructures have been pro-posed theoretically and realized experimentally [ 40–46]. The
majority of these consist of ferromagnetic (FM) insulators,but a few examples with antiferromagnetic (AFM) materialshave been considered [ 13,14,47–49]. Despite all this effort,
realization of the CI and AI phases in these heterostructures isstill quite challenging, and only very recently has observationof the QAHE in a FM /TI/FM trilayer heterostructure been
reported [ 50].
In this paper, we employ density functional theory (DFT)
to study the electronic and topological properties of anAFM/TI/AFM trilayer heterostructure, where the hexago-
nal manganese telluride (MnTe) semiconductor is used tomagnetize the Dirac surface states of two prototypical 3DTIs: Bi
2Se3and Bi 2Te3. The goal of this paper is to inves-
tigate the possibility of realizing both the CI and AI phaseswithin the same heterostructure and introduce efficient waysof switching between these two phases. MnTe is a materialthat had been extensively studied in the past [ 51], but it has
received great renewed attention recently for its relevanceto AFM spintronics [ 52,53]. Its magnetic structure consists
of FM hexagonal Mn planes which are antiferromagneticallycoupled along the c axis (growth direction). AFM insulatorshave an advantage over FM insulators in that their stray mag-netic field, which has detrimental effects at the interface, isconsiderably smaller than the field of FM insulators. Straymagnetic fields can also introduce spurious effects in the studyof the QAHE and the TME.
The trilayer heterostructure considered here allows us to
realize the two mentioned CI and AI phases shown in Fig. 1.
We find that the magnetic anisotropy energy due to the AFMlayers favors an out-of-plane easy axis (i.e., orthogonal to thesurface) with an energy barrier of /lessorequalslant1 meV . The resulting
orthogonal exchange field generates an energy gap of theorder of 10 meV at the DP of the surface states of the TIfilm, which should be viewed as fairly large for this type ofheterostructures. The topological analysis fully supports theexpectation that in the P configuration the system has Chernnumber C=1, whereas in the AP configuration C=0. Impor-
tantly, the band structures for both phases show the emergenceof nontopological surface states in the region of the DP,caused by the interface potential. This feature confirms thatengineering a heterostructure in which the magnetic layersonly provide gap-opening exchange fields remains a challeng-ing task. Nevertheless, a calculation of the AHC shows that,for the heterostructure considered here, such nontopologicalsurface states do not disrupt the transport properties that de-pend on the topological invariants. Atomistic TB calculations
FIG. 1. FM /TI/FM trilayer heterostructure in the two possible
magnetic configurations: (a) parallel (P) displaying the CI phase;(b) antiparallel (AP) displaying the AI phase.
with parameters extracted from the DFT results show that a
nanoribbon with width of ∼200 nm cut out of the heterostruc-
ture does support chiral edge states in the CI phase.
The paper is organized as follows: in Sec. II, we describe
the details of the computational procedure employed in thispaper. The results of the electronic structure, topological anal-ysis, and investigation of the chiral edge states are presentedin Sec. III. In Sec. IV, we discuss an electric mechanism based
on the spin-orbit torque (SOT), which can be used to promotetopological phase transitions from the CI to the AI phase in atopological memory device. Finally, in Sec. Vwe present our
conclusions.
II. COMPUTATIONAL DETAILS
A. First-principles DFT calculations
To study the electronic and topological properties of
MnTe/Bi2(Se, Te) 3/MnTe heterostructures, we have con-
structed a periodic supercell (no vacuum) consisting of sixquintuple layers (QLs) of Bi
2(Se,Te) 3TI sandwiched be-
tween two MnTe films with each film containing three unitcells of MnTe as shown in Fig. 2(a) to preserve the inversion
symmetry. The termination of the MnTe at the interface playsa critical role in magnetizing the TI surface states [ 48]. Since
the Mn atoms provide the exchange coupling to open the gapat the DP, we have placed the Mn layer of MnTe film closestto the TI material to have a stronger impact. However, theMn atoms can be coupled to the topmost TI layer in twodifferent configurations: (i) a top-site setup, where the Mnatom is aligned with the Se (Te) atoms [see Fig. 2(a)] and (ii) a
hollow-site setup, where Mn is aligned with the Se (Te) atomsat the second layer of the TI. We have found that the top-sitesetup is energetically favorable. Therefore, all self-consistentcalculations in this paper are performed using the top-sitesetup.
All DFT calculations are performed by employing the
V
IENNA AB INITIO SIMULATION PACKAGE [54,55] and using
the Perdew-Burke-Ernzerhof generalized gradient approxima-tion (PBE-GGA) for the exchange correlation functional [ 56].
We have first relaxed the crystal structure for both the cell
195308-2REALIZATION OF THE CHERN-INSULATOR AND … PHYSICAL REVIEW B 103, 195308 (2021)
FIG. 2. (a) The relaxed structure of MnTe /Bi2Se3/MnTe het-
erostructure. Comparison between the DFT bands and the low-energy bands extracted using W
ANNIER 90 around the DP without
SOC [(b), (c)] and with SOC [(d), (e)]. The bands with green, red,
and blue correspond to projected bands of Bi 2Se3, Mn, and Te,
respectively.
parameters and the atomic positions using a kmesh of size
6×6×1 until the stress on the cell and the average forces on
the atoms are 0.02 eV /Å.
The final relaxed structure is then used to study the elec-
tronic properties with the inclusion of the spin-orbit coupling(SOC); for this part, we use a larger kmesh of 10 ×10×1
to improve the accuracy of the calculations. To incorporatethe effect of correlations at the transition metal Mn atoms,all self-consistent calculations are performed using GGA +U
with U
eff=4e V .
B. Tight-binding models
For the topological studies of this system, we have con-
structed a real-space tight-binding (TB) Hamiltonian in thebasis of the Wannier states for the low-energy bands extractedby W
ANNIER 90 [57]. The accurate construction of the wave
functions (WFs) of this complex structure is computationallyvery challenging, particularly with SOC. For this reason, wehave implemented the following strategy. We have first calcu-lated the KS orbitals without SOC. Since the DFT calculationsshow that the states near the DP are predominantly the pstates
of Bi and Se (Te) atoms of the TI part of the heterostructure,we have projected the Bloch states onto these porbitals. Thus,
for our system we have constructed a total number of 90 WFsfor each of the two spin states (up and down). To ensure thatthese functions are atomiclike orbitals localized on their re-spective sites, we have used up to 260 bands, which resulted inWFs with a spread of about 3 Å
2or less, compatible with the
spreads of the pristine compounds. To reduce the numericalerror during the Wannierization, we have also implementeddisentanglement, a procedure to project out the contributionof the relevant bands from the unwanted bands [ 58]. The
condition that must be satisfied for this approach to work isthe possibility of disentangling the pbands of the TI from
the bands of the magnetic layer. This is possible with MnTebut not for transition-metal pnictides such as CrSb. Indeed,we also have tried other heterostructures, e.g., CrSb /Bi
2(Se,
Te) 3/CrSb, but were unable to disentangle the pbands of
the TI from the bands of the magnetic layer. Transition-metalpnictides like CrSb show negligible charge transfer [ 59]; as a
consequence, the dbands of the transition metals are at the
Fermi level. Moreover, these dbands cannot be disentangled
from the pbands of the pnictides [ 60].
Since the WFs are constructed from the DFT bands which
include the structural and magnetic effects of all atoms, theseWFs also include the effects of the MnTe films. We wouldalso like to mention that in this approach we have not im-plemented the procedure for maximally localized Wannierfunctions since such a procedure leads to a mixing of differentorbitals, making the use of atomic SOC constants in construct-ing the TB Hamiltonian with SOC unfeasible.
The effective TB Hamiltonian containing the effect of the
exchange field obtained from this calculation is then used toconstruct the Hamiltonian with SOC. The acceptable accuracyof the TB Hamiltonian is determined by the criterion thatthe bands obtained from this Hamiltonian should be a goodmatch with the corresponding bands obtained from the fullDFT calculations when SOC is also included. The inclusionof the SOC into the TB Hamiltonian affects the porbitals of
the Bi and Se (Te) atoms with a coupling, atomic in character,whose strength is given by possibly renormalized atomic SOCparameters λ
Bi, Se, Te . Our calculations show that a good match
of the bands can be obtained with λBi=1.60 and λSe=0.34
for Bi 2Se3TI. On the other hand, for Bi 2Te3TI, only a
small adjustment of the original atomic parameters is required(λ
Bi=1.2 eV and λTe=0.5 eV—note that the correspond-
ing atomic values of the SOC parameters are λBi=1.25 eV ,
λTe=0.49 eV , and λSe=0.22 [61]). Figures 2(b)–2(e) show
the comparison of the bands obtained from DFT and theW
ANNIER 90 TB calculations. A good match implies that the
topological properties can be reliably calculated from the TBHamiltonian.
C. Topological Chern numbers
The topological properties of the system with broken TRS
are characterized by the Chern number, which can be calcu-lated either by integrating the total Berry curvature /Omega1
xy(k)
over the 2D Brillouin zone (BZ) or by integrating the Berryconnection A(k) over the boundary of the 2D BZ:
C=1
2π/integraldisplay
BZd2k/Omega1xy(k)=1
2π/contintegraldisplay
BZdk·A(k), (1)
where
/Omega1xy(k)=ˆz·∇k×A(k), (2)
with
A(k)=i/summationdisplay
n=occ/angbracketleftunk|∇k|unk/angbracketright. (3)
Here unkare Bloch states with energies Enkand the sum in
Eq. ( 3) is over all occupied states, defined by putting the Fermi
energy inside the exchange gap.
195308-3N. POURNAGHA VI et al. PHYSICAL REVIEW B 103, 195308 (2021)
By introducing the velocity operator with components
vi(k)=∂H(k)/∂ki,i=x,y, the Berry curvature can be
rewritten as
/Omega1xy(k)=−2Im/summationdisplay
n=occ
n/prime=unocc/angbracketleftunk|vy(k)|un/primek/angbracketright/angbracketleftun/primek|vx(k)|unk/angbracketright
(Enk−En/primek)2,
(4)
where now the sum in Eq. ( 4) is over both occupied and
unoccupied states.
The expression of the Chern number given by the line
integral of A(k) can be reformulated in terms of Wannier
charge centers (WCCs), localized along the ydirection, which
are defined by [ 62]
¯yn(kx)=i
2π/integraldisplayπ/a
−π/adky/angbracketleftunk|∂
∂ky|unk/angbracketright. (5)
The approach in terms of WCCs is particularly convenient
for the numerical evaluation of the Chern number, as imple-mented in W
ANNIER TOOLS [63]; it is this procedure that we
have used in our work. Nonetheless, the alternative expressionof the Berry curvature given in Eq. ( 4) will be very useful for
the microscopic interpretation of the Chern number and forunderstanding the conditions of validity of the method.
Strictly speaking, the Chern number Cis a well-defined
integer only when the Fermi energy lies between Bloch statebands. In this case, the anomalous Hall conductance of 2Dsystems is equal to the quantized value Cin units of e
2/h.
As we explained in Sec. II.A, the first-principles DFT cal-culations reported here are carried out on bulk 3D systems,where the 2D heterostructure is periodically repeated in thezdirection. However, the topological Chern number is evalu-
ated by means of the effective 2D TB model, correspondingto selecting the k
z=0 plane in the BZ.
III. RESULTS
A. Electronic and magnetic properties
of the MnTe /Bi2(Se, Te) 3/MnTe heterostructures
In this paper, we have investigated two different
heterostructures, namely, MnTe-Bi 2Se3/MnTe and
MnTe/Bi2Te3/MnTe. Since the most important effect caused
by the AFM MnTe layers, namely, the magnetization of thesurface states and the opening of an energy gap at their DP,is expected to occur predominantly only when the exchangefield is orthogonal to the surface of the heterostructure, wewill assume for the time being that this direction of theexchange fields is the one that corresponds to the lowestenergy state of the system. At the end of this section, we willshow that, by calculating the magnetic anisotropy energy,the system indeed has an out-of-plane easy axis with a smallenergy barrier ∼1m e V .
The presence of the magnetic films is expected to open up
a gap at the DPs but it can also modify the orbital properties ofthe surface states, which can play a crucial role in determiningthe topological properties. Therefore, we first discuss how theoverall electronic structure and, in particular, the Dirac surfacestates of a pristine Bi
2Te3slab are modified by the magnetic
films at the two interfaces.
FIG. 3. Projected bands of different QLs showing how the pris-
tine bands of Bi 2Se3in (a) are modified due to the MnTe film in
(b). The gray circles in (b) are the projected bands of the MnTe film(P configuration). The linear dispersion region of the Dirac surface
states is shown in the box.
In Figs. 3(a)and3(b), we have plotted the band structure of
pristine Bi 2Se3and MnTe /Bi2Se3/MnTe, respectively, high-
lighting the contribution of different QLs (first, second, andthird) of Bi
2Se3. Note that only the first three top QLs of
the TI film QLs are shown, the second three being identicalby mirror symmetry with respect to the plane in the middleof the TI film. It is evident that the first and the second QLsclosest to the two interfaces provide most of the contributionto the Dirac surface states (note the linear dispersion regioninside the box) of pristine Bi
2Te3, with the largest contribution
coming from the first QL. The contributions from the thirdQL to the surface states are vanishingly small, and thereforethese QLs may be viewed to involve bulk states only. Inthe heterostructure, on the other hand, we find the followingoutstanding features:
(i) Due to the closest proximity to the MnTe film, the
atomic orbitals of the first QL hybridize with the magneticfilm and decouple from the surface states, which now residemostly on the atoms of the second QL.
(ii) Importantly, as a result of the shift of the surface states
from the first QL to the second, the DP, which was at the Fermilevel for the pristine TI film, moves below the Fermi level byapproximately /lessorequalslant0.2e V .
(iii) As a result of interface potential, a new nontopolog-
ical surface state appears, localized on the very first QL ofthe TI film. See top panel of Fig. 3(b). Given its immedi-
ate proximity with the magnetic layer, this surface state isstrongly spin polarized. At the /Gamma1, the energy band of this
state drops deeply inside the bulk states and has a gap of∼50 meV . Away from the /Gamma1point, this band extends inside
the bulk gap and couples strongly with the topological surface
195308-4REALIZATION OF THE CHERN-INSULATOR AND … PHYSICAL REVIEW B 103, 195308 (2021)
FIG. 4. (a) and (b) Schematic of the setup of the P and AP
configurations of the two MnTe AFM films at the two interfaces. (c),
(d) DFT band structure of the MnTe /Bi2Te3/MnTe heterostructure
with SOC for the P and AP configurations between the two magneticfilms, respectively. The bands with green, red, and blue correspond
to projected bands of the pstates of Bi
2Te3,dstates of Mn, and p
states of Te, respectively. (e), (f) Corresponding bands around theDP showing the breaking of degeneracy for the P configuration. (g),
(h) Expectation value of the spin showing the spiral spin texture at
E=−0.1eV for the P and AP configurations, respectively.
states. It is via this coupling that the topological surface states,
now not directly in contact with the magnetic layers, willacquire a gap (see below). We will also see that the presenceof this nontopological band inside the energy region of the DPrepresents an important challenge in realizing the CI and AItopological phases.
(iv) Very similar features are also found for the
MnTe/Bi
2Te3/MnTe heterostructure.
The space shift of the topological surface states from the
first to the second QL, and the emergence of a nontopologicalbound state at the interface as a result of the crystal symmetrybreaking at the interface, were already pointed out in Ref. [ 14]
for the study of the Bi
2Se3/MnSe heterostructure.
We now discuss the effect of magnetism on the topological
surface states due to the magnetic film. For each of these twoheterostructures, we have studied both the P and AP magneticconfigurations (see Fig. 1) between the exchange fields of the
MnTe films at the two interfaces [see Figs. 4(a)and4(b)]. Our
calculations show that the AP configuration is lower in energycompared to the P configuration by about 9 meV in Bi
2Se3and
by 1 meV in Bi 2Te3TIs. However, this difference in energy is
due to the exchange interaction between the magnetic films ofthe neighboring supercells in our bulklike approach, that is, itis not a property of the magnetic coupling of the two MnTefilms in the isolated heterostructure. To estimate the mag-netic interaction between two interfaces, we have performeda calculation by adding a vacuum of about 16 Å between
the two supercells, which shows that both the P and the APconfigurations are almost degenerate (AP is lower in energythan P by less than 0.1 meV).
In Figs. 4(c) and 4(d), we have plotted the band struc-
ture of MnTe /Bi
2Te3/MnTe for P and AP configurations.
The band structure of the MnTe /Bi2Se3/MnTe heterostruc-
ture (not shown here) is qualitatively identical [see Fig. 2(d)].
We observe the following salient features.
The band structure of the P and AP configurations are
qualitatively very similar, with one important distinction [seeFigs. 4(c) and4(d)]. The DP of Bi
2Te3(Bi2Se3) TI resides
at about 0.13 eV (0.17 eV) below the Fermi level due to thepresence of MnTe films at the two interfaces.
For both configurations, breaking of the TRS at the surface
of the TI by the MnTe film opens up a gap at the DP of theTIs, as shown in Figs. 4(e)–4(f), where we zoom in the region
around the DP. Although MnTe is an AFM insulator andtherefore has no net magnetization, it can provide an effectiveFM exchange coupling at the interface with the followingmechanism: Since the first FM layer of Mn atoms is the closestto the TI surface, its Mn dstates are exchange coupled with
thepstates of the surface electrons within the first QL of the
TI, resulting in a FM spin polarization of the surface states(residing on this first QL) along the magnetization directionof the magnetic film. An exchange gap then opens up at theDP due to the overlap between the spin-polarized nontopo-logical surface states [the states outside the box in Fig. 3(b)]
and the topological surface states residing on the second QL[14]. These spin-polarized surface states have a small but
non-negligible contribution to the topological surface states,which also contribute to the gap opening at the DP. The secondmagnetic layer, which is antiparallel to the first layer, can alsocontribute to the exchange field in the opposite sense. Butsince it is about 2.6 Å away from the first layer, its contributionis exponentially smaller than the other. Therefore, the contri-bution to the exchange gap at the DP essentially results fromthe closest Mn layer. Our calculations show that the gap at theDP of MnTe /Bi
2Te3/MnTe heterostructure is about 12 meV ,
whereas for MnTe /Bi2Se3/MnTe heterostructure, it is about
7m e V .
Although the band structures of the P and AP con-
figurations are quite similar, there is a noticeable qual-itative difference, albeit quantitative small, as shown inFigs. 4(e)–4(f)by zeroing inside the region of the split DP. The
energies in this region originate from two Dirac-cone surfacestates (one from the top and one from the bottom interface).We can see that these energies are exactly twofold degeneratefor the AP configuration, whereas for the P configuration thisdegeneracy is lifted. Disregarding for a moment the com-plications caused by the hybridization with the states of theAFM layers, we can explain this result in the following way.The exchange interaction at the two surfaces spin polarizesthe surface states around the /Gamma1point of the two Dirac cones
and opens a magnetic gap separating valence majority-spinstates (lower in energy) from conduction minority-spin states(higher in energy). In the P configuration, the valence bands ofthe two cones, degenerate in energy, also have the same spincharacter and so do the two conduction bands. Therefore, in athin film, any tunneling between the top and bottom surfaces
195308-5N. POURNAGHA VI et al. PHYSICAL REVIEW B 103, 195308 (2021)
can couple these degenerate states and lift the degeneracy
by creating bonding and antibonding states. In the AP con-figuration, the two valence and two conduction bands havean opposite spin character at the /Gamma1point and cannot couple
directly. In principle, the valence band of the top surface cancouple with the conduction band of the bottom surface andvice versa. However, apart from being a smaller effect sincethey have different energies, this coupling still generates pairsof perfectly symmetric splitting, and therefore the energiesremain twofold degenerate.
A careful inspection of the energy bands shows that in
the AI phase the spectrum of the whole heterostructure isexactly degenerate everywhere. This degeneracy stems fromthe perfect mirror symmetry (included the direction of themagnetization at the interfaces) with respect to a plane locatedin the middle of the heterostructure. Alternatively, the overallsymmetry behind this degeneracy can be viewed as the prod-uct of inversion I(with respect to the origin located in the
middle of the heterostructure) and time-reversal T. Clearly,
the AI phase is symmetric under T·I, whereas the CI is not.
The consequence of the degeneracy is also reflected into
the spin texture of the surface states, as shown in Figs. 4(g)
and4(h), where we have plotted the expectation value of the
spin along a closed loop in kspace around the /Gamma1-point at
constant energy E=−0.1 eV , just above the DP. It is evident
from the figure that for the P configuration the spin statesassociated with the top and the bottom surface states are splitin the momentum space and form a spiral spin texture withopposite helicity. On the other hand, for the AP configuration,these two helical spin states are degenerate in the momentumspace.
To gain insight into the gap difference for the two
heterostructures, we have calculated the potential energy dis-tribution across the interface, averaged over the xyplane as
a function of the distance along the growth direction ( z), and
have analyzed the effect of the interface potential on the wavefunction of the surface electrons. In Fig. 5, we have plotted the
potential profile and the wave function of a surface electronclose to the DP. We note that for Bi
2Se3the peak of the po-
tential is shifted slightly toward the bulk. However, for Bi 2Te3
the potential profile is rather uniform across different QLs.The uniformity is due to the fact that the Te atom is commonto both TI and magnetic film. Therefore, the structure maybe viewed as a continuation of the TI material with some Biatoms replaced by Mn atoms resulting in a weak interface ef-fect compared to Bi
2Se3. As a consequence, the wave function
of the surface electrons of Bi 2Se3is pushed toward the bulk
(extended up to the third QL), similar to the one obtained forthe Bi
2Se3/MnSe heterostructure [ 14]. For Bi 2Te3,t h ew a v e
function is essentially localized between the first and the sec-ond QLs, which is consistent with the surface wave functionof pure TI materials with no interface effect [ 64]. Since the
surface electrons in Bi
2Te3are closer to the magnetic film,
they experience a larger effective exchange field, resulting ina larger gap at the DP compared to that of Bi
2Se3.
Realization of the quantum phases CI and AI in mag-
netic TIs requires opening of an energy gap at the DP. Thesimplest Dirac model of the topological surface states withlinear dispersion predicts that the presence of a TRS breakingexchange field opens a gap only when the magnetization is
FIG. 5. (a) The potential energy distribution and the real-
space representation of a surface state near the DP of the
MnTe/Bi2Te3/MnTe heterostructure (only one interface is shown
here). The red and the green vertical lines show the position ofMn and Te (Se) atoms at the interface. The horizontal dotted lines
show the difference in the two successive potential energy peaks
close to the interface. (b) The same as that for (a) but for theMnTe/Bi
2Se3/MnTe heterostructure. For comparison, we have used
the same cutoff to plot the wave function using VESTA [ 65].
orthogonal to the surface. It turns out that it is also possi-
ble to open a gap when the magnetization is in plane bymeans of other mechanisms. When the magnetization lies inthe plane of the surface, it shifts the DP from the /Gamma1point
along a direction perpendicular to the magnetization. A gapthen opens at the shifted DP due to nonlinear correctionsto the simple Dirac model that are also responsible for thehexagonal warping of the dispersion [ 66,67]. However, such
a gap is typically smaller compared to the gap induced byan out-of-plane magnetization. Therefore, to assess whichmechanism is relevant for the heterostructures investigated inthis paper, we have calculated the magnetic anisotropy energyof the two heterostructures. Magnetic anisotropy calculationsare computationally very subtle because they involve smallenergy differences between the (large) ground state energiesfor different orientations of the magnetization. To attain therequired accuracy in the total energy for a given magnetizationorientation, we have used 16 ×16×1k-mesh for this calcu-
lation. Our calculations show that, for both Bi
2Te3and Bi 2Se3
heterostructures, the ground-state magnetization is orientedalong the out-of-plane direction, namely, we have an easy axisalong the zdirection, with an anisotropy barrier of 0.5 meV
and 0.6 meV , respectively. It is interesting to note that themagnetization of bulk MnTe lies in plane [ 53] with a small
anisotropy of 0.2 meV , but the interaction of the magnetic filmwith the surface electrons of the TI ( d-phybridization) and
the change in lattice constant with respect to the bulk MnTerotate the magnetization along the out-of-plane direction. Theobtained anisotropies, of the order of 0.3 meV per two Mnatoms are one of the largest values obtained in DFT; thereare very few systems like Fe /Pt where one can get higher
values. Actually, these anisotropies would allow the stability
195308-6REALIZATION OF THE CHERN-INSULATOR AND … PHYSICAL REVIEW B 103, 195308 (2021)
of the two phases at 4 K. Note that the majority of the ex-
periments on the QAHE in TI systems are done at sub-Kelvintemperatures.
B. Topological properties of MnTe /Bi2(Se, Te) 3/MnTe
heterostructures
In this section, we investigate whether the trilayer
AFM/TI/AFM heterostructure consisting of the AFM semi-
conductor MnTe and the TI Bi 2Se3or Bi 2Te3can provide a
platform to realize both the AI and CI phases within the samesystem, as a result of the effect of 2D magnetism on the TIsurface states. Topologically, these two phases are character-ized by a total Chern number C=0 and C=1, respectively.
Therefore, we have calculated the total Chern number Cfor
both the P and AP configurations. To set up the calculation, wefirst construct a set of Wannier functions from the DFT result,which are then used to construct the TB Bloch Hamiltonian.The details of this computational procedure are described inSec. II.
In applying this procedure to our two heterostructures, we
face the following problem. As we can see from the bandstructure plotted in Fig. 4, if we set the chemical potential μ
inside the exchange gap, the highest occupied valence band,which at the /Gamma1point is the highest band below μ, will even-
tually cross μat some finite k. Therefore, strictly speaking,
Cis not well defined. However, a close inspection shows
that the highest occupied band at the /Gamma1always maintains a
finite gap with the band immediately above (which is the firstunoccupied conduction band at the /Gamma1point) throughout the
BZ. Therefore, we can formally still apply the procedure givenin Eq. ( 4), and obtain a topological characterization of the
valence energy bands, by assuming that the highest occupiedand the lowest unoccupied bands are the ones defined at the/Gamma1point. Note that this procedure was used in Ref. [ 26]t o
calculate topological invariants for 3D TIs which present afinite direct energy gap throughout the BZ but a negative indi-rect gap due to band overlap. In agreement with expectations,our calculations show that C=1 for the P configuration and
C=0 for the AP configuration for both heterostructures. This
result demonstrates that the P and AP configurations give riseto two topologically distinct states which are consistent withthe CI phase and the AI phase, respectively.
To elucidate further the distinct topological character of
the P and AP configurations and support the claim that theAP configuration does correspond to the AI phase, we haveinvestigated the 3D AHC. The AHC can be calculated directlyfrom the Kubo formula in terms of eigenstates and eigenvaluesof the Bloch Hamiltonian [ 68]
σ
AHC
xy=e2
¯h/summationdisplay
n/negationslash=n/prime/integraldisplay
BZdk
(2π)3[f(Enk)−f(En/primek)]
×Im/angbracketleftunk|vy(k)|un/primek/angbracketright/angbracketleftun/primek|vx(k)|unk/angbracketright
(Enk−En/primek)2, (6)
where f(Enk) is the Fermi-Dirac distribution function for a
given chemical potential μ. From a comparison with Eq. ( 4),
we can see that σAHC
xy is proportional to the integral of the
Berry’s curvature /Omega1xy(k) over the 3D Fermi sea, and therefore
FIG. 6. Anomalous Hall conductivity versus chemical potential
for the MnTe /Bi2Te3/MnTe heterostructure in the (a) AP and (b) P
configurations of the MnTe films. The zero of the chemical potential
corresponds to the center of the nontrivial exchange gap. μ1andμ2
are the chemical potentials at which the two peaks appear close to
the gap edges. The inset in each plot shows the location of μ1andμ2
with respect to the gap.
it is directly related to the topological properties of the Bloch
states.
In Fig. 6, we have plotted σAHC
xy as a function of chemical
potential μ. Note that integral in Eq. ( 6) is over the 3D BZ;
the units of the 3D conductivity σAHC
xy are in ( e2/h)n m−1=
(25812 ×10−7)(/Omega1−1cm−1). The figure shows that when μ
lies inside the exchange gap, delimited by the values μ1and
μ2,σAHC
xy≈0 for the AP configuration [Fig. 6(a)], which
is consistent with the topology-controlled transport proper-ties expected for the AI phase. On the other hand, for theP configuration [Fig. 6(b)],σ
AHC
xy displays a constant value
≈−0.25e2/hnm−1when μis inside the exchange gap,
which would correspond to 2D conductance of the order ofthe quantized value e
2/hfor a few-nanometer-thick thin film.
It is quite remarkable that the presence of nontopologicalstates in the exchange gap region (albeit away from the DP)does not seem to disrupt these topological features for bothconfigurations, suggesting that these states do not contributeto the overall Berry’s curvature.
As the μreaches the top and bottom edge of the exchange
gap, we observe the emergence of two sharp peaks in σ
AHC
xyfor
both spin configurations. This occurrence can be ascribed tothekdependence of the Berry’s curvature, which is sharply
peaked at the /Gamma1point. Therefore, the main contribution to
195308-7N. POURNAGHA VI et al. PHYSICAL REVIEW B 103, 195308 (2021)
σAHC
xy in Eq. ( 6) involves the coupling between the occupied
and unoccupied states at k=0, with the small energy de-
nominator ( Enk=0−En/primek=0)=(μ2−μ1), which is possible
only when μis inside the exchange gap (see insets of Fig. 6).
Clearly, this contribution drops out precisely when μ/lessorequalslantμ1or
μ/greaterorequalslantμ2. Outside the gap region, σAHC
xy is identically zero for
the AP configuration; for the P configuration σAHC
xyis typically
finite but not quantized.
At this point, it is important to address the question of
whether the topological and transport properties presentedhere are an incontrovertible proof that both the CI and AI
phase are in fact realized in these heterostructures. The
nonzero Chern number and the quantized value of σ
AHC
xyfound
in the P configuration is a strong indication that this configura-tion is indeed a CI phase. The case for the AI phase in the APconfiguration is more subtle, since C=0 andσ
AHC
xy=0 could
as well correspond to a trivial insulator with TRS broken atthe surface. Here the only real proof for the AI phase would
be the existence of a half-quantized Hall conductance with
opposite signs at the top and bottom surface. We are unable
to carry out these calculations with the present version ofW
ANNIER TOOLS . Nevertheless, it is very unlikely that flipping
the magnetization from P to AP, which results in only minorchanges in the band structure, should bring about a topologicalphase transition from a CI to a trivial insulator. Therefore, we
believe that the case for the AI phase in the AP configuration
is equally quite strong.
A nontrivial gap at the DP, characterized by different
topological invariants for the CI and AI phases, has differ-ent observable consequences. For a heterostructure of finitewidth, the non-zero Chern number, characterizing the CIphase, results in one-dimensional (1D) dissipationless chiraledge states appearing on the sidewalls, leading to the QAHE(see Fig. 1). On the other hand, in the AI phase, these chi-
ral edge states are absent, and provided that no additionalnontopological 1D conducting states appear on the sidewallsand all the surfaces are insulating; this is one of the essential
condition for the realization of the quantized TME.
To ascertain some of these physical consequences in
the MnTe /Bi(Se,Te) /MnTe heterostructures, we have investi-
gated the heterostructures in a quasi-1D nanoribbon geometry,with a finite width of the order of several tensof nm. The goalis to see the emergence of chiral edge states on the sidewallsof the nanoribbon when the system is in the P configuration
corresponding to the CI phase. Unfortunately, the presence of
nontopological surface states in the gap region of the 2D bandstructure (localized in the first QL, see Fig. 3), when projected
along the edge of the nanoribbon, generates additional non-topological 1D states, blurring the detection of the topologicalchiral edge states.
We have then resorted to the less ambitious goal of assess-
ing the presence and nature of topological chiral edge statesin a magnetic TI nanoribbon in the CI phase, with the sameexchange gap found by DFT for the realistic heterostructure,where the additional nontopological states have been artifi-cially removed. Using an atomistic TB model, we have firstconstructed a pristine Bi
2Se3slab of six QLs. To break the
TRS, we have added an exchange field at the top and thebottom surface layers to mimic the effect of the magnetic filmon the surface states. Since the strength of the exchange field
determines the size of the gap at the DP, we have chosen anexchange field that opens a gap of ≈10 meV , the same order
of gap that we have obtained from the DFT calculations. Thisapproach allows us to obtain a clean surface gap throughoutthe BZ. The P configuration that results in the CI phase isrealized simply by aligning the exchange fields at the twosurfaces along the same direction.
To model the Hamiltonian of Bi
2Se3,w eh a v eu s e dt h e
following sp3TB model [ 39,64,69]:
HC=/summationdisplay
ii/prime,σαα/primetαα/prime
ii/primeeik·rii/primecσ†
iαcσ
i/primeα/prime
+/summationdisplay
i,σσ/prime,αα/primeλi<i,α,σ|L·S|i,α/prime,σ/prime>cσ†
iαcσ/prime
iα/prime
+/summationdisplay
i,σ,αMicσ†
iασσσ
zcσ
iα. (7)
In the first term, tαα/prime
ii/primeare the Slater −Koster parameters for
the hopping energies. cσ†
iα(cσ
iα) is the creation (annihilation)
operator for an electron with spin σand the atomic orbital
α∈(s,px,pypz) at site i.kis the reciprocal-lattice vector that
spans the BZ. i/prime/negationslash=iruns over all the neighbors of atom iin
the same atomic layer as well as the first- and second-nearest-neighbor layers in the adjacent cells ( r
ii/primerepresents the vector
connecting two neighbor atoms). In the second term, the on-site SOC is implemented in the intra-atomic matrix elements[70], in which |i,α,σ > are spin- and orbital-resolved atomic
orbitals. LandSare the orbital angular momentum and the
spin operators, respectively, and λ
iis the SOC strength [ 69].
The last term indicates the exchange field to break the TRS.We assume M=0.025 eV only for the surface atoms yielding
a 10 meV surface gap. Using this TB Hamiltonian, we haveplotted the band structure for a Bi
2Se3slab in Fig. 7. When a
ribbon geometry is constructed with a finite width along thexdirection, edge states (the red lines) appear in the gap as
shown in Fig. 7(b). These edge states are chiral and polarized
as shown in the inset. The black (magenta) arrows show spin-up states which are propagating at the right (left) edge withnegative (positive) velocity.
Since the magnetization is small and there is a small gap
where the edge states cross, there is a small contribution ofstates propagating in the opposite direction at each sidewall.However, by increasing the magnetization we get perfect chi-ral edge states. It is also evident from the figure that the bulkbands occupy the region around the surface gap. Therefore, itis also important to have a larger surface gap to distinguish theedge states from the bulk states. We have plotted the projectedwave functions for the conduction and the valence bandsin Figs. 7(c) and7(d), respectively, which clearly show the
edge character of these bands, localized on the sidewalls. Thepresence of gapless edge states in the CI phase crucially de-pends on the width of the system since the coupling betweenopposite sidewalls can open a gap at the edge states [ 39]. The
minimum width depends on the strength of the exchange field.In the present case, because of the coupling of the side walls,for a nanoribbon of width 125 nm (the largest that we havesolved numerically) there is still a small gap of ≈3 meV at
the/Gamma1point where the edge states cross. This gap decreases
195308-8REALIZATION OF THE CHERN-INSULATOR AND … PHYSICAL REVIEW B 103, 195308 (2021)
FIG. 7. (a) The TB bands of a Bi 2Se3TI slab, in the presence of
an exchange field of M=0.025 eV , which opens a gap of 10 meV
at the DP of the topological surface states. (b) Band structure of aTI nanoribbon with a thickness of six QLs and a width of 125 nm.
The red lines indicate the chiral edge states with polarized spins,
which are blown up in the inset. Black (magenta) arrows show thepolarized states at the right (left) edge. (c), (d) Modulus square of
the wave functions of typical chiral edge states belonging to the
conduction band (c) and the valence band (d) inside the exchange
gap, demonstrating their localization on the sidewalls of the nanorib-
bon. e) Evolution of the gap at the /Gamma1point of the edge states as
a function of the nanoribbon width for a six-QL nanoribbon with
M=0.025 eV . The inset shows the gap as a function of width in
logarithmic scale.
exponentially with increasing width, as shown in Fig. 7(e).F o r
the exchange field used in this calculation, we expect to get azero gap for a width of about 250 nm. This critical width forgapless edge states may be reduced by increasing the strengthof the exchange field.
IV . PROPOSAL FOR A TOPOLOGICAL MEMORY DEVICE
The different topological characters of the CI phase and AI
phase are manifested in their transport properties in Hall bargeometries [ 35–37]. A nonzero Chern number results in ro-
bust dissipationless conducting edge states in the CI phase andthe QAHE. On the other hand, the absence of edge states in theAI phase makes the system highly resistive, with zero longitu-dinal conductance and characteristic zero-field plateaus in theHall conductance. Therefore, realizing both the AI phase andCI phase within the same system can be utilized to developa robust spintronic memory device that exploits their verydifferent conducting properties. For device applications, it isclearly necessary to have a mechanism that induces efficienttransitions between these two phases. Since the MnTe filmused in this study is AFM, its magnetic states are largelyinsensitive to any external magnetic field, if excluding thepresence of remnant magnetizations coming from uncompen-
FIG. 8. Schematics of a setup of the topological memory device
demonstrating the transition from the AI phase to the CI phase using
the spin Hall effect (SHE) and the spin-orbit torque (SOT) effect. The
heterostructure is placed on a SHE substrate (any heavy metal withstrong SOC or a TI). (a) The AI phase with AP configuration be-
tween the magnetic films at the interface; no edge states are present.
(b) Charge current ( J
e) generating a spin current ( Js)b yt h eS H E ,
which gives rise to spin accumulation with in-plane spin polarization
(S) at the interface of the spin Hall substrate and the magnetic film.
The spin accumulation exerts a SOT on the bottom magnetic film,flipping its direction and resulting in the CI phase with chiral edge
state QAHE current J
H.T h e read line detects whether the device is
in|0/angbracketrightor|1/angbracketrightstate.
sated spins at the interfaces. Therefore, to control the relative
orientation of the magnetic order at the two interfaces, wepropose a mechanism based on a combination of the spin Halleffect (SHE) and SOT effect [ 71]. A schematic picture of the
proposed memory device is shown in Fig. 8.
According to this approach, the heterostructure is placed
on a spin Hall substrate, which in principle, can be any heavymetal with strong SOC or a TI. Among heavy metals, a largeSHE is predicted and observed in platinum [ 72–74] and gold
[75–77]. In recent years, there have also been suggestions of
large SHE in TIs such as Bi
2Se3[78] and BiSb [ 79], since a
large nonequilibrium spin accumulation is predicted at the TIsurfaces [ 80]. For our heterostructure, Bi
2Se3could be more
suitable since its lattice constant is a good match with the oneof MnTe. In a recent experiment, an efficient method of spincurrent generation has been demonstrated in WTe
2type-II
Weyl semimetal [ 81], which may also be utilized as a substrate
for the setup proposed here.
Since the AI and CI states are nearly degenerate, the device
can be in any of these two states, which are separated by anenergy barrier set by the small easy-axis magnetic anisotropyenergy. To determine whether the device is in the AI state,labeled as |0/angbracketright, or in the CI state, labeled as |1/angbracketright, we can use
theread line , which simply measures the QAHE current:
For the CI phase, the Hall resistance will be quantized, whilefor the AI phase it will be at a zero-field plateau. If the systemis in the |0/angbracketrightstate, it can be changed to the |1/angbracketrightstate by using
thewrite line as follows: if a charge current J
eis injected into
the substrate along, say the ydirection, then a spin current
Jsis generated along the zdirection via the SHE due to the
strong SOC of the substrate. Consequently, a nonequilibriumspin density of conduction electrons with spin Salong the x
direction is accumulated at the interface of the substrate and
195308-9N. POURNAGHA VI et al. PHYSICAL REVIEW B 103, 195308 (2021)
the bottom layer of MnTe film [Fig. 8(b)]. Since the local
moments of MnTe is perpendicular to S, a SOT is exerted to
the magnetic film. For a sufficiently large Je, the magnetic
orientation of the film can flip, resulting in a transition to theCI phase with the associate chiral edge-state QAHE currentJ
Halong −xdirection (for the opposite edge, JHis in the
opposite direction).
V . CONCLUSIONS
In conclusion, using DFT methods, we have studied
the electronic and topological properties of AFM /TI/AFM
MnTe/Bi2(Se, Te) 3/MnTe heterostructures, with the goal of
assessing the possible realization of both the CI and AItopological phases via the breaking of TRS induced by themagnetic proximity effect. Our calculations show that theelectronic structure of the pristine TI film is sensibly modifiedby the presence of the adjacent AFM films, causing a shiftof the topological surface states away from the interface andthe concomitant appearance of spin-polarized nontopologicalsurface states localized at the interface. An orthogonal andshort-range exchange field, stabilized by a small magneticanisotropy barrier, opens a gap of the order of 10 meV at theDP of the topological surface states in both heterostructures.The topological character of the exchange-induced gap, cap-tured by the Chern number, depends on the relative orientationof exchange fields at the two surfaces of the TI film (Fig. 1):
it is consistent with a CI phase for the parallel spin configura-tion and with the AI phase for the antiparallel configurations,respectively. Given the size of the attained exchange gap, 1Dchiral edge states responsible for the QAHE should becomediscernible on the sidewalls of these heterostructures in theCI phase when their lateral width is of the order of a fewhundred nm.
The strong coupling between the TI and the magnetic films
is a condition to achieve a sizable exchange gap but it comesat the cost of the unwanted presence of nontopological surfacestates in the gap (away from the DP) induced by the interfacepotential. However, our calculations show that such nontopo-logical states have a negligible effect on the conductivity ofboth the CI and the AI phase when the chemical potentiallies inside the exchange gap. Nonetheless, these states pose
an important challenge for the observation of edge states. Analternative that avoids this problem is to consider heterostruc-tures where the magnetic and TI films are coupled by thevan der Waals interaction. Typically, the resulting gaps tendto be small. However, trilayer van der Waals heterostructuresconsisting of a TI thin film sandwiched between recentlydiscovered 2D FM monolayers such as CrI
3[82] seems to be
a promising approach [ 83].
The use of AFM thin films in trilayer heterostructures
has significant advantages over the use of FM films. Theirshort-range TRS breaking field at the TI surface does notproduce stray fields and does not break TRS in the bulkof the TI film, which is a condition for the realization ofthe AI phase [ 32–34,36]. In particular, the AFM MnTe thin
films considered here are presently intensively investigated inspintronics [ 52,53], and progress is being made in the efficient
electrical manipulation of their spin texture and domain walls.In this context, we have proposed a spintronic mechanismbased on the SOT exerted on one of the AFM MnTe films ofthe heterostructure as a means to induce transitions betweenthe CI and AI topological phases, without the need of an ex-ternal magnetic field. This system could realize a topologicalmemory device where the two digital states are encoded andread out by means of the two topological phases.
ACKNOWLEDGMENTS
This work was supported by the Swedish Research Council
(VR) through Grants No. 621-2014-4785, No. 2017-04404,and by the Carl Tryggers Stiftelse through Grant No. CTS14:178. Computational resources have been provided by theLunarc Center for Scientific and Technical Computing atLund University and HPC2N at Umeå university. We alsoacknowledge A. Lau for useful suggestions. The work is par-tially supported by the Foundation for Polish Science throughthe International Research Agendas program co-financed bythe European Union within the Smart Growth OperationalProgramme. We acknowledge the access to the computingfacilities of the Interdisciplinary Center of Modeling at theUniversity of Warsaw, Grants No. G73-23 and No. G75-10.
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195308-12 |
PhysRevB.81.045210.pdf | Temperature-dependent electron and hole transport in disordered semiconducting polymers:
Analysis of energetic disorder
J. C. Blakesley, *H. S. Clubb, and N. C. Greenham
Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
/H20849Received 21 October 2009; revised manuscript received 9 December 2009; published 29 January 2010 /H20850
We have used space-charge limited current measurements to study the mobility of holes and electrons in two
fluorene-based copolymers for temperatures from 100 to 300 K. Interpreting the results using the standardanalytical model produced an Arrhenius-type temperature dependence for a limited temperature range only andmobility was found to be apparently dependent on the thickness of the polymer film. To improve on this,we have interpreted our data using a numerical model that takes into account the effects of the carrierconcentration and energetic disorder on transport. This accounted for the thickness dependence and gavea more consistent temperature dependence across the full range of temperatures, giving support to the ex-tended Gaussian disorder model for transport in disordered polymers. Furthermore, we find that the samemodel adequately describes both electron and hole transport without the need to explicitly include adistribution of electron traps. Room-temperature mobilities were found to be in the region of 4 /H1100310
−8
and 2 /H1100310−8cm2V−1s−1in the limit of zero field and zero carrier density with disorders of
110/H1100610 and 100 /H1100610 meV for polymers poly /H208539,9-dioctylfluorene-co-bis /H20851N,N/H11032-/H208494-butylphenyl /H20850/H20852bis/H20849N,
N/H11032-phenyl-1,4-phenylene /H20850diamine /H20854and poly /H208499,9-dioctylfluorene-co-benzothiadiazole /H20850, respectively.
DOI: 10.1103/PhysRevB.81.045210 PACS number /H20849s/H20850: 72.80.Le, 73.61.Ph, 73.40.Sx, 85.30.De
I. INTRODUCTION
Charge-carrier transport in many disordered organic semi-
conductors and, in particular, conjugated polymers occurs bythermally assisted tunneling via highly localized states and isconsequently strongly temperature dependent. In contrast tothe bandlike transport seen in many inorganic crystals, thisleads to room-temperature mobilities that are small enoughto be a limiting factor in the performance of devices such asorganic light-emitting diodes /H20849OLEDs /H20850and organic photo-
voltaics /H20849OPVs /H20850. In the case of OLEDs, much theoretical
work has been carried out to model charge transport and itseffect on efficiency.
1Generally, larger luminances can be
achieved when electron and hole mobilities are large and
well balanced. It has also been shown that the efficiency ofOPVs may be limited by the rate at which photogeneratedcarriers can be extracted from organic layers.
2Therefore, a
proper description of charge transport in organic semicon-ductors is necessary for understanding device performance.Despite this, there is still no consensus on the most appro-priate way to do this. In particular, it is not clear whethercharge transport is limited by trapping, energetic disorder, orpolaronic effects. Variations in electron transport betweendifferent conjugated copolymers have been attributed to thetrapping effects of impurities introduced during synthesis.
3
Previous studies of electron currents in conjugated polymershave included specific modeling of shallow traps
4or traps
combined with energetic disorder5in order to account for
apparent temperature and thickness dependence of mobility,with such models giving good agreement with experimentalmeasurements. The latter of these studies reconciles the dif-ference in electron and hole mobilities in a derivative ofpoly /H20849phenylene vinylene /H20850by assuming that electron transport
is limited by charge trapping. However, models based solelyon energetic disorder can also account for such effects asthey implicitly include a number of low-energy states thateffectively act as trapping centers.In this paper, we focus on using energetic disorder as a
method for modeling charge transport, showing that materi-als can be described simply by a single value of energeticdisorder, without needing to refer to an explicit distributionof trap states. Furthermore we show that the same model canbe used for both positive and negative charge carriers inconjugated polymers, leading to simplified modeling of bi-polar devices.
II. THEORY OF SPACE-CHARGE LIMITED CURRENTS
The polymers used is this work,
poly /H208539,9-dioctylfluorene-co-bis /H20851N,N/H11032-/H208494-butylphenyl /H20850/H20852bis
/H20849N,N/H11032-phenyl-1,4-phenylene /H20850diamine /H20854/H20849PFB /H20850and poly /H208499,9-
dioctylfluorene-co-benzothiadiazole /H20850/H20849F8BT /H20850/H20849see Fig. 1for
chemical structure /H20850are air-stable polymers based on poly-
fluorene. The fluorene units are copolymerized with hole- orelectron-transporting units, respectively. These polymershave been widely investigated for their use in both OLEDs/H20849Ref. 6/H20850and all-polymer OPVs,
7and are a useful materials
system for studying the physics of polymer devices.
Since most OLED and OPV devices consist of one or
more thin /H20849on the order of 100 nm /H20850layers of active material
C8H17N Nn
N
SNnPFB
F8BTC8H17
C8H17 C8H17
FIG. 1. Chemical structure of PFB and F8BT.PHYSICAL REVIEW B 81, 045210 /H208492010 /H20850
1098-0121/2010/81 /H208494/H20850/045210 /H208499/H20850 ©2010 The American Physical Society 045210-1sandwiched between metal contacts, it is appropriate to mea-
sure the mobility in devices with a similar configuration.Therefore charge transport is investigated using steady-statecurrent-voltage /H20849J-V/H20850measurements. This is achieved by se-
lecting metal contacts such that only holes /H20849electrons /H20850are
injected into the active layer. A build up of positive /H20849nega-
tive /H20850charge in the active layer then limits the total current
density. Such a device is called space-charge limited /H20849SCL /H20850.
At the simplest level, J-Vcurves in single-carrier SCL de-
vices can be interpreted by assuming a constant mobility andfitting
8with the Mott-Gurney relation9
J=9
8/H9262/H9255V2
d3, /H208491/H20850
where Jis the current density, Vis the applied voltage /H20849tak-
ing account of any built-in voltage /H20850,/H9262is the mobility, /H9255is
the permittivity of the material, and dis the film thickness. It
has been shown that this is an oversimplification as studiesusing time-of-flight,
10–12SCL current,13dark injection, and
admittance spectroscopy14have all shown that mobility tends
to change with the application of an electric field in organicsemiconductors. The field dependence of mobility has oftenbeen described by a Poole-Frenkel-type enhancement factor,
g
1/H20849E,T/H20850= exp ⌊/H9253/H20849T/H20850/H20881E⌋, /H208492/H20850
where Eis the electric field and /H9253/H20849T/H20850is a materials parameter
reflecting the strength of the electric field dependence. Thisparameter is generally found to decrease with increasingtemperature, T. A modified version of the Mott-Gurney equa-
tion by Murgatroyd
15is used to describe the J-Vcharacter-
istics of SCL devices with Poole-Frenkel-type mobilityenhancement,
7,16
J=9
8/H92620/H9255exp /H208490.89/H9253/H20881V/d/H20850V2
d3. /H208493/H20850
More recently, Tanase et al.17considered the effect of carrier
concentration on mobility. Devices with large carrier concen-trations, such as field-effect transistors, were found to havemuch higher mobility than those with low carrier concentra-tions. They demonstrated that the effect of carrier concentra-tion can be understood by including a second mobility en-hancement factor. Thus, rather than describing transport by aconstant mobility, it is more appropriate to consider mobilityas a function of a number of parameters,
/H9262/H20849T,E,n/H20850=/H92620/H20849T/H20850g1/H20849E,T/H20850g2/H20849n,T/H20850, /H208494/H20850
where g1andg2are the dimensionless mobility enhancement
factors for electric field, E, and carrier density n, respec-
tively. /H92620is the mobility as a function of temperature, T,i n
the low-field, low-carrier-density limit.
Modeling studies of charge transport have been carried
out in the framework of the Gaussian disorder model /H20849GDM /H20850
pioneered by Bässler.18A fundamental assumption of this
model is that carriers are highly localized to specific fixedsites within a medium. Carriers hop between these sites,which each have a random energy offset taken from a Gauss-ian distribution. The width /H20849standard deviation /H20850of the Gauss-
ian,
/H9268, quantifies the amount of energetic disorder in thematerial. Bässler’s Monte Carlo simulations of transport
have successfully described the field-dependent mobility en-hancement observed in experiments. However, Monte Carlosimulations become more computationally intensive as thecarrier density is increased, so they are generally carried outin the limit of low carrier densities, which gives no informa-tion on the second mobility enhancement factor. Instead,numerical
19and analytical20models have been used to make
predictions about the carrier-density dependence of mobility.
Of these two models, the Pasveer model,19which is re-
ferred to as the extended Gaussian disorder model /H20849EGDM /H20850,
obtains mobility functions by solving carrier transport in athree-dimensional medium with Gaussian disorder using amaster-equation approach. This approach allows the simula-tion of large carrier densities without the computational over-heads experienced with Monte Carlo simulations. The mo-bility is then parameterized using a simple equation thatallows a rapid approximation of the mobility enhancementfactors, g
1and g2without the need for detailed modeling.
The Fishchuk model20instead uses an analytical model to
derive the mobility as a function of carrier density. Thismodel involves integrating a hopping rate over the density ofoccupied and unoccupied states. Unlike the Pasveer model,the Fishchuk model does not derive mobility as a function ofelectric field.
Both of these models give similar results, predicting a
constant mobility at low carrier concentrations with a strongenhancement when the carrier concentration exceeds a cer-tain level. This critical concentration corresponds to the den-sity at which carriers essentially cease to behave indepen-dently of one another. The mobility enhancement is
dependent on the ratio
/H9268/kBT/H20849referred to as /H9268ˆ/H20850, with stron-
ger enhancement occurring as disorder is increased or tem-perature is reduced. The reason for the mobility enhancementwith carrier concentration is easy to understand. The Gauss-ian density of states includes a low-energy tail of states ex-tending well below the mean site energy. Carriers in theselow-energy sites are effectively trapped since the surround-ing sites are likely to have higher energies and hence thecarriers require significant thermal activation to hop to aneighboring site. At low carrier concentrations, carriers pre-
dominantly occupy the low-energy sites and thus the mobil-ity is low. When the carrier concentration exceeds a criticalvalue, the tail of the density of states is effectively filled andany new carriers added to the system must occupy higher-energy states from which hopping is easier, leading to anincreased mobility.
The temperature dependence of mobility is a critical fac-
tor in understanding the physics underlying charge transport.All three models mentioned above /H20849Bässler, Pasveer, and
Fishchuk /H20850based on the GDM predict the logarithm of mo-
bility will vary as
/H9268ˆ2in the low-carrier-density limit. Contra-
dictorily, many studies of mobility in SCL devices21show an
apparent temperature dependence closer to 1 /T. Rather than
contradicting the GDM, this temperature dependence hasbeen attributed to the effects of carrier density,
22which can
typically be on the order of 1016–1018cm−3in SCL devices.
Indeed, Fishchuk20has shown that the effective mobility is
expected to follow this behavior at high carrier densities.
Therefore, to completely understand experimental data, it
is necessary that the next generation of device simulationsBLAKESLEY, CLUBB, AND GREENHAM PHYSICAL REVIEW B 81, 045210 /H208492010 /H20850
045210-2include both field and concentration contributions to mobil-
ity. van Mensfoort and Coehoorn22have recently included
these effects /H20849using the Pasveer19model /H20850within a one-
dimensional continuum device model. The model has beenused to fit J-Vcurves of hole transport in a conjugated
polymer
23and successfully described the temperature and
film thickness dependence. By the inclusion of the concen-
tration effects, the predicted /H9268ˆdependence was reproduced.
In this paper, we use a similar device simulation to investi-gate charge transport in films of PFB and F8BT. We alsocompare the effects of interchanging the Fishchuk model andPasveer model for carrier-density dependence and of using aPoole-Frenkel field dependence. Although detailed simula-tions have shown that transport in disordered materials oc-curs through a three-dimensional network of microscopicfilaments, a recent study has shown that the use of suchone-dimensional continuum models is justified as long as thematerial thickness is greater than the typical filament size.
24
In practice, all SCL devices are likely to fall into this cat-egory.
III. EXPERIMENTAL METHOD
Single-carrier devices were fabricated by spin-coating
PFB and F8BT from solution in p-xylene. Indium-tin oxide
/H20849ITO /H20850coated glass was used for substrates. Substrates were
cleaned ultrasonically in acetone then propan-2-ol, thentreated with an oxygen plasma. For PFB, hole-only deviceswere fabricated by spin-coating poly /H208493,4-ethylene dioxy-
thiophene /H20850doped with poly /H20849styrene sulfonate /H20850as the bottom,
hole-injecting electrode. PFB was spin coated in ambientconditions, then gold was evaporated to form the top elec-trode. For F8BT devices, the ITO coating was first removedfrom the substrate by etching in HCl prior to cleaning. Alu-minum was evaporated onto the glass to create a bottomcontact. The aluminum-coated substrates remained in a pro-tective nitrogen environment while F8BT was spin coated.Finally, calcium capped with aluminum was evaporated as anelectron-injecting top electrode. For each polymer, devices ofthree different thicknesses ranging from 50 to 350 nm werefabricated by varying the concentration of the polymer solu-tions. Film thicknesses were measured using a Dektak sur-face profiler. Devices were mounted in a liquid-helium cry-ostat and steady-state current-voltage characterization wascarried out using a Keithley 6487 electrometer at tempera-tures ranging from 100 to 295 K. Care was taken to ensurethat steady-state current measurements were obtained freefrom hysteresis. This involved keeping sweep rates low/H20849typically about 1 V/min /H20850and introducing a 3 s delay be-
tween changing voltage and reading current. The devices re-mained in an atmosphere of low-pressure /H20849/H1101110 mbar /H20850he-
lium during measurements. Examples of temperature-dependent current-voltage curves are shown in Fig. 2.
IV. DRIFT-DIFFUSION MODEL
A one-dimensional continuum model was used to simu-
late steady-state current-voltage curves in single-carrier de-vices. The model is quantitatively very similar to the modelemployed by van Mensfoort and Coehoorn, although we use
a different iteration algorithm, more similar to that used byKoster et al.
25Current density, J, is described by drift and
diffusion of carriers in the standard way,
J=−/H9262neE+eDdn
dx, /H208495/H20850
dE
dx=e
/H9255n, /H208496/H20850
and
V=−/H20885
0d
Edx, /H208497/H20850
where xis the depth in the polymer film, Dis the diffusion
coefficient, and eis the charge of a single carrier. Equation
/H208495/H20850describes the current density in terms of the drift and
diffusion currents. Equation /H208496/H20850is Poisson’s equation while
Eq. /H208497/H20850describes the application of a bias voltage. The equa-
tions are solved simultaneously to reach a steady-state solu-tion in which dJ /dx=0.
The above equations are the standard drift-diffusion equa-
tions, which are commonly used to describe charge transportin organic semiconductors. In many previous studies, theseare solved by using a constant mobility and applying theBoltzmann approximation /H20849D=
/H9262kBT/e/H20850, which, for semicon-
ductors with a broad density of states, is accurate only in thelow-carrier-density limit. These solutions are more appropri-0 5 10 1510-1110-910-710-510-310-1Current density (A cm-2)
Bias (V)Al / 110nm F8BT/ Ca/ Al100K295K
a)
0 5 10 1510-1010-810-610-410-2Current density (A cm-2)
Bias(V)PEDOT:PSS/140nm PFB/Au100K295K
b)
FIG. 2. Example current-voltage curves for /H20849a/H20850a 110-nm-thick
F8BT electron-only device and /H20849b/H20850a 140-nm-thick PFB hole-only
device at temperatures from 100 to 295 K /H20849solid lines in order from
bottom to top: 100, 125, 150, 175, 200, 225, 250, 275, and 295 K /H20850.TEMPERATURE-DEPENDENT ELECTRON AND HOLE … PHYSICAL REVIEW B 81, 045210 /H208492010 /H20850
045210-3ate to crystalline semiconductors than disordered organic
semiconductors. Here, we aim to solve the equations in thecontext of the EGDM. This is done by making three modifi-cations to the standard drift-diffusion model. First, the shapeof the Gaussian density of states is taken into account byusing the generalized Einstein relation, as described by Ro-ichman and Tessler.
26This introduces a carrier-density-
dependent enhancement factor, g3, to the diffusion coeffi-
cient,
D/H20849n,T/H20850=g3/H20849n,T/H20850/H9262kBT
e. /H208498/H20850
In the case of a Gaussian density of states, this factor is
g3/H20849n,T/H20850=/H20885
−/H11009/H11009
exp/H20875−/H92642
2/H9268ˆ/H208761
1 + exp /H20849/H9264−/H9264F/H20850d/H9264
n, /H208499/H20850
where /H9264Fis the relative quasi-Fermi level divided by kBT,
which can be evaluated from the Fermi distribution,
n=/H20885
−/H11009/H11009
exp/H20875−/H92642
2/H9268ˆ/H20876exp /H20849/H9264−/H9264F/H20850
1 + exp /H20849/H9264−/H9264F/H20850d/H9264. /H2084910/H20850
Second, the carrier-concentration dependence of mobility is
included by introducing the density-dependent mobility en-hancement factor g
2/H20849n,T/H20850as in Eq. /H208494/H20850. We have tried using
both the Pasveer19and Fishchuk20models to describe this
factor. Both models assume that charge transport occurs byhopping with a Miller-Abrahams
27hopping rate between
sites with random energetic disorder. The Pasveer modelgives a parameterization of results of numerical simulationsof hopping behavior and is simple to apply
g
2/H20849n,T/H20850= exp/H208751
2/H20849/H9268ˆ2−/H9268ˆ/H20850/H208732n
N0/H20874/H9254/H20876, /H2084911/H20850
/H9254=2ln/H20849/H9268ˆ2−/H9268ˆ/H20850−l n /H20849ln 4 /H20850
/H9268ˆ2, /H2084912/H20850
where N0is the total density of states per unit volume. The
Fishchuk model is more complicated to apply. It works byperforming integrals of the hopping rate over the density ofstates in an effective-medium approach and is a function ofthe same parameters. Despite the different approaches, thetwo models produce quantitatively similar results. To reducecomputation time, the functions g
3/H20849n,T/H20850and g2/H20849n,T/H20850were
evaluated before the start of the simulations and applied aslook-up tables.
Finally, the electric field dependence of mobility is intro-
duced by including the field-dependent mobility enhance-ment factor, g
1/H20849E,T/H20850,a si nE q . /H208492/H20850. Pasveer et al.19derived
an alternative field-dependent expression in their model,which, as noted by Coehoorn et al. ,
28is quite dissimilar from
the Poole-Frenkel-type field dependence that we have used.As discussed later in this paper, we have tried using thisexpression for g
1but the agreement with our experimental
data was found to be considerably worse than with thePoole-Frenkel-type expression. The use of the Poole-Frenkel-type field dependence and the Fishchuk concentra-tion dependence are the most significant differences between
our model and van Mensfoort’s model, which uses the Pas-veer model for both g
1andg2.
We treat all contacts as Ohmic with no injection barriers
by fixing the carrier concentration at the boundaries accord-ing to thermal equilibrium /H20851n/H208490/H20850=n/H20849d/H20850=N
0/2/H20852. A significant
amount of band bending occurs implicitly at the contactswhen using these boundary conditions,
29causing carrier den-
sity to fall rapidly between the contact and the first few na-nometers of polymer. The film was discretized into 300 gridpoints, calculating the field, carrier density, and current den-sity at each point. The mobility is also calculated at each gridpoint according to the local electric field and carrier density.A site density of N
0=1021cm−3was assumed.
V. FITTING RESULTS
For each temperature, J-Vcurves were simulated using
the analytical expression for SCL current with Poole-Frenkelfield dependence /H20851Eq. /H208493/H20850/H20852for all thicknesses of device mea-
sured. A dielectric constant of 3.5 was assumed and thebuilt-in voltage was assumed to be zero. The curves werefitted to experimental data by varying the free parameters
/H92620
and/H9253using a least-squares fitting algorithm on a logarithmic
scale. The same values of /H92620and/H9253were shared between
devices with different thicknesses. The best-fit curves for295, 150, and 100 K temperatures are shown in Figs.3/H20849a/H20850–3/H20849c/H20850for PFB and Figs. 4/H20849a/H20850–4/H20849c/H20850for F8BT. While rea-
sonable fits are achieved at room temperature, an increas-ingly poor fit results as the temperature is reduced. In par-ticular, a clear thickness dependence of the fit is observed, asthe experimental current density is higher than predicted inthin polymer devices.
The fitting process was repeated using the drift-diffusion
simulation detailed above. The simulations were run usingboth the Pasveer concentration-dependence model and theFishchuk concentration dependence. Since each model re-quires an energetic disorder parameter,
/H9268, to be defined, the
fitting process was repeated using values of /H9268=75, 90, 100,
and 110 meV. The best-fit curves for 75 and 100 meV disor-der are shown in Fig. 3, and Figs. 4/H20849d/H20850–4/H20849f/H20850for the Fishchuk
model and Figs. 4/H20849g/H20850–4/H20849i/H20850for the Pasveer model.
In both models, an increase in the level of disorder leads
to increasing influence of the concentration-dependent term,g
2. This causes not only a large difference between best-fit
values for /H92620from different models at low temperatures but
also an enhancement in simulated current density in thinnerdevices due to the larger carrier densities found in them. Thelatter of these effects corrects for the thickness dependenceobserved with the analytical fit, resulting in very good agree-ment between simulation and experiment at all temperatures.The best fits were achieved with the Pasveer concentrationdependence with
/H9268=100 meV for F8BT and 110 meV for
PFB. The Fishchuk model has slightly weaker concentrationdependence, requiring slightly larger values of disorder /H20849by
/H1101110 meV /H20850for a good fit, although good fits could be
achieved with both models. Assuming that the carrier-concentration-dependent models are valid, the thickness de-pendence of the low-temperature J-Vcurves therefore pro-BLAKESLEY, CLUBB, AND GREENHAM PHYSICAL REVIEW B 81, 045210 /H208492010 /H20850
045210-4vides us with a tool for estimating the amount of disorder in
the polymers by choosing the value of disorder that gives usthe best fit across different thicknesses.
Interestingly, recent work by Steyrleuthner et al.
4was
able to successfully describe electron transport in F8BT us-ing an alternative model based on trap-limited transport in-stead of Gaussian disorder. The model
30assumes that the
material contains a number of trapping sites characterized byan exponential energy distribution. An apparent carrier-density dependence of mobility arises as trap sites becomefull at high concentrations. The characteristic energy of thetrap distribution used to best describe transport in F8BT inthis work was 100 meV, the same as the width of disorderused in our model to best describe the same material. Al-though the models are considerably different, they arrive atthe same conclusion that electron transport in F8BT is domi-nated by localized transport between states with a distribu-tion of energies of 100 meV characteristic width. The exactnature of transport between these sites appears to be lessimportant than the characteristic energy.
Implementations of the GDM with Miller-Abrahams hop-
ping rates predict a temperature dependence for mobility inthe low-carrier-concentration limit of the form
/H92620/H20849T/H20850=/H9262/H11009exp /H20851−c/H9268ˆ2/H20852, /H2084913/H20850
for mobility, where /H9262/H11009is the high-temperature limit of mo-
bility and cis a constant. The value of cis slightly dependent
on the degree of carrier localization used in thesimulations.
28Localization is characterized by the ratio ofthe carrier localization length, b, to the intersite distance, a.
Generally, for disordered organic semiconductors, it is as-sumed that b/ais about 1/10. Using this value, cis equal to
0.44 for Bässler’s model,
180.42 in Pasveer’s model19and
0.48 for Fishchuk’s.20Figure 5shows the best-fit values of
/H92620as a function of 1 /T2for different values of disorder. The
solid lines show best fits to expression /H2084913/H20850, with the param-
eters shown in Table I. The above temperature dependence is
indeed observed for the values of disorder that gave the best
fit to the thickness-dependent J-Vcurves over the complete
range of temperatures from room temperature to 100 K. Val-ues of cshown in Table Iare in reasonable agreement with
the above values for disorders of 100 and 110 meV, suggest-ing that the assumed ratio of b/a=1 /10 is reasonable for
modeling PFB and F8BT.
We found that a good fit to expression /H2084913/H20850was achieved
for a range of values of
/H9268, all giving reasonable values of c.
The temperature dependence alone is insufficient for deter-mining the amount of disorder in the materials. It is apparent,then, that both thickness-dependent measurements and awide range of temperatures are necessary for deriving
/H9268with
confidence. Even then, we qualify our values of /H9268with at
least 10 meV of uncertainty.
While mobilities derived from numerical fitting with the
GDM produced a T−2temperature dependence, the mobilities
derived by fitting with the analytical expression /H208493/H20850, shown
in Fig. 6/H20849a/H20850, instead have an Arrhenius temperature depen-
dence between about 150 K and room temperature. Activa-tion energies of 330 and 370 meV were obtained for PFB andF8BT, respectively. Craciun et al.
21also found experimen-10-710-510-310-1
100nm140nm
295K
experimental
analyticalCurrent density (A cm-2)
350nm
a)
10-710-510-310-1
100nm
295K Fishchuk
75meV disorder
100meV disorderCurrent density (A cm-2)
140nm
350nm
d)
0 5 10 1510-710-510-310-1
295K Pasveer
75meV disorder
100meV disorderCurrent density (A cm-2)
Voltage(V)100nm140nm
350nm
g)100K100nm
140nm
350nm
f)150K140nm
350nm100nm
e)150K140nm
350nm100nm
b)
0 1 02 03 04 05 0
Voltage (V)100K100nm
140nm
350nm
i)
0 1 02 03 04 0
Voltage (V)150K140nm
350nm100nm
h)100K100nm
140nm
350nm
c)
FIG. 3. /H20849Color online /H20850Best-fit curves for 100,
140, and 350 nm PFB devices. /H20849a/H20850–/H20849c/H20850fitted with
analytical model, /H20849d/H20850–/H20849f/H20850fitted by numerical
simulation using Fishchuk concentration-dependent mobility, and /H20849g/H20850–/H20849i/H20850numerical simu-
lation with Pasveer concentration dependence./H20849a/H20850,/H20849d/H20850, and /H20849g/H20850at 295 K; /H20849b/H20850,/H20849e/H20850, and /H20849h/H20850at 150
K, and /H20849c/H20850,/H20849f/H20850, and /H20849i/H20850at 100 K.TEMPERATURE-DEPENDENT ELECTRON AND HOLE … PHYSICAL REVIEW B 81, 045210 /H208492010 /H20850
045210-5tally an Arrhenius temperature dependence for mobility in a
wide range of materials when fitting data using the sameanalytical expression. The explanation is that the analyticalmodel measures an effective mobility,
23/H9262eff/H20849T/H20850
=/H92620/H20849T/H20850g2/H20849n,T/H20850, at a finite carrier concentration and is differ-
ent in definition from the zero-field, zero carrier-density con-centration mobility,
/H92620/H20849T/H20850, measured by fitting with the full
simulation. This behavior is expected according to the obser-vation of Fishchuk
20that the mobility tends to Arrhenius
behavior at high carrier concentrations in the GDM. Cautionshould be exercised when applying such an effective mobil-ity to different device configurations in which the carrier
density is likely to be very different. Figure 6/H20849b/H20850compares
the effective mobilities extracted analytically with the mo-bilities from the best-fitting numerical simulations. Due tothe enhancement effect of g
2, the former mobilities are con-
siderably higher than the latter.
By simulating J-Vcurves, we are able to assess the rela-
tive importance of the different mobility enhancement fac-tors. At room temperature, we found that the field-dependentterm, g
1, was generally of much greater importance than the
carrier-concentration dependence, g2. For example, in a 150-10-710-510-310-1
295K
experimental
analyticalCurrent density (A cm-2) 70nm 110nm
150nm
a)
150K70nm 110nm
150nm
b) 100K70nm110nm
150nm
c)
150K70nm 110nm
150nm
e) 100K70nm 110nm
150nm
f)
0 5 10 15 20
Voltage (V)150K70nm 110nm
150nm
h)
0 5 10 15 20
Volta ge(V)100K70nm 110nm
150nm
i)10-710-510-310-1
295K Fishchuk
75meV disorder
100meV disorderCurrent density (A cm-2)
150nm110nm70nm
d)
05 1 010-710-510-310-1
295K Pasveer
75meV disorder
100meV disorderCurrent density (A cm-2)
Voltage (V)150nm110nm70nm
g)FIG. 4. /H20849Color online /H20850Best-fit curves for 70,
110, and 150 nm F8BT devices. /H20849a/H20850–/H20849c/H20850fitted
with analytical model, /H20849d/H20850–/H20849f/H20850fitted by numerical
simulation using Fishchuk concentration-dependent mobility, and /H20849g/H20850–/H20849i/H20850numerical simu-
lation with Pasveer concentration dependence./H20849a/H20850,/H20849d/H20850, and /H20849g/H20850at 295 K; /H20849b/H20850,/H20849e/H20850, and /H20849h/H20850at 150
K, and /H20849c/H20850,/H20849f/H20850, and /H20849i/H20850at 100 K.
=7 5 m e V
90meV
100meV
110meVPFB Fishchuk model
b)
10-3010-2510-2010-1510-1010-5
= 75meV
90meV
100meV
110meVµ0(cm2V-1s-1)PFB Pasveer model
a)
2468 1 010-3010-2510-2010-1510-1010-5
= 75meV
90meV
100meV
110meVµ0(cm2V-1s-1)
105T-2(K-2)F8BT Pasveer model
c)
2468 1 0
= 75meV
90meV
100meV
110meV
105T-2(K-2)F8BT Fishchuk model
d)FIG. 5. Best-fit mobility values derived from
numerical simulations. /H20849a/H20850PFB with Pasveer con-
centration dependence, /H20849b/H20850PFB with Fishchuk
concentration dependence, /H20849c/H20850F8BT with Pas-
veer, and /H20849d/H20850F8BT with Fishchuk. Symbols: mo-
bilities derived for each temperature and eachvalue of disorder. Solid lines: best-fit lines ac-cording to expression /H2084910/H20850.BLAKESLEY, CLUBB, AND GREENHAM PHYSICAL REVIEW B 81, 045210 /H208492010 /H20850
045210-6nm-thick F8BT film, the inclusion of the g1term in the mo-
bility equation caused a factor of 8 increase in current den-sity at 1 V, whereas g
2only contributes a factor of 1.5 and g3
1.3. At low temperatures, the concentration dependence is
much more important, with current-density enhancements of11, 23, and 1.8 for g
1,g2, and g3, respectively for the same
situation at 200 K. Therefore, a study of the form of the fielddependence at high temperatures is of great importance.
Figure 7/H20849a/H20850shows the experimental current density in a
150-nm-thick F8BT device normalized by the square of themean electric field /H20849V/d/H20850plotted against the square root of
the mean electric field. According to the Murgatroyd expres-sion /H208493/H20850, the result should be a straight line if Poole-Frenkel-
type field dependence is observed and concentration-dependent effects are small. Here this behavior is obeyed atfields down to 5 /H1100310
4V/cm, below which anomalies due to
possible small finite built-in potential and carrier-injectioneffects cause some uncertainty. This is in agreement with thegeneral observation that this field dependence is observed inmost organic semiconductors over a wide field range. Solid
lines in Fig. 7/H20849a/H20850correspond to simulated J-Vcurves using
Pasveer concentration dependence with Poole-Frenkel fielddependence and 100 meV disorder. We repeated the fittingexperiment using the Pasveer field dependence instead ofPoole-Frenkel. In this case, the free parameter,
/H9253,i sn o
longer used. Instead, the Pasveer model is strongly depen-dent on the intersite distance, a, so we allowed this param-
eter to vary in our fitting /H20849with N
0=a−3/H20850. Best fits were ob-
tained with aof about 3 nm, a value which is rather large
compared with the hopping distances normally expected inconjugated polymers. The resulting curves are shown asdashed lines in Fig. 7/H20849a/H20850. It is clear that the Poole-Frenkel
expression results in a much better fit to our data.
Monte Carlo transport simulations using the GDM predict
Poole-Frenkel field dependence only at strong fields /H20849E/H110225
/H1100310
5V/cm/H20850. Instead, simulations using spatially correlated
disorder31–34have been found to predict Poole-Frenkel be-
havior extending to lower electric fields. Such models predicta temperature dependence of
/H9253/H11008T−/H9251, where /H9251varies be-
tween models but can be expected to lie between 1 and 2depending on the degree of correlation, with completely un-correlated disorder giving about 2. Figure 7/H20849b/H20850shows valuesTABLE I. Parameters describing the temperature dependence of mobility fitted with different levels of
disorder.
/H9268
/H20849meV /H20850/H9262/H11009PFB
/H20849cm2V−1s−1/H20850/H9262/H11009F8BT
/H20849cm2V−1s−1/H20850 cPFB cF8BT
Pasveer Fishchuk Pasveer Fishchuk Pasveer Fishchuk Pasveer Fishchuk
75 1.9 /H1100310−62.4/H1100310−64.7/H1100310−76.8/H1100310−70.53 0.56 0.49 0.53
90 6.0 /H1100310−67.6/H1100310−61.5/H1100310−61.7/H1100310−60.46 0.47 0.43 0.44
100 1.3 /H1100310−52.2/H1100310−54.5/H1100310−65.4/H1100310−60.42 0.45 0.41 0.43
110 6.8 /H1100310−53.8/H1100310−51.5/H1100310−51.2/H1100310−50.42 0.43 0.40 0.41
3.5 4.0 4.5 5.0 5.5 6.0 6.510-1310-1210-1110-1010-910-810-710-6
F8BT Ea=3 7 0 m e V
PFB Ea=3 3 0 m e Vµ0(cm2V-1s-1)
103T-1(K-1)analytical fit mobilitya)
1.5 2.0 2.5 3.0 3.5 4.0 4.510-1710-1510-1310-1110-910-7
analytical F8BT
analytical PFB
Pasveer F8BT
= 100meV
Pasveer PFB
= 110meVµ0(cm2V-1s-1)
105T-2(K-2)b)
FIG. 6. /H20849a/H20850Best-fitting mobilities used for fitting the analytical
Murgatroyd expression /H208493/H20850for SCL current on an Arrhenius plot for
F8BT /H20849solid line /H20850and PFB /H20849dotted line /H20850./H20849b/H20850Best-fit mobilities from
analytical model compared with best-fit mobilities from numericalsimulations using Pasveer concentration dependence and 100 meVdisorder for F8BT and 110 meV disorder for PFB, with best-fitcurves to expression /H2084910/H20850for F8BT /H20849solid line /H20850and PFB /H20849dashed
line /H20850200 400 600 80010-1710-1610-1510-1410-13J(V/d)2(AV-2cm-4)
(V/d)1/2(V1/2cm-1/2)a)
0.0 0.2 0.4 0.60.0000.0050.0100.015
F8BT
PFB
(cm1/2V-1/2)
103T-3/2(K-3/2)b)
FIG. 7. /H20849Color online /H20850/H20849a/H20850The field dependence of normalized
space-charge limited current in 150-nm-thick F8BT devices. Sym-bols are experimental results: squares 200 K, circles 225 K, uptriangles 250 K, down triangles 275 K, and diamonds 295 K. Solidlines are best fits with Poole-Frenkel field dependence and Pasveerconcentration dependence and dashed lines are best fits with the fullPasveer EGDM for field and concentration dependence. /H20849b/H20850Tem-
perature dependence of
/H9253derived from fitting F8BT with 100 meV
disorder and PFB with 110 meV disorder using Pasveer concentra-tion dependence and Poole-Frenkel field dependence. Solid lines:best fit with
/H9253/H11008T−1.5.TEMPERATURE-DEPENDENT ELECTRON AND HOLE … PHYSICAL REVIEW B 81, 045210 /H208492010 /H20850
045210-7of/H9253obtained from the simulated best fits. The figure shows
a reasonable fit to a temperature dependence of /H9251=1.5 /H20849solid
lines /H20850, suggesting that correlated disorder might explain the
observed field dependence in PFB and F8BT. Recently theEGDM was modified to include the effects of correlated dis-order using the same methods as Pasveer et al. , resulting in
the extended correlated disorder model /H20849ECDM /H20850.
34This
model was used to study the mobility of holes in a PFB-likepolymer,
35whereupon it was concluded that both EGDM and
ECDM provided a good fit to experiments, although theEGDM was the preferred model as the ECDM required anunrealistically high site density for a good fit. Our resultsappear to counter this view, as they suggest that some formof correlated disorder is necessary to fully explain the fielddependence. Between these two studies, the appropriate useof correlated disorder still remains an unresolved issue.
VI. CONCLUSIONS
We have implemented a model that can successfully pre-
dict the temperature and thickness dependence of SCLcharge transport in PFB and F8BT without needing to in-clude specific trapping sites. This is especially significant inthe case of electron transport in F8BT as most previous at-tempts to model electron transport in conjugated polymershave described steady-state currents as being a trap-limitedprocess.
4,5,36We found that hole transport in PFB could be
modeled using a Gaussian disorder of 110 /H1100610 meV and
electron transport in F8BT with 100 /H1100610 meV disorder. The
amount of disorder in F8BT is the same as the width of trapenergies used to describe electron transport in a previouswork,
4suggesting that trap-limited behavior and Gaussiandisorder share many features in the underlying physics. The
use of a Gaussian disorder reduces the effects of many pro-cesses /H20849for example, local energetic variations due to dipoles,
chain distortions, impurities, and trapping effects /H20850to a single
disorder parameter without describing them explicitly. Byreducing a number of unknown properties to a single value,it provides a simple yet adequate model for describing trans-port behavior in a range of complex disordered systems and
can be included in device models with the inclusion of aminimal set of empirical parameters.
We have confirmed that the EGDM works over a wider
range of temperatures than has previously been used. It wasalso found that both Pasveer and Fishcuk models of carrier-concentration dependence can be used. However, the Pasveerdescription of the field dependence of mobility did not workfor either material and a Pool-Frenkel-type field dependencewas used instead. This suggests that correlated disordermight play a role in transport in these materials.
We also confirmed the previously made observation
23that
a simple analytical analysis of SCL current densities mea-sures an effective mobility that has an Arrhenius temperaturedependence. However, this effective mobility varies as afunction of device thickness. Caution should be exercisedwith such effective mobilities, as real carrier mobilities candiffer by orders of magnitude when different electric fieldsand carrier densities are used.
ACKNOWLEDGMENTS
This work was supported by the European Commission
Framework Programme 6 project MODECOM /H20849Grant No.
NMP-CT-2006-016434 /H20850. Polymers were supplied by Cam-
bridge Display Technology Ltd.
*Present address: University of Potsdam, Institute of Physics and
Astronomy, Karl-Liebknecht Strasse 24-25, Building 28, D-14476Potsdam-Golm, Germany; james.blakesley@physics.org
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045210-9 |
PhysRevB.86.045313.pdf | PHYSICAL REVIEW B 86, 045313 (2012)
First-principles studies of the effect of (001) surface terminations on the electronic properties
of the negatively charged nitrogen-vacancy defect in diamond
H. Pinto,*R. Jones, and D. W. Palmer
School of Engineering, Mathematics and Physical Sciences, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom
J. P. Goss, Amit K. Tiwari, P. R. Briddon, Nick G. Wright, and Alton B. Horsfall
School of Electrical and Electronic Engineering, Newcastle University, Newcastle upon Tyne, England NE1 7RU, United Kingdom
M. J. Rayson and S. ¨Oberg
Department of Engineering Sciences and Mathematics, Lule ˚a University of Technology, Lule ˚a S-97187, Sweden
(Received 10 May 2012; published 16 July 2012)
Density functional calculations have been carried out on (001)-orientated slabs of diamond with different
surface terminations. A negatively charged nitrogen-vacancy defect (NV−) is placed in the middle of the slab
approximately 1 nm from each surface and the effect of the surface on the internal optical transition in NV−
investigated. The calculations show that the chemical nature of the surface is important. We find that although theclean surface does not lead to charge transfer between the defect and the surface, there is a splitting of the emptyexcited state, the final state in optical absorption, arising from a strong hybridization of the surface and defectbands. This leads to a broadening of the 1.945-eV transition of the NV
−defect. OH- and F-terminated surfaces
have no surface states in the band gap and again charge transfer between the defect and surface does not occur.The splitting of the elevels responsible for the optical transitions for OH or F termination is similar to that found
in periodic boundary condition simulations for bulk diamond where the defects are separated by 1 nm, and thus
the calculations show that hydroxylated or fluorinated surfaces give favorable optical properties.
DOI: 10.1103/PhysRevB.86.045313 PACS number(s): 71 .55.−i, 73.20.At, 78 .67.−n, 78.68.+m
I. INTRODUCTION
Nitrogen defects in diamond are of interest because they
exist in high concentration and can strongly influence theproperties of the material. One center of particular currentinterest is the nitrogen-vacancy (NV) center with C
3vpoint
group symmetry and illustrated schematically in Fig. 1.I n
nitrogen-rich samples, this center can be formed by irradiation,creating the vacancies, followed by an anneal at 700
◦C which
allows them to become mobile and be trapped by substitutionalnitrogen (N
s). For the negatively charged defect, the ground
state of NV is a spin triplet chiefly made from sp3orbitals
pointing towards the vacant site from the four atoms borderingthe vacancy. In a one-electron sense, these introduce into thefundamental gap an orbital singlet of symmetry a
1, occupied
by two electrons, lying below an orbital doublet ex,eyalso
occupied by two electrons but with parallel spin.1In terms of
spin orbitals, the spin-up exandeyorbitals are occupied and the
spin-down exandeyorbitals are empty. The optical transition
between the occupied spin-down a1level and the unoccupied
spin-down ex,eylevels is responsible for a zero-phonon line
at 1.945 eV (637 nm).2
Surprisingly, the process of optical excitation and decay
results in a preferential occupation of the Sz=0 ground-
state level3and this property, combined with a long spin-
lattice lifetime,4makes the defect of particular interest
for high-resolution magnetometry and quantum informationapplications.
5–7Since diamond is a biocompatible material
with an amiable surface, single diamond nanoparticles withembedded NV centers can be used for biomarking andtracking, an application helped by its room-temperaturephotostability.
8These applications require isolated defects,
and attempts have been made to form the defect in single-digitnanodiamonds (SNDs), i.e., those with a diameter smaller than
10 nm.
It is of interest to estimate the expected number of nitrogen
atoms in spherical nanodiamonds of diameters 10 and 5 nm.
If the bulk N atomic concentration is 300 ppm, with a meanseparation of N atoms of 2.6 nm, then the mean number ofcarbon atoms, carbon atoms on the surface, and nitrogen atomsare 93 000, 8600, and 28 for a 10-nm-diam nanodiamond.For a 5-nm SND, these numbers become 11 600, 2100, and3.5, respectively. Hence by decreasing the particle size the
effect of the surface on the optical properties becomes more
important. This could result in a reduced luminescence due to(a) more vacancies being lost to the surface, (b) a broadeningof the optical transition caused by hybridization of the surfaceand defect-related bands, (c) a broadening caused by straindue to the surface, and (d) a transfer of electrons from NV
−
to surface acceptor states. The acceptor and donor levels of
NV defects have been calculated and lie at Ec−3.3e V
andEv+1.5 eV , respectively,9with the acceptor level also
being determined experimentally10atEc−2.58 eV . Thus
any surface acceptor level below around Ev+2 eV would
be expected to cause charge transfer from NV−. This would
result in a complete loss of NV−defects and a dominance
of surface-related luminescent defects over NV−. Clearly, the
number of nitrogen atoms in the SND has to be sufficient to
occupy surface acceptors as well as creating NV defects andmaking them negatively charged.
Early findings suggested that the NV luminescence
11could
be found in nanodiamonds with sizes down to 50 nm but wasobscured by surface broadband emission for smaller diamonds.It was found that the intensity of the broadband could bereduced by a surface oxygen treatment,
12and luminescent
045313-1 1098-0121/2012/86(4)/045313(8) ©2012 American Physical SocietyH. PINTO et al. PHYSICAL REVIEW B 86, 045313 (2012)
FIG. 1. (Color online) A schematic illustration of the NV center
in diamond, comprising a vacancy surrounded by three carbon atoms
and a substitutional nitrogen impurity.
centers have recently been found in detonation-produced
SND.13–15
In one study, luminescence measurements were performed
on weakly coupled SNDs uniformly distributed in a quartzsubstrate.
13After the filtration of the surface-emission com-
ponent which has a short radiative lifetime, a broad peak at637 nm, corresponding to the zero-phonon line of NV
−,w a s
detected. The intensity of the luminescence due to NV−from
the SNDs was, however, about 50 times weaker than that from55-nm nanodiamonds of the same weight as the collection ofSNDs. This suggests a low production yield of luminescentNV
−centers in SNDs. A difference in nitrogen concentration
between the SND and bulk samples was excluded and thedifference in yields was attributed to the preferential trappingof the mobile vacancy by the surface of the SND rather thanby a nitrogen atom.
In another study, luminescent NV centers were observed by
confocal scanning fluorescence microscopy in noninteractingnanodiamonds with an average size of 7.5–8 nm.
14,15The
results showed that, for the irradiation doses used, about 16vacancies were introduced into the 10-nm diamond along withabout 0.16 NV
−defects. In this case, the yield of NV−centers,
about 1 defect per 100 vacancies, was approximately the sameas found in the bulk.
14The disagreement between the results
of these studies is presumably related to the different surfacetreatments or irradiation doses.
Bradac et al.
15reported intermittency in the luminescence
(blinking) from 5-nm nanodiamonds, possibly due to thepresence of surface functional groups and/or adsorbed specieson the nanodiamond surface which can act as exciton traps anddegrade the luminescence of the defect. The effect of surfaceson the luminescence of NV
−has been also demonstrated both
experimentally and theoretically.16–19
Further work has highlighted the importance of the surface.
Experimentally it was shown that, in the vicinity of anoxygen-terminated diamond surface, NV centers are foundin the negative charge state and can give rise to the 1.945-eVluminescence.
17,19However, the defect loses its electron, be-
coming neutral, when the oxygen at the surface is replaced byhydrogen.
17,18This can be attributed to the transfer of electrons
via redox reactions at the surface of hydrogenated diamond.20
The importance of the surface in the luminescence of NV−was
also studied by ion implantation studies, which show that thereis a decreasing yield of NV
−defects close to the surface.18Fluorine passivation is reported to be an efficient proce-
dure to control the properties of carbon materials.21Unlike
hydrogen, it is not expected to lead to a surface with negativeelectron affinity,
22and the type of redox reactions leading
to neutralization of NV−would not occur. In this paper the
influence of the clean, hydroxylated and fluorinated (001)-(2×1) diamond surface on the electronic levels of the NV
−
center when it lies about 1 nm from the surface is investigated
using density functional theory. The effect on defects lyingfurther from the surface cannot be accomplished at presentbecause of computing constraints.
II. METHOD
The electronic levels of the NV−defects have been
investigated in this work using the AIMPRO density functional
code.23,24The many-body effects were treated with the local
density approximation (LDA). To represent the core electronsof the atoms we used the Hartwigsen-Goedecker-Hutter(HGH) pseudopotentials
25and the orbitals of the valence
electrons consist of independent s-,p-,d-like Gaussian
functions centered on atoms.26The electronic levels were
filled using Fermi-Dirac statistics with kBT=0.01 eV . Matrix
elements of the Hamiltonian are determined using a plane-wave expansion of the density and Kohn-Sham potential
27
with a cutoff of 200 Ha.
The Brillouin zone was sampled with a grid of k-points
within the Monkhorst-Pack scheme ensuring convergence.28
The method was previously used successfully to study theelectronic structure of different diamond surfaces.
29
FIG. 2. (Color online) A slab of diamond (blue) of thickness 2.2
nm is inserted into the unit cell with 1.7 nm of vacuum separatingslabs. Each cell contains a single NV defect placed about 1.1 nm from
the surface. The minimum in-plane distance between NV defects in
different cells is 1.5 nm.
045313-2FIRST-PRINCIPLES STUDIES OF THE EFFECT OF ... PHYSICAL REVIEW B 86, 045313 (2012)
(a)
(b)
(c)
(b
)
(c
)
FIG. 3. (Color online) Schematic structure for the (a) clean,
(b) OH-terminated, and (c) F-terminated (001)-(2 ×1) diamond
surfaces.
The interaction between the defect and defects in neighbor-
ing cells was studied by placing a NV−defect in a simple cubic
supercell of size n×n×nconventional unit cells, resulting
in simulation cells containing 64, 216, and 512 host atoms forn=2, 3, and 4, respectively. The separation between defects
in the neighboring supercells is then 0.7, 1.0, and 1.4 nm,respectively.
To investigate the influence of the surface on the electronic
levels of the NV
−, a single defect was placed in the middle of a
diamond slab of thickness 2.2 nm (Fig. 2). In plane the length
was 1.5 nm. A vacuum layer of 1.7 nm separated the surfacesin adjacent cells in the zdirection to prevent the interactions
between them.
Clean, OH-terminated, and F-terminated (001)-(2 ×1)
diamond surfaces (illustrated in Fig. 3) were considered.
During the relaxation all the atoms were allowed to moveuntil the forces became negligible. The electronic bandstructure was then calculated along high-symmetry directionsand the nature of the wave functions of any gap levelsfound.
III. RESULTS
A. NV−centers in bulk diamond
Before studying the influence of the surface on the elec-
tronic levels of the NV−, it is important to have a detailed
picture of the electronic properties of the defect in bulkdiamond.
Figure 4shows the calculated band structure for NV
−
centers separated by 0.7 nm (2 ×2×2 cubic supercell). The
splitting of the edoublet is found to be 0.3 eV , with a total
dispersion of around 0.4 eV due to the interaction betweenNV
−defects in different cells. Thus, if NV−defects areFIG. 4. (Color online) Calculated spin-up (solid) and spin-down
(dashed) band structure in the vicinity of the band gap for an NV−
center in bulk diamond for a 2 ×2×2 conventional unit diamond
supercell. The zero of energy is set to the top of the valence band.Red lines denote occupied levels while blue lines show empty levels.
Theaandeorbitals are labeled.
present in diamond at a very high concentration such that
they are separated by such a distance, the optical transitionwill be expected to be broadened by about 0.3 eV . For manyapplications this is too large.
The wave function of the a
1level [Fig. 5(a)] can be
described as an antibonding combination of the N-related sp3
hybrid with the three C-related orbitals and is thus mainly
constituted from the N-lone-pair orbital in the electronicground state. The elevels [Figs. 5(b) and 5(c)] are linear
combinations of the three C-related sp
3orbitals with very
little amplitude on the N atom. These orbitals correspond tothose expected for the defect.
30
A3×3×3 conventional cell where the NV−defects are
separated by 1.0 nm leads to a smaller defect-defect interactionas indicated by the splitting of the elevel, which is reduced to
(a)
(b)
(c)
FIG. 5. (Color online) Wave-function isosurfaces for (a) the a1
and (b, c) the elevels of the NV−in bulk diamond. The light (red)
and dark (blue) surfaces represent surfaces of equal amplitude but
opposite sign.
045313-3H. PINTO et al. PHYSICAL REVIEW B 86, 045313 (2012)
FIG. 6. (Color online) Calculated spin-up (solid) and spin-down
(dashed) band structure in the vicinity of the band gap for an NV−
center in bulk diamond for a 3 ×3×3 conventional unit diamond
supercell. The zero of energy is set to the top of the valence band.Red lines denote occupied levels while blue lines show empty levels.
Theaandeorbitals are labeled.
50 meV (Fig. 6). The decreasing dispersive character of the e
levels as a function of increasing separation between defects
is in agreement with previous calculations.30For a 4 ×4×4
cell with a separation between defects of 1.4 nm, the elevels
are degenerate to within 8 meV (Fig. 7).
Even this is much larger than the separation of the elevels
found experimentally, which is about 10 GHz or 0.04 meVin the absence of any stress and is due to spin-orbit andspin-spin interactions.
31By fitting a power law decay to the
calculated splitting as a function of separation, we find thatthe separation of defects needs to be at least 4 nm to achievethe experimentally observed splitting.
B. Clean (001)-(2 ×1) diamond surface
We now turn to the (001)-(2 ×1) clean, carbon-terminated
surface. Figure 8(a) shows its geometry. The calculated C–C
dimer bond length and the relaxation of the underlying atomic
FIG. 7. (Color online) Calculated spin-up (solid) and spin-down
(dashed) band structure in the vicinity of the band gap for an NV−
center in bulk diamond for a 4 ×4×4 conventional unit diamond
supercell. The zero of energy is set to the top of the valence band.
Red lines denote occupied levels while blue lines show empty levels.
Theaandeorbitals are labeled.(a)
(b)
FIG. 8. (Color online) (a) Optimized atomic geometry and (b)
band structure in the vicinity of the band gap for the (001)-(2 ×1)
C-terminated diamond surface. The distances are given in ˚A. For the
band structure, the zero of the energy scale is set to the top of the
valence band. Solid (red) lines denote occupied levels while dotted(blue) levels are empty.
layers are found to be in excellent agreement with previous
calculations.29,32
The band structure of the (001)-(2 ×1) C-terminated
diamond surface is presented in Fig. 8(b). An occupied, broad
band of πcharacter lies close to the top of the valence band,
and an empty π∗band lies between around 2 and 3.5 eV above
the valence band, near the middle of the band gap. These bandsare mostly localized on the surface carbon dimers.
The presence of the band-gap surface states is important
when we consider the addition of the NV center to the diamondslab: the location of the a
1andestates around 1 eV and 2–3
eV above the valence band top (Fig. 7) coincides in energy in
particular with the π∗band.
Figure 9shows the band structure for NV−placed in the
middle of a diamond slab at a distance of 1.1 nm from the(001)-(2 ×1) C-terminated diamond surfaces. There is clearly
an overlap between the broad surface-related bands and thoseof the defect. However, the highest occupied spin-up levels ofthe defect fall just below the empty surface bands, showingthat there is no propensity for charge transfer with the clean
(a) (b)
FIG. 9. (Color online) Calculated (a) spin-up and (b) spin-down
band structure in the vicinity of the band gap for an NV−defect
placed 1.1 nm from (001)-(2 ×1) clean diamond surfaces. The zero of
energy is set to the top of the valence band. Occupied and unoccupied
electronic levels are indicated by solid (red) and dotted (blue) lines,
respectively. The aandeorbitals are labeled.
045313-4FIRST-PRINCIPLES STUDIES OF THE EFFECT OF ... PHYSICAL REVIEW B 86, 045313 (2012)
surface. However, despite the distance of the defect from its
periodic images, and from the reconstructed surface, the emptyelevels are split. This is a consequence of the overlap with the
surface bands.
In order to identify the defect-related electronic levels, the
wave functions at the ¯/Gamma1were calculated and compared with
those obtained for NV
−in bulk diamond. The wave function
shown in Fig. 10(a) corresponds to the a1level marked in
Fig. 9(b). Figures 10(b) and10(c) show the wave functions
associated with the two elevels marked in Fig. 9(b).
We note from the band structure (Fig. 9) that the empty,
spin-down elevels overlap in energy with the surface π∗band.
The interaction between the surface and the defect states causesa splitting of the two components, e
xandey, of approximately
0.2 eV , significantly greater than the splitting determined forthe interaction between NV
−and its images in the 3 ×3×3
conventional cell with intercenter distances of around 1.0 nm.Thus the (001)-(2 ×1) clean diamond surface is expected to
significantly broaden the luminescence of NV
−defects lying
within around 1 nm of the surface, the extent of the broadeningbeing in the range of hundreds of meV .
(a)
(b)
(c)
FIG. 10. (Color online) Wave functions of the (a) a1and the (b,
c) degenerate elevels of NV−in the middle of 2.2-nm diamond slab
terminated with clean (001)-(2 ×1) diamond surfaces.(a)
(b)
FIG. 11. (Color online) (a) A schematic showing the structure
of the optimized atomic geometry of the hydroxylated (001)-(2 ×1)
diamond surface, and (b) the corresponding calculated band structurein the vicinity of the band gap. In (a), the carbon atoms are represented
in gray, with the large and small (red and white) surface atoms
representing oxygen and hydrogen, respectively. Lengths are statedin˚A. In (b), the zero of the energy scale is set to the top of the valence
band, with occupied and empty levels indicated by solid (red) and
dotted (blue) lines, respectively.
C. Hydroxylated (001)-(2 ×1) diamond surface
We now move to the case of a chemically passivated
diamond surface and begin with termination by hydroxylgroups. Figure 11(a) shows the optimized structure for the
(001)-(2 ×1):OH surface in its stable “anti” configuration
form. Here the two OH bonds attached to the surface carbondimers are antiparallel to each other. Each surface carbonatom is saturated by an OH group, rendering all C atoms sp
3
hybridized.
The OH groups chemically modify the surface, removing
theπbond in the surface reconstructions. Consequentially,
theπandπ∗bands are removed from the band gap, leaving
an empty gap close in energy to the band gap found for bulkdiamond using the current method.
33
The band structure for NV−in the center of a 2.2-nm-thick
slab is shown in Fig. 12. The OH termination of the surface
(a) (b)
FIG. 12. (Color online) Calculated (a) spin-up and (b) spin-down
electronic band structure in the vicinity of the band gap for a NV−
defect 1.1 nm from (001)-(2 ×1):OH surfaces. The zero of the energy
is set to the top of the valence band. Occupied and unoccupied
electronic levels are indicated by solid (red) and dotted (blue) lines,
respectively. The aandeorbitals are labeled.
045313-5H. PINTO et al. PHYSICAL REVIEW B 86, 045313 (2012)
(a) (b)
FIG. 13. (Color online) Calculated (a) spin-up and (b) spin-down
electronic band structure in the vicinity of the band gap for an NV0
defect 1.1 nm from (001)-(2 ×1):OH surfaces. The zero of the energy
is set to the top of the valence band. Occupied and unoccupied
electronic levels are indicated by solid (red) and dotted (blue) lines,
respectively. The aandeorbitals are labeled.
removes the surface bands characteristic of the clean surface
capable of accepting an electron. This leaves only the a1- and
e-derived levels of NV−in the gap. Nevertheless we find that
there is a splitting of the elevels, but of just 50 meV . It is
also important to note that for the OH-terminated surface, thecalculated band gap is approximately 3.8 eV , which is about0.6 eV smaller than found for the bulk. This, we believe, is aconsequence of treating a charged defect. The use of periodicboundary conditions necessitates that the cell has to have anoverall neutral charge state to enable its total energy to befinite, and for the negatively charged NV center, this requiresa uniform compensating positive charge to be added to thecell. Part of the uniform positive charge resides in the vacuumseparating the slabs. This affects the position of the surface-related bands close to the conduction-band edge, since thesebands have wave functions which extend into the vacuum.
29
(a)
(b)
FIG. 14. (Color online) (a) A schematic showing the structure
of the optimized atomic geometry of the fluorinated (001)-(2 ×1)
diamond surface, and (b) the corresponding calculated band structure
in the vicinity of the band gap. In (a), the carbon atoms are represented
in gray, with the surface (green) atoms representing fluorine. Lengthsare stated in ˚A. In (b), the zero of the energy scale is set to the top of
the valence band, with occupied and empty levels indicated by solid
(red) and dotted (blue) lines, respectively.(a) (b)
FIG. 15. (Color online) Calculated (a) spin-up and (b) spin-down
band structure in the vicinity of the band gap for NV−1.1 nm from
(001)-(2 ×1):F surfaces. The zero of the energy scale is set to the
top of the valence band. Occupied and unoccupied electronic levels
are indicated by solid (red) and dotted (blue) lines, respectively. The
aandeorbitals are labeled.
This argument is supported by calculating the electronic
band structure for a NV neutral defect in the same cell withthe same (001)-(2 ×1):OH surface. Figure 13shows the band
structures for this case and, in spite of the large splitting of theelevels, due presumably to the filling of one of them leading
to a structural change, the gap is close to the bulk value.
D. The fluorinated (001)-(2 ×1) diamond surface
We turn now to the (001)-(2 ×1):F surface, shown schemat-
ically in Fig. 14(a) . The electronic band structure for this
surface is shown in Fig. 14(b) and can be compared with
the band structures of the clean and hydroxylated surfacesshown in Figs. 8(b) and11(b) , respectively. As expected, F
termination passivates the surface in the same way as OHgroups. The band gap is approximately 4.5 eV .
Figure 15shows the band structure of NV
−when placed
1.1 nm from (100)-(2 ×1):F diamond surfaces. It is perhaps
unsurprising to find that hydroxylated and fluorinated surfaceshave the same effect on the electronic levels of NV
−, and
we find the same splitting of about 50 meV . However, incontrast to the case of hydroxyl termination, we do not finda reduction in the band gap in the presence of the chargeddefect. This is a consequence of the relative electron affinitiesof the two surface terminations. OH termination leads to anegative electron affinity of around 0.55 eV .
29This places
the vacuum level below the conduction band. In contrast, theelectron affinity of fluorine-terminated surfaces is calculated
22
to be positive and around +2.1 eV , so that the vacuum level lies
far above the conduction-band minimum in energy. Therefore,although the charge-neutrality condition will lead to a positivecharge density in the vacuum, the associated states lie in theconduction band in Fig. 15.
IV . DISCUSSION AND CONCLUSIONS
Density functional calculations for NV−defects in bulk
diamond show an e-doublet splitting whose magnitude de-
creases with increasing separation of the defects. A summaryof the splitting is given in Table I. The splitting will cause
broadening of the 1.945-eV zero-phonon luminescence line
045313-6FIRST-PRINCIPLES STUDIES OF THE EFFECT OF ... PHYSICAL REVIEW B 86, 045313 (2012)
TABLE I. Splitting of unoccupied elevel for bulk and surfaces
(eV). Distances are indicated to either the nearest image in bulk
superlattices, or to the surface (nm).
System Distance Splitting
64 atoms in bulk diamond 0.7 0.3
216 atoms in bulk diamond 1.0 0.05512 atoms in bulk diamond 1.4 0.008
Clean (100)-(2 ×1) surface 1.1 0.2
OH (100)-(2 ×1) surface 1.1 0.05
F (100)-(2 ×1) surface 1.1 0.05
due to the e-to-a1electronic transition, and we predict that
the NV defects need to be separated by more than 4 nm(corresponding to a maximum NV concentration of 90 ppmfor a uniform defect distribution) to achieve a splitting of lessthan the 0.04 meV observed in high-quality bulk diamond ofvery low NV concentration and due to spin-spin and spin-orbitinteractions.
The effects of different surface terminations upon the
electronic levels of NV
−have also been investigated. A
wave-function analysis confirms that the levels in the middleof the band gap belong to the a
1- ande-derived levels of the
defect. The results presented here indicate that passivationof the (100)-(2 ×1) diamond surface by OH or F groups
removes the πbands from the band gap. Then, for a NV
−
defect placed 1.1 nm away from the OH- or F-terminated
surface, its electronic levels and charge state are qualitativelyindependent of the presence of the surface. However, becausethe electron affinities of OH and F terminations are not thesame, the locations of the a
1andelevels differ with respect to
the top of the valence band. Although the empty eorbitals
in all cases are split by the presence of the surface, thissplitting is greatest for the clean surface. This C-terminated,
reconstructed surface deserves further discussion. For suchtermination with NV
−∼1 nm away from the surface, we found
that the empty surface-related bands lie above the occupiedeorbitals of the defect and thus charge transfer from the
defect to the surface would be unlikely to occur. However,the empty elevels hybridized with the empty surface bands
and would lead to a 0.2-eV broadened optical emission. It wasfound that (100)-(2 ×1):OH and (100)-(2 ×1):F surfaces, in
contrast with the (100)-(2 ×1) clean surface, do not affect
significantly the electronic levels of the defect. The broadeningfound in these cases of 50 meV is the same as found for NV
−
in the bulk diamond when separated by around 1 nm. Thus thesplitting caused by the surface is smaller than that caused byinterdefect coupling.
We note that calculations of surface bands for the clean
(111), (100), and (110) surfaces and oxidized and hydro-genated surfaces all show broad, empty bands lying in thegap.
29These bands are expected to overlap the e-derived
levels of NV−, also leading to a broadened emission as
found for the (100) surface investigated here. There is alsothe possibility of an electron transfer from NV
−defects
to the surface. Such transitions would form neutral NV defects.The loss of the negative charge state is highly deleterious tothe exploitation of the spin polarization of the optical emissioncenter. Therefore, control of the surface is expected to be ofparamount importance for bright, controllable emission fromNV
−defects.
ACKNOWLEDGMENTS
This work is supported by BAE Systems and the Engineer-
ing and Physical Sciences Research Council (EPSRC) UKthrough the DHPA scheme.
*pinto@excc.ex.ac.uk
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045313-8 |
PhysRevB.76.085341.pdf | Incipient Wigner localization in circular quantum dots
Amit Ghosal,1,2A. D. Güçlü,1,3C. J. Umrigar,3Denis Ullmo,1,4and Harold U. Baranger1
1Department of Physics, Duke University, Durham, North Carolina 27708-0305, USA
2Physics Department, University of California Los Angeles, Los Angeles, California 90095-1547, USA
3Theory Center and Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA
4CNRS; Université Paris-Sud; LPTMS UMR 8626, 91405 Orsay Cedex, France
/H20849Received 2 March 2007; revised manuscript received 18 June 2007; published 27 August 2007 /H20850
We study the development of electron-electron correlations in circular quantum dots as the density is
decreased. We consider a wide range of both electron number, N/H3335520, and electron gas parameter, rs/H1135118,
using the diffusion quantum Monte Carlo technique. Features associated with correlation appear to developvery differently in quantum dots than in bulk. The main reason is that translational symmetry is necessarilybroken in a dot, leading to density modulation and inhomogeneity. Electron-electron interactions act to enhancethis modulation ultimately leading to localization. This process appears to be completely smooth and occursover a wide range of density. Thus there is a broad regime of “incipient” Wigner crystallization in thesequantum dots. Our specific conclusions are /H20849i/H20850the density develops sharp rings while the pair density shows
both radial and angular inhomogeneity; /H20849ii/H20850the spin of the ground state is consistent with Hund’s /H20849first/H20850rule
throughout our entire range of r
sfor all 4 /H33355N/H3335520; /H20849iii/H20850the addition energy curve first becomes smoother as
interactions strengthen—the mesoscopic fluctuations are damped by correlation—and then starts to showfeatures characteristic of the classical addition energy; /H20849iv/H20850localization effects are stronger for a smaller
number of electrons; /H20849v/H20850finally, the gap to certain spin excitations becomes small at the strong interaction
/H20849large r
s/H20850side of our regime.
DOI: 10.1103/PhysRevB.76.085341 PACS number /H20849s/H20850: 73.21.La, 73.23.Hk, 02.70.Ss, 73.63.Kv
I. INTRODUCTION
One of the fundamental quests in condensed matter phys-
ics research is to understand the effects of strong correlationbetween a system’s constituent particles. A simplified systemof particular interest is the “electron gas,”
1in which conduc-
tion electrons interact pairwise via Coulomb forces while theeffect of atoms is ignored. It is well known that the electrongas has a Fermi liquid ground state with extended wavefunctions in the limit of high electron density, while whenthe density is decreased, thereby increasing the interactionstrength, electrons become localized in space and orderthemselves in a “Wigner crystal” phase.
1The interaction
strength is often parametrized by the gas parameter rs
=/H20849cd/aB*/H20850/H208491/n/H208501/d, where nis the electron density, dis the
spatial dimension, aB*is the effective Bohr radius, and cdis a
dimension-dependent constant.2In two dimensions, rs
=1/aB*/H20849/H9266n/H208501/2. The physics at intermediate rscontinues to
offer puzzles, both from theory and experiments. There has
been numerical evidence in bulk two3–5and three6,7dimen-
sional /H208492D and 3D /H20850systems that a single transition takes
place at rsc,2D/H1101530–35 and rsc,3D/H11011100. However, recent work
in 2D has predicted more complex phases and associatedtransitions or crossovers around these critical values.
8–14
While the experimental evidence is largely inconclusive,15,16
and in particular the way in which the transition occurs is not
known experimentally, the problem has drawn a great deal ofattention due to the hope of uncovering fundamental aspectsof correlation effects.
Over the past decade or so, small confined systems, such
as quantum dots /H20849QD/H20850, have become very popular for experi-
mental study.
17,18Beyond their possible relevance for nano-
technology, they are highly tunable in experiments and intro-duce level quantization and quantum interference in acontrolled way. In a finite system, there cannot, of course, be
a true phase transition, but a crossover between weakly andstrongly correlated regimes is still expected. There are sev-eral other fundamental differences between quantum dotsand bulk systems: /H20849a/H20850Broken translational symmetry in a QD
reduces the ability of the electrons to delocalize. As a result,a Wigner-type crossover is expected for a smaller value of r
s.
/H20849b/H20850Mesoscopic fluctuations, inherent in any confined
system,17,19lead to a rich interplay with the correlation ef-
fects. These two added features make strong correlationphysics particularly interesting in a QD. As clean 2D bulksamples with large r
sare regularly fabricated these days in
semiconductor heterostructures,20it seems to be just a matter
of time before these systems are patterned into a QD, thusproviding an excellent probe of correlation effects.
Circularly symmetric quantum dots have been a focal
point of theoretical attention for several years.
21Early calcu-
lations using density functional theory /H20849DFT /H20850within the lo-
cal spin density approximation showed a spin density wave/H20849SDW /H20850signature for r
sas small as 2.22,23Unrestricted
Hartree-Fock /H20849UHF /H2085024yielded both SDW and charge density
wave /H20849CDW /H20850features for rs/H110111. These were initially identi-
fied as signatures of strong correlations related to the analogof a Wigner crystal in a finite system, often called a “Wignermolecule.” However, SDW and CDW both break the funda-mental rotational symmetry of the 2D circular dots. Indeed,later calculations confirmed that these effects are largely ar-tifacts of the approximations used.
23,25–30Projection methods
were then used to restore symmetries as a second stageof the UHF calculations.
31–35DFT has been pushed toward
the large rslimit through the average spin density
approximation.36However, the validity of all these
methods—those that reduce an interacting problem to an ef-fective noninteracting one—remain questionable, particu-larly for large r
s.PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
1098-0121/2007/76 /H208498/H20850/085341 /H2084915/H20850 ©2007 The American Physical Society 085341-1More demanding computational techniques, which treat
the correlations in an exact fashion, have also been applied tothis problem. For example, exact diagonalization /H20849ED/H20850is ac-
curate for small QDs,
26,30,37but becomes exponentially in-
tractable for dots with more than six electrons /H20849N/H333566/H20850and
rs/H333564. Path integral Monte Carlo /H20849PIMC /H20850has also been ap-
plied to large rscircular dots.38–42One study38found a cross-
over from Fermi liquid to Wigner molecule behavior at rs
/H110154, a value significantly smaller than the bulk rsc,2D.
Another,40using different criteria for the transition, found a
two-stage transition for rslarger than rsc,2D. Although PIMC
treats interactions accurately, it has its own systematic andstatistical problems; for instance, it generates a thermal av-erage of states with different Land Squantum numbers,
preserving only S
zsymmetry. Since the energy for low-lying
spin excitations becomes very small in the low density limit,the constraint on the temperature in order to access theground state becomes extremely stringent. In the absence ofa coherent picture, the role of correlations in QD remains an
open problem.
The most accurate method for treating the ground state of
strongly interacting quantum dots, in our opinion, is varia-tional Monte Carlo /H20849VMC /H20850followed by diffusion Monte
Carlo /H20849DMC /H20850. This has been carried out in a number of cases
for high to medium density quantum dots.
43–47We discuss
this method in more detail below. Briefly, while there is asystematic “fixed-node error,” it is considerably smaller thanthe systematic and statistical errors of PIMC.
In this paper, we present a systematic study of circular
parabolic QDs over a wide range of interaction strengthsusing an accurate VMC and DMC technique. We demon-strate how strong interactions bring out the interesting as-pects of the Wigner physics in QDs; a short report on someaspects appeared in Ref. 48. Our main results are the follow-
ing:
/H20849i/H20850The development of inhomogeneities as the interaction
strength increases is completely smooth, with no discerniblespecial value below r
s/H1101518 for all N/H1102120.
/H20849ii/H20850The density develops sharp rings but remains circu-
larly symmetric; the pair densities show both radial and an-gular inhomogeneity.
/H20849iii/H20850The spin of the ground state follows Hund’s /H20849first/H20850
rule throughout our entire range of r
sfor all 4 /H33355N/H3335520: it is
the maximum consistent with the shell structure, in contrastto several previous claims.
23,27,36,38,42We do find many vio-
lations of Hund’s second rule—the value of the orbital angu-lar momentum is not necessarily a maximum—but a modi-fied second rule holds for small r
s.
/H20849iv/H20850The addition energy curve first becomes smoother as
interactions strengthen—the mesoscopic fluctuations aredamped by correlation—and then starts to show featurescharacteristic of the classical addition energy, indicating in-cipient Wigner crystallization.
/H20849v/H20850Localization effects are stronger for a smaller number
of electrons.
/H20849vi/H20850Finally, the gap to certain spin excitations becomes
small at the strong interaction side of our regime.
The organization of the paper is as follows. In Sec. II, we
describe the model and parameters for the quantum dots westudy. Section III discusses our technical tools, VMC andDMC. We present our results in Sec. IV . We focus our atten-
tion first on the density and pair densities; it is these twoquantities that encode rich information on the correlation-induced inhomogeneities. We then discuss the energy of theground and excited states. The interesting issue of spin cor-relation in a QD in the large r
slimit is addressed at the end
of the section. Finally, we present our conclusions in Sec. V .
II. MODEL AND PARAMETERS
We consider quantum dots with Nelectrons confined in a
circularly symmetric harmonic potential, Vcon/H20849r/H20850=m*/H92752r2/2,
using the Hamiltonian
H=/H20858
i=1N/H20873−/H60362
2m*/H11612i2+Vcon„ri…/H20874+/H20858
i/H11021jNe2
/H92801
/H20841ri−rj/H20841, /H208491/H20850
where m*is the effective mass of the electrons and /H9280is the
dielectric constant of the medium. We consider 2D systems,so that r
2=x2+y2. This is because experimental dots made by
patterning GaAs/AlGaAs heterostructures have very strongconfinement in the zdirection so that they are essentially two
dimensional. The last term in the Hamiltonian is the pairwiseCoulomb repulsion between electrons. The strength of thisinteraction is characterized by the gas parameter, r
s/H20849in units
of the effective Bohr radius aB*/H20850, which is related to the av-
erage density of electrons, n¯/H11013/H20848n2/H20849r/H20850dr/N,b y rs/H11013/H20849/H9266n¯/H20850−1/2
/H20849in 2D /H20850. More physically, rsis essentially the ratio between
the potential and kinetic energy of the system, justifying itsidentification as the interaction strength. We tune r
sby vary-
ing/H9275inVcon/H20849r/H20850; this makes the confining potential more
narrow or more shallow, making the average density at fixed
Nlarger or smaller, thus controlling rs.
The confining potential prevents the electrons from flying
apart from each other, and thus an extra positive backgroundcharge is unnecessary for the stability of the dot. In the limitof weak interaction, the density and number of electrons inthe dot are related to the strength of the confinement by
/H92752
=e2//H20849/H9280m*rs3/H20881N/H20850. In general, the electron-electron interactions
tend to expand the dot, making the effective /H9275smaller than
the bare /H9275. This is quite significant for large rs, and so the
above simple relation between /H9275andrsbreaks down.
Circular parabolic confinement is a good description for
the experimental vertical dots, as well as for the few electronlateral dots where electrons sit in the central region far fromthe confining gates.
49In this paper, we assume that the cir-
cular geometry is preserved even at large rs, leaving the issue
of irregular dots47for future study.
Throughout this paper we use effective atomic units in
which the length unit aB*is/H9280/m*times the Bohr radius aB,
and the energy is given in effective Hartrees, H*=m*//H92802Har-
trees. For GaAs, for example, m*=0.067 meand/H9280=12.4,
leading to aB*=98 Å and H*=11.9 meV.
We have studied the Hamiltonian /H208491/H20850for a wide range of
parameters. We varied /H9275between 3.0 and 0.0075 for 2 /H33355N
/H3335520. For the range of /H9275considered, rslies in the range 0.4
to 18.GHOSAL et al. PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-2III. METHOD
Circular parabolic dots with noninteracting electrons are
described by single-particle orbitals, called Fock-Darwin/H20849FD/H20850orbitals, specified by principal quantum number nand
azimuthal quantum number l. The energy of the orbitals is
E
n,l=/H208492n+/H20841l/H20841+1/H20850/H6036/H9275. /H208492/H20850
The effective interaction between electrons can be built into
the single particle problem within the framework of a meanfield theory, such as the DFT or UHF methods mentioned inthe Introduction. These introduce some interaction effectsinto the single-particle wave functions—for instance, the re-sulting orbitals are more extended than the correspondingnoninteracting Fock-Darwin orbitals.
As a starting point of our calculation, we use Kohn-Sham
/H20849KS/H20850orbitals
/H9278/H9251/H20849r/H20850, from a DFT calculation within the local
density approximation /H20849LDA /H20850. We then construct configura-
tion state functions /H20849CSFs /H20850that are eigenstates of Lˆ,S2ˆ, and
Szˆ, each of which is a sum of a product of up- and down-spin
Slater determinants,
/H9021iL,S/H20849R/H20850=/H20858
j=1m
/H9252ijDj↑Dj↓. /H208493/H20850
R/H11013/H20853r1,r2,..., rN/H20854denotes collective coordinates of the N
electrons, and D↑,D↓are Slater determinants of spin-up and
spin-down KS orbitals, respectively. The coefficients /H9252ijare
determined by the requirement that /H9021L,S/H20849R/H20850is an eigenstate
ofLˆ,Sˆ2, and Szˆ.
The many-body trial wave function that we use is
/H9023TL,S/H20849R/H20850=J/H20858
i=1NCSF
ci/H9021iL,S/H20849R/H20850, /H208494/H20850
where the Jastrow factor J=JenJeeJeenis the product of
electron-nucleus, electron-electron, and electron-electron-nucleus factors. By “nucleus” here we mean the center of theharmonic external potential. The detailed form of the Jastrowfactor can be found in Ref. 50. The independent parameters
to be optimized are the c
iand the parameters in J. The linear
combination of CSFs builds in the near-degeneracy correla-tion in the wave function, whereas the Jastrow factor effi-ciently describes the dynamic correlation that would other-wise require a very large number of CSFs. The Jastrow factoris so effective that only a small number of CSFs is needed.
In the very weakly interacting limit, the ground state con-
sists of simply filling the lowest energy single-particle orbit-als. Both the circular and harmonic nature of the externalpotential cause degeneracies in the noninteracting spectrum,Eq. /H208492/H20850, and hence in the noninteracting many-body spectrum
as well. Thus there is a definite “shell structure” in the ener-gies of the many-particle states, much as in atoms. The shellsare full for N=2,6,12,20,...; for these values of Nthe
ground state clearly has L=0 and S=0. For intermediate val-
ues of N, the orbitals are filled so that the interaction effects
yield the lowest energy. For the total spin, Hund’s first rule,familiar from atomic physics, applies here as well: the elec-trons in the open shell arrange so as to have maximum pos-sible spin in order to gain exchange energy. For instance, the
ground state for N=9 has S=3/2.
For all of the results shown in this paper, we use only
CSFs constructed from the lowest energy shells consistentwith the desired LandS: no intershell excitations are used.
Thus for the ground state, the CSFs we include are those forthe/H20849possibly degenerate /H20850noninteracting ground state. For a
closed shell and L=S=0, for instance, there is only one such
CSF. For an open shell, there can be more than one CSFmeeting our criteria; as an illustration, consider the L=S=0
state of N=8. Six electrons fill the lowest two shells while
two are distributed among the third shell’s three levels, /H20849n
=1,l=0/H20850and /H20849n=0,l=±2 /H20850. The desired state can be made in
two ways, by either putting both electrons in the former levelor putting one in each of the latter /H20849in a singlet state /H20850.W e
always include both of these CSFs in our calculations for thisstate. For a given N, we consider all LandSwhich can be
obtained with CSFs built from orbitals in the lowest energyshell.
Selected cases are further checked by including more
CSFs. For N=3, 6, 7, 9, and 20, some /H20849L,S/H20850values were
studied with CSFs which included up to two intershell exci-
tations. For the range of r
sstudied here, the energies of these
states were not significantly changed by including these ad-ditional CSFs, giving confidence in the accuracy of our re-sults.
We perform both variational Monte Carlo /H20849VMC /H20850and dif-
fusion Monte Carlo /H20849DMC /H20850calculations. Both the VMC and
the DMC energies are upper bounds to the true ground stateenergy, E
GS. The VMC energy, EVMC/H33356EGS,i s
EVMC=/H20885/H9023T*/H20849R/H20850H/H9023T/H20849R/H20850dR
/H20885/H9023T*/H20849R/H20850/H9023T/H20849R/H20850dR
=/H20885dREL/H20849R/H20850P/H20849R/H20850/H11015/H20858
i=1NMC
EL/H20849Ri/H20850. /H208495/H20850
Here, /H20849a/H20850theNMCMonte Carlo configurations are sampled
from P/H20849R/H20850=/H20841/H9023T/H20849R/H20850/H208412//H20848/H20841/H9023T/H20849R/H20850/H208412dR, and /H20849b/H20850EL/H20849R/H20850
=/H9023T−1/H20849R/H20850H/H9023T/H20849R/H20850is the local energy which is constant and
equal to the true energy in the limit that /H9023Tis an exact
eigenstate. The variational parameters were optimized usingthe variance-minimization method,
51and some of the results
were checked by using two recently developed energy mini-mization methods.
52,53
We use diffusion Monte Carlo /H20849DMC /H20850to project out the
best estimate of EGSstarting with the optimized /H9023T. The
DMC method employs the importance-sampled Green’sfunction,
G/H20849R
/H11032,R,/H9270/H20850=/H9023T/H20849R/H11032/H20850/H20855R/H11032/H20841exp/H20849−H/H9270/H20850/H20841R/H20856//H9023T/H20849R/H20850, /H208496/H20850
to project out /H90230/H20849R/H20850/H9023T/H20849R/H20850, where /H90230/H20849R/H20850is the lowest en-
ergy state that has the same spatial and spin symmetry as
/H9023T/H20849R/H20850.A s/H9270becomes large, the amplitudes of higher energy
states decay exponentially compared to that of the ground
state. However, G/H20849R/H11032,R,/H9270/H20850is not known exactly for theINCIPIENT WIGNER LOCALIZATION IN CIRCULAR … PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-3Hamiltonian Hof Eq. /H208491/H20850, and a short time approximation to
exp/H20849−H/H9270/H20850using the Trotter formula is used repeatedly to
achieve the desired projection. We use a refined algorithm
following Ref. 54which has a very small time step error.
The mixed estimator for the DMC energy,
EDMC=/H20885/H90230*/H20849R/H20850H/H9023T/H20849R/H20850dR
/H20885/H90230*/H20849R/H20850/H9023T/H20849R/H20850dR, /H208497/H20850
equals the ground state energy E0. However, mixed estima-
torsODMC of operators that do not commute with the Hamil-
tonian, such as the density, have errors that are linear in theerror in /H9023
T. On the other hand, the extrapolated estimators
2ODMC−OVMC and ODMC2/OVMC have errors that are qua-
dratic in the error in /H9023T. All of the data shown in this paper
for such operators is based on the extrapolated estimators,labeled “QMC,” unless explicitly marked “VMC.”
The function which minimizes the energy in the absence
of any constraints has bosonic symmetry, but we are inter-ested in the lowest Fermionic state. Consequently, comparedto the Fermionic state of interest, the bosonic componentgrows exponentially fast. In the absence of statistical noise,one could still employ symmetry to obtain the fermionic en-ergy, but in a MC method the estimate for the energy has astatistical error that grows exponentially with
/H9270. The fixed-
node approximation prevents this catastrophe by imposing
the constraint that the nodes of /H90230/H20849R/H20850are the same as those
of/H9023T/H20849R/H20850. The result of this constraint is that the fixed-node
DMC energy is an upper bound to the true energy. For flex-
ible wave functions with well-optimized parameters thefixed-node error is typically very small.
In our circular quantum dots, since states with angular
momentum Lare degenerate with those of angular momen-
tum − L, we are free to construct real wave functions by
choosing the linear combinations, /H9023
TL,S+/H9023T−L,Sand/H9023TL,S
−/H9023T−L,S. In this case we can use the fixed-node approxima-
tion. If instead we choose to employ states with definite L
then the wave functions are complex, and we must use thefixed-phase approximation,
55,56the generalization of the
fixed-node method to complex wave functions. Usually thefixed-phase error is comparable to and slightly larger thanthe fixed-node error.
Technical considerations limit the present study to ap-
proximately r
s/H1135118. For larger rs, one expects that more
CSFs should be included in the trial wave function becauseexcitations across the shell gaps produced by the interactionsbecome more important. For example, for the N=6 ground
state /H20849L=S=0/H20850with
/H9275=0.01, inclusion of CSFs correspond-
ing to two intershell excitations lowers the energy from
0.689 228 /H208495/H20850to 0.689 202 /H208495/H20850. However, the variance optimi-
zation used for most of our results fails to lower EVMC or
EDMC if there are more than 3–4 CSFs in /H9023T/H20849though it does
lower the fluctuations of the energy /H20850. For a few cases, includ-
ing moderate and large rsand several N, we have done pre-
liminary calculations with higher orbitals by including alldeterminants involving promotion of two electrons across ashell gap /H20849e.g., 10 CSFs for N=20 /H20850, using two recently de-veloped energy optimization procedures.
52,53This, then, al-
lows for a change in the nodes of /H9023T/H20849R/H20850. We find that typi-
cally a multi-CSF calculation produces a slight decrease in
the energy for rs/H3335615, with larger decreases per electron for
smaller N. The change in the spatial structure, as in the den-
sity and pair-density discussed below, is even smaller thanthat in the energy.
IV . RESULTS
Examples of DMC energies from our calculations are pre-
sented in Table I. For the purpose of comparison, we also
include energies obtained using other techniques. The DMCenergies, which are an upper bound on the true ground stateenergies, are lower than those obtained from the other meth-ods, showing the accuracy of the method.
Further results are organized in six topical areas: the elec-
tron density, the pair density, real vs complex wave function,the addition energy of the dot, the ordering of the different/H20849L,S/H20850states in energy, and the nature of the spin correlations.
The sensitivity of these results to aspects of the
methodology—VMC vs DMC, and the type of single particleorbital—are discussed in the Appendix.
A. Spatial density profile
The evolution of the radial density of electrons, n/H20849r/H20850,a srs
varies is shown in Fig. 1. The ground state density at four
different rsis plotted for four values of N/H208497, 9, 16, and 20 /H20850.
With increasing rs, the electron-electron repulsion expands
the system spatially; note that the linear dimensions of thedot scale roughly linearly with r
s, as expected given the re-
lation between rsand the average density.
Figure 1shows that increasing rscauses the density pro-
file to change dramatically. For small rs, the density is fairly
smooth, with some weak structure coming from the nonin-teracting shell structure. We found that the weak structure inn/H20849r/H20850for small r
sis very similar to that obtained from Fock-
Darwin orbitals consistent with the shell filling. On the other
hand, large rsinduces strong modulation in n/H20849r/H20850, resulting in
the formation of radial rings in the density profile. The rings
become sharp with increasing rs. Strikingly, once rsbecomes
larger than 10–12, the number of rings is the same as in theclassical limit
57for that N: one ring for N/H333555, two rings for
6/H33355N/H3335515, and three rings for larger N’s up to 20. Further-
more, for these larger rs, the average number of electrons in
the outer ring /H20849obtained by simply integrating the density
over that region /H20850is mostly consistent with the classical
value.
If one employs wave functions of definite orbital angular
momentum quantum number L, both the total density and the
spin densities must be circularly symmetric /H20849in two dimen-
sions /H20850. Since our VMC and DMC methods do not break this
symmetry, we obtain circularly symmetric densities in allcases. However, at sufficiently large r
seven a small pertur-
bation of the circular symmetry would be sufficient to pinelectrons; in that case, the density would have the sharppeaks of a Wigner localized state. In the absence of a pertur-bation, the signature of localization is in the pair density, asGHOSAL et al. PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-4discussed in the next subsection, even when it is absent in
the density. Alternatively, a symmetry broken density profile
is obtained when Lis not fixed; this case is treated in Sec.
IV C.
We find that the formation of the rings and the increase of
their sharpness is completely smooth . To quantify this state-
ment, we define a quantity, the “fractional peak height”/H20849FPH /H20850, that tracks the sharpness of the rings in n/H20849r/H20850. Its defi-nition and dependence on r
sis presented in Fig. 2. The FPH
is the ratio of the depth of the valley in n/H20849r/H20850between the two
outer rings to the “average” height of the rings, AB/AC.I n
Fig. 2the FPH always increases with rsas expected /H20849see the
Appendix for the corresponding VMC result /H20850. However, the
increase is surprisingly smooth for the whole range of rs
studied. We thus infer that its increase is not associated with
any special value of rssignifying a threshold or “onset.” TheTABLE I. The ground state energy of a circular 2D quantum dot obtained by three different computational
methods: diffusion quantum Monte Carlo /H20849DMC /H20850, full configuration interaction /H20849CI/H20850, and path-integral quan-
tum Monte Carlo /H20849PIMC /H20850.N,L, and Sspecify the number of electrons in the dot, their angular momentum,
and their spin. The energy is given in units of /H6036/H9275, the characteristic energy of the external parabolic confining
potential. /H9261=1//H20881/H9275/H20849in atomic units /H20850characterizes the strength of the interactions.
N /H9261rs
/H20849approx. /H20850 LSDMC
/H20849this work /H20850CI
/H20849Ref. 37/H20850 SzPIMC
/H20849Ref. 38/H20850
6 8 12.5 0 0 60.3251 /H208493/H20850 60.64
1 1 60.4027 /H208493/H20850 60.71 1 60.37 /H208492/H20850
0 2 60.3520 /H208492/H20850 60.73
0 3 60.3924 /H208492/H20850 60.80 3 60.42 /H208492/H20850
10 16.3 0 0 68.9202 /H208495/H20850 69.74
1 1 69.0568 /H208497/H20850
0 2 68.9254 /H208496/H20850 69.81
0 3 68.9458 /H208494/H20850 69.86
7 4 5.2 0 1/2 53.7351 /H208493/H20850 54.69
2 1/2 53.7265 /H208492/H20850 54.68
1 3/2 53.8183 /H208492/H20850 54.78
0 5/2 53.8357 /H208492/H20850 54.93
3 7/2 54.1633 /H208491/H20850 55.20
8 12.4 0 1/2 80.3846 /H208494/H20850
2 1/2 80.4925 /H208494/H20850
1 3/2 80.4795 /H208494/H20850
0 5/2 80.4135 /H208493/H20850 5/2 80.45 /H208494/H20850
3 7/2 80.5146 /H208492/H20850 7/2 80.59 /H208494/H20850
8 2 2.1 0 0 46.8070 /H208494/H20850
2 0 46.8746 /H208494/H20850
4 0 46.7793 /H208493/H20850
0 1 46.6787 /H208493/H20850 1 46.5 /H208492/H20850
2 1 46.7560 /H208494/H20850
1 2 46.9170 /H208494/H20850 2 46.9 /H208493/H20850
0 2 47.4058 /H208494/H20850
3 3 47.4035 /H208493/H20850 3 47.4 /H208493/H20850
0 4 48.1810 /H208494/H20850 4 48.3 /H208492/H20850
8 12.2 0 0 102.9402 /H208494/H20850
2 0 102.9464 /H208494/H20850
4 0 103.0465 /H208494/H20850
0 1 102.9263 /H208494/H20850
2 1 102.9198 /H208494/H20850
1 2 102.9280 /H208494/H20850 2 103.08 /H208494/H20850
0 2 103.1965 /H208494/H20850
3 3 103.0185 /H208493/H20850 3 103.19 /H208494/H20850
0 4 103.0464 /H208494/H20850 4 103.26 /H208495/H20850INCIPIENT WIGNER LOCALIZATION IN CIRCULAR … PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-5continuous thin lines are the best fits of our QMC results to
power-law behavior. Even though the best fit curve is /H11011/H20881rs
for small N/H20849e.g., 7 and 9 /H20850, for larger N=16,20, it is nearly
linear. A large FPH signifies stronger radial localization asthe rings tend to decouple from each other. From Fig. 2,w e
see that for the larger values of r
s, small Ndots are more
strongly radially localized for a similar interaction strength.
The FPH is, by construction, a sort of “peak to valley
ratio,” and the construction used here is not unique. How-ever, we have checked for a few cases that a different con-struction /H20849e.g., the ratio between the height of the valley and
the outer peak /H20850leaves our conclusions unchanged.The smooth and featureless increase of FPH with r
sde-
velops naturally from the radial structure created by nonin-teracting shell effects. The latter can be physically thought ofas Friedel oscillations due to radial confinement. Our resultsshow that these oscillations smoothly grow into strong inho-mogeneities, finally leading to radial localization. This sug-
gests that the electron-electron interactions are acting onpreexisting oscillations, namely those caused by noninteract-ing interference effects, and smoothly amplifying them.
This point is further made by exploring explicitly the role
of electron-electron correlations in the density profile. Weshow in Fig. 3the density resulting from four successively
better treatments: noninteracting, DFT in the LDA approxi-mation, VMC, and finally fixed-node QMC /H20849extrapolated es-
timator /H20850. We use the N=20 ground state for r
s/H1101515. As ex-
pected, we see that the interaction significantly modifies n/H20849r/H20850
at this large rs, first by greatly expanding the dot at the DFT
level compared to the noninteracting description, and then by
amplifying the inhomogeneous structures /H20849e.g., rings /H20850,a s
seen by comparing the DFT and QMC results. We note thatthe difference between the DFT and QMC densities is largecompared to the corresponding difference for real atoms,
58
reflecting the fact that the dot is much more strongly corre-lated.
B. Pair density
The issue of correlation induced localization of the indi-
vidual electrons cannot be addressed by looking at the den-(a)
(b)
FIG. 1. Radial electron density n/H20849r/H20850=n↑/H20849r/H20850+n↓/H20849r/H20850in the ground
state for four values of Nand four interaction strengths rs./H20849a/H20850N
=7, /H20849L,S/H20850=/H208490,1/2 /H20850,/H20849b/H20850N=9, /H20849L,S/H20850=/H208490,3/2 /H20850,/H20849c/H20850N=16, /H20849L,S/H20850
=/H208490,2/H20850, and /H20849d/H20850N=20, /H20849L,S/H20850=/H208490,0/H20850.A s rsincreases, strong radial
modulation develops leading to ring structure in n/H20849r/H20850. The number
of rings for rs/H3335610 is the same as in the classical limit /H20849rs→/H11009/H20850:t w o
for the first two panels and three for the last two. /H20849The values of /H9275
used are for N=7 and 9, /H9275=0.8, 0.08, 0.02, and 0.01; for N=16,
/H9275=0.269, 0.06, 0.02, and 0.01; and for N=20,/H9275=0.8, 0.04, 0.02,
and 0.0075. /H20850FIG. 2. Evolution of the sharpness of the rings in n/H20849r/H20850with rs,
which measures the radial inhomogeneity in the dot. The sharpnessis quantified in terms of the “Fractional Peak Height” /H20849FPH /H20850, the
construction for which is shown in the inset, FPH=
AB/AC. The
main panel shows the rs-dependence of the FPH for the N= 7 ,9 ,1 6 ,
and 20 ground states /H20849same states as in Fig. 1/H20850. The lines show the
best fit of the data to power-law behavior. The FPH grows verysmoothly with r
sfor all N’s without any signature of a special
threshold value. Note that the rings sharpen faster for smaller N,
and that the FPH is nearly linear for the larger N’s.GHOSAL et al. PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-6sity alone, which is manifestly rotationally symmetric for our
Hamiltonian. The formation of radial rings in the densityimplies, as explained in the previous section, radial localiza-tion of electrons—a feature special to circularly symmetricconfined systems. This symmetry is, however, broken when
one of the electrons is held fixed at a particular position.Other electrons then organize themselves in the dot so as tominimize their mutual interaction and kinetic energy; in par-ticular, when the repulsion is strong enough, electrons local-ize at the classical positions. Therefore, to address the ques-tion of individual electron localization we turn to the pairdensity.
The spin-resolved pair density, g
/H9268/H9268/H11032/H20849r0;r/H20850, is defined as
the probability of finding an electron with spin /H9268/H11032at location
rwhen an electron with spin /H9268is held fixed at r0, and the
total pair density is gT=g↑↑+g↑↓. These quantities detect, in
addition to radial rings, any angular structure induced by the
interactions.
We present in Fig. 4g↑↓andg↑↑for the N=20 ground
state for weak, intermediate, and strong interaction strengths/H20849r
s/H110150.4, 5, and 15, respectively /H20850. The location r0of the up-
spin electron is fixed at the outermost local maximum of thedensity, i.e., on the center of the outer ring. The behavior ofg
/H9268/H9268/H11032/H20849r0;r/H20850for small rsis well known: in this weakly inter-
acting limit, the “correlation hole” /H20849the hole around r0ing↑↓/H20850
is much smaller than the “exchange hole” /H20849the similar hole in
g↑↑/H20850. This is because an opposite spin electron can come
arbitrary close to the fixed one, while a same spin electron isforbidden to be in its vicinity by the Pauli exclusion prin-ciple. We see that the correlation hole becomes bigger withr
s, becoming comparable in size to the exchange hole byintermediate interaction strength /H20849rs/H110155/H20850. At the same time
that the correlation hole becomes bigger, so do the closest
peaks in the antiparallel-spin pair density, but not those in theparallel-spin pair density. This is a reflection of the fact thatg
↑↓integrates to N/2 while g↑↑integrates to N/2−1.
Figure 4shows that increasing rsinduces weak angular
modulation in g/H9268/H9268/H11032/H20849r0;r/H20850, in addition to the radial ring struc-
tures. /H20849Sensitivity to the initial single-particle orbitals used is
presented in the Appendix. /H20850Because of the continuous de-
velopment toward sharper angular structure, we view thehigher r
sregime as showing “incipient Wigner localization.”
To focus further on this angular modulation, we plot
g/H9268/H9268/H11032/H20849r0;r/H20850along the outer ring on which the up spin is fixed
in Fig. 5. We see that the angular oscillations in gT,g↑↓, and
g↑↑are damped and weak in comparison with the radial
modulation. For the intermediate value of rs, notice the small
but clear oscillations of g↑↓andg↑↑all the way around the
ring; these oscillations are out of phase with each other, sothat there is a small amplitude spin pair-density wave. Theperiod of this wave is approximately /H9261
F/2 where the Fermi
wavelength /H9261Fis given by the effective 1D density in the
outer ring /H20851/H9261F/H208491D/H20850=4/n/H208491D/H20850/H20852.
The angular oscillations grow continuously as a function
ofrs, as in the case of radial modulation. We find that theFIG. 3. The effect of correlations on the radial density at differ-
ent stages of approximation for the N=20 ground state /H20849L=0, S
=0/H20850with rs/H1101115/H20849/H9275=0.01 /H20850. The noninteracting density nFD/H20849dashed /H20850
from Fock-Darwin orbitals is much too compact and has very weakradial structure. In the LDA result, n
LDA /H20849dotted /H20850, the mutual elec-
tronic repulsion makes the dot greatly expand in the radial directioncompared to n
FD. Both the VMC result /H20849dash-dotted line /H20850and QMC
extrapolated estimate /H20849solid /H20850show much stronger inhomogeneous
structure due to correlation that the QMC methods build in throughthe Jastrow factor.(−26.5,−26.5)(−26.5,−26.5)(−26.5,−26.5)(−26.5,−26.5)(2.75,2.75)(2.75,2.75)(2.75,2.75)(2.75,2.75)(−2.75,−2.75)(−2.75,−2.75)(−2.75,−2.75)(−2.75,−2.75)
(26.5,26.5)(26.5,26.5)(26.5,26.5)(26.5,26.5)
(80,80)(80,80)(80,80)(80,80)(−80,−80)(−80,−80)(−80,−80)(−80,−80)ω=3ω=3ω=3ω=3 (r =0.41)(r =0.41)(r =0.41)(r =0.41)ssss
(r =4.7)(r =4.7)(r =4.7)(r =4.7)ssssω=0.06ω=0.06ω=0.06ω=0.06
ω=0.01ω=0.01ω=0.01ω=0.01ssss(r =14.7)(r =14.7)(r =14.7)(r =14.7)ggggggggN=20N=20N=20N=20
L=0L=0L=0L=0
S=0S=0S=0S=0
FIG. 4. /H20849Color online /H20850Evolution of the spin resolved pair den-
sities, g/H9268/H9268/H11032/H20849r0;r/H20850, for the N=20 ground state /H20849L=S=0/H20850.rsincreases
from the top to the bottom; g↑↓is on the left while g↑↑is on the
right; r0is chosen near the outer ring of the dot. During the initial
increase of correlation until rs/H110115, the “correlation” and the “ex-
change” hole become fully developed. A further increase of rspro-
duces short-range order as seen from the bumps along the outer ringnear r
0. The range of these angular oscillations as well as their
amplitude increase gradually with rs, suggesting the term “incipient
Wigner localization” for this regime.INCIPIENT WIGNER LOCALIZATION IN CIRCULAR … PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-7total pair density is almost featureless even for rssignifi-
cantly greater than 1. Short-range correlations set in for rs
/H3335610; the period in the lower panel is approximately the in-
terparticle spacing of electrons in the outer ring /H20849or equiva-
lently 4 kF/H208491D/H20850/H20850. Even at the largest rsstudied here, the weak-
ness of these oscillations suggests that the electrons remainessentially delocalized on each ring.
One of the intriguing features in the spin resolved pair
density in Fig. 5is a bump at
/H9258=/H9266. At this position, which is
diametrically opposite to the fixed up electron, g↑↑decreases
while g↑↓increases compared to their average value /H20849the ef-
fect of the bump disappears from gT/H20850. We find a similar fea-
ture for several N’s with different L,Scombinations when
rs/H333565. The feature becomes more pronounced as rsincreases.
We lack a detailed understanding of this “bump” at this time;however, we suspect that its origin lies in the nature of thespin correlation between electrons, which will be discussedfurther in Sec. IV F.
We have also studied the pair density with the position r
0
of the fixed up electron at different locations. Figure 6shows
the spin-resolved pair densities for the N=20 ground state at
rs/H1101515 with r0chosen on the middle ring /H20849upper panel /H20850and
at the center59of the dot /H20849bottom panel /H20850. These plots, to-
gether with those of Fig. 4, suggest that the angular modula-
tion is produced primarily in the same ring as the fixed elec-tron, while the other rings remain little affected. This is anindication that even though the FPH at r
s/H1101515 is substan-
tially smaller than its maximum value of 1, the radial local-ization of the electrons is rather strong so that the rings es-sentially decouple from each other.
We have studied the pair density for several different N
over a wide range of r
s. As an example, the pair density for
a small dot, N=6, in its ground state /H20849a filled shell case /H20850isshown in Fig. 7for three values of rs./H20849The case N=9, cor-
responding to a half-filled shell, was shown in our previouspaper, Ref. 48./H20850In the classical limit /H20849r
s→/H11009/H20850, two rings are
expected for N=6—a single electron in the center and an
outer ring containing the remaining five. Note that the rs
/H1101516.3 result is quite consistent with this classical structure.
The way in which the total modulation is shared between g↑↓
and g↑↑near/H9258=/H9266is surprising /H20851see Fig. 7/H20849d/H20850/H20852: all of the
modulation is in g↑↓while g↑↑is constant and smaller. We
also notice that compared to N=20 for similar rs, the angular
oscillations in this case are stronger. From our extensivestudy, we find, using results for both pair density and densityFIG. 5. Evolution of the angular modulation in g/H9268/H9268/H11032/H20849r0;r/H20850along
the outer ring for the same cases as in Fig. 4.g↑↓is shown by the
dotted line and g↑↑by the dashed line. The solid line represents the
spin-summed pair-density gT. The location of the fixed electron r0
defines /H9258=0. Note the clear oscillations and the resulting spin den-
sity modulation in the middle panel. In the lower panel /H20849strongest
interactions /H20850, a strong feature at /H9258=/H9266is present.gggg ggggN=20N=20N=20N=20
L=0, S=0L=0, S=0L=0, S=0L=0, S=0
r =14.7r =14.7r =14.7r =14.7(ω=0.01)(ω=0.01)(ω=0.01)(ω=0.01)
ssss
(80,80)(80,80)(80,80)(80,80)(−80,−80)(−80,−80)(−80,−80)(−80,−80)
FIG. 6. /H20849Color online /H20850Spin resolved pair densities with the fixed
electron chosen on the middle ring /H20849top panel /H20850or in the center of the
dot /H20849bottom panel /H20850. The parameters are the same as in the bottom
panel of Fig. 4.
(c) (d)(c) (d)(c) (d)(c) (d)(b)(b)(b)(b) (a)(a)(a)(a)(55,55)
(55,55)
(55,55)
(55,55)(17,17)(17,17)(17,17)(17,17)ggggTTTT r = 4.5r = 4.5r = 4.5r = 4.5ssss ssssr = 16.3r = 16.3r = 16.3r = 16.3
N=6, L=0, S=0N=6, L=0, S=0N=6, L=0, S=0N=6, L=0, S=0
FIG. 7. /H20849Color online /H20850Pair densities for the N=6 ground state
/H20849L=S=0, a closed shell configuration /H20850.gTfor a fixed electron on the
outer ring is shown for /H20849a/H20850rs/H110154.5 and /H20849b/H20850rs/H1101516.3. The corre-
sponding angular modulation of the spin-resolved pair densities isshown in /H20849c/H20850and /H20849d/H20850. The magnitude of the angular modulation is
clearly larger than that for N=20 at similar values of r
s/H20849compare
with Fig. 4/H20850: electrons are more localized in smaller dots.GHOSAL et al. PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-8/H20849previous section /H20850, that modulation caused by correlation ef-
fects is stronger when Nis small.
C. Real vs complex trial wave-function
In the absence of a magnetic field, one has the choice, as
discussed in Sec. III, of working with either a complex or areal wave function. /H20849In fact, for a circularly symmetric po-
tential, one has that choice even in the presence of a mag-netic field. /H20850A real /H9023
Tis a superposition of the degenerate
states with angular quantum number Land − L,s o/H9023Tis no
longer an eigenfunction of Lˆ. As a result, the angular part of
the wave function has sin /H20849m/H9258/H20850and cos /H20849m/H9258/H20850terms instead of
exp/H20849im/H9258/H20850/H20849mis an integer /H20850. While the electron density com-
ing from a complex wave function with definite Lmust be
circularly symmetric, a real wave function may have angularstructure for states with L/HS110050.
A calculation using real wave functions which shows an-
gular modulation is presented in Fig. 8. The spin-resolved
density for N=19 electrons in the L=3, S=1/2 state is
shown using both real and complex orbitals. We emphasizethat the strong modulation seen in the top panels is notin-
trinsically related to strong correlation /H20849though we find that
the modulation amplitude increases with r
s/H20850; in particular, the
modulation is found even for small rs. Furthermore, the
modulation disappears in calculations using complex wavefunctions, even for large r
s, as it must. Oscillations of a simi-
lar nature are found in the pair density—any modulation inthe density must be tracked in the pair density. Thus weconclude that to identify the correlation induced inhomoge-neities in a L/HS110050 state, it is important to use a complex /H9023
T
that has a fixed L.
A disadvantage of using a complex wave function is,
however, that the systematic error is typically slightly larger/H20849fixed phase instead of fixed-node approximation /H20850. As a re-
sult, energy estimates are more accurate when /H9023
Tis real.
D. Addition energy
The addition energy, /H90042E/H20849N/H20850, is defined as the second
difference of the ground-state energy with respect to the
number of electrons, N, on the dot,
/H90042E/H20849N/H20850/H11013EGS/H20849N+1/H20850+EGS/H20849N−1/H20850−2EGS/H20849N/H20850. /H208498/H20850
The addition energy can be accessed experimentally through
the spacing between conductance peaks in a Coulomb block-ade transport measurement;
17–19,21of the various quantities
that the we discuss in this paper, /H90042Eis the simplest to
measure experimentally. The leading term in the additionenergy of quantum dots is the charging energy. Single par-ticle effects and corrections to the simple charging modelcause the addition energy to vary with N. For example, in the
noninteracting limit of our Hamiltonian /H208491/H20850, the spectrum
is given by Eq. /H208492/H20850, and thus the total many-body energy is a
sequence of straight line segments of increasing slope.One finds /H9004
2E/H20849N/H20850=/H6036/H9275for the “magic numbers”
N=2,6,12,20,30,... and zero otherwise. These special N’s
correspond to closed shell structures for which /H90042E/H20849N/H20850has
peaks above its baseline charging value because of the extra
stability provided by the gap between shells. Weak residualinteractions beyond charging produce further small peaks in/H9004
2Efor half-filled shells, N=4,9,16,25.... The general oc-
currence of such peaks is referred to as “mesoscopic fluctua-tions;” they are always present in a confined system.
The addition energy as a function of electron number is
shown in Fig. 9for three values of
/H9275.A sNincreases at fixed
/H9275, the dot grows larger but the confinement potential forces
the density to increase as well, causing rsto decrease slightly(80,80)(80,80)(80,80)(80,80)(−80,−80)(−80,−80)(−80,−80)(−80,−80)
nnnnnnnn
(ω=0.01)(ω=0.01)(ω=0.01)(ω=0.01)L=3, S=1/2L=3, S=1/2L=3, S=1/2L=3, S=1/2N=19N=19N=19N=19
FIG. 8. /H20849Color online /H20850Spin-resolved density calculated with a
real /H20849top/H20850or complex /H20849bottom /H20850wave function. Up- and down-spin
densities for N=19 with L=3, S=1/2 are on the left and right,
respectively. The real trial wave function mixes angular momenta L
and − Land so breaks the rotational symmetry, leading to angular
modulation in the density. The complex trial function has eigen-value Land shows no angular structure /H20849within our statistics /H20850. Thus
angular modulation occurs in the density if the Lsymmetry is bro-
ken by construction; such modulation is not due to correlationeffects.FIG. 9. Addition energy /H20849normalized by /H6036/H9275/H20850as a function of N
for three different /H9275and for the classical limit /H20849Ref. 57/H20850/H20849rs→/H11009,
multiplied by an arbitrary scale /H20850. As interactions strengthen because
of decreasing /H9275, the mesoscopic fluctuations in /H90042Ebecome
weaker. Note that this happens more readily for small N. Features in
the/H9275=0.01 trace at small Nare remarkably similar to those found
in the classical limit, indicating that electrons are nearly localizedfor small N.INCIPIENT WIGNER LOCALIZATION IN CIRCULAR … PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-9with increasing N. In the regime of current experimental
quantum dots /H20849/H9275=0.28 corresponding to rs/H110152/H20850, our result is
similar to previous studies:21a base value caused by charging
modulated by shell effects give strong /H20849weak /H20850peaks for filled
/H20849half filled /H20850configurations. Upon increasing the average rs
/H20849/H9275=0.04, middle trace /H20850, we find that the strength of these
peaks weakens, signaling a reduction in the mesoscopic fluc-tuations as a result of electronic correlations. In particular,notice that the peak at N=6 has disappeared completely. The
fact that the /H9004
2Ecurve is smoother for small Nindicates that
correlations are stronger in that regime, a result consistentwith our inference from densities and pair densities.
Upon further decreasing
/H9275, and so increasing rs,/H90042E/H20849N/H20850
becomes smooth at large N, with a weak remnant of the shell
effects. However, the behavior for small Nhas changed com-
pletely: it is no longer smooth. Note in particular a new peakthat appears at N=7. To show the origin of this peak, we plot
for comparison the addition energy in the classical limit us-ing the ground state energy data of Bedanov and Peeters.
57
Clearly, the nature of fluctuations in the classical limit is verydifferent from those in the noninteracting limit: the peak atN=7 can be understood as due to the extra stability of the
classical configuration. The remarkable similarity of our re-
sult for small N at r
s/H1101516with the classical result strongly
indicates that electrons in this regime are well localized.
We emphasize that these changes in the nature of the fluc-
tuations of the addition energies occur gradually, and happenover different r
sranges for small and large N. Our study
indicates that, first, the noninteracting fluctuations die outand, then, the classical fluctuations creep in.
E. Reordering of states in energy
The shell structure present in circular quantum dots in the
weakly interacting limit suggests, in analogy with atoms, thatthe orbitals within a shell should be filled in the ground stateso as to follow Hund’s rules: the spin should be the highestpossible /H20849first rule /H20850and the angular momentum should be the
maximum consistent with the first rule /H20849second rule /H20850. Implicit
in the rules is the definition of a shell, containing orbitals thatare either exactly or approximately degenerate in energy. Inatoms, orbitals with the same principal quantum number, n,
and the same angular quantum number, l, are exactly degen-
erate, while orbitals with the same nbut different lhave an
accidental degeneracy in the noninteracting limit because ofthe Coulomb potential. The self-consistent potential splitsthis degeneracy to favor states with low l. In quantum dots,
orbitals with the same radial quantum number, n, and the
same /H20841l/H20841are exactly degenerate, while orbitals with the same
value of 2 n+/H20841l/H20841have an accidental degeneracy in the nonin-
teracting limit because of the harmonic potential. The self-consistent potential splits this degeneracy to favor states withhigh /H20841l/H20841. In atoms, the gain in exchange energy is insufficient
to overcome the energy splitting and so orbitals with thesame nandlconstitute a shell for the purpose of defining
Hund’s rules. In contrast, for dots, for the range of r
sof
interest, the exchange energy overcomes the orbital splittingand so orbitals with the same 2 n+/H20841l/H20841define a shell. Since the
exchange interaction is present in both atoms and dots,agreement with Hund’s first rule is expected for weakly in-
teracting dots. However, there is no reason to expect Hund’ssecond rule to hold, but one can enunciate a modified Hund’ssecond rule—states that occupy orbitals with high /H20841l/H20841are
favored—which should hold for weakly interacting dots.
Taking N=9 as an example, we show in Table IIresults
for the low-lying states at three values of r
s; the level struc-
ture and pair densities are shown in Fig. 10. Note that the
ground state has S=3/2 as expected from Hund’s first rule.
At small rs, the higher spin states lie at progressively higher
energy because they involve promotion across one or moreshell gaps: the kinetic energy cost of such a promotion is toolarge for any interaction effects to overcome. For instance, atransition from the /H20849L,S/H20850=/H208490,3/2 /H20850state to the /H208490,7/2 /H20850state
involves promoting the spin-down electrons in the /H20849n,l/H20850
=/H208490,1/H20850,/H208490,−1 /H20850orbitals to spin-up electrons in the /H20849n,l/H20850
=/H208490,3/H20850,/H208490,−3 /H20850orbitals. The S=9/2 state requires a further
promotion of the spin-down electron in /H20849n,l/H20850=/H208490,0/H20850to the
/H208491,1/H20850orbital.
Note that /H20849L,S/H20850=/H208490,3/2 /H20850is the ground state for all three
values of r
s, in agreement with Hund’s rules. As expected,
there are numerous violations of Hund’s original second rulein the different spin cases but the modified rule discussedabove holds at small r
s. For example, /H208491/H20850forN=9 in the S
=1/2 sector /H20849Table II/H20850, the L=2 state /H20851with the /H20849n,l/H20850=/H208490,2/H20850
orbital doubly occupied and the /H20849n,l/H20850=/H208490,−2 /H20850orbital singly
occupied /H20852has a lower energy than the L=4 state /H20851with the
/H20849n,l/H20850=/H208490,2/H20850orbital doubly occupied and the /H20849n,l/H20850=/H208491,0/H20850or-
bital singly occupied /H20852, and, /H208492/H20850forN=8 /H20849see Table I/H20850, the
ground state at rs/H110152i s /H20849L,S/H20850=/H208490,1/H20850rather than /H208492,1/H20850.
As a function of rs, there are two examples in Table IIof
reordering of excitations: First, in the S=1/2 sector, the L
=0 and L=4 excited states interchange their position by rs
/H110156.7, and then by rs/H1101516 the L=0 state replaces L=2 as the
lowest energy state. Second, as the strength of the interac-tions increases, the S=7/2 excitation becomes progressively
of lower energy, interchanging with both S=5/2 states by
r
s/H110156.7 and then with the three S=1/2 states by rs/H1101516.
In general, we find that Hund’s first rule is very robust:
according to our data, it can be used for the ground state spinthroughout the range 4 /H33355N/H3335520 and
/H9275/H110220.01, corresponding
tors/H1135116. For instance, Tables I and II show that the first
rule works well for N=6–9, both at small and large rs.I n
this respect we disagree with the PIMC results of Ref. 38
which predicted violations for all these cases. We believe thelack of sufficient statistical accuracy in Ref. 38led to that
erroneous conclusion. Note, for instance, that the statisticalerror in the PIMC energies is often larger than the energydifferences between different Sstates that we find.
The only problematic case for Hund’s first rule is N=10.
Here for
/H9275/H333550.04, we find that a state with spin 0 /H20851the state
/H208490,0/H20850/H20852becomes essentially degenerate within the accuracy of
our calculation with the expected S=1 ground state /H208492,1/H20850.
Determining the true ground state in the low-density regimein this case must await further work. The near degeneracy ofthese two states has been noted previously.
43,46,60The reason
is clear from the level diagram in Fig. 10: the kinetic energy
difference in moving an electron between the n=1 and 0
orbitals /H20849due to the small splitting between states having dif-GHOSAL et al. PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-10ferent principal quantum numbers n/H20850very nearly equals the
exchange energy of one pair of spins. In fact, N=10 is the
only Nin our range for which promotion of an electron
across the intrashell gap results in a gain in the exchangeenergy of just one pair of electrons, explaining why it is theonly inconclusive case. The next such situation occurs at N
=28, when the fifth shell is nearly filled.
The single example of a clear violation of the first rule
that we have encountered is for N=3 when
/H9275/H333550.028: the
/H208490,3/2 /H20850state is lower in energy than the Hund’s rule ground
state /H208491,1/2 /H20850. Our results for the energies are essentially the
same as from a configuration interaction /H20849CI/H20850calculation37
and hence not shown here. /H20849For small N, CI calculations
produce very precise energies. /H20850The first rule violation in this
case is probably due to the fact that instead of paying theCoulomb cost of doubly populating the spatially localized l
=0 orbital, electrons prefer to populate the more delocalizedl=1 orbital.
We see a trend toward violation of the first rule at our
largest r
sfor small N; this may indicate an actual violation
forrslarger than was possible in this study. For example, for
N=9 and /H9275=0.01 in Table II, the highly spin-polarized state
/H208490,7/2 /H20850becomes degenerate with the usual Hund’s rule
ground state /H208490,3/2 /H20850, to within our numerical accuracy. It is
clear from the shell structure why S=7/2 is favored: once
the gain in exchange energy favors promotion of a spin-down electron in orbital /H208490,1/H20850across two shell gaps to a
spin-up electron in /H208490,3/H20850, the /H208490,−1 /H20850electron will be pro-
moted to /H208490,−3 /H20850, thus explaining why S=7/2 becomes lower
in energy than S=5/2. On the other hand, S=9/2 requires
promotion across three shell gaps and so is not necessarilyfavored. Similarly, it is expected that for N=6, the relative
energy of state /H20849L,S/H20850=/H208490,2/H20850would decrease substantially at
large r
scompared to both the Hund’s rule state /H208490,0/H20850and the
fully polarized state /H208490,3/H20850. For N=7, the corresponding state
is/H208490,5/2 /H20850. This is indeed the result seen in Table I.
Finally, we present the spin-summed pair densities, gT, for
different spin states for N=9 in Fig. 10. It is clear that the
more polarized states are better localized: as expected, ex-change acts to keep the electrons apart. This is a rather ge-
neric feature in our study for a wide parameter range.
F. Nature of spin correlation
The last topic we address is the nature of the spin corre-
lation in the dots for our larger values of rs. For values of rs
by which angular modulation in the pair density has devel-
oped, the radial localization is already strong, restricting themotion of the electrons in the radial direction. Thus, the elec-tronic behavior in this limit is perhaps best described interms of a quasi-one-dimensional /H208491D/H20850system on a circular
ring. Effective spin interactions and the resultant correlationshave been studied extensively in quasi-1D and 2D over theyears.
61–63Recently, for instance, Klironomos et al. have
shown that in a quasi-1D system “ring exchange” processesdominate at intermediate r
s, leading to novel ground states.63
Our results on circularly symmetric quantum dots suggest
that the the nature of the spin structure depends on the an-gular momentum quantum number L. We illustrate this byTABLE II. The energy /H20849in units of /H6036/H9275/H20850of several low-lying
states /H20849identified by LandS/H20850of a circular 2D quantum dot for N
=9 and three values of rs. Note the reordering of states in energy as
the interaction strength increases; in particular, the S=7/2 state be-
comes nearly degenerate with the ground S=3/2 state at large rs.
LS/H9275=3.0
/H20849rs/H110150.49 /H20850/H9275=0.04
/H20849rs/H110156.7/H20850/H9275=0.01
/H20849rs/H1101515.8 /H20850
0 1/2 32.4874 /H208491/H20850 96.3487 /H208494/H20850 146.5469 /H208499/H20850
2 1/2 32.3993 /H208491/H20850 96.3375 /H208494/H20850 146.5558 /H208499/H20850
4 1/2 32.4387 /H208491/H20850 96.4594 /H208494/H20850 146.5746 /H208498/H20850
0 3/2 32.3365 /H208491/H20850 96.2531 /H208494/H20850 146.4651 /H208498/H20850
3 3/2 33.0545 /H208491/H20850 96.4594 /H208494/H20850 146.5701 /H208499/H20850
1 5/2 34.3464 /H208491/H20850 96.6618 /H208494/H20850 146.6323 /H208498/H20850
4 5/2 33.5959 /H208491/H20850 96.4836 /H208493/H20850 146.5892 /H208498/H20850
0 7/2 34.7717 /H208491/H20850 96.4718 /H208493/H20850 146.4651 /H208497/H20850
1 9/2 36.6964 /H208492/H20850 96.8211 /H208492/H20850 146.6712 /H208497/H20850(0,3)(1,1)
(1,0)
(0,2)
(0,1)
(0,0)(1,1)
(0,3)
(0,2)(1,0)
(0,1)
(n,l) (n,l)
Second shell Third shell
(0,0)(1,−1)
(0,−3)
(0,−2)
(0,−1)(1,−1)
(0,−3)
(0,−2)
(0,−1)
Fourth shel l First shell
FIG. 10. /H20849Color online /H20850Pair density and level structure for N
=9 and /H9275=0.01 /H20849rs/H1101515.8 /H20850. The spin-summed pair density, gT,i s
shown for the lowest energy state in each of the five possible spinsectors; a spin-up electron is fixed on the outer ring of electrons.The states in the top row are the lowest in energy and degeneratewithin our statistical error /H20851/H208490,3/2 /H20850is the Hund’s rule state /H20852. For
each of these states, the filling of the orbitals in the trial wavefunction is shown. The classical configuration for this Nis 2 elec-
trons in the center with 7 in the outer ring; this configuration isclearly discernible in the pair densities of the higher spin states.INCIPIENT WIGNER LOCALIZATION IN CIRCULAR … PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-11taking the simplest example, a four-electron dot. Figure 11
shows the spin-resolved pair densities for three different low-lying states /H20849L,S/H20850=/H208490,1/H20850,/H208492,0/H20850, and /H208490,0/H20850forr
s/H1101118. The
fixed spin-up electron is at /H9258=0 on the single ring. We find
that the probability of finding the other electrons is maxi-mum at the classical locations—
/H9258=/H9266/2,/H9266, and 3 /H9266/2. But
the spin of the electron at these locations depends cruciallyon the quantum numbers LandSof the state. The ground
state is /H208490,1/H20850/H20849the Hund’s rule state /H20850, for which the uncon-
strained electrons /H20849two up and one down /H20850are equally likely
to occupy any of the remaining three classical positions.
More interesting situations occur for the states /H208492,0/H20850and
/H208490,0/H20850, for which there are an equal number of up and down
electrons and which differ only in total angular momentum.The L=2 state shows clear antiferromagnetic correlations.
The L=0 state is rather unusual: the
/H9258=/H9266position is occu-
pied by a down electron, while the remaining up and downelectron are equally distributed over locations
/H9258=/H9266/2 and
3/H9266/2. We interpret this equal weight of up and down elec-
trons as representing a spin lying in the plane; thus, this statecorresponds to a spin-density wave with wave vector
/H9266/2. It
is interesting to note that the L=0 state has lower energy than
theL=2 state /H20849energies are given in Fig. 11/H20850—the antiferro-
magnetic state is disfavored . The ordering of these two states
is opposite to that in the weak rslimit and is another example
of a violation of Hund’s second rule. How the value of Lis
responsible for the spin correlation /H20849for a given S=0/H20850is not
settled yet and will be pursued in future studies. Similar re-sults were found for other cases of small Nas well; for large
N, there is little angular localization of electrons on each
ring, making any conclusion unreliable. However, notice thatthe unusual spin correlation here results in an unexpectedsurplus of down spins directly opposite the fixed up spin; to
that extent, the feature here for N=4 is similar to that at
/H9258
=/H9266noted for N=20 in Sec. IV B above.
V . CONCLUSIONS
The emerging picture for the correlation-induced inhomo-
geneities in circular dots appears to be quite different fromthat in the bulk. First, the absence of translational symmetryin the radial direction introduces radial localization of theelectrons in rings well before individual electrons localize.This is clearly against the conventional notion of a singletransition or crossover from a weak to strong correlation re-gime. The radial localization can be tracked by the densitydue to the broken translational symmetry; angular localiza-tion is reflected in the pair density, which does not respectthe rotational symmetry of the Hamiltonian. It is clear fromour study that at a given r
s, radial localization is stronger
than angular. This is expected due to the circular geometry.We also note here that the rings in the density could in prin-ciple be directly observed using scanning tunneling micros-copy. In some of the low-density electron gas systems, reso-lution may be limited by the fact that the electrons are buriedbelow the surface.
20Systems have been developed, however,
in which the electron gas is near or at the surface,64–66and it
is these systems which provide the best opportunities forobservation of density rings.
Second, the transition between the weak and strong cor-
relation regimes is surprisingly broad. In fact, the completelysmooth evolution of the FPH with r
ssuggests the absence of
any crossover scale at all. Note that the “smoothness” goesfar beyond the the usual “rounding” of a phase transition in afinite system in which a distinct change of slope occurs in thecrossover region.
Third, the correlation strength depends not only on r
sbut
also on the number of electrons in the dot: it is typicallystronger for smaller N, as evident from our results for FPH,
pair density, and addition energy. In particular, the nature ofthe mesoscopic fluctuations in addition energy is very differ-ent for small and large r
s. While they are determined by the
noninteracting shell effects for small rs, the structure of the
classical configurations dictates the fluctuations for large rs.
Asrsincreases, first the noninteracting fluctuations are
smoothed out, and then the classical fluctuations set in. Ourresults indicate that the value of r
swhere the change takes
place depends on N. Thus for confined systems, the change
from the weak to strong correlation regime is not universal.
Fourth, the ground state spin is consistent with Hund’s
/H20849first/H20850rule throughout the range of our study, 4 /H33355N/H3335520 and
rs/H1135118. At large rs, the excitation energy of certain strongly
spin-polarized states become small but they do not becomethe ground state. It would be interesting to see if these po-larized states become energetically favorable for even largerr
s. The extent of inhomogeneity in the dot shows a clear
dependence on the spin state, with stronger localization oc-curring for larger spin polarization. While the degree of elec-tron localization is insensitive to the angular momentum Lof
the state /H20849for the small Lhere /H20850, the nature of the spin-
correlation /H20849its spatial pattern, for instance /H20850is closely con-
nected to Lin an intriguing manner.E=0.517267(5) E=0.519033(9)E=0.517267(5) E=0.519033(9)E=0.517267(5) E=0.519033(9)E=0.517267(5) E=0.519033(9) E=0.516821(6)E=0.516821(6)E=0.516821(6)E=0.516821(6)L=0L=0L=0L=0
S=0S=0S=0S=0 S=1S=1S=1S=1L=0L=0L=0L=0 L=2L=2L=2L=2
S=0S=0S=0S=0
(32,32)(32,32)(32,32)(32,32)(−32,−32)(−32,−32)(−32,−32)(−32,−32)gggg
gggg
FIG. 11. /H20849Color online /H20850Nature of spin correlation in the three
low-lying states for N=4 at large rs/H20849/H9275=0.01, rs/H1101518/H20850. Top and
middle panels show g↑↓andg↑↑, respectively. Notice that the prob-
ability of finding electrons at a given location depends crucially ontheLandSquantum numbers. The schematic spin correlation is
given in the lower panel. Interestingly, antiferromagnetic correlationoccurs for L=2,S=0 which has the highest energy among these
three states.GHOSAL et al. PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-12These conclusions disagree with much of the previous
literature, the majority of which predicts a single, smallcrossover scale accompanied by significant deviations fromHund’s rule. Because our method yields the lowest energiesfor quantum dots to date /H20849the method is strictly variational /H20850
and involves less severe approximations, we believe thepresent results to be more accurate.
Finally, we close with a comment on the connection be-
tween these results for circular dots and those for the bulktwo-dimensional electron gas. We speculate that with in-creasing N, the deep interior of the dot will behave much the
same as the bulk system while the radial rings will persistnear the boundary, reflecting the circular confinement. Be-cause of the rings, much of the physics discussed here willpersist locally near the boundary.
ACKNOWLEDGMENTS
We would like to thank X. Waintal for valuable conversa-
tions. This work was supported in part by the NSF /H20849Grant
Nos. DMR-0506953 and DMR-0205328 /H20850. A.G. was sup-
ported in part by funds from the David Saxon Chair atUCLA. D.U. and H.U.B. thank the Aspen Center for Physicsfor its hospitality.
APPENDIX: SENSITIVITY OF RESULTS
TO METHODOLOGY
In this appendix, we address two interrelated issues: /H208491/H20850
the difference between VMC, DMC, and QMC results/H20849where the latter denotes results obtained from the extrapo-
lated estimator /H20850, and /H208492/H20850the variation caused by using differ-
ent orbitals in the Slater determinant part of the trial wavefunction. We will see that the latter is a very small effect.With regard to the first issue, we show that the DMC step isdefinitely needed.
We start by comparing VMC and DMC energies. Table III
shows the energies for two cases, the ground state of N=6 at
r
s/H110154.5 /H20849a full shell case /H20850and the ground state of N=9 atrs/H1101516/H20849a half filled shell case /H20850. The improvement brought
about by the DMC step is substantial in the high rscase:
DMC energies for other spin states are given in Table II,
from which we see that the splittings among the low lyingspin and angular momentum states is substantially less thanthe improvement provided by DMC over the VMC energies.Thus, for getting the correct ordering of states in the strongcorrelation regime, the DMC step is critical.
This is further demonstrated in Fig. 12where the frac-
tional peak height /H20849FPH, see Sec. IV A for the definition /H20850
obtained from VMC is compared to the full QMC result fortheN=20 ground state. Interestingly, the VMC result does
show a feature at r
s/H110154, namely a kink in the FPH vs rs
curve, which goes away in the full result. This demonstrates
the insufficient accuracy of VMC in even the modest rs/H113515
regime.67
One may worry at this point whether the extrapolated es-
timator that we use for the density is breaking down, as thereappears to be such a large difference between VMC andTABLE III. Dependence of the ground state energy on the Slater determinant part of the trial wave
function. Three different types of orbitals were used in forming the Slater determinant: orbitals from a DFTcalculation in the local density approximation /H20849LDA /H20850, from a Hartree /H20849H/H20850or Hartree-Fock calculation /H20849HF/H20850,
and from the noninteracting problem /H20849Fock-Darwin orbitals, FD /H20850. The energy after the variational quantum
Monte Carlo /H20849VMC /H20850step and the final diffusion quantum Monte Carlo /H20849DMC /H20850energy are given. N,L, and S
specify the number of electrons in the dot, their angular momentum, and their spin. The energy is in units of/H6036
/H9275, the characteristic energy of the external parabolic confining potential. Note that the type of orbital used
makes little difference in the DMC energy.
N /H9275rs
/H20849approx /H20850 LSOrbital
type VMC DMC
6 0.07 4.5 0 0 LDA 3.02265 /H208499/H20850 3.018020 /H208491/H20850
HF 3.0243 /H208493/H20850 3.01872 /H208491/H20850
FD 3.0229 /H208495/H20850 3.01816 /H208491/H20850
9 0.01 15.8 0 3/2 LDA 146.759 /H208491/H20850 146.4726 /H208496/H20850
H 146.764 /H208491/H20850 146.4755 /H208496/H20850
FD 146.728 /H208491/H20850 146.4761 /H208496/H20850
FIG. 12. Growth of the fractional peak height /H20849FPH /H20850for the N
=20 ground state from the QMC extrapolated estimator /H20849solid sym-
bols /H20850compared to the less accurate VMC results /H20849open symbols /H20850.
Note that the VMC estimate alone shows a break point in the slopenear r
s/H110154. Thus, a DMC calculation, with its better treatment of
the interactions, is necessary to produce the smooth behavior of theFPH as a function of r
s.INCIPIENT WIGNER LOCALIZATION IN CIRCULAR … PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-13DMC for the FPH at rs/H110154. Figure 13addresses this point:
the electron density obtained in both VMC and DMC isshown at the point of maximum deviation in the FPH values.It is clear that the change in density is, in fact, reasonablysmall, and so we expect that the extrapolated estimator willbe accurate.
We now turn to the effect of using different single-particle
orbitals on our result. We have performed calculations usingthree types of orbitals to construct the Slater determinants inthe trial wave function Eq. /H208494/H20850: orbitals from a DFT calcula-
tion within LDA, orbitals from a Hartree or Hartree-Fockcalculation, and orbitals obtained by solving the noninteract-ing problem, the Fock-Darwin states. First, VMC and DMCenergies for these three types of orbitals are shown in TableIIIfor the two cases discussed above. Notice, first, that all
three types of orbitals give very good energies. The DMC
step reduces the spread in the energies. The final energydifferences are much smaller /H20849/H110215%/H20850than the splittings
between the low lying spin and angular momentum states.
The effect of changing the type of orbital is also negli-
gible in the spatially resolved quantities. As an example, weshow in Fig. 14the pair density obtained using LDA orbitals
/H20849our usual procedure /H20850and the difference obtained using Har-
tree or Fock-Darwin orbitals. The difference is clearly verysmall. Thus, for all of the conclusions in this paper, any ofthe three types of orbitals could have been used.
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(b)
(c)
FIG. 14. /H20849Color online /H20850Pair density /H20849total, from extrapolated
estimator /H20850using different types of orbitals in the trial wave func-
tion. The ground state for N=9 at rs/H1101516/H20849/H9275=0.01 /H20850is used. /H20849a/H20850:
LDA orbitals. /H20849b/H20850: Difference between using Hartree and LDA or-
bitals. /H20849c/H20850: Difference between using Fock-Darwin and LDA orbit-
als. All panels use the same horizontal and vertical scales.FIG. 13. Extrapolated estimator of the density /H20849labeled QMC /H20850
compared to the VMC and DMC results for the N=20 ground state
/H20849L=0,S=0/H20850with rs/H110114/H20849/H9275=0.085 /H20850. The change in density produced
by the DMC step is sufficiently small that we expect the extrapo-lated estimator to be accurate.GHOSAL et al. PHYSICAL REVIEW B 76, 085341 /H208492007 /H20850
085341-148D. G. Barci and L. E. Oxman, Phys. Rev. B 67, 205108 /H208492003 /H20850.
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085341-15 |
PhysRevB.85.125314.pdf | PHYSICAL REVIEW B 85, 125314 (2012)
Stationary phase approximation approach to the quasiparticle interference on the surface of
a strong topological insulator
Qin Liu,1,2Xiao-Liang Qi,2and Shou-Cheng Zhang2
1State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology,
CAS, Shanghai 200050, China
2Department of Physics, Stanford University, Stanford, California 94305, USA
(Received 24 August 2011; published 30 March 2012)
Topological insulators have surface states with unique spin-orbit coupling. With impurities on the surface,
the quasiparticle interference pattern is an effective way to reveal the topological nature of the surface states,which can be probed by scanning tunneling microscopy. In this paper, we present a general analytic formulationof the local density of states using the stationary phase approximation. The power laws of Friedel oscillationsare discussed for a constant energy contour with a generic shape. In particular, we predict unique signature ofmagnetic impurities in comparison with nonmagnetic impurities for a surface state trapped in a “magnetic wall.”
DOI: 10.1103/PhysRevB.85.125314 PACS number(s): 68 .37.Ef, 72.25.Dc, 73 .50.Bk, 73 .20.−r
I. INTRODUCTION
Topological insulators in three dimensions (3D) are band
insulators which have a bulk insulating gap and gaplesssurface states with an odd number of Dirac cones protectedby time-reversal symmetry (TRS).
1–3A family of 3D topo-
logical insulators (TI) with a large bulk gap and a singleDirac cone on the surface includes the compounds Bi
2Se3,
Bi2Te3, and Sb 2Te3, which have been theoretically predicted
and experimentally observed.4–7The surface state of these
materials can be described by the effective Dirac HamiltonianH
0=¯hvFˆz·(σ×k) [with k=(kx,ky) the momentum] when
the Fermi level is close to the Dirac point, which behaveslike a massless relativistic Dirac fermion with the spin lockedto its momentum.
8However, compared to the familiar Dirac
fermions in particle physics, those emergent quasiparticlesexhibit richer behaviors. In Bi
2Te3, an unconventional hexag-
onal warping effect appears due to the crystal symmetry,9
which means the constant energy contour (CEC) of the surfaceband evolves from a convex circle to a concave hexagon asthe energy moves away from the Dirac point. Although thetopological property of the surface states is not affected, suchkinds of deformation of the CEC do affect the behavior of thesurface states in the presence of impurities.
Quasiparticle interference (QPI) caused by impurity scat-
tering on the surface of 3D TIs is an effective way to revealthe topological nature of the surface states. The interferencebetween incoming and outgoing waves at momenta k
iand
kfleads to an amplitude modulation of the local density
of state (LDOS) at wave vector q=kf−ki, known as
the Friedel oscillation.10Nowadays, such modulation can
be studied by a powerful surface probe, scanning tunnelingmicroscopy (STM), which directly measures the LDOS. Theinformation in momentum space is obtained through Fouriertransform scanning tunneling spectroscopy (FT-STS). SeveralSTM measurements
11–19have been performed on the surface
of 3D TIs in the presence of nonmagnetic point and edgeimpurities, and the following features are shared in common.(i) The topological suppression of backward scattering fromnonmagnetic point and edge impurities is confirmed bythe observation of strongly damped oscillations in LDOS,together with the invisibility of the corresponding scatteringwave vector qin FT-STS. (ii) Anomalous oscillations are
reported in Bi
2Te3for both point and edge impurities when
the CEC becomes concave. These experimental facts havebeen interpreted theoretically by several groups.
19–24For
short-range point and edge impurities, the Friedel oscillationin an ordinary two-dimensional electron gas (2DEG) has thepower law of R
−1andR−1/2, respectively.25In comparison,
the Friedel oscillation in a helical liquid with a convex CECis dominated by the scattering between time-reversed points(TRP) and is thus suppressed to R
−2andR−3/2for point and
edge impurities separately. This result is the crucial reason ofthe invisibility of the scattering wave vector qin FT-STS, and
is the direct consequence of the suppression of backscatteringprotected by TRS in helical liquid. When the CEC becomesconcave, scattering between wave vectors, which are notconnected by TRP, can have a significant contribution andleads to a slower decay of the Friedel oscillation.
9,13
Motivated by these results, in this work we develop a
general theory of the QPI for a CEC of generic shape using
the stationary phase approximation approach.26This approach
has been applied successfully to the Ruderman-Kittel-Kasuya-Yosida interaction in 3D systems with nonspherical Fermisurfaces.
26In the stationary phase approximation, the long-
distance behavior of the Friedel oscillations is dominatedby the so-called “stationary points” on the CEC. Using this
approach, a complete result of the power-expansion series of
the LDOS and spin LDOS is obtained for both point- andedge-shaped nonmagnetic and magnetic impurities, which wemodel by δ-function potentials. The spin LDOS is the local
spin density at a given energy, which can be measured bya STM experiment with a magnetic tip. Our results dependonly on the TRS and the local geometry around the stationary
points on the CEC, which explain not only the usual R
−1
andR−1/2power laws in 2DEG, but also the R−2andR−3/2
oscillations in the helical liquid. With a generic shape of CEC,
a different power law can be obtained due to the presence ofadditional stationary points aside from the TRP, which canbe used to predict the result of STM and spin-resolved STMexperiments on the surface of other TI materials with more
complicated surface states. An important consequence of our
result is that an ordinary STM measurement can not distinguish
125314-1 1098-0121/2012/85(12)/125314(8) ©2012 American Physical SocietyQIN LIU, XIAO-LIANG QI, AND SHOU-CHENG ZHANG PHYSICAL REVIEW B 85, 125314 (2012)
magnetic and nonmagnetic impurities, although the former can
induce backscattering while the latter can not. To distinguishthe effect of magnetic and nonmagnetic impurities and observebackscattering induced by magnetic impurities, it is necessaryto use a magnetic tip to measure the spin LDOS.
The rest of this paper is organized as follows. In Sec. II,
we introduce an intuitive picture of the interference betweenhelical waves scattered by magnetic impurities. In Sec. III,
we present the general analytic formulation of LDOS forpoint and edge impurities, respectively, by focusing first onthose CEC where the stationary points are extremal points.We then generalize our results to the more generic CECwhere the stationary points are saddle points, with the firstnonzero expansion coefficient occurring at a higher power. Aconclusion and discussion are given in Sec. IV.
II. STANDING WAVE OF THE SPIN INTERFERENCE
BETWEEN TWO HELICAL WA VES
With the presence of TRS, the backscattering by nonmag-
netic impurities is known to be forbidden on the surface of3D TIs due to the πBerry’s phase associated with the full
rotation of electron spin.
27,28In experiments, this manifests in
the invisibility of the scattering wave vector 2 kFin FT-STS.18
It would then be interesting to ask how the surface states
respond differently to magnetic impurities, and what theircharacteristic signatures in STM measurements are. Withmagnetic impurities, naively one would expect to see anontrivial interference pattern since backscattering is alloweddue to the breaking of TRS. However, it turns out that theFriedel oscillation in the charge LDOS, which is measuredin an ordinary STM experiment with a nonmagnetic tip, isstill suppressed in the same way as nonmagnetic impurities.The broken TRS would only manifest itself in the spin LDOSmeasured by a spin-polarized STM tip.
29
To understand this result, we first present a simple picture
of the interference between two counterpropagating helicalwaves on the surface of a 3D TI, and then give a completetheoretical survey in the next section. Consider a magneticedge impurity placed along the xaxis on the surface. For
the effective Hamiltonian H
0=¯hvFˆz·(σ×k), the electron
state propagating along the ydirection perpendicular to the
impurity line has spin polarized to the xdirection, with the
wave function ψ1=1√
2eikFy(1 1)T. Here, the superscript “T”
indicates the transpose. This wave is then backscattered by
the magnetic edge and counterpropagates in the −ydirection.
For the same energy, the state with opposite kmust have
opposite spin, with the wave function ψ2=1√
2e−ikFy(−11 )T.
This situation is illustrated in Fig. 1. A simple calculation
shows that the interference of the two counterpropagatinghelical waves ψ(y)=
1√
2[ψ1(y)+ψ2(y)] leads to a constant
charge LDOS on the surface /angbracketleftρ/angbracketrightψ=|ψ†ψ(y)|=1 since
ψ1andψ2have orthogonal spin. However, the interference
leads to a spiral spin LDOS in the yzplane as /angbracketlefts/angbracketrightψ=
ψ†sψ=[0,−¯h
4sin(2kFy),−¯h
4cos(2kFy)], where s=¯h
2σis
the electron spin operator. Therefore, a STM experiment witha nonmagnetic tip will observe no interference pattern, whileone with a magnetic tip will observe the oscillation of the spindensity of states. Such a contrast between charge and spin
FIG. 1. (Color online) Illustration of charge and spin interference
patterns between two counterpropagating helical waves. The gray
block is a 3D TI with a magnetic edge impurity (green stripe withthick arrows pointing up) lying along the xaxis on the surface. An
incident helical wave along the ydirection with spin polarized in the
xdirection (the blue dashed line) is backscattered by the magnetic
edge and the spin is flipped (the red solid line). The interference of
the two orthogonal helical waves leads to a constant LDOS in the
charge channel, but a spiral LDOS in the spin channel (purple arrowsbetween the solid and dashed lines) in the yzplane.
density of states is a unique signature of the helical liquid,
which is a direct demonstration of the locking between spinand momentum.
To observe such a spin interference pattern, a more
convenient setup is a closed “magnetic wall” as shown inFig. 2. Consider a magnetic layer deposited everywhere on
the 3D TI surface except a hole in the middle with the diskshape. The magnetic layer can open a gap on the surface statesuch that the low-energy surface states are trapped in the holeregion and form a standing wave. Similar to the straight-linemagnetic impurity discussed above, the standing wave trappedby the magnetic barrier can be obtained by setting the boundarycondition of fixed spin at the boundary of the hole. For largeR(R/greatermuch1/k
F), the spin density of the standing wave has
the behavior of /angbracketleftsR/angbracketright∼sin(2kFR)√
R,/angbracketleftsz/angbracketright∼cos(2kFR)√
R,w i t h Rand
zstanding for longitudinal and perpendicular directions in a
spherical coordinate. A unique property of the helical surfacestates is the spin-charge locking.
8For the effective Hamilto-
nianH0=¯hvFˆz·(σ×k), the electric current operator in the
long-wavelength limit is j=∇kH0=¯hvFˆz×σ. Therefore,
there is a loop charge current jφ=−2evF/angbracketleftsR/angbracketrightalong the
azimuthal direction associated with the spin density.
FIG. 2. (Color online) Standing wave of spin interference be-
tween two helical waves inside a closed “magnetic wall” on top of a
3D TI surface. The magnetic wall is surrounded by a magnetic layer
deposited on top of the 3D TI surface, which opens a gap in the helicalsurface states and plays the role of a barrier. The out-of-plane spin
LDOS is exhibited by the colored (dark and bright) rings, and the
in-plane spin LDOS is indicated by the black arrows.
125314-2STATIONARY PHASE APPROXIMATION APPROACH TO ... PHYSICAL REVIEW B 85, 125314 (2012)
III. GENERAL FORMULATION OF STATIONARY PHASE
APPROXIMATION APPROACH TO QPI ON
THE SURFACE OF 3D TI
In this section, we obtain the general long-distance features
of charge and spin LDOS on the surface of a generic 3DTI induced by nonmagnetic and magnetic impurities usingthe stationary phase approximation method.
26We study both
pointlike and edgelike impurities. We shall focus first on thebehavior of a special kind of CEC where the stationary pointsare extremal points, and then generalize our results to genericCEC with higher-order nesting points.
A. Point impurity
We start by considering a point defect on the surface of a
3D TI. The Hamiltonian with a single impurity is
H=/integraldisplay
d2rψ†(r)[h0(k)+Vσμδ(r)]ψ(r), (1)
where σμ=1f o rμ=0 andσa=σx,y,zare the Pauli matrices
fora=1,2,3.k=−i∇is the momentum operator. For such
a potential, the LDOS can be expressed exactly. Using the σμ
matrices, the charge and spin LDOS are combined to the form
ρν(ω,R)=−1
πIm{tr[Gr(ω,R,R)σν]}, (2)
withGr(ω,R,R/prime) being the retarded Green’s function in real
space. Let ρν0(ω) be the LDOS of the unperturbed system with
V=0; the deviation of the LDOS from the background value
ρν0(ω) is then given by
δρμν(ω,R)≡ρν(ω,R)−ρν0(ω)
=−1
πIm/integraldisplayd2kd2k/prime
(2π)4ei(k−k/prime)·R
×tr/bracketleftbig
Gr
0(ω,k)Tμ(ω,k,k/prime)Gr
0(ω,k/prime)σν/bracketrightbig
.(3)
Here, Gr
0(ω,k) is the free retarded Green’s function gov-
erning the CEC under consideration. For the topologicalsurface states, G
r
0(ω,k)=[ω+iδ−h0(k)]−1.T h eTmatrix
Tμ(ω,k,k/prime) is defined by
Tμ(ω)=Vσμ/bracketleftbig
1−VσμGr
0(ω)/bracketrightbig−1, (4)
which is momentum independent when the impurity has a δ-
function potential, and we have denoted the real-space Green’s
function Gr
0(ω)=/integraltextd2k
(2π)2Gr
0(ω,k). As is required by the TRS,
Gr
0(ω) is always proportional to the identity matrix.
We first note that the spin LDOS induced by a nonmagnetic
impurity vanishes uniformly, i.e., δρ0a≡0f o r a=1,2,3.
This is a direct consequence of TRS because, under time-
reversal transformation /Theta1=iσy,w eh a v e /Theta1−1σa/Theta1=−σaT
and/Theta1−1Gr
0,k/Theta1=GrT
0,−k; then, the trace in Eq. (3)satisfies
tr[Gr
0,kT0Gr
0,k/primeσa]=−tr[Gr
0,−k/primeT0Gr
0,−kσa], where we have
abbreviated Gr
0(ω,k)≡Gr
0,k. By interchanging kand−k/primein
the integral in Eq. (3), one is led to the result δρ0a(ω,R)=0.
To obtain other components of the Tmatrix, we expandtheTmatrix into spin-dependent and spin-independent
parts as
Ta=Ta
aσa+Ta
0,Ta
a=V
1−V2Gr2
0(ω),
(5)
Ta
0=V2Gr
0(ω)
1−V2Gr2
0(ω),T0=V
1−VGr
0(ω),
where the fact that Gr
0(ω) is proportional to identity has been
used, and no summation over repeated indices is impliedthroughout the paper. Similar to the argument in the δρ
0a
case, we see that the contribution of Ta
ato the charge LDOS
of a magnetic impurity δρa0vanishes. Hence, we have
δρa0/δρ 00=Ta
0/T0. Therefore, in the following, we shall
focus only on δρ00andδρab.
To proceed, the measured LDOS in Eq. (3)is then
rewritten in the diagonal basis of the topological surface
bands. We define the unitary matrices Uksuch that U†
kh0(k)Uk
diagonalizes h0(k), and Eq. (3)becomes
δρμν(ω,R)=−1
πIm/integraldisplayd2kd2k/prime
(2π)4ei(k−k/prime)·Rtr/bracketleftbig/parenleftbig
U†
kGr
0,kUk/parenrightbig
×(U†
kTμUk/prime)(U†
k/primeGr
0,k/primeUk/prime)(U†
k/primeσνUk)/bracketrightbig
(6)
=−1
πIm/integraldisplayd2kd2k/prime
(2π)4ei(k−k/prime)·R
×/summationdisplay
nmγμν
nm(k,k/prime)/Sigma1ν∗
nm(k,k/prime)
(ω+iδ−εn)(ω+iδ−ε/primem), (7)
where εn(m)(k) are the energy eigenvalues of the bands |n(m)k/angbracketright,
and we have defined /Sigma1μ
nm(k,k/prime)=/angbracketleftnk|σμ|mk/prime/angbracketright,a sw e l la s
γμν
nm(k,k/prime)=/braceleftbiggT0/Sigma10
nm,μ =ν=0
Ta
a/Sigma1a
nm+Ta
0/Sigma10
nm,μ=a,ν=b.(8)
Following the standard process of density of states
calculations,26the integrations over kandk/primeare then converted
into coordinates dk=(dk⊥,dkφ)a s
δρμν(ω,R)=−1
πIm/contintegraldisplaydkφdk/prime
φei(k−k/prime)·R
(2π)4/integraldisplaydεndε/prime
m
|∇⊥εn∇/prime
⊥ε/primem|
×/summationdisplay
nmγμν
nm(k,k/prime)/Sigma1ν∗
nm(k,k/prime)
(ω+iδ−εn)(ω+iδ−ε/primem), (9)
where k⊥andkφare components of knormal and tangential
to the CEC, respectively.
To evaluate the loop integrals along the CEC, it is essential
to introduce the stationary phase approximation. For example,consider the LDOS at a point R=Rˆy(here and hereafter we
shall always take the ydirection for example), the phase factor
e
i(k−k/prime)·R=ei(ky−k/prime
y)R. Locally, one can write ky=ky(ε,kx)a s
a function of energy εandkx. For large distance Rfrom
the impurity, the phase factors eiky(ε,kx)Rande−ik/prime
y(ε/prime,k/prime
x)Rvary
rapidly with respect to kxandk/prime
xfor almost every point on the
CEC, so that most of the integrations cancel out exactly exceptfor the stationary points k
i,26which satisfy the condition
∂ky(ε,kx)
∂kx=∂k/prime
y(ε/prime,k/prime
x)
∂k/primex=0. (10)
125314-3QIN LIU, XIAO-LIANG QI, AND SHOU-CHENG ZHANG PHYSICAL REVIEW B 85, 125314 (2012)
q1q3
q1q1q2′
q2KM (b) (a) (c)
FIG. 3. (Color online) Schematic picture of CEC and stationary
points for point and edge impurities. (a) Convex CEC where there
is only one pair of stationary points connected by wavevector q1
(the red solid arrow) along any given direction for both point and
line impurities. (b) Concave CEC for point impurity where there are
multiple pairs of stationary points. Examples of nonstationary points
are shown by q2andq3(blue thick solid arrows). (c) The concave
CEC for edge impurities (brown shaded line) where the slopes (green
dashed lines) at the pair of stationary points are the same.
The stationary points defined above include (i) extremal points
such as the pairs connected by q1in Figs. 3(a)and3(b), where
the second derivative ∂2ky/∂k2
xis nonvanished; (ii) the turning
points such as the pair connected by q/prime
2in Fig. 3(b), where
the second derivative also vanishes. In the following, we firstfocus only on the extremal points, and leave the more generaldiscussions to Sec. III C.
Having identified the pairs of stationary points on the CEC
in direction R, the loop integrals in Eq. (9)at large distances
are then approximated by the summation of integrals in theneighborhood of all the stationary-point pairs, which is theessence of the method of the stationary phase approximation.To start with, we first change the integral variables as d
2k=
dεdk x/¯h|vyi|, where vyi=∂ε(k)/¯h∂kyi, and then expand the
CEC at the extremal points as ky=kyi−(kx−kxi)2/2ρxi,
where ρxi=−[∂2kyi(ε,kx)/∂2kxi]−1are the principal radii
of curvature of the CEC at the extremal points, which arepositive for maxima while negative for minima. Under thisapproximation, Eq. (9)becomes
δρ
μν(ω,R)/similarequal−1
πIm/summationdisplay
mn/summationdisplay
ij/integraldisplaydεn
(2π)21
ω+iδ−εneikyiR
¯h|vyi|
×/integraldisplaydε/prime
m
(2π)21
ω+iδ−ε/primeme−ik/prime
yjR
¯h|v/prime
yj|
×/integraldisplay∞
−∞dx e−ix2
2ρxiR/integraldisplay∞
−∞dx/primeeix/prime2
2ρ/prime
xjR
×γμν
nm(k,k/prime)/Sigma1ν∗
nm(k,k/prime), (11)
where we have denoted x=kx−kxi,x/prime=k/prime
x−k/prime
xj, and all
the quantities at the extremal points ( ij) still depend on
the energies εandε/prime. Now, the matrix element /Sigma1μ
nm(k,k/prime)
at the extremal points is in general some nonzero constantCμ
ni,mj(ε,ε/prime), except that it vanishes when μ=0 and the pair of
stationary points are time-reversal partners |nki/angbracketright=/Theta1ˆK|mk/prime
j/angbracketright.
Here, ˆKis the complex-conjugation operator. Examples are
shown as the pairs of stationary points connected by q1’s in
Figs. 3(a)and3(b) for convex and concave CEC, respectively.
To obtain the generic behavior of the interference pattern,the matrix element is expanded in the distance x,x
/primeto
the stationary points as /Sigma1μ
nm(x,x/prime)=Cμ
ni,mj+ax+a/primex/prime+o(x)+o(x/prime), where Cμ
ni,mj=0f o rμ=0 at TRP, and a nonva-
nishing but energy-dependent constant otherwise. Inserting theseries into Eq. (11), one can integrate first over xandx
/prime
by using the relations/integraltext∞
−∞dxeiCx2=√π/|C|eiπ
4sgn(C)and/integraltext∞
−∞dxx2eiCx2=√π/(2|C|3/2)e−iπ
4sgn(C), and then integrate
over the energies using the residue theorem by summation overthe integrand at the poles ε=ε
/prime=ω+iδ. Finally, by taking
the limit ω=εF,δ→0+, we get
δρμν(ω,R)/similarequal1
2π2¯h2RIm/summationdisplay
mn/summationdisplay
ijei(kyi−k/prime
yj)R|ρxiρ/prime
xj|1
2
|vyiv/prime
yj|
×/bracketleftBigg
ei(φi−φ/prime
j)/summationdisplay
sTμ
sCs
ni,mjCν∗
ni,mj+1
R
×(a/prime2ei(φi+φ/prime
j)/vextendsingle/vextendsingleρ/prime
xj/vextendsingle/vextendsingle+a2e−i(φi+φ/prime
j)|ρxi|)/bracketrightBigg
εF,
(12)
where φi=−π
4sgn(ρxi). This is the long-wavelength behavior
of LDOS induced by a point impurity. In the above result, wehaves=0 and T
0
0≡T0=V/[1−VGr
0(ω)] for the charge
LDOS of a nonmagnetic impurity δρ00. While for the spin
LDOS of a magnetic impurity δρab, the summation is over s=
a,0, where Ta
aandTa
0are, respectively, the spin-dependent
and spin-independent coefficients in the T-matrix expansion
introduced above.
There are several comments regarding this result. First, for
a pair of non-TRS stationary points such as q2in Fig. 3(b),
the leading power is given by the first term in Eq. (12), which
is ofR−1. While for a pair of TRS stationary points as q1
in Figs. 3(a) and3(b), the first nonvanishing contribution to
the power law is dominated by the second term in Eq. (12)
asR−2for nonmagnetic impurity, and for magnetic impurity
with ordinary tip. Such suppression of LDOS is a directconsequence of the absence of backscattering of helical wavesdue to TRS. Correspondingly, in the Fourier transform ofLDOS, there is a sharp peak at k=2k
Ffor the R−1power
law, which is absent for the R−2power law, as shown in Fig. 4.
For magnetic impurities with spin-polarized tip, the first termin Eq. (12) dominates no matter whether the pair of stationary
0 0.5 1 1.5 2 2.5 3 3.5 4−10123456
|k|/kFFourier transformation
FT of cos(2kFR)/R
FT of sin(2kFR)/R2
FIG. 4. (Color online) Fourier transformation of the LDOS with
R−1andR−2power laws.
125314-4STATIONARY PHASE APPROXIMATION APPROACH TO ... PHYSICAL REVIEW B 85, 125314 (2012)
TABLE I. Power laws of Friedel oscillations for point impurity.
Charge LDOS Spin LDOS
Nonmagnetic TRP R−2
Non-TRP R−1
Magnetic TRP R−2R−1
Non-TRP R−1R−1
points is TRS or not (due to the contribution of the s=a
term), which gives the visibility of the TRS scattering wavevector q
1. This distinct response of surface states to magnetic
impurities from that of nonmagnetic impurities provides acrucial criteria for the breaking of TRS on the surface ofTIs.
11Second, in the discussion above, we have assumed the
matrix element /Sigma1μ
nmto be nonzero if it is not forbidden by
time-reversal symmetry. There may be some other reasons forthe matrix element to vanish. For example, the states at twoTRS stationary points have opposite spin. If the impurity spinhappens to be parallel (or antiparallel) to their spin, the matrixelement /Sigma1
μ
nmcan vanish. For non-TRS stationary points, this
may occur accidentally, but generically the spin of the twostates nandmis not parallel, so that the matrix element
is nonvanished for any impurity spin. Since such zeros ofmatrix elements are at most only realized for some particulardirections of the impurity spin, in the following we will focuson the generic cases with nonzero matrix element as long asit is not forbidden by time-reversal symmetry. Third, in theintegral over energy, we have assumed v
yi,v/prime
yj/negationslash=0s ot h a tt h e
only poles in the complex energy plane are ε=ε/prime=ω+iδ.
However, in general, it is possible that there are other polesfromv
yi=0o rv/prime
yj=0, which means the stationary points
in CEC are also saddle points in the energy-momentumdispersion. In that case, we shall further expand v
yi(or
v/prime
yj) around ωasvyi(ε)=vyi(ω)+(∂vyi/∂ε)(ε−ω)+··· ,
and keep the first nonzero term. This will not modify thepower laws in spatial dependence.
30Finally, note that when
summation over the stationary-point pairs ( ij), we always
choose the pair such that one point has positive velocity vyi
and the other has negative velocity v/prime
yj. As a summary of the
discussion above, the power laws of LDOS for point impurityare concluded in Table I.
To provide further intuition on the result (12), we consider
some simple examples. The first example is a 2DES withoutspin-orbit coupling described by the familiar HamiltonianH
Q=¯h2k2/2m, which has two degenerate and isotropic Fermi
surfaces, as shown in Fig. 5(a). According to our theory,
the main contribution to the LDOS in this example comesfrom the intraband scattering of the same spin orientationbetween two extremal points, which we denote as “1” and“2.” At these points, we have k
y2=ρx2=kε,k/prime
y1=ρ/prime
x1=
−kε/prime,kε=(2mε/¯h2)1/2,vy2=¯hky2/m,v/prime
y1=¯hk/prime
y1/m, and
C0
11=C0
22=1. By inserting these quantities into Eq. (12)and
keeping only to the first-order expansion of the Tmatrix,
we get δρ(1)
00(ω,R ˆy)/similarequal−(Vm2/π2¯h4q) cos(2 qR)/R, which
hasR−1power law. Note that the interband contribution to
the LDOS in this example is from a pair of TRS extremalpoints, which has a R
−2power law. In contrast, in theFIG. 5. (Color online) Schematic CEC of (a) quadratic, (b) Dirac,
and (c) Rashba dispersions. The spin orientations for each degenerate
band are indicated respectively by the green (solid) and purple
(dotted) arrows. The stationary points are represented by red dotsand blue triangles respectively, which are connected by the scattering
vector qshown as dashed arrows. The intraband scattering occurs
between the stationary points with the same color (shape), whilethe interband scattering occurs between those with different colors
(shapes).
example of a 2D Dirac CEC, HD=γˆz·(σ×k), there is
only one nondegenerate band at a given energy due to thespin splitting, as shown in Fig. 5(b). Thus, only intraband
scattering between a pair of extremal TRP contributes tothe LDOS, and C
0
ni,mj=0. By inserting the quantities ky2=
ρx2=ε/γ,k/prime
y1=ρ/prime
x1=−ε/γ, and vy1(2)=γsgn[ky1(2)]/¯h
into Eq. (12), we get δρ(1)
00(ω,R ˆy)/similarequal(V/4π2γ2)s i n ( 2 qR)/R2,
which is consistent with our expectation.
In a recent STM measurement of the TI Bi 2Te3doped
with Ag,11clear standing waves and scattering wave vectors
are imaged through FT-STS when the Fermi surface is ofhexagram shape. It is observed that the high-intensity regionsare always along the ¯/Gamma1-¯Mdirection, but the intensity in
the¯/Gamma1-¯Kdirection vanishes. This observation can be well
understood using our stationary phase approximation theory.Among the wave vectors q
1,q2,q/prime
2, andq3shown in Fig. 3(b),
q1andq/prime
2correspond to scattering between stationary points,
while q3andq2do not. This explains why no standing waves
corresponding to q3are observed in FT-STS. Within the other
two, stationary points connected by q1are also TRP, which
shall contribute the power law of R−2according to our result.
Therefore, its intensity in FT-STS is too weak to be observed.For wave vectors q
2andq/prime
2along the ¯/Gamma1-¯Mdirection, q/prime
2is
stationary but non-TRS. Our result shows that this wave vectorcontributes an R
−1power law, which is responsible for the high
intensity reported in Ref. 11.
B. Edge impurity
Aside from point impurities, a one-dimensional line defect
in the form of step edge has also been observed on the surfaceof 3D TI.
13,17Magnetic edge defects can possibly be realized
by depositing a magnetic layer on top of a 3D TI. In thissection, we discuss the interference patterns of electronicwaves induced by magnetic and nonmagnetic edge defects.
We consider an edge defect along the the xdirection on top
of a 3D TI surface with the Hamiltonian V(r)=Vδ(y)σ
μ.A
magnetic edge defect has been illustrated in Fig. 1.T h em a i n
difference between an edge defect and a point defect is themomentum conservation along the edge impurity orientation,which means one of the loop integrations in Eq. (9)should be
125314-5QIN LIU, XIAO-LIANG QI, AND SHOU-CHENG ZHANG PHYSICAL REVIEW B 85, 125314 (2012)
removed. Following similar calculations as performed in the
case of a point impurity, the LDOS for the edge impurity isgiven by
δρ
μν(ω,R)=−1
πIm/integraldisplayd2kd2k/prime
(2π)4δkx,k/primexei(k−k/prime)·R
×tr/bracketleftbig
Gr
0(ω,k)Tμ(ω,kx)Gr
0(ω,k/prime)σν/bracketrightbig
,(13)
where Tμ(ω,kx)=Vσμ/[1−VσμGr
0(ω,kx)] with
Gr
0(ω,kx)=/integraltextdky
2πGr
0(ω,k). Similar to the case of a
magnetic point impurity, the Tmatrix for a magnetic edge
impurity can again be separated into a spin-dependent and aspin-independent term. However, in the following discussion,we shall keep only to the first-order expansion of the T
matrix Vσ
μ, which is spin dependent. This simplification is
appropriate for the weak impurity potential, and it will notaffect the qualitative conclusion of the Friedel oscillationpower laws, as we have learned from the case of pointimpurities.
In the presence of edge impurity, we are usually interested
in the LDOS in the direction perpendicular to the edgeorientation. Similar to the case of point impurity, the LDOSin Eq. (13) is first transformed into the diagonal basis of the
topological surface bands, and then converted into integrationsover normal and tangential components as in Eq. (9).B yu s i n g
the stationary phase approximation, now the main contributionto the loop integrals comes from such stationary points wheretheir momentum transfer qis normal to the edge orientation,
and the “slopes” of CEC at the two stationary points are thesame:
∂
∂kx[ky(ε,kx)−k/prime
y(ε/prime,kx)]=0. (14)
Compared with the stationary-point condition for point impu-
rity, the condition for edge impurity allows more possibilities.One such example is shown schematically as q
1in Fig. 3(c)
where the pair of stationary points has the same nonvanishedslope. Such a pair of scattering end points is not consideredas stationary points in the case of point impurities, but arestationary for edge impurities. Following the same logic as thediscussion of point impurity in the last section, the CEC is thenexpanded around the stationary points as k
y=kyi+αi(kx−
kxi)−(kx−kxi)2/2ρxi, and the LDOS is approximated by
δρ(1)
μν(ω,R)/similarequal−V
πIm/summationdisplay
mn/summationdisplay
ij/integraldisplaydεn
(2π)21
ω+iδ−εneikyiR
¯h|vyi|
×/integraldisplaydε/prime
m
(2π)21
ω+iδ−ε/primeme−ik/prime
yjR
¯h|v/prime
yj|
×/integraldisplay∞
−∞dxe−ix2
2ρxiR/integraldisplay∞
−∞dx/primeeix/prime2
2ρ/prime
xjReiαi(x−x/prime)δx,x/prime
×/bracketleftbig
Cμ
ni,mjCν∗
ni,mj+(ax+a/primex/prime)2/bracketrightbig
. (15)
Although Eq. (15) looks similar to Eq. (11) in the point
impurity case, the definition of stationary points for theedge impurity in Eq. (14) is quite different from that of the
point impurity. Therefore, a lot more terms should be includedin the summation of stationary-point pairs ( ij) here compared
with the point impurity case. By integrating out x(x
/prime) andTABLE II. Power laws of Friedel oscillations for edge impurity.
Ordinary Spin polarized
Nonmagnetic TRP R−3/2
Non-TRP R−1/2
Magnetic TRP R−3/2R−1/2
Non-TRP R−1/2R−1/2
energy variables, we get
δρ(1)
μν(ω,R)/similarequalV
(2π)2¯h2/radicalbigg
2
πRIm/summationdisplay
mn/summationdisplay
ij|Pij|1/2
|vyiv/prime
yj|
×ei(kyi−k/prime
yj)R/bracketleftbig
Cμ
ni,mjCν∗
ni,mjei/Phi1ij
+e−i/Phi1ij(a+a/prime)2Pij/R/bracketrightbig
εF, (16)
where Pij=ρxiρ/prime
xj/(ρ/prime
xj−ρxi) and/Phi1ij=−π
4sgn(Pij). In the
equation above, we have assumed vyi,v/prime
yj/negationslash=0 andρxi/negationslash=ρ/prime
xj.
In other words, this result is not applicable to the case wherethe CEC near the pair of stationary points is nested to thesecond-order expansion. If such nesting happens, the quadraticterms in the expansion of CEC near the stationary points cancelout exactly, and higher-order expansion should be employed.The power laws of Friedel oscillations for edge impurity aresummarized in Table II, which shall be used to explain the
STM measurements about edge impurities.
13,17
To have a feeling of how Eq. (16) works explicitly, again
we apply it to the examples of the 2DEG HamiltonianH
Qand 2D Dirac Hamiltonian HDdiscussed previously. A
few lines of calculations yield that for 2D quadratic disper-
sion,δρ(1)
00(ω,R ˆy)=(Vm2/2π2¯h4q3/2)s i n ( 2 qR−π
4)/√
πR,
which is consistent with the experimental observation
in 2DEG.25For the 2D Dirac fermion, δρ(1)
00(ω,R ˆy)=
(V/8π2γ2√πq)s i n ( 2 qR+π
4)/R3/2, which is a consequence
of the absence of backscattering in the helical liquid. Informa-tion in reciprocal space can be extracted via FT-STS similarlyto the point-impurity case exhibited in Fig. 4, where a notable
sharp peak is present at k=2k
Ffor a 2DEG, but is absent for
the helical liquid.
In an experiment by Gomes et al. , a nonmagnetic step is
imaged by STM topography in the Sb (111) surface.17The
Fermi surface consists of one electron pocket at ¯/Gamma1surrounded
by six hole pockets in the ¯/Gamma1-¯Mdirection, where the surface
dispersion has a Rashba spin splitting. The measured LDOSin the ¯/Gamma1-¯Mdirection is fitted by a single qparameter using
the zeroth order of Bessel function of the first kind [seeFig. 2(c) in Ref. 17], which agrees exactly with our result
in Table II. Along the ¯/Gamma1-¯Mdirection, the surface band can
be modeled by a Rashba Hamiltonian where the LDOS isdominated by interband scattering between a pair of non-TRSstationary points, as shown in Fig. 5(c). According to our
analysis, the Friedel oscillation has R
−1/2power law, which
is the asymptotic expansion of J0(qR) at large distances.
Another STM experiment studying the edge impurity byAlpichshev et al.
13is in Bi 2Te3where hexagonal warping
effect exists, and a nonmagnetic step defect is observed on acrystal surface. A strongly damped oscillation is reported whenthe bias voltage is at the energy with a convex Fermi surface
125314-6STATIONARY PHASE APPROXIMATION APPROACH TO ... PHYSICAL REVIEW B 85, 125314 (2012)
as shown in Fig. 3(a). Although no fitting of the experimental
data is estimated in this region, our results predict a R−3/2
power law. Pronounced oscillations at higher bias voltages
where the hexagon warping effect emerges are observed withR
−1fitting. Despite the quantitative difference with our result
ofR−1/2,t h i sR−1oscillation has been explained in several
other works20,21beyond our simple model.
The results summarized in Tables IandIIprovide a quanti-
tative description of the QPI by magnetic impurities in general,which include the interference between two orthogonal helicalwaves discussed in Sec. IIas a particular case. The interference
of helical waves corresponds to the scattering between twoTRS stationary points, such as the q
1’s in Figs. 3(a)–3(c).T h e
interesting thing is that the LDOS in charge and spin channelsfrom the very same pair of TRS stationary points have quitedistinct behavior. With magnetic impurities, the power laws ofcharge LDOS are R
−2andR−3/2for point and edge impurities,
respectively. As a result of TRS, the charge LDOS has higherpower indices than the R
−1andR−1/2modulations of the
corresponding spin-polarized LDOS, which manifests the TRSbreaking. To distinguish the response of topological surfacestates to magnetic impurities from that of the nonmagneticimpurities,
11spin-resolved STM experiments are essential.
C. Friedel oscillations for CEC with generic shape
In this section, we generalize the results obtained above and
obtain the most general formulation of the QPI on the surfaceof a 3D TI. In the discussion of point impurity in Sec. III A ,w e
have focused on the case of extremal points, around which theexpansion of the CEC has nonvanishing second derivatives.However, it is in general also possible that the principal radiiof the curvature of the CEC at the stationary points ρ
xidiverge
so that the third- or even higher-order expansions of the CEC atthe stationary points should be employed. For example, whenthe stationary points are also turning points on the CEC [seeq
/prime
2in Fig. 3(b)], the expansion of the CEC should be kept
to the third order. In the case of edge impurity presented inSec. III B, it is possible that ρ
xi,ρ/prime
xj/negationslash=0, butρxi=ρ/prime
xjso that
Pijdiverges. This happens when the CEC near the stationary
points is highly nested, and we need to go beyond the quadraticexpansion of the CEC until the first power at which the twosegments of the CEC are not nested.
To understand the LDOS behavior in ordinary and spin-
resolved STM experiments in these most general situations,we assume in general that the first nonvanishing coefficientsin the expansion of the CEC around the stationary pointshave the order landh, respectively, where l,h∈Na r e
generically different. Then, k
y(ε,kx) and k/prime
y(ε/prime,k/prime
x)o nt h e
CEC are expanded around the stationary points separately
asky=kyi+β(l)
i(kx−kxi)landk/prime
y=k/prime
yj+β/prime(h)
j(k/prime
x−k/prime
xj)h,
where the β’s are the first nonzero expansion coefficients
withβ(l)
i=(∂lky/∂kl
xi)/l! and similarly for β/prime(h)
j. Notice that
in the case of edge impurity, if l=h, one more constraint
β(l)
i/negationslash=β/prime(h)
jshould be further imposed on the expansion to
obtain a meaningful LDOS. Having analyzed the propertiesof the stationary points on the CEC, the same calculationprocedures as performed in Secs. III A andIII B for point and
edge impurities can be carried out in a straightforward way,which leads to the following most general results for pointTABLE III. General power laws of Friedel oscillations for point
impurity.
Ordinary Spin polarized
Nonmagnetic TRP R−(1
l+1
h)−2
min(l,h)
Non-TRP R−(1
l+1
h)
Magnetic TRP R−(1
l+1
h)−2
min(l,h) R−(1
l+1
h)
Non-TRP R−(1
l+1
h)R−(1
l+1
h)
impurity
ρ(1)
μν(ω,R)∝V
R1
l+1
hIm/summationdisplay
mn/summationdisplay
ij/braceleftBigg
ei(kyi−k/prime
yj)R
|vyiv/prime
yj|/vextendsingle/vextendsingleβ(l)
xi/vextendsingle/vextendsingle1
l/vextendsingle/vextendsingleβ/prime(h)
xj/vextendsingle/vextendsingle1
h
×/bracketleftBigg
Cμ
ni,mjCν∗
ni,mj+a2
/vextendsingle/vextendsingleβ(l)
xi/vextendsingle/vextendsingle2
lR2
l+a/prime2
/vextendsingle/vextendsingleβ/prime(h)
xj/vextendsingle/vextendsingle2
hR2
h/bracketrightBigg/bracerightBigg
εF,
(17)
and for edge impurity
ρ(1)
μν(ω,R)∝V
R1
max(l,h)Im/summationdisplay
mn/summationdisplay
ij/braceleftbiggei(kyi−k/prime
yj)R
|vyiv/prime
yj|
×/vextendsingle/vextendsingleβ(l)
xi−β/prime(h)
xj/vextendsingle/vextendsingle−1
max(l,h)/bracketleftbigg
Cμ
ni,mjCν∗
ni,mj
+(a+a/prime)2/parenleftbig
R/vextendsingle/vextendsingleβ(l)
xi−β/prime(h)
xj/vextendsingle/vextendsingle/parenrightbig−2
max(l,h)/bracketrightbigg/bracerightbigg
εF.(18)
These two equations complete the key results in this work. In
the above, we have used the notation min( l,h) and max( l,h)t o
represent taking the minimum or the maximum one betweenlandh. The corresponding power laws of the Friedel
oscillations in these most general cases are summarized inTables IIIand IV. We see that by taking l=h=2, these
results recover those exhibited in Tables IandIIobtained in
the last two sections.
IV. CONCLUSIONS
In conclusion, long-distance asymptotic behavior of the
LDOS for nonmagnetic and magnetic, point, and edge impu-rities on a generic shape CEC are derived in Eqs. (12) and
(16)–(18) using the stationary phase approximation approach.
The corresponding power laws of Friedel oscillations aresummarized in Tables I–IV. The QPI induced by surface
magnetic impurities is studied, in particular, to illustratethe fact that the interference patterns of charge intensities
TABLE IV . General power laws of Friedel oscillations for edge
impurity.
Ordinary Spin polarized
Nonmagnetic TRP R−3
max(l,h)
Non-TRP R−1
max(l,h)
Magnetic TRP R−3
max(l,h) R−1
max(l,h)
Non-TRP R−1
max(l,h) R−1
max(l,h)
125314-7QIN LIU, XIAO-LIANG QI, AND SHOU-CHENG ZHANG PHYSICAL REVIEW B 85, 125314 (2012)
are indistinguishable from those of nonmagnetic impurities,
while the spin LDOS shows distinct behavior from those ofnonmagnetic impurities. We propose a closed “magnetic wall”geometry, which manifests such a unique interference propertyof helical liquids. These results depend only on the TRS aswell as the local geometry around the stationary points on theCEC, which provide a systematic tool for the analysis of STMexperiments for generic surface states.ACKNOWLEDGMENTS
Q. Liu is supported by NKBRPC (Grant No.
2012CB927401), NSFC (Grants No. 11004212, No.11174309, and No. 60938004), and the STCSM (Grants No.11ZR1443800 and No. 11JC1414500). X. L. Qi and S. C.Zhang are supported by the Department of Energy, Office ofBasic Energy Sciences, Division of Materials Sciences andEngineering, under Contract No. DE-AC02-76SF00515.
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125314-8 |
PhysRevB.102.235428.pdf | PHYSICAL REVIEW B 102, 235428 (2020)
Phonon softening near topological phase transitions
Shengying Yue ,1Bowen Deng,1Yanming Liu ,1,2Yujie Quan,1Runqing Yang,1and Bolin Liao1,*
1Department of Mechanical Engineering, University of California, Santa Barbara, California 93106, USA
2School of Aerospace Engineering, Tsinghua University, Beijing 100084, China
(Received 27 September 2020; revised 1 December 2020; accepted 3 December 2020; published 21 December 2020)
Topological phase transitions occur when the electronic bands change their topological properties, typically
featuring the closing of the band gap. While the influence of topological phase transitions on electronic andoptical properties has been extensively studied, its implication on phononic properties and thermal transportremains unexplored. In this paper, we use first-principles simulations to show that certain phonon modes aresignificantly softened near topological phase transitions, leading to increased phonon-phonon scattering andreduced lattice thermal conductivity. We demonstrate this effect using two model systems: pressure induced topo-logical phase transition in ZrTe
5and chemical composition induced topological phase transition in Hg 1−xCdxTe.
We attribute the phonon softening to emergent Kohn anomalies associated with the closing of the band gap. Ourpaper reveals the strong connection between electronic band structures and lattice instabilities, and opens up apotential direction towards controlling heat conduction in solids.
DOI: 10.1103/PhysRevB.102.235428
I. INTRODUCTION
Topological materials possess electronic bands with non-
trivial topological indices [ 1]. Depending on whether a bulk
band gap exists, topological materials can be classified intotopological insulators with a finite band gap and topologicalsemimetals without a band gap. Due to the distinct topologicalindices in different insulating states (e.g., normal insulator,strong topological insulator, and weak topological insula-tor states), direct transitions between these states cannot beachieved by continuous tuning of material parameters and/orexternal conditions [ 2–4]. Instead, these transitions must go
through a semimetal phase associated with the closing ofthe band gap. Recently, these so-called topological phasetransitions (TPTs) have attracted significant research efforts.TPTs not only are of fundamental scientific interest, but alsosignal potential practical methods to drastically alter materialproperties with small changes of external parameters.
Previous studies have predicted or demonstrated TPTs
induced in different ways, e.g., by photoexcitation [ 5]o ra p -
plying an electric field [ 6], or by changing the strain [ 2,7,8],
the quantum well thickness [ 9], the chemical composition
[3,10–13], temperature [ 14], and pressure [ 15]. Most of the
previous studies focused on the electrical transport and op-tical properties across the TPTs. Some of these interestingproperties are enabled by the emergent topological semimetalphases, such as the Dirac fermions with ultrahigh mobilities[3,16–19], the Fermi arc surface states [ 20–22], and highly
temperature-sensitive optical properties [ 23]. On the other
hand, the phonon properties and phonon-mediated thermalconduction across TPTs have not been examined in detail.In a recent study [ 24], we reported the existence of ultrasoft
optical phonons in the topological Dirac semimetal cadmium
*bliao@ucsb.eduarsenide (Cd 3As2), which gives rise to strong phonon-phonon
scatterings and a low lattice thermal conductivity. The phononsoftening was attributed to potentially strong Kohn anomalies[25] associated with the Dirac nodes. In parallel, Nguyen
et al. reported experimental observation of Kohn anomalies
in a topological Weyl semimetal tantalum phosphide [ 26].
Therefore, it is interesting to see how the phonon propertiescontinuously evolve across the TPTs and whether the TPTsoffer a promising route towards sensitive control of the ther-mal conduction.
In this paper, we use first-principles simulations to examine
the phonon properties across TPTs in two model systems:pressure induced TPT in zirconium pentatelluride (ZrTe
5)
[2,27,28] and chemical composition induced TPT in mercury
cadmium telluride (Hg 1−xCdxTe) [3,4]. We show that certain
acoustic phonons and low-lying optical phonons significantlysoften near the TPTs when the semimetal phases emerge, atlocations in the Brillouin zone that match the characteristics of
Kohn anomalies. We further demonstrate that the phonon soft-
ening largely increases the phase space for phonon-phononscatterings and reduces the lattice thermal conductivity. Wenote that there is no structural phase transition associated withthe TPTs discussed in the two model systems, and thus theobserved change in phonon properties is solely due to the evo-lution of the electronic structure. Our findings point to thedirect connection between the electronic band topology andlattice instabilities, potentially adding to the understanding ofthe coincidence between topological materials and good ther-moelectric materials. Our results also suggest a route towardscontrolling solid-state thermal transport based on TPTs.
II. COMPUTATIONAL DETAILS
ZrTe 5is a van der Waals (vdW) layered material that
crystallizes in the base-centered orthorhombic Cmcm (D17
2h)
structure under ambient conditions [ 28]. In this structure,
2469-9950/2020/102(23)/235428(8) 235428-1 ©2020 American Physical SocietyYUE, DENG, LIU, QUAN, YANG, AND LIAO PHYSICAL REVIEW B 102, 235428 (2020)
FIG. 1. (a) The crystal structure of ZrTe 5with calculated lattice
constants and the corresponding first Brillouin zone, where the band
touching points under 5-GPa pressure are labeled with two cones.
(b) The supercell structure used and the corresponding Brillouin zoneof the Hg
1−xCdxTe alloy system.
ZrTe 3chains run along the aaxis, which are connected by
zigzag Te atoms to form two-dimensional (2D) sheets. These2D sheets are further stacked along the baxis. The crystal
structure and the conventional cell with the correspondingBrillouin zone are shown in Fig. 1(a), where /Gamma1-X,/Gamma1-Y, and
/Gamma1-Z are aligned with the a,b, and caxis, respectively. Both
HgTe and CdTe crystallize in the zinc-blende structure. TheHg
1−xCdxTe alloys with different Cd concentrations are sim-
ulated by randomly replacing Hg atoms with Cd atoms insupercells [Fig. 1(b)]. We use 2 ×2×2 supercells containing
64 atoms, allowing for compositions with Cd concentrationsdiffering by steps of 0.031 25. Different configurations of theCd atoms at each Cd concentration were calculated and we didnot notice significant differences in the electronic structure orthe phonon dispersion between different configurations thatare relevant to our discussion here.
All of the structural optimizations and electronic and
phonon properties were calculated based on the densityfunctional theory (DFT) by employing the Vienna ab initio
simulation package [ 29,30]. The projector augmented wave
method [ 31,32] and generalized gradient approximation(GGA) with the Perdew-Burke-Ernzerhof (PBE) exchange-
correlation functional were adopted [ 33]. For ZrTe
5, the vdW
corrected optB86b-vdw functional [ 34,35] was used through-
out all calculations because ZrTe 5is a layered material for
which the vdW correction is essential to obtain the cor-rect interlayer distance [ 8,36]. The plane-wave cutoff energy
was set to 350 eV for all materials. The Monkhorst-Pack k
meshes 9 ×9×4 were taken for ZrTe
5, and 8 ×8×8f o r
Hg1−xCdxTe. The k-mesh density and the energy cutoff of
augmented plane waves were checked to ensure the conver-gence. Full optimizations were applied to both structures withthe Hellmann-Feynman forces tolerance 0 .0001 eV /Å. The
spin-orbit coupling effect was included through all calcula-tions. For HgTe, it is known in previous literature that theGGA-level DFT does not generate the correct order of the/Gamma1
7and/Gamma16electronic bands [ 37]. While more advanced tech-
niques, such as GWand hybrid functionals [ 38], are known to
provide the correct band order, they are computationally tooexpensive for our current paper involving phonon calculationsusing supercells. However, for the purpose of this paper, theprecise location of the /Gamma1
7band does not have an impact since
it is positioned roughly 1 eV away from the Fermi level[39], which is much larger than the phonon energy scale.
Furthermore, the position of the /Gamma1
7band is not affected by
the Cd concentration [ 39]. Thus, we adopted the GGA-PBE
functional for all calculations in this paper.
To evaluate the phonon dispersions and lattice ther-
mal conductivities, we further calculated the second- andthird-order interatomic force constants (IFCs) using the finite-displacement approach [ 40]. In the calculations of the second-
and third-order IFCs, we adopted 3 ×3×1 supercells [ 36]
for ZrTe
5and 2 ×2×2 supercells for Hg 1−xCdxTe. The
phonon dispersions based on the second-order IFCs werecalculated using the
PHONOPY package [ 40]. The interactions
between atoms were taken into account up to sixth nearestneighbors in third-order IFC calculations. We calculated thelattice thermal conductivity by solving the phonon Boltz-mann transport equation (BTE) iteratively as implementedin
SHENGBTE [41,42]. The q-mesh samplings for the BTE
calculations of ZrTe 5and Hg 1−xCdxTe were 8 ×8×20 and
10×10×10, respectively. The convergence of the lattice
thermal conductivity with respect to the q-mesh density and
the interaction distance cutoff was checked. Our adoption ofthe finite-displacement approach implicitly determines thatonly the static, or adiabatic, Kohn anomalies can be cap-tured [ 43], while the dynamic effect [ 26] requires explicit
electron-phonon coupling calculations [ 44], which is compu-
tationally intractable for complex crystal structures such asthat of ZrTe
5. For simple crystal structures of HgTe and CdTe,
we used the EPW package [ 45] to explicitly calculate the dy-
namic electron-phonon scattering rates with the interpolationscheme based on the maximally localized Wannier functions(MLWFs). In the electron-phonon coupling calculations, weused the
QUANTUM ESPRESSO package [ 46] to evaluate the
electronic band structures and phonons in the Brillouin zone.Am e s hg r i do f2 0 ×20×20 was adopted for both materials
with norm-conserving pseudopotentials. The kinetic-energycutoff for wave functions was set to 30 Ry. The kinetic-energycutoff for charge density and potential is set to 120 Ry. The to-tal electron energy convergence threshold for self-consistency
235428-2PHONON SOFTENING NEAR TOPOLOGICAL PHASE … PHYSICAL REVIEW B 102, 235428 (2020)
is 1×10−10Ry. The crystal lattice is fully relaxed with a force
threshold of 1 ×10−4eV/Å. Applying the density functional
perturbation theory implemented in the QUANTUM ESPRESSO
package [ 46], the phonon dispersion and the electron-phonon
matrix elements are calculated on a coarse mesh of 6 ×6×6.
The electronic band structure, phonon dispersion relation, andelectron-phonon scattering matrix elements are subsequentlyinterpolated onto a fine mesh of 30 ×30×30 using a MLWFs
based scheme as implemented in the
EPW package [ 45]. We
checked the convergence of the phonon scattering rates as afunction of the fine sampling mesh density.
III. PRESSURE INDUCED TOPOLOGICAL PHASE
TRANSITION IN ZrTe 5
ZrTe 5has been intensively studied recently due to its close
proximity to a topological semimetal state. The nature of itsground state, however, remains controversial. While its 2Dmonolayer is predicted to be a quantum spin Hall insulator[27], conflicting views about its three-dimensional bulk band
structure have been reported. Early studies based on transport[47,48], optical [ 49], and photoemission [ 50] measurements
suggested that ZrTe
5is a topological Dirac semimetal with a
single Dirac node at the Brillouin-zone center. However, thevdW layered structure indicates that the Dirac node wouldlack the protection from additional crystalline symmetries asin other Dirac semimetals, e.g., Cd
3As2and Na 3Bi. Other
studies [ 2,8,51–56] concluded that ZrTe 5is a topological in-
sulator with a small band gap at the zone center. In particular,Mutch et al. reported [ 2] that the bulk band gap in ZrTe
5can
be closed by a small in-plane strain accompanying a TPT. Thelack of a consensus so far implies the extreme sensitivity ofthe electronic bands in ZrTe
5to sample quality and external
conditions.
Our calculated electronic band structure of relaxed ZrTe 5
is shown in Fig. 2(a), which is consistent with previous re-
ports of a bulk band gap at the zone center /Gamma1[2,56]. The
band gap calculated in our paper is roughly 150 meV andslightly higher than the experimental values below 100 meV.In addition to the in-plane strain induced TPT demonstratedby Mutch et al. [2], we find that applying a hydrostatic
pressure of 5 GPa can also drive a TPT and induce asemimetal phase [Fig. 2(b)]. Our calculation is consistent with
a previous report of pressure-induced superconductivity andmetal-insulator transition in ZrTe
5.[28] A recent study of
pressure-dependent transport and infrared transmission studyof ZrTe
5also signaled a pressure-induced band gap closing
near 5 GPa [ 57]. Interestingly, in ZrTe 5under 5 GPa of hy-
drostatic pressure, the conduction band and the valence bandtouch at near halfway along the /Gamma1-Y direction [highlighted
in Fig. 1(a)], which aligns with the layer-stacking baxis in
the conventional cell, agreeing with a previous calculation[28]. This is in contrast to the experimentally demonstrated
strain-induced TPT [ 2] and the theoretically predicted lattice
expansion induced TPT [ 8]i nZ r T e
5, where the band gap
closes at the /Gamma1point. In addition, we find that the electronic
energy dispersion near the touching points is linear alongthe out-of-plane direction ( /Gamma1-Y) with a Fermi velocity of
approximately 3 .3×10
5m/s. Along the in-plane directions,
however, we find the energy dispersion near the touching
FIG. 2. (a) The calculated electronic band structure of ZrTe 5
under zero pressure. (b) The calculated electronic band structure of
ZrTe 5along the /Gamma1-Y direction under different hydrostatic pressures.
The band gap is closed when ZrTe 5is under 5-GPa pressure. (c) The
quadratic electronic energy dispersion along the aandcaxis near the
band touching points in ZrTe 5under 5-GPa pressure.
points is anisotropic and quadratic [Fig. 2(c)]. Along the
a(c) axis, the conduction-band effective mass is 0.12 m0
(0.5 m0), and the valence-band effective mass is 0.18 m0
(0.28 m0), respectively, where m0is the free-electron mass.
This characteristic differs from that of the low-energy bandsnear the /Gamma1point under the ambient condition, where the in-
plane dispersions are determined to be close to linear while theout-of-plane dispersion is quadratic [ 58]. Our finding provides
a possible explanation for the increased electrical resistivity[57] along the aaxis under hydrostatic pressure as the energy
band contributing to the transport along aaxis evolves from
being linear to being quadratic.
235428-3YUE, DENG, LIU, QUAN, YANG, AND LIAO PHYSICAL REVIEW B 102, 235428 (2020)
FIG. 3. (a) The calculated phonon dispersions of ZrTe 5under
zero pressure and 5-GPa pressure. (b) The relative changes of thecalculated lattice thermal conductivity of ZrTe
5along different direc-
tions under 5-GPa pressure compared to those under zero pressure.
(c) The phonon-phonon scattering rates in ZrTe 5under zero pressure
and 5-GPa pressure, highlighting the increased scattering near the
softened phonons in pressured ZrTe 5.
The phonon dispersions of ZrTe 5under ambient condition
and 5-GPa pressure are shown in Fig. 3(a). The zero-pressure
phonon dispersion is in good agreement with previous cal-culations [ 36,59]. A prominent feature is the significant
softening of the transverse acoustic mode along the /Gamma1-Y direc-
tion when ZrTe
5is subjected to the 5-GPa pressure, as well as
a weaker kink and softening of the low-lying optical phononat the/Gamma1point.
The observed phonon softening locations ( /Gamma1and Y) sug-
gest that they are instances of Kohn anomalies [ 25]. Kohn
anomalies are the distortions of phonon dispersions observedin metals and semimetals caused by the resonance between
the Fermi surface and certain phonon modes. Specifically,when two electronic states on the Fermi surface are paral-lelly connected (“nested”) by a phonon momentum q,t h e
polarizability /Pi1(ω,q), which describes the collective re-
sponse of the conduction electrons to an external disturbancewith frequency ωand wave vector q, becomes nonanalytic.
Since phononic vibrations are screened by the conductionelectrons, the nonanalyticity of the polarizability at certainphonon momentum qleads to abrupt changes of the phonon
dispersion. The Fermi surface being two discrete nodes inZrTe
5under 5-GPa pressure gives rise to two possible types
of Kohn anomalies associated with intranode or internodeelectron-phonon scatterings, which were similarly observedin graphene [ 43,44]. In the intranode case, one electron
within the vicinity of one Dirac node is scattered to anotherelectronic state near the same node, mediated by a phononwith a wave vector q≈0. These processes are responsible
for phonon anomalies near the Brillouin-zone center, /Gamma1.I n
the internode case, one electron close to one Dirac nodeis scattered to another electronic state near the other node,mediated by a phonon with a wave vector matching the dis-tance between the two nodes, q≈2k
D, where kDmarks the
location of one Dirac node. In ZrTe 5under 5-GPa pressure,
since the band touching point is located near the midpointalong the /Gamma1-Y direction, the internode Kohn anomaly is
expected to occur near Y, which is consistent with our cal-culation.
Nguyen et al. showed analytically [ 26] that Kohn anoma-
lies associated with three-dimensional Dirac nodes are causedby strong singularities in the electronic polarizability function/Pi1(ω,q) that are similar to those in one-dimensional simple
metals, where Kohn anomalies are known to give rise tostructural instabilities (Peierls transitions). While the bandtouching points in ZrTe
5under 5-GPa pressure are not strictly
Dirac nodes due to the quadratic dispersion along the aand
caxis, the observed Kohn anomalies are strong and expected
to significantly enlarge the available phase space for phononscatterings and reduce the lattice thermal conductivity. Thestrong phonon softening at Y can also be responsible for theexperimentally observed structural phase transition in ZrTe
5
above 6 GPa [ 28].
In Fig. 3(b), we show the relative change of the calculated
lattice thermal conductivity of ZrTe 5induced by the 5-GPa
hydrostatic pressure, as a function of the temperature. Underzero pressure, the lattice thermal conductivities along threeaxes are all different, with the one along the layer-stackingdirection ( baxis) being the lowest (4 W /mK along the aaxis,
0.4 W/mK along the baxis, and 1.8 W /mK along the caxis).
Our results agree with the previous calculation and experi-ment by Zhu et al. [59]. Significant reductions of the lattice
thermal conductivity above 10% at room temperature alongall three directions are observed when ZrTe
5is under 5-GPa
pressure, particularly along the baxis. More interestingly, we
observed anomalous peaks in the phonon-phonon scatteringrates of the low-frequency acoustic modes in pressured ZrTe
5,
as shown in Fig. 3(c): one near 0.3 THz and the other near
0.7 THz, corresponding to the Kohn anomalies at Y and /Gamma1,
respectively, providing strong evidence that the Kohn anoma-lies in pressured ZrTe
5contribute to its low lattice thermal
235428-4PHONON SOFTENING NEAR TOPOLOGICAL PHASE … PHYSICAL REVIEW B 102, 235428 (2020)
FIG. 4. (a) The calculated electronic band structure near /Gamma1of the Hg 1−xCdxTe alloys, showing the band closing at the critical composition
x=0.16. The high-symmetry points correspond to those in the Brillouin zone of the supercell as shown in Fig. 1(b). (b) The evolution
of the calculated band gap of Hg 1−xCdxTe as a function of the composition. (c) The evolution of the calculated TO phonon frequency in
Hg1−xCdxTe as a function of the band gap. The inset shows examples of the calculated optical phonon dispersion. (d) The calculated lattice
thermal conductivity of Hg 1−xCdxTe is shown in the left panel. The calculated values are normalized by v2
sand shown in the right panel.
conductivity. Our study of ZrTe 5suggests potential means to
tune the thermal conductivity of topological materials throughinduced TPTs.
IV . CHEMICAL COMPOSITION INDUCED
TOPOLOGICAL PHASE TRANSITION IN Hg 1−xCdxTe
The HgTe /CdTe system is among the most studied model
systems with nontrivial topological properties [ 9]. HgTe is a
semimetal with inverted electronics bands: the s-type/Gamma16band
is below the p-type/Gamma18band in energy due to the strong spin-
orbit coupling in Hg. [ 60] In comparison, CdTe is a finite-gap
semiconductor with the normal band order ( /Gamma16above /Gamma18in
energy). Thus, by continuously substituting Cd for Hg, theelectronic band structure of the Hg
1−xCdxTe alloy will evolve
from an inverted semimetal to a normal insulator with dis-tinct topological indices [ 9], going through a TPT where a
three-dimensional Dirac dispersion near the /Gamma1point emerges
[3,4]. This transition is found experimentally to occur around
x=0.17. The large tunability of the Hg
1−xCdxTe alloy has
enabled its application in infrared photodetectors [ 61].
We show the calculated electronic band structure of
Hg1−xCdxTe alloys with different Cd concentration in
Fig. 4(a). The band gap of CdTe shrinks as the Cd con-
centration decreases, and completely closes when x=0.16
in our calculation. At the critical concentration when bandclosing occurs, linear conical bands emerge at the /Gamma1point.
The evolution of Hg 1−xCdxTe between inverted semimetal
and insulator phases is illustrated in Fig. 4(b). The emer-
gent band overlap at /Gamma1is expected to cause intranode Kohn
anomalies that affect phonon modes near the /Gamma1point. As
expected, our phonon calculation shows a clear drop of thetransverse optical (TO) phonon frequency at /Gamma1as Cd con-
centration decreases. To eliminate the influence of the atomicmass difference between Hg and Cd on the TO phononfrequency, we plot the calculated TO phonon frequency nor-
malized by a scaling factor
vs(HgTe)
vs(Hg 1−xCdxTe)in Fig. 4(c), where
vsis the speed of sound [ 62]. Without the normalization,
the TO phonon frequency would decrease even more withthe decreasing Cd concentration. This decreasing trend qual-itatively agrees with an infrared reflectivity measurement ofHg
1−xCdxTe [63]. A recent optical study of the Pb 1−xSnxSe
system also showed the decreasing optical phonon frequencyas the band gap shrinks to zero through a TPT [ 64]. The TO
phonon frequency further decreases past the TPT and into thesemimetal phase, reaching a minimum in HgTe, indicating thepersisting strong Kohn anomaly in the semimetal phase.
We further evaluate the lattice thermal conductivity of
Hg
1−xCdxTe at room temperature as a function of the Cd con-
centration, as shown in Fig. 4(d). Our calculated result agrees
well with experimental reports [ 39]. The calculated lattice
thermal conductivity is plotted in the left panel of Fig. 4(d),
235428-5YUE, DENG, LIU, QUAN, YANG, AND LIAO PHYSICAL REVIEW B 102, 235428 (2020)
FIG. 5. The calculated mode-resolved phonon scattering rates
due to electron-phonon interaction in HgTe (left) and CdTe (right).
The color represents the scattering rates of the specific phononmodes.
while the lattice thermal conductivity normalized by v2
sis
plotted in the right panel to eliminate the impact of the Hg /Cd
mass difference. Two major factors affect the lattice thermalconductivity of the Hg
1−xCdxTe alloys: strong phonon-defect
scattering due to alloying and increased phonon-phonon scat-tering due to softened TO mode. The alloy effect typicallyleads to the minimum thermal conductivity near x=0.5,
which is the case here in Hg
1−xCdxTe. However, the nor-
malized thermal conductivity of Hg 1−xCdxTe with small Cd
concentration remains lower compared to compositions withsmall Hg concentration, such that the normalized thermalconductivity curve shown in Fig. 4(d) right panel is slightly
skewed towards the Hg-rich region. This result indicates thepromise of Hg
1−xCdxTe near the TPT as a good thermoelec-
tric material due to high mobility of the Dirac bands, strongelectron-hole asymmetry for high Seebeck coefficients [ 38],
and low lattice thermal conductivity due to strong alloy scat-tering and soft optical phonons shown in this paper.
To confirm that the origin of the softened TO phonons
is Kohn anomalies induced by the strong electron-phononinteraction in the semimetal phase, we explicitly calculatedthe phonon scattering rates due to electron-phonon interac-tion [ 65,66] in HgTe and CdTe, as shown in Fig. 5(phonondispersions are scaled by the speed of sound). It is clear that
the TO phonon in HgTe is much more strongly scattered byelectrons than that in CdTe. The TO phonon softening dueto electron-phonon interaction in the Hg
1−xCdxTe system was
previously discussed in terms of “returnable” electron-phononinteraction [ 67], meaning the impact of electron scatterings on
phonons. Similar effects have been discussed in Pb
1−xSnxSe
and Pb 1−xSnxTe systems [ 64,68] with small or zero band gaps,
suggesting that the softened optical phonon is a universalfeature of materials near TPTs. Intuitively, this is caused bymore electron-phonon scattering channels when the electronicband gap is comparable to or smaller than the phonon energyscale.
V . CONCLUSION
In summary, we identified the existence of significant
phonon softening near TPTs by examining two modelsystems: pressure induced TPT in ZrTe
5and chemical com-
position induced TPT in Hg 1−xCdxTe through first-principles
simulations. We attributed the phonon softening to strongKohn anomalies associated with the band-gap closing at aTPT. We further evaluated the impact of the softened phononmodes on the thermal transport in these materials. Our paperalludes to deep connections between the electronic structureand lattice dynamics and exemplifies the rich phonon physicsin topological materials. Our results further suggest potentialroutes towards effective thermal conduction control and moreefficient thermoelectric devices based on topological materi-als.
ACKNOWLEDGMENTS
This work is based on research supported by the U.S.
Department of Energy, Office of Basic Energy Sciences,under Grant No. DE-SC0019244; the University of Califor-nia Santa Barbara NSF Quantum Foundry funded via theQ-AMASE-i program under Grant No. DMR-1906325 (forstudying topological materials); and the NSF under Grant No.CBET-1846927 (for studying phonon-electron scattering).Y.L. acknowledges support from the Tsinghua Scholarship forUndergraduate Overseas Studies. This work used the ExtremeScience and Engineering Discovery Environment, which issupported by NSF Grant No. ACI-1548562.
S.Y., B.D., and Y.L. contributed equally to this work.
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235428-8 |
PhysRevB.81.245302.pdf | Probing local electronic states in the quantum Hall regime with a side-coupled quantum dot
Tomohiro Otsuka,1,*Eisuke Abe,1Yasuhiro Iye,1and Shingo Katsumoto1,2
1Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan
2Institute for Nano Quantum Information Electronics, University of Tokyo, 4-6-1 Komaba, Meguro, Tokyo 153-8505, Japan
/H20849Received 25 March 2010; revised manuscript received 26 April 2010; published 2 June 2010 /H20850
We demonstrate a method for locally probing the electronic states in the quantum Hall regime utilizing a
side-coupled quantum dot positioned at an edge of a Hall bar. By measuring the tunneling of electrons from theHall bar into the dot, we acquire information on the local electrochemical potential and electron temperature.Furthermore, this method allows us to observe the spatial modulation of the electrostatic potential at the edgestate due to many-body screening effect.
DOI: 10.1103/PhysRevB.81.245302 PACS number /H20849s/H20850: 73.63.Kv, 73.43. /H11002f, 85.35. /H11002p
The edge states1formed in two-dimensional electron
gases under strong magnetic fields play important roles in thetransport properties in the quantum Hall /H20849QH /H20850effect.
2They
are formed as a consequence of the Landau quantization andthe confinement potential at the edges of the devices. In con-ventional experimental methods for the study of electronicstates in the QH effect, “voltage probes,” which are com-posed of macroscopic Ohmic contacts, are used. Althoughthe voltage difference between the contacts can be measured,this is not enough to explore microscopic properties of theedge states. Also these macroscopic contacts induce distur-bance of the electronic states and it is inevitable to changethe original electronic states in the QH effect.
3
For exploring the microscopic electronic states, local
probes utilizing liquid-helium films,4the Pockels effect,3cy-
clotron emission,5and scanning probe microscopy6–9have
been reported. They succeeded to show real-space images ofelectric fields or electron densities. Nevertheless local elec-trostatic and thermodynamic properties are still elusive. Toinvestigate the electronic states in the QH regime, artificialnanostructures can be effective as demonstrated, for instance,by Granger et al.
10In our study, we apply side-coupled quan-
tum dots /H20849QDs /H20850to obtain local electrochemical potential,
electron temperature, and spatial configuration of the edgestates. Similar structure has been used by Fève et al. for a
coherent single-electron source.
11The different points in our
work are the QD contains only a few electrons and use it toinvestigate properties of the QH states. Since it is easy toobtain a side-coupled QD in the few electron regime withkeeping tunneling probability to the edge,
12we can use a
well-defined single level in the QD and this enables high-energy resolution. Also the flow of electrons between theedge and the QD is regulated by the Coulomb blockade andcan be very small /H20849less than femtoampere /H20850and the measure-
ment has very small disturbance to the original electronicstates. Though the positions of the QDs are fixed, the high-energy resolution and the sensitivity to QD-edge distanceenable us to detect characteristic variation in the electrostaticpotential in the QH effect.
13,14
We measured two devices fabricated from a GaAs/
AlGaAs heterostructure wafer with the sheet carrier densityof 2.1/H1100310
15m−2and the mobility of 32 m2/V s. After the
formation of Ohmic contacts, 36 /H9262m/H11003108/H9262m-sized Hall
bars /H20849HBs /H20850were patterned by wet etching, followed by the
deposition of Au/Ti Schottky gates to define QDs. The sizeof the HBs is sufficiently large for observing the QH effect.
As depicted in Fig. 1/H20849a/H20850, one of them /H20849Device 1 /H20850h a saQ Di n
the middle of the right edge. The other device /H20849Device 2, not
shown /H20850has an additional QD placed 6 /H9262m apart from con-
tact C1 on the right edge. The devices were cooled with adilution refrigerator /H20849base temperature around 30 mK /H20850, and
the perpendicular magnetic field Bwas applied using a su-
perconducting solenoid.
Here we measure the local electronic states utilizing a QD
side coupled to the edge state. In the following, we brieflysummarize the technique, which is fully described in Refs.15–17. A QD and the edge state are separated by a potential
barrier formed by a Schottky gate and we detect tunnelingevents between them through changes in the number of elec-trons in the QD. This detection is realized by a remote chargedetector utilizing a quantum point contact /H20849QPC /H20850placed next
to the QD.
18–21
By applying square-wave voltages on gate P VP, we regu-
larly shift the chemical potential of the QD and form anenergy window with the width /H9004E/H20851Fig.1/H20849b/H20850/H20852. To minimize
the change in the tunnel barrier by the square wave, we chosegate P, which is placed at the opposite side of the tunnelingbarrier, for applying the square wave. When the electro-
(a)
(b)(c)
(d)
Isync
VPDC/c68E/c109
VPDCE
C2
C1
B
QD
QPCTP
S
Q
200 nm
HB QD/c68E /c109xy
z
FIG. 1. /H20849Color online /H20850/H20849a/H20850Schematic of Device 1. The scanning
electron micrograph shows the QD part of the device. /H20849b/H20850Energy
diagram when the electron shuttling occurs. /H20849c/H20850Shift of the energy
window /H20849the gray zone /H20850as a function of VPDC./H20849d/H20850Isyncas a function
ofVPDC. The solid /H20849broken /H20850line shows an expected result at low
/H20849high /H20850electron temperature Te.PHYSICAL REVIEW B 81, 245302 /H208492010 /H20850
1098-0121/2010/81 /H2084924/H20850/245302 /H208495/H20850 ©2010 The American Physical Society 245302-1chemical potential /H9262in the HB is in this window, the poten-
tial shift causes electron shuttling between the edge and theQD. The direct electrostatic coupling between gate P and theQPC leads to a synchronous current I
syncand the effect of the
electron shuttling appears as a decrease in Isync. As sweeping
the dc offset voltage on gate P VPDCand continuously shift-
ing the energy window /H20851Fig.1/H20849c/H20850/H20852, a dip of Isyncis observed
in the region in which /H9262and the energy window cross /H20851Fig.
1/H20849d/H20850/H20852. This dip structure contains information on the local
electronic states in the vicinity of the tunneling barrier of theQD.
Figure 2shows the observed I
syncof Device 1 as a func-
tion of VPDCand the bias voltage on the HB Vbias. We used
the square wave with the frequency of 833 Hz and the am-plitude of 40 mV. The conductance of the QPC was set in thetransition regions between plateaus to maximize the detec-tion sensitivity. The bias across the QPC was 700
/H9262V. We
measured Isyncusing lock-in amplifier with the reference fre-
quency of 833 Hz. Vbiaswas applied on one contact while the
other contact was connected to the ground. The left /H20849right /H20850
graphs show the results when Vbiasis applied on contact C1
/H20849C2/H20850. The number of electrons in the QD is set to zero or
one. The absolute number of electrons was determinedthrough the QPC charge detection.
20,21In the QH regime
with the filling factor /H9263=4 /H20849B=2.25 T /H20850/H20851Fig. 2/H20849a/H20850/H20852, the dip
positions /H20849bands in dark color /H20850are almost fixed when C1 isbiased while linear shifts are observed when C2 is biased.
Note that the survival of the QH effect in this bias range isconfirmed by measuring the longitudinal resistance with con-ventional voltage probes.
This asymmetry in the bias condition is reasonable con-
sidering that the system is in the QH regime, where the volt-age drop along ydirection is zero except at the hot spots
4,5
near contacts through which current flows. The result is also
viewed as a consequence of the chirality of the edge states.When the magnetic field is applied in + zdirection /H20851Fig.2/H20849a/H20850/H20852,
the electrons emitted from C2 enter the right-edge state.Then the electrochemical potential of the right-edge state isequal to zero when C2 is grounded and equal to V
biaswhen
C2 is biased. When the direction of the magnetic field isreversed, the electrochemical potential of the right edge fol-lows C1. The results shown in Figs. 2/H20849a/H20850and2/H20849b/H20850are in good
agreement with the above deduction. We attribute the smallshifts of the dip positions in the left graph of Fig. 2/H20849a/H20850and
the right graph of Fig. 2/H20849b/H20850to the contact resistances, which
induce small voltage changes at the contacts.
At zero magnetic field, the electrochemical potential var-
ies linearly along ydirection between the two contacts irre-
spective of the bias condition because the edge states whichsuppress the energy relaxation are not formed. Since the QDis positioned halfway between the contacts, the shift of thedip positions as observed in Fig. 2/H20849c/H20850left /H20849right /H20850is just half
of that in the QH regime in Fig. 2/H20849b/H20850left /H20851/H20849a/H20850right /H20852.
We now focus on the line shape of the dip in Fig. 2.I ti s
observed in Fig. 2/H20849c/H20850that the boundaries of the dip are pro-
gressively blurred as V
biasbecomes larger. On the contrary,
the boundaries are always sharp in the QH regime /H20851Figs. 2/H20849a/H20850
and2/H20849b/H20850/H20852. As the dip occurs when /H9262and the energy window
intersect, its sharpness reflects the sharpness of the electrondistribution around
/H9262and thus the local electron temperature
Te, as illustrated in Fig. 1/H20849d/H20850. The cross sections along the
white lines in Fig. 2are shown in Fig. 3.
For quantitative evaluation, we assume that the broaden-
ing follows the Fermi-Dirac distribution,
F/H20849E/H20850=/H20851exp /H20853/H20849E−/H9262/H20850/kBTe/H20854+1/H20852−1, /H208491/H20850
where kBis the Boltzmann constant. To apply Eq. /H208491/H20850, the
coefficient /H9251to convert VPDCinto the energy Eis necessary.
By modeling a current-carrying channel as a series circuit ofthe zero-resistance one-way conductor /H20849i.e., edge channel /H20850
and resistors at the two contacts, we obtain
/H9251=/H208491/t1
+1 /t2/H20850−1, where t1andt2are the tangents of the dips when
C1 and C2 are biased, respectively.22Therefore, /H9251=0.061 is
directly obtained from Fig. 2. The solid lines in Figs. 3/H20849a/H20850
and3/H20849b/H20850show the results of the fitting using Te,/H9262, an addi-
tional offset, and a magnitude factor as fitting parameters.For comparison, we also evaluate T
eat/H9263=4.5. At zero bias,
we obtain Te=523, 621, and 856 mK for B=0 T, /H9263=4, and
/H9263=4.5, respectively. At this stage, it is not certain what
causes the high Teand the differences between them. One
possible reason is the radiation of the noise from theQPC.
23–25But it is possible to analyze the effect induced by
Vbiasbecause the effect is large and we can extract that by
evaluating the increase in the electron temperature/H9004T
e/H20849Vbias/H20850=Te/H20849Vbias/H20850−Te/H20849Vbias=0/H20850./H9004Teis plotted in Fig. 3/H20849c/H20850.
5
0
-5/s86bias(mV)
-0.7 -0.6
/s86PDC(V)
-0.7 -0.6 432
/s73sync(arb. unit)
5
0
-5/s86bias(mV)
-0.7 -0.6
/s86PDC(V)
-0.7 -0.6 3 2
/s73sync(arb. unit)
5
0
-5/s86bias(mV)
-0.7 -0.6
/s86PDC(V)
-0.7 -0.6 3.02.52.0
/s73sync(arb. unit)
(b)
(c)C2
C1
C1C2(a)
C2
C1B
BC1:Bias
C2:GND
=4
+/c110
zC1:GND
C2:Bias
C1:Bias
C2:GND
=4
-/c110
zC1:GND
C2:Bias
C1:BiasC2:GND0TC1:GNDC2:Bias
FIG. 2. /H20849Color online /H20850Isyncas a function of VPDCandVbiasin the
/H20849a/H20850positive, /H20849b/H20850negative, and /H20849c/H20850zero magnetic fields. The filling
factor is /H9263=4 in both /H20849a/H20850and /H20849b/H20850. The left /H20849right /H20850graphs are the
results when Vbiasis applied on C1 /H20849C2/H20850. The schematics in the
right-hand side illustrate the direction of the edge states. The hori-zontal lines in /H20849a/H20850and /H20849c/H20850correspond to the data in Fig. 3.OTSUKA et al. PHYSICAL REVIEW B 81, 245302 /H208492010 /H20850
245302-2As increasing Vbias,/H9004Tein non-QH conditions becomes large
up to as high as 3 K. On the contrary, /H9004Teat/H9263=4 is nearly
zero /H20849under 140 mK /H20850, regardless of Vbias. This certifies the
lack of the scattering mechanisms that raise Tein the QH
regime.
Next we measure Device 2 in order to examine the posi-
tion and Bdependence of /H9262and/H9004Te. Figures 4/H20849a/H20850and4/H20849b/H20850,
respectively, show the change in /H9262with the change in Vbias,
/H9254/H11013/H9004/H9262//H9004Vbias, and/H9004Teas a function of BatVbias=4 mV at
the two QD positions. In the QH regimes /H20849gray regions /H20850, the
values of /H9254are very close to zero or one, depending on
whether the corresponding edge state is in equilibrium withthe grounded or biased contact. Also, /H9004T
eis very small /H20849un-
der 300 mK /H20850, as expected. It is consistent with the nature of
the edge states that these features do not depend on the po-sition along the device edge. Although negative /H9004T
eis ob-
served in some fields, we are not certain about the reason. Innon-QH regimes, we observe that
/H9254behaves in a manner
similar to that in the QH regimes although the values are notas close to zero or one as the latter. Furthermore,
/H9254at the
middle position is symmetric with regard to Bwhile /H9254near
C1 is asymmetric /H20849closer to zero or one in negative magnetic
fields /H20850. In the transition region between the QH regimes sup-
pression of backscattering and energy relaxation is lifted.This deviates
/H9254from zero or one. As for the asymmetry, it is
explained as a result of the difference in the degree of theenergy relaxation. In negative fields, the electrons from C1enter the QD near C1 without suffering the energy relax-ation. Thus, the values of
/H9254are very close to zero or one. On
the other hand, in positive fields, the electrons from C2 enterthe QD after large energy relaxation. In much the same rea-son, the asymmetry is also observed in /H9004T
efor the QD near
C1.
Note that the benefit of this method with a side-coupled
QD is the small disturbance17to the original electronic states.
It is different from the measurement with conventional volt-age probes in which disturbance by the probes is inevitable.With this property, it becomes possible to measure the degreeof the relaxation shown in Fig. 4.
So far, we have confirmed that our method is capable of
probing basic features of the electronic states in the QH re-gime, such as the chirality, and the absence of energy relax-ation. We now proceed to obtain more detailed informationon the edge states, namely, the spatial modulation of theelectrostatic potential. In the theory beyond the single-particle picture, the reconstruction of the electrostatic poten-tial leads to the formation of stepwise distributions /H20849along x/H20850
of the edge states at absolute zero temperature.
13Even at
finite temperature, this screening effect survives and makesthe gradient of the Landau level dE /dxnear
/H9262smaller than
that without screening /H20851Fig.5/H20849a/H20850/H20852.14While the signatures of
the screening effect have been observed in experiments oninteredge tunneling
27and AB-type oscillation in antidots,283.0
2.5
2.0/s73sync(arb. unit)
-0.64 -0.62 -0.60 -0.58 -0.56
/s86PDC(V)
4
3
2/s73sync(arb. unit)
-0.64 -0.62 -0.60 -0.58 -0.56
/s86PDC(V)
3
2
1
0∆ /s84e(K)
5 4 3 2 1 0
/s86bias(mV)0T
ν= 4.5
ν=4(a)
(b)
(c)0m V 5mV
0T/c110=4
FIG. 3. /H20849Color online /H20850Isyncas a function of VPDCat/H20849a/H20850/H9263=4 and
/H20849b/H20850in the zero magnetic field. The solid lines superposed on the data
are the results of the fitting using Teas a fitting parameter. From the
left to the right, Vbiaswas varied from 0 to 5 mV by 1 mV. /H20849c/H20850/H9004Te
as a function of Vbias.1.0
0.5
0.0δ1.0
0.5
0.0
-4 -2 0 2 4
/s66(T)
4
2
0/s68 /s84e(K)
4
2
0
-4 -2 0 2 4
/s66(T)(a)
(b)C1C2
BC2
C1B/c110=5 43 2
C1:Bias
C2:GND
C1:GND
C2:Bias
C1:Bias
C2:GND
C1:GND
C2:Bias
FIG. 4. /H20849Color online /H20850/H20849a/H20850/H9254/H11013/H9004/H9262//H9004Vbiasas a function of B/H20849Ref.
26/H20850. The triangles /H20849squares /H20850are the results of the QD at the center
/H20849near C1 /H20850. The upper /H20849lower /H20850graph shows the result when C1 /H20849C2/H20850
is biased. The gray regions are the QH regimes with /H9263as indicated.
The insets show schematics of the edge states in negative and posi-tive magnetic fields. /H20849b/H20850/H9004T
eas a function of B. The symbols are the
same as /H20849a/H20850.PROBING LOCAL ELECTRONIC STATES IN THE … PHYSICAL REVIEW B 81, 245302 /H208492010 /H20850
245302-3the present method gives more direct and detailed access to
this effect.
The idea is utilizing the relation between the gradient
dE /dxand the tunneling rate /H9003. Since /H9003depends exponen-
tially on the tunneling distance, it is highly sensitive to thechange in dE /dx/H20851arrows in Fig. 5/H20849a/H20850/H20852. The dip depth /H9004I
sync
and/H9003are related by the formula,15
/H9004Isync/H110081−/H92662
/H90032/4f2+/H92662, /H208492/H20850
where fis the frequency of the square wave. By adjusting f
to be comparable with /H9003, we can realize the condition in
which /H9004Isyncis sensitive to /H9003and consequently to dE /dx.
Note that in the preceding measurements, the condition/H9003/2f/H112711 was employed and /H9004I
syncwas nearly constant.
Here, the bottom of the dip is no longer flat but rather showsa buildup structure with the increase in V
PDCreflecting the
barrier thickness, as illustrated in Fig. 5/H20849b/H20850. It is expected that
the stronger screening results in a faster buildup of the bot-tom of the dip because of the faster decrease in /H9003. In this
way, we are able to investigate the electronic states below
/H9262.Figure 5/H20849c/H20850shows the observed Isyncas a function of VPDC.
The frequency of the square wave was 370 Hz in this mea-surement. The traces show the results at
/H9263=3,4,5 as well as
B=0 T. Isyncis normalized to make the deepest points equal
to −1. In the measurement, we canceled the change in /H9003with
Bby readjusting the voltage of gate T. This procedure is to
compensate the possible change in the QD-edge state dis-tance at the Fermi energy as Bis varied and to extract the
pure change induced by the modification of dE /dx.I ti so b -
served that the buildup of I
syncis faster at smaller /H9263. This
implies the stronger screening at smaller /H9263. In our method,
we are probing only the outermost channel because it has byfar the largest tunneling probability among the channels. Thedegeneracy in this channel becomes larger at higher fields.This could enhance the electron-electron interactions and fa-cilitate the redistribution of the electrons. This interpretationqualitatively explains the observed phenomena.
In conclusion, we have investigated the local electronic
states in the quantum Hall regime utilizing side-coupledquantum dots as local probes. We have observed the forma-tion of the edge states, and confirmed their chirality. We havesucceeded in determining the local electron temperature andconfirmed the suppression of energy relaxation in quantumHall regime. Finally, we have investigated the screening ef-fect in the edge states. Our results demonstrate the ability ofthe method to deduce the local information on the quantumHall states, which is not obtained through the conventionaltransport measurements. This method will be applicable toapproach the hotspots and edge states in the fractional quan-tum Hall effect.
Note added . Recently, we became aware of a paper by
Altimiras et al. about nonequilibrium edge-channel spectros-
copy utilizing a quantum dot between two edge channels.
29
We thank A. Endo, M. Kato, and T. Fujisawa for fruitful
discussions, and Y. Hashimoto for technical support. Thiswork was supported by Grant-in-Aid for Scientific Researchand Special Coordination Funds for Promoting Science andTechnology.
*t-otsuka@issp.u-tokyo.ac.jp
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4617 /H208492004 /H20850.-1.0-0.50.0/s73sync, normalized
-0.53 -0.52 -0.51
/s86PDC(V)ν=3
ν=4
ν=5
0T(a)
(b)(c)
/c109E
Isync
VPDCHB QD x/c71
/c68Isync
FIG. 5. /H20849Color online /H20850/H20849a/H20850Energy diagram near the device edge.
The solid /H20849broken /H20850line shows the Landau level without /H20849with /H20850
screening around /H9262./H20849b/H20850Isyncas a function of VPDCfor cases without
/H20849solid line /H20850and with /H20849broken line /H20850screening. /H20849c/H20850Normalized Isyncas
a function of VPDC.OTSUKA et al. PHYSICAL REVIEW B 81, 245302 /H208492010 /H20850
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26Although the QH effect was also observed in lower fields below
/H20841B/H20841=1.5 T, it appears that opening of the Landau gaps in the QH
regime is not large enough to allow for clear comparison withnon-QH regimes. At fields above /H20841B/H20841=4.5 T, the QDs were not
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245302-5 |
PhysRevB.100.121107.pdf | PHYSICAL REVIEW B 100, 121107(R) (2019)
Rapid Communications
Topological properties of multilayers and surface steps in the SnTe material class
Wojciech Brzezicki , Marcin M. Wysoki ´nski, and Timo Hyart
International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32 /46, PL-02668 Warsaw, Poland
(Received 14 December 2018; revised manuscript received 29 March 2019; published 16 September 2019)
Surfaces of multilayer semiconductors typically have regions of atomically flat terraces separated by atom-
high steps. Here we investigate the properties of the low-energy states appearing at the surface atomic stepsin Sn
1−xPbxTe1−ySey. We identify the important approximate symmetries and use them to construct relevant
topological invariants. We calculate the dependence of mirror- and spin-resolved Chern numbers on the numberof layers and show that the step states appear when these invariants are different on the two sides of thestep. Moreover, we find that a particle-hole symmetry can protect one-dimensional Weyl points at the steps.Since the local density of states is large at the step the system is susceptible to different types of instabilities,and we consider an easy-axis magnetization as one realistic possibility. We show that magnetic domain wallssupport low-energy bound states because the regions with opposite magnetization are topologically distinct inthe presence of nonsymmorphic chiral and mirror symmetries, providing a possible explanation for the zero-biasconductance peak observed in the recent experiment [Mazur et al. ,P h y s .R e v .B 100,041408(R) (2019 )].
DOI: 10.1103/PhysRevB.100.121107
Sn1−xPbxTe1−ySeysystems have attracted interest due to
the realization of a three-dimensional topological crystallineinsulator phase [ 1–5], prediction of two-dimensional (2D)
topological phases [ 6–10], and the appearance of low-energy
states at the defects [ 11,12]. Robust one-dimensional (1D)
modes were observed at the surface steps separating regions ofeven and odd number of layers [ 12] and interpreted as topo-
logical flat bands using a model obeying a chiral symmetry[13]. In a more accurate description step modes can have a
band width but the local density of states (LDOS) is large sothat the system is susceptible to the formation of correlatedstates [ 11,14,15]. Recent experiments indicate that an order
parameter emerges at low temperatures and it is accompaniedwith an appearance of a robust zero-bias peak (ZBCP) in thetunneling conductance [ 16,17]. The temperature and magnetic
field dependence of the energy gap are consistent with super-conductivity and under such circumstances the ZBCP is ofteninterpreted as an indication of Majorana zero modes [ 16–19],
which are intensively searched non-Abelian quasiparticles[20]. Thus, this finding calls for a critical study of different
mechanisms which may explain the appearance of the ZBCP.
We show that Sn
1−xPbxTe1−ySeymultilayers are a paradig-
matic system for realization of topological phases due toemergent symmetries of the low-energy theory, and ZBCP canappear in the absence of superconductivity. The important 2Dtopological invariants are the mirror-resolved Chern number
C
±(due to structural mirror symmetry) and spin-resolved
Chern number C↑(↓)(due to approximate spin-rotation sym-
metry). For odd number of layers Nthe mirror symmetry is
a point-group operation whereas for even Nit is a nonsym-
morphic (NS) symmetry (Fig. 1), so that adding one layer
can change the topology of the system [ 9,10]. We calculate
the dependence of the Chern numbers on Nand show that
the step states appear when these invariants are different onthe two sides of the step. The theory and experiment [ 12,13]
attribute step states only to odd-height steps, but we predict
FIG. 1. (a),(b) Schematic views of the Sn 1−xPbxTe1−ySeymulti-
layers stacked in the (001) direction. Green (brown) balls are (Sn,Pb)[(Te,Se)] atoms. For odd (even) number of layers Nthere exists
symmorphic (nonsymmorphic) mirror symmetry. (c),(d) The corre-
sponding edge-state spectra for N=3a n d N=4. (e) Surface atomic
steps describing an interface between three- and four-layer systems
and (f) a symmetrized atomic step. (g),(h) The corresponding spectra
for step modes. The panel below (e) shows local density of states ofthe step states. The width of the sample is N
y=600.
2469-9950/2019/100(12)/121107(6) 121107-1 ©2019 American Physical SocietyBRZEZICKI, WYSOKI ´NSKI, AND HYART PHYSICAL REVIEW B 100, 121107(R) (2019)
that also even-height steps can exhibit step states consistent
with another experiment [ 21].
We discuss the conditions under which a spontaneous
symmetry breaking gives rise to an energy gap at the step andstudy an easy-axis magnetic order as one possibility. We showthat magnetic domain walls (DWs) support low-energy boundstates because the regions with opposite magnetization aretopologically distinct in the presence of NS chiral and mirrorsymmetries. Due to the appearance of DWs the Fermi level ispinned to the energy of the DW states for a range of electrondensity providing an explanation for the ZBCP observed inthe experiment [ 16]. The observed temperature and magnetic
field dependencies are consistent with our theory.
Our starting point is a p-orbital tight-binding Hamilto-
nian describing a bulk topological crystalline insulator in theSn
1−xPbxTe1−ySey-material class [ 1]:
H(k)=m12⊗13⊗/Sigma1+t12/summationdisplay
α=x,y,z12⊗/parenleftbig
13−L2
α/parenrightbig
⊗h(1)
α(kα)
+t11/summationdisplay
α/negationslash=β12⊗/bracketleftbig
13−1
2(Lα+εαβLβ)2/bracketrightbig
⊗h(2)
α,β(kα,kβ)/Sigma1
+/summationdisplay
α=x,y,zλασα⊗Lα⊗18, (1)
where we have chosen a cubic unit cell with internal sites
at the corners labeled by i=1,..., 8[22],εαβis a Levi-
Civita symbol, Lα=−iεαβγare the 3 ×3 angular momentum
L=1 matrices, σαare Pauli matrices acting in the spin
space, /Sigma1is a diagonal 8 ×8 matrix with entries si=±1a t
the two sublattices [(Sn,Pb) /(Te,Se) atoms], and h(1)
α(kα) and
h(2)
α,β(kα,kβ)a r e8 ×8 matrices describing hopping between
the nearest-neighbor and next-nearest-neighbor lattice sites inthe directions ˆ αand ˆα+/epsilon1
αβˆβ, respectively [ 22]. We allow
the possibility to tune the spin-orbit coupling terms λα(α=
x,y,z) to be different from each other although in the real
material λα=λ. When not otherwise stated we use m=
1.65 eV , t12=0.9e V , t11=0.5 eV , and λ=0.3e V[ 23].
We focus on Hamiltonian HN(/vectork)f o rNlayers in the zdirec-
tion. The mirror symmetries for odd (even) Ncan be written as
Mo/e
z(kx)=σz⊗(2L2
z−1)⊗mo/e
z(kx), where mo
zis a point-
group reflection and me
z(kx) is a (momentum-dependent) NS
operation consisting of reflection and a shift by a half lattice
vector [Figs. 1(a) and1(b)]. To calculate C±we split HN(/vectork)
into two blocks in the Mo/e
z(kx) eigenspace. Since Mo/e
z(kx)
anticommutes with time-reversal symmetry (TRS) operatorT=Kσ
y⊗13⊗18, these blocks carry opposite Chern num-
bersC±[24]. We find that C+oscillates between +2 and −2
forN=2n+1 (we exclude N=1) and C±=0f o r N=2n
(n∈N)[22,25]. We point out that for even number of layers
the nonsymmorphic nature of the mirror symmetry guaranteesthat the Chern number calculated for the blocks must alwaysbe equal to zero. Therefore, one can equivalently concludethat the mirror-resolved Chern number does not exist as atopological invariant for even number of layers. The edge-state spectra for N=3 and N=4 are shown in Figs. 1(c)
and 1(d). As predicted by C
±=∓2 we see two pairs of
gapless edge modes in the case N=3, but surprisingly weFIG. 2. (a) Spin-resolved Chern numbers C↑as a function of
Nforλx=λy=0. Because we have switched off λxandλythe
effective spin-orbit coupling becomes smaller. In the case of odd
number of layers where C±is a useful topological invariant, we have
checked that this reduction of the spin-orbit coupling does not change
theC±so that it is reasonable to use λz=0.3 eV. The only exception
is the case N=3 where we have used renormalized λz=0.5e Vt o
keep C±fixed. (b) The spectrum for N=4 showing gapless edge
modes. (c) Gapless spectrum for a step between N=3a n d N=4
can be protected by a Z2invariant if λx=0.
find four pairs of edge modes if N=4. These edge modes are
consistent with C±=0 because there exist small gaps which
do not vanish by increasing system size.
To understand the existence of edge modes in the case
ofN=4 we notice that the momentum in the zdirection is
quantized and the low-energy degrees of freedom are asso-ciated with a motion within the ( x,y) plane [ 26]. Therefore,
the components of the spin-orbit coupling λ
α(α=x,y,z)
contribute differently to the spectrum, and the dominant effectcomes from λ
zσzLz. Hence, turning off λxandλyis a good
approximation [cf. Figs. 1(d) and2(b)] and this leads to a
spin rotation symmetry with respect to the zaxis. Thus, by
block-diagonalizing HN(/vectork) we can calculate C↑=−C↓as
a function of N[22]. The results are shown in Fig. 2(a)
and suggest that C↑grows linearly with Nand takes values
C↑=2+4m(m∈Z) for odd NandC↑=4m(m∈Z)f o r
even N. The careful study of whether this behavior persists for
arbitrary thickness goes beyond the scope of the current RapidCommunication, but importantly this numerical evidence isalready enough that we obtain a topological description of stepmodes for reasonably large systems, including all the systemsconsidered in this Rapid Communication. In particular, theresult for N=4 is consistent with th enumber of edge modes
in Figs. 1(d)and2(b).F o rλ
x=λy=0 the tiny gaps originally
present in the spectrum vanish completely.
These Chern numbers provide an interpretation for the
appearance of the step modes because they appear whenever
C↑is different on the two sides of the step [Figs. 1(e)–1(h)]
[22]. For a step separating even Nand odd N/Delta1C↑=2+4m
(m∈Z) and therefore at least two pairs of helical step modes
[27] exist at these steps in agreement with Refs. [ 12,13]
[Figs. 1(e) and1(g)]. These step modes are weakly gapped
because the spin-rotation symmetry is only present as anapproximate symmetry. However, we can use C
+to show that
121107-2TOPOLOGICAL PROPERTIES OF MULTILAYERS AND … PHYSICAL REVIEW B 100, 121107(R) (2019)
these gaps vanish in the limit N→∞ . Namely, by using a
symmetrized step construction shown in Figs. 1(f)and1(h)
and fixing Nso that C+=±2 on the different sides of the
step, we find that in a system containing steps on both surfacesthere exist |/Delta1C
+|=4 pairs of gapless edge modes protected
by the mirror symmetry. In the limit N→∞ the step modes at
the different surfaces are completely decoupled, and thereforeeach step supports two pairs of gapless edge modes. Interest-ingly, we find that (depending on N) also even-height steps
can exhibit |/Delta1C
+|=4o r|/Delta1C↑|=4|m|(m∈Z) pairs of step
modes consistent with the experiment [ 21].
In the case of a one-atom-high step and finite Nthe step
modes are weakly gapped even though C±are different on
two sides of the step because the step breaks the mirrorsymmetry and hybridizes the mirror blocks. However, the stepmodes can still be exactly gapless for certain widths N
yin
theydirection if λx=0 [Fig. 2(c)]. This effect comes from
the existence of an effective particle-hole symmetry whichtogether with a mirror symmetry gives rise to an antiunitarychiral symmetry S=Kσ
y⊗(2L2
x−1)⊗(i/Sigma1mx), where mx
is a mirror reflection with respect to the xplane interchanging
the sublattices [ 22]. Due to this symmetry the Hamiltonian
HN,Ny(kx) supports a Z2Pfaffian invariant, which protects a
1D Weyl point if it changes sign as a function of kx[28,29]. As
demonstrated in Fig. 2(c)this kind Weyl points can be realized
at the steps if λx=0.
Since we have now established the topological origin of the
step modes in the noninteracting system, we turn our attention
to the correlation effects (e.g., spin, charge, orbital, or su-
perconducting order), which are inevitably present due to the
large LDOS in the limit N/greatermuch1[11,14,15,22]. Our aim is to
show that there exists a mechanism for the appearance of theZBCP in the absence of superconductivity (without analyzing
the competition between different types of order [ 22]). We
require that the order parameter opens an energy gap, which
means that it breaks the symmetries associated with C
↑(↓)and
C±. Thus, we assume that there exists a magnetic instability
in the vicinity of the steps (due to magnetic impurities or
electron-electron interactions [ 22]) giving rise to a Zeeman
field HZ=h·/vectorσ. Because the step modes are approximately
spin polarized along the zdirection the directions of hwithin
the ( x,y) plane are efficient in opening an energy gap, and
due to spin-orbit coupling the gap depends on the direction of
hwithin the ( x,y) plane [ 22]. Therefore, the system realizes
an easy-axis ferromagnet and the topological defects are DWs
[30]. In the following we consider h=(hx,0,0).
To study the topological DW states we determine the low-
energy theory for a single step, the emergent symmetries,and the topological invariants. Although h
xbreaks TRS the
spectrum of a system with two steps still exhibits Kramersdegeneracy at k
x=π[Fig. 1(g)] due to a remaining NS TRS
T/prime(kx)=K12⊗(2L2
y−1)⊗g(kx)rz, where g(kx) is a diago-
nal matrix with entries e±ikx/2andrzdenotes πrotation with
respect to the zaxis [ 22].T/prime(kx) squares to +1(−1) at kx=0
(kx=π) which yields Kramers degeneracy only at kx=π.
We assign half of the states at kx=πto each step by selecting
from each Kramers’ doublet the state with larger projectionon each step. Our low-energy theory is obtained by expandingthe Hamiltonian around k
x=πin one of the subspaces of theprojected states [ 22]. By assuming λy=0 the system supports
ak-dependent mirror symmetry Mx(kx)=σx⊗(2L2
x−1)⊗
g(kx) and NS chiral symmetry S(kx)=iσy⊗13⊗/Sigma1mxg(kx)
[Fig. 3(a)][22].
We find that in the presence of these symmetries there
exist three topologically distinct phases shown in Fig. 3(b).
The trivial phase for |hx|<hcis separated from the two
nontrivial ones by the energy gap closings at hx=±hc.T h e
nontrivial phases are characterized by a NS chiral Z2invariant
ν=1[22,31], and the phases at hx<−hcandhx>hcare
topologically distinct because the band inversions occur inthe different mirror sectors M
x(π)=±1 [Fig. 3(b)]. It is not
ap r i o r i known whether the DWs between the topologically
distinct phases in this symmetry class support DW states [ 31].
However, our calculations show that sharp interfaces betweentrivial and nontrivial phases (two nontrivial phases) supportone (two) low-energy bound state(s) per DW [Figs. 3(c)–3(f)].
The number of low-energy states in each case is consistentwith the number of zero-energy DW states expected in thecase of smooth DWs with slowly varying spin textures [ 22].
Although these states resemble the zero-energy DW statesconsidered in the context of a Dirac equation [ 32] and the Su-
Schrieffer-Heeger (SSH) model [ 33–35], there is an important
difference because they are realized in a model belonging toa different symmetry class. Namely, the appearance of theDW breaks the symmetries, and therefore the energies ofthese states in the case of sharp DWs remain nonzero evenif the DWs are well separated [ 22]. Moreover, the energies
depend on the tight-binding parameters and in this sense theyresemble the topological DW states in systems with morecomplicated unit cells consisting of three or more atoms [ 36].
Nevertheless, for realistic system parameters the DW statesappear close to the zero energy [ 22].
Because the DW states have nonzero energy, in high-
resolution tunneling spectroscopy one would observe two splitpeaks in the conductance. However, already small broadeningof the energy levels leads to a single ZBCP [Figs. 3(g) and
3(h)][22]. Moreover, for sufficiently large density of DWs
the bound states hybridize and form a band inside the energygap leading to a single ZBCP where the height of the peakdepends on the density of DWs [Figs. 3(e) and3(h)]. The
ZBCP is robust against variations of the density because inanalogy to the SSH model [ 35] we expect that the DWs
are the lowest-energy charged excitations in the system, andtherefore small density of excess electrons (excess holes) isaccommodated in the system by increasing the number ofDWs, so that up to a critical variation of the density theFermi level is pinned to the energy of the DW states. Asimilar situation occurs in quantum Hall ferromagnets wherethe lowest-energy charged excitations are skyrmions [ 37]
which appear due to excess electrons and have been observedexperimentally [ 38]. Also the parametric dependencies of the
ZBCP and the energy gap are consistent with the experi-ment [ 16]. The increase of temperature suppresses the order
parameter and the energy gap. The external magnetic fieldbreaks the degeneracy of states with opposite magnetizationleading to a confinement between the DWs similarly as asymmetry-breaking term in the SSH model [ 35], so that the
number of DWs and the magnitude of the ZBCP decrease.
121107-3BRZEZICKI, WYSOKI ´NSKI, AND HYART PHYSICAL REVIEW B 100, 121107(R) (2019)
FIG. 3. (a) Schematic representation of symmetries Mx(kx)a n d S(kx) operating at a single step [ 22]. (b) The energies of the step states
atkx=πas a function of hxforNy=52 and λz=λx=0.5 eV . The states are colored according to the eigenvalues of Mx(π). There exist
three topologically distinct phases denoted as trivial phase ν=0 and nontrivial phases with ν=1±. The nontrivial phases are distinct from
the trivial phase due to NS chiral Z2invariant ν[22,31]. The lower index ±1 describes the subspace of Mx(π)=±1 where the band inversion
occurs. (c),(d) Spectrum and LDOS for a system with DWs separating trivial and nontrivial phases. The system supports four low-energy
bound states at the four DWs. The parameters are Nx=260, Ny=52, and λz=λx=0.5 eV . In the trivial (nontrivial) phase hx=0.003 eV
(hx=0.04 eV). (e),(f) Same for a system with DWs separating two nontrivial phases with opposite magnetizations. The system supports
eight low-energy bound states at the four DWs [blue dots in (e) and (f)]. If the number of DWs is increased to eight there exist16 low-energy
bound states [orange dots in (e)]. The parameters are Nx=160, Ny=140,|hx|=0.034 eV , λz=0.5e V ,a n d λx=0. (g),(h) The differential
conductance Gas a function of bias voltage Vdc[22] corresponding to spectra in (c) and (e), respectively. In all figures λy=0. The magnitude
of the gap and hxin our simulations are larger than in the experiment by Mazur et al. [16], but they necessarily decrease for larger Nbecause
the bulk gap decreases and the dispersion of the step states gets flatter [ 22].
Furthermore, we find that by increasing the Zeeman field the
energy gap of the system decreases [ 22]. Therefore, all the ob-
servations can be explained without requiring the existence ofsuperconductivity. Finally, the observation that magneticdopants enhance the ZBCP and the energy gap [ 16] makes
it more plausible that the effect originates from magneticinstability instead of superconductivity. The systematic anal-ysis of the correlated states which are consistent with theobservations [ 16] and the exploration of the possible common
origin of the zero-bias anomalies in various topological semi-conductors and semimetals [ 16–19] are interesting directions
for future research [ 22].
We thank T. Dietl, J. Tworzydło, Ł. Skowronek, G. Mazur,
M. Cuoco, R. Rechci ´nski, and R. Buczko for discussions.
The work is supported by the Foundation for Polish Sciencethrough the IRA Programme cofinanced by EU within theSG OP Programme. W.B. also acknowledges support byNarodowe Centrum Nauki (NCN, National Science Centre,Poland) Project No. 2016 /23/B/ST3/00839.
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121107-6 |
PhysRevB.81.224508.pdf | Pressure-induced superconductivity and Mott transition in spin-liquid /H9260-(ET) 2Cu2(CN) 3
probed by13C NMR
Y . Shimizu,1,2,*H. Kasahara,1T. Furuta,1K. Miyagawa,1K. Kanoda,1M. Maesato,2and G. Saito2,†
1Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
2Division of Chemistry, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
/H20849Received 29 November 2009; revised manuscript received 13 May 2010; published 9 June 2010 /H20850
Pressure-induced superconductivity and Mott transition in the spin-liquid Mott insulator /H9260-/H20849ET/H208502Cu2/H20849CN/H208503
have been investigated by13C NMR measurements. The Mott transition from the spin liquid to the Fermi
liquid is observed as a sudden decrease in 1 /T1T, where T1is the nuclear-spin-lattice relaxation time.
Pseudogap behavior is absent in the Fermi-liquid state, unlike in the metallic phase adjacent to the antiferro-magnetic Mott insulator. In the superconducting state, 1 /T
1have a cubic temperature dependence with no
Hebel-Slichter peak, which is consistent with non- s-wave superconducting pairing.
DOI: 10.1103/PhysRevB.81.224508 PACS number /H20849s/H20850: 74.20.Rp, 74.25.N /H11002, 74.70.Kn, 71.30. /H11001h
I. INTRODUCTION
Superconductivity in correlated electron systems usually
appears near the long-range magnetic order, as observed inthe copper oxide superconductors
1and heavy-fermion
systems.2On the other hand, the superconductivity emerging
from a quantum spin liquid has been sought since the pro-posal of a resonating valence bond state by Anderson.
3The
candidates have been geometrically frustrated systems, in-cluding triangular lattices, in which the superconductingpairing is theoretically nontrivial because of the depressedmagnetic correlation and the absence of the electron scatter-ing at preferential wave vectors on the circular Fermisurface.
4
One model material is the layered organic superconductor
/H9260-/H20849ET/H208502X, which has a half-filled triangular lattice. In the
distorted triangular lattice, the ground state of the Mott insu-lator has antiferromagnetic long-range order, as observed in
/H9260-/H20849d8-ET /H208502Cu/H20851N/H20849CN/H208502/H20852Br/H20851d8 denotes the full deuteration of
ET/H20852and/H9260-/H20849ET/H208502Cu/H20851N/H20849CN/H208502/H20852Cl. These materials have a Mott
transition at low temperatures and a superconducting transi-tion at T
c/H1101113 K under pressure.5For the ambient-pressure
superconductor /H9260-/H20849ET/H208502Cu/H20851N/H20849CN/H208502/H20852Br, the superconducting
order parameter is likely a spin-singlet dwave or an aniso-
tropic swave, according to the majority of experiments.6
Near the Mott boundary, the normal metallic state exhibits
the pseudogapped behavior in the13C nuclear-spin-lattice re-
laxation rate 1 /T1, demonstrating a possible precursor of
superconductivity.7In clear contrast, another Mott insulator
/H9260-/H20849ET/H208502Cu2/H20849CN/H208503, which has a more triangular lattice, ex-
hibits no long-range magnetic order down to 20 mK despitean antiferromagnetic exchange interaction J/H11011250 K.
8,9The
power-law temperature /H20849T/H20850dependence of 1 /T1at low tem-
peratures is indicative of low-lying spin excitations.9A finite
residual density of states has been observed by a specific-heat measurement,
10whereas thermal-conductivity measure-
ments support the opening of a spin gap.11The presence or
absence of a spinon Fermi surface is currently under activedebate.
12–16
Besides the nature of the spin-liquid state, the supercon-
ducting and metallic states emerging across the Mott transi-tion are not well understood, since the only known exampleis
/H9260-/H20849ET/H208502Cu2/H20849CN/H208503. In a pressure-temperature phase dia-
gram based on resistivity and1H NMR studies, a supercon-
ducting phase /H20849Tc/H110114K /H20850abuts the spin-liquid phase.17The
slope of the first-order Mott transition line in the phase dia-gram remains positive down to low temperatures,
17in con-
trast to the behavior of /H9260-/H20849ET/H208502Cu/H20851N/H20849CN/H208502/H20852Cl.18This im-
plies a larger spin entropy in the spin-liquid state than in theneighboring Fermi liquid. The superconducting pairing sym-metry in
/H9260-/H20849ET/H208502Cu2/H20849CN/H208503has been extensively theoretically
investigated.19–30The suppression of spin fluctuations dimin-
ishes superconductivity in a regular triangular lattice but thed
x2−y2pairing becomes favorable when the lattice is
distorted.19–22Even if a superconducting state appears, dx2−y2,
dx2−y2+idxy,dx2−y2+is,o r p-wave pairings are proposed for
the triangular lattice.23–28An exotic superconducting state
due to Amperean or triplet pairing may emerge from the spinliquid.
29,30
The previous1H NMR study under pressure could not
detect a superconducting state because the random orienta-tion of the polycrystalline sample suppressed the upper criti-
cal field H
c2.17Even in a superconducting state,1H NMR is
usually susceptible to vortex dynamics in layered
superconductors.31In this paper, we report13C NMR mea-
surements of a13C-enriched single crystal of
/H9260-/H20849ET/H208502Cu2/H20849CN/H208503under hydrostatic pressure. A large hyper-
fine coupling at13C sites and a precise alignment of the
magnetic field allowed us to observe quasiparticle excitationsof the superconducting state. We observed the T
3behavior of
1/T1in the superconducting state, as expected for an aniso-
tropic order parameter, and analogous to the other /H9260-/H20849ET/H208502X
members, whereas the Knight shift underwent a small butfinite decrease. The dynamic and static spin susceptibilitiesin the normal metallic state are also presented.
II. EXPERIMENTAL
To conduct13C NMR measurements of
/H9260-/H20849ET/H208502Cu2/H20849CN/H208503, the double-bonded carbon sites at the cen-
ter of the ET were selectively enriched with13C isotope to
98% concentration.32A single crystal 1 mm /H110031.5 mm
/H110030.05 mm in size was soaked into a pressure medium ofPHYSICAL REVIEW B 81, 224508 /H208492010 /H20850
1098-0121/2010/81 /H2084922/H20850/224508 /H208495/H20850 ©2010 The American Physical Society 224508-1Daphne 7373 and placed in a BeCu pressure cell. A pressure
of 0.4 GPa was applied at room temperature, slightly exceed-ing the critical pressure of the Mott transition.
17The pressure
cell was uniaxially rotated in the magnetic field with a pre-cision of 0.1°. The spin-echo signal was measured using a
/H9266/2-/H9270-/H9266pulse sequence under a static magnetic field H0
=2.0 or 3.5 T parallel to the conducting layer. The typical
length of the /H9266/2 pulse was 1 /H9262s and the interval between
the/H9266/2 and/H9266pulses, /H9270, was 20 /H9262s/H20849/H11270T1/H20850.T1was obtained
from the saturation recovery of the nuclear magnetization,which was well fitted by the single-exponential function, 1−M/H20849t/H20850/M/H20849/H11009/H20850=exp /H20849−t/T
1/H20850, where Mis the spin-echo inten-
sity integrated over all13C resonance lines to provide a suf-
ficient signal-to-noise ratio. The analysis did not qualitativelyaffect the Tdependence of 1 /T
1since T1averaging due to
spectral broadening and nuclear-spin-spin relaxation did notallow an independent determination of T
1of each constituent
line. The intensity of the spin-echo signal was much lowerthan that of the extrinsic ringing signal after the
/H9266pulse at
the low frequency of 21 MHz. We employed the time spec-trum soon after the saturation comb pulses /H20849comb 2 ms
/H9266/2-/H9270-/H9266/H20850as a background, which was effective in the
present experiments with T1of longer than 50 ms. The fre-
quency spectrum was obtained by Fourier transformation of
the time spectrum. The13C Knight shift Kwas defined as the
relative shift from the resonance frequency of the tetrameth-ylsilane reference sample.
III. RESULTS AND DISCUSSION
To probe the superconducting state by NMR, one should
align the magnetic field precisely parallel to conductinglayer, in order to minimize orbital pair breaking and maxi-mize H
c2. In weakly coupled superconducting layers, vorti-
ces are locked into the anion layer as Josephson vorticeswithout fluctuations of in-plane pancake vortices, whichwould cause an additional spin-lattice relaxation.
31We deter-
mined the parallel-field geometry in the superconductingstate by measuring the resonance frequency of the NMR tankcircuit,
/H9263LC,a tH0=0.2 and 2 T. Figure 1/H20849a/H20850shows /H9263LCplot-
ted against the angle between the magnetic field and the su-perconducting layer at 1.5 and 10 K, well below and aboveT
c, respectively. While /H9263LCis insensitive to the H0direction
at 10 K, a decrease in /H9263LCis observed in a narrow angle
range /H20849/H110110.2° bottom width /H20850at 1.5 K. Such a dip structure is
expected to appear in a parallel-field geometry /H20851the right
panel in Fig. 1/H20849a/H20850/H20852.33
/H9263LCwas measured on cooling the sample at various H0
parallel to the layers as shown in Fig. 1/H20849b/H20850. In the absence of
a static magnetic field, the sudden increase in /H9263LCbelow 3.8
K signifies a superconducting transition, in good agreementwith the resistivity measurement of T
c=4 K.17The applica-
tion of a magnetic field lowers Tcand reduces the /H9263LCin-
crease due to easier penetration of vortices.33,34The H0de-
pendence of the onset Tcmarked by arrows in Fig. 1/H20849b/H20850gives
theT-Hc2diagram for the in-plane magnetic field as shown
in the inset. An extrapolation of Hc2toT=0 yields Hc2/H208490/H20850
=10–12 T, somewhat exceeding the Pauli limiting field of7.5 T. NMR measurements were conducted at H
0=2 T and3.5 T, where Tc=3.5 K and 3.3 K, respectively.
Figure 2shows the Tdependence of13C NMR spectra
measured at H0=3.5 T /H20849Tc=3.3 K /H20850. There are four in-
equivalent13C sites: two ET /H20849labeled A and B in the right
panel of Fig. 2/H20850are inequivalent against the applied field and
each ET contains two inequivalent13C/H20849labeled 1 and 2 /H20850withH0
θsuperconducting
layerinsulating
layer
2T 10K2T 1.5K0.2T 1.5KνLC(arb. units)
θ(deg)0.2o
-2 -1 0 1 2
HC2||(T)
TC(K)νLC(arb. units)
12 3 456 7
T(K) (b)(a)
FIG. 1. /H20849Color online /H20850/H20849a/H20850Angular dependence of the resonance
frequency of the NMR tank circuit, /H9263LC, and a schematics configu-
ration of magnetic field. /H20849b/H20850Temperature dependence of /H9263LCat vari-
ous magnetic fields parallel to the layer. Inset: Hc2/H20648vsTc, as indi-
cated by the arrows in the main figure.
200 0 200 400 6002.4 K1.5 K
4.0 K
10 K
40 KSpin echo intensity (arb. units)
Shift from TMS (ppm)
ABA1B1 A2
B2H0
A1 A2+B1 B2
FIG. 2. /H20849Color online /H2085013C NMR spectra at 0.4 GPa with H0
=3.5 T parallel to the conducting layer. The colored vertical lines
are the centers of the fitting spectra /H20849the shaded spectra and solid
lines are the fit for each line and the sum, respectively /H20850. Bold ver-
tical lines indicate the /H9273spin=0 positions evaluated from the K-/H9273spin
plots in Fig. 3. Right figure: corresponding13C sites in ET
molecules.SHIMIZU et al. PHYSICAL REVIEW B 81, 224508 /H208492010 /H20850
224508-2different hyperfine couplings to electron spin. Each line
splits into a doublet due to nuclear dipole-dipole interaction/H20849Pake doublet /H20850, resulting in a total of eight lines. We ob-
served three doublets in the 40 K spectrum and the intensityof the middle doublet was twice as high as those of the otherdoublets because of accidental superposition of the A2 andB1 lines.
AsTis lowered from 40 to 10 K, the A2+B1 and B2
lines shift appreciably toward lower frequencies, reflectingtheTdependence of the spin susceptibility
/H9273spinin the nor-
mal state. Below 10 K, the spectra broadened, making thedefinition of Kambiguous. We defined Kby fitting the spec-
tra to a superposition of six Lorentzians /H20849colored solid and
dotted lines in Fig. 2/H20850with minimum parameters, assuming
that the spectral weights and nuclear dipole splitting are in-dependent of T. Spectrum analysis indicates that Kdecreases
for A2+B1 and B2 but slightly increases for A1, as shown inFig. 2. Therefore, the hyperfine coupling constant A
hfin-
creases in the order: Al, A2+B1, and B2.
Since the orbital contributions to Kare negligible in or-
ganic conductors that have well-separated orbital energy lev-els and negligible spin-orbit coupling, Kcan be expressed as
K=A
hf/H9273spin /N/H9262B+K0, where K0is the T-invariant component
including the chemical shift and the Pake doublet. To evalu-
ate/H9273spin, we determined Ahffor each13C site from K-/H9273spin
plots at ambient pressure with the same field direction as at
0.4 GPa, as shown in Fig. 3, using the known /H9273spinvalues.8
We assumed that Ahfis nearly independent of pressure, as
expected from the small lattice contraction of less than 4% at1.2 GPa in
/H9260-/H20849ET/H208502Cu/H20849NCS /H208502,35and from the absence of
symmetry breaking across the Mott transition. The linearityin the K-
/H9273spin plot gives Ahf=0.3 /H20849/H110060.03 /H20850T//H9262Band
0.5/H20849/H110060.05 /H20850T//H9262Bfor A2+B1 and B2, respectively, while
the intercept of the vertical axis gives chemical shifts of thePake doublet of 10 /H1100610 and 95 /H1100610 ppm, as marked by the
bold vertical lines in Fig. 2.
Using A
hf, we transformed Kat 0.4 GPa into /H9273spin,a s
shown in Fig. 4/H20849a/H20850. For comparison, /H9273spinof the Mott insula-
tors/H9260-/H20849ET/H208502Cu2/H20849CN/H208503and/H9260-/H20849d8-ET /H208502Cu/H20851N/H20849CN/H208502/H20852Br at am-
bient pressure are also displayed in Fig. 4/H20849a/H20850with the core
diamagnetic susceptibility already subtracted.7,8In the Mott
insulating phase at ambient pressure, /H9273spin of
/H9260-/H20849ET/H208502Cu2/H20849CN/H208503decreases smoothly with Tbelow 50 K,8although the values are much larger than those of
/H9260-/H20849d8-ET /H208502Cu/H20851N/H20849CN/H208502/H20852Br, which exhibits macroscopic
phase separation below 30 K into an antiferromagnetic insu-lating phase /H20849T
N=18 K /H20850and a metallic phase. At 0.4 GPa,
/H9273spinof/H9260-/H20849ET/H208502Cu2/H20849CN/H208503is remarkably insensitive to tem-
perature but continuously decreases across the Mott transi-tion at 10–20 K. Once
/H9273spinbecomes constant below 10 K, an
appreciable decrease is observed below Tc. While this could
be a signature of spin-singlet pairing, the decrease seemsinsufficient for conventional singlet pairing as shown in theinset of Fig. 4/H20849a/H20850. Note that the site-dependent Kshown in
Fig. 2rules out the effect of superconducting diamagnetic
shielding on the shift below T
c.
One possible reason for the insufficient decrease in Kis
the presence of triplet Cooper pairing. However, one mustconsider the rf heating problem that is sometimes encoun-tered in pulse NMR measurements of good conductors at lowtemperatures. In the superconducting state, the inelastic mo-tion of vortices can produce Joule heating. Although T
1mea-
surements would be unaffected by this heating due to thelong T
1/H20849/H110222 s below 4 K /H20850, the spin echo reflects the elec-
tronic temperature several tens to hundreds of microsecondafter the rf pulses, which is much shorter than T
1, and there-
fore could suffer from the heating problem. However, it was0 GPa
A1A2
+
B1B2
0200400600∆K(ppm)
χspin(emu/mol)0 24 6
FIG. 3. /H20849Color online /H20850Kvs/H9273spinplots for three13C sites, de-
fined in Fig. 2, at ambient pressure. The magnetic field was applied
in the same direction as in the experiments at 0.4 GPa.
0.00.10.20.30246
0.000.050.100.151/T1T(1/sK)
T(K)χ
spin(10-4emu/mol)
01 0 2 0 30 40 500.5 1.0 1.50.00.51.0
T/Tc(K)χ/χ(Tc)
Tc
Tcκ-(ET)2Cu2(CN)30.4GPa
κ-(d8-ET)2Cu[N(CN)2]Brκ-(d8-ET)2Cu[N(CN)2]Brκ-(ET)2Cu2(CN)30GPa
κ-(ET)2Cu2(CN)3
0GPaκ-(ET)2Cu2(CN)3
0.4GPa
(b)(a)
FIG. 4. /H20849Color online /H20850Temperature dependence of /H20849a/H20850spin sus-
ceptibility /H9273spinand /H20849b/H208501/T1T, for/H9260-/H20849ET/H208502Cu2/H20849CN/H208503at 0.4 GPa and
ambient pressure, and /H9260-/H20849d8-ET /H208502Cu/H20851N/H20849CN/H208502/H20852Br/H20849Ref. 7/H20850.T1at am-
bient pressure was obtained for the A site in a field normal to thelayer /H20849Ref. 9/H20850. The inset in /H20849a/H20850shows
/H9273spinvsTnormalized at Tc.PRESSURE-INDUCED SUPERCONDUCTIVITY AND MOTT … PHYSICAL REVIEW B 81, 224508 /H208492010 /H20850
224508-3difficult to determine the electronic temperature soon after
the rf pulse in the present experimental setup. One method ofminimizing rf heating is to measure, instead of the spin echo,the free-induction decay /H20849FID /H20850at low rf power to obtain
NMR spectra. Unfortunately, this method was difficult to ap-ply to the present measurements because the extrinsic rf ring-
ing signal obscured the weak FID signals with short T
2/H11569on
the order of several tens of microsecond. By separate experi-ments, however, we observed the rf power dependenceof the shift decrease below T
cin superconducting
/H9260-/H20849ET/H208502Cu/H20849NCS /H208502under pressure. Thus, the rf heating effect
was not ruled out even in the present measurements. Confir-mation of the possible mixing of triplet pairing requires fur-ther experiments at lower temperatures using lower rf power,which has not yet been accessible due to the small crystalsize. Another possible origin for the small change in Knightshift is sample inhomogeneity due to metal-insulator phaseseparation near the critical pressure of the Mott transition, as
observed in
/H9260-/H20849d8-ET /H208502Cu/H20851N/H20849CN/H208502/H20852Br below 30 K. As de-
scribed below, however, the profile of the nuclear relaxationcurve, which would reflect the phase mixture, remainedsingle exponential and unchanged across T
c, indicating no
appreciable phase separation.
The Tdependence of 1 /T1T, which measures the wave-
vector summation of the dynamical spin susceptibility, is dis-played in Fig. 4/H20849b/H20850. In the Mott insulating state, 1 /T
1Tin-
creases with decreasing T, indicating the growth of
antiferromagnetic correlations. At 0.4 GPa, 1 /T1Tdecreases
abruptly at 15 K, where the /H20849first-order /H20850insulator-metal tran-
sition was observed in the resistivity.17Hence the suppres-
sion of magnetic correlation coincides with the Mott transi-tion from a paramagnetic Mott insulator to a metal. Similarbehavior was observed in
/H9260-/H20849d8-ET /H208502Cu/H20851N/H20849CN/H208502/H20852Br at 30 K,
as also shown in Fig. 4/H20849b/H20850.7Once the system gets into the
metallic state, the Korringa’s relation, 1 /T1T=constant, is
preserved in the Trange between 4 and 10 K. This is in
contrast to a depression of 1 /T1Tin the metallic phase
of/H9260-/H20849d8-ET /H208502Cu/H20851N/H20849CN/H208502/H20852Br below 22 K.7Note that,
in the metallic state, the 1 /T1Tand/H9273spin values of
/H9260-/H20849ET/H208502Cu2/H20849CN/H208503 are larger than those of
/H9260-/H20849d8-ET /H208502Cu/H20851N/H20849CN/H208502/H20852Br. This can be attributed to a higher
density of states due to strong electron correlations and theabsence of a pseudogap. Nevertheless, the superconductingtransition temperature of
/H9260-/H20849ET/H208502Cu2/H20849CN/H208503is much
lower than those of /H9260-/H20849d8-ET /H208502Cu/H20851N/H20849CN/H208502/H20852Br and
/H9260-/H20849ET/H208502Cu/H20851N/H20849CN/H208502/H20852Cl under pressure. If the pseudogap be-
havior is related to short-range spin correlation in the neigh-boring Mott insulating phase, the absence of the pseudogapmay arise from frustrated spin correlation in the triangularlattice.
The low-temperature region of 1 /T
1is shown in Fig. 5,
where the upper inset shows the exponent /H9252in the stretched
exponential fit, exp /H20851−/H20849t/T1/H20850/H9252/H20852, to the recovery curve of the
nuclear magnetization. /H9252is close to unity and independent of
temperature across Tc, which rules out macroscopic phase
separation across the first-order Mott transition under thepresent pressure and magnetic fields. A slight deviation from
/H9252=1 stems from averaging of inequivalent13C lines with
differing T1values. With decreasing T,1 /T1decreases lin-
early at high temperatures, as shown in the lower inset, butbegins to decrease more steeply below Tcof 3.5 K. A coher-
ence peak just below Tc, which would indicate a full-gap
superconductor, is not corroborated within the current ex-perimental resolution. Instead, T
3behavior is seen below Tc
down to 1.5 K. The 1 /T1behavior is consistent with a
non-s-wave anisotropic superconducting gap with line nodes.
This feature, the T3dependence persisting up to Tc, was also
observed in CeIrIn 5/H20849Ref. 36/H20850and /H20849TMTSF /H208502ClO 4/H20849Ref. 37/H20850,
while 1 /T1of/H9260-/H20849ET/H208502Cu/H20851N/H20849CN/H208502/H20852Br/H20849Refs. 5and38/H20850and
YBa 2Cu3O7had a steep drop just below Tc. The difference
may be attributed to the superconducting coupling strength,/H9004/k
BTc, where /H9004andkBdenote the superconducting order
parameter and the Boltzmann constant, respectively. Assum-ingd-wave pairing, for example, the present data are close to
the calculations for /H9004/k
BTc=2.2, while /H9004/kBTc=4.0 gives a
sharp drop in 1 /T1near TcandT3behavior for T/Tc/H110210.5.4
IV. CONCLUSION
We reported13C NMR investigations of the Mott transi-
tion and superconducting state in /H9260-/H20849ET/H208502Cu2/H20849CN/H208503under
pressure, which is at ambient pressure the spin-liquid Mottinsulator with a half-filled triangular lattice. The Mott tran-sition from the spin-liquid state to the Fermi-liquid state wasaccompanied by a strong suppression of the dynamical spinsusceptibility probed by the nuclear-spin-lattice relaxationrate, but the static spin susceptibility was less affected, andmaintained a high value above the superconducting transitiontemperature. The Fermi-liquid metallic state followed Pauliparamagnetic behavior and Korringa’s law, without openinga pseudogap, unlike the metallic phase neighboring the anti-ferromagnetic insulator. In the superconducting state, thenuclear-spin-lattice relaxation rate followed a T
3depen-
dence, as expected for a nodal order parameter. The finite0.010.11
0 5 10 150.00.51.0
β
T(K)
024680.00.51.01/T1(1/s)
T(K)T31/T1(1/s)
T(K)11 0
FIG. 5. /H20849Color online /H20850Log plot of13C1 /T1at 0.4 GPa at H0
=2.0 T /H20849closed circle /H20850and 3.5 T /H20849open circles /H20850applied parallel to
the conducting layer. Broken and dotted lines are calculated resultsfor/H9004/T
c=2.2 and 4.0 in the d-wave pairing. Lower inset: a linear
plot of 1 /T1. Upper inset: the exponent /H9252of the stretched exponen-
tial fit to the recovery curve at 3.5 T. Circles and squares are ob-tained for A1 and all
13C lines.SHIMIZU et al. PHYSICAL REVIEW B 81, 224508 /H208492010 /H20850
224508-4decrease in spin susceptibility below Tcmay indicate spin-
singlet Cooper pairing While the possibility of mixed tripletpairing is intriguing, a possible extrinsic effect due to rf heat-ing should be ruled out by further experiments.
ACKNOWLEDGMENTS
We thank A. Kawamoto for synthesis of13C-substitutedET, and P. A. Lee and T. Senthil for valuable discussions.
This work was partially supported by Grants-in-Aid for Sci-entific Research in the Priority Area /H20849Grant No. 17071003 /H20850
and Innovative Area /H20849Grant No. 20110002 /H20850from the MEXT,
Grants-in-Aid for Scientific Research /H20849A/H20850/H20849Grant No.
20244055 /H20850and /H20849C/H20850/H20849Grant No. 20540346 /H20850from the JSPS,
and by Global COE Program: Global Center of Excellencefor the Physical Sciences Frontier /H20849No. G04 /H20850.
*Present address: Institute for Advanced Research, Nagoya Univer-
sity, Japan.
†Present address: Meijo University, Nagoya, Japan.
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224508-5 |
PhysRevB.88.045423.pdf | PHYSICAL REVIEW B 88, 045423 (2013)
Influence of oxygen and hydrogen adsorption on the magnetic
structure of an ultrathin iron film on an Ir(001) surface
Franti ˇsek M ´aca,1,*Josef Kudrnovsk ´y,1V´aclav Drchal,1and Josef Redinger2
1Institute of Physics ASCR, Na Slovance 2, CZ-182 21 Praha 8, Czech Republic
2Department of Applied Physics, Vienna University of Technology, Gusshausstrasse 25/134, 1040 Vienna, Austria
(Received 7 June 2013; published 15 July 2013)
We present a detailed ab initio study of the electronic structure and magnetic order of an Fe monolayer on
the Ir(001) surface covered by adsorbed oxygen and hydrogen. The results are compared to the clean Fe/Ir(001)system, where recent intensive studies indicated a strong tendency towards an antiferromagnetic order andcomplex magnetic structures. The adsorption of an oxygen overlayer significantly increases interlayer distancebetween the Fe layer and the Ir substrate, while the effect of hydrogen is much weaker. We show that theadsorption of oxygen (and also of hydrogen) leads to a p(2×1) antiferromagnetic order of the Fe moments,
which is also supported by an investigation based on a disordered local moment state. Simulated scanningtunneling images using the simple Tersoff-Hamann model hint that the proposed p(2×1) antiferromagnetic
order could be detected even by nonmagnetic tips.
DOI: 10.1103/PhysRevB.88.045423 PACS number(s): 75 .30.−m, 81.20.−n
I. INTRODUCTION
Magnetic overlayers, i.e., thin films of magnetic materials
on a nonmagnetic substrate, prepared by molecular beamepitaxy, represent a new type of material with potentialtechnological applications. Among a number of such systems,recently the fcc-Fe/Ir(001) overlayer system has attractedboth experimental and theoretical attention. Thin overlayerswere grown successfully with negligible Fe-Ir intermixing.A metastable unreconstructed monolayer (ML) of Fe on theIr(001) surface could be prepared
1,2and studied theoretically
byab initio methods.3,4Both theoretical investigations pre-
dicted a complex magnetic ground state. It should also be notedthat complex chiral magnetic structures were also observed fora related bcc-Fe/W(001) system.
5
Concerning adsorption, oxygen and hydrogen are the two
most likely adsorbates to exist on Fe/Ir(001) surfaces. Theinvestigation of their influence on the structural, electronic,and magnetic ground state is thus an interesting and relevantproblem. We like to mention here a related recent experimentaland theoretical study of an oxygen covered bcc-Fe(001)surface,
6where experiment confirms the formation of a well-
defined monolayer where the oxygens occupy the fourfoldcoordinated hollow positions and stabilize the ferromagnetic(FM) state. The situation is richer and even more interesting inthe present fcc-Fe/Ir(001) case because oxygen adsorption caninfluence the Fe-Ir distance and thus also the magnetic groundstate which sensitively depends on.
3In contrast, hydrogen,
which is likely to be present due to the preparation of theFe/Ir(001) samples,
1adsorbs and hybridizes differently from
oxygen.
The ground state structure including possible layer re-
laxations is determined by conventional ab initio tools (the
supercell V ASP method,7for more details see the next
section). The magnetic ground state is predicted on thebasis of the Heisenberg model with exchange interactionparameters determined from first principles. Two approachesare used: (i) in the first one, two effective pair interactions areobtained utilizing the total energies of the ferromagnetic (FM)and two antiferromagnetic (AFM) configurations [ c(2×2)
andp(2×1)] for the respective relaxed structural models
calculated by V ASP; and (ii) the disordered-local moment(DLM) state is used as a reference state for the extractionof exchange interactions as done in a previous paper
3for
the clean Fe/Ir(001) system. The input electronic structurefor the DLM state employs a realistic semi-infinite geometryand the tight-binding linear muffin-tin orbital (TB-LMTO)method in the Green function formulation. The DLM stateis simulated in the alloy analogy approximation (for moredetails see Ref. 3and references therein). We used the
geometry as obtained by the atomic force minimization ofV ASP and chose the radii of the atomic spheres in order tominimize their overlap. The above two approaches for theestimation of the exchange interactions are complementaryto each other, the former one is more accurate, but since itemploys only three magnetic configurations, its predictionsare limited to the simplest magnetic structures and morecomplex configurations may thus be missed. We demonstratethis for a monolayer of Fe on the Ir(001) surface. The latterapproach, based on the DLM method, does not assume anyspecific magnetic structure and takes into account a largenumber of exchange integrals (which in a two-dimensionalcase decay more slowly with distance than in the bulk case
8).
The DLM approach using the TB-LMTO method is, however,less accurate, in particular for the case of small interlayeradsorbate-iron distances. Finally, STM images simulating theconstant current measuring mode of STM were calculatedusing the Tersoff-Hamann approximation,
9where tip-sample
interactions are neglected and thus the tip is supposed to followthe contours of a constant density of states contained in anenergy interval given by E
Fand the applied bias voltage VBias.
II. COMPUTATIONAL DETAILS
First-principles density functional theory calculations were
performed using the Vienna ab initio simulation package
VA S P7using the projector augmented wave scheme10and
045423-1 1098-0121/2013/88(4)/045423(7) ©2013 American Physical SocietyM´ACA, KUDRNOVSK ´Y, DRCHAL, AND REDINGER PHYSICAL REVIEW B 88, 045423 (2013)
local density approximation (LDA) as given by Perdew-
Zunger (Ceperly-Alder).11Repeated asymmetric slabs with
seven layers of Ir and a single Fe monolayer on one sideand also symmetric slabs with 11 substrate layers andadsorbate layers on both sides were used, separated by at least19˚A vacuum. Three top layer distances have been relaxed,
the remaining interlayer distances were fixed to the bulkexperimental value for the Ir crystal (1.92 ˚A). The calculated
relaxations turned out to be essentially the same for bothsetups. We have tested the FM, c(2×2) AFM, and p(2×1)
AFM arrangements, all performed with four Fe/Ir atoms in thetwo-dimensional supercell to guarantee a reliable comparisonof total energies. Technically we have used a Brillouin zonesampling with 100–200 special kpoints in the irreducible
two-dimensional wedge. The allowed error in total energy was0.05 meV . The total force on single atoms was in every casesmaller than 2.5 meV /˚A.
The input electronic structure for the DLM state was deter-
mined for a realistic semi-infinite geometry in the frameworkof the TB-LMTO method and the Green function formulation.The same LDA functional as in V ASP calculations was used
11
together with layer relaxations as obtained by V ASP andatomic sphere radii chosen to minimize their overlap. Thevacuum above the overlayer was simulated by empty spheres(ES). Electronic relaxations were allowed in three emptysphere layers adjoining the oxygen overlayer, in the oxygenand iron layers, and in five adjoining Ir substrate layers. Thisfinite system was sandwiched self-consistently between thefrozen semi-infinite fcc-Ir(001) bulk and the ES vacuum spaceincluding the dipole surface barrier. For more details see Ref. 3
and references therein.
The effective exchange integrals J
1andJ2were estimated
in the total energy model from total energies calculated for
three different magnetic configurations as follows:
J1=1
8/bracketleftbig
Etot
c(2×2)(AFM) −Etot
c(2×2)(FM)/bracketrightbig
,
(1)
J2=1
8/bracketleftbig
Etot
p(2×1)(AFM) −Etot
p(2×1)(FM)−4J1/bracketrightbig
,
where corresponding energies were obtained for the calculated
equilibrium atomic structure and indicated magnetic phase.We note that on the square lattice the most probable magneticconfigurations are employed.
In the DLM model the exchange integrals J
Fe,Fe
i,j between
sitesi,jin the magnetic overlayer are expressed as follows
(the generalized Liechtenstein formula) (see, e.g., Ref. 12):
JFe,Fe
i,j=1
4πIm/integraldisplay
CtrL/bracketleftbig
/Delta1Fe
i(z)g↑
i,j(z)/Delta1Fe
j(z)g↓
j,i(z)/bracketrightbig
dz.
(2)
Here the trace extends over s, p, anddbasis sets, the quantities
/Delta1Fe
iare proportional to the calculated exchange splittings, and
the Green function gσ
i,jdescribes the propagation of electrons
of a given spin ( σ=↑,↓) between sites i,j, both in the
magnetic layer and via the Ir substrate.
Once the exchange interactions are known, we construct
the corresponding two-dimensional (2D) Heisenberg model inwhich only interactions between iron moments are includedexplicitly and the interactions with iridium atoms and with theadsorbate are included indirectly via self-consistent electronicstructure calculations. In the next step we determine its latticeFourier transform J( q) using calculated exchange interactions
up to 90 nearest neighbors. The wave vector q
0in the 2D
Brillouin zone at which it acquires its maximum determinesthe theoretically predicted magnetic ground state.
3
The STM simulations using the Tersoff-Hamman
approach9were done with V ASP using the optimum geometry
as determined before. A fine k-point mesh of 20 ×40×1o n
an/Gamma1-centered grid was used to sample the 2D Brillouin zone
and the energy cutoff was increased up to 600 eV to ensurea smooth numerical representation even for the small valuesof the charge density in the vacuum region. Energy slicesaccording to the applied bias voltage, i.e., states between E
F
andVBias, were considered. A typical charge density contour of
10−6e˚A−3puts a presumed tip at a distance of ∼3–4 ˚A above
the terminal O atoms, a value not too far from experimentaldistances. In order to simulate experimental conditions usingnonmagnetic tips, the images are obtained from spin averageddensities.
III. RESULTS AND DISCUSSION
A. Atomic structure
In the following we present the results for possible ground
state adsorbate structures. All calculations here were doneusing the V ASP-LDA method and the experimental bulklattice constant for fcc Ir to facilitate a direct comparisonwith our previous study.
3This choice of exchange-correlation
potential may lead to some underestimation of the absolutevalues of interlayer distances for the adsorbates since theexperimental value (3.84 ˚A) is slightly larger than the
theoretical one (3.82 ˚A). On the other hand, the trends
like, e.g., an increase/decrease of interlayer distances dueto the adsorbate are predicted correctly. In Table Iwe
summarize layer relaxations for various investigated systems.Only the results for configurations minimizing the total energyare presented.
It is well known that oxygen atoms favor the occupation of
the fourfold hollow Fe sites (see, e.g., Ref. 6and references
therein). This site preference is also reproduced by ourV ASP calculations, which put the O atoms directly abovethe substrate Ir atoms. Figure 1shows that for 1 ML oxygen
coverage a p(2×1) AFM magnetic order of Fe magnetic
moments is preferred ( E
c(2×2)
AFM>E FM>Ep(2×1)
AFM ,/Delta1E tot=
TABLE I. Calculated interlayer distances dijbetween top three
sample layers for ground state magnetic configurations of Fe/Ir(001),O on Fe/Ir(001), and H on Fe/Ir(001) in the AFM states. b
irepresents
the buckling of the atoms in layer i. Oxygen is adsorbed in a hollow
site and hydrogen on a favorable bridge position. LEED experimentalvalues are taken from Refs. 1and2.
d12(˚A)d23(˚A)d34(˚A)d45(˚A)b3(˚A)
c(2×2)Fe 1.55 1.97 1.88 1.92 0.00
1M LOo n p(2×1)Fe 0.59 1.98 1.86 1.90 0.00
0.5 ML O on p(1×2)Fe 0.72 1.74 1.91 1.92 0.17
0.5 ML H on p(2×1)Fe 1.12 1.59 1.92 1.92 0.17
Fe (expt.) 1.69 1.96 1.91 1.92 –
H on Fe (expt.) – 1.72 1.94 1.91 –
045423-2INFLUENCE OF OXYGEN AND HYDROGEN ADSORPTION ... PHYSICAL REVIEW B 88, 045423 (2013)
AFM p(2x1)
ΔE=38meVAFM c(2x2)
ΔE=25meVFM
ΔE=7meVAFM p(1x2)
ΔE=0meV0.5 MLAFM p(2x1)
ΔE=0meV1.0 MLAFM c(2x2)
ΔE=76meVFM
ΔE=38meV
FIG. 1. Schematic structure of 1 ML (top row) and 0.5 ML oxygen
(bottom row) on a magnetic Fe/Ir(001) surface. Squares mark thetopmost Ir substrate atoms, circles mark the Fe atoms (full/empty
denote up/down spin), and white triangles mark the oxygen atoms.
The energy difference per Fe atom with respect to the lowest energysolution is indicated beneath each configuration.
EFM−Ep(2×1)
AFM =38 meV /Fe atom). For a 0.5 ML oxygen
coverage diverse geometrical arrangements are possible.
We considered c(2×2)O and p(2×1)O adlayers and
compared minimum total energies ( Emin) for the FM and
three different AFM configurations: c(2×2) AFM, p(2×1)
AFM, and p(1×2) AFM magnetic ordering of Fe moments.
Oxygen atoms prefer a p(2×1) type overlayer structure
(/Delta1E tot=Ec(2×2)O
min −Ep(2×1)O
min =65 meV /O atom), the lowest
total energy has been found for the p(1×2) AFM magnetic or-
dering of Fe moments ( Ep(2×1)
AFM >Ec(2×2)
AFM>E FM>Ep(1×2)
AFM ,
/Delta1E tot=EFM−Ep(1×2)
AFM =7m e V /Fe atom).
In contrast to oxygen, hydrogen atoms prefer Fe-bridge sites
on the Fe/Ir(001) surface as shown schematically in Fig. 2.
Similar to oxygen adsorbates, different geometrical ar-
rangements are possible for a 0.5 ML coverage. In our modelcalculations we considered two H adlayer geometries, H atomsoccupying neighboring (N-H) or next-neighboring Fe-bridge(NN-H) sites. Again the energies for FM and three AFMordered states are compared, i.e., for a c(2×2) AFM, a
p(2×1) AFM, and a p(1×2) AFM magnetic ordering of Fe
moments. Our calculations show that H atoms prefer to occupynext-neighboring (NN-H) Fe-bridge sites ( /Delta1E
tot=EN-H
min−
ENN-H
min=58 meV /H atom) and the ground state is obtained
for ap(2×1) AFM magnetic ordering of Fe moments (H
atoms between parallel Fe moments). Figure 2shows the
magnetic ordering with Ep(1×2)
AFM >E FM>Ec(2×2)
AFM>Ep(2×1)
AFM ,
and/Delta1E tot=Ec(2×2)
AFM−Ep(2×1)
AFM =5m e V /Fe atom). The opti-
mized geometry further showed that an unsymmetrical (N-H)occupation of bridge sites on the Fe/Ir(001) surface inducesconsiderable buckling in the subsurface Ir layer ( b
3=0.17˚A).
We suppose that under real conditions the hydrogen
adsorption starts for lower coverage ( /Theta1≈0) with a random
occupation of bridge sites. By increasing the H coverage(/Theta1→1) the ordering of adsorbed H atoms starts to support
thep(2×1) magnetic structure. Nevertheless, the formation
of domains can be expected rather than a long-range order.NN-H N-H
ΔE=58meV
AFM p(2x1)
ΔE=0meVAFM c(2x2)
ΔE=5meVFM
ΔE=76meVAFM p(1x2)
ΔE=110meVΔE=0meV
0.5 ML
FIG. 2. Schematic structure of 0.5 ML hydrogen occupying
Fe-bridge sites on a magnetic Fe/Ir(001) surface. Squares mark thetopmost Ir substrate atoms, circles mark the Fe atoms (full/empty
denote up/down spin), and white triangles mark the hydrogen atoms.
Upper panel: Hydrogen atoms occupying either next-neighboring(NN-H) Fe-bridge sites (left frame) or less stable neighboring sites
(right frame) on a p(2×1) antiferromagnetic Fe/Ir(001) surface.
Lower panel: Different magnetic configurations for hydrogen occu-pying next-neighboring Fe-bridge sites. The energy differences /Delta1E
with respect to the minimum energy solution is given per H atom in
the upper and per Fe atom in the lower panel.
B. Electronic structure
Here we wish to discuss the changes induced by hydrogen
and oxygen adsorption on the Fe/Ir(001) surface as reflected inthe local density of states (LDOS). We only show results for thestructural and magnetic configurations with the lowest energyas determined in the previous subsection. The changes inducedby the H and O adsorption is best discussed by comparing withthe clean reference Fe/Ir(001) system for which in Refs. 3and4
a complex magnetic ground state (chiral magnetic structure)was predicted.
In Fig. 3we compare the LDOS for a c(2×2) AFM ordered
Fe on Ir(001) with the Fe/Ir(001) system in the DLM state,which can be considered as a “disordered” AFM state. In orderto exclude the effects of different interlayer distances, the samegeometry was used in both cases. We find a good overallagreement between both LDOS (main peaks, the widths),which means that if we find relevant changes between thereference Fe/Ir(001) system and its counterpart with adsorbedO or H atoms, these changes reflect mostly changes inducedby adsorbate-Fe hybridization
6rather than small differences
between similar magnetic configurations.
The effect of the oxygen adsorption on Fe/Ir(001) is shown
in Fig. 4for 1 ML O and it is compared with the reference
c(2×2) Fe on the Ir(001) surface (no oxygen). In both cases
fully relaxed geometries were used. The most important effectshown in Fig. 4is a strong change of the LDOS features
due to the oxygen-iron hybridization. The effect is strongerfor majority states as compared to minority states similar tooxygen-covered bcc-Fe(001).
6In Fig. 5we show the LDOS
for an ordered half-monolayer of oxygen.
While the overall shape of the LDOS is similar for the two
oxygen coverages, some changes can be traced down in thevicinity of the Fermi level (see insets in Figs. 4and5showing
045423-3M´ACA, KUDRNOVSK ´Y, DRCHAL, AND REDINGER PHYSICAL REVIEW B 88, 045423 (2013)
-4-2024
-3 -2 -1 0 1LDOS [states/eV]
Energy [eV]majority
minority
FIG. 3. Spin resolved local density of states (LDOS) of an Fe
atom in Fe/Ir(001): (i) c(2×2) AFM (full lines), and (ii) DLM solu-
tion (dotted lines) for the experimental geometry and EFset to zero.
in detail the density for minority states). The changes in the
LDOS due to the oxygen-iron wave function hybridization as
-4-2024
-3 -2 -1 0 1LDOS [states/eV]
Energy [eV]majority
minority
FIG. 4. (Color online) Spin resolved local density of states
(LDOS) of an Fe atom of antiferromagnetic p(2×1) Fe/Ir(001)
covered by an oxygen monolayer (full lines) compared to the clean
antiferromagnetic c(2×2) Fe/Ir(001) system (dotted lines). The inset
shows the vicinity of the Fermi level ( EF=0) for minority states.-4-2024
-3 -2 -1 0 1LDOS [states/eV]
Energy [eV]majority
minority
FIG. 5. (Color online) The same as in Fig. 4but for a p(1×2)
ordered half-monolayer oxygen coverage (full lines).
described above are a precursor of differences in their magnetic
stability (see Fig. 8below).
The effect of hydrogen adsorption on the LDOS is similar
to oxygen as shown for the adsorption of 0.5 ML H in Fig. 6.
-4-2024
-3 -2 -1 0 1LDOS [states/eV]
Energy [eV]majority
minority
FIG. 6. (Color online) Spin resolved local density of states
(LDOS) of an Fe atom of antiferromagnetic p(2×1) Fe/Ir(001)
covered by half a monolayer of hydrogen (full line) compared to
the clean antiferromagnetic c(2×2) Fe/Ir(001) system (dotted lines)
andEFset to zero.
045423-4INFLUENCE OF OXYGEN AND HYDROGEN ADSORPTION ... PHYSICAL REVIEW B 88, 045423 (2013)
-6-4-2 0 2 4J(q) [mRy]
q vectorΓ⎯X⎯M⎯Γ⎯VASP E 2
LMTO E 2
LMTO Li 2
LMTO Li 90
FIG. 7. (Color online) Lattice Fourier transform of the real-space
exchange interactions JFe,Fe
ij,J(q/bardbl), for the clean reference Fe/Ir(001)
system in the relaxed minimum energy geometry obtained from total
energy and DLM approaches. Total energy approach using V ASP(V ASP E 2) and TB-LMTO (LMTO E 2). DLM approach using TB-
LMTO and just two exchange integrals (LMTO Li 2) or TB-LMTO
with 90 exchange integrals (LMTO Li 90).
We again see mostly a broadening of LDOS peaks due to the
Fe-H interaction.
C. Magnetic phase stability
We study the magnetic phase using the lattice Fourier
transform of the real-space exchange integrals estimated usingthe total energy and DLM models, as shown in Fig. 7.W e
start with the reference case of the Fe/Ir(001) system studiedpreviously in the framework of the DLM model,
3but compare
it here with its total energy model counterparts [see Eq. (1)].
The corresponding total energies have been obtained both fromV ASP and TB-LMTO calculations. The DLM approach isshown here for two cases with vastly different numbers ofexchange integrals included in the lattice Fourier transform,namely for 90 integrals and for just the two leading terms (twoshells of neighbors). It should be noted that the DLM exchangeintegrals correspond to bare ones while those obtained fromtotal energies are effective ones which in some sense combineall of them into two terms.
We see an overall good agreement between both ap-
proaches, nevertheless in the full DLM approach (90 shells)the ground state moves from the antiferromagnetic c(2×2)
state (the ¯M-ordering vector) to a more complex magnetic
state with an ordering vector lying on the ¯X-¯Mline. We also
note that both the p(2×1) AFM state (the ¯X-ordering vector)
andc(2×2) AFM state are energetically close, while the
ferromagnetic state (the ¯/Gamma1-ordering vector) is energetically
higher [due to the adopted convention for the HeisenbergHamiltonian
3the lowest energy corresponds the maximum
of the J(q/bardbl) curve].-6-4-2 0 2 4J(q) [mRy]
q vectorΓ⎯X⎯M⎯Γ⎯1 ML O E 2
0.5 ML O E 2
0 ML O E 2
FIG. 8. (Color online) Lattice Fourier transform of real-space
exchange interactions JFe,Fe
ij,J(q/bardbl), for clean antiferromagnetic
c(2×2) Fe/Ir(001) (0 ML O E 2) and covered by half a monolayer
of oxygen (0.5 ML O E 2) and an oxygen monolayer (1 ML O E 2).The oxygen atoms occupy fourfold Fe hollow sites.
1. O and H adsorption: Total energy model
The lattice Fourier transform for the oxygen adsorption is
shown in Fig. 8and compared with the reference oxygen-free
case.
We see that 1 ML oxygen induces a p(2×1) antiferromag-
netic state (the ¯X-ordering vector) in contrast to the c(2×2)
antiferromagnetic state (the ¯M-ordering vector) originally
considered for the clean surface. The high stability of thep(2×1) state is strongly reduced for a half-monolayer oxygen
coverage. Interestingly the ferromagnetic state is energeticallyrather close to the ground state, which can be ascribed tothe interplay between Fe-O hybridization which promotes ap(2×1) state and the increase of the Fe-Ir distance upon O
adsorption. The ferromagnetic state is the ground state forlarge Fe-Ir interlayer distances in the Fe/Ir(001) system.
3
The energetic ordering can be understood, at least qualita-
tively, by applying the Kanamori-Goodenough-Anderson13–15
rules in a simple way. These rules predict that Fe-O-Fe
bond angles of 90◦favor a ferromagnetic (FM) coupling
of the Fe atoms, while Fe-O-Fe bond angles of 180◦favor
an antiferromagnetic (AFM) one. We apply now these rulesfor the 1 ML O overlayer as depicted in the top panel ofFig. 1. The preferred magnetic coupling is a result of magnetic
interactions between neighboring Fe atoms (Fe-O-Fe bondangles 85
◦) and next-neighboring Fe atoms (Fe-O-Fe bond
angles 146◦). For the present purpose, the values of 85◦
and 146◦for the calculated minimum energy geometry are
sufficiently close to the ideal values of 90◦and 180◦. Summing
up neighboring and next-neighboring Fe couplings one easilyfinds that for the most stable AFM p(2×1) state one half
of the 85
◦couplings are favorable while the other half are
unfavorable, whereas both 146◦couplings are favorable. In
case of the FM configuration the favorable 85◦couplings are
offset by unfavorable 146◦FM couplings. For AFM c(2×2)
045423-5M´ACA, KUDRNOVSK ´Y, DRCHAL, AND REDINGER PHYSICAL REVIEW B 88, 045423 (2013)
-4-2 0 2 4J(q) [mRy]
q vectorΓ⎯
X⎯
M⎯
Γ⎯0 ML H E 2
0.5 ML H E 2
FIG. 9. (Color online) Lattice Fourier transform of real-space
exchange interactions JFe,Fe
ij,J(q/bardbl), for the Fe/Ir(001) system covered
by half a monolayer of hydrogen (0.5 ML H E 2). The magnetic
ground state is obtained for an antiferromagnetic p(2×1) ordering of
Fe the moments. The hydrogen-free clean antiferromagnetic c(2×2)
Fe/Ir(001) system (0 ML H E 2) is shown for comparison.
all couplings are unfavorable, indicating a predominance of
Fe-O-Fe induced couplings over the substrate related c(2×2)
AFM coupling of the Fe atoms, which is also offset by theincreased Fe/Ir distance upon O adsorption. In summary onecan safely state that oxygen adsorption on the Fe/Ir(001)surface strongly influences the magnetic ordering in the Femonolayer. Turning now to the influence of hydrogen adsorbedat the preferred Fe-bridge sites on the magnetic state, the resultsfor the lattice Fourier transform are summarized in Fig. 9.
A weak preference for the antiferromagnetic p(2×1)
state ( ¯X-ordering vector) is seen, as expected from the total
energy calculations but the antiferromagnetic c(2×2) state
(¯M-ordering vector) is energetically quite close. These results
suggest that the effect of H adsorption on the magnetic stateof Fe/Ir(001) is weaker as compared to the O adsorption. Thisis also accompanied by a weaker influence on the geometricalstructure.
2. O adsorption: DLM model
In this subsection we present results for the magnetic
stability of the oxygen monolayer as determined from theDLM model. We wish to point out that the TB-LMTO-DLMmodel employs the spherical charge approximation which isless accurate as compared to the full potential methods likeV ASP. We have shown, however, that for layer relaxationsup to 15% of the interlayer distance [the case of Fe/Ir(001)
3]
the results obtained using the total energy and TB-LMTOmethods are in good agreement (see Fig. 7and discussion
therein). However, the situation for an O monolayer is worseas the O-Fe interatomic distance amounts to only 2.01 ˚A( a s
compared to the host interatomic distance of 2.72 ˚A). The
problem in our TB-LMTO model consists of the division of-8-6-4-2 0 2 4 6 8J(q) [mRy]
q vectorΓ⎯
X⎯
M⎯
Γ⎯1 ML O E 2
1 ML O Li 2
1 ML O Li 90
FIG. 10. (Color online) Lattice Fourier transform of the real-
space exchange interactions JFe,Fe
ij,J(q/bardbl) for the Fe/Ir(001) covered
by the 1 ML of oxygen determined in the framework of the DLMapproach is compared to results from a total energy approach using
V ASP (1 ML O E 2). Compared are two different TB-LMTO
based DLM approaches, one with just two exchange integrals(LMTO Li 2) and the other with all 90 calculated exchange integrals
(LMTO Li 90).
the space into vacuum, oxygen, and iron spheres, which is
not unique. We have therefore used the same approach as inour previous work.
3A similar approach used recently for the
oxygen adsorption on bcc-Fe(001) has lead to a qualitativeagreement with experiment. In Fig. 10we present results for
1 ML O on Fe/Ir(001) and compare them with the results ofthe supercell V ASP approach as presented in Fig. 8.
Like in Fig. 7, we tested the dependence of the magnetic
stability on the number of shells included in calculations ofthe lattice Fourier transform. The most important conclusion,namely, that the oxygen monolayer stabilizes the antiferromag-neticp(2×1) ground state for Fe/Ir(001), was unambiguously
obtained in both approaches. In addition, even quantitativeagreement is satisfactory although the differences for variousnumber of shells included in the DLM method are larger ascompared to the oxygen free case (see Fig. 7). On the other
hand, there is no indication of a more complex magnetic stateif the number of included exchange interactions is increased.
D. Scanning tunneling microscopy
Surprisingly, the simulated constant current STM images
shown in Fig. 11resolve the p(2×1) AFM magnetic ordering
of the Fe atoms even for a simulation assuming an unpolarizedtip in the Tersoff-Hamann model. Although a spin polarizedtip could directly deal with the different orientation of the Femoments, the AFM ordering of the Fe atoms as imprintedon the O atoms by hybridization may be also detected by anonmagnetic tip. For small bias voltages, ±100 meV around
E
F, the consequences of magnetic Fe-O hybridization are
clearly visible in the corrugation of the O atoms. Scanning
045423-6INFLUENCE OF OXYGEN AND HYDROGEN ADSORPTION ... PHYSICAL REVIEW B 88, 045423 (2013)
113
7
113
7
420
12
420
12
016
8
016
8
113
7
113
7
(a) (d)
(b) (c)VBias=-100mV VBias=+100mV
VBias=-20mV VBias=+20mV
FIG. 11. (Color online) (a)–(d) Simulated (Tersoff-Hamann
model) constant current STM images of 1 ML O on an antiferro-
magnetic p(2×1) Fe/Ir(001) surface for bias voltages VBiasaround
EF. Red spheres denote the positions of the O atoms, blue and green
spheres mark Fe atoms with up and down moments, respectively.Simulated line scans across the oxygen rows are drawn outside each
image. A different corrugation for scans either parallel to FM or AFM
ordered Fe atom rows is clearly visible, highlighting the p(2×1)
AFM ordering. The chosen charge density contour leads to tip-sample
distances of ∼3–4 ˚A (above the O atoms).
above oxygen rows either parallel to FM oriented Fe rows or
AFM oriented Fe rows, leads to corrugation differences of up to8 pm, as shown in the simulated line scans in Fig. 11. Resolving
such differences is certainly possible for experimental STMscans and thus should yield direct information on the predictedmagnetic ordering.IV . CONCLUSIONS
We have investigated the effect of oxygen and hydrogen
adsorption on the structural and magnetic properties of anFe/Ir(001) system from first principles. While the structuralpart was solved using an supercell V ASP approach, we usedtwo complementary approaches for the prediction of themagnetic state: A total energy model using supercell V ASPcalculations and the TB-LMTO-DLM method working withthe semi-infinite geometries. The emphasis was put on theinfluence of the O and H adsorption on the magnetic stability.The following main conclusions can be drawn: (i) Oxygenadsorbs on the Fe/Ir(001) surface at fourfold hollow Fe sitesand influences the atomic geometry (interlayer distances) ofthe system more strongly than hydrogen adsorbing at Fe-bridgepositions. In particular, the adsorption of an oxygen monolayerstrongly increases the interlayer distance between Fe andthe top Ir layer. (ii) The oxygen-iron hybridization mainlybroadens density of states features as compared to the cleansurface. This change of the electronic structure manifests itselfin corresponding modifications of exchange interactions andthe STM current. (iii) The magnetic stability is influencedby oxygen adsorption, and we predict an antiferromagneticp(2×1) magnetic ground state with a ¯X-ordering vector as
obtained by two complementary approaches, namely supercellV ASP and DLM model. (iv) The ¯Xordering is weakened by
decreasing oxygen coverage and changes into a complex mag-netic ground state for oxygen-free Fe/Ir(001). (v) Hydrogenadsorption leads to a weak stabilization of an antiferromagneticp(2×1) order. (vi) STM images for nonmagnetic tips reflect
thep(2×1) AFM ordering of the Fe moments not directly,
but rather by hybridization with the O atoms, best visible forsmall bias voltages.
ACKNOWLEDGMENTS
F.M., J.K., and V .D. acknowledge support of the Czech Sci-
ence Foundation (Projects IAA100100912 and P202/09/0775).J.R. is grateful for financial support from the Austrian ScienceFund (FWF) SFB ViCoM F4109-N13 and for computersupport of the Vienna Scientific Cluster (VSC).
*maca@fzu.cz
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P h y s .R e v .L e t t . 101, 027201 (2008).
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A. Ernst, W. Hergert, I. Mertig, W. Wulfhekel, and J. Kirschner,P h y s .R e v .B 81, 195410 (2010).7G. Kresse and J. Furthm ¨uller, P h y s .R e v .B 54, 11169
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045423-7 |
PhysRevB.100.134505.pdf | PHYSICAL REVIEW B 100, 134505 (2019)
Editors’ Suggestion
Exceptional points in tunable superconducting resonators
Matti Partanen ,1,2,*Jan Goetz,1Kuan Yen Tan,1Kassius Kohvakka,1Vasilii Sevriuk,1Russell E. Lake,1,3
Roope Kokkoniemi,1Joni Ikonen,1Dibyendu Hazra,1Akseli Mäkinen,1Eric Hyyppä,1Leif Grönberg,4
Visa Vesterinen,1,4Matti Silveri,1,5and Mikko Möttönen1,4,†
1QCD Labs, QTF Centre of Excellence, Department of Applied Physics, Aalto University, P .O. Box 13500, FI-00076 Aalto, Finland
2Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, DE-85748 Garching, Germany
3National Institute of Standards and Technology, Boulder, Colorado 80305, USA
4VTT Technical Research Centre of Finland Ltd, P .O. Box 1000, FI-02044 VTT, Finland
5Research Unit of Nano and Molecular Systems, University of Oulu, P .O. Box 3000, FI-90014 Oulu, Finland
(Received 17 May 2019; revised manuscript received 9 August 2019; published 7 October 2019)
Superconducting quantum circuits are potential candidates to realize a large-scale quantum computer. The
envisioned large density of integrated components, however, requires a proper thermal management and controlof dissipation. To this end, it is advantageous to utilize tunable dissipation channels and to exploit the optimizedheat flow at exceptional points (EPs). Here, we experimentally realize an EP in a superconducting microwavecircuit consisting of two resonators. The EP is a singularity point of the effective Hamiltonian, and correspondsto critical damping with the most efficient heat transfer between the resonators without back and forth oscillationof energy. We observe a crossover from underdamped to overdamped coupling across the EP by utilizing photon-assisted tunneling as an in situ tunable dissipative element in one of the resonators. These methods can be used
to obtain fast dissipation, for example, for initializing qubits to their ground states. In addition, these results pavethe way for thorough investigation of parity-time symmetry and the spontaneous symmetry breaking at the EPin superconducting quantum circuits operating at the level of single energy quanta.
DOI: 10.1103/PhysRevB.100.134505
I. INTRODUCTION
Systems with effective non-Hermitian Hamiltonians have
been actively studied in various setups in recent years [ 1–7].
They show many intriguing properties such as singularities intheir energy spectra [ 8–13]. A square-root singularity point
in the parameter space of a non-Hermitian matrix is referredto as an exceptional point (EP) if the eigenvalues coalesce[12,13]. Previously, EPs have been shown to emerge, for
example, in nonsuperconducting microwave circuits, laserphysics, quantum phase transitions, and atomic and molecularphysics [ 12,13]. The fascinating effects of EPs include the
disappearance of the beating Rabi oscillations [ 14], chiral
states in microwave systems [ 15], and spontaneous symmetry
breaking in systems with parity- and time-reversal ( PT)
symmetry [ 16–19]. In the quantum regime, PT-symmetric
systems may show features that are different from the semi-classical predictions, such as new phases owing to quantumfluctuations [ 19,20]. Despite the active research on EPs, they
have not been sufficiently investigated in superconductingmicrowave circuits to date [ 21].
*matti.t.partanen@aalto.fi
†mikko.mottonen@aalto.fi
Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.Superconducting microwave circuits provide an ideal plat-
form to realize various quantum technological devices, suchas ultrasensitive photon detectors and counters [ 22–25], and
potentially even a large-scale quantum computer [ 26,27],
or a quantum simulator [ 28] in the framework of circuit
quantum electrodynamics [ 29,30]. Notably, superconducting
qubits have been shown to approach the required coherencetimes [ 31,32] for quantum error correction [ 33,34]. However,
despite the tremendous interest in superconducting microwavecircuits in recent years [ 35–37], there are still many issues
to be solved before a fully functional quantum computer isrealizable. For example, the precise engineering of energyflows between different parts of the circuit in scalable archi-tectures is of utmost importance since heat is a typical sourceof decoherence in qubits [ 38,39]. In many error correction
codes, qubits are repeatedly initialized, which requires fastand efficient cooling schemes [ 40–42]. It is also of great
importance to reset the qubit readout resonators, for example,using pulse shaping [ 43,44]. One very promising method
for absorbing energy and, hence, initializing qubits to theirground state is based on resonators with tunable dissipation[45].
The recently developed quantum-circuit refrigerator
(QCR) [ 46,47] provides great potential for both qubit
initialization and thermal management since it enablestunability of energy dissipation rates over several orders ofmagnitude in a superconducting microwave resonator [ 48].
The operation of the QCR relies on inelastic tunneling ofelectrons through a normal-metal–insulator–superconductor(NIS) junction [ 49]. The tunneling electrons can absorb
2469-9950/2019/100(13)/134505(17) 134505-1 Published by the American Physical SocietyMATTI PARTANEN et al. PHYSICAL REVIEW B 100, 134505 (2019)
R2R1
SQUID QCRCC
1mmR1 R2
CCκext
κ1ω1
κ2(Vb)ω2(Φ)
Port 2
Port 1
R2
R1
S
QU
I
D
QCR
C
C
1
mm
R1
R2
C
C
κ
ext
κ
1
ω
1
κ
2
(
V
b
V
V
)
ω
2
(Φ)
P
or
t
2
P
or
t
1NS
S
S
SNIS CN
4µm VthVb(a)
(b)(c)
SeVbΔIth
ΔE(d)g
IN I S
FIG. 1. Sample structure. (a) The sample consists of two ca-
pacitively coupled resonators, R1 and R2, which are presented as
analogous cavities with coupling strength g. The primary resonator
R1 has a fixed dissipation rate κ1and angular frequency ω1whereas
the dissipative resonator R2 has a tunable rate κ2(Vb) controlled by
a QCR, and angular frequency ω2(/Phi1) tuned with a SQUID. The
coupling strength to external ports is denoted by κext. (b) Optical mi-
crograph of the sample. The transmission coefficient S21is measured
from port 1 to port 2. (c) False-color scanning electron micrographof the QCR together with a schematic control circuit. The QCR
consists of normal-metal (N) and superconducting (S) electrodes
separated by an insulator (I). The QCR is operated with bias voltage
V
b, and the electron temperature of the normal metal is obtained
from voltage Vthand current Ith. (d) The operation principle of a
SINIS junction. The occupied states in the superconductor densityof states are shown in blue, the occupation of the normal metal is
given by the Fermi distribution shown in orange, and the empty states
are shown in gray with energy Eon the vertical axis. The Fermi
levels of the superconducting electrodes (dashed lines) are shifted by
applying a voltage V
b. The black arrows indicate elastic tunneling,
and blue arrows inelastic tunneling with photon absorption. The reddashed arrows show photon emission that is suppressed due to lack of
available unoccupied states on the other side of the tunneling barrier.
or emit photons to a resonator which allows to control
the coupling strength to a low-temperature bath in situ .
This tunable coupling strength arising from a broadbandenvironment has also been shown to induce a Lamb shift [ 48].
In this work, we utilize tunable dissipation to realize EPs,
which correspond to critical coupling between two supercon-ducting microwave resonators. To this end, we investigate acircuit consisting of two coupled resonators, one of which isequipped with NIS junctions and a flux-tunable resonance fre-quency (Fig. 1). We denote the NIS junctions and the normal-
metal island that is capacitively coupled to the resonator asa QCR. Thanks to the voltage-tunable dissipation within theQCR and the flux-tunable resonance frequency, an EP arisesin the Hamiltonian that describes the modes of the coupledresonator system. We investigate the emergence of the EPusing frequency and dissipation as control parameters (Fig. 2)
and verify its properties experimentally by measuring themicrowave transmission coefficient (Fig. 3). The maximum
-10-505
310
4 50-5 5-16-14-12-10-8-6-4
-15-10
3-5
4
50 5 -5-10-50510Re(λ)/(2π)(MHz)
(MHz)
δ/(2π)( M H z )κ2/gIm(λ)/(2π)(a)
(b)
Re(λ)/(2π) (MHz)
δ/(2π)( M H z )κ2/gIm(λ)/(2π)( M H z )
FIG. 2. Eigenvalues of the effective Hamiltonian and the excep-
tional point. (a) Calculated mode frequency shift Re( λ) with respect
to the uncoupled mode frequency of R1, and (b) negative value
of the mode decay rate −Im(λ) as functions of the decay rate κ2
and frequency detuning δ. The figure shows both λ+andλ−calcu-
lated according to Eq. ( 2) with the experimental coupling strength
g/(2π)=7.2 MHz and decay rate κ1/(2π)=260 kHz. The EP is
located approximately at κ2=4g,a n dδ=0 as indicated with the
red circle. The panels (a) and (b) can be compared with each other
with the help of the colors which denote Im( λ) in (a) and Re( λ)i n( b ) .
dissipation rate given by the coupling strength can be reached
at the EP (Fig. 4). Furthermore, we demonstrate temporal
control of the damping rates by utilizing voltage pulses atthe QCR (Fig. 5). By adding voltage-tunable dissipation to a
system of two resonators with tunable coupling, our approachprovides a conceptually higher level of control comparedto the commonly studied systems which only have tunablecoupling between the resonators [ 42,50–57]. Hence, our work
demonstrates a platform to control the dissipation and, con-sequently, local heat transport between neighboring nodesin a quantum electrical circuit. In addition to thermal man-agement within superconducting multiqubit systems, thesemethods may be applicable to thermally assisted quantumannealing [ 58] and to studies of the eigenstate thermalization
hypothesis in many-body quantum problems [ 59]. Further-
more, our work is an important step toward thorough in-vestigation of PT-symmetric systems at the quantum level
134505-2EXCEPTIONAL POINTS IN TUNABLE SUPERCONDUCTING … PHYSICAL REVIEW B 100, 134505 (2019)
Experiment Theory
1 01.0
1.3
1.7
|S21|eVb
2Δ=0.7κ2
4g=0.7
4.2
9.9
0.42 0.44 0.4 0.42 0.44 0.40.5
0
-0.5
Φ/Φ0 Φ/Φ00.5
0
-0.5
0.5
0
-0.5
0.5
0
-0.50.5
0
-0.50.5
0
-0.50.5
0
-0.50.5
0
-0.5(a) (b)f−f1(MHz)
f−f1(MHz)1.3
FIG. 3. Scattering parameter of sample A. (a) Experimental and
(b) theoretical transmission amplitudes as functions of frequency
and magnetic flux through the SQUID. The panels in (a) show thecrossing of the second mode of R1 and the first mode of R2 at
different bias voltages, and the panels in (b) show the corresponding
theoretical results at different coupling strengths, as indicated in thefigure. The EP is obtained at eV
b/(2/Delta1)≈1, where the ratio κ2/(4g)
crosses unity. The maximum in each panel is normalized to unity,
and the frequency axes are shifted by the resonance frequency f1=
5.223 GHz in good agreement with the simulated frequency f1=
5.2 GHz. The input power at port 1 is approximately −100 dBm.
that can be realized with circuit quantum electrodynamics
architectures [ 21,60].
This paper is structured as follows: We introduce the
experimental samples in Sec. II, describe the EPs in Sec. III,
and present the experimental results in Sec. IV. The results are
discussed in Sec. V. Extended measurement results are shown
in Appendix A, electron tunneling through NIS junctions is
described in Appendices BandC, quantum-mechanical and
classical models for our samples are discussed in detail inAppendices D–I, and experimental techniques are presented
in Appendices J–L.
II. EXPERIMENTAL SAMPLES
Our samples consist of two coplanar waveguide resonators,
R1 and R2, which are capacitively coupled to each other,as depicted in Figs. 1(a) and1(b). The resonator R1 has a
fixed fundamental frequency at 2 .6 GHz. This mode does not
couple strongly to the resonator R2 owing to the voltage nodein the middle of the resonator R1 where the coupling capacitor
C
Cis located. Therefore, we focus on the second mode of
R1, with frequency f1=ω1/(2π)=5.2 GHz, which has a
voltage antinode at the coupling point of the resonators. Theresulting capacitive coupling between the resonators has astrength g/(2π)=7.2 MHz. In contrast to R1, the resonator
R2 has a flux-tunable resonance frequency ω
2(/Phi1)o w i n gt oa
superconducting quantum interference device (SQUID), anda voltage-tunable loss rate κ
2(Vb) owing to the QCR. Here,
/Phi1andVbare the magnetic flux threading the SQUID loop
and the voltage bias of the QCR, respectively. The theoretical01234107108109Sample A
Sample B
Theory
4g
EP
01234106107108109 Sample A
Sample B
Theory
2g
EPκ2/(2π) (Hz) κeff/(2π) (Hz)
eVb/(2Δ)eVb/(2Δ)κQCR
0.6
eVb/(2Δ)κeff/(2π)( H z )
106107108
0.8 1 1.2R2 mode
R1 modeR1, R2(a)
(b)
FIG. 4. Transition rates. (a) Extracted decay rates of the bare
resonator R2, κ2, for samples A and B as functions of the bias
voltage together with the theoretical model (Appendix H). The EP
is obtained at the intersection of κ2and the critical coupling 4 g.
Furthermore, the figure shows the theoretical coupling strength of the
QCR, κQCR, without taking dephasing and other voltage-dependent
losses into account. Here, the flux is approximately /Phi1//Phi1 0=0.4,
and the frequency 5 .2 GHz. The probe power at the input of the
device is −100 dBm for sample A and −115 dBm for sample B. The
uncertainty of the data points is of the same order as the marker size.
(b) Experimental and theoretical effective decay rates of the coupled
system κeff=−2I m (λ±) obtained from κ2at zero detuning using
Eq. ( 2). The two branches at high voltages correspond to λ+and
λ−with the modes located predominantly in one of the resonators
as indicated. The maximum decay rate for R1 is obtained at the
EP. The damping rates of the modes are equal at eVb/(2/Delta1)<1 due
to hybridization. The inset shows the effective damping rate in thevoltage regime near the exceptional point.
background for the QCR is discussed in Appendix Band
Ref. [ 49]. The inductance of the SQUID and, hence, also the
resonance frequency of R2 are periodic in the flux with aperiod of a flux quantum /Phi1
0=h/(2e). Consequently, due to
the coupling of the resonators, R1 also shows flux-dependentfeatures. We show the QCR in Fig. 1(c) and schematically
present its operation principle in Fig. 1(d). The difference
in photon absorption and emission rates originates from thegap of 2 /Delta1in the density of states of the superconductor, and
the difference can be utilized to cool down quantum circuits[46,49]. We study two samples with different R2 resonator
lengths: sample A (12 mm) and sample B (13 mm). TheR1 resonator has a length of 24 mm in both samples. See
134505-3MATTI PARTANEN et al. PHYSICAL REVIEW B 100, 134505 (2019)
- 1 012345610−210−1
QCR pulse R1 drive0240.20.30.4A/A0
t(µs)eVb/(2Δ)κeff,1/(2π)( M H z )100
FIG. 5. Energy dissipation of sample B as a function of time.
Measured voltage amplitude A(t) of the resonator R1 normalized
with A(0)=A0(markers). We apply microwave drive to R1 at the
resonance frequency for 40 μs and switch it off at time t=0. At
t≈0.5μs, we send nominally rectangular 2- μs-long voltage pulse
with the amplitude eVb/(2/Delta1)≈2.4 to the QCR, which results in in-
creased energy dissipation. After the pulse, the dissipation decreases
approximately to its original value, as shown by the model (solid
line). The magnetic flux through the SQUID is shifted from themode crossing by approximately 0 .015×/Phi1
0yielding a detuning
of 200 MHz (see Appendix I). The inset shows the experimental
(markers) and theoretical (solid line) decay rate of the R1 modeduring the pulse as a function of the pulse amplitude. The input
power is approximately −105 dBm.
Appendices E,J, and Kfor sample parameters, fabrication,
and measurement setup, respectively.
III. EXCEPTIONAL POINTS
To study exceptional points, we utilize two control param-
eters in the effective Hamiltonian of the system: we use thevoltage-tunable dissipation rate κ
2(Vb) and the flux-tunable
detuning between the resonators δ(/Phi1)=ω2(/Phi1)−ω1.W e
study the system consisting of the two resonators in a framerotating with a frequency corresponding to the uncoupledmode frequency of R1. Thus, the excitations of the system canbe described with the effective non-Hermitian Hamiltonian inmatrix form operating on a vector ψ=(A,B)
Twhere Aand
Bare field amplitudes in R1 and R2, respectively,
H=/parenleftbigg−iκ1
2g
gδ−iκ2
2/parenrightbigg
, (1)
where κ1is the decay rate of the resonator R1. This ef-
fective Hamiltonian is analogous with the actual quantum-mechanical Hamiltonian operating on the correspondingHilbert space, and it can be derived from the Lindblad masterequation (see Appendix Dand Refs. [ 56,61]). The eigenvalues
ofHcan be written as
λ
±=1
4(2δ−iκ1−iκ2±s), (2)and the corresponding eigenvectors are
ψ±=/parenleftbigg−2δ−iκ1+iκ2±s
4g,1/parenrightbiggT
, (3)
where s=/radicalbig
(2δ+iκ1−iκ2)2+16g2. Thus, the eigenvalues
and eigenvectors coalesce when the square-root term svan-
ishes resulting in an EP. Consequently, there is only a singleeigenvalue and, importantly, there is also only a single eigen-vector. The EP occurs at |κ
2−κ1|=4g, andδ=0. In our
samples κ1/lessmuchκ2, which we verify by measuring the internal
quality factor of the primary resonator R1 with R2 far detunedat/Phi1=/Phi1
0/2. From the quality factor, we extract a loss rate
κ1/lessorsimilar2π×260 kHz for both samples which is substantially
lower than the extracted value of κ2, as discussed below.
Consequently, the condition for the EP simplifies to κ2=4g.
To visualize the system singularity, i.e., the EP, we show
the real and imaginary parts of λ±in Fig. 2. The eigenvalues
form a self-intersecting Riemann surface in the parameterspace of κ
2andδ. The imaginary part corresponds to mode
decay, and real part to mode frequency deviation from theuncoupled mode frequency of R1. Our system consistingof the resonators R1 and R2 can be considered as a sin-gle damped harmonic oscillator, where the energy oscillatesbetween the two resonators, as discussed in Appendix D.
In the underdamped case κ
2<4g, the modes have an equal
decay rate at zero detuning, and there is an anticrossing ofthe mode frequencies. In contrast, in the overdamped caseκ
2>4g, there is an anticrossing in the mode decay rates as a
function of the detuning, and the mode frequencies are equalat zero detuning. Consequently, one of the modes remainslossy whereas the other one has a low decay rate at differentdetunings.
Let us connect the meaning of this critical point to the
efficiency of energy transfer between the two resonators. Interms of coupled dissipative systems, the EP separates thesystem between the overdamped and underdamped regimesbeing the point of critical coupling. It follows from the dy-namics of the coupled system that at this point, the energy istransferred between the two resonators optimally fast withoutback and forth oscillation [ 42,56]. In particular, the rate of
heat transfer at zero detuning is given by κ
eff=−2I m (λ±)≈
κ2{1∓Re[/radicalbig
1−(4g/κ2)2]}/2. Here, the branch with the up-
per signs corresponds to a mode located predominantly in theprimary resonator R1, and the branch with the lower ones inthe dissipative resonator R2. Consequently, by reaching theEP at κ
2=4g, we operate our sample at a point of opti-
mally efficient heat transfer out of R1. The effective couplingstrength in the limit κ
1→0 is also relevant for Purcell filters
[62].
IV . EXPERIMENTAL OBSERVATIONS
To explore the dissipative dynamics of the two coupled
resonators, we measure the flux- and frequency-dependentscattering parameter S
21describing the transmission from
port 1 to port 2 for different bias voltages using a vectornetwork analyzer. We tune the magnetic flux in a rangewhere the frequency of R2 crosses that of R1. As shownin Fig. 3(a), we observe a transition from an anticrossing
134505-4EXCEPTIONAL POINTS IN TUNABLE SUPERCONDUCTING … PHYSICAL REVIEW B 100, 134505 (2019)
into a single mode already indicating the presence of an EP
between these regimes. A broader range of bias voltages isshown in Appendix A. To generate a quantitative description
of our system, which is required for the investigation ofEPs, we simulate the scattering coefficient using an analyticalcircuit model, as shown in Fig. 3(b). Here, we model the
SQUID as a flux-tunable inductor, and the QCR as an effectiveresistance R
effdescribing the dissipation (see Appendix I).
The theoretical model in Fig. 3is in very good agreement
with the experimental results. Consequently, we can extractthe experimental damping rates of the dissipative resonatorR2 as a function of the bias voltage using the circuit model.In addition to the damping rates, we extract the couplingcapacitance C
Cand the critical current of the SQUID, Ic,f r o m
the theoretical model (see Appendices F,G, and H).
To demonstrate the presence of an EP, we show the ex-
tracted damping rates of the bare resonator R2 in the absenceof R1 in Fig. 4(a)as functions of the bias voltage for samples
A and B. The damping rate κ
2is approximately equal in both
samples and its value can be tuned by approximately twoorders of magnitude. At low bias voltages eV
b/(2/Delta1)<1, the
rateκ2is below 4 g, and at eVb/(2/Delta1)>1 the damping rate
exceeds 4 g. The continuity of the theoretical curve guarantees
a crossing at the critical coupling and, thus, the existence ofthe EP. We describe the origin of the tunable damping ratesusing a theoretical model that contains the photon absorptionand emission at the QCR given by the rate κ
QCR,a sw e l la s
constant internal losses κint,2, and voltage-dependent residual
losses κr,2. We have designed our sample in such a way that
κQCR covers the critical damping rate, and thus the losses
originating from the QCR are sufficient to realize the EP.The underlying analysis is based on the excellent agreementbetween theory and experiment found for similar devices inearlier experiments [ 48]. We can accurately obtain the damp-
ing rate κ
QCR shown in Fig. 4using the measured electron
temperatures of the normal-metal island (Appendix C). At
higher voltages, κ2is dominated by the residual damping coef-
ficient, as discussed in Appendix G. This damping coefficient
includes dephasing, quasiparticle losses, and resistive losses,and we extract its value based on the experimental rates.
In Fig. 4(b), we show the damping rates of the coupled
circuit defined using Eq. ( 2)a sκ
eff=−2I m (λ±). The max-
imum energy decay rate for the mode in R1 is obtainedat the EP as discussed above, and it is given by κ
max,1=
max[−2I m (λ+)]≈κ2/2=2g. Thus, in the optimal case, the
decay rate is limited by the coupling strength between theresonators, a scenario related to the quantum speed limit. Wedo not observe a distinctively large increase in the dampingrate of R1 when approaching the EP with increasing biasvoltages since κ
2is already relatively close to the optimal
value 4 gat low voltages. Furthermore, κ2is very sensitive to
the voltage near the EP.
In Fig. 5, we study the time dependence of the damping
rates to demonstrate the temporal control we have over thesystem. To this end, we operate the system with a finitedetuning δ. For our samples, the finite detuning provides a
convenient measurement point with relatively low dampingrates, thus allowing us to clearly observe the changes in theirvalues even when the system is operated at the single-photonlevel. Furthermore, at this operation point the measureddamping rates are less affected by the experimental inaccuracy
of the magnetic flux compared to zero detuning. To extractthe time- and voltage-dependent damping rates, we measurethe ring-down of R1 in time domain. During the ring-down,we apply a nominally rectangular 2- μs-long voltage pulse to
the QCR to study the effect of the induced dissipation fromR2. The finite detuning reduces the sensitivity of the dampingrates to the nonidealities in the pulse reaching the sample.The pulse increases the dissipation rate, as expected. Afterthe pulse, the dissipation returns approximately to an equallevel with that before the pulse. The instantaneous responseof the damping rates to the applied voltages shows that wecan control the dissipation on a timescale substantially shorterthan 1 μs. The QCR can be operated even at timescales in
the range of 10 ns [ 63]. Furthermore, the dependence of the
damping rate on the pulse amplitude follows the theory, asshown in the inset. The position of the peak is shifted tohigher voltages as compared to Figs. 4(a) and4(b) owing to
the detuning. The good agreement of the time traces and thefact that the system returns back to its intrinsic relaxation ratequickly after turning off the voltage of the QCR shows thatwe can reliably control the dissipative dynamics within theparameter space spanned by κ
2andδ.
V . SUMMARY AND DISCUSSION
We have experimentally realized an exceptional point (EP)
in a superconducting microwave circuit consisting of two cou-pled resonators. We study the presence of the EP by observinga transition from an avoided crossing to single modulatingresonance frequency. This point corresponds to the maximumheat transfer between the two resonators without back andforth oscillation of the energy. The measurement results are invery good agreement with our theoretical model. The circuitis based on a QCR, which operates as a voltage-tunable dissi-pator, thus enabling the investigation of the crossover from anunderdamped to critically damped and further to overdampedcircuit. In addition to the realization of an EP, the circuit alsobehaves as a frequency- and voltage-tunable heat sink forquantum electric circuits that can be applied, for example, inquantum information processing for initializing qubits to theirground state by absorbing energy [ 45]. The tunability of the
damping rate enables one to obtain the fastest possible photonabsorption allowed by a given coupling coefficient.
In the future, it is interesting to further investigate the
EP by modifying the circuit design. By introducing tunneljunctions to both resonators, one obtains a continuous lineof EPs instead of an isolated singularity point. Incorporatingqubits also enables the investigation and utilization of the EPwith single energy quanta. Furthermore, one can investigateresonators with an additional coupling circuit realized with aSQUID or a qubit [ 52,54] in addition to the tunable coupling
realized with frequency detuning and a tunable damping real-ized with a QCR that are employed here. Moreover, the useof several microwave resonators will result in a more versatileparameter space [ 64], and hence yields an interesting platform
for studying fundamental physics. Dynamic encircling of theEP with topological energy transfer [ 3,9] can be realized
with superconducting resonators in a straightforward mannerusing standard microwave techniques. It requires fast tuning
134505-5MATTI PARTANEN et al. PHYSICAL REVIEW B 100, 134505 (2019)
of the magnetic field, which can be realized by fabricating a
flux bias line on the chip. Topological energy transfer withmicrowave pulses may provide an asset for applications inquantum information processing and other quantum techno-logical devices. In addition, EPs are suitable for investigatingPT symmetry on the level of single microwave photons.
Here, superconducting circuits provide an attractive architec-ture owing to the ability to design system parameters yielding,for example, ultrastrong- and deep-strong-coupling regimes[21,65]. Furthermore, we envision EPs as candidates to realize
nonreciprocal signal routing beneficial for active quantumcircuits [ 60,66,67].
Note added. Recently, we became aware of Ref. [ 68]
investigating EPs in superconducting circuits with a differentoperation principle compared to our system. Our work is fullyindependent of this reference.
ACKNOWLEDGMENTS
We thank A. A. Clerk for discussions, and J. Govenius
and M. Jenei for assistance. We acknowledge the provisionof facilities and technical support by Aalto University atOtaNano-Micronova Nanofabrication Centre. We acknowl-edge the funding from the European Research Council un-der Consolidator Grant No. 681311 (QUESS), and MarieSkłodowska-Curie Grant No. 795159, the Academy of Fin-land through its Centres of Excellence Program (ProjectsNo. 312300 and No. 312059) and Grants No. 265675, No.305237, No. 305306, No. 308161, No. 312300, No. 314302,and No. 316551), the European Union via the QuantumFlagship project QMiCS (Grant No. 820505), the Vilho, Yrjöand Kalle Väisälä Foundation, the Technology Industries ofFinland Centennial Foundation, the Jane and Aatos ErkkoFoundation, the Alfred Kordelin Foundation, and the EmilAaltonen Foundation.
APPENDIX A: MEASUREMENTS IN A BROAD RANGE OF
VOLTAGES AND FLUXES
We show experimentally and theoretically obtained scat-
tering parameter |S21|over a broad range of bias voltages
applied at the QCR and magnetic fluxes through the SQUID inFigs. 6and7. The measurements are in good agreement with
the theoretical model.
APPENDIX B: QUANTUM-CIRCUIT REFRIGERATOR
We use a QCR to absorb and emit photons in the resonator
R2. The resonator transition rate from the occupation numbermtom
/primecan be written as [ 49]
/Gamma1m,m/prime(V)=M2
mm/prime2RK
RT/summationdisplay
τ=±1→
F[τeV+¯hω2(m−m/prime)],(B1)
where V=Vb/2,RTis the single-junction tunneling re-
sistance, Mmm/primeis the corresponding matrix element, RK=
h/e2≈25.8k/Omega1is the von Klitzing constant, and the normal-
ized rate for forward tunneling is given by
→
F(E)=1
h/integraldisplay∞
−∞dE/primenS(E/prime)[1−f(E/prime,TS)]f(E/prime−E,TN),
(B2)
Experiment Theoryf−f1(MHz)
-0.500.5
f−f1(MHz)
-0.500.5
0.4 0.42 0.44
Φ/Φ0-0.5
Φ/Φ0
|S21| 1 000.5-0.500.5-0.500.5-0.500.5-0.500.5-0.500.5
-0.500.5
-0.500.5
-0.500.5
-0.500.5
-0.500.5
-0.500.5
0.4 0.42 0.44
FIG. 6. Scattering parameter of sample A. Measured and simu-
lated scattering parameter |S21|as a function of frequency and flux
for different bias voltages. The bias voltages from top to bottom are
eVb/(2/Delta1)=0.0, 1.0, 1.1, 1.4, 1.7, 2.5, and 3.5. Maximum value in
each panel is normalized separately to unity. The input power at port1 is approximately −100 dBm. See Fig. 11for the fitted parameters
and Fig. 4for the corresponding transition rates.
where f(E,T)=1/{exp[E/(kBT)]+1}is the Fermi-Dirac
distribution, kBis the Boltzmann constant, and the density of
states in a superconductor can be expressed with the help ofthe Dynes parameter γ
Das
nS(E)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleRe/parenleftBigg
E//Delta1+iγ
D/radicalbig
(E//Delta1+iγD)2−1/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (B3)
The matrix element describing the transition can be written in
terms of the generalized Laguerre polynomials Ll
n(ρ)a s[ 49]
M2
m,m/prime=/braceleftBigg
e−ρρm−m/primem/prime!
m!/bracketleftbig
Lm−m/prime
m/prime(ρ)/bracketrightbig2,m/greaterorequalslantm/prime
e−ρρm/prime−mm!
m/prime!/bracketleftbig
Lm/prime−m
m/prime(ρ)/bracketrightbig2,m<m/prime(B4)
where ρ=πα2
c/(ω2Clx2RK) is an environmental parameter,
Clis the capacitance per unit length of the coplanar waveg-
uide, 2 x2is the length of the resonator R2, and the capacitance
fraction αcis given in terms of the capacitance between
the normal-metal island and the center conductor CN, and
the single-junction capacitance Cjasαc=CN/(CN+4Cj).
In the equations above, we have neglected the effects owing to
134505-6EXCEPTIONAL POINTS IN TUNABLE SUPERCONDUCTING … PHYSICAL REVIEW B 100, 134505 (2019)
Experiment Theoryf−f1(MHz)
f−f1(MHz)
Φ/Φ0 Φ/Φ0
|S21| 1 00.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2-101-101-101-101-101-101-101
-10101-101-101-101-101-101
-1
FIG. 7. Scattering parameter of sample B. Measured and simu-
lated scattering parameter |S21|as a function of frequency and flux
for different bias voltages. The bias voltages from top to bottom
areeVb/(2/Delta1)=0.0, 0.6, 1.1, 1.6, 2.4, 3.7, and 6.2. The maximum
value in each panel is normalized to unity. The input power at port1 is approximately −115 dBm. At eV
b/(2/Delta1)=6.2, the effective
resistance obtains the value of 30 /Omega1and the critical current Ic/Ic,0≈
0.3. See Fig. 11for the the other fitted parameters and Fig. 4for the
corresponding transition rates.
the charging of the normal-metal island since the capacitance
of the island is relatively large. Furthermore, the rates forsingle-photon transitions can be expressed as [ 49]
/Gamma1
m,m−1=κQCR(NQCR+1)m,
/Gamma1m,m+1=κQCRNQCR(m+1),(B5)
where κQCRdenotes the coupling strength of the QCR, and the
Bose-Einstein distribution at the effective temperature of the
electron tunneling TQCRis given by
NQCR=1
exp/parenleftbig¯hω2
kBTQCR/parenrightbig
−1, (B6)
where
TQCR=¯hω2
kB⎧
⎨
⎩ln⎡
⎣/summationtext
τ=±1→
F(τeV+¯hω2)
/summationtext
τ=±1→
F(τeV−¯hω2)⎤
⎦⎫
⎬
⎭−1
.(B7)These equations are derived by defining
κQCR=/Gamma1m,m−1
m−/Gamma1m,m+1
m+1. (B8)
APPENDIX C: ELASTIC TUNNELING IN
NORMAL-METAL–INSULATOR–SUPERCONDUCTOR
JUNCTIONS
Typically, the elastic tunneling is the dominating tunneling
process in NIS junctions. It can be utilized for temperaturecontrol of the normal-metal electrons [ 69–71], and for precise
thermometry down to millikelvin temperatures [ 71,72]. Re-
cently, NIS junctions have also been utilized in a realizationof a quantum heat valve [ 73], and phase-coherent caloritronics
[74]. Figure 8shows the measured current-voltage character-
istics and normal-metal temperatures.
The electric current through a single NIS junction can be
written as [ 71,75]
I(V)=1
eRT/integraldisplay∞
0nS(E)[f(E−eV,TN)
−f(E+eV,TN)]dE, (C1)
where TNdenotes the normal-metal temperature, and Vis the
voltage across the junction. For a symmetric SINIS structure,we apply a voltage V
b=2V. Importantly, this equation has
a monotonic dependence on the temperature of the normalmetal but only a very weak dependence on the temperature ofthe superconductor. Thus, we may use NIS junctions as ther-mometers measuring the electron temperature of the normalmetal.
The tunneling electrons transfer heat through the insulating
barrier. The average power is given by [ 71]
P=1
e2RT/integraldisplay∞
−∞nS(E)(E−eV)[f(E−eV,TN)−f(E,TS)]dE.
(C2)
Based on this equation, we can reduce and increase the
temperature of the normal metal. The applied voltage at theSINIS junction produces a total Joule heating power P=V
bI,
which is unequally divided between the N and S electrodes.
APPENDIX D: QUANTUM-MECHANICAL MODEL
Here, we analyze the temporal evolution of the coupled
resonators following Ref. [ 56]. Alternatively, the dynamics of
the system can be solved using, for example, a reconstructionmethod [ 61]. The Hamiltonian can be written in the rotating-
wave approximation as
ˆH
RWA=¯hω1ˆa†ˆa+¯hω2ˆb†ˆb+¯hg(ˆaˆb†+ˆa†ˆb). (D1)
The first term describes the energy of the primary resonator
R1 with annihilation operator ˆ a, the second term the energy of
R2 with annihilation operator ˆb, and the third term describes
the coupling between the resonators. Here, we have neglecteddriving. Furthermore, this equation is valid only for a linearresonator. The effects owing to nonlinearity are discussedbelow. To solve the dynamics of the system, we utilize theLindblad master equation for the density matrix of the coupled
134505-7MATTI PARTANEN et al. PHYSICAL REVIEW B 100, 134505 (2019)
-2 -1 0 1 2
eVb/(2Δ)-30-20-100102030I(nA)
10 mK
100 mK
200 mK
05 1 0
eVb/(2Δ)0100200300400500600T(mK)
10 mK
100 mK
200 mK
0 0.2 0.4 0.6
T0(K)00.10.20.30.4Vth(mV)Measurement
Linear
Theory(a) (b) (c)
FIG. 8. Current-voltage characteristics and temperature for sample B. (a) Electric current through a SINIS junction as a function of bias
voltage at different bath temperatures. (b) Measured thermometer voltage Vthas a function of bath temperature T0at fixed bias current Ith=
17 pA, and Vb=0. The theoretical curve is calculated using Eq. ( C1). We extract the electron temperatures of the normal-metal island using
a linear voltage-to-temperature conversion below 300 mK, and above that we extract the temperatures from the voltages corresponding to
the different experimental bath temperature points. At high temperatures, the low sensitivity reduces the reliability of the extracted islandtemperatures. (c) Electron temperature of the normal-metal island as a function of bias voltage at different bath temperatures.
system ˆ ρas
dˆρ
dt=−i
¯h[ˆHRWA,ˆρ]+κ1L[ˆa]ˆρ+κ2L[ˆb]ˆρ, (D2)
where the Lindblad superoperator is defined by L[ˆx]ˆρ=
ˆxˆρˆx†−1
2{ˆx†ˆx,ˆρ}. We can write the resulting equations of
motion as [ 56]
d/angbracketleftˆa/angbracketright
dt=−iω1/angbracketleftˆa/angbracketright−ig/angbracketleftˆb/angbracketright−κ1
2/angbracketleftˆa/angbracketright, (D3)
d/angbracketleftˆb/angbracketright
dt=−iω2/angbracketleftˆb/angbracketright−ig/angbracketleftˆa/angbracketright−κ2
2/angbracketleftˆb/angbracketright. (D4)
We define the resonator fields as /angbracketleftˆa/angbracketright=Aexp(−iω1t),/angbracketleftˆb/angbracketright=
Bexp(−iω1t). Consequently, the equations assume the form
dA
dt=−igB−κ1
2A, (D5)
dB
dt=−iδB−igA−κ2
2B, (D6)
where δ=ω2−ω1. These equations can be written in a
matrix form as a time-dependent Schrödinger equation
d
dtψ=−iHψ, (D7)
where ψ=(A,B)T, and
H=/parenleftbigg−iκ1
2g
g−iκ2
2+δ/parenrightbigg
, (D8)
as given in Eq. ( 1). Here, His an effective non-Hermitian
Hamiltonian scaled by ¯ h. We note that Hoperates here on
classical field amplitudes instead of quantum states. Never-theless, Eq. ( D7) is analogous to the quantum mechanical
Schrödinger equation. The eigenvalues and eigenvectors of H
are given by Eqs. ( 2) and ( 3) in the main text, respectively, and
Fig.9shows them as a function of κ
2for zero detuning.Equations ( D5) and ( D6) can also be written as a second-
order differential equation
d2A
dt2+/parenleftbiggκ1+κ2
2+iδ/parenrightbiggdA
dt+/parenleftbigg
g2+iδκ1
2+κ1κ2
4/parenrightbigg
A=0.
(D9)
When the resonators are tuned into resonance, δ=0, we can
express Eq. ( D9)a s
d2A
dt2+κ1+κ2
2dA
dt+/parenleftbigg
g2+κ1κ2
4/parenrightbigg
A=0. (D10)
This equation describes a damped harmonic oscillator corre-
sponding to energy transfer between the resonators R1 and R2
at an angular frequency/radicalbig
g2+κ1κ2/4. Due to the asymmetric
damping rates in the two resonators, the total dissipation rateof the system is time dependent and reaches its maximumvalue when the excitations are in R2. The damping ratio isgiven by
ξ=κ
1+κ2
2/radicalbig
4g2+κ1κ2. (D11)
Here, κ2is a function of voltage Vb, which allows us to
examine the transition from an underdamped system ξ<1
through critical damping ξ=1 to an overdamped system
ξ>1. Critical damping is obtained when |κ2−κ1|=4g.T h e
total damping rate of R1 is given by κ1=κint,1+κext, and
of R2 by κ2(Vb)=κint,2+κQCR(Vb)+κr,2(Vb), where κint,1/2
denote the internal losses, κextthe losses to the external
measurement circuit, κQCR(Vb) the photon-assisted tunneling
in Eq. ( B8), andκr,2(Vb) the residual voltage-dependent losses
in R2. In our samples, κ2(Vb)/greatermuchκ1and g/greatermuchκint,1/greatermuchκext,
as discussed below. Therefore, we obtain an approximatecondition for the critical damping as
κ
2=4g. (D12)
The critical damping, which corresponds to the EP, is obtained
ateVb/(2/Delta1)≈1 where the photon number remains low and,
134505-8EXCEPTIONAL POINTS IN TUNABLE SUPERCONDUCTING … PHYSICAL REVIEW B 100, 134505 (2019)
23456-505
EP
23456-20-15-10-5
EP
234560.20.40.60.8
EP
234560.20.40.60.8
EPRe(λ)/(2π) (MHz)
Im(λ)/(2π) (MHz)
κ2/g κ2/g
κ2/g κ2/gλ−λ+
λ−λ+
A−A+
B−B+|A|2
|B|2(a)
(c) (d)(b)
FIG. 9. Eigenvalues and eigenvectors of the effective Hamiltonian. (a) The real part of the eigenvalues corresponding to the frequency
shifts from the bare resonator R1 mode frequency as a function of the decay rate κ2at zero detuning. (b) The imaginary part of the eigenvalues,
the absolute values of which correspond to the decay rates. (c) The squared absolute value of the eigenvector component corresponding to the
resonator R1. Here, the amplitude of the eigenvector /Psi1±=(A±,B±)Tis normalized to unity. (d) As (c) but for the eigenvector component
corresponding to R2.
therefore, the slight nonlinearity caused by the SQUID is neg-
ligible. However, at eVb/(2/Delta1)>1, the QCR generates ther-
mal photons that result in photon-number-dependent losses,as discussed below.
APPENDIX E: SAMPLE PARAMETERS
The main parameters for the samples are summarized in
Table I. The coupling strength between the resonators can
be estimated as [ 76]g=CCV1V2/¯h≈2π×7.2 MHz, where
the voltages are given by Vi=√¯hω1/(2xiCl),i=1,2, the
angular frequency of the second mode of the resonator R1 isω
1/(2π)≈5.223 GHz for samples A and B, and Clis the ca-
pacitance per unit length. Consequently, the critical dampingis obtained with κ
2=4g≈2π×29 MHz. The external qual-
ity factor corresponding to the leakage from the resonator R1to the transmission line through the capacitances C
TLis given
by [77]Qext=2x1Cl/(4ZLω1C2
TL)≈9×105. Consequently,
the corresponding damping rate is κext=ω1/Qext≈2π×
6 kHz. The loaded quality factor of the second mode of R1 isapproximately Q
L=2×104, when the resonators are far de-
tuned (Fig. 10). Thus, the internal losses in R1 dominate over
the losses to the transmission line Qint≈QL. Furthermore, we
obtain the damping rate κ1=ω1/QL≈2π×300 kHz. The
real part of the complex wave propagation coefficient, γ=
α+iβ, describes the damping in the waveguide, and it can becalculated as [ 77]α=nmπ/(4x1Qint)≈7×10−3m−1, where
nmis the mode number such that nm=2 denotes the second
mode. The internal losses without the photon-assisted tun-neling in the QCR are somewhat higher in the resonator R2than in R1 since the design and fabrication of the QCR andthe SQUID have not been optimized for low loss rates. Theinternal loss rate for R2 can be extracted at zero detuningand zero-bias voltage from the saturation level of extractedκ
2values since κint,1/lessmuchκ2(0), and hence the losses in R2
dominate over those in R1. We obtain from the circuit modelthe internal loss rate for R2 as κ
int,2≈κ2(0)=2π×16 MHz.
We measure a slight temperature and power dependence ofthe quality factor of R1 as expected in the case of two-levelfluctuators dominating the losses [ 78] (Fig. 10).
The photon number inside R1 when R2 is far detuned can
be estimated as [ 79]n=4/Omega1
2
d/κ2
1≈10, where the driving
strength is given by /Omega1d=CTLVinV1/¯h, the input voltage is
obtained from the input power as Vin=√PinZL, and the input
power is Pin≈−115 dBm. The input power is −100 dBm
for sample A in Figs. 3and6, and−115 dBm for sample B
in Fig. 7. We also measure the resonators at different power
levels. When R1 and R2 are in resonance at Vb=0, the total
photon number is approximately equally divided betweenthe resonators if κ
1≈κ2. However, in our samples κ1<κ 2,
especially at Vb>2/Delta1/e, and therefore the number of coherent
photons is lower in R2 than in R1. When the Qfactor of the
134505-9MATTI PARTANEN et al. PHYSICAL REVIEW B 100, 134505 (2019)
-130 -120 -110 -100 -90
P(dBm)12345678QL104
10
50
100
200
400
600
800
0 0.2 0.4 0.6 0.8
T(K)12345678QL104
-87
-89
-91
-93
-95
-97-100
-105
-110
-115
-120
-125
-130(a) (b)
FIG. 10. Measured quality factor of sample B. (a) QLas a function of power at different bath temperatures as indicated in mK. The flux is
/Phi10/2. (b) As (a) but the data are presented as a function of the bath temperature at different powers as indicated in dBm. The lines are guides
for the eye.
resonator is reduced to 200, which is of the order of the critical
damping, photon numbers close to unity are obtained with
an input power Pin≈−85 dBm. Consequently, the photon
number at the EP is well below unity in our experiments.
APPENDIX F: SHIFT OF RESONANCE
In Figs. 3,6, and 7, the crossing of the mode frequencies as
a function of the flux shifts slightly toward lower flux valueswith increasing bias voltage. We attribute this shift to heatingof the SQUID, which results in a reduced critical current of theSQUID, and hence a larger inductance and lower resonancefrequency of R2. The extracted critical currents are shown in
Fig.11. In principle, we vary also the Lamb shift [ 48], which,
however, is not resolved since the resonance of R2 is verybroad and, therefore, we neglect it in our model.For simplicity, we do not take the frequency shift orig-
inating from the sample heating into account in the time-domain measurements in Fig. 5. The dependence of the
damping rates on the frequency detuning decreases withincreasing detuning, i.e., the surface in Fig. 2is relatively
flat with large δ. Therefore, the effect of the heating is
expected to be weak especially at the lowest pulse ampli-tudes. We note, however, that we observe a slight increasein the damping rate before the pulse with increasing pulseamplitudes. It may originate from an insufficient thermal-ization time between repeated measurements, resulting in anincreased quasiparticle concentration in the sample. Neverthe-less, the damping rate before the pulse increases from 2 π×
0.16 MHz only up to 2 π×0.21 MHz, which is substan-
tially below the maximum damping rate of 2 π×0.39 MHz
in Fig. 5. Hence, the heating of the sample is of minor
01234
eVb/(2Δ)102103104Reff(Ω)
Sample A
Sample B
01234
eVb/(2Δ)0.50.60.70.80.91Ic/Ic,0
Sample A
Sample B
01234
eVb/(2Δ)0246810n(a) (b) (c)
FIG. 11. Extracted parameters for the transition rates for samples A and B. (a) The effective resistance in the circuit model as a function
of the bias voltage. (b) Critical current Icnormalized with the maximum critical current Ic,0. (c) Estimated average photon number in the
dissipative resonator R2.
134505-10EXCEPTIONAL POINTS IN TUNABLE SUPERCONDUCTING … PHYSICAL REVIEW B 100, 134505 (2019)
TABLE I. Sample parameters. The parameters for sample B
that differ from those for sample A are given in parenthesis. SeeAppendices E–Iand Fig. 14(b) for details. The resonance frequency
of the second mode of the resonator R1, f
1, is a measured value, the
characteristic impedances of the transmission lines in the resonator,
Z0, and in the external measurement circuit, ZL, are nominal values.
The lengths of the resonator sections x1andx2are design values,
and the effective permittivity is calculated as [ 77]√εeff=c/(2f1x1),
where cis the speed of light in vacuum. We obtain the values for the
capacitance per unit length Cl, and the capacitance CTLfrom finite-
element method (FEM) simulations. The capacitance CCis obtained
by fitting the circuit model to the measured scattering parameter
|S21|in good agreement with FEM simulations, and the capacitances
CN,CS,a n d Cjare calculated using a parallel-plate model. The
coupling strength gis obtained from CC. The loaded quality factor
of the second mode of R1 Qint,1and the tunneling resistance RTare
measured values, and the Dynes parameter γDis estimated as the
ratio of the asymptotic resistance and the zero-voltage resistance. The
critical current at zero bias Ic,0is given by the flux corresponding to
the crossing of the modes in the circuit model in good agreement with
a control sample with slightly smaller junction area and a critical
current of approximately 200 nA. The damping rate κ1is given by
the ratio ω1/Qint,1, and the damping rate κint,2is extracted from the
saturation value of κ2at zero bias. The proportionality coefficient for
the residual losses ωr,totis a fitted value.
Parameter Value
f1 5.223 GHz
Z0 50/Omega1
ZL 50/Omega1
x1 12 mm
x2 6.0 (6.5) mm
εeff 5.73
Cl 155 pF /m
CTL 0.8 fF
CC 3.8 fF
CN 98 fF
CS 460 fF
Cj 6.2 fF
g/(2π) 7.2 MHz
Qint,1 2.7×104(2.0×104)
RT 8.4 (9.5) k /Omega1
γD 1×10−4
Ic,0 340 (300) nA
κ1/(2π) 190 (260) kHz
κint,2/(2π)1 6 M H z
ωr,tot/(2π)2 2 M H z
importance in the measurements of the damping rates during
the QCR pulse.
APPENDIX G: RESIDUAL LOSSES IN
THE RESONATOR R2
We attribute the residual voltage-dependent losses to de-
phasing, and to dissipation sources such as quasiparticlegeneration in the superconductors and resistive losses in thenormal metal. First, the resonator R2 is slightly nonlinearowing to the SQUID and, hence, an increasing incoherentphoton number results in dephasing. Dephasing can be addedin Eq. ( D2) with a term κ
φL[ˆb†ˆb]ˆρ, where the dephasing rate
κφdepends on the number of thermal photons in the resonator.
Similarly, in the case of superconducting qubits, the dephasing
can be written as κφL[ˆσz]ˆρ, where ˆ σzis a Pauli operator. The
factor κφcauses a similar effect as κ2in Eqs. ( D2)–(D12)
although it does not decrease the total photon number inthe resonators. The photon-number variance for a thermalstate is of the form [ 38,39]n(n+1) and, therefore, thermal
photons cause more dephasing than the coherent photonswith a variance of n, where nis the average photon number.
Consequently, we assume that κ
φ=ωφn(n+1), where ωφis
a proportionality coefficient. Furthermore, as discussed above,the number of the coherent photons is low in R2 due to therelatively high loss rate. The steady-state photon number inR2 can be estimated as [ 49]
n=κ
QCRNQCR
κQCR+κint,2, (G1)
where we assume that the photon number of the effective
bath, to which R2 is coupled through κint,2, vanishes owing to
the very low cryostat temperatures of approximately 10 mK.The photon number ndepends linearly on the bias voltage at
voltages above the superconductor energy gap, as shown inFig.11.
Second, we take the quasiparticle losses into account. The
critical temperature of Nb is approximately 9 K and, there-fore, the quasiparticle density remains low in it. However,the critical temperature of Al approximately 1.2 K, whichenables higher quasiparticle density than in Nb. We observe adecrease in the critical current of the SQUID, which indicatesincreased temperature in the Al leads of the SQUID, andhence heat dissipation. The quasiparticle loss rate [ 80,81]
κ
qp∝nqp∝√
P, where Pis the absorbed power. The Al
leads at the NIS junctions receive half of the Joule powerP=IV
bat high voltages, whereas the other half is absorbed
to the normal metal. Thus, the power is quadratic in voltage,which is linear in the estimated photon number. Therefore,the expected quasiparticle losses are linear in photon numberκ
qp=ωqpn, where ωqpis a proportionality coefficient. The dc
power dissipated in the junctions is substantially higher thanthe microwave input power. At eV
b/(2/Delta1)=2, the dc power
is approximately 30 pW compared to a microwave power of−100 dBm =0.1 pW. The normal metal in the QCR acts as an
effective quasiparticle trap [ 81] minimizing the quasiparticle
losses. However, some fraction of the power dissipated at theQCR leaks to the SQUID.
There is an approximately 10- μm-long section of normal
metal between the actual Nb resonator and the NIS junctions,which may cause some losses. The loss rate κ
resat the resistor
depends on the current profile of the microwave mode, whichcan depend on the voltage V
b. Nevertheless, we assume these
losses to be small since there is a layer of superconductingAl below the normal metal due to the shadow evaporationtechnique. This Al layer decreases the current in the resistor,and hence also the resistive losses. The very weak resistivelosses are quadratic in the voltage amplitude of the microwaveresonator which is linear in photon number if the modeprofile does not change. Thus, the loss rate per photon κ
res
is approximately constant.
134505-11MATTI PARTANEN et al. PHYSICAL REVIEW B 100, 134505 (2019)
Consequently, the total voltage-dependent losses in R2
including the dephasing, quasiparticle losses in the super-conductors and the resistive losses are given by κ
r,2=κφ+
κqp+κres.We expect the dephasing to dominate over the
quasiparticle and resistive losses. Therefore, in the numericalanalysis, we take the photon-number-dependent losses intoaccount as
κ
r,2=ωr,totn(n+1), (G2)
with only one fitting parameter ωr,toteffectively describing
the different loss methods discussed above. From the exper-imental damping rates of the dissipative resonator R2, weextract the the coefficient ω
r,tot≈2π×22 MHz. The good
agreement with the experimental damping rate κ2and the
model with the quadratic residual losses κr,2in Fig. 4(a)gives
further support for the approximation in Eq. ( G2). We do not
take this loss rate into account in Eq. ( G1) for simplicity, and
also due to the fact that pure dephasing does not decrease thephoton number.
The odd modes of R1 do not show flux dependence as
expected due to the voltage node at the coupling capacitor.However, they do show some dependence on the voltage V
b.
Similar dependence can be observed also for the even modesat/Phi1//Phi1
0=0.5 where the inductance of the SQUID ideally
vanishes and thus decouples the QCR from the resonator R1.We attribute this observation to unintentional asymmetry ofthe sample. Furthermore, the QCR may be weakly coupled tothe input and output microwave fields through some spuriousmode of the sample holder. The very broad resonance athigh-bias voltages enables the coupling to the spurious modes.We note that the spurious modes may be partially responsiblefor the κ
r,2. However, we do not quantitatively model these
losses. Instead, they are effectively included in the parameterω
r,totin Eq. ( G2).
APPENDIX H: FULL MODEL FOR κ2AND κeff
The decay rate of the resonator R2, κ2, and the effective
damping rate of the coupled system, κeff, are obtained as
follows. First, we extract the effective resistance correspond-ing to the QCR by fitting the classical circuit model tothe experimentally obtained scattering parameter S
21using
a least-squares algorithm. In addition to the effective resis-tances, we extract the critical current of the SQUID from thetheoretical model (Fig. 11). From the same fit, we also extract
the coupling capacitance C
Cat zero-bias voltage and assume it
to be voltage independent throughout this work. The couplingcapacitance is found to be 3.8 fF which agrees well with thefinite-element-method simulation that yields approximately5 fF. The internal losses of R1 are extracted separately withR2 far detuned. Second, we calculate the quality factor ofthe resonator R2, Q
R2, for the obtained effective resistance,
as discussed below. The coupling rate is related to the qualityfactor as κ
2=ω2/QR2. The full model shown in Fig. 4(a) is
obtained by fitting
κ2(Vb)=κQCR(Vb)+κr,2(Vb)+κint,2 (H1)
to the experimental transition rates according to Eqs. ( B8),
(G1), and ( G2). Here, we use ωr,totas the only fitting parameter
since we fix κint,2to the saturation value at zero bias, asdiscussed above, and κQCR is obtained from Eq. ( B8) with
parameters extracted from independent measurements.
Subsequently, we may proceed to the effective damping
ratesκeff=−2I m (λ±), which can be obtained from κ2with
the help of Eq. ( 2). The damping rate of R2 above the critical
damping κ2>4gresults in the two branches of the effective
damping rate κeffat bias voltages Vb/greaterorsimilar2/Delta1/e.W eu s eE q .( 2)
for both the experimental effective damping rates and themodel in Fig. 4(b).
APPENDIX I: CLASSICAL CIRCUIT MODEL
To simulate the scattering parameter S21, we use a classical
circuit model similar to the one presented in Ref. [ 57]. We
analyze the samples using standard microwave circuit theory[82]. The input impedance of the resonator R2 is
Z
R2=ZC+Z0/braceleftbig
ZS+Z0tanh(γx2)+Z0[Reff+Z0tanh(γx2)]
Z0+Refftanh(γx2)/bracerightbig
Z0+tanh(γx2)/braceleftbig
ZS+Z0[Reff+Z0tanh(γx2)]
Z0+Refftanh(γx2)/bracerightbig,
(I1)
where the impedance of the SQUID and the capacitors be-
tween the SQUID and the center conductor is given by ZS=
iωLS+2/(iωCS), the impedance of the coupling capacitor
between the resonators by ZC=1/(iωCC),γis the complex
propagation coefficient discussed above, and the terminat-ing impedance consisting of the effective resistance of theNIS junctions and the capacitor between the normal-metalisland and the center conductor is modeled as an effectiveresistor with resistance R
eff. The inductance of the SQUID
is calculated as LS(/Phi1)=/Phi10/[2πI0|cos(π/Phi1//Phi1 0)|],where the
maximum supercurrent through the SQUID is I0, and the flux
quantum is /Phi10=h/(2e).
The scattering parameter S21describing the voltage trans-
mission from port 1 to port 2 can be calculated using thetransmission matrix method as [ 82]
S
21=2
Am+Bm/ZL+CmZL+Dm, (I2)
where ZLis the characteristic impedance of the external
measurement cables, and
/parenleftbigg
AmBm
CmDm/parenrightbigg
=M1M2M3M2M1, (I3)
with
M1=/parenleftbigg11
iωCTL
01/parenrightbigg
, (I4)
M2=/parenleftBigg
cosh(γx1) Z0sinh(γx1)
1
Z0sinh(γx1) cosh( γx1)/parenrightBigg
, (I5)
M3=/parenleftbigg10
1
ZR21/parenrightbigg
. (I6)
We analyze the losses in the resonator R2 also in the absence
of coupling to R1 as shown in Fig. 12. In this case, we omit
matrices M1andM2from Eq. ( I3). The resonator R2 causes
a dip in the amplitude of the transmission coefficient S21,
whereas R1 causes a peak. The quality factor can be estimateddirectly from the ratio of the center frequency and the width ofthe peak or dip. Alternatively, more advanced methods can be
134505-12EXCEPTIONAL POINTS IN TUNABLE SUPERCONDUCTING … PHYSICAL REVIEW B 100, 134505 (2019)
-1
Φ/Φ0f(GHz)|S21|
1.000
0.998
-0.5 0 0.5 134567
FIG. 12. Simulation of the scattering parameter of resonator R2.
The scattering parameter is presented as a function of magnetic fluxand frequency. We omit R1 in the simulation by letting C
TL→∞ .
For clarity, we show the flux-independent resonance frequency of
R1 as the dashed line. Near the mode crossing at /Phi1≈0.4×/Phi10,
a flux detuning of 0 .01×/Phi10results in a frequency detuning of
approximately 140 MHz. The simulation parameters correspond to
sample B.
used [ 83]. In addition, these simulations yield the frequency
detuning as a function the magnetic flux.
APPENDIX J: SAMPLE FABRICATION
The samples are fabricated on a Si wafer with a thickness
of 500 μm and a diameter of 100 mm. First, a 300-nm-thick
layer of SiO 2is thermally grown on the wafer with resistivity
ρ> 10 k/Omega1cm. Subsequently, a 200-nm-thick layer of Nb is
sputtered on top of the oxide. The resonators are patternedon the Nb layer with optical lithography and reactive ionetching. We cover the complete wafer with a 40-nm-thicklayer of Al
2O3fabricated using atomic-layer deposition. This
oxide layer serves as an insulating barrier in the parallelplate capacitors and separates the QCR lines from the groundplane. The nanostructures are defined using electron beamlithography and two-angle shadow evaporation followed by alift-off process. The SQUID consists of two Al layers withthicknesses of 40 nm each. The first Al layer is oxidizedin situ in the evaporation chamber at 1 .0 mbar for 5 min. TheSINIS junctions consist of Al (40 nm) and Cu (40 nm), and the
Al layer is similarly oxidized as in the SQUID. The shadowevaporation technique results in overlapping metal layers.Scanning electron micrographs of the samples are shown inFig.13.
APPENDIX K: MEASUREMENT SETUP
The measurement setup is schematically presented in
Fig. 14. The samples are measured in a commercial dry
dilution refrigerator with a base temperature of approximately10 mK. To characterize the samples, we use standard mi-crowave techniques that are well established in the field of cir-cuit quantum electrodynamics. The scattering parameters aremeasured with a vector network analyzer which contains boththe microwave source and the detector. The microwave signalis attenuated at different temperature stages to avoid heatleakage from higher temperatures to the sample. We employamplifiers at 4 K and at room temperature. The NIS junctionsare controlled by applying a bias voltage or current throughcontinuous thermocoax cables from room temperature downto the base temperature. Magnetic flux for the SQUID isproduced using a superconducting coil with a bias current.
The time-domain measurements are carried out using a
field programmable gate array with a measurement setupresembling the one used in Ref. [ 23]. The voltage pulses to
the QCR are produced with an arbitrary waveform generator.We obtain an accurate in situ calibration of the attenuation in
the measurement cables leading to the QCR from the rapidincrease of the damping rate with increasing pulse amplitudesateV
b/(2/Delta1)=1.
APPENDIX L: NORMALIZATION OF
SCATTERING PARAMETERS
All measured scattering parameters S21are normalized.
Initially, we compensate for the phase winding originatingfrom the electrical delay τ≈50 ns in the measurement setup
outside the sample by multiplying the measured transmis-sion coefficient by exp( iωτ). Consequently, the resonance
produces a circle on the complex plane as the frequency isincreased over the resonance. We transform this circle to itscanonical position where max |S
21|is on the positive real axis
and the circle intersects the origin. Finally, we normalize theamplitude to unity by dividing with max |S
21|.
5µm 40µm 3µm 5µmR1
Port 1R1
R2(a) (b) (c) (d)
FIG. 13. False-color scanning electron micrographs showing sample structure. (a) Normal-metal island with four NIS junctions. (b) SQUID
with two Josephson junctions. (c) Coupling capacitance between R1 and port 1. (d) Coupling capacitance between R1 and R2.
134505-13MATTI PARTANEN et al. PHYSICAL REVIEW B 100, 134505 (2019)
4K
700 mK
10 mK20 dB
20 dB
40 dB300 K
MagnetPort 1Port 2
10 dB
SAMPLEPort 2
Port 1QCR
linesImIthVth
Vb
Z0x1
x1 Z0Z0 Z0ZL
ZLLSQCR
lines
CN CS CSCCCTL
CTLResonator R1Resonator R2
x2 x2(a)(b)
Sample
holder
Magnetic
shield
FIG. 14. Measurement setup and circuit diagram of the sample. (a) Simplified measurement setup showing the attenuators, and amplifiers
at different temperatures. We measure the sample response to microwave signal from port 1 to port 2. Magnetic field for the SQUID is generated
using a coil with current Im. A bias voltage Vband bias current for thermometry Ithare applied to the NIS junctions. The temperature of the
normal metal is deduced from voltage Vthmeasured with an applied bias current Ith. (b) Sample structure presented as an electrical circuit
diagram. The transmission lines of the resonators have characteristic impedances Z0, and the external transmission lines ZL. The sections of the
resonators have lengths x1andx2. The capacitances at the external ports are denoted by CTL, between the resonators by CC, between the SQUID
with inductance LSand the center conductor of the transmission line by CS, and between the normal-metal island and the center conductor
byCN.
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PhysRevB.97.201110.pdf | PHYSICAL REVIEW B 97, 201110(R) (2018)
Rapid Communications
Giant anisotropic magnetoresistance and planar Hall effect in the Dirac semimetal Cd 3As2
Hui Li,1Huan-Wen Wang,2Hongtao He,3Jiannong Wang,1,*and Shun-Qing Shen2,†
1Department of Physics, the Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China
2Department of Physics, the University of Hong Kong, Pokfulam Road, Hong Kong, China
3Department of Physics, Southern University of Science and Technology, Shenzhen, Guangdong 518055 ,China
(Received 9 March 2018; revised manuscript received 30 April 2018; published 14 May 2018)
Anisotropic magnetoresistance is the change tendency of resistance of a material on the mutual orientation
of the electric current and the external magnetic field. Here, we report experimental observations in the Diracsemimetal Cd
3As2of giant anisotropic magnetoresistance and its transverse version, called the planar Hall effect.
The relative anisotropic magnetoresistance is negative and up to −68% at 2 K and 10 T. The high anisotropy and
the minus sign in this isotropic and nonmagnetic material are attributed to a field-dependent current along themagnetic field, which may be induced by the Berry curvature of the band structure. This observation not onlyreveals unusual physical phenomena in Weyl and Dirac semimetals, but also finds additional transport signaturesof Weyl and Dirac fermions other than negative magnetoresistance.
DOI: 10.1103/PhysRevB.97.201110
Anisotropic magnetoresistance (AMR) was first discovered
by Thomson in 1857 and was observed in many ferromagnetic
metals [ 1]. It is closely associated with changes of magneti-
zation relative to the current. Its transverse version shows atransverse current or voltage in response to the longitudinalcurrent flow and an applied in-plane magnetic field, calledthe planar Hall effect (PHE) [ 2,3]. Usually, the relative AMR
is weak, less than 1% or at most up to a few percent insome ferromagnetic metals and half-metallic ferromagnets [ 4].
Recently, negative longitudinal magnetoresistance and non-saturated linear out-of-plane perpendicular magnetoresistanceand in-plane transverse magnetoresistance were reported ina series of newly discovered topological semimetals [ 5–16].
While negative magnetoresistance is possibly associated withthe chiral anomaly of the Weyl fermions in an electric fieldand a magnetic field [ 17–20], linear out-of-plane perpendicular
magnetoresistance and in-plane transverse magnetoresistanceillustrate the high anisotropy of magnetotransport in topolog-ical semimetals. This is a rare property for a paramagneticmetal. The AMR and PHE have started to attract a lot oftheoretical studies in topological semimetals [ 21,22] and other
topological materials [ 23–25].
We denote transverse resistivity by ρ
⊥(B) when the mag-
netic field Bis perpendicular to the electric current density j,
i.e.,j·B=0, and longitudinal resistivity by ρ/bardbl(B) when the
magnetic field is parallel to the electric current density, i.e.,B/bardblj. Usually, transverse resistivity is larger than longitudinal
resistivity, i.e., ρ
⊥(B)/greaterorequalslantρ/bardbl(B), in Weyl and Dirac semimetals.
The equality holds only for B=0. Thus, the resistivity is very
sensitive to the angle between the electric current density andthe magnetic field. In general, the AMR and PHE can be welldescribed by a formula between the electric field Eand the
*Authors to whom correspondence and requests for materials should
be addressed: phjwang@ust.hk
†sshen@hku.hkelectric current density in a vector form as below,
E=ρ⊥j+(ρ/bardbl−ρ⊥)BB·j
B2+ρ⊥χB×j, (1)
where χis the mobility of the charge carriers. Assuming j
andBconstruct an x−yplane [see Fig. 1(a)], the in-plane
field-dependent resistivity ρij=ρ⊥δij+(ρ/bardbl−ρ⊥)BiBj/B2,
withi,j=x,y. In-plane diagonal or longitudinal resistiv-
ity is highly anisotropic as a function of the angle ϕbe-
tween the magnetic field and electric current density, ρxx=
ρ/bardbl+ρ⊥
2+ρ/bardbl−ρ⊥
2cos 2ϕ. In-plane off-diagonal resistivity leads
to a nonzero electric field that is normal to the electric currentdensity but parallel to the magnetic field. In-plane off-diagonalresistivity is called planar Hall resistivity, ρ
xy=ρ/bardbl−ρ⊥
2sin 2ϕ.
It is worth emphasizing that ρxx(ϕ) andρxy(ϕ) have identical
forms to that for ferromagnetic metals [ 1].
In our previous magnetotransport study of Dirac semimetal
Cd3As2microribbons [ 8], we have successfully observed the
carrier density dependence of nonsaturating positive magne-toresistance in out-of-plane perpendicular magnetic fields andnegative longitudinal magnetoresistance in parallel magneticfields. Here, we present further in-plane magnetotransportresults on the AMR and PHE in Dirac semimetal Cd
3As2
microribbons. Our experimental results are in excellent agree-ment with the relation between the magnetic field and electriccurrent density in Eq. ( 1), which can be derived from the
field-dependent current induced by the chiral anomaly of Weyland Dirac fermions, or the Berry curvature in conventional andtopological metals in the semiclassical theory. This implies thatthe observed AMR and PHE in our Cd
3As2microribbons is
associated with the physics of Berry curvature intrinsic to theDirac semimetal Cd
3As2.
Detailed growth and structural charaterizations of Cd 3As2
microribbons can be found in our earlier work [ 8]. In brief,
aC d 3As2microribbon was grown by the chemical vapor
deposition method on Si (001) substrates and Ar gas wasused as a carrier gas. The furnace was gradually heated up to
2469-9950/2018/97(20)/201110(6) 201110-1 ©2018 American Physical SocietyLI, W ANG, HE, W ANG, AND SHEN PHYSICAL REVIEW B 97, 201110(R) (2018)
FIG. 1. Cd 3As2microribbon devices and magnetotransport char-
acteristics. (a) The optical image of the Cd 3As2microribbon device.
The current probes Iand voltage probes VxxandVxyare labeled in
orange, green, and purple colors, respectively. The current is appliedalong the xdirection as indicated, and the microribbon is rotated in the
x−yplane. (b) AFM image and (c) the height profile of the Cd
3As2
microribbon in (a). (d) The in-plane longitudinal magnetoresistance
(MR xx) measured at 2 K with the applied magnetic field ( B) direction
changing from parallel ( ϕ=0◦) to transverse ( ϕ=90◦) to the applied
current ( I) direction in the x−yplane. (e) In-plane longitudinal
magnetoresistance (MR xx) measured at the different temperatures
indicated in parallel fields in the x−yplane.
750 °C in 20 min and the Ar flow was kept as 100 sccm (sccm
denotes cubic centimeter per minute at standard temperatureand pressure) during the growth process. The duration of thegrowth is 60 min, and then it cools down to room temperaturenaturally. To study the magnetotransport properties, a Hallbar structure was fabricated with standard e-beam lithography
and lift-off processes. Al /Au/Cr electrodes with thicknesses
of 500 nm /175 nm /25 nm were deposited using thermal
evaporation and e-beam evaporation methods. The transport
properties of the devices were then measured in a QuantumDesign physical properties measurement system (PPMS) withthe highest magnetic field up to 14 T.
Figure 1(a) shows the optical image of a typical Cd
3As2
device studied in this work. The width wis about 5 μm,
and the intervoltage-probe distance for VxxandVxyis about
10 and 2 μm, respectively. The ribbon thickness tis about
592 nm according to the atomic force microscopy measurementshown in Fig. 1(b) and the measured height profile in Fig. 1(c).
The current Iis applied along the longitudinal direction ( x
direction) of the Cd
3As2microribbon, as indicated in Fig. 1(a).
The carrier concentration and mobility is in the order of10
17/cm3and 104cm2/V s, respectively, for temperatures
below 50 K. The Fermi energy EF, defined as the energy
difference between the Fermi level and the Dirac point, isestimated to be about 88 meV above the Dirac point basedon the known Fermi velocity v
F∼106m/sf o rC d 3As2[26]
FIG. 2. Planar Hall effect (PHE) in the Cd 3As2microribbon.
(a) Schematic of the PHE in the Cd 3As2microribbon devices. (b)
The magnetic field dependence of the Rxymeasured at 2 K with
the applied B-field direction changing from parallel ( ϕ=0◦)t o
transverse ( ϕ=90◦) to the applied current direction in the x−yplane.
(c) The symmetrized angular dependence of the Rxy(top panel) and
Rxx(bottom panel) measured at 2 K and 5 T. The red lines are the
fitting curves using the inset equations, where ϕis the angle between
theIandBfield in the x−yplane; R/bardblandR⊥are the resistance when
ϕis equal to 0° and 90°, respectively; γis the ratio of the width to
the length of the Cd 3As2microribbon device.
(see Sec. A in the Supplemental Material [ 27] for details).
Figure 1(d) shows the in-plane longitudinal magentoresistance
(MR xx=Rxx(B)−Rxx(0)
Rxx(0)×100%) measured at T=2Kw i t h
various angles ϕbetween the applied magnetic field and current
directions in the x−yplane (see the inset). When the magnetic
field ( Bfield) is parallel to the applied current ( I), i.e.,ϕ=0◦,a
pronounced negative MR xxis observed in low magnetic fields.
When ϕincreases, negative MR xxvanishes and an evident
positive MR xxis observed. It reaches the maximum value of
∼275% at 14 T when the Bfield is transverse to I(ϕ=90°).
Figure 1(e) shows the MR xxcurves measured at ϕ=0◦at
the indicated temperatures. Negative MR xxhas been observed
in a wide temperature range, and the largest negative MR xx
of∼−21% is observed at T=50 K and B=7 T. However,
negative MR xxis not observable when the temperature further
increases to above 200 K. Similar MR xxbehaviors have been
observed in other Cd 3As2microribbon devices. Such negative
MRxxhas been studied in detail and attributed to the chiral
anomaly in our previous work [ 8]. Similar observations of
negative MR xxwith B/bardblIhave also been reported and are
considered as a signature of the chiral anomaly of topologicalsemimetals [ 5–7,9–12].
According to Eq. ( 1), the longitudinal resistance (diagonal)
R
xxand the planar Hall resistance (off-diagonal) Rxy, defined
asRxy=Vxy
I, change systematically as a function of the
rotating angle ϕ. Figure 2(a) is a schematic illustration of
the PHE in the Cd 3As2microribbon devices, the current I
is applied along the longitudinal direction of the Cd 3As2
microribbon, and the Bfield is rotated in the x−yplane. In
the experiment, a misalignment between the actual rotationplane and the Cd
3As2microribbon plane may exist. This could
result in a finite ordinary Hall resistivity component in themeasured planar Hall resistivity R
xy. Fortunately, the ordinary
201110-2GIANT ANISOTROPIC MAGNETORESISTANCE AND … PHYSICAL REVIEW B 97, 201110(R) (2018)
Hall component is antisymmetric to the B-field directions and
can be readily eliminated by summing the measured Rxyin
both positive and negative B-field directions. Figure 2(b) shows
the symmetrized B-field dependence of the Rxymeasured at
different rotating angles ϕat 2 K. When ϕvaries from 0° to
90°, the magnitude of Rxyincreases first and then decreases,
as expected in the PHE discussions of Eq. ( 1).
Figure 2(c) shows the symmetrized angular-dependent Rxy
andRxxmeasured at 2 K and 5 T. Both the measured Rxy
andRxxshow a 180° periodic angular dependence, which is
not expected for a nonmagnetic and isotropic solid but is inagreement with Eq. ( 1). We fit the R
xyusing the equation
Rxy=γR/bardbl−R⊥
2sin 2ϕderived from Eq. ( 1), where R/bardbland
R⊥is the longitudinal resistance when ϕis equal to 0° and
90°, respectively, and γis the geometric ratio of the width to
the length of the Hall bar device. Remarkably, as indicatedby the red line in the top panel in Fig. 2(c), the measured
R
xycan be well fitted by the equation, demonstrating the
existence of the PHE in this nonmagnetic material. Moreover,the measured R
xxcan also be well fitted by the equation
Rxx(B,ϕ)=R/bardbl+R⊥
2+R/bardbl−R⊥
2cos 2ϕ, as indicated by the red
line in the bottom panel in Fig. 2(c). Such a periodic resistance
oscillation is a peculiar characteristic of the AMR, as will bediscussed below. The same PHE and AMR features have beenobserved in other Cd
3As2microribbon devices, as can be seen
in Sec. B in the Supplemental Material [ 27].
Prior to discussing the AMR effect, we revisit the Rxx−B
curves at different rotating angles ϕ, as shown in Fig. 3(a).
According to Eq. ( 1), the longitudinal resistance Rxxfollows
Rxx(B,ϕ). Theoretically, we can calculate the Rxxvalue at any
arbitrary angle ϕif the Rxxvalues at ϕ=0◦(R/bardbl) and 90°
(R⊥) are known. As shown in Fig. 3(a), the measured Rxx−B
curves at ϕ=0◦, 30°, 60°, and 90° in a magnetic field range
of±5 T are plotted. The red lines are the calculated results
ofRxx(B,ϕ) using the measured curves at ϕ=0◦and 90°.
The yielded ϕis 27° and 55°, respectively, which is very close
to the experimental set values of 30° and 60°. The differencemay be caused by the misalignment between the actual rotationplane and the Cd
3As2microribbon plane. Figure 3(b) shows the
symmetrized angular-dependent Rxxat the Bfields indicated
and 2 K. The AMR effect can be seen clearly at different B
fields and can be well described by Rxx(B,ϕ) [red lines in
Fig. 3(b)].
More remarkably, the observed AMR here shows anoma-
lously larger R⊥thanR/bardbl. Thus, the AMR ratio, defined as
R/bardbl−R⊥
R⊥×100% [ 4], is negative for the Cd 3As2microribbon
device at 2 K, as shown in Fig. 3(c). With increasing Bfields,
the magnitude of the AMR ratio increases monotonically andsaturates at ∼68% around B=10 T. In Fig. 3(c), indicated
as solid symbols, the saturated AMR ratios for ferromagneticmetals CoMnAl, NiFe, and Fe
4N and half-metallic ferromag-
net La 0.7Sr0.3MnO 3are replotted from Ref. [ 4] for comparison.
As can be seen, the saturated AMR ratio for the Cd 3As2
microribbon device is one or two orders of magnitude largerthan that for ordinary ferromagnetic metals and half-metallicferromagnets. This giant and negative AMR is a striking featurein topological Weyl and Dirac semimetals. Moreover, the AMRamplitude follows a quadratic B-field dependence at a small
B- fi e l dr e g i m e( B< 1.0 T), as theoretically expected (see
FIG. 3. Anisotropic magnetoresistance (AMR) in the Cd 3As2
microribbon. (a) The symmetrized longitudinal resistance ( Rxx)
measured at 2 K at ϕindicated, where ϕis the angle between the
IandBfield in the x−yplane. (b) The angular dependence of
theRxxat 2 K and Bfields indicated. The red lines in (a) and (b)
are fitting curves using Rxx=R/bardbl+R⊥
2+R/bardbl−R⊥
2cos 2ϕ.( c )T h eA M R
ratio of the Cd 3As2microribbon devices at 2 K as a function of
theBfields. The black line is a guide to the eyes. The AMR ratio
of ferromagnetic metals CoMnAl, NiFe, Fe 4N, and half-metallic
ferromagnet La 0.7Sr0.3MnO 3from Ref. [ 23] is also indicated for
comparison. (d) The angular dependence of the in-plane longitudinalconductance G
xxat 2 K and Bfields indicated. The red lines are the
fitting curves using Eq. ( 3).
Sec. C in the Supplemental Material [ 27] for details). However,
the AMR amplitude deviates from the quadratic function at ahigher Bfield above 3.5 T, which may be caused for multiple
reasons and is still under theoretical investigation.
Figure 3(d) shows the angular-dependent magnetoconduc-
tance. The longitudinal magnetoconductance is not simplyproportional to cos
2ϕ. The nonsinusoidal feature becomes
more evident when increasing the Bfield from 1 to 10 T,
demonstrating that the measured AMR increased with themagnetic field. This effect can be well understood from theAMR and PHE [ 21]. In this case, the conductance Gis given
asR
−1, and the relative longitudinal magnetoconductance is
then given by
G−G⊥
G⊥=cos2ϕ
R/bardbl
R⊥−R/bardbl+sin2ϕ. (2)
The correction of sin2ϕin the denominator reflects the in-
plane transverse voltage induced by the applied Bfield or
the PHE. This is another peculiar feature of the AMR. Weplug the experimental value of R
/bardblandR⊥into Eq. ( 2)t o
reproduce the angular-dependent magnetoconductance [redlines in Fig. 3(d)], which shows very good agreement with the
experimentally measured ones [open circles in Fig. 3(d)]. Thus
the PHE is attributed to this angle narrowing effect. This anglenarrowing effect was also observed in a previous experiment[6], which implies the existence of PHE in Na
3Bi.
201110-3LI, W ANG, HE, W ANG, AND SHEN PHYSICAL REVIEW B 97, 201110(R) (2018)
The AMR and PHE can be attributed to a B-field-dependent
current given by
jB=α(B·E)B. (3)
The key feature of Eq. ( 3) is that the current density is
parallel to the magnetic field instead of the electric field.Several mechanisms may produce this type of current: (1)The chiral magnetic effect of Weyl fermions gives a nonzerocurrent density which is proportional to the magnetic field B,
j
CME=e2
4π2¯h2μ5B, where μ5is the chemical potential differ-
ence between the two Weyl nodes according to the quantumfield theory [ 28,29]. However, when the Weyl fermions are
subject to both an electric field Eand a magnetic field B,t h e
chiral anomaly equations for Weyl fermions induce a nonzero
valueμ
5≈e2¯hv3
FE·Bτv
μ2 , where τvis the relaxation time, vFis the
Fermi velocity, and μis the chemical potential. Substituting
μ5into the current equation of the chiral magnetic effect,
one obtains α=e2
4π2¯he2v3τv
μ2.αis anticipated to be a constant
in weak magnetic fields [ 18] and become field dependent at
strong fields [ 30]. (2) In the second-order semiclassical theory
[31], the Berry curvature in a conventional metal without
chiral anomaly can also produce a current along the directionof the Bfield. It is attributed to the Fermi surface property
that highly depends on the geometric quantities such as theorbital magnetic moment. For nonmagnetic metals in thesemiclassical regime, the leading-order magnetoresistivity isquadratic of Bdue to the constraint of time-reversal symmetry
and the Onsager’s relation [ 32]. (3) Other possible mechanisms
are also proposed in conventional and topological conductors.For example, the electric and magnetic field can produce ahelicity imbalance leading to the field-dependent current in aDirac-like material [ 33].
In the presence of a magnetic field, the Lorentz force, which
deflects the motion of charged particles in a magnetic field, isalso one of the main sources to produce magnetotransport ina solid. Considering the drift velocity of charge carriers in amagnetic field and the field-dependent correction to the chargecurrent, the charge current density jcan be expressed as
j−χj×B=σ
DE+α(E·B)B, (4)
where σDis the isotropic conductivity, and χis the mobility.
The resulting resistivity is a tensor instead of a scalar afterj
Bis included. When the magnetic field is transverse to the
electric current density, i.e., j·B=0, the transverse resis-
tivity is ρ⊥=1
σD. When the magnetic field is parallel to the
electric current density, i.e., B/bardblj, the longitudinal resistivity
isρ/bardbl(B)=1
σD+αB2. The parameter αcan then be expressed in
terms of ρ/bardblandρ⊥asα=(ρ−1
/bardbl−ρ−1
⊥)/B2. In practice, ρ/bardbland
ρ⊥are two physical quantities to be measured experimentally.
As a result, αB2may give rise to a negative magnetoresistivity
defined as δρ/bardbl=ρ/bardbl(B)−ρ/bardbl(0). In addition, Eq. ( 1) can be
explicitly derived from Eq. ( 4) by the vector calculation. All
the parameters in Eq. ( 1) are measurable experimentally.
The excellent agreement between the measured AMR and
PHE and that described by Eqs. ( 1) and ( 4) reveals the existence
of the field-dependent current [given by Eq. ( 3)] in our Cd 3As2devices. For the Dirac semimetal Cd 3As2, the conduction and
valence bands are inverted near the /Gamma1point to form Dirac
points and the Lifshitz point, but a linear dispersion persistseven when the Fermi energy E
Fmoves up to 250 meV above
the Dirac point [ 26]. Since the Lifshitz energy is relatively
small [ 34], it is believed that a strong coupling between the
conduction and valence bands produces the Berry curvature,which can induce the field-dependent current according to thesemiclassical theory [ 31,33]. If the two bands just touch at
one point [ 35], the chiral anomaly could provide a reasonable
mechanism to produce the AMR and PHE [ 22].
Large in-plane transverse magnetoresistance has not been
well understood up to now. In this case, the effect of chiralanomaly is ruled out as the current is normal to the magneticfield. Although Abrisokov found a linear magnetoresistance ata screened Coulomb potential in the quantum limit [ 36], a linear
magnetoresistance was observed even at relatively weak fieldsin many Weyl and Dirac semimetals, especially those with highmobility [ 37]. Therefore, the true physical mechanism is still
unclear [ 38]. In this work, a quadratic in-plane transverse mag-
netoresistance is only measured in very weak fields B< 1T as
shown in Fig. 1(d). Large positive in-plane transverse magne-
toresistance clearly deviates from the quadratic behaviors when
B> 2 T. This cannot be simply explained in the framework of
the Drude theory assuming that the relaxation time is indepen-dent of field. However, the observed large magnetoresistanceindicates that the relaxation time has a large correction in afinite magnetic field. Negative longitudinal magnetoresistancehas been discussed in the previous paper [ 8]. It is worth
noting that negative magnetoresistance can be also inducedby some mechanisms other than chiral anomaly [ 31,33,39–43].
Whether these mechanisms can also induce the AMR and PHEis still an open question and deserves further study.
To conclude, we have observed giant and negative
anisotropic magnetoresistance and the planar Hall effect innonmagnetic Cd
3As2microribbons. Our experimental results
are in excellent agreement with the theoretical descriptionsand formulas of AMR and PHE. The puzzle of the anglenarrowing effect which was first observed in Na
3Bi is also
resolved according to the theory of AMR and PHE. Therefore,our work not only reveals unusual physical phenomena inWeyl and Dirac semimetals, but also finds additional transportsignatures of Weyl and Dirac fermions other than negativemagnetoresistance. The observed giant AMR and PHE intopological semimetals might also have potential applicationsin magnetic sensors.
Recently, we became aware of a work on the measurement
of the planar Hall effect by Wu et al. [44].
This work was supported in part by the Research Grants
Council of the Hong Kong Special Administrative Regionin China under Grants No. 16305215, No. 17301116, No.C6026-16W, and No. AoE/P-04/08, and in part by the NationalNatural Science Foundation of China (No. 11574129) andthe Natural Science Foundation of Guangdong Province (No.2015A030313840). The electron-beam lithography facility issupported by the Raith-HKUST Nanotechnology Laboratoryat MCPF.
H.L. and H.-W.W. contributed equally to this work.
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201110-6 |
PhysRevB.72.075360.pdf | Charge-carrier statistics at InAs/GaAs quantum dots
O. Engström1,*and P. T. Landsberg2
1Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
2Faculty of Mathematical Studies, University of Southampton, Southampton S09 5NH, United Kingdom
/H20849Received 9 November 2004; revised manuscript received 10 May 2005; published 26 August 2005 /H20850
The statistics of thermal electron emission from InAs/GaAs quantum dots with base/height dimensions of
20/10 /H20849nm/H20850are developed. The quantum dots considered are assumed to have two electron energy levels. For
the electrons captured in the ground state, this gives the possibility of two different emission paths. Startingfrom a grand canonical ensemble and using an idea for “truncated cascade capture,” we derive “effectivethermal emission rates” corresponding to experimental quantities. From experimental data of the capture crosssections, we demonstrate that the thermal emission path for electrons is shifted when the temperature ischanged. In an Arrhenius plot for electron emission rates from the ground state, this is manifested as atransition region with varying slope which does not give any information about activation energies. Theposition on a temperature scale of this transition region depends on the internal relaxation time for electrons togo from the excited to the ground states. Due to limitations of experimental setups normally used for measuringactivation energies, such measurements are done within a very limited temperature range. Erroneous interpre-tations of measured data therefore may occur if the possibility of a change in emission path is not taken intoaccount. A method to avoid this problem in an experimental situation is pointed out in the discussion.
DOI: 10.1103/PhysRevB.72.075360 PACS number /H20849s/H20850: 73.21.La, 73.22.Dj
I. INTRODUCTION
For quantum dots /H20849QDs /H20850embedded in a semiconductor
matrix, the emission and capture of charge carriers have aresemblance with the corresponding phenomena of recombi-nation centers.
1–5Their electron potentials confine the charge
carriers to orbitals similar to those in impurity atoms.6–8The
most extensively investigated system, the InAs/GaAs com-bination, has InAs dots with base/height dimensions of about20/10 /H20849nm/H20850embedded in a GaAs matrix and gives rise to
discrete energy levels with only a few electrons captured. For
the InAs/GaAs system, theory and experiments have dem-onstrated that in most cases two different electron shells existinside the dots. The eigen energies of their electron states aredependent on dot dimension, shape, orientation, and chemi-cal composition. In general, for 20/10 /H20849nm/H20850dots, discussed
in the present work, an electron shell with scharacter is
found at an energy level of about 150 meV from the GaAsconduction band, and a pshell occurs about 50 meV above
thesground states. The sshell can accept two electrons with
opposite spins. Due to the flattened shape of the dots, one oftheporbitals receives an eigen energy in or very close to the
conduction band,
6–8and cannot normally be experimentally
observed. The other two porbitals capture four electrons
together. When all six electrons are present in the two shells,as a result of the difference in eigen energies, the thermalemission rate of electrons in the pstates will be much higher
than for the selectrons. Furthermore for the two selectrons,
there is a Coulomb-induced energy difference of a few mil-lielectron volts that was omitted from our previous work.
9As
the normal experimental temperature range for investigationsof thermal electron emission from the sshell
12–14is not lower
than about 60 K, this interval is within kT. Therefore, from a
statistical point of view, it is reasonable to lump these twoenergy levels together in the following treatment and con-sider the sshell to have the same energy level for both the
one- and two-electron states. This gives the electron energyscheme shown in Fig. 1. For the thermal emission from the s
level of this arrangement, two paths exist: direct emissionfrom the level to the conduction band, or a two-step processacross the plevel.
In this paper, we present a theory for thermal emission of
electrons from QDs, where the dots are treated using a grandcanonical ensemble. By specializing in InAs/GaAs dots withbase/height dimensions of 20/10 /H20849nm/H20850and taking into ac-
count the two possible thermal emission paths, expressions
for measurable thermal emission rates are derived. We dem-onstrate that the thermal emission of electrons from thequantum dot sshell to the host material gradually changes
from being dominated by a two-step excitation process viathepshell at the lower temperatures, to a direct emission at
the higher temperatures. The transition temperature region isdetermined by the time it takes for an electron excited to the
FIG. 1. Energy scheme for an InAs/GaAs quantum dot with
base/height dimensions 20/10 nm. For larger dots, a level with d
character appears closer to the conduction band at Ec. Arrows /H20849a/H20850
and /H20849c/H20850–/H20849e/H20850indicate the two possible thermal emission paths for s
electrons while arrows /H20849b/H20850and /H20849d/H20850–/H20849f/H20850show the corresponding cap-
ture paths.PHYSICAL REVIEW B 72, 075360 /H208492005 /H20850
1098-0121/2005/72 /H208497/H20850/075360 /H208497/H20850/$23.00 ©2005 The American Physical Society 075360-1penergy level to relax back to the slevel. The influence of
this effect on experimental data is pointed out.
II. QUANTUM DOTS IN THE GRAND CANONICAL
ENSEMBLE
Treating QDs with l=/H208531,2,3,…,/H20854eigen state configura-
tions and capable of capturing Melectrons, counted by i
=/H208531,2,3,…,/H20854in a grand canonical ensemble, the probability
P/H20849r/H20850of a QD to capture relectrons is given by
P/H20849r/H20850=/H9261rZr
/H20858
i=0M
/H9261iZi, /H208491/H20850
where
/H9261= exp/H20873/H9262
kT/H20874. /H208492/H20850
Here,/H9262is the Fermi level, kis Boltzmann’s constant, and T
is absolute temperature. Zris the partition function for a
canonical ensemble
Zr=/H20858
lexp/H20875−E/H20849l,r/H20850
kT/H20876, /H208493/H20850
where E/H20849l,r/H20850is the eigen energy for configuration lwith r
electrons captured.
If the electron states merge into degenerate energy levels,
for each rthe summation over configurations lcan be limited
to the configurations available at every specific degeneratelevel E/H20849r/H20850. Furthermore, assuming that the degenerate energy
levels are separated by a number of kTunits, the summation
over lin Eq. /H208493/H20850can be approximated by a product of the
number of permutations g
rofrelectrons among the number
of available states, and a Boltzmann factor
Zr=grexp/H20875−E/H20849r/H20850
kT/H20876. /H208494/H20850
The ratio P/H20849r−1/H20850/P/H20849r/H20850, is then given by Eqs. /H208491/H20850and /H208494/H20850as
P/H20849r−1 /H20850
P/H20849r/H20850=1
/H9261gr−1
grexp/H20877E/H20849r/H20850−E/H20849r−1 /H20850
kT/H20878. /H208495/H20850
Taking eras the thermal emission rate of electrons, from a
QD with rQD electrons captured, to the conduction band,
andcras the capture rate for an electron in the conduction
band to become the r-th captured electron in the QD, we
have at thermal equilibrium
erP/H20849r/H20850=crnP/H20849r−1 /H20850. /H208496/H20850
Here, n=Ncexp /H20851−/H20849Ec−/H9262/H20850//H20849kT/H20850/H20852is the concentration of elec-
trons in the conduction band, Ncis the effective density of
states in the conduction band, and Ecis the energy position
of the conduction band edge. Using Eq. /H208495/H20850in Eq. /H208496/H20850,w eg e t
er=gr−1
grcrNcexp/H20877−Ec−/H20851E/H20849r/H20850−E/H20849r−1 /H20850/H20852
kT /H20878. /H208497/H20850
It should be noticed that the difference E/H20849r/H20850−E/H20849r−1/H20850is the
energy added to the ensemble when adding one electron.Assuming, for example, that the first ielectrons are captured
to the same degenerate energy level, all energy differencesE/H20849r/H20850−E/H20849r−1/H20850up to r=ihave the same value. For
InAs/GaAs QDs with base/height dimensions of about
20/10 nm, where two energy levels with sandpcharacters,
respectively, occur,
8thesshell is two-fold, and the pshell is
fourfold degenerate. This means that
E/H20849r/H20850−E/H20849r−1 /H20850=/H20877Es,r= 1,2,
Ep,r= 3,4,5,6, /H20878 /H208498/H20850
The thermal emission rate for the selectrons when relec-
trons are captured in the QDs, therefore, can be expressed as
es,r=gs,r−1
gs,rcs,rNcexp/H20877−/H9004Es
kT/H20878,r= 1,2, /H208499/H20850
and for the pelectrons
ep,r=gp,r−1
gp,rcp,rNcexp/H20877−/H9004Ep
kT/H20878,r= 1,2,3,4, /H2084910/H20850
where /H9004Es=Ec−Esand/H9004Ep=Ec−EpandEcis the conduc-
tion band edge level. In the following, we assume that /H9004Es
−/H9004Epis larger than a few kTunits. This enables one to treat
the two energy levels as independent2,5and motivates count-
ingrf r o m1t o4i nE q . /H2084910/H20850instead of using the total number
/H20849i.e.,r=3,…,6/H20850.
Due to the different physical properties of the QD crystal
and the host, the electron potential, and thus the electroneigen energies, may be influenced by lattice strain and by theenergy band-gap off-set values. A certain influence on energyeigen values by temperature is then expected
/H9004E
x=/H9004Ex0−/H9251xT,x=s,p. /H2084911/H20850
If the /H9251xcoefficient is constant, /H9004Ex0is the energy eigen
value at zero temperature. In practice, /H9251xis often temperature
dependent, which means that /H9004Ex0can be taken as the eigen-
value extrapolated to zero temperature from a linearized tem-perature region of Eq. /H2084911/H20850. Understanding the factor
/H9251xas an
entropy contribution from the lattice, the total entropy /H9004Sx,r
associated with a captured electron can be expressed by
/H9004Sx,r=/H9251x+klngx,r−1
gx,r. /H2084912/H20850
Introducing a free energy, /H9004Fx,r=/H9004Ex0−/H9004Sx,rT, the thermal
emission rates in Eqs. /H208499/H20850and /H2084910/H20850can now be expressed in
two alternative ways depending on the energy quantity usedin the Boltzmann factor
e
x,r=Xx,rcx,rNcexp/H20877−/H9004Ex0
kT/H20878/H11013cx,rNcexp/H20877−/H9004Fx,r
kT/H20878,
/H2084913/H20850
where
Xx,r/H11013exp/H20873/H9004Sx,r
k/H20874 /H2084914/H20850
is an entropy factor expressing the influence of the total en-
tropy change connected with electron emission and capture.O. ENGSTRÖM AND P. T. LANDSBERG PHYSICAL REVIEW B 72, 075360 /H208492005 /H20850
075360-2The first part of Eq. /H2084913/H20850is to be used in the interpretation of
activation plots while the second part should be used for datataken at constant temperature, e.g., when the Fermi level isused as a probe to detect an energy level.
10
The degeneracy factors gx,r−1/gx,rin Eq. /H2084912/H20850are different
for different combinations of xandrdepending on the car-
rier configuration from which the emission takes place. TableI demonstrates the possibilities available, assuming that welimit the study to a maximum of three captured electrons.
III. THERMAL EMISSION RATES FOR InAs/GaAs
QUANTUM DOTS
In the treatment below, we will take into account the
emission of two selectrons and one pelectron, which means
that cases /H20849a/H20850–/H20849c/H20850of Table I are treated. This limitation is
consistent with what normally is found in experiments. Asthe energy separation between the two levels is normallylarger than a few kT, the thermal emissions of electrons from
each of the two shells are regarded as independent.
2,5
After emission of the pelectrons, the two selectrons left
in the QD may each be emitted through two different chan-nels as depicted in Fig. 1. The first s electron emitted may godirectly to the conduction band or via one of the pstates, as
arrows /H20849a/H20850and /H20849c/H20850indicate in the figure. The second selec-
tron has the same possibilities but based on different magni-tudes of the quantum statistical parameters. The electronemission, therefore, consists of an independent thermal pro-cess from the pshell /H20851arrow /H20849e/H20850in Fig. 1 /H20852and from a coupled
two-electron system in the sshell /H20851arrows /H20849a/H20850and /H20849c/H20850–/H20849e/H20850in
Fig. 1 /H20852. For the analysis of such a system we use the argu-
ments developed for “truncated cascade capture” as dis-cussed in Ref. 11 and slightly altered in Ref. 5. For thistreatment, the different emission rates associated with differ-ent emission channels are lumped into one effective emission
ratee
e,ras shown for the two electrons in Fig. 2 as discussed
below.
The process marked ep,rin Fig. 1 is much faster than the
process labeled es,rdue to the difference in binding energy
and is given byep,3=cp,3NcXp,3exp/H20873−/H9004Ep0
kT/H20874. /H2084915/H20850
This process is followed by the emission of the selectrons.
We consider the energy level system in Fig. 1 and let itrepresent the situation when emitting an electron thermallyfrom the sstates of the quantum dots studied here.
For the transfer between the sandplevels we introduce
two time constants: t
r/H11032for the excitation from stopandtrfor
the relaxation from ptos. The portion of the total cycle time,
tr+tr/H11032, spent by the electrons in the pstates is given by the
ratio tr//H20849tr+tr/H11032/H20850. This is the relative time slot open for emis-
sion of an electron in the pstate to the conduction band. The
total emission time tt,rfor an electron stepping from stop
and from there to the conduction band is thus given by
tt,r=tr/H11032+/H20875tr
tr+tr/H11032ep/H20876−1
,r= 1,2. /H2084916/H20850
The effective emission rate can now be expressed as the sum
of the thermal emission rate es,rfor excitation from the s
level to the conduction band and the inverse of tt,r
ee,r=es,r+tt,r−1. /H2084917/H20850
At thermal equilibrium, we expectTABLE I. Possible configurations and associated degeneracy factors for a quantum dot with a maximum of three electrons. The ratios
gx,r−1/gx,rare obtained by taking into consideration that the possible permutation number for an empty level is 1, for a single electron on the
slevel it is 2, for two electrons on the slevel it is 1, and for a single electron on the plevel it is 4.
FIG. 2. Energy scheme illustrating the electron “effective emis-
sion rates” from a single-level two-electron system. The first elec-tron leaves with a rate e
e,2from the sshell, occupied by two elec-
trons, and turns the QD into a one-electron system /H20849dashed arrow /H20850.
This increases the concentration of one-electron systems before thesecond electron leaves with a rate e
e,1.CHARGE-CARRIER STATISTICS AT InAs/GaAs … PHYSICAL REVIEW B 72, 075360 /H208492005 /H20850
075360-3tr−1Pp,r=/H20849tr/H11032/H20850−1Ps,r, /H2084918/H20850
where Ps,randPp,rare the probabilities for an electron to
occupy the sandplevels, respectively. These two quantities
are related by /H20849see Sec. 4.1 in Ref. 5 /H20850
Ps,r
Pp,r=Xp,r
Xs,rexp/H20873/H9004Es0−/H9004Ep0
kT/H20874. /H2084919/H20850
From Eqs. /H2084918/H20850and /H2084919/H20850, assuming that /H9004Es0−/H9004Ep0is large
enough compared to kT,w efi n d tr/H11270tr/H11032so that Eq. /H2084916/H20850can
be approximated as
tt,r=tr/H11032+tr/H11032
tr1
ep,r. /H2084920/H20850
From Eq. /H208499/H20850together with Eqs. /H2084917/H20850–/H2084920/H20850, we find the effec-
tive emission rate
ee,r=/H20849cs,r+/H9008rcp,r/H20850Xs,rNcexp/H20873−/H9004Es0
kT/H20874, /H2084921/H20850
where
/H9008r=/H208491+trep,r/H20850−1/H2084922/H20850
is a “sticking probability” expressing the tendency for an
electron to stay on the plevel.11
Considering Eq. /H2084921/H20850, one observes that the effective ther-
mal emission process from the sshell is made up of two
parts. The first part, with the capture rate cs,rfor electrons
into the sstate in the preexponential factor, corresponds to
the direct transitions between the slevel and the conduction
band /H20851arrows /H20849a/H20850and /H20849b/H20850in Fig. 1 /H20852. The second part ex-
presses the two-step transitions between the sstate and the
conduction band via the pstate, involving the capture rate to
theplevel and the sticking probability /H9008r/H20851arrows /H20849c/H20850–/H20849e/H20850
and /H20849d/H20850–/H20849f/H20850in Fig. 1 /H20852.
IV. THERMAL EMISSION OF ELECTRONS FROM A
SINGLE-LEVEL TWO-ELECTRON SYSTEM
With the effective emission rates from the sshell defined,
we can now treat the system as the single-level two-electronsystem, shown in Fig. 2. The concentration n
s2of QDs which
have two electrons in the sshell has a time evolution deter-
mined by
dns,2
dt=−ee,2ns,2. /H2084923/H20850
Every electron emitted from the two-electron system leaves
behind a single-electron system and increases the concentra-tionn
s1of QDs occupied by one electron. Taking further into
account the emission of the last electron gives the followingtime evolution of n
s,1:
dns,1
dt=ee,2ns,2−ee,1ns,1. /H2084924/H20850
As boundary conditions we take ns,1/H208490/H20850=0 and ns,2/H208490/H20850=NT,
where NTis the total concentration of QDs. We find for the
time dependence of the electron occupationns,2/H20849t/H20850=NTexp /H20849−ee,2t/H20850, /H2084925/H20850
ns,1/H20849t/H20850=ee,2
ee,2−ee,1NT/H20851exp /H20849−ee,1t/H20850− exp /H20849−ee,2t/H20850/H20852./H2084926/H20850
The concentration pTof empty dots and dots filled by one
and two electrons, respectively, sum up to the total concen-tration of dots, N
T
pT/H20849t/H20850+ns,1/H20849t/H20850+ns,2/H20849t/H20850=NT. /H2084927/H20850
In deep-level transient spectroscopy /H20849DLTS /H20850,12–14the tran-
sients expressed by Eqs. /H2084925/H20850and /H2084926/H20850are measured as
changes of charge in a semiconductor depletion region. Dur-ing the process of emitting electrons from the dots, each dotoccupied by one electron gives rise to a change of one el-ementary charge and each empty dot gives rise to a changeof two elementary charges in the space charge region. Theconcentration of elementary charges /H9004p
Cproduced in the
space charge region during the emission process is, by usingEq. /H2084927/H20850
/H9004p
C/H20849t/H20850=2pT+ns,1+/H20849NT−np,3/H20850=3NT−ns,1/H20849t/H20850−2ns,2/H20849t/H20850
−np,3/H20849t/H20850, /H2084928/H20850
where the time dependencies of ns,1andns,2are given by
Eqs. /H2084925/H20850and /H2084926/H20850, respectively, while np3is determined by
the same kinetics as ns,2in Eq. /H2084925/H20850
np,3=NTexp /H20849−ep,3t/H20850, /H2084929/H20850
and where ep,3is given by Eq. /H2084915/H20850.
As long as the charge represented by /H9004pCis much smaller
than the total charge in the space charge region, it is propor-tional to the amplitude of the capacitance transient obtainedfrom a DLTS measurement.
12–14
V. ACTIVATION PLOTS AND EMISSION TRANSIENTS
For calculation purposes, we rewrite Eq. /H2084921/H20850in the fol-
lowing way:
ee,r=/H20849Ys,r+/H9008rYp,r/H20850T2exp/H20873−/H9004Es
kT/H20874,r=1 , 2 /H2084930/H20850
where
Ys,r=A/H9268s,rXs,r,r= 1,2, /H2084931/H20850
Yp,r=A/H9268p,rXs,r,r= 1,2. /H2084932/H20850
Here, we have used the relation cs,r=/H20855v/H20856/H9268s,r, where the aver-
age thermal electron velocity /H20855v/H20856, is set to /H20855v/H20856=/H208493kT/m*/H208501/2
with m*representing the electron effective mass in the GaAs
conduction band while /H9268s,rand/H9268p,rare the capture cross
section for electrons of the sand the plevels, respectively.
For the effective density of states in the GaAs conductionband we have used the relation N
c=4.45/H110031017/H20849T/300 /H208503/2
cm−3which gives A=3.51/H110031024/H20851s−1K−2/H20852.
For our calculations of the emission rates we use values
for the capture cross section obtained experimentally in anearlier study,
10where we determined the values for the firstO. ENGSTRÖM AND P. T. LANDSBERG PHYSICAL REVIEW B 72, 075360 /H208492005 /H20850
075360-4captured electron into the QDs. It has been demonstrated in
intraband spectroscopy that the capture at low temperatureoccurs by a two-step process, first into the pshell and from
there to the sshell.
15The values of the capture cross sections
measured in Ref. 10, therefore, should be taken as those ofthepshell as given in Table II. As the direct transition from
the conduction band to the sshell has not been observed in
luminescence, it is reasonable to assume that the capturecross sections are smaller for such a process than for thecapture into the pshell. In the calculations below, we set
capture cross sections for direct transitions to the sshell one
order of magnitude smaller than those to the pshell. The
internal relaxation time t
rhas been investigated in a number
of published works on InAs/GaAs QDs. It seems to dependon the physical properties of the dot structures and variesbetween different works
16–19in the interval 10−12−10−9s.
Therefore, the calculations below have been done with tr
varying in this region.
Figure 3 shows the sticking probability /H90081, given by Eq.
/H2084922/H20850as a function of inverse absolute temperature, 1000/ T,
for four different values of the internal relaxation time t1./H90082
has a similar shape. At higher temperatures, /H90081approaches
zero because of the high value of the emission rate ep,1.
When the temperature is lowered and ep,1decreases, /H90081in-
creases within a limited temperature interval depending onthe value of t
1and approaches unit value. This influences the
effective thermal emission rates ee,1, given by Eq. /H2084921/H20850,a s
shown in Fig. 4. The change in sticking probability separatestwo different regions of the activation curve for the emissionrates. At the higher temperatures the direct emission processfrom the sshell to the conduction band dominates, while for
the lower temperatures, the emission is dominated by thetwo-step excitation from the sto the pshell and further to the
conduction band. The slopes of the two sections in Fig. 4 arethe same because the two different emission paths require thesame energy as seen from Eq. /H2084923/H20850and illustrated by Fig. 1.
The results of the calculation presented in Fig. 4 depend
on the values chosen for the entropy factor X
s,rwhich in turn
depends on the degeneracy factors and the lattice entropycontribution
/H9251sas expressed by Eq. /H2084912/H20850. The degeneracy
factors depend on the possible electron configurations asshown in Table I. The actual configurations for X
s,1andXs,2
to be used in Eq. /H2084921/H20850are those labeled /H20849b/H20850and /H20849c/H20850in Table
II, respectively, while the configuration /H20849a/H20850is valid for Xp,3to
be inserted into Eq. /H2084915/H20850. A recent theoretical investigation20
on the influence of temperature on lattice strain for
InAs/GaAs QDs of the present size, indicates that the QDelectron potential and the energy eigenvalues are only mar-ginally influenced by a temperature variation between zeroand 100 K.
The time dependence of the probabilities for electron oc-
cupation, P
s1=ns1/H20849t/H20850/NTandPs2=ns2/H20849t/H20850/NT, respectively, of
thesshell during an emission cycle is shown in Fig. 5. The
transient Ps2falls off exponentially and feeds the increase of
Ps1before the remaining single-electron occupation de-
creases as expressed by Eqs. /H2084927/H20850and /H2084928/H20850and illustrated by
Fig. 2. The total change of elementary charges in the QDswhen emitting one pelectron followed by two selectrons, as
given by Eq. /H2084930/H20850, is shown in the semilogarithmic graph of
Fig. 6. The transient is valid for 1000/ T=12, at the low-
temperature side of the transition region in Fig. 4 betweenthe two-step and the direct s to conduction band emission.Due to the combination of emission from the pshell and the
two electrons in the sshell, the transient has a steep nonex-
ponential shape at the beginning of the time scale for timessmaller than about 0.5
/H9262s. For times longer than about 3 /H9262s,
a straight line is approached revealing an exponential devel-opment. This part reflects the emission of the last electron/H20849s1/H20850leaving the QD. In the intermediate region between 0.5
and 3
/H9262s, the nonexponential behavior originates from a mixTABLE II. Quantities used in the numerical calculations of Figs.
3 and 4.
/H9268s,1
/H20849cm2/H20850/H9268s,2
/H20849cm2/H20850/H9268p,3
/H20849cm2/H20850Xs,1Xs,2Xs,3/H9004Es
/H20849meV /H20850/H9004Ep
/H20849meV /H20850
2/H1100310−122/H1100310−122/H1100310−111/2 2 1/4 150 100
FIG. 3. Sticking probability of the /H20849s,1/H20850electron as a function of
reciprocal temperature as obtained by Eq. /H2084922/H20850for different values
of the time t1it takes for an electron to relax from the plevel to the
slevel when the plevel is filled by one electron and the slevel is
empty.
FIG. 4. Thermal emission rate ee,1of an electron from the s
level when occupied by one electron as obtained by Eq. /H2084913/H20850. For
the lower temperatures, the two-step emission path across the p
level dominates the emission, while the direct process from the s
level to the conduction band dominates for higher temperatures.Between these two temperature regimes, there is a transition region,where the slope of the Arrhenius plot origins from a combination ofthe two emissions.CHARGE-CARRIER STATISTICS AT InAs/GaAs … PHYSICAL REVIEW B 72, 075360 /H208492005 /H20850
075360-5of emissions by the two selectrons. These kinds of tran-
sients, composed of multielectron emission are expected tobe reflected in spectra obtained from deep-level transientdata. The two-electron character of the slevel and the emis-
sion from the plevel have been observed in such experimen-
tal spectra.
13,14
VI. DISCUSSION AND CONCLUSIONS
The temperature dependence of the sticking probability,
shown in Fig. 3 is strongly influenced by the internal relax-
ation time tr/H11032of electrons from the pshell to the sshell
indicated by an arrow /H20849d/H20850in Fig. 1. It is related to the so
called “phonon bottleneck,”16–19often mentioned in the lit-
erature, and depends on the availability of phonon energiesfitting to the energy difference between the different energylevels in the QD. The relaxation time, therefore, may beexpected to be dependent on the sizes, shapes, and chemicalcompositions of dots. This has a consequence for the inter-pretation of activations plots for the thermal emission ratesof charge carriers. The calculated set of curves in Fig. 4,demonstrates how the relaxation time influences the transi-tion from stopshell to conduction band at lower tempera-
tures to approach direct emission from the sshell to the
conduction band for the higher temperatures. Experimentalsetups for DLTS, normally used for measuring emissionrates, have response times which for the present system limitthe measurements to the temperature region below about 100K. Therefore, the slope of an activation curve may be influ-enced by the change due to the change of emission path. Forthe largest values of the relaxation time, there is an obviousrisk to misinterpret the values of activation energies and cap-ture rates. In order to avoid such erroneous interpretations,additional and independent measurements of, for example,the capture cross sections may be necessary. For the dataused in the present analysis, this was done in earlier workwhich supported an assumption that the relaxation time wasshort enough for the transition region to be outside the mea-sured activation plot.
10,13,20In DLTS, capacitance transients transformed into tem-
perature spectra are used to find the thermal activation ener-gies related to charge carrier emission. When emissions takeplace by more than one electron with close lying activationcurves, the spectral peaks may interfere and give rise to mis-leading conclusions. The emission of the two selectrons in
the present system has been investigated experimentally inthe low temperature range where two-step processes acrossthepshell take place
13and the peaks occurring in DLTS
spectra could be explained by the spread in QD size.
Thermal emission rates are often measured by DLTS,
where the emission sources are placed in a space chargeregion of a p-njunction or a Schottky diode. Especially for
QDs, with the electron eigen energies relatively shallow, inthe region of 200 mV and lower, a nonvanishing tunnelingrate may add to the thermal emission rate studied for higherapplied voltages. This is clearly the case for some of the datain Refs. 12–14 and has a significance for higher voltages andlower temperatures. Tunneling was not included in thepresent description. Taking into account the tunneling pathsfrom both the sand the pshells, these processes can be
added to the thermal processes expressed in Eq. /H2084921/H20850in order
to obtain the full expression for electron emission. In thiscontext it should be observed that when a realistic potentialfor the QD electrons is taken into account
9the tunneling
rates deviate by at least one order of magnitude from thosegiven in the classical paper by Korol,
21often used in the
literature. In this latter case, the calculation was done fordeep impurities represented by a delta function and canhardly be used for QDs as demonstrated in Ref. 9.
Grundmann and Bimberg had earlier developed statistics
for the capture of electrons into quantum dots.
22It should be
noted that their treatment is valid under nonequilibrium situ-ations and for a high concentration of excess charge carriersin the semiconductor energy bands, i.e., for example, forluminescence experiments. In the present work, we havegiven the statistics for an experiment where electrons arethermally emitted from quantum dots in a semiconductordepletion region. This is the common situation when measur-ing thermal emission rates by DLTS. The two treatments,therefore, describe charge-carrier traffic at quantum dots fortwo different and complementary experimental cases.
FIG. 6. Logarithm of the charge /H9004pcin units of elementary
charge, created per quantum dot in a semiconductor depletion re-gion as it would appear in a DLTS measurement for t
1=t2
=10−10s and 1000/ T=12 mK−1as in Fig. 5
FIG. 5. Emission transients for the first /H20849s,2/H20850and second /H20849s,1/H20850
electrons leaving the quantum dot. The curves show the probabili-tiesP
s,1=ns,1/NTandPs,2=ns,2/NTthat the sshell is occupied by
one and two electrons, respectively. Calculation has been done fort
1=t2=10−10s and 1000/ T=12 mK−1O. ENGSTRÖM AND P. T. LANDSBERG PHYSICAL REVIEW B 72, 075360 /H208492005 /H20850
075360-6*Corresponding author. Email address:
olof.engstrom@mc2.chalmers.se
1W. Shockley and W. T. Read, Phys. Rev. 87, 835 /H208491952 /H20850.
2P. T. Landsberg, Recombination in Semiconductors /H20849Cambridge
University Press, Cambridge, 1992 /H20850.
3M. J. Kirton and M. J. Uren, Adv. Phys. 38, 367 /H208491989 /H20850.
4O. Engström and A. Alm, J. Appl. Phys. 54, 5240 /H208491983 /H20850.
5P. T. Landsberg and O. Engström, in Handbook of Semiconduc-
tors, V ol. 1, edited by P. T. Landsberg /H20849Elsevier, New York,
1992 /H20850, p. 197.
6M. Grundmann, O. Stier, and D. Bimberg, Phys. Rev. B 52,
11969 /H208491995 /H20850.
7L.-W. Wang, J. Kim, and A. Zunger, Phys. Rev. B 59, 5678
/H208491999 /H20850.
8P. Hawrylak and A. Wojs, Semicond. Sci. Technol. 11, 1516
/H208491996 /H20850.
9Y . Fu, O. Engström, and Y . Lou, J. Appl. Phys. 96, 6477 /H208492004 /H20850;
Y . Fu, Y . Lou, and O. Engström /H20849unpublished /H20850.
10O. Engström, M. Kaniewska, Y . Fu, J. Piscator, and M.
Malmkvist, Appl. Phys. Lett. 85, 2908 /H208492004 /H20850.
11P. T. Landsberg and S. R. Dhariwal, Phys. Rev. B 39,9 1 /H208491989 /H20850.12C. M. A. Kapteyn, F. Heinrichsdorff, O. Stier, R. Heitz, M.
Grundmann, N. D. Zakharov, and D. Bimberg, Phys. Rev. B 60,
14265 /H208491999 /H20850.
13O. Engström, M. Malmkvist, Y . Fu, H. Ö. Olafsson, and E. Ö.
Sveinbjörnsson, Appl. Phys. Lett. 83, 3578 /H208492003 /H20850.
14S. Schulz, S. Schnull, Ch. Heyn, and W. Hansen, Phys. Rev. B
69, 195317 /H208492004 /H20850.
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16F. Adler, M. Geiger, A. Bauknecht, F. Scholz, H. Schweizer, and
M. H. Pilkuhn, J. Appl. Phys. 80, 4019 /H208491996 /H20850.
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075360-7 |
PhysRevB.85.075108.pdf | PHYSICAL REVIEW B 85, 075108 (2012)
Hole depletion of ladders in Sr 14Cu24O41induced by correlation effects
V . Ilakovac,1,2,*C. Gougoussis,3M. Calandra,3N. B. Brookes,4V . Bisogni,4S. G. Chiuzbaian,1J. Akimitsu,5
O. Milat,6S. Tomi ´c,6and C. F. Hague1
1Laboratoire de Chimie Physique Mati `ere et Rayonnement, UPMC, CNRS, F-75231 Paris Cedex 05, France
2Universit ´e de Cergy-Pontoise, F-95031 Cergy-Pontoise, France
3Institut de Min ´eralogie et de Physique des Millieux Condens ´ees, UPMC, CNRS, F-75252, Paris Cedex 05, France
4ESRF , Bo ˆıte Postale 220, F-38043 Grenoble Cedex, France
5Departement of Physics, Aoyama-Gakuin University, Setagaya, Tokyo 157-8572, Japan
6Institut za fiziku, P .O. Box 304, HR-10001 Zagreb, Croatia
(Received 9 November 2011; published 7 February 2012)
The hole distribution in Sr 14Cu24O41is studied by low-temperature polarization-dependent O Knear-edge x-ray
absorption fine-structure measurements and state-of-the-art electronic structure calculations that include core-holeand correlation effects in a mean-field approach. Contrary to all previous analysis, based on semiempirical models,we show that correlations and antiferromagnetic ordering favor the strong chain-hole attraction. For the remainingsmall number of holes accommodated on ladders, leg sites are preferred to rung sites. The small hole affinityof rung sites explains naturally the one-dimensional to two-dimensional crossover in the phase diagram of(La,Y ,Sr,Ca)
14Cu24O41.
DOI: 10.1103/PhysRevB.85.075108 PACS number(s): 78 .70.Dm, 71 .15.Mb, 74 .72.Gh, 74 .78.−w
I. INTRODUCTION
The quasi-one-dimensional spin chain and ladder
(La,Y ,Sr,Ca) 14Cu24O41compounds have attracted consider-
able interest since the discovery of a quantum critical phasetransition in their phase diagram.
1These compounds are the
first superconducting copper oxide materials with a nonsquarelattice. They are composed of alternately stacked chain andladder planes. The chains are made up of CuO
2linear edge-
sharing CuO 4squares, and the ladder planes consist of two
zigzag strings of corner-sharing Cu 2O3squares (see Fig. 1).
The parent compound Sr 14Cu24O41is naturally doped with six
holes per formula unit (f.u.).
Determining exactly how the holes are distributed in the
system is hindered by the complexity of the crystal structureof Sr
14Cu24O41and electron correlation effects.2,3Room-
temperature optical conductivity,4near-edge x-ray absorption
fine-structure (NEXAFS),5x-ray emission spectroscopy,6and
Hall coefficient measurements7on Sr 14Cu24O41estimated that
at least five holes per f.u. reside in chains and at most one
resides in ladders. The high density of holes in chains iscompatible with the low-temperature T
C≈200 K8charge
ordering with the fivefold chain periodicity accompanying
the antiferromagnetic (AF) spin dimerization, observed byinelastic neutron scattering.
9–11In ladders, a gapped spin liquid
and the charge-density wave (CDW) appear in the ground statein the spin and charge sectors, respectively. Whereas the spinliquid and its collective spin excitations (triplons) are wellunderstood theoretically and experimentally,
12–14the low hole
density in the charge sector is at variance with the fivefold
periodicity in ladder cell units (Wigner hole crystal) observed
by low-temperature resonant x-ray diffraction measurementsat the O Kedge if a 4 k
FCDW picture is assumed.15On the
other hand, a revisited interpretation of NEXAFS spectra16
claimed a distribution of 2.8 holes in ladders and 3.2 holesin chains, which satisfactorily explains the CDW in ladders,although it fails to explain AF dimer order in chains.
The apparently contradictory results are mainly related
to the lack of a suitable theoretical model. All publishedNEXAFS analyses are based on semiempirical models that
do not take explicitly into account the complex interplaybetween spins and holes in ladders and chains. Thus the basicquestion of where the holes reside in Sr
14Cu24O41has remained
unresolved up to now.
The hole distribution between chains and ladders is the
key to stabilization of interdependent electronic phases inthese two subsystems. It is a prerequisite to understandingthe low-energy physics of the system and such open issuesas the occurrence of Zhang-Rice singlets,
17the evolution of
spin order, and the origin of the superconductivity in theCa-doped compound.
2We bridge the gap between theory
and experiment by performing low-temperature electron-yieldpolarization-dependent O K-preedge NEXAFS measurements
and state-of-the-art electronic structure calculations. In thetheoretical modeling we consider the full 316-atom AF unitcell, we included the core-hole effects, core level shift, andcorrelations in a DFT +U framework. This method has proven
to be fairly successful in reproducing x-ray absorption spectrain correlated metals with well-localized orbitals.
19,23We show
that, contrary to previous claims, correlation effects and AForder stabilize holes on chains and induce hole depletion inladders, where rung sites become less populated than leg sites.
II. EXPERIMENTAL AND CALCULATION DETAILS
High-quality single crystals of Sr 14Cu24O41were grown by
the traveling-solvent floating-zone method and characterizedby x-ray diffraction measurements. A well-oriented samplewas cleaved in situ along the ( a,c) plane under a pressure
of 10
−10mbars. The O K-edge NEXAFS measurements were
performed using the helical undulator Dragon beam line ID08at the European Synchrotron Radiation Facility (ESRF) inthe total-electron-yield mode with 200-meV resolution. Theincident light was normal to the sample surface, and itspolarization was parallel to the a/csample axis.
NEXAFS calculations are performed by the
XSPECTRA18–20
code based on density functional theory (DFT)21and
075108-1 1098-0121/2012/85(7)/075108(6) ©2012 American Physical SocietyV . ILAKOV AC et al. PHYSICAL REVIEW B 85, 075108 (2012)
FIG. 1. (Color online) 3D view along the ccrystal axis of four
316-atom AF unit cells of Sr 14Cu24O41. The structure is composed
of alternately stacked chain layers, constituted of CuO 2linear edge-
sharing CuO 4squares, and ladder planes, made up of two zigzag
strings of Cu 2O3corner-sharing squares. They are parallel to the ( a,
c) plane and are separated by Sr atoms. Cu, O, and Sr atoms are
colored in red (medium gray), blue (dark gray), and yellow (light
gray).
all-electron wave-function reconstruction.22Correlation ef-
fects are simulated in a mean-field DFT +U framework23,24
using a Hubbard on-site energy of Udd=10 eV for copper and
Upp=4 eV for oxygen. The chain and ladder atomic positions
were taken from Gotoh et al.25without imposing any super-
structure modulations, either along the chains or along theladders. We performed a specific NEXAFS calculation foreach of the five inequivalent oxygen sites existing in the
structure, two in the ladders and three in the chains. We find anegligible core level shift.
III. RESULTS AND DISCUSSION
A. NEXAFS measurements
The low-temperature (150 K; see Fig. 2)OKNEXAFS
preedge spectra have two distinct structures, Ae, centered
at 527.9 eV , and Be, centered at 529.5 eV ( estands for
experiment), in agreement with previously published room-temperature spectra.
5,6,16The shape of the two structures is
dependent on the polarization of the incident photon: for E/bardbla,
Aeis stronger than Be, while for E/bardblc(direction along the
chains and the ladders) it is the opposite. For E/bardblcstructure Ae
has a second, well-resolved peak at 528.5 eV , A/prime
e. The intensity
of this structure is only weakly affected by the charge-orderingtransition (see Fig. 3). The E/bardblaspectra are even less affected1.5
1.0
0.5
0.0Intensity (arb. units)
532 531 530 529 528 527 526
Photon energy (eV)1.6 eV
0.6 eV
AeA'eBe 150 K
E||a
E||c XAS
O K edge
Experiment
Pol. dep.
FIG. 2. (Color online) Polarization-dependent O KNEXAFS
preedge intensity for E/bardblaandE/bardblcmeasured at 150 K. Subscript
estands for experiment.
atTC(see Fig. 3, inset). The attribution of the features in these
spectra has so far been very controversial.5,15,16
B. Calculation of the NEXAFS spectra
Theoretical modeling using the AF unit cell with fivefold
periodicity in chains is shown in Fig. 4. The two structures,
AeandBe, are well reproduced in the calculations (labeled
AcandBc, where cis for the calculated spectra), although
their separation is only 0.7 eV , compared to 1.6 eV for theexperiment. This discrepancy is attributed to the well-knownunderestimation of the Hubbard gap in DFT +U simulations.
The agreement between theory and experiment is particularly
2
1
0
530 528 526
h (eV)E||a
2.5
2.0
1.5
1.0
0.5
0.0Intensity (arb. units)
532 531 530 529 528 527 526
Photon Energy (eV) E||c
150 K
200 K
230 K
300 K XAS
O K edge
Experiment
Temp. dep.
FIG. 3. (Color online) Temperature dependence of the O K
NEXAFS preedge intensity for E/bardblc, showing that for this polarization
spectra change below and above the charge-ordering temperature
(≈200 K). The inset shows that the E/bardblaspectra temperature change
is negligible.
075108-2HOLE DEPLETION OF LADDERS IN Sr 14Cu24O... PHYSICAL REVIEW B 85, 075108 (2012)
0.20
0.15
0.10
0.05
0.00Cross section (arb. units)
3 2 1 0 -1
Energy (eV)0.3 eV0.7 eV
AcBcA'c E||a
E||c XAS
O K edge
Calculation
5-fold
chain per.E||a
1 0
Energy (eV) E||c
C
L
R
FIG. 4. (Color online) Calculated O KNEXAFS preedge cross
section on the AF unit cell with fivefold periodicity in chains for E/bardbla
andE/bardblc. The insets show the contribution of chain (C), ladder-leg
(L), and ladder-rung (R) oxygens for E/bardbla(top inset) and E/bardblc(bottom
inset). Zero energy corresponds to a photon energy of 527.4 eV .
Subscript cstands for calculated.
good for the E/bardblcpolarization because in the cdirection the
charges are less localized. For E/bardblc, (i) the Bccross section
is stronger than the Accross section, and (ii) the high-energy
shoulder A/prime
cis clearly visible. For E/bardblapolarization, the Bc
structure has a smaller cross section than the Acpeak.
The two insets in Fig. 4show the calculated cross-section
contribution of chain (C), ladder-leg (L), and ladder-rung (R)oxygens to the NEXAFS spectra. The contribution from ladderrungs is small or negligible for all features except for the B
c
peak in the E/bardblageometry. This result is in line with the analysis
by N ¨ucker et al.5and is in strong disagreement with the results
of Rusydi et al.16where the contribution of the ladder rungs
was overestimated. Our results demonstrate that ladder rungs
are hole depleted in Sr 14Cu24O41.
For both polarizations the low-energy Acpeak is equally
composed of chain and ladder-leg states, with a minorcontribution from ladder rungs in the E/bardblageometry. Its high-
energy A
/prime
cshoulder is composed of both ladder-leg and chain
contributions. This result is in stark disagreement with allprevious claims based on semiempirical NEXAFS analysis
5,16
that attribute the A/prime
cstructure solely to ladders. It points to the
fact that the O K-edge resonant diffraction results, performed
at the photon energy corresponding to A/prime
c, could be related
to the modulation reflections due to the chain-ladder latticemismatch.
26The Wigner-hole crystallization analysis of the
OK-edge resonant soft x-ray scattering data in Ref. 15now
becomes questionable.
Calculations using the AF unit cell with fourfold periodicity
in chains27are shown in Fig. 5. The two insets show the
calculated cross-section contribution of chain (C), ladder-leg(L), and ladder-rung (R) oxygens, equivalent to those in Fig. 4.
The experimental results are less well reproduced compared tothe fivefold chain-periodicity unit cell. First, the A
c-Bcenergy
separation is smaller. Further, for the E/bardblcgeometry AcandE||a
1 0
Energy (eV) E||c
C
L
R0.20
0.15
0.10
0.05
0.00Cross section (arb. units)
3 2 1 0 -1
Energy (eV)0.6 eV
AcBc
A'c0.33 eV
E||a
E||c XAS
O K edge
Calculation
4-fold
chain per.
FIG. 5. (Color online) Calculated O KNEXAFS preedge cross
section on the AF unit cell with fourfold periodicity in chains for E/bardbla
andE/bardblc. The insets show the contribution of chain (C), ladder-leg
(L), and ladder-rung (R) oxygens for E/bardbla(top inset) and E/bardblc(bottom
inset). Zero energy corresponds to a photon energy of 527.4 eV .
Bcare almost equal in intensity, contrary to the experiment.
Finally, A/prime
cappears as a low-energy shoulder to structure Bc
and not as a high-energy shoulder to Ac. For these reasons,
the fourfold chain-periodicity unit cell can be eliminated as amodel for chain spin ordering.
C. Density of states
To understand the role of core-hole effects we also
performed DOS calculations for the system without a corehole (see Fig. 6). Contrary to previous theoretical works,
28–30
performed on a minimal cell without AF ordering, we find
the occurrence of a 0.25-eV gap appearing in ladder states.This finding is in agreement with the insulating behavior ofthe ladders
31–33and points out the need for including the full
AF crystal structure in the simulation. The occurrence of a0.25-eV Hubbard gap in ladder states results in a strong holedepletion of this subsystem. Indeed, we find that only about0.4 holes reside in ladders, while the chains accommodateabout 5.3 holes per f.u. A more detailed analysis of holedistribution is presented in Table I. The mean value of
the probability of accommodating a hole on one atom isschematically presented in Fig. 7by the thickness of the
contour surrounding atoms of each group (chain, ladder leg,and ladder rung).
The role of correlations is even more evident when analyz-
ing hole and spin order along the chains in the paramagneticDFT calculations and in the DFT including U and AF ordering.When correlation effects are switched on, the number of holesin the ladder is decreased by a factor of 5. Hole depletion ofladders is thus a pure correlation effect.
The distribution of more than five holes in chains and
fewer than one in ladders is compatible not only with the AFunit cell with fivefold chain periodicity but also with the AFmodel with fourfold chain periodicity.
27But the latter model
turns out to be irrelevant since in this case the NEXAFS
075108-3V . ILAKOV AC et al. PHYSICAL REVIEW B 85, 075108 (2012)
FIG. 6. (Color online) (top) Total DOS of Sr 14Cu24O41without
a core hole, calculated on the AF unit cell with fivefold chainperiodicity. The Fermi level E
Fcorresponds to 0 eV . (bottom) Local
chain and ladder DOS projected to Cu [red (medium gray) line]
and O [light blue (light gray) and dark blue (dark gray) lines]. Thesecond dotted line indicates where E
Fwould be if the system had six
additional electrons per f.u., i.e., no holes ( E/prime
F=EF+/Delta1E,w h e r e
/Delta1E=0.31 eV corresponds to an additional 6 ×4=24 electrons).
calculations could not reproduce the experimental data (see
Sec. III B). Including core-hole effects and the treatment of all
inequivalent atoms is thus of major importance for determiningthe correct AF ordering. Moreover, the fourfold chain-spinperiodicity destroys the natural fivefold hole ordering, whilethe fivefold spin periodicity preserves it, as shown in Fig. 8.
We would like to stress once more that taking into accountthe proper long-range AF order is crucial to achieve a correctdescription of O K-edge NEXAFS spectra.
It is worthwhile to note that our DOS is in good agreement
with experiments as we find the occurrence of a strong−2 eV band with mixed chain and ladder contributions and
large Cu-ladder weight, in accordance with angle-resolvedphotoemission spectroscopy (ARPES) measurements.
34Fur-
thermore the two ladder-DOS structures at −0.25 and −0.1e V
have also been observed in a recent ARPES35experiment and
were identified as the quasi-one-dimensional underlying Fermisurface.
TABLE I. Calculated fraction nof six holes per f.u. residing in
chain, ladder, and Sr interlayers of Sr 14Cu24O41. Also given are the
number of atoms in each group Ngand the distribution of holes
between Cu and O sites in chains and ladders ng.
Chain Ladder Sr interlayer
n 5.3 0.4 0.3
Atom Cu O Cu O (leg) O (rung) Sr
Ng 10 20 14 14 7 14
ng 2.75 2.55 0.21 0.14 0.05 0.3
FIG. 7. (Color online) The thickness of the contour at O/Cu sites
symbolizes the mean hole density per site in (top) chains and (bottom)
ladders as determined by ab initio calculations.
Finally, the tendency of hole-depleting ladder rungs found
in our work provides a satisfactory explanation for the two-dimensional (2D) to one-dimensional (1D) crossover when thenumber of holes in the system is reduced, as in underdopedcompounds,
3or at low temperature, where the less localized
ladder subsystem suffers from the migration of holes tochains (back transfer) as reported in the NMR.
36On the fully
FIG. 8. (Color online) Chain-site-dependent hole (spin) density:
(a) Chains have natural fivefold hole periodicity in the paramagnetic
cell without correlations. (b) When the fourfold chain spin periodicityis imposed, the fivefold hole periodicity is destroyed. (c) In the case of
the fivefold spin periodicity, the natural hole periodicity is preserved
on oxygen as well as on copper sites.
075108-4HOLE DEPLETION OF LADDERS IN Sr 14Cu24O... PHYSICAL REVIEW B 85, 075108 (2012)
doped side we suggest that the Ca doping has a twofold
effect: it destroys long-range AF order in chains9,37,38and
reduces hole depletion in ladders. Increased population ofthe rung sites reinforces the 2D character of the ladders. Forsufficient Ca-induced ladder doping the CDW ground state isthen suppressed in favor of the 2D superconductivity underpressure.
1,39
IV . CONCLUSION
In conclusion, our experimental NEXAFS spectra and
first-principles theoretical modeling including the complete316-atom AF unit cell, core hole, and correlation effectsdemonstrate that holes mainly reside on chains, and thusladders are hole depleted. This finding resolves long-standingcontroversial interpretations by several authors based onsemiempirical models. The analysis by N ¨ucker et al.
5is in
agreement with our findings based on an ab initio approach.
It is clear, however, that their arguments were based on atoo limited description since correlations and long-range AFbehavior consisting of a fivefold chain periodicity must beincluded to obtain the correct interpretation of the experimentalspectra. Further, the inclusion of the core-hole effects turnsout to be as important as including the Hubbard U forunderstanding the physics of this compound via the O K
NEXAFS spectra.
In ladders, the majority of holes reside in leg sites,
while only a tiny minority populates rung sites. Furtherexperiments and calculations are under consideration to verifyour suggestion that the small affinity of rung sites in theparent compound explains the 1D-2D crossover going fromthe underdoped to the fully doped side in the phase diagramof (La,Y ,Sr,Ca)
14Cu24O41.
Our work unambiguously answers the outstanding question
of where the holes reside in Sr 14Cu24O41. This is crucial to
understanding all phenomena occurring in the phase diagramof this family of compounds, ranging from charge-orderedantiferromagnetism to superconductivity, and is a prerequisiteto all investigations based on correlated models of spin ladderand chain compounds.
2
ACKNOWLEDGMENTS
M.C. and C.G. acknowledge discussions with F. Mauri.
Calculations were performed at the IDRIS superconductingcenter (Project No. 96053). S.T. acknowledges support fromthe Croatian Ministry of Science, Education and Sports underGrant No. 035-0000000-2836.
*vita.ilakovac-casses@upmc.fr
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075108-6 |
PhysRevB.86.125423.pdf | PHYSICAL REVIEW B 86, 125423 (2012)
Endstates in multichannel spinless p-wave superconducting wires
M.-T. Rieder, G. Kells, M. Duckheim, D. Meidan, and P. W. Brouwer
Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit ¨at Berlin, Arnimallee 14, 14195 Berlin, Germany
(Received 13 July 2012; published 13 September 2012)
Multimode spinless p-wave superconducting wires with a width Wmuch smaller than the superconducting
coherence length ξare known to have multiple low-energy subgap states localized near the wire’s ends. Here
we compare the typical energies of such endstates for various terminations of the wire: A superconducting wirecoupled to a normal-metal stub, a weakly disordered superconductor wire and a wire with smooth confinement.Depending on the termination, we find that the energies of the subgap states can be higher or lower than for thecase of a rectangular wire with hard-wall boundaries.
DOI: 10.1103/PhysRevB.86.125423 PACS number(s): 74 .78.Na, 74 .20.Rp, 03 .67.Lx, 73 .63.Nm
I. INTRODUCTION
In the current search for Majorana fermions in nanowire
geometries1,2an important theoretical challenge is to un-
derstand the multiplicity of possible fermionic bound statesthat can form at the ends of the wire and how a possibleMajorana bound state can be identified among them. Thisis particularly relevant for multichannel geometries, in whichfermionic states localized near the ends of the wire are expectedto occur at energies much smaller than the excitation gap forbulk excitations if the wire width is much smaller than thesuperconducting coherence length. In this article we explorethe dependence of these subgap endstates on the details of thetermination of the wire and on impurity scattering.
The interest in isolating Majorana fermions arises be-
cause their nonlocal properties and non-Abelian braidingstatistics render them potentially useful for fault tolerantquantum computation.
3–9Majorana fermions occur—at least
theoretically—at the ends of one-dimensional spinless p-wave
superconductors.10Recent proposals suggest ways of engi-
neering solid-state systems that effectively behave as spinlessp-wave superconductors by combining an s-wave supercon-
ductor and a topological insulator,
11,12a semiconductor13–16
or ferromagnet.17–21Building on the proposals of Refs. 15
and16, two experimental groups have reported an enhanced
tunneling density of states at the ends of InAs and InSb wires inproximity to a superconductor, consistent with the existence ofMajorana bound states at the ends of these wires,
22,23whereas
a number of other groups claim the observation of Majoranabound states using different methods.
24–26
Whereas the original proposals for Majorana fermions in
wire geometries focused on one-dimensional systems, it isby now well established that the topological superconductingphase with Majorana end states may persist in a quasi-one-dimensional multichannel setting.
27–37A difference between
the quasi-one-dimensional and one-dimensional settings is,however, that a possible zero-energy Majorana state localizedat the wire’s end may coexist with other fermionic subgapstates, analogous to those found in vortex cores of bulksuperconductors.
38For the case of an N-channel spinless
p+ipsuperconductor with a rectangular geometry and with
width Wmuch smaller than the superconducting coherence
length ξ, three of us recently showed that the number of
such fermionic subgap states is ∼N/2, and that their typical
energy is εtyp∼/Delta1(W/ξ )2,/Delta1being the superconducting gapsize.27The lowest-lying and highest-lying fermionic subgap
states have energies εmin∼εtyp/NlnNandεmax∼Nεtyp,
respectively. The fermionic subgap states also exist in anontopological phase without zero-energy Majorana endstate,thus posing a potential obstacle for the identification of thetopological phase through the observation of an enhanceddensity of states near zero energy.
In a recent article, Potter and Lee
28observe that the
dependence of the energy of the lowest-lying fermionicsubgap state on system parameters changes qualitatively if therectangular geometry of Ref. 27is replaced by a geometry with
rounded ends. They point out that the calculation of the energyof the fermionic subgap state for the rectangular geometry isplagued by a subtle cancellation, which does not appear for
a generic wire ending. In particular it was found in Ref. 28
that the lowest-lying fermionic subgap state has an energysignificantly above the prediction of Ref. 27for a wire with
width W∼ξand rounded ends.
Motivated by these observations we present here a detailed
investigation of the effect that the wire termination has on theenergies of the fermionic subgap states for the multichannel
spinless p+ipsuperconductor. Remarkably, we find that,
depending on the details of the wire ending, the energies ofthe fermionic subgap states can be significantly above, as wellas below, the rectangular-wire case of Ref. 27.F o rW/lessmuchξ,
which is the regime in which all subgap states have energywell below the bulk gap /Delta1, we find an increase of the energies
of the subgap states if an arbitrarily-shaped normal layer isattached to the wire’s end, the magnitude of the increase being
consistent with the estimate of Ref. 28for a wire with rounded
ends. On the other hand, the presence of impurities—weakenough to preserve the topological phase
39,40—on average
reduces the energies of the fermionic endstates below the
estimate of Ref. 27, while a smooth confinement (with a slowly
increasing potential energy providing the confinement alongthe wire’s axis) leads to even smaller energies of the fermionic
subgap states.
Our results are derived for the two-dimensional spinless
p+ipsuperconducting strip of width W. The model of a
spinless p+ipsuperconductor is an effective low-energy de-
scription for the various proposals to realize one-dimensionalor quasi-one-dimensional topological superconductors, pro-vided the number Nof propagating channels at the Fermi level
is chosen equal to the number of spin-polarized channels inthe case of the semiconductor or ferromagnet proposals, which
125423-1 1098-0121/2012/86(12)/125423(10) ©2012 American Physical SocietyRIEDER, KELLS, DUCKHEIM, MEIDAN, AND BROUWER PHYSICAL REVIEW B 86, 125423 (2012)
may be smaller than the total number of transverse channels
in the wire. (The edges of a topological insulator always haveN=1, so that a multichannel p+ipmodel is not relevant
in that case.) In the appendix we give a mapping between thespinless p-wave model and the semiconductor-wire proposals,
valid for magnetic fields with a Zeeman splitting that exceedsboth the spin-orbit and the pairing energy. A large numberof subgap states can be found in wide semiconductor wires,where different transverse modes are nearby in energy. Further-more, even for three-dimensional experimental realizationsof smaller width one may occasionally face the case of asmall but finite number of channels N> 1 due to accidental
degeneracies of different transverse modes. In the context ofproposals involving half metals, the spinless p+ip-model
provides a good basis since, by definition, the Zeeman energyis the largest energy scale in the system.
The remainder of this article is organized as follows: In
Sec. IIwe briefly review the symmetries of the model (2)and
the reason for the appearance of multiple low-lying states ifthe wire width Wis much smaller than the superconducting
coherence length ξ. In Sec. IIIwe describe a scattering theory
of fermionic subgap states with arbitrary wire endings. SectionIVdiscusses the p+ipmodel with weak disorder, while the
effect of a smooth potential at the wire’s end is discussed in Sec.V. We conclude in Sec. VI. In the two appendices we discuss
the mapping between the p+ipmodel and the semiconductor
models, as well as the case W∼ξof comparable wire width
and coherence length.
II.p+ipMODEL
Our calculations are performed for a two-dimensional
spinless p+ipsuperconductor, which is described by the
two-component Bogoliubov-de Gennes Hamiltonian, whichwe write as
H=H
0+Hy+HV, (1)
with
H0=/parenleftbiggp2
2m−μ/parenrightbigg
τz+/Delta1/primepxτx,
Hy=−/Delta1/primepyτy,H V=V(r)τz. (2)
Hereτx,τy, andτzare Pauli matrices in particle-hole space,
/Delta1/primespecifies the p-wave superconducting order parameter,
μ=¯h2k2
F/2mandmare the chemical potential and electron
mass, and V(r) a potential that describes the confinement at
the ends of the wire as well as the scattering off impurities. Thetwo-dimensional coordinate r=(x,y), where 0 <y<W ,
with hard-wall boundary conditions at y=0 and y=W.
The superconducting order parameter derives from proximity
coupling to a bulk superconductor, so that no self-consistency
condition for /Delta1
/primeneeds to be employed.
Hypothetical endstates are localized within a distance of the
order of the superconducting coherence length ξ=¯h(/Delta1/primem)−1
from the wire’s ends. For thin wires with W/lessmuchξit is a
good starting point to analyze the Hamiltonian H=H0+HV
without the term Hy. The Hamiltonian H0has a chiral
symmetry,41τyH0τy=−H0, and there exist
N=int/bracketleftbig
(W/π )/radicalBig
k2
F−ξ−2/bracketrightbig
(3)FIG. 1. (Color online) Schematic picture of the spectrum of low-
energy excitations of a p+ipwire as a function of its width W.
The gap for bulk excitations closes at those values of Wfor which
(W/π )/radicalbig
k2
F−ξ−2is an integer. When the bulk gap is finite, there are
low-energy subgap states localized near the ends of the wire. In thetext, we use ε
minto denote the energy of the lowest-lying fermionic
subgap state, εtypfor the typical energy of a subgap state, and εmaxfor
the energy of the highest-lying fermionic subgap state.
Majorana bound states at each end of the wire.27–30The
stepwise increase of the number of Majorana endstates for
wire widths Wsuch that ( W/π )/radicalBig
k2
F−ξ−2is an integer is
accompanied by a closing of the bulk excitation gap of H0.
Inclusion of the potential term HVdoes not lift the degeneracy
of the Majorana endstates, since HVpreserves the chiral
symmetry, although it may change the boundaries of thephases with different NifH
Vis nonzero in the bulk of the
wire. In contrast, the term Hybreaks the chiral symmetry
and couples the NMajorana bound states, giving rise to
(generically) int ( N/2) fermionic states at each end and a
single Majorana endstate if Nis odd. If W/lessmuchξthe splitting
of the endstates is small in comparison to the bulk energy gap/Delta1=/Delta1
/prime¯hkF, and the resulting fermionic states cluster near zero
energy.27,28
A schematic picture of the endstate spectrum as a function
ofWis shown in Fig. 1. The endstates are characterized by the
energy εminof the lowest-lying fermionic end state, the typical
endstate energy εtyp, and the energy εmaxof the highest-lying
endstate. For small Nthese three energy scales are comparable,
but for large Nthey may differ considerably. The energy εmin
serves as the “energy gap” protecting the topological state and
sets the required energy resolution if the presence or absenceof a Majorana endstate is detected through a tunneling densityof states measurement.
The specific case of a rectangular wire geometry, with hard-
wall boundary conditions at each end of the wire and withoutdisorder, was investigated in Ref. 27. We now investigate two
other possible terminations, as well as the effect of disorderon the energies of subgap endstates in multichannel spinlessp-wave superconducting wires.
III. NORMAL-METAL STUB
In this section, we consider a quasi-one-dimensional spin-
lessp+ipsuperconductor without disorder and coupled to
a normal-metal stub at its end. We choose coordinates suchthat the spinless superconductor occupies the space x> 0,
0<y<W , see Fig. 2. Such a wire ending is relevant, e.g., for
the experimental geometry of Ref. 22, in which a topological
phase is induced in a semiconductor nanowire by laterally
125423-2ENDSTATES IN MULTICHANNEL SPINLESS p-WA VE ... PHYSICAL REVIEW B 86, 125423 (2012)
(a)
(b)
FIG. 2. (Color online) Schematic drawing of a spinless p-wave
superconducting wire ( S) coupled to a normal-metal ( N) stub at one
end. The top panel shows a rectangular stub, the bottom panel shows
a chaotic cavity attached to the superconducting wire.
coupling it to a superconductor, while a part of the wire sticks
out from under the superconductor and is pinched off by a gateat a finite distance.
We take the Hamiltonian of the normal stub to be real and
symmetric, in order to preserve the chiral symmetry of theHamiltonian H
0. Following Ref. 27we first solve for the wave
functions ψ(j)of the NMajorana modes for the Hamiltonian
H0and then treat Hyin perturbation theory. The potential term
HVis set to zero throughout this calculation.
The Majorana states have support in the normal stub as well
as in a segment of the superconducting wire of length ∼ξ.I n
the superconducting region x> 0 the wave functions ψof the
Majorana states can be written as
ψ(r)=/summationdisplay
nan−φn−(r)+an+φn+(r), (4)
where the basis states φn±,n=1,2,..., N , read
φn±(r)=/parenleftbigg
eiπ/4
e−iπ/4/parenrightbigg/radicalBigg
2m
W¯hkne±iknx−x/ξsin/parenleftbiggnπy
W/parenrightbigg
, (5)
with
kn=/radicalBig
k2
F−(nπ/W )2. (6)
The basis states φn±have been normalized to unit flux. The
above expressions for the basis states and their normalizationare valid up to corrections of order ( W/ξ )
2, which we neglect
throughout this calculation.
The coupling to the normal-metal stub imposes boundary
conditions on the coefficients an±, which we express in terms
of the scattering matrix Snn/primeof the normal stub,
an+=/summationdisplay
n/primeSnn/primean/prime−,a n−=/summationdisplay
n/primeS∗
nn/primean/prime+. (7)
Because the Hamiltonian of the normal stub is real and
symmetric, the scattering matrix Snn/primeis unitary and symmetric,
Snn/prime=Sn/primen, which ensures that the 2 Nequations (7)have
Nindependent solutions, corresponding to the NMajorana
endstates.
For finding an explicit representation of the NMajorana
states ψ(j),j=1,2,..., N we use the fact that the scatter-
ing matrix Sand the Wigner-Smith time-delay matrix42,43
Q=i¯hS†∂S/∂μ of the normal stub can be simultaneously
decomposed as
S=UTU, Q =U†diag (τ1,..., τ N)U, (8)where Uis an N×Nunitary matrix and the τi>0,i=
1,2,..., N , are the so-called “proper time delays.” With this
decomposition, a solution to the boundary conditions (7)is
given by
a(j)
n+=Unj,a(j)
n−=U∗
nj,j=1,2,..., N. (9)
TheNstates that are defined through these coefficients,
˜ψ(j)(r)=/summationdisplay
na(j)
n−φn−(r)+a(j)
n+φn+(r), (10)
are Majorana modes (they satisfy ˜ψ(j)∗=τx˜ψ(j)), but they
are not necessarily orthonormal. In order to construct anorthonormal set, we first calculate the scalar product M
jlof
the modes ˜ψ(j),
Mjl=/integraldisplay∞
0dx/integraldisplayW
0dy˜ψ∗(j)(r)˜ψ(l)(r)
+/integraldisplay
stubdr˜ψ∗(j)(r)˜ψ(l)(r)
=/summationdisplay
n2mξ
¯hknRe/parenleftbigg
UnjU∗
nl+UnjU∗
nl
1−iknξ/parenrightbigg
+2τjδjl,(11)
Here we used the relation between the Wigner-Smith time
delay matrix and the normalization of scattering states inorder to perform the integration over the sub, see Ref. 43.T h e
overlap matrix Mis real, positive definite, and symmetric.
It is manifestly diagonal if the scattering matrix Sand the
time-delay matrix Qare diagonal or in the “large-stub limit,”
which is defined as the limit in which the mean inversedwell time ¯ h/¯τis much smaller than the superconducting
gap. In both cases, one obtains an orthonormal basis fortheNMajorana modes by setting ψ
(j)=˜ψ(j)//radicalbigMjj.I nt h e
general case, Mis not diagonal, however, one has to construct
an orthonormal system with the help of the orthogonaltransformation Othat diagonalizes M, i.e., O
TMO=λ2,
where λ=diag (λ1,λ2,..., λ N) is a diagonal matrix with
positive elements. The corresponding orthonormal basis setone thus obtains reads
ψ
(j)(r)=N/summationdisplay
n=1˜ψ(n)(r)Onjλ−1
n. (12)
Inclusion of Hy, which breaks the chiral symmetry, gives
rise to a splitting of the Ndegenerate Majorana endstates
constructed above. With respect to the unnormalized states
˜ψ(j), this splitting is described by the N×Nmatrix
˜H(1)
jl=/angbracketleft˜ψ(j)|Hy|˜ψ(l)/angbracketright
=4i/Delta1/primem
W/summationdisplay
nn/primenn/prime[1−(−1)n+n/prime]
(n/prime2−n2)√knkn/prime
×/summationdisplay
±/bracketleftbigg
ImUnjU∗
n/primel
kn/prime±kn+2
ξReUnjU∗
n/primel
(kn/prime±kn)2/bracketrightbigg
,(13)
where we neglected corrections smaller by a factor of order
(W/ξ )2. The matrix ˜H(1)
jlis antisymmetric and purely imag-
inary, which ensures the existence of a single zero-energybound state if Nis odd. In order to find a true effective
Hamiltonian H
(1), the eigenvalues of which represent the
125423-3RIEDER, KELLS, DUCKHEIM, MEIDAN, AND BROUWER PHYSICAL REVIEW B 86, 125423 (2012)
energies of the fermionic endstates, one should transform to
the basis of orthogonal states ψ(j)introduced in Eq. (12),
H(1)=1
λOT˜H(1)O1
λ. (14)
In the special case N=2, this transformation can be carried
out for an arbitrary scattering matrix S, and the energy of the
resulting single fermionic bound state is
ε=/vextendsingle/vextendsingle˜H(1)
12/vextendsingle/vextendsingle
/radicalBig
M11M22−M2
12. (15)
We now discuss two particular realizations of a metal stub in
detail.
A. Rectangular stub
First, we consider a rectangular stub of length Lattached to
the spinless p-wave superconducting wire, see Fig. 2(a).F o r
this geometry, both the scattering matrix Sand the Wigner-
Smith time-delay matrix Qare diagonal,
Snn/prime=−e2iknLδnn/prime,Q nn/prime=2mL
¯hknδnn/prime, (16)
withkngiven by Eq. (6). Since there is no mixing between
different channels, the zero energy modes ˜ψ(j)already form an
orthogonal basis. The effective Hamiltonian in the normalized
basisψ(j)=˜ψ(j)//radicalbigMjjhasH(1)
jl=0i fj+lis even and
H(1)
jl=4i/Delta1y¯hjl
W(ξ+2L)(l2−j2)/summationdisplay
±/braceleftbiggsin[L(kj±kl)]
kl±kj
+2W
πξcos[L(kj±kl)]
(kl±kj)2/bracerightbigg
(17)
ifj+lis odd, up to corrections smaller by a factor of order
(W/ξ )2.
The second term in the effective Hamiltonian (17)is smaller
than the first one by a factor of order W/ξ . However, only this
second term contributes in the limit L=0 in which there is
no normal metal stub.27This is a variation of the cancellation
effect pointed out by Potter and Lee.28We now analyze the
eigenvalues of the effective Hamiltonian H(1)for finite L,
when the first term between brackets dominates.
Since no closed-form expressions for the eigenvalues of
H(1)could be obtained, we numerically diagonalized H(1)
and investigated the dependence of the minimum, typical, and
maximal positive eigenvalues on the ratio L/W as well as the
channel number N.F o rL∼W, this analysis gives
εtyp∼εm≡W/Delta1
ξ, (18)
see Fig. 3. The maximum and minimum energies of the subgap
states scale as εmin∼εtyp/N,εmax∼εtyp. A similar analysis
fork−1
F/lessmuchL/lessmuchWgives estimates for εmin,εtyp, and εmax
which are smaller by a factor L/W , whereas for L/lessmuchk−1
F,t h e y
are smaller by a factor L3k2
F/W. A crossover to the results of
Ref. 27takes place for L/lessorsimilar(W2/k2
Fξ)1/3. In the limit of large
Nthe energies of the fermionic subgap states are best described
through their level density, which is shown in Fig. 4.2 4 6 8 100.20.40.60.81.0
FIG. 3. (Color online) Typical and maximal energies of fermionic
subgap states in a spinless p-wave superconductor with a rectangular
normal-metal stub of length Las a function of L/W for different
channels numbers ( N=15,27,55,99). The maximal energies εmax
have a finite- Ncorrection of order εm/√
N, which is why the curves
forεmaxstill show an Ndependence for large N.
B. Chaotic Cavity
As a second example, we consider a chaotic cavity attached
to the end of the superconducting wire, see Fig. 2(b).I nt h i s
case, the unitary matrix Uis randomly distributed in the unitary
group,44whereas the proper delay times have the probability
distribution45
P(τ1,...,τ N)∝N/productdisplay
j=1θ(τj)τ−3N/2−1
j e−N¯τ/2τj/productdisplay
i<j|τi−τj|,(19)
with the average delay time ¯ τ. In this case, the matrix Uis
not diagonal, and the prescription of Eq. (14) needs to be
used in order to construct the effective Hamiltonian H(1)for
the low-energy subgap states. As in the previous case, wecould not obtain closed-form expressions for the energies ofthe fermionic subgap states and had to resort to a numericalanalysis, in which the unitary matrices Uwere generated ac-
cording to the Haar measure on the unitary group and the timedelays τ
iaccording to the probability distribution given above,
following the method described in Ref. 46. This analysis gives
different results for the limiting cases of a “small cavity” anda “large cavity,” corresponding to the inverse mean dwell time¯h/¯τlarge or small in comparison to the superconducting gap /Delta1.
Small-cavity limit. In the small-cavity limit, the normal-
ization of the NMajorana states ψ
(j)is dominated by the
integration over the superconducting wire. Not counting the
2 4 6 80.020.040.06
1 2 3 4 50.100.20
FIG. 4. (Color online) Level density of fermionic subgap states
for a rectangular stub in the limit of large N,f o rL/W=0.1 (left)
andL/W=3 (right). The level density is measured in units of νm=
N/ε m.
125423-4ENDSTATES IN MULTICHANNEL SPINLESS p-WA VE ... PHYSICAL REVIEW B 86, 125423 (2012)
1.0 2.0 3.00.100.20
1 2 3 4 50.100.200.30
FIG. 5. (Color online) Level density of fermionic subgap states
in the small-cavity limit (left) and large-cavity limit (right). The level
density is measured in units of νs=N/ε sandνl=N/ε lfor the left
and right panels, respectively.
Majorana states, the excitation spectrum of the cavity has a
gap comparable to the bulk excitation gap /Delta1. Upon including
Hyone obtains Nfermionic subgap states, which have a typical
energy
εtyp∼εs≡W/Delta1
ξ, (20)
andεmax∼εtyp,εmin∼εtyp/N. The precise location of the
subgap states depends on the precise scattering matrix of thecavity. The mean level density for an ensemble of cavities isshown in the left panel of Fig. 5.
Large-cavity limit. In the large-cavity limit, the overlap
matrix M
jlis dominated by the in-cavity parts of the wave
functions, so that the Majorana modes ˜ψ(j)are already orthog-
onal and the effective Hamiltonian H(1)
jl=˜H(1)
jl//radicalbig4τjτl, with
˜H(1)
jlgiven in Eq. (13). Not counting the Majorana states, the
cavity’s excitation spectrum has a gap of order ET=¯h/π¯τ,47
where ¯ τis the mean dwell time in the cavity. In this case, the
typical energy of the fermionic subgap states is
/epsilon1typ∼εl≡ETW
ξ, (21)
while εmax∼εtypandεmin∼εtyp/N. The mean level density
of the subgap states for an ensemble of cavities is shown in theright panel of Fig. 5.
IV .p+ipMODEL WITH DISORDER
Whereas strong disorder is known to destroy the topological
superconducting phase in the p+ipmodel in one dimension,
weak disorder with mean free path l>ξ / 2 preserves the
topological phase.39,40In this section we investigate the effect
of weak disorder on the energies of the fermionic subgapstates in a multichannel rectangular p+ipmodel. Because
the disorder is necessarily weak (strong disorder suppressesthe topological phase), the effect of disorder can be treated inperturbation theory.
The starting point of our analysis is the chiral-symmetric
Hamiltonian H
0, which has Nnormalized Majorana bound
states |ψ(j)/angbracketright,j=1,2,..., N at each end of the wire. We
take a rectangular geometry, with a wire end and hard-wallboundary conditions at x=0, and take the potential V(r)t o
be a Gaussian white noise potential with mean /angbracketleftV(r)/angbracketright=0 and
variance
/angbracketleftV(r)V(r
/prime)/angbracketright=v2
F
kFlδ(r−r/prime), (22)where lis the mean free path and vF=¯hkF/mthe Fermi
velocity. In our perturbative analysis we treat both the impuritypotential Vand the transverse superconducting order as
perturbations and write
H=H
0+U, (23)
where U=Hy+HVcontains the superconducting correla-
tions coupling to pyas well as the impurity potential.
The effective Hamiltonian Heffdescribing the splitting of
theNMajorana states into fermionic subgap states can be
found using the degenerate perturbation theory of Kato48
and Bloch.49(For additional details on this methodology see
also Refs. 50and51.) Defining P=/summationtext|ψ(j)/angbracketright/angbracketleftψ(j)|as the
projector onto the zero-energy subspace and Q=1−P,w e
can then write using Bloch’s method
Heff=PUP −PUQ
H0UP+PUQ
H0UQ
H0UP
−1
2/parenleftbigg
PUQ
H2
0UPUP +PUPUQ
H2
0UP/parenrightbigg
.(24)
It is essential to note that the disorder potential V(r) alone
cannot lift the degeneracy of the Majorana endstates at anyorder of the perturbation theory. This can be understooddirectly from the observation that the disorder potentialV(r) does not break the chiral symmetry of the unperturbed
Hamiltonian H
0that is responsible for the N-fold degeneracy.
On the level of perturbation theory this can be understoodimmediately through the particle-hole symmetry present inthe Majorana bound states and the knowledge that for eachperturbative diagram that connects Majoranas through thepositive energy bulk states there is a canceling path throughthe negative energy states.
Keeping terms to first order in H
yand up to second order
inHVonly, we obtain
Heff=H(1)+H(2)+H(3a)−H(3b), (25)
with
H(1)
jl=/angbracketleftψ(j)|Hy|ψ(l)/angbracketright,
H(2)
jl=− /angbracketleftψ(j)|HyQ
H0HV+HVQ
H0Hy|ψ(l)/angbracketright,
H(3a)
jl=/angbracketleftψ(j)|HyQ
H0HVQ
H0HV|ψ(l)/angbracketright+permutations ,
H(3b)
jl=1
2/summationdisplay
k/parenleftbig
V(2)
jkH(1)
kl+H(1)
jkV(2)
kl/parenrightbig
, (26)
where
V(2)
jl=/angbracketleftψ(j)|HVQ
H2
0HV|ψ(l)/angbracketright. (27)
The effective Hamiltonian Heffis antisymmetric, which im-
plies that the diagonal elements of all the above terms arezero. The first-order term H
(1)describes how the transverse
superconducting correlations lift the degeneracy of the N
Majorana modes in the absence of disorder. The second-ordertermH
(2)is linear in the disorder potential. Its elements are
random variables with zero mean and standard deviation thatdoes not appreciably change with ξ. The third order term
125423-5RIEDER, KELLS, DUCKHEIM, MEIDAN, AND BROUWER PHYSICAL REVIEW B 86, 125423 (2012)
contains two terms, the first of which is also a random variable
with zero mean and with a root-mean-square proportional ξ.
The term H(3b)contains corrections to the effective Hamil-
tonian arising from the renormalization and reorthogonaliza-tion of wave functions at the first order of the perturbationtheory. Since this term is a weighted sum of first order elements
H
(1)
jl, it is the only one of the higher-order perturbation
corrections that gives a systematic dependence of energies onthe disorder strengths. To see this in more detail, it is instructiveto separate the diagonal and the off-diagonal elements of V
(2)
in the expression for H(3b),
H(3b)
jl=1
2/parenleftbig
V(2)
jjH(1)
jl+H(1)
jlV(2)
ll/parenrightbig
+1
2/summationdisplay
k/negationslash=j/parenleftbig
V(2)
jkH(1)
kl+H(1)
jkV(2)
kl/parenrightbig
. (28)
The first term here is the most important because the weights
V(2)
kkare positive definite random variables. A simple scaling
analysis predicts that these variables have both mean andstandard deviation proportional to the ratio ξ/lof coherence
length and mean free path. This term effectively renormalizesthe entire first order contribution, on average driving theenergies of the fermionic subgap states towards zero. Thesecond term, which contains the contribution from the off-diagonal elements of V
(2), is less important because the
disorder potential here connects different Majorana modes.These matrix elements are therefore randomly distributedwith zero mean and a root mean square proportional to thecoherence length.
Motivated by these observations, we write the effective
Hamiltonian in the form
H≈/Delta1
/prime
y/bracketleftbigg/parenleftbigg
1−cξ
l/parenrightbigg
H(1)+H/prime/bracketrightbigg
, (29)
where c=(l/Nξ )/summationtext
kV(2)
kkis a number of order unity, and
H/prime=H(2)+/parenleftbigg
H(3a)−1
2/braceleftbigg
V(2)−cξ
l,H(1)/bracerightbigg/parenrightbigg
. (30)
The correction H/primehas zero mean.
We have numerically diagonalized a lattice version of
the Hamiltonian (2)in order to provide numerical evidence
for the applicability of the above arguments. Details of therelationship between the continuum and lattice models canbe found in Ref. 27. Results of the numerical simulations
are shown in Fig. 6. For weak disorder, the perturbation
H
(2)dominates the response of the fermionic subgap states,
and the energies of the fermionic subgap states may bothincrease or decrease, depending on the specific realizationof the disorder potential. While large fluctuations persist, forstronger disorder the quadratic-in-disorder perturbation H
(3b)
leads to a systematic decrease of the energies of the fermionic
subgap states, which is well described by a linear dependenceonξ/l, consistent with the first term in Eq. (29).
V . SMOOTH POTENTIAL AT WIRE’S END
In this section we consider a wire which is terminated
by a smooth potential V(x), as shown schematically in
the inset of Fig. 7. In order to address this scenario weFIG. 6. (Color online) Distribution of energies of fermionic
subgap endstates in a spinless p-wave superconductor with N=7
channels (dots), as a function of ξ/l. For small amounts of disorder
the term H(2)dominates, pushing the subgap energies up or down
with equal probability. At stronger disorder the term in H(3b)∝ξ/l
eventually dominates and pulls all energy levels towards zero. The
red lines indicate the mean calculated from the local distribution
of eigenvalues. The black lines, which are a linear fit to the meanvalues in red, share a common intercept at the horizontal axis
atξ/l=c
−1=2.2. Dotted black lines indicate the unperturbed
energies. Energies are measured in units of ε0
typ=/Delta1W2/ξ2.T h e
lattice parameters used in the numerical calculation correspond to
kFW≈23 and kFξ≈320.
solve the Bogoliubov-de Gennes Hamiltonian in the WKB
approximation. Without the transverse pairing term Hythere
areNMajorana states ψ(j)with wave function
ψ(j)(r)=/radicalbigg
2
W/parenleftbigg
eiπ/4
e−iπ/4/parenrightbigg
χj(x)s i n/parenleftbiggnπy
W/parenrightbigg
, (31)
FIG. 7. (Color online) Energies of fermionic subgap endstates,
in a six-channel p+ipwire as a function of the adiabaticity
parameter σ(red). The flat black lines indicate the energies for
hard-wall boundary conditions ( σ=0). Energies are measured in
units of ε0
typ=/Delta1W2/ξ2. The lattice parameters used in the numerical
calculation correspond to kFW∼19 and kFξ∼90,a=5μ.T h e
inset shows the potential profile used in the calculations.
125423-6ENDSTATES IN MULTICHANNEL SPINLESS p-WA VE ... PHYSICAL REVIEW B 86, 125423 (2012)
where the functions χj(x) take the form
χj(x)=⎧
⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩e−x/ξ+/integraltextx
xjdx/primeκj(x/prime)
2/radicalbig/Omega1jκj(x)ifx<x j,
e−x/ξcos[/Phi1j(x)]/radicalbig/Omega1jkj(x)ifx>x j,(32)
where /Phi1j=π/4−/integraltextx
xjdx/primekj(x/prime),
kj(x)=/radicalBigg
2m/parenleftbigg
μ−V(x)−π2n2
2mW2/parenrightbigg
−1
ξ2,
κj(x)=/radicalBigg
1
ξ2−2m/parenleftbigg
μ−V(x)−π2n2
2mW2/parenrightbigg
,
/Omega1jis the normalization constant, and xjis the transverse-
mode-dependent classical turning point, defined as the solutionofk
j(xj)=0. Inclusion of the transverse pairing Hamiltonian
Hylifts the degeneracy of the Nzero energy Majorana end-
states, where the energy splitting is given by the eigenvalues
of the antisymmetric matrix with elements H(1)
jl=0i fj−l
is even and
H(1)
jl=16/Delta1/prime
Wlj
j2−l2/parenleftbig
X(1)
jl+X(2)
jl+X(3)
jl/parenrightbig
(33)
ifj−lis odd, with, for j<l ,
X(1)
jl=/integraldisplayxj
−∞dxe−2x/ξ−/integraltextx
xjdx/primeκj(x/prime)−/integraltextx
xldx/primeκl(x/prime)
4/radicalbig/Omega1j/Omega1lκjκl,
X(2)
jl=/integraldisplayxl
xjdxe−2x/ξ−/integraltextx
xlκlcos[/Phi1j(x)]
2/radicalbig/Omega1j/Omega1lkjκl, (34)
X(3)
jl=/integraldisplay∞
xldxe−2x/ξcos[/Phi1j(x)] cos[ /Phi1l(x)]/radicalbig/Omega1j/Omega1lkjkl.
Figure 7shows numerical calculations for a lattice version
of the spinless p+ipmodel, with a potential V(x)=
ae−x2/2σ2, turning the hard-wall ending at x=0 effectively
into a smooth end. The parameter σtunes the length scale over
which the potential is turned on. The case σ→0 corresponds
to a hard-wall boundary. The prefactor ahas the dimension
of energy and determines the height of the potential. For thecalculations shown in the figure, we chose a=5μ. The results
of the figure show a clear exponential dependence on σfor
states on the wire end terminated by the smooth function V,
allowing for energies of the subgap states that are significantlybelow the (already small) estimates for a rectangular geometrywith hard-wall boundary conditions.
The origin of the anomalously small energy splittings
shown in Fig. 7is the smoothness of all terms in the
Hamiltonian H.I ft h e p+ipwire is coupled to a normal-metal
stub, as in Sec. III, and the superconducting order parameter
/Delta1jumps at the interface x=0, no reduction of the endstate
energies is found, even if the normal-metal stub is terminatedby a smooth potential. (This scenario is well described bythe calculation of Sec. III.) On the other hand, one finds
a suppression very similar to that shown in Fig. 7if the
superconducting order parameter goes to zero smoothly atthe interface. We refer to Ref. 52for a discussion of the effect
of a smooth confinement in the semiconductor model.
VI. CONCLUSION
In this paper, we have investigated fermionic subgap states
localized near the end of a spinless p-wave superconducting
wires for two terminations of the wire—a normal-metal stuband a smooth confining potential—and in the presence ofweak disorder. The three scenarios give qualitatively differentestimates for the energies of the subgap states. However, theyshare the common feature that a wire with Ntransverse
channels with a width Wthat is much smaller than the
superconducting coherence length ξhas int ( N/2) fermionic
endstates, all with energy much below the bulk excitation gap/Delta1. These states appear for the topological phase (which has
a Majorana fermion at the wire’s end), as well as for thenontopological phase (which does not).
The appearance of low-energy fermionic endstates poses
an obstacle to the identification of Majorana fermions througha measurement of the tunneling density of states at thewire’s end, unless the energy resolution of the experimentis good enough to resolve the splitting between the fermionicendstates. The corresponding energy scale ε
minscales pro-
portional to /Delta1/k Fξ∼/Delta12/μin the most favorable scenario
we considered (wire’s end coupled to a small normal metalstub), which is the same dependence as the subgap states in avortex core.
38The important difference with the subgap states
in a vortex core is, however, that the number of fermionicendstates is limited, so that there exists a maximum energyε
max, whereas no such maximum energy exists in a vortex.
Other terminations, such as a rectangular end with or withoutdisorder, or a smooth confinement potential, give significantlysmaller values for ε
min, and, hence, lead to stricter requirements
for the energy resolution required to separate an eventualMajorana state from fermionic endstates.
The recent experiments that reported the possible observa-
tion of a Majorana fermion involved semiconductor nanowireswith proximity-induced superconductivity.
22,23Effectively, the
induced superconductivity in these wires is of spinless p-wave
type. However, it should be emphasized that this does not implythat the effective description of such a semiconductor wirewithNtransverse channels is a p+ipmodel with the same
number of transverse channels. Instead, only those channelsin the semiconductor that are effectively spinless (i.e., spinpolarized or helical, depending on the relative strength of theapplied magnetic field and the spin-orbit coupling) appear inthe effective description in terms of a p+ipmodel. (This
latter distinction was overlooked in Ref. 30.) Typically, this
number is smaller than the number of transverse channels in thesemiconductor. In particular, the nanowires of the experimentsof Refs. 22and23are believed to be thin enough that they
map to a single-channel p+ipmodel. Hence, we do not
expect that the mechanism for the generation of fermionicendstates we consider applies to those experiments. However,it will apply to nanowires with a larger diameter, which we thusexpect to exhibit a clustering of low-energy fermionic states inthe topologically trivial as well as the topologically nontrivialphases. In this context, it is important to note that the conditionthatW/lessmuchξdoes not a priori prevent the applicability of
125423-7RIEDER, KELLS, DUCKHEIM, MEIDAN, AND BROUWER PHYSICAL REVIEW B 86, 125423 (2012)
our analysis to thicker wires, because the effective pairing
potential /Delta1may decrease with Wfor proximity-induced
superconductivity in the limit of thick wires (see Ref. 17for
an example in which /Delta1∝W−1).
ACKNOWLEDGMENTS
We gratefully acknowledge discussions with Felix von
Oppen and Inanc Adagedeli. This work is supported by theAlexander von Humboldt Foundation in the framework ofthe Alexander von Humboldt Professorship, endowed by theFederal Ministry of Education and Research.
APPENDIX A: RELATIONSHIP BETWEEN THE p+ipAND
PROXIMITY COUPLED SEMI-CONDUCTOR MODELS
A practical realization of a the p+ipmodel can be found
in semiconducting nanowires with strong spin-orbit coupling,laterally coupled to a standard s-wave superconductor and
subject to a Zeeman field. In the following we discuss theprecise relationship between the models. A related discussioncan also be found in Ref. 14.
In two dimensions, and without coupling to the supercon-
ductor, the Hamiltonian for this system reads
H
N=p2
2m−μ+Bσx+ασypx−α/primeσxpy, (A1)
where αandα/primeset the strength of the spin-orbit coupling
andB> 0 is the Zeeman energy of the applied magnetic
field. In the limit of a narrow wire (width Wmuch smaller
than the coherence length ξof the induced superconductivity),
subgap states as well as the above-gap quasiparticle states havea vanishing expectation value of the transverse momentumk
y, which allows us to treat the transverse spin obit term
as a perturbation, initially setting α/prime=0. Without the term
proportional to α/primedifferent transverse channels do not couple
to each other and the eigenfunctions of the Hamiltonian HN
are of the form
ψ±
n,k(r)∝/parenleftbigg
e−iθk
±1/parenrightbigg1√
Weikxsinπny
W, (A2)
where the angle θkis defined as
sinθk=αk√
B2+α2k2,cosθk=B√
B2+α2k2,(A3)
and the corresponding energies are
ε±
k,n=¯h2k2
2m+¯h2π2n2
2mW2−μ±/radicalbig
B2+α2k2. (A4)
Upon laterally coupling the semiconductor wire to an
s-wave superconductor, the excitations are described by the
Bogoliubov-de Gennes Hamiltonian
HBdG=/parenleftbigg
HN/Delta1σy
/Delta1σy−H∗
N/parenrightbigg
=/parenleftbiggp2
2m−μ+Bσx+ασypx/parenrightbigg
τz
−α/primeσxpy+/Delta1σyτx, (A5)
where τx,τy, and τzare Pauli matrices in electron-hole
space. Without the transverse spin-orbit coupling α/prime,t h e
Bogoliubov-de Gennes Hamiltonian has a chiral symmetry,τ
yHτy=−H. In the basis of the normal-state eigenfunctionsψ±
n,k, the Bogoliubov-de Gennes Hamiltonian (A5) takes the
form
HBdG=/parenleftbigg¯h2k2
2m+¯h2π2n2
2mW2−μ/parenrightbigg
τz+σzτz/radicalbig
B2+α2k2
+/Delta1σyτxcosθk+/Delta1σzτxsinθk
−α/primepy(σzcosθk+σysinθk). (A6)
In the limit, when both /Delta1and the spin orbit energy are
smaller than the Zeeman splitting, the s-wave pairing term
proportional to σyis ineffective, and each transverse channel
separately maps to two spinless p-wave superconductors, one
forψ+
n,kand one for ψ−
n,k. Neglecting |αk|in comparison to
B, the corresponding pairing term /Delta1sinθkσzτx≈−/Delta1/primekσzτx,
with
/Delta1/prime=−α/Delta1
B. (A7)
Without the term proportional to α/prime, the transverse channels
in Eq. (A6) can be treated independently (at least in the bulk
of the wire, see below). If μ<B , only the “ −” channels
(eigenspinors of σzwith eigenvalue −1) in Eq. (A6) are
topologically nontrivial and can possibly have endstates.5
Projecting the Bogoliubov-de Gennes Hamiltonian in therotated basis (A6) onto these channels, one finds an effective
Hamiltonian of the form
H
eff
BdG=/parenleftbigg¯hk2
2m+¯h2π2n2
2mW2−μ−B/parenrightbigg
τz+/Delta1/prime¯hkτx+α/primepy,
(A8)
Without the transverse spin-orbit coupling α/prime, the effective
Hamiltonian (A8) has chiral symmetry and NMajorana
endstates at each end of the wire. The chiral symmetry isbroken by the transverse spin-orbit coupling α
/prime. Because of
the particle-hole symmetry of the Majorana modes, the matrixelements of this perturbation between the NMajorana endstate
ofH
eff
BdGwithα/prime=0 are the same as the matrix elements of the
p-wave superconducting pairing Hyof Eq. (2), if we identify
/Delta1/prime=α/primein the expression for Hy.
If the condition μ<B is not met, the relation between the
semiconductor and p+ipmodels is more complicated. For
transverse channels for which ¯ h2π2n2/2mW2<μ−Bthe
wire ends represent a chiral-symmetry-preserving perturbationthat gaps out eventual Majorana endstates, so that suchchannels may be disregarded when considering low-energyendstates. For transverse channels for which
μ−B<¯h
2π2n2
2mW2<μ+B, (A9)
the Majorana endstate in the “ −band” (eigenspinors of σz
at eigenvalue −1 in the rotated basis) is protected in the
presence of the chiral symmetry, and only perturbations thatlift the chiral symmetry can lead to a splitting of theseendstates. Projecting the Bogoliubov-de Gennes Hamiltonianin the rotated basis (A6) onto these channels, one finds again
an effective Hamiltonian of the form Eq. (A8) , but with
the additional restriction that only those transverse channelsthat meet the condition (A9) are considered. The number
Nof transverse channels that meet this condition may be
smaller than the original number of propagating channels inthe semiconductor.
125423-8ENDSTATES IN MULTICHANNEL SPINLESS p-WA VE ... PHYSICAL REVIEW B 86, 125423 (2012)
FIG. 8. (Color online) Top: Distribution of energies of fermionic
subgap endstates in a spinless p-wave superconductor with N=
7 channels, as a function of ξ/l. The red lines indicate the mean
calculated from the local distribution of eigenvalues. The black lines,
which are a linear fit to the mean values in red, share a common
intercept at the horizontal axis at ξ/l=c−1=0.7. Dotted black lines
indicate the unperturbed energies. Energies here are measured inunits of the bulk gap. The lattice parameters used in the numerical
calculation correspond to k
FW=kFξ=≈23.5. Bottom: Energies of
fermionic subgap endstates, in a five-channel p+ipwire as function
of the adiabaticity parameter σ(red). (See Sec. Vfor the definition of
σ.) The flat black lines indicate the energies for hard-wall boundary
conditions ( σ=0). New endstates (blue dashed) are formed just
below the bulk gap, availing of the smaller local gap near the wire’s
end. All energies are measured in units of the bulk gap. The lattice
parameters used in the numerical calculation correspond to kFW=
kFξ≈14.75 and a=5μ.APPENDIX B: THE CASE W∼ξ
The results of the main text were derived for the case
W/lessmuchξ, wire width much smaller than the superconduct-
ing coherence length. This is the appropriate limit if theinduced superconductivity is weak. It applies, e.g., in thesemiconductor model if there is a barrier interface between
the superconductor and the semiconductor nanowire, which
suppresses the strength of the induced superconductivity and,hence, increases ξ. The limit W/lessmuchξensures that all int ( N/2)
fermionic subgap states have energy well below the bulk gap/Delta1. For wider wires, the condition W/lessmuchξmay be violated
(but this is not necessarily so, see Sec. VI), although the
lowest-energy states remain localized near the wire’s ends as
long as W/lessorsimilarξ.( F o rW/greatermuchξ, the lowest-energy subgap states
are extended along the wire’s edges.
27,36)
In this Appendix we now discuss how our results are
modified when Wandξbecome comparable. Our discussion
must be limited to εmin, because the energies of the fermionic
states extend up to and into the bulk spectrum if W/greaterorsimilarξ,s o
that the energy scales εtypandεmaxhave lost their meaning.
For the case of a p+ipsuperconductor with a normal-metal
stub, which was discussed in Sec. III, we have verified that our
estimates for εminremain qualitatively valid up to W∼ξ.
A numerical investigation similar to that of Fig. 6shows
that the lowest fermionic subgap levels in a p+ipmodel
with a rectangular ending are only weakly dependent ondisorder, and on the average decrease with a common factor(1−cξ/l ), although the value numerical constant cdiffers
from that obtained in the limit W/lessmuchξof the main text.
An example is shown in the top panel of Fig. 8. Finally,
with a smooth confinement at the wire’s end, the energiesof the lowest-lying fermionic subgap states are stronglysuppressed, in a way very similar to what is shown in Fig. 7
for the case W/lessmuchξ, see the bottom panel of Fig. 8for a
representative example. An essential difference with the caseW/lessmuchξ, however, is the absence of a region of the spectrum
in which there are no states. Instead, discrete endstates appearthroughout the entire subgap range 0 <ε</Delta1 , and whereas
all endstates lower their energy upon making the confinementsmoother, new endstates are formed just below the bulkgap, utilizing the reduced value of the local gap near thewire’s end that results from the termination with a smoothpotential.
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125423-10 |
PhysRevB.80.195110.pdf | Temperature dependence of the electronic structure of the Jeff=1
2Mott insulator Sr 2IrO 4studied
by optical spectroscopy
S. J. Moon,1Hosub Jin,2W. S. Choi,1J. S. Lee,1,*S. S. A. Seo,1,†J. Yu,2G. Cao,3T. W. Noh,1and Y . S. Lee4,‡
1ReCOE and FPRD, Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
2CSCMR and FPRD, Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
3Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA
4Department of Physics, Soongsil University, Seoul 156-743, Korea
/H20849Received 14 May 2009; revised manuscript received 26 August 2009; published 17 November 2009 /H20850
We investigated the temperature-dependent evolution of the electronic structure of the Jeff=1
2Mott insulator
Sr2IrO 4using optical spectroscopy. The optical conductivity spectra /H9268/H20849/H9275/H20850of this compound has recently been
found to exhibit two d-dtransitions associated with the transition between the Jeff=1
2andJeff=3
2bands due to
the cooperation of the electron correlation and spin-orbit coupling. As the temperature increases, the two peaksshow significant changes resulting in a decrease in the Mott gap. The experimental observations are comparedwith the results of first-principles calculation in consideration of increasing bandwidth. We discuss the effect ofthe temperature change in the electronic structure of Sr
2IrO 4in terms of local lattice distortion, excitonic effect,
electron-phonon coupling, and magnetic ordering.
DOI: 10.1103/PhysRevB.80.195110 PACS number /H20849s/H20850: 78.20. /H11002e, 71.30. /H11001h, 71.20. /H11002b
I. INTRODUCTION
Transition metal oxides /H20849TMOs /H20850have been the topic of
much recent research due to their numerous intriguing phe-nomena, such as Mott transition, superconductivity, colossalmagnetoresistance, and spin/orbital ordering.
1–3The large
variety of phenomena originates from various interactions,including on-site Coulomb, hopping, electron-phonon, andspin orbit, which mediate the coupling among spin, charge,orbital, and lattice degrees of freedom. Especially, the spin-orbit coupling /H20849SOC /H20850is an important consideration in de-
scribing the physical properties of 5 dTMOs. While the
electron-electron interaction becomes weaker, the SOC be-comes about 0.4–0.5 eV for 5 dTMOs, which is much larger
than those of 3 d/4dTMOs.
4As the SOC energy scale is
comparable to those of other fundamental interactions, itcould give rise to some interesting properties of 5 dTMOs.
Sr
2IrO 4h a saK 2NiF 4-type layered perovskite structure
with five valence electrons in the 5 dt2gstates. As the 5 d
orbitals are spatially more extended than 3 dand 4 dorbitals,
their bandwidth /H20849Coulomb interactions /H20850could be larger
/H20849smaller /H20850than those of 3 dand 4 dorbitals. Thus 5 dTMOs
are expected to have a metallic ground state. Contrary to theconventional understanding of the electron correlation,Sr
2IrO 4was found to be an insulator with weak ferromag-
netic ordering below TM=240 K.5,6It has recently been
found that the peculiar insulating state could be due to thecooperation of the SOC and on-site Coulomb interaction, U,
and that the electronic ground state should be the J
eff=1
2Mott
state.7,8
The Jeff=1
2state can be expressed as
/H20841Jeff=1
2,mJeff=/H110061/2/H20856
=1
/H208813/H20849/H11006/H20841xy/H20856/H20841/H11006/H20856+/H20841yz/H20856/H20841/H11007/H20856/H11006i/H20841zx/H20856/H20841/H11007/H20856/H20850, /H208491/H20850
where /H20841+/H20856and /H20841−/H20856represent spin-up and spin-down states,
respectively. The Jeff=1
2state is strikingly different from thed-orbital states of the 3 dor 4dTMOs. In 3 dor 4dTMOs, in
which the SOC is small, the electronic state is determined bythe change in the crystal field and the resulting electronicstate is described by real wave functions in the forms of thet
2g/H20849xy,yz, and zx/H20850and eg/H20849x2-y2and 3 z2-r2/H20850orbitals.2,9,10
However, as shown in the Eq. /H208491/H20850, the SOC-induced Jeff=1
2
state is the complex state composed of the xy,yz, and zx
orbitals and the spins. This state can be a basis for interestingemergent phenomena, such as the low-energy Hamiltonian ofthe Kitaev model
11and the room-temperature quantum spin
Hall effect in a honeycomb lattice.12Therefore, investigating
the characteristic properties of the Jeff=1
2state and establish-
ing a territory of the strongly correlated electron systems areimportant.
In this paper, we investigated the temperature-dependent
optical conductivity spectra
/H9268/H20849/H9275/H20850ofJeff=1
2Mott insulator
Sr2IrO 4. The two-peak structure of the d-dtransition in /H9268/H20849/H9275/H20850,
which is associated with the Hubbard bands of the Jeff=1
2
state and the Jeff=3
2bands, showed significant temperature
dependence. Spectral weight /H20849SW /H20850redistribution occurred
between the two optical transitions, and the Mott gap drasti-cally decreased with increasing temperature. We performed afirst-principles calculation and found that our experimentalobservation could be associated with the change in the band-width of the J
effstates. We discussed possible effects of the
magnetic ordering and electron-phonon coupling to the un-usual temperature dependence of
/H9268/H20849/H9275/H20850.
II. EXPERIMENTAL AND THEORETICAL METHODS
Single crystals of Sr 2IrO 4were grown using flux
technique.5We measured near-normal incident abplane re-
flectance spectra R/H20849/H9275/H20850in the energy region between 5 meV
and 6 eV as a function of temperature, and between 6 and 20eV at room temperature. In the temperature range between10 and 300 K, a continuous-flow liquid-helium cryostat wasused. For higher temperature measurements up to 500 K, aPHYSICAL REVIEW B 80, 195110 /H208492009 /H20850
1098-0121/2009/80 /H2084919/H20850/195110 /H208495/H20850 ©2009 The American Physical Society 195110-1custom-built sample holder with a heating system was
used.13We used a Fourier transform spectrometer /H20849Bruker
IFS66v/S /H20850and a grating-type spectrophotometer /H20849CARY 5G /H20850
in the photon energy region of 0.1–1.2 eV and 0.4–6.0 eV ,respectively. In the deep ultraviolet region above 6.0 eV ,we used synchrotron radiation from the normal incidencemonochromator beamline at Pohang Light Source. The cor-responding
/H9268/H20849/H9275/H20850were obtained using the Kramers-Kronig
transformation.14
The density functional theory code OPENMX was used for
the theoretical calculation on the electronic structure. Thecalculation is based on a linear combination of pseudoatomicorbitals method, whereby both the local densityapproximation+ Umethod and SOC /H20849LDA+ U+SOC /H20850are in-
cluded via a relativistic J-dependent pseudopotential scheme
in the noncollinear density functional formalism. Details ofthe calculation method can be found in Ref. 15.
III. RESULTS AND DISCUSSION
A. Temperature dependence of /H9268(/H9275)
Figure 1/H20849b/H20850shows /H9268/H20849/H9275/H20850of Sr 2IrO 4at various tempera-
tures. The sharp spikes below 0.1 eV are due to the opticalphonon modes. The spectra show two optical transitionsabove 0.2 eV , labeled as
/H9251and/H9252in Fig. 1/H20849b/H20850. As the charge
transfer transition from O 2 pto Ir 5 dorbital states are lo-
cated above 3 eV , the peaks /H9251and/H9252should be d-d
transitions.6This two-peak structure is associated with the
SOC-induced Mott insulating state.8The strong SOC of 5 dIr
ion splits the t2gorbital states into the total angular momen-
tum Jeff=1
2andJeff=3
2states. The Jeff=1
2bands are partially
occupied by a single electron and the Jeff=3
2bands are fully
occupied by four electrons. The effect of Usplits the par-
tially filled Jeff=1
2bands into a lower Hubbard band /H20849LHB /H20850
and an upper Hubbard band /H20849UHB /H20850, opening a Mott gap as
shown in Fig. 1/H20849a/H20850. The sharp peak /H9251corresponds to the
optical transition from the LHB and to the UHB of the Jeff
=1
2states and the peak /H9252corresponds to that from the Jeff
=3
2bands to the UHB.
Notably, the two peaks show significant temperature de-
pendence. As the temperature increases, the sharp peak /H9251
becomes broader and shifts to lower energy. While theheights of both peaks decrease, the SW of the peak
/H9251/H20849SW/H9251/H20850
increases and that of the peak /H9252/H20849SW/H9252/H20850decreases with in-
creasing temperature. These spectral changes result in theblurring of the two-peak structure and a gradual decrease inthe SOC-induced Mott gap. These observations are quiteconsistent with the resistivity behavior of Sr
2IrO 4. As tem-
perature increases, the resistivity decreases continuouslywithout exhibiting any discernible anomaly indicating thegradual change in the electronic structure.
16The temperature
dependence of the resistivity was found to follow the vari-able range hopping model
5and the increase in the hopping
rate with increasing temperature could be associated with thedecrease of the optical gap.
To obtain quantitative information on the spectral
changes, we fitted
/H9268/H20849/H9275/H20850using the Lorentz oscillator model.
As shown in Fig. 2/H20849a/H20850, the energy of the peak /H9251,/H9275/H9251, de-
creases from about 0.54 eV at 10 K to 0.44 eV at 500 K. Atthe same time, the width of the peak
/H9251,/H9253/H9251, increases from
about 0.15 eV at 10 K to 0.49 eV at 500 K. In addition tothese changes, as shown in Fig. 2/H20849b/H20850, the SW
/H9251increases and
the SW /H9252decreases with temperature. The decrease in the
SW/H9252is almost the same as the increase in the SW /H9251, satisfy-
ing the optical sum rule below 1.5 eV . It is also noted that
/H9268/H20849/H9275/H20850shows little temperature dependence at higher energies.
These experimental observations imply that the main change
in electronic structure occurs in the Jeff=1
2andJeff=3
2bands.
The broadening of the peaks and the SW redistribution
cause the decrease in the SOC-induced Mott gap. We esti-mated the optical gap energy, /H9004
opt, by taking the crossing
point of the line at an inflection point of the /H9268/H20849/H9275/H20850with the
abscissa. As the temperature increases from 10 to 500 K, /H9004opt
changes from 0.41 to 0.08 eV . Note that the change in /H9004opt,
which is more than 4500 K, is much larger than that of thetemperature. To get more insight, we compared our findingwith the case of semiconductors, such as Ge, InAs, andHg
3In2Te6, whose band gaps are comparable to that of
Sr2IrO 4.17,18The optical band gap of the semiconductors has
been known to follow the empirical Varshni relation, i.e.,E
g/H20849T/H20850=E0-aT2//H20849T+b/H20850, where Egis the energy gap, E0is the
gap value at 0 K, and aandbare constants.17We fitted /H9004opt
of Sr 2IrO 4and found that the constant a, which is related toFIG. 1. /H20849Color online /H20850/H20849a/H20850Schematic band diagram of the elec-
tronic structure of Sr 2IrO 4.EFrepresents the Fermi level. Peak /H9251
corresponds to the optical transition from the lower Hubbard band
to the upper Hubbard band of the Jeff=1
2states. Peak /H9252corresponds
to the optical transition from the Jeff=3
2band to the upper Hubbard
band of the Jeff=1
2states. /H20849b/H20850Temperature-dependent optical con-
ductivity spectra /H9268/H20849/H9275/H20850of Sr 2IrO 4. As temperature increases, the
peaks /H9251and/H9252become broader and the Mott gap decreases. The
sharp spikes below the optical gap energy are due to optical phononmodes. The inset shows the result of Lorentz oscillator fit for
/H9268/H20849/H9275/H20850
at 10 K. Open circle and red line represent experimental and fitted
/H9268/H20849/H9275/H20850, respectively.MOON et al. PHYSICAL REVIEW B 80, 195110 /H208492009 /H20850
195110-2the rate of decrease in the gap with temperature, is about four
to five times larger than those of the semiconductors.17,18
This indicates that a simple thermal effect could not be suf-
ficient for explaining the strong temperature dependence ofthe Mott gap of Sr
2IrO 4.
It is worthwhile to compare the temperature-dependent
evolution in /H9268/H20849/H9275/H20850of the 5 dMott insulator Sr 2IrO 4with those
of 3dMott insulators. Previous optical studies on 3 dMott
insulators, such as LaTiO 3,19YTiO 3,20and Yb 2V2O7,21
showed that their electronic structure exhibited little tem-perature dependence. In particular, we note that LaTiO
3
showed a structural transition at its magnetic transition tem-perature. Structural studies using x-ray and neutron diffrac-tion techniques indicated a structural transition near the mag-netic transition temperature,
22,23and an optical study on
LaTiO 3showed that the additional phonon modes appeared
below the magnetic transition temperature due to the struc-tural transition.
19However, the electronic structure and Mott
gap of LaTiO 3hardly changed with temperature. The behav-
ior of /H9268/H20849/H9275/H20850in LaTiO 3is quite distinguished from that of
Sr2IrO 4, where /H9268/H20849/H9275/H20850of Sr 2IrO 4showed strong temperature
dependence without a structural transition.24The strong tem-
perature dependence of /H9268/H20849/H9275/H20850without the structural transition
is likely to be associated with the strong hybridization andresulting sensitivity to the electron-phonon interaction of 5 d
TMOs due to the extended 5 dorbitals.
B. Theoretical calculation on electronic structure with
bandwidth change
We now discuss possible origins of the significant elec-
tronic structure change in relation to the change in the band-width of Ir Jeffstates. The electronic bandwidth of layered
perovskite is controlled by the in-plane metal–oxygen–metalbond angle, that is, the rotation of the metal-oxygen octahe-dron about the caxis, which can induce an insulator-metal
transition.
25Generally, the active thermal fluctuation favors a
high-symmetry phase of crystal structure, and hence the in-plane Ir–O–Ir bond angle can increase with temperature.Given the strong electron-phonon interaction of the 5 d
TMOs, the change in the bond angle could lead to the largechanges in the electronic structure. To simulate the electronicstructure changes according to the variation in bandwidth,we preformed LDA+ U+SOC calculation using different
values of the bond angle. It was reported that the Ir–O–Irbond angle is about 157° at 10 K.
24We gradually increased
the bond angle from 157° to 170° and obtained the corre-sponding density of states /H20849DOS /H20850. Figure 3shows the DOS
of Sr
2IrO 4between −1.5 and 1.0 eV where the Ir 5 dt2gor-
bital states are the main contributors. The DOS above the
Fermi energy, EF, corresponds the UHB of the Jeff=1
2states.
The DOS between −1.5 and −0.5 eV is from the Jeff=3
2
bands, and that between −0.5 and 0.0 eV /H208490.0 and 0.5 eV /H20850is
from the LHB /H20849UHB /H20850of the Jeff=1
2states.
The calculated electronic structure changes as a function
of the bond angle are qualitatively similar with the changesin the experimental
/H9268/H20849/H9275/H20850. As the Ir–O–Ir bond angle in-
creases, the DOS of the Jeff=3
2bands decreases and that of
theJeff=1
2bands increases, indicating a DOS redistribution
between the two bands. At the same time, the Mott gap be-
tween the LHB and UHB of the Jeff=1
2states decreases. In
the experimental /H9268/H20849/H9275/H20850, the SW transfer from the peak /H9252to
the peak /H9251occurred and the Mott gap decreased with in-
creasing temperature. Note that the experimental /H9004optde-
creases by about 0.15 eV when temperature changes from 10to 300 K as shown in Fig. 2/H20849c/H20850. The gap in the calculation
decreases by about 0.13 eV when the bond angle changesfrom 157° to 170°. These results indicate that the change inbond angle might be larger than 13° to reproduce thetemperature-induced change in /H9004
opt. However, a previous
structural study on Sr 2IrO 4showed that the bond angle in-
creased by about 1° as temperature increased from 10 K toroom temperature,
26which is much smaller than that used inFIG. 2. /H20849a/H20850Temperature-dependent changes in energy /H20849solid
square /H20850and width /H20849open circle /H20850of the peak /H9251./H20849b/H20850Temperature-
dependent changes in the optical spectral weight of the peaks /H9251
/H20849solid triangle /H20850and/H9252/H20849open triangle /H20850./H20849c/H20850Temperature-dependent
change in the optical gap /H9004opt. The inset shows the value of the gap
differentiated with respect to temperature.FIG. 3. /H20849Color online /H20850Total densities of states with variation in
the Ir–O–Ir bond angle. EFrepresents the Fermi level. The inset
shows the rotation of the two neighboring octahedra. The largegreen and small blue circles represent Ir and O atoms, respectively.
/H9258denotes the Ir–O–Ir bond angle.TEMPERATURE DEPENDENCE OF THE ELECTRONIC … PHYSICAL REVIEW B 80, 195110 /H208492009 /H20850
195110-3the calculation. Although the structural studies at higher tem-
peratures are needed for direct comparison with /H9268/H20849/H9275/H20850,i ti s
hardly expected that the change in the bond angle could belarge enough to induce the observed electronic structurechanges. This suggests that the simple lattice distortion couldnot explain the large change in the bandwidth of Sr
2IrO 4with
temperature variation.
C. Comparison with optical spectra of La 2CuO 4: Excitonic
effect and electron-phonon coupling
It is interesting to note that the temperature dependence of
/H9268/H20849/H9275/H20850of Sr 2IrO 4is quite similar with that of La 2CuO 4. Falck
et al. reported the temperature-dependent optical spectra of
La2CuO 4.27A narrow charge transfer peak was observed in
the optical spectra of La 2CuO 4. The narrowness of the charge
transfer peak was attributed to the electron-hole interaction,i.e., excitonic effect, which can dramatically enhance matrixelements for interband transition in two-dimensionalsystem.
27As temperature increased, the charge transfer peak
became broader and shifted to lower energy. At the low tem-peratures below 100 K, the peak energy did not show dis-cernible change and it decreased continuously with increas-ing temperature above 100 K. These temperature-inducedchanges in
/H9268/H20849/H9275/H20850were explained in terms of the coupling of
charge carriers to a optical phonon mode, i.e., electron-phonon coupling. /H20849The absence of temperature dependence
below 100 K was explained in terms of the freeze-out ofoptical phonons. /H20850It should be noted that the temperature
dependences of the peak
/H9251and/H9004optin/H9268/H20849/H9275/H20850of Sr 2IrO 4,
shown in Fig. 2, are quite similar with that of the charge
transfer peak of La 2CuO 4. The similarity of the changes in
the optical spectra of La 2CuO 4and Sr 2IrO 4and the two-
dimensional character suggest that some excitonic effect andelectron-phonon coupling could play important roles for theobserved changes in
/H9268/H20849/H9275/H20850of Sr 2IrO 4.
D. Phonon dynamics
To gain some insight into the temperature-dependent lat-
tice dynamics, we examined the phonon spectra of Sr 2IrO 4.
Figure 4shows the temperature-dependent phonon spectra.
The phonon modes are classified into three groups: external,bending, and stretching modes.
28,29Figures 4/H20849a/H20850–4/H20849c/H20850show
the external, bending, and stretching modes, respectively.While the external mode is related to the vibrations of the Srions against IrO
6octahedra, the bending and stretching
modes are related to the modulation of the Ir–O–Ir bondangle and the Ir–O bond length, respectively.
As shown in Figs. 4/H20849a/H20850–4/H20849c/H20850, the change in the phonon
modes with temperature is rather gradual and no clear split ofthe phonon modes is observed, indicating the absence of astructural transition.
24,26To be interesting, the bending mode
phonon shows the strongest temperature dependence amongthe observed phonon modes. Figure 4/H20849d/H20850shows the phonon
frequency
/H9275phof each mode normalized to /H9275ph/H2084910 K /H20850. While
the/H9275phof the other phonon modes changes less than 1%, that
of the higher-frequency bending mode, which is related tothe modulation of the in-plane bond angle,
28changes by
about 3%. /H20849The change in the /H9275phis about 1 cm−1for theexternal and lower-frequency bending modes and 3 cm−1for
the stretching mode, respectively. The change in the /H9275phof
the higher-frequency bending mode is about 10 cm−1./H20850
These observations imply that the electronic structure mightbe coupled to the bending mode phonon.
E. Correlation between magnetic ordering and
electronic structure
Finally we check the possible effects of the magnetic or-
dering on the electronic structure. Sr 2IrO 4shows weak fer-
romagnetic ordering below TM=240 K, which originates
from the canted antiferromagnetic ordering of the Jeff=1
2mo-
ments in the in-plane.5,16It is noted that the SOC-induced
Jeff=1
2state is the complex state composed of the orbitals and
the spins. As the electronic orbitals are locked-in by the lat-
tice, the canting of the Jeff=1
2moment follows the rotation of
the IrO 6octahedra.11Therefore, the magnetic property of the
Jeff=1
2state is susceptible to the local lattice change, such as
the rotation of the IrO 6octahedra. Conversely, a long range
magnetic ordering in the Jeff=1
2state can make the IrO 6ro-
tation stiff. In this respect, the magnetic and electronic statescould be closely coupled to each other.
The temperature dependence of /H9004
optsuggests that the
electronic structure and magnetic ordering are closelycoupled. As shown in Fig. 2,/H9004
optchanges little below 200 K
and its decrease becomes faster above 200 K, which is nearthe magnetic transition temperature. To see the rate of thechange in /H9004
optmore clearly, we differentiated /H9004opt/H20849T/H20850with
respect to temperature. As shown in the inset of Fig. 2/H20849c/H20850, the
change in /H9004optis largest near the magnetic transition tem-
perature as indicated by the gray rectangle. Optical studiesFIG. 4. /H20849Color online /H20850Temperature-dependent phonon spectra
corresponding to /H20849a/H20850external, /H20849b/H20850bending, and /H20849c/H20850stretching modes.
The insets of /H20849b/H20850show the atomic displacements corresponding to
the bending modes /H20849Ref. 28/H20850. The large green and small blue circles
represent Ir and O atoms, respectively. /H20849d/H20850Peak positions of the
phonon modes /H9275phnormalized to those at 10 K.MOON et al. PHYSICAL REVIEW B 80, 195110 /H208492009 /H20850
195110-4under high magnetic fields could provide further information
on the coupling of the electronic and magnetic structures of
theJeff=1
2Mott state.30
IV. SUMMARY
We investigated the temperature-dependent optical con-
ductivity spectra of the Jeff=1
2Mott insulator Sr 2IrO 4.W e
observed significant electronic structure changes, which wereassociated with the decrease of the spin-orbit coupling-induced Mott gap. First-principles calculation with variedIr–O–Ir bond angles demonstrated the change in electronicstructure with the bandwidth control, which appears to beconsistent with our experimental findings. We also discussedthe effect of excitonic effect, electron-phonon coupling, and
magnetic ordering on the electronic structure of the J
eff=1
2
state. It should be emphasized that these results exhibit char-acteristic features of 5 dtransition metal oxides. The ex-
tended character of 5 dorbitals provides the strong electron-
phonon coupling. Furthermore, the spin-orbit couplingpresents the coupling among spin, orbital, and lattice. Ourstudy clearly demonstrates that these characteristic featuresmake the electronic structure of the 5 dtransition metal oxide
be affected by the electron-phonon interaction and bymagnetic ordering.
ACKNOWLEDGMENT
This research was supported by Basic Science Research
Program through the National Research Foundation of Koreafunded by the Ministry of Education, Science and Technol-ogy /H20849Grant No. 2009-0080567 /H20850, and the Korean Science and
Engineering Foundation through the ARP /H20849Grant No. R17-
2008-033-01000-0 /H20850. Y .S.L. was supported by the Soongsil
University Research Fund.
*Present address: Department of Applied Physics, Multiferroics
Project, Exploratory Research for Advanced Technology, Japan
Science and Technology Agency, University of Tokyo, Tokyo 113-8656, Japan.
†Present address: Materials Science and Technology Division, Oak
Ridge National Laboratory, Oak Ridge, TN 37831, USA.
‡ylee@ssu.ac.kr
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PhysRevB.85.035410.pdf | PHYSICAL REVIEW B 85, 035410 (2012)
Photoisomerization for a molecular switch in contact with a surface
J¨org Henzl,1Peter Puschnig,2Claudia Ambrosch-Draxl,2Andreas Schaate,3Boris Ufer,3Peter Behrens,3and
Karina Morgenstern3,*
1Institut f ¨ur Festk ¨orperphysik, Abteilung ATMOS, Leibniz Universit ¨at Hannover, Appelstrasse 2, D-30167 Hannover, Germany
2Lehrstuhl f ¨ur Atomistic Modelling and Design of Materials, Montanuniversit ¨at Leoben, Franz-Josef-Strasse 18, A-8700 Leoben, Austria
3Institut f ¨ur Anorganische Chemie, Leibniz Universit ¨at Hannover, Callinstrasse 9, D-30167 Hannover, Germany
(Received 9 December 2011; published 6 January 2012)
The photoisomerization properties of an azobenzene derivative on a thin insulating NaCl layer on Ag(111)
are investigated with a low-temperature scanning tunneling microscope and density functional calculations.Illumination with UV light at 365 nm induces the reversible direct isomerization of the adsorbed species, whilevisible light does not lead to any changes. This unexpected behavior cannot be explained by the change of theelectronic structure upon adsorption on the inert surface. It is rationalized in terms of electrostatic interactionscaused by the atomistic details of the surface.
DOI: 10.1103/PhysRevB.85.035410 PACS number(s): 68 .37.Ef, 68.43.Hn, 68 .65.−k, 82.37.Gk
I. INTRODUCTION
Future nanoelectronics and nanomachines demand active
elements on a molecular scale to trigger and to control theirfunctionality.
1Photochromic molecules are highly attractive
candidates for this purpose because light can reversibly switchthem between two different states.
2Among photochromic
molecules, azobenzene derivatives are extensively studied
because of their robust photoisomerization in solution phase2
and on top of self-assembled monolayers.3For a potential
implementation into real devices, however, the active elements,including the photochromic molecules, have to work underdifferent environments, especially in contact with a surface.This contact with a surface is expected to alter the photoisomer-ization properties substantially, e.g., by bond formation, steric
hindrance, or electronic quenching. Indeed, the photoisomer-
ization of native azobenzene molecules directly adsorbed on ametal substrate is suppressed
4presumably because of a shorter
lifetime of the photoexcited state on the metal surface. As aconsequence, azobenzene molecules have been functionalizedwith tert-butyl spacer groups, which are intended to lift the
azobenzene scaffold away from the surface, while keeping the
molecular plane parallel to it. Although exposure to light led
to isomerization of the azobenzene derivative on Au(111),
4–6
this was not related to the direct photoisomerization of the
molecule. Rather, the light was first absorbed by the d-band
of the metal and then the isomerization was induced by anelectron transfer from the molecule to the thus created hole inthed-bands.
5
Another attempt for decoupling of a dye molecule from a
Cu(111) surface was based on physisorption.7In this study, the
molecules could be switched between the isomers by exposureto UV light, while usually on this surface the isomerization iscompletely quenched. The position of the d-bands of copper
suggested that the substrate-mediated process is not accessible.
Here, we report the direct photoisomerization of a bistable
azobenzene molecule, 4 ,4
/prime-dihydroxyazobenzene [DHA, see
Figs. 1(a)and1(b)], in contact with a surface using a mercury
arc lamp. We achieve this by decoupling the molecule from themetal substrate by a thin insulating layer of sodium chloride(NaCl).
8–10We investigate the two isomer configurations and
the isomerization process by STM at 5 K. By wavelength-dependent measurements, we reveal that photoisomerization
is possible via one out of two possible pathways only. In theliquid phase, the rotation pathway follows excitation of theπ-π
∗transition [cf. Fig. 2(h), inset, below] by UV
light and the inversion pathway follows the excitation ofthen-π
∗transition by visible light.11On the NaCl surface, the
latter pathway is suppressed. Possible origins of this quenchingare discussed based on density functional calculations.
II. EXPERIMENTAL AND THEORETICAL METHODS
The measurements are performed with a scanning tun-
neling microscope at 5 K under UHV conditions. The 4 ,4/prime-
dihydroxyazobenzene molecules are deposited at 17 K ontoNaCl islands grown on a clean Ag(111) surface. The sampleis illuminated in situ the STM by a 100 W mercury lamp with
different filters. The angle of incident is 85
◦and the spot size
is approximately 1.1 cm2. The intensities on the sample are
80 mW cm−2for the UV filter and 442 mW cm−2for the
vis filter. During illumination, the STM tip is retracted as faras possible away from the sample and the incoming light (upto 250 nm vertically and up to 450 nm horizontally) in orderto minimize the shadowing effect of the tip. The absorptionspectra in 1,4-dioxane were determined by means of a Cary5E UV-vis-NIR spectrophotometer using a quartz cuvette withthe solvent as the reference standard. The irradiation of thesolution is done by an 8 W laboratory UV lamp of type NU-8KL with an emission maximum at 366 nm.
Adsorption geometry and electronic structure of both DHA
isomers are calculated within density functional theory. Ourstructural model is based on a (4 0, 2 6) overlayer structureof NaCl on an Ag(111) surface with the lattice vectors a=
(1.157, 0, 0) nm and b=(1.446, 1.503, 0) nm. With an
experimental lattice parameter of a
Ag=0.409 nm for silver
and a bulk lattice constant of aNaCl=0.564 nm, this results
in 2.5% and 4.5% strains in the NaCl layers along the twounit-cell vectors, respectively. We use an asymmetric repeatedslab approach with three layers of Ag, two NaCl layers, and theDHA molecule adsorbed on the top side. For the determinationof the adsorption geometry, we allow all structural degreesof freedom to be relaxed except for the positions of thebottom two Ag layers. We treat exchange-correlation effects
035410-1 1098-0121/2012/85(3)/035410(5) ©2012 American Physical SocietyJ¨ORG HENZL et al. PHYSICAL REVIEW B 85, 035410 (2012)
010203040
h (pm)
0 12 3 4x(n m )50 1.3 nm (d)
1n mcisAg(111)
NaCl(c)
transcis
trans
1.3 nm
1.3 nm(a)
(b)
FIG. 1. (Color online) 4 ,4/prime-dihydroxyazobenzene molecules on
a NaCl double layer on Ag(111). (a), (b) Ball-and-stick models of
(a) the trans - and (b) the cis-isomer as optimized in gas phase. (c)
STM image of a trans -a n da cis-isomer. The black area on top of the
image is the Ag(111) surface, the gray area is the NaCl surface, and the
white protrusions are molecules. I=30 pA, V=306 mV . (d) Line
scans across a trans - (solid line) and a cis-isomer (dashed line) similar
to the ones shown in (c) and measured across the highest points. The
apparent heights of the trans -configuration and cis-configuration are
only (29 ±5) and (41 ±5) pm, respectively.
by the generalized gradient approximation12and take into
account van der Waals interactions by an empirical correctionscheme.
13We use the V ASP code with the projector augmented
wave method,14a plane-wave cutoff of 285 eV , and a kgrid of
3×2×1.
III. RESULTS AND DISCUSSION
Scanning tunneling microscopy (STM) investigations have
proven the principal feasibility of isomerization of azobenzenederivatives on metal surfaces by electrons or by the electricfield of the STM tip and have established the appearance ofthis molecule in STM images.
15–17The trans -configuration
of other azobenzene derivatives on metal surfaces is imagedby STM as a double protrusion with a distance between theprotrusions in the region of 0.5–1.5 nm.
15–17Indeed, double
protrusions with a distance of (1 .31±0.08) nm are observed
after the deposition of DHA on NaCl [see Fig. 1(c)and the line
scan in Fig. 1(d)]. This length fits to a planar adsorbed molecule
with the end groups being the most prominent features in theSTM images [see the ball-and-stick model in Fig. 1(a)]a s
expected for a physisorbed molecule.
18
We illuminate such trans -isomers with UV light (365 nm)
for up to 18 h. The large-scale images in Figs. 2(a) and2(b)
demonstrate that we are able to image the same area of thesurface after this illumination. The trans -isomer is converted
into a single protrusion [Figs. 2(c) and2(d)]
19with a slightly
asymmetric shape and a larger apparent height as compared tothe double protrusion. A line scan of a similar protrusion asfound directly after adsorption is shown in Fig. 1(d). Qualita-
tively similar ellipsoidal protrusions were identified before asthecis-configuration of unsubstituted azobenzene molecules
on Au(111) (Ref. 16) and of amino-nitro-azobenzene on
NaCl/Ag(111).
20Note that the interpretation of the STM
images is in line with a multitude of previous studies ofazobenzene derivatives.
21
Thecis-configuration can be converted back into the trans -
configuration by illumination with UV light (365 nm), whichis evidenced by the STM images in Figs. 2(e) and2(f)taken
before and after light exposure, respectively.
(d) (c)
trans
2n m(e) (f)
(g)cis
cistrans(a) (b)
200 300 400 500 600 700 800
(nm)absorbance(a.u.)
I (a.u.)-*
n- *
cistrans
E
n* (h)
(g)
FIG. 2. (Color online) Photoisomerization of 4 ,4/prime-
dihydroxyazobenzene on NaCl. (a) and (b) Large-scale imagesshowing diffusion of molecular parts and isomerization as magnified
in (c) and (d) Trans-cis isomerization with enhanced contrast. (e) and
(f)Cis-trans isomerization from different experiment. (a), (c), (e)
Before and (b), (d), (f) after light exposure for 18 h. Surface areas on
the bottom right are higher because of a buried Ag step beneath the
NaCl layer. I=26 pA, V=196 mV . Note that images (a,b) are cut
outs of even larger images and that the NaCl island terminates on the
left-hand side. (g) 3D false color image of another example close to
the NaCl edge; Ag(111) is colored red, NaCl bilayer yellow to blue,and isomers purple. (h) Absorption spectrum of trans - (black line)
andcis- (black dotted line) dihydroxyazobenzene in 1,4-dioxane and
spectrum of Hg lamp with different filters. Gray: UV filter. Lightgray: vis filter. Upper inset shows corresponding orbitals for native
azobenzene. Lower inset shows simultaneous atomic resolution of
chlorine lattice imaged with functionalized tip, 28 pA, 193 mV .
The long exposure times in connection with the thermal
drift induced by the temperature increase during illuminationdid not allow us to observe the reversibility of the process
035410-2PHOTOISOMERIZATION FOR A MOLECULAR SWITCH IN ... PHYSICAL REVIEW B 85, 035410 (2012)
++(b) (a) (c)
2n m
transcis
cis
FIG. 3. Electron-induced isomerization: Cross indicates tip po-
sition during manipulation with −800 mV , 100 pA for 2 s. (a) and
(b)Cistotrans isomerization. (b) and (c) Trans tocisisomerization.
I=30 pA, V=306 mV .
at the same molecule.22We thus demonstrate reversibility
based on manipulation experiments with the STM (Fig. 3).
These manipulations trigger the reversible isomerization byinelastically tunneling electrons (cf. Ref. 20). Both, trans-cis
andcis-trans isomerization can be induced with the same bias
voltages above 0.8 V in both polarities. A threshold in thisregion corresponds to the height of the energy barrier in theground state between the two isomers.
21
We continue by discussing possible mechanisms of the UV-
light-induced isomerization demonstrated in Fig. 2concluding
that it is triggered via direct light absorption by the molecule.Thermal isomerization is excluded because the temperatureduring light exposure rises by less than 5 K. The correspondingthermal energy at 10 K is far below the ground-state iso-merization barrier of ≈0.8V .
23Furthermore, annealing the
sample at 10 K for the longest illumination time of 18 hdoes not lead to any changes. Indirect, i.e., substrate-mediated,isomerization is unlikely due to the presence of the 0.56-nm-thick NaCl spacer layer, which reduces the transfer probabilityof the photoexcited substrate electrons to a negligible value.Furthermore, the energy of the UV light (365 nm, 3.4 eV)is not sufficient to excite Ag(111) d-band electrons (band
edge≈4 eV below the Fermi energy), which triggered the
above-discussed isomerization of tetra- tert-butyl-azobenzene
on Au(111).
5
The direct photoexcitation at 365 nm leads to isomerization
of approximately 40% and 71% of the trans -molecules after
13 and 15 h, respectively. After 18 h, corresponding to aphoton dose of 1 .4×10
22cm−2, nearly all trans -isomers
are converted into cis-isomers (82 ±18)%. The statistics is
based on roughly 50 molecules. The stationary state is thusreached after only a five-times-higher photon dose than forthe substrate-mediated photoisomerization of tetra- tert-butyl-
azobenzene on Au(111) [ ≈3×10
21cm2(Ref. 24)], for which
a 70:30 ratio could be reached.
In order to further elucidate the isomerization process, we
analyze its wavelength dependence. First, the wavelength ofthe UV filter (365 nm) is compared to the absorbance spectraof the DHA molecules in 1,4-dioxane [Fig. 2(h)]. It matches
theπ-π
∗absorption peak of the trans -configuration. This
transition is followed by the so-called rotation pathway, inwhich the trans -to-cisisomerization is achieved by rotating
the molecule around the N-N double bond.
The probability for the reversed switching is only about 5%
within 18 h in our experiment. The smaller probability of cis-
trans isomerization as compared to trans-cis isomerization is
in agreement with the reduced intensity of the π-π
∗adsorption
peak for the cis-isomer [Fig. 2(g)].On the other hand, it is known from liquid phase experi-
ments that exciting the n-π∗transition by visible light results in
the inversion pathway.2In this case, the cis-trans isomerization
is accomplished by an inversion of the N-C bond with thenitrogen’s lone pair orbital. We thus illuminate the samplewith the vis filtered light. Although we use a three-times-largerphoton dose of 5 ×10
22cm−2, no isomerization is observed
for the adsorbed DHA. From our experiments, we concludethat the rotation pathway is active also for the adsorbedmolecule while the inversion pathway is efficiently suppressedby adsorption of the molecule onto the NaCl.
We now discuss possible reasons for this suppression of the
inversion pathway. Adsorption onto the surface could affectthree processes. First, the switching of a dye molecule isinitiated by absorbance of the light . Second, the excitation
should live long enough to allow for the much slower change in
nuclear positions. Finally, the motion of the different molecularparts should not be hindered . To identify the affected process,
we calculate the electronic structure of adsorbed DHA by DFTand compare it to that of the isolated molecule. The adsorptionsite used as a starting point in the calculations is supported fromexperiment by employing a functionalized tip,
20which allows
us to image the molecules (though with a different shape) andthe chlorine lattice simultaneously [lower inset in Fig. 2(g)].
Though it is not obvious how the molecule is imaged, weutilize the symmetry of the molecule to identify the adsorptiontip. The symmetry implies that the azo-group is positioned inthe middle between these two circles. The common motif ofall adsorption sites is the position of the azo-group above afourfold hollow site, in between two chlorine and two sodiumatoms similar to the adsorption of amino-nitro-azobenzeneon the same surface.
20As the understanding of imaging with
functionalized tips is still at its infancy, these determinationsare not beyond all doubt. However, the calculation showedthat the position of the azo-group is of minor importance asbinding is through the hydroxyl end groups.
Figures 4(a) and4(b) display top and side views of the
relaxed trans -DHA adsorption geometry, respectively, while
Figs. 4(d) and 4(e) show the corresponding cis-adsorption
geometry. We rationalize the computed adsorption sites andgeometries in terms of electrostatic interactions between thehydroxy end groups and the ionic NaCl surface. For the trans -
isomer, the negatively charged oxygen atoms are pulled towardthe Na cations leading to an overall bending of the moleculewith an O-Na distance of 0.28 nm while the azo-group is 50 pmhigher [Fig. 4(b)]. Also the cis-isomer attaches with its oxygen
atoms pointing toward the Na ions while the azo-group is liftedoff the surface [Fig. 4(e)]. Thus, both adsorption geometries
differ from the adsorption geometries on metals, on whichthe azo-group pulls the molecule toward the surface.
25The
nature of the bonding on NaCl is visualized by plotting theelectrostatic potential of the molecule and of the uncoveredNaCl surface in Figs. 4(a)and4(d), respectively.
We further analyze the electronic structure of the adsorbed
molecules in terms of the projected density of states (pDOS)[Fig. 4(c)]. The pDOS reveals negligible differences in the
energetic position and spread of the adsorbed molecule’s π,n,
andπ
∗orbitals as compared to those of the same isomer in the
gas phase. Thus, a suppression of the isomerization inducedby visible light is not due to a change in optical transitions.
035410-3J¨ORG HENZL et al. PHYSICAL REVIEW B 85, 035410 (2012)
FIG. 4. (Color online) Adsorption geometry of trans -DHA
(a), (b) and cis-DHA (d), (e) on NaCl/Ag(111). Coloring of atoms:
Ag: gray, Cl: green, Na: yellow, C: brown, O: red, N: blue. (a) Top
view of trans -DHA with an added electron density isosurface of the
molecule color-coded with electrostatic potential. (b) Side view of the
trans -DHA adsorption geometry. (c) Partial density of states (pDOS)
fortrans -DHA (top) and cis-DHA (bottom). The DOS projected on
silver atoms (gray) and the DHA molecule (black) is indicated. For
comparison, the pDOS of free DHA molecules is added. The Fermi
level is shown as a red line, energies are with respect to the vacuum
level. (d) Top view of cis-DHA with an added electrostatic potential
map of the uncovered NaCl surface at a distance of 0.28 nm above theNaCl surface. (e) Side view of the trans -DHA adsorption geometry.
Particularly, the norbital, which is localized at the azo-group,
remains almost unaffected upon adsorption due to its compara-bly large distance from the surface. Furthermore, such a minorchange also makes a faster deexcitation within the moleculeunlikely. Also, the charge-transfer probability to the metalsurface across the 0.56-nm-thick NaCl is negligible on the timescale of isomerization. We conclude that electrostic repulsion
is the likely cause for suppressing the inversion pathway.
However, why should one ring be hindered to move
in parallel to the surface? Here, we should keep in mindthe polar nature of the OH end groups as visualized bythe electrostatic isosurface map in Fig. 4(a). In a possible
inversion pathway, the OH group would have to slide over
the Cl anion where electrostatic interactions clearly providean additional energy barrier, thereby preventing this pathway.This is illustrated in Fig. 5(a). In contrast, the rotation pathway
(a)
(b)
FIG. 5. (Color online) Illustration of two possible isomerization
pathways starting from trans -DHA and ending at cis-DHA; top view,
atom colors are identical to those of Fig. 4: (a) Snapshots along the
inversion pathway; the C-N-N bond angle is indicated in the top left
corners of the panels. (b) Snapshots along the rotation pathway; the
dihedral angle is indicated.
allows for an energetically more advantageous situation as
visualized in Fig. 5(b). One oxygen atom can remain at its
favorable adsorption site during the complete isomerizationpath; the other oxygen atoms slides along a Na column, therebyexperiencing only minor electrostatic barriers.
We are aware of the fact that our above arguments
are based on the potential energy landscape of adsorbedmolecules in their ground state, while a full treatment ofthe photoisomerization process would require the knowledgeof the potential energy surface in an optically excited state.Unfortunately, this is out of reach for molecules adsorbed atsurfaces using present-day ab initio methods. Nevertheless,
we believe that the ground-state energetics correctly capturesthe molecule-substrate interactions, which we identify to beresponsible for the preference of the rotation over the inversionpathway since the dominant electrostatic interactions of thehydroxy end groups with Na and Cl surface ions would belikewise present in optically excited state.
IV . CONCLUSION
In summary, we have demonstrated a bistable photo switch
in contact with a substrate. Switching is feasible via therotation pathway of trans-cis isomerization but not via the
inversion pathway because of electrostatics. This demonstratesthat the atomistic details of an inert surface have to beconsidered because they might significantly influence thenanoscale functionality of single molecules.
ACKNOWLEDGMENTS
J.H. and K.M. acknowledge financial support from the
Deutsche Forschungsgemeinschaft (DFG). P.P and C.A.-D.acknowledge financial support from the Austrian Science Fund(FWF), projects P23190-N16 and S97-14.
*Corresponding author: morgenstern@fkp.uni-hannover.de
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035410-5 |
PhysRevB.82.075326.pdf | Adsorption and desorption kinetics of Ga on GaN(0001): Application of Wolkenstein theory
Giovanni Bruno *and Maria Losurdo
Institute of Inorganic Methodologies and of Plasmas (IMIP)-CNR, via Orabona, 4-70126 Bari, Italy
Tong-Ho Kim and April Brown
Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA
/H20849Received 11 April 2010; revised manuscript received 9 June 2010; published 26 August 2010 /H20850
The kinetics of Ga adsorption/desorption on GaN /H208490001 /H20850surfaces is investigated over the temperature range
of 680–750 °C using real-time spectroscopic ellipsometry. The adsorption and desorption kinetics are de-scribed in the framework of the Wolkenstein theory, which considers not only the equilibrium between Gaadsorbed on the surface and Ga in the gas phase but also the electronic equilibrium at the surface. It is shownthat, because of the fixed polarization charge existing at the GaN /H208490001 /H20850surface, Ga adsorption and desorption
processes involve neutral and charged Ga states. By considering the GaN surface charge involved in thesurface processes, we demonstrate that a second-order kinetics more accurately describes Ga desorption, incomparison with conventional models, and yields an apparent activation energy of 2.85 /H110060.02 eV for Ga
desorption consistent with experiments.
DOI: 10.1103/PhysRevB.82.075326 PACS number /H20849s/H20850: 68.43.Mn, 68.47.Fg
I. INTRODUCTION
The Ga adlayer present on the surface of evolving
GaN /H208490001 /H20850films during molecular beam epitaxy /H20849MBE /H20850is
critical to the ultimate film morphology and dopant incorpo-ration as a result of its, impact on adatom surface diffusionand incorporation kinetics.
1Numerous studies have found
that good morphology and properties GaN /H208490001 /H20850growth
occurs under Ga-rich or near-Ga-rich conditions, suggestingthat GaN surfaces are stabilized by Ga atoms. Such experi-mental findings have motivated research on the behavior ofGa on GaN /H208490001 /H20850surfaces including adsorption and desorp-
tion dynamics and the variety of surface reconstructions re-alized under these conditions. Various surface reconstruc-tions evolve with increasing Ga on the surface.
2The
pseudo- /H208491/H110031/H20850structure of GaN /H208490001 /H20850isthe most Ga-rich
structure and is comprised of /H110112.3 monolayers /H20849MLs /H20850
/H208491 ML=1.14 /H110031015atoms /cm2/H20850/H20849Ref. 3/H20850of Ga residing on
top of a Ga-terminated bilayer on the surface.4It has also
been established that the unreconstructed /H208491/H110031/H20850pattern cor-
responds to 1 ML of Ga tightly bound to GaN, with addi-tional Ga adatoms on top of this layer that are responsible forsurface reconstruction.
4–6
The adsorption7and desorption kinetics of Ga adlayers
have been investigated in previous studies8,9in a wide tem-
perature range, Ga fluxes range, and in presence or absenceof nitrogen during the experiment, aiming at understandingthe kinetics of Ga adsorbate at GaN surfaces. A variety oftechniques including mass spectrometry,
10grazing-incidence
x-ray scattering,11and reflection high-energy electron dif-
fraction /H20849RHEED /H20850.7,12,13However, limitations exist with
such techniques. For example, mass spectrometry samplesGa in the gas phase but not directly on the surface. Addition-ally, Ga coverage can only be indirectly deduced withRHEED since the Ga adlayer on the GaN surface does notdiffract the electron beams directly but causes an attenuationof the RHEED beam intensity due to the disorder inherent inthe layer. On the other hand, spectroscopic ellipsometry /H20849SE/H20850
has unique advantages over other in situ monitoring tech-niques and is well suited to this particular problem. It is an
optical technique and is hence noninvasive and can directlyinterrogate the surface and formation of overlayers withmonolayer sensitivity by exploiting the real-time variation inthe GaN pseudodielectric function.
14,15
Indeed, an aspect that has been neglected in the previous
studies is the interface charge transfer occurring during ad-sorption of the adsorbate layer /H20849Ga in the present case /H20850on
polar semiconductors.
We, as well as other authors, have previously reported an
analysis and study of the kinetics of gallium adlayer adsorp-tion and desorption dynamics on polar and nonpolar GaNsurfaces in the “standard” framework of the Langmuirtheory.
14Indeed, from a summary of the literature on the
topic, it has become clear that the adsorption/desorption ki-netics of Ga on GaN cannot well described by Langmuirkinetics. As already reported by Adelman et al. ,
7activation
energy and prefactor for Ga adsorption are not constant andmay vary with Ga surface coverage.
Indeed, it should be considered that GaN is a piezoelectric
material with fixed polarization charge at the surface. Duringchemisorptions, electronic charge transfer between the GaN
semiconductor and the chemisorbed species may occur,modifying the electronic structure at the surface and in theadjacent space charge region, which in return affects thechemisorption process in a nonlinear manner. This conceptof electron transfer between the adsorbate and the surfacehas been previously considered for adsorption on ZnO.
16
Unlike the Langmuir’s approach, Wolkenstein’s theory of
chemisorption takes into account these electronic interac-tions between the GaN semiconductor surface and the Gaadsorbate and their effect on the adsorptivity ofsemiconductors.
17
In this work, we develop a more complete model that
includes the role of electronic equilibrium at the GaN sur-face. The extension of this model in the framework of theWolkenstein theory gives us the opportunity to further under-stand the complex interactions and role of the GaN evolvingsurface and its interaction with the Ga adlayer during gas-PHYSICAL REVIEW B 82, 075326 /H208492010 /H20850
1098-0121/2010/82 /H208497/H20850/075326 /H208497/H20850 ©2010 The American Physical Society 075326-1phase synthesis. Within this new framework, we examine
both the adsorption and desorption kinetics of Ga on aGaN /H208490001 /H20850template in a regime far from Ga-droplets forma-
tion.
In situ spectroscopic ellipsometry is used to monitor the
Ga surface coverage in real time. For Ga adsorption onGaN /H208490001 /H20850, the relative variation in the imaginary part of the
GaN pseudodielectric function is directly proportional to theGa coverage in the monolayer and submonolayer range.
18
Exploiting the ability of SE to directly monitor the surface,we uncover a distinct surface phenomena related to chargetransfer between the adlayer and thin film during Ga adsorp-tion and desorption. Interpreting the kinetics of Ga adsorp-tion and desorption on the GaN /H208490001 /H20850surface in the frame-
work of the Wolkenstein theory,
17considers both chemical
equilibrium between the surface and the gas phase and elec-tronic equilibrium at the GaN surface. These processes in-volve Ga in two states on the GaN /H208490001 /H20850surface, specifically
neutral and charged Ga. It is shown that when electronicequilibrium is considered and Ga desorption is described bya second-order kinetics, all of the desorption profiles ob-tained at various Ga coverages and surface temperatures re-sult in an apparent activation energy of 2.85 /H110060.02 eV for
desorption.
II. EXPERIMENTAL
Studies were performed in a VEECO Gen II MBE system
equipped with RHEED and spectroscopic ellipsometry/H20849UVISEL, Jobin Yvon /H20850operating in the 1.5–6.5 eV photon
energy range. A HVPE /H20849hydride vapor phase epitaxy /H20850n-type
GaN /H208490001 /H20850template was used as the substrate for all of the
experiments. The Ga flux was fixed at 9.63 /H1100310
−8Torr, with
Ga pulse times varying from 5 to 180 s. Temperatures of 680,710, 730, and 750 °C were investigated. Combinations ofthe above parameters resulted in Ga coverage values on theGaN surface below or equal to a bilayer,
14and far from the
accumulation of Ga droplets regime also according to theGa/GaN adsorption diagram reported in Ref. 8.
The Ga adsorption/desorption kinetics was monitored re-
cording in real time the variation in the real, /H20855/H9255
1/H20856, and imagi-
nary, /H20855/H92552/H20856, parts of the GaN pseudodielectric function at 32
photon energies in the range 1.5–6.5 eV with a time reso-lution o f 1 s using a phase-modulated spectroscopic ellip-
someter /H20849UVISEL, Horiba Jobin-Yvon /H20850. The Ga surface cov-
erage and/or equivalent thickness was estimated from themodeling of ellipsometric spectra knowing the dielectricfunction of GaN and Ga as described in Ref. 14.
III. RESULTS AND DISCUSSION
Figure 1shows the variation in the imaginary part, /H20855/H92552/H20856,
of the GaN pseudodielectric function during Ga adsorptionand its desorption when the Ga shutter is closed /H20849OFF /H20850with
varying pulse time and varying temperature. The starting /H20855/H9255
2/H20856
value is representative of the GaN template dielectric func-tion; and the increase in /H20855/H9255
2/H20856is proportional to the Ga cov-
erage. The adsorption of Ga results in a monotonic increasein /H20855/H9255
2/H20856, indicating that the Ga surface coverage evolvescontinuously1up to a critical value, which depends on tem-
perature and Ga flux. The ellipsometric data, or value of /H20855/H92552/H20856,
can be converted to an equivalent Ga thickness /H20849in MLs /H20850by
modeling /H20855/H92552/H20856with a simple two-layer optical model consist-
ing of the layered GaN substrate/Ga in air and knowing thedielectric function of GaN and Ga.
15Using this method, we
also report corresponding Ga coverage in the figure. It
should be noted that the initial GaN substrate value is re-stored upon desorption of the Ga adlayer, indicating that theGa desorption is complete.
Figure 2shows the variation in /H20855/H9255
2/H20856during Ga adsorption
with increasing Ga pulse duration and during the subsequentcorresponding desorption cycle. Interestingly, different phe-nomena can be inferred from the variation in the shapes ofthe desorption curves and depend upon the Ga pulse time/H20849i.e., the Ga surface coverage /H20850. For temperatures of 680 and
710 °C, we found in earlier work using SE that the Ga ad-layer critical thickness at the steady state is 4.8 Å,
15which
agrees well with the thickness estimated for a laterally con-tracted Ga bilayer.
2The average vertical separation between
the GaN and the first Ga layer has been calculated to be
FIG. 1. /H20849Color online /H20850/H20849a/H20850Variation in the imaginary part, /H20855/H92552/H20856,
of the GaN pseudodielectric function during Ga adsorption and itsdesorption at the Ga shutter OFF, at a constant Ga flux of9.63/H1100310
−8Torr for temperature and Ga pulse time of /H20849a/H20850680 °C
and 45 s; /H20849b/H20850710 °C and 90 s; /H20849c/H20850730 °C and 180 s; /H20849d/H20850750 °C
and 180 s. The dotted line represents the Ga bilayer level. /H20849b/H20850
Expanded region of adsorption curves. The lines represent the fit-ting to obtain the neutral adsorption rate constant in Table I.BRUNO et al. PHYSICAL REVIEW B 82, 075326 /H208492010 /H20850
075326-22.47–2.54 Å, while an average vertical separation of
2.37 Å has been calculated between the first and second Galayers /H20850.
2As shown in Fig. 1/H20849b/H20850, the critical Ga thickness
decreases with increasing temperature: an observation thatcan be rationalized by a decrease in the adsorption rate con-stant k
1and an increase in desorption rate constant k−1with
the increase in the temperature /H20849see discussion below /H20850.A t
higher temperatures of 730 and 750 °C, the critical Gacoverage corresponding to the bilayer can be reached by
increasing the incident Ga flux from 9.63 /H1100310
−8to
1.86/H1100310−7Torr, as demonstrated in our previous
works,14and consistently with the phase diagram in Ref. 8.
Herein, we show that experimental data of Ga adsorption
and desorption behavior on Ga-polar GaN can be rational-ized using the Wolkenstein theory for adsorption on semi-conductors in which the surface Fermi level and/or the sur-face charge plays a dominant role in adsorbatechemisorption. According to Wolkenstein’s theory,
17local-
ized electronic states are created into the semiconductor bandgap by chemisorbed species. These states serve as traps forelectrons or holes /H20849acceptorlike or donorlike states, respec-
tively /H20850, depending on their nature. The cardinal feature of
Wolkenstein’s theory is that adsorbed species, depending onelectron transitions between those states and semiconductorsbands, may be chemisorbed on the semiconductor surface inthree ways: /H208491/H20850“weak” chemisorption involving a neutral
adsorbate species: in this case free carriers /H20849electron or holes /H20850
from the substrate do not participate in the adsorption pro-cess; /H208492/H20850“strong acceptor chemisorption” occurring when an
electron from the surface is captured by the adsorbate speciesand denoted as CeL /H20849where eL denotes the free electron par-
ticipating from the substrate /H20850;/H208493/H20850“strong donor chemisorp-
tion” occurring when an hole is captured by the adsorbatespecies and denoted as CpL /H20849pL is the free hole from the
substrate /H20850.
The possibility of those different types of chemisorption
stems from the ability of the chemisorbed specie to draw ordonate a free electron and/or hole from/to the substrate lat-tice with consequent variation in surface band bending.
A diagram representing these three forms of chemisorp-
tion is shown in Fig. 3. Furthermore, one form of the chemi-sorption may change to another depending upon temperature,
pressure, surface coverage and other factors, such as thepresence of impurities.
The surface of Ga-polar GaN is characterized by the pres-
ence of a negative fixed surface polarization charge inducinga strong upward surface band bending.
19Figure 3also shows
the energy-band diagram for the case of depletive chemisorp-tion of an acceptorlike univalent particle on a n-type GaN
surface. At the beginning of chemisorption, the upward bandbending exists. A neutral Ga atom from the gas phase /H20851de-
signed as Ga /H20849gas /H20850/H20852approaching the surface may become
chemisorbed as a neutral adsorbate /H20849denoted as Ga
0or Ga/H11569L
in the chemical reactions below /H20850. This is the neutral form of
chemisorption, which is referred to as the weak form inWolkenstein’s theory. When chemisorbed Ga captures a freeelectron /H20849denoted as in Fig. 3and as eL in the chemical
reactions below /H20850at the GaN surface, it becomes a chemi-
FIG. 2. /H20849Color online /H20850Variation in /H20855/H92552/H20856during Ga adsorption at
increasing Ga pulse duration and during corresponding desorption.The Ga flux is fixed at 9.63 /H1100310
−8Torr.
FIG. 3. /H20849Color online /H20850/H20849a/H20850Sketch of the various forms of chemi-
sorption according to the Wolkenstein theory for the GaN/Ga.Energy-band diagram for depletive chemisorption of an acceptor-like adsorbate /H20849gallium /H20850onn-type GaN: /H20849b/H20850at the beginning of
chemisorption /H20849zero coverage /H20850;/H20849b/H20850after electron transfer from GaN
to adsorbate forming charged Ga. Ga
gas,G a0, and Ga−designate a
free Ga in the gas phase, a neutral Ga adsorbate, and a negativelycharged adsorbed Ga, respectively. E
C,EV, and EFare the bulk
conduction band, valence band, and the Fermi level, respectively.The superscripts “b” and “s” denote bulk and surface properties,respectively. E
ssindicate the energy level of surface charge.ADSORPTION AND DESORPTION KINETICS OF Ga ON … PHYSICAL REVIEW B 82, 075326 /H208492010 /H20850
075326-3sorbed negatively charged Ga−/H20849denoted as Ga-eL in the
chemical reaction below /H20850. This is the charged form of chemi-
sorption, referred to as the “strong” form in Wolkenstein’stheory. Figure 3shows that the binding energy of the charged
chemisorbed Ga is E
B/H20849Ga−/H20850=/H20849ECs−Ess/H20850+eVscharged, where ECs
is the conduction band edge at the surface and eVschargedis the
chemisorption-induced surface band bending, /H20849eis the elec-
tron charge and Vsis the surface potential /H20850. Noteworthy, eVs
is a function of the coverage of chemisorbed species and
therefore the adsorption heat of chemisorption of the chargedform depends on its coverage. This coverage variation in theadsorption heat would be consistent with a nonconstant de-sorption barrier and pre-exponential factor described byAdelmann.
7
A previous independent observation of the existence of
“Ga islands different states” on the GaN /H208490001 /H20850surface
comes from Zheng et al. ,20who using scanning tunneling
microscopy demonstrated that there is a form of Ga islandsthat is converted to another form by voltage application, i.e.,charge injection. Therefore, this work indicated that there is achange in phase of Ga islands involving charges.
Thus, a novel interpretation of the Ga adsorption/
desorption on GaN is presented, which consider not only theequilibrium between the chemisorbed specie Ga and the gasphase but also the electronic equilibrium at the surface, ac-cording to the Wolkenstein theory, discussing the three re-gions that can be discerned in Figs. 1and2:/H20849i/H20850the adsorp-
tion; /H20849ii/H20850the steady state; and /H20849iii/H20850the desorption.
Noteworthy, the present model, based on the copresence ofcharged and neutral Ga, is consistent with the bilayer forma-tion reported from previous authors since our data indicatethe 2.5 ML formation and all the analysis deals with fitting abilayer adsorbed thickness. In fact, the presence of two dif-ferent charge states of Ga might be argued to be a possiblereason for the two different pseudomorphic layer and con-tracted layer formation. In fact, the various forms of chemi-sorption differ also in the strength of bonding between thechemisorbed particle and the lattice: when no electron orhole of the semiconductor participate in the bond, we havethe so called “weak form” since capturing of an electron byan acceptor level /H20849or of a hole by a donor level /H20850always leads
to strengthening of the chemisorptions, i.e., “charged strongform,” with a continuous dynamic exchange between thecharged and neutral Ga forms in the bilayer.
A. Ga adsorption
In studying kinetics of chemisorption on semiconductor, it
is true that surface electronic equilibrium is to be achieved atthe steady state of adsorption /H20849gas phase-surface /H20850equilib-
rium; however, it is not achieved at the beginning of thisprocess. The 5 s-10 s profiles in Fig. 2show a faster revers-
ible monoexponential desorption as soon as the Ga shutter isclosed. It has also been reported
21that only the neutral form
of chemisorptions participates in the exchange with the gasphase /H20849to maintain the electroneutrality equilibrium of the
gas phase /H20850. Therefore, it can be inferred that at the initial
stage of adsorption, Ga in the neutral form adsorbs “ weakly ”
on the GaN surface. Thus, the following equilibrium betweenthe neutral Ga and the gas phase applies:Ga/H20849gas /H20850+L↔
k−1k1
Ga/H11569L, /H208491/H20850
where L denotes a lattice surface site and Ga/H11569L denotes ad-
sorbed neutral Ga.
Without electronic transitions between GaN and Ga, and
far from the electronic equilibrium, the ordinary Langmuirtheory applies in this region of low coverage, i.e., the rate ofadsorption, r
ads, of neutral Ga, N0, can be expressed as
r/H6023ads=dN0/H20849t/H20850
dt=k1/H20851N/H110090−N0/H20849t/H20850/H20852−k−1N0/H20849t/H20850, /H208492/H20850
where N/H110090is the number of adsorption centers for unit area
or, the same, the maximum number of atoms which can beadsorbed on the surface /H20849also in terms of ML /H20850,k
1is the rate
constant of adsorption, and k−1is the rate constant of desorp-
tion of neutral Ga /H20849k1=1 //H9270, where /H9270is the lifetime of ad-
sorbed Ga /H20850. In this region of low coverage, the neutral Ga
form predominates; phenomenologically, this can be due tothe fact that the strong charged form has a higher energy,which is equal to the increased band bending and surfacepotential /H20849see Fig. 3/H20850, although we cannot exclude also the
existence of an additional energy barrier between the twoforms. The neutral specie instantaneous concentration isgiven by
N
0/H20849t/H20850=N/H110090·k1·t. /H208493/H20850
This is consistent with the linear dependence of the /H20855/H92552/H20856in-
crease /H20849proportional to the Ga coverage /H20850with time in Fig.
1/H20849b/H20850. Therefore, the adsorption rate constant k1can be de-
rived as a function of temperature, and data are reported inTable I, which shows that the rate constant for Ga adsorption
decreases with the increase in temperature, i.e., the lower thetemperature the more rapidly the surface/gas phase equilib-rium is established, indicating that Ga adsorption onGaN /H208490001 /H20850is nonactivated. This is consistent with a weak
form of adsorption, and is reasonable considering that anatomic specie is adsorbing /H20849no energy for breaking bonds in
molecules is needed /H20850.
Figure 4shows that although reasonable fit is achieved
using the Langmuir theory for neutral Ga adsorption for pro-file at 750 °C /H20849which is characterized by lower coverage /H20850,
the same Langmuir theory does not apply as well to profile at680 °C. Therefore, it is supposed that with the increase insurface coverage, a certain fraction of the weak Ga adsorbedpasses to a “ strong acceptor ” chemisorption through an elec-
tron transition from the GaN surface /H20849where excess of free
electron exists /H20850, i.e., according to Wolkenstein /H20849see Fig. 3/H20850:TABLE I. Dependence of the adsorption constant on
temperature.
T/H20849°C/H20850 680 710 730 750
k1/H20849s−1/H20850 0.0736 0.0686 0.0627 0.0584BRUNO et al. PHYSICAL REVIEW B 82, 075326 /H208492010 /H20850
075326-4Ga/H11569L
weak+e L↔
k−k0
Ga-eL
strong, /H208494/H20850
where Ga-eL denotes a charged form of Ga, and k0=1 //H92700
denotes the probability of charging of the chemisorbed neu-
tral Ga /H20849or/H92700is the lifetime of the chemisorbed Ga in the
neutral state /H20850.
At the steady state, for each temperature, adsorption equi-
librium between the surface and the gas phase is establishedalong with the electronic equilibrium at the surface with adynamic exchange between the neutral and charged Gaforms, according to Eq. /H208494/H20850, where k
−=1 //H9270−denotes the
probability of neutralizing the chemisorbed charged Ga /H20849or
/H9270−is the lifetime of the chemisorbed Ga in the charged state /H20850.
Let us denote with N0andN−the density of the neutral
/H20849weak /H20850and charged /H20849strong /H20850chemisorbed Ga at equilibrium,
being N/H20849t/H20850=N0/H20849t/H20850+N−/H20849t/H20850. Wolkenstein demonstrated17that
the instantaneous concentration of the neutral and chargedforms of Ga is given by
N
0/H20849t/H20850=N/H110090
/H20849/H92702−/H92701/H20850/H9270/H20877/H92702/H20849/H92702−/H92701/H20850/H208751 − exp/H20873−t
/H92702/H20874/H20876+/H92701/H20849/H92702−/H9270/H20850
/H11003/H208751 − exp/H20873−t
/H92701/H20874/H20876/H20878,N−/H20849t/H20850=N/H11009−
/H92702−/H92701/H20877/H92702/H208751 − exp/H20873−t
/H92702/H20874/H20876−/H92701/H208751 − exp/H20873−t
/H92701/H20874/H20876/H20878,
/H208495/H20850
where /H92701and/H92702are complex functions of /H9270/H208491//H9270=k−1is the
desorption probability /H20850,/H92700and/H9270−.22These expressions for
N/H20849t/H20850=N0/H20849t/H20850+N−/H20849t/H20850fits well all profiles in Figs. 1,2, and 4,
validating the applicability of Wolkenstein electronic chemi-sorption to Ga.
B. Ga desorption
Figures 1and2show that different desorption kinetics are
observed depending on Ga coverage. In particular, for verylow surface coverage /H20849i.e., 5 s Ga pulse /H20850as soon as the Ga
shutter is closed, desorption of neutral Ga starts immediatelyand it follows a first-order monoexponential decay, i.e.,
N/H20849t/H20850=N
/H110090exp /H20849−k−1·t/H20850. /H208496/H20850
This is consistent with the presence of the neutral Ga only at
low coverage, and Arrhenius analysis of desorption data as afunction of temperature /H20849for low Ga coverage /H20850yielded an
activation energy for desorption of neutral Ga of 2.8eV .
17
This value is well in agreement with the activation energy fordesorption of Ga from liquid Ga.
7Indeed, with the increase
in Ga coverage /H20849i.e., Ga pulse time /H20850, specifically when the Ga
coverage is higher than 1 ML /H20849as deduced by the modeling of
/H20855/H92552/H20856spectra /H20850, the decay is not monoexponential /H20849as it can be
seen from data at 20 s Ga pulse in Fig. 4/H20850, supporting the
hypothesis that more than one Ga forms are present at theGaN surface. Moreover, when the steady-state value corre-sponding to the Ga bilayer thickness is reached, immediatedesorption is not observed at the Ga shutter OFF differentlyfrom the short /H208495s ,1 0s /H20850Ga pulses. A desorption delay time
is observed, resulting in S-shape desorption profile /H20849see Figs.
1and 2/H20850. These data indicate that desorption is coverage
dependent contrarily to the simplifying assumption that ad-sorption and desorption were independent of coverage madeby Brandt et al.
3
We derived /H9270=83 s /H11271/H92700=1.96 s /H11271/H9270−=0.18 s, which sat-
isfy the Wolkenstein condition /H9270/H11271/H92700,/H9270−; therefore, the elec-
tronic equilibrium during desorption is maintained and de-sorption is complete /H20849see Fig. 1/H20850. In particular, the electronic
equilibrium is maintained through two balancing processes:desorption of neutral Ga /H20849weak adsorption /H20850, which violates
the equilibrium, and the discharging of charged Ga, whichrestores the equilibrium. This desorption mechanism can besummarized as
Ga-eL
Strong+p L→k2
Ga/H11569L
weak→k−1
Ga/H20849gas /H20850+L , /H208497/H20850
where k2andk−1are the rate constants of the discharge and
desorption processes, respectively, and pL denotes a hole atGaN surface.
Thus,
r
des/H6024=−dN/H20849t/H20850
dt=k−1·N0/H20849t/H20850/H20849 8/H20850
according to reaction /H208497/H20850, we can write
FIG. 4. /H20849Color online /H20850/H20849a/H20850and /H20849b/H20850fit according to the Langmuir
theory for neutral Ga adsorption for profiles at 750 °C—180 s and680 °C—45 s, and /H20849c/H20850fit according to the Wolkenstein theory for
charged Ga for profile at 680 °C—45 s.ADSORPTION AND DESORPTION KINETICS OF Ga ON … PHYSICAL REVIEW B 82, 075326 /H208492010 /H20850
075326-5dN0/H20849t/H20850
dt=k2·N−/H20849t/H20850·/H20855pL/H20856/H20849t/H20850−k−1·N0/H20849t/H20850, /H208499/H20850
where /H20855pL/H20856indicates the surface charge density involved in
the strong chemisorption and electronic equilibrium /H20851/H20855pL/H20856/H20849t/H20850
indicates its time dependence /H20850. According to Eqs. /H208497/H20850–/H208499/H20850, the
rate of desorption depends on the availability of free holes atGaN surface.
Therefore, by applying the steady state to N
0/H20849t/H20850, the fol-
lowing rate equation for Ga desorption can be derived:
rdes/H6024=−dN/H20849t/H20850
dt=k2·N−/H20849t/H20850·/H20855pL/H20856/H20849t/H20850/H20849 10/H20850
which is a second-order kinetic equation. Integration of Eq.
/H2084910/H20850with the boundary conditions that for desorption at
t=0 /H20849when Ga shutter is closed /H20850,Nt=0=N/H11569,/H20849i.e., the value
determined by ellipsometric monitoring at the steady state /H20850,
and /H20855pL/H20856t=0=/H20855pL/H20856/H11569, results in the following expression for the
Ga desorption:
N/H20849t/H20850=N/H11569−N/H11569·/H20855pL/H20856/H11569·/H208531 − exp /H20851/H20849/H20855pL/H20856/H11569−N/H11569/H20850·k2·t/H20852/H20854
/H20855pL/H20856/H11569+N/H11569exp /H20851/H20849/H20855pL/H20856/H11569−N/H11569/H20850·k2·t/H20852.
/H2084911/H20850
Figure 5shows examples of the fit goodness of desorption
profiles according to Eq. /H2084911/H20850.
The second-order rate constant values derived for desorp-
tion of Ga according to process /H208497/H20850are shown in the Arrhen-
ius plot of Fig. 6, which yields an apparent activation energy
Ea=2.85/H110060.02 eV for Ga desorption, in agreement with the
value for desorption from liquid Ga, and supporting that onlyneutral Ga is desorbing from the GaN /H208490001 /H20850surface.
This model is consistent with Adelman et al. ,
7who intro-
duced a temperature-dependent activation energy. Consider-ing that the surface coverage changes by changing the tem-perature, the Adelman work represented advancement tomodeling chemisorption with a nonconstant binding energybetween the adsorbate and adsorbent. In fact, in the case of
chemisorption on semiconductors, where charge transfer isinvolved, the binding energy /H20849adsorption heat /H20850varies with
the degree of coverage of chemisorbed species due to thestrong electronic interaction between the adsorbate andadsorbent.
23
Thus, the following analysis shows that when surface
electronic equilibrium is considered in the adsorption/desorption process of Ga a second-order kinetic analysis re-sults in a unique value of apparent activation energy for Gadesorption independent of surface coverage. Conversely, inthe simplifying assumption of first-order desorption widelyapplied in the previous literature resulted in a wide range ofvalues for the activation energy.
IV. CONCLUSIONS
In summary, the Ga adsorption and desorption kinetics on
GaN /H208490001 /H20850surface have been directly monitored in real time
using spectroscopic ellipsometry in the temperature range680–750 °C. Results indicate that charge transfer betweenthe Ga-polar GaN and the Ga adsorbate occurs stabilizing thebilayer at the GaN /H208490001 /H20850surface. The Ga adsorption and
desorption kinetics have been modeled in the frame of theWolkenstein theory of chemisorption on semiconductors,which also consider electronic equilibrium at the surface. Inthe frame of this theory, neutral and charged Ga states areinvolved in the adsorption on Ga-polar GaN. This can beunderstood considering that for ionic polar surface to be sta-bilized, the surface charge density must be modified so as tobalance the polarization electric field, and the charge com-pensation is operated by the Ga surface metallization.
ACKNOWLEDGMENT
The authors acknowledge Soojong Choi for her contribu-
tion to discussion and collection of data.
FIG. 5. /H20849Color online /H20850Example of the fit goodness of the
680 °C—45 s desorption profile according to Eq. /H208499/H20850/H20849see text /H20850
from Wolkenstein theory.
FIG. 6. /H20849Color online /H20850Arrhenius plot of the second-order rate
constant values derived for desorption of Ga according to process/H208496/H20850from which activation energy is derived. ln k
2=A−B/T/H20849K/H20850.BRUNO et al. PHYSICAL REVIEW B 82, 075326 /H208492010 /H20850
075326-6*Corresponding author. FAX: /H1100139-0805443562;
giovanni.bruno@ba.imip.cnr.it
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/H92701=/H9261/H208491+/H208811−/H9262
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/H92702=/H9261/H208491−/H208811−/H9262
/H92612/H20850;/H9261
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2/H208491
/H9270+1
/H92700+1
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/H92701
/H9270−.
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075326-7 |
PhysRevB.94.075427.pdf | PHYSICAL REVIEW B 94, 075427 (2016)
Noble-metal intercalation process leading to a protected adatom in a graphene hollow site
M. Narayanan Nair,1M. Cranney,1,*T. Jiang,2S. Hajjar-Garreau,1D. Aubel,1F. V onau,1A. Florentin,1E. Denys,1
M.-L. Bocquet,2and L. Simon1,*
1Institut de Sciences des Mat ´eriaux de Mulhouse IS2M, UMR 7361, CNRS, UNISTRA, and UHA, 3 bis rue A. Werner, 68093 Mulhouse, France
2Universit ´e de Lyon, Laboratoire de Chimie, Ecole Normale Sup ´erieure de Lyon, CNRS, 46 all ´ee d’Italie, 69007 Lyon, France
(Received 23 May 2016; revised manuscript received 9 July 2016; published 22 August 2016)
In previous studies, we have shown that gold deposited on a monolayer (ML) of graphene on SiC(0001)
is intercalated below the ML after an annealing procedure and affects the band structure of graphene. Herewe prove experimentally and theoretically that some of the gold forms a dispersed phase composed of singleadatoms, being intercalated between the ML and the buffer layer and in a hollow position with respect to C atomsof the ML on top. They are freestanding and negatively charged, due to the partial screening of the electrontransfer between SiC and the ML, without changing the intrinsic n-type doping of the ML. As these singleatoms decouple the ML from the buffer layer, the quasiparticles of graphene are less perturbed, thus increasingtheir Fermi velocity. Moreover, the hollow position of the intercalated single Au atoms might lead to spin-orbitcoupling in the graphene layer covering IC domains. This effect of spin-orbit coupling has been recently observedexperimentally in Au-intercalated graphene on SiC(0001) [D. Marchenko, A. Varykhalov, J. S ´anchez-Barriga,
T h .S e y l l e r ,a n dO .R a d e r , Appl. Phys. Lett. 108,172405 (2016 )] and has been theoretically predicted for heavy
atoms, like thallium, in a hollow position on graphene [C. Weeks, J. Hu, J. Alicea, M. Franz, and R. Wu, Phys.
Rev. X 1,021001 (2011 ); A. Cresti, D. V . Tuan, D. Soriano, A. W. Cummings, and S. Roche, Phys. Rev. Lett.
113,246603 (2014 )].
DOI: 10.1103/PhysRevB.94.075427
I. INTRODUCTION
A prerequisite for the development of graphene-based
nanoelectronics is the precise control of its functionalization.In the case of epitaxial graphene grown on SiC(0001) (hereEG), the graphitic layer on top exhibits a gapless graphenelikeelectronic structure with n-type doping. This graphene layeris decoupled from its SiC substrate by a graphitic layerat the interface, called the buffer layer (BuL), which iscomposed of one-third C atoms covalently bonded to theSi atoms of the substrate [ 1–4]. Despite the presence of the
BuL, the SiC substrate still alters the electronic propertiesof EG by decreasing the mobility of its quasiparticles (QPs)and is responsible for its intrinsic n-type doping [ 5–7]. One
way to functionalize EG is to deposit atoms on top ofit, which may intercalate between the different C sheetsin this layered structure. This was studied experimentallyfor different elements, from alkali metals to halogens andlanthanides [ 1,8–29]. In the case of noble metals, we have in-
vestigated the functionalization of EG by deposition of gold inthorough studies using scanning tunneling microscopy (STM)techniques (STM-STS) and by photoelectron spectroscopy(ARPES) [ 12,30–32]. Upon a specific preparation procedure,
gold atoms intercalate below the ML, forming two phases.One phase, labeled AuF, corresponds to the intercalation of acontinuous ML of gold. This continuous layer of Au inducesp-type doping of the graphene layer on top [ 12]. The other
phase, labeled IC, is due to the formation of small intercalatedgold dots almost regularly distributed. We have shown thatthese intercalated gold dots do not dope the graphene layer ontop but still modify its band structure, with a 20% increase in
*Corresponding authors: marion.cranney@uha.fr;
laurent.simon@uha.frthe Fermi velocity ( vf) of the QP, with a mass renormalization
around the Dirac point, and with a strong extension of itsvan Hove singularities. In this study, focusing on the ICdomain, we show, with the help of density functional theory(DFT) simulations and complementary experiments (x-rayphotoelectron spectroscopy; XPS), that Au dots are due to theaggregation of single Au atoms, freestandingly intercalatedbetween the ML and the BuL. As they partially screen theelectron transfer from SiC to the ML on top, they are negativelycharged. The huge increase in the Fermi velocity of the QP isdiscussed and attributed to an increase in the nearest-neighborhopping potential γ
0. Indeed, the presence of single Au atoms
intercalated between the ML and the BuL decouple the MLfrom its SiC substrate, which is experimentally proved bymeasuring a strong reduction in the roughness of the graphenelayer on IC domains. Moreover, computational methods showthat the intercalated single Au atoms are in hollow positionswith respect to the graphene atoms on top. This mightinduce spin-orbit coupling in the graphene layer. Indeed,spin-orbit coupling has been recently studied experimentallyin Au-intercalated graphene on SiC(0001) [ 13] and has been
theoretically predicted for heavy atoms, such as thallium, inhollow positions on graphene [ 33,34]. This might lead to the
observation of the quantum spin Hall effect in our system,covered mainly by IC domains.
II. EXPERIMENTAL METHODS
Graphene samples were prepared in situ in ultrahigh
vacuum (UHV) on n-doped 6H-SiC(0001) as described pre-viously [ 12,30,32]. The majority of the surface was ML
graphene. The deposition of gold on top of these sampleswas done in UHV as described elsewhere [ 12,30,32]. Samples
were further prepared by several cycles of annealing at 1000 Kfor 2 min. In-house physical characterizations were performed
2469-9950/2016/94(7)/075427(8) 075427-1 ©2016 American Physical SocietyM. NARAYANAN NAIR et al. PHYSICAL REVIEW B 94, 075427 (2016)
FIG. 1. (a) STM picture showing the two phases (i.e., IC and
AuF) due to the intercalation of gold below the graphene monolayer(ML) after several cycles of annealing. (b) Zoom-in of an IC domain,
showing that the majority of gold dots are due to the aggregation
of three clusters, as shown in the STM image (c). Inset in (b)Its corresponding self-correlation image, showing a quasiperiodic
hexagonal arrangement of Au dots. (d) Graph of the distribution of
the number of clusters per Au dots, from five STM images totalingmore than 500 dots. (a) 86 ×86 nm
2,−1.5 V; (b) 26 ×26 nm2,
−2.2V , ∂z/∂x -derivative representation of the topography;
(c) 2.3×2.7n m2,−1.5 V (image processing using Gwyddion and
WSxM software [ 46]).
in UHV by scanning tunneling microscopy and by photoelec-
tron spectroscopy techniques (XPS). STM experiments wereperformed in situ with an LT-STM from Omicron at 77 K at a
base pressure in the 10
−11mb range. Images were acquired in
constant-current mode with bias voltage applied to the sampleand employing chemically etched W tips. XPS measurementswere performed in situ using a VG Scienta R3000 spectrometer
equipped with a monochromatized Al K αx-ray source
(1486.6 eV) and a hemispherical analyzer. At this high photonenergy, our measurements probe not only the surface but alsothe SiC substrate, and thus the C 1 sand the Si 2 pspectra have
higher SiC bulk signals than in the literature [ 4,7,35–38]. The
electron energy analyzer operated at a 100-eV pass energy.Spectra were measured at both grazing and normal (not shownhere) incidence. Shirley background was subtracted from theC1s,S i2 p, and Au 4 fspectra. We used a Lorentzian
asymmetric lineshape (LF) for metallic and graphitic corelevel peaks and a symmetric V oigt lineshape (GL) for the otherpeaks in the fitting procedure using CasaXPS software [ 4,35].
For XPS measurements, we used homogeneous samples witha surface covered by 5% ±3% ML, 10% ±5% AuF, and
82%±5% IC domains, as shown in Figs. 1(a) and1(b) of
Nair et al. [32]. Some three-dimensional islands of Au arestill present on the surface of the sample after the annealing
procedure, but they are scarce. The gold intercalated samplesare extremely stable (no change in the gold structures) atroom temperature for several months and they remain quiteclean even after several weeks in an ambient atmosphere. Thecontamination after several months in an ambient atmosphereis completely removed by annealing in UHV at 900 K forseveral hours. This means that the graphene layer protects theintercalated gold structures from contamination, degradation,sand desorption, even when heated at 900 K.
III. METHODOLOGY OF CALCULATIONS
Calculations were carried out within the framework of DFT
implemented in the Vienna Ab initio Simulation Package(V ASP) [ 39,40]. The local density approximation was con-
sidered to describe the exchange and correlation energy. Inorder to get a good understanding of the EG substrate, ourmodel was based on a (13 ×13) graphene on a (6√
3×6√
3)
SiC substrate, instead of a smaller model without the properlarge commensurability like (2 ×2) graphene on a (√
3×√
3)
SiC substrate, which gives 8% extra strain due to the mismatchof graphene and SiC lattice constants. The SiC substratewas modeled by two SiC bilayers, the top face being Siand the bottom being C saturated with hydrogen atoms,including 432 atoms. All structures were relaxed until the totalforces were lower than 0.02 eV /˚A. STM images at constant
current were simulated by means of the Tersoff-Hamanntheory [ 41,42]. An implementation [ 43,44] was used in order to
correctly reproduce the exponential decay of wave functionsin the vacuum region: above a given height (approximately23˚A from the outermost atoms of the sample), the analytical
expression of the wave function for a flat potential in vacuumwas considered.
IV. RESULTS
The STM image in Fig. 1(a) is a scan area representative
of the EG surface after the intercalation of gold atoms. AuFcorresponds to the intercalation of a monolayer of Au andIC is due to the formation of small intercalated Au dots.These dots are two-dimensional (2D) and lie flat, parallelto the surface below the ML [ 12]. As shown in the STM
images in Figs. 1(a) and1(b), the dots are homogeneously
distributed all over the IC domains, even when adjoining arim or a AuF domain. They form a quasiperiodic hexagonalarrangement of aggregates of clusters, with a measured meandistance of 2 .25±0.07 nm between the centers of mass of two
neighboring dots and of 4 .17±0.05˚A between two clusters
inside a dot (see Fig. 5as well). These distances are always
identical and not related to the deposited quantity of Au or thesize of the IC domain (see Supplemental Material Fig. 1 [ 45]).
Most of the dots are formed by three clusters, as shown inFigs. 1(b) to1(d). An individual cluster is shown in Fig. 2(a).
As can be seen, an individual cluster has an almost-triangularshape and it affects the contrast of 9 C atoms on top. Inorder to understand in more detail the composition of theseclusters and their positions below the ML, a theoretical studywas performed to compare the experimental STM imagesand the simulated STM images, as shown in Fig. 2. Several
075427-2NOBLE-METAL INTERCALATION PROCESS LEADING TO . . . PHYSICAL REVIEW B 94, 075427 (2016)
FIG. 2. Direct comparison of (a) the experimental STM image
with (b) the simulated STM image of an individual Au atom
intercalated between the ML and the BuL. Image (a), 2 .1×2.1n m2,
wa made at −1.1 V . Image (b), simulated at −1.3 V , is the same size
as (a), with the C atoms of the ML on top of the Au atom displayed
in green. (c) The hollow position of the intercalated Au atom.
situations were tested, changing the size of the 2D Au clusters
(from Au 6clusters to single Au atoms) and changing their
positions, i.e., intercalated between the ML and the BuL orintercalated below the BuL (see simulated STM images of allthese possible cases in Supplemental Material Fig. 2 [ 45]).
Among all the simulated situations, only that of a singleAu atom intercalated between the ML and the BuL showsa simulated STM image very similar to the experimental one,with the same shape and the same number of affected C atomson top, as shown in Fig. 2(b). Thus we may infer that IC
domains are made of single Au atoms intercalated betweenthe BuL and the ML. Moreover, the calculations show thatthe Au atom is nearly in a hollow position with respect toC atoms on top, i.e., nearly in the center of the hexagon asdepicted in Figs. 2(c) and3(a). The measured mean distance
of 4.17±0.05˚A between two Au atoms inside a dot means that
all Au atoms remain single and are in hollow positions insidea dot. The hollow position of the single Au atoms is highlyimportant for the graphene functionalization and, notably, forthe observation of the quantum spin Hall effect in such system.Indeed, such an adatom position might lead to the inductionof spin-orbit coupling in the graphene layer, as theoreticallypredicted for heavy atoms, like thallium [ 33,34].
The freestanding nature of the intercalated single Au atom
is shown by the calculated differential charge density plot [seeFig.3(b)]: there is no significant charging of either BuL or ML
and no shared electrons between Au atom and C or Si atomcounterparts. This points to the absence of a chemical bondbetween Au and ML, BuL, or SiC. The charge redistributionafter Au intercalation between the ML and the BuL showsa dual behavior of the single Au atom: the diffuse sshell
gains electrons, while the contracted out-of-plane dshells
lose electrons. The net charge on the intercalated Au atomcan be roughly evaluated by comparing the atomic charge ofAu before and after intercalation: it amounts to 0 .25e. Hence
the DFT analysis shows that a single Au atom intercalatedbetween the BuL and the ML is freestanding and slightlynegatively charged. In order to support this assertion, weperform in situ XPS measurements on samples prior to and
after Au deposition (covered then by 82% IC domains). Thereshould be no formation of Au-C covalent bonds, as Au andC atoms have almost the same electronegativity, and henceonly Au-Si bonds can be formed. Gierz et al. came to the
conclusion that Au atoms intercalated below the BuL are
FIG. 3. (a) Side view of the DFT optimized structure of a single
Au atom intercalated between the BuL and the ML. The Au atom
is displayed in yellow, the C in gray, and the Si in blue. Inset:
Corresponding top view, showing the position of the intercalatedsingle Au atom: nearly at a hollow site with respect to the ML
(distance of 2.40 ˚A; C atoms displayed in light gray) and nearly
at a top site with respect to the BuL (distance of 2.06 ˚A; C atoms
in dark gray). The BuL-to-ML distance varies from 3.20 to 4.20 ˚A.
(b) Charge density difference upon intercalation of a single Au atom
between the BuL and the ML. Red (blue) regions mark accumulation
(depletion) of electrons. Units are 10
−3e.
bonded to Si atoms of the SiC substrate, thus altering the
BuL or decoupling it from the SiC substrate [ 47]. However,
the deposition of Au was done in their case before the fullformation of an ML. We present in Fig. 4the C 1 sand the Si
2pspectra obtained both for the pristine EG samples prior to
Au deposition (measured spectra in red) and for our samplescovered by 82% IC domains (measured spectra in black) forcomparison. The deconvolutions of the different spectra areshown in Supplemental Material Fig. 3 [ 45] and the positions
of the deconvoluted peaks are listed in Supplemental MaterialTable 1 [ 45]. Note that the positions of the peaks of
pristine EG are in good agreement with the data reportedin Refs. [ 4,7], and [ 35–38]. As shown directly in Fig. 4,
the C 1 sand Si 2 pspectra of pristine EG and of IC
domains are nearly identical and not shifted in binding energy(BE) as expected, as Au dots do not dope the graphenelayer [ 12,32]. The insets in the C 1 sand Si 2 pspectra in
Fig. 4present the deconvoluted peaks related to the BuL
(i.e., the ones labeled S1 and S2 in the C 1 sspectra related
to the sp
3andsp2C atoms of the BuL, respectively, and the
one labeled 6√3f o rt h eS i2 pspectra), whose positions and
areas are unchanged prior to and after Au intercalation. Thismeans that the BuL is not altered and not decoupled fromits SiC substrate by intercalated Au atoms. Moreover, we cansafely exclude the formation of gold silicide, as this should
075427-3M. NARAYANAN NAIR et al. PHYSICAL REVIEW B 94, 075427 (2016)
90 88 86 84 82
Binding Energy (eV)104 102 100Intensity (arb. units)288 286 284 282
Au 4fC 1s
Si 2p287 286 285 284
103 102 101
86 85 84Au dots
Au ML6√√√√3S1S2
FIG. 4. C 1 sand Si 2 pspectra measured on pristine EG
(experimental data are displayed in red) and in IC domains (in
black) are presented for direct comparison. Insets: Deconvoluted
peaks related to the BuL for both spectra. We also show two Au 4 f
spectra, the one in red being measured on a clean Au(111) surface as
a reference [ 50] and the other one, in black, measured on IC domains.
Inset: The two deconvoluted peaks from the IC-domain spectrum, onerelated to the intercalated monolayers of Au from the AuF domains
(Au ML) and the other to the intercalated Au 2D dots (Au dots). The
deconvolutions of all spectra are shown in Supplemental MaterialFig. 3 [ 45] and the positions of all deconvoluted peaks are listed in
Supplemental Material Table 1 [ 45]. See text for details.
drastically change the peaks related to the BuL in C 1 sand Si
2pspectra of IC domains (i.e., a strong decrease in their areas)
and as another peak should appear in the Si 2 pspectrum
at a BE of around 99.9 eV , following Refs. [ 48] and [ 49].
We present in Fig. 4a detailed analysis of the chemical
states of deposited gold by comparing two Au 4 fspectra,
the one in red being measured on a clean Au(111) surfaceas a reference [ 50] and the other one, in black, measured in
IC domains. The deconvolutions of the different spectra areshown in Supplemental Material Fig. 3 [ 45] and the positions
of the deconvoluted peaks are listed in Supplemental MaterialTable 1 [ 45]. The reference Au 4 f
7/2spectrum and the one
measured in IC domains differ only at a BE around 85 eV dueto the presence in the IC-domain spectrum of two additionalpeaks that are probably related to the formation of intercalatedAu nanoparticles and not due to the formation of Au silicide,in agreement with Ref. [ 51]. Indeed, when decreasing the size
of a Au nanoparticle, its related Au 4 f
7/2peak shifts towards
a higher BE due to two coexisting effects: the initial-state andthe final-state effects [ 51–56]. Thus, the peak at a lower BEis related to MLs of Au in AuF domains, and the peak at a
higher BE to Au dots in IC domains, as shown in the inset inFig.4. Therefore, we can conclude that single Au atoms of IC
domains are freestanding, being intercalated between the BuLand the ML.
As gold atoms have a high positive electron affinity and a
relatively high first ionization potential, they have a tendencyto attract electrons, leading to screening of the transfer ofcharge from the SiC to the graphene layer. Thus p-type dopingof graphene is expected, as measured experimentally [ 11,12].
Previous STS and ARPES measurements have shown thatintercalated Au dots did not dope the graphene layer, as therewas no change in the position of the Dirac point E
Dprior to
versus after Au intercalation [ 12,32,57]. In addition, ARPES
measurements revealed an increase in the Fermi velocity(v
f) of the QPs in IC domains from 0 .99±0.08×106to
1.24±0.2×106m·s−1[32]. Even if the screening of the
charge transfer from the SiC to the graphene layer by singleAu atoms is not effective enough to dope it, Au atoms attractelectrons from SiC. This was confirmed by calculations [seethe differential charge density plot in Fig. 3(b)]. We estimate
the transfer of charge per intercalated Au atom by using thevariation of the density of electrons n
eat the Fermi energy in
relation to EDandvf[58]:
ne=E2
D
π/planckover2pi12v2
f. (1)
Following Eq. ( 1), the estimated electron density at the
Fermi energy is 6 .22×1012cm−2for a pristine ML and
2.57×1012cm−2for an IC domain, respectively, using the
values of EDandvfobtained from our ARPES measure-
ments [ 32]. This means that the transfer of electrons from
SiC to intercalated single Au atoms is 3 .65×1012cm−2.I f
we assume homogeneous coverage of the IC domain, with ahexagonal arrangement of Au dots (distance between Au dotsof 2.25 nm) consisting of three single Au atoms, then eachAu atoms has attracted 5 .33×10
−2electrons. This estimated
electron transfer is higher than those for Bi and Sb atoms [ 11],
which is expected, as Au has a higher electronegativity than Sband Bi. From our experiments, we can conclude that freestand-ing single Au atoms of IC domains intercalated between theBuL and the ML are negatively charged. This raises the ques-tion of the driving force of their self-organization, for example,possible Coulomb interactions, as they are freestanding.
We obtain further information about the interactions be-
tween negatively charged Au dots by studying the topographicSTM images, as shown in Fig. 5. At first glance, the distribution
of Au dots in IC domains may appear random, but self-correlation images, such the insets in Figs. 1(b) and 5(a),
show some order. There is a clear sixfold pattern due to thesuperimposition of an isotropic ring on a hexagonal patternof six spots, as in the case of a hexatic phase in melted2D crystals [ 59–61]. This phase is between a 2D crystal
and an isotropic liquid, showing quasi-long-range order inthe orientation of nearest-neighbor pairs of Au dots (as ina 2D solid) and short-range positional order of Au dots(as in a 2D liquid). In our case, the exponential decay[∝exp(−0.997r)] of the pair distribution function f(r)
obtained using Fiji [ 62,63] prove that the positional order
075427-4NOBLE-METAL INTERCALATION PROCESS LEADING TO . . . PHYSICAL REVIEW B 94, 075427 (2016)
0123456789 1 0C(r) (arb. units)
r (nm)0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160,00,51,01,52,0f(r)
r (nm)
(a) (b)
(c)
FIG. 5. (a) STM picture of an IC domain, with its corresponding
self-correlation image in the inset, showing a quasiperiodic hexagonal
arrangement of Au dots. (b) Pair distribution function, f(r), whose
envelope decays exponentially [ ∝exp(−0.997r) ;fi ts h o w nb yd a s h e d
red line]. (c) Graph of the self-correlation function of the atomic
positions, C(r), which stems from a radially averaged self-correlation
image of an STM picture. From C(r), we measure a mean distance
between the centers of mass of two neighboring Au dots of 2 .25±
0.07 nm. These measurements, using Fiji [ 62,63]f o rf(r) and WSxM
forC(r), were made on 14 STM images at different bias voltages,
using different tips and on different samples. Image (a): 22 .7×55.6
nm2,−1.5V .
is indeed only short range [see Fig. 5(b)]. The first peak in
the averaged self-correlation function of the atomic positionsC(r) (obtained using WSxM) corresponds to the mean nearest-
neighbor distance between Au dots. This large distance,2.25±0.07 nm, implies that the interactions between Au dots
are long-range, like repulsive dipole-dipole electrostatic inter-actions [ 64,65] or indirect electrostatic interactions mediated
by electrons from the environment (from graphene, the BuL,or SiC) [ 61,66–69]. We perform classical molecular dynamics
simulations using LAMMPS software [ 70] to check whether
the spatial distribution of Au dots is due only to direct interac-tions between them. The simulations are performed assumingLennard-Jones and electrostatic (Coulomb, charge-dipole,dipole-dipole) interactions between the Au dots only, whichare treated as a microcanonical ensemble and approximated asspherical pseudoatoms carrying a charge and/or a point dipolemoment [ 71]. The simulations generate the pair distribution
function f(r), directly compared to the experimental one
displayed in Fig. 5(b). Two criteria must be fulfilled in order
to judge the level of reliability of the simulation: the positionsof the peaks of f(r) (which correspond to the representative
distances between Au dots) and the amplitudes of the peaks(which represent the probability of measuring those specificdistances) must be the same as the experimental values.Whatever the temperature of the thermostat, we never obtain apair distribution function exactly the same as the experimentalone using realistic values (based on our ARPES experiments)
of charge (from 5 .33×10
−2to 0.1 electron per Au atom) and
of dipole moment (from 0 to 8 D). Thus we cannot excludeany influence of the simulated direct repulsive long-rangeinteractions on the observed distances between Au dots, butthe environment (graphene, the BuL, and/or SiC) should alsohave an effect on the repartition of Au dots. There must indeedbe an influence of the BuL on the repartition of Au dots, asthe measured distance of 2 .25±0.07 nm is quite close to the
length of the hexagons (2.13 nm) due to the (6√
3×6√
3) R30-
SiC reconstruction of the BuL [ 5,72] and not directly related
to the periodicity of the SiC substrate or of the graphene layer.
In our previous studies, we have found that intercalated
single Au atoms have a great impact on the band structure ofEG, with a strong extension of the van Hove singularities and a20% increase in v
f[30–32]. There are two possible reasons for
this increase in vf, either an increase in the lattice parameter aG
of the ML on top or an increase in the nearest-neighbor hopping
energy γ0, i.e., the amplitude of the probability of a QP’s
tunneling between two neighboring lattice sites [ 73]. Indeed,
vfis proportional to both parameters in the first approximation
(a)
ML
ICRim
010203040
BuL
QFMLGML BL IC TL(c)
with atomic corrug.
withoutRMS roughness (pm)
01 0 2 0 3 0 4 0 5 00,00,20,40,60,8z (Å)(b)
ML ICRim
x (nm)
FIG. 6. (a) STM picture showing an IC domain separated from
a pristine ML by a rim. (b) Profile curve corresponding to thedashed blue line in the STM image. One can directly see that the
graphene layer in the IC domain is almost flat between intercalated Au
dots. (c) Root-mean-square roughness of different graphene layers:
BuL, pristine ML; QFMLG, ML with intercalated H atoms [ 75];
BL, pristine bilayer graphene; and TL, pristine trilayer graphene.Some were measured without removing the atomic corrugation (black
squares), whereas red triangles represent measurements of the “real”
roughness of the layer. This figure is adapted from Refs. [ 6,74],
and [ 75]. Image (a): 45 .4×27.2n m
2,−1.3V .
075427-5M. NARAYANAN NAIR et al. PHYSICAL REVIEW B 94, 075427 (2016)
as given by Eq. ( 2):
vf=√
3
2/planckover2pi1γ0aG. (2)
We measured aGin a pristine ML and in IC domains in the
same STM images at different bias voltages and found novariation of a
Ggreater than 4%. Consequently, we attribute
this increase in vfto an increase in γ0. This means that
the intercalation of single Au atoms between the ML andthe BuL induces fewer perturbations of the QPs of theML on top, which is counterintuitive, as we have alreadyshown that QPs are scattered in IC domains at a specificenergy range [ 30]. The fact that the QPs of the ML are
less perturbed is actually due to the better decoupling ofthe graphene layer on top of the IC domains from the BuL,a ss h o w ni nF i g . 6. Indeed, Figs. 6(a) and6(b) display the
BuL-induced corrugation of a pristine ML [ 5,6] and that of an
IC domain, showing that the roughness of IC domains is muchreduced, the graphene layer in IC domains being almost flatbetween Au dots (the roughness here is due only to the atomiccorrugation). Figure 6(c)shows that the measured roughness of
IC domains is even below that of ML graphene decoupled froma SiC substrate by hydrogen intercalation below BuL (labeledQFMLG) [ 75] and equal to that of pristine trilayer graphene
on SiC(0001) within measurement uncertainty [ 6,74] (see
Supplemental Material Fig. 4 [ 45] for a detailed analysis of the
root-mean-square roughnesses of a pristine ML, an IC domain,and an ML between Au dots in an IC domain depending on theapplied bias voltage). Negatively charged single atoms of Au,
freestandingly intercalated between the BuL and the ML, thus
improve the electronic properties of graphene by decouplingit from its SiC substrate. We report here a new way to obtaina quasi-ideal freestanding graphene on a SiC(0001) substrate.
V. CONCLUSIONS
The deposition and subsequent annealing of Au on EG lead
to the intercalation of Au atoms below the ML. Some of themform a specific type of domain, called the IC domain, madeof almost regularly distributed Au dots that do not dope thegraphene layer on top. From experiments and computational
methods, we have shown that these Au dots are made ofsingle Au atoms (usually three), freestandingly intercalatedbetween the BuL and the ML, and are negatively charged,as they partially screen the electron transfer from SiC to thegraphene layer on top. Their distribution between the BuLand the ML is not random but probably due to interactionsmediated by electrons from the environment, particularlywith the BuL. Former ARPES measurements have shown anincrease in v
fdue to the presence of intercalated Au dots.
This effect has finally been attributed to an increase in thenearest-neighbor hopping potential γ
0due to the decoupling
of the ML from the BuL, which has been experimentallyproved. We report here a new way to decouple graphene fromits SiC(0001) substrate, which leads to an improvement inits electronic properties. Moreover, computational methodsshow that the intercalated single Au atoms are in hollowpositions with respect to the graphene atoms on top, whichmight induce spin-orbit coupling in the graphene layer. Thiseffect of spin-orbit coupling has been proved experimentallyon Au-intercalated graphene on SiC(0001) [ 13] and has been
theoretically predicted for heavy atoms, such as thallium, inhollow positions on graphene [ 33,34]. This might lead to the
observation of the quantum spin Hall effect in our system,covered mainly by IC domains.
ACKNOWLEDGMENTS
We thank L. Daukiya for the preparation of epitax-
ial graphene samples for x-ray photoelectron spectroscopymeasurements. M.C. thanks J. Renard (N ´eel Institut) and,
particularly, I. Deroche (IS2M) for very useful discussionsabout molecular dynamics simulations using LAMMPS. Shethanks also A. Kohlmeyer and S. Plimpton, LAMMPS devel-opers at Sandia National Laboratories and Temple University,respectively, for their corrections of the simulation’s program.This work was supported by the R ´egion Alsace and the CNRS.
The Agence Nationale de la Recherche supported this workunder the ANR Blanc program, reference ANR-2010-BLAN-1017-ChimiGraphN.
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075427-8 |
PhysRevB.100.155143.pdf | PHYSICAL REVIEW B 100, 155143 (2019)
Simplification of the electron-ion many-body problem: N-representability
of pair densities obtained via a classical map for the electrons
M. W. C. Dharma-wardana*
National Research Council of Canada, Ottawa, Canada, K1A 0R6
(Received 25 July 2019; revised manuscript received 4 October 2019; published 28 October 2019)
The classical map hypernetted-chain (CHNC) method for interacting electrons uses a kinetic energy functional
in the form of a classical-fluid temperature. Here, we show that the CHNC generated two-body densities andpair-distribution functions (PDFs) correspond to N-representable densities. Comparisons of results from CHNC
with quantum Monte Carlo (QMC) and path-integral Monte Carlo (PIMC) are used to validate the CHNC results.Since the PDFs are sufficient to obtain the equation of state or linear response properties of electron-ion systems,we apply the CHNC method for fully classical calculations of electron-ion systems in the quantum regime, usinghydrogen at 4000 K and 350 times the solid density as an example since QMC comparisons are available. We alsopresent neutral pseudoatom (NPA) calculations which use rigorous density functional theory (DFT) to reducethe many nuclear problem to an effective one-ion problem. The CHNC PDFs and NPA results agree well withthe ion-ion, electron-ion, and electron-electron PDFs from QMC, PIMC, or DFT coupled to molecular dynamicssimulations where available. The PDFs of a two-dimensional (2D) electron-hole system at 5 K are given as anexample of 2D “warm dense” matter where the electrons and the counterparticles (holes) are all in the quantumregime. Basic methods such as QMC, PIMC, or even DFT become prohibitive while CHNC methods, beingindependent of the number of particles or the temperature, prove to be easily deployable.
DOI: 10.1103/PhysRevB.100.155143
I. INTRODUCTION
T h ew a v ef u n c t i o n /Psi1({/vectorri},{σi},{/vectorRj})o fa n N-electron
quantum system depends on 3 Nspace coordinates /vectorri,s p i nσi,
and the coordinates {/vectorRj}specifying the positions of the nuclei.
In the following, at first the ions are treated as passively pro-viding an “external potential” to the electrons, so that we onlyhave an electron system placed in the external potential of theion subsystem. Subsequently, the two-component system ofinteracting electrons and nuclei is treated.
In the general case with electrons with a one-body density
n(/vectorr) and nuclei at a density ρ(/vectorr), we have three many-body
interaction terms in the Hamiltonian, viz., V
ee,Vei, and Vii.
Density functional theory (DFT) can be used to reduce each ofthese three terms to three effective one-body interactions andthree corresponding exchange-correlation functionals. That is,a full DFT reduction of the electron-ion many-body problemwill involve the familiar eeexchange-correlation functional
F
ee
xcas well as two additional functionals Fei
xcand Fii
xc. Since
ions behave as classical particles under normal conditions,F
ii
xc[ρ(r)] is simply a correlation functional Fii
c, with exchange
effects being normally of no importance, even if the ionswere fermions or bosons. However, if the positively chargedparticles are not nuclei, but “holes” in semiconductor energybands at ambient temperatures, positrons, or muons, exchangeeffects have to be included.
In a complete DFT description of, say, a fluid of electrons
and carbon nuclei, the many-body problem would be reducedto a single “Kohn-Sham electron” interacting with a single
*chandre.dharma-wardana@nrc-cnrc.gc.ca“Kohn-Sham carbon ion” via Coulomb interactions and threecoupled xc functionals. In effect, the total electron-ion prob-lem can be reduced to an effective “hydrogenic” problem.Such a model is already available, mainly for extended sys-tems like fluids and metallic solids, in the implementationof the neutral pseudoatom (NPA) model that was initiallyintroduced into solid-state physics [ 1] using linearly screened
ions. Ions screened by a DFT-generated nonlinear electrondensity were introduced by Dagens [ 2], and then into plasmas
and fluids [ 3] mainly using cluster expansion models for
dilute superposed densities of ions, but without recourse toan ion-ion correlation functional. The NPA was reformulatedusing more systematic DFT arguments in Ref. [ 4] where
explicit ion-ion and electron-ion correlation functionals wereintroduced.
A further level of simplification, going beyond the NPA
model, is to replace the quantum electrons by an equivalentclassical representation of electrons [ 5]. In effect, such models
replace the quantum kinetic energy functional by a functional
for an “effective temperature” of an “equivalent” classical
system. Then, the pair-distribution functions (PDFs) can bedetermined using classical molecular dynamics (MD) simula-tions or classical integral equations rather than those of quan-tum systems which involve fermion sign problems, problemsof evaluation of multicenter integrals, and large basis sets.Thus, even at zero temperature, even DFT calculations scale
nonlinearly with the number of particles, and rapidly become
prohibitive at finite temperatures, while classical methodsremain feasible.
Most DFT calculations only use the eexc functional and
make no recourse to F
ei
xcand Fii
c. This is perhaps because the
construction of an xc functional even for electrons has been a
2469-9950/2019/100(15)/155143(13) 155143-1 ©2019 American Physical SocietyM. W. C. DHARMA-W ARDANA PHYSICAL REVIEW B 100, 155143 (2019)
major task since the inception of density functional theory.
However, the three needed xc functionals can be formallyexpressed in terms of the three pair-distribution functionsg
aa/prime(/vectorr,/vectorr/prime) where astands for particle species e,o ri. This de-
fines fully nonlocal functionals most appropriate to the givencalculation. In fact, if the three pair-distribution functions(PDFs) were known, allstatic quantum properties and thermo-
dynamic properties as well as linear transport properties (e.g.,electrical conductivities) can be determined without havingto know the many-body wave function. Hence, it would bea great simplification of the electron-ion many-body problemif there were accurate methods for the direct determination ofthe PDFs via “orbital-free” methods.
The objective of the paper is to demonstrate a method
where (a) it is shown that calculations using a classical mapfor electrons provide results for the three PDFs having anaccuracy quite competitive with those from DFT in “difficult”regimes like warm dense matter; (b) the PDFs obtained us-ing the classical map hypernetted-chain technique satisfy thecriteria for N-representability [ 6,7].
N-representability stipulates that any directly determined
PDF or two-body density must be such that it is the resultof integrating out the space and spin coordinates of all buttwo of the particles from the square of a many-body wavefunction (which is unknown). Of course, N-representability of
a two-body density does not guarantee that it is the one withminimum energy, but only that the PDF or two-body densityis a reduction of an N-body wave function.
It can perhaps be argued that current rapid developments
in computer technology have made attempts at simplifica-tions of the electron-ion problem less relevant. However,even today, DFT calculations are unable to provide all threePDFs g
a,a/prime(/vectorr,/vectorr/prime) even for uniform systems like fluids or
electron-hole layers, even at zero temperature. QuantumMonte Carlo (QMC) and path-integral Monte Carlo (PIMC)methods become major calculational projects when appliedeven to electron-proton systems at finite T. In this study, we
present examples of competitively accurate calculations usinga classical map approach capable of reproducing such heavycomputations “on a laptop” in minutes, without the use ofad hoc models and using only the temperature, density, and
nuclear charge as inputs, for a wide class of problems wherethese methods apply.
II. SURVEY OF THE THEORY
Usually, the ions behave “classically,” while the electrons
are quantum mechanical. We seek to represent the electronseven at the extreme quantum limit of T=0 by an “equiva-
lent” classical Coulomb fluid (CCF), only in the limited senseof having the same pair-distribution functions (PDFs) as theelectron system.
The many-electron wave function contains significantly
more information than necessary for calculating measurableproperties of physical systems. As Nbecomes large, the
solution of the many-particle Schrödinger or Dirac equation,or their QMC pr PIMC implementations, become numericallyprohibitive. A way out is presented by the Hohenberg-Kohntheorem of density functional theory (DFT), which assertsthat the ground-state energy Eof the N-particle system isa functional of just the one-body electron density n(/vectorr,σ)
[8–10]:
n(/vectorr
1,σ1)=/Sigma1N
i=2/integraldisplay
d/vectorr2...d/vectorrN|/Psi1(/vectorr1,...,/vectorrN,{σi})|2.(1)
From now on, we consider a paramagnetic electron fluid and
suppress spin indices unless we consider effects explicitlydependent on spin resolution.
The Hohenberg-Kohn theorem is a counterintuitive result
since the many-particle Hamiltonian
H=T+V(/vectorr)+/Sigma1
i<jVee(/vectorri,/vectorrj)( 2 )
explicitly contains the electron-electron Coulomb potential, a
two-body interaction. The many-body effects of this interac-tion, as well as corrections arising from the kinetic energyoperator Tacting on the many-body wave function /Psi1,a r e
contained in a one-body energy functional known as the xcfunctional of DFT, viz., E
xc([n]). Kohn and Sham modeled
the xc functional using results for the uniform electron liquid(UEL) that we consider here. The Hartree energy of theUEL is zero, and the exchange energy component E
xof the
Hartree-Fock energy is known explicitly. In fact, it is givenat arbitrary temperatures in parametrized form in Ref. [ 11].
The correlation energy E
cis usually defined as including all
contributions to the total energy ETbeyond the Hartree-Fock
energy EHF(cf. Eq. 6.107 of Ref. [ 10]):
Ec=ET−EHF. (3)
The grouping of Exand Ectogether is needed (especially for
free-electron systems like metals and plasmas at finite T)t o
accommodate important cancellations between the two terms[11]. At finite Twe replace the internal energies Ein Eq. ( 3)
by their Helmholtz free energies F. Furthermore, F
x,Fcoccur
as the sum in direct evaluations of xc energies from the PDF[12]. Thus, F
x,Fcare grouped together in the xc functional
Fxcwhose functional derivative with respect to the one-body
density gives the Kohn-Sham one-body xc potential.
A. Kohn-Sham theory, one-body and two-body densities
The Hohenberg-Kohn theory posits that the exact ground-
state one-body density n(/vectorr) is precisely the one which min-
imizes the ground-state energy within a constrained searchscheme. Its extension to finite T[13] states that the Helmholtz
free energy Fof the system is a functional of the one-body
density, and that Fis a minimum for the true density. The
finite- Ttheory is considered to be more robust than the T=0
theory, e.g., when magnetic fields are included [ 14,15].
It was recognized prior to DFT that the ground-state energy
can be expressed entirely via the two-body density n(/vectorr
1,/vectorr2),
but a reduction to a one-body functional was not suspected.The two-body density matrix is obtained by integrating all buttwo of the space and spin variables of the N-body density,
i.e.,|/Psi1({/vectorr
i},{σi})|2. This is also known as the two-particle
reduced density matrix (2-RDM), and identifies with the PDFitself (depending on the prefactors used). The pair-distributionfunction g(/vectorr
1,/vectorr2) reduces to g(r) for a uniform system, and
gives the probability of finding a second particle at the radialdistance r, with the first particle at the origin.
155143-2SIMPLIFICATION OF THE ELECTRON-ION MANY-BODY … PHYSICAL REVIEW B 100, 155143 (2019)
The one-body density in a system where the origin of
coordinates is attached to one of the particles automaticallybecomes a two-body density in the laboratory frame, andhence the PDF of homogeneous systems, e.g., a uniformCoulomb fluid, can be used to display the inherent particlecorrelations in a uniform fluid:
n(/vectorr
1=0,/vectorr2=/vectorr)=¯ng12(r).
While placing the origin on a classical particle is possible, so
identifying a specific electron in the quantum problem is notpossible.
B.N-representability condition on the two-body density matrix
Mayer proposed [ 16] to compute the ground-state energy
ofN-electron systems variationally as a functional of the
two-electron RDM, i.e., the PDF, instead of the many-bodywave function. Both /Psi1and the 2-RDM are unknown, but,
unlike the wave function, the 2-RDM has the advantage thatits application scales polynomially with the number Nof
electrons. However, the two-electron RDM must be a reduc-tion of an N-body wave function for it to be a physically
acceptable 2-density. Otherwise, the ground-state energy forN>2 can even fall below the true ground-state energy during
a variational calculation. So, the two-electron RDM must beconstrained to represent an N-electron wave function. Cole-
man called these constraints N-representability conditions [ 6].
The Hohenberg-Kohn minimization must be constrained tosatisfy the requirements of N-representability [ 17,18]. We
do not present the mathematical constraints enumerated bymathematicians here as we will not use them. Instead, wepropose to directly link our CHNC methods to an underlyingN-representable density.
The N-electron problem when treated in the grand canon-
ical formalism uses a chemical potential μand an explicit
number of electrons is used only in the canonical ensemble.The passage to a canonical ensemble requires an “inversion”of thermodynamic functions from the μrepresentation (see
Ref. [ 11]). Such issues are irrelevant to the N-representability
problem as we can choose to work entirely in the canonicalensemble.
C. Kinetic energy functionals
The Hohenberg-Kohn method works directly with the den-
sity and does not use a Kohn-Sham equation. The imple-mentation of DFT used in Kohn-Sham theory [ 9] maps the
interacting electrons to a set of noninteracting electrons attheinteracting density , and calculates the Kohn-Sham one-
electron wave functions φ
j(r) using some approximation,
e.g., a local-density approximation (LDA) to the exchange-correlation potential based on the uniform electron fluid.Hence, the corresponding many-body wave function is asingle Slater determinant at T=0, and the Kohn-Sham theory
gives rise to the N-representable density n(r) given by
n(r)=/summationdisplay
j|φj(r)|2f(/epsilon1j). (4)AtT=0, the Fermi occupation factors f(/epsilon1j) are unity or zero
for occupied and unoccupied states. Hence, the summation atT=0 is over occupied states.
At finite Tthe many-body wave function /Psi1can be written
as a sum over a set of Slater determinants as done in themethod of configuration interactions (CI). Thus, if there are n
electrons, we need N/greatermuchnorthonormal functions, e.g., Kohn-
Sham (KS) functions φ
jsuch that the corresponding Fermi
occupation factor of the highest-energy state used is deemedto be negligible. This is the statistical average over manyconfigurations, and the actual occupancies of the one-bodystates in each electronic configuration are unity or zero. Wecan construct N!/{n!(N−n)!}determinants out of the one-
body states. All these determinants contain (or omit) orbitalsthat have unit (or zero) occupations. However, the squaresof the coefficients of these determinants, and the occurrenceof the orbitals in the determinants, give rise to the fractionalFermi factors contributing to the density in Eq. ( 4).
Alternatively, the Fermi factors are simply given as
the statistical weights of the diagonal elements of the N-
representable 2-RDM constructed from /Psi1in the basis of φ
j
one-body functions. Hence, the finite-temperature case can
also be restated as a discussion in terms of properties of Slaterdeterminants, as is the case for T=0. In practical calculations
at finite T, the required basis sets become rapidly prohibitive
asTincreases. Thus, plane-wave basis sets cut of at 500–
1000 eV are needed, even with ultrasoft pseudopotentials, intypical applications of DFT for warm dense matter [ 19]. CI
calculations using /Psi1become impossible in such cases.
The classical map simplifies the DFT problem further
and works with a classical electron system at a classicalfluid temperature T
cf. The latter is constructed to include the
physical temperature Tas well as a kinetic energy quantum
correction brought in via a quantum temperature Tqto be
discussed below.
The KS φj(r),/epsilon1jhave the physical meaning of being the
eigenstates and eigenenergies of the fictitious noninteractingelectron map of the interacting electron system, rather thanthose of the original interacting electron system. The Kohn-Sham procedure guarantees the N-representability of the den-
sity by treating the kinetic energy operator explicitly, withoutusing a kinetic energy (K.E.) functional as in Hohenberg-Kohn DFT.
The simplest K.E. functional is used in Thomas-Fermi
theory. Extensions of Thomas-Fermi theory under the name of“orbital-free” DFT, as well as practical applications, continueto be relevant [ 10,20–24]. Many formulations use the von
Weizsäcker ansatz where just one orbital, viz., φ(r)=√n(r),
is used in a Schrödinger-type equation to obtain the kineticenergy. However, the nonlocal nature of the K.E. operatorcontinues to be a great stumbling block. The excellent reviewby Carter [ 20], though littered with many acronyms, shows
the highly heuristic nature of the search for a K.E. functionalthat has continued for some four score years.
Several exact requirements on the K.E. functional (such
as positivity) and their violation in various implementationshave been noted [ 25,26]. However, whether “orbital-free” for-
mulations lead to N-representable densities, or non-negative
electron-electron pair-distribution functions, etc., do not seemto have been studied. In any case, it is known that calculations
155143-3M. W. C. DHARMA-W ARDANA PHYSICAL REVIEW B 100, 155143 (2019)
using K. E. functionals are far less accurate than KS calcula-
tions. Furthermore, energies from such calculations may fallbelow the exact energies, as the approximate K.E. functionals
may not satisfy N-representability constraints. In fact, even
some Kohn-Sham calculations that use generalized gradientapproximations show such anomalies [ 27].
D. Neutral pseudoatom model
A kinetic energy functional is unnecessary for simple
“one-center” calculations which are very rapid, and typical inatomic physics or with the neutral-pseudoatom (NPA) model,originally proposed for solids [ 2], and adapted to finite- T
metallic fluids and plasmas [ 4,28–31]. The NPA has been
formulated in a number of different ways [ 32–35]. Here,
we follow the model of Ref. [ 29] which is a simplification
of [4] and adapted to multicomponent finite- Tcalculations.
In these NPA models of electron-ion systems, the many-ionproblem is replaced by a one-ion problem together with thecorresponding ion-ion correlation functional, while the many-electron problem is replaced by a single-electron KS problem.
However, in simulations done with codes like the
V ASP [36]
orABINIT [37] the many-ion problem is notreduced. Instead,
they explicitly use some 100–200 nuclear centers, say NI, and
even up to N=Ne∼1000 electrons in thousands of steps of
KS and molecular-dynamics (MD) calculations. Hence, suchmethods are extremely expensive and become prohibitivefor many problems in warm dense matter, materials science,and biophysics. However, they provide useful benchmarks insimplified limits. Such N
I-ion quantum calculations can be
greatly simplified as follows:
(1) By the use of an explicit electron kinetic energy func-
tional of the one-body electron density n(/vectorr) if an adequate
K.E. functional were available.
(2) Using a neutral pseudoatom approach where the NI
nuclei are replaced by a one-body ion density ρ(/vectorr)[4,28],
while the electrons are treated as usual as a functional of n(/vectorr)
from KS theory. Since ions are normally classical particles,an ion is chosen as the origin of coordinates with no loss ofgenerality. Two coupled KS equations for the two subsystems(e-i) arise on functional differentiation of F:
δF([n],[ρ])
δ[n]=μe,δF([n],[ρ])
δ[ρ]=μI. (5)
The electron and ion chemical potentials appear on the right-
hand side. The first equation reduces to a one-center Kohn-Sham equation for the electrons in the field of the ion atthe origin, while the second equation defines a classicaldistribution around the origin containing an ion-correlationfunctional, and reduces to a hypernetted-chain (HNC) typeintegral equation [ 4,28]. If there are many types of ions, a
coupled set of one-center HNC equations appear [ 29].
This reduction of the electron-ion problem does not invokethe Born-Oppenheimer (BO) approximation, but BO can beimplemented by neglecting F
ei
xc[n,ρ]. The solution of such
one-center equations is numerically extremely rapid, evenat finite T. Such calculations reproduce the PDFs g
cc(r)o f ,
say, molten carbon (or silicon) containing a complex bond-ing structure that are only exposed by taking “snapshots”in lengthy and expensive DFT-MD simulations. That is, theone-center NPA calculations include sufficiently good ion-ion
classical correlation functionals such that they are able, e.g.,to reproduce the peak in the g
cc(r) that corresponds to the
1.4–1.5 Å C-C covalent bond as well as the peaks in the g(r)
due to the hard-sphere-like packing effects seen in DFT-MDsimulations. This is demonstrated in Refs. [ 30,31].
(3) The NPA approach can also be further simplified by
the use of a K.E. functional; but, the NPA calculation is sorapid that little is gained on using approximate K.E. functionalwith their own errors.
Several models use the the name “neutral pseudoatom,”
but there are significant differences. Thus, Chihara uses aneutral-pseudoatom construction where he begins from theHNC equation and identifies a “quantum” Ornstein-Zernike(OZ) equation applicable to electrons as well [ 35]. Its validity
for quantum electrons is debatable. Thus, Anta and Louis [ 33]
in their implementation of an NPA using Chihara’s “quantalHNC (qHNC)” scheme cautiously avoid the use an eeqHNC
equation. The NPA approach proposed by the present authorand Perrot [ 4] simply uses DFT for both electrons and ions,
and invokes the HNC diagrams, bridge diagrams, and theOrnstein-Zernike equation only to construct an ion-ion cor-relation functional [ 4,29].
A simplification of the effort to construct a kinetic energy
functional is to look for a classical description of the electrons.This is possible when the bound states have already beentreated using some complementary approach like the NPA, orwhen there are no bound states because the system is highlycompressed or at a temperature where such effects can beneglected. If the system is a fluid or plasma, classical integralequations or classical molecular dynamics can be used toobtain the PDFs of the classical map electrons that are notplagued by fermion sign problems.
E. Classical map hypernetted-chain scheme
The study of the electron distribution in a uniform electron
liquid (UEL) when a “test electron” is placed at the originleads to the question of the direct calculation of the physicallyvalid g
ee(r) of the UEL rather than for a “test particle.” Here,
the electron kinetic energy functional must satisfy the requiredconstraints, and also avoid any selection of an “electron” held
at the origin whereby it is made into a specific test particle.Such a problem does not arise for classical electrons [ 5].
We recapitulate the classical map hypernetted-chain
(CHNC) scheme for the convenience of the reader. It hasbeen used successfully [ 5,38–41] for a number of uniform
systems, namely, 3D and 2D UELs, electron-proton plasmas[42], warm dense matter [ 43], double quantum wells [ 44]),
etc. We present arguments to show that the pair densitiesobtained via the classical map technique are N-representable.
Consider an N-electron system in a volume Vsuch that
N/V=¯n, forming a uniform electron liquid in the presence
of a neutralizing positive uniform background. The electroneigenfunctions for the self-consistent field problem (Hartreeas well as Hartree-Fock models) are simple plane waves:
φ
j(r)=φ/vectorkσ(/vectorr)=(¯n/N)1/2exp( i/vectork·/vectorr)ζσ. (6)
Here, jis an index carrying any relevant quantum numbers
including the spin index σassociated with the spin function
155143-4SIMPLIFICATION OF THE ELECTRON-ION MANY-BODY … PHYSICAL REVIEW B 100, 155143 (2019)
ζ, with σ=1,2 or “up, down,” specifies the two possible
spin states. Some of the vector notation will be suppressed forsimplicity, as appropriate for uniform liquids with sphericalsymmetry in 3D and planar symmetry in 2D. The spin indexmay also be suppressed where convenient.
F. Noninteracting pair-distribution function g0(r)
The many-electron wave function for noninteracting elec-
trons in a uniform system, as well as for Hartree-Fock (mean-field) electrons, is a normalized antisymmetric product ofplane waves [ 45], i.e., a Slater determinant D(φ
1,...,φ j)o f
N-plane waves. Its square is the N-particle density matrix,
while the PDF is the two-particle density matrix [ 46]. In the
following, we assume Hartree atomic units with |e|=¯h=
me=1, where standard symbols are used:
gσ1σ2(/vectorr1,/vectorr2)=V2/Sigma1σ3...σ N/integraldisplay
d/vectorr3.../vectorrND(φ1,...,φ j).(7)
If the spins are antiparallel, then the noninteracting PDF,
g0
u,d(r), is unity for all /vectorr. Denoting ( /vectorr1−/vectorr2)b y/vectorr, and ( /vectork1−
/vectork2)b y/vectork, we have, for parallel spins,
g0
σ,σ(/vectorr)=2
N2/Sigma1/vectork1,/vectork2f(k1)f(k2)[1−exp( i/vectork·/vectorr)], (8)
f(k)=[1+exp{(k2/2−μ0)/T}]−1. (9)
Here, we have generalized the result to finite T, where the
temperature is measured in energy units. Thus, the noninter-acting PDFs, i.e., g
0(r), are explicitly available at T=0, and
numerically at finite T:
g0
σ,σ(r)=1−F2(r), (10)
F(r)=/parenleftbig
6π2/k3
F/parenrightbig/integraldisplay
f(k)sin(kr)
rkd k
2π2, (11)
3D, zero T,=3sin(x)−xcos(x)
x3,x=kFr. (12)
The equations contain the Fermi momentum kFwhich is
defined in terms of the mean density ¯ nand the correspond-
ing electron Wigner-Seitz radius rs. Here, we have assumed
equal amounts of up and down spins (paramagnetic case) anddefined the Fermi wave vector k
F:
kF=1/(αrs),rs=[3/(4π¯n)]1/3,α=(4/9π)1/3.(13)
Similar expressions can be developed for the 2D electron layer
[38], two coupled 2D layers [ 44], or a two-valley 2D layer
[47] relevant to silicon-metal oxide field effect transistors.
The method has also been used successfully to obtain thelocal-field factors of 2D layers at zero and finite T[48], and
for the study of thick 2D layers which are of technologicalinterest [ 49].
III.N-REPRESENTABILITY OF PAIR DENSITIES FROM
THE CLASSICAL MAP
We first discuss the noninteracting pair density and
then use its manifest N-representability to establish the N-
representability of the interacting map.Noninteracting electron gas
The PDFs g0
σ,σ/prime(r)=1−δσ,σ/primeF(r) calculated in the pre-
vious section were derived from the Slater determinantD(φ
1,...,φ N) and hence manifestly N-representable. At this
stage, irrespective of where it came from, we regard g0(r)
as a classical pair-distribution function for classical electronsinteracting by a classical pair potential βP(r) where βis the
inverse temperature. This is the first step in our classical map,and we may now identify one of the classical particles as beingat the origin, without loss of generality, in a classical pictureof the PDF. Clearly, for antiparallel spins, i.e., σ/negationslash=σ
/prime,t h e
pair potential βP(r) is zero, while it is finite and creates the
well-known “exclusion hole” in the PDF of two parallel-spinparticles. Hence, P(r) has been called the “Pauli exclusion
potential” and should not be confused with the Pauli kineticpotential that appears in the theory of the kinetic energyfunctional.
Lado was the first to present an extraction of βP(r)f o r
3D electrons at T=0 using the hypernetted-chain (HNC)
equation and the Ornstein-Zernike (OZ) equation [ 50]. Only
the dimensionless potential βP(r) is determined from the
equations. Although the physical temperature Tof the quan-
tum fluid is zero, the temperature of the classical fluid in-voked by the map is left undetermined (but nonzero) in the“noninteracting” system. The Pauli exclusion potential for 2Delectrons at arbitrary Twas derived in Ref. [ 38]. Although the
quantum electrons are not interacting via a Coulomb potential,βP(r) becomes a classical manifestation of entanglement in-
teractions which scale as r/r
s, and hence extend to arbitrarily
large distances [ 51]. Assuming that g0(r) can be written as an
HNC equation, we have
g0(r)=exp[−βPr+h0(r)−c0(r)], (14)
h0(r)=c0(r)+¯n/integraldisplay
d/vectorr/primeh0(|/vectorr−/vectorr/prime|)c0(/vectorr/prime), (15)
h0(r)=g0(r)−1. (16)
The first of these is the HNC equation, while the second
equation is the Ornstein-Zernike relation. These contain thedirect correlation function c
0(r) and the total correlation
function h0(r). It should be noted that we have ignored the
two-component character of the electron fluid (two spin types)in the equations for simplicity, but the full expressions aregiven in, say, Ref. [ 5]. These equations can be solved by taking
their Fourier transforms, and the Pauli exclusion potential canbe obtained by the inversion of the HNC equation. The “Pauliexclusion potential” (PEP) βP(r) is given by
βP(r)=− log[g
0(r)]+h0(r)−c0(r). (17)
The PEP is a universal function of rkForr/rs. It is long
ranged and mimics the exclusion effects of Fermi statistics.At finite Tits range is about a thermal de Broglie wavelength
and is increasingly hard-sphere-like as r→0. The Fourier
transform βP(q) in 3D behaves as ∼1/qfor small q, and as
∼c
1/q2+c2/q4for large q. Plots of βP(r) and g0(r)f o ra3 D
UEL are given in Fig. 1.
We note that the HNC or MHNC integral equation, to-
gether with the OZ equation, may be regarded as a transforma-tion where, given the dimensionless pair potential βφ
ij(r), the
155143-5M. W. C. DHARMA-W ARDANA PHYSICAL REVIEW B 100, 155143 (2019)
FIG. 1. The exclusion potential ( 17) and the noninteracting PDF
g0
σ,σatt=T/EF=0 (solid line) and at t=2 (dashed line). They
are universal functions of r/rs.T h eP D F g0
σ/negationslash=σ/prime(r)=1 as there is no
exclusion effect for σ/negationslash=σ/prime.
corresponding PDF, i.e., gij(r) is generated. Similarly, given
thegij(r), HNC inversion is the process which extracts the
corresponding βφ ij(r). The value of g(r)f o r the full range of
ror additional constraints are needed to obtain an unequivocal
HNC inversion to extract a valid pair potential from a PDF[52,53].
IV . INTERACTING SYSTEM AND ITS CLASSICAL MAP
In the previous section, we reviewed a classical fluid whose
g0(r) exactly recovers the PDFs of the noninteracting quantum
UEL at any density, spin polarization, or temperature. Fromnow on, for simplicity we consider a paramagnetic electronliquid (equal amounts of up spins and down spins) althoughspin-dependent quantities will be indicated where needed forclarity. Although the quantum liquid was “noninteracting,”the classical map already contains the pair potential βU
ij=
βP(r).
On addition of a Coulomb interaction βVij(r) the total pair
potential becomes
βUij(r)=βP(r)+βVcou(r). (18)
The temperature T=1/βoccurring in Eq. ( 18)i sa sy e t
unspecified. In quantum systems the Coulomb interaction isgiven by the operator 1/|/vectorr
1−/vectorr2|which acts on the eigenstates
of the interacting pair. It can be shown (e.g., by solving the rel-evant quantum scattering equation) that the classical Coulombinteraction V
cou(r),r=|/vectorr1−/vectorr2|acquires a diffraction correc-
tion for close approach. Depending on the temperature T,a n
electron is localized to within a thermal de Broglie wave-length. Thus, following earlier work on diffraction-correctedpotentials (e.g., in Compton scattering in high-energy physics)
or in plasma physics as in, e.g., Minoo et al. [54], we use a
“diffraction-corrected” potential
V
cou(r)=(1/r)[1−e−r/λth];λth=(2πmT cf)−1/2.(19)
Here, mis the reduced mass of the interacting electron pair,
i.e.,m∗(rs)/2 a.u., where m∗(rs) is the electron effective mass.
It is weakly rsdependent, e.g., ∼0.96 for rs=1. In this
work we take m∗=1. The “diffraction correction” ensures the
correct quantum behavior of the interacting g12(r→0) for all
rs. The essential features of the classical map are as follows:
(1) The use of the exact noninteracting quantum PDFs
g0
σ,σ/prime(r) as inputs.
(2) A diffraction-corrected Coulomb interaction.(3) The specification of the temperature of the classical
Coulomb fluid T
cf(rs)=1/βas the one that recovers the
quantum correlation energy Ec(rs).
The selection of Tcfis a crucial step. This is guided by
the Hohenberg-Kohn-Mermin property that the exact mini-mum free energy is determined by the true one-body electrondensity n(r). Since we are dealing with a uniform system, the
Hartree energy E
His zero. The exchange energy Exis already
correctly accounted for by the construction of the classicalmap g
0(r) to be identical with the quantum g0(r)a ta n y Tor
spin polarization. Hence, the only energy to account for is Ec.
So, we choose to select the temperature Tcfof the classical
Coulomb fluid to recover the known DFT correlation energyE
cat each rsatT=0. Since this is most accurately known
for the spin-polarized electron liquid, Tcfis best determined
from Ec(rs) for full spin polarization. A trial temperature is
selected and the interacting g(r,λ) is determined for various
values of the coupling constant λin the interaction λVc(r)t o
calculate a trial Ecat the given rsfrom the coupling constant
integration. The temperature is adjusted until the Ec(rs,Tcf)
obtained from the classical fluid g(r) agrees with the known
quantum Ec(rs,T=0). Given an electron fluid at T=0, the
temperature of the classical fluid with the same Ecis called its
quantum temperature T q. This was parametrized as
Tq/EF=1.0/(a+b√rs+crs). (20)
For the range rs=1t o1 0 , Tq/EFgoes from 0.768 to 1.198.
The values of the parameters a,b,care given in Ref. [ 5].
There is no ap r i o r i reason that the n(r), i.e., ¯ ng(r),
obtained by this procedure would agree with the quantum
¯ng(r), except for the Hohenberg-Kohn theorem that requires
n(r) to be the true density distribution when the energy
inclusive of the xc energy is correctly recovered. Many well-known and often very useful quantum procedures (e.g., thatof Singwi et al. [55,56]) for the PDFs lead to negative g(r)
asr
sis increased beyond unity even into the “liquid metal” rs
range.
However, as shown in Refs. [ 5,38,57], etc., the classical
map HNC g(r) is an accurate approximation to the QMC
PDFs then available only at T=0. Correlations are stronger
in reduced dimensions. The classical map for the 2D UELwas constructed using the modified-HNC (MHNC) equationwhere a hard-sphere bridge function was used, with the hard-sphere radius determined by the Gibbs-Bogoliubov criterion,as given by Lado, Foils, and Ashrcoft (LFA) [ 58]. Other
workers [ 39–41] have examined different parametrizations
155143-6SIMPLIFICATION OF THE ELECTRON-ION MANY-BODY … PHYSICAL REVIEW B 100, 155143 (2019)
than our Eq. ( 20). Datta and Dufty [ 59] examined the classical
map approach and the method of quantum statistical potentials[60,61] within a grand-canonical formalism. They proposed
using additional conditions (aside from the requirement thatE
cis reproduced by Tcf) to constrain the classical map for
warm dense electrons, a topic recently reviewed by Dornheimet al. [62].
Although E
cvalues at T=0 were available when the
classical map for the UEL was constructed, no reliable xcfunctional (beyond RPA) was available for the finite- Telec-
tron liquid. Hence, we proposed the use of the Tansatz:
T
cf=/parenleftbig
T2
q+T2/parenrightbig1/2(21)
as a suitable map for the finite- TUEL. This was based on
the behavior of the heat capacity and other thermodynamicproperties of the UEL. Furthermore, using Eq. ( 21) it became
possible to predict the xc-free energy F
xc(rs,T)a sw e l la s
the finite- TPDFs of the UEL at arbitrary temperatures and
spin polarizations. These were found to agree closely with theF
xc(rs,T) and PDFs resulting from the restricted path-integral
Monte Carlo (RPIMC) simulations reported 13 years later byBrown et al. [63]. The Brown et al. data have been used
by Liu and Wu [ 41] to construct a direct fit of a T
cfthat
avoids the model used in Eq. ( 21), by using temperature-
dependent parameters a,b, and cin Eq. ( 20). The RPIMC
data have been parametrized by Karasiev et al. [64]. However,
Groth et al. [65] presented a new ab initio parametrization of
Fxc(rs,T) using accurate data from recently developed finite-
Tfermionic PIMC methods that deal with the sign problem
more carefully and also compensate more systematically forfinite-size effects [ 62]. These agree even more closely with the
CHNC data.
Calculations of F
xcusing the finite- Tparametrization given
by Perrot and Dharma-wardana [ 57] are compared with the
Karasiev et al. parametrized results, and those of Groth et al.
in Fig. 2. The classical temperature ansatz of Eq. ( 21) recovers
the highly accurate Groth et al. results to within 94%, i.e., with
an error of at most 6%. The parametrizations given by PDW[57], Iyatomi and Ichimaru, and subsequent parametrizations
incorporate the high- TDebye-Hückel limit of F
c, the high-
Tbehavior of Fx(T), as well as the behavior at the T=0
limit. The PDW fit to the CHNC data fall below the Grothet al. data near T=0 partly because older T=0 data were
used in the CHNC parametrizations. The CHNC method hasalso been used to construct F
xc(T) for 2D electron layers and
used to calculate finite- Tlocal-field factors, PDFs, and related
quantities relevant to double quantum wells, metal-oxide fieldeffect transistors, and nanostructures; but, no finite- TQMC or
PIMC benchmarks are currently available for the 2D electronsystem.
N-representability of the interacting g(r) of the classical map
The conditions n(r)=¯ng(r)>0, and/integraltext
n(/vectorr)d/vectorr=Nare
always satisfied by the classical map. Furthermore, the clas-sical map becomes more accurate as t=T/E
Fis increased,
or when rsis increased, since quantum electrons become
increasingly classical in those limits.
We present two types of arguments to conclude that the
g(r) of the interacting UEL obtained by the classical map is0 0.5 1 1.5 2
T/EF0.60.811.21.4Fxc(T) / Ex(0)
rs=1
rs=3
rs=10
GDS fit to QMC 2017
---- KSD fit to RPIMC 2014Lines with symbols, CHNC 2000
FIG. 2. Finite- Texchange and correlation free energy fxc(rs,T)
scaled by the exchange energy EX=FXatT=0 as a function of
the reduced temperature t=T/EFin units of the Fermi energy is
displayed. The lines with symbols are results from CHNC calcula-
tions [ 57]. The restricted path-integral Monte Carlo (RPIMC) data
of Brown et al. [63], as parametrized by Karasiev et al. [64], are
shown as dashed lines. The thick continuous lines are from the Groth
et al. parametrization of very accurate fermionic PIMC data. The
temperature range t<1 is relevant to WDM studies.
N-representable. One of them is a formal argument based on
CHNC being a“well-behaved” transformation of the alreadyN-representable noninteracting density. The second is a prac-
tical demonstration of the implementability of CHNC methodand the close agreement with results from QMC, PIMC, andother more microscopic benchmark calculations. Finally, wegive an example of a CHNC calculation for electron-holelayers at finite temperatures, as an example of topical techno-logical interest for which QMC, PIMC, and even DFT seemto be quite prohibitive at present.
(1)Argument based on the HNC equation being an N-
representability conserving transformation. Once the g
0(r)
of the quantum fluid is evaluated, we consider a classicalfluid which has the same g
0(r). The noninteracting g0(r) and
the corresponding n0(r)=¯ng0(r) of the classical fluid are
generated from the homogeneous density ¯ nby a transfor-
mation where the origin of coordinates is moved to one ofthe particles. The corresponding transformation of the densityprofile is written as
n
0(r)=T0(r)¯n, (22)
T0(r)=exp[βP(r)+h0(r)+c0(r)]. (23)
155143-7M. W. C. DHARMA-W ARDANA PHYSICAL REVIEW B 100, 155143 (2019)
The so generated n0(r)i sN-representable by its construction
from a Slater determinant. Then, in a next step the interactingg(r) is generated from the N-representable noninteracting
g
0(r) by a transformation which can be written as
g(r)=T1(r)g0(r), (24)
T1(r)=e[βVcou(r)+{h(r)−h0(r)}+{c(r)−c0(r)}]
=exp[β{Vcou(r)+VMF(r)+Vxc(r)}]. (25)
In effect, the uniform density ¯ nhas been transformed (by a
selection of the origin of coordinates, and by switching on theCoulomb interaction) by a single composite transformationT=T
1T0with its components acting one after the other.
In Eq. ( 25)w eu s e Vxc(r) to indicate the exchange-
correlation correction to the mean-field potential VMF(r)a s
discussed in Ref. [ 4] where explicit expressions for these
classical xc potentials in the HNC approximation are given.These potentials are expected to be well-behaved functions.The diffraction-corrected classical Coulomb potential V
cou(r)
has a finite value at r=0, and not singular, unlike the point-
Coulomb potential 1 /rwhich is not used in CHNC. Hence, we
may regard the above transformation as being mathematicallyequivalent to a type of smooth, or “well-behaved” coordinatetransformation of /vectorrto another variable /vectors:
d/vectors=T(r)¯nd/vectorr=n(r)d/vectorr. (26)
That is, the initial plane-wave states (¯ n/N)
1/2exp( i/vectork·/vectorr)a r e
transformed to a new set (n(/vectorr)/N)1/2exp[ i/vectorq·/vectors(r) ] .I ti se a s i l y
shown that they form a mutually orthogonal complete set.For instance, consider the initial plane-wave state used inthe Slater determinant, i.e., φ
j(/vectorr)=φk(/vectorr), and consider its
transformed state ˜φk(/vectorr) given below:
φk(/vectorr)=(¯n/N)1/2exp( i/vectork·/vectorr), (27)
˜φk(/vectorr)=(n(/vectorr)/N)1/2exp[ i/vectork·/vectors(/vectorr)]. (28)
We regard /vectorkas an arbitrary kvector and hence it is sufficient to
transform /vectorr, while the theory can also be constructed entirely
inkspace in an analogous manner. The transformed wave
functions ˜φk(/vectorr) have the following properties:
/integraldisplay
˜φ∗
k/prime(/vectorr)˜φk(/vectorr)d/vectorr=/integraldisplayn(/vectorr)
Nei(/vectork/prime−/vectork)d/vectorr (29)
=1
N/integraldisplay
exp{i(/vectork/prime−/vectork)}d/vectors
N(30)
=(2π)3
Nδ3(/vectork/prime−/vectork). (31)
Furthermore,
/integraldisplay
˜φ∗
k(/vectorr)˜φk(/vectorr/prime)d/vectork
(2π)3=δ3(/vectorr−/vectorr/prime)
N. (32)
Hence, the transformed functions ˜φk(/vectorr) form a complete
orthogonal set. This implies that the initial Slater determi-nant D(φ
k1,...,φ kN) of the noninteracting electron system
transforms to the determinant D(˜φk1,..., ˜φkN) of the interact-
ing system, explicitly showing the N-representability of the
n(r)=¯ng(r) obtained via the classical map which consists ofthe application of the two transformations T1T0. Furthermore,
the transformations commute, in the sense that one may firstapply only the diffraction-corrected Coulomb potential to
noninteracting fermions to generate a g
c(r) for a Coulomb
fluid, and then apply the Pauli exclusion potential to generatethe fully interacting classical map inclusive of exchange-correlation effects, or vice versa . This is equivalent to iterating
the HNC equations from the noninteracting state via twodifferent paths, and indeed the two different procedures T
1T0
andT0T1lead to the same final g(r).
In the above demonstration we have appealed to the con-
cept of “well behavedness” of the potentials. These have beendefined by Lieb to be based on a Hilbert space of potentialsV=L
3/2+L∞[66]. These are potentials that do not become
singular or include infinite barriers, discontinuities, etc. Thepotentials used in CHNC satisfy such concepts of “wellbehavedness.”
What if a phase transition, e.g., a Wigner crystallization,
intervenes in passing from the noninteracting to the inter-acting system? The CHNC procedure for a uniform systemwill smoothly proceed to the best interacting fluid state (anupper bound to the ground-state energy) and not the lower-energy Wigner crystal state. This is still consistent with thevariational principle and N-representability.
The above analysis confirms the N-representability of the
pair densities of the interacting uniform electron liquid gen-erated by the classical map presented here. A similar analysiscan be given if MD were used to generate the interacting g(r)
via CHNC potentials, instead of using the HNC equation.
(2)Argument based on the N-representability of the QMC
density. The diffusion quantum Monte Carlo (DQMC) cal-
culations use a Slater determinant together with Jastrowfactors, and hence the DQMC procedure is based on anexplicit many-body wave function whose variation producesa minimum energy and a corresponding E
c(rs). Hence, its
two-particle reduced density matrix, i.e., the electron-electronPDF is N-representable; the correlation energy E
cassociated
with the N-representable two-body density is the correlation
contribution to the best approximation to the energy mini-mum as per Hohenberg-Kohn theorem since the minimumis achieved only for the true density. The CHNC electron-electron PDFs agrees with the DQMC- g(r) with no attempt
at fitting the PDFs. The only input is the single number E
c(rs)
at each density introduced via the classical temperature Tcf,
a classical kinetic energy. The CHNC electron density ¯ ng(r)
agrees closely at every rwith those of the N-representable
density ¯ ngQMC(r). Furthermore, the CHNC energies Exc(rs,ζ)
at arbitrary spin polarizations ζthat were not included in any
fits agree with microscopic calculations. At finite T, an ansatz
is used for Tcfand yet the agreement is good to within 94%.
Hence, we conclude that the CHNC n(r)i sa s N-representable
as the DQMC procedure.
The classical pair potential U(r)=P(r)+Vcou(r) may be
used in a classical molecular dynamics simulation to generatethe interacting g(r) instead of using the HNC equations. Such
a procedure can reveal crystal ground states and go beyondthe liquid model inherent in the usual HNC equations. It isalso possible to generate the dynamics of fluid states, e.g.,determine S(k,ω) by a classical simulation. However, the
Pauli exclusion interaction is really a kinematic quantum
155143-8SIMPLIFICATION OF THE ELECTRON-ION MANY-BODY … PHYSICAL REVIEW B 100, 155143 (2019)
effect and not a true “interaction.” It is not known at present
whether such a classical map S(k,ω) agrees in detail with the
quantum S(k,ω) for an interacting electron fluid.
V . CHNC METHOD FOR SYSTEMS OF INTERACTING
ELECTRONS AND IONS
In this section the UEL model is extended to two interact-
ing subsystems, namely, electrons and ions, or possibly forelectrons and holes. The application of the CHNC method tocoupled electron-ion system will be illustrated by calculationsof hydrogen plasmas where the CHNC results are comparedwith NPA calculations, as well as recent QMC, PIMC, DFT-MD, and other N-center simulation methods. The CHNC,
NPA, and quantum-simulation methods are in good agree-ment. This agreement is the basis of our second argument fortheN-representability of the pair densities obtained from the
CHNC method.
Consider the two coupled density functional equations of
the NPA, viz., Eqs. ( 5). As shown in the Appendix to Ref. [ 4]
these equations for n(r),ρ(r), when applied to classical par-
ticles, reduce to classical Kohn-Sham equations which areBoltzmann-type distributions. In the classical case they canbe reduced to two coupled HNC-like equations for the elec-trons and ions. The HNC equation (with or without a Bridgeterm) for the electrons when replaced by their classical mapgives the CHNC equation which now includes an electron-ioncontribution to the potential of mean force. Similarly, theHNC-like equation for the ions will contain contributionsfrom the electrons. That is, the electron screening of the ions,or ion screening of the electrons, is controlled by the CHNCequations, which only needs the basic pair interactions. Ifthe particles are in the quantum regime, a Pauli exclusionpotential is needed, be it for electrons or for protons (or holesin semiconductor applications).
As an example, we take a system of ions of mean charge ¯Z,
mass M, with a mean density ¯ ρinteracting with a system of
electrons at a mean density ¯ n=¯Z¯ρ. The electron mass is unity
(atomic units). For simplicity, we assume that there is justone kind of ion, and that ¯Z=1 as for a hydrogenic system.
We denote the ion species by p =H
+. The coupled CHNC
equations are given in Eq. ( 33) and in Refs. [ 42,43]. They are
discussed below.
The densities ¯ ρand ¯ nare equal since the ion charge
¯Z=1. Consider a fluid of total density nto t, with three species,
electrons of two types of spin, ns=¯n/2f o r s=1,2f o rt h e
two spin species, and s=3,n3=¯ρfor the ions, denoted here
as “p.” The physical temperature is T, while the classical-
fluid temperature of the electrons Tcf=Tee=1/βee, with
1/βee=√(T2+T2
q). For the ion p =H+, no quantum cor-
rection is usually needed and Tpp=1/βppisT. Otherwise,
an ion-quantum temperature Tqpis defined as before, using
Eq. ( 20), but using the Fermi energy EFpof the ions (or holes).
If the densities are those typical of stellar densities, thenthe calculation will automatically be for quantum hydrogenions as appropriate. Normally, treating the positively chargedcounterparticles of the electrons quantum mechanically is notneeded except in semiconductor structures.
Thus, the classical map converts the system with the
physical temperature Tinto a system with two temperaturesT
ee,Tppassociated with the two different subsystems. It should
be noted that the temperature is not an observable in purequantum systems as there is no operator associated with it.But, it is a Lagrange multiplier in quantum statistical systems
and ensures the conservation of the energy of each subsystemwhen coupled to classical heat baths where it is measurable.Thus, two Lagrange multipliers are implied in the electron-ionsystem. However, the cross subsystem temperature T
epis not
easily defined as there is no uniform density or specific energyassociated with the cross interactions, and no conserved quan-tity to define a Lagrange multiplier. It can, however, be unam-biguously extracted from an NPA calculation or constructedfrom a suitable physical model, as discussed below. So, usingT
ij=1/βijandφijfor the interparticle temperatures and pair
potentials, the coupled CHNC and OZ equations for the PDFsare
g
ij(r)=e[−βijφij(r)+hij(r)−cij(r)+Bij(r)], (33)
hij(r)=cij(r)+/Sigma1sns/integraldisplay
dr/primehi,s(|r−r/prime|)cs,j(r/prime).(34)
The pair potential φij(r) between electrons is just the
diffraction-corrected Coulomb potential Vcou(r) added
to the Pauli exclusion potential which vanishes as thede Broglie radius of the particles becomes negligible inapproaching the classical regime. The interaction between twoions is also a Coulomb potential with a diffraction correctionwith a very small length scale ( ∼1/√
M) due to the mass M
of the ions.
The HNC is sufficient for the uniform three-dimensional
(3D) electron liquid for a range of rs,u pt o rs=50, as shown
previously [ 57]. But, the bridge term is important for the
ions at high compressions and low temperatures. Hence, weneglect the eebridge corrections in this study, but retain them
for ions.
The construction of β
ep,φep(r) requires attention. Only the
product βepφij(r) is unambiguously available from the theory,
and that is necessary and sufficient for CHNC calculations.However, simple electron-ion interaction models may also beused. Thus, Bredow et al. [43] examined the applicability of
a very simple electron-ion interaction and a simple modelfor the intersubsystem temperature of the two-temperatureclassical map:
φ
0
ep(r)=U(r){1−exp(−r/λep)}, (35)
U(r)=(−¯Z/r)f(r),f(r)=1 (36)
λep=(2πmT ep)−1/2,m=1 (37)
Tep=(TeeTpp)1/2. (38)
The ep de Broglie length λepmoderates the r→0 behavior
of the electron-proton interaction. The latter includes a pseu-dopotential U(r) where the form factor f(r) is set to one for
the linear response regime. For ions with a bound core ofradius r
c, the form factor can be chosen to have the Heine-
Abarankov form. In such cases, the diffraction correctionbecomes irrelevant.
Equation ( 38)f o r T
epas a geometric mean of the in-
tersubsystem temperatures is justifiable in the large- ror
155143-9M. W. C. DHARMA-W ARDANA PHYSICAL REVIEW B 100, 155143 (2019)
0 0.5 1 1.5 2 2.5
r [a.u.]00.511.522.5gij(r)gep CHNC
gee CHNC
gpp CHNC
gpp NPA η=0.475
gpp QMC Liberatore et al
rs = 0.4473 T=0.34 eV
Hydrogen
FIG. 3. Pair-distribution functions gij(r),i=e,p, using the
simple CHNC model Tep,φepfrom Eq. ( 35), from the NPA, and
from QMC (Liberatore et al. [69]), for fully ionized hydrogen. The
ion density is ¯ ρ=1.8×1025cm−3(i.e.,∼350 times the density
of solid hydrogen) with T=0.34 eV. The NPA, CHNC, and QMC
agree very well for gppwhen Bridge corrections specified by the
hard-sphere packing fraction ηare included in the NPA. See Fig. 4
for more details on gep.
small- klimit using compressibility sum-rule arguments
[67,68]. However, for small r, binary collisions dominate
and the ansatz becomes less valid. Furthermore, the electrondensity near the nucleus is large, and the effective T
eeincrease
from the bulk value. Hence, a single Tepis not strictly possible
but found to work quite well, as shown below. The ion-ionPDF is insensitive to the choice of T
ep, and hence the model
(35) proves to be very convenient.
We display the results from the simple model ( 35)i nF i g . 3
and compare them with the results from NPA calculationsas well as with highly computer-intensive QMC simulationsby Liberatore et al. [69]. Liberatore et al. assume a linear
response form for the proton-electron interaction followingthe Hammerberg-Ashcroft model of 1974 [ 70]. Such a model
is normally questionable for protons in an electron gas, asthe proton-electron interaction is highly nonlinear [ 71,72]. A
calculation inclusive of all nonlinear effects is available fromthe NPA model, and displayed in Fig. 4(a) for a single proton,
confirming that linear response is accurate at this density.Figure 3shows that the NPA and CHNC agree accurately with
each other and with quantum simulations. While the nonlinearCHNC procedure gives good agreement with the QMC resultsofg
pp, bridge corrections are needed for the form of the ion-
ion pair potential used in the NPA, when excellent agreementis obtained, both for the positions of the peaks and the peakheights. The Gibbs-Bogoliubov LFA criterion determines η=
0.475 for this ∼350 times compressed hydrogen fluid. The
g
eeand gepare insensitive to bridge corrections. However,
as expected, the accuracy of gepdepends on the choice of
βepφep(r). In Fig. 4(b), we see that the ansatz given by Eq. ( 35)
works well even halfway into the Wigner-Seitz (WS) sphereof the electron with the WS radius r
s=0.4473. Hence, the
classical map is quite accurate for equation of state, transport,and other calculations of compressed hydrogen in a highlyquantum regime.0.01 0.1 1r [a.u.]
2345nnpa(r)NPA one-H+
mean density
Linear Resp.
0.4 0.8 1.2
r [a.u.]11.5gep(r)gep [one-H+] NPA
gep [fluid] NPA η =0.475
gep [fluid] NPA η =0
gep [fluid] CHNC, Eq. 35T = 0.34 eV
rs = 0.4473(a)
(b)
rs
FIG. 4. (a) The free-electron density nNPA(r) calculated from the
NPA model at a proton in a hydrogen plasma, T=0.34 eV, with an
ion density of ¯ ρ=1.8×1025cm−3, i.e., rs=0.4473. At this high
density, linear response theory (dashed curve) is accurate. (b) Thedensity displacement can be used to define g
ep(r) for the one-proton
system and its generalization to the fluid. Results for gepobtained
from a simple model of Eq. ( 35) within the CHNC, and calculations
from the NPA for hydrogen fluid using the MHNC equation are
displayed.
The assumptions in Eq. ( 35) that Tep=√TeeTii,φep=
Z{1−exp(−r/λ)}/r, can be avoided if NPA inputs can be
used. For instance, in Ref. [ 42] the free-electron NPA density
n(r→0 )w a su s e dt ofi x λep. A more complete approach
is also possible. Thus, if the free-electron density incrementaround one H
+ion as calculated from the NPA is /Delta1nNPA(r),
then we define the gep[one H+](q) as follows and invert it by
the HNC equation to obtain the effective ep potential βepφep:
hep[one H+](q)=/Delta1nNPA(q,T)/¯n, (39)
gep(r)=1+hep(r)=1+/Delta1nNPA(r)/¯n, (40)
βeiφep(r)=HNC inversion of gep(r). (41)
In Eq. ( 41) we imply that the gep(r) is now interpreted as a
classical PDF in a system containing protons and electrons. Itis HNC inverted to obtain the classical pair potential β
epφep(r)
in a manner analogous to the extraction of the Pauli potentialfrom g
0
ee(r). However, in the regime of high compressions, the
model of Eq. ( 35) seems to be sufficient.
While the NPA and CHNC calculations agreed with the
QMC results of Liberatore et al. in the linear response regime,
we show that similar agreement is found in regimes wherelinear response does not hold. In Fig. 5we show that the
PDFs obtained using our single-center approaches agree verywell with highly computer-intensive quantum simulations,e.g., those of Morales et al. [73] using coupled electron-ion
Monte Carlo calculations with 54 protons in the simulationcell. The g
pp(r) of such quantum simulations are limited by
the simulation cell size, while the NPA calculations capturethe effect of many Friedel oscillations in the potentials. Ourresults imply that the ion-ion correlation functionals used in
155143-10SIMPLIFICATION OF THE ELECTRON-ION MANY-BODY … PHYSICAL REVIEW B 100, 155143 (2019)
01234 5
r/rs00.511.5gpp(r)
NPA
QMC-Morales
0.01 0.1 1 10
r/rs012345nep(r) / n10,000K NPA
6,000K NPA6,000K
10,000Krs = 1.05
Hydrogen
FIG. 5. (a) The fractional free-electron density nNPA(r)/¯norgep
at a single proton calculated from the NPA model at a proton in
a hydrogen plasma at rs=1.05 a.u. This is effectively unchanged
between T=6000 and 10 000 K. (b) The NPA density displacement
can be used to define a pseudopotential for the one-proton system and
the fluid gppis calculated using an ion-correlation functional which
reduces to the HNC diagrams, as the bridge corrections are negligiblein this case. Results agree closely with the coupled electron-ion
quantum Monte Carlo simulations of Morales et al. [73] which are,
however, limited by the size of the simulation box. Only the caser
s=1.05 is shown.
the single-center NPA method to incorporate many-ion effects
are successful. This has also been verified in many othercalculations during past decades, even with respect to complexfluids like warm dense carbon, silicon, etc., where there arecovalent bonding effects as well [ 30,31]. A similar verifica-
tion is available in the context of ion-dynamical calculations[74,75]. The NPA and CHNC methods are not limited by
the Born-Oppenheimer approximation, as the static effectsof the electron-nuclear coupling can be incorporated in theelectron-ion correlation functionals [ 76].
The quantum simulation method used by Morales et al. is
described by them as “a QMC-based ab initio method devised
to use QMC electronic energy in a Monte Carlo simulationof the ionic degrees of freedom. … Specifically, the use oftwist-averaged boundary conditions (TABCs) on the phase ofthe electronic wave function, together with recently developedfinite-size correction schemes, allows us to produce energiesthat are well converged to the thermodynamic limit with 100atoms”. More detailed calculations using CHNC, NPA, andcomparisons of the resulting thermodynamic data (calculatedfrom the PDFs using coupling constant integrations) will betaken up elsewhere.
VI. SYSTEMS WHERE ALL PARTICLES ARE IN THE
QUANTUM REGIME
If both subsystems, viz., ions and electrons, are in the quan-
tum regime, this poses no additional difficulty for the CHNCmethod. However, in practice, such quantum corrections forions in the liquid state are possible only under extreme com-pressions, even for hydrogen, and such compressions are onlyavailable in astrophysical settings.TABLE I. Characteristic quantities for electrons and holes in a
typical GaAs /GaAlAs in a double-quantum-well structure. These
are examples for two-dimensional warm dense matter states where
the electron and its positive counterparticle behave quantum me-
chanically. The material dielectric constant ε=12.9 for both layers
separated by an AlAs barrier of 10-nm width and held at a tempera-
ture T=5 K. Each layer has its effective Bohr radius and effective
Hartree unit. So, rs,EF, etc., are expressed in the respective effec-
tive units. The classical strong coupling parameter /Gamma1=1/(rsTcf)i s
given.
Item Electron layer Hole layer
Particle density (cm−2)4 ×10114×1011
Effective mass m∗
s=ms/m0 0.0670 0.3350
Layer width (nm) 15.00 20.00Effective r
s 2.768 13.84
Effective EF 0.1304 0 .5218×10−2
Degeneracy parameter T/EF 0.3015 1.508
Classical fluid temp. Tcf/EF 1.966 2.188
Inverse de Broglie length 1.180 0.1807
Plasma coupling parameter /Gamma1 1.408 6.327
On the other hand, in electron-hole systems as found in
semiconductor materials, the quantum nature of both types ofparticles must be included as the hole masses (empty statesin the valence band) are usually within a factor of 10 of theelectron mass. Furthermore, if the electrons and holes areconfined in two quantum wells separated by a thin insulatingbarrier, the spontaneous recombination of electrons and holesis suppressed. Such systems can be fabricated and are knownas double quantum wells (DQWs) where the electrons occupythe lowest conduction subband in one of the wells, whilethe holes occupy the highest valence subband in the otherwell. They form two interacting but spatially separated 2Delectron fluids. Properties of such DQWs in the symmetriccase (i.e., for the case where the election and hole massesm
e,mh, well widths we,wh, layer dielectric constants, and
temperatures are equal) have already been studied using theCHNC method [ 44]. However, in typical GaAs-GaAl
xAs1−x
DQWs, the electron and hole masses are, typically, 0.067 and
0.335, respectively, while the material dielectric constant εis
taken as ∼12.9 for the whole structure since the aluminum
alloy content xis small. The barrier dielectric constant is
typically about 12.62. The effective Bohr radius is given bya
∗
B=¯h2ε/(m0e2m∗
s) where m0is the free-electron mass while
m∗
sis the effective mass of the species “ s.” Typical well widths
wand barrier widths are 10–30 nm. Given these material
properties, a DQW with equal densities of electrons and holesat a temperature of even 5 K is found to be a two-componentinteracting partially degenerate system where the r
s,EF
values, and hence the degeneracies are widely different (see
Table I). In fact, these DQWs provide excellent laboratory
examples of two-dimensional interacting warm dense matter.They contain four interacting subsystems as the electrons,and holes are spin-
1
2fermions for GaAs /GaAlAs DQWs, but
there is no exchange interaction between particles in separatelayers. Such DQWs can be made with two electron layers, twohole layers, or an electron layer coupled to a hole layer. Formore computational details, see Ref. [ 44].
155143-11M. W. C. DHARMA-W ARDANA PHYSICAL REVIEW B 100, 155143 (2019)
048 1 2 16 20
r/rse 00.40.8gij(r)geegehghh
T = 5 K
GaAs/GaAlAs electron-hole DQW.
FIG. 6. The ee, eh ,a n d hhpair-distribution functions g(r)f o ra
an electron layer interacting with a hole layer in a GaAs /GaAlAs
double quantum well maintained at 5 K. The xaxis is in units of
the electron Wigner-Seitz radius rse=2.768 in units of the effective
electron atomic unit of length (materials details are given in Table I).
This is an example of quasi-two-dimensional warm dense matter,realized at 5 K.
These systems are of great interest in nanostructure physics
as the transport properties, plasmon dispersion, energy relax-ation, etc., depend on the corresponding structure factors andlocal-field corrections which enter into response functions andeffective potentials. All such quantities can be extracted fromCHNC calculations [ 5,44]. Currently, the CHNC technique
is the only method available for treating such systems atarbitrary layer degeneracies, spin polarizations, and arbitraryeffective masses at zero to finite temperatures. The pair-distribution functions for the spin-unpolarized ehDQW sys-
tem described in Table Iare displayed in Fig. 6. Unfortunately,
QMC or other microscopic calculations for such systems arebelieved to be too prohibitive at present, and no comparisonsare available.
N-representability and v-representability of CHNC densities
for electron-ion systems
The N-representability of the electron-electron gee(r) ob-
tained from the CHNC method for electron-ion systems needsto be examined. This too can be approached as in the pre-
vious section. It appears that N-representability is preserved
in this case too, where we have merely made the electronsto interact with the “external potential” of the ions whichis, however, self-consistently adjusted in the two-componentproblem, with no invoking of the Born-Oppenheimer approx-imation. The BO corrections come through the electron-ioncorrelation potentials (HNC-like diagrams) contained in theHNC equation describing the g
ep(r) pair distribution function
[76].
We may also note that the v-representability of the den-
sities generated by CHNC or via the NPA can be treatedusing standard methods since we are dealing entirely withCoulombic systems and spherical charge densities. For suchsystems, Kato’s theorem [ 77] applies, and the methods based
on spherical densities due to Theophilou, Nagy can be used[78,79].
VII. CONCLUSION
A review of the classical map hypernetted-chain procedure,
which is a way of side stepping the construction of a quantumkinetic energy functional for density functional theory, ordoing quantum calculations is presented. General argumentssupporting the N-representability of the classical map, and
specific demonstrations of N-representability using the agree-
ment of CHNC results with QMC and other benchmark resultsare given. The application of the CHNC method to generalelectron-ion systems is reviewed. Computationally demand-ing quantum systems like warm dense matter become numer-ically very simple and rapid within the CHNC method. Theclassical map may be used without the HNC procedure, viaclassical MD simulations. Thus, within certain limits, entirelyclassical calculations which are very rapid and independentof the number of particles and the system temperature arepossible for a wide class of quantum problems.
ACKNOWLEDGMENTS
The author thanks Professor S. Trickey and Prof. J. Dufty
for raising the question of the N-representability of the CHNC
procedure at the CECAM workshop at Lausanne, Switzerland.
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155143-13 |
PhysRevB.91.075104.pdf | PHYSICAL REVIEW B 91, 075104 (2015)
Topological semimetal-to-insulator phase transition between noncollinear and noncoplanar
multiple- Qstates on a square-to-triangular lattice
Satoru Hayami and Yukitoshi Motome
Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
(Received 23 November 2014; published 6 February 2015)
Noncollinear and noncoplanar magnetic orders lead to unusual electronic structures and transport properties.
We investigate two types of multiple- Qmagnetically ordered states and a topological phase transition between
them in two dimensions. One is a coplanar but noncollinear double- Qstate on a square lattice, which is a
semimetal accommodating massless Dirac electrons. The other is a noncoplanar triple- Qstate on a triangular
lattice, which is a Chern insulator showing the quantum anomalous Hall effect. We discuss the peculiar electronicstructures in these two multiple- Qstates in a unified way on the basis of the Kondo lattice model, which suggests
a quantum phase transition between the two states in a continuous change of lattice geometry between the squareand triangular lattices. We systematically examine the possibility of such a transition by using the mean-fieldapproximation for the ground state of the periodic Anderson model. After clarifying the parameter region in whichthe double- Q(triple- Q) state is stabilized on the square (triangular) lattice, we show that a continuous topological
phase transition indeed takes place between the double- QDirac semimetal and the triple- QChern insulator on
the square-to-triangular lattice. The nature of the transition is discussed by the topologically protected edge statesas well as the bulk magnetic and electronic properties. The results indicate that unusual critical phenomena mayoccur at finite temperature related with multiple- Qchiral spin-liquid states.
DOI: 10.1103/PhysRevB.91.075104 PACS number(s): 71 .10.Fd,03.65.Vf,71.27.+a,75.47.−m
I. INTRODUCTION
The search for topologically nontrivial states has attracted
growing interest in condensed matter physics [ 1,2]. Among
them, a topological insulator is a new quantum state of matter[3–7]. In this state, although the bulk of the system is insulating
with an energy gap opened by the spin-orbit coupling, aspin current flows along the edge/surface of the system. Thecurrent flow is dissipationless and gives rise to a quantizedspin Hall response. Such a quantum spin Hall effect hasbeen experimentally observed in two-dimensional quantumwells, such as HgTe/CdTe [ 8] and InAs/GaSb [ 9,10], and in
three-dimensional bulk systems, such as Bi
2Se3,B i 2Te3, and
Sb2Te3[11–14].
On the other hand, topologically nontrivial states are also
realized in magnetically ordered systems with broken timereversal symmetry. Such topological insulators with magneticordering are called magnetic Chern insulators, as they areclassified according to nonzero values of the Chern number[15]. Remarkably, the magnetic Chern insulators exhibit a
quantum anomalous (topological) Hall effect even in theabsence of an external magnetic field. The origin of theanomalous Hall effects has been discussed in terms of thespin scalar chirality and associated Berry phase in spin space[16–19].
Such noncoplanar scalar chiral orders have been theoret-
ically explored in the Kondo lattice model on several frus-trated lattice structures: a face-centered-cubic [ 20], triangular
[21–28], checkerboard [ 29], pyrochlore [ 30], and kagome
lattices [ 19,31–34]. These chiral ordered states will be
important for potential applications to spintronics, as thetopological nature can be controlled by external magnetic fieldand pressure as well as temperature. Furthermore, they arebeneficial owing to the possibility of acquiring a wide energygap determined by the exchange interaction rather than thespin-orbit coupling. To stimulate experimental exploration ofsuch exotic states, it is desired to systematically study how
and when the noncoplanar magnetic orders become stable. Itis also interesting to study how the topological nature changesin magnetic phase transitions.
In the present study, we investigate the stability of magnetic
Chern insulators and their phase transitions, with focusing on
multiple- Qmagnetically ordered states. The multiple- Qstates
are characterized by more than one ordering wave vector, andtend to accommodate noncollinear and noncoplanar magneticorders. Hence, the multiple- Qstates in itinerant electron
systems provide an ideal prototype for the magnetic Cherninsulators. In fact, some of the above-mentioned states inthe Kondo lattice model are the multiple- QChern insulators
[19–22,29,33].
Specifically, we focus on a triple- Qstate on a triangular
lattice and a double- Qstate on a square lattice. The former
exhibits a ferroic order of the spin scalar chirality associatedwith four-sublattice noncoplanar magnetic order, leadingto a magnetic Chern insulating state [ 21,22]. Meanwhile,
the latter shows a coplanar but noncollinear four-sublattice
order, resulting in a semimetallic state with massless Dirac
electrons [ 35–37]. First, we discuss the topologically nontrivial
electronic structures for these two states in the Kondo latticemodel. We show that a topological phase transition betweenthe two states can occur while changing the lattice geometry.Next, we discuss the stability of the multiple- Qstates and the
possibility of such a topological phase transition by mean-field
calculations for the periodic Anderson model. We show that
the double- Qand triple- Qstates are robustly realized on the
square and triangular lattices, respectively, in a wide rangeof parameters. Furthermore, we find that the model indeedexhibits a continuous topological phase transition between thetwo states by changing the lattice geometry. We discuss thedetailed nature of the transition by presenting the change ofthe topologically protected gapless edge states as well as the
bulk physical properties.
1098-0121/2015/91(7)/075104(9) 075104-1 ©2015 American Physical SocietySATORU HAY AMI AND YUKITOSHI MOTOME PHYSICAL REVIEW B 91, 075104 (2015)
The organization of this paper is as follows. In Sec.
II, after introducing the double- Qand triple- Qstates on
the square and triangular lattices, we discuss the peculiarelectronic structures for bulk and edge states and an anticipatedquantum phase transition between them by using the Kondolattice Hamiltonian. In Sec. III, by using the mean-field
approximation for the periodic Anderson model, we showthat the system exhibits a continuous quantum phase transitionbetween the double- QDirac semimetal and the triple- QChern
insulator. Section IVis devoted to a summary and concluding
remarks.
II. MULTIPLE- QSTATES AND THEIR ELECTRONIC
STRUCTURE
In this section, we examine the fundamental magnetic and
electronic nature of the multiple- Qstates that we study in this
paper. In Sec. II A, we introduce the definition of multiple- Q
magnetic orders. In Sec. II B, we present the Kondo lattice
Hamiltonian for the double- Qand triple- Qstates in a unified
way. We show their electronic structures for both bulk andpeculiar gapless edge states in Sec. II C. In Sec. II D, we discuss
an anticipated phase transition between these two states bymodifying the lattice geometry.
A. Multiple- Qstates
In this study, we consider double- Qand triple- Qstates on
square, triangular, and their intermediate lattices. The orderparameter for the double- Qstate is written by
/angbracketleftS
i/angbracketright∝(cos Q1·ri,cosQ2·ri,0), (1)
while that for the triple- Qstate is given by
/angbracketleftSi/angbracketright∝(cos Q1·ri,cosQ2·ri,cosQ3·ri). (2)
Here, Q1,Q2, and Q3stand for the wave vectors charac-
terizing the multiple- Qstates; riis the position vector of
the site i. Equation ( 1) defines a noncollinear but coplanar
spin configuration as exemplified on the square lattice inFig. 1(a) [36], while Eq. ( 2) gives a noncoplanar configuration
as illustrated on the triangular lattice in Fig. 1(b) [21,22]; both
of them are represented by superpositions of single- Qstates.
)c( )b( )a(
A B
C DA B
C DC ABD
FIG. 1. (Color online) Schematic pictures of (a) a double- Qstate
with a coplanar spin configuration on a square lattice and (b) a triple- Q
state with a noncoplanar spin configuration on a triangular lattice. The
arrows denote the directions of local magnetic moments. The inset of
(b) shows the directions of magnetic moments at the four-sublatticesites in the triple- Qstate. (c) The square lattice with diagonal bonds,
which is topologically equivalent to the triangular lattice in (b). tand
t
/primerepresent the hopping amplitudes in the models in Eqs. ( 3)a n d( 8).
We call this the square-to-triangular lattice.In the following, we treat these two multiple- Qstates in a
unified way by regarding the triangular lattice as a topologi-cally equivalent square lattice with diagonal bonds, as shown inFig. 1(c). We call this structure the square-to-triangular lattice.
Then, the double- Qstate is given by the superposition of
(cosπr
x
i,0,0) and (0 ,cosπry
i,0), which is described by taking
Q1=(π,0) and Q2=(0,π)i nE q .( 1). Here, ri=(rx
i,ry
i),
and we set the lattice constant as unity. Similarly, the triple- Q
state is given by Q1=(π,0),Q2=(0,π), and Q3=(π,π)
on the square-to-triangular lattice in Fig. 1(c). We note that, as
shown in Figs. 1(a) and 1(b), all the relative angles between
neighboring moments are cos−1(0)=90◦for the double- Q
state and cos−1(−1/3)∼109◦for the triple- Qstate.
B. Kondo Hamiltonian
In order to investigate the influence of the noncollinear and
noncoplanar magnetic ordering on the electronic structure,we consider the coupling of the magnetic moments tononinteracting electrons through the local exchange coupling.A minimal model to describe the situation is the Kondo latticemodel, whose Hamiltonian is given by
H=−t/summationdisplay
/angbracketlefti,j/angbracketrightσ(c†
iσcjσ+H.c.)−t/prime/summationdisplay
/angbracketleft/angbracketlefti,j/angbracketright/angbracketrightσ(c†
iσcjσ+H.c.)
−J/summationdisplay
iσσ/primec†
iσσσσ/primeciσ/prime·Si, (3)
where c†
iσ(ciσ) is a creation (annihilation) operator of itinerant
electrons at site iand spin σ. The first term in Eq. ( 3) represents
the hopping of conduction electrons on the nearest-neighborbonds on the square lattice, while the second term is for thediagonal bonds introduced to connect the square lattice tothe triangular lattice [see Fig. 1(c)]. Note that t
/prime=0(t/prime=t)
corresponds to the square (triangular) lattice. Hereafter, wetaket=1 as an energy unit. The third term in Eq. ( 3) describes
the exchange coupling between conduction and localizedspins. Here, σ=(σ
x,σy,σz) is the vector of Pauli matrices,
Siis a localized spin at site i, andJis the exchange coupling
constant (the sign is irrelevant for the following mean-fieldtype arguments).
By replacing S
iby Eqs. ( 1) and ( 2), the Hamiltonian in Eq.
(3) is written in the momentum representation,
H=/summationdisplay
kσεkc†
kσckσ−Jm/summationdisplay
kσσ/primeηc†
kσση
σσ/primeck+Qησ/prime, (4)
where c†
kσandckσare the Fourier transform of c†
iσandciσ,
respectively; εkis the energy dispersion for free electrons,
εk=− 2t(coskx+cosky)−2t/primecos(kx−ky). Here, the sum
ofηin the second term is taken for η=1 and 2 for the double-
Qstate, and η=1, 2, and 3 for the triple- Qstate. We set
the normalization factor mso that |/angbracketleftSi/angbracketright| = 1:m=1/√
2 and
1/√
3 for double- Qand triple- Qstates, respectively.
In the momentum representation, the Hamiltonian is di-
vided into two irreducible parts as
H=c†
I˜HcI+c†
II˜HcII, (5)
075104-2TOPOLOGICAL SEMIMETAL-TO-INSULATOR PHASE . . . PHYSICAL REVIEW B 91, 075104 (2015)
)b( )a(
-4-2 0 2 4
-4-2 0 2 4
-6Energy
Energy
)0,0( )0,0()0,0( )2/π,2/π()0,2/π()0,0( (π/3,-π/3) (π/2,0)
FIG. 2. Energy dispersions of (a) the noncollinear double- Qstate
on the square lattice ( t/prime=0) and (b) the noncoplanar triple- Qstate
on the triangular lattice ( t/prime=1) atJ=2, shown along the symmetric
lines in the magnetic Brillouin zone. In (a), two-dimensional massless
Dirac nodes appear at ( π/2,π/2) at 1 /4a n d3 /4 fillings. Meanwhile,
an energy gap opens at 1 /4a n d3 /4 fillings in (b).
where
˜H=⎛
⎜⎜⎜⎝ε
k /Delta1 −i/Delta1 α/Delta1
/Delta1ε k+Q1−α/Delta1 i/Delta1
i/Delta1 −α/Delta1 ε k+Q2/Delta1
α/Delta1 −i/Delta1 /Delta1 ε k+Q3⎞
⎟⎟⎟⎠. (6)
Here, c†
I=(c†
k↑,c†
k+Q1↓,c†
k+Q2↓,c†
k+Q3↑),c†
II=(c†
k↓,c†
k+Q1↑,
−c†
k+Q2↑,−c†
k+Q3↓), and /Delta1=Jm. We introduce the parame-
terαto connect the double- Qand triple- Qstates: α=0f o rt h e
pure double- Qstate and α=1 for the pure triple- Qstate. We
note that this Hamiltonian with α=1 is formally common to
the four-sublattice triple- Qorders on the triangular and cubic
lattices [ 21,38]. In the two-dimensional triangular lattice case,
the triple- Qmagnetic order induces Chern insulating states
[21,22][ s e eF i g . 2(b)], while, in the three-dimensional cubic
lattice case, it leads to semimetallic states with massless Diracdispersions [ 38].
C. Electronic structure
The energy dispersion of the Hamiltonian in Eq. ( 5)i s
shown in Fig. 2(a) att/prime=0,J=2(/Delta1=√
2), and α=0,
which corresponds to the double- Qstate on the square lattice.
All the bands are doubly degenerate; the degeneracy comesfrom the fact that c
Iand cIIare related by a combination
of lattice translation and spin rotation, which leaves ˜H
unchanged. In Fig. 2(a), a peculiar structure is found near the
(π/2,π/2) point; the band dispersions are linearly dependent
onkand cross with each other at the ( π/2,π/2) point, resulting
in two-dimensional conelike structures. This is a signature oftwo-dimensional massless Dirac electrons appearing at 1 /4
and 3/4 fillings of conduction electrons [ 35,36,39–41]. In fact,
by expanding the Hamiltonian around the ( π/2,π/2) point
and performing the unitary transformations, the Dirac-typeequation is obtained:
˜H/similarequal˜H
2D
±=±√
2/Delta1σ 0±√
2t(κxσz+κyσx), (7)
where σ0is the unit matrix and κ=(κx,κy) is the wave vector
measured from ( π/2,π/2).(b) (c)
-4-2 0 2 4
-6 -4-2 0 2 4Energy
Energy
π 0 π 0xy(110) surface(a)
unit cellBA
DCC
DA A
A A
A AD
D D
DB
CB
C
B
CB
C
B
C
π/2 π/2
FIG. 3. (Color online) (a) Schematic picture of the system with
the (110) edges. The sites in the dashed box represent the unit cell
used for the calculations of the edge states. The number of sites
in the unit cell is 130. The relation between ( kx,ky)a n d( k⊥,k/bardbl)i s
also shown. Energy dispersions for the systems at J=2 for (b) the
double- Qstate on the square lattice ( t/prime=0a n d α=0) and (c) the
triple-Qstate on the triangular lattice ( t/prime=1a n dα=1).
On the other hand, the energy dispersion for the triple- Q
state on the triangular lattice is shown in Fig. 2(b): the result is
calculated at t/prime=1,J=2(/Delta1=2/√
3), and α=1i nE q .( 6).
In the triple- Qstate, the system becomes insulating at 1 /4
and 3/4 fillings. The opening of the energy gap is due to
the combination of the diagonal hopping t/primeand the triple- Q
magnetic order parameter α/Delta1. In fact, the energy gap does not
open when either t/prime=0o rα=0. The insulating states at 1 /4
and 3/4 fillings are topologically nontrivial Chern insulators
[21,22]. Consequently, they exhibit the quantization of the
Hall conductivity and gapless chiral edge states, as discussedbelow.
Let us discuss the electronic state in the multiple- Qstates
more closely, with emphasis on the peculiar edge statesassociated with their topological nature. We here consider thesystem with the (110) open edges, in which both edges consistof A and C sublattice sites, as shown in Fig. 3(a) [42]. In the
(1¯10) direction, we adopt the periodic boundary condition.
Figure 3(b) shows the band dispersions of the system with
the (110) edges for the double- Qstate on the square lattice.
In this case, the Dirac nodes at the ( π/2,π/2) point in the
bulk system are projected onto k
/bardbl=0 and π. See Fig. 3(a)
for the relation between ( kx,ky) and ( k⊥,k/bardbl). Interestingly,
there appear bands connecting the Dirac nodes for both 1 /4
and 3/4 fillings. These are the chiral edge states appearing
075104-3SATORU HAY AMI AND YUKITOSHI MOTOME PHYSICAL REVIEW B 91, 075104 (2015)
0.00 0.02 0.04 0.06
0.0 0.2 0.4 0.6 0.8 1.0double- Q triple- Q
FIG. 4. (Color online) t/primedependence of the energy difference
between the noncollinear double- Qand noncoplanar triple- Qstates
at 1/4 filling and J=2.
in the two-dimensional Dirac semimetal. The emergence of
the bands connecting the Dirac nodes is similar to the casein zigzag-edged graphene nanoribbons; the bands, however,are dispersive in the present system, while they have no k
/bardbl
dependence in the nanoribbons [ 43–45].
On the other hand, the triple- Qstate on the triangular lattice
shows gapless chiral edge states traversing the gaps, as shownin Fig. 3(c). They have crossing points at k
/bardbl=π/2 with linear
dispersions. These are the gapless edge states emergent in theChern insulating states, resulting in the nonzero quantized Hallconductivity [ 21,22].
D. Topological phase transition between the double- Qand
triple- Qstates
We here consider the stability of the multiple- Qstates from
the energetic point of view. The noncollinear double- Qstate
on the square lattice ( t/prime=0) and the noncoplanar triple- Q
state on the triangular lattice ( t/prime=1) were shown to appear
at 1/4 filling in the Kondo lattice model by the Monte
Carlo simulation combined with the exact diagonalization[23,36]. Furthermore, the instability toward these multiple- Q
orders was explained by the perturbation in the weak-couplinglimit [ 24,46]. Hence, the double- Qand triple- Qstates are
considered to be realized in the ground state, at least, at t
/prime=0
andt/prime=1, respectively.
Then, let us discuss how these two multiple- Qstates
are connected with each other when we modify the latticegeometry by changing t
/primefrom 0 to 1. We simply compare the
energy for the double- Qand triple- Qstates while changing t/prime.
Figure 4shows the energy difference between the double-
Qstate, E(double), and the triple- Qstate, E(triple),a tJ=
2;E(double)andE(triple)are calculated for α=0 and 1,
respectively. When t/prime=0, the energy in the double- Qstate
is lower than that in the triple- Qstate. As increasing t/prime,t h e
energy difference becomes smaller, and finally, the energy forthe triple- Qstate becomes lower for t
/prime/greaterorsimilar0.89. Namely, the
result indicates that a topological phase transition is expectedbetween the Dirac semimetal and the Chern insulator att
/prime∼0.89. In the present simple energy comparison, however,
we ignore a modulation of localized moments from theperfectly ordered double- Qand triple- Qstates. We will
examine whether this topological transition takes place forthe periodic Anderson model, in which modulations of anglesand lengths of the local moments are allowed.
III. STABILITY OF MULTIPLE- QSTATES AND
TOPOLOGICAL PHASE TRANSITION
In this section, we discuss the stability of multiple- Q
states in a systematic way on the square-to-triangular lattice.We adopt a fundamental model for describing the couplingbetween conduction and localized electrons, the periodicAnderson model. After showing the Hamiltonian and thecalculation method in Sec. III A , we first present the ground-
state phase diagram on the square and triangular lattices inSecs. III B and III C , respectively. We then discuss the phase
transition between the noncollinear double- Qand noncoplanar
triple-Qstates on the square-to-triangular lattice in Sec. III D .
We also show the change of the edge states in the band structureassociated with the phase transition.
A. Model and method
The Hamiltonian for the periodic Anderson model is written
as
H=−t/summationdisplay
/angbracketlefti,j/angbracketrightσ(c†
iσcjσ+H.c.)−t/prime/summationdisplay
/angbracketleft/angbracketlefti,j/angbracketright/angbracketrightσ(c†
iσcjσ+H.c.)
−V/summationdisplay
iσ(c†
iσfiσ+H.c.)+U/summationdisplay
inf
i↑nf
i↓+E0/summationdisplay
iσnf
iσ,
(8)
where f†
iσ(fiσ) is the creation (annihilation) operator of
localized felectrons with spin σat site i, andnf
iσ=f†
iσfiσ.
The first and second terms represent the kinetic energy ofconduction celectrons as in Eq. ( 3), the third term the on-site
hybridization between candfelectrons, the fourth term the
on-site Coulomb interaction for felectrons, and the fifth term
the atomic energy of felectrons. The periodic Anderson
model is reduced to the Kondo lattice model in Eq. ( 3)i n
the large Ulimit with one felectron per site; felectrons give
localized moments, which couple with conduction electronsvia the Kondo coupling J∝V
2/U [47]. We focus on the
commensurate filling, ntot=(1/N)/summationtext
iσ/angbracketleftc†
iσciσ+f†
iσfiσ/angbracketright=
3/2, which corresponds to the 1 /4-filling case in the Kondo
lattice model. Hereafter, we take t=1 andE0=− 4.
In order to study the ground state of the model in Eq. ( 8), we
employ the Hartree-Fock approximation for the Coulomb U
term, which preserves the SU(2) symmetry of the interactionterm as
nf
i↑nf
i↓∼nf
i↑/angbracketleftbig
nf
i↓/angbracketrightbig
+/angbracketleftbig
nf
i↑/angbracketrightbig
nf
i↓−/angbracketleftbig
nf
i↑/angbracketrightbig/angbracketleftbig
nf
i↓/angbracketrightbig
−/angbracketleftf†
i↑fi↓/angbracketrightf†
i↓fi↑
−f†
i↑fi↓/angbracketleftf†
i↓fi↑/angbracketright+/angbracketleftf†
i↑fi↓/angbracketright/angbracketleftf†
i↓fi↑/angbracketright. (9)
Here, /angbracketleft ···/angbracketright is the statistical average with respect to the
one-body mean-field Hamiltonian. In the calculations, weadopt a 2 ×2-site unit cell to determine the phase diagram.
We confirm that the phase diagram is not qualitatively alteredin the calculations by using an enlarged 4 ×4-site unit cell.
075104-4TOPOLOGICAL SEMIMETAL-TO-INSULATOR PHASE . . . PHYSICAL REVIEW B 91, 075104 (2015)
FIG. 5. (Color online) Ground-state phase diagram of the peri-
odic Anderson model in Eq. ( 8) on the square lattice at ntot=3/2
obtained by the mean-field calculations for E0=− 4a n d t/prime=0.
Schematic pictures of the ordering patterns in localized electrons areshown in the bottom panel. The sizes of the spheres denote local
electron densities, and the arrows represent local spin moments.
AF, Ferro, CO, and NM stand for antiferromagnetic (collinearN´eel-type), ferromagnetic, charge-ordered, and nonmagnetic metallic
states, respectively. Single- Qcorresponds to Q=(0,π) and double-
Q(0,π), (π,0). The c-double- Qrepresents the double- Qstate with
spin canting.
B. Square lattice
First, we discuss the stability of the double- Qstate with
massless Dirac electrons by investigating the ground-statephase diagram of the periodic Anderson model in Eq. ( 8)o nt h e
square lattice ( t
/prime=0). Figure 5shows the result at ntot=3/2
obtained by the mean-field approximation. The phase diagramis mainly divided into three regions: multiple- Qregion for
largeUand small V(forU/greaterorsimilarV
2), charge-ordered region for
intermediate UandV, and nonmagnetic region for small Uand
largeV(forV/greaterorsimilar2U). Schematic pictures for each magnetic
and charge ordered states of localized electrons are presentedin the bottom of Fig. 5.
In the large Uand small Vregion, we find the double- Q
state as one of the dominant magnetic phases. This phaseaccommodates the two-dimensional massless Dirac electronsin the electronic band structure as described in Sec. II A.D i r a cnodes also appear at n
tot=1/2, 5/2, and 7 /2. We note that a
similar double- Qstate was found by Monte Carlo simulation
for the Kondo lattice model at 1 /4 filling [ 36], corresponding
tontot=3/2i nt h el a r g e Ulimit with keeping V2/U at a
nonzero constant. Interestingly, the double- Qstate in Fig. 5
appears around U∝2V2.
Other dominant instabilities in the phase diagram in
Fig. 5are the charge-ordered insulators. Interestingly, al-
though the periodic Anderson model include neither bareoff-site repulsive Coulomb interaction nor electron-phononinteractions, there are three charge-ordered phases in theintermediate UandVregion: the ferromagnetic, N ´eel-
type antiferromagnetic, and double- Qcharge-ordered phases.
Among them, the charge-ordered state with N ´eel-type an-
tiferromagnetic state was found also in the Kondo lat-tice model [ 48]. Note that similar charge order insta-
bilities were also pointed out in three-dimensional cubicsystems [ 49,50] and infinite-dimensional systems [ 51–53].
On the other hand, the double- Qcharge-ordered state has
a noncollinear spin state; a similar but noncoplanar triple- Q
charge order was found on a cubic lattice [ 50].
C. Triangular lattice
Next, we discuss the stability of the multiple- Qstate on
the triangular lattice ( t/prime=1). Figure 6shows the ground-
state phase diagram at ntot=3/2 obtained by the mean-field
approximation. We find that the triple- Qstates are stabilized
in the three regions: small Uand large V(around V∼6U),
intermediate UandV(around U∼2V2), and large Uand
smallV(forU/greaterorsimilar3 andV/lessorsimilar1) regions. We here focus on the
former two, as the last one is replaced by other magneticallyordered phase when we include other additional interactionsand hoppings in the model (not shown here).
The stabilization mechanism of the triple- Qs t a t ei nt h e
smallUand large Vregion is attributed to the perfect nesting of
the Fermi surface as explained below. For large V, the energy
bands are split into the bonding and antibonding ones by thehybridization between candfelectrons, and the energetically
lower bonding band is partially filled at n
tot=3/2. The
effective filling in the bonding band is 3 /4. As pointed out
in Ref. [ 21], the Fermi surface at 3/4 filling on the triangular
lattice is perfectly nested, leading to the instability towardthe triple- Qorder. This nesting instability is the origin of the
triple-Qorder found in the small Uand large Vregion.
On the other hand, the stabilization mechanism is different
for the triple- Qstate in the intermediate UandVregion. In
this case, the origin is understood by considering the large U
limit with keeping V
2/U at a constant, where the periodic
Anderson model at ntot=3/2 is reduced to the Kondo lattice
model at 1 /4 filling. Indeed, the local density of localized
electrons is approximately half filling in the large Uregion in
this triple- Qstate, which is considerably different from that
in the triple- Qstate in the small Uand large Vregion. In the
Kondo lattice model, a similar triple- Qorder was found at and
near 1 /4 filling [ 22,23], and the origin was attributed to the
(d−2)-dimensional instability of the Fermi surface [ 24,46]. A
similar mechanism may work in stabilizing the triple- Qstate
in the intermediate UandVregion in the periodic Anderson
model.
075104-5SATORU HAY AMI AND YUKITOSHI MOTOME PHYSICAL REVIEW B 91, 075104 (2015)
FIG. 6. (Color online) Ground-state phase diagram of the peri-
odic Anderson model in Eq. ( 8) on the triangular lattice at ntot=3/2
obtained by the mean-field calculations for E0=− 4a n d t/prime=1.
Schematic pictures of the ordering patterns in localized electrons are
shown in the bottom panel. The notations are similar to those in Fig. 5.
We note that both triple- Qstates are insulating and
topologically nontrivial with nonzero Chern numbers. As inthe Kondo lattice case [ 21,22], both states exhibit the quantum
anomalous Hall effect.
D. Connection between square and triangular lattices
In the previous two sections, we found that the double- Q
state on the square lattice and the triple- Qstate on the
triangular lattice appear in the similar parameter region (alongU∼2V
2)a tntot=3/2. Now let us discuss the connection be-
tween the two states by changing t/primeon the square-to-triangular
lattice. Figure 7shows the ground-state phase diagram at
U=10 as a function of Vandt/prime. As described in Sec. III B ,
there are two regions at t/prime=0: the multiple- Qmagnetically
ordered region for small Vand the charge-ordered region for
largeV. These two regions exhibit distinct responses to the
diagonal hopping t/prime>0 as described below.
In the large Vregion where the charge-ordered states
are realized at t/prime=0, the system shows a continuous phase
transition accompanied by the melting of charge order at arelatively small t
/prime∼0.1. The resultant ferromagnetic state
without charge disproportionation is extended to the triangularlattice case with t
/prime=1.
FIG. 7. (Color online) Ground-state phase diagram of the peri-
odic Anderson model at ntot=3/2f o rE0=− 4a n dU=10.t/prime=0
corresponds to the isotropic square lattice system and t/prime=1t ot h e
isotropic triangular lattice system.
On the other hand, in the small Vregion, the phase transition
between multiple- Qstates occurs in a complicated manner. We
focus on the double- Qstate in the region of 2 .2/lessorsimilarV/lessorsimilar2.9.
While increasing t/prime, as shown in Fig. 7, the double- Qstate
extends up to t/prime∼0.7, and shows a phase transition to the
triple-Qstate, which continues to the triangular case with
t/prime=1.
The phase transition is a topological one between the
double- QDirac semimetal and the triple- QChern insulator.
Figure 8shows the changes of various physical quantities
while changing t/primeatV=2.5: the magnitude of local mo-
ments mf(c)
i=√
/angbracketleftsf(c)
i,x/angbracketright2+/angbracketleftsf(c)
i,y/angbracketright2+/angbracketleftsf(c)
i,z/angbracketright2[sf(c)
i,μ is the μ
component of the spin operator for f(c) electron], the local
electron density nf(c)
i=/angbracketleft/summationtext
σf†
iσfiσ(c†
iσciσ)/angbracketright, relative angles
between nearest and next-nearest neighbor moments, thespin scalar chirality per plaquette for fmoments defined
byχ=(1/N
α)/summationtext
{i,j,k}∈αsf
i·(sf
j×sf
k)/(|sf
i||sf
j||sf
k|)[sf
i=
(sf
i,x,sf
i,y,sf
i,z),i,j, andkdenote the sublattice index, and α
labels each triangle plaquette; Nαrepresents the number of
triangle plaquettes], and the energy gap.
The magnitudes of the local spins and charge densities do
not change significantly in the entire t/primeregion, as shown in
Figs. 8(a) and8(b). On the other hand, the magnetic structure,
scalar chirality, and energy gap show drastic changes at thephase transition, as shown in Figs. 8(c)–8(e).F o r0 <t
/prime/lessorsimilar0.7,
the magnetic structure does not change so much from thedouble- Qstate at t
/prime=0, and the energy gap remains zero. With
further increasing t/prime(t/prime/greaterorsimilar0.7), the fmoments are gradually
canted in the perpendicular direction to the coplanar plane ofthe double- Qorder, as shown in Fig. 8(c). Accordingly, the
spin scalar chirality takes a nonzero value and the energy gapopens, as shown in Figs. 8(d) and8(e).A st
/prime→1, the magnetic
order approaches to the pure triple- Qone; the both relative
angles between nearest-neighbor and next-nearest-neighbor
075104-6TOPOLOGICAL SEMIMETAL-TO-INSULATOR PHASE . . . PHYSICAL REVIEW B 91, 075104 (2015)
0.0 0.1 0.2 0.3 0.4
0.6 0.7 0.8 0.9
0 60 120 180
f-nn
f-nnn
0.0 0.2 0.4 0.6 0.8 1.0t(a)
(b)
(c)
(d)
0.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8gap 1.0(e)
′
FIG. 8. (Color online) t/primedependencies of (a) the magnitude of
local moments, (b) local electron densities, (c) relative angles between
moments, (d) spin scalar chirality, and (e) energy gap. The data are
taken at ntot=3/2,V=2.5,U=10, and E0=− 4. In (c), f-nn
andf-nnn mean the relative angles between nearest and next-nearest
neighbor moments for felectrons, respectively. In (c) and (d), the
horizontal dashed line shows the values in the perfectly triple- Q
ordered states.moments become cos−1(−1/3)∼109◦, as shown in Fig. 8(c).
At the same time, the value of the spin scalar chiralityapproaches 4 /(3√
3) for the perfect triple- Qstate. All these
results coherently indicate that the phase transition occurscontinuously between the double- QDirac semimetallic state
and the triple- QChern insulating state at t
/prime∼0.7.
Finally, we examine the topological transition between the
multiple- Qstates from the viewpoint of the electronic structure
by focusing on the edge states. We here consider the systemwith the (110) edges, as in Fig. 3in Sec. II C. Figure 9shows
the band structure of the system with the (110) edges at V=
2.5 with varying t
/prime.I nF i g s . 9(a) and 9(b) corresponding to
the square lattice case ( t/prime=0), the edge states appear with
connecting the Dirac nodes projected onto k/bardbl=0 andπ.T h i s
is similar to the case in the Kondo lattice model as shown inFig. 3(b). With increasing t
/prime, the band structure does not show
a significant change within the double- Qstate, as shown in
Figs. 9(c) and 9(d). Once the transition to the triple- Qstate
occurs upon increasing t/primefort/prime/greaterorsimilar0.7, the dispersions show
qualitatively different behavior. In the triple- Qstate, while the
Dirac nodes for bulk disappear by opening an energy gap, thegapless dispersions from the edge states traverse the gap andcross with each other at k
/bardbl=π/2, as shown in Figs. 9(e)–9(j).
Namely, the degeneracy of edge states connecting the Diracnodes in the double- Qstate is lifted according to the gap
opening, and each edge state continuously develops to thechiral one in the Chern insulating state.
IV. SUMMARY AND CONCLUDING REMARKS
To summarize, we have investigated a phase transition
between the noncollinear double- Qstate and the noncoplanar
triple-Qstate in two-dimensional itinerant magnets. This is
a topological transition between the Dirac semimetal andthe Chern insulator. The result was shown for the Kondolattice model and the periodic Anderson model on thesquare-to-triangular lattice. In particular, for the latter model,we explicitly showed that, by the mean-field approximation,the system exhibits a continuous phase transition accompaniedby the growth of the spin scalar chirality and the energygap while changing of lattice geometry. We examined thetopological nature of the transition by the development ofpeculiar edge states.
Our results provide a reference to further studies for the
multiple- Qstates and the phase transitions between them.
Especially, the phase transitions between double- Q, triple- Q,
and paramagnetic states at finite temperatures will be stimu-lating. As a continuous symmetry cannot be broken at finitetemperatures in two-dimensional systems if the interactionsare short range [ 54], the magnetic long-range orders will be
destroyed in both the double- Qand triple- Qstates at finite
temperatures. Nevertheless, the scalar chirality in the triple- Q
state has a discrete symmetry, which can be broken at a finitetemperature even in two dimensions; hence, it is expected thatthe system exhibits a scalar chiral spin-liquid phase at finitetemperatures [ 23]. On the other hand, in the double- Qregion,
the vector chirality remains as an active continuous degree offreedom at finite temperatures, which might lead to an exotic
075104-7SATORU HAY AMI AND YUKITOSHI MOTOME PHYSICAL REVIEW B 91, 075104 (2015)
-2-1 0
-2-1 0
-3-2-1
-3-2-1
-3-2-1-6-4-2 0 2 4 6Energy
π 0
Energy
π 0Energy
π 0Energy
π 0Energy
π 0Energy
π 0(a) (b)
)d( )c(
(e) (f)
(g) (h)-6-4-2 0 2 4 6Energy
π 0
-6-4-2 0 2 4 6Energy
π 0
-6-4-2 0 2 4 6Energy
π 0
-6-4-2 0 2 4 6Energy
π 0-8(i) (j)2/π 2/π
2/π 2/π
2/π 2/π
2/π 2/π
2/π 2/π
FIG. 9. Energy dispersions for the system with the (110) edges at (a) t/prime=0.0, (c)t/prime=0.6, (e)t/prime=0.7, (g)t/prime=0.8, and (i) t/prime=1.0. The
data are taken at V=2.5,U=10, and E0=− 4. (b), (d), (f), (h), and (j) are enlarged figures of (a), (c), (e), (g), and (i) in the vicinity of the
Fermi level at ntot=3/2, respectively.
075104-8TOPOLOGICAL SEMIMETAL-TO-INSULATOR PHASE . . . PHYSICAL REVIEW B 91, 075104 (2015)
phase transition, such as the Berezinskii-Kosterlitz-Thouless
[55–57] and the Z2vortex transition [ 58]. Thus, it is intriguing
to clarify how the two multiple- Qstates develop in the finite-
temperature region and how the system behaves around thequantum critical point between the two states. To investigatesuch exotic phenomena, it is necessary to perform an analysisbeyond the mean-field approximation used in the present study,such as an effective field theory and numerical simulation. Thisis left for a future study.ACKNOWLEDGMENTS
The authors acknowledge Y . Akagi for fruitful discussions
and careful reading of the manuscript. They also thankT. Misawa and Y . Yamaji for helpful comments. S.H. issupported by a Grant-in-Aid for JSPS Fellows. This workwas supported by a Grant-in-Aid for Scientific Research (No.24340076), the Strategic Programs for Innovative Research(SPIRE), MEXT, and the Computational Materials ScienceInitiative (CMSI), Japan.
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075104-9 |
PhysRevB.85.205212.pdf | PHYSICAL REVIEW B 85, 205212 (2012)
Multiscale calculations of thermoelectric properties of n-type Mg 2Si1−xSnxsolid solutions
X. J. Tan,1W. Liu,2H. J. Liu,1,*J. Shi,1X. F. Tang,2and C. Uher3
1Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education and School of Physics and Technology,
Wuhan University, Wuhan 430072, China
2State Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Wuhan University of Technology,
Wuhan 430070, China
3Department of Physics, University of Michigan, Ann Arbor, Michigan, 48109, USA
(Received 13 April 2012; published 31 May 2012)
The band structure of Mg 2Si1−xSnxsolid solutions with 0.250 /lessorequalslantx/lessorequalslant0.875 is calculated using the first-
principles pseudopotential method. It is found that the low-lying light and heavy conduction bands convergeand the effective mass reaches a maximum value near x=0.625. Using the semiclassical Boltzmann transport
theory and relaxation-time approximation, we find that the system with x=0.625 exhibits both higher Seebeck
coefficient and higher electrical conductivity than other solid solutions at intermediate temperatures. By fittingfirst-principles total energy calculations, a modified Morse potential is constructed, which is used to predicate thelattice thermal conductivity via equilibrium molecular dynamics simulations. Due to relatively higher power factorand lower thermal conductivity, the Mg
2Si0.375Sn0.625is found to exhibit enhanced thermoelectric performance
at 800 K, and additional Sb doping is considered in order to make a better comparison with experiment results.
DOI: 10.1103/PhysRevB.85.205212 PACS number(s): 72 .15.Jf, 71.15.Mb, 71 .20.−b, 66.70.Df
I. INTRODUCTION
As fossil fuels are being exhausted and their burning
contributes to the global warming, it is urgently needed todevelop clean and renewable sources of energy. Thermoelec-tric materials have attracted much attention because they candirectly convert heat into electricity and vice versa. Suchmaterials utilize the Seebeck effect for power generationand the Peltier effect for refrigeration. The efficiency of athermoelectric material is determined by the so-called figureof merit
ZT=S
2σT/κ, (1)
where Sis the Seebeck coefficient, σis the electrical
conductivity, Tis the absolute temperature, and κis the
thermal conductivity which contains both the lattice ( κl) and
electronic components ( κe). A high ZTvalue indicates good
thermoelectric performance, and one therefore should try toincrease the power factor ( S
2σ) and/or decrease the thermal
conductivity ( κ=κe+κl).1
Intermetallic compounds Mg 2Si, Mg 2Sn, and their solid
solutions could be promising thermoelectric materials atintermediate temperature range between 500–800 K. Thesecompounds contain nontoxic and environmentally friendlyconstituent elements that are inexpensive and abundant in theEarth’s crust.
2,3Mg 2Si can be readily n-type doped, and the
usual donors are Bi and Sb. For example, Tani and Kido4
fabricated the Mg 2SiBi 0.02sample which exhibited a ZTvalue
of 0.86 at 862 K. Akasaka et al.5obtained a ZTvalue of 0.65
at 840 K for the Mg 2Si:Bi 0.01.B u x et al.6found a ZT value
of 0.7 at 775 K for the Mg 2SiBi 0.0015.Y o u et al.7prepared
the Mg 2Si:Bi 0.02sample with a ZT value of 0.7 at 823 K.
The Sb-doped sample Mg 2SiSb 0.02has aZTvalue of 0.56 at
862 K,8while the Mg 2Si0.9Sb0.1shows a ZTvalue of 0.55 at
750 K.9On the other hand, Zaitsev et al.10investigated n-type
Mg 2Si1−xSnxsolid solutions in a broad range of compositions,
and found that the large atomic mass difference between Si andSn dramatically reduces the thermal conductivity of the solid
solutions, which leads to a high- ZT value of about 1.1 near
800 K. Isoda et al.11obtained a ZTvalue of 0.87 at 630 K for
the Bi-doped Mg 2Si0.5Sn0.5. Tani and Kido12found that the
Al-doped Mg 2Si0.9Sn0.1shows a ZTvalue of 0.68 at 864 K,
which is six times larger than that of the undoped solid solution.Luo et al.
13fabricated the Mg 2Si1−xSnx(0/lessorequalslantx/lessorequalslant1.0) solid
solutions and enhanced the ZTvalue to 0.1 at 490 K when x=
0.2. Liu et al.14prepared the low-cost Sb-doped Mg 2Si0.6Sn0.4,
and the ZTvalue can be reached to 1.11 at 860 K. Gao et al.15
reproducibly obtained ZT values larger than 0.9 at 780 K
for their Sb-doped Mg 2Si0.5Sn0.5. Based on the symmetry of
energy bands of Mg 2Si and Mg 2Sn, Zaitsev et al.10reasoned
that for some particular content of Si and Sn, the two low-lying conduction bands of Mg
2Si1−xSnxsolid solutions will
coincide in energy but did not consider consequences of sucha band convergence. Very recently, the positive effect of bandconvergence on the Seebeck coefficient and thus the overallthermoelectric performance of Mg
2Si1−xSnxsolid solutions
was experimentally demonstrated by our team16and yielded a
ZTvalue of 1.3 around 750 K.
The above survey indicates that the measured ZT value
of Mg 2Si-based materials varies in the range of 0.1–1.3 and
depends on the condition of synthesis, alloying proportions,and doping levels. However, the bulk of these works werefocused on experimental studies with little theoretical in-
sight and guidance provided. In this work, we investigate
the thermoelectric properties of n-type doped Mg
2Si1−xSnx
(0.250/lessorequalslantx/lessorequalslant0.875) solid solutions by a multiscale approach,
which includes first-principles calculations, semiclassicalBoltzmann transport theory, and empirical molecular dynam-ics (MD) simulations. We shall see that the Mg
2Si1−xSnxwith
x=0.625 exhibits the highest ZT value at 800 K due to
the highest power factor and the lowest thermal conductivity,
which suggests its promising thermoelectric applications.
The rest of this paper is organized as follows. Sec. II
gives the computational details of our multiscale approach.
205212-1 1098-0121/2012/85(20)/205212(10) ©2012 American Physical SocietyTAN, LIU, LIU, SHI, TANG, AND UHER PHYSICAL REVIEW B 85, 205212 (2012)
In Sec. III, we discuss the electronic, transport, and thermo-
electric properties of n-type Mg 2Si1−xSnx(0.250/lessorequalslantx/lessorequalslant0.875)
solid solutions. A summary of our work is given in Sec. IV.
II. METHOD OF CALCULATIONS
A. First-principles calculations and Boltzmann theory
The energy band-structure calculations have been per-
formed using a first-principles plane-wave pseudopotentialformulation
17–19within the framework of density functional
theory (DFT). The exchange-correlation energy is in theform of Perdew-Wang-91 (Ref. 20) with generalized gradient
approximation (GGA). Ultrasoft pseudopotentials are used for
the Mg, Si, and Sn atoms and the cutoff energy is set as 151 eV .During the geometry optimizations, both the atomic positionsand the lattice constants are fully relaxed until the magnitudeof the force acting on all atoms becomes less than 0.01 eV /˚A,
which also converges the total energy within 1 meV . Theirreducible Brillouin zone (IBZ) is sampled with 10 ×10×10
Monkhorst-Pack kmeshes. The Mg
2Si1−xSnxsolid solutions
are modeled by a rhombohedral 2 ×2×2 supercell which
has eight equivalent positions for the Si or Sn atoms. The Snconcentration xcan thus be 0.250, 0.375, 0.500, 0.625, 0.750,
and 0.875, which are comparable to those measured in Ref. 16.
In a microscopic model of transport process, the Seebeck
coefficient S,the electrical conductivity σ,and the electronic
thermal conductivity κ
ecan be derived from the calculated
band structure in the IBZ. Here, we use a semiclassical ap-proach by solving the Boltzmann’s equation in the relaxation-time approximation.
21The kernel is to find the so-called
transport distribution, which can be expressed as
/Xi1=/summationdisplay
/arrowrighttophalfk/arrowrighttophalfv/arrowrighttophalfk/arrowrighttophalfv/arrowrighttophalfkτ/arrowrighttophalfk, (2)
where/arrowrighttophalfv/arrowrighttophalfkandτ/arrowrighttophalfkare the group velocity and relaxation
time at state/arrowrighttophalfk, respectively. In principle, the relaxation time is
energy dependent. However, it is usually reasonable to assumea constant τfor transport properties since they involve the
derivative of the Fermi function, which is significant only in asmall energy window around the Fermi level and τusually does
not change much over this interval.
22The Seebeck coefficient
Sand the electrical conductivity σare given by
σ=e2/integraldisplay
dε/parenleftbigg
−∂f0
∂ε/parenrightbigg
/Xi1(ε), (3)
S=ekB
σ/integraldisplay
dε/parenleftbigg
−∂f0
∂ε/parenrightbigg
/Xi1(ε)ε−μ
kBT, (4)
where f0is the equilibrium Fermi function, kBis the
Boltzmann’s constant, and μis the chemical potential. The
electronic thermal conductivity κeis calculated according to
the Wiedemann-Franz law23
κe=LσT, (5)
where Lis the Lorenz number.
B. Empirical potential and molecular dynamics
To study the phonon transport in the Mg 2Si1−xSnxsolid
solutions, an accurate interatomic potential is necessary. We
FIG. 1. (Color online) The crystal structure of antifluorite com-
pounds Mg 2Si and Mg 2Sn.
thus develop a modified Morse potential which contains the
two-body bond and three-body angle interactions
U=U1+U2. (6)
Here, the two-body potential U1is in the form of24
U1=D{[1−e−a(r−r0)]2−1}, (7)
where Dis the depth of the potential well, ais the bond
elasticity, rrepresents the interatomic separation, and r0is the
corresponding equilibrium distance. The three-body potentialU
2is given by24
U2=1
2k(cosθ−cosθ0)2, (8)
where kis the force constant, θrepresents the bond angle, and
θ0is the corresponding angle at equilibrium. These potential
parameters can be determined by fitting the energy surfacefrom first-principles calculations.
The lattice thermal conductivity κ
lis then predicted using
MD simulations combined with the Green-Kubo autocorre-lation decay method.
24,25Such an approach can deal with
phonon-phonon scattering and it is possible to obtain anaccurate κ
lfrom the periodic unit cell much smaller than the
phonon mean-free path. In this work, the time step is set as0.5 fs, and the constant temperature simulation with periodicboundary conditions runs for 4 000 000 steps, giving a totaltime of 2.0 ns. The system reaches an equilibrium state andthe temperature stabilizes around the set value after 0.5 ns.The data collection is performed from 0.5 to 2.0 ns and the
TABLE I. Calculated lattice constant a(in unit of ˚A), effective
massm∗(in terms of electron mass m0), and the energy difference /Delta1E
(in units of eV) between the two low-lying conduction bands (CB L
and CB H) for the Mg 2Si1−xSnxsolid solutions.
m∗
xa CB L CB H /Delta1E
0.000 6.35
0.250 6.46 0.19 0.38 0.300.375 6.52 0.24 0.43 0.18
0.500 6.59 0.26 0.43 0.11
0.625 6.64 0.20 0.61 0.0380.750 6.70 0.25 0.52 0.15
0.875 6.76 0.28 0.37 0.27
1.000 6.81
205212-2MULTISCALE CALCULATIONS OF THERMOELECTRIC ... PHYSICAL REVIEW B 85, 205212 (2012)
corresponding heat flux Jis given by
J=/summationdisplay
iεivi+1
2/summationdisplay
i<j[fij·(vi+vj)]rij, (9)
where εiis the site energy of atom i,viandvjare, respectively,
the velocities of atoms iandj,fijis their interaction force,
andrijis their separation. The lattice thermal conductivity κl
is thus obtained by
κl=1
3kBT2V/integraldisplay∞
0/angbracketleftJ(0)·J(t)/angbracketrightdt, (10)
where Vis the volume of the simulation system, and
/angbracketleftJ(0)·J(t)/angbracketrightis the so-called heat-flux autocorrelation func-
tion. Our results are carefully tested with respect to thetotal MD simulation time and the size of the simulationbox.III. RESULTS AND DISCUSSIONS
We begin our discussions with the crystal structures of
Mg 2Si/Mg 2Sn shown in Fig. 1. Pristine Mg 2Si (Mg 2Sn) has
a face-centered-cubic antifluorite lattice with Fm¯3mspace
group. The unit cell contains four primitive cells with theMg and Si (Sn) atoms located at the 8c: (0.25, 0.25, 0.25)and 4a: (0, 0, 0) sites, respectively. The calculated latticeconstants are a=6.35 and 6.81 ˚Af o rt h eM g
2Si and
Mg 2Sn, respectively. These values are very close to those
found experimentally.26The optimized lattice constants of a
series of Mg 2Si1−xSnx(0.250 /lessorequalslantx/lessorequalslant0.875) solid solutions
are summarized in Table I, which increases almost linearly
from 6.47 to 6.76 ˚A as the Sn concentration xis increased.13,27
In particular, the calculated lattice constant ( a=6.59 ˚A) for
the Mg 2Si0.5Sn0.5agrees well with the measured value ( a=
6.56 ˚A),13which confirms the reliability of our theoretical
calculations.
FIG. 2. (Color online) Calculated energy band structures for a series of Mg 2Si1−xSnxsolid solutions. The red and blue lines correspond to
the light and heavy conduction bands, respectively. The Fermi level is at 0 eV .
205212-3TAN, LIU, LIU, SHI, TANG, AND UHER PHYSICAL REVIEW B 85, 205212 (2012)
A. Energy band structure
Figures 2(a)–2(f) show the calculated energy band struc-
tures for a series of Mg 2Si1−xSnxsolid solutions. As known,
both Mg 2Si and Mg 2Sn are indirect gap semiconductors. The
valence-band maximum (VBM) is located at the /Gamma1point,
while the conduction-band minimum (CBM) at the Xpoint.
However, we see from Fig. 2that all the Mg 2Si1−xSnxsolid
solutions become direct gap semiconductors where both theVBM and the CBM appear at the /Gamma1point. This is due to the
fact that we are using a 2 ×2×2 supercell rather than the
primitive cell and the bands are thus folded. It should be notedthat DFT tends to underestimate the energy gap, therefore, allthe calculated band gaps of the Mg
2Si1−xSnxsolid solutions
are corrected in this work. We find that the difference betweenexperimental
26and calculated gap is almost the same for the
Mg 2Si and Mg 2Sn. For simplicity, we thus make a uniform
shift of the calculated band gaps for all the solid solutions.If we focus on the two low-lying conduction bands aroundthe/Gamma1point, we see that with the Sn content xincreasing
from 0.250 to 0.875, the heavy conduction band (CB
H, blue)
shifts down while the light conduction band (CB L, red) shifts
up. As a result, these two bands converge at x=0.625. We
have calculated the band-decomposed charge-density contour(not shown here), and find that the lowest conduction bandhas Si character for the Mg
2Si, while Sn character for the
Mg 2Sn. In the case of the Mg 2Si1−xSnx, the lowest conduction
band exhibits both Si and Sn character; however, the majorcontribution comes from Si when 0 .25/lessorequalslantx< 0.625, and
from Sn when 0 .625<x/lessorequalslant0.875. This is consistent withthe fact that the lowest conduction band of Mg
2Si1−xSnxis
the light band for 0 .25/lessorequalslantx< 0.625 but is the heavy band
for 0.625<x/lessorequalslant0.875 (see Fig. 2). Atx=0.625, the light
and heavy bands converge and the charge-density contourindicates that the contributions from Si and Sn are nearlyequal to each other. Overall speaking, the effective mass ofthe lowest conduction band increases with the Sn content x,
reaches a maximum at x=0.625, and then decreases. Both
the convergence of the conduction bands and the increasedeffective mass lead to a high absolute value of the Seebeckcoefficient,
1,28which is very beneficial to the thermoelectric
performance. We will come back to this point later. The energydifference between the CB
Hand CB Las well as their effective
masses are summarized in Table I.
B. Relaxation time and electrical conductivity
Based on the calculated band structures, we are able to eval-
uate the electronic transport coefficients of the Mg 2Si1−xSnx
solid solutions by using the semiclassical Boltzmann theory
and the rigid-band approach.29To get reliable results, we use
a very dense kmesh up to 1000 points in our calculations. It
should be mentioned that the electrical conductivity σcan
only be calculated with respect to the electron relaxationtimeτ, that is, what we actually get is σ/τ. The relaxation
time is then determined by comparing the experimentallymeasured electrical conductivity σ(Ref. 16) at a particular
carrier concentration and temperature. The fitted relaxationtime for the solid solutions is summarized in Table II.W es e e
TABLE II. Determining the electron relaxation time τfor the Mg 2Si1−xSnxsolid solutions by comparing the experimentally measured
electrical conductivity at different carrier concentration and temperature.
τ(fs)
xn (1020cm−3) 300 K 400 K 500 K 600 K 700 K 800 K
0.25 1.70 11.4 9.28 7.87 6.26 5.22 4.31
2.10 12.3 10.1 8.13 6.85 5.7 4.43
(Mean value) 11.8 9.7 8.0 6.6 5.5 4.4
0.375 1.35 6.74 5.54 4.85 4.17 3.54 2.95
1.80 6.32 5.27 4.31 3.76 3.23 2.74
2.35 6.44 5.41 4.46 3.62 3.10 2.59
(Mean value) 6.5 5.4 4.5 3.8 3.3 2.8
0.5 1.90 5.43 4.62 3.84 3.39 2.74 2.37
2.39 3.99 3.4 2.86 2.57 2.15 1.96
2.84 4.41 3.82 3.26 2.74 2.28 1.99
(Mean value) 4.6 3.9 3.3 2.9 2.4 2.1
0.625 0.63 7.65 6.71 5.58 4.55 4.09 3.73
1.68 6.97 5.80 4.75 4.18 3.59 2.98
(Mean value) 7.3 6.3 5.1 4.4 3.8 3.4
0.75 1.57 6.34 5.09 4.06 3.51 2.83 2.61
1.69 6.99 5.67 4.56 3.93 3.39 2.921.72 6.78 5.96 4.79 3.83 3.30 2.84
2.13 6.95 5.58 4.45 3.86 3.13 1.69
(Mean value) 6.8 5.6 4.5 3.8 3.2 2.5
0.875 1.70 7.87 6.44 5.14 4.06 3.41 2.87
2.30 8.21 6.69 5.52 4.58 3.73 2.99
(Mean value) 8.0 6.6 5.3 4.3 3.6 2.9
205212-4MULTISCALE CALCULATIONS OF THERMOELECTRIC ... PHYSICAL REVIEW B 85, 205212 (2012)
that the relaxation time does not vary much in the carrier-
concentration range considered. In contrast, there is obvioustemperature dependence. As a reasonable approximation, inthe following discussions we use an average relaxation time ateach temperature. For a particular x, we see from Table IIthat
the relaxation time decreases with increasing temperature. Thisis reasonable since the electron scattering is more frequent athigh temperatures. At a particular temperature, we see that therelaxation time decreases before reaching a minimum value atx=0.5 and then increases, suggesting that the solid solution
atx=0.5 is of highest disorder as far as the carrier transport
is concerned.
Figure 3(a) plots the calculated electrical conductivity σas
a function of temperature for the Mg
2Si1−xSnxsolid solutions,
where the doping level (or carrier concentration) of eachsystem is fixed at 1.9 ×10
20cm−3. In the temperature range of
300–800 K, we see that the electrical conductivity decreaseswith increasing temperature. This is believed to be causedby the reduced electron mobility at high temperatures, whilethe carrier (electron) concentration remains almost constantwith temperature. It is interesting that the calculated σof
Mg
2Si0.375Sn0.625decreases relatively slower with increasing
temperature and is higher than those of the other solid solutionsexcept x=0.250 and 0.875 at intermediate temperatures.
On the other hand, we plot in Fig. 3(b) the calculated
electrical conductivity σas a function of carrier concentration,
FIG. 3. (Color online) (a) Calculated electrical conductivity σof
Mg 2Si1−xSnxsolid solutions as a function of temperature. All systems
have a carrier concentration of 1.9 ×1020cm−3. (b) Calculated σas
a function of carrier concentration n(1019–1021cm−3) at 800 K.where the temperature is fixed at 800 K. As expected, the
electrical conductivity σincreases with increasing carrier con-
centration n. The electrical conductivity of Mg 2Si0.750Sn0.250
and Mg 2Si0.375Sn0.625are higher than those of the other
solid solutions with Mg 2Si0.625Sn0.375and Mg 2Si0.500Sn0.500
exhibiting the lowest values.
C. Seebeck coefficient and power factor
Figure 4(a) shows the calculated Seebeck coefficient Sas
a function of temperature for the Mg 2Si1−xSnxsolid solutions
at a carrier concentration of 1.9 ×1020cm−3. It is found that
the absolute value of the Seebeck coefficient increases withincreasing temperature. Compared with the experimentallymeasured result of Mg
2Si0.500Sn0.500, the calculated Sexhibits
a similar variation with temperature, but the absolute value issomehow underestimated. In Fig. 4(b), we plot the calculated
Seebeck coefficient at 800 K as a function of carrier concentra-tion in the range of 10
19–1021cm−3. We see the absolute value
of Seebeck coefficient increases initially, reaches a maximum,and then decreases as the carrier concentration increases. Asmentioned above, when the two conduction bands convergeand the effective mass increases, one can obtain a verylarge absolute value of Seebeck coefficient. Indeed, exceptat very low carrier concentrations, Mg
2Si0.375Sn0.625exhibits
the highest absolute value of Samong these solid solutions.
FIG. 4. (Color online) (a) Calculated Seebeck coefficient Sof
Mg 2Si1−xSnxsolid solutions as a function of temperature. All systems
have a carrier concentration of 1.9 ×1020cm−3, and the measured
result of Mg 2Si0.500Sn0.500is also shown. (b) Calculated Sas a function
of carrier concentration n(1019–1021cm−3) at 800 K.
205212-5TAN, LIU, LIU, SHI, TANG, AND UHER PHYSICAL REVIEW B 85, 205212 (2012)
Although there is some underestimation of the calculated
Seebeck coefficient, it is still more than 250 μV/Kf o rt h e
Mg 2Si0.375Sn0.625at a carrier concentration of 1020cm−3,
which is close to those measured previously.10,14,30We want
to mention that the underestimation of the calculated Seebeckcoefficient indicated in Fig. 4(a) can be attributed to the fact
that in the experiment,
16the sample prepared deviates a little
from the standard nominal formula of Mg 2Si0.5Sn0.5and has
a real composition of Mg 2.11Si0.52Sn0.48Sb0.006.Here, the Mg
excess was found to increase the absolute value of the Seebeckcoefficient.
33
Figure 5(a) shows the calculated power factor S2σas a
function of temperature for the Mg 2Si1−xSnxsolid solutions
at a carrier concentration of 1.9 ×1020cm−3. As discussed
above, the Mg 2Si0.375Sn0.625exhibits the highest absolute value
of Seebeck coefficient Sdue to the band convergence and
enhanced effective mass. It also has a relatively higher elec-trical conductivity σ,especially at intermediate temperatures.
Consequently, the power factor of Mg
2Si0.375Sn0.625is much
higher than those of the other solid solutions in the wholetemperature range from 300 to 800 K. On the other hand,we plot in Fig. 5(b) the power factor at 800 K as a function
of carrier concentration in the range of 10
19–1021cm−3.A s
known, the Seebeck coefficient decreases with increasing car-rier concentration, while the electrical conductivity increases.In a broad range of carrier concentrations, the power factor
FIG. 5. (Color online) (a) Calculated power factor S2σof
Mg 2Si1−xSnxsolid solutions as a function of temperature. All systems
have a carrier concentration of 1.9 ×1020cm−3. (b) Calculated S2σ
as a function of carrier concentration n(1019–1021cm−3) at 800 K.should therefore increase initially, reach a maximum value, and
then decrease. Within the carrier concentration considered inthis work, we see from Fig. 5(b) that the power factor increases
monotonously. The calculated value of Mg
2Si0.375Sn0.625at
800 K is about 3.1 ×10−3W/mK2with a carrier concentration
of 1020cm−3, which is somehow higher than that of other
Mg 2Si-based thermoelectric materials.15,31,32
D. Electronic thermal conductivity
We now discuss the electronic thermal conductivity κeof
the Mg 2Si1−xSnxsolid solutions. As mentioned before, the
κecan be derived from σusing Eq. (5). Here, the Lorenz
number Lfor the Mg 2Si1−xSnxsolid solutions is estimated
based on the Fermi-Dirac statistics noted in Ref. 11which
attains the fully degenerate value of 2.45 ×10−8V2/K2as the
carrier concentration reaches typical metallic densities. For theMg
2Si1−xSnxsolid solutions, the Lorenz number is found to
be 1.8 ∼1.9×10−8V2/K2around the carrier concentration
of 1020cm−3(Refs. 11and 33) and is therefore used in our
calculations. Figure 6(a) shows the temperature dependence
of the calculated κefor the Mg 2Si1−xSnxseries at a carrier
concentration of 1.9 ×1020cm−3. The general trend observed
is an initial mild increase followed by a rather undistinguishedmaximum and finally a slowly decreasing conductivity. The
FIG. 6. (Color online) (a) Calculated electronic thermal conduc-
tivityκeof Mg 2Si1−xSnxsolid solutions as a function of temperature.
All systems have a carrier concentration of 1.9 ×1020cm−3.( b )C a l -
culated κeas a function of carrier concentration n(1019–1021cm−3)
at 800 K.
205212-6MULTISCALE CALCULATIONS OF THERMOELECTRIC ... PHYSICAL REVIEW B 85, 205212 (2012)
calculated κeof Mg 2Si0.375Sn0.625is 1.17 W /mK at 300 K and
1.31 W /mK at 800 K.
A ss h o w ni nF i g . 6(b), the carrier-concentration dependence
of the electronic thermal conductivity κeat 800 K mimics
the behavior of the electrical conductivity σ, i.e., it increases
monotonously with increasing carrier concentration. TheMg
2Si0.750Sn0.250and Mg 2Si0.375Sn0.625solid solutions exhibit
higher κethan the others in the whole carrier-concentration
range from 1019to 1021cm−3. Compared with the power
factor shown in Fig. 5(b), the electronic thermal conductivity
of Mg 2Si0.375Sn0.625decreases faster with decreasing carrierconcentration naround 1020cm−3. For example, as nde-
creases from 2.0 to 1.0 ×1020cm−3at 800 K, the κeof
Mg 2Si0.375Sn0.625falls by 48%, while the power factor S2σis
reduced by only 21%. In the following discussions, we shall seethat such a trend is important for optimizing the thermoelectricperformance of these solid solutions.
E. Lattice thermal conductivity
To deal with the phonon transport, we have performed MD
simulations where a modified Morse potential is constructed
TABLE III. Fitted parameters in the modified Morse potential for the Mg 2Si, Mg 2Sn, and Mg 2Si1−xSnxsolid solutions.
Two-body Three-body
D(eV) a(˚A−1) r0(˚A) k(eV) θ0(◦)
Mg 2Si Mg-Mg 0.789 0.853 3.17 Si-Mg-Si 1.223 109.47
Mg-Si 0.628 0.399 2.75 Mg-Si-Mg 1.120 109.47
Si-Si 0.460 0.991 4.49 Mg-Si-Mg 1.120 70.53
x=0.250 Mg-Mg 0.765 0.952 3.232 Si/Sn-Mg-Si/Sn 0.958 109.47
Mg-Si 0.548 0.689 2.799 Mg-Si/Sn-Mg 0.958 109.47
Mg-Sn 0.545 0.686 2.799 Mg-Si/Sn-Mg 0.942 70.53
Si-Si 0.437 0.836 4.571
Sn-Sn 0.436 0.837 4.571
Si-Sn 0.434 0.835 4.571
x=0.375 Mg-Mg 0.898 0.924 3.257 Si/Sn-Mg-Si/Sn 1.106 109.47
Mg-Si 0.639 0.576 2.821 Mg-Si/Sn-Mg 1.106 109.47
Mg-Sn 0.635 0.574 2.821 Mg-Si/Sn-Mg 1.102 70.53
Si-Si 0.481 0.873 4.606
Sn-Sn 0.481 0.872 4.606
Si-Sn 0.480 0.872 4.606
x=0.500 Mg-Mg 0.810 0.904 3.284 Si/Sn-Mg-Si/Sn 1.168 109.47
Mg-Si 0.559 0.561 2.844 Mg-Si/Sn-Mg 1.168 109.47
Mg-Sn 0.555 0.560 2.844 Mg-Si/Sn-Mg 1.164 70.53
Si-Si 0.467 0.862 4.644
Sn-Sn 0.469 0.861 4.644
Si-Sn 0.468 0.858 4.644
x=0.625 Mg-Mg 0.735 0.789 3.316 Si/Sn-Mg-Si/Sn 1.144 109.47
Mg-Si 0.569 0.536 2.872 Mg-Si/Sn-Mg 1.144 109.47
Mg-Sn 0.568 0.538 2.872 Mg-Si/Sn-Mg 1.140 70.53
Si-Si 0.461 0.862 4.489
Sn-Sn 0.462 0.859 4.489
Si-Sn 0.460 0.858 4.489
x=0.750 Mg-Mg 0.658 0.866 3.350 Si/Sn-Mg-Si/Sn 1.236 109.47
Mg-Si 0.490 0.723 2.901 Mg-Si/Sn-Mg 1.236 109.47
Mg-Sn 0.489 0.723 2.901 Mg-Si/Sn-Mg 1.228 70.53
Si-Si 0.431 0.862 4.738
Sn-Sn 0.430 0.861 4.738
Si-Sn 0.428 0.863 4.738
x=0.875 Mg-Mg 0.883 0.894 3.380 Si/Sn-Mg-Si/Sn 1.328 109.47
Mg-Si 0.681 0.455 2.927 Mg-Si/Sn-Mg 1.328 109.47
Mg-Sn 0.682 0.452 2.927 Mg-Si/Sn-Mg 1.324 70.53
Si-Si 0.453 0.658 4.780
Sn-Sn 0.454 0.661 4.780
Si-Sn 0.453 0.660 4.780
Mg 2Sn Mg-Mg 0.742 0.797 3.41 Sn-Mg-Sn 1.158 109.47
Mg-Sn 0.597 0.356 2.95 Mg-Sn-Mg 1.140 109.47
Sn-Sn 0.329 1.050 4.82 Mg-Sn-Mg 1.140 70.53
205212-7TAN, LIU, LIU, SHI, TANG, AND UHER PHYSICAL REVIEW B 85, 205212 (2012)
FIG. 7. (Color online) Calculated lattice thermal conductivity κl
of Mg 2Si1−xSnxand Sb-doped Mg 2Si0.375Sn0.625solid solutions at a
temperature range from 300 to 800 K.
by fitting the energy surface from first-principles calculations.
Table IIIlists all the fitted potential parameters used in
this work. Our calculated lattice thermal conductivity κlof
Mg 2Si and Mg 2Sn is found to be 5.98 and 4.90 W /mK at
300 K, respectively. These values are consistent with thosemeasured experimentally
13,26and thus confirm the reliability
of our MD simulations. The calculated κlfor the Mg 2Si1−xSnx
series is plotted in Fig. 7in a wide temperature range.
We see that the lattice thermal conductivity does not showstrong temperature dependence. At any given temperature,the thermal conductivity initially decreases with increasingcontent of Sn. It reaches a clear minimum at x=0.625
and then increases. Note that the minimum lattice thermalconductivity is obtained at x=0.625 rather than at x=
0.5 where the Si and Sn should be most disordered. Thecalculated κ
lfor Mg 2Si0.375Sn0.625is only 1.62 W /mK at
300 K, representing merely 27% of that of the pure Mg 2Si.
Such significant reduction in the lattice thermal conductivity isprimarily a result of the large atomic mass difference betweenthe Sn and Si that gives rise to a strong mass defect scattering.Our calculated variation of the lattice thermal conductivity asa function of Sn content is similar to the previously reportedresults.
13,16,26
F. Optimized ZTvalue
With all transport coefficients available, we can now eval-
uateZTvalues of the Mg 2Si1−xSnxsolid solutions according
to Eq. (1). Figure 8(a) shows the carrier-concentration depen-
dence of the calculated ZTvalues at 800 K. As noted before,
good thermoelectric performance requires a power factor S2σ
as high as possible and a thermal conductivity κ(=κe+κl)
as low as possible. With higher S2σand lower κthan the other
solid solutions, we see that Mg 2Si0.375Sn0.625exhibits superior
thermoelectric performance in the whole carrier-concentrationrange ( n=10
19–1021cm−3). In particular, a ZT value of
1.11 can be achieved at 800 K with a carrier concentration of1.2×10
20cm−3. This is consistent with the conjecture
that band convergence and the maximum effective masswill lead to enhanced thermoelectric performance. The ZT
FIG. 8. (Color online) (a) Calculated ZTvalue of Mg 2Si1−xSnx
and Sb-doped Mg 2Si0.375Sn0.625solid solutions as a function of carrier
concentration n(1019–1021cm−3) at 800 K. (b) Calculated ZT
value at optimal carrier concentration of Mg 2Si1−xSnxand Sb-doped
Mg 2Si0.375Sn0.625as a function of temperature.
values of other Mg 2Si1−xSnxsolid solutions, together with the
optimal carrier concentration n, the power factor S2σ, and the
electronic and lattice thermal conductivities κeandκlare listed
in Table IV.
Figure 8(b) plots the ZTvalues of these solid solutions as a
function of temperature at optimal carrier concentrations. Wesee that the ZTvalues increase from 300 to 800 K, which is
consistent with those found previously.
12,14–16Moreover, all
the Mg 2Si1−xSnxsolid solutions exhibit higher ZTvalues at
800 K than at lower temperatures, which indicates that theycould be applied as thermoelectric materials at intermediatetemperatures. The optimized ZT value of 1.11 achieved at
800 K with Mg
2Si0.375Sn0.625is by far the highest value among
all the solid solutions.
Although a ZT value of 1.11 at 800 K is close to
the maximum experimental value reported,10,14,16,33this is a
very conservative value and we believe the thermoelectricperformance of n-type Mg
2Si1−xSnxsolid solutions could
be further enhanced. It should be mentioned that up to now,we were considering electron and phonon transport in pristineMg
2Si1−xSnxsolid solutions. In real experiments, however, the
different carrier concentrations are actually realized by dopingthese systems with Bi or Sb. According to previous experi-mental works,
9,16,30,32appropriate doping not only enhances
205212-8MULTISCALE CALCULATIONS OF THERMOELECTRIC ... PHYSICAL REVIEW B 85, 205212 (2012)
TABLE IV . Optimized ZT values of Mg 2Si1−xSnxas well as Sb-doped Mg 2Si0.375Sn0.625solid solutions at 800 K. The corresponding
optimal carrier concentration n, the power factor S2σ, and the electronic and lattice thermal conductivity κeandκlare also indicated.
xn (1020cm−3) S2σ(10−3W/mK2) κe(W/mK) κl(W/mK) ZT
0.250 4.5 5.20 3.54 3.79 0.57
0.375 3.6 3.32 1.86 4.00 0.45
0.500 2.7 2.54 1.18 3.66 0.420.625 1.2 3.24 0.85 1.48 1.11
0.625 (Sb-doped) 1.1 3.15 0.78 1.25 1.24
0.750 2.1 2.83 1.16 2.27 0.660.875 2.0 3.10 1.38 2.77 0.60
the power factor S2σ, but can also significantly reduce the
lattice thermal conductivity κlof Mg 2Si-based thermoelectric
materials. To make a better comparison between our theoreticalpredications and the experimental results, we have done addi-tional calculations where the lattice thermal conductivity κ
lof
properly Sb-doped Mg 2Si0.375Sn0.625is explicitly calculated.
Indeed, it is found that κlis decreased by about 15% (see
Fig. 7). As a result, the optimal carrier concentration is
shifted to a value somewhat smaller than 1.1 ×1020cm−3
as indicated in Fig. 8(a). At such carrier concentration, the
power factor S2σis slightly reduced from 3.24 to 3.15 ×
10−3W/mK2. However, there is a faster decrease of the
corresponding electronic thermal conductivity κe(from 0.85 to
0.78 W /mK), leading to a higher ZTvalue of 1.24 as indicated
in both Fig. 8(a) and Table IV. We want to emphasize that
the lattice thermal conductivity κlof the Mg 2Si1−xSnxsolid
solutions could be further reduced by many other means suchas isotope doping
34and embedded nanoinclusions,35–37which
leave the power factor S2σless affected. If the lattice thermal
conductivity could be reduced to ∼0.8 W/mK,16the optimized
ZT values of Mg 2Si0.375Sn0.625should reach 1.61 at 800 K,
which would be very competitive, especially given the low costand minimal environmental impact of this material system.
IV . SUMMARY
In summary, we have studied the thermoelectric properties
of Mg 2Si1−xSnx(0.250/lessorequalslantx/lessorequalslant0.875) solid solutions using a
multiscale approach. The convergence of the two conductionbands and the increased electron effective mass lead to ahigh value of the Seebeck coefficient for x=0.625, which
has been confirmed by explicit calculations of the electronictransport coefficients. On the other hand, Mg
2Si0.375Sn0.625
exhibits the lowest lattice thermal conductivity among all the
solid solutions considered due to the high alloy disorder and alarge Sn/Si mass difference scattering. Our theoretical resultsindicate that the maximum ZT value of Mg
2Si1−xSnxsolid
solutions is ∼1.24 at 800 K for x=0.625 with a carrier
concentration of n=1.1×1020cm−3. Moreover, we find that
theZTvalue is limited by the high lattice thermal conductivity,
which for some solid solutions is as high as 4.0 W /mK.
Therefore, there is still room to improve the thermoelectricperformance of the Mg
2Si1−xSnxsolid solutions. In any case,
then-type Mg 2Si0.375Sn0.625is a very promising thermoelectric
material, and major effort should be directed to developingcomparatively effective p-type Mg
2Si1−xSnxsolid solutions
so that efficient thermoelectric modules based on this inex-pensive and environmentally friendly material system couldbe realized.
ACKNOWLEDGMENTS
This work was supported by the “973 Program” of China
(Grant No. 2007CB607501), the National Natural ScienceFoundation (Grant No. 51172167), and the Program for NewCentury Excellent Talents in University. C. Uher is supportedby the CERC-CVC U.S.-China Program of Clean Vehicleunder Award Number DE-PI0000012. All the calculationswere performed in the PC Cluster from Sugon Company ofChina.
*Author to whom correspondence should be addressed:
phlhj@whu.edu.cn
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205212-10 |
PhysRevB.81.075407.pdf | Many-body corrections to cyclotron resonance in monolayer and bilayer graphene
K. Shizuya
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
/H20849Received 25 October 2009; published 5 February 2010 /H20850
Cyclotron resonance in graphene is studied with focus on many-body corrections to the resonance energies,
which evade Kohn’s theorem. The genuine many-body corrections turn out to derive from vacuum polariza-tion, specific to graphene, which diverges at short wavelengths. Special emphasis is placed on the need forrenormalization, which allows one to determine many-body corrections uniquely from one resonance to an-other. For bilayer graphene, in particular, both intralayer and interlayer coupling strengths undergo infiniterenormalization; as a result, the renormalized velocity and interlayer coupling strength run with the magneticfield. A comparison of theory with the experimental data is made for both monolayer and bilayer graphene.
DOI: 10.1103/PhysRevB.81.075407 PACS number /H20849s/H20850: 73.22.Lp, 73.43.Lp, 76.40. /H11001b
I. INTRODUCTION
Graphene, a monolayer graphite, attracts great attention
for its unusual electronic transport1–6as well as its potential
applications. It supports as charge carriers massless Diracfermions, which lead to such exotic phenomena as the half-integer quantum-Hall /H20849QH/H20850effect and minimal conductivity.
The multispinor character of the electrons in graphene
derives from the sublattice structure of the underlying hon-eycomb lattice, and this immediately implies, in the low-energy effective theory of graphene, the quantum nature ofthe vacuum state;
7the conduction and valence bands are re-
lated by charge conjugation and, in particular, the latter actsas the Dirac sea. Graphene in a magnetic field Bthus gives
rise to a particle-hole symmetric “relativistic” pattern of Lan-
dau levels, with spectra
/H9280n/H11008/H11006/H20881/H20841n/H20841/H20881Bunequally spaced, to-
gether with four characteristic zero-energy Landau levels/H20849whose presence has a topological origin
8/H20850.
This nontrivial vacuum structure is the key feature that
distinguishes graphene and its multilayers from conventionalQH systems. In particular, bilayer graphene
9supports, as a
result of interlayer coupling, massive fermions, which, in amagnetic field, again develop a particle-hole symmetrictower of Landau levels, with an octet of zero-energy levels.
10
Bilayer graphene has a unique property that its band gap isexternally controllable.
11–14
Graphene and its multilayers give rise to rich spectra of
cyclotron resonance, with resonance energies varying fromone transition to another within the electron band or the holeband, and, notably, even between the two bands. This is insharp contrast with conventional QH systems with a para-bolic dispersion, where cyclotron resonance /H20849optically in-
duced at zero-momentum transfer k=0/H20850takes place between
adjacent Landau levels, hence at a single frequency
/H9275c=eB /m/H11569which, according to Kohn’s theorem,15is unaf-
fected by Coulomb interactions. Nonparabolicity16of the
electronic spectra in graphene evades Kohn’s theorem andoffers the possibility to detect the many-body corrections tocyclotron resonance, as discussed theoretically for mono-layer graphene.
17,18
Experiment has already studied via infrared spectroscopy
cyclotron resonance in monolayer19,20and bilayer21
graphene, and verified the characteristic features of the asso-
ciated Landau levels. Data generally show no clear sign ofthe many-body effect, except for one19on monolayer
graphene.
The purpose of this paper is to study the many-body effect
on cyclotron resonance in graphene, by constructing an ef-fective theory of cyclotron resonance within the single-modeapproximation. It is shown that the genuine many-body cor-rections arise from vacuum polarization, specific tographene, which actually diverges logarithmically at shortwavelengths and requires renormalization. Our approach inpart recovers results of earlier studies
17,18on monolayer
graphene but essentially differs from them in this handling ofcutoff-dependent corrections by renormalization, which al-lows one to determine many-body corrections uniquely fromone resonance to another. Our analysis also reveals that forbilayer graphene both intralayer and interlayer couplingstrengths undergo renormalization.
22,23We compare theory
with the experimental data for monolayer and bilayergraphene.
In Sec. IIwe briefly review the effective theory of
graphene and, in Sec. III, study cyclotron resonance in
monolayer graphene, with focus on the many-body correc-tions and renormalization. In Sec. IVwe extend our analysis
to bilayer graphene. Section Vis devoted to the summary
and discussion.
II. MONOLAYER GRAPHENE
Graphene has a honeycomb lattice which consists of two
triangle sublattices of carbon atoms. The electrons ingraphene are described by a two-component spinor field/H20849
/H9274A,/H9274B/H20850on two inequivalent lattice sites /H20849A,B/H20850. The elec-
trons acquire a linear spectrum near the two inequivalentFermi points /H20849KandK
/H11032/H20850in the Brillouin zone, with the “light
velocity” v0=/H20849/H208813/2/H20850aL/H92530//H6036/H11015106m/s related to the intra-
layer coupling/H92530/H11013/H9253AB/H110152.9 eV /H20849with aL=0.246 nm /H20850.
Their low-energy features are described by an effective
Hamiltonian of the form7
H0=/H20885d2x/H20851/H9274†H+/H9274+/H9274/H11032†H−/H9274/H11032/H20852,
H/H11006=v0/H20849/H92681/H90161+/H92682/H90162/H11007/H9254m/H92683/H20850−eA0, /H208492.1/H20850
where/H9016i=−i/H11509i+eAi/H20851i=/H208491,2 /H20850or/H20849x,y/H20850/H20852involve coupling to
external electromagnetic potentials A/H9262=/H20849Ai,A0/H20850. HerePHYSICAL REVIEW B 81, 075407 /H208492010 /H20850
1098-0121/2010/81 /H208497/H20850/075407 /H2084910/H20850 ©2010 The American Physical Society 075407-1/H9274/H11013/H20849/H92741,/H92742/H20850t=/H20849/H9274A,/H9274B/H20850tstands for the electron field at one /H20849or
K/H20850valley, and/H9274/H11032=/H20849/H9274B/H11032,/H9274A/H11032/H20850tto one at another valley, with A
andBreferring to the associated lattice sites. /H9254mdenotes a
possible tiny asymmetry in sublattices.
Let us place graphene in a uniform magnetic field
B/H11036=B/H110220; we set Ai→B/H20849−y,0/H20850. The electron spectrum then
forms an infinite tower of Landau levels Lnof energy
/H9280n=sn/H9275c/H20881/H20841n/H20841+/H20849/H9254m/H208502/H51292/2 /H208492.2/H20850
labeled by integers n=0,/H110061,/H110062,..., and px
/H20849ory0/H11013/H51292pxwith the magnetic length /H5129/H110131//H20881eB/H20850. Here
sn/H11013sgn/H20853n/H20854=/H110061 specifies the sign of the energy /H9280nand
/H9275c/H11013/H208812v0//H5129/H1101536.3/H11003v0/H20851106m/s/H20852/H20881B/H20851T/H20852meV /H208492.3/H20850
is the basic cyclotron frequency; v0/H20851106m/s/H20852stands for v0
in units of 106m/s and B/H20851T/H20852stands for a magnetic field in
tesla.
Suppose, without loss of generality, that /H9254m/H110220.
Then the n=0 level at the Kvalley has positive energy
/H92800+=v0/H9254m/H110220 while the n=0 level at the K/H11032valley has nega-
tive energy/H92800−=−v0/H9254m. In general, the spectra at the two
valleys are related as /H9280n/H20841K=−/H9280−n/H20841K/H11032. With the electron spin
taken into account /H20849and Zeeman splitting ignored for sim-
plicity /H20850, each Landau level is thus fourfold degenerate, ex-
cept for the doubly degenerate n=0/H11006levels split in valley.
With this feature in mind, we shall set the asymmetry
/H9254m→0 in what follows.
The Coulomb interaction is written as
HCoul=1
2/H20858
pvp:/H9267−p/H9267p:, /H208492.4/H20850
where/H9267pis the Fourier transform of the electron density
/H9267=/H9274†/H9274+/H9274/H11032†/H9274/H11032/H20849here/H9274†/H9274, e.g., is summed over spinor and
spin indices /H20850;vp=2/H9266/H9251//H20849/H9280b/H20841p/H20841/H20850is the Coulomb potential with
the fine-structure constant /H9251=e2//H208494/H9266/H92800/H20850/H110151/137 and the
substrate dielectric constant /H9280b.
The Landau-level structure is made explicit by
passing to the /H20841n,y0/H20856basis, with the expansion
/H9274/H20849x,t/H20850=/H20858n,y0/H20855x/H20841n,y0/H20856/H9274n/H20849y0,t/H20850./H20849For conciseness, we shall
only display the /H9274sector from now on. /H20850The Hamiltonian H0
is thereby rewritten as
H0=/H20885dy0/H20858
n=−/H11009/H11009
/H9274n†/H20849y0,t/H20850/H9280n/H9274n/H20849y0,t/H20850/H20849 2.5/H20850
and the charge density /H9267−p/H20849t/H20850=/H20848d2xeip·x/H9274†/H9274as24
/H9267−p=/H20858
k,n=−/H11009/H11009
/H9267−pkn=/H20858
k,n=−/H11009/H11009
gpknRpkn,
Rpkn=/H9253p/H20885dy0/H9274k†/H20849y0,t/H20850eip·r/H9274n/H20849y0,t/H20850, /H208492.6/H20850
where/H9253p=e−/H51292p2/4; and r=/H20849rx,ry/H20850=/H20849i/H51292/H11509//H11509y0,y0/H20850stands for
the center coordinate with uncertainty /H20851rx,ry/H20852=i/H51292. The
charge operators Rpknobey the W/H11009algebra25/H20851Rkmm/H11032,Rpnn/H11032/H20852=/H9254m/H11032nek†p/2Rk+pmn/H11032−/H9254n/H11032mep†k/2Rk+pnm/H11032,/H208492.7/H20850
where k†p=k·p−ik/H11003pwith k/H11003p/H11013kxpy−kypx. This actu-
ally consists of two W/H11009algebras associated with intralevel
center-motion25and interlevel mixing26of electrons.
The coefficient matrix gpknis given by
gpkn=1
2bkbn/H20849fp/H20841k/H20841−1,/H20841n/H20841−1+sksnfp/H20841k/H20841,/H20841n/H20841/H20850, /H208492.8/H20850
where bn=1 for n/HS110050 and b0=/H208812:
fpkn=/H20881n!
k!/H20873−/H5129p
/H208812/H20874k−n
Ln/H20849k−n/H20850/H208731
2/H51292p2/H20874 /H208492.9/H20850
fork/H11350n/H113500 and fpnk=/H20849f−pkn/H20850†;p=px−ipy. Note that gpknare
essentially the same at the two valleys, i.e., gpkn/H20841K/H11032=gpkn/H20841Kand
gpk0−/H20841K/H11032=gpk0+/H20841Kfor /H20841k/H20841/H113501 and /H20841n/H20841/H113501; one simply needs to
specify n=0/H11006accordingly.
III. CYCLOTRON RESONANCE
In this section we study cyclotron resonance in monolayer
graphene. Let us first note that the charge operator
/H9267−pkn=gpknRpknin Eq. /H208492.6/H20850annihilates an electron at the nth
level Lnand creates one at the kth level Lk. One may thus
associate it with the interlevel transition Ln→Lkor regard it
as an interpolating operator for the exciton consisting of ahole at L
nand an electron at Lk.
To describe such inter-Landau-level excitations one can
make use of a nonlinear realization of the W/H11009algebra, as is
familiar from the theory of quantum-Hall ferromagnet.27One
may start with a given ground state /H20841G/H20856and describe an
excited collective state /H20841G˜/H20856over it as a local rotation in
/H20849n,y0/H20850space
/H20841G˜/H20856=e−iO/H20841G/H20856, /H208493.1/H20850
where the operator e−iOwith
O=/H20858
p/H9253p−1/H9021pknRpkn/H208493.2/H20850
locally rotates /H20841G/H20856by “angles”/H9021pkn, which define textures in
/H20849n,y0/H20850space. /H20849Remember here that /H9267p/H11011Rpknare diagonal in
spin so that/H9021pkncarry the spin index as well, though it is
suppressed. In principle, one has to retain in Oall possible
pairs/H9021pknand /H20849/H9021pkn/H20850†=/H9021−pnkcontributing to the Ln→Lktran-
sition. /H20850
Repeated use of the charge algebra in Eq. /H208492.7/H20850
allows one to express the texture-state energy
/H20855G˜/H20841H/H20841G˜/H20856=/H20855G/H20841eiOHe−iO/H20841G/H20856with H=H0+HCoulas a functional
of/H9021pknor its x-space representative /H9021kn/H20849x,t/H20850. The kinetic
term for/H9021knis supplied from the electron kinetic term and
one can write the effective Lagrangian for /H9021as
L/H9021=/H20855G˜/H20841/H20849i/H11509t−H/H20850/H20841G˜/H20856=/H20855G/H20841eiO/H20849i/H11509t−H/H20850e−iO/H20841G/H20856./H208493.3/H20850
This representation systematizes the single-mode
approximation25/H20849SMA /H20850within a variational framework.26
The present theory thus embodies the nonperturbative fea-
tures of the SMA.K. SHIZUYA PHYSICAL REVIEW B 81, 075407 /H208492010 /H20850
075407-2Indeed, for a transition from the filled level Ln/H20849with den-
sity/H9267¯n/H20850to the empty level Lk, one finds
L/H9021=/H9267¯n/H20858
p/H9021−pnk/H20849i/H11509t−/H9280pexc/H20850/H9021pkn+¯ /H208493.4/H20850
with the excitation spectrum given by the SMA formula
/H9280pexc=/H20855G/H20841/H20851/H9267pnk,/H20851H,/H9267−pkn/H20852/H20852/H20841G/H20856//H20855G/H20841/H9267pnk/H9267−pkn/H20841G/H20856, /H208493.5/H20850
i.e., as the oscillator strength divided by the static structure
factor
/H20855G/H20841/H9267pnk/H9267−pkn/H20841G/H20856//H9024=/H9267¯n/H9253p2/H20841gpkn/H208412, /H208493.6/H20850
where/H9024=/H20848d2x.
Let us first consider the n=0/H11006→n=1 transitions at filling
factor/H9263=2 with all n/H113490/H11006levels filled and all n/H113501 levels
empty. A laborious direct calculation of Eq. /H208493.5/H20850yields
/H9280kexc=/H92801−/H92800+/H9004/H9280k10with
/H9004/H9280k10=/H9267¯0vk/H9253k2/H20841gk10/gk00/H208412+/H20858
pvp/H9253p2Ip,k,
Ip,k=/H20858
n/H113490/H20849/H20841gp0n/H208412−/H20841g−p1n/H208412/H20850−cp,kgp11g−p00, /H208493.7/H20850
where/H9253k2=e−/H51292k2/2andcp,k=cos /H20849/H51292p/H11003k/H20850.
When the n=0/H11006level is partially filled, /H9004/H9280k10involves the
following contribution:
/H9254/H9280k10=/H20858
pvp/H9253p2/H20851/H20841gp01/gp00/H208412s¯/H20849p+k/H20850//H9253p+k2
+/H20849cp,kgp11/gp00−1/H20850s¯/H20849p/H20850//H9253p2/H20852, /H208493.8/H20850
where s¯/H20849p/H20850stands for the projected static-structure factor de-
fined as /H20855G/H20841/H9267−p00/H9267p00/H20841G/H20856//H9024=/H9267¯0/H20853/H9267¯0/H9254p,0+s¯/H20849p/H20850/H20854. The determina-
tion of s¯/H20849p/H20850is a highly nontrivial task which requires an
exact diagonalization study of model systems16and is not
attempted here. We instead focus on the case of integer fill-
ing, for which s¯/H20849p/H20850is taken to vanish.
Equation /H208493.7/H20850, together with Eq. /H208493.8/H20850, essentially agrees
with the result of MacDonald and Zhang28for the L0→L1
transition in the standard QH system. The key difference is
that quantum fluctuations in Ip,know involve a sum
/H20858n/H11349−1/H20849¯/H20850over infinitely many Landau levels in the valence
band /H20849or the Dirac sea /H20850. Actually, one can verify that the
SMA expressions /H208493.7/H20850and /H208493.8/H20850equally apply to a general
interlevel transition La→Lbif one sets the superscripts
0→aand 1→b, in an obvious fashion, and takes the sum
/H20858nover filled levels.
To be precise, the 0 →1 transition of our interest consists
of four channels, /H208490+→1/H20850/H20841Kand /H208490−→1/H20850/H20841K/H11032each with spin
sz=/H110061/2. One therefore has to consider mixing of four /H902110
to determine/H9280kexc. Actually only the first term /H11008vk/H20841gk10/H208412in Eq.
/H208493.7/H20850, which comes from the direct Coulomb exchange, is
responsible for such mixing17,18because the rest of terms are
diagonal in spin and valley. Such direct terms /H11008vk/H20841g−kba/H208412
/H20849with b/HS11005a/H20850in general vanish for k→0 and mixing thus
takes place only for k/HS110050. In what follows we focus on the
k=0 excitation energies /H9280k=0excwith no mixing taken into ac-
count. In addition, we make no distinction between the 0 /H11006levels because gpknare essentially the same at the two valleys,
as noted in Sec. II.
The cyclotron-resonance energy for a general La→Lb
transition with the Landau levels filled up to n=nfis written
as/H9280k=0exc=/H9280b−/H9280a+/H9004/H9280k=0b←awith
/H9004/H9280k=0b←a=/H20858
pvp/H9253p2/H20875/H20858
n/H11349nf/H20849/H20841g−pan/H208412−/H20841gpbn/H208412/H20850−gpbbg−paa/H20876./H208493.9/H20850
This is the basic formula we use in what follows. Note that
nf=1,0 +,−1,−2,−3,... correspond to the filling factors
/H9263=4nf+2=6,2,−2,−6,−10,..., respectively. The
/H20858n/H11349nf/H20849/H20841gpan/H208412−/H20841g−pbn/H208412/H20850term refers to the change in quantum
fluctuations, via the a→btransition, of the filled states. Its
structure is easy to interpret physically: as an electron is
excited from LatoLb,/H20841n=a,y0/H20856→/H20841n=b,y0/H11032/H20856, virtual transi-
tions from any filled levels to the /H20841b,y0/H11032/H20856state are forbidden
while those to the newly unoccupied /H20841a,y0/H20856state are allowed
to start.
For standard QH systems this correction /H9004/H9280k=0b←avanishes
for each transition to the adjacent level, Ln→Ln+1, according
to Kohn’s theorem.15Indeed, one can verify, for the 0 →1
transition, the relation
/H20841g−p00/H208412−/H20841gp10/H208412−gp11g−p00=0 /H208493.10 /H20850
/H20849with gp00→1 and gp10→−/H5129p//H208812/H20850and analogous ones for
other Ln→Ln+1as well.
Interestingly, it happens that Eq. /H208493.10 /H20850also holds for the
0/H11006→1 transition in graphene, with gp00=1,gp10=−/H5129p/2, and
gp11=1− /H51292p2/4. Any nonzero shift /H9004/H9280k=01←0for the 0 →1 cy-
clotron resonance therefore comes from the quantum fluctua-tions of the Dirac sea and actually diverges logarithmicallywith the number N
Lof filled Landau levels in the sea
/H9004/H928001←0=/H20858
pvp/H9253p2/H20858
−NL/H11349n/H11349−1/H20849/H20841g−p0n/H208412−/H20841gp1n/H208412/H20850=VcCN,
CN/H11015/H20849/H208812/8/H20850/H20849logNL− 1.017 /H20850, /H208493.11 /H20850
where
Vc/H11013/H9251//H20849/H9280b/H5129/H20850/H11015/H20849 56.1 //H9280b/H20850/H20881B/H20851T/H20852meV. /H208493.12 /H20850
CNagrees29with the result of earlier works17,18obtained by a
different method.
This divergence in CNderives from short-wavelength
vacuum polarization and is present even for B=0. To see this
one may evaluate the Coulomb exchange correction in freespace /H20849with
/H9251→/H9251//H9280b/H20850, using the instantaneous photon and
fermion propagators vk=2/H9266/H9251//H20849/H9280b/H20841k/H20841/H20850andiS/H20849p/H20850=/H9268ipi//H208492/H20841p/H20841/H20850:
/H20858
kvkiS/H20849p+k/H20850=/H9268·p/H9251
8/H9280blog/H20849C/H90112/p2/H20850/H20849 3.13 /H20850
with momentum cutoff /H20841k/H20841/H11349/H9011 and some constant C.
This divergent correction causes infinite renormalization30
of velocity v0in the electron kinetic term /H11008v0/H9268i/H11509i; Eq. /H208493.13 /H20850
agrees with an earlier result of Ref. 30. It vanishes at p=0
but, for B/HS110050, turns into a nonvanishing energy gap, with
p2→2eB=2 //H51292. Actually, this diverging piece precisely
agrees with that in Eq. /H208493.11 /H20850, if one simply chooses theMANY-BODY CORRECTIONS TO CYCLOTRON RESONANCE … PHYSICAL REVIEW B 81, 075407 /H208492010 /H20850
075407-3“Fermi momentum” /H9011so that the Dirac sea accommodates
the same number of electrons as in the B/HS110050 case,
Nsea=/H90112/4/H9266/H11015NL/2/H9266/H51292, i.e.,/H90112/H110152NL//H51292.
Since such an infinite correction is already present
forB=0, it does not make sense to discuss the magnitude of
the cutoff-dependent number CNin Eq. /H208493.11 /H20850. The legitimate
procedure is to renormalize v0by rescaling
v0=Zvv0ren/H208493.14 /H20850
and put reference to the cutoff into Zvwith v0renregarded as
an observable quantity.
The renormalized velocity v0renis defined by referring to a
specific resonance. Let us take the 0 →1 resonance and
choose to absorb the entire O/H20849Vc/H20850correction at some refer-
ence scale /H20849e.g., at magnetic field B0/H20850into Zv, i.e., we write
/H9280k=01←0=/H9263=2
/H92801+/H9004/H9280k=01←0=/H208812v0ren/H20841B//H5129/H11013/H9275cren/H20841B /H208493.15 /H20850
by setting
Zv=1−/H9251
/H208812v0/H9280bCN=1−/H9251
8v0/H9280blog/H90112
/H92602, /H208493.16 /H20850
where/H92602=/H20849const. /H20850eB0. The renormalized velocity then de-
pends on Bor runs with B
v0ren/H20841B=v0ren/H20841B0−/H9251
8/H9280blog/H20849B/B0/H20850, /H208493.17 /H20850
decreasing slightly for B/H11022B0; actually the correction is
rather small /H20849about 3% for B/B0/H110112 and/H9280b/H110115 with
v0ren/H11011c/300 /H20850. With such Bdependence in mind, we denote
v0ren/H20841Basv0renand/H9275cren/H20841Bas/H9275crenfrom now on.
The divergences in /H9280kexc=/H9280k−/H9280n+/H9004/H9280kfor all other
resonances, as illustrated in Fig. 1/H20849a/H20850, are taken care
of by this velocity renormalization. The finite corrections/H11008V
cafter renormalization then make sense as genuine ob-
servable corrections. In particular, for several intrabandchannels L
n→Ln+1at filling factor /H9263=4nf+2, direct calcula-
tions yield/H9280k=02←1=/H9263=6
/H20849/H208812−1 /H20850/H20853/H9275cren− 0.264 Vc/H20854,
/H9280k=03←2=/H9263=10
/H20849/H208813−/H208812/H20850/H20853/H9275cren− 0.358 Vc/H20854,
/H9280k=04←3=/H9263=14
/H20849/H208814−/H208813/H20850/H20853/H9275cren− 0.419 Vc/H20854,
/H9280k=05←4=/H9263=18
/H20849/H208815−/H208814/H20850/H20853/H9275cren− 0.464 Vc/H20854, /H208493.18 /H20850
where Vc=/H9251//H20849/H9280b/H5129/H20850. The Coulomb corrections, shown nu-
merically here, are analytically calculable. The excitationspectra
/H9280kin the hole band are essentially the same
/H9280k−n←−/H20849n+1/H20850/H20841nf=−/H20849n+1/H20850/H9263=−/H208494n+2/H20850=/H9280kn+1←n/H20841nf=n/H9263=4n+2/H208493.19 /H20850
reflecting the particle-hole symmetry.
Figure 1/H20849b/H20850shows some of the momentum profiles /H9253p2/H20851¯/H20852
in/H9004/H9280k=0n+1←nof Eq. /H208493.9/H20850, which, when integrated over
/H5129/H20841p/H20841, give/H9004/H9280k=0n+1←n//H20849/H20881n+1− /H20881n/H20850in units of Vc. It is clearly
seen that the slowly decreasing high-momentum
tails /H11011/H20849/H208812/4/H20850//H20849/H5129/H20841p/H20841/H20850are responsible for the ultraviolet /H20849UV/H20850
divergence and that the finite observable corrections /H11008Vcare
uniquely determined from the profiles in the low-momentumregion /H5129/H20841p/H20841/H1135115.
A look into the structure of the total current operator tells
us that the optically induced cyclotron resonance /H20849fork=0/H20850
in graphene is governed by the selection rule /H9004/H20841n/H20841=/H110061, in
contrast to the “nonrelativistic” rule /H9004n=/H110061. In particular,
there are two classes of transitions /H20849i/H20850−n→/H11006/H20849n−1/H20850and /H20849ii/H20850
/H11006/H20849n−1/H20850→n/H20849with n/H113501/H20850, which are distinguished
31by use
of circularly polarized light /H20849/H11008Ax/H11006iAy/H20850; see Appendix B
As a result, graphene supports interband cyclotron reso-
nances. The lowest channels are open at /H9263=−2 with
/H9280k=02←−1=/H9263=−2
/H20849/H208812+1 /H20850/H20853/H9275cren+ 0.122 Vc/H20854,
/H9280k=01←−2=/H9263=−2
/H20849/H208812+1 /H20850/H20853/H9275cren+ 0.155 Vc/H20854. /H208493.20 /H20850
Some other interband channels yield
/H9280k=01←−2=/H9263=−6
/H20849/H208812+1 /H20850/H20853/H9275cren+ 0.084 Vc/H20854,
/H9280k=03←−2=/H9263=−6
/H20849/H208812+/H208813/H20850/H20853/H9275cren+ 0.058 Vc/H20854,
/H9280k=02←−3=/H9263=−6
/H20849/H208812+/H208813/H20850/H20853/H9275cren+ 0.114 Vc/H20854,
/H9280k=02←−3=/H9263=−10
/H20849/H208812+/H208813/H20850/H20853/H9275cren+ 0.044 Vc/H20854. /H208493.21 /H20850
It is now clear that cyclotron resonance is best analyzed
by plotting the rescaled energies /H9280k=0b←a//H20841sb/H20881/H20841b/H20841−sa/H20881/H20841a/H20841/H20841as a
function of /H20881BorB. The Coulombic many-body effect will
be seen as a variation in the characteristic velocity
v0ren/H208511+O/H20849Vc/H20850/H20852from one resonance to another and a deviation
of/H9275crenfrom the /H20881Bbehavior would indicate the running of
v0renwith B.2 4 6 8 10 12 140.050.100.150.20
FIG. 1. /H20849Color online /H20850/H20849a/H20850Cyclotron resonance; circularly polar-
ized light can distinguish between two classes of transitions indi-cated by different types of arrows. /H20849b/H20850Momentum profiles of the
many-body corrections /H9004
/H9280k=0n+1←n//H20849/H20881n+1− /H20881n/H20850in units of Vcfor
n= 0 ,1 ,2 ,a n d3 .K. SHIZUYA PHYSICAL REVIEW B 81, 075407 /H208492010 /H20850
075407-4Figure 2/H20849a/H20850shows such plots for some intraband and in-
terband channels, using v0ren/H20841B=10 T /H110151.13/H11003106m/s which
fits the 0 →1 resonance data, and, as a typical value,
Vc=/H9251//H20849/H9280b/H5129/H20850/H1101512/H20881B/H20851T/H20852meV /H20849/H9280b/H110115/H20850.
Actually, experiment19has already observed a small de-
viation of the 1: /H208491+/H208812/H20850ratio of/H9280k=01←0to/H9280k=02←−1well outside
of the experimental errors under high magnetic fieldsB=/H208496–18 /H20850T; the data are apparently electron-hole symmet-
ric,
/H9280k=01←0/H11015/H9280k=00←−1. Figure 2/H20849a/H20850includes such data reproduced
from Ref. 19. A small increase in v0renin/H9280k=02←−1//H20849/H208812+1 /H20850, rela-
tive to/H9280k=01←0, is roughly consistent with Eq. /H208493.20 /H20850which
suggests a 0.122 Vc//H9275cren/H110114% increase in v0ren/H20849since
Vc//H9275cren/H110110.3/H20850.
This feature is clearer from Fig. 2/H20849b/H20850, which plots the /H9280k=01←0
and/H9280k=02←−1//H20849/H208812+1 /H20850data as a function of Bin units of
/H9275c=/H208812v0//H5129/H11008/H20881B/H20849with v0→v0ren/H20841B=10 T /H20850. The deviation of the
/H20849−1→2/H20850resonance data is more pronounced. In the figure a
dotted curve represents a possible profile of the running of
v0renwith B, and, especially, the /H20849−1→2/H20850data /H20849with smaller
error bars /H20850suggests such running.
It is too early to draw any definite conclusion from the
present data alone but the data is certainly consistent /H20849in sign
and magnitude /H20850with the present estimate of the many-body
effect. In this connection, let us note that an earlier experi-ment on thin epitaxial graphite
32also observed the /H208490→1/H20850
and /H20849−1→2/H20850resonances with apparently no deviation from
the 1: /H208491+/H208812/H20850ratio. This measurement was done under rela-
tively weak magnetic fields B=/H208490.4–4 /H20850T, and it could be
that a small deviation, under larger error bars, simply es-caped detection, apart from the potential difference betweenthin graphite and graphene.
More precise measurements of cyclotron resonance, espe-
cially in the high Bdomain where the Coulomb interaction
becomes sizable, would be required to pin down the many-body effect in graphene. In this respect, the comparison be-
tween interband and intraband resonances from the same ini-
tial state, e.g., − n→/H11006/H20849n−1/H20850at
/H9263=2−4 nwith n=2,3,...,
would provide a clearer signal for the many-body effect, withthe influence of other possible sources reduced to a mini-mum. From Eqs. /H208493.18 /H20850–/H208493.21 /H20850one can read off the varia-
tions in
v0ren
/H9004R/H20849−2→/H110061/H20850/H11015/H9263=−6
0.34Vc//H9275cren/H1101110%,
/H9004R/H20849−3→/H110062/H20850/H11015/H9263=−10
0.40Vc//H9275cren/H1101112%, /H208493.22 /H20850
which imply that a comparison of the /H20849−2→/H110061/H20850resonances
and that of the /H20849−3→/H110062/H20850resonances would find variations
inv0, about three times larger than the /H110114% variation for
/H9280k=02←−1//H20849/H208812+1 /H20850vs/H9280k=00←−1at/H9263=−2.
IV. CYCLOTRON RESONANCES IN BILAYER
GRAPHENE
In this section we consider cyclotron resonance in
bilayer graphene. In bilayer graphene the electrons are de-scribed by four-component spinor fields on the four in-equivalent sites /H20849A,B/H20850and /H20849A
/H11032,B/H11032/H20850in the bottom and top
layers, arranged in Bernal A/H11032Bstacking. Interlayer
coupling33/H92531/H11013/H9253A/H11032B/H11011/H208490.3–0.4 /H20850eV modifies the intralayer
linear spectra/H11006v0/H20841p/H20841to yield, in the low-energy branches
/H20841/H9280/H20841/H11021/H92531, quasiparticles with a parabolic dispersion.10They, in
a magnetic field, lead to a particle-hole symmetric tower ofLandau levels /H20853L
n/H20854/H20849n=0/H11006,/H110061,... /H20850with spectrum
/H9280n=sn/H9275c/H9257n/H20851/H20849/H92531//H9275c/H208502/H20852, /H208494.1/H20850
where/H9257n/H20849x/H20850=/H20853/H20849an+x−/H20881x2+2anx+1/H20850/2/H208541/2with an=2/H20841n/H20841−1;
see Appendix A The sequence of low-lying levels is madeclearer in the form
/H9280n=sn/H9275bi/H20881/H20841n/H20841/H20849/H20841n/H20841−1/H20850//H9264n/H20849w/H20850/H20849 4.2/H20850
with the characteristic cyclotron energy
/H9275bi/H11013/H9275c2//H92531=2v02//H20849/H92531/H51292/H20850/H110115B/H20851T/H20852meV, /H208494.3/H20850
where/H9264n/H20849w/H20850=/H20853/H208491+anw+/H208811+2anw+w2/H20850/2/H208541/2and w
/H11013/H20849/H9275c//H92531/H208502=/H9275cbi//H92531/H110110.01B/H20851T/H20852/H110211;/H9264n/H208490/H20850=1. The high-
energy branches /H20841/H9280/H20841/H11022/H92531of the spectra give rise to another
tower of Landau levels with spectrum
/H9280n+=sn/H92531/H9264n/H20849w/H20850, /H208494.4/H20850
where n=/H110061,/H110062,....
Note that/H9264n/H20849w/H20850=1+ O/H20849eB //H92531/H20850. As a result,/H9280nrises lin-
early with Bat low energies /H20841/H9280n/H20841/H11021/H92531and turns into a /H20881Brise
for /H20841/H9280n/H20841/H11271/H92531. Both/H9280nand/H9280n+approach/H11006/H20881/H20841n/H20841/H9275cfor /H20841n/H20841→/H11009
since the bilayer turns into two isolated layers at short wave-lengths.
In the bilayer there arise four zero-energy levels
/H9280n=0
with n=/H208490/H11006,/H110061/H20850per spin. At one valley /H20849say, K/H20850they are
electron levels with n=/H208490+,1/H20850and, at another valley, they are
hole levels with n=/H208490−,−1/H20850; this feature is made explicit
with a weak layer asymmetry, such as an interlayer voltage012345050100150200
0 5 10 15 200.80.91.1.1
1.1.2
FIG. 2. /H20849Color online /H20850/H20849a/H20850Rescaled cyclotron-resonance ener-
gies as a function of /H20881Bwith v0ren/H110151.13/H11003106m/s/H20849atB=10 T /H20850
and Vc/H1101512/H20881B/H20851T/H20852meV /H20849/H9280b/H110115/H20850. The experimental data on /H9280k=01←0
and/H9280k=02←−1//H20849/H208812+1 /H20850are quoted from Ref. 19with error bars inferred
from the symbol size in the original data. /H20849b/H20850The same data
plotted in units of /H9275c=/H208812v0//H5129/H20849with v0→v0ren/H20841B=10 T /H20850as a function
ofB. The dotted curve represents a possible profile of the
running of v0ren/H20841B, normalized to 1 at B=10 T, with
v0ren/H20841B=10 T /H110151.13/H11003106m/s and/H9280b/H110155.MANY-BODY CORRECTIONS TO CYCLOTRON RESONANCE … PHYSICAL REVIEW B 81, 075407 /H208492010 /H20850
075407-5which opens up a /H20849tunable /H20850band gap.11–14With a nonzero
band gap, the zero-energy levels evolve into two quartets ofnearly degenerate levels /H20849separated by the gap /H20850, i.e.,
“pseudozero-mode” levels, which are expected to supportpseudospin waves
34,35as characteristic collective excitations.
For simplicity, we here turn off such a layer asymmetry as
well as Zeeman splitting and the effect of trigonal warping/H20849coming from
/H92533/H11013/H9253AB/H11032/H11021/H92531/H20850. In view of the small layer
separation, we do not distinguish between the intralayer and
interlayer Coulomb interactions. Each Landau level Lnis
thus treated as fourfold degenerate, except for the zero-modelevels /H20849L
0+,L1/H20850or/H20849L0−,L−1/H20850which are fourfold degenerate at
each valley.
The effective Hamiltonian for the electrons in bilayer
graphene takes a 4 /H110034 matrix form which, for studying the
properties of the low-lying levels, may be reduced to an ap-proximate 2/H110032 form.
10Actually, the bending of the
spectrum/H9280nwith Bis appreciable in the high- Bdomain,
B=/H2084910–20 /H20850T, where cyclotron resonance in bilayer
graphene has been studied experimentally. Accordingly weemploy the full four-component spinor description of the bi-layer system; see Appendix A for details.
The charge density
/H9267−p/H20849for each spin and valley /H20850takes
the same form as Eq. /H208492.6/H20850with gpknreplaced by
gpkn=DkDn/H20851fp/H20841k/H20841,/H20841n/H20841+/H9252k/H208492/H20850/H9252n/H208492/H20850fp/H20841k/H20841−2,/H20841n/H20841−2+/H20849/H9252k/H208493/H20850/H9252n/H208493/H20850
+/H9252k/H208494/H20850/H9252n/H208494/H20850/H20850fp/H20841k/H20841−1,/H20841n/H20841−1/H20852; /H208494.5/H20850
see Appendix A for the coefficients /H20853/H9252n/H20849i/H20850/H20854andDn. The sets
/H20849/H9280n,gpkn/H20850at the KandK/H11032valleys are related as
/H9280n/H20841K/H11032=−/H9280−n/H20841K,gpkn/H20841K/H11032=gp−k,−n/H20841K. /H208494.6/H20850
Actually, for zero band gap, gpknare essentially the same at
the two valleys since one further finds that
gpkn/H20841K/H11032=gpkn/H20841K,gpk,−1/H20841K/H11032=gpk,1/H20841K,gpk,0−/H20841K/H11032=gpk,0+/H20841K
/H208494.7/H20850
for /H20841k/H20841/H113502 and /H20841n/H20841/H113502.
One can now use the SMA formula /H208493.9/H20850to calculate the
interlevel excitation energies /H9280kexc=/H9280b−/H9280a+/H9004/H9280kb←a. The result
applies to both valleys if one specifies the zero-mode levelsaccordingly. It is important to remember that for bilayergraphene the sum /H20858
nover filled levels involves two branches
/H9280−nand/H9280−n+in the valence band.
Cyclotron resonance in bilayer graphene again obeys the
selection rule31/H9004/H20841n/H20841=/H110061; see Appendix B and Fig. 3/H20849a/H20850. The
Coulombic corrections /H9004/H9280k=0b←aare diagonal in spin and valley
/H20849while mixing arises for k/HS110050/H20850. The vacuum-polarization ef-
fect again makes /H9004/H9280k=0b←acutoff dependent.
For renormalization let us first look into the B=0 case.
One can construct the electron propagator and, as in themonolayer case, calculate the Coulombic quantum correc-tions. It turns out that not only
v0but also/H92531undergo infinite
renormalization and, rather unexpectedly, the divergent termsare the same for both of them to O/H20849V
c/H20850at least; they also
coincide with the divergent term in the monolayer case; seeAppendix C for details. To be precise, the divergences areremoved, to O/H20849V
c/H20850of our present interest, by rescalingv0=Zv0ren,/H92531=Z/H92531ren/H208494.8/H20850
with a common factor Z.
This scaling tells us how to carry out renormalization in
the presence of a magnetic field B. Let us write, as in Eq.
/H208493.18 /H20850of the monolayer case, the excitation energy for the
Ln→Lktransition in the form
/H9280k=0k←n=/H20849sk/H9257k−sn/H9257n/H20850/H20849/H208812v0//H5129+cknVc/H20850/H20849 4.9/H20850
with/H9257n=/H9257n/H208491/w/H20850. Note first that v0//H92531=v0ren//H92531renis invariant
under renormalization; it is therefore finite and does not run
with B. Similarly, w=/H20849/H9275c//H92531/H208502/H11008/H20849v0ren//H92531ren/H208502Bis invariant
and is linear in B. This means that /H9257n/H208491/w/H20850remain unrenor-
malized and finite. Equation /H208494.9/H20850then reveals a remarkable
structure of the Coulombic corrections ckn: The divergent
pieces are common to all cknand are removed by a single
counterterm/H11008/H20849Z−1/H20850v0ren.
Figure 3/H20849b/H20850depicts the momentum profiles /H9253p2/H20851¯/H20852of
/H9004/H9280k=0k←n//H20849sk/H9257k−sn/H9257n/H20850for some typical resonances. For com-
parison the profile for the monolayer resonance /H20849/H9004/H9280k=01←0/H20850mono
is also included there. The gradually decreasing high-
momentum tails, common to all, numerically demonstratethe validity of the scaling in Eqs. /H208494.8/H20850and /H208494.9/H20850. This further
verifies that the leading logarithmic velocity renormalizationis formally the same for both monolayer and bilayergraphene.0 5 10 15 20020406080100
(-12)(-8)(-4)
(-16)2 4 6 8 10 120.10.20.30.40.5
(4)(8) 0
FIG. 3. /H20849Color online /H20850/H20849a/H20850Cyclotron resonance in bilayer
graphene. /H20849b/H20850Momentum profiles of the bilayer many-body
corrections/H9004/H9280k=0n+1←n//H20849/H9257n+1−/H9257n/H20850forn=1, 2, and 3, with vˆ0=1.15
and/H9253ˆ1=3.5; real curves at B=10 T and dashed curves at
B=16 T. A dotted curve refers to the profile of the monolayer
/H208490→1/H20850resonance. /H20849c/H20850Resonance energies as a function of B; real
curves, with vˆ0=1.15,/H9253ˆ1=3.5, and Vc=0; dotted curves,
with vˆ0=1.15,/H9253ˆ1=3.8, and Vc/H110155.6/H20881B/H20851T/H20852meV /H20849or/H9280b/H1101510/H20850. The
experimental data with error bars are reproduced from Ref. 21. Note
that the low-lying n=2 spectrum/H92802/H20849equal to the/H9263=4 curve /H20850
significantly deviates from the approximate spectrum /H92802/H11015/H208812/H9275cbi
with/H92642/H20849w/H20850→1/H20849dashed line/H11008B/H20850forB/H1102210 T.K. SHIZUYA PHYSICAL REVIEW B 81, 075407 /H208492010 /H20850
075407-6For renormalization let us refer to a specific resonance,
e.g., the −3 →−2 resonance at /H9263=−8, and define v0renso as to
absorb its entire O/H20849Vc/H20850correction
/H9275cren/H11013/H208812v0ren/H20841B//H5129=/H208812v0//H5129+c−2,−3Vc. /H208494.10 /H20850
One then has, for general n→kchannels
/H9280k=0k←n=/H20849sk/H9257k−sn/H9257n/H20850/H20849/H9275cren+/H9004cknVc/H20850. /H208494.11 /H20850
Here/H9004ckn/H11013ckn−c−2,−3are now free from the UV
divergence and are uniquely fixed as genuine quantumcorrections. In terms of the bilayer cyclotron frequency
/H20849
/H9275bi/H20850ren/H11013/H20849/H9275cren/H208502//H92531ren, this also reads
/H9280k=0k←n=/H20849sk/H9256k−sn/H9256n/H20850/H20853/H20849/H9275bi/H20850ren+/H9004ckn/H20881wV c/H20854/H20849 4.12 /H20850
with/H9256n=/H20881/H20841n/H20841/H20849/H20841n/H20841−1/H20850//H9264n/H20849w/H20850andw=/H20849/H9275bi/H20850ren//H92531ren.
The quantum corrections /H9004ckn, unlike those of the mono-
layer case, are not pure numbers and, actually, are functions
of/H20881w=/H9275cren//H92531ren. This is seen if one notes that gpknare func-
tions of /H20881wand/H5129pso that cknare functions of /H20881wand the
cutoff NL/H11008/H90112//H20849eB/H20850; the cutoff-independent corrections /H9004ckn
thus depend on walone. Let us set /H92531ren=/H9253ˆ1/H11003100 meV and
v0ren=vˆ0/H11003106m/s so that 1 //H20881w=/H92531ren//H9275cren/H110152.75Gwith
G=/H9253ˆ1//H20849vˆ0/H20881B/H20851T/H20852/H20850;G=1 for/H9253ˆ1=3.5 and vˆ0/H110151.107 at
B=10 T. It turns out that /H9004ck,n, when plotted in G, behave
almost linearly around G=1.
The way v0renruns with Bis determined from
v0ren/H20841B=v0ren/H20841B0+/H9254c−2,−3/H9251//H20849/H208812/H9280b/H20850, /H208494.13 /H20850
where/H9254c−2,−3/H11013c−2,−3/H20841B−c−2,−3/H20841B0. Numerically /H9254c−2,−3is
nearly twice as large as the monolayer expression
−/H20849/H208812/8/H20850log/H20849B/B0/H20850over the range 0.5 /H11021B/B0/H110212 around
G=1. The decrease in v0ren/H20841Bwith Bis larger in bilayer
graphene and may amount to about 7% for B/B0/H110112/H20849and
/H9280b/H110115/H20850. In this way, the renormalized velocity v0renis in gen-
eral different, in magnitude and running with B, for mono-
layer and bilayer graphene; it reflects their low-energy fea-tures as well.
We are now ready to look into some typical channels of
cyclotron resonance. We use Eq. /H208494.11 /H20850and evaluate
/H9004c
k,n/H11013/H9004ck←nnumerically; for the bilayer the filling factor
/H9263=4/H20849nf+1/H20850fornf/H11349−2 while/H9263=4nffornf/H113501. For intraband
channels one finds
/H9004c/H110061,−2=/H9263=−4
0.7270 + 0.5484 /H9254G,
/H9004c−2,−3=/H9263=−8
0,
/H9004c−3,−4=/H9263=−12
− 0.1521 − 0.0453 /H9254G,
/H9004c−4,−5=/H9263=−16
− 0.2496 − 0.0797 /H9254G, /H208494.14 /H20850
where/H9254G=G−1 with G=/H9253ˆ1//H20849vˆ0/H20881B/H20851T/H20852/H20850. Similarly, for inter-
band resonances one obtains
/H9004c3←−2=/H9263=−4
0.3922 − 0.0023 /H9254G,/H9004c2←−3=/H9263=−4
0.4794 + 0.0706 /H9254G,
/H9004c2←−3=/H9263=−8
0.3872 + 0.0552 /H9254G,
/H9004c3←−4=/H9263=−12
0.2961 + 0.00145 /H9254G; /H208494.15 /H20850
also/H9004c4←−3=/H9263=−4
0.41+¯and/H9004c4←−3=/H9263=−8
0.29+¯. These lin-
earized expressions are numerically precise with errors ofless than 3% over the range 0.3 /H11021G/H110211.5. The many-body
effect is thus expected to be sizable in bilayer graphene. An
effective variation in
v0renwould amount to about
0.7Vc//H9275cren/H1101120% for/H9280k=01←−2/H20841/H9263=−4 and about −5% for
/H9280k=0−3←−4/H20841/H9263=−12, in comparison with /H9280k=0−2←−3/H20841/H9263=−8.
As for experiment, Henriksen et al.21measured, via IR
spectroscopy, cyclotron resonance in bilayer graphene inmagnetic fields up to 18 T. They observed intrabandtransitions, which are identified with
/H20853
/H9280k=02←1/H20841/H9263=4,/H9280k=03←2/H20841/H9263=8,/H9280k=04←3/H20841/H9263=12,/H9280k=05←4/H20841/H9263=16/H20854and the correspond-
ing hole resonances listed in Eq. /H208494.14 /H20850, together with an
appreciable asymmetry between the electron and hole data.
Figure 3/H20849c/H20850reproduces the electron data of Ref. 21. There
the real curves represent the resonance energies in Eq. /H208494.11 /H20850
forVc=0, with vˆ0/H110151.15 deduced from the /H9263=4 data and/H9253ˆ1
taken to be 3.5, as supposed in Ref. 21. They poorly fit the
/H9263=8,12,16 data. Unfortunately, inclusion of the O/H20849Vc/H20850cor-
rections scarcely improves the fit, as seen from the dottedcurves.
The situation becomes clearer if one, in view of Eq.
/H208494.11 /H20850, reorganizes the experimental data in the form
/H9280k=0k←n//H20849sk/H9257k−sn/H9257n/H20850and plots them in units of /H9275c=/H208812v0//H5129
/H20849with v0=1.15/H11003106m/s/H20850. Figure 4/H20849a/H20850shows such a plot for
the electron data; for clarity the data points for differentchannels, originally at B=/H2084910,12,14,16 /H20850T, are slightly
shifted in B. It is to be contrasted with Fig. 4/H20849b/H20850, which
illustrates how each resonance would behave with B, accord-
ing to Eq. /H208494.11 /H20850, for V
c/H1101510/H20881B/H20851T/H20852meV /H20849or/H9280b/H110155.6/H20850;i n
particular, the/H9263=8 curve represents the running of v0ren/H20841B
according to Eq. /H208494.13 /H20850. In Fig. 4/H20849a/H20850the/H9263=8,12,16 reso-
nances are apparently ordered in a way opposite to Fig. 4/H20849b/H20850,
and an appreciable gap between the /H9263=4 resonance and the
rest is not very clear. The /H9263=8,12,16 data show a general
trend to decrease with B, consistent with possible running of
v0ren/H20841Bbut at a rate faster than expected. It is rather difficult to
interpret these features, but they, in part, could be attributedto possible quantum screening
35of the Coulomb interaction
in bilayer graphene such that /H9280bis effectively larger36for
lower B. Note, in this connection, Fig. 4/H20849c/H20850which shows that
the same data may suggest a Coulombic gap for a choice
/H92531/H110154 favored in Ref. 33. We further remark that, in spite of
an asymmetry in electron and hole data, the hole data sharesessentially the same features; see Fig. 4/H20849d/H20850.
No data are available for interband cyclotron resonance in
bilayer graphene at present. They are highly desired becausethe comparison of interband and intraband resonances fromthe same initial states would provide a clearer signal for themany-body effect. We record the ratiosMANY-BODY CORRECTIONS TO CYCLOTRON RESONANCE … PHYSICAL REVIEW B 81, 075407 /H208492010 /H20850
075407-7/H9004R/H20849−3→/H110062/H20850/H11015/H9263=−8
0.39Vc//H9275cren/H1101115%,
/H9004R/H20849−4→/H110063/H20850/H11015/H9263=−12
0.45Vc//H9275cren/H1101118%, /H208494.16 /H20850
which imply that a close look into the /H20849−3→/H110062/H20850resonances
and the /H20849−4→/H110063/H20850resonances would find a sizable variation
/H1101115% in v0ren.
V. SUMMARY AND DISCUSSION
Graphene supports charge carriers that behave as Dirac
fermions, which, in a magnetic field, lead to a characteristicparticle-hole symmetric pattern of Landau levels. Accord-ingly, unlike standard QH systems, there is a rich variety ofcyclotron resonance, both intraband and interband reso-nances of various energies, in graphene.
In this paper we have studied many-body corrections to
cyclotron resonance in graphene. We have constructed aneffective theory using the SMA and noted that genuine non-zero many-body corrections /H20849not due to fine splitting in spin
or valley /H20850derive from the quantum fluctuations of the
vacuum /H20849the Dirac sea /H20850. Such quantum corrections are intrin-sically ultraviolet divergent and, as we have emphasized, it is
necessary to carry out renormalization of velocity
v0/H20849and,
for bilayer graphene, interlayer coupling /H92531as well /H20850to deter-
mine the many-body corrections uniquely in terms of physi-cal quantities. As a result, the observable intralayer and in-
terlayer coupling strengths
v0ren/H11008/H92530renand/H92531renin general run
with the magnetic field B.
Experimental data on cyclotron resonance generally have
sizable error bars, which make a clear identification of themany-body effect difficult. In this respect, we have presenteda way to analyze the data, as in Figs. 2/H20849b/H20850and4, with the
effect of renormalization properly taken into account.
For monolayer graphene a piece of data
19which compares
some leading interband and intraband resonances is appar-ently consistent with the presence of many-body correctionsroughly in magnitude and sign, and also in the running of
v0renwith B. For bilayer graphene the existing data are only
for intraband resonances and are rather puzzling, as dis-cussed in Sec. IV. They generally appear to defy good fit by
theory but certainly suggest nontrivial features of many-bodycorrections, such as running with B.
More precise measurements of cyclotron resonances are
highly desired. Of particular interest are experiments whichcompare interband and intraband resonances from the sameinitial states, as listed in Eqs. /H208493.22 /H20850and /H208494.16 /H20850, which would
clarify the many-body effect with minimal uncertainties.
ACKNOWLEDGMENTS
This work was supported in part by a Grant-in-Aid for
Scientific Research from the Ministry of Education, Science,Sports and Culture of Japan /H20849Grant No. 21540265 /H20850.
APPENDIX A: LANDAU LEVELS IN BILAYER GRAPHENE
This appendix summarizes the effective Hamiltonian and
its eigenfunctions for bilayer graphene in a magnetic field B.
The bilayer Hamiltonian with interlayer coupling /H92531/H11013/H9253A/H11032B
is written, at one /H20849K/H20850valley, as10
Hbi=/H208980 v0/H9016†
0 v0/H9016
v0/H9016†0/H92531
v0/H9016/H92531 0/H20899, /H20849A1/H20850
which acts on an electron field of the form
/H9023K=/H20849/H9274A,/H9274B/H11032,/H9274A/H11032,/H9274B/H20850tin obvious notation; /H9016=/H9016x−i/H9016yand
/H9016†=/H9016x+i/H9016ywith Ai→B/H20849−y,0/H20850.
The energy eigenvalues obey the equation
/H20849/H20841n/H20841−1−/H9280/H110322/H20850/H20849/H20841n/H20841−/H9280/H110322/H20850−/H9253/H110322/H20849/H9280/H11032/H208502=0 , /H20849A2/H20850
where/H9280/H11032/H11013/H9280n//H9275c,/H9253/H11032/H11013/H92531//H9275c, and/H9275c=/H208812v0//H5129. This leads to
the two branches of spectra /H20849/H9280n,/H9280n+/H20850in Eqs. /H208494.1/H20850and /H208494.4/H20850.I n
particular, zero energy /H9280n=0 is possible for /H20841n/H20841=0 or
/H20841n/H20841=1 while /H20841/H9280/H110061+/H20841/H11022/H92531. A weak interlayer voltage
1
2/H20849/H9254V/H20850diag /H208511,−1,−1,1 /H20852, added to Hbi, reveals that the zero
modes actually have n=0+andn=1 for/H9254V/H110220.
The corresponding eigenfunctions for n=/H110062,/H110063,...
take the form8 1 01 21 41 61 80.80.91.1.11.21.3
10 12 14 16 180.91.1.11.2
8 10 12 14 1 6 180.80.91.01.11.21.38 1 01 21 41 61 80.80.91.1.11.21.3
FIG. 4. /H20849Color online /H20850Experimental data of Ref. 21, reorganized
in the form/H9280k=0k←n//H20849sk/H9257k−sn/H9257n/H20850and plotted in units of /H9275c=/H208812v0//H51292
/H20849with v0=1.15/H11003106m/s/H20850./H20849a/H20850Electron data, analyzed with
vˆ0=1.15 and/H9253ˆ1=3.5; for clarity the data points, originally
atB=/H2084910,12,14,16 /H20850T, are slightly shifted in B./H20849b/H20850Theoretical
expectation according to Eq. /H208494.11 /H20850with vˆ0=1.14,/H9253ˆ1=3.5, and
Vc//H9275c/H110150.24 /H20849or/H9280b/H110155.6/H20850./H20849c/H20850Electron data, reanalyzed with
/H9253ˆ1=4. /H20849d/H20850Hole data, analyzed with vˆ0=1.02 and/H9253ˆ1=3.5.K. SHIZUYA PHYSICAL REVIEW B 81, 075407 /H208492010 /H20850
075407-8/H9023n=Dn/H20898/H20841/H20841n/H20841/H20856
/H9252n/H208492/H20850/H20841/H20841n/H20841−2/H20856
/H9252n/H208493/H20850/H20841/H20841n/H20841−1/H20856
/H9252n/H208494/H20850/H20841/H20841n/H20841−1/H20856/H20899,
/H9252n/H208492/H20850=/H20881/H20841n/H20841−1
/H9280/H11032/H9252n/H208493/H20850,/H9252n/H208493/H20850=−1
/H9253/H11032/H20841n/H20841−/H9280/H110322
/H20881/H20841n/H20841, /H20849A3/H20850
/H9252n/H208494/H20850=/H9280/H11032
/H20881/H20841n/H20841,Dn=1
/H208812/H20881/H20841n/H20841/H20849/H20841n/H20841−1−/H9280/H110322/H20850
/H20841n/H20841/H20849/H20841n/H20841−1/H20850−/H9280/H110324, /H20849A4/H20850
where only the orbital eigenmodes are shown using the stan-
dard harmonic-oscillator basis /H20853/H20841n/H20856/H20854. These expressions for
/H9023nare equally valid for both the low- and high-energy
branches/H9280nand/H9280n+of Landau levels, depending on /H9280/H11032one
employs.
The zero-energy eigenmodes are given by
/H90230+=/H20849/H208410/H20856,0,0,0 /H20850t,
/H90231=D1/H20849/H208411/H20856,0,− /H208491//H9253/H11032/H20850/H208410/H20856,0/H20850t, /H20849A5/H20850
with D1=/H208491+1 //H9253/H110322/H20850−1 /2.
At another /H20849K/H11032/H20850valley the Hamiltonian is given by Eq.
/H20849A1/H20850with v0→−v0and acts on a field of the form
/H9023K/H11032=/H20849/H9274B/H11032,/H9274A,/H9274B,/H9274A/H11032/H20850t. Accordingly one finds that
/H9280n/H20841K/H11032=−/H9280−n/H20841K,Dn/H20841K/H11032=D−n/H20841K,/H9252n/H208492/H20850/H20841K/H11032=/H9252−n/H208492/H20850/H20841K,
/H9252n/H208493/H20850/H20841K/H11032=−/H9252−n/H208493/H20850/H20841K,/H9252n/H208494/H20850/H20841K/H11032=−/H9252−n/H208494/H20850/H20841K. /H20849A6/H20850
The zero-energy levels now have n=0−andn=−1.
APPENDIX B: COUPLING TO CURRENT
Consider a weak time-varying vector potential
/H20851Ax/H20849t/H20850,Ay/H20849t/H20850/H20852coupled to the total current in graphene. For the
effective Lagrangian L/H9021in Eq. /H208493.4/H20850this yields coupling of
Aito/H9021k=0kn=/H20848d2x/H9021knof the form
HA=−ie/H5129/H9275c
/H208812/H20881/H92670dn/H20853A/H9021k=0/H11006/H20849n−1/H20850,−n+A†/H9021k=0n,/H11006/H20849n−1/H20850/H20854+ H.c.,
/H20849B1/H20850
where dn=/H110061/2 for n/H113502 and d1=/H110061//H208812;A=Ax−iAy; and
/H92670=1 //H208492/H9266/H51292/H20850. The cyclotron resonance thus obeys the selec-
tion rule/H9004/H20841n/H20841=/H110061. In particular, the − n→/H11006/H20849n−1/H20850transi-
tions and the /H11006/H20849n−1/H20850→ntransitions /H20849n/H113501/H20850are
distinguished31by use of circularly polarized light /H11008Ax/H11006iAy.Equation /H20849B1/H20850/H20849with n/H113502/H20850applies to the case of bilayer
graphene as well if one sets /H9275c→/H9275cbi,A†→−A†,dn=/H20881n−1
forn/H113503, and d2=/H208812, apart from terms of O/H20851/H20849/H9275c//H92531/H208502/H20852.
APPENDIX C: PROPAGATORS
In this appendix we derive the electron propagator for
bilayer graphene in free space. Let us set /H9016→px−ipyinHbi
of Eq. /H20849A1/H20850and consider the propagator
/H20855/H9023/H20849x/H20850/H9023†/H20849x/H11032/H20850/H20856=/H20855x/H208411//H20849i/H11509t−Hbi/H20850/H20841x/H11032/H20856/H20849 C1/H20850
with /H20841x/H20856/H11013/H20841x/H20856/H20841t/H20856. We divide the 4 /H110034 matrix Hbiinto a 2/H110032
block form and invert /H20849i/H11509t−Hbi/H20850. In Fourier /H20849p,/H9275/H20850space the
propagator reads, in 2 /H110032 block form
/H20855/H9023/H9023†/H2085611=i
D/H20853/H9275/H20849/H92752−v02p2−/H925312/H20850+/H92531v02P/H92681P/H20854,
/H20855/H9023/H9023†/H2085621=i
D/H20849/H92752−v02p2+/H9275/H92531/H92681/H20850v0P,
/H20855/H9023/H9023†/H2085612=i
Dv0P/H20849/H92752−v02p2+/H9275/H92531/H92681/H20850,
/H20855/H9023/H9023†/H2085622=i/H9275
D/H20849/H92752−v02p2+/H9275/H92531/H92681/H20850, /H20849C2/H20850
where D=/H20849v02p2−/H92752/H208502−/H92752/H925312=/H20849/H92752−E+2/H20850/H20849/H92752−E−2/H20850 with
E/H11006=/H20881/H925312/4+v02p2/H11006/H92531/2; P=p†/H9268++p/H9268− and P/H92681P
=/H20849p†/H208502/H9268++p2/H9268−with p=px−ipy,p†=px+ipy, and/H9268/H11006
=/H20849/H92681/H11006i/H92682/H20850/2.
This leads to the instantaneous propagator
/H20848/H20849d/H9275/2/H9266/H20850/H20855/H9023/H9023†/H20856, giving
/H20855/H9023/H9023†/H20856/H20841t=t/H11032=1
4Dp/H20873/H92531P/H92681P/p22v0P
2v0P/H92531/H92681/H20874 /H20849C3/H20850
withDp=/H20881/H925312/4+v02p2.
To calculate the Coulomb exchange correction one
may replace, in Eq. /H208493.13 /H20850,iS/H20849p/H20850by this propagator.
Note that v0P//H208492Dp/H20850approaches, for p→/H11009, the monolayer
propagator iS/H20849p/H20850=/H9268ipi//H208492/H20841p/H20841/H20850 /H20849apart from an inessential mis-
match/H92682→−/H92682in notation /H20850. As a result, setting
iS/H20849p/H20850→v0P//H208492Dp/H20850forv0andiS/H20849p/H20850→/H92531//H208494Dp/H20850for/H92531and
carrying out the kintegration, as in Eq. /H208493.13 /H20850, yield the
same amount of logarithmic divergence /H11011/H20849/H9251/8/H9280b/H20850log/H90112as
in the monolayer case; it thus renormalizes v0and/H92531simul-
taneously as in Eq. /H208494.8/H20850.
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075407-10 |
PhysRevB.76.045421.pdf | Atomic-scale structure of the SrTiO 3„001 …-c„6Ã2…reconstruction:
Experiments and first-principles calculations
C. H. Lanier,1,2A. van de Walle,3N. Erdman,4E. Landree,5O. Warschkow,6A. Kazimirov,7K. R. Poeppelmeier,2,8
J. Zegenhagen,9,10M. Asta,11and L. D. Marks1,2
1Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA
2Institute for Catalysis in Energy Processes, Northwestern University, Evanston, Illinois 60208, USA
3Engineering and Applied Science Division, California Institute of Technology, Pasadena, California 91125, USA
4JEOL USA, Inc., 11 Dearborn Road, Peabody, Massachusetts 01960, USA
5RAND Corporation, Arlington, Virginia 22202, USA
6School of Physics, The University of Sydney, Sydney, New South Wales 2026, Australia
7Cornell High Energy Synchrotron Source, Ithaca, New York 14953, USA
8Department of Chemistry, Northwestern University, Evanston, Illinois 60208, USA
9Max-Planck Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
10European Synchrotron Radiation Facility (ESRF), Boîte Postale 220, F-39043 Grenoble, France
11Chemical Engineering and Materials Science Department, University of California at Davis, Davis, California 95616, USA
/H20849Received 27 January 2007; revised manuscript received 15 May 2007; published 23 July 2007 /H20850
The c/H208496/H110032/H20850is a reconstruction of the SrTiO 3/H20849001/H20850surface that is formed between 1050 and 1100 °C in
oxidizing annealing conditions. This work proposes a model for the atomic structure for the c/H208496/H110032/H20850obtained
through a combination of results from transmission electron diffraction, surface x-ray diffraction, direct meth-ods analysis, computational combinational screening, and density functional theory. As it is formed at hightemperatures, the surface is complex and can be described as a short-range-ordered phase featuring micro-scopic domains composed of four main structural motifs. Additionally, nonperiodic TiO
2units are present on
the surface. Simulated scanning tunneling microscopy images based on the electronic structure calculations areconsistent with experimental images.
DOI: 10.1103/PhysRevB.76.045421 PACS number /H20849s/H20850: 68.35.Bs, 71.15.Mb, 61.10.Nz, 68.37.Ef
I. INTRODUCTION
Strontium titanate /H20849SrTiO 3/H20850has received considerable
attention over the last decade because of its numerous tech-
nological applications,1including use as a substrate for thin
film growth2and as a candidate crystalline gate dielectric
in silicon-based devices.3,4Furthermore, the surface of
SrTiO 3plays an important role in surface reactions and
catalysis.5,6Many of these applications are governed by in-
terfacial processes, which motivates a continuing interest inthe surfaces of SrTiO
3, but despite extensive research into
the surface properties, there remain many important unan-swered questions. Only recently, the /H208492/H110031/H20850andc/H208494/H110032/H20850re-
constructions on SrTiO
3/H20849001/H20850have been solved by direct
methods.7,8Other reconstructions have been observed on
pure SrTiO 3/H20849001/H20850, including the /H208491/H110031/H20850,/H208492/H110032/H20850,c/H208492/H110032/H20850,
/H208494/H110034/H20850,c/H208494/H110034/H20850,/H208496/H110032/H20850,c/H208496/H110032/H20850,/H20849/H208815/H11003/H208815/H20850R26°, and
/H20849/H2088113/H11003/H2088113/H20850R33.7°.6,9–19Models have been proposed for
many of these structures, though they are often inconsistent
with one another, and theoretical models have also been de-veloped; however, these too remain contradictory.
18,20–22
One structure which has proven especially difficult to de-
termine is the SrTiO 3/H20849001/H20850-c/H208496/H110032/H20850surface reconstruction.
The main challenge, as will be shown, is the fact that a single
reconstruction is unable to adequately describe the surface,which probably is related to the high annealing temperature/H208491050–1100 °C /H20850required to form the surface. Instead, the
equilibrium c/H208496/H110032/H20850surface at the formation temperature is
found to be short-range ordered, consisting of microscopic
domains of four related structural motifs. Upon sufficientlyrapid cooling, the surface structure is quenched, and the do-
mains of the four motifs persist.
The c/H208496/H110032/H20850has been previously reported by Jiang and
Zegenhagen with scanning tunneling microscopy /H20849STM /H20850and
low-energy electron diffraction
23,24and by Naito and Sato
with reflection high-energy electron diffraction /H20849RHEED /H20850.17
The STM results are included here, and newly available
x-ray diffraction results are also utilized. The c/H208496/H110032/H20850stud-
ied by RHEED17was found to coexist with domains of
/H20849/H2088113/H11003/H2088113/H20850R33.7° and may likely be different from the sur-
face studied here, as the surface preparation, which is known
to play a large role, was different. As mentioned earlier, a/H208496/H110032/H20850overlayer has also been observed on Nb doped
SrTiO
3/H20849001/H20850/H20849Ref. 19/H20850; however, this structure is not the
same as the c/H208496/H110032/H20850reported here since the /H208496/H110032/H20850surface
unit cell is not centered and thus has a different symmetry
and structure.
Direct methods for surfaces based on diffraction data
have been employed to solve numerous structures /H20849for more
information, see Refs 25and 26/H20850, including two other
surface reconstructions on SrTiO 3/H20849001/H20850, the /H208492/H110031/H20850and
c/H208494/H110032/H20850,7,8as well as the SrTiO 3/H20849106/H20850surface27and the
/H20849/H208815/H11003/H208815/H20850R26.6° reconstruction on LaAlO 3/H20849001/H20850.28In some
cases, e.g., the /H208492/H110031/H20850and c/H208494/H110032/H20850reconstructions on
SrTiO 3, direct methods can be used to find all of the atoms in
the surface structure. However, even with ideally perfectdata, sometimes direct methods fail to resolve the atomicpositions of certain atoms, particularly weakly scattering el-ements. Moreover, if disorder or twinning is present on thesurface, structure completion /H20849finding the full structure fromPHYSICAL REVIEW B 76, 045421 /H208492007 /H20850
1098-0121/2007/76 /H208494/H20850/045421 /H208499/H20850 ©2007 The American Physical Society 045421-1an initial fragment /H20850becomes exceedingly difficult. In this
work on the c/H208496/H110032/H20850reconstruction, direct methods alone did
not result in a structure solution, but instead a combinatorial
approach was taken that merged a variety of experimentaland computational techniques and resulted in a model of theSrTiO
3/H20849001/H20850-c/H208496/H110032/H20850surface that is consistent with all avail-
able experimental reports.
In more detail, the approach used in this work is to apply
direct methods on a set of transmission electron and x-raydiffraction data
25,26in order to determine the approximate
positions of the surface cations. Since the weak scatteringof oxygen ions prevented conclusive determination of theirpositions from diffraction methods alone, computationalcombinatorial screening methods were used along with first-principles calculations to identify candidate oxygen configu-rations. First principles calculations were also used to moreaccurately determine the surface cation positions. Thesestructural configurations were then used as input for structurerefinement using surface x-ray data with the help of the
SHELX-97 /H20849Ref. 29/H20850program, and simulated STM images
from the output of the ab initio calculations were also com-
pared with available experimental STM images as a finalcross-check. The proposed surface structure for the c/H208496/H110032/H20850
reconstruction is consistent with all of the available experi-
mental and computational evidence.
II. EXPERIMENTAL AND COMPUTATIONAL METHODS
Transmission electron diffraction /H20849TED /H20850experiments
were conducted on samples prepared from single crystal, un-doped SrTiO
3/H20849001/H20850wafers /H2084910/H1100310/H110035m m3, 99.95% pure /H20850.
The wafers were cut into 3 mm diameter disks using a rotarydisk cutter, mechanically thinned to /H11011120
/H9262m, polished,
dimpled, and ion milled to electron transparency with a4.8 kV Ar
+ion beam. Samples were annealed for 2–5 h in a
tube furnace at 1050–1100 °C under a flow of high purityoxygen at atmospheric pressure in order to produce the re-constructed surface. Transmission electron microscopy im-ages and off-zone diffraction patterns were obtained on theHitachi ultrahigh vacuum /H20849UHV /H20850H9000 electron micro-
scope, operated at 300 kV. A series of off-zone diffractionpatterns were recorded with exposure times ranging from0.5 to 120 s. The negatives were scanned with a 25
/H9262m
pixel size and digitized to 8 bits with an Optronics P-1000microdensitometer. The diffraction intensities were then av-eraged with the c2mm Patterson plane group symmetry,
yielding 58 independent intensities.
Surface x-ray diffraction /H20849SXRD /H20850experiments were per-
formed at the BW2 wiggler beamline at the Hamburg Syn-chrotron radiation laboratory using radiation of 8 keV,monochromatized and sagittally focused by a pair of Si /H20849111/H20850
crystals. Two single crystal SrTiO
3/H20849001/H20850samples were an-
nealed at the Max-Planck Institut in Stuttgart at 1100 °C inflowing oxygen for about 2 h. The samples were stored in anoxygen atmosphere container and shipped to another labora-tory where they were characterized at room temperature bySXRD in air. One of the samples was measured in air a fewdays after the preparation. The second crystal was reloadedinto a UHV chamber, exposed to a mild annealing in UHV at/H11011300 °C, and loaded into a small portable UHV chamber
which was mounted on the diffractometer for the SXRDmeasurements. The acquisition of the diffraction data tookapproximately 3 days for each of the two samples. The sta-bility of the surface over the acquisition period was ascer-tained by checking the stability of the /H20849080/H20850reflection at
regular intervals, and integrated intensities were recorded for263 in-plane reflections and 32 rods. The data were correctedfor footprint and polarization, had reflections below the criti-cal angle discarded, and were averaged using C2mmspace
group symmetry. The data taken for the two differentlyhandled samples /H20849oxygen annealed and oxygen and UHV
annealed /H20850were used separately for the structure refinement.
See Ref. 30for a copy of the SXRD data.
STM images were obtained using an Omicron “micro-
STM” system operating under UHV conditions. TheSrTiO
3/H20849001/H20850-c/H208496/H110032/H20850sample, which was prepared outside
the system by annealing at 1100 °C in a flow of oxygen, was
loaded into the UHV-STM system and annealed for approxi-mately 10–15 min at 800 °C in order to generate enoughoxygen vacancies in the bulk to allow imaging by STM.Tungsten tips were used, and the STM scanner was cali-
brated with the use of the well-known Si /H20849111/H20850-/H208497/H110037/H20850recon-
struction. Images were obtained in constant current topogra-
phy mode, and the sample was biased positively with respectto the tip, thus tunneling occurred into the empty states of thesample.
Direct methods were used to determine the scattering po-
tential map of the surface structure based on the transmissionelectron and x-ray diffraction data. Direct methods solve thephase problem by utilizing probability relationships betweenthe amplitude and the phase of the diffracted beams. A set ofphases is determined with the lowest figures of merit mostconsistent with scattering from discrete atoms and is com-bined with the measured beam amplitudes. By this approach,scattering potential maps and candidate structures can begenerated from the diffraction data without the need for astructure guess.
First-principles /H20849ab initio /H20850density functional theory
/H20849DFT /H20850calculations were performed using the Vienna ab ini-
tiosimulation package /H20849
V ASP /H20850,31–34which solves the DFT
equations within the plane-wave-pseudopotential formalism.The SrTiO
3/H20849001/H20850surface was represented by a surface slab
model as illustrated in Fig. 1, with all atomic positions re-
laxed except for the center atomic layer which was held fixedat bulk positions and lattice parameters /H20851determined in a
separate bulk local density approximation /H20849LDA /H20850calcula-
tion/H20852. The calculated lattice parameter /H208493.827 Å /H20850is about 2%
smaller than the experimental lattice parameter at room tem-
perature /H208493.905 Å /H20850, which is typical for LDA calculations.
Core electrons were represented by Vanderbilt-type ultrasoft
pseudopotentials
35,36/H20849V ASP library pseudopotentials “Ti,”
“Sr,” and “O /H6018s”/H20850, and electron exchange and correlation were
treated in the LDA /H20849Ceperley and Adler37/H20850. The plane-wave
basis set was cut off at 270 eV. Simulated STM images wereproduced from the output of the ab initio calculations in the
Tersoff-Hamann approximation,
38which assumes that the
pointlike STM tip follows an isosurface of the local densityof states within a specified energy window around the Fermilevel. A relatively high isodensity surface lying very close toLANIER et al. PHYSICAL REVIEW B 76, 045421 /H208492007 /H20850
045421-2the surface was used, thus enabling us to use a smaller
“vacuum” region in the supercell calculation. Simulated im-ages were created using the integrated density of unoccupiedstates between 0 and +2.1 V relative to the Fermi level.
Surface x-ray diffraction data structure refinements were
performed using the
SHELX-97 code,29which is a widely used
structural refinement program used in many fields includingcrystallography. The atomic positions for each of the plau-sible structures generated by DFT were input into the
SHELX-
97program and refined primarily against the experimental
data obtained in air. Since LDA calculations underestimatethe lattice parameters, we scaled all atomic positions isotro-pically until the calculated lattice parameters matched theexperimental value. This approach is preferable to imposingthe experimental in-plane lattice parameters in the calcula-tions, since the system would then contract perpendicular tothe surface, resulting in an unphysical distortion that wouldbe difficult to correct. The data were decomposed into 33batches, 1 for the in-plane set and 32 for each of the rods,and each batch was given a different scale factor to accountfor experimental error in the data collection owing tochanges in the sample-detector geometries upon measure-ments of different rods. Refinement parameters are given inthe input /H20849.ins/H20850file and are described in the
SHELX-97 manual.
See Ref. 30for a copy of the input /H20849.ins/H20850file and final /H20849Fc/H208502
values.
III. RESULTS
A. Transmission electron microscopy
Dark field transmission electron microscopy images and
off-zone diffraction patterns were obtained for the c/H208496/H110032/H20850
surface, as shown in Fig. 2. For the sample preparation tech-niques employed here, the c/H208496/H110032/H20850surface reconstruction is
highly reproducible and was found to be air stable over a
period of months. The dark field image in Fig. 2shows a flat,
faceted surface with large terraces separated by step bunches,and the c/H208496/H110032/H20850surface reconstruction was found to cover
the entire surface. V oids are also visible in the near-surface
region of the sample, and similar morphologies have beenobserved for other reconstructed SrTiO
3/H20849001/H20850surfaces.39
B. Direct methods
Direct methods provided the scattering potential maps
shown in Fig. 3based on surface x-ray diffraction data /H20851Figs.
3/H20849a/H20850–3/H20849c/H20850/H20852and transmission electron diffraction data /H20851Fig.
3/H20849d/H20850/H20852. Further analysis, based on symmetry and difference
maps, indicated that the dark spots were titanium atom sitesand that the surface contained no strontium atoms. Numerousattempts were made to refine a single structure with reason-able oxygen sites, but no single structure yielded good re-sults. This occurred, as will be shown, because the surface isreally a mixture of four different structural motifs. While thepositions of the titanium atoms averaged over the four struc-tural motifs could be determined in projection from the elec-tron diffraction data and in three dimensions from the x-raydiffraction data, the positions of the surface oxygen atomscould not be determined owing to larger variation of theoxygen positions among the four motifs. This conclusionwas reached by applying a combinational screening methodin conjunction with first-principles methods to identify plau-sible oxygen configurations, with the averaged positions ofthe titanium atoms from the direct methods analysis used asthe input for the screening method.
C. Computational screening
The determination of the minimum energy oxygen con-
figuration represented a challenging optimization problem,given the large configuration space that needed to besampled and the presence of an enormous number of local
FIG. 1. /H20849Color online /H20850Geometry employed in the ab initio cal-
culations, with the primitive c/H208496/H110032/H20850surface unit cell outlined /H20849rep-
resentative structure shown /H20850. Large red spheres are oxygen, small
light gray spheres are titanium, and medium dark gray spheres arestrontium. The geometries of the two lowest-energy structures ateach composition were also reoptimized using a thicker slab /H20849in-
cluding four strontium layers instead of two /H20850in which the middle
layer /H20849containing Ti and O /H20850was kept frozen.
FIG. 2. /H20849Color online /H20850Dark field image and transmission elec-
tron diffraction data /H20849inset /H20850from the c/H208496/H110032/H20850surface. Primitive re-
ciprocal unit cells for the two surface domains unit cell are outlined.Adapted from Ref. 39.ATOMIC-SCALE STRUCTURE OF THE SrTiO
3/H20849001/H20850-… PHYSICAL REVIEW B 76, 045421 /H208492007 /H20850
045421-3minima in the system’s potential energy surface, i.e., the en-
ergy of the system as a function of all atomic coordinates.Each local minimum is surrounded by a basin of attraction ,
where the set of all points connected to that local minimumfollows a continuous path along which the energy decreases.The screening approach divided the optimization probleminto /H20849i/H20850a discrete outer optimization problem over the differ-
ent basins of attraction and /H20849ii/H20850a continuous inner optimiza-
tion problem over atomic coordinates within each basin. Theouter optimization problem scanned over basins and pro-vided suitable starting points for the inner continuous opti-mization problem. In effect, each basin was represented bythe selection of one starting point or starting configuration
within it. The specific starting configuration in a basin wassomewhat arbitrary since the inner continuous optimizationproblem should find the same local minimum regardless ofthe starting configuration used.
Starting configurations were constructed via enumeration
of every possible placement combination of oxygen atomson a lattice of plausible candidate sites. These candidatesites, shown in Fig. 4, are located at the midpoint of /H208491/H20850
every pair of titanium atoms separated by /H333554.25 Å and /H208492/H20850
every triplet of titanium atoms separated by /H333554.25 Å. Four-
coordinated oxygen sites were not considered, because theyeither produced redundant sites or required at least one of thefour titanium-oxygen bonds to be longer than 2.3 Å. Onefoldcoordinated oxygen sites on top of each of the four sym-metrically distinct surface titanium atoms were considered aswell.
The total number of possible ways to place oxygen atoms
on the candidate sites was 2
40, as there were 40 candidateoxygen sites in the asymmetric unit of the surface unit cell.
All of these configurations possess, by construction, theC2mmspace group determined from the experimental SXRD
data. Since it would have been prohibitively computationallyexpensive to calculate the minimum energy within each ba-sin associated with each of these starting configurations viafirst-principles methods, a hierarchy of increasingly precisecriteria was utilized instead to screen out high-energy con-figurations.
At the coarsest level a simple geometric criteria was used,
discarding configurations /H208491/H20850with an oxygen deficiency ex-
ceeding two oxygen atoms per primitive surface unit cell, /H208492/H20850
with oxygen-oxygen bonds shorter than 1.8 Å, or /H208493/H20850con-
taining a titanium atom with a coordination number less than3 or more than 6. These simple criteria reduced the numberof plausible configurations to 17 095. While this number re-mained too large to be handled via ab initio methods, it was
easily manageable using a simple electrostatic pair potentialmodel, where the species Sr, Ti, and O take the nominalcharges 2+, 4+ and 2−, respectively, which could be used toefficiently identify the most promising configurations.
The electrostatic energy was calculated for each of the
17 095 candidate starting configurations previously identi-fied. Note that the atomic positions were not relaxed in thesecalculations, otherwise the system would have collapsed to apoint, since there were no short-range repulsive componentsin the interatomic forces. For nonstoichiometric structures/H20849stoichiometry of TiO
2−x, where x/H110220/H20850, the charges of all
titanium atoms were reduced to 4−2 xto maintain charge
balance, since the Fermi level would lie within the titaniumbands under oxygen-deficient conditions.
At the end of this screening step, /H1101175 structures with the
lowest electrostatic energy were retained, at each of the threesurface stoichiometries considered /H20849from zero to two oxygen
vacancies per primitive surface unit cell /H20850. Fully relaxed LDA
calculations were then performed for each of these /H1101175
structures using the
V ASP code. A representative structural
geometry is illustrated in Fig. 1. Given the large number of
candidate geometries, a relatively thin slab was used to rep-resent the surface and the Brillouin zone was sampled at the/H9003point only, in order to limit the computational costs. After
the structural relaxations, some of the starting configurationsthat were initially distinct actually converged to the sameconfiguration: the /H1101175 starting configurations produced 64
distinct relaxed geometries. The convergence of some of the
FIG. 3. /H20849Color online /H20850/H20851 /H20849a/H20850–/H20849c/H20850/H20852Electron density maps for the
centered c/H208496/H110032/H20850unit cell from SXRD direct methods at z=3.6 Å,
z=2.8 Å, and z=2.0 Å above the first bulklike TiO 2layer, respec-
tively. Regions of high electron density /H20849possible atomic sites /H20850are
yellow /H20849light /H20850./H20849d/H20850Scattering potential map /H20849projected /H20850for the cen-
tered c/H208496/H110032/H20850unit cell from TED direct methods. Regions of high
scattering potential /H20849possible atomic sites /H20850are black.
FIG. 4. /H20849Color online /H20850/H20849a/H20850Geometric rules used to generate can-
didate O sites shown in /H20849b/H20850. The top panel is view toward the sur-
face, while the bottom panel is a side view with the free surfacepointing upward.LANIER et al. PHYSICAL REVIEW B 76, 045421 /H208492007 /H20850
045421-4configurations toward each other was an indication that the
initial partitioning of configuration space was sufficientlyfine, as the “distance” between any two starting configura-tions was, on average, slightly smaller than the size of thetypical basin of attraction.
The lowest-energy configurations, i.e., structural motifs,
thus identified for each of the three stoichiometries areshown in Fig. 5. These geometries were reoptimized using a
thicker slab /H20849twice the thickness shown in Fig. 1/H20850and a finer
k-point mesh /H208494/H110034/H110031/H20850to yield more accurate energies. At
the TiO
2stoichiometry, the lowest-energy structure is labeled
as “RumpledStoichiometric” /H20851Fig. 5/H20849a/H20850/H20852. The next lowest-
energy structure, labeled as “FlatStoichiometric” /H20851Fig. 5/H20849b/H20850/H20852,
is 0.37 eV less stable /H20849per primitive unit cell /H20850.
At an oxygen content corresponding to one oxygen va-
cancy per primitive surface unit cell, the screening algorithmidentified the “RumpledVacancy” as the lowest-energy struc-ture /H20851Fig. 5/H20849c/H20850/H20852. Visual inspection of that structure revealed
that slight displacements along the surface normal of thetitanium atoms near the center of the cell changed their co-ordination from fourfold to fivefold, resulting in anotherplausible structure, labeled “FlatVacancy” /H20851Fig. 5/H20849d/H20850/H20852. Given
that the positions determined from direct methods were av-
eraged over the four structural motifs, it is not entirely sur-prising that some adjustments would be required for the cat-ion positions of any one of the individual structures. Whilethe combinational screening did not find this structure auto-matically, as it assumed the positions of the Ti atom to beexact, it did identify a sufficiently similar structure to enableits discovery. During our initial lower-precision screening,the FlatVacancy structure appeared to have a lower energythan the RumpledVacancy structure. However, our more ac-curate reoptimization of the geometries revealed that theRumpledVacancy structure is the ground state at that compo-sition, with an energy 0.26 eV/unit cell lower than the Flat-Vacancy structure.
At the composition corresponding to two oxygen vacan-
cies per primitive surface unit cell, the “DoubleVacancy”structure was identified as the lowest-energy structure /H20851Fig.
5/H20849e/H20850/H20852. The second most stable structure is more than
3 eV/unit cell less stable than the DoubleVacancy structureand can thus be ruled out.
The relative surface energy per primitive unit cell for each
of these structural motifs was calculated and plotted as afunction of oxygen chemical potential in Fig. 6. It is noted
that the DoubleVacancy structure has a potential that is sohigh that its corresponding line lies far above the range of thefigure and is therefore unlikely to be present on the surface.Since the exact surface energies are also a function of the Tiand Sr chemical potentials /H20849which are difficult to infer from
experimental conditions /H20850, we plot the surface energies rela-
tive to the RumpledStoichiometric surface energy. This dif-ference in surface energies is sufficient to assess the relativestability of the motifs and offers the advantage that the con-tributions of the Ti and Sr chemical potentials cancel outexactly /H20849because all motifs have the same number of Ti or Sr
atoms /H20850. In contrast, the dependence on the O chemical po-
tential cannot be similarly eliminated because the differentmotifs have different oxygen contents.
The range of chemical potentials considered corresponds
to temperatures ranging from 0 to 1300 K. The oxygenchemical potential /H20849in O
2at atmospheric pressure /H20850was ob-
tained from the equation
FIG. 5. /H20849Color online /H20850Candidate surface reconstructions show-
ing side view, top view /H20849showing only atoms in the topmost surface
layer /H20850, and simulated STM image. Large red spheres are oxygen,
small light gray spheres are titanium, and medium dark grayspheres are strontium.-0.500.511.52
-1.6-1.4-1.2 -1-0.8-0.6-0.4-0.2 0RelativeSurface Energy (eV/unitc ell)
Oxygen Chemical Potential (eV/atom)RumpledStoichiometricFlatStoichiometricRumpledVacancyFlatVacancyµOat 1300K µOat 0K
FIG. 6. Relative surface energy per primitive surface unit cell
of the four proposed surface motifs as a function of oxygen chemi-cal potential. The surface energies are given relative to theRumpledStoichiometric structure and the chemical potential is rela-tive to its value at 0 K.ATOMIC-SCALE STRUCTURE OF THE SrTiO
3/H20849001/H20850-… PHYSICAL REVIEW B 76, 045421 /H208492007 /H20850
045421-5/H9262O/H20849T/H20850=/H208491/2/H20850/H9262O2/H20849T/H20850=/H208491/2/H20850/H20851HLDA+H/H20849T/H20850−H/H208490/H20850−TS/H20849T/H20850/H20852,
where HLDA=−9.676 eV /H20849from a LDA calculation of an iso-
lated O 2molecule /H20850and the following tabulated thermody-
namic values from Ref. 40 were used: H/H208491300 K /H20850
=33 344 J/mol, H/H208490K/H20850=−8683 J/mol, and S/H208491300 K /H20850
=252.878 J/ /H20849mol K /H20850.
It is expected that the actual relative surface energies can
be read off from Fig. 6at a value of the oxygen chemical
potential lying somewhere between the calculated extremesshown in the figure. At T=0 K, the calculations have as-
sumed zero entropy and therefore overstabilize the stoichio-metric phases, while at T=1300 K, the calculations only ac-
count for the entropy of the gas phase and, since the freeenergy change of the solid phases may partially offset the O
2
chemical potential change, probably result an overstabiliza-tion of the gas phase and of the nonstoichiometric phases.
The surface energies of the four structural motifs
considered /H20849RumpledStoichiometric, FlatStoichiometric,
FlatVacancy, and RumpledVacancy /H20850lie within 0.4 eV/unit
cell of each other for chemical potentials slightly below the1300 K value. The actual energy range is likely to be evensmaller than our calculated range of 0.4 eV because our re-sults neglect the contribution of lattice vibrations to the freeenergy. Structures that are very stable /H20849low in energy /H20850tend to
be stiffer and therefore have a lower vibrational entropy anda more positive free energy. Conversely, vibrational effectstend to lower the free energies of high-energy structures,resulting in a reduction of the spread in the free energies.Thus, the surface energy differences lie in a range that islikely to be somewhat smaller than 0.4 eV, and thus compa-rable in magnitude to k
BTat 1300 K /H20849about 0.12 eV /H20850, mak-
ing it quite plausible for the equilibrium surface structure toconsist of a disordered mixture of these four structural mo-tifs.
The accuracy of the approach we used to obtain the oxy-
gen chemical potential is limited by the fact that the LDAtends to poorly predict the energy of an isolated molecule.An alternative approach, following Ref. 20, that avoids cal-
culating isolated molecule energies yielded qualitativelysimilar conclusions: All points where the different surfaceenergies intersect lie between the values of the O chemicalpotential at 0 and 1300 K.
The four structural motifs RumpledStoichiometric,
FlatStoichiometric, FlatVacancy, and RumpledVacancy canbe described using four atomic layers. Starting at the bottomfor all motifs /H20849in reference to the geometry shown in Fig. 5/H20850,
there is a bulklike TiO
2layer followed above with a bulklike
SrO layer, and these two layers are nearly identical in all fourmotifs. The next TiO
2layer up is similar in all structural
motifs and has a rumpled bulklike structure, with relaxationsalong the direction normal to the surface of at most/H110110.12a
bulk. Finally, the topmost layer is different for each of
the four motifs in the number and placement of the oxygenatoms: the top layer has a Ti
20O40stoichiometry in the sto-
ichiometric structure centered unit cell and has a Ti 20O38
stoichiometry in the vacancy structure centered unit cell.Note that the titanium positions are nearly identical in allstructures. See Ref. 30for the atomic positions of the four
structural motifs.
For each structural motif, the topmost layer contains
a zigzag along the b/H20849short axis /H20850direction of fivefold
coordinated titanium atoms in the form of truncated octahe-dra. In the centered unit cell, the two zigzags are located at
approximately
1
4and3
4along the length of the long /H20849a/H20850axis
/H20849see Fig. 5/H20850, and the relative orientation of the truncated oc-
tahedra along the zigzags is the same for three of the fourstructures and is reversed in the RumpledVacancy structure.In the rumpled structures /H20849RumpledStoichiometric and
RumpledVacancy /H20850, the zigzag is elevated normal to the sur-
face relative to the center of the unit cell, and in the flatstructures /H20849FlatStoichiometric and FlatVacancy /H20850, the center
of the unit cell is at approximately the same elevation as thezigzag. Accordingly, the titanium atoms at the center of theunit cell /H20849not part of the zigzag /H20850in the rumpled structures are
coordinated to the bulklike layer below, while in the flatstructures they are not. The coordination of the titanium at-oms at the center of the cell is the driving force for theplacement of the singly coordinated oxygen /H20849if any /H20850in the
various structures. In the structures containing a singly coor-dinated oxygen, i.e., the RumpledStoichiometric, FlatSto-ichiometric, and RumpledVacancy structures, the singly co-ordinated Ti-O bonds are 1.68, 1.65, and 1.65 Å long,respectively, indicating double bond /H20849titanyl /H20850character. Es-
sentially, the differences among the four structures lie in therelative orientation of the truncated octahedra in the zigzagchain, the elevation and coordination of the titanium atomslocated in the center of the unit cell, and the placement of thesingly coordinated oxygen /H20849if any /H20850at the surface.
D. STM experiment and simulation
In STM images taken under empty-state bias conditions
/H20849Fig. 7/H20850, the c/H208496/H110032/H20850reconstruction appears as bright rows
with a spacing of 11.7 Å /H20851cf. with 11.715 Å for1
2thec/H208496
/H110032/H20850long axis length, 23.43 Å /H20852. Confirmed to be c/H208496/H110032/H20850by
low-energy electron diffraction, the reconstruction was found
to cover the surface uniformly wherever probed by the STM.In large-scale images /H20849not shown /H20850, the rows appear to be
FIG. 7. High resolution STM image of the c/H208496/H110032/H20850surface re-
construction /H20849Vs=2.1 V, I=0.28 nA /H20850. The c/H208496/H110032/H20850centered unit
cell is outlined. Adapted from Ref. 23.LANIER et al. PHYSICAL REVIEW B 76, 045421 /H208492007 /H20850
045421-6aligned with equal probability along the /H20851100/H20852or/H20851010/H20852crys-
tal directions, and in addition to the rows, bright protrusionssituated on the rows can be seen randomly distributed overthe surface with a density of approximately one for everythree c/H208496/H110032/H20850centered unit cells. It is noted that sufficient
conductivity in SrTiO
3is achieved with an overall carrier
density due to oxygen vacancies smaller than 1018e/cm3,
i.e., roughly 1 out of every 30 neighboring oxygen atomsmissing. It is expected that the density of oxygen vacancieson the surface may be slightly higher, but still low comparedto the density of observed contrast variations. Furthermore,preliminary experimental studies in which SXRD data werecollected on samples used for STM and low-energy electrondiffraction have evidenced that the UHV anneal prior toSTM measurements has a minimal effect on the c/H208496/H110032/H20850
structure.
The simulated STM images, shown in Fig. 5for each of
the structural motifs considered, confirm that in empty stateonly titanium atoms image brightly, while oxygen atoms aredark, and thus the experimentally observed rows are, in fact,the zigzags of truncated octahedra discussed earlier. Notethat the pointlike tip approximation and the tracing of a rela-tively high isodensity surface resulted in simulated STM im-ages of higher resolution /H20849sharper /H20850than the experimental im-
age. Upon detailed investigation of the experimental image,changes in the relative orientation of the zigzags can be seenoccasionally from one row to another, evidence of domainboundaries between different structural motifs.
Upon inspection of the simulated STM images from the
structural motifs alone, the bright protrusions observed in theexperimental STM images are not accounted for. Basedon the previous observation that the STM is imaging tita-nium atoms, it was determined that the contrast of the brightprotrusion is due to excess nonperiodic titanium atoms alongthe zigzag. Upon studying plausible structures, a likely loca-tion for the titanium atom is readily apparent in theRumpledStoichiometric structure. This plausible geometry issuggested by the fact that the two singly coordinated oxygenatoms are at just the right position so that an additional TiO
2
unit could be placed on the surface, and the inserted titaniumatom would have a fourfold coordination and the insertedoxygen atoms would complete the octahedral coordination ofthe truncated octahedra in the zigzag. To clarify the nature ofthese bright protrusions, a simulated STM image was gener-ated of the RumpledStoichiometric surface with an addi-tional TiO
2unit located on the zigzag /H20851see Fig. 5/H20849f/H20850/H20852, and the
calculated STM image of this surface is in qualitative agree-ment with the experimentally observed bright protrusions.Note that the final surface stoichiometry is Ti
21O42for one
unit added per centered unit cell, and thus TiO 2is added to
the structure in a stoichiometric manner. See Ref. 30for the
DFT refined positions of the TiO 2unit.
E. Structure refinement
To substantiate the proposed c/H208496/H110032/H20850surface structure
model, refinement with XRD data was carried out by means
of the SHELX-97 refinement program.29Use of this program
allowed for the refinement of the complicated, multido-mained c/H208496/H110032/H20850structure through partial occupancies of
atom sites. Figures of merit including weighted Rvalues
/H20849wR2/H20850and goodness of fit were employed as a gauge for the
quality of the refinement, and the Hamilton R-factor ratio41
was utilized to compare wR2values for structural refine-
ments with various numbers of parameters. The absolute val-ues of the figures of merit do not hold much meaning outsideof this study, as this is not a standard
SHELX structural refine-
ment, but rather the figures of merit are used to comparemodels relative to one another. Further, it is important to notethat one should not expect a perfect fit between the DFT-calculated positions and the refined positions. Both methodsinvoke approximations: notably, the refinement process relieson partial occupancies to model disorder, and the DFT cal-culations neglect thermal expansion, which could affect theaverage positions of atoms in low-symmetry environmentsand have an accuracy limited by the unavoidable approxima-tion of the exchange-correlation functional and, to a lesserextent, by the finite k-point mesh and energy cutoff.
The four DFT-relaxed structural motifs were refined inde-
pendently for 25 least squared cycles, and the structures hadthree bulklike layers below the surface atoms, as illustratedin the cartoons of Fig. 5. Additionally, in order to better
represent the surface from which the data were acquired, allfour structural motifs were combined and refined simulta-neously for 25 least squared cycles. In this case, the com-bined structure had the same three bulklike layers as theother structures but had a surface containing the atoms fromall four structural motifs. The occupancies for the surfaceatoms representing the four motifs, FlatStoichiometric,FlatVacancy, RumpledStoichiometric, and RumpledVacancy/H20849x
FS,xFV,xRS, and xRV, respectively /H20850, were constrained such
that the sum of the four occupancies is 1, and initially eachmotif was assigned an occupancy of 25%.
A TiO
2unit was placed on top of the surface’s zigzag
with occupancy xTiO2to correlate with the bright protrusions
in the experimental STM images. Owing to the symmetry
constraints of the refinement, the TiO 2was added in a peri-
odic fashion, because adding a single TiO 2unit in the unit
cell would require a reduction in the symmetry, thereforeincreasing the number of parameters /H20849p/H20850, which is undesir-
able. Thus, to model the nonperiodic nature of the TiO
2unit,
the occupancy /H20849xTiO2/H20850was allowed to vary as an independent
variable.
Table Ishows the figures of merit for each of the
structural refinements: four motifs combined plus theTiO
2unit, four motifs combined without TiO 2unit,
RumpledStoichiometric, FlatStoichiometric, FlatVacancy,and RumpledVacancy. It is important to note that the posi-tions relaxed by the DFT calculations did not change muchupon refinement, providing strong evidence that they are ap-propriate models. Using the Hamilton R-factor ratio,
41the
structure with the four motifs combined fits the data betterthan any of the other individual models with greater than90% certainty. Other models were tested, including struc-tures composed of combinations of two or three of the struc-tural motifs and structures incorporating the DoubleVacancymotif; however, these refinements tended to be inferior andsupported the four structural motif model.ATOMIC-SCALE STRUCTURE OF THE SrTiO 3/H20849001/H20850-… PHYSICAL REVIEW B 76, 045421 /H208492007 /H20850
045421-7The figures of merit for the individual structure
refinements are similar for the FlatVacancy,RumpledStoichiometric, and FlatStoichiometric structuresand showed a worse fit for the RumpledVacancy structure,all in qualitative agreement with the relative surface energyvalues. For the four motifs combined structure, the final val-ues for x
FS,xFV,xRS, and xRVeach remained close to 25%,
i.e., each structural motif is present on approximately1
4of
the surface. The TiO 2unit /H20849xTiO2/H20850is situated on roughly
15%–45% of the c/H208496/H110032/H20850surface unit cells, which agrees
well with the experimental STM measurement of approxi-
mately 33%. Data from the second sample, also annealed inO
2at 1100 °C but subsequently annealed in UHV at 300 °C,
also gave similar occupancies for xFS,xFV,xRS,xRV, and xTiO2in the four motifs combined structure, which is expected
since the oxygen chemical potential at 1000 °C in O 2and at
300 °C in UHV are similar /H20849−3.2 and −2.57 eV, respec-
tively /H20850.
IV . DISCUSSION
A model for the structure of the c/H208496/H110032/H20850reconstruction
has been proposed, and unlike the /H208492/H110031/H20850andc/H208494/H110032/H20850recon-
structions on SrTiO 3/H20849001/H20850, the c/H208496/H110032/H20850structure solution
was not explicitly provided from direct methods analysis
alone. Of the three reconstructions, the c/H208496/H110032/H20850forms at the
highest temperature, 1050–1100 °C, compared to
850–930 °C for c/H208494/H110032/H20850and 950–1050 °C for /H208492/H110031/H20850, and
is therefore, not surprisingly, the most complex structure.
The surface is composed of short-range-ordered domains offour related structures, ranging from stoichiometric toslightly reduced /H20849one oxygen vacancy per primitive surface
unit cell /H20850, each present on approximately
1
4of the total sur-
face area. At the temperature and oxygen partial pressurerequired for the formation of the c/H208496/H110032/H20850surface reconstruc-
tion, the formation energies for these structures are quite
comparable, and the surface thus takes the form of a random/H20849although short-range-ordered /H20850mixture of these four struc-
tural motifs. A rough approximation for the entropy of mix-ing is 1.39 kTper unit cell area, which at 1100 °C is
0.164 eV. This value represents the upper bound, as it ne-glects domain boundary energy and assumes that the struc-ture of one unit cell does not influence the structure of neigh-boring cells. Additionally, the TiO
2unit, which is present
nonperiodically on the surface, also results in an entropicfree energy gain for the surface.The proposed c/H208496/H110032/H20850structure, while the most compli-
cated reconstruction on SrTiO
3, shows similarities to the /H208492
/H110031/H20850andc/H208494/H110032/H20850structures.7,8All three structures are termi-
nated with a Ti yOxsurface layer, that is, there are no stron-
tium atoms on the surface. The c/H208494/H110032/H20850and /H208492/H110031/H20850recon-
structions are composed of a single TiO 2-stoichiometry
overlayer above bulklike TiO 2, and the difference between
thec/H208494/H110032/H20850and /H208492/H110031/H20850structures is the distribution of the
surface Ti among the possible sites. The c/H208496/H110032/H20850,o nt h e
other hand, has a thicker /H20849more than one /H20850TiO xoverlayer
above the bulklike TiO 2layer. Furthermore, the c/H208494/H110032/H20850and
/H208492/H110031/H20850structures have titanium cations present on the sur-
face solely in the form of fivefold, truncated octahedra, and
while the c/H208496/H110032/H20850reconstruction does have titanium cations
in fivefold truncated octahedra, titanium cations are also
present in the surface structure with fourfold coordination.The most striking difference is the fact that the c/H208496/H110032/H20850re-
construction is composed of multiple related, but different,
structural domains, while the c/H208494/H110032/H20850and /H208492/H110031/H20850reconstruc-
tions are single-structure surfaces. Finally, TiO
2units are sta-
bilized on the surface of SrTiO 3/H20849001/H20850-c/H208496/H110032/H20850, but no evi-
dence exists for this type of behavior on the c/H208494/H110032/H20850or
/H208492/H110031/H20850surfaces.
It is believed that the c/H208496/H110032/H20850surface is likely to be the
most catalytically active surface of SrTiO 3/H20849001/H20850. With tita-
nium atoms present in multiple coordination geometries and
oxidation states, the surface would likely be able to bindreactant molecules and promote redox-type reactions. Thec/H208496/H110032/H20850reconstruction /H20851as well as the /H208492/H110031/H20850/H20852contains
TivO/H20849titanyl /H20850groups which have recently been implicated
with catalytic activity on the /H20849011/H20850surface of rutile TiO
2.42
Furthermore, the presence of the TiO 2unit suggests the abil-
ity of the surface to stabilize reaction intermediates, and re-search is currently under way to investigate the adsorption,desorption, and reactivity of methyl radicals on the variousreconstructions of SrTiO
3/H20849001/H20850.
V . CONCLUSIONS
In conclusion, a model for the atomic-scale structure of
the SrTiO 3/H20849001/H20850-c/H208496/H110032/H20850surface reconstruction has been
proposed. The surface reconstruction is formed at high tem-
peratures /H208491050–1100 °C /H20850in oxidizing conditions and is
highly stable and reproducible. The surface is composed of
domains of similar but distinct structures, and additionally,TABLE I. Figures of merit for refinement of DFT-relaxed structures against SXRD data.
ModelNo. of
LSNo. of
data /H20849n/H20850No. of
parameters /H20849m/H20850 wR2Goodness
of fit
Four motifs combined, with TiO 2unit 25 848 286 0.65 5.67
Four motifs combined, without TiO 2unit 25 848 280 0.65 5.65
RumpledStoichiometric only 25 848 158 0.74 6.27FlatStoichiometric only 25 848 158 0.74 6.33RumpledVacancy only 25 848 157 0.77 6.69FlatVacancy only 25 848 157 0.73 6.25LANIER et al. PHYSICAL REVIEW B 76, 045421 /H208492007 /H20850
045421-8TiO 2units are randomly distributed on the surface. While the
structure solution method was not conventional, we have ac-quired the maximum amount of information through a com-bination of techniques. Transmission electron diffraction andsurface x-ray diffraction provided the positions of the surfacetitanium atoms averaged over the four structural motifs, andtheab initio screening technique proved to be indispensable
for the determination of oxygen positions, as well as thetitanium positions along the zdirection. Adaptation of the
SHELX-97 program for structure refinement against surface
x-ray data merged theory with experiment to corroborate themodel, and finally STM simulations confirmed consistencywith experimental observations.
ACKNOWLEDGMENTS
C.H.L., K.R.P., and L.D.M. were supported by the Chemi-
cal Sciences, Geosciences and Biosciences Division, Officeof Basic Energy Sciences, U.S. Department of Energy Officeof Science Grant No. DE-FG02-03ER15457, and N.E. and
O.W. were supported by the EMSI program of the NationalScience Foundation and the U.S. Department of Energy Of-fice of Science Grant No. CHE-9810378, all at the North-western University Institute for Environmental Catalysis.A.v.d.W and M.A. were supported by the National ScienceFoundation under program NSF-MRSEC /H20849DMR-00706097 /H20850
and through TeraGrid computing resources provided byNCSA and SDSC. E.L. was supported by the National Sci-ence Foundation via Grant No. DMR-9214505. A.K. and J.Z.were supported by the German BMBF under Contracts Nos.05SE8GUA5 and 05KS1GUC3. The work of A. v. d. W., N.E., E. L., and M. A. was performed while at NorthwesternUniversity, Department of Materials Science and Engineer-ing. The work of O.W. was performed while at NorthwesternUniversity, Department of Physics and Astronomy. The workof A. K. was performed while at Max-Planck-Institute fürFestkörperforschung.
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045421-9 |
PhysRevB.83.075110.pdf | PHYSICAL REVIEW B 83, 075110 (2011)
Surface and edge states in topological semimetals
Rui-Lin Chu, Wen-Yu Shan, Jie Lu, and Shun-Qing Shen
Department of Physics and Center of Computational and Theoretical Physics, The University of Hong Kong,
Pokfulam Road, Hong Kong
(Received 22 September 2010; revised manuscript received 10 November 2010; published 15 February 2011)
We study the topologically nontrivial semimetals by means of the six-band Kane model. Existence of surface
states is explicitly demonstrated by calculating the local density of states (LDOS) on the material surface. Inthe strain-free condition, surface states are divided into two parts in the energy spectrum, one part is in thedirect gap, the other part including the crossing point of surface state Dirac cone is submerged in the valenceband. We also show how uniaxial strain induces an insulating band gap and raises the crossing point fromthe valence band into the band gap, making the system a true topological insulator. We predict the existenceof helical edge states and spin Hall effect in the thin-film topological semimetals, which could be tested byfuture experiments. Disorder is found to significantly enhance the spin-Hall effect in the valence band of thethin films.
DOI: 10.1103/PhysRevB.83.075110 PACS number(s): 73 .20.At, 73 .61.At
I. INTRODUCTION
Topological insulators (TIs) have been recognized as novel
states of quantum matter.1They are of tremendous interest
to both fundamental condensed matter physics and potentialapplications in spintronics as well as quantum computing.TIs are insulating in the bulk which is usually due todirect bulk band gaps. However, on the boundary thereare topologically protected gapless surface or edge states.Usually in these materials the spin-orbital coupling is verystrong such that the conduction band and valence bandare inverted. Such phenomena have already been reportedin the literature of the 1980s.
2–6However, the topological
nature behind these phenomena was not revealed at thetime. With the recent development of topological bandtheories, a series of materials including HgTe quantumwell, Bi
xSb1−x,B i 2Se3, and Bi 2Te3have been theoretically
predicted and experimentally realized as two-dimensional(2D) and three-dimensional (3D) TIs.
7–9,11–15The search
for TIs has been extended from these alloys and binarycompounds to ternary compounds. Very recently a largefamily of materials, namely the Heusler-related and Li-basedintermetallic ternary compounds, have been predicted to bepromising 3D TIs through first principle calculations.
16–22
The enormous variety in these compounds provides wide
options for future material synthesizing of TIs. However,despite the fact that an insulating bulk is a critical prerequisiteto the TI theory, none of these newly found materials arenaturally band insulators. Many of them are semimetals oreven metals. It has been shown that the band structure andband topology of many of these compounds closely resemblethat of the zinc blende structure binary compound HgTe andCdTe.
16–22
In this work we study the evolution of surface states in these
topologically nontrivial compounds whose band structuresclosely resemble 3D HgTe by means of the six-band Kanemodel. By studying the local density of states (LDOS) on thematerial boundary, we demonstrate explicitly the existence ofsurface states in the direct band gap. Furthermore, we showthat in the strain-free condition, the surface states are separatedinto two parts in the band structure. One part exists in the
direct band gap, the other part of surface states (including thecrossing point of the Dirac cone) submerges in the valencebands. The latter surface states have distinct momentum-dependent spatial distribution from the former. By applyinguniaxial strains, the crossing point can be raised up intothe strain-induced insulating gap. In the thin films made ofthese materials, topologically protected helical edge stateswill emerge on the sample edges. In this way we show acrossover between 3D TIs and 2D TIs. In the strain-freecondition, the bulk band gap can be controlled by tuningthe film thickness. Meanwhile, we found that in the thinfilms, disorders or impurities can significantly enhance thespin-Hall effect (SHE) when the Fermi level is in the valenceband.
II. MODEL HAMILTONIAN
3D HgTe and CdTe share the same zinc blende structure.
The band topology of these two materials is distinguished bythe band inversion at the /Gamma1point, which happens in HgTe
but not CdTe. This causes HgTe to be topologically nontrivialwhile CdTe is trivial.
3,11,28The essential electronic properties
of both are solely determined by the band structure near theFermi surface at the /Gamma1point, where the bands possess /Gamma1
6
(s-type, doubly degenerate), /Gamma18(p-type, j=3/2, quadruply
degenerate), and /Gamma17(p-type, j=1/2, doubly degenerate)
symmetry.25The band inversion in HgTe takes place because
the/Gamma16bands appear below the /Gamma18band, whereas in the normal
case (such as CdTe) /Gamma16is above /Gamma18(see Fig. 1).23,24In this
work, we use a six-band Kane model Hamiltonian which takesinto account the /Gamma1
6and/Gamma18band. The spin-orbit split off /Gamma17
band usually appears far below the /Gamma16and/Gamma18bands and hence
can be neglected because it does not affect the low-energyapproximation. The six-band Kane model describes the bandstructure near the /Gamma1point well and adequately captures the
band inversion story.
3,28As expected, we find gapless surface
states exist in the direct gap of 3D HgTe while absentin CdTe.
075110-1 1098-0121/2011/83(7)/075110(9) ©2011 American Physical SocietyRUI-LIN CHU, WEN-YU SHAN, JIE LU, AND SHUN-QING SHEN PHYSICAL REVIEW B 83, 075110 (2011)
The six-band Kane Hamiltonian is
H0=⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝T 0 −
1√
2Pk+/radicalBig
2
3Pkz1√
6Pk− 0
0 T 0 −1√
6Pk+/radicalBig
2
3Pkz1√
2Pk−
−1√
2Pk− 0 U+V 2√
3¯γBk −kz√
3¯γBk2
− 0
/radicalBig
2
3Pkz−1√
6Pk−2√
3¯γBk +kz U−V 0√
3¯γBk2
−
1√
6Pk+/radicalBig
2
3Pkz√
3¯γBk2
+ 0 U−V −2√
3¯γBk −kz
01√
2Pk+ 0√
3¯γBk2
+−2√
3¯γBk +kz U+V⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (1)
where
B=¯h2
2m0,P=/radicalBigg
¯h2Ep
2m0,k2
/bardbl=k2
x+k2
y,
k±=kx±iky,T=Eg+B(2F+1)/parenleftbig
k2
/bardbl+k2
z/parenrightbig
,
U=−Bγ 1/parenleftbig
k2
/bardbl+k2
z/parenrightbig
,V=−B¯γ/parenleftbig
k2
/bardbl−2k2
z/parenrightbig
,
where m0is the electron mass. For the simplicity of the
physical picture, we have taken the axial approximation
¯γ=(γ2+γ3)/2 which makes the band structure isotropic
in the kx,kyplane. Eg,Ep,F,γ1,γ2, andγ3are material-
specific parameters. The basis functions are denoted as(|ψ
1/angbracketright,|ψ2/angbracketright)/Gamma16,(|ψ3/angbracketright,|ψ4/angbracketright,|ψ5/angbracketright,|ψ6/angbracketright)/Gamma18. Here we take the pa-
rameters of HgTe just for an illustration of the physics(Table I). The same physics should also happen in the recently
discovered Heusler and Li-based ternary compounds whichshare similar band topologies.
16–22
III. TWO TYPES OF SURFACE STATES
IN 3D SEMIMETALS
Without strain, HgTe is a semimetal with an inverted band
structure. At the /Gamma1point, the conduction band [ /Gamma18light hole
(LH)] touches the valence band [ /Gamma18heavy hole (HH)] and
the/Gamma16is below the /Gamma18band (see right part of Fig. 1). In the
Hamiltonian H0the band topology is solely determined by
the parameter Eg.A tt h e /Gamma1point, LH and HH are degenerate
at energy E=0, and /Gamma16is at position E=Eg. So when
Eg>0, the band structure is normal and insulating with a
positive energy gap /Delta1=Eg. When Eg<0, the band structure
is inverted /Gamma16appears below LH and HH. In this case, the
system is semimetal since the bulk band gap /Delta1is always
zero because of the LH/HH degeneracy. We will show thissemimetal is topologically nontrivial by showing that gaplesssurface states exist in the direct gap of LH and HH. Andby lifting the degeneracy of LH and HH by strain or finiteconfinement which makes /Delta1nonzero, the system instantly
becomes a topological insulator. Whereas in trivial semimetals,
TABLE I. Band structure parameters of HgTe at T=0K .25
Eg EP=2m0P2/¯h2Fγ 1 γ2 γ3
−0.3 eV 18.8 eV 0 4.1 0.5 1.3lifting the zero band gap only turns the systems into trivial band
insulators.
To study the topological properties of this Hamiltonian we
transform it into a tight-binding model on a cubic lattice, wheresuch approximation substitutions are used:
k
i→1
asin(kia),k2
i→2
a2[1−cos(kia)], (2)
herekirefers to kx,ky, andkz,ais the lattice constant which
is taken to be 4 ˚A; in this work. This approximation is valid in
the vicinity of the /Gamma1point.
Surface states reside only on the system surface, which will
project larger LDOS on the surface than the bulk states. Thusthe surface LDOS can be studied to identify the existenceof surface states. The surface LDOS is given by ρ(k)=
−
1
πTrImG00(k), where G00is the retarded Green’s function
for the top layer of a 3D lattice. In a semi-infinite 3D systemthe surface Green’s function can be obtained by means of thetransfer matrix,
G
00=(E−H00−H01T)−1, (3)
T=(E−H00−H01T)−1H†
01, (4)
where H00andH01are the matrix elements of the Hamiltonian
within and between the layers (or supercells) and Tis the
transfer matrix. Usually Eq. ( 4) can be calculated iteratively
FIG. 1. (Color online) A schematic illustration of the band
inversion between /Gamma16and/Gamma18the left is the normal case where the blue
curve represents the LH and HH of the /Gamma18valence band, the right is
the inverted case where the LH flips up and becomes the conductionband, the /Gamma1
6appears below the HH band.
075110-2SURFACE AND EDGE STATES IN TOPOLOGICAL SEMIMETALS PHYSICAL REVIEW B 83, 075110 (2011)
FIG. 2. (Color online) (a) Surface LDOS of 3D HgTe without strain, bright line in the direct gap between LH and HH /Gamma18bands indicates
the first-type surface state, bright regions in the valence band indicates the second-type surface state, E/Gamma16<E 0<E HH=ELH.( b )A n
insulating band gap is opened with strain T/epsilon1=0,U/epsilon1=0,V/epsilon1=− 0.224 eV , the first- and second-type surface states become connected,
E/Gamma16<E HH=E0<E LH. (c) Same as (b) at T/epsilon1=0.1e V ,U/epsilon1=− 0.05 eV , V/epsilon1=− 0.25 eV , note that the /Gamma16band has been inverted with the
HH/Gamma18band and thus appears in the middle of /Gamma18bands (LH and HH), i.e., EHH<E /Gamma16<E 0<E LH. A highest LDOS limit is set and higher
data points have been filtered to this limit for a clearer view of the whole spectrum.
until Tconverges, which is quite time consuming. Here we
use a fast converging algorithm proposed by Sancho et al.
to calculate the transfer matrix.26In Fig. 2(a) we present the
LDOS on an infinite xysurface, where the zdimension is semi-
infinite. As expected, the LDOS clearly shows the existenceof surface states between the LH and HH bands. Throughfurther checking we find that these states indeed reside onlyon the surface boundary [see Fig. 3(a)]. Its spatial distribution
is in the decaying form beneath the surface. Interestingly,we also find another kind of surface state submerging in thevalence bands [bright crossing regions in the valence bandof Fig. 2(a)]. It shows up between the inverted /Gamma1
6and HH
/Gamma18bands. We confirm its surface-state nature by checking its
spatial distribution along the zdirection [Figs. 3(b) and3(c)].
It is found that the spatial distribution of this surface state hasa very distinct form from that of the LH and HH bands. It bearsthe oscillating feature of LDOS of bulk states [see Fig. 3(d)]
but is also clearly decaying beneath the surface. We call theformer surface state as the first type and the latter the secondtype. Both types of surface-state project much larger LDOS onthe surface than the bulk states. However, the first-type surfacestate is clearly decoupled from the bulk states while the secondtype is coupled with the valence bulk conduction states.
The spatial distribution can be obtained by enlarging the
supercell when calculating the surface Green’s function G
00.
For example, take the first 50 layers as a unit cell, we canobtain the LDOS distribution in the first 50 layers. Figure 3(a)
shows the spatial distribution of the first-type surface state atvariant k. It shows that the closer to the /Gamma1point the wider the
wavefunction distributes in space. Only away from the /Gamma1point,
the first-type surface state shows strong localization on thesurface. Close to the /Gamma1point, the LDOS distribution is
bulk-like and can barely be recognized as surface state. InFig. 2the bright line indicating the first-type surface has been
highlighted for a clearer view. The Fermi surface of 3D HgTeis close to the point where the conduction band touches thevalence band.
18,29The first-type surface states at the Fermi
surface distribute widely in the space. They are expected tomake no significant contribution to the transport propertieseither. Without appropriate doping or gating it is difficult to
detect the first-type surface states in 3D HgTe experimentally
FIG. 3. (Color online) Surface state (ss) distribution along the
zdirection. (a) First-type ss at k=0.005 (red, plus sign dot),
0.01 (green, cross dot), 0.02 (blue, star dot), 0.035 (pink, empty squaredot) 1/˚A. (b) Second-type ss below the crossing point at k=0.0066
(red, plus sign dot), 0.01 (green, cross dot), 0.014 (blue, star dot),
0.017 (pink, empty square dot), 0.022 (cyan, filled square dot) 1 /˚A.
(c) Second-type ss above the crossing point at k,s a m ea si n( b ) .( d )
Bulk states at k=0.0066 1 /˚AandE=− 0.4 eV (red, plus sign dot),
−0.1 eV (green, cross dot), 0.2 eV (blue, star dot).
075110-3RUI-LIN CHU, WEN-YU SHAN, JIE LU, AND SHUN-QING SHEN PHYSICAL REVIEW B 83, 075110 (2011)
either through angle-resolved photoemission spectroscopy
(ARPES) or transport measurements.
Unlike the first type surface states which show up as well-
defined sharp lines in the energy spectrum, the second-typesurface state appears above the boundary of the /Gamma1
6band as
two crossing bright regions submerging in the valence bandsin the energy spectrum. Its LDOS on the surface is the highestclose to the /Gamma1point. Away from the /Gamma1point, the surface
LDOS becomes smaller and finally those states merge intothe bulk states. Also unlike the first-type surface state whosedistribution width becomes very large when approaching the/Gamma1point, the second-type surface state’s distribution width in
space does not significantly depend on the momentum. Atthe/Gamma1point, the second-type surface-state energy and wave
function is exactly solvable from the Hamiltonian. We willshow that this point is the crossing point of the surface stateDirac cone (see text below). Beneath the surface, the LDOSquickly decays in an oscillatory way. For comparison, wealso plot out the bulk states LDOS which is oscillatory and
extended all over the space without decaying [Fig. 3(d)].
Since from the LDOS the second-type surface state canbe easily distinguished from the bulk states, we believethey may be easily detected from the ARPES measurementsas well.
IV . ORIGIN OF SURFACE STATES AND STRAIN-INDUCED
BAND GAP
The origin of the two types of surface states can be
understood intuitively in the following way. If we divide theHamiltonian into two subspaces which are spanned by twogroups of basis: P
1={ψ1,ψ4,ψ2,ψ5}andP2={ψ3,ψ6},i ti s
amazing that the effective Hamiltonian in the subspace P1
is quite similar to the effective model proposed for a 3D
topological insulator.14The Hamiltonian after arranging the
basis as [ P1(ψ1,ψ4,ψ2,ψ5),P2(ψ3,ψ6)] reads
H/prime
0=⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝T/radicalBig
2
3Pkz 01√
6Pk− −1√
2Pk+ 0
/radicalBig
2
3Pkz U−V −1√
6Pk− 02√
3¯γBk +kz√
3¯γBk2
−
0 −1√
6Pk+ T/radicalBig
2
3Pkz 01√
2Pk−
1√
6Pk+ 0/radicalBig
2
3Pkz U−V√
3¯γBk2
+−2√
3¯γBk −kz
−1√
2Pk−2√
3¯γBk −kz 0√
3¯γBk2
− U+V 0
0√
3¯γBk2
+1√
2Pk+−2√
3¯γBk +kz 0 U+V⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (5)
Without considering the coupling between P1andP2sub-
spaces, P1gives the LH (conduction) and /Gamma16(valence) band,
plus a gapless Dirac cone of surface states between theirgap;P
2gives the HH (valence) which overlaps with the
surface-state Dirac cone and /Gamma16band. After turning on the
coupling between the two subspaces, the surface-state Diraccone becomes separated into two parts. One part appears inthe direct gap between LH and HH, the other submerges intothe HH of valence band. In this sense, our results are verysimilar to an earlier work which discusses interface states inHg
1−xCdxTe heterojunctions.3
The/Gamma18degeneracy at the /Gamma1point makes 3D HgTe a
semimetal. To lift the degeneracy and open an insulatinggap at the Fermi energy, which makes the system a true 3DTI, we consider applying a uniaxial strain along the (001)axis.
11,28According to Ref. 23, the additional strain induced
Hamiltonian is introduced as [in the original basis order as inEq. ( 1)]
H
s=⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎝T
/epsilon100 0 0 0
0T/epsilon1 0000
00 U/epsilon1+V/epsilon1 000
00 0 U/epsilon1−V/epsilon1 00
00 0 0 U/epsilon1−V/epsilon1 0
00 0 0 0 U/epsilon1+V/epsilon1⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎠,where the T
/epsilon1,U/epsilon1, andV/epsilon1are strain-induced interaction terms.
To illustrate the physical picture more clearly, we have madethese terms artificially large in this work. At the /Gamma1point,
the boundary for the /Gamma1
6band is E/Gamma16=Eg+T/epsilon1,f o rt h eH H
band the boundary is EHH=U/epsilon1+V/epsilon1, and for the LH band
the boundary is ELH=U/epsilon1−V/epsilon1. In the strain-free condition,
E/Gamma16<E LH=EHH=0. However, under strain, the relative
position of the three is subject to change and is determined bythe strain terms.
Surface state at the crossing point of the Dirac cone is
easily solvable directly from the Hamiltonian H
0+Hs, where
kx=ky=0.30Using the boundary condition ψ(z=0)=0
andψ(z=− ∞ )=0, we get the eigenenergy for the surface
state:
E0=C+D1M
B1
=(Eg+T/epsilon1)(γ1+2γ)+(U/epsilon1−V/epsilon1)(2F+1)
γ1+2γ+2F+1,(6)
where
C=Eg+T/epsilon1+U/epsilon1−V/epsilon1
2,M =Eg+T/epsilon1−U/epsilon1+V/epsilon1
2,
D1=B
2(2F+1−γ1−2γ), (7)
075110-4SURFACE AND EDGE STATES IN TOPOLOGICAL SEMIMETALS PHYSICAL REVIEW B 83, 075110 (2011)
B1=−B
2(2F+1+γ1+2γ).
And the wave function is
ϕ=1√
2⎛
⎜⎝i/radicalBig
D+
B1
/radicalBig
−D−
B1⎞
⎟⎠c(eλ1z−eλ2z),
with the basis ψ1andψ4orψ2andψ5, where
λ2
1,2=D2±/radicalBig
D2
2−D3
2D+D−,with
D2=−/bracketleftbig
A2
1+D+(E0−L1)+D−(E0−L2)/bracketrightbig
,
D3=4D+D−(E0−L1)(E0−L2),
D±=D1±B1,L1=C+M,L2=C−M, and A1=/radicalBig
2
3P. For the surface-state solution to exist, it is required
that
MB 1>0.
This condition is easily satisfied in an inverted band structure
without strain, where Eg<0.
In Figs. 2(b) and2(c), the spectrum after adding the strain
interaction is plotted. An insulating gap is opened betweenthe LH and HH /Gamma1
8bands. Meanwhile, both the first- and
second-type surface states go through an evolution. At a criticalpoint [Fig. 2(b)] the crossing point can move to the top of the
valence band at the /Gamma1point and both types of surface states
become connected with each other. On further increasing thestrain strength, the crossing point jumps out of the valenceband and sits in the band gap. At this stage, we find onlyone kind of surface state appearing in the insulating gap[Fig. 2(c)]. The spectrum of surface state forms a gapless
Dirac cone at the /Gamma1point and the system becomes a typical 3D
TI. Notice that Fig. 2(c) closely resembles the band structure
of 3D TI Bi
2Se3and Bi 2Te3.14,15Although Fig. 2is obtained
from the tight binding model on a lattice, we find the energyof the crossing point agrees with the exact solution quiteprecisely.
V . EDGE STATES IN 2D THIN FILMS
A. Quasi-2D lattice model
It is, however, technically difficult applying strong enough
strains to make a semimetal insulating in experimental condi-tions. The finite-size effect can open a gap in the bulk states atthe/Gamma1point but will also open a gap in the surface states.
27,30
Instead of considering the surface states, we make the 3D
topologically nontrivial semimetals into thin films and studytheir edge effects in the strain-free condition. The same latticemodel is used as in the 3D case, but now the zdimension is
of finite thickness L. We study the LDOS on the side surface
of a thin film where xis infinite and yis semi-infinite. In
this case, only k
xis a good quantum number. In Fig. 4,t h e
LDOS on the film edge surface for various film thickness isplotted. In the thick film limit, the band gap opened by thefinite-size effect is not obvious. The spectrum still resemblesthat of the 3D case [Figs. 4(a) and 4(b)]. Surface states still
FIG. 4. (Color online) LDOS at the edge of thin films at
different thickness calculated with the 3D lattice model. (a) L=
116 ˚A; (b) L=76˚A; (c) L=36˚A; (d) L=28˚A; (e) L=20˚A;
(f)L=12˚A;.
show residuals on the spectrum. The finite confinement of
zcauses energy-level discretization as in a quantum well,
which induces a series of sub-bands appearing in the energyspectrum as layered structures. When the film is thinned downthe discretized energy level spacing increases. A pair of edgestates with linear dispersion are found in the band gap [Fig. 4(c)
and 4(d)]. The system becomes a 2D topological insulator.
When the film is thinner than 20 ˚A, another transition happens,
the system becomes trivial. The edge states disappear from the
band gap.
B. Quantum well approximation and 2D lattice model
When the film is thin enough, the finite-size-caused band
gap becomes obvious. In this case the finite-confinement-induced sub-bands are far away from the low-energy regime.We can then use the quantum well approximation /angbracketleftk
z/angbracketright=
0,/angbracketleftk2
z/angbracketright/similarequal(π/L )2.31Using these relations in the Hamiltonian
in Eq. ( 1), and choosing the basis set in the sequence ( |ψ1/angbracketright,
|ψ3/angbracketright,|ψ5/angbracketright,|ψ2/angbracketright,|ψ6/angbracketright,|ψ4/angbracketright), we can obtain a two-dimensional
six-band Kane model
H(k)=/parenleftbiggh(k)0
0h∗(−k)/parenrightbigg
, (8)
075110-5RUI-LIN CHU, WEN-YU SHAN, JIE LU, AND SHUN-QING SHEN PHYSICAL REVIEW B 83, 075110 (2011)
where
h(k)=⎛
⎜⎜⎝Eg+B(2F+1)(k2
/bardbl+/angbracketleftk2
z/angbracketright) −1√
2Pk+1√
6Pk−
−1√
2Pk− −(γ1+γ)Bk2
/bardbl−(γ1−2γ)B/angbracketleftk2
z/angbracketright√
3γBk2
−
1√
6Pk+√
3γBk2
+ −(γ1−γ)Bk2
/bardbl−(γ1+2γ)B/angbracketleftk2
z/angbracketright⎞
⎟⎟⎠. (9)
The system keeps time-reversal symmetry, and the represen-
tation of the symmetry operation in the new set of basesis given by T=K·iσ
y⊗I3×3, where Kis the complex
conjugation operator, σyandIdenote the Pauli matrix and
unitary matrix in the spin and orbital space, respectively. Wecan study the two blocks separately since they are time-reversalcounterparts of each other. Here we focus on the upperblock first. At k
x=0, the boundaries of /Gamma16, LH, and HH
are atE=Eg+B(2F+1)/angbracketleftk2
z/angbracketright,E=− (γ1−2γ)B/angbracketleftk2
z/angbracketright, and
E=− (γ1+2γ)B/angbracketleftk2
z/angbracketright, which are controllable by choosing
film thickness L. When Ldecreases from the thick limit down
toL≈30˚Athe/Gamma16band flips up and exchanges position with
HH, the system is still nontrivial. Further down to L≈20˚A,
/Gamma16flips up and exchanges with the conduction band (LH)
[see Fig. 5(e)]. The band structure then becomes trivial just
as the illustrated case in Fig. 1. This rough picture serves as
an intuitive understanding of the topological transition andedge-state formation in the thin films. It also agrees with theresult we obtained with the 3D lattice model in Fig. 4.I nt h e
thick film limit, the bulk band gap is always between the LH
FIG. 5. (Color online) LDOS at the edge of thin films at
different thickness using the Hamiltonian ( 9) with 2D lattice model.
(a)L=28˚A; (b) L=25˚A; (c) L=20˚A; (d) L=16˚A; (e) illustra-
tion of the position change of LH, HH, and /Gamma16bands as film thickness
is varied.and HH derived states while in thin limit it is between the /Gamma16
and LH derived states. In the thick-film limit, surface states
(on top and bottom surfaces) and edge states (on side surfaces)coexist. The first-type surface states on the top/bottom surfacesbecome the effective bulk states of the film appearing as aconduction band minimum. The edge states in the thinnerfilms actually evolve out from the surface states on the sidesurfaces, whose nature is 2D instead of 1D.
Using the same approximation as in Eq. ( 2), we can
transform h(k) into a tight-binding model on a 2D lattice.
In Fig. 5we show the LDOS on the edge of a semi-infinite
film for h(k). The finite-size gap agrees with that obtained with
the 3D lattice in Fig. 4well. When L> 20˚A, the edge states
are found connecting the valence and conduction bands. Afterthe system becomes trivial when L< 20˚A, the edge states do
not cross the band gap anymore, instead they only attach to theconduction band. At the critical point L=20˚A, the valence
band and conduction band touch and form a linear Dirac coneat the low-energy regime, which is shown in both Figs. 4and5.
This shows that by controlling the film thickness, it is possibleto obtain a single-valley Dirac cone for each spin block withoutusing the topological surface states.
32Notice that in Fig. 5
we can also see the edge states submerging in the valencebands, which is similar to the case of second-type surfacestates discussed previously. We call them the second-type edgestates.
C. Chern number description of thin-film band topology
In the quantum-Hall effect (QHE), bulk-edge correspon-
dence tells us that a nonzero Thouless–Kohmoto–Nightingale–den Nijs (TKNN) integer, summed by the Chern number ofoccupied bands, is closely related to the presence of edgestates on the sample boundaries.
33,34For the time-reversal
invariant systems which belong to the universality class ofzero-charge Chern number, the Z
2index is then introduced
to characterize the topologically nontrivial states.7,8However,
when the Hamiltonian of the system is composed of time-reversal diagonal blocks,
9theZ2invariant can be identified
with the parity of the Chern number for each block.11Focusing
on the upper block of the thin-film Hamiltonian Eq. ( 8) [i.e.
h(k)], we can discuss the topology of the band structure. To
identify the existence of edge states, we can calculate the firstChern number of each band and sum up for all occupied bands.The Berry curvature for each band is defined as
35,36
/Omega1n(k)=i/parenleftbigg/angbracketleftbigg∂un,k
∂kx/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u
n,k
∂ky/angbracketrightbigg
−/angbracketleftbigg∂un,k
∂ky/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u
n,k
∂kx/angbracketrightbigg/parenrightbigg
.
Then the Chern number is computed by integrating the
Berry curvature over the first Brillouin Zone (BZ): Cn=
1
2π/integraltext
BZd2k/Omega1n(k). For each block of the 2D lattice model, the
075110-6SURFACE AND EDGE STATES IN TOPOLOGICAL SEMIMETALS PHYSICAL REVIEW B 83, 075110 (2011)
Chern number can be given to each of the three bands. Tuning
the thickness of the quantum well Lcan inverse the band
structure near the /Gamma1point, resulting in the change of Chern
number for the touching bands. Several critical thicknessesforLcan be found when there is a band touching at the
/Gamma1point: L≈30˚AandL≈20˚A. The numerical result of
this integral divides the parameter space into three regions:forL∈(30˚A,+∞ ), Chern number C=1,0,−1 arranged
in sequence from the lowest energy band to the highest(the same for the latter two cases); for L∈(20˚A,30˚A),
Chern number C=2,−1,−1, and for L∈(0,20˚A), Chern
number C=2,−2,0. When the Fermi level lies between
the highest and the middle energy band, the TKNN integerN=1f o r L∈(20˚A,+∞ ) and N=0f o r L∈(0,20˚A).
The former case is topologically nontrivial [referred to inFigs. 5(a) and
5(b)] and the latter is trivial [referred to in
Fig. 5(d)].
VI. QSHE AND DISORDER-ENHANCED
SPIN-HALL EFFECT
The edge state picture in thin films can be verified with
explicit Landauer-B ¨uttiker calculations.37,38We construct a
four-terminal Hall bar device using the tight binding version ofthe Hamiltonian ( 8). Because the system keeps time reversal
symmetry (TRS), we need only consider a single block ofthe 6×6 Hamiltonian. The transverse charge conductance of
each sub-block is defined as G
tc=G12−G14.Gijdenotes
the conductance between terminal iandj, which is given by
Gij=e2
hTr[/Gamma1iGR/Gamma1jGA]./Gamma1i=i[/summationtextR
i−/summationtextA
i] is the spectrum
function of lead i.GR/Ais the retarded and advanced Green’s
function of the sample which has taken into account the four
semi-infinite leads through the self-energy/summationtextR/A
i:GR/A=
[E−Hc−/summationtext4
i=1/summationtextR/A
i]−1.Hcdenotes the Hamiltonian of the
shaded region of the sample.38–40In a real system with TRS, the
transverse charge conductance calculated here corresponds tothe transverse “spin” conductance, which is defined as G
ts=
G↑
tc−G↓
tc. The longitudinal conductance is defined as Gl=
G↑
13+G↓
13. However, it should be noted that the “spin” up or
down here is only an index of each Hamiltonian sub-blockand does not mean real spin. The bases of each sub-blockare hybrids of j=± 1/2 and j=± 3/2 states of /Gamma1
6and/Gamma18
bands.9
In Fig. 6, we show the result for a film whose thickness
is 25 ˚A. In the clean limit, i.e., without disorder, in the
finite-size induced band gap, the Glvanishes, Gtsis quantized
asGts=2e2/h(withG↑
12=G↓
14=e2/h,G↑
14=G↓
12=0),
which proves the existence of helical edge states in thethin films and indicates the existence of quantum spin-Halleffect (QSHE).
41–45In the conduction band, the edge states
coexist with the bulk states. Gtsthere is not quantized, which
implies that the bulk states themselves carry transverse spinconductance that partly cancels out the quantized spin-Hallconductance carried by the edge state channels. In the valenceband, similarly, the second-type edge states coexist with bulkstates. Notice that the confinement in the x,ydimension also
generates a finite-size gap in the edge state itself, which isindicated by the arrow in Fig. 6where G
tsvanishes.
FIG. 6. (Color online) Four terminal Landauer-B ¨uttiker calcula-
tions for thin films. Four identical semi-infinite leads are attached
to the cross-bar device, which is shown schematically in the inset.
Disorders are put on the central area (shaded areas). 200 disordersamples are taken for each disorder strength W. The arrow indicates
the position of finite-size gap of the helical edge states.
Disorder is recently known to play an important role in
some exotic phenomena in TIs.46,47In real materials, dirty
impurities are inevitable. It is even possible to control themartificially.
48,49Here we consider the effect of disorder impu-
rities to the transport properties of the thin films. Anderson-type white noise is introduced on the lattice model, whichare spin independent on-site random potentials in the range[−W/2,W/2]. TRS is not violated by disorder. In Fig. 6we
show the transverse spin conductance G
tsof the four-terminal
device at various disorder strength W. The quantized Gts
in the band gap shows robustness against modest disorder
strength, which is expected because the disorder we applydoes not couple the TRS sub-blocks of the Hamiltonian andthus brings no backscattering between different spin-edgechannels.
7,9Interestingly, Gtsis significantly enhanced in both
conduction and valence bands even with weak disorder. WhileG
tsquickly becomes suppressed in the conduction band when
disorder increases, it still remains enhanced in the valenceband. Notice that with modest disorder strength (3 eV inFig. 6), the longitudinal conductance G
lalready vanishes in
the low-energy regime, but the transverse-spin conductanceremains enhanced in the valence band. Therefore with modestdisorder strength, we expect a strong-disorder-enhanced spin-Hall effect in the valence band, which would manifest itselfthrough strong signals in nonlocal measurements. In a veryrecent experiment a similar observation of the SHE differencein valence and conduction bands has been reported with HgTequantum wells.
50
VII. SUMMARY
We have demonstrated that the six-band Kane model can
be utilized to study the topological properties of both 3Dand thin-film realistic materials. By choosing proper modelparameters, it describes the band structure well in the low-energy regime and captures the essential band topological
075110-7RUI-LIN CHU, WEN-YU SHAN, JIE LU, AND SHUN-QING SHEN PHYSICAL REVIEW B 83, 075110 (2011)
features sufficiently. By calculating the LDOS on the system
boundary, the existence of topological surface states is explic-itly demonstrated. Using the model parameters of HgTe, weshow that even though the system in semimetal surface statesalready exists in the material as long as the band structureis inverted. We also find in the strain-free condition, surfacestates are divided into two parts in the spectrum, each ofdifferent characteristics. We demonstrate that uniaxial strainscan generate an insulating gap and transform the semimetalsinto true TIs, in which the gapless Dirac cone of surfacestates is found in the bulk band gap. Because of the similarband structures and band topologies, we expect that the samephysics applies to the recently discovered Heusler-related andLi-based intermetallic ternary compounds, which are mainlytopologically nontrivial semimetals or metals.We also demonstrate a crossover from a 3D topological
semimetal to 2D QSHE insulators. In the thin films madeof these materials, we predict the existence of helical edgestates and QSHE in the strain-free condition. And finally weshow that disorder plays an important role in the transportproperties in the thin films. It significantly enhances the SHEin the valence bands.
ACKNOWLEDGMENTS
This work was supported by the Research Grant Coun-
cil of Hong Kong under Grant Nos. HKU 7037/08P,HKUST3/CRF/09 and in part by a Hong Kong UGC SpecialEquipment Grant (SEG HKU09).
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PhysRevB.92.020407.pdf | RAPID COMMUNICATIONS
PHYSICAL REVIEW B 92, 020407(R) (2015)
Giant voltage modulation of magnetic anisotropy in strained heavy
metal/magnet/insulator heterostructures
P. V . Ong,1Nicholas Kioussis,1,*D. Odkhuu,1P. Khalili Amiri,2K. L. Wang,2and Gregory P. Carman3
1Department of Physics and Astronomy, California State University Northridge, Northridge, California 91330, USA
2Department of Electrical Engineering, University of California, Los Angeles, California 90095, USA
3Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, California 90095, USA
(Received 21 December 2014; revised manuscript received 12 April 2015; published 20 July 2015)
Ab initio electronic structure calculations reveal that epitaxial strain has a dramatic effect on the voltage-
controlled magnetic anisotropy (VCMA) in Ta/FeCo/MgO junctions. Strain can give rise to a wide range of novelVCMA behaviors where the MA can change from a ∨-t oa∧-shape electric-field dependence with giant VCMA
coefficients which are asymmetric under voltage reversal. The underlying mechanism is the interplay of thestrain- and electric-field-induced changes in the spin-orbit coupled dstates at the interfaces and the strain-induced
modification of the dielectric constant of MgO. These findings demonstrate the feasibility of highly sensitiveVCMA through strain engineering, which may provide a viable avenue for tailoring magnetoelectric propertiesfor spintronic applications.
DOI: 10.1103/PhysRevB.92.020407 PACS number(s): 75 .85.+t,73.20.−r,75.30.Gw,75.70.Cn
Even though the switching of magnetization in magnetic
random access memory (RAM) bits using spin-polarizedcurrents via the spin transfer torque (STT) effect has provenvery successful [ 1], it requires high current densities and
hence high power consumption. Furthermore, the switchingenergy per bit ( ∼100 fJ) of STT-RAM is still around two
orders of magnitude higher compared with complementary
metal-oxide semiconductor switching energies of ∼1f J[ 2].
A highly promising approach for developing ultralow power,highly scalable, and nonvolatile spin-based RAMs is thevoltage-controlled magnetic anisotropy (VCMA) of heavymetal/ferromagnet/insulator (HM/FM/I) nanojunctions via themagnetoelectric effect (MeRAM) [ 3–6] where the nonmag-
netic HM contact electrode (Ta, Pd, Pt, and Au) has strongspin-orbit coupling (SOC). In the low-bias regime the VCMAis proportional to the electric field (E field) in the insulator,VCMA =βE
I=βEext
ε, where βis the VCMA coefficient,
εis the dielectric constant of the I, and Eextis the external
E field. The challenge for achieving a switching bit energybelow 1 fJ and a write voltage below 1 V requires largeperpendicular MA (PMA) [ 4,7] andβ/greaterorequalslant200 fJ V
−1m−1[2].
The VCMA of HM/FM/I junctions exhibits a wide range
of behavior ranging from linear with positive or negative β
to nonmonotonic ∨-shape or inverse- ∨-(∧-) shape E-field
dependence with asymmetric β’s. On the experimental side,
a linear VCMA was observed in epitaxial Au/FeCo/MgO [ 8]
and in Ta /Co40Fe40B20/MgO [ 9] tunnel junctions with βof
−38 and −33 fJ V−1m−1, respectively, where the convention
of the positive E field corresponds to the accumulation ofelectrons in the FM/I interface. On the other hand, a ∨-
shape VCMA was found in Au/Fe/MgO [ 5]. Furthermore, in
Ta/Co
40Fe40B20/MgO/Co40Fe40B20junctions the coercivity
of the bottom FM electrode decreases linearly with voltagewithβ≈50 fJ V
−1m−1whereas that of the top FM electrode
exhibits a ∨-shape voltage behavior [ 6].
*nick.kioussis@csun.eduRecent experiments in Pd/FePd/MgO [ 10], V/Fe/MgO [ 11],
and MgO/FeB/MgO/Fe [ 12] junctions show linear ∧- and
∨-shape VCMAs with giant βvalues of about 600, 1150,
and 100 fJ V−1m−1, respectively. Although in general the
underlying mechanism remains unresolved, a possible originof the nonmonotonic VCMA in V/Fe/MgO may be due tothe internal E field caused by charges trapped by defects
in MgO [ 11]. On the theoretical side, ab initio electronic
structure calculations of Fe/MgO [ 13] and Au/Fe/MgO [ 14]
junctions with in-plane lattice constants of Fe and MgO,respectively, predicted a linear VCMA with βof about +130
and+70 fJ V
−1m−1, respectively.
The fairly large lattice mismatch (4% to 5%) among the I,
the FM, and the HM layers in these heterostructures invitesseveral intriguing and important questions which remainunexplored. What is the effect of strain on the: (1) theVCMA behavior and its coefficients, (2) the dielectric constantand hence the E
I, and (3) the critical field where the MA
reaches its maximum or minimum value? The purpose of thisRapid Communication is to employ ab initio electronic struc-
ture calculations to understand the effect of epitaxial strainon the VCMA behavior of the Ta/FeCo/MgO junction andreconcile the origin of the experimental controversies. Wedemonstrate that the strain can selectively tune the VCMAfrom the ∨to∧shape with giant VCMA coefficients which
are asymmetric upon E-field switching.
The ab initio electronic structure calculations have
been carried out within the framework of the projectoraugmented-wave formalism [ 15] as implemented in the Vienna
ab initio simulation package (
V ASP )[16,17] with the general-
ized gradient approximation [ 18] for the exchange-correlation
functional. The slab supercell for the Ta/FeCo/MgO (001)junction along [001] consists of three monolayers (MLs) ofbcc Ta on top of three MLs of B2-type FeCo on top of
seven MLs of rock-salt MgO and a 15- ˚A-thick vacuum region.
The O atoms at the FeCo/MgO interface are placed atop Featoms. We denote with Fe1 and Fe2 [the inset in Fig. 1(a)]
the atoms at the Fe/MgO and Fe/Ta interfaces, respectively.The expansive strain η
FeCo on the FeCo film is varied from
1098-0121/2015/92(2)/020407(5) 020407-1 ©2015 American Physical SocietyRAPID COMMUNICATIONS
P. V . ONG et al. PHYSICAL REVIEW B 92, 020407(R) (2015)
FIG. 1. (Color online) (a) Strain dependence of zero-field MA
of the Ta/FeCo/MgO junction where the expansive (compressive)strain η
FeCo (ηMgO) is shown along the bottom (top) ordinate.
(b) Strain dependence of the relative in-plane (solid squares) and
out-of-plane (solid circles) components ( ε/bardblandε⊥) of the dielectric
tensor of bulk MgO and of ε⊥for the MgO thin film (open circles).
(c)–(e) Energy- and k-resolved distributions of the orbital character
of minority-spin bands along /Gamma1M for the interfacial Fe1 atom dstates
forηFeCo=0%, 2%, and 4%, respectively. (f)–(h) MA( k) (in meV) in
the two-dimensional (2D) Brillouin zone (BZ) for ηFeCo=0%, 2%,
and 4%, respectively. Numerals in panels (c)–(h) refer to BZ k/bardblpoints
(BZPn, n=1–3) where there are large changes in MA.
0% to 4% to simulate the effect of in-plane strain under
experimental conditions [ 19]. At each strain, the magnetic
and electronic degrees of freedom and atomic positions along[001] are relaxed in the presence of the E field . Employing a
31×31×1k-point mesh, the MA per unit interfacial area A
is determined from MA =[E
[100]−E[001]]/A, where E[100]
andE[001] are the total energies with magnetization along the
[100] and [001] directions, respectively.
Effect of strain on zero-field MA . Figure 1(a) shows the
strain dependence of the zero-field MA of the Ta/FeCo/MgOjunction. We find that the MA decreases almost linearlywith increasing strain and undergoes a transition from anout-of-plane to an in-plane magnetization at ∼η
FeCo=+ 2.5%.
From the strain dependence of the MA (see the SupplementalMaterial [ 20]) we find that the effective interfacial uniaxial
magnetocrystalline anisotropy K
i
2=1.41 erg /cm2and the
interfacial magnetoelastic coefficient Bi
1=− 18.81 erg /cm2.
Furthermore, epitaxial strain has a large effect on the di-
electric constant of MgO which in turn controls the magnitudeofE
Iat the FM/I interface. Employing density-functionalcalculations as implemented in the PWSCF package [ 21], we
display in Fig. 1(b) the out-of-plane ( ε⊥) and in-plane ( ε/bardbl)
components of the relative dielectric tensor of bulk MgOversus in-plane strain. We find that ε
⊥increases exponentially
from its zero-strain value of 9.8 with increasing compressiveη
MgO strain indicating a decrease in EI. This result is further
corroborated by independent V ASP slab calculations for the
Ta/FeCo/MgO junction, where ε⊥(open circles) is determined
from the ratio of the E field in vacuum to that in MgO wherethe latter is determined from the change in the planar averageelectrostatic potential.
To elucidate the mechanism of the strain effect on the
zero-field MA, we show in Figs. 1(c)–1(e) the energy- and
k-resolved distribution of the minority-spin band of the Fe1-
derived d
xy,dxz/yz , anddx2−y2states along the /Gamma1M symmetry
direction for ηFeCo=0%, 2%, and 4%, respectively. We find
that the strain-induced change in the zero-field MA arisesprimarily from changes in the band structure of the Fe1interfacial atom. We have employed both the total energymethod and the force theorem [ 22]M A =/summationtext
kMA(k)t o
calculate the effect of strain on the MA. The k-resolved
MA(k)≈/summationtext
n∈occ[ε(n,k)[100]−ε(n,k)[001]] in the 2D BZ is
shown in Figs. 1(f)–1(h) forηFeCo=0%, 2%, and 4%,
respectively. Here, ε(n,k)[100]([001])are the eigenvalues of
the Hamiltonian for magnetization along the [100] ([001])direction. Using the force theorem we find that MA =1.98,
0.69, and −0.83 erg /cm
2forηFeCo=0%, 2%, and 4%,
respectively. These values agree well with those obtained fromtotal energy calculations of 1.43, 0.61, and −0.83 erg /cm
2
for the corresponding strain values. Within second-order
perturbation theory the MA can be expressed as [ 23]
MA∝ξ2/summationdisplay
o,u|/angbracketleft/Psi1↓
o|ˆLz|/Psi1↓
u/angbracketright|2−| /angbracketleft/Psi1↓
o|ˆLx|/Psi1↓
u/angbracketright|2
E↓
u−E↓
o,(1)
where /Psi1↓
o(E↓
o) and /Psi1↓
u(E↓
u) are the one-electron occupied
and unoccupied minority-spin states (energies) of band indexnand wave vector k(omitted for simplicity), ξis the SOC, and
ˆL
x(z)is thex(z) component of the orbital angular momentum
operator. This expression is valid when the majority band is fulland the SOC between states of opposite spin can be ignored.Analysis of the density of states and the energy- and k-resolved
distribution of dorbitals shows that the majority-spin states
of the interfacial Fe1 and Fe2 atoms are well below 0.5 eVfrom the Fermi level. Therefore, the spin-flip contributionis negligible. In the analysis below the wave functions areprojected on the dorbitals.
At zero strain, the maxima of MA( k
/bardbl)i nF i g . 1(f) occur
around /Gamma1along the /Gamma1M (BZP1) and /Gamma1X (BZP2) directions
and around1
2/Gamma1M (BZP3). The underlying origin of the MA
maxima at BZP1 and BZP2 is the SOC between the minority-spin interfacial Fe1-derived occupied d
xyanddxz(yz)states
with the unoccupied dx2−y2anddyz(xz)states, respectively, in
Fig. 1(c), through the ˆLzoperator. On the other hand, the MA
maximum at BZP3 is due to the large SOC /angbracketleftx2−y2|ˆLz|xy/angbracketright
and/angbracketleftxz(yz)|ˆLz|yz(xz)/angbracketrightof Fe1.
Overall, an increase in strain induces large downward shifts
of the minority-spin bands of the Fe1-derived dxyanddxz(yz)
states and upward energy shifts of the dx2−y2bands along
020407-2RAPID COMMUNICATIONS
GIANT VOLTAGE MODULATION OF MAGNETIC . . . PHYSICAL REVIEW B 92, 020407(R) (2015)
the/Gamma1M direction. This leads to substantial rearrangement of
occupied and unoccupied bands and hence large changes in thematrix elements of the ˆL
zand ˆLxoperators throughout the 2D
BZ. Thus, at 2% strain the SOC /angbracketleftx2−y2|ˆLx|yz(xz)/angbracketrightat BZP1
becomes dominant rendering the k-resolved MA(BZP1) <0
[blue ring around /Gamma1in Fig. 1(g)]. Furthermore, the increase
in energy splitting between the occupied dxy-derived and
unoccupied dx2−y2-derived bands as well as between the
occupied dxz(yz)-derived and unoccupied dyz(xz)-derived bands
at BZP3 leads to a decrease in the k-resolved MA(BZP3).
Note, that under 2% strain the rearrangement of bands shiftsthe MA maximum from BZP1 to BZP1
/primedue to large SOC
/angbracketleftxy|ˆLz|x2−y2/angbracketright[Fig. 1(g)]. Under 4% strain, the Fe1 dxy
shifts further down in energy [Fig. 1(e)] leading to a reduction
in MA at BZP1/primeand BZP3. The Fe1 SOC /angbracketleftyz(xz)|ˆLx|x2−y2/angbracketright
around /Gamma1becomes dominant resulting in the out-of-plane to
in-plane magnetization transition.
Effect of strain on VCMA . The variation in the MA as a
function of the E field in MgO is shown in Figs. 2(a)–2(c)
forηFeCo=0%, 2%, and 4%, respectively. The results reveal
that strain can have a dramatic effect on the VCMA, whichchanges from: (i) a ∨shape at zero strain with giant β
values of −648 (486) fJ V
−1m−1for a negative (positive)
E field, to (ii) a symmetric ∧shape at 2% strain with β
values of +252 (−241) fJ V−1m−1for a negative (positive)
E field, and to (iii) an asymmetric ∧shape at 4% strain with
βvalues of +189 (−238) fJ V−1m−1. Note that at 4% the
MA reaches its maximum value at EI=0.93 V/nm, which is
close to the breakdown voltage of the MgO film ( ∼1V/nm).
Consequently, the experimentally measured VCMA appearslinear. The underlying origin presumably arises from the factthat the interface bands depend on the magnetization direction
due to the Rashba effect. The Rashba coupling, which isproportional to the netelectric field E
zat the interface, has
contributions from both the internal and the external fields
FIG. 2. (Color online) (a)–(c) MA versus E field in MgO for
different strain values. (d)–(f) Orbital moment difference /Delta1mo=
m[001]
o−m[100]
oof the Fe1 and Fe2 interfacial atoms versus the E field
for the same strain values.[24]. The critical field where the MA reaches its maximum
or minimum value depends on the interplay between the twoE fields where the internal E field can be tuned via strain.Interestingly, recent experiments have reported the influenceof the internal E field at the FM/I interface on the voltage-dependent tunneling anisotropic magnetoresistance [ 25].
These VCMA coefficient values are the highest reported
today and are larger by about an order of magnitude comparedto most experimental values, except for those reported inRefs. [ 10,11] where charged defects may play a role. Fur-
thermore, our predicted β’s are close to or larger than the
value of ∼200 fJ V
−1m−1required to achieve a switching bit
energy below 1 fJ in the next-generation MeRAMs. Thus, our
results demonstrate that the VCMA and its coefficients canbe selectively tuned via proper epitaxial strain engineering.The even dependence of the MA on the E field is not onlyof potential interest for MeRAM, but also for STT-RAM withPMA, where a ∧-shaped VCMA can symmetrically reduce the
switching current in both directions.
Figures 2(d)–2(f) show the difference between the out-of-
plane and in-plane orbital moments /Delta1m
o=m[001]
o−m[100]
ofor
the Fe1 and Fe2 interfacial atoms as a function of the E field forη
FeCo=0%, 2%, and 4%, respectively. The E-field variation
in/Delta1mofor the Co and Ta1 atoms is not shown because it
is much weaker. For single atomic species FMs with largeexchange splitting the MA is related to the orbital moment
anisotropy via the Bruno expression MA =ξ/Delta1 m
o/(4μB)
[26]. This expression needs to be modified for structures
consisting of multiple atomic species with strong hybridization[27]. Nevertheless, overall the results show that the E-field
dependence of /Delta1m
ofor Fe2 (and to a lesser degree for Fe1)
correlates well with that of the MA for all strain values.Furthermore, /Delta1m
o>0 for Fe1 whereas /Delta1mo<0 for the Fe2
atom (except for E >0 at 4%), indicating that the Fe2 /Ta
interface favors in-plane MA in agreement with experiment[28]. This is due to the fact that, in contrast to Fe1, the
Fe2-derived d
x2−y2density of states around the Fermi energy
is low leading to a decrease in /angbracketleftxy|ˆLz|x2−y2/angbracketright. Consequently,
the SOC between the occupied dxy-derived and the unoccupied
dxz(yz)-derived Fe2 states through ˆLxbecomes dominant.
In order to understand the VCMA behavior under zero
strain we show in Fig. 3(a) the field-induced /Delta1MA(k)=
MA(k,E)−MA(k,E=0) along symmetry directions under
EI=± 0.37 V/n m .W ea l s os h o wi nF i g s . 3(b) and3(c),t h e
shift in the zero-field minority-spin band structures (dottedcurves) under fields of −0.37 V/nm (green curves) and
+0.37 V /nm (red curves), respectively. The E-field-induced
/Delta1MA(k) in the 2D BZ is displayed in Figs. 3(d) and 3(e)
for fields of −0.37 and +0.37 V /nm, respectively. Integration
of the /Delta1MA(k) over the 2D BZ for negative and positive
fields yields induced MA values consistent with the ∨shape
of the VCMA at zero strain in Fig. 2(a). Since /Delta1MA>0t h e
analysis below is focused on the positive peaks in Fig. 3(a).
The/Delta1MA(k)>0 under E
I<0 in the vicinity of /Gamma1(peaks 1
and 2) is due to the downward energy shift in the unoccupiedFe1d
x2−y2-derived bands in contrast to the occupied Fe1
dxy-derived bands which remain unshifted. The coupling of
these states via ˆLzand the decrease in the denominator in
Eq. ( 1) result in /Delta1MA(k)>0. Around1
3/Gamma1M the negative field
020407-3RAPID COMMUNICATIONS
P. V . ONG et al. PHYSICAL REVIEW B 92, 020407(R) (2015)
FIG. 3. (Color online) Zero strain: (a) E-field-induced /Delta1MA(k)
along symmetry directions for EI=± 0.37 V/nm. (b) and (c) Shift
in zero-field minority-spin band structures (dotted curves) under−0.37 V/nm (solid green curves) and +0.37 V /nm (solid red
curves) fields, respectively. We show the dominant Fe1 and Ta
d-derived states. E-field-induced /Delta1MA(k) (in meV) in the 2D BZ for
(d)−0.37 V/nm and (e) +0.37 V /nm, respectively. The numbered
vertical lines in (b) and (c) correspond to the numbered peaks in (a),(d), and (e).
has a negligible effect on the Ta1 occupied dxy-derived states
whereas it induces a significant shift in the Ta1 unoccupiedd
x2−y2-derived states coupled via ˆLz, resulting in /Delta1MA(k)>0
(peak 3).
Under +0.37 V /nm both the Ta1 occupied dxy-derived
and the Ta1 unoccupied dx2−y2-derived bands around1
3/Gamma1M
do not shift [Fig. 3(c)], and hence /Delta1MA(k)→0 [Fig. 3(a)].
On the other hand, a new peak in /Delta1MA(k) develops around
2
3/Gamma1X [peak 4 in Figs. 3(a) and3(e)] due to the fact that both
Fe1 occupied dxy-derived and unoccupied dxz-derived bands
(coupled through ˆLx) shift up in energy rendering the dxystates
partially unoccupied. Thus, the out-of-plane contribution toMA(k) is enhanced.
Atη
FeCo=2% the E-field-induced /Delta1MA(k) is plotted
along the high-symmetry directions in Fig. 4(a) forEI=
±0.58 V/nm. The shift in the zero-field minority-spin bands
(dotted curves) are shown in Figs. 4(b) and4(c) for an E field
of−0.58 V/nm (green curves) and +0.58 V (red curves),
respectively. Figures 4(d) and 4(e) display /Delta1MA(k)i nt h e
2D BZ for fields of −0.58 V/nm and (e) +0.58 V /nm,
FIG. 4. (Color online) The same as Figs. 3(a)–3(e) but for 2%
strain and EI=± 0.58 V/nm. We show only the dominant Fe1 d-
derived states in (b) and (c).
respectively. Integration of /Delta1MA(k) over the 2D BZ for neg-
ative and positive fields yields induced MA values consistentwith the ∧shape of the VCMA at 2% strain in Fig. 2(b). Since
for both field directions /Delta1MA<0, we focus on the main
negative /Delta1MA(k) troughs (1 and 2) in Figs. 4(a),4(d), and 4(e)
around
1
3/Gamma1X and near X. Their origin lies on the field-induced
shift in the Fe1 dx2−y2-derived bands from above to below the
Fermi level [Figs. 4(b) and 4(c)] and the concomitant SOC
of these states with the unoccupied Fe1 dxz,yz bands via ˆLx,
resulting in /Delta1MA<0.
To summarize, we have demonstrated that epitaxial strain,
which is ubiquitous in many HM/FM/I trilayers, has a dramaticeffect on the VCMA. It can change the VCMA from a∨-shape to a ∧-shape E-field dependence with giant VCMA
coefficients. Furthermore, the critical field where the MAreaches its maximum or minimum value can be controlledselectively via strain tuning. These findings, which are generalfor other HM/FM/I junctions with different HM caps [ 29],
open interesting prospects for exploiting strain engineeringto harvest higher efficiency VCMA for the next generationMeRAM devices.
This research was supported by NSF under Grant No.
1160504 NSF Nanosystems Engineering Research Center(ERC) for Translational Applications of NanoscaleMultifer-roic Systems (TANMS).
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020407-5 |
PhysRevB.99.205422.pdf | PHYSICAL REVIEW B 99, 205422 (2019)
Disordered Si:P nanostructures as switches and wires for nanodevices
Amintor Dusko,1,*Belita Koiller,2and Caio Lewenkopf1
1Instituto de Física, Universidade Federal Fluminense, 24210-346 Niterói, Rio de Janeiro, Brazil
2Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21941-972 Rio de Janeiro, Brazil
(Received 1 February 2019; revised manuscript received 31 March 2019; published 17 May 2019)
Atomically precise placement of dopants in Si permits creating substitutional P nanowires by design. High-
resolution images show that these wires are a few atoms wide with some positioning disorder with respect tothe substitutional Si structure sites. Disorder is expected to lead to electronic localization in one-dimensional(1D)-like structures. Experiments, however, report good transport properties in quasi-1D P nanoribbons. Weinvestigate theoretically their electronic properties using an effective single-particle approach based on a linearcombination of donor orbitals (LCDO), with a basis of six orbitals per donor site, thus keeping the ground statedonor orbitals’ oscillatory behavior due to interference among the states at the Si conduction band minima.Our model for the P positioning errors accounts for the presently achievable placement precision allowing us tostudy the localization crossover. In addition, we show that a gatelike potential may control its conductance andlocalization length, suggesting the possible use of Si:P nanostructures as elements of quantum devices, such asnanoswitches and nanowires.
DOI: 10.1103/PhysRevB.99.205422
I. INTRODUCTION
The approaching breakdown of Moore’s law has trig-
gered a strong research effort to avoid compromising theminiaturization spiral in electronics. One of the promisingstrategies to keep it evolving consists in transferring currentdevice functionalities to nanostructures prepared with atomic-scale control. Given the ubiquity of silicon integrated circuitspresently in use, atomic implantation of dopants in Si hostsconstitutes a very attractive road towards achieving suchstructures. This requires effective control of donor positioningat preassigned sites, i.e., fabricating devices at the atomiclevel by design [ 1–3]. Reports of successful placement of P
arrays in Si suggest that this arrangement could, in principle,play the role of nanowires connecting different componentsof nanodevices, similar to a metallic wire in regular chips[1–7].
The adequacy of P nanochains and nanoribbons in Si to
serve as channels for electronic transport in devices raisessome questions. In principle, a perfectly ordered array doesprovide the desired connections [ 1–7]. However, in real sam-
ples the positioning uncertainties, inherent to the current fab-rication processing standards, may spoil the desired conduc-tance features: Due to the well known property that electronicstates in disordered one-dimensional (1D) materials are local-ized [ 8], disordered nanowires can become insulators, with
negligible electronic transport. Since the nanowires of interesthere are finite, the electronic conductance is significant, aslong as the electronic localization length is comparable to orlarger than the system length itself [ 9].
Here we investigate these questions theoretically, modeling
P nanochains and nanoribbons by a tight-binding description
*amintor.dusko@gmail.comwith six orbitals per P substitutional site, corresponding tothe combinations of the six minima in the Si conductionband, symmetrized according to the tetrahedral crystal fieldpotential at the donor site, see the Appendix. The sixfolddegenerate levels split into states that have the symmetry ofthe different irreducible representations of the T
dgroup [ 10].
This leads to a singlet with A 1symmetry, a triplet with T 2
symmetry, and a doublet with E symmetry. Starting from anideal target configuration for the P sites, the actual positionsare individually chosen according to a Gaussian distributionof lattice positions centered at each target site.
In this multiorbital scenario, we systematically study how
the choice of the device geometry, namely the interdonordistance and the wire dimensions (width and length), affect thesystem’s electronic conductance and localization. In addition,we show that such generated nanostructures can serve asnanoswitches controlled by an external gate potential.
This paper is organized as follows: In Sec. IIwe sum-
marize the theoretical LCDO scheme, the atomistic modelconsidered here, and the Landauer-Büttiker approach forquantum coherent transport. In Sec. IIIwe outline the lo-
calization length calculation scheme and compare the mainfeatures of the different disorder intensities scenarios. InSec. IVwe investigate the sensitivity of the localization
length parameter to an external gate potential and in Sec. V
we analyze the corresponding effects on the nanostructureconductance. Our conclusions and summary are presented inSec. VI.
II. MODEL AND METHODS
The full set of electronic states that describe mesoscopic
nanostructures formed by donors in a Si host correspond to aHilbert space whose size is typically larger than 10
6atomic
orbitals. As demonstrated in Refs. [ 9,11], the Hilbert space
2469-9950/2019/99(20)/205422(11) 205422-1 ©2019 American Physical SocietyDUSKO, KOILLER, AND LEWENKOPF PHYSICAL REVIEW B 99, 205422 (2019)
FIG. 1. (a) Target Si:P nanostructure fragment, along the Si [110] crystalline direction, for different widths W, specifying the geometric
parameters RWandRL. The rectangles define a neighboring region around a reference site (orange sphere). The number of neighbors sites
changes with W.F o r W>2 one can define edge and bulk sites, with five and eight neighbors, respectively. (b) Sketch of the model system:
disordered Si:P sample connected to semi-infinite leads with translational symmetry, subjected to a back gate potential. (c) Probability
distribution Pof an implanted donor to occupy the aimed position as a function of σd. Two cases are studied in this paper (blue squares):
σd=0.1 and 0.2 nm, corresponding to a deposition matching the aimed position 90% and 50% of the time, respectively. Inset: Graphical
representation of the disorder cutoff radius δ. Here green spheres indicate the Si structure and the red circled sphere is the target position.
can be effectively represented by a reduced basis formed by
a linear combination of donor orbital (LCDO). In this hybridmethod each donor orbital is accounted for by a multivalleycentral cell effective mass approach that incorporates the Sihost effects in the donor orbital itself.
We characterize the nanostructures by four geometric pa-
rameters, namely width ( W), length ( L), transversal donor
distance ( R
W), and longitudinal donor distance ( RL), see
Fig. 1(a). Considering the placement process to occur along
the Si /angbracketleft110/angbracketrightdirection, the target P donors form a rectangular
lattice with lattice parameters defined by RWandRL.
The model multiorbital Hamiltonian written in the LCDO
basis [ 9] reads
H=/summationdisplay
i,lεi,lni,l+/summationdisplay
/angbracketlefti,j/angbracketright,l,mt(i,l)(j,m)c+
i,lcj,m, (1)
where c+
i,l(ci,l) are creation (annihilation) operators of elec-
trons at the orbital lcentered at the ith site, ni,l=c+
i,lci,lis the
corresponding number operator, εi,lis the onsite energy, and
t(i,l)(j,m)the hopping term. In this equation /angbracketlefti,j/angbracketrightcomprises
the sum over pairs of sites for which the hopping terms arenot negligible: The summation is performed over sites insiderectangular regions like the ones in Fig. 1(a).F o r W=1,2,
and 3 we take up to two, five, and eight neighbors, respec-tively. The parameters were calculated within the LCDOscheme. In order to improve the reliability of the electroniccalculations at smaller interdonor distances, we extend thetreatment presented in Ref. [ 11] by including multiorbitals
and three-center corrections due to neighboring cores in thehopping energies. A detailed presentation is found in theAppendix. These developments allow us to accurately addressthe nanoribbon model ( W/greaterorequalslant2 sites) placement parameters(R
Land RW) of the order of 3 nm. We keep the isotropic
approximation.
The model system we study consists of a central region,
corresponding to the disordered Si:P nanostructure coupledto leads in thermal and chemical equilibrium with electronicreservoirs, see Fig. 1(b). The leads are semi-infinite, trans-
lational invariant, and define the electronic bands density ofstates coupled to the system of interest [ 12]. In addition, we
investigate the effect of a uniform back gate potential andstudy its applicability to control the nanostructure transportproperties. The gate potential V
Gis included in the model as
a correction to the onsite energy, namely, εi,l(VG)=εi,l(0)+
UG.H e r e UGis the shift in the electronic states energy and
εi,l(0) is the unbiased onsite energy calculated within the
LCDO scheme. The energy gained by the electron is UG=
ηeVG∝−VG, where VGis the gate potential, eis the electron
charge, and ηis a sample-dependent constant incorporating
the Si dielectric screening, geometry, and the capacitive cou-pling of the donor electron with other leads in the system. Weexpect that some trends for a lateral gate potential as the onepresent in Refs. [ 2,3] can be inferred by comparing different
nanoribbon widths, as a confining lateral potential decreasesthe effective W.
For sufficiently large values of V
Gone expects that higher
orbitals (such as 2 porbitals) influence the results. We estimate
the validity of our approach as follows: Previous calculations[13] show that the lower energy states of a single donor
remain in the A
1,T2, E symmetry manifold. Remarkably, the
corresponding energy levels remain almost constant up to afield strength of about 20 keV /cm, which are within the order
of magnitude required to generate the U
Gvalues we discuss.
We study the impact of positional disorder in such systems
using a Gaussian disorder model. The disorder is quantified by
205422-2DISORDERED Si:P NANOSTRUCTURES AS SWITCHES … PHYSICAL REVIEW B 99, 205422 (2019)
two parameters, namely a cutoff radius δ[14] around a target
substitutional site and the position standard deviation σd.F o r
simplicity, we choose δ=0.4 nm, in which case each donor
can be placed at five different Si sites. The degree of disorderis controlled by σ
d. Figure 1(c) gives the dependence of the
distribution of the implanted ion positions on σd.T h em a i n
panel shows the probability distribution Pof an implanted
donor to occupy the aimed position as a function of σdand
the inset gives a graphical representation of the disorder cutoffradius δ.T h eσ
dvalues considered in this work, namely σd=
0.1 nm and σd=0.2 nm, are indicated by the blue squares.
These values are within state-of-the-art precision of STMatomic placement techniques [ 1,6,7].
We calculate the nanostructure linear conductance using
the Landauer-Büttiker formula [ 15],
G
AB=2e2
h/integraldisplay∞
−∞dE/parenleftbigg
−∂f
∂E/parenrightbigg
TAB(E), (2)
given in terms of the Fermi-Dirac distribution function
f(E)=[1+e(E−μ)/kBT]−1and the electronic transmission
TAB(E)=tr[/Gamma1B(E)Gr(E)/Gamma1A(E)Ga(E)] [16]. In Eq. ( 2),
Gr(Ga) is the retarded (advanced) Green’s function of the
complete system (nanoribbon and leads), which we computeusing the recursive Green’s function approach, implementedas in Refs. [ 12,17,18]. The nth line or decay width, matrix ele-
ments /Gamma1
n=i[/Sigma1r
n−(/Sigma1r
n)†], are obtained from the embedding
self-energy /Sigma1r
n=V†
nGr
nVn, where Vncontains the coupling
matrix elements of the sample with the nth lead, while Gr
n
is the contact Green’s function. There are several ways to
calculate the latter [ 19–22]; we compute Gr
nby a standard
decimation procedure based on renormalization-group ideas[23,24].
We cast the nanostructures transport properties in terms
of the localization length ξ, formally defined by the wave
function asymptotic behavior, /Psi1(x)∝exp(−|x|/ξ). In this
work, we infer the localization length by the analysis of theconductance at zero temperature.
III. TRANSPORT AND PLACEMENT
Si:P nanostructures are multipath systems due to their
multiorbital nature. The hopping term in this multiorbitalframework plays an extremely nontrivial role, opening andclosing channels depending on the system parameters. To im-prove the understanding of such a system towards applicationsin nanodevices control, we investigate how the disorder andplacement parameters affect conductance and localization.
According to the localization theory in disordered systems
[25], the conductance is expected to decrease exponentially
with the ratio between the sample length Land the lo-
calization length ξ. Hence, we extract ξfrom the relation
/angbracketleftlnG
AB(L)/angbracketright∝− L/ξ, where /angbracketleft.../angbracketrightis an ensemble average [ 25]
(here typically over 103...104realizations). For each set of
parameters, our algorithm selects iteratively a range of systemlengths, from the order of 10
1to 102sites, to accurately de-
termine the localization length ξfrom the linear dependence
of/angbracketleftlnGAB(L)/angbracketrightwith L. Figure 2shows few representative
examples of /angbracketleftlnGAB/angbracketrightversus Land the corresponding linear
fit that gives ξ.(a)
(b)
FIG. 2. Conductance G(in units of G0=2e2/h) for disordered
nanochains ( W=1) as a function of the system length L(in units
ofRLor sites) for a few representative target interdonor distances RL
for (a) σd=0.1n ma n d( b ) σd=0.2 nm. The results correspond to
an average over 104disorder realizations. In all cases the standard
deviations are smaller than the markers.
In Fig. 3we present the localization length for W=1
(nanochains) behavior with RLfor two levels of disorder. As
expected, increasing RLor the disorder level lowers ξ.N o t e
that for small RLwe observe an enhanced sensitivity of ξ
withσd. For these two disorder levels, ξshows an abrupt fall
around RL=5.7n m .
In order to represent the combined effect of the geomet-
ric parameters RLand RWin the transport trends of our
system, we calculate the localization length ξ(RL,RW)f o r
3.0n m/lessorsimilar[RL,RW]/lessorsimilar6.5 nm. The range of values was chosen
to represent realistic geometries of the current experimentalrealizations [ 1–3]. For technical reasons, our analysis is lim-
itedR
W,RL/greaterorsimilar3 nm, as discussed in the Appendix.
Figure 4presents ξas a function of RLandRW, where
for each pair of parameters, ξis represented by the given
color code. In Figs. 4(a) and4(b) we show ξ(RL,RW)f o r
disorder σd=0.1 nm and in Figs. 4(c) and 4(d) forσd=
0.2 nm. The frames on the left refer to W=2 and on the
FIG. 3. Localization length ξ(in units of RLor sites) as a func-
tion of the interdonor target separation RLfor nanochains ( W=1)
and disorder intensities σd=0.1 and 0.2 nm.
205422-3DUSKO, KOILLER, AND LEWENKOPF PHYSICAL REVIEW B 99, 205422 (2019)
FIG. 4. Localization length ξfor nanoribbons of W=2 and 3 as a function of the interdonor target separation RLandRW. Graphs with
the same σdpresent the same color bar. (a) σd=0.1n ma n d W=2, (b)σd=0.1n ma n d W=3, (c)σd=0.2n ma n d W=2, and (d) σd=
0.2n ma n d W=3.
right to W=3. The results suggest a metal-insulator phase
diagram with a very similar overall behavior for both disorderintensities presented. In Figs. 4(a) and4(c) our simulations
reveal a relatively small region in the investigated parameterspace with nonmonotonic behavior, roughly R
W/greaterorsimilar4.5n m
andRL/lessorsimilar5.0 nm. In particular, ξis peaked at RL≈3.5n m
andRW≈5.4,6.1, and 6 .5 nm. Outside this nontrivial re-
gion, by increasing RLor decreasing RWthe electronic states
tend to become more localized. In Figs. 4(b) and4(d) we find
an overall increase of ξand a wider region with nontrivial
extended states, corresponding to the parameter range definedbyR
W/lessorsimilar6.0 nm and RL/lessorsimilar4.2 nm. In particular, ξshows
peaks for RL≈3.5 nm and RW≈5.4,6.1, and 6 .5n m .O u t
of this nontrivial region, increasing RLor decreasing RW
favors localization.
Comparing W=1,2, and 3 we observe an overall increas-
ing in localization length with the system width, consistentwith the increasing in the maximum number of transportchannels, respectively, 6 ,12, and 18. The sensitivity of ξ
on the disorder intensity seems to become stronger for largervalues of W.
IV . TUNING LOCALIZATION LENGTH
The nonmonotonic behavior of the localization length with
the lattice geometry, namely RLandRW, suggests that onecan tune it and hence control the system’s conductance G
by a suitable external handle. In what follows we show thata back gate potential, as described in Sec. II, is capable of
dramatically modifying the transport properties of disorderedSi:P nanowires. We recall that for electrons, U
G∝−VG.
In order to get some insight on the gate control over lo-
calization lengths in nanoribbons, we start with the nanowirecase, W=1. Results for ξunder a gate bias from 0 down
to−250 meV are presented in Fig. 5for two degrees of
disorder. For a fixed interdonor distance, according to thesmaller (larger) degree of disorder ξoscillates in a larger
(smaller) range in the graph truncated to 100 (50) nm. Thisnontrivial interference driven oscillatory behavior can be ex-plained as follows: V
Grigidly shifts the nanowire energy
spectrum. Hence, VGdrives localized and extended states, as
well as small and large density of states of the disorderedsystem across the Fermi energy fixed by the contacts. Giventhat the P donor in Si lower energy levels are 45 meV belowthe bottom of the Si conduction band edges, applying abias of U
G=45 meV would ionize the donors completely
inside the active (sample) region. A wider range of control isprovided for negative values of U
Gwhich increases separation
of the P electrons levels to the Si conduction band edge, thusremaining operational for the wide U
Grange shown in the
figures. Therefore we restrict our results to UG<0(VG>0).
The parameter range for a conducting behavior ( ξ/L/greaterorsimilar1)
205422-4DISORDERED Si:P NANOSTRUCTURES AS SWITCHES … PHYSICAL REVIEW B 99, 205422 (2019)
FIG. 5. Localization length ξfor nanochains ( W=1) as a func-
tion of the gate energy UGand the interdonor target separation RLfor
(a)σd=0.1n ma n d( b ) σd=0.2n m .
shrinks for increasing values of RL, consistent with the drop
in the mean value of the hopping matrix elements.
The case W=2 and σd=0.1 nm is illustrated in Fig. 6.
The simulations indicate an overall increase of ξas a function
ofUGfollowed by an oscillatory pattern. As in the W=1 case
we observe that the UGrange corresponding to conducting
behavior shrinks with RL. In addition, a similar feature can
be observed for increasing values of RW. By varying UG
for different RWvalues, we find the formation of a gap—a
region of negligible values of ξ—followed by a “reactivation”
in the localization phase diagram for larger values of RL.
This gap is highlighted in Figs. 6(a)–6(c)where the threshold
RLvalues are 4 .6 nm, 5 nm, and 5 .4 nm, respectively. The
gapRLthreshold value continues to increase monotonically
along Figs. 6(d)–6(f). In the last three panels [Figs. 6(g)–6(i)],
the gap closes resembling the signature of the W=1 case.
Although it is reasonable to recover a phase diagram similar to
W=1 case while increasing RW, we observe an enhancementin the overall localization length values and VGrange leading
to conducting behavior.
The wider ribbon case, W=3, is given in Fig. 7.A si n
W=1 and 2 cases, one observes an increase in ξwith UG
followed by an oscillatory pattern and that the UGrange lead-
ing to conducting behavior shrinks with RLandRWinterdonor
distances. For smaller RWvalues, Figs. 7(a)–7(c)show a larger
gap than in the W=2 case and the opening of a second gap.
Throughout Figs. 7(d)–7(f)we observe that this second gap
is short lived compared to the first one. In summary, we findthat both ξandU
Grange leading to conducting behavior are
overall larger than in the W=1 and 2 cases. We have also
performed calculations for σd=0.2 nm, for both W=2 and
3, not shown since all properties follow the trends identifiedin the previous cases.
V . CONDUCTANCE CONTROL
In this section, we investigate the use of a gate potential
VG(UG∝−VG) as an external control of conductance GAB
for Si:P nanostructures. Here we set L=60 sites for the
purpose of investigating a nanoswitch implementation in alength comparable to the experimental realizations [ 1–3]o f
higher P density.
Figure 8shows the average conductance G
ABas a function
ofUGandRLfor nanochains ( W=1). The results show os-
cillations in GABas a function of both RLandVG. A minimum
inGABoccurs around RL≈4.6 nm, which should be avoided
in practical implementations of the system as a nanoswitch.Oscillations due to U
Gstand out for smaller RLvalues. In line
with the localization length analysis, an increase of RLcauses
the range of UGvalues corresponding to a conducting behavior
to shrink. For RL≈3.1 nm, introducing a gate potential, we
observe an increase in GABof approximately 50% and 100%
forσd=0.1 nm and 0 .2 nm, respectively.
The conductance for W=2 sites nanoribbons and σd=
0.1 nm results, presented in Fig. 9, show a rapidly oscil-
latory behavior as a function of UGfor small RWvalues,
see Figs. 9(a) and9(b). For larger RWvalues however [see
Figs. 9(e) and9(f)] the oscillations are strongly damped for
small UG. In all cases, it is possible to observe a UGtransition
edge between larger and smaller GABvalues regimes. There
is also a minimum in GABaround RL≈3.5 nm; the feature is
more pronounced in the cases shown in Figs. 9(b)–9(e).W e
observe a very subtle gap opening in GABwhile increasing RL.
The RLvalue corresponding to this opening increases with
RW.I nF i g s . 9(a),9(c) and9(e) the corresponding gapping
opening value is RL≈4.2,5.0, and 5 .4 nm, respectively. As
in the W=1 case, introducing a gate potential induces an
increase of approximately 50% in GAB.
The results for nanoribbons of W=3 sites are presented in
Fig.10. Some similarities with W=2 case can be observed:
Rapidly oscillating GABspectrum with a clear change in
overall behavior in a given transition edge, for an example seeFig. 10(a) atR
L≈3.1 nm and UG=−350 meV. In contrast
with the W=2 case we observe two gap openings and an
overall minimum in GABvalues around RL≈3.9 nm. The first
gap can be observed in Figs. 10(a) ,10(c) and10(e) forRL≈
5.0,5.4, and 5 .9 nm, respectively. The second gap is more
205422-5DUSKO, KOILLER, AND LEWENKOPF PHYSICAL REVIEW B 99, 205422 (2019)
FIG. 6. Localization length ξas a function of VGandRL(RWfixed) for nanoribbons of W=2a n d σd=0.1 nm. (a)–(i) correspond to
different values of RW.
subtle but can be observed in Fig. 10(d) forRL≈5.9n m ,f o r
example.
In summary, by considering nanostructures with increasing
width, W=1,2, and 3, we observe a corresponding increase
in: (i) the overall GABvalues, (ii) the window of VGvalues
leading to a conducting behavior, and (iii) in the number
of gap openings. We also observe a change in the RLvalue
corresponding to an overall minimum in GAB. We also find
that the overall behavior of the conductance on the latticeparameters does not depend on the disorder strength. Thiscan be explicitly seen for the W=1 case by comparing the
simulations for σ
d=0.2 nm and 0 .1n m .
Finally, let us stress the sharp VGdriven metal-insulator
transition appearing for any given choice of RLin all cases we
analyze in this work. This remarkable feature strongly sug-gests that Si:P nanostructures can act as switches by properlytuning the gate potential.
VI. DISCUSSIONS AND CONCLUSIONS
In this work we extended the LCDO formalism [ 9,11]
to include Gaussian disorder, a multiorbital description, andtechnical improvements, specified in the Appendix, whichresults in a more realistic description of P nanochains and to
access nanoribbons of arbitrary widths. Our simulations treatthe problem considering realistic system sizes and disorder.We also have put forward a proposal for an external controlof transport properties such as localization length and con-ductance suggesting a new path of investigations for futureexperimental implementations.
We have found nontrivial features of the electronic trans-
port properties due to system fabrication specifications stillremaining robust against disorder. Specific values of place-ment parameters and nanostructure width provide optimizedlocalization length, favoring high conductance. Our calcula-tions indicate that a similar behavior is expected for differentdisorder levels.
We further analyze the effects of an external back gate
potential V
Gto localization length and conductance. Properly
tuning VGone can control localization lengths, allowing donor
nanowires to keep current-carrying wave functions even forrelatively long samples, serving as efficient connectors amongnanodevices parts. In addition, it is possible to increase thenanostructure conductance, or decrease it by using this ex-ternal potential, which suggests the use of such structures asnanoswitches. Both connectors and switches provide state-
205422-6DISORDERED Si:P NANOSTRUCTURES AS SWITCHES … PHYSICAL REVIEW B 99, 205422 (2019)
FIG. 7. Localization length ξas a function of VGandRL(RWfixed) for nanoribbons of W=3a n d σd=0.1 nm. (a)–(i) correspond to
different values of RW.
of-the-art resources contributing to nanodevices technology
development.
ACKNOWLEDGMENTS
The authors acknowledge the financial support
of the Brazilian funding agencies CL CNPq (GrantNo. 308801 /2015-6) and FAPERJ (Grant No. E-
26/202.882 /2018); BK CNPq (Grant No. 304869 /2014-7)
and FAPERJ (Grant No. E-26 /202.767 /2018). This study was
also financed in part by the Coordenação de Aperfeiçoamentode Pessoal de Nível Superior - Brazil (CAPES) - FinanceCode 001.
APPENDIX: MICROSCOPIC MODEL—
TECHNICAL DETAILS
The multiorbital approach employed in this study rep-
resents a significant generalization of the method used inour previous works, where only one A
1orbital per site was
considered [ 9,11]. Here, effects due to neighboring donor
potentials, known to change the ground-state symmetry forsmall interdonor distances, are correctly treated. Nonetheless,the first order perturbation theory breaks down for sufficientlysmall values of the interdonor distances R
Land RW.W e
estimate the lower bound to be around 3 nm.
1. Linear Combination of Dopant Orbitals (LCDO)
Following the well established Kohn and Luttinger pre-
scription [ 10,26,27] for shallow donors in Si, we consider a
basis of six donor orbitals per site, corresponding to the sixminima in Si conduction band. Valley orbit coupling, includedby first order perturbation theory for degenerate states [ 28,29],
renders donor orbitals as superpositions of pure valley statesobtained by the effective mass approach:
/Psi1
l
i(r)=1
Nl6/summationdisplay
μ=1al
μFμ(r−Ri)φμ(r−Ri), (A1)
where lrefer to the donor iorbitals pinned to the donor
coordinates Ri. The constants Nlandal
μstand for the normal-
ization and valley population (presented in Table I),Fμ(r)=
F(r)=(πa∗3)−1/2e−r/a∗is for simplicity approximated as
an isotropic hydrogenlike envelope function, with a speciesdependent effective Bohr radius a
∗(1.106 nm for Si:P), and
φμ(r)=eikμ·ruμ(r) are the Bloch functions of the six Si con-
duction band degenerate minima ( μ=1,..., 6). The latter
205422-7DUSKO, KOILLER, AND LEWENKOPF PHYSICAL REVIEW B 99, 205422 (2019)
FIG. 8. Conductance GAB(in units of G0=2e2/h) as a func-
tion of a gate potential VGand interdonor target separation RLfor
nanochains ( W=1),L=60 sites, and (a) σd=0.1n m ,( b ) σd=
0.2n m .
are located along the equivalent directions ±x,±y,±zat
|kμ|=k0=0.85(2π/aSi), where aSiis the conventionally
called Si lattice parameter [ 30]. The effective Bohr radius is
obtained by incorporating screening effects due to the Si hostcharge carriers in the donor singular potential.
Screening effects are included through a potential that
interpolates the expected behavior for large and small valuesofr, namely,
V(r)=−e
2
4πr/bracketleftbigg1
/epsilon1Si+/parenleftbigg1
/epsilon10−1
/epsilon1Si/parenrightbigg
e−r/r∗/bracketrightbigg
, (A2)
the screening length r∗defines the transition between
a bare V(r→0)=−e2/4π/epsilon10rand a screened
V(r→∞ )=−e2/4π/epsilon1Sirpotential. Here /epsilon10and/epsilon1Si
are, respectively, the free space and the static relative
permittivities.As in previous works [ 9,11] the Hamiltonian terms are
calculated by the atomistic Hamiltonian ˆH=ˆHi+ˆH/prime, where
ˆHiis the single donor Hamiltonian and ˆH/primeis the perturbation
due to neighboring donor cores. We project the donor orbitalto this atomistic Hamiltonian to extract the onsite and hoppingterms. For the onsite term we obtain,
ε
i,l=/angbracketlefti|ˆHi|i/angbracketright+/angbracketleft i|ˆH/prime|i/angbracketright≈− El+/summationdisplay
k/angbracketlefti|ˆVk|i/angbracketright,(A3)
where Elis the single donor level energy given in Table I,
which contains valley-orbit corrections.
Similarly the hopping reads,
t(i,l)(j,m)=/angbracketleftj|ˆHi|i/angbracketright+/angbracketleft j|ˆH/prime|i/angbracketright (A4a)
≈−E0/angbracketleftj|i/angbracketright+/summationdisplay
k/angbracketleftj|ˆVk|i/angbracketright=Tij(R)/Theta1lm(R)
/Theta1lm=1
NlNm6/summationdisplay
μ,ν=1al
μam
νeikμ·R(A4b)
Tij(R)=E0Sij+Tijj+/summationdisplay
kTik j (A4c)
Sij(R)=/angbracketleftF(Rj)|F(Ri)/angbracketright (A4d)
Tik j=/angbracketleftF(Rj)|V(Rk)|F(Ri)/angbracketright, (A4e)
where R=Rj−Riis the interdonor distance, E0is the
donor ground state energy, /Theta1lmcomes from the valley inter-
ference, and Tij(R) depends on the envelope overlap function
Sijand on two-centers ( Tijj) and three-centers ( Tik j) envelope
function integrals. The Tijjintegrals have a closed analytical
solution [ 11], while the Tik jare calculated numerically. The k
labels all cores in the neighborhood of the iandjdonors, see
Fig.1(a).
Comparisons with experiments show that this multivalley
central cell corrected dopant approximation gives an accu-rate description of the single impurity spectrum [ 29] and
the corresponding wave functions [ 31], as well as the two
impurities spectra in ionized [ 32] and neutral excited states
[33]. The computational advantage is clear: By incorporating
the Si matrix explicitly in the orbitals, this approach allowsthe investigation of shallow donor systems of mesoscopicdimensions, a prohibitive task for a full atomistic approach.
2. Gaussian Expansion—Three-center Integrals
In this paper we consider hopping terms due to all neigh-
boring cores. Since the straightforward calculation of thesethree-center integrals is computationally expensive, we writethe envelope orbitals and the Coulomb potential, as a Gaussianexpansion, namely
F(r)=
NG/summationdisplay
n=1cF
ne−sF
nr2, (A5)
V(r)=−e2
4πr/bracketleftBigg
1
/epsilon1Si+/parenleftbigg1
/epsilon10−1
/epsilon1Si/parenrightbiggNG/summationdisplay
n=1cV
ne−sV
nr2/bracketrightBigg
,(A6)
where the coefficients cF
n,cV
n,sF
n, and sV
nare obtained by a
standard least square fit and presented in Table II. We find
205422-8DISORDERED Si:P NANOSTRUCTURES AS SWITCHES … PHYSICAL REVIEW B 99, 205422 (2019)
FIG. 9. Conductance GAB(in units of G0=2e2
h) for disordered nanoribbons of W=2a n d L=60 sites with σd=0.1 nm as a function of
a gate potential VGand interdonor target separations RLandRW. (a)–(f) correspond to different values of RW.
FIG. 10. Conductance GAB(in units of G0=2e2
h) for nanoribbons of W=3a n d L=60 sites with σd=0.1 nm as a function of a gate
potential VGand interdonor target separations RLandRW. (a)–(f) correspond to different values of RW.
205422-9DUSKO, KOILLER, AND LEWENKOPF PHYSICAL REVIEW B 99, 205422 (2019)
TABLE I. Valley population al
μ, normalization constant Nl,a n d
P0donor energy Elfor the six donor orbitals l.
lal
x al
−x al
y al
−y al
z al
−z Nl El(meV)
A1 11 1 1 1 1√
6 −45.58
Tz
2 00 0 0 1 −1√
2
Ty
2 00 1 −10 0√
2 −33.90
Tx
2 1−10 0 0 0√
2
Exy11 −1−10 0 2 −32.60
Ez11 1 1 −2−2√
12
that by taking NG=13 Gaussian terms, the expansions agree
within 10−8accuracy for all values of rwhere the target
function satisfies f(r)/greaterorsimilar10−20.
3. Gaussian Coulomb Integrals—Product Rule
Let us now show the main derivation steps to obtain
very simple expressions for the Gaussian integrals introducedabove. The Gaussian expansion of the Coulomb three-centerintegral T
acbreads
Tacb=/angbracketleftF(Rb)|V(Rc)|F(Ra)/angbracketright (A7)
=/summationdisplay
m,ncF
mcF
n/integraldisplay
Vdre−sF
mr2
be−sF
nr2
aV(rc),
where rn=|r−Rn|is the relative position to donor n.
Let us now use the Gaussian product rule, i.e.,
e−sF
mr2
be−sF
nr2
a=e−ηmnR2
bae−umnr2
u, (A8)
where the constants umn=sF
m+sF
nandηmn=sF
msF
n/umnare
the total and reduced exponents, while Rba=|Ra−Rb|and
ru=(sF
mrb+sF
nra)/umnare the relative and the Gaussian cen-
ter of mass positions. Equation ( A8) expresses the product
of two Gaussians in a new product where the first term isa constant and only the second term depends on r. In other
TABLE II. Gaussian expansion coefficients for the envelope
function F(r) and exponential in the screened Coulomb potential
V(r).
F(r) V(r)
cF
n sF
n(nm−2) cV
n sV
n(nm−2)
9.26×10−24.10×10−11.91×10−13.82×101
8.56×10−29.19×10−11.76×10−18.60×101
7.31×10−21.89×10−11.52×10−11.76×101
6.71×10−22.15 1 .38×10−12.02×102
4.80×10−25.30 9 .86×10−24.99×102
3.26×10−21.38×1016.68×10−21.31×103
3.13×10−28.99×10−26.53×10−28.35
2.12×10−23.89×1014.34×10−23.70×103
1.33×10−21.20×1022.72×10−21.15×104
8.03×10−34.20×1021.63×10−24.05×104
4.68×10−31.78×1029.47×10−31.74×105
3.84×10−34.26×10−28.03×10−33.96
3.73×10−31.54×1047.45×10−31.53×106words, the problem is reduced to a two-center integral
Tacb=/summationdisplay
m,ncF
mcF
ne−ηmnR2
ba/integraldisplay
Vdre−umnr2
uV(rc). (A9)
When V(r) is a screened Coulomb potential, this two-
centers integral can be decomposed in two terms, i.e.,T
F(ru,rc)=TSi(ru,rc)+Tsc(ru,rc). The Gaussian expan-
sion of the exponential term in the V(r)g i v e s
TSi=−e2
4π/epsilon1Si/summationdisplay
m,ncF
mcF
ne−ηmnR2
baISi
mn,
ISi
mn=/integraldisplay
Vdre−umnr2
u1
rc,
Tsc=−e2
4π/parenleftbigg1
/epsilon10−1
/epsilon1Si/parenrightbigg/summationdisplay
m,n,ocF
mcF
ncV
oe−ηmnR2
baIsc
mno,
Isc
mno=/integraldisplay
Vdre−umnr2
ue−sF
or2
c
rc.
The next step consists of writing r−1
cas a Gaussian integral,
namely, r−1
c=π−1/2/integraltext∞
−∞dt e−t2r2
c. After rearranging the inte-
grals and applying the product rule in the scterm, one obtains
ISi
mn=1√π/integraldisplay∞
−∞dt/integraldisplay
Vdre−umnr2
ue−t2r2
c,
Isc
mno=1√πe−νmnoR2
uc/integraldisplay∞
−∞dt/integraldisplay
Vdre−vmnor2
ve−t2r2
c,
where vmno=(umn+sF
o),νmno=umnsF
o/vmno,Ruc=|Rc−
Ru|, and rv=|r−Rv|where Rv=(umnRu+sF
oRc)/vmno.
Applying the product rule, as in Eq. ( A8), we find
ISi
mn=1√π/integraldisplay∞
−∞dt e−/parenleftBig
umnt2
umn+t2/parenrightBig
R2
uc/integraldisplay
Vdre−(umn+t2)r2
p
where rp=(umnru+t2rc)/(umn+t2) and
Isc
mno=e−νmnoR2
uc
√π/integraldisplay∞
−∞dt e−/parenleftBig
vmnot2
vmno+t2/parenrightBig
R2
vc/integraldisplay
Vdre−(vmno+t2)r2
q
where rq=(vmnorv+t2rc)/(vmno+t2) and Rvc=|Rc−Rv|.
As in Eq. ( A8), the spatial integrals depend only on the
Gaussian center of mass rpandrq.
Finally, adjusting the integration limit in the remaining
integrals, we obtain the simple expressions
ISi
mn=2√π/integraldisplay∞
0/parenleftbiggπ
umn+t2/parenrightbigg3/2
e−/parenleftBig
umnt2
umn+t2/parenrightBig
R2
ucdt,
Isc
mno=2e−νmnoR2
uc
√π/integraldisplay∞
0dt/parenleftbiggπ
vmno+t2/parenrightbigg3/2
e−/parenleftBig
vmnot2
vmno+t2/parenrightBig
R2
vc.
By introducing the change of variables q2
u=t2/(umn+t2)
and q2
v=t2/(vmno+t2) the integrals are conveniently
written as
ISi
mn=2π/integraldisplay1
0dque−umnRucq2
u=2πF0/bracketleftbig
umnR2
uc/bracketrightbig
,
Isc
mno=2πe−νmnoR2
uc/integraldisplay1
0dqve−vmnRvcq2
v (A10)
=2πe−νmnoR2
ucF0/bracketleftbig
vmnR2
vc/bracketrightbig
,
205422-10DISORDERED Si:P NANOSTRUCTURES AS SWITCHES … PHYSICAL REVIEW B 99, 205422 (2019)
TABLE III. Fitting coefficients of Boys function, see Eq. ( A11).
Coefficient Fitted value
s1 6.70×10−5
B0 1.00
B1 −3.33×10−1
B2 9.94×10−2
B3 −2.28×10−2
B4 3.81×10−3
B5 −3.99×10−4
B6 2.15×10−5
s2 6.01×10−5where F0is called zero degree Boys function [ 34,35]. To
optimize computational resources we choose to solve theintegral once, with high precision and in a range coveringsmall and large values, and to adjust a curve that interpolateswith rapidly decaying exponentials the expected behavior inall domains.
F
adj
0(x)=e−s1x66/summationdisplay
n=0Bnxn+/parenleftbig
1−e−s2x6/parenrightbig1
2/radicalbiggπ
x(A11)
The coefficients of the fitted curve are presented in Table III.
For the domain we considered, x∈[10−8,104], we find that
|Fadj
0−F0|≈10−7, confirming the fitting quality.
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205422-11 |
PhysRevB.98.214115.pdf | PHYSICAL REVIEW B 98, 214115 (2018)
Pressure-dependent intermediate valence behavior in YbNiGa 4and YbNiIn 4
Z. E. Brubaker,1,2R. L. Stillwell,2P. Chow,3Y . Xiao,3C. Kenney-Benson,3R. Ferry,3D. Popov,3S. B. Donald,2P. Söderlind,2
D. J. Campbell,4J. Paglione,4K. Huang,5R. E. Baumbach,5R. J. Zieve,1and J. R. Jeffries2
1Department of Physics, University of California, Davis, Davis, California 95616, USA
2Lawrence Livermore National Laboratory, Livermore, California 94550, USA
3HP-CAT, X-ray Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
4Center for Nanophysics and Advanced Materials, Department of Physics, University of Maryland, College Park, Maryland 20742, USA
5National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32313, USA
(Received 4 October 2018; published 28 December 2018)
We report a comprehensive structural and valence study of the intermediate valent materials YbNiGa 4and
YbNiIn 4under pressures up to 60 GPa. YbNiGa 4undergoes a smooth volume contraction and shows steady
increase in Yb valence with pressure, though the Yb valence reaches saturation around 25 GPa. In YbNiIn 4,a
change in pressure dependence of the volume and a peak in Yb valence suggest that a pressure-induced electronictopological transition occurs around 10–14 GPa. In the pressure region where YbNiIn
4and YbNiGa 4possess
similar Yb-Yb spacings, the Yb valence reveals a precipitous drop. This drop is not captured by density functionaltheory calculations and implies that both the lattice degrees of freedom and the chemical environment play animportant role in establishing the valence of Yb.
DOI: 10.1103/PhysRevB.98.214115
I. INTRODUCTION
Strongly correlated rare-earth materials have been heavily
studied due to the exotic physical properties that they exhibit.
Many of the rare earths form compounds in intermediatevalence states, which will naturally dictate the magnetic prop-erties of these materials and which can readily be tuned viathe application of pressure. While in Ce compounds pressuregenerally favors a nonmagnetic Ce
4+state, in Yb compounds
pressure favors the magnetic Yb3+state [ 1]. The strong
electron correlation in rare-earth bearing materials originates
from 4 felectrons that at normal conditions are localized on
the atom. However, if the rare-earth atoms are sufficientlyclose, due to application of pressure or within a suitablecrystal structure, they can interact, displaying behavior thatadopts delocalized character. Thus, by choosing appropriaterare-earth compounds, intermediate valence behavior (degree
of localization) can be achieved and tuned via the application
of pressure. Among Yb-based compounds, one of the onlyknown superconductors, β-YbAlB
4, shows valence fluctua-
tions at 20 K with an effective valence of n=2.75, which
has sparked interest in better understanding the relationshipbetween intermediate valence behavior, quantum criticality,and superconductivity [ 2,3]. The effect of the Yb-valence state
on magnetic properties has also been studied in YbNi
3Ga9,
which forms a nonmagnetic state at low temperature with aneffective valence of n=2.6 under ambient conditions [ 4].
With the application of pressure, the valence is increased to2.9, allowing a magnetic ground state to develop.
Because the rare-earth valence plays such a crucial role
in determining magnetic properties, it is imperative to un-derstand both its cause and how to tailor materials to exhibitthe desired valence configuration. With this goal in mind, wehave performed density functional theory (DFT) calculations
as well as a comprehensive structural and valence study of theorthorhombic YbNiGa
4and YbNiIn 4system under pressure
at room temperature. Previous work reported a valence ofn=2.40 in YbNiIn
4,n=2.48 in YbNiGa 4, andn=2.66 in
YbNiAl 4, showing a general trend of increasing Yb valence
with decreasing size of the group IIIb ions [ 5–7]. The Yb
valence in YbNiGa 4was measured up to a pressure of 25 GPa,
revealing a steady increase in valence up to a maximum valueofn=2.7[7].
Our DFT calculations are consistent with the overall trend
of the Yb valence at ambient pressure in this system, but sug-gest a stronger dependence on interatomic spacing. In orderto determine if the valence is a simple function of interatomicspacing, we have determined the Yb valence in YbNiGa
4and
YbNiIn 4under pressures up to 40 GPa. Rather than being a
continuous function of atomic volume or lattice spacing, wefind that the Yb valence is sensitive to both the lattice degreesof freedom and the chemical environment. Density functionaltheory does well to reproduce the pressure dependence of theIn compound up to a pressure that generates lattice spacingscomparable to those of the Ga variant at ambient pressure.However, the substitution of Ga for In results in a precipitousdrop in valence at fixed lattice size, an effect not captured byour DFT calculations and one that implies a more prominentrole for the 4 fhybridization with specific pstates than might
be conventionally expected.
II. EXPERIMENTAL AND THEORETICAL METHODS
Polycrystalline samples with nominal compositions of
YbNiGa 4and YbNiIn 4were grown via arc melting in argon
atmosphere. Due to the low boiling temperature of ytterbium,
2469-9950/2018/98(21)/214115(7) 214115-1 ©2018 American Physical SocietyZ. E. BRUBAKER et al. PHYSICAL REVIEW B 98, 214115 (2018)
a 5% excess of ytterbium was necessary to account for the
mass loss during melting. Each sample was melted and flippedsix times. Samples were subsequently annealed at 625
◦Cf o r
ten days and powder x-ray-diffraction (PXRD) measurementswere performed both before and after the annealing procedure.There was no detectable mass loss during the annealing pro-cess. Powder x-ray-diffraction analysis indicated phase puri-ties greater than 96% for YbNiGa
4and YbNiIn 4, with YbGa 2
and YbIn 3being the main impurity phases, along with less
than 1% of Yb 2O3. The Yb compounds order in the Cmcm
structure, with Yb and Ni occupying the 4 cWyckoff position
and the Ga or In atoms occupying the 4 a,4c, and 8 fWyckoff
positions. The only intermediate composition we successfullysynthesized was YbNiGa
3In, with a phase purity of 93%.
Refinement of YbNiGa 3In suggests that indium has a strong
site preference and almost exclusively occupies the 4 asite [ 8].
Once this site is fully occupied, the pseudobinary alloy rangeappears to be truncated, evidenced by nominal compositionsof YbNiGaIn
3and YbNiGa 2In2not yielding a significant
phase fraction of the desired 114 phase. Previous reports sug-gest YbNiAl
4can be grown via similar methods, but attempts
via (1) arc melting, (2) tetra-arc melting, (3) induction melt-ing, and (4) Al-flux growth with a variety of starting composi-tions and annealing procedures failed to provide specimens ofsatisfactory quality [ 9]. In the cases of (1) and (2), Yb
3Ni5Al19
was the dominant impurity phase, typically 15%–25%, with
less than 3% YbAl 3. Annealing did not improve phase purity
and in some instances increased the 3-5-19 phase. Attempts togrow YbNiAl
4via (3) induction melting as well as attempts to
anneal ingots from (1) in Yb atmosphere resulted in a seriesof new peaks appearing, which are comparable in intensity tothe 114 peaks and have not been successfully indexed [ 8]. In
the case of (4) we grew only single crystals of Yb
3Ni5Al19,
consistent with previous results [ 10].
Ambient pressure x-ray-diffraction (XRD) patterns were
collected with a standard D8 diffractometer and were usedto determine the lattice parameters and atomic positions.Refinement of the ambient pressure data suggests all sites arewithin 2% of being fully occupied. Pressure-dependent XRDstudies were performed at sector 16-BMD of the AdvancedPhoton Source (APS) using a 30-keV x-ray beam. Powderedsamples were loaded into a diamond anvil cell (DAC) witha rhenium gasket and pressurized with neon. The pressurewas determined via copper powder mixed with the samplesand confirmed with ruby spectroscopy at select pressures[11]. The XRD patterns were collected via an area detector
and converted to powder patterns using
FIT2D[12]. A CeO 2
sample was used as a calibrant to determine the instrument pa-
rameters used in the fitting, which was performed using GSAS -I
[13,14]. The instrument parameters and atomic positions
were then held constant for all subsequent refinements; onlylattice parameters, broadening due to strain, and preferredorientation parameters were allowed to vary under pressure.Because of peak broadening that occurs under pressure, somepeaks merge and become difficult to index under pressure.If merging peaks prevent an adequate fit, they are removedfor the pressure space in which they overlap. In instanceswhere including and excluding the peaks over various pres-sure ranges cause inconsistencies in the lattice parameters,they are removed for all XRD spectra.X-ray absorption spectroscopy (XAS) at the ytterbium
L
IIIedge was performed at sector 16-IDD of the APS using
partial fluorescence yield (PFY) from the Lαemission line.
The incident energy was scanned with a Si (111) fixed-exit, double-crystal monochromator and the L
αemission was
recorded using three Si (400) analyzers. Samples were loadedinto a DAC with a beryllium gasket for XAS measurementsand mineral oil was used as a pressure transmitting medium.Pressure was measured via ruby-fluorescence spectroscopy.For both diffraction and spectroscopy measurements, pressurewas increased using a gas-driven membrane. Resonant x-ray-emission spectroscopy (RXES) was fit using the Kramers-Heisenberg formula for photon-atom scattering [ 15,16]. The
Yb volume was found by calculating the V oronoi cell, i.e., thevolume closer to one atom than any other. For this calculation,the atomic positions are held constant under pressure and theatoms are weighted by covalent radii.
All calculations are performed within the framework of
DFT and similar to a recent study on the rare-earth elementalmetals [ 17]. The necessary assumption for the unknown elec-
tron exchange and correlation functional is chosen to be thatof generalized gradient approximation. The implementation isdone for a full-potential linear muffin-tin orbital (FPLMTO)method [ 18]. The orbital polarization (OP) is included in the
FPLMTO as a parameter-free scheme where an energy termproportional to the square of the orbital moment is addedto the total energy functional to account for intra-atomicinteractions. It is an approximate method that is analogous tothe mean-field approximation for the spin-polarization energy.Anf
natomic configuration involves intra-atomic interactions
such as (vector model) si·sjandli·lj(electron ispin and
angular momenta). Here we replace the energy associatedwith the angular momenta −
1
2/summationtextli·ljwith a mean-field
expression −1
2(/summationtextlz
i)(/summationtextlz
j)(zcomponent of vector l). This
term is proportional to L2in analogy to the Stoner energy
for spin polarization −1
2(/summationtextsz
i)(/summationtextsz
j), which is proportional
toM2.H e r e LandMare the total orbital and spin moments,
respectively. In the OP scheme this then provides for a one-electron eigenvalue shift proportional to −Lm
l(for each
stateml) that enhances the orbital polarization over the spin-
orbit coupling only case. The connection between OP andthe local-density approximation plus Hubbard-type Coulombinteraction (LDA +U) methodologies was discussed recently
[19]. One distinct advantage with the OP scheme over the
LDA+U method is that the former does not depend on a
parameter whose pressure dependence is unknown.
III. RESULTS
A. Density functional theory
The results of the DFT calculations are shown in Fig. 1and
compared to previously published work. The DFT predictsa continuous increase in Yb valence with decreasing inter-atomic distance and reproduces the general trend of the pre-viously published experimental data. The DFT calculations,however, suggest a stronger dependence on Yb-Yb spacingthan observed, thus predicting a smaller valence for YbNiIn
4,
but predicting a larger than observed valence for YbNiGa 4,
214115-2PRESSURE-DEPENDENT INTERMEDIATE V ALENCE … PHYSICAL REVIEW B 98, 214115 (2018)
FIG. 1. Summary of DFT calculations and previously published
work. The DFT reproduces the general behavior, but implies astronger dependence on Yb-Yb spacing.
implying that electron correlation effects beyond what is
included in the present DFT calculations may be responsible.
B. Structural studies
X-ray-diffraction measurements were performed on
YbNiGa 4and YbNiIn 4up to a pressures of 63 and 45 GPa,respectively. Figure 2shows that YbNiGa 4contracts without
any sign of a structural transition and can be well describedby the Birch-Murnaghan (BM) equation of state (EOS)withB=76.7 GPa and B
/prime=5.5[20]. YbNiIn 4shows a
plateau in volume between 12 and 14 GPa, which is causedby plateaus in the aandbaxes in this pressure range [Fig.
2(d)]. Below 12.5 GPa, the BM EOS yields B=54.2G P a
andB
/prime=7.0, and the high-pressure region above 17 GPa
yields B=63 GPa and B/prime=5.0, values that more closely
resemble those of YbNiGa 4. To better determine the origin of
the plateau near 13 GPa, we have performed a linearization ofthe BM EOS as described in Ref. [ 21] and plot the resulting
reduced pressure Hvs the Eulerian strain f
Ein Fig. 3.
Plotting the reduced pressure vs Eulerian strain should belinear for any stable compound, while a change in slope maybe indicative of an electronic topological transition (ETT).As shown in Fig. 3,Y b N i I n
4shows a sudden spike in the
reduced pressure at a Eulerian strain of 0.05 (correspondingto 12.5 GPa), which is accompanied by a change in slope,which may indicate the presence of an ETT.
C. Valence measurements
1. X-ray-absorption spectroscopy
X-ray-absorption spectroscopy is sensitive to the valence
of the Yb ions because the 4 fstates (4 f13and 4f14) both
FIG. 2. The PXRD spectra at select pressures for (a) YbNiGa 4and (b) YbNiIn 4. In (b) the spectra shown for 29.8 and 39.4 GPa were
acquired for a measurement separate from those presented for pressures below, which accounts for some of the difference in intensities of the
sample peaks. (c) V olume contraction as a function of pressure. (d) Contraction of lattice parameters as a function of pressure. The aandb
axes appear to contract similarly among the compounds, but the caxis displays significantly different behavior. The uncertainties are taken
from the uncertainty in GSAS fitting and are generally smaller than the markers. The solid lines in (c) represent a fit to the Birch-Murnaghan
equation of state. The solid lines in (d) are guides to the eye, and the aandbaxes are offset by 0.04 and 0.02, respectively.
214115-3Z. E. BRUBAKER et al. PHYSICAL REVIEW B 98, 214115 (2018)
FIG. 3. Reduced pressure Hplotted against Eulerian strain for
YbNiIn 4. The YbNiIn 4displays a sharp spike as well as a change in
slope just above a Eulerian strain of 0.05, suggestive of an ETT. The
solid black lines represent linear fits below a Eulerian strain of 0.05
and above 0.06. The shaded area represents the transition region.
experience different screening. Each valence state (Yb3+and
Yb2+) will result in a distinct absorption peak in the XAS
spectra, which are separated by approximately 8–12 eV . Bycalculating the weighted average of the peak intensities, theeffective Yb valence can be determined. The XAS spectracan be fit by describing each valence state with a Gaussianand error function. As reported in several papers studying Ybvalence in other materials, we observed a splitting of the Yb
3+
peak, which is likely due to the crystal field splitting of the
unoccupied 5 dstates [ 22–25]. Previous work measuring the
valence of YbNiGa 4and YbNiIn 4was performed in transmis-
sion mode and lacked the resolution to fit both Yb3+peaks,
resulting in a minor difference in the determined valence andpressure dependence thereof compared to the work reportedherein [ 6,7]. Figure 4shows the details of our fit for the
ambient pressure data, as well as several XAS spectra at selectpressures.
For YbNiGa
4there exists a clear decrease in the intensity
of the Yb2+peak and an increase in intensity of the Yb3+peak
up to a pressure of 25 GPa. Above 25 GPa, the ratio of absorp-tion peak to fluorescence decreases, but the valence remainslargely unchanged. For YbNiIn
4the ambient pressure mea-
surement reveals a larger contribution from the fluorescentregion than the subsequent pressure measurements, resultingin an apparent increase of both the Yb
2+and Yb3+peaks
from ambient to 6.8 GPa. Nonetheless, the ratio of amplitudesof these valence peaks results in an increase in valence withpressure, following the trend observed for all the measuredpressures. Summarizing, the valence determination via XASand adding previous valence determinations for YbNiGa
4
yields Fig. 5[7].
The Yb valence of YbNiGa 4increases up to about 20–
25 GPa, at which point the valence saturates at n=2.68. The
Yb valence in YbNiIn 4may be approaching saturation at the
highest measured pressures, but there is also a peak in valenceclose to 10 GPa. This peak is likely another manifestation ofthe ETT which was observed in the structural measurementsand could be the result of the changing electronic density ofstates near the ETT.
FIG. 4. X-ray-absorption spectra at select pressures and the fit
functions for (a) YbNiGa 4and (b) YbNiIn 4. The spectra are normal-
ized to an edge jump of unity. The solid lines of the fit correspondto the Gaussian functions associated with the valence peaks and
the dashed lines are their respective error functions accounting
for entering the fluorescent region. The Yb
2+and Yb3+peaks are
indicated. With increasing pressure, the Yb3+peak gains intensity,
while the Yb2+peak loses intensity.
2. Resonant x-ray-emission spectroscopy
Resonant x-ray-emission spectroscopy is a powerful tool
for fully describing the valence state of a given material,which scans the emission energy in addition to the incidentenergy. Converting the emitted energy to transferred energyand combining this into a single plot results in the RXESspectra shown in Fig. 6for YbNiGa
4and in [ 8]f o rt h e
single pressure measured for YbNiIn 4. As in the case of PFY
measurements, the amplitudes of the absorption peaks allowfor determination of the valence.
YbNiGa
4begins with a rather broad peak due to the
overlap of the three distinct contributions of the measuredvalence peaks, but with increasing pressure the valence state isshifted away from the Yb
2+and towards the Yb3+state. This
results in only a weak Yb2+structure remaining at 42 GPa.
The overall trend of the valence determined from RXES andXAS is the same for YbNiGa
4. The valence peaks in YbNiIn 4
have a larger separation resulting in more distinct peaks andagreeing with the valence determined via XAS.
IV . DISCUSSION
While both the structural and spectroscopic data of
YbNiIn 4are suggestive of an ETT, it is important to note
214115-4PRESSURE-DEPENDENT INTERMEDIATE V ALENCE … PHYSICAL REVIEW B 98, 214115 (2018)
FIG. 5. The Yb valence as a function of pressure determined
via XAS for (a) YbNiGa 4and (b) YbNiIn 4. The inset shows the
XAS spectra of YbNiIn 4around 10 GPa. The valence for YbNiGa 4
increases up to about 25 GPa, above which the valence appears tobe saturated. For YbNiIn
4, the valence reveals a peak near 10 GPa,
consistent with the ETT proposed from structural results. Aside from
this peak, the valence increases steadily up to the maximum pressure
P=27 GPa, though the highest measured pressure points suggest
that the valence may be reaching saturation. The uncertainties were
calculated from weighted fitting in IGOR .
that previous work measuring ETTs did not observed a peak
in reduced pressure or conversely did not show a plateau involume near the ETT [ 21,26,27]. We speculate that this is
due to the dual nature of the Yb 4 felectrons, which display
both local and itinerant character in these intermediate valencestates observed in YbNiIn
4. Previous pressure-induced ETT
has been identified in weakly correlated itinerant systems, im-plying compressibility changes arising only from the bondingchanges driven by subtle changes near the Fermi level. In thecase of YbNiIn
4, the dual nature of the 4 f-electron subsystem
yields consequences not only for the electronic structure nearthe Fermi level, as with the itinerant systems, but also for thelocal corelike states, which have ramifications for the ionicvolume of the Yb atoms independent of the physics at theFermi level. The physics that drives the dual nature of the 4 f
electrons in YbNiIn
4inherently couples the local part of the
wave function to the ETT, which may be expected to yieldmore pronounced effects on compressibility and volume thantypically seen in weakly correlated, itinerant systems.
FIG. 6. (a)–(d) The RXES spectra and (e)–(h) the corresponding
fit for YbNiGa 4. With increasing pressure, the low energy 2 +peak
decreases in intensity. The gray lines correspond to the XAS PFYscans described above. Intensities are normalized to the maximum
intensity of the 3 +peak of the experimental data.
Figure 7(a) summarizes the valence determined via each
of the described methods and includes the valence determinedfor YbNiAl
4from Ref. [ 5]. YbNiGa 4appears to reach satu-
ration at n=2.68 above P=25 GPa and surpasses the Yb
valence measured in YbNiAl 4. Both YbNiGa 4and YbNiIn 4
have comparable Yb valence at ambient pressure, but inYbNiGa
4the valence appears to be more sensitive to pressure.
As shown in Fig. 7(b), the unit cell volume fails to describe the
overlapping region of these materials, though both materialsindividually reveal the expected trend of increasing valencewith decreasing unit cell volume.
Figure 8shows Yb valence vs Yb-Yb spacing, which
shows behavior similar to that for the unit cell volume.Most of the data for YbNiIn
4, the high-pressure region of
YbNiGa 4, and YbNiAl 4at ambient pressure appear to fol-
low a smooth valence vs Yb-Yb spacing curve. However,in the region where these compounds have similar Yb-Ybspacing, the Yb valence reveals a precipitous drop, indicatingthat Yb-Yb spacing does not capture the entirety of theunderlying physics. Figure 8also includes DFT calculations
for the Yb valence in the YbNiGa
4and YbNiIn 4systems.
While the zero-pressure value of valence in YbNiIn 4predicted
by the DFT is lower than the experiments, the pressure-dependent trend of the valence as predicted by the DFT isin good agreement with the experimental observations. For
214115-5Z. E. BRUBAKER et al. PHYSICAL REVIEW B 98, 214115 (2018)
FIG. 7. Summary of valence measurements of this system plot-
ted against (a) pressure and (b) unit cell volume. With increasingpressure, the Yb valence increases, though in YbNiGa
4the Yb
valence appears to reach saturation above 25 GPa. Unit cell volume
is insufficient to fully describe the Yb-114 system.
YbNiGa 4the behavior is also reproduced well for smaller
lattice spacings, while the drastic drop at the larger latticespacings is not predicted by the theory. It is particularly the
FIG. 8. The Yb valence plotted against the Yb-Yb spacing and
DFT calculations for Yb valence vs Yb spacing. The DFT does not
reproduce the sharp drop when transitioning from YbNiIn 4under
pressure to YbNiGa 4under ambient conditions.
FIG. 9. The Yb valence plotted against the Yb volume. There is
no convincing trend and YbNiAl 4does not fit within this framework.
measured valence at the largest Yb-Yb spacing (4.07 Å) that
deviates from theory (2.44 vs 2.54) and the reason is notclear. We speculate that electron correlation effects beyondwhat is included in the present DFT calculations may bethe cause.
As an attempt to account for the effect of substituting In
and Ga, we calculated the Yb volume, i.e., the space availableto the Yb atoms, for each measured pressure. The results areshown in Fig. 9. Consideration of the Yb volume does not
provide a satisfactory result, and YbNiAl
4does not fit into this
scheme. In the case of Yb volume, the In and Ga variants arecomparable and exhibit similar slopes, but this still does notfully capture the evolution of the Yb valence. This, combinedwith the Yb spacing and unit cell volume data, suggests thatstructural parameters alone are insufficient to fully describethe valence behavior of this system. This implies that the Ybvalence is also sensitive to the chemical environment, i.e., thehybridization between the Yb 4 fand group IIIb pstates, an
effect not captured by DFT calculations.
V . CONCLUSION
In summary, the Yb valence in the Yb-114 system can be
readily modified by pressure, but that valence is not simplydescribed by nearest-neighbor bond distances. By using par-tial fluorescence yield measurements, we have improved theresolution of the valence determination in YbNiGa
4, which
reveals a steady increase in valence from n=2.44 up to
n=2.68 near P=25 GPa, saturating shortly thereafter. The
Yb valence of YbNiIn 4shows similar overall behavior, but we
have also observed a sharp valence enhancement in YbNiIn 4
just above 10 GPa. This peak coincides with a plateau involume, which we speculate is the result of an electronictopological transition. The Yb valence is most closely relatedto the Yb-Yb spacing in this structure, though this parameteris insufficient to describe the valence across the entirety of theYbNi(Ga ,In)
4system. The hybridization resulting from the
Yb-In, Yb-Ga, and Yb-Al bonds appears to be dependent onatomic species and not just the natural bond lengths set byionic sizes.
214115-6PRESSURE-DEPENDENT INTERMEDIATE V ALENCE … PHYSICAL REVIEW B 98, 214115 (2018)
ACKNOWLEDGMENTS
This work was performed under LDRD (Tracking Code
18-SI-001) and under the auspices of the U.S. Department ofEnergy by Lawrence Livermore National Laboratory (LLNL)under Contract No. DE-AC52- 07NA27344. Part of the fund-ing was provided through the LLNL Livermore GraduateScholar Program. Portions of this work were performed atHPCAT (Sector 16), Advanced Photon Source (APS), Ar-gonne National Laboratory. HPCAT operations are supportedby DOE-NNSA’s Office of Experimental Sciences. The Ad-vanced Photon Source is a U.S. Department of Energy (DOE)Office of Science User Facility operated for the DOE Officeof Science by Argonne National Laboratory under ContractNo. DE-AC02-06CH11357. Beamtime was generously pro-vided through the GUP system and through CDAC. Thismaterial was based upon work supported by the NationalScience Foundation under Grant No. NSF DMR-1609855.
R.E.B. and K.H. performed crystal synthesis experiments atthe National High Magnetic Field Laboratory, which is sup-ported by National Science Foundation Cooperative Agree-ments No. DMR-1157490 and No. DMR-1644779 and thestate of Florida. R.E.B. and K.H. were supported in part bythe Center for Actinide Science and Technology, an EnergyFrontier Research Center funded by the U.S. DOE, Officeof Science, BES, under Award No. DE-SC0016568. D.J.Cacknowledges support from the U.S. Department of Energy,Office of Science, Office of Workforce Development forTeachers and Scientists, Office of Science Graduate StudentResearch program, administered by the Oak Ridge Institutefor Science and Education for the DOE under Contract No.DESC0014664. J.P. and D.J.C. acknowledge support fromthe Gordon and Betty Moore Foundation’s EPiQS Initiativethrough Grant No. GBMF4419.
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214115-7 |
PhysRevB.74.024408.pdf | Orbital magnetization in crystalline solids: Multi-band insulators, Chern insulators, and metals
Davide Ceresoli,1T. Thonhauser,2David Vanderbilt,2and R. Resta3
1Scuola Internazionale Superiore di Studi Avanzati (SISSA/ISAS) and DEMOCRITOS, via Beirut 2-4, 34014 Trieste, Italy
2Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
3Dipartimento di Fisica Teorica, Università di Trieste and DEMOCRITOS, strada Costiera 11, 34014 Trieste, Italy
/H20849Received 7 December 2005; revised manuscript received 6 March 2006; published 12 July 2006 /H20850
We derive a multi-band formulation of the orbital magnetization in a normal periodic insulator /H20849i.e., one in
which the Chern invariant, or in two dimensions /H208492D/H20850the Chern number, vanishes /H20850. Following the approach
used recently to develop the single-band formalism /H20851Thonhauser, Ceresoli, Vanderbilt, and Resta, Phys. Rev.
Lett. 95, 137205 /H208492005 /H20850/H20852, we work in the Wannier representation and find that the magnetization is comprised
of two contributions, an obvious one associated with the internal circulation of bulklike Wannier functions inthe interior and an unexpected one arising from net currents carried by Wannier functions near the surface.Unlike the single-band case, where each of these contributions is separately gauge invariant, in the multi-bandformulation only the sumof both terms is gauge invariant. Our final expression for the orbital magnetization
can be rewritten as a bulk property in terms of Bloch functions, making it simple to implement in modern codepackages. The reciprocal-space expression is evaluated for 2D model systems and the results are verified bycomparing to the magnetization computed for finite samples cut from the bulk. Finally, while our formal proofis limited to normal insulators, we also present a heuristic extension to Chern insulators /H20849having nonzero Chern
invariant /H20850and to metals. The validity of this extension is again tested by comparing to the magnetization of
finite samples cut from the bulk for 2D model systems. We find excellent agreement, thus providing strongempirical evidence in favor of the validity of the heuristic formula.
DOI: 10.1103/PhysRevB.74.024408 PACS number /H20849s/H20850: 75.10.Lp, 73.20.At, 73.43. /H11002f
I. INTRODUCTION
During the last decade, charge and spin transport phenom-
ena in magnetic materials and nanostructures have attractedmuch interest due to their important role for spintronicdevices.
1An adequate description of magnetism in these ma-
terials, however, should not only include the spin contribu-tion, but also should account for effects originating in theorbital magnetization. In light of this, it is surprising that thetheory of orbital magnetization has long remained underde-veloped. Earlier attempts to develop such a theory usedlinear-response methods, which allow calculations of magne-tization changes,
2–5but not of the magnetization itself.
Just recently, a new approach using Wannier functions
/H20849WFs /H20850has been proposed,6,7which nicely parallels the analo-
gous case of the electric polarization. The primary difficultyin both cases is that the position operator ris not well de-
fined in the Bloch representation. Since WFs are exponen-tially localized in an insulator, this difficulty disappears if theproblem is reformulated in the Wannier representation. Forthe polarization, this approach lead to the development of themodern theory of polarization in the early 1990s.
8,9Simi-
larly, in the case of the orbital magnetization, where the cir-culation operator r/H11003vis ill defined in the Bloch representa-
tion, the Wannier representation was used to derive a theoryfor the orbital magnetization of periodic insulators.
7
While the formalism developed in Ref. 7lays a firm foun-
dation for the orbital magnetization, its application is limitedto certain systems, such as single-band models and insula-tors. In this paper we expand the applicability to a muchwider class of systems by developing a corresponding multi-band formalism, essential for most “real” materials. This ex-tension is nontrivial and the corresponding proof of gaugeinvariance is much more complex than for the single-band
case. We proceed in two steps. First, we carry out a deriva-tion for the case of an insulator with zero Chern invariant.Second, we give heuristic arguments for an extension of ourformalism to metals and Chern insulators, i.e., systems witha nonzero Chern invariant, arriving at a formula identical tothat proposed by Xiao, Shi, and Niu
10on the basis of semi-
classical arguments. Chern insulators have been introducedinto the theoretical literature by means of model Hamilto-nians in two dimensions /H208492D/H20850which break time-reversal
/H20849TR/H20850symmetry without breaking translational symmetry,
11
i.e., maintaining a vanishing macroscopic magnetic field. De-
spite the absence of a macroscopic field, Chern insulatorsshare several properties with quantum-Hall systems, mostnotably the quantization of the transverse conductivity in2D.
11To the best of our knowledge, there is no known ex-
perimental realization of a Chern insulator /H20849in zero field /H20850in
either 2D or 3D, and the search for such a system remains afascinating challenge.
Our extensions to metals and Chern insulators are heuris-
tic and notbased on an analytical proof. The fact that our
final formula is identical to the one derived from the semi-classical wave packet treatment
10is reassuring, but neither of
these approaches can yet be said to constitute a “derivation”of the formula in the fully quantum context. Nevertheless,we provide strong numerical evidence of their validity, thusposing a theoretical challenge: how to provide an analyticproof of the heuristic formula, beyond the range of the semi-classical approximation, for both the metallic and Chern-insulating cases.
Before proceeding, we emphasize that the present work
only addresses the question of how to compute the orbitalmagnetization for a given independent-particle Hamiltonian.Many interesting questions remain concerning which flavorPHYSICAL REVIEW B 74, 024408 /H208492006 /H20850
1098-0121/2006/74 /H208492/H20850/024408 /H2084913/H20850 ©2006 The American Physical Society 024408-1of density-functional theory /H20849DFT /H20850or which exchange-
correlation /H20849XC/H20850functional might give the most accurate or-
bital magnetization. While exact Kohn-Sham /H20849KS/H20850density
/H20849or spin-density /H20850functional theory is guaranteed to yield the
correct charge /H20849or spin /H20850density,12there is no reason to expect
it to yield the correct orbital currents. The orbital magnetiza-tion, being defined in terms of surface currents, is not guar-anteed to be correct either. A prescription that seems moresuited to the present situation is that of Vignale and Rasolt,
13
in which the spin-labeled density and current /H20853n/H9268/H20849r/H20850,j/H9268/H20849r/H20850/H20854
are connected to corresponding scalar and vector potentials
/H20853V/H9268/H20849r/H20850,A/H9268/H20849r/H20850/H20854. However, it is an open question whether
an approximate Vignale-Rasolt XC functional exists that can
give improved values of magnetization in practice. Whiletime-dependent density functional theory is more devel-oped,
14this theory only establishes a connection between
n/H20849r,t/H20850and V/H20849r,t/H20850, and a knowledge of n/H20849r,t/H20850is only suf-
ficient to determine the longitudinal part of j/H20849r,t/H20850, not
the transverse part upon which the orbital magnetization
depends. An alternative approach worthy of explorationis time-dependent current-density functional theory/H20849TDCDFT /H20850,
15in which /H20853n/H20849r,t/H20850,j/H20849r,t/H20850/H20854is connected to
/H20853V/H20849r,t/H20850,A/H20849r,t/H20850/H20854. However, the present problem is essentially
a static problem, and it is therefore unclear whether TD-
CDFT would provide any practical advantage over theVignale-Rasolt theory. Finally, it is worth remembering thateven in standard DFT, the mapping from interacting densityto non-interacting potential is sometimes pathological /H20849e.g., a
KS metal can represent an interacting insulator /H20850.I nt h e
present work, we bypass all these interesting issues, and onlyconsider how to compute the magnetization for a givenKohn-Sham Hamiltonian arising from some unspecified ver-sion of DFT in the context of broken TR symmetry.
We have organized this paper as follows. In Sec. II we
derive the multi-band theory of orbital magnetization in crys-talline solids. After some definitions and generalities, westart by considering the orbital magnetization of a finitesample. The resulting expression is then transformed to re-ciprocal space and its gauge invariance is demonstrated. Wethen give a heuristic extension of our formalism to metalsand Chern insulators. In Sec. III numerical results for theorbital magnetization are presented for several different sys-tems. We conclude in Sec. IV. Some details concerning thefinite-difference evaluation of the magnetization and certainproperties of the nonAbelian Berry curvature are deferred totwo appendixes.
II. THEORY
A. Generalities
Our basic starting point is a single-particle KS
Hamiltonian12having the translational symmetry of the crys-
tal, but having no TR symmetry: as said above, translationalsymmetry of the Hamiltonian implies vanishing of the mac-roscopic magnetic field. There may, however, be a micro-scopic magnetic field Bthat averages to zero over the unit
cell, and we assume that a particular magnetic gauge hasbeen chosen once and for all to represent this magnetic field.Wave vector kis a good quantum number under these con-
ditions. This could be realized, for example, in systems inwhich the TR breaking comes about through the spontaneousdevelopment of ferromagnetic order or via spin-orbit cou-pling to a background of ordered local moments.
11,16–19No-
tice that we carefully avoid referring to an externally appliedfield; such concept is legitimate only for a finite sample,free-standing in vacuo. Indeed, for a finite sample, the rela-tionship between the externally applied field and the “inter-nal” /H20849or screened /H20850one depends on the sample shape. For an
extended sample in the thermodynamic limit, the only legiti-mate and measurable field is the screened Bfield which is
present inside the material. In the present work, the cell-average of this field is assumed to vanish.
As usual, we let
/H9280nkand /H20841/H9274nk/H20856be the Bloch eigenvalues
and eigenvectors of H, respectively, and unk/H20849r/H20850=e−ik·r/H9274nk/H20849r/H20850
be the corresponding eigenfunctions of the effective Hamil-
tonian Hk=e−ik·rHeik·r. We choose to normalize them to one
over the crystal cell of volume /H9024.
The notation is intended to be flexible as regards the spin
character of the electrons. If we deal with spinless electrons,then nis a simple index labeling the occupied Bloch states;
factors of two may trivially be inserted if one has in minddegenerate, independent spin channels. In the context of thelocal spin-density approximation in which spin-up and spin-down electronic states are separate eigenstates of spin-up andspin-down Hamiltonians, one may let nrange over both sets
of bands, but with the understanding that inner products ormatrix elements between spin-up and spin-down bands al-ways vanish. Of more realistic interest here is the case of afully noncollinear treatment of the magnetism, as for the caseof a Hamiltonian containing the spin-orbit operator. In thiscase, nlabels bands that are neither purely spin up nor spin
down, /H20841u
nk/H20856must be understood to be a spinor wave function,
and the contraction over spin degrees of freedom is under-stood to be included in the definition of inner products suchas/H20855u
nk/H20841un/H11032k/H20856and matrix elements such as /H20855unk/H20841Hk/H20841un/H11032k/H20856.
A key issue in the present work is the additional “gauge
freedom” in which the occupied Bloch orbitals at fixed kare
allowed to be transformed among themselves by an arbitraryunitary transformation. In fact, any KS ground-state elec-tronic property should be uniquely determined by the sub-
space of occupied orbitals as represented by the one-particle
density matrix; the occupied orbitals just provide a conve-nient orthonormal representation for this subspace. More-over, when it comes to the formulation of Wannier functions/H20849WFs /H20850for composite energy bands, the nth WF is generally
not simply the Fourier transform of the nth band of Hamil-
tonian eigenvectors, but instead, of a manifold of states /H20841u
nk/H20856
which are related to the eigenstates by a k-dependent unitary
transformation.20Thus, in what follows, we allow /H20841unk/H20856to
refer to this generalized interpretation of the nklabels unless
otherwise specified. In addition, we introduce a generalized“energy matrix”
E
nn/H11032k=/H20855unk/H20841Hk/H20841un/H11032k/H20856, /H208491/H20850
which reduces to Enn/H11032k=/H9280nk/H9254nn/H11032in the special case of the
“Hamiltonian gauge” in which the /H20841unk/H20856areeigenstates of
Hk.CERESOLI et al. PHYSICAL REVIEW B 74, 024408 /H208492006 /H20850
024408-2A key quantity characterizing a three-dimensional KS in-
sulator in absence of TR symmetry is the /H20849vector /H20850Chern
invariant21
C=i
2/H9266/H20885
BZdk/H20858
n/H20855/H11509kunk/H20841/H11003/H20841/H11509kunk/H20856, /H208492/H20850
with the usual meaning of the cross product between three-
component bra and ket states. Here and in the following thesum is over the occupied n’s only, the integral is over the
Brillouin zone /H20849BZ/H20850, and
/H11509k=/H11509//H11509k. The Chern invariant is
gauge-invariant in the above generalized sense /H20849as will be
shown in Sec. II D /H20850and—for a three-dimensional crystalline
system—is quantized in units of reciprocal-lattice vectors G.
In Secs. II B–II D we assume that we are working with in-sulators with zeroChern invariant; the more general case will
be discussed only later in Secs. II E and II F.
Owing to the zero-Chern-invariant condition, the Bloch
orbitals can be chosen so as to obey /H20841
/H9274nk+G/H20856=/H20841/H9274nk/H20856/H20849the so-
called periodic gauge /H20850, which in turn warrants the existence
of WFs enjoying the usual properties. /H20849For a Chern insulator,
it is not clear whether a Wannier representation exists. /H20850We
shall denote as /H20841nR/H20856thenth WF in cell R. These WFs are
related via
/H20841unk/H20856=/H20858
Reik·/H20849R−r/H20850/H20841nR/H20856,
/H20841nR/H20856=/H9024
/H208492/H9266/H208503/H20885
BZdkeik·/H20849r−R/H20850/H20841unk/H20856, /H208493/H20850
to the Bloch-like orbitals /H20841unk/H20856defined in the generalized
sense discussed just above Eq. /H208491/H20850.
B. The magnetization of a finite sample
We start by considering a macroscopic sample of Nccells
/H20849with Ncvery large but finite /H20850cut from a bulk insulator,
having Nboccupied bands, with “open” boundary conditions.
The finite system then has N/H11229NcNboccupied KS orbitals.
Suppose we perform a unitary transformation upon them, byadopting some localization criterion. By invariance of thetrace the orbital magnetization of the finite system is writtenin terms of the localized orbitals /H20841w
i/H20856as
M=−1
2c/H9024Nc/H20858
i=1N
/H20855wi/H20841r/H11003v/H20841wi/H20856, /H208494/H20850
where the velocity is defined as
v=i/H20851H,r/H20852. /H208495/H20850
In the case of density-functional implementations, it should
be noted that vmay differ from p/mbecause of the presence
of microscopic magnetic fields /H20849which introduce p·Aterms
in the Hamiltonian /H20850, spin-orbit interactions, or semilocal or
nonlocal pseudopotentials. In the case of tight-bindingimplementations, the matrix representations of Handrare
assumed to be known /H20849ris normally taken to be diagonal /H20850in
the tight-binding basis and vis then defined through Eq. /H208495/H20850.
We divide the sample into an “interior” and a “surface”
region, in such a way that the latter occupies a nonextensivefraction of the total sample volume in the thermodynamic
limit. The orbitals /H20841w
i/H20856which are localized in the interior
region converge exponentially to the WFs /H20841nR/H20856of the peri-
odic infinite system; for instance, if the Boys22localization
criterion is adopted, they become by construction theMarzari-Vanderbilt
20maximally localized WFs. Therefore
the interior is composed of an integer number Niof replicas
of a unit cell containing NbWFs each. Note that this choice
is not unique; there is freedom both to shift all of the R’s by
some constant vector /H20849effectively changing the origin of the
unit cell /H20850, or to shift any one of the WFs by a lattice vector,
or to carry out a unitary remixing of the bands. We insistonly that some consistent choice is made once and for all.
The remaining N
slocalized orbitals residing in the surface
region need not resemble bulk WFs; we denote them as /H20841ws/H20856
and continue to refer to them as “WFs” in a generalizedsense. We thus partition the entire set of NWFs of the finite
sample into N
iNbones belonging to the interior and Nsones
in the surface region. Correspondingly, the contribution tothe orbital magnetization Mcoming from the interior orbitals
will be denoted as M
LC/H20849for “local circulation” /H20850, while that
arising from the surface orbitals will be referred to as MIC
/H20849for “itinerant circulation” /H20850. We will take the thermodynamic
limit in such a way that Nsgrows more slowly with sample
size than does Ni, so that Ns/Ni→0. Because of the ambigu-
ities discussed in the previous paragraph, we do not expectM
LCand MICto be separately gauge invariant. However,
their sum, Eq. /H208494/H20850, must be gauge invariant.
Since the interior orbitals are bulklike, we have, following
Eq. /H208494/H20850,
MLC=−1
2c/H9024Nc/H20858
nR/H20855nR/H20841/H20849r−R/H20850/H11003v/H20841nR/H20856, /H208496/H20850
where the number of Rvectors in the sum is smaller than Nc
only by a nonextensive fraction, and we have used that
/H20858n/H20855nR/H20841v/H20841nR/H20856=0. Because of the zero-Chern-invariant condi-
tion the WFs enjoy the usual translational symmetry, and we
finally find that
MLC=−1
2c/H9024/H20858
n/H20855n0/H20841r/H11003v/H20841n0/H20856/H20849 7/H20850
in the thermodynamic limit.
We now consider the contribution from the Nssurface
orbitals, whose centers we denote as rs=/H20855ws/H20841r/H20841ws/H20856:
MIC=−1
2c/H9024Nc/H20858
s=1Ns
/H20849/H20855ws/H20841/H20849r−rs/H20850/H11003v/H20841ws/H20856+rs/H11003/H20855ws/H20841v/H20841ws/H20856/H20850.
/H208498/H20850
The first term in parenthesis clearly vanishes in the thermo-
dynamic limit, while the second term—owing to the pres-ence of the “absolute” coordinate r
s—does not. At first sight,
this second term in MICappears to depend on surface details;
instead, we are going to prove that even this term can berecast in terms of bulk Wannier functions. Remarkably, both
M
LCandMICare genuine bulk properties in the thermody-
namic limit, and can eventually be evaluated as BZ integrals.ORBITAL MAGNETIZATION IN CRYSTALLINE ¼ PHYSICAL REVIEW B 74, 024408 /H208492006 /H20850
024408-3We consider a surface facing in the + xˆdirection, and iden-
tify a surface region given by x/H11022x0as in Fig. 1. There is
then a contribution to the macroscopic surface current K
flowing at the surface that is given by
K=−1
A/H20858
s/H11032/H11032/H20855ws/H11032/H20841v/H20841ws/H11032/H20856, /H208499/H20850
where the primed sum is taken over the surface WFs whose
yzcoordinates are within one surface unit cell of area A.
Because /H20855ws/H11032/H20841v/H20841ws/H11032/H20856decays exponentially to zero with dis-
tance from the surface, it is straightforward to capture the
entire surface current by letting the width of the surface re-gion diverge slowly /H20849say, as the 1/4 power of linear dimen-
sion /H20850in the thermodynamic limit, so that x
0is moved arbi-
trarily deep into the bulk.
It is now expedient to use the identity
/H20855wi/H20841v/H20841wi/H20856=/H20858
jv/H20855j,i/H20856, /H2084910/H20850
where
v/H20855j,i/H20856=2I m /H20855wi/H20841r/H20841wj/H20856/H20855wj/H20841H/H20841wi/H20856/H20849 11/H20850
has the interpretation of a current “donated from WF /H20841wj/H20856to
WF /H20841wi/H20856,” and exploit the fact that the total current carried by
any subset of WFs can be computed as the sum of all v/H20855j,i/H20856for
which iis inside and jis outside the subset. Applying this to
the piece of surface region considered above, we get
K=−1
A/H20858
s/H11032/H11032/H20858
s/HS11005s/H11032v/H20855s,s/H11032/H20856. /H2084912/H20850
Setting the boundary deep enough below the surface to be in
a bulklike region and invoking the exponential localizationof the WFs and of related matrix elements, we can identify/H20841w
s/H20856and /H20841ws/H11032/H20856with the bulk WFs /H20841mR/H20856and /H20841nR/H11032/H20856, re-
spectively. Exploiting translational symmetry, v/H20855mR,nR/H11032/H20856
=v/H20855m0,n/H20849R/H11032−R/H20850/H20856, Eq. /H2084912/H20850becomes
K=−1
A/H20858
Rx/H11021x0/H20858
Rx/H11032/H11022x0/H11032/H20858
mnv/H20855m0,n/H20849R/H11032−R/H20856/H20850, /H2084913/H20850
where the lattice sum is still restricted to the R/H11032vectors
whose yzcoordinates are within the surface unit cell. The
number of terms in the lattice sum of Eq. /H2084913/H20850having a given
value of R/H11032−Ris just /H20849Rx/H11032−Rx/H20850A//H9024if/H20849Rx/H11032−Rx/H20850/H110220 and zero
otherwise. With a change of summation index, Eq. /H2084913/H20850be-
comesK=−1
2/H9024/H20858
RRx/H20858
mnv/H20855m0,nR/H20856, /H2084914/H20850
where the factor of 2 enters because the sum has been ex-
tended to all R. Notice that the surface-cell size has eventu-
ally disappeared.
Evidently the corresponding surface current on a surface
with unit normal nˆis then
K/H9251/H20849nˆ/H20850=/H20858
/H9252G/H9251/H9252n/H9252, /H2084915/H20850
where
G/H9251/H9252=−1
2/H9024/H20858
R/H20858
mnv/H20855m0,nR/H20856,/H9251R/H9252. /H2084916/H20850
Now for a sample of size Lx/H11003Ly/H11003Lz, the left and right faces
carry currents of ± LyLzGyxseparated by a distance Lx, and
thus contribute to the magnetic moment per unit volume asG
yx/2c; similarly, the front and back faces contribute as
−Gxy/2c. Together they contribute to Mzas −GxyA/c, where
G/H9251/H9252A=1
2/H20849G/H9251/H9252−G/H9252/H9251/H20850, /H2084917/H20850
is the antisymmetric part of the Gtensor. Deriving corre-
sponding expressions for MxandMyby permutation of indi-
ces, the contribution of the surface current in Eq. /H2084914/H20850to the
magnetization can thus be cast in a coordinate-independentform and evaluated for the whole sample surface in the ther-modynamic limit as
M
IC=−1
4c/H9024/H20858
mnRR/H11003v/H20855m0,nR/H20856. /H2084918/H20850
Note that Eq. /H2084918/H20850describes the current circulating in the
surface WFs, while the expression on its right-hand side in-
volves only bulk WFs.
This is quite remarkable, and indeed it is one of the cen-
tral results of this paper, as well as of Ref. 7. It implies that
even MICis a bulk property, as anticipated above. This may
appear counterintuitive, but indeed closely parallels a well-known /H20849and equally counterintuitive /H20850feature of the quantum-
Hall effect, where the Hall current is accomodated by chiraledge states.
23,24Nevertheless, these edge currents are com-
pletely determined by bulk properties of the system, and canbe evaluated by adopting toroidal boundary conditions inwhich the sample has no edges. Such a finding, in fact, is oneof the most remarkable results of the quantum-Hall theoret-ical literature.
21,25–27We also notice that the bulk nature of
MICguarantees that our general expressions, valid in the
thermodynamic limit, apply regardless of whether surfacestates are present in bounded samples, and if they arepresent, regardless of their character.
It might be thought that the surface currents Kmust flow
parallel to the surface, and thus that the diagonal elementsG
xxandGyymust vanish, or more generally, that the sym-
metric component
G/H9251/H9252S=1
2/H20849G/H9251/H9252+G/H9252/H9251/H20850/H20849 19/H20850
FIG. 1. Horizontal slice from a sample that extends indefinitely
in the vertical direction. Vertical dashed lines delimit bulk and sur-face regions in which WFs are labeled by sands
/H11032, respectively.CERESOLI et al. PHYSICAL REVIEW B 74, 024408 /H208492006 /H20850
024408-4of the Gtensor should vanish. This turns out notto be true.
In some of our tight-binding model calculations, we haveexplicitly computed the right-hand side of Eq. /H208499/H20850and con-
firmed the existence of a surface-normal component of K.
The explanation is that K, as defined by Eq. /H208499/H20850, is only
one contribution to the physical macroscopic surface current.There is an additional contribution arising from the fact that,when TR symmetry is broken, the second-moment spreads
20
of the WFs are notgenerally stationary with respect to time.
For example, if the WFs are in the process of expanding,then electron charge is in the process of spilling out of thesurface. To formalize this notion, we introduce a symmetricCartesian tensor
W
/H9251/H9252=−1
2/H9024Nc/H20858
i/H20855wi/H20841r/H9251v/H9252+v/H9251r/H9252/H20841wi/H20856/H20849 20/H20850
that is a kind of symmetric analog of the antisymmetric ex-
pression for Mgiven in Eq. /H208494/H20850.I fW/H9251/H9252is nonzero, then we
would expect surface currents of the form K/H9251/H20849nˆ/H20850=/H20858/H9252W/H9251/H9252n/H9252.
If present, these would violate continuity. However, they are
not present, because we can write
W/H9251/H9252=−1
2/H9024Ncd
dt/H20858
i/H20855wi/H20841r/H9251r/H9252/H20841wi/H20856. /H2084921/H20850
Noting that the trace of any operator /H20849here r/H9251r/H9252/H20850must be
independent of time in any stationary state /H20849here the ground
state of the finite sample /H20850, it follows that W/H9251/H9252=0. Neverthe-
less, if we were to follow a route parallel to that used for thetreatment of Mearlier in this section, we could decompose
Winto a “local spread” part W
LSand an “itinerant spread”
part WIS. The former is
WLS,/H9251/H9252=−1
2/H9024/H20858
n/H20855n0/H20841r/H9251v/H9252+v/H9251r/H9252/H20841n0/H20856
=−1
2/H9024d
dt/H20858
n/H20855n0/H20841r/H9251r/H9252/H20841n0/H20856, /H2084922/H20850
which is just related to the rate of spread of the bulk WFs in
one bulk unit cell, while the latter is just WIS,/H9251/H9252=G/H9251/H9252Sof Eq.
/H2084919/H20850. Because the total W/H9251/H9252must vanish, we conclude that
the nonphysical current that we were concerned about, aris-
ing from G/H9251/H9252Sin Eq. /H2084919/H20850, is exacly cancelled by another
non-physical one arising from the spreading of the bulk WFs.Thus, in the end, the physical edge current has pure circulat-ing character and is related only to antisymmetric Cartesiantensors.
C. Reciprocal-space expressions
The above two final expressions /H208497/H20850and /H2084918/H20850are given in
terms of bulk WFs. Therefore the total orbital magnetizationM=M
LC+MICof the finite sample converges in the thermo-
dynamic limit to a bulk, boundary-insensitive, material prop-erty. Next, using the WF definition, Eq. /H208493/H20850, we are going to
transform M
LCandMICinto equivalent expressions involv-
ing BZ integrals of Bloch orbitals. Specifically, we are goingto prove the two identitiesM
LC=1
2c/H208492/H9266/H208503Im/H20858
n/H20885
BZdk/H20855/H11509kunk/H20841/H11003Hk/H20841/H11509kunk/H20856,
/H2084923/H20850
MIC=1
2c/H208492/H9266/H208503Im/H20858
nn/H11032/H20885
BZdkEn/H11032nk/H20855/H11509kunk/H20841/H11003/H20841/H11509kun/H11032k/H20856.
/H2084924/H20850
These two expressions generalize to the multi-band case our
previous finding for the case of a single occupied band.7
There is an important difference, however; while in thesingle-band case Eqs. /H2084923/H20850and /H2084924/H20850areseparately gauge
invariant, only their sumis gauge invariant in the multi-band
case, as we shall see in Sec. II D.
We carry the derivation in reverse, starting from Eqs. /H2084923/H20850
and /H2084924/H20850and showing that they reduce to Eqs. /H208497/H20850and /H2084918/H20850.
First, using Eq. /H208493/H20850,w eg e t
/H20841
/H11509kunk/H20856=−i/H20858
Reik·/H20849R−r/H20850/H20849r−R/H20850/H20841nR/H20856,
Hk/H20841/H11509kunk/H20856=−i/H20858
Reik·/H20849R−r/H20850H/H20849r−R/H20850/H20841nR/H20856. /H2084925/H20850
Since the velocity operator is v=i/H20851H,r/H20852=i/H20851H,/H20849r−R/H20850/H20852, and
exploiting /H20849r−R/H20850/H11003/H20849r−R/H20850=0, we may express Eq. /H2084923/H20850as
MLC=−1
2c/H9024Nc/H20858
nR/H20855nR/H20841/H20849r−R/H20850/H11003v/H20841nR/H20856, /H2084926/H20850
where the number of cell Nchere is formally infinite, and
appears because the /H20841unk/H20841are normalized differently from the
WFs. Since we limit ourselves to the case of an insulatorwith zero Chern invariant, the WFs enjoy the usual transla-tional symmetry, and Eq. /H2084926/H20850is indeed identical to Eq. /H208497/H20850.
Next, we address Eq. /H2084924/H20850, whose second factor in the
integral is
/H20855
/H11509kunk/H20841/H11003/H20841/H11509kun/H11032k/H20856
=1
Nc/H20858
R·R/H11032eik·/H20849R/H11032−R/H20850/H20855nR/H20841/H20849r−R/H20850/H11003/H20849r−R/H11032/H20850/H20841n/H11032R/H11032/H20856
=1
Nc/H20858
R·R/H11032eik·/H20849R/H11032−R/H20850/H20849R/H11032−R/H20850/H11003/H20855nR/H20841r/H20841n/H11032R/H11032/H20856, /H2084927/H20850
where the last line follows because only the cross terms sur-
vive from the product /H20849r−R/H20850/H11003/H20849r−R/H11032/H20850. We then exploit
/H20855n/H11032R/H11032/H20841H/H20841nR/H20856=/H9024
/H208492/H9266/H208503/H20885
BZdkeik·/H20849R/H11032−R/H20850En/H11032nk /H2084928/H20850
in order to rewrite Eq. /H2084924/H20850asORBITAL MAGNETIZATION IN CRYSTALLINE ¼ PHYSICAL REVIEW B 74, 024408 /H208492006 /H20850
024408-5MIC=Im
2c/H9024Nc/H20858
R·R/H11032
mn/H20849R/H11032−R/H20850/H11003/H20855mR/H20841r/H20841nR/H11032/H20856/H20855nR/H11032/H20841H/H20841mR/H20856.
/H2084929/H20850
Since the matrix elements only depend on the relative WF
coordinate R/H11032−R, Eq. /H2084929/H20850is transformed into
MIC=1
2c/H9024Im/H20858
mnRR/H11003/H20855m0/H20841r/H20841nR/H20856/H20855nR/H20841H/H20841m0/H20856./H2084930/H20850
Using Eq. /H2084911/H20850, it is then easy to check that Eq. /H2084930/H20850is indeed
identical to Eq. /H2084918/H20850.
This completes our proof. Our final expression for the
macroscopic orbital magnetization of a crystalline insulatoris
M=1
2c/H208492/H9266/H208503Im/H20858
nn/H11032/H20885
BZdk/H20855/H11509kun/H11032k/H20841/H11003/H20849Hk/H9254nn/H11032+En/H11032nk/H20850
/H11003/H20841/H11509kunk/H20856. /H2084931/H20850
Owing to the occurrence of HkandEnn/H11032kwith the same sign
/H20849in contrast to the magnetization of an individual wave
packet discussed in Ref. 28/H20850, Eq. /H2084931/H20850does not appear at first
sight to be invariant with respect to translation of the energyzero. However, the zero-Chern-invariant condition—com-pare Eq. /H2084931/H20850to Eq. /H208492/H20850—enforces such invariance. As for
the gauge invariance of Eq. /H2084931/H20850, this will be demonstrated in
the next subsection.
D. Proof of gauge invariance
Here we prove the gauge invariance in the multi-band
sense of the Chern invariant /H208492/H20850and of our main expression
for the macroscopic magnetization /H2084931/H20850. While these expres-
sions are BZ integrals, we will actually prove that even theirintegrands are gauge invariant. To this end, we will show
that both integrands can be expressed as traces of gauge-invariant one-body operators acting on the Hilbert space oflattice-periodical functions.
Our key ingredients are the effective Hamiltonian H
k, the
ground-state projector
Pk=/H20858
n/H20841unk/H20856/H20855unk/H20841, /H2084932/H20850
and its orthogonal complement Qk=1− Pk. These three op-
erators are obviously unaffected by any unitary mixing of the/H20841u
nk/H20856among themselves at a given k, and therefore any ex-
pression built only from these ingredients will be a mani-festly multi-band gauge-invariant quantity. In particular, wedefine the three quantities
f
k,/H9251/H9252=t r/H20853/H20849/H11509/H9251Pk/H20850Qk/H20849/H11509/H9252Pk/H20850/H20854, /H2084933/H20850
gk,/H9251/H9252=t r/H20853/H20849/H11509/H9251Pk/H20850QkHkQk/H20849/H11509/H9252Pk/H20850/H20854, /H2084934/H20850
hk,/H9251/H9252=t r/H20853Hk/H20849/H11509/H9251Pk/H20850Qk/H20849/H11509/H9252Pk/H20850/H20854, /H2084935/H20850
where /H11509/H9251=/H11509//H11509k/H9251and the trace is over electronic states. We
are going to show that the Chern invariant and the magneti-zation can be expressed as integrals of fkand of gk+hk,
respectively.
From Eq. /H2084932/H20850it follows that
/H11509/H9251Pk=/H20858
n/H20849/H20841/H11509/H9251unk/H20856/H20855unk/H20841+/H20841unk/H20856/H20855/H11509/H9251unk/H20841/H20850 /H20849 36/H20850
so that
/H20849/H11509/H9251Pk/H20850Qk/H20849/H11509/H9252Pk/H20850=/H20858
nn/H11032/H20841unk/H20856/H20855/H11509/H9251unk/H20841Qk/H20841/H11509/H9252un/H11032k/H20856/H20855un/H11032k/H20841.
/H2084937/H20850
We now specialize to the “Hamiltonian gauge” in which the
Bloch functions /H20855unk/H20856are eigenstates of Hkwith eigenvalues
/H9280nk. Inserting Eq. /H2084937/H20850into Eqs. /H2084933/H20850and /H2084935/H20850and using a
similar approach for Eq. /H2084934/H20850, the three quantities can be
written as
fk,/H9251/H9252=/H20858
n/H20855/H11509/H9251unk/H20841/H11509/H9252unk/H20856−/H20858
nn/H11032/H20855/H11509/H9251unk/H20841un/H11032k/H20856/H20855un/H11032k/H20841/H11509/H9252unk/H20856,
/H2084938/H20850
gk,/H9251/H9252=/H20858
n/H20855/H11509/H9251unk/H20841Hk/H20841/H11509/H9252unk/H20856−/H20858
nn/H11032/H9280n/H11032k/H20855/H11509/H9251unk/H20841un/H11032k/H20856
/H11003/H20855un/H11032k/H20841/H11509/H9252unk/H20856, /H2084939/H20850
and
hk,/H9251/H9252=/H20858
n/H9280nk/H20855/H11509/H9251unk/H20841/H11509/H9252unk/H20856−/H20858
nn/H11032/H9280nk/H20855/H11509/H9251unk/H20841un/H11032k/H20856/H20855un/H11032k/H20841/H11509/H9252unk/H20856.
/H2084940/H20850
Regarded as 3 /H110033 Cartesian matrices, Eqs. /H2084933/H20850–/H2084935/H20850are
clearly Hermitian, so that the antisymmetric parts of Eqs./H2084938/H20850–/H2084940/H20850are all pure imaginary. Thus, the information con-
tent of the antisymmetric part of f
k,/H9251/H9252is contained in the
gauge-invariant real vector quantity
f˜k,/H9251=−I m /H9255/H9251/H9252/H9253fk,/H9252/H9253, /H2084941/H20850
where /H9255/H9251/H9252/H9253is the antisymmetric tensor. We define g˜k,/H9251and
h˜k,/H9251in the corresponding way in terms of gk,/H9252/H9253and hk,/H9252/H9253,
respectively. Looking at the second term in Eq. /H2084938/H20850and
using /H11509/H9251/H20855unk/H20841un/H11032k/H20856=/H11509/H9251/H9254nn/H11032=0, we find that its antisymmetric
part vanishes, and in fact f˜kis nothing other than the Berry
curvature. We thus recover the Chern invariant of Eq. /H208492/H20850in
the form
C=1
2/H9266/H20885
BZdkf˜k. /H2084942/H20850
Next, inspecting the second terms of Eqs. /H2084939/H20850and /H2084940/H20850,
we find that neither of these terms vanishes by itself underantisymmetrization. However, the sum of these two terms
does vanish under antisymmetrization. Using the sum only,and comparing with Eq. /H2084931/H20850, we find that the magnetization
may be written in the manifestly gauge-invariant formCERESOLI et al. PHYSICAL REVIEW B 74, 024408 /H208492006 /H20850
024408-6M=−1
2c/H208492/H9266/H208503/H20885
BZdk/H20849g˜k+h˜k/H20850. /H2084943/H20850
/H20849The sign reflects the fact that the electron has negative
charge. /H20850This completes the proof that the integrand in Eq.
/H2084931/H20850is multi-band gauge invariant.
Notice that if we take the first term only in Eq. /H2084939/H20850and
antisymmetrize, we get the integrand in MLC/H20849times a multi-
plicative constant /H20850; the same holds for Eq. /H2084940/H20850and MIC.
However, the second terms in Eqs. /H2084939/H20850and /H2084940/H20850have non-
zero antisymmetric parts which are essential to their gaugeinvariance. Therefore, M
LCandMICas defined above are not
separately gauge invariant, except in the single-band case.7
On the other hand, it is possible to regroup terms and
write M=M˜LC+M˜IC, where
M˜LC=−1
2c/H208492/H9266/H208503/H20885
BZdkg˜k /H2084944/H20850
and
M˜IC=−1
2c/H208492/H9266/H208503/H20885
BZdkh˜k /H2084945/H20850
areindividually gauge invariant, even in the multiband case.
This raises the fascinating question as to whether these twocontributions to the orbital magnetization are, in principle,independently measurable. On the one hand, Berry has em-phasized in his milestone paper
29that any gauge-invariant
property should be potentially observable. On the other hand,any measurement of orbital magnetization—or, equivalently,of dissipationless macroscopic surface currents—will onlybe sensitive to their sum. At the present time we have noinsight into how to propose an experiment that could distin-guish them, and we therefore leave this as an open question.
In Appendix A, we show how to compute f
˜k,g˜k, and h˜kon a
3Dkmesh using finite-difference methods to approximate
the derivatives in Eqs. /H2084933/H20850–/H2084935/H20850.
E. Heuristic extension to metals and Chern insulators
All of the above results are derived under the hypothesis
that the crystalline system is a KS insulator in which theChern invariant, Eq. /H208492/H20850, is zero. These conditions, in fact,
are essential for expressing any ground-state property interms of WFs. Nonetheless the integrand in our final
reciprocal-space expression /H2084931/H20850is gauge invariant. This sug-
gests a possible generalization to Chern insulators /H20849defined
as insulators with nonzero Chern invariant /H20850and even to KS
metals.
We notice that Eq. /H2084931/H20850is somehow reminiscent of the
Berry-phase formula appearing in the modern theory of elec-trical polarization.
8,9There is an important difference, how-
ever. In the electrical case, the integrand is notgauge invari-
ant, and the formula corresponding to our Eq. /H2084931/H20850only
makes sense when integrated over the whole BZ, i.e., for aKS insulator. Indeed, macroscopic polarization is a well-defined bulk property only for insulating materials.
30Instead,
orbital magnetization is a phenomenologically well-definedbulk property for both insulating and metallic materials.
Therefore, it is worthwhile to investigate heuristically thevalidity of an extension of Eq. /H2084931/H20850to the metallic case, even
though we cannot yet provide any formal proof. Additionally,we also heuristically investigate Chern insulators. Metals andChern insulators share the property that their magnetizationhas a nontrivial dependence on the chemical potential
/H9262.
We already observed that Eq. /H2084931/H20850is invariant by transla-
tion of the energy zero, but this owes to the facts that theintegration therein is performed over the whole BZ, and thatthe Chern invariant is zero. If we abandon either of theseconditions, the formula has to be modified in order to restorethe invariance. To this end, we first need to restrict our for-mulation to the “Hamiltonian gauge,” where the energy ma-trix is diagonal: E
nn/H11032k=/H9280nk/H9254nn/H11032. The /H20855unk/H20856are therefore eigen-
states of Hk, and the only gauge freedom allowed is now the
arbitrary choice of their phase.
In the general case, including metals and Chern insulators,
we propose to generalize Eq. /H2084931/H20850to
M=1
2c/H208492/H9266/H208503Im/H20858
n/H20885
/H9280nk/H33355/H9262dk/H20855/H11509kunk/H20841/H11003/H20849Hk+/H9280nk−2/H9262/H20850
/H11003/H20841/H11509kunk/H20856, /H2084946/H20850
where /H9262is the chemical potential /H20849Fermi energy /H20850. Equation
/H2084946/H20850has the desirable invariance property addressed above.
Furthermore, in the metallic case, Eq. /H2084946/H20850provides a mag-
netization dependent on /H9262, as it should. In the insulating
case, when /H9262is varied in the gap, Mchanges linearly only if
the Chern invariant is nonzero, and remains constant other-wise. In fact, Eqs. /H208492/H20850and /H2084946/H20850imply that
dM
d/H9262=−1
c/H208492/H9266/H208502C /H2084947/H20850
for any insulator and /H9262in the gap.
The modification from Eq. /H2084931/H20850to Eq. /H2084946/H20850is the minimal
one enjoying the desired properties. Furthermore, in thesingle-band case it is essentially identical to a formula re-cently proposed by Niu and co-workers,
10whose derivation
rests upon semiclassical wave packet dynamics. We providestrong numerical evidence that this formula retains its valid-ity well beyond the semiclassical regime, and is in fact theexact quantum-mechanical expression for the orbital magne-tization /H20849in a vanishing macroscopic Bfield /H20850.
An expression related—though not identical—to Eq. /H2084946/H20850
occurs in the theory of the Hall effect. Upon replacement ofthe quantity in parenthesis with the identity, one obtainssomething proportional to the integral of the Berry curvatureover occupied portions of the BZ. This quantity correspondsto the entire Hall conductivity in quantum-Hall systems
25,26
/H20849which are in fact two-dimensional Chern insulators31/H20850and
the so-called “anomalous” Hall term in metals with brokenTR symmetry. The theory of the anomalous Hall effect hasattracted much attention in the recent literature.
17,19,32
F. The two-dimensional case
In two dimensions, the magnetization is a pseudoscalar
M, and the Chern invariant is the Chern number C/H20849a dimen-ORBITAL MAGNETIZATION IN CRYSTALLINE ¼ PHYSICAL REVIEW B 74, 024408 /H208492006 /H20850
024408-7sionless integer /H20850.21Our heuristic formula /H2084946/H20850, then becomes
M=1
2c/H208492/H9266/H208502Im/H20858
n/H20885
/H9280nk/H33355/H9262dk/H20855/H11509kunk/H20841/H11003/H20849Hk+/H9280nk−2/H9262/H20850
/H11003/H20841/H11509kunk/H20856. /H2084948/H20850
The two-dimensional analogue of Eq. /H2084947/H20850is
dM
d/H9262=−C
2/H9266c. /H2084949/H20850
The physical interpretation of this equation is best under-
stood in terms of the chiral edge states of a finite sample cutfrom a Chern insulator. Owing to the main equation /H11633/H11003M
=j/c, a macroscopic current of intensity I=cMcirculates at
the edge of any two-dimensional uniformly magnetizedsample, hence Eq. /H2084949/H20850yields
dI
d/H9262=−C
2/H9266. /H2084950/H20850
This is just what is to be expected: raising the chemical po-
tential by d/H9262fills dk/2/H9266states per unit length, i.e., dI
=−vdk/2/H9266; but the group velocity is just v=d/H9262/dk. Thus,
Eq. /H2084950/H20850follows with the interpretation that Cis the excess
number of chiral edge channels of positive circulation overthose with negative circulation. Remarkably, the above equa-tions state that the contribution of edge states is indeed abulk quantity, and can be evaluted in the thermodynamiclimit by adopting periodic boundary conditions where thesystem has no edges. As already observed, this feature maylook counterintutitive, but a similar behavior has been knownfor more than 20 years in the theory of the quantum-Halleffect.
23–26
In contrast to our case, a magnetic field is usually present
in the standard theory of the quantum-Hall effect, although itis not strictly needed.
11The role of chiral edge states is elu-
cidated, for example,23,24by considering a vertical strip of
width l, where the currents at the right and left boundaries
are ± I. The net current vanishes insofar as /H9262is constant
throughout the sample. When an electric field Eis applied
across the sample, the right and left chemical potentials dif-fer by /H9004
/H9262=Eland the two edge currents no longer cancel.
Our Eq. /H2084950/H20850is consistent with the known quantum-Hall re-
sults. In fact, according to Eq. /H2084950/H20850, the net current is /H9004I
/H11229−C/H9004/H9262/2/H9266, while the transverse conductivity is defined by
/H9004I=/H9268TEl. We thus arrive at /H9268T=−C/2/H9266/H20849or, in ordinary
units, /H9268T=−Ce2/h/H20850, which is indeed a celebrated re-
sult.21,25–27We stress that the Chern number Cis a bulk
property of the system, and can be evaluated by adoptingtoroidal boundary conditions, where the edges appear to playno role.
III. NUMERICAL TESTS
In a previous paper7we tested Eq. /H2084948/H20850for the insulating
C=0 single-band case on the Haldane model Hamiltonian,11
described below /H20849Sec. III C /H20850. In this special case, Eq. /H2084948/H20850is
notheuristic, since we provided an analytical proof. We ad-
dressed finite-size realizations of the Haldane model, cutfrom the bulk; our analysis confirmed that MLCarises en-
tirely from the magnetization of bulk WFs in the thermody-namic limit, whereas M
ICarises from current-carrying sur-
face WFs. Both terms have also been evaluated in terms ofbulk Bloch orbitals, by means of Eq. /H2084948/H20850, confirming that
the orbital magnetization is indeed a genuine bulk quantity.
Here we extend this program of checking the correctness
of our analytic formulas by carrying out numerical tests onour new multi-band formula /H2084931/H20850, derived for the C=0 insu-
lating case. This is done using a four-band model Hamil-tonian on a square lattice as described below /H20849Sec. III A /H20850.
Furthermore, we perform computer experiments to assesswhether our hypothetical Eq. /H2084948/H20850, proposed to cover also the
metallic and the C/HS110050 insulating cases, is consistent with
calculations on finite samples. We do this for metals in Sec.III B using the same square lattice as in Sec. III A, but atfractional band filling. We then do this in Sec. III C forChern insulators using the Haldane model
11in a range of
parameters for which C/HS110050.
Numerical implementation of Eqs. /H2084931/H20850,/H2084946/H20850, and /H2084948/H20850is
quite straightforward once one has in hand an efficientmethod for evaluating the kderivatives of the Bloch orbitals.
There are several possible approaches to doing this. One pos-sibility is to evaluate /H20855
/H11509/H9251unk/H20856by summation over states as
/H20841/H11509/H9251unk/H20856=/H20858
m/HS11005n/H20841umk/H20856/H20855umk/H20841v/H9251/H20841unk/H20856
/H9280mk−/H9280nk. /H2084951/H20850
This is very practical in the context of tight-binding calcula-
tions, where the sum over conduction bands runs only over asmall number of terms, and we adopted this for the test-casecalculations reported below. However, in first-principles cal-culations the sums over conduction states can be quite te-dious, and one has to be careful to use the correct form forthe velocity operator in the matrix elements /H20851see discussion
following Eq. /H208495/H20850/H20852. Alternatively, the needed derivatives of
/H20855u
nk/H20856can be obtained from finite difference methods by mak-
ing use of the discretized covariant derivative33,34as dis-
cussed in Appendix A. It may also be possible to use stan-dard linear-response methods
35to compute /H20855/H11509/H9251unk/H20856, as this is
an operation which is already implemented as part of com-
puting the electric-field response in several modern codepackages.
A. Normal insulating case
We present in this section numerical tests using a nearest-
neighbor tight-binding Hamiltonian on a 2 /H110032 square lattice
in which the primitive cell comprises four plaquettes, asshown in Fig. 2. This results in a four-band model. The
modulus tof the /H20849complex /H20850nearest-neighbor hopping ampli-
tude is set to 1, thus fixing the energy scale. TR breaking isachieved by endowing some of the hopping amplitudes witha complex phase factor e
i/H9278. This amounts to threading a pat-
tern of magnetic fluxes through the interiors of the fourplaquettes, as shown in Fig. 2, in such a way that the thread-
ing flux /H9021
iis just the sum of the phase factors associated
with the four bonds delineating plaquette i, counted with
positive signs for counterclockwise-pointing bonds and mi-CERESOLI et al. PHYSICAL REVIEW B 74, 024408 /H208492006 /H20850
024408-8nus signs for clockwise ones. The constraint of vanishing
macroscopic magnetic field corresponds to /H90211+/H90212+/H90213
+/H90214=2/H9266/H11003integer. We found that not all flux patterns break
TR symmetry. For instance, for the flux patterns/H20849/H9021
1,/H90212,/H90213,/H90214/H20850=/H20849+/H9278,+/H9278,−/H9278,−/H9278/H20850and /H20849+/H9278,−/H9278,+/H9278,−/H9278/H20850,
TR symmetry is restored by some spatial symmetry /H20849an ad-
ditional translational symmetry and a mirror symmetry, re-spectively /H20850; the orbital magnetization then vanishes for any
value of the parameter
/H9278. On the other hand, the flux pattern
/H208492/H9278,−/H9278,0,−/H9278/H20850violates inversion and mirror symmetry, and
therefore realizes TR symmetry breaking.
The on-site energies /H20849EA,EB,EC,ED/H20850have been set to the
values /H20849−3,0,−3,0 /H20850. This choice results in an insulator with
two groups of two entangled bands as shown in Fig. 3.
Switching on the fluxes splits the bands along the X-Lline,
which are otherwise twofold degenerate. The kderivative of
Bloch orbitals was computed by the sum-over-states formula/H2084951/H20850. We treated the two lowest bands as filled and we veri-
fied that the multiband Chern number is zero.
We built square finite samples, cut from the bulk, made of
L/H11003Lfour-site unit cells and having 2 L+1 sites on each
edge. Their orbital magnetization /H20849dipole per unit area /H20850M/H20849L/H20850
is straightforwardly computed as in Eq. /H208494/H20850. We expect the
L→/H11009asymptotic behaviorM/H20849L/H20850=M+a/L+b/L
2, /H2084952/H20850
where Mis the bulk magnetization according to Eq. /H2084948/H20850.
The terms a/Landb/L2account for edge and corner correc-
tions, respectively.
We performed calculations up to L=14 /H20849841 lattice sites /H20850.
The resulting orbital magnetization as a function of the pa-rameter
/H9278is shown in Fig. 4. We independently computed
the bulk orbital magnetization Mfrom a discretization of the
reciprocal-space formula /H2084948/H20850. We get well converged results
/H20849to within 0.1% /H20850f o ra5 0 /H1100350k-point mesh in the full BZ.
So far, we have studied a model multi-band insulator, hav-
ing zero Chern number. For this specific case we providedabove a solid analytic proof of our reciprocal-space formula,which holds in the thermodynamic limit. Indeed, the numeri-cal results confirm the correctness of the k-space formula,
while also providing some information about actual finite-size effects and numerical convergence.
B. Metallic case
In the previous section we addressed the case of a TR-
broken multiband insulator, by treating the two lowest bandsas occupied. Here we are going to extend our analysis to themetallic case. We are using the same model Hamiltonian asin the previous section, but we allow the Fermi level to spanthe energy range roughly from −5.45 to 2.45 energy units,namely, from the bottom of the lowest band to the top of thehighest one. In order to smooth Fermi-surface singularities,and to obtain well converged results, we adopt the simpleFermi-Dirac smearing technique, widely used in first-principle electronic-structure calculations. This amounts toreplace, the /H20849integer /H20850Fermi occupation factor /H9008/H20849
/H9262−/H9280nk/H20850
with a suitable smooth function f/H9262/H20849/H9280nk/H20850. We therefore replace
in Eq. /H2084948/H20850:
/H20858
n,/H9280nk/H11021/H9262→/H20858
nf/H9262/H20849/H9280nk/H20850. /H2084953/H20850
Reasoning in terms of a fictitious temperature, one may
choose a Fermi-Dirac distribution
FIG. 2. 2 /H110032 four-site square lattice used in the numerical tests.
The absolute value of the hopping parameter tis set to 1. /H90211¯4are
the threading fluxes through the four plaquettes.
FIG. 3. Band structure of the square lattice for /H9278=/H9266/10. The
flux pattern is /H20849/H90211,/H90212,/H90213,/H90214/H20850=/H208492/H9278,−/H9278,0,−/H9278/H20850, and the on-site
energies are /H20849EA,EB,EC,ED/H20850=/H20849−3,0,−3,0 /H20850/H20849see also Fig. 2/H20850. The
two lower bands are treated as occupied.
FIG. 4. Orbital magnetization of the square-lattice model as a
function of the parameter /H9278. The two lower bands are treated as
occupied. Open circles: extrapolation from finite-size samples.Solid line: discretized k-space formula /H2084931/H20850.ORBITAL MAGNETIZATION IN CRYSTALLINE ¼ PHYSICAL REVIEW B 74, 024408 /H208492006 /H20850
024408-9f/H9262/H20849/H9280/H20850=1
1 + exp /H20851/H20849/H9280−/H9262/H20850//H9268/H20852. /H2084954/H20850
In all subsequent calculations, we set /H9268=0.05 a.u., which
provides good convergence.
We compute the orbital magnetization as a function of the
chemical potential /H9262with/H9278fixed at /H9266/3. Using the same
procedure as in the previous section, we compute the orbitalmagnetization by the means of the heuristic k-space formula
/H2084948/H20850and we compare it to the extrapolated value from finite
samples, from L=8 /H20849289 sites /H20850toL=16 /H208491089 sites /H20850.W e
verified that a k-point mesh of 100 /H11003100 gives well con-
verged results for the bulk formula /H2084948/H20850.
The orbital magnetization as a function of the chemical
potential for
/H9278=/H9266/3 is shown in Fig. 5. The resulting values
agree to a good level, and provide solid numerical evidencein favor of Eq. /H2084948/H20850, whose analytical proof is still lacking.
The orbital magnetization initially increases as the filling ofthe lowest band increases, and rises to a maximum at a
/H9262
value of about −4.1. Then, as the filling increases, the first/H20849lowest /H20850band crosses the second band and the orbital mag-
netization decreases, meaning that the two bands carryopposite-circulating currents giving rise to opposite contribu-tions to the orbital magnetization. The orbital magnetizationremains constant when
/H9262is scanned through the insulating
gap. Upon further increase of the chemical potential, the or-bital magnetization shows a symmetrical behavior as a func-tion of
/H9262, the two upper bands having equal but opposite
dispersion with respect to the two lowest bands /H20849see Fig. 3/H20850.
C. Chern insulating case
In order to check the validity of our heuristic Eq. /H2084948/H20850for
a Chern insulator, we switch to the Haldane modelHamiltonian
11that we used in a previous paper7to address
theC=0 insulating case. In fact, depending on the parameter
choice, the Chern number Cwithin the model can be either
zero or nonzero /H20849actually, ±1 /H20850.
The Haldane model is comprised of a honeycomb lattice
with two tight-binding sites per cell with site energies ± /H9004,
real first-neighbor hoppings t1, and complex second-neighbor
hoppings t2e±i/H9272, as shown in Fig. 6. The resulting Hamil-tonian breaks TR symmetry and was proposed /H20849forC=±1 /H20850
as a realization of the quantum Hall effect in the absence ofa macroscopic magnetic field. Within this two-band model,one deals with insulators by taking the lowest band as occu-pied.
In our previous paper
7we restricted ourselves to C=0 to
demonstrate the validity of Eq. /H2084948/H20850, which was also analyti-
cally proved. In the present work we address the C/HS110050 insu-
lating case, where instead we have no proof of Eq. /H2084948/H20850yet.
We are thus performing computer experiments in order toexplore uncharted territory.
Following the notation of Ref. 11, we choose the param-
eters/H9004=1, t
1=1, and /H20841t2/H20841=1/3. As a function of the flux
parameter /H9278, this system undergoes a transition from zero
Chern number to /H20841C/H20841=1 when /H20841sin/H9278/H20841/H110221//H208813.
First we checked the validity of Eq. /H2084948/H20850in the Chern
insulating case by treating the lowest band as occupied. Wecomputed the orbital magnetization as a function of
/H9278by Eq.
/H2084948/H20850at a fixed /H9262value, and we compared it to the magneti-
zation of finite samples cut from the bulk. For the periodicsystem, we fix
/H9262in the middle of the gap; for consistency,
the finite-size calculations are performed at the same /H9262
value, using the Fermi-Dirac distribution of Eq. /H2084954/H20850. The
finite systems have therefore fractional orbital occupancyand a noninteger number of electrons. The biggest samplesize was made up of 20 /H1100320 unit cells /H20849800 sites /H20850. The com-
parison between the finite-size extrapolations and the dis-cretized k-space formula is displayed in Fig. 7. This heuris-
tically demonstrates the validity of our main results, Eqs./H2084946/H20850and /H2084948/H20850, in the Chern-insulating case.
Next, we checked the validity of Eq. /H2084948/H20850for the most
general case, following the transition from the metallic phaseto the Chern insulating phase as a function of the chemicalpotential
/H9262. To this aim we keep the model Hamiltonian
fixed, choosing /H9278=0.7/H9266; for/H9262in the gap this yields a Chern
insulator. The behavior of the magnetization while /H9262varies
from the lowest-band region, to the gap region, and then tothe highest-band region is displayed in Fig. 8, as obtained
from both the finite-size extrapolations and the discretizedk-space formula. This shows once more the validity of our
heuristic formula. Also notice that in the gap region the mag-netization is perfectly linear in
/H9262, the slope being determined
by the lowest-band Chern number according to Eq. /H2084949/H20850.
FIG. 5. Orbital magnetization of the square-lattice model as a
function of the chemical potential /H9262for/H9278=/H9266/3. The shaded areas
correspond to the two groups of bands. Open circles: extrapolationfrom finite-size samples. Solid line: discretized k-space formula
/H2084948/H20850.
FIG. 6. Four unit cells of the Haldane model. Filled /H20849open /H20850
circles denote sites with E0=−/H9004/H20849+/H9004/H20850. Solid lines connecting near-
est neighbors indicate a real hopping amplitude t1; dashed arrows
pointing to a second-neighbor site indicates a complex hopping am-plitude t
2ei/H9278. Arrows indicate sign of the phase /H9278for second-
neighbor hopping.CERESOLI et al. PHYSICAL REVIEW B 74, 024408 /H208492006 /H20850
024408-10IV . CONCLUSIONS
We present here a formalism for the calculation of the
orbital magnetization in extended systems with broken TRsymmetry, in the case of vanishing /H20849or commensurate /H20850mac-
roscopic Bfield. This extends our previous work of Ref. 7to
the multiband case, essential for realistic calculations.
First, we consider the case of zero Chern invariant, where
we provide an an analytic proof, based upon the Wannierrepresentation. Our main result /H2084931/H20850takes the form of a BZ
integral of a gauge-invariant quantity, which can easily becomputed using reciprocal-space discretization. We providenumerical tests for a two-dimensional model, where our dis-cretized formula is checked against calculations performedfor finite samples cut from the bulk, with “open” boundaryconditions. Our numerical tests appear to confirm that indeedEq. /H2084931/H20850is the correct expression for the orbital magnetiza-
tion in a periodic system.
Second, we propose a heuristic extension of Eq. /H2084931/H20850to
the case of nonzero Chern invariant, based on the observa-tion that the integrand in Eq. /H2084931/H20850is gauge invariant, con-
trary to the analogous electrical case, where only the BZintegral is gauge invariant, notthe integrand.
8,9On the basis
of general considerations /H20849namely, invariance by translationof the energy zero /H20850, the minimal modification extending Eq.
/H2084931/H20850to the nonzero-Chern-number case yields Eq. /H2084946/H20850. Re-
markably, Eq. /H2084946/H20850is essentially identical to a recent expres-
sion derived by Xiao et al.10in the context of a semiclassical
approximation. We check the full quantum-mechanical valid-ity of Eq. /H2084946/H20850on a two-dimensional model by means of
numerical tests, comparing to finite size calculations asabove. The agreement is excellent, thus providing strongsupport for our formula, well beyond the semiclassical re-gime, even though we cannot yet provide an analytic proofof it.
Third, since our heuristic Eq. /H2084946/H20850is well-defined for a KS
metal, we also check the validity of Eq. /H2084946/H20850using the same
two-dimensional model as for the metallic case, this timeallowing the chemical potential
/H9262to be varied through the
bands. Once more the agreement is excellent, providing anumerical demonstration of the validity of Eq. /H2084946/H20850.
The electrical analog of the present theory is the modern
theory of polarization,
8,9developed in the 1990s, and valid
for insulators only. When comparing that theory with thepresent one, in the insulating case, there is an important dif-ference which is worth stressing. In the electrical case, thewhole electronic contribution to the macroscopic polariza-tion canbe expressed in terms of the electric dipoles of the
bulk WFs. This has a precise counterpart here, where thelocal-circulation contribution can in fact be expressed interms of the magnetic dipoles of the bulk WFs. However, wehave shown that in the magnetic case there is an additional“itinerant-circulation” contribution which has noelectrical
analog. When analyzing finite samples, the latter contribu-tion appears to be due to chiral currents circulating at thesample boundaries. Nonetheless, one of our major findings isthat even this contribution can be expressed as a bulk,boundary-insensitive term.
Both our original expression /H2084931/H20850and its heuristic exten-
sion /H2084946/H20850for the orbital magnetization of a crystalline solid
can easily be implemented in existing first-principles elec-tronic structure codes, making available the computation ofthe orbital magnetization in crystals, at surfaces and in re-duced dimensionality solids such as nanowires.
ACKNOWLEDGMENTS
This work was supported by ONR Grant No. 00014-03-
1-0570, NSF Grant No. DMR-0233925, and Grant No. PRIN2004 from the Italian Ministry of University and Research.
APPENDIX A: FINITE DIFFERENCE EV ALUATION OF
THE CHERN INV ARIANT AND MAGNETIZATION
Using the definition of the covariant derivative33,34
/H20841/H11509˜/H9251unk/H20856=Qk/H20841/H11509/H9251unk/H20856. /H20849A1/H20850
Equations /H2084933/H20850–/H2084935/H20850can be rewritten as
fk,/H9251/H9252=/H20858
n/H20855/H11509˜/H9251unk/H20841/H11509˜/H9252unk/H20856, /H20849A2/H20850
gk,/H9251/H9252=/H20858
n/H20855/H11509˜/H9251unk/H20841Hk/H20841/H11509˜/H9252unk/H20856, /H20849A3/H20850
FIG. 7. Orbital magnetization of the Haldane model as a func-
tion of the parameter /H9278. The lowest band is treated as occupied.
Open circles: extrapolation from finite size samples. Solid line: Eq./H2084948/H20850. The system has nonzero Chern number in the region in be-
tween the two vertical lines.
FIG. 8. Orbital magnetization of the Haldane model as a func-
tion of the chemical potential /H9262for/H9278=0.7/H9266. The shaded areas
correspond the position of the two bands. Open circles: extrapola-tion from finite-size samples. Solid line: discretized k-space for-
mula /H2084948/H20850.ORBITAL MAGNETIZATION IN CRYSTALLINE ¼ PHYSICAL REVIEW B 74, 024408 /H208492006 /H20850
024408-11hk,/H9251/H9252=/H20858
nn/H11032Enn/H11032k/H20855/H11509˜/H9251un/H11032k/H20841/H11509˜/H9252unk/H20856. /H20849A4/H20850
We assume that the occupied wave functions /H20841unk/H20856have been
computed on a regular mesh of kpoints, and we let b1,b2,
andb3be the primitive reciprocal vectors that define the k
mesh. Then the covariant derivative in mesh direction ican
be defined as
/H20841/H11509˜iunk/H20856=bi/H9251/H20841/H11509˜/H9251unk/H20856/H20849 A5/H20850
/H20849sum over /H9251implied /H20850. Inserting this into Eqs. /H20849A2/H20850–/H20849A4/H20850and
taking the antisymmetric imaginary part as in Eq. /H2084941/H20850,w e
obtain
f˜k=1
v/H9280ijlbi/H20858
nIm/H20855/H11509˜junk/H20841/H11509˜lunk/H20856, /H20849A6/H20850
g˜k=1
v/H9280ijlbi/H20858
nIm/H20855/H11509˜junk/H20841Hk/H20841/H11509˜lunk/H20856, /H20849A7/H20850
h˜k=1
v/H9280ijlbi/H20858
nn/H11032Enn/H11032kIm/H20855/H11509˜jun/H11032k/H20841/H11509˜lunk/H20856, /H20849A8/H20850
where a sum over ijkis implied and vis the volume of the
unit cell of the k-space mesh. On this mesh, the BZ integral
in Eq. /H2084942/H20850becomes a summation
C=1
2/H9266/H20858
k/H9280ijlbi/H20858
nIm/H20855/H11509˜junk/H20841/H11509˜lunk/H20856/H20849 A9/H20850
and similarly for the magnetization in Eq. /H2084943/H20850.
The appropriate finite-difference discretization of the co-
variant derivative in mesh direction iis33,34
/H20841/H11509˜iunk/H20856=1
2/H20849/H20841u˜n,k+bi/H20856−/H20841u˜n,k−bi/H20856/H20850, /H20849A10 /H20850
where /H20841u˜n,k+q/H20856is the “dual” state, constructed as a linear
combination of the occupied /H20841un,k+q/H20856at neighboring mesh
point q, having the property that /H20855un/H11032k/H20841u˜n,k+q/H20856=/H9254n/H11032n. This en-
sures that /H20855un/H11032k/H20841/H11509˜iunk/H20856=0 consistent with Eq. /H20849A1/H20850, and is
solved by the construction33,34
/H20841u˜n,k+q/H20856=/H20858
n/H11032/H20849Sk,k+q−1/H20850n/H11032n/H20841un/H11032,k+q/H20856, /H20849A11 /H20850
where
/H20849Sk,k+q/H20850nn/H11032=/H20855unk/H20841un/H11032,k+q/H20856. /H20849A12 /H20850Equations /H20849A6/H20850–/H20849A12 /H20850provide the formulas needed to
calculate the three gauge-invariant quantities f˜k,g˜k, and h˜kon
each point of the kmesh. By summing these as in Eq. /H20849A9/H20850it
is straightforward to obtain C,M˜LC, and M˜IC, respectively.
Since we have derived this finite-difference representationusing gauge-invariant quantities at each step, it is not surpris-
ing that we obtain the gauge-invariant contributions M˜LCand
M˜IC, as opposed to the gauge-dependent MLCandMIC, from
this approach.
APPENDIX B: THE NON-ABELIAN BERRY CURV ATURE
It has been noticed in Sec. II D that the vector quantity f˜k
is the Berry curvature. From Eqs. /H2084938/H20850and /H2084941/H20850, this can be
regarded as the trace of the Nb/H11003Nbmatrix Fkhaving vector
elements
Fk,nn/H11032=i/H20855/H11509kunk/H20841/H11003/H20841/H11509kun/H11032k/H20856−i/H20858
m/H20855/H11509kunk/H20841umk/H20856/H11003/H20855umk/H20841/H11509kun/H11032k/H20856.
/H20849B1/H20850
This quantity is known within the theory of the geometric
phase as the non-Abelian Berry curvature,36and character-
izes the evolution of an Nb-dimensional manifold /H20849here, the
states /H20841unk/H20856/H20850in a parameter space /H20849here, kspace /H20850. The non-
Abelian curvature is gauge covariant, meaning that if thestates are unitarily transformed as
/H20841u
nk/H20856→/H20858
n/H11032Unn/H11032/H20849k/H20850/H20841un/H11032k/H20856, /H20849B2/H20850
then the matrix Fktransforms as
Fk,nn/H11032→/H20858
mm/H11032Unm†/H20849k/H20850Fk,mm/H11032Um/H11032n/H11032/H20849k/H20850. /H20849B3/H20850
This implies that the invariants of the matrix Fk, such as its
trace f˜k, are gauge invariant. In fact, as discussed in Sec.
II D, f˜kbehaves similar to a standard /H20849i.e., Abelian /H20850curva-
ture.
We also notice that the energy matrix Ek, Eq. /H208491/H20850, is also
gauge covariant in the sense of Eq. /H20849B3/H20850. It is then easy to
verify that the trace /H20849over the band indices /H20850of the matrix
product EkFkis a gauge-invariant quantity. In fact, this trace
is identical to h˜kas defined in Sec. II D, whose gauge-
invariance we proved in a different way. The special Nb=1
case was previously dealt with in Ref. 7, where the analog of
h˜ktakes the form of the product of energy times curvature,
both gauge-invariant quantities. The present finding showsthat, in the multi-band case, this must be generalized as thetrace of the /H20849matrix /H20850product E
ktimes Fk, both gauge-
covariant quantities.
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024408-13 |
PhysRevB.91.115202.pdf | PHYSICAL REVIEW B 91, 115202 (2015)
Stable kagome lattices from group IV elements
O. Leenaerts,1,*B. Schoeters,1,2,†and B. Partoens1,‡
1Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium
2IMEC, Kapeldreef 75, B-3001 Leuven, Belgium
(Received 14 November 2014; revised manuscript received 18 February 2015; published 10 March 2015)
A thorough investigation of three-dimensional kagome lattices of group IV elements is performed with
first-principles calculations. The investigated kagome lattices of silicon and germanium are found to be of similarstability as the recently proposed carbon kagome lattice. Carbon and silicon kagome lattices are both direct-gapsemiconductors but they have qualitatively different electronic band structures. While direct optical transitionsbetween the valence and conduction bands are allowed in the carbon case, no such transitions can be observedfor silicon. The kagome lattice of germanium exhibits semimetallic behavior but can be transformed into asemiconductor after compression.
DOI: 10.1103/PhysRevB.91.115202 PACS number(s): 61 .66.Bi,61.50.−f,71.15.Mb,71.20.Mq
I. INTRODUCTION
In recent years, some new bulk phases of well-known
elements such as carbon and silicon have been proposed.A remarkable example is silicite. This layered allotrope ofsilicon was reported [ 1] recently and can be formed by
stacking dumbbell layers of silicene [ 3], which have been
experimentally observed [ 2]. Each Si atom in this three-
dimensional (3D) structure has fourfold coordination, but the
bond angles deviate considerably from the ideal 109 .5
◦. Silicite
has nonetheless an impressive cohesion of about 0.18 eV /atom
less than its cubic diamond counterpart [ 1].
Very recently, a new three-dimensional allotrope of carbon
was predicted with some interesting properties [ 4]. This
material, which was called a carbon kagome lattice (CKL),consists of 1D zigzag carbon chains locked together in a 2D
kagome lattice structure [see Fig. 1(a)], resulting in a 3D
material. Although this structure contains some 60
◦bonds
between C atoms, it was also found to be remarkably stable,comparable to C
60fullerene molecules. The presence of C
triangles in the structure gives rise to orbital frustration, whichcauses a direct gap in the electronic band spectrum. Chenet al. [4] explicitly demonstrated the origin of this band gap by
considering another related material, called an interpenetrated
graphene network (IGN), which is pictured in Fig. 1(b).
The IGN is semimetallic and it shows no orbital frustration.However, when the structure is compressed in one directionthe orbital frustration increases and the IGN transforms intothe CKL. This process is accompanied by the opening of adirect band gap in the electronic spectrum and is thereforestrong evidence that the orbital frustration is indeed the origin
of the gap. The unusual origin of the band gap makes the
electronic spectrum of the carbon kagome lattice rather uniquefor a bulk elemental semiconductor. This is illustrated by itsoptical properties, which are similar to those of direct-gapsemiconductors such as GaN and ZnO rather than otherelemental semiconductors such as diamond and silicon [ 4].
The presence of frustration in kagome lattices is not unique
to this case, but has a longer history in the condensed-matter
*ortwin.leenaerts@uantwerpen.be
†bob.schoeters@uantwerpen.be
‡bart.partoens@uantwerpen.beliterature. In fact, kagome lattices are well-known structures tostudy the physics of frustration [ 5–7]. An illustrative example
is the spin frustration in kagome lattice structures that givesrise to the formation of spin ices, liquids, and glasses [ 8,9].
In this work we expand the study of carbon kagome
lattices to the other group IV elements, silicon and germanium.Heavier elements are not considered here because we foundthem to be metallic. First we examine the dynamical stabilityof these structures through their phonon spectrum. The relative
phase stability of the silicon and germanium kagome lattices
(SiKL and GeKL) is also investigated by comparing theircohesive energies to other elemental crystals such as 3Ddiamond and 2D graphenelike crystals. We perform a detailedexamination of the electronic structure of SiKL and stressthe differences with CKL. The character of the valence andconduction band states are determined and the possible opticaltransitions between these states are investigated. The electronic
properties of germanium kagome lattices are also thoroughly
studied. We find GeKL to be semimetallic and thereforeinvestigate its behavior under strain.
II. COMPUTATIONAL DETAILS
We performed first-principles calculations within the
density functional theory formalism as implemented in
V ASP [10,11]. The generalized gradient approximation of
Perdew, Burke, and Ernzerhof (PBE-GGA) [ 12] was used
for the exchange-correlation functional together with themore advanced Heyd-Scuseria-Ernzerhof hybrid functional
(HSE06) [ 13] for band structure calculations. We made use
of the projector augmented wave method [ 14–16] to reduce
the size of the plane-wave basis set. The energy cutoff wasset to 600 eV , 400 eV , and 300 eV for the cases of C, Si, andGe systems, respectively. Integrations over the Brillouin zonewere done with a 9 ×9×16 Monkhorst-Pack k-point grid [ 17]
for structural relaxations and band structure calculations, while
a finer grid of 45 ×45×80 was used for the calculation of the
electronic density of states (DOS).
III. RESULTS
We start our study of the group IV kagome lattices by a
detailed investigation of their structural properties. The crystalstructure of the kagome lattice is shown in Fig. 1(a).T h e
1098-0121/2015/91(11)/115202(6) 115202-1 ©2015 American Physical SocietyO. LEENAERTS, B. SCHOETERS, AND B. PARTOENS PHYSICAL REVIEW B 91, 115202 (2015)
FIG. 1. (Color online) Top (top) and side (bottom) views of the
crystal structure of (a) CKL, (b) IGN, (c) SiKL, and (d) ISiN. Theunit cell is indicated with black lines.
kagome lattice consists of zigzag chains of atoms with their
axes vertically aligned. These chains are bundled in groupsof three in such a way that their atoms form alternativelysmall and larger triangles, as pictured in Fig. 1(a). Picturing
every chain as a lattice point, the zigzag chains form a 2Dtriangular lattice in which every fourth lattice site is missing,i.e., a kagome lattice. Every atom in the structure has fourfoldcoordination, which makes this crystal structure well suitedfor group IV elements. We calculated the structure parametersof the first three group IV KLs with both the PBE-GGA andHSE06 functional and show the obtained results in Table I.
The PBE-GGA results for CKL are almost indistinguish-
able from those obtained by Chen et al. [4] because we
used the same DFT code with similar settings. The structuralparameters calculated with the hybrid functional are within1% of the PBE-GGA results for C and Si, while the differenceTABLE I. Structural properties of group IV kagome lattices. The
interatomic bond lengths in the zigzag chains, dzz, and in the triangles,
dtr,a r eg i v e ni n ˚A. The bond angles in the zigzag chain, θzz,i nt h e
triangles, θtr, and between the zigzag chains and the triangles, θzt,a r e
given in degrees (◦). Values in parentheses are HSE06 results.
CKL SiKL GeKL
a 4.457 (4.422) 6.941 (6.880) 7.442 (7.316)
c 2.530 (2.516) 3.909 (3.892) 4.145 (4.083)
dzz 1.532 (1.517) 2.368 (2.349) 2.544 (2.498)
dtr 1.499 (1.491) 2.333 (2.318) 2.479 (2.442)
θzz 117.7 (117.7) 118.2 (118.1) 118.4 (118.4)
θtr 115.1 (115.0) 113.8 (114.2) 113.5 (113.5)
θzt 60.0 (60.0) 60.0 (60.0) 60.0 (60.0)
for the GeKL is about 2%. The bond lengths obtained with the
HSE06 functional are always smaller and are believed to bemore accurate [ 18,19]. Note that the geometry of the various
KLs is rather similar although their size is very different. TheSiKL and GeKL are approximately 1.55 and 1.65 times largerthan the CKL, respectively, which corresponds roughly to theratios of the covalent radii of these atoms (1.53 and 1.62),calculated in a diamond structure. All the KLs contain bondangles of 60
◦in their structure, which is very small for fourfold
coordinated atoms. As shown below, this does not prevent thestructures from having large cohesive energies [ 4].
The stability of carbon kagome lattices was already demon-
strated by Chen et al. [4], but it is interesting to see whether the
Si and Ge counterparts are stable too. Therefore, we investigatethe dynamical and relative phase stability of these differentstructures. We start with a study of their structural stability bycalculating their phonon spectra. The phonon band structuresof CKL, SiKL, and GeKL are given in Fig. 2. The absence
of negative (imaginary) phonon frequencies for CKL andSiKL [see Figs. 2(a) and2(b), respectively] demonstrates the
dynamical stability of these structures. Some small imaginaryfrequencies around the /Gamma1point are observed in the phonon
spectrum of GeKL. Since these imaginary frequencies belongto an optical mode, this observation indicates an instability ofthe GeKL structure rather than a computational inaccuracy.The negative phonon mode corresponds to a synchronizedrotation of the two triangles in the GeKL unit cell and can beremoved by a rotation of these triangles over approximately12
◦in either direction [see Fig. 3(b)]. This distortion leads
(a) (b) (c)
Γ A H K Γ M ΓycneuqerF( T H z )50
40
30
2010
0
-10
Γ A H K Γ M Γ20
15
10
50
-5Frequency (THz)
Γ A H K Γ M Γ10
8
6
42
0
-2Frequency (THz)
FIG. 2. (Color online) The phonon band spectra of (a) CKL, (b) SiKL, and (c) GeKL as calculated with the GGA functional. The negative
frequencies around /Gamma1in case of GeKL indicate an instability of the structure.
115202-2STABLE KAGOME LATTICES FROM GROUP IV ELEMENTS PHYSICAL REVIEW B 91, 115202 (2015)
(a)
Γ A H K Γ M Γ8
6
4
2
0Frequency (THz)(b)
(c)
FIG. 3. (Color online) Top (top) and side (bottom) view of the
undistorted (a) and (b) GeKL structure. (c) The phonon band spectra
of the distorted GeKL.
to an energy gain of only 5 meV /atom and results in a
dynamically stable structure, as demonstrated by the absence
of negative phonon frequencies in the phonon band spectrum ofthe distorted structure [Fig. 3(c)]. The smallness of the energy
gain makes the distortion unstable at normal temperatures: Anab initio molecular dynamics (AIMD) simulation at 500 K
needs less than 1 psto switch the direction of the distortion.
We also did not observe substantial changes in the electronicproperties upon distortion.
To investigate the relative phase stability of the KLs, we
calculate their cohesive energies and compare them with thoseof various related structures. The studied group IV elements allhave very stable 3D diamondlike allotropes, but they can alsoform 2D hexagonal lattice structures called graphene [ 20,21],
silicene, and germanene, respectively [ 22]. These 2D struc-
tures can also be used to make interpenetrated networks (IN),as shown in Figs. 1(b) and1(d). The interpenetrated graphene
network (IGN) is different from the interpenetrated siliceneand germanene networks (ISiN and IGeN) in that the latterhave a more buckled structure [indicated by the red circle inFig. 1(d)], just like their 2D counterparts [ 22]. The IGN was
proposed by Chen et al. as an intermediate structure to create
CKL because it is more stable than CKL and can be trans-formed into the latter by compressing the system [ 4]. We also
compare the KL with the recently proposed layered dumbbell(LD) structure of silicite [ 1]. The cohesive energies of all the
different elements and structures are given in Table II. These
cohesive energies are defined as the energy (per atom) neededto separate the atoms of the system. We also show the differ-ence in cohesive energy between the various allotropes and thekagome lattice structure, /Delta1E
coh=Ecoh(X)−Ecoh(KL).
Compared to the diamond structures the kagome lattices of
all investigated elements have a similar cohesive energy, i.e.,they are about 300 meV less stable. SiKL and GeKL are there-fore also realistic candidates to realize 3D kagome lattices.TABLE II. The cohesive energy Ecohof various C, Si, and Ge
allotropes. The energies are given in eV .
CS i G e
Ecoh /Delta1E coh Ecoh /Delta1E coh Ecoh /Delta1E coh
KL 7.438 0.000 4.225 0.000 3.412 0.000
diamond 7.721 0.282 4.553 0.328 3.722 0.310
x-ene 7.853 0.414 3.914 −0.311 3.243 −0.169
IN 7.613 0.175 4.080 −0.145 3.357 −0.055
LD 7.212 −0.226 4.370 0.145 3.552 0.140
The fact that the cohesive energy of CKL is 300 meV smaller
than diamond could be more or less expected by comparingthe cohesive energy of cyclohexane to cyclopropane (C
6H12
and C 3H6). The 60◦bond angles in the latter induce a strain
energy of 368 meV per strained bond angle. This calculatedvalue of the difference in the ring strain between C
6H12and
C3H6compares well to the experimental value of 0.39 eV [ 23]
and is of the same order as the difference in cohesive energybetween diamond and CKL.
Somewhat surprisingly, the experimentally realized sil-
icene [ 24] has smaller cohesive energy than the Si kagome
lattice by as much as 311 meV . This fact means that oncecreated the SiKL is more stable than silicene, but does notimply that it is easier to create the KL. Indeed, it is harder
to imagine an easy route towards the creation of SiKL than
silicene. A similar observation can be made for Ge. Germanenehas a cohesive energy that is 169 meV smaller than that ofGeKL. The Si and Ge cases should be contrasted to the Ccase where graphene has larger cohesive energy than CKLand even diamond. The reason can be found in the preferredfourfold coordination of Si and Ge atoms. sp
2hybridization
is not favorable for these elements as demonstrated by thebuckled structure of the 2D hexagonal lattices they form [ 22].
They favor sp
3hybridization and consequently prefer fourfold
over threefold coordination. This has also consequences forthe stability of the interpenetrated network structures. The INscontain a mixture of threefold and fourfold coordinated atoms,rendering ISiN (IGeN) more stable than silicene (germanene)but less stable than SiKL (GeKL). This excludes the INstructure as a possible intermediate material to create kagomelattices of Si and Ge. Finally, the same trends can be observedfor the layered dumbbell structures of Si and Ge. Thesematerials contain fourfold coordinated atoms but have lessdistorted bond angles than the KL structures. Therefore, theircohesive energy is in between those of diamond and the KLs.The carbon DL is somewhat exceptional in that it is less stablethan the CKL. This can probably be explained by the instabilityof C dumbbell configurations [ 25].
Now that we have demonstrated the stability of the different
group IV KLs, we investigate their electronic properties inthe next part. The electronic band structure of the differentKLs as calculated with the HSE06 functional is shown inFig. 4. CKL and SiKL show semiconducting behavior with
band gaps of 3.427 and 0.616 eV , respectively, while GeKLremains gapless. The band gap of CKL (3.4 eV) is considerablysmaller than that of diamond (5.3 eV [ 18]) and graphane
(4.4 eV [ 26]) at the same level of computation (HSE06), but it
is still rather large. Similarly, the band gap of SiKL (0.6 eV)
115202-3O. LEENAERTS, B. SCHOETERS, AND B. PARTOENS PHYSICAL REVIEW B 91, 115202 (2015)
-10-50510
SiKL
-4-202
Γ AH K Γ MAEnergy (eV)
GeKLCKL
-4-202
FIG. 4. (Color online) The electronic band structure of CKL,
SiKL, and GeKL as calculated with the HSE06 functional. The
valence band maximum is put to 0 for CKL and SiKL, while the
Fermi level is taken as the origin for GeKL.
is substantially smaller than that of silicon (1.15 eV [ 18])
and silicane (2.91 eV [ 27]). We can conclude from these
observations that the 3D kagome lattices tend to a have smallerband gap than their diamondlike counterparts. Consequently, itcomes as no surprise that GeK has overlapping bands becausethe bulk germanium has a band gap of only 0.8 eV [ 18].
We examine the electronic properties of CKL and SiKL
first and postpone the discussion of GeKL to the last part ofthis work. Except for the quantitative difference in the bandgap, the electronic band structure of CKL and SiKL look rathersimilar at first sight. However, further inspection shows thatthey are qualitatively different. In Fig. 5, the orbital projected
density of states and band structure as calculated within theGGA are shown. We make a distinction between s,p
x/y, and
pzorbital character. Both CKL and SiKL are found to be
direct-gap materials, which should be contrasted to the otherC and Si allotropes that show indirect gaps. The valence bandstates show strong p
x/ycharacter for both CKL and SiKL,
but the conduction bands have different symmetry. In thecase of CKL, the conduction bands have also p
x/ycharacter,
making this material into a true direct-gap material [ 4]. The
conduction bands of SiKL, on the other hand, have mixedsymmetry (predominantly slike). This qualitative difference
between the band structures of SiKL and CKL is explicitlydemonstrated by the imaginary part of the dielectric function,depicted in Fig. 5(c). The dipole-allowed transitions in CKL
start from the first conduction band, but no such transitionsare observed for SiKL. For SiKL the optical transitions onlystart from the fifth conduction band (including degeneracies),which has the same symmetry as the first conduction band ofCKL (i.e., p
x/y).
FIG. 5. (Color online) The orbital projected electronic band
structure and density of states of CKL (a) and SiKL (b) calculated
with the PBE-GGA xcfunctional. The energy corresponding to the
valence band maximum is put to zero. (c) The imaginary part of thedielectric function, /epsilon1
2of CKL (left) and SiKL (right).
The effective masses of the various possible charge carriers
in CKL and SiKL are given in Table III. Note that there is
a large difference between the GGA-PBE and HSE06 resultsfor the effective masses in SiKL. This is related with the largequantitative difference in the electronic band gap, 0.144 vs0.616 eV , resulting from the two xcfunctionals. The HSE06
results are obviously more reliable [ 28,29]. We can make a
distinction between the direction along the zigzag chains inthe kagome lattice (perpendicular) and the plane normal tothese chains (parallel). In CKL, the electrons and holes have
TABLE III. Electronic properties of C and Si kagome lattices.
The band gap is given in eV and the effective masses are given in unit
of the free electron mass.
CKL SiKL
GGA-PBE HSE06 GGA-PBE HSE06
Egap 2.257 3.427 0.144 0.616
m/bardbl
e 0.141 0.156 1.338 2.523
m/bardbl
lh −0.119 −0.127 −0.072 −0.182
m/bardbl
hh −0.511 −0.469 −0.132 −0.351
m⊥
e 1.229 1.063 0.116 0.230
m⊥
h −0.913 −0.819 −1.023 −1.889
115202-4STABLE KAGOME LATTICES FROM GROUP IV ELEMENTS PHYSICAL REVIEW B 91, 115202 (2015)
light masses in the parallel direction while their masses are
free-electron-like in the direction along the zigzag chain. Thesame is true for the holes in SiKL, but the electrons showopposite behavior in that case. This can be explained by thedifferent orbital symmetries of the respective bands. The holestates consist of p
x/yorbitals which give rise to large dispersion
(small effective mass) in the xyplane and little dispersion
(large effective mass) in the direction perpendicular to thisplane because of the size of the orbital overlap. The sameis true for the electron states in CKL which also have p
x/y
symmetry. The electron states of SiKL have mixed orbital
character with no obvious preference in direction (mostly s
like and similar contributions of px/yandpz). However the
bonds along the zigzag chain are significantly shorter than thein-plane bonds (see Table I) and lead consequently to lower
effective masses.
In the last part of this work, we treat the electronic properties
of GeKL to demonstrate the role of orbital frustration on theelectronic properties. We showed before that GeKL showssemimetallic behavior. In their recent study, Chen et al.
showed how the gapped state in CKL originates from theorbital frustration that is built in into the kagome latticestructure [ 4]. This physical origin suggests that an increase in
the orbital frustration might induce an electronic gap in GeKL.This can be achieved by increasing the orbital interaction bycompressing the GeKL crystal. In Fig. 6we show the evolution
of the electronic band structure, calculated with the HSE06
functional, when going from 0 to 6% uniform compression,in which 1% compression corresponds to a reduction of thelattice parameters with 1%. Since we are only interested in theelectronic properties, we neglect the distortions in the GeKLstructure. It can be seen that a gap appears at a compressionof 4%. Although this large compression is probably not verypractical to achieve in experiment, the opening of the band
-3-2-1012
0%
2%
4%
6%-3-2-1012
-3-2-1012Energy (eV)
-3-2-1012
Γ AH K Γ MA
FIG. 6. (Color online) The band structure of GeKL under various
compressions calculated with the HSE06 functional.0123456-0.20.00.20.40.6
0123456-0.6-0.4-0.20.00.2Energy (eV)
compression (%)VB1+2
CB1+2
CB3
Egap(eV)
compression (%)(a) (b)
FIG. 7. (Color online) The positions of the valence and conduc-
tion bands relative to the middle of the band gap (a) and the electronic
band gap (b) as a function of compression.
gap demonstrates the importance of orbital frustration in the
process.
In Fig. 7, we show the position of the relative positions of
the valence band maxima (VBM) and conduction conductionband minima (CBM), together with the size of the electronicband gap as a function of compression. The band gap is takennegative in the semimetallic case and it can be seen to be acontinuous function of the compression [see Fig. 7(b)]. Note
that there is a kink at higher compressions, which is due toa change of bands that contribute to the band gap. As can beseen in Fig. 7(a), this change occurs at approximately 5.5%.
Because the behavior of the two bands (CB1 +2 and CB3)
under compression is similar, the band gap will converge to avalue of 0.3 eV for compressions larger than 5.5%.
IV . SUMMARY AND CONCLUSIONS
We performed an in-depth ab initio study of the structural
and electronic properties of 3D kagome lattices of group IVelements. We found that such kagome lattices are stable for C,Si, and Ge, and have cohesive energies that are about 30 meVlower than the corresponding diamond crystals. Heavier groupIV elements give rise to metallic behavior and we did notinvestigate them any further. The structural parameters andelectronic properties were calculated with both GGA andhybrid functionals. The kagome lattices made from carbonand silicon were found to be semiconductors with band gapsof 3.4 and 0.6 eV , respectively. We investigated the symmetriesof the valence and conduction band states and found them tobe very different for CKL and SiKL. Although both materialsare direct band gap semiconductors, CKL allows for directdipole transitions between the VBM and CBM while this isnot the case for SiKL. The reason for this is the differentsymmetries of the lower conduction bands.
Germanium kagome lattices were shown to be dynamically
unstable to synchronized rotations of the two triangles in theunit cell over 12
◦. They exhibit semimetallic behavior but
an electronic band gap can be opened by an increase of theorbital frustration interaction by compression. The gap opensat a compression of 4% and converges to a value of 0.3 eV .
ACKNOWLEDGMENTS
This work was supported by the Fonds Wetenschappelijk
Onderzoek (FWO-Vl). The computational resources and ser-vices used in this work were provided by the VSC (FlemishSupercomputer Center), funded by the Hercules Foundationand the Flemish Government – department EWI.
115202-5O. LEENAERTS, B. SCHOETERS, AND B. PARTOENS PHYSICAL REVIEW B 91, 115202 (2015)
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115202-6 |
PhysRevB.72.054125.pdf | Pressure-induced structural phase transition in NaBH 4
C. Moysés Araújo,1R. Ahuja,1A. V . Talyzin,2and B. Sundqvist2
1Condensed Matter Theory Group, Department of Physics, Uppsala University, Box 530, SE-751 21 Uppsala, Sweden
2Department of Physics, Umeå University, S-901 87 Umeå, Sweden
/H20849Received 23 February 2005; revised manuscript received 2 May 2005; published 26 August 2005 /H20850
We present a combined experimental and theoretical study of the technologically important NaBH 4com-
pound under high pressure. Using Raman spectroscopy at room temperature, we have found that NaBH 4
undergoes a structural phase transformation starting at 10.0 GPa with the pure high-pressure phase beingestablished above 15.0 GPa. In order to compare the Raman data recorded under high pressure with thelow-temperature tetragonal phase of NaBH
4, we have also performed a cooling experiment. The known order-
disorder transition from the fcc to the tetragonal structure was then observed. However, the new high pressurephase does not correspond to this low-temperature structure. Using first-principle calculations based on thedensity functional theory, we show that the high-pressure phase corresponds to the
/H9251-LiAlH 4–type structure.
We have found a good agreement between the measured and calculated transition pressures. Additionally, wepresent the electronic structure of both the fcc and the high-pressure phases.
DOI: 10.1103/PhysRevB.72.054125 PACS number /H20849s/H20850: 64.70.Kb, 61.50.Ks, 71.15.Mb
I. INTRODUCTION
The complex alkali hydrides /H20849MXH4with M=Na, Li and
X=Al, B /H20850have attracted considerable attention, mainly for
being promising hydrogen storage materials due to their highhydrogen content /H20849up to 18.4 wt. % in LiBH
4/H20850.1,2However,
in these systems the hydrogen is held by strong covalent and
ionic bonds resulting in high dissociation temperatures.These features limit their practical applications to hydrogenstorage. To overcome such limitations, a great deal of re-search has been conducted to study the role of catalyticagents on their thermodynamic properties, which is moti-vated by the discovery that addition of a Ti-based catalyst inNaAlH
4can significantly lower the hydrogen desorption
temperature.3In addition to these attempts to make the com-
plex alkali hydrides more suitable as hydrogen storage ma-terials, much effort has also been paid to understand theirbehavior under high pressure.
4–6These compounds are ex-
pected to undergo several structural phase transformationsdisplaying novel characteristics. For instance, it was recentlypredicted theoretically
5and verified experimentally6that
LiAlH 4undergoes a phase transition at 2.6 GPa. Theory pre-
dicted a huge volume collapse of 17%, which would signifi-cantly increase its hydrogen packing density. In this area,first-principles calculations have not only been shown to bereliable but have actually played an important role, since it isvery difficult to determine the hydrogen positions by high-pressure diffraction techniques.
Recently, NaBH
4slurry was suggested as a promising
system for applications in fuel cell technology.7–10It is a very
efficient storage system /H208499.2 wt. % /H20850, and also provides a
simple way for generating hydrogen through the exothermalreaction NaBH
4+2H 2O→NaBO 2+4H 2. This reaction is ac-
tivated by adding a proper catalyst agent and can operate atambient conditions. Furthermore, the by-product NaBO
2can
be recycled into NaBH 4through a fuel recovery reaction that
makes the hydrolysis above a reversible process.8Addition-
ally, NaBH 4itself is also a promising hydrogen storage ma-
terial since it has one of the highest gravimetric hydrogendensities /H2084910.5 wt. % /H20850among the alkali metal hydrides.
1De-spite such recognized technological importance only little is
known about its fundamental physical properties. Further-more, no high pressure study has been reported yet.
In this work, we present a combined experimental and
theoretical study of NaBH
4under high pressure. Using Ra-
man spectroscopy, we show that NaBH 4undergoes a struc-
tural phase transformation starting at 10.0 GPa with the purehigh-pressure phase being established above 15.0 GPa. Thistransition is shown to be reversible on pressure release. Fromthe analysis of the Raman spectra, the high pressure phase issuggested to be either orthorhombiclike or monocliniclikewith the BH
4cluster displaying lower symmetry. In order to
compare the Raman data recorded under high pressure withthe possible low-temperature tetragonal phase of NaBH
4,w e
have also performed a cooling experiment. The known order-disorder transition from the fcc to the tetragonal structurewas then observed. However, the new high-pressure phasedoes not correspond to this low-temperature structure. Usingab initio theoretical methods we have studied six closely
related potential structures. They are namely tetragonal- P
−42
1c/H20849low-temperature phase /H20850,/H9251-NaBH 4/H20849cubic- F43m/H20850,/H9251
-LiAlH 4/H20849monoclinic, P21/c/H20850,/H9251-NaAlH 4/H20849tetragonal- I41/a/H20850,
KGaH 4/H20849orthorhombic- Cmc 21/H20850,/H9251-LiBH 4/H20849orthorhombic-
Pnma /H20850. We show that the high-pressure phase corresponds to
the/H9251-LiAlH 4type structure. The calculated transition pres-
sure of 19.0 GPa agrees quite well with the experimentalvalue of 15.0 GPa. A subsequent phase transition from
/H9251
-LiAlH 4–t o/H9251-LiBH 4–type structure is expected to occur at
33.0 GPa. In addition, we discuss the electronic structure ofboth the
/H9251-NaBH 4and the high-pressure phase.
II. EXPERIMENTAL AND COMPUTATIONAL METHODS
NaBH 4of 97% purity was purchased from Merck and
studied using a diamond-anvil cell /H20849DAC /H20850with 0.4-mm flat
culets. Using an argon-filled glove box the sample wasloaded into a 0.2-mm hole in the DAC steel gasket togetherwith a ruby chip used for pressure calibration, without anypressure-transmitting medium. The accuracy of the pressurePHYSICAL REVIEW B 72, 054125 /H208492005 /H20850
1098-0121/2005/72 /H208495/H20850/054125 /H208495/H20850/$23.00 ©2005 The American Physical Society 054125-1measurements is estimated as ±0.5 GPa. An x-ray diffraction
/H20849XRD /H20850test of the starting material confirmed good structural
quality. Only peaks due to the NaBH 4fcc structure were
observed and the calculated cell parameter a=6.16 Å is in
good agreement with literature data.11A Renishaw Raman
1000 spectrometer with a 514 nm excitation laser and a reso-lution of 2 cm
−1was used in these experiments. Raman spec-
tra were recorded in situ through the diamond anvils using a
long focus 20/H11003objective during both compression and de-
compression, using pressure steps of 1–1.5 GPa. Spectra forthe starting material and for the sample loaded into the DACwere identical. The pressure was increased gradually withRaman spectra recorded on every step during compressionand decompression. Usually, the total time spent at eachpressure step was 1–2 h. For the low-temperature measure-ments a Linkam FDCS 196 cold stage cryogenic cell cooledby cold nitrogen gas was used. Again, samples were loadedinto the cell under Ar gas in a glovebox with sub-ppm levelsof H
2O and O 2.
The electronic structure and the total energy were calcu-
lated within the framework of the generalized gradient ap-proximation /H20849GGA /H20850to density functional theory /H20849DFT /H20850/H20849Ref.
12/H20850using projected augmented wave /H20849PAW /H20850/H20849Ref. 13 /H20850
method as implemented in the
V ASP code.14We have used the
exchange-correlation potential of Perdew et al.15The PAW
potentials with the valence states 2 pand 3 sfor Na, 3 sand
3pfor B, and 1 sfor H were used. For sampling the irreduc-
ible wedge of the Brillouin zone we used k-point grids of
7/H110037/H110037 for the geometry optimization and 9 /H110039/H110039 for the
final calculation at the equilibrium volume, which were gen-erated according to the Monkhost-Pack scheme. The relax-ations of the structures were performed by using theconjugated-gradient algorithm
16and by considering the mini-
mization of both the forces and stress tensors. In all calcula-tions, self-consistency was achieved with a tolerance in thetotal-energy of at least 0.1 meV/atom. The density of states/H20849DOS /H20850for static configurations was calculated by means of
the modified tetrahedron method of Blöchl et al.
17
The equilibrium volume /H20849V0/H20850, bulk modulus /H20849B0/H20850at am-
bient pressure and its first derivative /H20849B0/H11032/H20850were estimated
through a least-square fit of calculated total energy- /H20849E-/H20850vol-
ume /H20849V/H20850set to the Murnaghan equation of states. The hydro-
static pressure as a function of the volume were calculated
from the first derivative of the E−Vcurve using the obtained
values for V0,B0, and 2B0/H11032, whereas the transition pressure is
determined by evaluating the enthalpy /H20849E+PV,a tT=0/H20850for
the two different phases. At a given pressure, the stable struc-ture is the one that has the lowest enthalpy.
III. RESULTS AND DISCUSSIONS
Raman spectra of NaBH 4at normal conditions exhibit
mostly internal modes of the /H20851BH 4/H20852−ionic clusters. In the
region of B-H stretching vibrations /H208492100–2500 cm−1/H20850the
spectrum is dominated by the symmetric stretching mode /H92631
at 2332 cm−1. The bending modes /H92632/H208491121 cm−1/H20850and
/H92634/H208491277 cm−1/H20850are relatively weak, and these modes were
impossible to observe in our DAC experiments since they
were too close to the diamond peak from the anvils. How-ever, the combination band /H92632+/H92634and the overtones 2 /H92634/H20849A1/H20850
and 2 /H92634/H20849F2/H20850are reasonably strong and can usually be ob-
served close to /H92631. Unfortunately, specific defects/impurities
in our diamonds also resulted in broad Raman peaks above2400 cm
−1. As a result of this unfortunate coincidence only
the evolution of the main peaks /H92631,2/H92634/H20849A1/H20850, and 2 /H92634/H20849F2/H20850
could be studied as a function of pressure. Figure 1 shows
the pressure dependence of the Raman spectrum of NaBH 4
in the pressure region up to 16.2 GPa recorded during com-pression. Peaks originating from the diamonds are indicatedby asterisks in the spectrum recorded at 0.3 GPa. Analysis ofFig. 1 shows that NaBH
4clearly undergoes a phase transition
above 10.0 GPa. Peaks from a new phase can be recognizedalready in the spectrum recorded at 10.8 GPa, at 13.5 GPapeaks from both low-pressure and high-pressure modifica-tions can be seen simultaneously and finally at 14.8 GPa onlypeaks from the new high pressure phase can be observed. Atthe highest pressure achieved in this experiment /H2084916.2 GPa /H20850,
peaks of the new phase are situated at 2315, 2413, 2449, and2500 cm
−1.
The evolution of the peak positions during compression is
shown in Fig. 2. The positions shift almost linearly withpressure above 2.0 GPa. Such a linear shift, but with a dif-ferent slope, is also observed in the pressure region 0-2 GPawhere only three individual points were measured. The exactnature of the observed change in slope is not clear, but wespeculate that it might be connected with the ordering tran-sition observed at low temperature or freezing of local mo-lecular motion. We also note from Fig. 1 that the relativeintensities of some peaks are changed near 2.0 GPa.
The sample was held for several days at maximum pres-
sure without detection of any changes in the Raman spec-trum. The decompression part of the experiment is shown inFigs 3 and 4. As can be seen from these figures, the phasetransition is reversible and the high-pressure modification re-verts back to normal
/H9251-NaBH 4during pressure release. The
phase transition from the high-pressure phase to the low-pressure phase can be observed to start at 10.8 GPa. At 8.3
FIG. 1. Raman spectra of NaBH 4recorded during the compres-
sion part of experiment. Broad peaks from diamond anvil impuritiesare marked with asterisks on the spectrum obtained at 0.3 GPa. Aspectrum of the original NaBH
4recorded outside the DAC is shown
for comparison /H20849bottom /H20850.ARAÚJO et al. PHYSICAL REVIEW B 72, 054125 /H208492005 /H20850
054125-2GPa we observe peaks from both phases and at 7.0 GPa only
peaks from the low-pressure phase can be observed.
The structure of the high-pressure phase found to exist
above 10–15 GPa could not be determined from our Ramandata. Only a few main peaks could be followed in our ex-periments due to the unfortunate coincidence of some peakpositions from NaBH
4with DAC-related Raman peaks. Nev-
ertheless, it can be concluded from our data that the symme-try of the BH
4ionic cluster is reduced in the high pressure
phase. Instead of the single strong peak /H92631due to symmetric
B-H stretching in /H9251-NaBH 4the high-pressure phase exhibits
three similarly strong peaks in the same region. Such a spec-trum can be consistent with, for example, an orthorhombicstructure /H20849as in LiBH
4, for example /H20850or a monoclinic one.
In principle, the new phase could be identical to the
known low-temperature tetragonal phase of NaBH 4.18The
phase transition between the tetragonal and fcc structures isof first order and occurs at /H11011186–190 K.
19According to
NMR data this phase transition is due to an order-disordertransformation,
20and it is also interesting to note that the twophases were observed to coexist in a narrow temperature
interval. This phase transition is relatively little studied andRaman spectra for the low-temperature phase have been pub-lished only for NaBD
4. Order-disorder phase transitions,
which occur at low temperatures, are often observed at roomtemperature upon pressure increase, and one well-studied ex-ample of such a phase transformation is the orientationalordering transition in C
60.21
In order to compare Raman spectra of the low-
temperature tetragonal phase of NaBH 4with our high-
pressure data a cooling experiment was performed. Ramanspectra recorded upon cooling are shown in Fig. 5 and theevolution of the peak positions is shown in Fig 6. As can beseen in Fig. 5 the phase transition from fcc to the low tem-perature tetragonal phase occurs in the temperature range193–183 K. Additional peaks and a significant sharpening ofthe original Raman peaks are observed below the transitiontemperature, in good agreement with the known order-disorder nature of this transformation.
20If we compare the
Raman spectra of the tetragonal phase with the spectra re-corded at high pressure conditions, it is clear that the new
FIG. 2. Peak positions found in the Raman spectra of NaBH 4on
compression. A phase transition occurs in the interval 12.8–14.8GPa. Peaks from the high-pressure phase are indicated by triangles.
FIG. 3. Raman spectra recorded on decompression of
NaBH 4.
FIG. 4. Peak positions of the Raman spectra of NaBH 4on
decompression.
FIG. 5. Raman spectra of NaBH 4recorded upon a cooling from
293 down to 83 K.PRESSURE-INDUCED STRUCTURAL PHASE … PHYSICAL REVIEW B 72, 054125 /H208492005 /H20850
054125-3high pressure phase observed in our experiments above 11
GPa is very different from tetragonal NaBH 4.
Our first-principles total energy calculations were per-
formed at zero temperature. We have therefore found thatthe NaBH
4ground-state corresponds to the tetragonal /H20849P
−42 1c/H20850phase in good agreement with the experimental find-
ings for the low-temperature structure. Furthermore, we have
found that this structure remains stable in whole pressurerange considered in this work. It means that no pressure-induced phase transition is expected to occur at low tempera-ture.
Since high-pressure experiments are performed at room
temperature, we have taken the experimentally observedroom temperature phase namely fcc phase as ground state tocompare it with four other closely related phases. In Fig. 7,
we show the calculated cohesive energy as a function of thevolume for all possible phases, except for KGaH
4type
/H20849orthorhombic- Cmc 21/H20850phase that has shown to be much less
stable than the others. As can be seen from Fig. 7, the fcc/H20849
/H9251-NaBH 4/H20850structure is stable at ambient pressure and at 19
GPa it shows a transition to the /H9251-LiAlH 4–type structure
/H20849/H9252-NaBH 4phase /H20850. A subsequent transformation occurs from
the/H9252-NaBH 4to the /H9251-LiBH 4type structure /H20849/H9253-NaBH 4
phase /H20850at 33 GPa. In order to show more clearly these tran-
sitions, we display, in Fig. 8, the enthalpy difference for the
/H9252and/H9253phases as well as for the NaAlH 4-type phase as a
function of pressure with reference to /H9251-NaBH 4.I no u r ab
initio calculations, we cannot treat the coexisting phases.
Thus, all calculated transitions are for the pure phases. Weshould therefore compare the theoretical value of 19 GPawith the experimental finding of 15 GPa, which are in goodagreement. Furthermore, in the
/H9251-LiAlH 4–type structure the
nearest B-H bond lengths vary between 1.22 and 1.23 Å,yielding a slightly distorted tetrahedral configuration. Such adeformation agrees with the experimental observation thatthe symmetry of BH
4ionic cluster is reduced at high pres-
sure.
The total DOS for the /H9251-NaBH 4phase at ambient pres-
sure is presented in Fig. 9 /H20849a/H20850. It displays an insulator behav-ior characterized by a band gap of nearly 6.5 eV , which is
quite close to the recent theoretical value of 6.2 eV forLiBH
4.22The valence band is split into two regions. The
low-energy band is composed of B 2 sa n dH1 sstates with a
bandwidth of 1.33 eV , whereas, the high-energy band is com-posed mainly by hybridization between H 1 sa n dB2 pstates
with a bandwidth of 2.08 eV . These are consistent with thedirectional covalent bond that occurs between the boron andhydrogen atoms. The bottom of the conduction band justabove the Fermi energy is composed primarily by the Na p
andsantibonding states which is consistent with the ionic
bonding between the Na and the BH
4unit. In Fig. 9 /H20849b/H20850,w e
present the total DOS for the /H9252-NaBH 4. The main features
are very similar to those described above for /H9251-NaBH 4. The
main difference is a narrower band gap of 6.3 eV . The originof this reduction is found to be due to the broadening of theNa antibonding states in the conduction band, which is asso-
FIG. 6. /H20849Color online /H20850Temperature dependence of the Raman
modes of NaBH 4.
FIG. 7. The energy-volume relation for all structures considered
in the calculation.
FIG. 8. The difference in enthalpy for LiAlH 4, LiBH 4, and
NaAlH 4type structures with reference to /H9251-NaBH 4as a function of
pressure.ARAÚJO et al. PHYSICAL REVIEW B 72, 054125 /H208492005 /H20850
054125-4ciated with the shortening of Na-nearest neighbors bond
length. The low and high energy regions of the valence bandhave the same bandwidth as in
/H9251-NaBH 4. This is consistent
with the slight change in the nearest B-H bond lengths.
Figures 9 /H20849c/H20850and 9 /H20849d/H20850show the correspondent DOS for the
/H9251- and /H9252-NaBH 4phases at a compressed volume V/V0
=0.6, where the V0is the equilibrium volume for each phase.
As one can observe, the bands become broader and the fun-
damental band gap tends to increase.IV. CONCLUSIONS
In summary, we present a combined experimental and the-
oretical study of NaBH 4under high pressure. Using Raman
spectroscopic technique, we show that NaBH 4undergoes a
pressure-induced phase transition from its fcc structure to anew high-pressure phase. The transformation starts above 10GPa with the pure high-pressure phase being establishedabove 15 GPa, and it is shown to be reversible with respectto pressure release. A transition back to the
/H9251phase occurs at
the somewhat lower pressures of 11–8 GPa. From the analy-sis of the Raman spectra, the new high-pressure phase isexpected to be either orthorhombic or monoclinic, with theBH
4cluster displaying lower symmetry. Through a cooling
experiment, we also show that the new high pressure phasedoes not correspond to the low-temperature tetragonal /H20849P
−42
1c/H20850phase. The theoretical investigation was performed
within the framework of GGA to DFT using PAW method.
We find that the new high-pressure phase corresponds to the
/H9251-LiAlH 4–type structure. The calculated transition pressure
of 19 GPa agrees quite well with the experimental finding of15 GPa. Furthermore, a slight distortion observed in the BH
4
supports the lower symmetry expected for this cluster. A sub-sequent phase transition is predicted to occur from
/H9252-t o/H9253
-NaBH 4at 33 GPa. This is a prediction. We welcome new
experiments to confirm this prediction. In addition, we alsodiscuss the electronic structure of both
/H9251- and/H9252-NaBH 4. The
main change corresponds to the band gap narrowing in the
/H9252-NaBH 4, which is explained to be due to the broadening of
Na antibonding states in the conduction band.
ACKNOWLEDGMENTS
This work was supported in part by Swedish Research
Council /H20849VR/H20850and by the Swedish Foundation for Interna-
tional Cooperation in Research and Higher Education/H20849STINT /H20850.
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FIG. 9. Density of states for /H9251-NaBH 4phase /H20849a/H20850and /H20849c/H20850and for
/H9252-NaBH 4phase /H20849b/H20850and /H20849d/H20850. The Fermi level is set at zero energy.PRESSURE-INDUCED STRUCTURAL PHASE … PHYSICAL REVIEW B 72, 054125 /H208492005 /H20850
054125-5 |
PhysRevB.99.224512.pdf | PHYSICAL REVIEW B 99, 224512 (2019)
Signatures of nonadiabatic superconductivity in lithium-decorated graphene
Dominik Szcz¸ e´sniak1and Radosław Szcz¸ e´sniak1,2
1Institute of Physics, Jan Długosz University in Cz¸ estochowa, Avenue Armii Krajowej 13 /15, 42-200 Cz¸ estochowa, Poland
2Institute of Physics, Cz¸ estochowa University of Technology, Avenue Armii Krajowej 19, 42-200 Cz¸ estochowa, Poland
(Received 21 February 2019; published 18 June 2019)
Recent studies confirmed that the long-searched-for phonon-mediated superconducting phase in graphene
may be induced via lithium deposition. However, it is also suggested that lithium-decorated graphene (LiC 6)
behaves as a canonical superconductor, despite the moderate electron-phonon coupling constant and lowFermi energy in this material. Herein, this issue is addressed within the isotropic Eliashberg formalism inthe adiabatic and nonadiabatic regimes to account for the potentially pivotal effects hitherto not captured. Theconducted analysis indicates signatures of the nonadiabatic superconductivity in LiC
6and the resulting effects
on the selected thermodynamic properties. It is shown that although the critical temperature obtained in thenonadiabatic regime is similar to the earlier estimates ( ∼6 K), its value corresponds to the smaller magnitude
of the Coulomb pseudopotential than previously expected. Moreover, the characteristic ratio of the pairinggap and the inverse temperature is found to notably exceed estimates of the BCS theory. In what follows,the obtained results attempt to present coherent interpretation of the superconducting phase in LiC
6and vital
insight into its fundamental properties. Moreover, they allow us to draw potential directions for future researchon low-dimensional superconductivity by reveling that many other graphene-related superconductors may besubject to nonadiabatic effects. This aspect appears to be essential for the design of new two-dimensionalhexagonal superconductors aimed at relatively high critical temperature values.
DOI: 10.1103/PhysRevB.99.224512
I. INTRODUCTION
Since its discovery, the two-dimensional (2D) carbon al-
lotrope known as graphene [ 1] has been established as an
extraordinary solid-state platform for implementation of mul-tiple electronic quantum phenomena [ 2]. However, a no-
table exception to these features constitutes superconductiv-ity, which is prevented in pristine graphene by the inherent
nature of its charge carriers [ 3]. Still, if it can be induced by
some means, superconductivity promises yet another break-through at the atomic scale associated with a substantialcross-disciplinary impact [ 3–6]. Therefore, multiple graphene
modifications were proposed in recent years to bypass thelimitations and allow induction of the conventional [ 6–9]
or unconventional [ 10–15] superconducting phase. Among
them, the decoration of graphene with lithium atoms appears
to be a particularly promising one [ 3,6,7]. This approach
allows us to promote a superconducting condensate which ismediated by the electron-phonon interaction [ 6,7], thus a con-
ventional mechanism with a mature theoretical background[3,16]. Moreover, the discussed scenario can be executed
via the well-established concept of using dopant atoms [ 17],
and it is feasible according to a recent experiment [ 7]. In
what follows, lithium-decorated graphene (LiC
6) emerges as
one of the potential case study materials in fundamental andapplied research on 2D superconductivity [ 4,5]. In particular,
general useful guidelines for future investigations on low-dimensional superconductivity are provided not only by theinitial discovery of the superconducting phase in LiC
6but also
by the follow-up studies which consider broader characteri-
zation [ 18–20], better preservation [ 21–23], or enhancement
[22–24] of the discussed state. As a result, the mentionedinvestigations already inspired multiple similar proposals of
conventional superconductivity in LiC 6derivatives [ 25] and
also in other graphene-related materials such as phosphorene[26], silicene [ 27], hexagonal boron nitride [ 28], borophene
[29], and antimonene [ 30].
In the context of the above facts, careful investigations
and proper in-depth characterization of the superconductingphase in LiC
6are of immense importance to the field of 2D
superconductivity. Nonetheless, it is argued here that despitemany earlier studies [ 6,7,18–25,31], there are still important
unexplored aspects of the superconducting state in LiC
6.O f
special attention are the recent experimental and theoreticalreports on the subject [ 7,19] which propose a consensus that
LiC
6is a canonical superconductor, as described by the BCS
theory [ 32,33]. Yet this observation is made by presuming
the canonical behavior of the pairing gap, i.e., postulating inadvance the universal BCS value of the characteristic pairinggap ratio [2 /Delta1(0)/k
BTC=3.5, where /Delta1(0) is the gap half
width at 0 K, kBdenotes the Boltzmann constant, and TCis
the critical temperature] [ 7,19]. Moreover, such behavior is
suggested regardless of the electron-phonon coupling constantin LiC
6(λ/similarequal0.6) [6,7,19], which exceeds the weak-coupling
limit ( λ/similarequal0.5) [16,34]. In what follows, the full many-body
and retardation effects (or, in general, the strong-couplingeffects) may be overlooked since they cannot be completelycaptured within the mean-field BCS theory [ 16,35]. That is to
say, the exclusion of the mentioned effects can cause prob-lems for the proper interpretation of some thermodynamicproperties in LiC
6or can even hinder observation of other
important phenomena within the present and future relatedstudies.
2469-9950/2019/99(22)/224512(7) 224512-1 ©2019 American Physical SocietySZCZ ¸ E´SNIAK AND SZCZ ¸ E´SNIAK PHYSICAL REVIEW B 99, 224512 (2019)
In fact, it appears that the mentioned canonical interpreta-
tion of superconductivity (which excludes the strong-couplingeffects) does not allow us to perceive the emerging conse-quences of the relatively narrow conduction band in LiC
6.I n
other words, the energy scales of the electrons and phonons inLiC
6are not very far from each other, meaning that the Fermi
energy EFand Debye frequency ωDdo not satisfy the fol-
lowing relation: EF/greatermuchωD(in LiC 6, the ratio ωD/EF∼0.15)
[6,7]. In the framework of the Eliashberg theory for the
strong-coupling superconductors [ 16,35], observed behavior
indicates the possible breakdown of Migdal’s theorem [ 36].
As a result, the adiabatic picture of the superconducting phaseis no longer preserved, and the nonadiabatic effects mayarise [ 37–40]. These effects are represented by the vertex
corrections to the bare electron-phonon interaction definedin the framework of the BCS theory. The vertex correctionsoriginate from the interplay of the electron-induced latticepolarization and the specific particle-hole excitations [ 41],
and to describe them consistently, the strong-coupling effectsshould also be recognized [ 40]. As a result, the nonadiabatic
effects can have an important meaning when theory attemptsto explain observed experimental data. For example, the nona-diabatic regime clarifies the high critical temperature valuein fullerene-based [ 42] and MgB
2[43] superconductors; it
allows us to understand the behavior of the groundbreakingH
3S superconductor near the Lifshitz phase transition [ 44] and
is even expected to apply in the case of cuprates [ 39] (includ-
ing when the nonphononic mechanism is considered [ 40]).
Therefore, the nonadiabatic superconductivity corresponds tothe relevant and physically observable effects in materialswhich do not behave exactly in line with the canonical pictureof the BCS theory [ 45].
Motivated by the above reasoning, herein the supercon-
ducting phase in the LiC
6superconductor is analyzed beyond
the canonical BCS picture to account for the effects hith-erto not considered. In this manner, the presented theoreticalanalysis attempts to address inconsistencies observed in the
previous interpretations of the superconducting phase in LiC
6
by recognizing the importance of the strong-coupling andnonadiabatic effects. This is done by tracing their impact onthe selected thermodynamic properties in LiC
6, such as the
temperature-dependent order parameter /Delta1(T), the Coulomb
pseudopotential μ∗, and the transition temperature TC. These
parameters are widely considered to be sensitive against thestrong-coupling and nonadiabatic effects and can be conve-niently explored within the experiment for future comparisonpurposes [ 16,45]. To this end, the discussion is given in the
context of the recent theoretical and experimental studieson the LiC
6superconductor, but the consequences of the
presented findings are also reviewed in terms of the entirefamily of graphene-related superconducting systems.
II. THEORETICAL MODEL
The convenient theoretical models which allow us to ap-
propriately discuss the parameters of interest are the Migdal-Eliashberg equations [ 16,35,36] and the Eliashberg equations
with vertex corrections to the electron-phonon interaction[37–40]. The former formalism is the strong-coupling gen-
eralization of the BCS theory, whereas the latter techniqueadditionally allows us to consider the nonadiabatic effects.In detail, the Eliashberg equations are introduced here withinthe isotropic approximation in order to conveniently referto the available isotropically averaged experimental data [ 7].
Moreover, since the occurrence of nonadiabatic effects inLiC
6is not certainly related to the breakdown of the Fermi
liquid picture, the vertex corrections are described withinthe perturbation scheme [ 38]. Hence, the general form of
the Eliashberg equations on the imaginary axis for the orderparameter function φ
n=φ(iωn) and the wave function renor-
malization factor Zn=Z(iωn) can be written as
φn=πkBTM/summationdisplay
m=−MKn,m−μ⋆
m/radicalbig
ω2mZ2m+φ2mφm−Vπ3(kBT)2
4EFM/summationdisplay
m=−MM/summationdisplay
m/prime=−MKn,mKn,m/prime/radicalBig/parenleftbig
ω2mZ2m+φ2m/parenrightbig/parenleftbig
ω2
m/primeZ2
m/prime+φ2
m/prime/parenrightbig/parenleftbig
ω2
−n+m+m/primeZ2
−n+m+m/prime+φ2
−n+m+m/prime/parenrightbig
×[φmφm/primeφ−n+m+m/prime+2φmωm/primeZm/primeω−n+m+m/primeZ−n+m+m/prime−ωmZmωm/primeZm/primeφ−n+m+m/prime]( 1 )
and
Zn=1+πkBT
ωnM/summationdisplay
m=−MKn,m/radicalbig
ω2mZ2m+φ2mωmZm−Vπ3(kBT)2
4EFωn
×M/summationdisplay
m=−MM/summationdisplay
m/prime=−MKn,mKn,m/prime/radicalBig/parenleftbig
ω2mZ2m+φ2m/parenrightbig/parenleftbig
ω2
m/primeZ2
m/prime+φ2
m/prime/parenrightbig/parenleftbig
ω2
−n+m+m/primeZ2
−n+m+m/prime+φ2
−n+m+m/prime/parenrightbig
×[ωmZmωm/primeZm/primeω−n+m+m/primeZ−n+m+m/prime+2ωmZmφm/primeφ−n+m+m/prime−φmφm/primeω−n+m+m/primeZ−n+m+m/prime], (2)
where the parameter Vdistinguishes between the Migdal-
Eliashberg equations (M-E; when V=0) and the Eliashberg
equations with the vertex corrections to the electron-phononinteraction (E-V; when V=1). As a reminder, k
Bis the
Boltzmann constant, Tdenotes temperature, and EFstands for
the Fermi energy. Moreover, ωnrepresents the nth Matsubarafrequency: ωn=πkBT(2n+1). In what follows, Mconsti-
tutes the cutoff value for the calculations, which is equal to1100 in order to ensure numerical stability for T>1K .T h e
Coulomb pseudopotential which depends on the Matsubarafrequency (and models the depairing correlations) is given asμ
⋆
n=μ⋆θ(ωC−|ωn|), where θis the Heaviside function and
224512-2SIGNATURES OF NONADIABATIC SUPERCONDUCTIVITY … PHYSICAL REVIEW B 99, 224512 (2019)
50 100 150 20000.30.60.9
50 100 150 20000.20.42F()
(meV)(a)
(meV)(b)
FIG. 1. The (a) theoretically and (b) experimentally derived
Eliashberg [ α2F(ω)] functions employed in the present analysis,
adopted from [ 6,7], respectively.
ωCrepresents the cutoff frequency. Note that it is assumed
thatωC=3ωmax, where ωmaxis the maximum value of the
phonon frequency ωin LiC 6. Finally, the electron-phonon
pairing kernel is defined in the following manner:
Kn,m≡2/integraldisplayωD
0dωω
ω2+4π2(kBT)2(n−m)2α2F(ω),(3)
where α2F(ω) is the electron-phonon spectral function (the
Eliashberg function), given as
α2F(ω)=1
2πρ(EF)/summationdisplay
qνδ(ω−ωqν)γqν
ωqν. (4)
In Eq. ( 4),αis the average electron-phonon coupling, F(ω)
represents the phonon density of states, and γqνis written in
the following manner:
γqν=2πωqν/summationdisplay
ij/integraldisplayd3k
/Omega1BZ|gqν(k,i,j)|2
×δ(Eq,i−EF)δ(Ek+q,j−EF), (5)
where ωqνgives values of the phonon energies and γqν
denotes the phonon linewidth. In this context, the electron-
phonon coefficients are represented by gqν(k,i,j), and Ek,i
stands for the electron band energy. Note that only the first-
order corrections are included in the Eliashberg equationsdue to the moderate λω
D/EFratio in LiC 6, which allows
truncation of the high-order terms [ 46]. Moreover, the mo-
mentum dependence of the electron-phonon matrix elementsis neglected (the local approximation) in order to followthe previously assumed isotropic character of the Eliashbergequations. Consequently, the Eliashberg equations allow usto calculate the order parameter of the form /Delta1
n(T,μ∗)=
φn/Zn, which is the central property employed to estimate the
thermodynamics of interest.
To this end, the details of the electron-phonon processes in
LiC 6are encoded here within one of the two representative
and isotropically averaged α2F(ω) functions, namely, the
function obtained theoretically in [ 6] by using the density
functional theory and the function derived directly from theangle-resolved photoemission spectroscopy experiment in [ 7]
(see Fig. 1for their graphical representation). In this manner,
it is possible to conveniently provide discussion of the LiC
6
thermodynamics on the same footing with respect to theavailable experimental and theoretical predictions. For the
completeness of the present analysis it is also important tonote that the pairing gap in LiC
6is actually predicted to have a
single anisotropic structure [ 7,19]. Nonetheless, the isotropic
approximation still provides an average picture which is suf-ficient for the purposes of the present study, as reinforced bythe results provided later in the text.
III. RESULTS AND DISCUSSION
In the present analysis, the described Eliashberg equations
are solved by using numerical procedures developed in house,originally employed in [ 20,47–49]. The main results of the
computations are summarized in Fig. 2, which presents the
behavior of the order parameter as a function of the Coulombpseudopotential (top row) and the temperature (bottom row).For transparency, the obtained results are additionally dividedinto two sets which correspond to the calculations conductedfor the theoretically [ 6] (left column) and experimentally [ 7]
(right column) derived Eliashberg functions. Moreover, theestimates of the Eliashberg equations with (E-V) and without(M-E) vertex corrections are conveniently marked by the openand solid symbols, respectively.
In this context, Figs. 2(a) and 2(b) correspond to the
initial computations, which are aimed at the determination ofthe critical value of the Coulomb pseudopotential μ
∗
C.N o t e
that this part of the analysis is carried out at T=3.5Kt o
coincide with the temperature for which the value of the orderparameter [ /Delta1(3.5)∼0.9 meV] was experimentally measured
in [7]. For convenience, in Figs. 2(a) and 2(b) the/Delta1(3.5)
value given in [ 7] is marked by the vertical solid blue line,
whereas its error bars are depicted as the vertical dashed linesabove and below the referential value, to present the experi-mentally assessed region (shaded green). In this manner, μ
∗
C
corresponds to the value for which the Eliashberg equations
yield the μ∗-dependent order parameter /Delta1(3.5,μ∗) equal to
the/Delta1(3.5) estimate given in [ 7]. In other words, μ∗
Cmarks the
point where the results of the Eliashberg equations cross thevertical blue line in Figs. 2(a)and2(b), as depicted by the open
black circles (M-E results) and triangles (E-V results). Finally,Figs. 2(c) and2(d) present the order parameter as a function
of the temperature for μ
∗equal to the previously determined
μ∗
Cvalues. Therein, the solid lines mark the central results
[/Delta1(T,μ∗)], whereas dashed lines describe their deviation
values [ /Delta1+(T,μ∗) and/Delta1−(T,μ∗)], which correspond to the
error bars in Figs. 2(a) and2(b). In what follows, the critical
values of the temperature are determined by using the follow-ing relation: /Delta1(T
C,μ∗
C)=0.
According to the estimates depicted in Figs. 2(a) and
2(b), it can be qualitatively observed that both sets of results
exhibit the same general behavior. Specifically, the inclusionof the vertex corrections leads to a notable decrease of theμ
∗value by ∼10% in the entire range of the /Delta1(T,μ∗)
function, regardless of the employed Eliashberg function type.Naturally, the results in Figs. 2(a) and2(b) differ from each
other, which should be attributed to the different shapes of the
Eliashberg functions assumed for calculations. This differenceamounts to around 10% between the results obtained withinthe same approach but for different Eliashberg functions. Thelower values correspond to the calculations based on the
224512-3SZCZ ¸ E´SNIAK AND SZCZ ¸ E´SNIAK PHYSICAL REVIEW B 99, 224512 (2019)
0.10 0.12 0.14 0.160.40.60.81.01.21.4
0480.00.40.81.20.10 0.12 0.14 0.160.40.60.81.01.21.4
0480.00.40.81.2Order parameter (meV)
Coulomb pseudopotentialM-E
E-V
Experiment
T=3.5 K (a)
*=*
CΔ(T,*):
M-E
E-V
Δ+(T,*):
M-E
E-V
Δ-(T,*):
M-E
E-V
*=*
COrder parameter (meV)
Temperature (K)Δ(T,*):
M-E
E-V
Δ+(T,*):
M-E
E-V
Δ-(T,*):
M-E
E-V
(c)Coulomb pseudopotentialM-E
E-V
Experiment
T=3.5 K (b)
Temperature (K)(d)
FIG. 2. The functional dependency of the order parameter in LiC 6on (a) and (b) the Coulomb pseudopotential μ∗at 3.5 K and (c) and
(d) the temperature Tat the critical values of μ∗. Results obtained within the Eliashberg formalism with (E-V) and without (M-E) the first-order
vertex corrections to the electron-phonon interaction are marked by the open and solid symbols, respectively. The first column presents
estimates obtained for the theoretically calculated electron-phonon spectral function given in [ 6], whereas the second column corresponds
to the function experimentally derived in [ 7]. For convenience, the referential experimental value of the order parameter at 3.5 K [ 7] (solid blue
line) and its error bars (dashed blue lines) are depicted in (a) and (b). In (c) and (d), the corresponding order parameter /Delta1(T,μ∗) values and
their deviations [ /Delta1+(T,μ∗)a n d/Delta1−(T,μ∗)] are marked by the solid and dashed lines, respectively.
experimentally derived α2F(ω) function, which is possibly
related to the imperfections of the experimental sample (e.g.,
random Li coverage) not present in the theoretical model.Nonetheless, the origin of the reduction behavior due to thevertex corrections appears to be the same for both sets ofthe results, i.e., the increased role of the dynamical effects inthe Eliashberg equations (stronger dependence on the fre-quency) caused by the nonadiabatic character of the super-conducting phase in LiC
6.
An additional physical meaning of the presented results
can be derived from the critical values of μ∗, despite the
fact that μ∗
Cis often considered a fitting parameter within
the Eliashberg formalism, which allows us to match modelpredictions with the experimental observables. In LiC
6,t h e
obtained μ∗
Cvalues are equal to 0.126 (0.140) and 0.114
(0.129), as calculated within the Eliashberg equations with(without) vertex corrections for the theoretical [ 6] and ex-
perimental [ 7]α
2F(ω) functions, respectively. In general,
for most phonon-mediated superconductors the values of μ∗
Cshould be of the order of 0.1–0.14 [ 50]. Yet it can be expected
thatμ∗
Cwill approach lower limits of this regime when the
electron-phonon constant is close to the weak-coupling limit(λ∼0.5), like in LiC
6. In other words, lower μ∗
Cimplies
weaker destructive electron correlations in the system, whichcan be realistically compensated by moderate λto allow the
induction of the superconducting state. Consequently, in thecase of the LiC
6superconductor, the predictions of the E-V
model appear to be more refined and realistic than theiradiabatic counterparts, even though the energy separationbetween the electronic and phonon scales is relatively low.This observation is even more coherent when noting thatthe experimentally derived Eliashberg function, which givesslightly lower λthan the theory [ 6,7], also leads to the previ-
ously observed lower values of μ
∗, including critical values.
Finally, to assess the accuracy of the obtained μ∗
Cestimates, it
should be noted that the Coulomb pseudopotential is sensitiveto the assumed cutoff frequency. However, even when theinitial ω
Cis increased up to the widely accepted threshold
224512-4SIGNATURES OF NONADIABATIC SUPERCONDUCTIVITY … PHYSICAL REVIEW B 99, 224512 (2019)
value of 10 ωmax[46,51],μ∗
Cdecreases consistently within all
employed approaches by only a few percent in comparisonto the results presented above. At the same time, the ω
Cvalue
does not influence the temperature-dependent order parameterand the corresponding critical temperature estimates, whichare discussed later in this section. Therefore, the analysisis continued for the initially assumed criterion ω
C=3ωmax,
which simultaneously allows us to lower the computationalneeds of the numerical procedures.
At this point, it is also instructive to compare re-
sults in Figs. 2(a) and 2(b) with the predictions of the
anisotropic Migdal-Eliashberg equations, provided by Zhengand Margine [ 19]. Such inspection allows us to note that
the M-E estimates in Fig. 2(a) are virtually the same as the
ones suggested in [ 19], whereas the remaining μ
∗
Cvalues
are evidently lower. The observed partial convergence of theresults can be related to the qualitative similarities of the elec-tronic and vibrational properties, as calculated theoreticallyin the isotropic [ 6] and anisotropic studies [ 19] discussed
here. In what follows, the results given here support the factthat the isotropic description of the paring gap in LiC
6well
approximates its realistic anisotropic structure, at least to theextent required in the present analysis. On the other hand,the visible and well-defined differences from other calculatedresults reinforce the suggestion that the nonadiabatic effectsplays an important role in LiC
6and cannot be simply omitted.
To better understand the influence of the vertex corrections
on the thermodynamics of the LiC 6superconductor, their
impact on the critical temperature is discussed next. Again,notable changes due to the vertex corrections can be noticedfor the order parameter as a function of temperature forμ
∗=μ∗
C[see Figs. 2(b) and 2(d)]. In general, the vertex
corrections cause a decrease of the TCvalue and an increase
of the /Delta1(0,μ∗
C) level for all considered cases. Interestingly,
in the framework of the M-E formalism the TCestimates for
both considered Eliashberg functions are the same and equalto 6.2 K. Likewise, the inclusion of the vertex correctionsdifferentiates both values only slightly and gives 5.8 and5.9 K for the theoretically and experimentally derived α
2F(ω)
functions, respectively. In this context, the M-E estimatesare slightly higher than the ones reported in [ 7,19], which
equals ∼5.8 K, whereas the E-V calculations yield practically
the same value as the one in the mentioned studies. Theformer disagreement is due to the fact that /Delta1(0,μ
∗
C) is not
presumed here to be equal to the experimentally extracted/Delta1(3.5) value, as was ambiguously done in [ 7,19]. On the
other hand, the latter convergence of the results is due tothe interplay between the lack of the above presumptionand the inclusion of the vertex corrections. Note, however,that although the discussed convergence occurs, only resultspresented here simultaneously give physically relevant μ
∗
C
values. In fact, the observed behavior of the critical tempera-
ture allows for additional insight into the previously calculatedcritical values of the Coulomb pseudopotential. Note that inorder to obtain the same predictions for T
Cwithin the M-E and
E-V models, the discrepancies between the corresponding μ∗
C
values will increase in comparison to the already calculated
ones. Therefore, the vertex corrections act as an effective μ∗
C
which can be mimicked in the adiabatic models only when
assuming much higher depairing correlations. That is to say,the same values of μ∗
Cin the M-E and E-V models will
lead to an even more evident decrease of the TCvalue in the
discussed material. In general, such a decrease of the TCvalue,
observed in the present paper, resembles the situation forweakly correlated materials [ 52], which is in agreement with
the previously calculated low μ
∗
Cvalues in LiC 6. Therefore, it
can be argued that this behavior is caused by the fact that thenegative part of the vertex function becomes most relevant,contrary to the situation in fullerene-based [ 42] and MgB
2
[43] superconductors.
Finally, the effects of the vertex corrections are also vis-
ible when inspecting the /Delta1(0,μ∗
C) levels, as noted above.
In particular, already the M-E formalism predicts here thatLiC
6is not a canonical example of the phonon-mediated
superconductor, contrary to suggestions made in [ 7,19]. The
2/Delta1(0,μ∗
C)/kBTCratio equals 3.72 and 3.71 within the M-E
model for the theoretical and experimental Eliashberg func-tions, respectively. Therefore, presented estimates notablyexceed the canonical value of 3.5, which is predicted bythe BCS theory [ 32,33]. Note that one can expect a slight
decrease of these values when the anisotropic effects are in-cluded in the calculations. However, based on previous similarcomparison studies [ 53], this decrease is not expected to be
more than a few percent. Thus, even when the anisotropiceffects are included, the 2 /Delta1(0,μ
∗
C)/kBTCratio is not likely
to approach the BCS limit. Moreover, the discrepancy be-tween the BCS predictions and the results presented here iseven more visible in the case when the vertex correctionsare included in the calculations. Specifically, the E-V modelincreases the 2 /Delta1(0,μ
∗
C)/kBTCratio to values of 4.57 and
4.37 for the theoretical and experimental Eliashberg functions,respectively. It is important to note here that the obtainedlevels are surprisingly high, which can be related to the highnumerical sensitivity of the E-V solutions at low temperatures.Nonetheless, the obtained results, even if qualitatively, suggestagain an increased role of the strong-coupling effects andreinforce previous observations on the importance of thenonadiabatic effects in the LiC
6superconductor.
IV . SUMMARY AND CONCLUSIONS
In summary, the analysis presented here within the adia-
batic and nonadiabatic regimes of the Eliashberg equationssuggests that the phenomenology of the superconducting statein LiC
6is more complex than previously expected. Specif-
ically, it is predicted that LiC 6exhibits a strong-coupling
superconducting phase mediated by the nonadiabatic pairing,rather than the canonical BCS state. This conclusion derivesfrom the observed impact of nonadiabatic effects on theselected thermodynamic properties in LiC
6, such as the criti-
cal temperature, the Coulomb pseudopotential, and the char-acteristic paring gap ratio. In detail, the inclusion of the men-tioned effects yields the critical temperature value of ∼6K ,i n
agreement with available reports [ 7,19]. However, this value
is obtained for a much smaller Coulomb pseudopotential (i.e.,smaller depairing correlations) and higher-order parameter at0 K than in previous studies. The suggested new characterof the superconducting phase is additionally reinforced bythe fact that the inclusion of the new effects explains consis-tently the coexistence of the relatively high electron-phonon
224512-5SZCZ ¸ E´SNIAK AND SZCZ ¸ E´SNIAK PHYSICAL REVIEW B 99, 224512 (2019)
TABLE I. The bare (ωD
EF) and dressed ( λωD
EF) nonadiabatic ratios for previously proposed graphene-related superconductors, as calculated
by employing available theoretical values of the Fermi energy EF, the Debye frequency ωD, and the electron-phonon coupling constant λ.
The first three data columns are adopted from previous studies based on density functional theory calculations. Presented data are sorted with
respect to the λvalues.
λω D(meV) EF(meV)ωD
EFλωD
EF
LiC 6[6] 0.61 193 1333 0.145 0.088
LiC 6(on a h-BN support) [ 23] 0.67 182 1353 0.134 0.089
LiC 6(10% tensile biaxial strain) [ 24] 0.73 132 1804 0.122 0.089
LiC 6(bilayer) [ 22] 0.86 182 1577 0.115 0.099
Silicene (5% tensile biaxial strain) [ 27] 1.04 63 667 0.095 0.099
Graphane (8% pdoped) [ 8] 1.44 174 969 0.180 0.259
Phosphorene (4% tensile biaxial strain) [ 26] 1.6 47 183 0.258 0.412
h-CN (4% pdoped, 15.6% tensile biaxial strain) [ 54] 3.35 133 3251 0.041 0.137
coupling and small difference between the electron and
phonon energy scales in LiC 6.
As a consequence, the outcomes described above allow
us to better understand the fundamental physics behind thesuperconducting phase in lithium-decorated graphene. Thissets a new context for future investigations, including researchaimed at the additional evidence for the nonadiabatic pairing
in LiC
6. In this respect, an interesting open problem is the
combination of nonadiabatic and anisotropic effects withinthe Eliashberg formalism, allowing simultaneous assessmentof these effects and potential improvement of the accuracyof theoretical predictions. However, obtained results also in-
dicate that the nonadiabatic effects may play an important
role in other low-dimensional superconductors. In particular,the literature review shows an entire group of the potentialgraphene-related superconductors which exhibits the nonzeroω
D/EFratio, similar to the discussed LiC 6system. In Table I,
such superconductors are summarized in terms of their elec-
tron and phonon energy scales as well as the electron-phononcoupling strength. The most important predictions for suchmaterials can be related to the impact of the nonadiabaticeffects on their critical temperature values. Specifically, the
other graphene-related materials may behave similarly to LiC
6
and, consequently, be subject to the reduction of their TC
values due to the nonadiabatic effects. In fact, by arguing that
the strength of the nonadiabatic effects scales as λωD/EF,
materials with higher λare likely to present the strongest
reduction of TC(e.g., the graphane, phosphorene, and h-CN
materials in Table I). This is clearly a negative effect which
should be considered when searching for higher critical tem-perature values in graphene-related materials. However, it alsoprovides a hint that the design of graphene-related materials
with a reduced role of nonadiabatic effects may be more
beneficial, e.g., by exploring systems with broader conductionbands.
ACKNOWLEDGMENTS
D.S. would like to acknowledge financial support for this
work under a Jan Długosz University Research Grant forYoung Scientists (Grant No. DSM /WMP/5517/2017).
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