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2401.11065v1.Extremely_strong_spin_orbit_coupling_effect_in_light_element_altermagnetic_materials.pdf | Extremely strong spin-orbit coupling effect in light element altermagnetic materials
Shuai Qu1,3, Ze-Feng Gao1,2,3, Hao Sun2, Kai Liu1,3, Peng-Jie Guo1,3,∗and Zhong-Yi Lu1,3†
1. Department of Physics and Beijing Key Laboratory of Opto-electronic Functional
Materials &Micro-nano Devices. Renmin University of China, Beijing 100872, China
2. Gaoling School of Artificial Intelligence, Renmin University of China, Beijing, China and
3. Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education),
Renmin University of China, Beijing 100872, China
(Dated: January 23, 2024)
Spin-orbit coupling is a key to realize many novel physical effects in condensed matter physics,
but the mechanism to achieve strong spin-orbit coupling effect in light element antiferromagnetic
compounds has not been explored. In this work, based on symmetry analysis and the first-principles
electronic structure calculations, we demonstrate that strong spin-orbit coupling effect can be real-
ized in light element altermagnetic materials, and propose a mechanism for realizing the correspond-
ing effective spin-orbit coupling. This mechanism reveals the cooperative effect of crystal symmetry,
electron occupation, electronegativity, electron correlation, and intrinsic spin-orbit coupling. Our
work not only promotes the understanding of light element compounds with strong spin-orbit cou-
pling effect, but also provides an alternative for realizing light element compounds with an effective
strong spin-orbit coupling.
Introduction. Spin-orbit coupling (SOC) is ubiquitous
in realistic materials and crucial for many novel physical
phenomena emerging in condensed matter physics, in-
cluding topological physics [1–3], anomalous Hall effect
[4], spin Hall effect [5, 6], magnetocrystalline anisotropy
[7] and so on. For instance, quantum anomalous Hall
(QAH) insulators are characterized by non-zero Chern
numbers [8]. The Chern number is derived from the inte-
gration of Berry curvature over the occupied state of the
Brillouin zone (BZ). For collinear ferromagnetic and an-
tiferromagnetic systems, the integral of Berry curvature
over the occupied state of the Brillouin zone must be zero
without SOC due to the spin symmetry {C⊥
2T||T}. Here
theC⊥
2andTrepresent the 180 degrees rotation perpen-
dicular to the spin direction and time-reversal operation,
respectively. Therefore, QAH effect can only be real-
ized in collinear magnetic systems when SOC is included
[9, 10]. On the other hand, strong SOC may open up a
large nontrivial bandgap, which is very important to re-
alize QAH effect at high temperatures. In general, strong
SOC exists in heavy element compounds. Unfortunately,
the chemical bonds of heavy element compounds are
weaker than those of light element compounds, which
leads to more defects in heavy element compounds. Thus,
the stability to realize exotic functionalities in heavy el-
ement compounds is relatively weak.
An interesting question is whether the strong SOC ef-
fect can be achieved in light element compounds. Very
recently, Li et al. demonstrated that the SOC can be en-
hanced in light element ferromagnetic materials, which
derives from the cooperative effects of crystal symme-
try, electron occupancy, electron correlation, and intrin-
sic SOC [11]. This provides a new direction for the design
of light element materials with strong effective SOC.
∗guopengjie@ruc.edu.cn
†zlu@ruc.edu.cnVery recently, based on spin group theory, altermag-
netism is proposed as a new magnetic phase distinct from
ferromagnetism and conventional collinear antiferromag-
netism [12, 13]. Moreover, altermagnetic materials have
a wide range of electronic properties, which cover met-
als, semi-metals, semiconductors, and insulators [13, 14].
Different from ferromagnetic materials with s-wave spin
polarization, altermagnetic materials have k-dependent
spin polarization, which results in many exotic physical
effects [12, 13, 15–21]. With spin-orbit coupling, similar
to the case of ferromagnetic materials, the time-reversal
symmetry-breaking macroscopic phenomena can be also
realized in altermagnetic materials [10, 22–24]. Never-
theless, altermagnetism is proposed based on spin group
theory and the predicted altermagnetic materials basi-
cally are light element compounds [13, 14]. Therefore,
it is very important to propose a mechanism to enhance
SOC in light element compounds with altermagnetism
and predict the corresponding compounds with strong
SOC effect.
In this work, based on symmetry analysis and the first-
principles electronic structure calculations, we predict
that the light element compound NiF 3is an i-wave al-
termagnetic material with extremely strong SOC effect.
Then, we propose a mechanism to enhance SOC effect
in light element compounds with altermagnetism, which
reveals the cooperative effects of crystal symmetry, elec-
tron occupation, electronegativity, electron correlation,
and intrinsic SOC. We also explain the weak SOC effect
in altermagnetic materials VF 3, CrF 3, FeF 3, CoF 3.
Results and discussion. The NiF 3takes rhombohedral
structure with nonsymmorphic R −3c (167) space group
symmetry, as shown in Fig. 1 (a)and(b). The corre-
sponding elementary symmetry operations are C 3z, C1
2t
and I, which yield the point group D 3d. The t repre-
sents (1/2, 1/2, 1/2) fractional translation. To confirm
the magnetic ground state of NiF 3, we consider six dif-
ferent collinear magnetic structures, including one ferro-arXiv:2401.11065v1 [cond-mat.mtrl-sci] 19 Jan 20242
FM AFM2 AFM4
AFM3 AFM5
AFM1
a
b
c
Ni
F
a
b
c
140
°(a) (c)(e)(g)
(b) (d)
(f)
(h)
a
b
c
FIG. 1. The crystal structure and six collinear magnetic structures of NiF 3.(a)and(b)are side and top views of the crystal
structure, respectively. The cyan arrow represents the direction of easy magnetization axis. (c)-(h)are six different collinear
magnetic structures including one ferromagnetic and five different collinear antiferromagnetic structures. The bond angle of
Ni−F−Ni for the nearest neighbour Ni ions is 140 degrees. The primitive cell of NiF 3is shown in (d). The red and blue
arrows represent spin-up and spin-down magnetic moments, respectively.
magnetic and five collinear antiferromagnetic structures
which are shown in Fig. 1 (c)-(h) . Then we calculate
relative energies of six magnetic states with the varia-
tion of correlation interaction U. With the increase of
correlation interaction U, the NiF 3changes from the fer-
romagnetic state to the collinear antiferromagnetic state
AFM1 (Fig. 2 (a)). The AFM1 is intralayer ferromag-
netism and interlayer antiferromagnetism (Fig. 1 (d)). In
previous works, the correlation interaction U was selected
as 6.7eV for Ni 3dorbitals [25, 26]. Thus, the magnetic
ground state of NiF 3is the AFM1 state, which is consis-
tent with previous works[14]. On the other hand, since
the bond angle of Ni −F−Ni for the nearest neighbour
Ni ions is 140 degrees, the spins of the nearest neigh-
bour and next nearest neighbour Ni ions are in antipar-
allel and parallel arrangement according to Goodenough-
Kanamori rules [27], respectively. This will result in NiF 3
being the collinear antiferromagnetic state AFM1. Thus,
the results of theoretical analysis are in agreement with
those of theoretical calculation.
Indeed, the structure of AFM1 is very simple and the
corresponding magnetic primitive cell only contains twomagnetic atoms with opposite spin arrangement which
is shown in Fig. 2 (b). From Fig. 2 (b), the two Ni
atoms with opposite spin arrangement are surrounded by
F-atom octahedrons with different orientations, respec-
tively. Thus, the two opposite spin Ni sublattices cannot
be connected by a fractional translation. Due to two Ni
ions located at space-inversion invariant points, the two
opposite spin Ni sublattices cannot be either connected
by space-inversion symmetry. However, the two opposite
spin Ni sublattices can be connected by C1
2t symmetry.
Thus, the NiF 3is an altermagnetic material. The BZ of
altermagnetic NiF 3is shown in Fig. 2 (c)and both the
high-symmetry and non-high-symmetry lines and points
are marked. In order to display the altermagnetic prop-
erties more intuitively, we calculate polarization charge
density of altermagnetic NiF 3, which is shown in Fig.
2(d). From Fig. 2 (d), the polarization charge densi-
ties of two Ni ions with opposite spin arrangement are
anisotropic and their orientations are different, result-
ing from F-atom octahedrons with different orientations.
The anisotropic polarization charge densities can result in
k-dependent spin polarization in reciprocal space. More-3
FM
AFM1
AFM2
AFM3
AFM4
AFM5
ΓFT
L
SHSଶmSଶHଶ(a)
(c) (d)(b)
abc
FIG. 2. The magnetic ground state of NiF 3and the cor-
responding properties. (a)Relative energies of six differ-
ent magnetic states with the variation of correlation inter-
action U. (b)and(c)are the magnetic primitive cell of NiF 3
and the corresponding Brillouin zone, respectively. The red
and blue arrows represent spin-up and spin-down magnetic
moments, respectively. The high-symmetry and non-high-
symmetry lines and points are marked in the BZ. (d)The
anisotropic polarization charge densities. The red and blue
represent spin-up and spin-down polarization charge density,
respectively.
over, according to different spin group symmetries, the
k-dependent spin polarization can form d-wave, g-wave,
ori-wave magnetism [12].
Without SOC, the nontrivial elementary spin sym-
metry operations in altermagnetic NiF 3have{E||C3z},
{C⊥
2||M1t},{E||I}, and {C⊥
2T||T}. The spin symme-
tries{C⊥
2||M1t},{T||TM 1t}, and {E||C3z}make alter-
magnetic NiF 3being an i-wave magnetic material, as
shown in Fig. 3 (a). Moreover, the spins of bands are
opposite along non-high-symmetry S 2−Γ and Γ −m(S 2)
directions, reflecting features of i-wave magnetism (Fig.
3(b)).
In order to well understand the electronic properties,
we also calculate the electronic band structures of alter-
magnetic NiF 3along the high-symmetry directions. Ig-
noring SOC, the NiF 3is an altermagnetic metal. There
are four bands crossing the Fermi level due to spin de-
generacy on the high-symmetry directions (Fig. 3 (c)).
Especially, these four bands are degenerate on the Γ −T
axis. In fact, any kpoint on the Γ −T axis has nontriv-
ial elementary spin symmetry operations {E||C3z}and
{C⊥
2||M1t}. And the spin symmetry {E||C3z}has one
one-dimensional irreducible real representation and two
one-dimensional irreducible complex representations. Al-
though the time-reversal symmetry is broken, altermag-
(d)Cଷ
CଶୄMଵt
TT Mଵt(c)
(b)(a)
Ni-3d
F-2p
ΓTHଶ|HLΓS |SଶFΓ-0.50-0.250.000.250.50Energy (eV)
Ni-3d
F-2p
ΓTHଶ|HLΓS|SଶFΓ-2-1012Energy (eV)
Γ Sଶ mSଶ-0.50-0.250.000.250.50Energy (eV)FIG. 3. Schematic diagram of the i-wave magnetism
and electronic band structures of altermagnetic NiF 3.(a)
Schematic diagram of the i-wave magnetism. The red and
blue parts represent spin up and down, respectively. (b)The
electronic band structure without SOC along the non-high-
symmetry directions. The red and blue lines represent spin-
up and spin-down bands, respectively. (c)and(d)are the
electronic band structures without and with SOC along the
high-symmetry directions.
netic materials have equivalent time-reversal spin sym-
metry {C⊥
2T||T}. The spin symmetry {C⊥
2T||T}will
result in two one-dimensional irreducible complex repre-
sentations to form a Kramers degeneracy. Meanwhile,
the spin symmetry {C⊥
2||M1t}protects the spin degen-
eracy. Therefore, there is one four-dimensional and one
two-dimensional irreducible representations on the Γ −T
axis. The quadruple degenerate band crossing the Fermi
level is thus protected by the spin group symmetry. Fur-
thermore, the orbital weight analysis shows that these
four bands are contributed by both the 3dorbitals of Ni
and the porbitals of F (Fig. 3 (c)). As is known to all,
the F atom has the strongest electronegativity among all
chemical elements, but the 2porbitals of F do not fully
acquire the 3d-orbital electrons of Ni, which is very in-
teresting.
In our calculations, the number of valence electrons of
NiF 3is 74, which makes the quadruple band only half-
filled. This is the reason why the porbitals of F do not
fully acquire the d-orbital electrons of Ni. When SOC is
included, the spin group symmetry breaks down to mag-
netic group symmetry. The reduction of symmetry will
result in the quadruple band to split into multiple bands.
Since the F atom has the strongest electronegativity, the
2porbitals of F will completely acquire the 3d-orbital
electrons of Ni. This will result in altermagnetic NiF 3to
transform from metal phase to insulator phase. In order
to prove our theoretical analysis, we calculate the elec-
tronic band structure of altermagnetic NiF 3with SOC.
The calculation of the easy magnetization axis and sym-
metry analysis based on magnetic point group are shown
in Supplementary Material[28]. Just like our theoretical
analysis, the 2porbitals of F indeed fully acquire the4
3d-orbital electrons of Ni and altermagnetic NiF 3trans-
forms into an insulator with a bandgap of 2 .31eV (Fig.
3(d)). In general, the SOC strength of Ni is in the order
of 10meV, so the SOC strength of altermagnetic NiF 3is
two orders of magnitude higher than that of Ni. Thus,
the SOC effect of altermagnetic NiF 3is extremely strong.
On the other hand, we also examine the effect of cor-
relation interaction in altermagnetic NiF 3. We calcu-
late the electronic band structures of altermagnetic NiF 3
along the high-symmetry directions without SOC under
correlation interaction U = 3 ,5,7eV, which are shown in
Fig. 4 (a),(b), and (c), respectively. From Fig. 4 (a),
(b)and(c), the correlation interaction has a slight ef-
fect on the band structure around the Fermi level with-
out SOC, due to the constraints of spin symmetry and
electron occupancy being 74. When including SOC, al-
termagnetic NiF 3transforms from a metal phase to an
insulator phase under different correlation interaction
U. Moreover, the bandgap of altermagnetic NiF 3in-
creases linearly with the correlation interaction U. Thus,
the correlation interaction can substantially enhance the
bandgap opened by the SOC of altermagnetic materials.
(a) (c)
(b)(d)ΓTL ΓFΓ-0.50-0.250.000.250.50Energy (eV)
HଶHSSଶ ΓTL ΓFΓ-0.50-0.250.000.250.50Energy (eV)
HଶHSSଶ
ΓTLΓ FΓ-0.50-0.250.000.250.50Energy (eV)
HଶHSSଶ3 4 5 6 7
U (eV)1.01.52.02.5Bandgap (eV)
FIG. 4. The electronic properties of altermagnetic NiF 3
under different correlation interaction U. (a),(b)and(c)
are the electronic band structures along the high-symmetry
directions without SOC under correlation interaction U =
3,5,7eV, respectively. (d)The bandgap as a function of cor-
relation interaction U under SOC.
Now we well understand the reason for the extremely
strong SOC effect in altermagnetic NiF 3. A natural ques-
tion is whether such a strong SOC effect can be realized
in other altermagnetic materials. According to the above
analysis, we propose four conditions for realizing such an
effective strong SOC in light element altermagnetic ma-
terials: First, the spin group of altermagnetic material
has high-dimensional (greater than four dimensions) irre-
ducible representation (crystal symmetry groups are pre-
sented in the Supplementary Material[28]); Second, the
band with high-dimensional representation crossing the
Fermi level is half-filled by valence electrons; Third, non-
metallic elements have strong electronegativity; Fourth,the altermagnetic material has strong electron correla-
tion. To verify these four conditions, we also calcu-
late the electronic band structures of four i-wave alter-
magnetic materials (VF 3, CrF 3, FeF 3and CoF 3), which
have the same crystal structure and spin group sym-
metry as NiF 3[14]. The calculations show that none of
the four altermagnetic materials meets the second con-
dition, and the SOC effect is very weak (Detailed calcu-
lations and analysis are presented in the Supplementary
Material[28]). On the other hand, since high-dimensional
irreducible representations can be protected by spin space
group in two-dimensional altermagnetic systems, the pro-
posed mechanism is also applicable to two-dimensional
light element altermagnetic materials, which may be ad-
vantage for realizing quantum anomalous Hall effect at
high temperatures [10].
The mechanism for enhancing the SOC effect that
we propose in altermagnetic materials is different from
that in ferromagnetic materials [11]. First, the high-
dimensional representation of the symmetry group is 2 or
3 dimensions in ferromagnetism, while in altermagnetism
the high-dimension representation is 4 or 6 dimensions, so
their symmetry requirements are entirely different. Sec-
ond, the band with high-dimensional representation in
ferromagnetism comes from dorbitals, while the band
with high-dimensional representation in altermagnetism
can come from the combination of porbitals and dor-
bitals. Third, the enhancement of SOC effect derives
from correlation interaction for ferromagnetic materials,
but from both correlation interaction and electronegativ-
ity of nonmetallic element for altermagnetic materials.
Due to one more degree of freedom to enhance the SOC
effect, a stronger SOC effect can be achieved in the al-
termagnetic materials. Moreover, if electronegativity of
nonmetallic element is weak, different topological phases
may be realized in altermagnetic materials when includ-
ing SOC. On the other hand, the mechanism for enhanc-
ing SOC effect in altermagnetic materials can be also
generalized to conventional antiferromagnetic materials.
Due to the equivalent time-reversal symmetry, more spin
groups with conventional antiferromagnetism have high-
dimensional irreducible representations. Moreover, con-
ventional antiferromagnetic materials are more abundant
than altermagnetic materials, thus conventional antifer-
romagnetic materials of light elements with strong SOC
effect remain to be discovered.
Summary. Based on spin symmetry analysis and
the first-principles electronic structure calculations, we
demonstrate that extremely strong SOC effect can be re-
alized in altermagnetic material NiF 3. Then, we propose
a mechanism to enhance SOC effect in altermagnetic ma-
terials. This mechanism reveals the cooperative effect
of crystal symmetry, electron occupation, electronega-
tivity, electron correlation, and intrinsic spin-orbit cou-
pling. The mechanism can explain not only the extremely
strong SOC effect in altermagnetic NiF 3, but also the
weak SOC in altermagnetic VF 3, CrF 3, FeF 3, CoF 3.
Moreover, the mechanism for enhancing SOC effect can5
be also generalized to two-dimensional altermagnetic ma-
terials.ACKNOWLEDGMENTS
This work was financially supported by the Na-
tional Key R&D Program of China (Grant No.
2019YFA0308603), the National Natural Science Foun-
dation of China (Grant No.11934020, No.12204533,
No.62206299 and No.12174443) and the Beijing Natural
Science Foundation (Grant No.Z200005). Computational
resources have been provided by the Physical Laboratory
of High Performance Computing at Renmin University of
China.
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1101.1268v3.Quantum_phase_transitions_in_a_strongly_entangled_spin_orbital_chain__A_field_theoretical_approach.pdf | arXiv:1101.1268v3 [cond-mat.str-el] 31 May 2011Quantum phase transitions in a strongly entangled spin-orb ital chain:
A field-theoretical approach
Alexander Nersesyan
The Abdus Salam International Centre for Theoretical Physi cs, 34100, Trieste, Italy
Andronikashvili Institute of Physics, Tamarashvili 6, 017 7, Tbilisi, Georgia
Center of Condensed Mater Physics, ITP, Ilia State Universi ty, 0162, Tbilisi, Georgia
Gia-Wei Chern and Natalia B. Perkins
Department of Physics, University of Wisconsin, Madison, W isconsin 53706, USA
Motivated by recent experiments on quasi-1D vanadium oxide s, we study quantum phase transi-
tions in a one-dimensional spin-orbital model describing a Haldane chain and a classical Ising chain
locally coupled by the relativistic spin-orbit interactio n. By employing a field-theoretical approach,
we analyze the topology of the ground-state phase diagram an d identify the nature of the phase
transitions. In the strong coupling limit, a long-range N´ e el order of entangled spin and orbital
angular momentum appears in the ground state. We find that, de pending on the relative scales
of the spin and orbital gaps, the linear chain follows two dis tinct routes to reach the N´ eel state.
First, when the orbital exchange is the dominating energy sc ale, a two-stage ordering takes place
in which the magnetic transition is followed by melting of th e orbital Ising order; both transitions
belong to the two-dimensional Ising universality class. In the opposite limit, the low-energy orbital
modes undergo a continuous reordering transition which rep resents a line of Gaussian critical points.
On this line the orbital degrees of freedom form a Tomonaga-L uttinger liquid. We argue that the
emergence of the Gaussian criticality results from merging of the two Ising transitions in the strong
hybridization region where the characteristic spin and orb ital energy scales become comparable.
Finally, we show that, due to the spin-orbit coupling, an ext ernal magnetic field acting on the spins
can induce an orbital Ising transition.
I. INTRODUCTION
Over the past decades, one-dimensional spin-orbital
models have been a subject of intensive theoretical stud-
ies. The interest is to a large extent motivated by exper-
imental discovery of unusual magnetic properties in vari-
ous quasi-one-dimensional Mott insulators.1,2The inter-
dependence of spin and orbital degrees of freedom is usu-
ally described by the so-called Kugel-Khomskii Hamil-
tonian in which the effective spin exchange constant de-
pends on the orbital configuration and vice versa.3,4An-
other mechanism of coupling spin and orbital degrees of
freedom is the on-site relativistic spin-orbit (SO) interac-
tionλL·S, whereListhe orbitalangularmomentumand
λis the coupling constant. In compounds with quenched
orbital degrees of freedom, the presence of the SO term
usuallyleadstothesingle-ionspinanisotropy DS2
zwhere
D∼λ2/∆ and ∆ denotes the energy scale of the crystal
field which lifts the degenerate orbital states.
For systems with residual orbital degeneracy, on the
other hand, the effect of the SO term is much less
explored compared with the Kugel-Khomskii-type cou-
pling. Due to the directional dependence of the orbital
wave functions, the SU(2) symmetry of the Heisenberg
spin exchange is expected to be broken in the presence of
theSOinteraction. Theresultantspinanisotropyislikely
to induce a long-range magnetic order in the spin sector.
Amoreintriguingquestioniswhathappenstotheorbital
sector. To answer this question, one needs to consider
the details of the interplay between the orbital exchange
and the SO coupling. Here we consider the simplestcase of a two-fold orbital degeneracy per site. Specifi-
cally, the two degenerate states could be the dyzanddzx
orbitals in a tetragonal crystal field observed in several
transition-metal compounds. We introduce pseudospin-
1/2 operators τa(a=x,y,z) to describe the doublet
orbital degrees of freedom assuming that τz=±1 corre-
spond to the states |yz/an}bracketri}htandi|zx/an}bracketri}ht, respectively. Alterna-
tively, one can also realize the double orbital degeneracy
in the Mott-insulating phase of a 1D fermionic optical
lattice where the eigenvectors of τzrefers to pxandpy
orbitals in an anisotropic potential.5,6Restricted to this
doublet space, the orbital angular momentum operator
L= (0,0,τx). This can be easily seen by noting that the
eigenstatesof τxcarryanangularmomentum /an}bracketle{tLz/an}bracketri}ht=±1.
The exchange interaction between localized orbital de-
grees of freedom is characterized by its highly direc-
tional dependence: the interaction energy only depends
on whether the relevant orbital is occupied for bonds of
a given orientation. This is particularly true for inter-
actions dominated by direct exchange mechanism. De-
noting the relevant orbital projectors on a given bond as
P= (1+τβ)/2, where τβ/2isanappropriatepseudospin-
1/2 operator ( τβbeing a Pauli matrix), the orbital inter-
action is thus described by an Ising-type term τβ
iτβ
j. The
well studied orbital compass model and Kitaev model
both belong to this category.7,8The quantum nature of
these models comes from the fact that different operators
τβ, which do not commute with each other, are used for
bonds of different types. To avoid unnecessary compli-
cations coming from the details of orbital interactions,
we assume that there is only one type of bond in our 1D2
system and the orbital interaction is thus governed by a
classical Ising Hamiltonian.
We incorporate these features into the following toy
model of spin-orbital chain ( Js,Jτ>0):
H=HS+Hτ+HSτ (1)
=JS/summationdisplay
nSn·Sn+1+Jτ/summationdisplay
nτz
nτz
n+1+λ/summationdisplay
nτx
nSz
n.
Motivated by the recent experimental characterizations
of quasi-1D vanadium oxides,14–19here we focus on the
case of quantum spin with length S= 1. The above
model thusdescribesaHaldanechainlocallycoupled toa
classical Ising chain by the SO interaction HSτ. The role
of theλ-term is two-fold: firstly it introduces anisotropy
tothespin-1subsystem, andsecondlyit endowsquantum
dynamics to the otherwise classical Ising chain.
Before turning to a detailed study of the phase dia-
gram of model (1), we first discuss its connections to
real compounds. As mentioned above, the interest in
the toy model is partly motivated by the recent experi-
mental progresson vanadium oxides which include spinel
ZnV2O414–17and quasi-1D CaV 2O4.18,19In both types
of vanadates, the two delectrons of V3+ions have a
spinS= 1 in accordance with Hund’s rule. In the low-
temperature phase of both vanadates, the vanadium site
embeddedinaflattenedVO 6octahedronhasatetragonal
symmetry. This tetragonal crystal field splits the degen-
eratet2gtriplet into a singlet and a doublet. As one of
the twodelectrons occupies the lower-energy dxystate,
a double orbital degeneracy arises as the second electron
could occupy either dzxordyzorbitals. The fact that
thedxyorbital is occupied everywherealso contributes to
theformationofweaklycoupledquasi-1Dspin-1chainsin
these compounds.20On the other hand, the details of the
orbital exchange depends on the geometry of the lattice
and in the case of vanadium spinel the orbital interaction
is of three-dimensional nature. The Ising orbital Hamil-
tonian in Eq. (1) thus should be regarded as an effective
interaction in the mean-field sense. Nonetheless, the toy
model provides a first step towardsunderstanding the es-
sentialphysics introducedby the SO coupling. Moreover,
many conclusions of this paper can be applied to the case
of quasi-1Dcompound CaV 2O4where the vanadium ions
form a zigzag chain.
It is instructive to first establish regions of stable mas-
sive phases. In the decoupling limit, λ→0, our model
describes two gapped systems: a quantum spin-1 Heisen-
bergchainand aclassicalorbitalIsingchain. Theground
state of the spin sector is a disordered quantum spin
liquid with a finite spectral gap21∆S, whereas the or-
bital ground state is characterized by a classical N´ eel
order along the chain: /an}bracketle{tτz
n/an}bracketri}ht= (−1)nηz. Quantum ef-
fects in the orbital sector induced by the SO coupling
play a minor role. Obviously, just because of being
gapped, both the spin-liquid phase and the orbital or-
dered state are stable as long as λremains small. Con-
sider now the opposite limit, λ≫JS,Jτ. In the ze-
roth order approximation, the model is dominated by thesingle-ionterm HSτwhose doubly degenerateeigenstates
|±/an}bracketri}ht=|Sz=±1/an}bracketri}ht⊗|τx=±1/an}bracketri}htrepresent locally entangled
spin and orbital degrees of freedom. Switching on small
JSandJτleads to a staggered ordering of the |+/an}bracketri}htand
|−/an}bracketri}htstates alongthe chain. Physically, the large- λground
state can be viewed as a simultaneous N´ eel ordering of
spin and orbital angular momentum characterized by or-
der parameters ζandηxsuch that /an}bracketle{tSz
n/an}bracketri}ht= (−1)nζand
/an}bracketle{tLz
n/an}bracketri}ht=/an}bracketle{tτx
n/an}bracketri}ht= (−1)nηx. The Ising order parameter ηz
vanishes identically in this phase.
These observations naturally lead to the following
questions. How is the magnetically ordered N´ eel state
at large λconnected to the disordered Haldane phase
asλ→0 ? What is the scenario for the orbital re-
orientation transition ηz→ηx, which is of essentially
quantum nature ? In this paper we employ the field-
theoretical approach to address these questions. We first
note that the one-dimensional model (1) is not exactly
integrable. As a consequence, the regime of strong hy-
bridization of the spin and orbital excitations, which is
the case when Jτ,JSandλare all of the same order,
stays beyond the reach of approximate analytical meth-
ods. We thus will be mainly dealing with limiting cases
Jτ≫JSandJτ≪JS, in which one can integrate out
the “fast” variables to obtain an effective action for the
“slow” modes. Following this approach, we establish the
topology and main features of the ground-state phase
diagram in the accessible parts of the parameter space
of the model. We were able to unambiguously identify
the universality classes of quantum criticalities separat-
ing different massive phases. Using plausible arguments
we comment on some features of the model in the regime
of strong spin-orbital hybridization.
We demonstrate that the aforementioned reorientation
transition ηz→ηxcan be realized in one of two possi-
ble ways. In the limit of large Jτ, we find a sequence
of two quantum Ising transitions and an intermediate
massive phase, sandwiched between these critical lines,
in which both ηzandηxare nonzero. This is consis-
tent with the recent findings22based on DMRG calcu-
lations and some analytical estimations. In the oppo-
site limit, when the Haldane gap ∆ Sis the largest en-
ergy scale, integrating out the spin excitations yields an
effective lowest-energy action for the orbital degrees of
freedom, which shows that the ηz→ηxcrossover takes
place as a single Gaussian quantum criticality. At this
critical point, the orbital degrees of freedom display an
extremelyquantumbehaviour: they aregaplessand form
a Tomonaga-Luttinger liquid. This is the main result of
this paper. We bring about arguments suggesting that
the emergence of the Gaussian critical line is the result
of merging of the two Ising criticalities in the region of
strong spin-orbital hybridization.
Any field-theoretical treatment of the model (1) must
be based on a properly chosen contiuum description of
the spin-1 antiferromagnetic Heisenberg chain. Its prop-
erties have been thoroughly studied, both analytically
and numerically (see for a recent review Ref. 23). In3
what follows, the spin sector of the model (1) will be
treated within the O(3)-symmetric Majorana field the-
ory, proposed by Tsvelik:24
HM=/summationdisplay
a=1,2,3/bracketleftbiggiv
2(ξa
L∂xξa
L−ξa
R∂xξa
R)−imξa
Rξa
L/bracketrightbigg
+Hint.
(2)
Hereξa
R,L(x) is a degenerate triplet of real (Majorana)
Fermi fields with a mass m, the indices RandLlabel
the chirality of the particles, and
Hint=1
2g/summationdisplay
a(ξa
Rξa
L)2
is a weak four-fermion interaction which can be treated
perturbatively. The continuum theory (2) adequately de-
scribesthelow-energypropertiesofthegeneralizedspin-1
bilinear-biquadratic chain
HS→¯HS=JS/summationdisplay
n/bracketleftBig
Sn·Sn+1−β(Sn·Sn+1)2/bracketrightBig
.(3)
in the vicinityofthe criticalpoint β= 1.25Thisquantum
criticality belongs to the universality class of the SU(2) 2
Wess-Zumino-Novikov-Witten(WZNW)modelwithcen-
tral charge c= 3/2.
At small deviations from criticality the Majoranamass
m∼JS|β−1|determines the magnitude of the triplet
gap, ∆ S=|m| ≪JS. The theory of a massive
triplet of Majorana fermions is equivalent to a system
of three degenerate noncritical 2D Ising models, with
m∼(T−Tc)/Tc. This is one of the most appealing fea-
tures of the theory because the most strongly fluctuating
physical fields of the S= 1 chain, namely the staggered
magnetization and dimerization operators, have a simple
local representation in terms of the Ising order and dis-
order parameters.24,26,27It is this fact that greatly sim-
plifies the analysis of the spin-orbital model (1). While
the correspondence between the models (2) and (3) is
well justified at |β−1| ≪1, it is believed that the Ma-
jorana model (2) captures generic properties of the Hal-
danespin-liquidphaseofthe spin-1chain, eventhoughat
large deviations from criticality ( |β−1| ∼1,∆S∼Js)
all parameters of the model should be treated as phe-
nomenological.
The remainderofthe paperis organizedasfollows. We
start our discussion with Sec. II which contains a brief
summary of known facts about the Majorana model24
that will be used in the rest of the paper. In Sec. III
we consider the limit Jτ/∆S≫1 and by integrating
out the ‘fast’ orbital modes, show that on increasing the
SO coupling λthe system undergoes a sequence of two
consecutive quantum Ising transitions in the spin and or-
bital sectors, respectively. In section IV we analyze the
opposite limiting case, Jτ/∆S≪1, and, by integrating
over the ‘fast’ spin modes, show that there exists a sin-
gle Gaussian transition in the orbital sector accompaniedNeel
ζ=0/
Orbital
Ising orderIIII
IIPath 1Orbital Ising order
xτxS Path 2
ζ=0 Haldane spin liquid
xzη =0, η =0//η =0, η =0z x
xz//η =0, η =0
FIG. 1: Schematic phase diagram of the model on the ( xS,
xτ)-plane, where xS= ∆S/λandxτ=Jτ/λ.
by a Neel ordering of the spins. We then conjecture on
the topology of the ground-state phase diagram of the
model. In Sec. V we show that spin-orbital hybridiza-
tion effects near the orbital Gaussian transition lead to
the appearance of a non-zero spectral weight well below
the Haldane gap which can be detected by inelastic neu-
tron scattering experiments and NMR measurements. In
Sec. VI we comment on the role of an external magnetic
field. We show that, through the SO interaction, a suf-
ficiently strong magnetic field affects the orbital degrees
of freedom and can lead to a quantum Ising transition
in the orbital sector. Sec. VII contains a summary of
the obtained results and conclusions. The paper has two
appendices containing certain technical details.
II. SOME FACTS ABOUT MAJORANA
THEORY OF SPIN-1 CHAIN
In this Section, we provide some details about the
O(3)-symmetric Majorana field theory,24Eq. (2), which
represents the continuum limit of the biquadratic spin-1
model (3) at |β−1| ≪1.
In the continuum description, the local spin density of
the spin model (3) has contributions from the low-energy
modes centered in momentum space at q= 0 and q=π:
S(x) =IR(x)+IL(x)+(−1)x/a0N(x) (4)
The smooth part of the local magnetization, I=IR+
IL, is a sum of the level-2 chiral vector currents. The
SU(2)2Kac-Moody algebra of these currents is faithfully
reproduced in terms of a triplet of massless Majorana
fields28ξ= (ξ1,ξ2,ξ3):
Iν=−i
2(ξν×ξν),(ν=R,L) (5)4
This fact is not surprising because, as already men-
tioned, the central charge of the SU(2) 2WZNW theory
isc= 3/2, whereas that of the theory of a massless Ma-
jorana fermion (equivalently, critical 2D Ising model) is
c= 1/2. At smalldeviationsfromcriticality( |β−1| ≪1)
thefermionsacquireamass. Stronglyfluctuatingfieldsof
the spin-1 chain, the staggered magnetization N(x) and
dimerization operator ǫ(x) = (−1)nSn·Sn+1, are nonlo-
cal in terms of the Majorana fields but admit a simple
representation in terms of the order, σ, and disorder, µ,
operators of the related noncritical Ising models:
N∼(1/α)(σ1µ2µ3, µ1σ2µ3, µ1µ2σ3),
ǫ∼(1/α)σ1σ2σ3, (6)
whereα∼a0is a short-distance cutoff of the contin-
uum theory. These expressions together with their duals
(i.e. their counterparts obtained by the duality trans-
formation in all Ising copies, σa↔µa) determine the
vector and scalar parts of the WZNW 2 ×2 matrix field
ˆgwhich is a primary scalar field with scaling dimension
3/8. It has been demonstrated in Ref. 28 that using
the representation (6) and the short-distance operator
product expansions for the Ising fields, one correctly re-
produces all fusion rules of the SU(2) 2WZNW model.
An equivalent way to make sure that this is indeed the
case is to consider the four-Majorana representation of
the weakly coupled spin-1/2 Heisenberg ladder26,27and
take the limit of a infinite singlet Majorana mass to map
the low-energy sector of the model on the O(3) theory
(2).
In the spin-liquid phase of the spin chain (3), which
is the case β <1, the Majorana mass mis positive,
implying that the degenerate triplet of2D Isingmodels is
inadisorderedphase: /an}bracketle{tσa/an}bracketri}ht= 0,/an}bracketle{tµa/an}bracketri}ht /ne}ationslash= 0 (a= 1,2,3). In
particular, this implies that the O(3) symmetry remains
unbroken, /an}bracketle{tN/an}bracketri}ht= 0, and the ground state of the system
is not spontaneously dimerized, /an}bracketle{tǫ/an}bracketri}ht= 0.
The representation (6) proves to be very useful for cal-
culatingthe dynamicalspincorrelationfunctionsbecause
the asymptotics of the Ising correlators /an}bracketle{tσ(x,τ)σ(0,0)/an}bracketri}ht
and/an}bracketle{tµ(x,τ)µ(0,0)/an}bracketri}htarewellknownbothatcriticalityand
in a noncritical regime. In the disordered phase ( m >0),
the leading asymptotics of the Ising correlators are:
/an}bracketle{tµ(r)µ(0)/an}bracketri}ht ∼(a/ξS)1/4/bracketleftBig
1+O(e−2r/ξS)/bracketrightBig
,
/an}bracketle{tσ(r)σ(0)/an}bracketri}ht ∼(a/ξS)1/4/radicalbig
ξS/r e−r/ξS(7)
whereξS=v/mis the correlation length, and r=√
x2+v2τ2. (By duality, in the ordered phase ( m <0)
the asymptotics of the correlators in (7) must be inter-
changed.) Correspondingly, the dynamical correlation
function
/an}bracketle{tN(r)N(0)/an}bracketri}ht ∼(a/ξS)3/4/radicalbig
ξS/r e−r/ξS.(8)
Its Fourier transform at q∼πand small ωdescribes a
coherent excitation – a triplet magnon with the mass gapm:
ℑm χ(q,ω)∼m
|ω|δ/parenleftBig
ω−/radicalbig
(q−π)2v2+m2/parenrightBig
.(9)
Since the single-ion anisotropy Hanis=D/summationtext
n(Sz
n)2
lowers the original O(3) symmetry down to O(2) ×Z2,
one expects24that in the continuum theory it will induce
anisotropy in the Majorana masses
m1=m2/ne}ationslash=m3,
as well as in the coupling constants parametrizing the
four-fermion interaction:
Hint→1
2/summationdisplay
a/negationslash=bgab(ξa
Rξa
L)/parenleftbig
ξb
Rξb
L/parenrightbig
, g13=g23/ne}ationslash=g12.
This can be checked by using the correspondence (4)
and short-distance operator product expansions (OPE)
for the physical fields. There will also appear anisotropy
in the velocities, v1=v2/ne}ationslash=v3, but we will systematically
neglect this effect. Thus, we have Hanis=/integraltext
dxHanis,
with
Hanis=Dα/integraldisplay
dx/bracketleftbig
I3(x)I3(x+α)+N3(x)N3(x+α)/bracketrightbig
,(10)
whereα∼ais a short-distance cutoff of the continuum
theory. Using (5) and keeping only the Lorentz invariant
terms (i.e. neglecting renormalization of the velocities)
we can replace ( I3)2by 2I3
RI3
L. To treat the second term
in the r.h.s. of (10), we need OPEs for the products of
Ising operators:29
σ(z,¯z)σ(w,¯w)
=1√
2/parenleftbiggα
|z−w|/parenrightbigg1/4/bracketleftbig
1−π|z−w|ε(w,¯w)/bracketrightbig
,(11)
µ(z,¯z)µ(w,¯w)
=1√
2/parenleftbiggα
|z−w|/parenrightbigg1/4/bracketleftbig
1+π|z−w|ε(w,¯w)/bracketrightbig
.(12)
Hereε=iξRξLis the energy density (mass bilinear) of
the Ising model, z=vτ+ ixandw=vτ′+ ix′are
two-dimensional complex coordinates, ¯ zand ¯ware their
conjugates. From the above OPEs it follows that
N3(x)N3(x+α) = i(π/α)/parenleftbig
ξ1
Rξ1
L+ξ2
Rξ2
L−ξ3
Rξ3
L/parenrightbig
−(π2C)[(ξ1
Rξ1
L)(ξ2
Rξ2
L)−(ξ1
Rξ1
L)(ξ3
Rξ3
L)−(ξ2
Rξ2
L)(ξ3
Rξ3
L)],
whereC∼1 is a nonuniversal constant. As a result,
Hanis=−i/summationdisplay
a=1,2,3δmaξa
Rξa
L+1
2/summationdisplay
a/negationslash=bδgij(ξa
Rξa
L)/parenleftbig
ξb
Rξb
L/parenrightbig
,(13)
where
δm1=δm2=−δm3=−(πC)D (14)5
are corrections to the single-fermion masses, and δg12=
(2−π2C)Dα, δg 13=δg23=π2CDαare cou-
pling constants of the induced interaction between the
fermions. Smallness of the Majorana masses ( |m|α/v≪
1) implies that the additional mass renormalizations
caused by the interaction in (13) are relatively small,
m(Dα/v)ln(v/|m|α)≪D, so that the main effect of
the single-ion anisotropy is the additive renormalization
of the fermionic masses, ma=m+δma, withδmagiven
by Eq.(14).
The cases D >0 andD <0 correspond to an easy-
plane and easy-axis anisotropy, respectively. The spin
anisotropy (18) induced by the spin-orbit coupling is of
the easy-axis type. At D <0 the singlet Majorana
fermion, ξ3, is the lightest, m3< m1=m2. Increas-
ing anisotropy drives the system towards an Ising crit-
icality at D=−D∗, where m3= 0. At D <−D∗
the system occurs in a new phase where the Ising dou-
blet remains disordered while the singlet Ising system
becomes ordered. It then immediately follows from the
representation (6) that the new phase is characterized by
a N´ eel long-range order with /an}bracketle{tN3/an}bracketri}ht /ne}ationslash= 0. Transverse spin
fluctuations, as well as fluctuations of dimerization, are
incoherent in this phase.
III. TWO ISING TRANSITIONS IN THE
∆S≪JτLIMIT
Nowwe turn to ourmodel (1). Let us considerthe case
when, in the absence of spin-orbit coupling, the orbital
gapisthelargest: Jτ≫Js. Theorbitalpseudospinsthen
represent the ‘fast’ subsystem and can be integrated out.
Assuming that λ≪Jτ, we treat the spin-orbit coupling
perturbatively. In this case, the zero order Hamiltonian
H0=HS+Hτdescribes decoupled spin and orbital sys-
tems, while the spin-orbit interaction HSτdenotes per-
turbation. Defining the interaction representation for all
operators according to A(τ) =eτH0Ae−τH0(hereτde-
notes imaginary time), the interaction term in the Eu-
clidian action is given by
SSτ=λ/summationdisplay
n/integraldisplay
dτ τx
n(τ)Sz
n(τ). (15)
The first nonvanishing correction to the effective action
in the spin sector is of the second order in λ:
∆Ss=−λ2
2/summationdisplay
nm/integraldisplay
dτ1dτ2/angbracketleftbig
τx
n(τ1)τx
m(τ2)/angbracketrightbig
τSz
n(τ1)Sz
m(τ2).
(16)
Averaging in the right-hand side of (16) goes over config-
urations of the classical Ising chain Hτ. The correlation
function /an}bracketle{tτx
n(τ1)τx
m(τ2)/an}bracketri}htτis calculated in Appendix A. It
is spatially ultralocal (because there are no propagating
excitations in the classical Ising model) and rapidly de-
caying at the characteristic time ∼1/Jτ, which is muchshorter than the spin correlation time ∼1/∆0:
/an}bracketle{tτx
n(τ1)τx
m(τ2)/an}bracketri}htτ=δnmexp(−4Jτ|τ1−τ2|).(17)
Passing to new variables, τ= (τ1+τ2)/2 andρ=τ1−τ2,
and integrating over ρyields a correction to the effec-
tive spin action which has the form of a single-ion spin
anisotropy. Thus in the second order in λ, the spin
Hamiltonian acquires an additional term
Hani=−λ2
4Jτ/summationdisplay
n(Sz
n)2. (18)
The anisotropy splits the Majorana triplet into a doublet
(ξ1,ξ2) and singlet ( ξ3), with masses
m1=m2=m+πCλ2
4Jτ, m3=m−πCλ2
4Jτ,(19)
whereC∼1 is a nonuniversal positive constant. The
anisotropy is of the easy-axis type, so that the singlet
mode has a smaller mass gap.
As long as all the masses maremain positive, the sys-
tem maintains the properties of an anisotropic Haldane’s
spin-liquid. The dynamical spin susceptibilities calcu-
lated at small ωandq∼π(see Sec. II),
ℑm χxx(q,ω) =ℑm χyy(q,ω) (20)
∼m1
|ω|δ/parenleftbigg
ω−/radicalBig
(q−π)2v2+m2
1/parenrightbigg
,
ℑm χzz(q,ω)∼m3
|ω|δ/parenleftbigg
ω−/radicalBig
(q−π)2v2+m2
3/parenrightbigg
,
indicate the existence of the Sz=±1 andSz= 0 optical
magnons with mass gaps m1andm3, respectively. In-
creasing the spin-orbital coupling leads eventually to an
Isingcriticalityat λ=λc1= 2/radicalbig
Jτm/πC, wherem3= 0.
Atm3<0thesystemoccursinalong-rangeorderedN´ eel
phase with staggered magnetization /an}bracketle{tSz
n/an}bracketri}ht= (−1)nζ(λ),
in whichthe Z2-symmetryofmodel (18) is spontaneously
broken. Using the Ising-model representation (6) of the
staggered magnetization of the spin-1 chain, we find that
at 0< λ−λc1≪λc1the order parameter ζ(λ) follows a
power-law increase:
ζ(λ)∼/parenleftbiggλ−λc1
λc1/parenrightbigg1/8
. (21)
The transverse spin fluctuations become incoherent in
this phase. The situation here is entirely similar to that
in the spontaneously dimerized massive phase of a two-
chain spin-1/2 ladder27,30, where the dimerization kinks
make spin fluctuations incoherent. In the present case,
the spontaneouslybroken Z2symmetry ofthe Neel phase
leadstotheexistenceofpairsofmassivetopologicalkinks
contributing to a broad continuum with a threshold at
ω=m1+|m3|(the details of calculation can be found in
Ref.27):
ℑm χxx(q,ω) (22)
∼1/radicalbig
m1|m3|θ(ω2−(q−π)2v2−(m1+|m3|)2)/radicalbig
ω2−(q−π)2v2−(m1+|m3|)2.6
In the N´ eel phase, the orbital sector acquires quantum
dynamicsbecauseantiferromagneticorderingofthe spins
generates an effective transverse magnetic field which
transforms the classical Ising model Hτto a quantum
Ising chain. At λ > λ c1the spin-orbit term takes the
form
HSτ=−h/summationdisplay
n(−1)nτx
n+H′
Sτ, (23)
whereh=λζ(λ) andH′
Sτ=−λ/summationtext
n(Sz
n−/an}bracketle{tSz
n/an}bracketri}ht)τx
nac-
counts for fluctuations. Since both the orbital and spin
sectors are gapped, the main effect of this term is a
renormalization of the mass gaps and group velocities.
The transverse field hgives rise to quantum fluctuations
which decrease the classical value of ηzand, at the same
time, lead to a staggered ordering of the orbital pseu-
dospins in the transverse direction. Since the orbital sec-
tor has a finite susceptibility with respect to a transverse
staggered field, in the right vicinity of the critical point
ηxfollows the same power-law increase as ζbut with a
smaller amplitude:
ηx∼/parenleftbiggh
Jτ/parenrightbigg
∼/radicalbigg
∆S
Jτ/parenleftbiggλ−λc1
λc1/parenrightbigg1/8
.(24)
This result is in a good agreement with previously
obtained numerical results for order parameters (See
Fig. 4(a) in Ref. 22).
Performing an inhomogeneous π-rotation of the pseu-
dospins around the y-axis,τx,z
n→(−1)nτx,z
n,τy
n→τy
n,
we find that at λ > λc1the effective model in the orbital
sectorreducesto a ferromagneticIsing chainin auniform
transverse (pseudo)magnetic field:
Hτ;eff=−Jτ/summationdisplay
nτz
nτz
n+1−h/summationdisplay
nτx
n.(25)
Notice that the restriction λ≪Jτ, which was imposed
in the derivation of the effective Hamiltonian in the spin
sector, now can be released because the spin sector is
assumed to be in the N´ eel phase.
Ath=Jτ, i.e. at λ=λc2whereλc2satisfies the
equation
λc2ζ(λc2) =Jτ, (26)
the model (25) undergoes a 2D Ising transition27,31to a
massive disordered phase with /an}bracketle{tτz
n/an}bracketri}ht= 0. This quantum
critical point can be reached when λis further increased
in the region λ > λc1. It is clear from (26) that λc2is of
theorderoforgreaterthan Jτ. Itisreasonabletoassume
that for such values of λthe N´ eel magnetization is close
to its nominal value, ζ∼1, implying that λc2∼Jτ. We
see that the two Ising transitions are well separated:
λc2/λc1∼(Jτ/∆S)1/2≫1. (27)
Thus, in the limit Jτ≫∆S, the ground-state phase
diagram of the model (1) consists of three gapped phasesFIG. 2: Schematic diagram of order parameters as functions
of the SO coupling constant λ. (a) Two Ising transitions in
theJτ≫∆Slimit. (b) A single Gaussian transition in the
∆S≫Jτlimit. These two scenarios correspond to path-1
and path-2 in the phase diagram (Fig. 1), respectively.
separated by two Ising criticalities, one in the spin sector
(λ=λc1) and the other in the orbital sector ( λ=λc2).
At 0< λ < λ c1the spin sector represents an anisotropic
spin-liquid while in the orbital sector there is a N´ eel-like
ordering of the pseudospins: ( −1)n/an}bracketle{tτz
n/an}bracketri}ht ≡ηz(λ)/ne}ationslash= 0.
Atλc1< λ < λ c2the orbital degrees of freedom reveal
their quantum nature: the onset of the spin N´ eel order
(ζ/ne}ationslash= 0) is accompanied by the emergence of the trans-
versecomponent of the staggered pseudospin density:
(−1)n/an}bracketle{tτx
n/an}bracketri}ht ≡ηx(λ)/ne}ationslash= 0. Upon increasing λ, the stag-
gered orbital order parameter ηundergoes a continuous
rotationfrom the z-directionto x-direction. At λ=λc2a
quantum Ising transition takes place in the orbital sector
whereηzvanishes. At λ > λ c2both sectors are long-
range ordered, with order parameters ζ,ηx/ne}ationslash= 0. The de-
pendenceoforderparameterson λisschematicallyshown
in Fig. 2(a); this picture is in full qualitative agreement
with the results of the recent numerical studies.22
The crossover between the small and large λlimits
studied in this sectioncorrespondsto path 1 onthe phase
diagram shown in Fig. 1. The path is located in the re-
gionJτ≫∆S. Starting from the massive phase I and
moving along this path we first observe the spin-Ising
transition (I →II) to the N´ eel phase. Long-range or-
dering of the spins induces quantum reconstruction of
the initialy classical orbital sector (i.e. generation of a
nonzeroηx). Theorbital-Ising transition (II →III) takes
place inside the spin N´ eel phase. Of course, feedback ef-
fects (that is, orbitaffecting spin) become inreasinglyim-
portant upon deviating from the critical curve ∆ SJτ∼1
into phases II and III, especially in the vicinity of the
orbital transition where the spin-orbit coupling is very
strong,λ∼Jτ. In this region the behavior of the spin
degrees of freedom is not expected to follow that of an
isolated anisotropic spin-1 chain in the N´ eel phase since
the effect of an “explicit” staggered magnetic field ∼ληx
becomes important. We will see a pattern of such be-
havior in the opposite limit of “heavy” spins, which is
discussed in the next section.7
IV. GAUSSIAN CRITICALITY AT J τ≪∆S
In this section we turn to the opposite limiting case:
∆S≫Jτ. Now the spin degrees of freedom constitute
the “fast” subsystem and can be integrated out to gen-
erate an effective action in the orbital sector. We will
showthat, in this regime, the intermediate massivephase
wherethe orbitalorderparameter ηundergoesacontinu-
ousrotationfrom η= (0,0,ηz) toη= (ηx,0,0)nolonger
exists. Going along path 2, Fig. 1, which is located in the
region ∆ S≫Jτ, we find that the two massive phases, I
and III, are separated by a single Gaussian critical line
characterized by central charge c= 1. On this line the
vectorηvanishes, the orbital degrees of freedom become
gapless and represent a spinless Tomonaga-Luttinger liq-
uid characterized by power-law orbital correlations.
Atλ= 0 the spin-1 subsystem represents a disordered,
isotropic spin liquid. Therefore the first nonzero correc-
tiontothelow-energyeffectiveactionintheorbitalsector
appears in the second order in λ:
∆S(2)
τ=−1
6/an}bracketle{tS2
Sτ/an}bracketri}htS (28)
=−1
2λ2/summationdisplay
nm/integraldisplay
dτ1/integraldisplay
dτ2/an}bracketle{tSn(τ1)Sm(τ2)/an}bracketri}htSτx
n(τ1)τx
m(τ2),
where/an}bracketle{t···/an}bracketri}htSmeans averaging over the massive spin de-
grees of freedom. According to the decomposition of the
spin density, Eq. (4), the correlation function in (29) has
the structure:
/an}bracketle{tSl(τ)S0(0)/an}bracketri}ht= (−1)lf1(r/ξS)+f2(r/ξS).(29)
Hereξs=vs/∆Sis the spin correlation length and
r= (vsτ,x) is the Euclidian two-dimensional radius-
vector.f1andf2aresmooth functions with the following
asymptotic behaviour27
f1(x) =C1x−1/2e−x, f2(x) =C2x−1e−2x(x≫1),(30)
whereC1andC2are nonuniversal constants. DMRG
calculations show32thatC2≪C1; for this reason the
contribution of the smooth part of the spin correlation
function can be neglected in (29).
Integrating over the relative time τ−=τ1−τ2we find
that the spin-orbit coupling generates a pseudospin xx-
exchange with the following structure:
H′
τ=/summationdisplay
n/summationdisplay
l≥1(−1)l+1J′
τ(l)τx
nτx
n+l (31)
Here the exchange couplings exponentially decay with
the separation l,J′
τ(l)∼(λ2/∆S)exp(−la0/ξS), so the
summation in (31) actually extends up to l∼ξS/a0. In
the Heisenberg model ξSis of the order of a few lattice
spacings, so for a qualitative understanding it would be
sufficient to consider the l= 1 term as the leading one
and treat the l= 2 term as a correction. Making a π/2rotation in the pseudospin space, τz
n→τy
n,τy
n→ −τz
n,
we passto the conventionalnotationsand write down the
effective Hamiltonian for the orbital degrees of freedom
as a perturbed XY spin-1/2 chain:
Heff
τ=/summationdisplay
n/parenleftbig
Jxτx
nτx
n+1+Jyτy
nτy
n+1/parenrightbig
+H′
τ.(32)
where
H′
τ=−J′
x/summationdisplay
nτx
nτx
n+2+···. (33)
HereJy=Jτ,Jx=J′
τ(1)>0 andJ′
x=J′
τ(2)>0. By
order of magnitude J′
x< Jx∼λ2/∆S.
In the absence of the perturbation H′
τ, the model (32)
represents a spin-1/2 XY chain which for any nonzero
anisotropy in the basal plane ( Jx/ne}ationslash=Jy) has a N´ eel long-
range order in the ground state and a massive excitation
spectrum. This follows from the Jordan-Wigner trans-
formation
τz
n= 2a†
nan−1, τ+
n=τx
n+iτy
n= 2a†
neiπ/summationtext
j<na†
jaj(34)
which maps the XY chain onto a model of complex spin-
less fermions with a Cooper pairing:33
Heff
τ= (Jx+Jy)/summationdisplay
n/parenleftbig
a†
nan+1+h.c./parenrightbig
+ (Jx−Jy)/summationdisplay
n/parenleftBig
a†
na†
n+1+h.c./parenrightBig
.(35)
By increasing λ(equivalently, decreasing Jτ) the model
(35) can be driven to a XX quantum critical point, Jx=
Jy(1), i.e. λ=λc∼√Jτ∆S, where the the system
acquires a continuous U(1) symmetry. At this point the
Jordan-Wignerfermionsbecomemasslessand thesystem
undergoes a continuous quantum transition.
The transition is associated with reorientation of the
pseudospins. Away from the Gaussian criticality the
effective orbital Hamiltoian is invariant under Z2×Z2
transformations: τx
n→ −τx
n,τz
n→ −τz
n. In massive
phasesthis symmetry is spontaneouslybroken. Making a
back rotation from τytoτzwe conclude that at Jy> Jx
(λ < λ c)ηz/ne}ationslash= 0,ηx= 0, while at Jy< Jx(λ > λ c)
ηz= 0,ηx/ne}ationslash= 0. Both ηzandηxvanish at the critical
point, so contrary to the case Jτ≫∆S, here there is no
region of their coexistence.
The passage to the continuum limit for the model (32)
based on Abelian bosonization is discussed in Appendix
B. There we show that the perturbation H′
τadds a
marginal four-fermion interaction g=J′
x(2)/πv≪1
to the free-fermion model (B3). In the spin-chain lan-
guage, this is equivalent to adding a weak ferromag-
neticzz-coupling. In the limit of weak XY anisotropy,
|λ−λc|/λc≪1, the low-energy properties of the orbital
sector are described by a quantum sine-Gordon model
(all notations are explained in Appendix B)
H=u
2/bracketleftbigg
KΠ2+1
K(∂xΦ)2/bracketrightbigg
+2γ
παcos√
4πΘ,(36)8
where
γ∼Jτ/parenleftbiggλ−λc
λc/parenrightbigg
, K= 1+2g+O(g2).(37)
The U(1) criticality is reached at λ=λcwhere, due to a
finite value of g, the orbital degrees of freedom represent
a Tomonaga-Luttingerliquid. Close to the criticality, the
spectral gap in the orbital sector scales as the renormal-
ized mass of the sine-Gordon model (36):
Morb∼/vextendsingle/vextendsingle/vextendsingleλ−λc
λc/vextendsingle/vextendsingle/vextendsingleK
2K−1. (38)
Strongly fluctuating physical fields acquire coupling
dependentscalingdimensions. Inparticular,accordingto
the bosonization rules,27the staggered pseudospin den-
sities are expressed in terms of the vertex operators,
(−1)nτx
n≡nx(x)∼sin√πΘ(x),
(−1)nτz
n≡nz(x)∼cos√πΘ(x),(39)
both with scaling dimension d= 1/4K. This anomalous
dimension determines the power-lawbehaviourof the av-
erage staggered densities close to the criticality:
ηz(λ)∼(λc−λ)1/4K, λ < λ c
ηx(λ)∼(λ−λc)1/4K, λ > λ c. (40)
A finite staggered pseudospin magnetization ηxat
λ > λ cgenerates an effective external staggered mag-
netic field in the spin sector:
HS→¯H=HS+H′
S, H′
S=−hS/summationdisplay
n(−1)nSz
n,(41)
wherehS=−ληx. The spectrum of the Hamiltonian ¯H
is always massive. This can be easily understood within
the Majorana model (2). According to (6), in the con-
tinuum limit, the sign-alternating component of the spin
magnetization, N3∼(−1)nSz
n, canbeexpressedinterms
of the order and disorder fields of the degenerate triplet
of 2D disordered Ising models: N3∼µ1µ2σ3. In the
leading order, the magnetic interaction H′
Sgives rise to
an effective magnetic field h3=hS/an}bracketle{tµ1µ2/an}bracketri}htapplied to the
third Ising system: h3σ3. The latter always stays off-
critical.
Since in the Haldane phase the spin correlations are
short-ranged, close to the transition point the induced
staggered magnetization ζcan be estimated using linear
response theory. Therefore, at 0 < λ−λc≪λc,ζfollows
the same power-law increase as that of ηxbut with a
smaller amplitude:
ζ∼hS
∆S∼/parenleftbiggJτ
∆S/parenrightbigg1/2/parenleftbiggλ−λc
λc/parenrightbigg1/4K
(42)
So, in the part of the phase C, Fig. 1, where ∆ S≫Jτ,
theηx-orbital order, being the result of a spontaneousbreakdown of a Z2symmetry τx
n→ −τx
n, acts as an ef-
fective staggered magnetic field applied to the spins and
inducestheir N´ eel alignment. This fact is reflected in a
coupling dependent, nonuniversal exponent 1 /4Kchar-
acterizing the increase of the staggered magnetization at
λ > λ c. The order parameters as functions of λin the
∆S≫Jτlimit is schematically shown in Fig. 2(b).
As already mentioned, the absence of a small parame-
ter in the regime of strong hybridization, Jτ∼JS∼λ,
makestheanalysisofthephasediagraminthisregionnot
easily accessible by analytical tools. Nevertheless some
plausible arguments can be put forward to comment on
the topology of the phase diagram. It is tempting to
treat the curve Jτ∆S/λ2∼1 as a single critical line go-
ing throughout the whole phase plane ( Jτ/λ,∆S/λ). If
so, we then can expect that there exists a special sin-
gular point located in the region Jτ∆S/λ2∼1. This
expectation is based on the fact that at Jτ≫∆slimit
the transition is of the Ising type and the spontaneous
spin magnetization below the critical curve follows the
lawζ∼(λ−λc1)1/8with auniversal critical exponent,
whereas at Jτ≪∆sthe spin magnetization has a differ-
ent,nonuniversal exponent, ζ∼(λ−λc)1/4K. Continu-
ity considerations make it very appealing to suggest that
at the special point the Tomonaga-Luttinger liquid pa-
rameter takes the value K= 2, and the two power laws
match. Since the central charges of two Ising and one
Gaussian criticalities satisfy the relation 1 /2+1/2 = 1,
the singular point must be a point where the two Ising
critical curves merge into a single Gaussian one.
V. DYNAMICAL SIN SUSCEPTIBILITY AND
NMR RELAXATION RATE IN THE VICINITY
OF GAUSSIAN CRITICALITY
It may seem at the first sight that, in the regime
∆S≫J, the spin degrees of freedom which have been
integrated out remain massive across the orbital Gaus-
sian transition, and the spectral weight of the staggered
spin fluctuations is only nonzero in the high-energy re-
gionω∼∆S. However, this conclusion is only correct
for the zeroth-order definition of the spin field N0(x),
given by Eq. (6), with respect to the spin-orbit inter-
action. In fact, the staggered magnetization hybridizes
with low-energyorbitalmodes viaSO couplingalreadyin
the first order in λand thus acquires a low-energy pro-
jection which contributes to a nonzero spectral weight
displayed by the dynamical spin susceptibility at ener-
gies well below the Haldane gap.
Tofind the low-energyprojectionofthe field Nz(r), we
must fuse the local operator Nz
0(r) with the perturbative
part of the total action. Keeping in mind that close to
and at the Gaussian criticality most strongly fluctuating
fields are the staggered components of the orbital po-
larization, we approximate the SO part of the Euclidian9
action by the expression
SSτ≃λa0
vS/integraldisplay
d2rNz(r)nx(r), (43)
wherer= (vSτ,x) is the two-dimensional radius vector
(hereτis the imaginary time). We thus construct
Nz
P(r) =/an}bracketle{te−SSτNz(r)/an}bracketri}ht
=Nz
0(r)−λa0
vS/integraldisplay
d2r1/an}bracketle{tNz
0(r)Nz
0(r1)/an}bracketri}htSnx(r1)
+O(λ2), (44)
where averaging is done over the unperturbed, high-
energy spin modes. For simplicity, here we neglect the
anisotropy of the spin-liquid phase of the S=1 chain and
use formula (8). The spin correlation function is short-
ranged. Treating the spin correlation length ξS∼vS/∆S
as a new lattice constant (new ultraviolet cutoff) and be-
ing interested in the infrared asymptotics |r| ≫ξS, we
can replace in (44) nx(r1) bynx(r). The integral
/integraldisplay
d2ρ/an}bracketle{tNz
0(ρ)Nz
0(0)/an}bracketri}htS (45)
∼1
a2
0(a/ξS)3/4/integraldisplay∞
0dρ ρ/radicalbig
ξs/ρ e−ρ/ξS∼(ξS/a0)5/4.
So the first-order low-energy projection of the staggered
magnetization is proportional to
Nz
P(r)∼λ
∆S/parenleftbiggξS
a0/parenrightbigg1/4
nx(r). (46)
Thisresultclarifiestheessenceofthehybridizationeffect:
close to the Gaussian criticality the spin fluctuations ac-
quire a finite spectral weight in the low-energy region,
ω≪∆S,q∼π, which is contributed by orbital fluctu-
ations and can be probed in magnetic inelastic neutron
scattering experiments and NMR measurements.
Away from but close to the Gaussian criticality the
behavior of the dynamical spin susceptibility ℑmχ(q,ω)
is determined by the excitation spectrum of the sine-
Gordon model for the dual field, Eq.(36). Since K >1, it
consists of kinks, antikinks carrying the mass Morb, and
their bound states (breathers) with masses (see e.g. Ref.
27)
Mj= 2Morbsin(πj/2ν),
j= 1,2,...ν−1, ν= 2K−1 (47)
SinceK−1 = 2gis small, there will be only the first
breather in the spectrum, with mass M1= 2Morb(1−
2π2g2). The sine-Gordon model is integrable, and the
asymptotics of its correlation functions in the massive
regime have been calculated using the form-factor ap-
proach (see for a recent review 35). Here we utilize
some of the known results. At λ < λ cthe operator
nx∼sin√πΘ has a nonzero matrix element betweenthe vacuum and the first breather state. This form-
factor contributes to a coherent peak in the dynamical
spin susceptibility at frequencies much smaller than than
the Haldane gap:
ℑmχ(q,ω,T= 0) = A(λ/∆S)2δ[ω2−(q−π)2v2−M2
1]
+ℑmχcont(q,ω,T= 0). (48)
HereAis a constant and the second term is the contribu-
tion of a multi-kink continuum of states with a threshold
atω= 2Morb. Atλ > λ cthe spectral properties of
the operator cos√πΘ coincide with those of the opera-
tor sin√πΘ atλ < λ c. For symmetry reasons35, this
operator does not couple to the first breather, so that
atλ > λcℑmχ(q,ω) will only display the kink-antikink
scattering continuum.
Weseethat, due tospin-orbithybridizationeffects, the
spin sector of our model loses the properties of a spin liq-
uid alreadyin anoncriticalorbitalregime. Thistendency
gets strongly enhanced at the orbital Gaussian criticality
(Morb→0) where all multi-particle processesmerge, and
the spin correlationfunction exhibits an algebraicallyde-
caying asymptotics
/an}bracketle{tNz(r)Nz(0)/an}bracketri}ht ≃ /an}bracketle{tNz
P(r)Nz
P(0)/an}bracketri}ht ∼/parenleftbiggλ
∆S/parenrightbigg2/parenleftBiga
r/parenrightBig1
2K,
(49)
implying that the spin sector of the model becomes remi-
niscent of Tomonaga-Luttinger liquid. In this limit (here
for simplicity we consider the T= 0 case) the dynamical
spin susceptibility is given by34
ℑmχ(q,ω,T= 0)∼(λ/∆S)2/bracketleftbig
ω2−v2(q−π)2/bracketrightbig1
4K−1.
(50)
The NMR relaxation rate probes the spectrum of local
spin fluctuations
1
T1=A2Tlim
ω→01
ω/summationdisplay
qℑmχzz(q,ω,T)
whereAis an effective hyperfine constant. In spin-
liquid regime of an isolated spin-1 chain, the existence of
a Haldane gap makes 1 /T1exponentially suppressed36:
1/T1∼exp(−2∆S/T). The admixture of low-energy or-
bital states in the spin-fluctuation spectrum drastically
changedthisresult. Asimplepowercountingargument37
leads to a power-law temperature dependence of the
NMR relaxation rate:
1
T1∼A2/parenleftbiggλ
∆s/parenrightbigg2
T1
2K−1(51)
This result is valid not only exactly at the Gaussian crit-
icality but also in its vicinity provided that the tempera-
ture is larger than the orbital mass gap. By construction
(see the preceding section) K≥1. This means that the
exponent 1 /2K−1 isnegative and the NMR relaxation
rateincreases on lowering the temperature. It is worth10
noticingthatsuchregimesarenotunusualforTomonaga-
Luttingerphasesoffrustratedspin-1/2ladders.38Forour
model, such behavior of 1 /T1would be a strong indica-
tion of an extremely quantum nature of the collective
orbital excitations.39
VI. BEHAVIOR IN A MAGNETIC FIELD:
QUANTUM ISING TRANSITION IN ORBITAL
SECTOR
We have seen in Sec.III that, due to spin-orbit cou-
pling, the N´ eel ordering of the spins is accompanied by
the emergence of quantum effects in the orbital sector:
the classical orbital Ising chain transforms to a quan-
tum one. In this section we briefly comment on a similar
situation that can arise upon application of a uniform
external magnetic field h.
Since the spin-1 chain is massive, it will acquire a finite
ground-statemagnetization /an}bracketle{tSz/an}bracketri}htonly when the magnetic
field,h, is higher than the critical value hc1∼∆S, corre-
sponding to the commensurate-incommensurate (C-IC)
transition. According to the definition (5), a uniform
magnetic field along the z-axis,Hmag=−hIz, mixes
up a pair of Majorana fields, ξ1andξ2, and splits the
spectrum of Sz=±1 excitations (the Sz= 0 modes are
unaffected by the field). At h=hc1the gap in the spec-
trum of the Sz= 1 excitations closes, and at h > h c1
these modes condense giving rise to a finite magnetiza-
tion. Once /an}bracketle{tSz/an}bracketri}ht /ne}ationslash= 0, the effective Hamiltonian of the
τ-chain becomes
¯Hτ=Jτ/summationdisplay
nτz
nτz
n+1−∆τ/summationdisplay
nτx
n,∆τ=λ/an}bracketle{tSz/an}bracketri}ht.(52)
Here we ignore the fluctuation term that couples τx
nto
∆Sz
n=Sz
n−/an}bracketle{tSz
n/an}bracketri}ht.
One should keep in mind that there exists the sec-
ond C-IC transition at a higher field hc2associated with
full polarization of the spin-1 chain. To simplify further
analysis, let us assume that the range of magnetic fields
hc1< h < h c2, where an isolated spin-1 chain has an in-
commensurate,gaplessgroundstate,issufficientlybroad.
This can be easily achieved in the biquadratic model (3)
withβ∼1, in which case the Haldane gap – and hence
hc1– is small, and the effects associated with the second
C-IC transition can be neglected.
Now, by increasing the magnetic field hin the region
h > h c1, the effective orbital chain (52) can be driven
to an Ising criticality. The induced transverse “magnetic
field” ∆ τis proportional to a nonzero magnetization of
the spin-1 chain. If λ/Jτis large enough, then upon in-
creasing the field the effective quantum Ising chain (52)
can reach the point ∆ τ(h∗) =Jτwhere the Ising transi-
tion occurs. This will happen at some field h=h∗> hc1.
In the region |h−h∗|/h∗≪1 the quantum Ising τ-chain
will be slightly off-critical. Due to the SO coupling, thesemassive orbital excitations will interact with the gap-
lessSz=±1 spin modes. However, this interaction can
only give rise to the orbital mass renormalization (i.e. a
small shift of the Ising critical point) and a group veloc-
ity renormalization of the spin-doublet modes. For this
reason we do not expect the aforementioned spin-orbital
fluctuation term to cause any qualitative changes.
The above discussion reveals an interesting fact: a suf-
ficiently strong magnetic field acting on the spin degrees
of freedom can affect the orbital structure of the chain
anddriveit to aquantum Isingtransition. Thedifference
with the situation discussedin Sec.III is that the external
magnetic field induces a uniform spin polarization which,
in turn, gives rise to a uniform transverse orbital order-
ing/an}bracketle{tτx
n/an}bracketri}ht /ne}ationslash= 0. Thus, the classical long-range orbital order
/an}bracketle{tτz
n/an}bracketri}ht= (−1)nηz, present at h < h∗, disappears in the
regionh > h∗, where the orbital degrees of freedom are
characterized by a transverse ferromagnetic polarization,
/an}bracketle{tτx/an}bracketri}ht /ne}ationslash= 0.
VII. CONCLUSION AND DISCUSSION
In this paper, we have proposed and analyzed a 1D
spin-orbital model in which a spin-1 Haldane chain is lo-
cally coupled to an orbital Ising chain by an on-site term
λτxSzoriginating from relativistic spin-orbit (SO) in-
teraction. The SO term not only introduces anisotropy
to the spin sector, but also gives quantum dynamics to
the orbital degrees of freedom. We approach this prob-
lem from well defined limits where either the spin or the
orbital sector is strongly gapped and becomes a ‘fast’
subsystem which can be integrated out. By analyzing
the resultant effective action of the remaining ‘slow’ de-
grees of freedom, we have identified the stable massive
and critical phases of the model which are summarized
in a schematic phase diagram shown in Fig. 1.
Inthe limit dominatedbyalargeorbitalgap, i.e. Jτ≫
∆S, integrating out the orbital variables gives rise to an
easy-axis spin anisotropy D(Sz)2whereD∼ −λ2/Jτ.
Asλincreases, the disordered Haldane spin liquid un-
dergoes an Ising transition into a magnetically ordered
N´ eel state. The presence of antiferromagnetic spin order
ζin the N´ eel phase in turn generates an effective trans-
verse field h∼λζacting on the orbital Ising variables.
The orbital sector which is described by the Hamiltonian
of a quantum Ising chain reaches criticality when h=Jτ.
In between the two Ising critical points lies an interme-
diate phase (phase II in Fig. 1) where both Ising order
parameters ηxandηzare nonzero. Such a two-stage or-
deringscenarioillustratedbypath1inthephasediagram
(Fig.1)hasbeenconfirmednumericallybyrecentDMRG
calculations.22Interestingly, the orbital Ising transition
can also be induced by applying a magnetic field to the
spin sector. As the field strength is greater than the Hal-
dane gap, a field-induced magnon condensation results
in a finite magnetization density /an}bracketle{tSz/an}bracketri}htin the linear chain.
Thanks to the SO coupling, the orbital sector again ac-11
quires a transverse field h∼λ/an}bracketle{tSz/an}bracketri}htand becomes critical
whenh=Jτ.
A distinct scenario of the orbital reorientation tran-
sitionηz→ηxoccurs in the opposite limit ∆ S≫Jτ.
This time we integrate out the fast spin subsystem and
obtain a perturbed spin-1/2 XY Hamiltonian for the or-
bital sector. The effective exchange constants are given
byJx∼λ2/∆SandJy=Jτ. Asλis varied, the orbital
sector reaches a Gaussian critical point when Jx=Jy,
at which the system acquires an emergent U(1) sym-
metry. The orbital order parameter goes directly from
η= (0,0,ηz)to(ηx,0,0)inthissingle-transitionscenario
(illustrated by path 2 in Fig. 1). Both order parameters
ηxandηzvanish at the critical point. We have shown
that spin-orbital hybridization effects near the Gaussian
transition lead to the appearance of a non-zero spectral
weight of the staggered spin density well below the Hal-
dane gap – the effect which can be detected by inelastic
neutronscatteringexperimentsandNMRmeasurements.
The stability analysis of the orbital Gaussian criti-
cality in the original lattice model (1), done in Ap-
pendix B, has shown that this critical regime is pro-
tected by the τz→ −τzsymmetry of the underlying
microscopic model. This symmetry will be broken in
the presence of an orbital field δ/summationtext
nτz
nwhich removes
degeneracy between the local orbitals dzxanddyzand
adds a ”magnetic” field along the y-axis in the effective
XY model (32). Such perturbation will drive the orbital
sector away from the Gaussian criticality. The same
argument applies to a perturbation with the structure
β/summationtext
nSz
nτz
nwhich also breaks the aforementioned sym-
metry. Integrating over the spins will generate an extra
term∼λβ/summationtext
n(τx
nτy
n+1+τy
nτz
n+1) which, in the contin-
uum limit, translates to λβsin√
4πΘ. As explained in
Appendix B, such perturbation will keep the orbital sec-
tor gapped with coexisting ηxandηzorderings.
Since the analysispresentedin thispaper isdonein the
limiting cases, precise predictions on the detailed shape
of the phase diagram or on the behavior of correlation
functions in the regime of strong hybridization of spin
and orbital degrees of freedom, where all interactions in-
cluded inthe modelareofthe sameorder, arebeyondour
reach and require further numerical calculations. On the
other hand, the continuity and scaling analysis allow us
to believe that the global topology of the phase diagram
and character of critical lines are given correctly. Finally
the spin-orbital model Eq. (1) can be generalized to the
zigzag geometrical where two parallel spin-1 chains are
coupled to a zigzag Ising orbital chain via on-site SO in-
teraction. The zigzag case is closely related to the quasi-
1D compound CaV 2O4. While the two-Ising-transitions
scenario is expected to hold in the Jτ≫∆Sregime, the
counterpart of Gaussian criticality in the zigzag chain
remains to be explored and will be left for future study.Acknowledgements
The authors are grateful to Andrey Chubukov, Fabian
Essler, Vladimir Gritsev, Philippe Lecheminant and
Alexei Tsvelik for stimulating discussions. A.N. grate-
fully acknowledges hospitality of the Abdus Salam Inter-
national Centre for Theoretical Physics, Trieste, where
part of this work has been done. He is also supported
by the grants GNSF-ST09/4-447 and IZ73Z0-128058/1.
G.W.C. acknowledges the support of ICAM and NSF
grant DMR-0844115. N.P. acknowledges the support
from NSF grant DMR-1005932 and ASG ”Unconven-
tional magnetism”. G.W.C. and N.P. also thank the hos-
pitality of the visitors program at MPIPKS, where the
part of the work on this manuscript has been done.
Appendix A: Ising correlation function
In this Appendix we estimate the correlation func-
tion Γxx
nm(τ) =/an}bracketle{tτx
n(τ)τx
m(0)/an}bracketri}ht, where the averaging is per-
formed over the ground state of the Ising Hamiltonian
Hτ=Jτ/summationtext
nτz
nτz
n+1, andτx
n(τ) =eτHττx
ne−τHτ.
It proves useful to make a duality transformation:
τz
nτz
n+1=µx
n, τx
n=µz
nµz
n+1.
The new set of Pauli matrices µa
nrepresents disorder op-
erators. The Hamiltonian and correlation function be-
come:
H→Jτ/summationdisplay
nµx
n, (A1)
Γzz
nm(τ)→ /an}bracketle{tµz
n(τ)µz
n+1(τ)µz
m(0)µz
m+1(0)/an}bracketri}ht.(A2)
The most important fact about the dual representation
is the additive, single-spin structure of the Hamiltonian:
the latter describes noninteracting spins in an external
“magnetic field” Jτ. Notice that by symmetry /an}bracketle{tµz
n/an}bracketri}ht= 0.
Therefore the correlationfunction in (A2) has an ultralo-
cal structure:
Γxx
nm(τ) =δnmY2(τ), Y(τ) =/an}bracketle{tµz
n(τ)µz
n(0)/an}bracketri}ht.(A3)
The time-dependence of the disorder operator can be ex-
plicitly computed,
µz
n(τ) =eτJτµx
nµz
ne−τJτµx
n=µz
ncosh(2Jττ)−iµy
nsinh(2Jττ).
Therefore (below we assume that τ > τ′)
Y(τ−τ′) = cosh2 Jτ(τ−τ′)+/an}bracketle{tµx/an}bracketri}htsinh2Jτ(τ−τ′)
= exp[−2|Jτ|(τ−τ′)]. (A4)
Hereweusedthefactthat, inthegroundstatetheHamil-
tonianHτ,/an}bracketle{tµx/an}bracketri}ht=−sgnJτ. Thus, as expected for the
1D Ising model, the correlation function Γxx
nm(τ) is local
in real space and decays exponentially with τ:
Γxx
nm(τ) =δnmexp(−4J⊥|τ|). (A5)12
Appendix B: Perturbed XY chain, Eq. (32)
In this Appendix we analyze the perturbation (33) to
the XY spin chain (32) and show that at the XX point
it represents a marginal perturbation which transforms
the free-fermion regime to a Gaussian criticality describ-
ing a Luttinger-liquid behavior of the orbital degrees of
freedom.
Using the Jordan-Wigner transformation (34) we
rewrite (33) as H′
τ=H′
1+H′
2, where
H′
1=J′
x(2)
2/summationdisplay
n(a†
nan+2+h.c.)(a†
n+1an+1−1
2),(B1)
H′
2=J′
x(2)
2/summationdisplay
n(a†
na†
n+2+h.c.)(a†
n+1an+1−1
2).(B2)
Assuming that |Jx−Jy|,J′
x≪Jx+Jy, we pass to a con-
tinuum description of the XY chain in terms of chiral,
right (R) and left (L), fermionic fields based on the de-
composition (to simplify notations we set here a0= 1):
an→(−i)nR(x)+inL(x).Then the Hamiltonian density
of the XY model takes the form:
HXY(x) =−iv/parenleftbig
R†∂xR−L†∂xL/parenrightbig
−2iγ/parenleftbig
R†L†−h.c./parenrightbig
,
(B3)
whereγ=Jx−Jy. Standard rules of Abelian
bosonization27transform(B3)toaquantumsine-Gordon
model:
HXY(x) =v
2/bracketleftBig
Π2+(∂xΦ)2/bracketrightBig
+2γ
παcos√
4πΘ,(B4)
wherev= 2(Jx+Jy)a0is the Fermi velocity, Π( x) =
∂xΘ(x) is the momentum conjugate to the scalar field
Φ(x) = Φ R(x) + ΦL(x), and Θ( x) =−ΦR(x) + ΦL(x)
is the field dual to Φ( x). Here Φ R,L(x) are chiral
components of the scalar field. Using the fact that
the fermions are spinless, one can impose the condi-
tion [Φ R(x),ΦL(x′)] =i/4 and thus make sure that the
bosonization rules correctly reproduce the anticommuta-
tion relations {R(x),L(x′)}={R(x),L†(x′)}= 0. An
explicit introduction of the so-called Klein factors be-
comes necessary when bosonizing fermions with an inter-
nal degree of freedom, such as spin 1/2, chain index etc,
which is not the case here.
Let is find the structure of the perturbation (33) in the
continuum limit. First of all we notice that
a†
n+1an+1−1/2≡:a†
n+1an+1:
→(:R†R: + :L†L:)+(−1)n+1(R†L+L†R)
=1√π∂xΦ+(−1)n
παsin√
4πΦ. (B5)
Similarly
a†
nan+2+h.c.
→ −2/bracketleftbig
(:R†R: + :L†L:)+(−1)n(R†L+L†R)/bracketrightbig
=−2/bracketleftbigg1√π∂xΦ−(−1)n
παsin√
4πΦ/bracketrightbigg
. (B6)Dropping Umklapp processes R†(x)R†(x+α)L(x+
α)L(x) +h.c.∼cos√
16πΦ as strongly irrelevant (with
scaling dimension 4) at the XXcriticality and ignoring
interaction of the fermions in the vicinity of the same
Fermi point, we find that
(a†
nan+2+h.c.)(a†
n+1an+1−1/2)/vextendsingle/vextendsingle/vextendsingle
smooth
→ −8 :R†R::L†L:= 2/bracketleftbig
Π2−(∂xΦ)2/bracketrightbig
.(B7)
We see that the perturbation H′
1generates a marginal
four-fermion interaction to the free-fermion model (B3),
thus transforming the model (32) to an XYZ model with
a weak ferromagnetic( zz)-coupling. This interaction can
be incorporated into the Gaussian part of the bosonic
theory (B4) by changing the compactification radius of
the field Φ:
H=HXY+H′
1
=u
2/bracketleftbigg
KΠ2+1
K(∂xΦ)2/bracketrightbigg
−2γ
παcos√
4πΘ.(B8)
Hereuis the renormalized velocity and Kis the inter-
action constant which at J′
x≪(Jx+Jy) is given by
K= 1+2g+O(g2),whereg=J′
x(2)a0/πv≪1.
Now we turn to H′
2. We have:
a†
na†
n+2+h.c. (B9)
→ −/bracketleftbig
R†(x)L†(x+α)+L†(x)R†(x+α)+h.c./bracketrightbig
+(−1)n/bracketleftbig
R†(x)R†(x+α)+L†(x)L†(x+α)+h.c./bracketrightbig
.
Bosonizing the smooth term in the r.h.s. of (B10) one
obtains∂xΦcos√
4πΘ. Bosonizing the staggered term
yields sin√
4πΦcos√
4πΘ. Using the OPE
sin√
4πΦ(x)sin√
4πΦ(x+α)
= const−πα2(∂xΦ)2−1
2cos√
16πΦ,
we find that, in the continuum limit, the Hamiltonian
densityH′
2is contributed by the operatorscos√
4πΘ and
(∂xΦ)2cos√
4πΘ (as before, we drop corrections related
to Umklapp processes). The former leads to a small ad-
ditive renormalization of the fermionic mass γand thus
produces a shift of the critical point. The latter repre-
sents an irrelevant perturbation (with scaling dimension
3) at the XX criticality. In a noncritical regime it renor-
malizes the mass and four-fermion coupling constant g.
Considering the structure of the remaining terms in
the expansion (31) one arrives at similar conclusions.
Here a remark is in order. The only dangerous perturba-
tion which would dramatically affect the above picture
is sin√
4πΘ. The presence of two nonlinear terms in
the Hamiltonian, γcos√
4πΘ+δsin√
4πΘ, would make
the fermionic mass equal to/radicalbig
(λ−λc)2+δ2. The Gaus-
sian criticality in this case would never be reached, the13
model would always remain massive, and nonzero stag-
gered pseudospin densities, ηzandηx, would coexist in
the whole parameter range of the model.
Fortunately, the appearance of the operator sin√
4πΘ
is forbidden by symmetry. The initial Hamiltonian (1)
is invariant under global pseudospin inversion In the z-
component only: τz
n→ −τz
n. After rotation τz→τythis translates to τy
n→ −τy
n. Using the bosonized ex-
pressions (39) for the staggered pseudospin densities we
find that the corresponding transformation of the dual
field is Θ →√π−Θ and so the bosonized Hamiltonian
density must be invariant under this transfomation. This
explainswhythe operatorsin√
4πΘcannotappearin the
effective continuum theory.
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1106.5667v2.Phase_separation_in_a_polarized_Fermi_gas_with_spin_orbit_coupling.pdf | Phase separation in a polarized Fermi gas with spin-orbit coupling
W. Yi and G.-C. Guo
Key Laboratory of Quantum Information, University of Science and Technology of China,
CAS, Hefei, Anhui, 230026, People's Republic of China
(Dated: November 20, 2018)
We study the phase separation of a spin polarized Fermi gas with spin-orbit coupling near a wide
Feshbach resonance. As a result of the competition between spin-orbit coupling and population
imbalance, the phase diagram for a uniform gas develops a rich structure of phase separated states
involving topologically non-trivial gapless super
uid states. We then demonstrate the phase sepa-
ration induced by an external trapping potential and discuss the optimal parameter region for the
experimental observation of the gapless super
uid phases.
PACS numbers: 03.75.Ss, 03.75.Lm, 05.30.Fk
Spin-orbit coupling (SOC), common in condensed mat-
ter systems for electrons, has been considered a key in-
gredient for many interesting phenomena such as topo-
logical insulators [1], quantum spin Hall e ects [2], etc.
The recent realization of synthetic gauge eld and hence
spin-orbit couplings in ultracold atomic systems opens
up exciting new routes in the study of these phenomena
[3, 4], allowing us to take advantage of the features of the
ultracold atoms, e.g. clean environment and highly con-
trollable parameters. In particular, with the Feshbach
resonance technique, the e ective interaction strength
between atoms can be tuned [5, 6]. This technique has
been applied to study various interesting topics, e.g. the
BCS-BEC crossover [7], polarize Fermi gases [8], itinerant
ferromagnetism [9], etc. The introduction of spin-orbit
coupling may shed new light on these strongly correlated
systems.
Spin-orbit coupled Fermi gas near a Feshbach reso-
nance has recently attracted much theoretical attention
[10{16]. The SOC has been shown to enhance pairing
on the BCS side of the Feshbach resonance [12, 14, 15].
Furthermore, for a polarized Fermi gas, the SOC intro-
duces competition against population imbalance, which
can lead to topologically non-trivial phases [13, 16]. Re-
cently, the phase diagrams for a polarized Fermi gas with
spin-orbit coupling near a Feshbach resonance have been
reported for a uniform gas [16]. The phase boundaries
have been calculated by solving the gap equation and
the number equations self-consistently. However, simi-
lar to the case of a polarized Fermi gas near Feshbach
resonance [17], due to the competition between di er-
ent phases, the solutions of the gap equation may corre-
spond to metastable or unstable states. By considering
the compressibility criterion [16], the unstable solutions
are correctly discarded, while the metastable solutions
may survive, rendering the resulting phase boundaries,
in particular those representing rst order phase transi-
tions, unreliable.
In this paper, we examine in detail the zero temper-
ature phase diagrams for a polarized Fermi gas with
Rashba spin-orbit coupling near a wide Feshbach reso-nance for both the uniform and the trapped cases. To
avoid getting metastable or unstable solutions, instead
of solving the gap equation, we minimize the thermody-
namic potential directly as in Ref. [17]. For the uniform
gas, we nd larger stability regions for the phase sepa-
rated state at unitarity as compared to the results in Ref.
[16]. More interestingly, we nd that SOC may induce
more complicated phase separated states involving gap-
less super
uid phases that are topologically non-trivial,
in addition to the typical phase separated state composed
of normal (N) and gapped super
uid (SF) phases. We
calculate the stability region for the various phase sepa-
rated states as well as for the gapless super
uid states, SF
state and normal state. We show that there are two dis-
tinct gapless phases that di er by the number of crossings
their excitation spectra have with the zero energy in mo-
mentum space, consistent with previous results [13, 16].
These novel gapless phases are stabilized by intermedi-
ate SOC strengths; whereas for large enough SOC, the
system always becomes a gapped super
uid of `rashbons'
[12]. We show how these phases can be characterized by
their di erent excitation spectra and momentum space
density distributions. We then discuss the phase separa-
tion in an external trapping potential, where the various
phases naturally phase separate in real space. By exam-
ining their respective stability regions, we demonstrate
the optimal parameter region to observe the gapless su-
per
uid states in the presence of a trapping potential.
For all of our calculations in the paper, we adopt the
BCS-type mean eld treatment. Although the mean eld
theory does not give quantitatively accurate results near
a wide Feshbach resonance, it is a natural rst step for
us to qualitatively estimate what phases may be stable,
as well as to understand their respective properties. We
also note that we have neglected the Fulde-Ferrell-Larkin-
Ovchinnikov (FFLO) phase in our calculations. This is
motivated by the fact that the FFLO phase is stable only
in a narrow parameter region in the absence of SOC due
to competition against other phases [8]. As SOC intro-
duces new gapless phases into this competition, we do
not expect a signi cant increase in its stability region.arXiv:1106.5667v2 [cond-mat.quant-gas] 30 Sep 20112
We rst consider a uniform three dimensional polarized
Fermi gas with Rashba spin-orbit coupling in the plane
perpendicular to the quantization axis z. The model
Hamiltonian takes the form [13, 14, 16]
H X
N=X
k;ka†
k;ak;
+h
2X
k
a†
k;#ak;# a†
k;"ak;"
+U
VX
k;k0a†
k;"a†
k;#a k0;#ak0;"
+X
k k?
e i'ka†
k;"ak;#+h:c:
; (1)
wherek=k , with the kinetic energy k=~2k2
2m;
=f";#gare the atomic spins; Ndenotes the to-
tal number of particles with spin ;ak;(a†
k;) annihi-
lates (creates) a fermion with momentum kand spin;
=h=2 is the chemical potential of the correspond-
ing spin species, and Vis the quantization volume. The
Rashba spin-orbit coupling strength can be tuned via
parameters of the gauge- eld generating lasers [4], while
k?=q
k2x+k2yand'k= arg (kx+iky). In writing
Hamiltonian (1), we assume s-wave contact interaction
between the two fermion species, with the bare interac-
tion rateUrenormalized following the standard relation
1
U=1
Up 1
VP
k1
2k[7]. The physical interaction rate is
given asUp=4~2as
m, whereasis the s-wave scattering
length between the two fermionic spin species.
To diagonalize the Hamiltonian, we make the trans-
formation: ak;"=1p
2ei'k(ak;++ak; ),ak;#=
1p
2(ak;+ ak; ), whereak;are the annihilation op-
erators for the dressed spin states with di erent helic-
ities () [12{16]. Taking the pairing mean eld =
U
VP
kha k;#ak;"ias in the standard BCS-type theory,
we may diagonalize the mean eld Hamiltonian in the
basis of the dressed spins:n
ak;+;a†
k;+;ak; ;a†
k; oT
.
The thermodynamic potential is then evaluated from
= 1
ln tr
e (H P
N)
, with = 1=kBT. In
this paper, we will focus on the zero temperature case,
for which the thermodynamic potential has the form
=1
2X
k;=( Ek;) Vjj2
U; (2)
with the quasi-particle excitation spectrum Ek;=r
2
k+ 2k2
?+jj2+h2
42q
(h2
4+ 2k2
?)2
k+h2
4jj2.
Before proceeding, let us examine the quasi-particle
excitations rst and study the conditions for possible
gapless phases. We see that at the points in the
momentum space where Ek; crosses zero, the quasi-
particle excitation becomes gapless while the pairing
gap remains nite. The SOC, together with the
population imbalance re-arranges the topology of the
0 0.5 1−0.2−0.15−0.1Ω/h
∆/h0 0.5 1−0.4−0.35−0.3
∆/hΩ/h
0 0.5 1−0.23−0.22−0.21−0.2
∆/hΩ/h
0 0.5 1−0.32−0.3−0.28−0.26
∆/hΩ/h(a)
(c)(b)
(d)N
GP2SF
SFFIG. 1. Illustration of typical shapes of the thermodynamic
potential
=has a function of order parameter =hfor var-
ious phases at unitarity: (a) =h = 0:52, kh=h= 0:1; (b)
=h= 0:7, kh=h= 0:1; (c)=h= 0:52, kh=h= 0:3, (d)
=h= 0:52, kF=h= 0:6. The chemical potential his taken
to be the energy unit, while the unit of momentum khis de-
ned through~2k2
h
2m=h.
Fermi surfaces of the spin species [13, 16]. The points
of gapless excitations lie on the kzaxis withk?= 0,
and are symmetric with respect to the kz= 0 plane.
More speci cally, for 0, the excitation spectrum
has two gapless points 2m
~2
+q
h2
4 jj21
2
,
so long asjhj
2>p
2+jj2. For > 0,
the excitation spectrum has four gapless points(
2m
~2
+q
h2
4 jj21
2
;2m
~2
q
h2
4 jj21
2)
,
withjj<jhj
2<p
2+jj2; two gapless points
2m
~2
+q
h2
4 jj21
2
, withjhj
2>p
2+jj2. We
identify the super
uid states with two excitation points
(GP1) and those with four excitation points (GP2) as
di erent topological phases [13, 16].
We illustrate in Fig. 1 typical shapes of the thermo-
dynamic potential as a function of with di erent pa-
rameters. Notably, due to the competition between dif-
ferent phases, a double-well structure appears (see Fig.
1(a-c)). Hence the solutions to the gap equation may
correspond to the metastable states (local minimum) or
the unstable states (local maximum). To make sure that
the ground state is achieved, we directly minimize the
thermodynamic potential [17].
Another complication comes from the existence of the
phase separated state, which must be considered explic-
itly for a uniform gas. As in the case of polarized Fermi
gases without SOC [8], we introduce the mixing coe-
cientx(0x1), and the thermodynamic potential
becomes
=x
( 1) + (1 x)
( 2); (3)3
00.2 0.4 0.6 0.8 11.200.20.40.60.81
α kF/EFPGP1
N+SF GP2+SF
SFGP2N
GP2+GP2
FIG. 2. Zero temperature phase diagram for a uniform Fermi
gas with population imbalance at ( kFas) 1= 0. Within the
bold phase boundaries are the various phase separated states
(see text). These phase separated states can be connected
with the non-phase separated states by rst order phase tran-
sitions (solid bold curve). The thin curves represent various
second order phase transitions (see text). Here kF= (32n)1
3,
EF=~2k2
F
2m, andnis the total density of the system.
where i(i= 1;2) is the pairing gap for the ith com-
ponent state. Note that due to SOC, we now have the
possibility of a phase separated state of two distinct su-
per
uid states (see Fig. 1(c)). The number equations of
the phase separated state become
N=x@
@
= 1+ (1 x)@
@
= 2: (4)
Minimizing the thermodynamic potential Eq. (3) with
respect to iandxwhile implementing the number con-
straints Eq. (4), we map out the phase diagram for a uni-
form polarized Fermi gas with SOC at ( kFas) 1= 0. Fig.
2 illustrates the resulting phase boundaries in the plane of
(P; kF=EF), where the polarization P=N" N#
N"+N#. When
the SOC is o ( = 0), the system remains in a phase
separated state of normal and gapped super
uid (PS1)
up toP0:93 before it becomes a normal state via a
rst order phase transition. This is consistent with pre-
vious mean eld calculations for a polarized Fermi gas
[8, 19], while di erent from the result in Ref. [16]. As
the SOC strength increases, a rich structure of dif-
ferent phases shows up, e.g. gapped super
uid phase
(SF), gapless super
uid phases with di erent Fermi sur-
face topology (GP1 and GP2), and notably, various phase
separated states. These phase separated states are con-
ned by a phase boundary of rst order phase transition
(bold curve in Fig. 2). In addition to the typical PS1
phase, we now have a phase separated state with GP2
and SF phases (PS2), and a phase separated state of two
distinct GP2 phases (PS3). As increases, the system
can undergo second order phase transitions from PS1 to
PS2 and then to PS3 for intermediate Pand . As-
sumingj1j<j2j, the phase boundaries between them
−1 0 101
kz/khEk,−/h
0 1 200.51
kz/khDensity
0 200.51
kz/khDensity
0 200.51
kz/khDensity0121
k⊥/kh0120.51
k⊥/kh
0121
k⊥/khSF
GP1
GP2(a) (b)
(d) (c)FIG. 3. Typical excitation spectrum and momentum space
density distribution for di erent phases. (a) Lower branch of
the excitation spectra for GP1 (solid), GP2 (dashed) and SF
(dash-dotted) phases; (b-d) Density distribution in momen-
tum space for spin-up (solid) and spin-down (dashed) species
alongk?= 0 andkz= 0 (inset), for (b) kh=h= 0:35,
=h = 0:5 (SF); (c) kh=h= 0:35,=h = 0:45 (GP1); (d)
kh=h= 0:45,=h= 0:43 (GP2), respectively.
can be determined by imposing 1= 0 (PS1 and PS2)
andh
2=j2j(PS2 and PS3), respectively. These phase
separated states nally become unstable and give way to
single component super
uid phases as becomes large.
The phase boundaries between these single component
states are determined by settingh
2=jj(SF and GP2),
h
2=p
2+jj2(GP2 and GP1), and = 0 (N and
GP1), respectively. When is large enough, the stabil-
ity region of the GP2 phase decreases and nally vanishes
at a tri-critical point ( = 0), beyond which only GP1,
SF and normal phase may exist. Note that beyond the
tri-critical point, the chemical potential becomes neg-
ative, and the phase boundary between GP1 and SF will
bend upwards so that in the large limit the SF phase
becomes dominant in the phase diagram.
To characterize the properties of the di erent
phases, we calculate the excitation spectrum and
number distribution in momentum space for SF,
GP1 and GP2 states (see Fig. 3). Several interest-
ing observations are in order. Firstly, the gapless
phases leave their signatures in the momentum
space density distribution. For k?= 0 andjkzj 2"
min
0; r
q
h2
4 jj2!
;r
+q
h2
4 jj2#
,
the minority spin component vanishes, and pairing does
not occur in this region. This is reminiscent of the
momentum space phase separation of a breached pairing
phase in the polarized Fermi gas [18], though now the
unpaired region lies only on the kzaxis. Away from
kzaxis, the occupation of the minority spin recovers
from zero gradually, leaving a signature which may be
detected in the time of
ight imaging experiment [16].
Secondly, for nite , both the gapless and the gapped4
0 0.5 1 1.5−0.500.5
αkh/hµ/h0.42 0.450.430.46
αkh/hµ/h
SF GP1 NGP2
V
FIG. 4. Phase diagram in the ( =h; k h=h) plane at
(khas) 1= 0. While the second order phase transitions are
in dashed thin curves, the rst order phase transitions are
shown in solid bold curves, which end at the point where the
double-well structure in the thermodynamic potential disap-
pears (inset). The boundary for vacuum (V) is determined
by setting the chemical potential of the majority spin species
to vanish in the normal phase.
super
uid phases can support population imbalance,
which can be seen from the density distribution along
k?(see Fig. 3 insets). Indeed as we will see later, for
large enough , we may expect no phase separation even
in the presence of a harmonic trapping potential. The
atoms in the trap will all be in the super
uid phase
induced by SOC.
To understand the spatial distribution of the various
phases in a trapping potential, we calculate the phase
diagram as a function of ( =h; kh=h) at unitarity (Fig.
4), wherekhis de ned in the caption of Fig. 1. Under
the Local Density Approximation (LDA) while assuming
both spin species experience the same harmonic poten-
tial, the local chemical potential (r) can be related to
that at the center of the trap as(r) = V(r),
whereV(r) gives the trapping potential. Thus a down-
ward vertical line in Fig. 4 represents a trajectory from
a trap center to its edge, with the chemical potential at
the trap center xed by that at the starting point of the
line. In Fig. 4, consistent with Fig. 2, the GP2 phase
only exists in a small parameter region in the trap, while
there appears to be considerable stability regions for the
GP1 phase. When is small, the Fermi gas in the trap
will phase separate into two regions, SF at the core, nor-
mal phase (N) towards the edge. At intermediate , the
gapless phases GP2 and GP1 may appear either near the
center of the trap or as a ring between the SF core and
the normal edge, depending on the chemical potentials.
Note that the boundary of the rst order phase transi-
tion between PS3 and GP2 (dotted thin curve in Fig. 2)
corresponds to a small scale structure here (Fig. 4 inset),
where a rst order phase transition (bold black curve) ex-
0 1 2012
αkh/hµ/h
00.20.40.6−0.5−0.4−0.3−0.2
αkh/hµ/hGP2
GP1SF
GP1SF(b) (a)
N
VN
VFIG. 5. Phase diagram in the ( =h; k h=h) plane at (a)
(khas) 1= 1 and (b) ( khas) 1= 0:5. First order phase
transitions are shown in solid bold curves, while second order
phase transitions are in dashed thin curves.
ists between two distinct gapless super
uids, both in the
GP2 phase. However, this region is found to be small at
unitarity and only increases slightly towards the BCS side
of the resonance. It is therefore dicult to observe this
phase transition in the trapped Fermi gas in the parame-
ter region that we considered. For large SOC beyond the
tip of the GP1-SF phase boundary, there is only the SF
phase in the phase diagram, and hence we will have only
the SOC induced SF phase in the trap for large enough
SOC.
We have also calculated the phase diagram in
(=h; kh=h) plane away from the resonance point. On
the BCS side (Fig. 5(a)), the stability region for the GP2
phase increases considerably. It is therefore desirable to
prepare the system on the BCS side of the resonance for
the observation of GP2 phases. On the BEC side (Fig.
5(b)), the GP2 phase vanishes from the phase diagram
altogether for <0, consistent with our previous discus-
sion.
In summary, we have calculated in detail the phase di-
agrams near a wide Feshbach resonance for a polarized
Fermi gas with Rashba spin-orbit coupling. We nd that
the competition among pairing, polarization and SOC
gives rise to a rich structure of phases and phase separa-
tions involving topologically non-trivial phases. From the
phase diagrams for both uniform and trapped systems,
we nd that the interesting gapless super
uid phases are
most likely to be observed in an experiment with moder-
ate polarization and SOC strength.
We would like to thank L.-M. Duan for helpful discus-
sions. This work was supported by NFRP 2011CB921200
and 2011CBA00200, NNSF 60921091, and The Fun-
damental Research Funds for the Central Universities
WK2470000001.
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2012.13914v1.Microwave_spectroscopy_of_spin_orbit_coupled_states__energy_detuning_versus_interdot_coupling_modulation.pdf | arXiv:2012.13914v1 [cond-mat.mes-hall] 27 Dec 2020Microwave spectroscopy of spin-orbit coupled states: ener gy detuning versus interdot
coupling modulation
G. Giavaras1and Yasuhiro Tokura1,2
1Faculty of Pure and Applied Sciences, University of Tsukuba , Tsukuba 305-8571, Japan
2Tsukuba Research Center for Energy Materials Science (TREM S), Tsukuba 305-8571, Japan
We study the AC field induced current peaks of a spin blockaded double quantum dot with
spin-orbit interaction. The AC field modulates either the in terdot tunnel coupling or the energy
detuning, and we choose the AC field frequency range to induce two singlet-triplet transitions
giving rise to two current peaks. We show that for a large detu ning the two current peaks can be
significantly stronger when the AC field modulates the tunnel coupling, thus making the detection
of the spin-orbit gap more efficient. We also demonstrate the i mportance of the time dependence of
the spin-orbit interaction.
I. INTRODUCTION
The singlet-triplet states of two electron spins in
tunnel-coupled quantum dots can be used to define spin-
qubits in semiconductor devices.1,2In the presence of a
strongspin-orbitinteraction(SOI)anappliedACelectric
field can give rise to singlet-triplet transitions and spin
resonance can be achieved.3,4In a double quantum dot
which is tuned to the spin blockade regime,5the transi-
tions can be probed by the AC induced current peaks.
It has been experimentally demonstrated that the two-
spin energy spectra can be extracted by examining the
magneticfielddependent positionofthecurrentpeaks.3,4
The exchange energy, the strength of the SOI, as well
as theg-factors of the quantum dots can then be esti-
mated. Microwavespectroscopy has also been performed
for the investigation of charge qubits,6,7as well as other
hybrid spin systems.8–10Charge localization in quantum
dot systems can be controlled with AC fields,11–13while
various important parameters of the spin and/or charge
dynamics can be extracted from AC induced interference
patterns.14–17
In this work, we study the current through a double
dot(DD) fortwodifferentcasesoftheACelectricfield; in
the first case the AC field modulates the interdot tunnel
coupling of the DD, and in the second case the AC field
modulates the energy detuning of the DD. We consider
a specific energy configuration and AC frequency range
which involve two SOI coupled singlet-triplet states, and
a third state with (mostly) triplet character. The two
SOI-coupled singlet-triplet states form an anticrossing
point,3,4,18,19and in this work we focus on this point.
Specifically, we perform electronic transport calcula-
tions and demonstrate that for a large energy detuning
the tunnel coupling modulation results in stronger AC-
induced current peaks than the corresponding peaks in-
duced by the detuning modulation. The stronger peaks
canofferasignificantadvantagewhenthespectroscopyof
the coupled spin system is performed by monitoring the
magnetic field dependence of the position of the peaks.
When the peaks are suppressed no reliable information
can be extracted.
The tunnel coupling modulation offers a similar ad-vantage when the transitions involve only the two states
forming the anticrossing point.20This finding together
with the results of the present work demonstrate that
modulating the tunnel coupling of a DD with an AC
electric field is a robust method to perform spectroscopy
of spin-orbit coupled states. Furthermore, in the present
work, we explore the time dependent role of the SOI, and
specify the regime in which this time dependence should
be taken into accountbecause it can drasticallyaffect the
AC-induced current peaks.
In some experimental works the interdot tunnel cou-
pling has been accurately controlled and transport
measurements have been performed.21,22For instance,
Bertrand et al[Ref. 22] have demonstrated that the tun-
nel coupling can be tuned by orders of magnitude on the
nanosecond time scale. Therefore, our theoretical find-
ings could be tested with existing semiconductor technol-
ogy.
II. DOUBLE QUANTUM DOT MODEL
We focus on the spin blockade regime5for two serially
tunnel-coupled quantum dots. In this regime the quan-
tum dot 1 (dot 2) is coupled to the left (right) metallic
lead and with the appropriate bias voltage current can
flow through the DD when the blockade is partially or
completely lifted. Each quantum dot has a single orbital
level and dot 2 is lower in energy by an amount equal to
the charging energy which is assumed to be much larger
thanthetunnelcoupling. Consequently,fortheappropri-
ate bias voltage a single electron can be localized in dot
2 during the electronic transport process.5If we use the
notation (n,m) to indicate nelectrons on the dot 1 and
melectronsonthedot2thenelectronictransportprocess
throughthe DDtakesplaceviathe chargecycle: (0 ,1)→
(1,1)→(0,2)→(0,1). For the DD system there are
in total six two-electron states but in the spin blockade
regime the double occupation on dot 1 can be ignored
because it lies much higher in energy and does not affect
the dynamics. Therefore, the relevanttwo-electronstates
are the triplet states |T+/angbracketright=c†
1↑c†
2↑|0/angbracketright,|T−/angbracketright=c†
1↓c†
2↓|0/angbracketright,
|T0/angbracketright= (c†
1↑c†
2↓+c†
1↓c†
2↑)|0/angbracketright/√
2 and the two singlet states2
|S02/angbracketright=c†
2↑c†
2↓|0/angbracketright,|S11/angbracketright= (c†
1↑c†
2↓−c†
1↓c†
2↑)|0/angbracketright/√
2. The
fermionic operator c†
iσcreates an electron on dot i= 1,
2 with spin σ=↑,↓, and|0/angbracketrightdenotes the vacuum state.
In this singlet-triplet basis the DD Hamiltonian20is
HDD= ∆[|T−/angbracketright/angbracketleftT−|−|T+/angbracketright/angbracketleftT+|]−δ|S02/angbracketright/angbracketleftS02|
−√
2Tc|S11/angbracketright/angbracketleftS02|+∆−|S11/angbracketright/angbracketleftT0|+H.c.
−Tso[|T+/angbracketright/angbracketleftS02|+|T−/angbracketright/angbracketleftS02|]+H.c.(1)
Here,δis the energy detuning, Tcis the tunnel coupling
between the two dots, and Tsois the SOI-induced tunnel
coupling causing a spin-flip.23,24The magnetic field is
denoted by Bwhich gives rise to the Zeeman splitting
∆i=giµBB(i= 1, 2) in each quantum dot. Then
∆ = (∆ 2+∆1)/2, and the Zeeman asymmetry is ∆−=
(∆2−∆1)/2. To a good approximation, in the transport
process of a spin-blockaded DD only the c†
2↑|0/angbracketright,c†
2↓|0/angbracketright
single electron states are important, and HDDcan be
also derived using a standard two-site Hubbard model.25
In the present work, we consider two cases for the AC
field. Specifically, in the first case the AC field modu-
lates the energy detuning of the DD, thus we consider
the following time dependence:
δ(t) =ε+Adsin(2πft), (2)
whereAdis the AC amplitude and fis the AC frequency.
Theconstantvalueofthedetuningisdenotedby ε. Inthe
second case, the AC field modulates the interdot tunnel
coupling, thus we assume the time dependent terms
Tc(t) =tc+Absin(2πft),
Tso(t) =tso+xsoAbsin(2πft).(3)
The AC amplitude is Aband in general Ab/negationslash=Ad. For
most calculations we assume that xso=tso/tc, so at any
time theratio Tso(t)/Tc(t) isatimeindependent constant
equaltoxso. We alsoaddressthe casewhere xso/negationslash=tso/tc,
but for simplicity we assume no phase difference between
the tunnel couplings Tc(t) andTso(t). For the numerical
calculations the DD parameters are taken to be tc=
0.2 meV,tso= 0.02 meV,g1= 2 andg2= 2.4. The
basic conclusions of this work are general enough and
not specific to these numbers.
III. RESULTS
In this section we present the basic results of our work.
Wedeterminethe ACinducedcurrentforeachcaseofthe
two AC fields Eq. (2) and Eq. (3). The DD eigenenergies
EisatisfyHDD|ψi/angbracketright=Ei|ψi/angbracketrightwithAd=Ab= 0, and are
shown in Fig. 1(a) at B= 1 T. When tso= 0 andg1=g2
singlet and triplet states are uncoupled. The energy lev-
elsE2,E3andE4correspond to the pure triplet states
|T+/angbracketright,|T0/angbracketrightand|T−/angbracketrightrespectively. These levels are detun-
ing independent and are Zeeman-split due to the applied
magnetic field. The energy levels E1,E5correspond to-0.6-0.4-0.2 0 0.2
0 0.5 1 1.5E (meV)
ε (meV)E1E2E3E4E5(a)
0 0.01 0.02
0 0.5 1 1.5 2 2.5 3 0 2 4 6∆so (meV)
∆so (GHz)
ε (meV)*
*(b)
FIG. 1: (a) Two-electron eigenenergies as a function of the
energydetuningfor themagnetic field B= 1T. The levels E4,
E5anticross at ε= 0.5 meV due to the spin-orbit interaction.
The two vertical arrows indicate possible transitions whic h
can be induced by the AC fields defined in the main text
Eq. (2) and Eq. (3). (b) Spin-orbit gap (∆ so=E5−E4at the
anticrossing point) as a function of the detuning. In this ca se
the magnetic field is detuning dependent. The AC induced
current is computed for the marked points.
pure singlet states which are |S11/angbracketright,|S02/angbracketrighthybridized due
to the tunnel coupling tcand are independent of the field
as can be seen from HDD. Importantly, the singlet levels
E1,E5define a two-level system and for a fixed tcthe
hybridization is controlled by the energy detuning. The
two levelsE1,E5anticross at ε= 0 where the hybridiza-
tion is maximum. This is the only anticrossing point in
the energy spectrum for tso= 0. However, according to
HDDwhentso/negationslash= 0 the polarized triplets |T±/angbracketrightcouple to
the singlet state |S02/angbracketright. Therefore, as seen in Fig. 1, at
ε≈0.5 meV the levels E4andE5form an anticrossing
point due to the SOI. Another SOI-induced anticross-
ing point is formed at ε <0 between the energy levels
E1andE2, but here we consider ε >0 and as in the
experiments3,26–28we taketso<tc.
BecauseoftheSOI( tso/negationslash= 0)andthedifferenceinthe g-
factors (g1/negationslash=g2) the DD eigenstates |ψi/angbracketrightare hybridized
singlet-triplet states and can be written in the general
form
|ψi/angbracketright=αi|S11/angbracketright+βi|T+/angbracketright+γi|S02/angbracketright+ζi|T−/angbracketright+ηi|T0/angbracketright.(4)3
The coefficients denoted by Greek letters determine the
character of the states, and are sensitive to the detuning.
One method to probe the SOI anticrossing point is to
focus on the AC frequency range 0 <hf/lessorsimilarE5−E4and
determine the position of the AC induced current peak.
This method has been theoretically studied in Ref. 20.
Another method to probe the anticrossing point is to fo-
cus on the AC frequency range E4−E2/lessorsimilarhf/lessorsimilarE5−E2,
and determine the positions of the two AC induced cur-
rentpeaks. The presentworkisconcernedwith the latter
method and the main subject of the present work is to
compare the current peaks induced separately by the two
AC fields; the tunnel barrier modulation and energy de-
tuningmodulation. InRef.3bothmethodshavebeenex-
perimentally investigated under the assumption that the
AC field modulates the energy detuning of the DD. The
case where the AC field modulates simultaneouslythe in-
terdot tunnel coupling and the energy detuning might be
experimentally relevant,29but this case is not pursued in
the present work.
In Fig. 1 the SOI anticrossing is formed at ε≈0.5
meV forB= 1 T. A lower magnetic field shifts the SOI
anticrossing point at larger detuning, and the degree of
hybridization due to the SOI decreases. The reason is
that asεincreases the |S02/angbracketrightcharacter in the original sin-
glet state ( tso= 0) is gradually replaced by the |S11/angbracketright
character. As a result, the SOI gap ∆ so=E5−E4,
defined at the anticrossing point, decreases with εas
shown in Fig. 1(b). For the parameters considered in
this work, the SOI gap can be analytically determined
from the expression30∆so= 2tso/radicalbig
(1−cosθ)/2, with
θ= arctan(2√
2tc/ε).
In ourpreviouswork20weexaminedthe transitionsbe-
tween the two singlet-triplet states |ψ4/angbracketrightand|ψ5/angbracketright, whose
energy levels form the SOI anticrossing point (Fig. 1).
These transitions give rise to one current peak which is
suppressedneartheanticrossingpoint, inagreementwith
an experimental study.3In the present work, we focus on
the transitions between the two pairs of states |ψ5/angbracketrightand
|ψ2/angbracketrightas well as |ψ4/angbracketrightand|ψ2/angbracketright. Here, |ψ4/angbracketrightand|ψ5/angbracketrightare
strongly hybridized singlet-triplet states, whereas |ψ2/angbracketright
has mostly triplet character provided the detuning is
large.
We compute the AC-induced current flowing through
the double dot within the Floquet-Markov density ma-
trix equation of motion.31,32In this approach we treat
the time dependence of the AC field exactly, taking ad-
vantage of the fact that the DD Hamiltonian is time pe-
riodic and thus it can be expanded in a Fourier series.
The model uses for the basis states of the DD density
matrix the periodic Floquet modes,33and consequently
it is applicable for any amplitude of the AC field. In
most calculations we take the parameter xso= 0.1 unless
otherwise specified.
To study the AC current spectra we choose two values
fortheenergydetuning ε(2, 0.5meV),anddeterminethe
magnetic field at which |ψ4/angbracketrightand|ψ5/angbracketrightanticross. At this
specific field we plot in Fig. 2 the AC-induced current as 0.05 0.1 0.15 0.2 0.25
17 17.5 18 18.5 19 19.5 20Ι (pA)
f (GHz)Ab = 10 µeV
Ad = 10 µeV (a)
0.4 0.8 1.2 1.6
58 59 60 61 62 63 64 65Ι (pA)
f (GHz)Ab = 10 µeV
Ad = 10 µeV (b)
FIG. 2: Current as a function of AC frequency, when the
AC field modulates the tunnel barrier with the AC amplitude
Ab= 10µeV, and the energy detuning with Ad= 10µeV.
The constant value of the detuning is ε= 2, 0.5 meV and the
corresponding magnetic field is B= 0.3, 1 T for (a) and (b)
respectively. These fields define the singlet-triplet antic ross-
ing point for each value of the detuning.
a function of the AC frequency. As the energy detuning
decreases the magnetic field defining the corresponding
anticrossing point increases. This in turn means that the
AC frequencyhasto increaseto satisfythe corresponding
resonance condition hf=E5−E2(orhf=E4−E2).
This increase in the frequency explains the different fre-
quency range in Fig. 2. Furthermore, the off-resonant
current is larger for ε= 0.5 meV due to the stronger SOI
hybridization.25
In the two cases shown in Fig. 2 two peaks are formed;
one peak is due to the transition between the eigenstates
|ψ2/angbracketrightand|ψ4/angbracketright, and the second peak is due to the transi-
tion between |ψ2/angbracketrightand|ψ5/angbracketright. Therefore, the distance be-
tween the centres of the two peaks is equal to the singlet-
triplet energy splitting E5−E4. For the specific choice
of magnetic field this energy splitting is equal to the SOI
gap of the anticrossing point. For example, for ε= 0.5
meV the gap is ∆ so≈3.5 GHz, and for ε= 2 meV the
gap is ∆ so≈1.1 GHz. These numbers are in agreement
with those derived from the exact energies of the time
independent part of the Hamiltonian HDD. According to
Fig. 2, for a given energy detuning and driving field the
two peaks are almost identical. This is due to the fact,
that at the anticrossing point the states |ψ4/angbracketright,|ψ5/angbracketrighthave
identical characters when the driving field is off, and the
relevant transition rates are almost equal. In contrast,4
0 0.5 1 1.5 2
B (T) 0.5 1 1.5 2ε (meV)
0 0.3 0.6
qb (µeV)(b) 0 0.5 1 1.5 2 0.5 1 1.5 2ε (meV)
0 0.25 0.5
qd (µeV)(a)
FIG. 3: (a) Absolute value of the coupling parameter qdas
a function of the energy detuning and magnetic field for the
AC amplitude Ad= 10µeV. The dotted curve defines the
anticrossing point for each εandB. (b) The same as (a) but
forqbwithAb= 10µeV.
0 10 20 30 40
0 0.5 1 1.5 2 2.5 3 3.5qb/qd
ε (meV)0.00.020.040.060.080.1xso = 0.12
FIG. 4: The ratio qb/qddefined in Eq. (9) as a function of
detuning for different values of xsoandAd=Ab.
away from the anticrossing point the two peaks can be
very different.33
The results in Fig. 2 demonstrate that the two driving
fields Eq. (2) and Eq. (3) induce different peak magni-
tudes. Specifically, the peaks due to the tunnel barrier
modulation are stronger than those due to the detuningmodulation. As an example, for ε= 0.5 meV [Fig. 2(b)]
the tunnel barrier modulation induces a relative peak
height of about 1 pA, whereas the relative peak height is
only 0.1 pA for the energy detuning modulation. Some
insight into this interesting behavior can be obtained by
inspecting the time-scale (“Rabi” frequency) of the co-
herent transitions between the eigenstates |ψ2/angbracketrightand|ψ4/angbracketright.
When the AC field modulates the tunnel coupling, the
transitionscanbestudiedwithintheexactFloqueteigen-
value problem, but for simplicity we here employ an ap-
proximate approach.20This two-level approach gives the
transition frequency qb/h, with
qb=hfhb
24
(hb
22−hb
44)J1/parenleftbiggAb(hb
22−hb
44)
hf/parenrightbigg
,(5)
whereJ1(r) is a Bessel function of the first kind, and the
argument is r=Ab(hb
22−hb
44)/hf, withhf=E4−E2
and
hb
ij=−γj(√
2αi+xsoβi+xsoζi)
−γi(√
2αj+xsoβj+xsoζj), i,j= 2,4(6)
When the AC field modulates the energy detuning the
time-scale of the coherent transitions between the eigen-
states|ψ2/angbracketrightand|ψ4/angbracketrightis approximately qd/h. The cou-
pling parameter qdis found by qbwith the replacements
Ab→Adandhb
ij→hd
ij, where
hd
ij=−γiγj, i,j= 2,4 (7)
In general, qbcan be very different from qd, even when
Ab=Ad. Therefore, the two driving fields are expected
to induce current peaks with different width and height.
To quantify the two parameters qb,qdwe plot in Fig. 3
qb,qd, as a function of the energy detuning and the mag-
netic field. Here, Ad=Ab= 10µeV, andhf=E4−E2
[in Eq. (5)] is magnetic field as well as detuning depen-
dent, and is determined by the energies of HDD. If we
denote byBanthe field at which the anticrossing point is
formed, then as seen in Fig. 3 both qbandqdare largefor
B > B an, but vanishingly small for B≪Ban. The rea-
son is that the state |ψ4/angbracketrightis singlet-like for B >B an, but
triplet-like for B <B an, whereas |ψ2/angbracketrighthas mostly triplet
characterindependent of B, providedεisawayfromzero.
Transitions between triplet-like states are in general slow
leading to vanishingly small qb,qdforB≪Ban. In
contrast, if we choose hf=E5−E2, then both qband
qdare large for B < B an. For large enough detuning
where the two spins are in the Heisenberg regime, the
exchange energy is approximately 2 t2
c/εthereforeBan
satisfies (g1+g2)µBBan/2≈2t2
c/ε.
Most importantly Fig. 3 demonstrates that qb> qd
whenε/greaterorsimilar0.2 meV. To understand this result we focus
on the anticrossing point where r<1, then from Eq. (5)
qb≈hb
24Ab/2 becauseJ1(r)≈r/2, and similarly qd≈
hd
24Ad/2. Moreover, away from zero detuning the state
|ψ2/angbracketrighthas mostly triplet character, therefore
hb
24≈ −γ4(√
2α2+xsoβ2)−γ2(√
2α4+xsoζ4),(8)5
and the ratio qb/qdis
qb
qd=Ab
Ad/parenleftbigg√
2α2
γ2+xsoβ2
γ2+√
2α4
γ4+xsoζ4
γ4/parenrightbigg
.(9)
Asεincreasesβ2→1,γ2≪1 and, considering absolute
values, the second term in Eq. (9) dominates
β2
γ2≫α2
γ2,α4
γ4,ζ4
γ4. (10)
Consequently, qbcan be much greater than qd, especially
at largeε, and for a fixed tunnel coupling tcthe exact
value of the ratio qb/qddepends sensitively on xso. This
demonstrates the importance of the time dependence of
the spin-orbit coupling. The conclusions derived from
the parameters qb,qdassume that there is no ‘multi-
level’ interference and only the levels Ei,Ejsatisfying
hf=|Ei−Ej|are responsible for the current peaks.
The approximate results are more accurate when the ar-
gumentrof the Bessel function is kept small.
To examine the xso-dependence, we consider Ab=Ad
and plot in Fig. 4 the ratio qb/qdversus the detuning
at the anticrossing point, and for different values of xso.
By increasing εand for large values of xsothe coupling
parameters qb,qdcan differ by over an order of mag-
nitude;qb/qd>10. This leads to (very) different cur-
rent peaks with the tunnel barrier modulation inducing
stronger peaks. The special value xso= 0 corresponds to
atimeindependent SOItunnelcoupling[seeEq.(3)], and
thespecialvalue xso= 0.1correspondstoatimeindepen-
dent ratioTso/Tc= 0.1. Although, the ratio qb/qdcan
be computed at any ε, the regime of small ε(<0.2 meV)
is not particularly interesting in this work. The reason
is that with decreasing εthe character of the state |ψ2/angbracketright
changes from triplet-like to singlet-triplet, which even-
tually becomes approximately equally populated to |ψ4/angbracketright
and|ψ5/angbracketright. Therefore, the current peaks induced by both
driving fields are suppressed even when qborqdis large.
In Fig. 4 the maximum value of the detuning is chosen
to giveε/tc≈17.5 which can be easily achieved in dou-
ble quantum dots. Some experiments1,3,4have reported
values greater than ε/tc≈100, thusqbcan be even two
orders of magnitude greater than qd.
According to the above analysis if qb/qd≈1 then the
current peaks induced by the two driving fields should
approximately display the same characteristics. As an
example, consider the two sets of current peaks shown
in Fig. 2 both for xso= 0.1 andε= 2 meV, ε= 0.5
meV respectively. Focusing on xso= 0.1 in Fig. 4, we
see that at ε= 2 meVqb/qd≈19 and atε= 0.5 meV
qb/qd≈4.9. These numbers suggestthat if at ε= 2 meV
we choose for the AC amplitudes the ratio Ab/Ad≈1/19
then the detuning and the barrier modulation should in-
duce approximately the same peak characteristics. Like-
wise atε= 0.5 meV the ratio should be Ab/Ad≈1/4.9.
These arguments are quantified in Fig. 5 where we plot
the current peaks for the two driving fields for different
AC amplitudes satisfying the condition qb/qd≈1. The 0.05 0.1 0.15 0.2 0.25
17 17.5 18 18.5 19 19.5 20Ι (pA)
f (GHz)Ad = 190 µeV
Ab = 10 µeV (a)
0.4 0.8 1.2 1.6
58 59 60 61 62 63 64 65Ι (pA)
f (GHz)Ad = 49 µeV
Ab = 10 µeV (b)
FIG. 5: As in Fig. 2, but (a) Ab= 10µeV andAd= 19Ab,
(b)Ab= 10µeV andAd= 4.9Ab. The value of Adis chosen
so that to approximately induce the same current peaks as
those induced by Ab.
0.05 0.1 0.15 0.2 0.25
17.5 18 18.5 19 19.5Ι (pA)
f (GHz)
FIG. 6: Current as a function of AC frequency, when the AC
field modulates the tunnel barrier, with the AC amplitude
Ab= 10µeV. The detuning is ε= 2 meV and from the upper
to the lower curve the parameter xso= 0.1, 0.04, 0.02, 0.
results confirm that the induced current peaks display
approximately the same characteristics.
Inducing strong current peaks can be advantageous in
order to perform spectroscopy of the singlet-triplet levels
and extract the SOI anticrossing gap. However, an im-
portant aspect is that the SOI gap cannot be extracted
from the positions of the current peaks at arbitrary large
AC amplitudes. In particular, by increasing the AC am-
plitude the two peaks start to overlap and eventually6
the resonant pattern of the current changes drastically.33
Therefore, the distance between the two peaks cannot
accurately predict the SOI gap. This effect has been
theoretically studied for the case of a time dependent
energy detuning,33and it can be readily shown that sim-
ilar trends occur for a time dependent tunnel coupling.
The driving regime where the two current peaks strongly
overlap is not considered in the present work, since it is
not appropriate for the spectroscopy of the SOI gap.
Finally, in Fig. 6 we plot the current peaks when the
AC field modulatesthe tunnel barrierwith the amplitude
Ab= 10µeVandtheconstantdetuning ε= 2meV.With
decreasing xsothe two peaks gradually weaken and for
xso= 0 the peaks arevanishingly small; for this value the
peaks are of the same order as the peaks induced by the
detuning modulation with the same amplitude Ad= 10
µeV (for clarity these peaks are not shown). The small
difference between the left and the right peaks, for exam-
ple whenxso= 0.02,can be understood byinspecting the
different values of qb[Eq. (5)] which involvedifferent ma-
trixelementsandfrequencies. Theoveralltrendsindicate
the important role of the time dependent spin-orbit term
andareconsistentwiththeresultsshowninFig.4. As xso
decreases the coupling parameter qbdecreases too, thus
the time scale of the singlet-triplet transitions becomes
longer leading to smaller peaks. Moreover, by decreasing
xso,qbbecomes approximately equal to qd, therefore the
tunnel barrier modulation and the detuning modulationresult in approximately the same current peaks.
IV. SUMMARY
In summary, we considered a double quantum dot in
the spin blockade regime and studied the AC induced
current peaks for a specific energy configuration which
involves two hybridized singlet-triplet states as well as
a third state with mostly triplet character. The two
AC induced transitions which rely on the spin-orbit in-
teraction, result in two current peaks. We found that
for a large energy detuning the two peaks are stronger
when the time periodic field modulates the interdot tun-
nel coupling (barrier)instead ofthe energy detuning. We
demonstrated that a time dependence in the spin-orbit
coupling can significantly modify the peak characteris-
tics, and should be taken into account even when the
actual spin-orbit coupling is small. Our work suggests
an efficient way of probing the spin-orbit energy gap in
two-spin states based on transport measurements.
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Part of this work was supported by CREST JST (JP-
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2003.13181v1.Intrinsic_orbital_moment_and_prediction_of_a_large_orbital_Hall_effect_in_the_2D_transition_metal_dichalcogenides.pdf | Intrinsic orbital moment and prediction of a large orbital Hall e ect in the 2D
transition metal dichalcogenides
Sayantika Bhowaland S. Satpathy
Department of Physics & Astronomy, University of Missouri, Columbia, MO 65211, USA
Carrying information using generation and detection of the orbital current, instead of the spin
current, is an emerging eld of research, where the orbital Hall e ect (OHE) is an important ingredi-
ent. Here, we propose a new mechanism of the OHE that occurs in non-centrosymmetric materials.
We show that the broken inversion symmetry in the 2D transition metal dichalcogenides (TMDCs)
causes a robust orbital moment, which
ow in di erent directions due to the opposite Berry cur-
vatures under an applied electric eld, leading to a large OHE. This is in complete contrast to the
inversion-symmetric systems, where the orbital moment is induced only by the external electric eld.
We show that the valley-orbital locking as well as the OHE both appear even in the absence of the
spin-orbit coupling. The non-zero spin-orbit coupling leads to the well-known valley-spin locking
and the spin Hall e ect, which we nd to be weak, making the TMDCs particularly suitable for
direct observation of the OHE, with potential application in orbitronics .
Orbital Hall e ect (OHE) is the phenomenon of trans-
verse
ow of orbital angular momentum in response to
an applied electric eld, similar to the
ow of spin angu-
lar momentum in the spin Hall e ect (SHE). The OHE
is more fundamental in the sense that it occurs with or
without the presence of the spin-orbit coupling (SOC),
while in presence of the SOC, OHE leads to the addi-
tional
ow of the spin angular momentum resulting in
the SHE. In fact, the idea of OHE has already been in-
voked to explain the origin of a large anomalous and spin
Hall e ect in several materials [1{3]. Because of this and
the fact that OHE is expected to have a larger magnitude
than its spin counterpart, there is a noticeable interest
in developing the OHE [4{7], with an eye towards future
\orbitronics" device applications.
In this work, we propose a new mechanism of the OHE
that occurs in non-centrosymmetric materials and explic-
itly illustrate the ideas for monolayer transition metal
dichalcogenides (TMDCs) which constitute the classic
example of 2D materials with broken inversion symme-
try. In complete constrast to the centrosymmetric mate-
rials [4, 6], where orbital moments are quenched due to
symmetry and a non-zero moment develops only due to
the symmetry-breaking applied electric eld, here an in-
trinsic orbital moment is already present in the Brillouin
zone (BZ) even without the applied electric eld. Unlike
the centrosymmetric systems, the physics here is dom-
inated by the non-zero Berry curvatures, which deter-
mines the magnitude of the OHE. Our work emphasizes
the intrinsic nature of orbital transport in contrast to the
valley Hall e ect [8{12], for example, which can only be
achieved by extrinsic means (doping, light illumination,
etc.).
We develop the key physics of the underlying mecha-
nism of the OHE using a tight-binding (TB) model as
well as from density-functional calculations. The e ect
is demonstrated for the selected members of the family
bhowals@missouri.edu
MXxy
X(a) (b)
x2-y2+i xy
x2-y2-i xy
3z2-1
3z2-1
K (t = +1) K' (t = -1)
MX2(d)
K K'
Eky
kxkz=0Mz < 0
Mz >0E
v =(e/ħ) E X Ωv
v
Mz < 0
Mz >0a
10
5
0
-5
-10(c)
FIG. 1. Illustration of OHE in monolayer MX 2. (a) Crystal
structure of MX 2, showing the triangular network of transi-
tion metal M atoms as viewed from top. The two out-of-plane
chalcogen atoms X occur above and below the plane. (b) The
band structure near K( 4=3a;0) andK0(4=3a;0), show-
ing the valley dependent spin and orbital characters. (c) The
orbital moment Mz(~k) in the BZ and the anomalous veloci-
tiesv, indicated by the blue and the red arrows. (d) Orbital
moments
ow in the transverse direction leading to the OHE.
of monolayer TMDCs, viz., 2H-Mo X2(X= S, Se, Te),
where we nd a large OHE and at the same time a negli-
gible intrinsic spin Hall e ect, making these materials an
excellent platform for the direct observation of the OHE.
The basic physics is illustrated in Fig. 1, where we have
shown the computed intrinsic orbital moments in the BZ
as well as the electron \anomalous" velocities at the K,
K0valleys. Symmetry demands that in the presence
of inversion (I), orbital moments satisfy the condition
~M(~k) =~M( ~k), while if time-reversal ( T) symmetry is
present, we have ~M(~k) = ~M( ~k). Thus for a non-zero
~M(~k), at least one of the two symmetries must be broken.arXiv:2003.13181v1 [cond-mat.mtrl-sci] 30 Mar 20202
In the present case, broken Ileads to a nonzero ~M(~k),
while its sign changes between the KandK0points due
to the presence of T. The Berry curvatures ~
(~k) follow
the same symmetry properties leading to the non-zero
anomalous velocity ~ v= (e=~)~E~
~k[13] which has op-
posite directions at the two valleys, and thus leads to
the OHE. These arguments are only suggestive, and one
must evaluate the magnitude of the e ect from the cal-
culation of the orbital Berry curvatures [13] as discussed
below.
Tight Binding results near the valley points { The val-
ley points ( K/K0) have the major contributions to the
OHE in the TMDCs and this can be studied analytically
using a TB model. Due to the broken I[see Fig. 1 (a)],
the chalcogen atoms must be kept along with the transi-
tion metal atom (M) in the TB basis set; However, their
e ect may be incorporated via the L owdin downfolding
[14] producing an e ective TB Hamiltonian for the M-
dorbitals with modi ed Slater-Koster matrix elements
[15]. The e ective Hamiltonian, valid near the KandK0
valley points reads
H(~ q) = (~d~ )
Is+
2(z+ 1)
sz; (1)
where only terms linear in ~ q=~k ~Khave been kept,
ignoring thereby the higher-order trigonal warping [16],
which are unimportant for the present study. Here ~ sand
~ are respectively the Pauli matrices for the electron spin
and the orbital pseudo-spins, jui= (p
2) 1(jx2 y2i+
i jxyi) andjdi=j3z2 r2i.Isis the 22 identity
operator in the electron spin space, is the SOC con-
stant, and the valley index =1 for theKandK0
valleys, respectively. The TB hopping integrals appear
in the parameter ~d, withdx= tqxa;dy= tqya;and
dz= =2, whereais the lattice constant, is the
energy gap at the K(K0) point, and tis an e ective
inter-band hopping, determined by certain d dhopping
matrix elements. We note that Eq. (1) is consistent with
the Hamiltonian derived earlier [10] using the kpthe-
ory. The TB derivation has the bene t that it directly
expresses the parameters of the Hamiltonian in terms of
the speci c hopping integrals.
The magnitude of the orbital moment ~M(~k) can be
computed for a speci c band of the Hamiltonian (1) using
the modern theory of orbital moment [13, 17], viz.,
~M(~k) = 2 1Im[h~rku~kj(H "~k)j~rku~ki]
+ Im[h~rku~kj(F "~k)j~rku~ki]; (2)
where"~kandu~kare the band energy and the Bloch wave
function, and the two terms in (2) are, respectively, the
angular momentum ( ~ r~ v) contribution due to the self-
rotation and due to the motion of the center-of-mass of
the Bloch electron wave packet. Diagonalizing the 4 4
Hamiltonian (1), we nd the energy eigenvalues: "
=
2 1[ (( )2+ 4t2a2q2)1=2], where=1 are
the two spin-split states within the conduction or valence
band manifold, denoted by the subscript . The wave
K'(a)
(b)3z2-r2 x2-y2 - i xyxz + i yz
xz - i yz
x2-y2 + i xyM
zDFTTB ModelE
F
x2-y2 - i xy3z2-r2 DEnergy (eV)M
z (eV Å2
)FIG. 2. (a) Density-functional band structure together with
the orbital characters near the valley points and (b) the com-
puted sum of the orbital moments ( Mz) over all occupied
bands along selected symmetry lines.
functions in the basis set ( ju"i;jd"i;ju#i;andjd#i)
are
ju=1
(q)i=Nh
1 (Dp
(D)2+d2)=d0 0iT
;
ju= 1
(q)i=Nh
0 0 1 (Dp
(D)2+d2)=diT
(3)
whereD= ( )=2,d=ta( qxiqy),d2=
t2a2(q2
x+q2
y), andNis the appropriate normalization
factor. With these wave functions, the orbital moments
can be evaluated exactly within the TB model from Eq.
2. For the two valence bands ( =1), the result is
Mz(~ q) = m0D(D )
2[(D)2+t2q2a2]3=2(4a)
m0[1 +(3 2)=](1 6m0q2=); (4b)
wherem0= 1t2a2, only the out-of-plane ^ zcomponent
of the orbital moment is non-zero, and the second line is
the expansion for small qand, both .
Note the important result (4) that a large orbital mo-
mentMzexists at the valley points ( ~ q= 0) and its sign al-
ternates between the two valleys ( =1) (valley-orbital
locking ). Furthermore, it exists even in absence of the
SOC (= 0). For typical parameters, t= 1:22 eV, =
1.66 eV, and = 0:08 eV, relevant for the monolayer
MoS 2,m09:1 eV. A22:4B(~=e). As seen from
Eq. 4 (b), there is only a weak dependence on .
In fact it is interesting to note that the valley-
dependent spin splitting [Fig. 1 (b)] directly follows from
the valley orbital moments due to the h~L~Siterm, which
favors anti-alignment of spin with the orbital moment [3].
Thus for the valence bands, the spin- #band is lower in
energy atK, while the spin-"band is lower at K0, with
a spin splitting of about 2 . Therefore, the well-known
spin polarization of the bands at the valley points can be
thought of to be driven by the robust orbital moments
via the perturbative SOC.3
The orbital moment is the largest at the valley points
K;K0, as seen from Eq. (4), falling o quadratically with
momentum q. This is also validated by the DFT results
shown in Fig. 2. The orbital moment at the center of the
BZ ( ) vanishes exactly due to symmetry reasons, and
therefore is expected to be small in the neighborhood of
as seen from Fig. 2 (b) as well.
It is easy to argue that under an applied electric eld,
the electrons in the two valleys move in opposite direc-
tions, so that a net orbital Hall current is produced. To
see this, we rst realize that only the Berry curvature
term in the semi-classical expression [13] for the electron
velocity _~ rc=~ 1[~rk"k+e~E~
(~k)]~kcis non-zero for
the two valleys. Furthermore, only the ^ zcomponent of
the Berry curvature survives, which we evaluate near the
K;K0valleys within the TB model using the Kubo for-
mula below. The result is
z
n(~ q) = 2~2X
n06=nIm
hun~ qjvxjun0~ qihun0~ qjvyjun~ qi
("n0~ q "n~ q)2
=2Mz(~ q)
+( 2)2 m0
2( + 2 6m0q2):(5)
Clearly,
zhas opposite signs for the two valleys, so that
~ v/~E~
is in opposite directions for the Kand the
K0valley electrons. Thus the positive orbital moment of
theKvalley moves in one direction, while the negative
orbital moment of K0moves in the opposite direction,
leading to a net orbital Hall current.
The magnitude of the orbital Hall conductivity (OHC)
may be calculated using the Kubo formula by the mo-
mentum sum of the orbital Berry curvatures [4, 6], viz.,
;orb
= e
NkVcoccX
n~k
;orb
n; (~k); (6)
where ; ;
are the cartesian components, jorb;
=
;orb
E is the orbital current density along the direc-
tion with the orbital moment along
, generated by the
electric eld along the direction. In the 2D systems, Vc
is the surface unit cell area, so that the conductivity has
the dimensions of ( ~=e) Ohm 1.
The orbital Berry curvature
;orb
n; in Eq. 6 can be
evaluated as
;orb
n; (~k) = 2 ~X
n06=nIm[hun~kjJ
;orb
jun0~kihun0~kjv jun~ki]
("n0~k "n~k)2;
(7)
where the orbital current operator is J
;orb
=1
2fv ;L
g,
withv =1
~@H
@k is the velocity operator and L
is the
orbital angular momentum operator.
It turns out that due to the simplicity of the TB Hamil-
tonian (1), valid near the valley points, the orbital and
the standard Berry curvatures are the same, apart from
a valley-dependent sign, viz.,
z;orb
n;yx(~ q) =
z
n(~ q): (8)To see this, we take the momentum derivative of (1) to
get
~vx(~ q) =2
640 ta 0 0
ta 0 0 0
0 0 0 ta
0 0 ta 03
75=ta x
Is;(9)
and, similarly, ~vy(~ q) = tay
Isandvz(~ q) = 0.
Furthermore, in the subspace of the TB Hamiltonian,
Lx=Ly= 0, andLz= ~(z+ 1)
Is. By matrix mul-
tiplication, we immediately nd that Jz;orb
= ~v and
Jx;orb
=Jy;orb
= 0, which leads to the result (8). The
expression for the orbital Berry curvature then follows
from Eqs. (5) and (8), viz.,
z;orb
;yx(~ q) =2 Mz(~ q)
+( 2); (10)
whereMz(~ q) is the orbital moment in Eq. (4). At a
generalkpoint, the full expression (7) must be evaluated
to obtain the OHC.
This is a key result of the paper, which shows that the
orbital Berry curvatures near the KandK0points are
directly proportional to the respective orbital moments,
and, more importantly, they have the same sign at the
two valleys as both andMzchange signs simultane-
ously. Thus, the contributions from these two valleys
add up, leading to a non-zero OHC. Another important
point is that
z;orb
;yx exists even without the SOC, and
it has only a weak dependence on as seen from Eq.
(10). Neglecting the dependence, we see that at both
valley points, the contribution to the OHC is given by
z;orb
;yx = 2t2a2=2. In fact, the momentum sum in OHC
can be performed analytically in this limit by integrating
up to the radius qc(q2
c=
BZ) to yield the result
z;orb
yx = 2e
(2)2X
=1Zqc
0d2q
z;orb
;yx(~ q)
= e
h
1 q
2+ (32t2=p
3)i
+O(2=2);
(11)
which is consistent with the anticipated result that the
larger the parameter t2=2, the larger is the OHC, pri-
marily because the orbital moment Mzincreases.
We pause here to compare the OHE with the related
phenomenon of the valley Hall e ect, which has been pro-
posed in the gapped graphene as well as in the TMDCs
[11, 12]. In the valley Hall e ect, electrons in the two
valleys
ow in opposite directions, leading to a charge
current and additionally to an orbital current (the valley
orbital Hall e ect [12]), if there is a valley population im-
balance (e.g., created by shining light). This is in com-
plete contrast to the OHE, which is an intrinsic e ect
without any need for population imbalance between the
valleys. More interestingly, unlike the valley Hall e ect,
the OHE described here does not have any net charge4
MK K'(a)
K K'
GM
1.00.50-0.5
1.5G
-505101520(b)
00
FIG. 3. (a) Orbital and (b) spin Berry curvatures (in units of
A2), summed over the occupied states, on the kz= 0 plane for
2H-MoS 2. The contours correspond to the tick values on the
color bar and the zero contours have been indicated explicitly.
current but there exists only a pure orbital current. Fur-
thermore, in the valley Hall e ect, the non-zero valley
orbital magnetization [11] explicitly breaks the Tsym-
metry, which is preserved in the present case. In this
sense the OHE studied here is completely di erent from
the valley Hall e ect proposed earlier.
Density functional results { We now turn to the DFT
results for the monolayer TMDCs. Orbital moments were
computed using pseudopotential methods [18] and the
Wannier functions as implemented in the Wannier90 code
[19, 20] [see Supplementary Materials [21] for details].
The complementary mun-tin orbitals based method
(NMTO) [22] was used to compute the orbital moment
as well as the orbital and the spin Hall conductivities.
In the latter method, e ective TB hopping matrix ele-
ments between the M- dorbitals are obtained for several
neighbors, which yields the full TB Hamiltonian valid
everywhere in the BZ, using which all quantities of inter-
est are computed. The BZ sums for the OHC and spin
Hall conductivity (SHC) were computed with 400 400
kpoints in the 2D zone. The computed orbital moments
using the Wannier90 or the NMTO method agree quite
well.
The DFT band structure and the corresponding or-
bital moments are shown in Fig. 1 (c) and Fig. 2 for
TABLE I. DFT results for the OHC of the monolayer TMDCs,
including the partial contributions ( z;orb
yx =K+ +rest),
K, , andrestbeing the contributions, respectively, from
the valley, -point, and the remaining regions of the BZ. OHC
are in units of 103(~=e)
1, while the SHC are in units of
(~=e)
1.
Materials K restz;orb
yxz;spin
yx
MoS 2 -9.1 1.7 -3.2 -10.6 1.0
MoSe 2 -8.0 1.7 -3 -9.3 1.8
MoTe 2 -9.1 1.1 -2.5 -10.5 3.0
WTe 2 -8.6 1.0 -2.6 -10.2 9.4MoS 2. As shown in Fig. 2 (b), the orbital moments com-
puted from the Hamiltonian (1) near the valley points
agree quite well with the DFT results. Note that the to-
tal orbital moment (summed over the BZ) vanishes due
to the presence of T, though it is non-zero at individual
kpoints. From the TB model (1), we had studied the or-
bital moment and the OHE near the valley points. From
the DFT calculations, we can compute the same over the
entire BZ, the result of which is shown in Fig. 3 (a). As
seen from the gure, the dominant contribution comes
from thekspace near the valley points K,K0. Since
the intrinsic orbital moment near the point is absent,
the orbital Berry curvature in this region takes a non-
zero value only due to the orbital moments induced by
the applied electric eld in the Hall measurement, similar
to the centrosymmetric case [4]. This results in a small
contribution to the net OHC, as seen from Table I,
which lists the partial contributions to the OHC coming
from di erent parts of the BZ. Note that there is only one
independent component of OHC, viz., z;orb
yx = -z;orb
xy.
Spin Hall E ect { For a material to be a good can-
didate for the detection of the OHE, the SHC must be
small, as both carry angular momentum. To this end, we
compute the SHC, rst from the model Hamiltonian and
then from the full DFT calculations. Analogous to the
calculation of the OHC, the SHC can be obtained by the
sum of the spin Berry curvatures,
z;spin
;yx (~k), evaluated
by replacing the orbital current operator with the spin
current operator J
;spin
=1
4fv ;s
gin Eq. 7. For the
two spin-split valence bands near the valley points in the
TB model, we nd
z;spin
;yx (~ q) =Mz(~ q)
+( 2)=
2
z;orb
yx(~ q):(12)
Note that
z;spin
;yx (~ q) has opposite signs for the two spin-
split bands and in the limit of = 0, they exactly cancel
everywhere producing a net zero SHC. For a non-zero
, these two contributions add up to produce a small
net SHC. Calculating the contributions from the valley
points with a similar procedure as Eq. (11), we obtain the
resultz;spin
yx e() 1;in the limit . This is
clearly much smaller than the OHC (11), by a factor of
=. From the DFT results (see Table I), we do indeed
nd that the SHC is about three orders of magnitude
smaller than the OHC. Even in doped samples though
the SHC is expected to be higher than the undoped sam-
ple, the typical values [8] are nevertheless still an order of
magnitude smaller than the computed OHC. These ar-
guments suggest the TMDCs to be excellent candidates
for the observation of OHE, since the intrinsic SHC is
negligible in comparison.
In conclusion, we examined the intrinsic OHE in non-
centrosymmetric materials and illustrated the ideas for
the monolayer TMDCs. The broken Iin TMDCs pro-
duces a robust momentum-space intrinsic orbital moment
~M(~k), present even in the absence of . Due to the op-
posite Berry curvatures at the valley points KandK0,
these orbital moments
ow in opposite directions, leading5
to a large OHC (104~=e
1). The vanishingly small
intrinsic SHC in these materials make them particularly
suitable for the direct observation of the OHE, which can
be measured by detecting the orbital torque generated by
the orbital Hall current [5]. Magneto-optical Kerr e ect
may also be used to detect the orbital moments accumu-
lated at the edges of the sample due to the OHE [23].
Furthermore, the valley-orbital locking can be probed in
photon polarized angle-resolved photoemission measure-ments [24]. In addition, it may be possible to tune the
OHC by applying a transverse electric eld [25, 26]. Ex-
perimental con rmation of the OHE in the TMDC's may
open up new avenues for the realization of orbitronics de-
vices.
We thank the U.S. Department of Energy, Oce of
Basic Energy Sciences, Division of Materials Sciences
and Engineering for nancial support under Grant No.
DEFG02-00ER45818.
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2105.04172v1.Self_Bound_Quantum_Droplet_with_Internal_Stripe_Structure_in_1D_Spin_Orbit_Coupled_Bose_Gas.pdf | arXiv:2105.04172v1 [cond-mat.quant-gas] 10 May 2021Self-Bound Quantum Droplet with Internal Stripe Structure in 1D
Spin-Orbit-Coupled Bose Gas
Yuncheng Xiong and Lan Yin∗
Peking University
(Dated: May 11, 2021)
We study the quantum-droplet state in a 3-dimensional (3D) B ose gas in the presence of 1D
spin-orbit-coupling and Raman coupling, especially the st ripe phase with density modulation, by
numerically computing the ground state energy including th e mean-field energy and Lee-Huang-
Yang correction. In this droplet state, the stripe can exist in a wider range of Raman coupling,
compared with the BEC-gas state. More intriguingly, both sp in-orbit-coupling and Raman coupling
strengths can be used to tune the droplet density.
PACS numbers: 03.75.Hh; 03.75.Mn; 05.30.Jp; 31.15.Md;
Introduction. Ultracold atoms have been excellent
platforms for investigating many-body quantum phe-
nomena since the experimental realization of Bose-
Einstein condensation(BEC) [ 1–3]. In most cases, Gross-
Pitaevskii (GP) equations, derived by minimizing mean-
field energy functional with respect to condensation
wavefunction, provide a good description for the BEC
state of trapped Bose gases [ 4]. The next-order cor-
rection to the gound state energy, i.e. the Lee-Huang-
Yang(LHY) energy, is usually negligible in the dilute
limit. However in a binary boson mixture, it was found
that when the attractive inter-species coupling constant
g↑↓is a little larger in magnitude than the geometric av-
erageofthe repulsiveintra-speciescoupling constants g↑↑
andg↓↓, the repulsive LHY energy overtakes the attrac-
tive MF ground-state energy, and the system becomes
a self-bound quantum droplet [ 5]. The quantum droplet
wasfirstobservedindipolarBosegases[ 6–10]andlaterin
binary boson mixtures [ 11–13]. The self-binding mech-
anism of a single-component dipolar Bose gas is simi-
lar to that of a boson mixture except that the residual
MF attraction arises from the counterbalance between
attractive dipole-dipole interaction and repulsive contact
interaction. Theoretically, the quantum droplet has been
investigated with various methods, including variational
HNC-EL method [ 14], ab initial diffusion Monte-Carlo
[15], and extended GPEs with the LHY correction in-
cluded [16].
On the other hand, Raman-induced spin-orbit-
coupling(SOC)hasbeenrealizedexperimentallyinrecent
years both in bosonic [ 17,18] and fermionic [ 19,20] sys-
tems. Alternative scheme of SOC which is immune from
heating problem has been theoretically [ 21–23] and ex-
perimentally [ 24] investigated. In a two-component Bose
gas with a one-dimensional (1D) SOC, a stripe struc-
ture appears when the inter-species coupling constant
is smaller than geometric average of intra-species cou-
pling constants below a critical Raman coupling (RC)
[17,25–30]. In previous studies, the stripe state has
been investigated in the BEC-gas region with repulsive
MF ground-state energy. Recently, theoretically stud-ies [31,32] reveal that the stripe state can also exist in
the quantum droplet regime. In Ref. [ 31], the quantum
droplet was found in a two-dimensional Bose gas with
very weak SOC, where the LHY energy density was ap-
proximated by that of a uniform system without SOC. In
Ref. [32] the LHY energy of a three-dimensional system
with SOC was calculated numerically, and its fitted form
wasusedintheextended(GP)equation. Thephasetran-
sition between a stripe gas and a stripe liquid was found
by tuning coupling constants and RC. In their calcula-
tion, the ultraviolet divergence in the expression of LHY
energy was removed by dimensional regularization. In
this work, we apply the standard regularization scheme
to treat the ultraviolet divergence in the LHY energy of
a three-dimensional system with SOC. We found that
the droplet density can be easily tuned by RC and SOC,
even to the zero limit. Compared to the case with a re-
pulsive inter-species interaction, in the quantum droplet
the stripe phase can exist in a bigger regime of RC and
SOC.
Boson mixture with SOC. Westudyatwo-component
Bose gas system with total particle number Nand vol-
umeV. In momentum space, its Hamiltonian is given
by
H=/summationdisplay
k/summationdisplay
ρρ′ˆφ†
ρk/parenleftbigg(k−krexσz)2
2+Ω
2σx/parenrightbigg
ρρ′ˆφρ′k
+1
2V/summationdisplay
k1,k2,q/summationdisplay
ρρ′gρρ′ˆφ†
ρk1+qˆφ†
ρ′k2−qˆφρ′k2ˆφρk1,(1)
whereˆφρkandˆφ†
ρkare the annihilation and creation op-
eratorsofthe ρ-component boson with the momentum k,
{ρ,ρ′}={↑,↓},krand Ω are the strengths of SOC and
RCrespectively, existhe unit vectorin x-directionwhich
is the SOC direction. For convenience, we set ¯ hand the
boson mass to be one. In this paper, we focus on the
Ω<4Erregime where the lower excitation spectrum of
the single-particle Hamiltonian has two degenerate min-
ima [17,28]. For simplicity, the interactions are chosen
to be symmetric g↑↑=g↓↓=g. The droplet regime is
set by the condition g↑↓<∼−g[5].2
To implement Bogoliubov approximation to obtain ex-
citation spectra, and thereby LHY correction, we need
to know ground state(GS) wavefunction. To this end, we
determine GS by variationally minimizing MF energy.
We choose GS ansatz to be superposition of plane waves
[27,28],
φ(r) =/parenleftbiggφ↑
φ↓/parenrightbigg
=/radicalbigg
N0
V/summationdisplay
m/parenleftbiggφ↑m
−φ↓m/parenrightbigg
eimk1·r,(2)
whereN0is the particle number of the condensate, k1≡
(γkr,0,0),γis a variational parameter to be determined.
In the lowest order, only m=±1 components corre-
sponding to the two minima of the lower single-particle
spectrum are relevent [ 27]. However, the periodic stripes
induced by the condensation of ±k1will lead to the cou-
plings between the momenta differing from each other by
reciprocal lattice vectors. Therefore, it is necessary to
include all the components with momenta K±k1in the
higher order approximations [ 28], where K= 2sk1withs= 0,±1,±2,..., arereciprocallattice vectors. In short,
the summation is over all the odd integer m= 2s±1 in
the region −C1≤m≤C1where the cutoff C1is a posi-
tive odd number. The normalization relation is given by/summationtext
ρ,m|φρm|2= 1.
In the BEC state, following the Bogoliubov prescrip-
tion, we replace ˆφ(†)
ρmk1by√N0φ(∗)
ρm+ˆφ(†)
ρmk1and keep
terms up to the quadratic order. The first-order terms
vanish due to the minimization of MF energy. The
number of atoms in the condensation is given by N0=
N−/summationtext′′
ρ,kˆφ†
ρkˆφρkwhich can be used to rewrite N0in
terms of the total atom number N. The MF energy
per particle εMFand the Bogoliubov Hamiltonian HBare
given by
εMF=/summationdisplay
m/summationdisplay
ρρ′/parenleftbiggk2
r
2(mγ−σz)2−Ω
2σx/parenrightbigg
ρρ′φ∗
ρmφρ′m
+/summationdisplay
m+l=i+j/summationdisplay
ρρ′gρρ′n
2φ∗
ρmφ∗
ρ′lφρ′iφρj,(3)
HB=EMF+/summationdisplay
ρρ′/summationdisplay
k/parenleftbigg(k−krexσz)2
2+Ω
2σx−µˆI/parenrightbigg
ρρ′ˆφ†
ρkˆφρ′k
+/summationdisplay
ρ,q/summationdisplay
m+l=α+βgρρl
2/bracketleftbigg
2φ∗
ρmφ∗
ρlˆφραk1−qˆφρβk1+q+2φ∗
ρmφρ−l(ˆφ†
ρ−αk1+qˆφρβk1+q+ˆφ†
ρ−αk1−qˆφρβk1−q)+H.c./bracketrightbigg
+/summationdisplay
ρ/negationslash=ρ′,q/summationdisplay
m+l=α+βgρρ′l
2/bracketleftbigg
(−φ∗
ρmφ∗
ρ′l)(ˆφρ′αk1−qˆφρβk1+q+ˆφρ′αk1+qˆφρβk1−q)+(−φ∗
ρmφρ′−l)×
(ˆφ†
ρ′−αk1+qˆφρβk1+q+ˆφ†
ρ′−αk1−qˆφρβk1−q)+φ∗
ρmφρ−l(ˆφ†
ρ′−αk1+qˆφρ′βk1+q+ˆφ†
ρ′−αk1−qˆφρ′βk1−q)+H.c./bracketrightbigg
,(4)
wherem,l,i,j,α,βare all odd integers with
−C1≤m,n,i,j ≤C1and−C2≤α,β≤C2,C2
is another cutoff necessary for numerical diagonaliza-
tion of Bogoliubov Hamiltonian, qxis in the first Bril-
lioun zone, 0 < qx< k1,nis total particle density,
andµ=/summationtext
m/summationtext
ρ,ρ′/parenleftbigg
k2
r
2(mγ−σz)2−Ω
2σx/parenrightbigg
ρρ′φ∗
ρmφρ′m+
/summationtext
m+l=i+j/summationtext
ρ,ρ′gρρ′nφ∗
ρmφ∗
ρ′lφρ′iφρjis, in nature, the MF
chemical potential, satisfying µ=∂EMF/∂N.
In Hamiltonian Eq.( 4) there are not only terms equiv-
alent to periodic potentials, but also off-diagonal terms
such as ˆφραk1+qˆφραk1−q. The quasiparticle spectra are
characterized by band index and quasimomentum.
We define a column operator
ˆAq≡/parenleftbig
···,ˆφ†
↑αk1−q,ˆφ†
↓αk1−q,ˆφ↑αk1+q,ˆφ↓αk1+q,···/parenrightbigT
withα=±1,±3,...,±C2, and rewrite Eq.( 4) in a com-pact form
HB=EMF+E1+/summationdisplay
qˆA†
qHqˆAq,
where
E1=−/summationdisplay
m,q,±/bracketleftbigg(mk1−q±kr)2
2−µ+gn+g↑↓n
2/bracketrightbigg
.(5)
Thematrix Hqcanbe obtainedfromEq.( 4)andsubse-
quentlydiagonalizedtoobtainquasiparticlespectra. The
diagonalized Bogoliubov Hamiltonian is given by
HB=EMF+E1+/summationdisplay
α,q(E↑−
α(q)+E↓−
α(q))
+/summationdisplay
α,q,±/summationdisplay
ρEρ±
α(q)ˆ˜φ†
ραk1±qˆ˜φραk1±q, (6)
where/parenleftbig
···,ˆ˜φ†
↑αk1−q,ˆ˜φ†
↓αk1−q,ˆ˜φ↑αk1+q,ˆ˜φ↓αk1+q,···/parenrightbigT=
MqˆAq,Mqthe Bogoliubov transformation matrix sat-
isfyingMqΣM†
q= Σ, and Σ is a diagonal matrix with3
everyfourdiagonalmatrixelementsgivenby −1,−1,1,1.
The quasi-particle energy Eρ±
α(q) can be solved from
the generalized secular equation |Hq−λΣ|= 0.
Before we write down the expression of LHY en-
ergy, we need to rewrite gρρ′in terms of scattering
lengthaρρ′through regularization relation gρρ′=Uρρ′+
(U2
ρρ′/V)/summationtext
k1/k2[33] whereUρρ′= 4πaρρ′. The LHY
energy is therefore given by
ELHY=E1+/summationdisplay
α,q/bracketleftbigg
(E↑−
α(q)+E↓−
α(q))+/parenleftbigg/summationdisplay
m+l=i+j
/summationdisplay
ρ,ρ′(Uρρ′n)2
2φ∗
ρmφ∗
ρ′lφρ′iφρj/parenrightbigg/parenleftbigg/summationdisplay
±1
(αk1±q)2/parenrightbigg/bracketrightbigg
,(7)
wherethe summation convergesquicklyfor largemomen-
tum due to regularization. In contrast, the divergence of
LHY energy was removed by dimensional regularization
in Ref. [32].
Self-bound quantum droplet with stripe With the ex-
plicit expressions of MF energy Eq.( 3) and LHY cor-
rection Eq.( 7), we are ready to investigate the inter-
play between SOC, RC and interactions in the forma-
tion of droplet. We take Er≡k2
r/2 andkras en-
ergy and momentum units respectively. The dimension-
less version of MF and LHY energies are given in ap-
pendix. In the following numerical calculations, we use
the parameters a↑↑=a↓↓≡a= 89.08a0,a↑↓=−1.1a
wherea0is the Bohr radius. Correspondingly, we de-
fineU≡4πa=U↑↑=U↓↓. And before the effect
of SOC is considered, the recoil momentum is fixed at
kr= 2π×106m−1(or equivalently akr≈0.0296).
When implementing the numerical calculations, we
introduce two cutoffs: C1is for the ground-state
ansatz, Eq.( 2), andC2is for the diagonalization of the
otherwise infinite-dimensional Bogoliubov Hamiltonian,
Eq.(4). We have numerically verified the convergence of
wavefunctionsandLHYenergies,andfindthatthechoice
ofC1= 9 and C2= 39 can produce sufficiently accurate
results. The condensate fraction at m= 9 is about 10−14
of the total density. In our calculations, the two charac-
teristic momenta,√gnand√
Ω, are at most of the same
order of the recoil momentum kr. The momentum cut-
offC2kris much larger than any of them. Therefore,
throughout our calculations, we set C1= 9 and C2= 39.
We first minimize the mean-field energy at fixed to-
tal density nto determine variational parameters φρm
andγ. It shows that, at low densities, the mean-field
energy per particle εMFshows linear dependence on den-
sity, and can be fitted by εMF=c0+c1(Un/E r). The
density-independent background energy appears due to
Raman energy in Eq.( 3), and thus c0depends strongly
on the strength of RC, while the proportionality coef-
ficientc1, as shown in Fig. 1, weakly relies on the RC
strength in the considered regime. Both c0andc1are
irrelevant to the strength of SOC, kr, since the MF en-
ergy has been rescaled in the unit of Er. Moreover, c1isnegative throughout the considered regime, which indi-
cates the tendency to collapse in the mean field and thus
higher-order correction is necessary to stabilize such a
system.
In the low density region, the mean-field density dis-
tribution exhibits stripes as in the low RC limit in re-
pulsive BEC-gas [ 27]. As has been studied [ 17,27,29],
in experiments on87Rb atoms, stripe phase exists only
for very small Ω ( <∼0.2Er), and stripes cannot be de-
tected directly in the absoption imaging. In contrast,
the stripe phase of the quantum droplet can survive in
a much larger range of RC, of the order of several Er.
This result can be obtained in the variational theory [ 27]
of a Bose gas with SOC, where the stripe phase and the
plane-wave phase can all be described. In the low den-
sity limit with strong SOC, there is a transition between
these two phases at a critical RC, ΩI-II= 4Er/radicalBig
2γ
1+2γ
whereγ= (g−g↑↓)/(g+g↑↓) (see Eq.(12) in [ 27]). Con-
sequently, we can reach a conclusion that in a quantum
droplet with Ω <4Erand strong SOC, stripe phase is
favored. Compared to the BEC case with repulsive iner-
species interaction where ΩI-II<∼0.2Er, one can draw
the conclusion that it is the strong inter-species attrac-
tion that significantly enlarges the region of stripe phase.
We compute the LHY energy by solving excitation
spectra and numerically performing the integration in
Eq.(7). As in the case without SOC [ 5], the lowest exci-
tation spectrum at qx≈0 and 2krhas small imaginary
part. When performing the numerical integration, we
keep all the real and imaginary contributions in the exci-
tation energies. Even though the resulting LHY energy
is complex, the imaginary part is at least three orders
smaller in magnitude than the real part in the parameter
region that we are considering, i.e. with low density and
strong SOC. Consequently, the imaginary parts can be
safely omitted in the ensuing calculations as in the case
without SOC [ 5].
Notice that LHY energy is a function of three di-
mensionless parameters, Un/E r, Ω/Erandakr(see
Eq.(A.9)) whereas the MF energy depends only on the
first two parameters as can be seen in Eq.( A.8). In the
dilute limit, in terms of Un/E r, the LHY energy per par-
ticle can be fitted by the formula εLHY=c2(Un/E r) +
c3(Un/E r)3/2, wherec2andc3arepositivefittingparam-
eters. Compared to the case without SOC, in addition
to the usual term proportional to n3/2, a linear repulsive
term appears in the LHY energy which will be discussed
later.
The coefficient c2, as shown in Fig. 1, has a roughly
quadratic dependence on RC strength while c3remains
constant in the range of 0 <Ω<3Er, in agreement with
the preceding work [ 32]. Also, such a behavior occurs in
the Rabi-coupled case [ 34] where the coefficient of usual
n3/2term in LHY energy is free of Ω. For Ω >3Er,
only low density region can be sampled, and while the4
0 1 2 3 40.000.010.020.030.04
Ω/Er|c1|
c2
c3
FIG. 1. Three fitting coefficients c1,c2andc3in MF en-
ergy and LHY energy vs the strength of Raman coupling Ω
atakr≈0.0296.c1is negative and has been shown by its
magnitude for convenient comparison with c2.
fittedc1andc2remains reliable, the value of c3, which
determines the behavior of the LHY energy in the higher
density region, can not be trusted due to the deficiency
of sampling.
0.001 0.010 0.100 1-0.008-0.006-0.004-0.0020.0000.002
Un/Er
/Er
=1/2 =3/2=5/2=7/2
FIG. 2. Sampled points (dots) and fitting functions (solid
lines) of total energy per particle ε=εMF+εLHYfor several
Raman coupling strengths at akr≈0.0296. The constant
energy background from the MF energy has been subtracted
for comparison.
Including both MF and LHY energies, the total en-
ergy per particle in the droplet regime is shown in Fig. 2.
Near the collapse point of the MF energy, contrary to
the monotonously decreasing tendency of the MF energy
with density, the total energy has a minimum, because
the repulsive LHY energy overcomes the attractive MF
energy at larger densities. As discussed above, the MF
wavefunction shows density modulation. Therefore, in
the dropletregime, the self-bound stripe phase exists and
can survive for even larger Raman coupling compared to
the BEC gas regime [ 17,27,29].
Although the general analytic total energy per particle0.050.100.150.200.250.301017101810191020
akrnd/m-3
Ω=1/2
Ω=3/2
Ω=5/2
FIG. 3. Droplet density ndversus SOC strength krfor several
RC strengths. With RC fixed, the stronger the SOC is, the
smaller the equilibrium density of droplet becomes. When
SOC is strong enough, for example at akr≈0.32 for Ω =
1/2Er, the droplet state disappears and the system expands
without an external trap.
is inaccessible in the presence of SOC and RC, it can be
approximatedbyaddingtogetherthe fitted MF andLHY
energies, ε=c0+(c2−|c1|)(Un/E r)+c3(Un/E r)3/2with
c0,1,2,3all fitted numerically. The equilibrium density of
self-bound droplet can obtained by solving zero-pressure
condition, i.e. P=∂ε/∂V= 0, yielding Und/Er=
4(|c1|−c2)2/9c2
3for|c1|−c2≥0.
We now discuss the role played by RC strength on
droplet formation. The dependence of droplet on RC
is similar to the case in uniform Rabi-coupled binary
mixture which has been reported in 3D [ 34] and lower
dimensions [ 35]. The similarity stems from the gapped
single-particlespectrum. Without RC,thesingle-particle
spectra of both components are gapless. The finite RC
induces coupling between the two components leading to
the new quasi-particle spectra, with the lower one gap-
lessandthe higheronegapped. Due tothe gappedmode,
LHY correction per particle acquires a positive term lin-
ear in density nin addition to the n3/2term, as men-
tioned above.
As Ω increases, the rapid increase of c2results in the
decreasing of |c1| −c2in magnitude, which is mani-
fest in Fig. 1, withc3remaining almost constant, mak-
ing it easier to counterbalance the attractive MF energy.
Thus the equilibrium density of droplet is smaller for
larger Ω as shown in Fig. 2. A critical point is reached
at|c1|=c2, as shown in Fig. 1where Ω c≈3.5Erat
akr≈0.0296. Above this point, the total energy in-
creases monotonously with density since both c2− |c1|
andc3are positive, and thereby no self-bound droplet
can exist. A similar droplet-gas transition has been re-
ported theoretically in Rabi-coupled binary mixture in
both 3D [ 34] and lower dimensions [ 35].
ItiseasiertoconsiderthedependenceonSOCstrength
askr(Er≡k2
r/2) serves as the momentum (energy)
unit. From the dimensionless expression of MF and LHY
energies Eq.( A.8) and Eq.( A.9), it’s easy to see that the
MF energy only depends on Un/E rand Ω/Er, and so5
does the excitation energy Eρ±
α(q). With fixed reduced
interaction Un/E rand Raman coupling Ω /Er, the LHY
energy is proportional to kr, and as a consequence, the
fitting parameters in LHY energy, c2andc3are both
linearly proportional to kr. Since c1is independent of
kr, increasing SOC strength krhas the same effect as
increasing RC strength Ω. Although the energy unit Er
is also increased in the same process, the overall effect of
increasing SOC strength is, as shown in Fig. 3, decreasing
equilibrium density of droplet. And finally above some
specific value, no droplet can exist any more.
Conclusion and Discussion In current experiment on
39K, the droplet state has been realized in the mixture of
hyperfine states |1,−1/an}bracketri}htand|1,0/an}bracketri}htby tunning scattering
lengths [5,11–13], but the artificial SOC has not been re-
alizedinthissystem. Incontrast,in87Rbsystems[ 17,18]
the SOC has been realized and the stripe state has been
observed, but tunning the interactions in this system has
not been achieved. Our results could be tested experi-
mentallyiftheinteractionscanbetunedandtheartificialSOC can be generated in the same system.
In conclusion, we have studied the quantum droplet
state of a uniform binary Bose gas in the presence of 1D
spin-orbit-coupling and Raman coupling, and find that
groundstate can display density modulation of the stripe
phase in the low Ω regime [ 27,28]. The density modula-
tion can survive for much larger Ω than in the BEC gas
state with inter-species interaction. Compared to the
case without SOC, the droplet density can be tuned by
changing the strength of SOC and RC. With the increase
of SOC and RC, the droplet density can be reduced by
several orders of magnitude, and eventually to the zero
limit at a critical kror Ω. We plan to study the finite-
size effect of the quantum droplet with SOC in the future
work.
Appendix
Dimensionless MF and LHY energies per particle are
given by
EMF/(NEr) =/summationdisplay
m/summationdisplay
ρρ′/parenleftbigg
(mγ−σz)2−Ω
2σx/parenrightbigg
ρρ′φ∗
ρmφρ′m+/summationdisplay
m+l=i+j/summationdisplay
ρρ′Uρρ′n
2φ∗
ρmφ∗
ρ′lφρ′iφρj,(A.8)
ELHY/(NEr) = (akr)1
π2(Un)/summationdisplay
α/integraldisplayγ
0/integraldisplayγC2
−γC2/integraldisplayγC2
−γC2dqxdqydqz/bracketleftbigg
(E↑−
α(q)+E↓−
α(q))−/summationdisplay
±/parenleftbigg
(αγ−qx±1)2+q2
y
+q2
z−µ+Un+U↑↓n
2/parenrightbigg
+/parenleftbigg/summationdisplay
m+l=i+j/summationdisplay
ρρ′Uρρ′n
2φ∗
ρmφ∗
ρ′lφρ′iφρj/parenrightbigg/parenleftbigg1/2
(αγ±qx)2+q2y+q2z/parenrightbigg/bracketrightbigg
, (A.9)
where Ω, Uρρ′nand excitation spectra Eρ−
αare in the
unit ofEr.
∗yinlan@pku.edu.cn
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2203.12263v1.Effects_of_a_single_impurity_in_a_Luttinger_liquid_with_spin_orbit_coupling.pdf | E ects of a single impurity in a Luttinger liquid with spin-orbit coupling
M. S. Bahovadinov1, 3and S. I. Matveenko2, 3
1Physics Department, National Research University Higher School of Economics, Moscow, 101000, Russia
2L. D. Landau Institute for Theoretical Physics, Chernogolovka, Moscow region 142432, Russia
3Russian Quantum Center, Skolkovo, Moscow 143025, Russia
(Dated: March 24, 2022)
In quasi-1D conducting nanowires spin-orbit coupling destructs spin-charge separation, intrinsic
to Tomonaga-Luttinger liquid (TLL). We study renormalization of a single scattering impurity
in a such liquid. Performing bosonization of low-energy excitations and exploiting perturbative
renormalization analysis we extend the phase portrait in K Kspace, obtained previously for
TLL with decoupled spin-charge channels.
I. INTRODUCTION
Low-energy excitations of the one-dimensional con-
ductors have collective phonon-like character, formalized
within the Tomonaga-Luttinger liquid (TLL) theory [1{
4]. The TLL formalism correctly predicts algebraic decay
of single-particle correlations with exponents depending
on the strength of the (short-range) electron-electron (e-
e) interactions. Due to the collective nature of these ex-
citations the interparticle interactions drastically change
the low-energy physics leading to fractionalization of con-
stituent carriers [5{7], vanishing density of states and
power law singularity of the zero-temperature momen-
tum distribution at the Fermi level [8{10]. Another sur-
prising prediction of the TLL theory is related to the
e ects of a single scattering impurity on transport prop-
erties at low temperatures, presented in prominent se-
ries of works in Refs. [11{13]. Perturbative renormaliza-
tion group (RG) studies have shown that if the electron-
electron interactions of the single-channel TLL is attrac-
tive (with the TLL parameter K > 1), impurity is irrele-
vant in the RG sense due to the pinned superconducting
uctuations. On the other hand, if the e-e interactions
have a repulsive character ( K < 1), the backscattering
e ects are relevant and grow upon the RG integration
of short-distant degrees of freedom. At T= 0 the sys-
tem is e ectively decoupled into two disjoint TLLs for
arbitrarily weak impurity potentials. The latter implies
a metal-insulator transition with vanishing two-terminal
dc-conductance G= 0 atT= 0, whereas at nite
temperatures it exhibits power-law dependence on T, i.e
GT2=K 2. These results are in sharp constrast with
the result in the non-interacting limit ( K= 1), where
the impurity is marginal in the RG sense (in all orders of
RG) and the transmission coecient Tdepends on the
impurity strength with Landauer conductance G=Te2
h.
Experimental con rmation of these e ects was performed
recently, using remarkable quantum simulator based on
a hybrid circuit [14].
In TLLs composed of spin-1/2 electrons [15{18] spin
and charge degrees of freedom are decoupled [19{23], in
contrast to higher dimensional counterparts. Each chan-
nel has individual bosonic excitations with the proper
velocity and carry the corresponding quantum number
0.5 2
KS 0.52KC
0.5 2
KS 0.52KCweak barrier strong barrier (a) (b)
IIII
IIIV
IIIII
IIVFIG. 1. Phase portrait at T= 0 for TLL with separated spin
and charge channels. Results are obtained from perturbative
RG studies of (a) a weak scattering barrier or (b) a weak link.
In region I the backscattering of carriers from a weak barrier
(tunneling through a link) is relevant (irrelavant) and
ows
to the strong-coupling (to the weak-coupling) xed point in
both channels. The opposite occurs in the regions IV. In the
remained regions the mixed phase is realized, where one of
the channels is insulating and the other is conducting.
separately. However, this decoupling of modes is not
characteristic for all TLLs. Particularly, the earlier stud-
ies have shown that spin-charge coupling (SCC) occurs
in the TLL subjected to a strong Zeeman eld [24] or
with strong spin-orbit interactions [25{27]. The latter is
typically expected in nanowires, where electrons in trans-
verse directions are con ned, whereas in the other direc-
tion they move freely. Spin-orbit coupling (SOC) in these
systems plays an important role on the realization of spin-
tronic devices [28, 29]. Thus, an interplay of (large) SOC
and (large) e-e interactions in the TLL regime is an in-
teresting question and has gained both theoretical and
experimental attention in recent years [30{32].
Impurity e ects in conventional TLLs with decoupled
channels were studied in the original works [12, 13], and
can be summarized by phase portrait at T= 0, presented
in Fig. 1. RG studies imply that electrons scattering on a
weak potential barrier [Fig. 1(a)] are fully re
ected when
Kc+Ks<2 (region I). In region II (III) charge (spin)
quanta are fully re
ected, while the spin (charge) ones
are transmitted. The backscattering terms scale to thearXiv:2203.12263v1 [cond-mat.str-el] 23 Mar 20222
ξ(k) ξ(k)
kυ1
kF,1υ2
-kF,2-kF,1
kF,2υ1 υ2υ
υ
kF (a) Polarized TLL
-kF (b) TLL with SOC
kυ
υ
FIG. 2. Linearized electronic excitation spectrum corre-
sponding to the (a) TLL in a strong Zeeman eld and (b)
TLL with SOC. In both models spin-charge separation is vi-
olated by the mixing terms given in Eq. (11) and Eq. (30).
weak-coupling xed point in region IV, so electrons fully
transmit through the barrier. The same physical picture
is obtained from the perturbative analysis of the tun-
neling events between disconnected wires [Fig. 1(b)]. In
region I (1
Kc+1
Ks<2;Kc<2;Ks<2) all hopping events
are irrelevant, so the system renormalizes into two dis-
connected wires. In region II (region III) only tunneling
of spin (charge) is relevant. Numerical con rmation of
these results was presented using path-integral Monte-
Carlo methods [33].
The modi cation of the phase portrait in the presence
of SCC has not yet been considered, although there are
several studies which partially addressed this question in
di erent aspects [34, 35]. Particularly, impurity e ects
on transport properties in the case of the SCC caused by
SOC were recently studied [31, 32]. As our main moti-
vation of this work we consider modi cation of the phase
portrait in the presence of SOC. We also show that the
spin- ltering e ect conjectured for this model in Ref. [35],
is not exhibited.
The paper is organized as follows: In Section II we
present the model of our study and set the necessary for-
malism and conventions. We next present a generalized
approach to tackle the impurity problem for a nite SOC
in Section III. In Section IV we present our main results
and discussions. Concluding remarks are given in Section
V.
II. MODEL AND METHODS
In this section we set up our used conventions and ter-
minology and present the model of our study. We ap-
proach the problem using the standard Abelian bosoniza-
tion technique with the consequent perturbative RG
analysis of impurity terms.
Bosonization of low-lying modes relies on the assump-
tion of linear electronic spectrum with the corresponding
right (R) and left (L) movers for carriers with both spin
projections (see Fig. 2). Fermionic eld operators foreach branch and spin component can be expressed via
the bosonic displacement and phase elds:
;=^Fp
2aeikF;xeip( (x) (x))(1)
with2f" +1;# 1gand2fR+1;L 1g.
The dual elds satisfy the commutation relations:
[ (x);(x0)0] =i;0(x x0); (2)
where (x) =r(x) is a conjugate to (x) momen-
tum. The UV cuto aof the theory is on the order
of inter-atomic distances. As in the standard litera-
ture, the Klein factors ^Fguarantee anti-commutation
of fermionic elds with di erent spin orientations.
The second exponent can also be expressed in terms of
constituent bosonic ladder operators in momentum space
(lis the length of the system)
ip( (x) (x)) =X
q>0Aq
bq;eiqx by
q;e iqx
(3)
withAq=q
2
ljqje ajqj=2and [bq;by
q0] =q;q0,q=2
l.
We hereafter use the spin and the charge basis as the
canonical basis, i.e we work with the elds c;s= " #p
2
and c;s="#p
2, which also satisfy Eq. (2). Hereafter,
the subbscripts "s" and "c" stand for spin and charge
degrees of freedom. The
uctuations of the spin and the
charge density are given by,
s(x) =r
2
X
@x (4)
and
c(x) =r
2
X
@x : (5)
To get similar expressions for the current, one uses the
transformation @x !@x. In the following, we imply
normal ordering with respect to Dirac sea, whenever it is
needed and avoid : () : symbol.
Model
The important e ect of SOC in quantum nanowires is
band distortion, which is usually considered within the
two-band model [36]. This distortion causes the veloc-
ity di erence = v1 v2, pronounced in Fig. 2(b) in
the approximation of linearized spectrum. Initially pro-
posed in Refs.[25, 26], the model can be realized by tun-
ing chemical potential, lling only the lowest subband.
It should be noted that the spin orientation of carriers
moving in the same direction can be tuned also from the
parallel [37] to anti-parallel [25] by band- lling [32]. In3
this work we consider anti-parallel spin orientation, as
shown in Fig. 2(b).
The non-interacting Hamiltonian of the model with lin-
earized excitation spectrum is
H0= iv1Z
dx
y
R;#(x)@x R;#(x) y
L;"(x)@x L;"(x)
iv2Z
dx
y
R;"(x)@x R;"(x) y
L;#(x)@x L;#(x)
;
(6)
with distinct Fermi velocities v1andv2. As it is clear
from the electronic spectrum, the model has broken chiral
symmetry, but the time-reversal symmetry is preserved.
Corresponding excitations can be rewritten equiva-
lently in the bosonic language bq(neglecting zero-energy
modes)
H0=v1 X
q>0jqjby
q;#bq;#+X
q<0jqjby
q;"bq;"!
+v2 X
q>0jqjby
q;"bq;"+X
q<0jqjby
q;#bq;#!
:(7)
Collecting the common terms, we obtain
H0=vF
2X
q6=0;jqjby
q;bq;+
2X
q6=0;qby
q;bq; (8)
with =v1 v2andvF=v1+v2. The second term
vanishes for vanishing SOC and has the following form
in the spin-charge basis:
Hmix=
2X
q6=0q
by
q;sbq;c+by
q;cbq;s
: (9)
This term with nite 6= 0 violates the spin-charge sep-
aration and demands more general approach.
We consider the interacting theory within the general-
ized g-ology approach and do not impose any constraint
on TLL parameters K,2(s;c). Within this general-
ization, the coordinate space representation of the inter-
acting Hamiltonian in the spin-charge basis has quadratic
Gaussian form:
HSOC =X
=s;cv
2Z1
K(@x )2+K(@x)2
dx+Hmix:
(10)
The SCC term is expressed as follows:
Hmix=
2Z
[@x s@xc+@x c@xs]dx: (11)
For vanishing SOC, the chiral symmetry is restored, with
fullSU(2) symmetry and Ks= 1.
In principle, one has to include also the backscattering
term to the Hamiltonian,
HBS=gs
2(a)2Z
dxcos(p
8 s): (12)In the parameter space of its relevancy, this term opens
a gap in the spin channel via the Berezinskii-Kosterlitz-
Thouless mechanism. There are several works which have
addressed the relevancy of this term for repulsive e-e in-
teractions [25, 38{40]. However, in a recent experiment
on InAs nanowires with a strong SOC, no sign of spin
gap is observed [31]. Based on the phenomenology, we
neglect all such terms within the whole parameter space.
We also emphasize that impurity e ects in the system
with gapped spin channel were also previously studied
[40, 41].
For perturbative analysis of impurity e ects, the
imaginary-time Euclidean actions for the displacement
elds can be obtained by integrating out the quadratic
phase elds. In the ~ x= (x; ) space it takes the following
form:
S
=1
2vZ
dxd
(@ )2+ ~v2
(@x )2
(13)
withv=vsKs=vcKc, guaranteed by Galilean invari-
ance of the model with = 0. The renormalized veloci-
ties are:
~v2
=v2
d; (14)
d=
1 2
4v2
: (15)
The contribution from the mixing term is,
S
mix=i
2vZ
dxd (@x s@ c+@x c@ s): (16)
Euclidean actions for the phase eld
uctuations can
be obtained in a similar fashion,
S
=v
2Z
dxd 1
v2(@ )2+d (@x)2
(17)
along with the mixing -term
S
mix=iv
2Z
dxd 1
v2c(@xs@ c) +1
v2s(@xc@ s)
:
(18)
For = 0, one correctly obtains the standard expres-
sions for actions S
andS
.
Impurity bosonization
We consider a single impurity embedded at the origin
x= 0. This impurity causes backscattering of carriers
Hbs=VbsX
y
R; L;+h:c:
; (19)
whereVbsis the Fourier component of the backscattering
potentialV(kF;1+kF;2). The forward-scattering term
can be always gauged out [3].4
The bosonized expression of the backscattering term
leads to the boundary sine-Gordon model in the spin-
charge basis:
Hbs=2Vbs
acos( I s(x= 0)) cos( I c(x= 0)) (20)
with 2
I= 2. This term also couples spin and charge
degrees of freedom at the impurity point. We note that in
the bosonization process of this term we do not take into
account Klein factors ^F, since their e ect is irrelevant
within our model with all terms included terms in this
work. The corresponding backscattering action takes the
following form:
Sb=2Vbs
aZ
0d cos( I s(0; )) cos( I c(0; )) (21)
=1
Tis the inverse temperature and should not be
confused with I. We hereafter assume Vbs= 1, where
in the e ective bandwidth for both channels.
For a complete analysis, pertubative study of the
backscatteting action Eq. (21) should be accompanied
with the RG study in the strong barrier limit. For this
purpose, we consider two disjointed wires and treat in-
terwire tunneling term perturbatively. The most relevant
tunneling term has the following contribution to the total
action,
St=t
aZ
0d cos( Is(0; )) cos( Ic(0; ));(22)
wheretis the bare tunneling amplitude through the weak
link.
III. EFFECTIVE BATHS ACTIONS
At this point it is worth noting that the previous
bosonization studies of TLL with SCC [24, 32, 35] were
performed using basis rotation approach, where one
transforms the initial basis to the new spin-charge basis
with decoupled channels and new renormalized velocities
andKparameters. This approach usually results on
redundant expressions for model parameters and com-
plicates further analysis. Instead, we follow a generic
approach [3] without performing a basis rotation. As
we show below, this procedure is not required. We next
trace out all space degrees of freedom except for the im-
purity point. The low-lying excitations of the elds away
from the impurity act as a dissipative bath reducing the
problem to an e ective 0D eld theory of a single Brow-
nian quantum particle in a 2D harmonic impurity poten-
tial [42].
We follow the standard procedure [3, 12] to get e ective
bath actions for the displacement
uctuation. The same
procedure applies for the phase elds.The integration of the bulk degrees of freedom is done
using the standard trick of introducing Lagrange multi-
pliers with the real auxilliary elds
Z =Z
D sD cDcDsDsDce ST(23)
with
ST=S[ c; s] +iZ
0d X
[( ) (0; )]( ):
(24)
One needs consequently integrate out the andto
get the nal baths actions:
S
=X
!nj!nj ^F
det(F)!
j(!n)j2(25)
with the bosonic Matsubara frequencies !n=2n
;(n2
Z). The mixing term is:
S
mix= X
!nj!nj ^Fmix
det(F)!
s( !n)c(!n);(26)
where theFmatrix and ^Fare given in Eqs. (A.1)-(A.3).
Importantly, the mixing action S
mixvanishes (see Ap-
pendix) and one is left with the decoupled set:
S
=X
!nj!nj
~K
j(!n)j2(27)
with ~K
given by Eq. (A.5) and 2(s;c). The resulted
Caldeira-Legett type actions are common and describe
dynamics of a single quantum Brownian particle in the
regime of Ohmic dissipation [42, 43]. These actions rep-
resent the weak-coupling xed point of pure Luttinger
liquid.
Similarly, the e ective baths actions for elds can
be obtained as
S
=X
!nj!nj~K
j(!n)j2(28)
with the new TLL parameters ~K
given by Eq. (A.6).
The mixing action also vanishes in this case.
Vanishing mixing terms and decoupled baths map the
problem onto the one with conventional TLL baths,
but with the new parameters ~Kandv ;+given by
Eqs. (A.5)-(A.7). Hence, the usual basis-rotation proce-
dure is excessive here. Obviously, for the vanishing spin-
charge mixing term = 0, the standard TLL parameters
are correctly recovered, ~K!K. The characteristic
velocitiesv+!max(vs;vc) andv !min(vs;vc). This
implies that in the presence of SOC, one has new modes,
with carriers composed of spin and charge quanta. These
new modes have excitation velocities v ;+and posses new
TLL parameters ~K. Prior to the discussion of our main5
results, we emphasize two important features arisen al-
ready at this stage of analysis.
Mode freezing . As the strength of SOC is increased, v+
monotonically increases, whereas v similarly decreases
and vanishes at the critical c
. As it was mentioned
in previous works [25, 27], at the critical SOC strength
"freezing" of the corresponding mode (phase separation)
with diverging spin (or charge) susceptibility is exhibited.
For repulsive e-e interactions, it was shown [25] that it is
the spin susceptibility which diverges at the critical SOC
as,
= 0
1
cs2! 1
(29)
with spin-susceptibility 0at zero SOC and c
s= 2=Ks.
Similar divergence can be observed in the charge channel,
if one considers full Kc Ksspace. These divergences in-
dicate a phase transition, which occurs in the spin/charge
channel (see Ref. [27] and references therein). In this
work, we consider SOC strengths causing the velocity dif-
ference =v < 0:8, which restricts the parameter space
toK<5=2.
Absence of spin- ltering e ect . In the case of the
spin-charge mixing caused by the strong Zeeman eld
[Fig. 2(a)] the mixing bath actions Smixdoes not vanish,
due to the di erent type of SCC mechanism. The mix-
ing term Eq. (11) in the Hamiltonian takes the following
form:
Hmix=B
2Z
[@x s@x c+@xs@xc]dx (30)
with B=v" v#. As it was demonstrated earlier in
Refs. [24, 35], a single impurity as in Eq. (19) embedded
to such TLL causes polarization of the spin current with
a ratio of tunneling amplitudes,
t"
t#=T
; (31)
and with a nite exponent (B) for nite B.
Observation of this e ect in our model was also pro-
posed earlier in Ref. [35]. However, an important conse-
quence of vanishing Smixaction terms of Eq. (26) (and
similar term for eld) is the absence of such spin-
ltering e ect, albeit the spin-charge separations is vi-
olated.
IV. RESULTS AND DISCUSSIONS
Weak potential barrier
Once the e ective bath actions are obtained as in
Eq. (27) and Eq. (28) , the standard pRG analysis in
the limit of weak and strong impurity potential can be
performed.We rst consider the scattering of electrons on a weak
barrier. In this limit, the partition function is,
Z=Z
DsDce ST; (32)
with the total action,
ST=S
c+S
s+Vm;n
aZ
0d cos(m Ic) cos(n Is):
(33)
To analyze the relevance of the last term within the Kc
Ksparameter space, we generalize the impurity term to
take di erent values of mandnwith the amplitudes
Vm;n , since such terms are necessarily generated
during the RG process.
Treating the backscattering terms perturbatively, one
obtains the standard set of rst-order RG equations [12,
13],
dV1;1(l)
dl=
1 ~K
s+~K
c
2!
V1;1(l); (34)
dV0;2(l)
dl=
1 2~K
s
V0;2(l); (35)
dV2;0(l)
dl=
1 2~K
c
V2;0(l) (36)
withdl=d
.
The rst equation for V1;1Vbscorresponds to the
kF;1+kF;2backscattering of a single electron, whereas
the second and the third equations correspond to the
backscattering of spin or charge (electron pair) degree of
freedom. The last processes have the fermionic expres-
sions y
R;" L;" y
L;# R;#+h:c:and y
R;" L;" y
R;# L;#+
h:c:, respectively. The sketches of these scattering pro-
cesses in the momentum space are presented in Fig. 3(b)-
(c). All higher-order decendent terms are neglected, since
the regions of their relevancy are covered with the ones
of the last two equations, even for the nite . Thus,
the main low-temperature processes are dictated by these
three equations Eqs.(34)-(36).
For vanishing SOC = 0 ( ~K!K), the marginal
xed line de ned in Eq. (34) is determined by the con-
ditionKs+Kc= 2. The region I in Fig. 3(a) corre-
sponds to a parameter space where the backscattering of
a single electron in a weak potential becomes a relevant
process and the backscattering amplitude V1;1
ows to
the strong-coupling regime. This leads to the blocked
transport in both charge and spin channels and the elds
s;care pinned on the minima of cosine functions. Qual-
itatively, in this region either single electron ( "or#) is
fractionalized by e-e interactions, which is responsible for
full backscattering of single carrier and hence, charge and
spin carriers, too. Thus, one has vanishing conductance
G= 0 atT= 0.6
(b)
(c)charge backscattering
spin backscattering(a)
IV
I
IIIII
KF,1
KF,2
KF,2K
K
KF,1ξ(k)
ξ(k)
0 0.5 1 1.5 2 2.5
Ks00.511.522.5Kc∆=0
∆=0.4
∆=0.8
FIG. 3. (a) The modi cation of the phase portrait for the
TLL with nite SOC. For >0 the phase boundary of region
I is modi ed. In (b) and (c) the backscattering processes of
charge and spin are sketched, which correspond to Eqs. (35)-
(36), respectively.
As dictated by Eqs. (35)-(36), the backscattering pro-
cesses of the charge/spin carriers also become relevant
whenKc=s<0:5. For the charge backscattering, this im-
plies the charge quanta to carry smaller than 0 :5 unit, to
make it a relevant process. In the region of relevance II
(III for spin channel), the charge (spin) channel is insu-
lating since the the c(s) is pinned. The spin (charge)
fully transmit the barrier in this region, implying the re-
alization of mixed phases in these regions. In region IV
both channels have dominating superconducting
uctu-
ations [27]. We emphasize again that in this region with
attractive e-e interactions, other type of instabilities in
the bulk may arise, which usually open a gap in one of
the channels. Here, we neglect all such processes and
present the simplest picture.
To investigate the e ect of SOC on backscattering of
carriers, we represent the new TLL parameters in a sym-
metric way:
~K
=Kv
~v~vc+ ~vs
v++v
; (37)
where ~vandvare given in Eq. (14) and Eq. (A.7),
respectively. In the presence of SOC the rst e ect is
the renormalization of spin and charge velocities, which
is manifested in the rst factor of Eq. (37). The second
factor is due to the emerged modes with velocities v+; .
As it was mentioned in the previous section, we limit
considiration of the parameter space K<5
2and =v<
0:8.
For non-interacting electrons, the e ect of SOC is lim-
ited to the breaking of chiral symmetry and the renor-
malization of excitation velocities. Indeed, one has ~K
=
K
= 1, since renormalizing factors in Eq. (37) cancel
each other. As shown in Fig. 3a this is also valid for
weakly-interacting electrons, K1.
For the given nite SOC strengths, where 0 <=v <
0:8, the second factor
~vs+~vc
v +v+
1 is xed for both
K, whereas the rst one de nes scattering of carriersand determines the boundary of region I. The strongest
e ect of SOC is exhibited when in one of the channels
K 1. In this limit one can lock the corresponding
eld to the minimum of cosine function and reexpress
the scaling dimension of the impurity term for m= 1
andn= 1 as follows:
21;1=K+K (1 +2
8v2
): (38)
The resulting marginality line is presented in Fig. 3(a)
for =v= 0:4 and =v= 0:8. At the critical K=5
2,
the excitations in the spin/charge channel become frozen
(v = 0) and the bulk of the channel becomes insulating.
The e ect of nite SOC on the boundaries of regions
II/III with the region IV is negligibally small. The largest
correction to the scaling dimension 0;1(1;0) is of the
order of 10 2for the largest =v= 0:8. Thereby it can
be safely neglected.
Finally, one can straightforwardly generalize the ex-
pressions for corrections to (bulk) conductances obtained
in Refs. [12][13] to the case with nite SOC by K!~K:
G=e2
hX
m;ncm;njVm;nj2T(m2~Kc+n2~Ks)=2 2(39)
with dimensionless coecients cm;n.
Strong barrier
For analysis of tunneling term in Eq. (22) one considers
the following total action ST,
ST=S
c+S
s+tm;n
aZ
0d cos(m Ic) cos(n Is):
(40)
Similar to the previous case, the impurity term is gener-
alized for di erent mandn.
The RG transformation of the impurity term leads to
the following set of equations,
dt1;1(l)
dl=
1 1
21
~Kc+1
~Ks
t1;1(l); (41)
dt2;0(l)
dl=
1 2
~Kc
t2;0(l); (42)
dt0;2(l)
dl=
1 2
~Ks
t0;2(l) (43)
with the TLL prameters rewritten as:
1
~K=1
Kv
^v^vc+ ^vs
v +v+
; (44)7
KSKC
0 0.5 1 1.5 2 2.5IIVIII
II∆=0
∆=0.4
∆=0.8
00.511.522.5
FIG. 4. (a) The modi cation of the phase portrait obtained
from the renormalization analysis of tunneling events. The
e ect of SOC is manifested on phase boundaries between the
regions I-II and I-III. In the vicinity of mode freezing, hopping
events of a single ( ")=(#) carrier become irrelevant. All shaded
regions correspond to disconnected wires with no transport of
carriers.
and
^v=vq
1 2
4v2: (45)
An amplidute t1;1corresponds to the interwire hopping
event of a single ( ")=(#) electron, while t2;0andt0;2are
tunneling amplitude of charge and spin, respectively.
Based on the set of equations, one obtains phase por-
trait atT= 0. It consists of four regions, exhibited also
in the limit of weak scattering potential and it is shown
in Fig. 4(a). For vanishing SOC ~K!K, the bound-
aries of regions are de ned by1
Kc+1
Ks= 2;Kc= 2,
andKs= 2. In region I (IV), hopping event of a single
(")=(#) electron is irrelevant (relevant) and one eventu-
ally renormalizes onto the xed point of disjoined (con-
nected) wires in the RG process. In region II, the tunnel-
ing amplitude t0;2for spin is relevant and grows upon RG
transformation, however charge carriers can not tunnel.
An opposite situation with a conducting charge channel
and an insulating spin channel occurs in the region III.
We note that the results obtained in the opposite limits
of weak and strong barrier are consistent.
For relatively weak SOC ( = 0 :4) the only pro-
nounced e ect is modi cation of boundaries between re-
gions I-II and I-III, as shown in Fig. 4(a). These e ects
are dictated by Eqs. (42)-(43). The regions II and III
are extended towards the region I with the new bound-ariesK1:8<2. The area of the extended region is
larger for the larger value of = 0 :8 with boundaries
de ned via K1:6. Remarkably, these e ects are also
exhibited in the weak barrier analysis, in Fig. 3(a).
At the largest = 0 :8, the tunneling of single carri-
ers with amplitude t1;1becomes irrelevant near the phase
separation point K= 5=2. This is consistent with the
picture of mode freezing, discussed in the previous sec-
tions. On the other hand, for weakly-interacting elec-
trons withK1 the e ect of SOC is negligibely small,
even for large values of , also compatible with the re-
sults of the previous section.
Conductance for insulating regions at nite tempera-
tureTcan be also generalized as follows:
G=e2
hX
m;ndm;nt2
m;nT2(m2=~Kc+n2=~Ks 2)(46)
with dimensionless coecients dm;n.
Experimentally relevant values of =v0:2 [25, 26,
31] are smaller than the maximum value considered in
this work. For this range of values, one can neglect
mode-freezing e ects safely. The results of the RG anal-
ysis on the last sections arm that for weak and mod-
erate e-e interactions e ects of SOC are negligible small.
This results are in accrodance with recent theoretical and
expertimental studies [31, 32].
V. CONCLUSIONS
We studied carrier scattering e ects upon a single im-
purity embedded to a Luttinger liquid with spin-orbit
coupling using Abelian bosonization and pertubative
renormalization techniques. Spin-orbit interaction de-
grades spin-charge separation and renormalizes the TLL
parameters Ks;cand the excitation velocities vs;c. We
demonstrated that the scaling dimension of impurity op-
erator is identical for both ( ")=(#) carriers. This im-
plies the absence of conjectured spin- ltering e ect in
Ref.[35]. The strongest e ects of spin-orbit coupling are
pronounced for strong e-e interactions, whereas these ef-
fects are negligibly small for moderate e-e interactions.
Our main results are summarized by the phase portrait
modi cations presented in Figs (3)-(4).
ACKNOWLEDGMENTS
We thank F. Yilmaz and V. I. Yudson for useful com-
ments and discussions.8
Appendix: Expressions for new TLL parameters
and excitations velocities
TheFmatrix has the following form,
F=
2^Fc^Fmix
^Fmix 2^Fs
: (A.1)
The matrix elements ^F=j!jFare expressed via the
following integrals,
F=Z+1
1dk
2
[G
] 1
det(Q)!
(A.2)
and for the mixing element,
Fmix= Z+1
1dk
2
[G
mix] 1
det(Q)!
(A.3)
where [G
] 1are the inverse propogators of Eqs. (13)-
(16) in~ q= (k;!) space, and the Qmatrix is de ned
as,
Q=
2[G
c] 1[G
mix] 1
[G
mix] 12[G
s] 1
: (A.4)For the mixing parameter Fmixthe numerator of the
kernel is odd function of k, i.e [G
mix] 1=k!, whereas
the denominator is an even function of k. This leads to
the vanishing action S
mix. Similarly, S
mixvanishes and
one is left with fully decoupled actions [Eqs. (27)-(28)].
Evaluation of integrals leads to the following results for
new TLL parameters for displacement
uctuation elds,
^K
=v
(v++v )~v2
v+v + 1
; (A.5)
and for the phase elds,
1
^K=v2
v(v +v+)v2
~v
v2~v + 1
: (A.6)
The sound velocities for the newly emerged modes are
given by,
v2
=1
2h
~v2
c+ ~v2
s+ 2p
(~v2c ~v2s+ 2)2+ 4~v2s2i
:
(A.7)
For vanishing SOC = 0, the sound velocities in the
spin/charge channel are recovered.
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0809.5119v1.Multi_terminal_Spin_Transport__Non_applicability_of_linear_response_and_Equilibrium_spin_currents.pdf | arXiv:0809.5119v1 [cond-mat.mes-hall] 30 Sep 2008Multi-terminal Spin Transport: Non applicability of linea r response and Equilibrium
spin currents
T. P. Pareek
Harish Chandra Research Institute
Chhatnag Road, Jhusi, Allahabad - 211019, India
We present generalized scattering theory for multi-termin al spin transport in systems with broken
SU(2) symmetry either due to spin-orbit interaction,magne tic impurities or magnetic leads. We
derive equation for spin current consistent with charge con servation. It is shown that resulting spin
current equations can not be expressed as difference of poten tial pointing to non applicability of
linear response for spin currents and as a consequence equilibrium spin currents(ESC) in the l eads
are non zero. We illustrate the theory by calculating ESC in t wo terminal normal system in presence
of Rashba spin orbit coupling and show that it leads to spin re ctification consistent with the non
linear nature of spin transport.
PACS numbers: 75.60.Jk, 72.25-b,72.25.Dc, 72.25.Mk
Spin transport has emerged as an important subfield
of research in bulk condensed matter system as well in
mesoscopic and nano system[1]. In macroscopic systems,
the very definition of spin currents is still debated due to
non conservation of spin in presence of SO interaction[2]
. On the other hand in mesoscopic hybrid system since
current is defined in the leads where SO coupling is
absent,therefore, it has been assumed that Landauer-
B¨ uttiker formula for charge current[3] (Eq. (9) in this
manuscript), which determines current in leads in terms
ofapplied voltagedifference multiplied by totaltransmis-
sionprobability, can be straightawaygeneralizedforspin
currents by replacing total transmission probability with
some particular combination of spin resolved transmis-
sion probabilities[4, 5, 6]. This simple generalization has
been widely used in the literature to study spin depen-
dent phenomena in nanosystems[4, 5, 6]. The Landauer-
B¨ uttiker formula in its widely used form has inbuilt cur-
rent conservation , i.e., total current is divergence less
(divj=0) which follows from basic Maxwell equations of
electrodynamics. Physically this implies that total cur-
rent has neither sources nor sinks this would be true for
spin currents as well if spin is conserved. However, in
presence of spin-orbit interaction, magnetic impurities or
non-collinear magnetization in leads, spin is no longer a
conserved quantity, hence the spin currents can not be
divergence less. Therfore a straight forward generaliza-
tion of Landauer-B¨ uttiker charge current formula to spin
currents can not be correct for spin non-conserving sys-
tems.
In view of the above discussion in this work we de-
velopaconsistentscatteringtheoreticformulationofcou-
pled spin and charge transport in multiterminal systems
with broken SU(2) symmetry in spin space following
B¨ uttiker’s work on charge transport [3]. The SU(2) sym-
metry in spin space can be broken due to either SO inter-
action, magnetic impurities or non-collinear magnetiza-
tion in leads[7, 8]. Our analysis provides a correct gener-
alization of Landauer-B¨ uttiker theory for spin transport.
In particular we derive a spin currents equation (Eq. (8)inthismanuscript)consistentwithchargecurrentconser-
vation. However, the resulting spin current equation can
not be cast in terms of spin resolved transmission and re-
flectionprobabilitiesmultiplied byvoltagedifferenceasis
the casefor chargecurrent(Eq. (9) in this manuscript)[3].
Therefore, equilibrium spin currents are generically non
zero.Moreover, it implies that linear response theory
with respect to electric field is not applicable to the spin
currents (equilibrium as well non-equilibrium).Thus spin
currents are intrinsically non-linear in electrical circu its.
However, this is not surprising since linear response is
valid for thermodynamically conjugate variable. In an
electrical circuit thermodynamically conjugate variable
to electric field is charge current not the spin currents.
We illustrate the theory by calculating ESC analytically
for two-dimensional electron system with Rashba SO in-
teraction in contact with two unpolarized metallic con-
tacts. Our analytical formula for ESC clearly demon-
strates that it is transfer of angular momentum per unit
time from SO coupled sample to leads where SO inter-
action is zero. Therefore,it is truly a transport current
in contrast to equilibrium spin currents in macroscopic
Rashba medium(see ref.[2, 9]).
To formulate scattering theory for spin transport we
consider a mesoscopicconductor with brokenSU(2) sym-
metry in spin space connected to a number of ideal mag-
netic and non-magnetic leads (without SO interaction)
which in turn are connected to electron reservoirs. To
include the effect of broken SU(2) symmetry, it is neces-
saryto write spin scattering state in eachlead along local
spin quantizationaxis. For magnetic leadslocal magneti-
zation direction provides a natural spin quantization axis
which we denote in a particular lead αbyˆmα(ϑα,ϕα)
whereϑαandϕαis polar and azimuthal angle respec-
tively. For nonmagnetic leads since there is no preferred
spin quantization axis, hence we choose an arbitrary spin
quantization axis ˆu(θ,φ) which is same for all nonmag-
netic leads. Thus the most general spin scattering state
in leadαwhich can be either magnetic or nonmagnetic2
is given by,
ˆΨσ
α(r,t) =/integraldisplay
dENσ
α(E)/summationdisplay
n=1Φαn(r⊥)χα(σ)/radicalbig
2π/planckover2pi1vσαn(E)
(aσ
αn(E)eikσ
αn(E)x+bσ
αn(E)e−ikσ
αn(E)x) (1)
where Φ αn(r⊥) is transverse wavefunction of channel n
andχα(σ) is corresponding spin wave function along
chosen spin quantization axis, ˆuorˆmαsuch that S·
ˆuχ(σ)=(σ/planckover2pi1/2)χ(σ) orS·ˆmαϕ(σ)=(σ/planckover2pi1/2)ϕ(σ) withσ=
σ(σ=±1,representing local up or down spin compo-
nents) for nonmagnetic and magnetic leads respectively.
HereS= (/planckover2pi1/2)σis a vector of Pauli spin matrices and
Nσ
αisnumberofchannelswith spin σinleadα. The rela-
tion between spin dependent wavevector kσ
αn(E) and en-
ergyEis specified by, E=/bracketleftbig
/planckover2pi12k2
αnσ/2m+εαn+σ∆α/bracketrightbig
,
whereεαnis energy due to transverse motion, ∆ αis
stoner exchange splitting in the magnetic lead α. The
stoner exchange splitting is zero for nonmagnetic leads.
The operators aσ
αnandbσ
αnare annihilation operator for
incoming and outgoing spin channels in lead αand are
related via the scattering matrix,
bσ
αm=/summationdisplay
βnσ′Sσσ′
αm;βnaσ′
βn (2)
The scattering matrix elements Sσσ′
αm;βnprovides scatter-
ing amplitude between spin channel nσ′in leadβto spin
channelmσin leadα. These scattering matrix elements
will be function of energy E as well angles, ϑandϕ.
The angular dependence of scattering matrix elements
on polar and azimuthal angle arise due to broken SU(2)
symmetry. Note that for noncollinear magnetization in
leads and in absence of SO interaction and magnetic im-
purities, the angular dependence is purely of geometric
origin and is related to the angular variation of various
magnetoresistance phenomena[10].
The current in spin channel σalong longitudinal direc-
tionˆx(through a cross section of lead α) and the local
spin quantization axis ˆuis defined as,
ˆIˆxσ
αˆu(t) =/planckover2pi1
2mi/integraldisplay/bracketleftBig
ˆΨ†σ
α(S·ˆu)∇xˆΨσ
α−∇xˆΨ†σ
α(S·ˆu)ˆΨσ
α/bracketrightBig
dr⊥.
(3)
Substituting for ˆΨσ
αfrom Eq.(1) into Eq.(3, we get an ex-
pression for spin current in terms of creation and annihi-
lation operators. On the resulting expression we perform
quantum statistical averages and after a lengthy algebra
we obtain followingexpression for averagecurrentin spin
channelσ(for brevity of notation we suppress the super-
scriptˆxwritten in Eq. (3),
/an}b∇acketle{tIσ
αˆu/an}b∇acket∇i}ht=g
h/integraldisplay∞
0dE/summationdisplay
βfβ(E)/bracketleftBigg
Nσ
αδαβ−/summationdisplay
σ′mnS†σ′σ
βm;αnSσσ′
αn;βm/bracketrightBigg
(4)
Wherefβ= 1/exp[(E−µβ)/kT]+1 is Fermi distribu-
tion function with chemical potential µβand the pre-
factorgequalsσ/planckover2pi1/2. The summation over σ′in Eq. (4)can take on values ±σcorresponding to two spin pro-
jections along local spin quantization axis. The second
term of Eq.(4) can be written explicitly in terms of spin
resolved reflection and transmission probabilities as,
/summationdisplay
βσ′;mnS†σ′σ
βm;αnSσσ′
αn;βm=/summationdisplay
σ′;mnS†σ′σ
αm;αnSσσ′
αn;αm
+/summationdisplay
β/negationslash=ασ′;mnS†σ′σ
βm;αnSσσ′
αn;βm
≡/summationdisplay
σ′Rσσ′
αα+/summationdisplay
β/negationslash=ασ′Tσσ′
αβ(5)
WhereRσσ′
ααandTσσ′
αβare spin resolved reflection and
transmission probability in the same probe and between
different probes respectively. In Eq.5 on right hand side
spin resolvedreflectionandtransmissionprobabilitiesare
summed over all possible input modes for a fixed output
spin mode σin leadα. Because partial scattering matrix
in spin subspaceis not unitary due to non conservationof
spin hence this summation need not to be equal to num-
ber of spin σchannels in lead α,i.e.Nσ
α, rather it can
have any value lying between zero and Nσ
α. To determine
Nσ
αin terms of spin resolved reflection and transmission
probabilities, consider a situation where current is in-
jected from reservoir only in spin channels σin leadα.
In this casechargeconservationrequiresthat this current
should leavethe spin channel σthrough all other possible
channels in the same lead as well in differing leads, which
implies,
Nσ
α=/summationdisplay
σ′Rσ′σ
αα+/summationdisplay
β/negationslash=ασ′Tσ′σ
βα. (6)
As we can see that Eq.(6) differs from Eq.(5) in a sub-
tle way and are not equal because in general spin re-
solved transmission or reflection probabilities can not be
related among themselves by interchanging spin indices,
i.e.,Tσ′σ
αβ/ne}ationslash=Tσσ′
βαandRσ−σ
αα/ne}ationslash=R−σσ
αα( we will discuss
constraints due to time reversal symmetry below). If we
demand that sum in Eq.(5) also equals to Nσ
αthen it
would imply spin conservation which is incorrect in pres-
ence of spin flip scattering or broken SU(2) symmetry.
The inadvertent use of this charge conservation sum rule
for spin degrees of freedom in Ref. [5, 6, 11] has led to
incorrect spin current equation. Though the partial scat-
tering matrix in spin subspace is not unitary,however,
the full scattering matrix is unitary,i.e., SS†=S†S=I,
therefore, if we sum over σalso in Eq.(5) or Eq.(6) then
it should give total number of channels in leads α,i.e.
N=Nσ
α+N−σ
α, and as a result we get the following sum
rule for total transmission probability,
Tβα=/summationdisplay
σ′σTσ′σ
βα=/summationdisplay
σσ′Tσσ′
αβ=Tαβ (7)
whereTαβis total tranmission probability.
The net spin current flowing in lead αis defined as
IS
ˆuα=/an}b∇acketle{tIσ
ˆuα/an}b∇acket∇i}ht+/an}b∇acketle{tI−σ
ˆuα/an}b∇acket∇i}htwhile the net charge current flowing3
is given by sum of absolute values, i.e., Iq
ˆuα=| /an}b∇acketle{tIσ
ˆuα/an}b∇acket∇i}ht |+|
/an}b∇acketle{tI−σ
ˆuα/an}b∇acket∇i}ht |with pre-factor greplace by the electronic charge
ein Eq. (4). Using Eqs.( 5),(6) in Eq.(4) we obtain net
spin and charge current as,
Is
αˆu= (/planckover2pi1
2h)/integraldisplay∞
0dE2fα(E)(R−σσ
αα−Rσ−σ
αα)+
/summationdisplay
β/negationslash=ασ′/bracketleftBig
fα(E)(Tσ′σ
βα−Tσ′−σ
βα)−fβ(E)(Tσσ′
αβ−T−σσ′
αβ)/bracketrightBig
(8)
Iq
αˆu= (e
h)/integraldisplay∞
0dE/summationdisplay
β/negationslash=ασ′[fα(E)−fβ(E)]Tαβ(9)
Equation (8) is the central result of this work. We stress
that Eqs. (8) and (9) are valid under most general condi-
tions as we have not made any assumptions about sym-
metries of the scattering region . It is instructive to note
that in general Tσ′σ
αβ/ne}ationslash=Tσ′−σ
αβandRσ−σ
αα/ne}ationslash=R−σσ
ααthere-
fore, spin current equation can not be simplified further
and written in terms of difference of Fermi function mul-
tiplied by transmission or reflection probabilities as is
the case for charge current in Eq. (9) which is stan-
dard Landauer-B¨ uttikerresult[3]. Hence the spin current
given by Eq. (8) will be nonzero even when all the leads
areatequilibrium, i.e., fα(E,µα) =f(E,µ),∀α, whereµ
is equilibrium chemical potential. For sake of complete-
ness we mention that equilibrium chargecurrentvanishes
as is evident from Eq. (9). The preceding discussion im-
plies that linear response for spin currents is not appli-
cable in an electrical circuit where external perturbation
is applied voltages which is conjugate to charge currents
and not to the spin currents as discussed in introduction.
Therefore,the most widely used equation for spin cur-
rent,see Ref.[4, 6] obtained by a generalization of charge
current Eq. (9) has to regarded as incorrect. In view of
this the theoretical study of spin dependent phenomena
in mesoscopic systems needs to be re-investigated.
We can gain further insight into spin current by con-
sidering non-equilibrium situation such that the chemi-
cal potential at the different leads differ only by a small
amount so that we can expand the Fermi distribution
function around equilibrium chemical potential µas,
fβ(E,µβ) =f(E,µ)+(−df/dE)(µβ−µ). In this case we
can immediately notice from Eq. (8) that total spin cur-
rent in non-equilibrium situation will have equilibrium as
well non equilibrium parts of spin current. For ESC the
full Fermi sea of occupied levels will contribute. There-
fore even in non-equilibrium situation ESC cannot be
neglected.
Equilibrium spin currents in time reversal symmetric
two terminal system: In time reversalsymmetric systems
spin resolved transmission and reflection probabilities in
Eq. (8) obey following relations i.e.,Rσσ′
αα=R−σ′−σ
ααand
Tσσ′
αβ=T−σ′−σ
βα[7]. In this case the spin currents Eq. (8)
further simplifies to (here we denote left and right termi-nals byLandRrespectively),
Is,eq
L,ˆu= (/planckover2pi1
2h)/integraldisplay∞
0dE2f(E,µ)/bracketleftbig
(R−σσ
LL−Rσ−σ
LL)
+(Tσ−σ
RL−T−σσ
RL)+(T−σ−σ
RL−Tσσ
RL)/bracketrightbig
,(10)
above equation gives spin current in Left terminal. Spin
current in right terminal are obtained from the same
equation by interchanging L↔R. On right hand side
in Eq. (10), σand−σrefers to up and down spin states
alongˆu. From the above equation and previous discus-
sion it is evident that even in time reversal symmetric
two terminal systems ESC are non zero. Incase SU(2)
symmetry in spin space is preserved, the spin resolved
transmission and reflection probabilities obey a further
rotational symmetry in spin space, i.e, Tσσ′
αβ=T−σ−σ′
αβ,
Rσσ′
αα=R−σ−σ′
ααand spin flip components are zero, which
implies that spin currents are identically zero for all ter-
minals as is evident from Eq. (10). This conclusion re-
mains valid even for systems without time reversal sym-
metry as can be seen easily from Eq. (8).
The expression in Eq. (10) can be cast in a more useful
form as (the details will be provided in Ref.[14]),
Is,eq
αˆu=1
2π/integraldisplay
Trσ[{ΓαGrΓβGa+
(ΓαgrΓαga)}(σ·u)]f(E,µ)dEdk/bardbl.(11)
In the above equation all symbols represents 2 ×2 matri-
ces in spin space( in σ·ubasis) and trace is taken over
spin space. Where Γ α,βrepresents broadening matrices
due to contacts, Gr(a)are retarded and advanced Green
function and gr,ais a off diagonal matrix in spin space
defined as gr,a= [{0,Gr,a
σ−σ},{Gr,a
−σσ,0}]. First term in
Eq. (11) corresponds to spin resolved transmission while
the second and third term give spin resolved reflection
probabilities as required by Eq. (10). Notice that the
above formula can not be simply written in terms of
transmission matrix and it is reminiscent of the charge
currentformulaforinteractingsystemderivedinRef.[12].
In our case this happens for spin current because in pres-
ence of SO interaction spin can not be described as a
non-interacting object.
Equilibrium spin currents in two terminal Rashba sys-
tem:We now apply Eq.(11) to study ESC(zero tempera-
ture)inafinite sizeRashbasampleoflength L, contacted
by two ideal and identical unpolarized leads. The Hamil-
tonian for two-dimensional electron system with Rashba
SO interaction and short-range spin independent disor-
der isH=/planckover2pi12k2/(2m∗)I+λso(σxky−σykx) +U(x,y),
whereλsois Rashba SO coupling strength, U(x,y) is the
random disorder potential and Iis 2×2 identity ma-
trix. Neglecting weak localization effects, the disorder
averaged retarded Green function including the effect of
leads is given by[13],
Gr,a(E,k) =E−/planckover2pi12k2
2m+iη(k)+λso(σxky−σykx)
[(E−/planckover2pi12k2
2m∗+iη(k))2−(λsok)2]
(12)4
withη(k) = (2γ(k) +/planckover2pi1
τ(k)) where τ(k) is momentum
relaxation time due to elastic scattering caused by im-
purities and γ(k) broadening due to leads. The contact
broadeningmatricesinEq.(11)arediagonalinspinspace
and defined as Γ 1,2= [{γ(k),0},{0,γ(k)}]. Physically
significance of γ(k) is that it represents /planckover2pi1/2 times the
rate at which an electron placed in a momentum state
kwill escape into left lead or right lead, hence as a first
approximation we can write, γ(k) =/planckover2pi1vx(k)
L≡/planckover2pi12kcos(φ
mL,
whereφis angle with respect to xaxis. The impurity
scattering time can be approximated as1
τ(k)≈/planckover2pi1k
lel, where
lelis elasticmean free path. With these inputs we can in-
tegrate Eq. (11) over transverse momentum (multichan-
nel case) and energy to obtain an analytical expression
for equilibrium spin current. We find that ESC with
spin parallel(antiparallel) to the ˆzorˆzaxis and flowing
to thexdirection vanishes in both leads( Iˆxs
ˆz≡σzvx=0,
Iˆxs
ˆx≡σxvx=0). The ESC with spin parallel(antiparallel)
to theˆyaxis are nonzero and given by,
I−ˆx,s
L,+ˆy=I+ˆx,s
R,−ˆy≅m∗λsoEFL
32π/planckover2pi12(2+(L
lel)2).(13)
In left lead ESC is polarizedalong+ ˆydirectionand flows
outwards from sample to lead,i.e.,along −ˆxdirection,
while in the right lead ESC is polarized along −ˆyand
flows outwards from sample to lead,i.e.,along + ˆxdirec-
tion. Physically this implies that spin angular momen-
tum is generated in sample with SO coupling which then
flows outwards in the regions where SO coupling is zero,
i.e., the left and right leads. This implies a spin rectifi-
cation effect which can only occur if the transport is non
linear and we see that this consistent with nonlinear na-
ture of spin currents as remarked earlier. It is important
to note that due to ESC there is no net magnetization in
the total system (sample+leads) which is consistent with
the Kramer’s degeneracy.
We can gain a deeper understanding of the above ex-
pression if we analyze the systems using additional sym-
metries. The disorder averaging establishes reflection
symmetry with respect to x(y)axis and the system has
a symmetry related with the operator σyRy. (σxRx).
As a result the total symmetry operator(time rever-
sal+reflection) for the system is UT R=ItσxRxσyRy=
It(iσz)RxRy, whereItis time reversal operator. Under
this symmetry operation the disorder averaged system
is invariant and the spin current operators σxvx,σyvx
are even while σzvxis odd. Therefore the spin current
alongˆzdirectionvanishes while in-planespin currentcanbe non zero. However as we have seen above that only
theycomponent of spin current survives after integrat-
ing over momentum. Fig.(1) illustrate the conservation
ofσyvxunder these symmetry operation. The depen-
dence of ESC on λsocan also be inferred from symmetry
consideration. As we can check easily that the Rashba
SO interaction changes sign under reflection along ˆz
axis(λso(σxky−σykx)/mapsto→ −λso(σxky−σykx)). Physi-
cally it corresponds to reversing the asymmetry of con-
fining potential along ˆzaxis. Therefore the spin currents
xy
zspin parallel to
+ y direction
flow parallel to
−ve x directionspin parallel to
−ve y directionflow parallel to
+ve x direction−
FIG. 1: Fig 1: The figure illustrate the conservation of spin
currentσyvxif th system is rotated by πalongˆzaxis, which
represents reflection along and y axis respectively. Under t his
transformation configuration in left and right goes over int o
each other hence the spin current remains invariant.
(equilibrium as wellnon-equilibrium) can only depend on
the odd powers of spin orbit coupling constants λso. Ac-
cording to Eq. (13) ESC are proportional to λsowhich is
consistent with these symmetry consideration and ESC
vanishes if λsozero as expected on physical grounds. It
is worth noting that even the local ESC in macroscopic
Rashba medium, discussed in Ref.[2] are proportional to
λ3
soandaswehaveseenthisisaconsequenceofsymmetry
consideration. Moreover ESC are proportional to length
of SO region because SO region acts as source of these
currents. Note that the right hand-side of Eq. (13) has
dimension of angular momentum per unit time signifying
that these currents are truly transport current.
To conclude, we have derived spin current formula
for multiterminal spin transport for system with broken
SU(2) symmetry in spin space. We have demonstrated
that spin currents are fundamentally different from the
charge currents and the ESC are generically non zero. In
view of this the spin transport phenomena in mesoscopic
system needs a fresh look. Further it will also be inter-
esting and desirable to study different magnetoresistance
phenomena from the perspective of spin currents.
I acknowledge helpful discussion with M. B¨ uttiker and
A. M. Jayannavar.
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1705.04427v1.Analytical_slave_spin_mean_field_approach_to_orbital_selective_Mott_insulators.pdf | Analytical slave-spin mean- eld approach to orbital selective Mott insulators
Yashar Komijani1;, and Gabriel Kotliar1,
1Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey, 08854, USA
(Dated: October 29, 2021)
We use the slave-spin mean- eld approach to study particle-hole symmetric one- and two-band
Hubbard models in presence of Hund's coupling interaction. By analytical analysis of Hamiltonian,
we show that the locking of the two orbitals vs. orbital-selective Mott transition can be formulated
within a Landau-Ginzburg framework. By applying the slave-spin mean- eld to impurity problem,
we are able to make a correspondence between impurity and lattice. We also consider the stability of
the orbital selective Mott phase to the hybridization between the orbitals and study the limitations
of the slave-spin method for treating inter-orbital tunnellings in the case of multi-orbital Bethe
lattices with particle-hole symmetry.
INTRODUCTION
Iron-based superconductors are the subject of inten-
sive study in the pursuit of high-temperature super-
conductivity [1{7] . These systems are interacting via
Coulomb repulsion and Hund's rule coupling and they
require the consideration of multiple bands with crys-
tal eld and inter-orbital tunnelling [8, 9]. Early DMFT
studies, pointed out the importance of the corrlations [10]
and Hund's rule coupling [11], and reported a notice-
able tendency towards orbital di erentiation, with the
dxyorbital more localized than the rest [12]. They also
demonstrated orbital-spin separation [13{15]. Note that
the orbital di erentiations has been recently observed in
experiments [16].
Another perspective on the electron correlations in
these materials is that the combination of Hubbard in-
teraction and Hunds coupling place them in proximity to
a Mott insulator [17] and, correspondingly, the role of the
orbital physics is provided by the orbital selective Mott
picture [18, 19]. Ref. [18] demonstrated an orbital selec-
tive Mott phase in the multi-orbital Hubbard models for
such materials, in the presence of the inter-orbital kinetic
tunneling. In such a phase, the wavefunction renormal-
ization for some of the orbitals vanishes. Such a phase
has been observed in angle-resolved photo-emission spec-
troscopy (ARPES) experiments [20, 21]. Although desir-
able, these e ects have not been understood analytically
in the past, partly due to the fact that an analytical
study is dicult for realistic models. However, there are
simpler models, capable of capturing part of the relevant
physics, which are amenable to such analytical under-
standing, and this is what we study in this paper.
The mean- eld approaches to study these problems
rely on various parton constructions or slave-particle
techniques. The latter include slave-bosons [22, 23],
Kotliar-Ruckenstein four-boson method [24] and its rota-
tionally invariant version [25], slave-rotor [26], Z2slave-
spin [27{30] and its U(1) version [18, 31], slave spin-1
method [32] and the Z2mod-2 slave-spin method [33, 34].
For a comparison of some of these methods see Appendix
A. While these methods are all equivalent in the sensethat they are exact representation of the partition func-
tion if the degrees of freedom are taken into account ex-
actly, di erent approximation schemes required for an-
alytical tractability, lead to di erent nal results and
therefore they have to be tested against an unbiased
method like the dynamical mean- eld theory (DMFT)
[36{44] in large dimensions or density function renormal-
ization group (DMRG) [45] in one dimension.
We use the Z2slave-spin [27{30] in the following to
study the orbital selectivity with and without Hund's
coupling. We brie
y go through the method for the sake
of completeness and setting the notations. By study-
ing the free energy analytically we develop a Landau-
Ginzburg theory for the orbital selectivity. A Landau-like
picture has been useful in understanding the Mott tran-
sition in in nite dimensions. Using a Landau-Ginzburg
approach, we show how the interaction in the slave-spin
sector tend to lock the two bands together in absence of
Hund's coupling and that the Hund's coupling promotes
orbital selectivity. We also apply the method to an im-
purity problem ( nite- UAnderson impurity) and its use
as an impurity show that the slave-spin mean- eld re-
sult can be understood as the DMFT solution with an
slave-spin impurity solver. This puts the method in per-
spective by showing that the mean- eld result is a subset
of DMFT. Additionally, we study the e ect on the orbital
selective Mott phase produced by inter-orbital kinetic
tunnelling and point out to some of the limitations of
the slave-spin for treating such inter-orbital tunnelling in
particle-hole symmetric Bethe lattices. Finally, we study
study the instability of the orbital selective Mott phase
by including hybridization between the two orbitals.
Z2Slave-spin method
We consider the Hamiltonian H=H0+Hint, where
H0=X
hiji t
ijdy
i dj (1)
We must demand t
ij= [t
ji]for this Hamiltonian to
be Hermitian. Unless mentioned explicitly, is a super-arXiv:1705.04427v1 [cond-mat.str-el] 12 May 20172
index that contains both spin and orbital degrees of free-
dom. We replace the d-fermions with the parton con-
struction [27]
dy
i = ^zi fy
i ; ^zi = x
i : (2)
i ,=x;y;z are SU(2) Pauli matrices acting on an
slave-spin subspace per site/spin/
avour, that is intro-
duced to capture the occupancy of the levels. Slave-spin
statesj*i iandj+i icorrespond to occupied/unoccupied
states of orbital/spin at sitei, respectively. Away from
half- lling, [28] has shown that x
i has to be replaced
with +
i =2+c
i =2 wherecis a gauge degree of freedom
and is determined to give the correct non-interacting re-
sult. Here, for simplicity we assume p h(particle-hole)
symmetry and thus maintain the form of Eq. (2). Note
that this parton construction has a Z2gauge degree of
freedom x;y! x;yandf! f, thus the name Z2
slave-spin. The representation (2) increases the size of
the Hilbert space. Therefore, the constraint
2fy
i fi = z
i + 1; (3)
to imposed to remove the redundancy and restrict the
evolution to the physical subspace. Using Eqs. (2,3) it
can be shown that the standard anti-commutation rela-
tions ofd-electron are preserved.
Plugging Eq. (2) in H0, and imposing the constraint
(on average) via a Lagrange multiplier, we have
H0=X
hiji t
ijfy
i fj ^zy
i ^zj i [fy
i fi ( z
i + 1)=2]
On a mean- eld level, the transverse Ising model of slave-
spins can be decoupled from fermions. The decoupling
is harmless in large dimensions [46] as the leading op-
erator introduced by integrating over the fermions be-
comes irrelevant at the critical point of the transverse
Ising model. Therefore, writing H0Hf+H0S, we have
H0S=X
hiji J
ijh
^zy
i ^zj Q
iji
+X
i z
i =2;
Hf=X
hiji ~t
ijfy
i fj i (fy
i fi 1=2) (4)
where ~t
ij=t
ijQ
ijwithQ
ij=h^zy
i ^zj iis the renor-
malized tunnelling and J
ij=t
ijhfy
i fj iis an Ising
coupling between slave-spins. The advantage of the par-
ton construction (2) is that the interaction Hintf gcan
be often written only in terms of the slave-spin variables,
so thatH=Hf+HSandHS=H0S+Hint.
Particle-hole symmetry - p hsymmetry on the orig-
inal Hamiltonian is de ned as ( nis a site index)
dn !( 1)ndy
n ; dy
n !( 1)ndn (5)
On a bipartite lattice, the nearest neighbor tunnelling
term preserves p hsymmetry, even in presence of inter-
orbital tunnelling. So, if the system is at half- lling theHamitonian is invariant under p hsymmetry. We have
to decide what p hsymmetry does to our slave-spin
elds. We choose
fn !( 1)nfy
n ; x
n ! x
n ; z
n ! z
n (6)
So, we see that if the original Hamiltonian had p h
symmetry, we necessarily have i = 0.
Single-site approximation - The Hamiltonian HSis a
multi-
avour transverse Ising model which is non-trivial
in general. Following [27{34] we do a further single-site
mean- eld for the Ising model, exact in the limit of large
dimensions:
^zy
i ^zj h^zy
i i^zy
j + ^zy
i h^zj i h^zy
i ih^zj i;(7)
The last term together with the second term of Eq. (8)
contributes a 2P
hiji J
ijQ
ij. We de ne zi =h^zi i
andZi =jzi j2as the wavefunction renormalization of
orbital at sitei. The slave-spin Hamiltonian becomes
(using the symmetry of J
ij)
H0S=X
i (h
i ^zi +h:c:); hi =X
j J
ijzj (8)
In translationally invariant cases hi andzi become in-
dependent of the site index and J
ijdepends on the dis-
tance between sites iandj. Therefore, we can simply
writeh =P
J z where
J X
(i j)J
(i j)=X
jt
ijD
fy
i fj E
:
In absence of inter-orbital tunnelling, Jis a diagonal ma-
trix, corresponding to individual orbitals, where for each
orbitalJ =RD
D d ()f()is the average kinetic en-
ergy and depends only on bare parameters, una ected
by the renormalization factor z. For semicircular band
(Bethe lattice),J= 0:2122D, while for a 1 Dtight-
binding modelJ1D= 0:318DwithD= 2t. Since the
operator ^z = x
is Hermitian, we can write the slave-
spin Hamiltonian (for each site) as [47]
HS=X
a x
+Hint (9)
wherea = 2P
J z (at half- lling). The only non-
trivial part of computation is the diagonalization of HS.
This is a 4Mdimensional matrix where Mis the number
of orbitals. The free energy (per site) is
F= 1
X
nkTr log[ G 1
f(k;i!n)] 2X
nJ z
z
1
logn
Trh
e HSio
: (10)
Here = 1=Tis the inverse temperature and the second
part comes from two constants introduced in Eqs. (4) and3
(7). At zero temperature, the rst term is just J z
z
and the last term is ESwhich depends on zviaa. Hence,
F= X
J z
z +ES(fag): (11)
ONE-BAND MODEL
In the one-band case the interaction is Hint=
UP
i~ni"~ni#where ~nini 1=2. Representing the
latter with z
i=2 and using translational symmetry we
obtainHint!(U=4) z
" z
#. Since we are in the para-
magnetic phase ( a"=a#), only sum of the two spins
2~T=~ "+~ #enter (the singlet decouples) and the Hamil-
tonain can be written as HS= 2aTx+U
2(Tz)2 U=4,
creating a connection to the spin-1 representation of [32].
Furthermore, we can form even and odd linear combina-
tions of the empty and lled states and at the half- lling,
only the even linear super-positions enters the the Hamil-
tonian. Thus, choosing atomic states of HSas
j 0i=j*ij+ip
2;j 1i=j*+ijOip
2(12)
withE0= U=4 andE1=U=4, the Hamiltonian
can be written as HS= 2a x+ (U=4) zwhere~ are
Pauli matrices acting between j +0iandj +1i, i.e. it
reduces to the Z2mod-2 slave-spin method [33, 34]. In
writing the states in Eq. (12) we have used a short-hand
notation (also used in the next section) j*"+#i ! j*i
andj+"*#i!j+i ,j*"*#i!j*+i and so on. The in-
set of Fig. (1b) shows a diagrammatic representation of
the slave-spin Hamiltonian and two states decouple. The
ground state of HSis that of a two-level system
ES= U
4p
1 + (4 =U)2 (13)
with the level-repulsion = 2aand the zero-temperature
(free) energy is given by [factor of 2 sdue to spin]
F= 2sjJjz2+ES(z) (14)
The free energy is plotted in Fig. (1a) and it shows a
second-order phase transition as Uis varied. Close to
the the transition !0, we can approximate ES
2 2=U+ 8 4=U3. Writing the rst term of the free en-
ergy as + 2=8jJj, we can read o the critical interaction
UC= 16jJj. Minimization of the free energy gives the
Gutzwiller projecion fomrula of Brinkman and Rice [48]
Z=1 u2u<1
0u>1(15)
withu=U=UCand is plotted in Fig. (1b). At nite
temperature this procedure gives a rst order transition
terminating at a critical point [34].
FIG. 1: (color online) (a) Free energy (at T= 0) as a
function of zshowing a second-order phase transition as
U=UCis varied. (b) Wavefunction renormalization
Z=jzj2as a function of Uhas the Brinkman-Rice
form. Inset: Diagrammatic representation of the
slave-spin Hamiltonian. Each dot denotes on atomic
state. Two states decouple and HSis equivalent to that
ofZ2mod-2 slave-spin.
Spectral function - The Green's functions of the d-
fermionsGd( )
Td( )dy
(0)
factorizes
Gd;( )
Tf( )fy
(0)
hT x
( ) x
(0)i (16)
to thef-electron and the slave-spin susceptibility and
thus the spectral function is obtained from a convolu-
tion with the slave-spin function Ad(!) =Af(!)AS(!),
in whichAfis a semicircular density of states with the
widthZand within single-site approximation ASis
AS(!) =Z(!) +1 Z
2[(!+ 2ES) +(! 2ES)] (17)
The spectral density has the correct sum-rule (in contrast
to the usual slave-bosons [22, 23]) since the commuta-
tion relations of the slave-spins are preserved. However,
the single-site approximation does not capture incoherent
processes, and this re
ects in sharp Hubbard peaks in the
Mott phase ( Z= 0) where Af=(!). Also, the spatial
independence of the self-energy implies that the inverse
e ective-mass of \spinons" m=~m=Z[1 + (m=kF)@k]
is zero in the Mott phase. This is again an artifact of
the single-site approximation. Both of these problems
are remedied, e.g. by doing a cluster mean- eld calcu-
lation [28, 33] or including quantum
uctuations around
the mean- eld value within a spin-wave approximation
to the slave-spins [33].
The fact that (beyond single-site approximation)
spinons disperse in spite of h xi! 0 and they carry a
U(1) charge as seen by Eq. (2), implies that vanishing of
h xidoes not generally correspond to the Mott phase in
nite dimensions. However, in large dimensions, this is
correct [34] and that is what we refer to in the following.4
TWO-BAND MODEL
In absence of inter-orbital tunnellings, the free-energy
is
F=a2
1=2jJ1j+a2
2=2jJ2j+ES(a1;a2) (18)
whereESis the ground state of the slave-spin Hamilto-
nian. For two bands we have the interaction
Hint=U(~n1"~n1#+ ~n2"~n2#)
+U0(~n1"~n2#+ ~n1#~n2")
+ (U0 J)(~n1"~n2"+ ~n1#~n2#) +HXP (19)
where ~n nf 1=2 = z
=2. The spin-
ip and pair-
tunnelling terms are
HXP= JX[dy
1"d1#dy
2#d2"+dy
1#d1"dy
2"d2#]
+JP[dy
1"dy
1#d2#d2"+dy
2"dy
2#d1#d1"]:(20)
This term mixes the Hilbert space of f-electron with that
of slave-spins. Following [27{29] we include this term ap-
proximately by dy
! +
andd !
substitution so
that it acts only in the slave-spin sector. The justi cation
is that such a term captures the physics of spin-
ip and
pair-hopping. Using the spherical symmetry U0=U J
this can be written as
Hint=U
2(~n1"+ ~n1#+ ~n2"+ ~n2#)2 U
2+HXP
J[~n1"~n2#+ ~n1#~n2"+ 2~n1"~n2"+ 2~n1#~n2#] (21)
ForJX=JandJP= 0 it has a rotational symmetry
[49]. Alternatively, U0=U 2JandJX=JP=J
has rotational symmetry. The choice does not a ect the
discussion qualitatively. We keep the former values in
the following.
Atomic orbitals - We start by diagonalizing the atomic
Hamiltonian in absence of the hybridizations. Close to
half- lling the doubly-occupied states have the lowest en-
ergy and are given by
j 0i=j*1*2ij+ 1+2ip
2; E0= U J=2;
j 1i=j*1+2ij+ 1*2ip
2; E1= U+J=2JX;
j 2i=j*+ 1;O2ijO1*+2ip
2; E2= U+ 3J=2JP;
These 3 doublets become the 6-fold degenerate ground
state when J!0. The 1;3-particle states are then next
j 3i=j*+ 1ij*2ij+ 2ip
2; E3=1;
j 4i=jOi1j*2ij+ 2ip
2; E4= 1
j 5i=j*1ij+ 1ip
2j*+ 2i; E5=2;
j 6i=j*1ij+ 1ip
2jOi2; E5= 2;and nally, there are two (empty and quadruple occu-
pancy) states at the top of the ladder
j 7i=j*+i1j*+i2; E 7=1+2+ 3U 3J=2;
j 8i=jOi1jOi2; E 8= 1 2+ 3U 3J=2:
No Hund's rule coupling - The hybridization causes
transition among atomic states. In the case of no Hund's
coupling we can block diagonalize HSinto several sectors
and diagrammatically represent it as shown in Fig. (2).
Therefore, the calculation can be reduced from 16 16 to
55. The larger the level-repulsion, the lower the ground
state energy in each sector. The fact that the slave-spins
decouple into several sectors brings about the possibility
of possible ground-state crossings between various sectors
as the parameters a1anda2are varied. Here, however,
it can be shown that the sector Chas the lowest ground
state energy for arbitrary parameters.
FIG. 2: Diagrammatic representation of the slave-spin
Hamiltonian HSin the two-band model with J= 0 and
1=2= 0. Each dot represents an atomic state with a
certain energy, denoted on the left, whereas the connecting
lines represent o -diagonal elements of the Hamiltonian
matrix, all assumed to be real. We have used the short-hand
notationp
2 S=D
a;b a b. Also note that ai= 2Jizi. The
Hamiltonian factorizes into several sectors.
Numerical minimization of the free-energy leads to
Fig. (3) which reproduces the results of [27]. For t2=t1>
0:2 the metal-insulator transition happens at the same
criticalUfor the two bands and we refer to it as the
locking phase , whereas for t2=t1<0:2 the critical Ufor
the bands are di erent U2< U 1and we refer to it as
orbital selective Mott (OSM) phase .
In order to have the result analytically tractable we do
one further simpli cation and that is to project out the
zero and quartic occupancies per site, by dropping the
high energy site at the apex of sector C. We expect such
an approximation to be valid close to the Mott transition
of the wider band, but invalid at low U. As a result5
FIG. 3: (color online Wavefunction renormalizations Z1
(blue) andZ2(green) as a function of U=UC1in absence of
Hund's rule coupling J= 0. The states at the bottom row
correspond to doubly occupied sites. The middle-row states
have occupancy of 1 or 3 and the states at the top row
correspond to zero or four-electron llings. (a) Moderate
bandwidth anisotropy t2=t1= 0:5 shows locking. (b) Large
bandwidth anisotropy t2=t1= 0:15 can unlock the bands and
cause OSM transition (OSMT). We also reproduce the kink
in the wider-bandwidth (blue) band as the narrow band
transitions to the Mott phase [27], marked with an arrow. In
the OSM phase, the wavefunction renormalization of the
wider band follows the Brinkman-Rice formula (solid line).
the sectorCdecouples into two smaller sectors C, each
equivalent to a two-level system with the level-repulsions
=q
a2
1(3=2 +p
2) +a2
2(3=2 p
2)
q
a2
1(3=2 p
2) +a2
2(3=2 +p
2):(22)
The ground state energy of the slave-spin sector is deter-
mined with +inserted in the ESexpression (13) (after
an inert U=4 energy shift). Note that this ground state
has theZ2symmetrya1$a2of the Hamiltonian HS.
ES( +) as a function of ( a2
1 a2
2)=(a2
1+a2
2), is mini-
mized fora1=a2. Discarding empty and lled states
corresponds to truncating part of the Hilbert space and
thus leads to reduced wavefunction renormalization at
U0. In Fig. (4) we have compared our analytical so-
lution to that of the exact result. When a2= 0, Eq. (22)
gives !2a1as in the single-band case and there-
fore, same critical interaction UC1= 16jJ1jis obtained.
But for symmetric bands a1=a2, it gives = 2p
3a.
Following similar analysis as before, the free energy is
a2=jJj 2 2=Uand we obtain UC= 24jJj= 1:5UC1in
agreement with [27, 29].
Locking vs. OSM phase - We formulate the locking vs.
OSM question as the following. Under what condition,
a1>0 buta2= 0 can be a minima of the Free energy. As
mentioned before, setting a2= 0, in Eq. (22) reduces to
the one-band !2a1. Therefore, the Mott transition for
the wide band happens at the same critical Uas before.
To have a non-zero a1solution, we must have U < Uc1.
The pointa2= 0 always satis es dF=da 2= 0. To ensure
that it is the energy minima we need to check the second
FIG. 4: (color online) A comparison of numerical
minimization of the free energy vs. the analytical two-level
system. Discarding the empty and full occupancy states
leads to underestimation of ZasU!0 but close to the
Mott transition the approximation is accurate.
derivative
d2F
da2
2
a2=0=1
jJ2j 5
jJ1j>0; (23)
which gives the condition jJ2=J1j<0:2.
We can better understand the transition by using an
order parameter. The trouble with the expression of is
that it cannot be Taylor expanded when a1anda2are
both small. However, we may assume a2=ra1, with
ras an order parameter replacing a2, and write down
(a1;a2) =a1 (r) where (r) = +(a1!1;a2!r). A
niterclose to the transition implies locking whereas r=
0 orr=1implies OSM phase. Close to the transition
of both bands 0 and we can write ES 2 2=U+
8 4=U3and Eq. (18) becomes
F(a1;r) =a2
1W
2jJ1j+O(a4); Wx(r;u) = 1 +xr2 2(r)
4u
Here,x=jJ1=J2j, andu=U=UC1. The metal-insulator
transition for a1happens when the mass coecient W
changes sign. For negative W,a2
1>0 and we still have to
minimize the free energy with respect to r. At smallr, we
can expand (r)2 + 5r2. To zeroth order in r, theW-
sign-change happen at u= 1. Another transition from
r= 0 tor > 0 happens when the corresponding mass
term (x 5=u)r2changes sign, giving the same critical
bandwidth ratio xc= 5 as we had before. So we have two
equationsW(r;u) = 0 and@rW(r;u) = 0. The function6
Wis plotted in the gure and the transition from locking
r>0 to OSM phase r= 0 are shown.
FIG. 5: The coecient W(r;u) is shown for various uas
function of r=a2=a1. Equations W= 0 and@rW= 0 are
satis ed at the minimum of the red curve, which is (a) at a
niter= 1 in the Locking phase, jJ1j=jJ2j. (b) and zero
r= 0 in the OSM phase, jJ1j5jJ2j.
Large Hund's coupling - In presence of Hund's cou-
pling the slave-spin Hamiltonian is modi ed to the dia-
gram shown in Fig. (6).
FIG. 6: Diagrammatic representation of the slave-spin
Hamiltonian HSin the two-band model at half- lling with
in presence of Hund's rule coupling J. Various degeneracies
are lifted by J-interaction. In the limit of large Hund's
couplingJ=U!1=4 we may only keep sector Cand neglect
all the gray lines.
The ground state still belongs to the sector C. In the
limit of large J=U!1=4, we may ignore all the gray lines
on the block Cand nd that the ground state is that of
a two-level system, Eq. (13) with the level-repulsion
= 2q
a2
1+a2
2 (24)
It is remarkable that the (orbital) rotational invariance
of the model (even though absent in HS) is recovered in
this ground state. When the two bands have the same
bandwidth, this formula predicts UC=UC1. SinceESno longer depends on a2
1 a2
2, there is no more compe-
tition between the two terms and an slight bandwidth
asymmetry lead to OSM phase. This can be formulated
again, following previous section, in terms of stability of
aa16= 0 buta2= 0 solution. We can check that
d2F
da2
2
a2=0=1
jJ2j 1
jJ1j>0; (25)
which givesjJ2j<jJ1jas the sucient condition for
OSMT, i.e., any di erence in bandwidth drives the sys-
tem to the OSM phasse. Alternatively, by expanding the
level-repulsion in this case (r)2 +r2and plugging
it intoW(r;u), we nd that the critical bandwidth ratio
xc=jJ1=J2jis equal to one.
TUNNELLING BETWEEN THE ORBITALS
A very interesting question is about the fate of or-
bital selective Mott phase upon turning on an inter-
orbital tunnelling. The band in Mott insulating phase
has one electron per site forming localized magnetic mo-
ment. There is a large entropy associated with this phase
and it is natural to expect that it would be unstable to-
ward possible ordering. A possible mechanism that can
compete with magnetic ordering, is the Kondo screening
of the insulating band by the itinerant band, leading to
conduction in the former and opening a hybridization gap
in the latter band (e ectively a new locking e ect com-
ing from Kondo screening). Within single-site approxi-
mation, however, the form of the renormalized coupling
~t
ij=z
i t
ijzj implies that once an orbital goes to the
Mott phase, it automatically shuts down its coupling to
all the other orbitals. We speculate that this e ect might
be responsible for the orbital selective Mott transition so-
lution found in [18]. However, it is still a valid question
whether or not the critical interactions UCfor a Mott
transition are modi ed by inter-orbital tunnelling, which
we explore in the following.
Before treating inter-orbital tunnelling, we discuss
how the slave-spin method can be applied to the impurity
problem, and its relation to the lattice.
Impurity vs. Lattice and the DMFT loop
We can also apply the slave-spin method to an im-
purity problem. In particular, we can use the slave-spin
(as well as any other slave-particle) method as an im-
purity solver for the DMFT. We show in the following
that the slave-spin mean- eld result corresponds to such
a DMFT solution with the corresponding slave-spin im-
purity solver. This puts the method on rm ground and
allows comparison between various methods.
First, consider a generic p hsymmetric impurity
model described by the Hamiltonian H=H0+Hint7
where
H0= X
k t
k(dy
ck +h:c:) +X
k
kcy
k ck :(26)
Again ; are superindices that include both orbital
and spin. We have assumed that the bath is diagonal
and discarded any local `crystal eld' dy
1d2for simplic-
ity. In the simple case of single-orbital impurity Hint=
U~nd"~nd#. Via a substitution of Eq. (2), the hybridiza-
tion term becomes H0= P
k t
k(fy
x
ck +h:c:).
This problem can be written in a similar way as be-
foreHHf+HSwhereHSis exactly what we had
in single-band lattice case. However, since the fy
x
ck
interaction happens only on the impurity site, we do not
need the second single-site approximation here, and ob-
taina = 2P
k t
khfy
k ck i. In order to have a gen-
eral formalism that applies to both impurity and lattice,
as well as scenarios with inter-orbital tunnelling for which
J renormalizes and is dicult to compute, we regard a
andzas independent variables and write the free energy
of Eq. (12) as [47]
F(fz;ag) =Ff(fzg) +FS(fag) X
a z : (27)
The saddle-point of Fwith respect to aandzgives the
correct mean- eld equations. Ffis the free energy of
thef-electron given by Ff= TP
nTr log[ G 1
f(i!n)]
whereG 1
f(i!n) =i!n1 zy(i!n)zwith(i!n) =P
ktyGc(k;i!n)t, the hybridization function. The slave-
spin part is given by FS= Tr[e HS] where for a single-
orbital Anderson impurity, HS= 2a x+U z=4, as we
had in the single-band case before.
The mean- eld equations w.r.t zandaare, respec-
tively
a =1
z Zd!
f(!)!Im
G
f(!+i)
; (28)
z =dFS
da : (29)
The rst equation provides a relation between aand
zthat generalizes a = 2P
J z (see appendix D).
Having expressions for ES(a) we can eliminate ain fa-
vor ofz, or vice versa, which is equivalent to a Legendre
transformation. In the appendix C, we apply these equa-
tions to the (single-orbital) nite- UAnderson impurity
problem and show the `transition' to the Kondo phase as
the temperature is lowered.
In a lattice, the free energy has the same
form as Eq. (27) with the di erence that Ff=
TP
k;nTr log[ G 1
f(k;i!n)] where the Green's func-
tion isGf(k;!+i) = [(!+i)1 zyEkz] 1. It can
be shown that exactly same mean- eld equations are ob-
tained ifGfin Eq. (28) is replaced with G
f(!+i)!P
kG
f(k;!+i). Therefore, we conclude that the twoproblems (lattice and impurity) are equivalent provided
that the hybridization function in the impurity problem
is chosen such that the impurity Green's function and the
local Green's function of the lattice are equal, i.e.
[i!n1 zy(i!n)z] 1=X
k[(!+i)1 zyEkz] 1:(30)
which is the DMFT consistency equation. Therefore,
slave-spin mean- eld is equivalent to a DMFT solution
using the slave-spin method as the impurity solver. Also,
note that a lattice problem in the OSM phase, corre-
sponds to an impurity problem in which the hybridiza-
tion of one of the orbitals to the bath has been turned
o [50]. See also Appendix B.
Inter-orbital tunnelling
Slave spins have been used to study Iron-based su-
perconductors [18] where the inter-orbital tunnelling are
important. We study this tunnelling e ect in the speci c
case withp hsymmetry and without orbital-splitting
(which allows for analytic calculations). The cases that
go beyond such conditions, as arising in the models for
the iron-based superconductors [18], remain to be ex-
plored and are left for future work.
A troublesome feature of the slave-spins is that
they break the rotational symmetry among the orbitals.
Within the p hsymmetric Bethe lattices that we study
here, this rotational variation leads to ambiguities in
presence of inter-orbital tunnellings, as we point out here.
Let us consider a 1D chain with two orbitals H0=
P
n(Dy
nTDn+1;+h:c:), no Hund's coupling in Hint,
and a dispersion
Ek= 2Tcosk;T=t11t12
t12t22
(31)
We have chosen t12=t21and all the elements real (and
positive)to preserve the p hsymmetry. Strictly speak-
ing, in 1D the mean- eld factorization that led to Eq. (4)
and the consequent single-site approximation are both
unjusti ed. The choice of dimensionality, here, is only
for the ease of discussion and not essential to the con-
clusions. As long as the dispersion matrix can be diago-
nalized with a momentum-independent unitary transfor-
mation (as well as any Bethe lattice, see the appendix
D), the following discussion applies. Diagonalizing the
tunnelling matrix gives E
k= 2tcoskwith
t=t11+t22
2rt11+t22
22
detT (32)
Including renormalization just changes t !~t . We
can simply use the diagonalized form of the tunnelling8
matrix to calculate F0atT= 0. Assuming det T>0,
Ff=X
=Zdk
2E
kf(E
k)
! (~t++~t )Z=2
=2dk
cos(k) = 2(~t11+~t22)=
Note thatt12does not enter the free energy. Inserting
this expression into Eq. (27) and setting dF=dzi= 0, we
can remove aiin favor of zi. This seems to imply that
there is a nite threshold (topological stability) for inter-
orbital tunnelling: as long as det T>0, introducing t12
does not change anything in the problem and it simply
drops out and OSM phase is stable against inter-orbtital
tunnelling. For large t12eventually det T<0. So, we get
t+>0 andt <0 and second band is inverted and F0
becomes
Ff! 2(~t+ ~t )== 4
s
~t11 ~t22
22
+ ~t12 2(33)
Hence,t12has non-trivial e ects on renormalization.
On the other hand, we could have used the rotational
invariance of Hintand done a rotation in d1 d2ba-
sis to band-diagonalize H0with the bandwidths T!
diagft+;t g, before using slave-spins to treat the inter-
actions. It is clear then that t12always has non-trivial
e ects by modifying t. For example we could start in
the locking phase where t =t+>0:2, and by increasing
t12slightly get to the OSMT phase t =t+<0:2, without
changing the sign of det T. This paradox exist for any
p hsymmetric lattice with diagonalizable tunnelling
matrix. The root of the problem is that our expression
in Eq. (9) is not invariant under rotations between vari-
ous orbitals. Therefore, the critical value where the OSM
phase persists, is basis-dependent. This ambiguity calls
for the use of unbiased techniques to understand the role
of inter-orbital tunnelling on OSMT. It might be that
the model we studied analytically here is a singular limit
which can be avoided by breaking p hsymmetry and
inclusion of crystal eld in more realistic settings [18].
This remains to be explored in a future work.
As discussed in [25], the way to achieve rotational-
invariance is to liberate the f-electrons that describe
quasi-particles from the physical d-electrons. This is
achieved by a d !P
^z f representation which leads
to a wavefunction-renormalization matrix z =h^z i
with o -diagonal elements. So far, we have not been
able to generalize the slave-spin to a rotationally invari-
ant form and we leave it as a future project.
ON-SITE INTER-ORBITAL HYBRIDIZATION
Even though models for the Iron-based superconduc-
tors have nite crystal level splitting and no on-site hy-
bridization, it is interesting to introduce a hybridizationbetween the two orbitals within the current formalism
[27]. This is interesting, because the on-site hybridiza-
tion, does not su er from the singe-site approximation
h^zi ^zi i 6=h^zi ih^zi i, as opposed to the inter-orbital
tunnelling andh^zi ^zi iappears as an independent or-
der parameter, which leads to the emergence of Kondo
screening as we show in this section.
We can include a termP
n;(v12dy
n;1dn;2+h:c:) to
the Hamiltonian. In order to preserve the p hsymmetry,
v12has to be purely imaginary. The modi cations to the
mean- eld Hamiltonians are
Hf=X
n;(~v12fy
n;1fn;2+h:c:) 2sA12Z12(34)
HS=X
A12 x
1 x
2 (35)
where ~v12=v12Z12withZ12=h x
1 x
2iandA12=
v12P
nhfy
n;1f2i+h:c:.Z12andA12are are related to
each other via the Hamiltonian above and they are inde-
pendent of in the paramagnetic regime. Alternatively,
we can regard them as independent and impose the mean-
eld equation Z12=@FS=@A 12to eliminate A12by a
Lagrange multiplier. Assuming a small A12we can com-
pute the change in slave-spin energy using second-order
perturbation theory. The result is of the form ES=
(A12)2where
is (in absence of Hund's coupling) a
positive constant which contains all the matrix elements
and the inverse gaps
=P
j0h 0j x
1 x
2j ji(Ej
E0) 1h jj x
10 x
20j 0iwhereEjandj jiare the eigen-
value/states of the HSsolved in the previous section.
Eliminating A12in favor ofZ12we nd that the free en-
ergy of the system is
F(z1;z2;Z12) = 2s
X
knTr log~k1 i!niZ0
12
iZ0
12 ~k2 i!n
+E0
S(z1;z2) +(Z0
12)2
0(36)
HereE0
Sis the value of ES(a1;a2) P
iaiziin absence
of hybridization v12in whicha1anda2are eliminated in
favor ofz1andz2. Also, we have rede ned jv12jZ12!
Z0
12and
jv12j2!
0.
Eq. (36) is nothing but the free energy of a Kondo
lattice at half- lling [51] with renormalized dispersions
~k1and ~k2. In a Kondo lattice, this form of the free
energy appears using Z0
12as the Hubbard-Stratonovitch
eld that decouples the Kondo coupling
0~S2dy
1~ d1. Here
S2=dy
2~ d2is the spin of the Mott-localized band and
0plays the role of the Kondo coupling. As a result of
this coupling, a new energy scale TKDexp[ 1=
0] ap-
pears, with D2~t11the bandwidth of the wider band,
below which the Kondo screening takes place which in
thep hsymmetric case gaps out both bands but away
fromp hsymmetry mobilizes the Mott localized band.
Either way, we conclude that orbital selective Mott in-
sulating phase is unstable against hybridization between9
the two orbitals in agreement with [27]. However, even
though a true selective Mottness is unstable, orbital dif-
ferentiation, re
ected as large di erence in e ective mass
can exist [16].
CONCLUSION
In conclusion, we have used slave-spin mean- eld
method to study two-band Hubbard systems in presence
of Hund's rule coupling. We have developed a Landau-
Ginzburg theory of the locking vs. OSMT. We discussed
the relation between slave-spins and the KR boson
methods (Appendix). We have also applied the method
to impurity problems and shown a correspondence
between the latter and the single-site approximation
of the lattice using the DMFT loop. Finally, we have
discussed the limitations of the slave-spin method for
multi-orbital models with both particle-hole symmetry
and inter-orbital tunnelling and shown that the orbital
selective Mott phase is unstable against on-site hy-
bridization between the two orbitals.
We appreciate valuable discussions with P. Coleman,
T. Ayral, M. Metlitski, L. de'Medici, K. Haule and
C.-H. Yee, and in particular, a detailed reading of the
manuscript and constructive comments by Q. Si. The
authors acknowledge nancial support from NSF-ONR.
After completion of this manuscript, we became aware
of another work [52] which contains a Landau-Ginzburg
theory of OSMT in presence of the inter-orbital tun-
nelling. The conclusions of the two work agrees wherever
there is an overlap.
APPENDIX
A. Various slave-particle methods
For a one band model, KR introduces four bosons and
uses the representation ^ zy
=P+[py
e+dyp ]P , where
py
,eyanddyare (hardcore) bosonic creation operators
for-spinon, holon and doublon, respectively and Pare
projectors that depend on the occupations of the bosons
and are introduced to normalize the probability ampli-
tudes over the restricted set of physical states. On the
other hand, a SU(2) spin-variable ~ can be represented
by two Schwinger bosons a andb satisfying the con-
straintay
a +by
b = 1 (hardcore-ness), via
z
=by
b ay
a ; x
=ay
b +by
a (37)
On an operator level, the two methods have the same
Hilbert space as depicted in Table (I) for the case of oneay
"a"by
"b"ay
#a#by
#b#
1 0 1 0
0 1 1 0
1 0 0 1
0 1 0 1eyepy
"p"py
#p#dyd
10 0 0
01 0 0
00 1 0
00 0 1
TABLE I: Comparison of the Schwinger boson
representation of the slave-spin (left) and
Kotliar-Ruckenstein slave-bosons (right).
orbital. Average polarization of the spin along various di-
rection in the Bloch sphere corresponds to condensation
ofaandbbosons.
A trouble with the slave-spin representation is that
thef-quasi-particles carry the charge of the d-electron
and thus the disordered phase of the slave-spins (in which
thef-electrons still disperse beyond single-site approxi-
mation) is not a proper description of the Mott phase.
As a remedy, it has been suggested [31] to replace xin
Eq. (2) with +and xing the problem of non-unity Z
in the non-interacting case by applying ne-tuned pro-
jectors ^zy=P+ +P . We note that this looks quite
similar to KR.
ForMspinful orbitals, KR requires introducing 4M
bosons (only one of them occupied at a time) whereas
only 2Mslave-spins are required (each with the Hilbert
space of 2). Thus the size of the two Hilbert spaces are
the same 22M= 4M.
B. General low-energy considerations
Generally for a lattice we can expand the self-energy
Gd(k;!) = [!1 Ek d(k;!)] 1; (38)
Expanding the self-energy
d;lat(k;!) =(0;0) +~k@~k(0;0) +!@!(0;0) +
Within single-site approximation, the second term is
zero. Denoting the third term as @!d1 Z 1and
assuming Z=zzywe can write
Gd(k;!)z[!1 zyEkz] 1zy; (39)
which simply means Gf(!) = [!1 ~Ek] 1and the corre-
lation functions of the slave-particles are just decoupled
h^zi ( )^zy
j i!z
z within single-site approximation also
discarding any time dynamics. For the tunnelling ma-
trix, we simply have ~t=zytz. Similarly, for an impurity
we have
i!n1 d;imp (i!n) =G 1
d;imp(i!n); (40)
Gd;loc(i!n) =X
k[i!n Ek d;lat(k;i!n)] 1(41)10
Denoting the interaction part of the self-energy
d;imp (i!n) =(i!n) +d;I(i!n), the DMFT approxi-
mation identi es d;I(i!n) =d;lat(k;i!n). Again ex-
pandingd;I(!)(1 Z 1)!we have
Gd;imp (i!n) =z[i!n1 ~(i!n)] 1zy; (42)
with ~(i!n) = zy(i!n)zin agreement with
Gf;imp (i!n) = [i!n1 ~(i!n)] 1. Using the same
approximation for Gd;locleads to
G 1
d;loc(i!n)!zX
k[i!n ~Ek] 1z (43)
the DMFT self-consistency loop equation is Gf;loc(i!n) =
Gf;imp (i!n) or
X
k[i!n ~Ek] 1= [i!n1 ~(i!n)] 1: (44)
Within the slave-spin approach there are no interactions
f;imp =zy(i!n)z;f;I= 0;f;lat= 0 (45)
and Eq. (44) is satis ed as it does for any non-interacting
problem.
Rotation - Using the vector Dfor thed-electrons, in
presence of inter-orbital tunnelling we may sometimes
be able to eliminate such inter-orbital tunnelling by a
rotation to D=UD. SinceD=zF, we assume the
same rotation in the F-spaceF=UF(otherwise they
would contain inter-orbital tunnelling) and the two z-s
are related by z=UyzU. Assuming that Uis a SO(2)
matrix, and zis diagonal, we nd
z=z++z
21 z+ z
2
cos 2 sin 2
sin 2 cos 2 !
(46)
which has o -diagonal elements. Note that if one of the
zelements vanishes, e.g. z = 0, we can factorize z
z=z+
cos
sin !
cos sin
: (47)
Then, it can be seen that Z=zzy!z+zhas the same
form. This basically means one linear combination of f
electrons is decoupled (localized) and the itinerant spinon
band carries characters of both d1andd2bands. This
basis-dependence of the orbital Mott selectivity is again
an artefact due to lack of rotational invariance.
C. Finite- UAnderson model
The slave-spin part of the Hamiltonian is as we had
in the one band case. We can use Eq. (27) to eliminate a
in favour of z. In the wide band limit for the conductionband, we have Gf(i!n) = [i!n iKsign(!n)] 1where
K=t2z2, and the free energy is
F(z) = 2sZD
Dd!
f(!)Im [log (iK !)] +E0
S(z):(48)
E0
Sis obtained by eliminating afromES(a) 2sazpart
of the free energy in Eqs. (13) and (27) and is equal to
E0
S= U
4p
1 z2. Here, we have done a simpli cation to
replaceFSwith its zero temperature value (ground state
energy) while maintaining the temperature dependence
of theFf. We expect this approximation to be valid in
the large-Ulimit especially close to the transition. The
mean- eld equation w.r.t zis
zZD
Dd!f(!)Re1
! iK
+U
4t2zp
1 z2= 0 (49)
Close to the transition, the second term is e ectively like
az=J withJ(z)(4t2=U)p
1 z2. At zero tempera-
ture the left-side simpli es
zlogK
D+z
Jp
1 z2=zlogK
TK(z)= 0; (50)
whereTK(z) =De 1=J(z). So to have non-zero zwe
must have K=TKwhich determines z. Also, we can
go to non-zero temperature. We just replace the log-term
in above expression with its nite-temperature expression
from Eq. (49)
zReh
~ (iK) ~ (D)i
+z
J(z)= 0 (51)
This is solved numerically and the result shown in
Fig. (7). It shows a Kondo phase z>0 forT <T0
K.
FIG. 7: (color online) The order parameter zfor the
Anderson model calculated from numerical evaluation of
Eq.(53).T0
K=De 1=J, whereJ= 4t2=U. Note that this is
o with a factor of 4, an artifact of slave-spin method. We
have usedD= 100Uwhereast=Uis varied.11
D. Stability of OSMT against interorbital tunnelling
in a Bethe lattice
Using equations of motion, the coecient J de ned
Eq. (4) can be related to the correlation function of the
electrons at the same site, The result is
J =t X
[~t 1] Zd!
2f(!)!A
ii(!) (52)
This together with a = 2P
J z leads to Eq. (27).
In a Bethe lattice we can use recursive methods [53] to
computeA
ii. When the tunnelling matrix is hermitian
and there is no chemical potential or crystal eld, the
procedure is especially simple. We diagonalize the renor-
malized tunnelling matrix ~t=U~tDU 1. Then the re-
tarded and the spectral functions are
GR(!) =~t 1U(!)U 1;A(!) =UAD(!)U 1(53)
where diagonal matirix contains-elements that sat-
isfyi+ 1
i=!=ti
Dwith the retarded boundary condi-
tion.ADis diagonal matrix of semicircular density states
whose width are given by the eigenvalues of ~t. By plug-
ging this into Eq. (53) and (52) and using
Zd!
2f(!)!AD
ii(!) = 0:21222 ~tD
we see that if the eigenvalues of the matrix ~tall have
the same sign, then U(j~tDj=~tD)U 1=~t. This is the
generalization of the protection of OSM phase against
inter-orbital tunnelling, discussed in the 1D case in the
paper. For the case of two bands,
det~t>0) J = 0:21222t ;
det~t<0) J = 0:21222t R (54)
i.e. for det ~t>0, theJ-matrix does not have any o -
diagonal elements and the diagonal elements are propor-
tional to the bare diagonal hoppings (as before), but but
if det ~t<0, there is a matrix R=U zU 1multiplying
element-by-elements of the J-matrix which does depend
on renormalization.
Again in this problem, one could have done the ro-
tation ind -sector before using the slave-spins, in which
case, inter-orbital tunnelling would have an e ect and
could cause OSM transition. Therefore, the stability
found above is basis-dependent. This ambiguity is ab-
sent whenp hsymmetry is broken and the tunnelling
matrix cannot be diagonalized independent of the mo-
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1705.05826v3.Theory_of_electron_spin_resonance_in_one_dimensional_topological_insulators_with_spin_orbit_couplings.pdf | Theory of electron spin resonance in one-dimensional topological insulators with
spin-orbit couplings: Detection of edge states
Yuan Yao,1,Masahiro Sato,2, 3Tetsuya Nakamura,4Nobuo Furukawa,4and Masaki Oshikawa1
1Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
2Department of Physics, Ibaraki University, Mito, Ibaraki 310-8512, Japan
3Spin Quantum Recti cation Project, ERATO, Japan Science and Technology Agency, Sendai 980-8577, Japan
4Department of Physics and Mathematics, Aoyama-Gakuin University, Sagamihara, Kanagawa 229-8558, Japan
Edge/surface states often appear in a topologically nontrivial phase when the system has a bound-
ary. The edge state of a one-dimensional topological insulator is one of the simplest examples.
Electron spin resonance (ESR) is an ideal probe to detect and analyze the edge state for its high
sensitivity and precision. We consider ESR of the edge state of a generalized Su-Schrie er-Heeger
model with a next-nearest neighbor (NNN) hopping and a staggered spin-orbit coupling. The
spin-orbit coupling is generally expected to bring about nontrivial changes on the ESR spectrum.
Nevertheless, in the absence of the NNN hoppings, we nd that the ESR spectrum is una ected by
the spin-orbit coupling thanks to the chiral symmetry. In the presence of both the NNN hopping and
the spin-orbit coupling, on the other hand, the edge ESR spectrum exhibits a nontrivial frequency
shift. We derive an explicit analytical formula for the ESR shift in the second-order perturbation
theory, which agrees very well with a non-perturbative numerical calculation.
I. INTRODUCTION
In recent decades, topological phases have become a
central issue in condensed matter physics. An important
class of topological phases is topological insulators and
topological superconductors1{4.
In condensed matter and statistical physics, one-
dimensional (1-D) systems, which are amenable to
several powerful analytical and numerical methods, of-
ten provide useful insights. 1-D topological phases are no
exceptions. One of the simplest 1-D models possessing
nontrivial topological nature is the Su-Schrie er-Heeger
(SSH) model5, which has been used to describe the lattice
structure of polyacetylene [C 2H2]n. The SSH model can
be also applied to the 1-D charge density wave systems,
such as quasi-one-dimensional conductors like TTF-
TCNQ (tetrathiofulvalinium-tetracyanoquinodime-
thanide) and KCP (potassium-tetracyanoplatinate)6.
While the SSH model had been studied intensively
much earlier than the notion of topological phases
was conceived, there is a renewed interest from the
viewpoint of topology. In fact, distinct phases of the
SSH model are classi ed by the Zak phase7which is
a topological invariant, and the bulk winding number
of the momentum-space Hamiltonian8. In this sense,
the SSH model can be regarded as a 1-D topological
insulator.
An important nontrivial signature of many topologi-
cal phases is edge states. The SSH model indeed pos-
sesses zero-energy edge states that are protected by a
chiral symmetry8. The number of edge states at a do-
main wall is equal to the bulk winding number. This is
known as the bulk-boundary correspondence in the spin-
less inversion-symmetric SSH model8. Experimentally,
1-D systems with boundaries or edges can be realized by
adding impurities to the material so that the system is
broken to many nite chains. However, the edge statesare often experimentally dicult to observe, since they
are localized near the boundaries or the impurities and
their contribution to bulk physical quantities is small.
Given this challenge, electron spin resonance (ESR) pro-
vides one of the best methods to probe the edge states,
thanks to its high sensitivity. In fact, the edge states
of theS= 1 Haldane chain were created by doping im-
purities and then successfully detected by ESR9,10. Fur-
thermore, combined with near-edge x-ray absorption ne-
structure experiments, ESR was applied successfully to
probe the magnetic edge state in a graphene nanoribbon
sample11,12. Such a strategy could also be applied to 1-
D topological insulators, which are described by the SSH
model.
Another intriguing nature of ESR is that it is highly
sensitive to magnetic anisotropies, such as the anisotropic
exchange interaction, single-spin anisotropy, and the
Dzyaloshinskii-Moriya (DM) interaction. The e ect of
magnetic anisotropies on ESR is well understood only
in limited circumstances, and there remain many open
issues13,14. These magnetic anisotropies are often con-
sequences of spin-orbit (SO) coupling which generally
breaks spin-rotation symmetry. The e ects of magnetic
anisotropies and SO couplings also play important roles
in magnetic dynamics in higher-dimensional topological
phases1{4,15,16. Thus it is of great interest to study
the e ect of SO coupling on ESR directly. However, this
question has not been explored in much detail so far.
An obstacle for the potential experimental ESR study of
SO coupling is the electromagnetic screening in metallic
systems. This problem does not exist in insulators. Un-
fortunately, band insulators are generally non-magnetic
and we cannot expect interesting ESR properties. On
the other hand, Mott insulators can have interesting
magnetic properties. However, strong correlation e ects,
which are essential in Mott insulators, make theoretical
analysis dicult.
In this context, the 1-D topological insulator providesarXiv:1705.05826v3 [cond-mat.str-el] 18 Nov 20172
a unique opportunity to study the e ects of SO coupling
on ESR. This would be of signi cant interest in several
aspects. Experimentally, the insulating nature makes
the observation of edge states by ESR easier. Theoret-
ically, the interesting e ects of anisotropic SO coupling
on ESR can be studied accurately for the SSH model
of non-interacting electrons. Moreover, the chiral sym-
metry, which is essential for the well-de ned topological
insulator phase, is often broken explicitly in realistic sys-
tems. When we introduce a chiral-symmetry breaking
perturbation to the 1-D SSH model, the energy eigen-
values of the edge states generally deviate from zero en-
ergy. However, the edge states are expected to still sur-
vive and be localized near the edge if the perturbation is
small enough. As we will demonstrate, the ESR of the
edge state can detect the breaking of the chiral symme-
try. The purpose of this paper is to present a theoretical
analysis on ESR of the edge states in 1-D topological
insulators, based on a generalized SSH model with SO
couplings. We demonstrate several interesting aspects of
ESR, which will hopefully stimulate corresponding ex-
perimental studies.
The paper is organized as follows. In Sec. II, we present
the model of interest and review the basic topological na-
ture of the SSH model. The next three sections are the
main part of this paper. The properties of edge states
are discussed in detail in Sec. III. In Sec. IV, we obtain
a compact analytical formula of the ESR frequency shift
in perturbation theory with respect to SO coupling. Sec-
tion V is devoted to a direct numerical calculation of the
ESR frequency shift, which is independent of the pertur-
bative approach in Sec. IV. We nd that the perturba-
tion theory agrees with the numerical results very well.
Finally, we present conclusions and future problems in
Sec. VI.
II. THE MODEL
A. A generalized SSH model
First let us consider a generalized SSH model with SO
coupling
H0= +1X
j=1
t
1 + ( 1)j0
cy
j+1exp
i( 1)j
2~ n~
cj
+h.c.g (1)
wherecjis the two-component electron annihilation op-
eratorcj[cj";cj#]Tat thej-th site,t > 0 is the
nearest-neighbor (NN) electron hopping amplitude, and
101 is the bond-alternation parameter. The
angle and axis~ n(which is a unit vector) parametrizes
the SO coupling on the NN bond. The angle denotes
the ratio of the SO coupling to the hopping amplitude on
the bond. In this paper, we assume that is suciently
small (j j 1), which is the case in many real materials.Expanding to rst order, we obtain a standard form
with so-called intrinsic and Rashba SO couplings17.
In our model Eq. (1), SO coupling is assumed to be
staggered along the chain. This would be required, in
the limit of 0= 0, if the system had site-centered inver-
sion symmetry. In general, other forms of SO coupling
including the uniform one along the chain are also pos-
sible. In this paper, however, we focus on the particular
case of the staggered SO coupling to demonstrate its in-
teresting e ects on the ESR spectrum.
B. The SSH model and its topological properties
In the limit = 0, our model is reduced to the standard
SSH model
HSSH= Xn
t
1 + ( 1)j0
cy
j+1cj+ h.c.o
:(2)
Let us rst consider a system of 2 Nsites (Nunit cells)
with the periodic boundary condition (PBC). It is then
natural to take the momentum representation
c2j;=1p
NX
kak;exp (ik(2j)); (3)
c2j+1;=1p
NX
kbk;exp (ik(2j+ 1)); (4)
where the summation of kis in the reduced Brillouin zone
[0;) withk=n=N andn= 0;;N 1. Correspond-
ing to the each sublattice (even and odd), there are two
avors of fermions, aandb. The Hamiltonian can then
be written as
HSSH=X
k(ay
k;by
k)hSSH(k)
ak
bk
; (5)
with
hSSH(k)dx(k) x+dy(k) y; (6)
where x;y;zare Pauli matrices acting on the
avor space,
anddx;y(k) are real numbers
dx(k) = 2tcos(k);dy(k) = 2tsin(k)0: (7)
The spin indices are again suppressed in the Hamiltonian.
From this expression, the single-electron energy reads
(k) =q
dx(k)2+dy(k)2=2tq
cos2k+02sin2k:
(8)
The gap is closed at k==2 when0= 0, while the sys-
tem has a gap 4 tj0jwhenever the bond alternation does
not vanish ( 06= 0). The gapless point can be regarded
as a quantum critical point separating the two gapped
phases,0<0 and0>0. Throughout this paper, we
consider the half- lled case with 2 Nelectrons. The bulk
mode near this gap-closing point has a linear dispersion
relation indicated in Fig. 1, and it can be described by a
one-dimensional Dirac fermion3.3
FIG. 1. Band structure of the SSH model in Eq. (2) with
a periodic boundary condition.1The solid and dashed lines
respectively represent band structures of the insulating case
with06= 0 and the gapless point at 0= 0. The low-energy
physics around k==2 can be described by one-dimensional
Dirac fermion model.
It is evident from the Hamiltonian that each of the
gapped phases is simply a dimerized phase. Neverthe-
less, we can identify them as a trivial insulator phase
and a \topological insulator" phase. This can be under-
stood by considering the system with the open boundary
condition. Let us consider the chain of 2 Nsites labeled
withj= 1;2;:::; 2N, and the open ends at sites j= 1
and 2N. For0>0 (0<0), sitesj= 2nandj= 2n+1
(j= 2n 1 andj= 2n) form a dimerized pair, re-
spectively. As a consequence, for 0>0 the end sites
j= 1 andj= 2Nremain unpaired. The electrons at
these unpaired sites give rise to S= 1=2 edge states.
In contrast, for 0<0, there are no unpaired sites and
thus no edge states. In this sense, 0>0 is a topo-
logical insulator phase and 0<0 is a trivial insulator
phase. Of course, considering the equivalence of the two
phases in the bulk, such a distinction involves an arbi-
trariness. That is, if we consider the an open chain of
Nsites withj= 0;1;:::; 2N 1, the edge states appear
only for0<0. It is still useful to identify the gapless
point at0= 0 as a quantum critical point separating
the topological insulator and the trivial insulator phase.
The particular shape of the Hamiltonian also implies
the existence of a chiral symmetry:
fhSSH; g= 0; (9)
where z. The chiral symmetry turns out to be im-
portant for the distinction of the two phases. In the con-
text of the general classi cation of topological insulators,
the present system corresponds to the \AIII" class with
particle number conservation and the chiral symmetry in
one spatial dimension18,19.
In a general one dimensional free fermion system, we
can de ne a topological invariant called the Zak phase7
for each band as follows:
Zak=iI
BZh (k)jOkj (k)i; (10)
FIG. 2. Amplitude j jof the edge-state wave function of the
SSH model in Eq. (2) under an open boundary condition.8
Blue and red colors respectively represent the spatial distri-
bution of the existing probability for left and right localized
edge states. The total site number is set to be even, the
dimerization parameter 0>0, and symbols vandwdenote
hopping amplitudes t(1 0) andt(1 +0), respectively. The
wave-function amplitude decays exponentially into the bulk4.
wherej (k)iis the Bloch wavefunction of the band with
the momentum k. In the presence of the chiral symmetry,
Zakis quantized to integral multiples of , if the band
is separated from others by gaps20.
For the present two-band SSH model in Eq. (5) with
the chiral symmetry, we can compute the Zak phase using
the explicit Bloch wavefunction. For the lower band,
j (k)i=1p
2
exp ( i k)
1
; (11)
with karctan[dy(k)=dx(k)]. As a result, we nd
Zak== 1 for 0 < 01 in which the system is a
topological insulator. In the other case 10<0,
where the system is a trivial insulator,
Zak== 0. In
this case, we can see that the Zak phase can be also iden-
tifed20with a winding number of the Hamiltonian as
Zak
=i
I
BZdkOkln [dx(k) idy(k)]: (12)
When
Zak== 1, there is an edge state localized at each
end of an open nite chain as shown in Fig. 2. This is
the bulk-boundary correspondence21in the SSH model.
In general, this topological number can take arbitrary
integer values, corresponding to Zclassi cation of BDI
or AIII class in d= 1 dimension. However, in the present
SSH model, its value is restricted to 0 or 1.
The existence of the edge states in the SSH model can
be demonstrated by an explicit calculation for a nite-
size chain. In Fig. 3, we can see that, when 0decreases
from 1 to -1, the edge states merge into the bulk spectrum
as0!0+. In addition, when the thermodynamic limit,
N!+1, is taken in the open-end SSH model, the edge
states are strictly at zero energy and topologically stable
against any local adiabatic deformation that respects the
chiral symmetry8.4
−1 −0.5 0 0.5 1−2−1012
δ0/tEnergy spectrum EEnergy spectrum with di fferentδ0
FIG. 3.0dependence of the energy spectrum of the SSH
model with t= 1 andN= 40 under an open boundary con-
dition. As 0!0+, two localized edge states at the ends
merge into the bulk. For the limit N!+1, the edge states
are strictly at zero energy as 0 < twhich are protected
by the chiral symmetry.8
C. ESR of edge states
Let us consider ESR of the 1-d half- lled topological in-
sulator phases at the low-temperature and low-frequency
limitj!j;T j0jton which we focus in this paper. The
ESR contribution from bulk excitations is negligible in
this limit since there is a large bond-alternation driven
band gap 4j0jt. On the other hand, spin-1/2 edge states
are located at the (nearly) zero energy point in the band-
gap regime. Therefore, ESR is dominated by the edge
state contribution.
When the chiral symmetry is preserved and SO cou-
pling is absent, the edge spin is precisely equivalent to a
freeS= 1=2. In this case, the edge ESR spectrum is triv-
ial, which means that it just consists of the delta function
at the Zeeman energy. However, breaking of the chiral
symmetry and introduction of SO couplings can bring
a nontrivial change on the edge ESR spectrum. In the
following, we shall analyze this e ect theoretically.
In ESR, absorption of an incoming electromagnetic
wave is measured under a static magnetic eld. Thus
we introduce the Zeeman term for the static, uniform
magnetic eld
HZ= H
2X
j=1cy
j(~ ~ nH)cj; (13)
where~ nHis a unit vector representing the direction of
the magnetic eld, H > 0 is its magnitude, and ~ =
(x;y;z).
In the paramagnetic resonance of independent electron
spins, absorption occurs for the oscillating magnetic eld
perpendicular to the static magnetic eld, which is mea-sured in the standard Faraday con guration. Therefore,
in this paper, we assume that the oscillating magnetic
eld is perpendicular to the static magnetic eld ~ nH. The
frequency of the electromagnetic wave is denoted by !.
In an electron system with the SO interaction, the
electric current operator contains a \SO current" that
involves the spin operator. Since the electric current
couples to the oscillating electric eld, in the actual
setting of the ESR experiment, the optical conductiv-
ity due to the SO current also contributes to the ab-
sorption of the electromagnetic wave with a spin
ip.
This e ect is called Electron Dipole Spin Resonance
(EDSR)22{25. The EDSR contribution is generically
larger than ESR if SO coupling is at the same order
as the Zeeman splitting26,27, as their relative contribu-
tions are of ( a=C)2106, wherea10 10m is the
lattice spacing and C10 13m the Compton length
of the electron25. In general, EDSR requires a sepa-
rate consideration from ESR as they involve di erent
operators25. Nevertheless, in the low-temperature/low-
frequency regime, only the two spin states of the edge
state are involved. Thus, although EDSR contributes to
the absorption intensity di erently from ESR, the reso-
nance frequency is identical between the ESR and EDSR.
With this in mind, we do not consider EDSR explicitly
in the rest of the paper. It should be noted that, for a
higher temperature or a higher frequency, the EDSR con-
tribution to the absorption spectrum is rather di erent
from the ESR one, as the absorption spectrum involves
bulk excitations.
We now consider the ESR in the system with the
Hamiltonian H0+Hz. Within the linear response the-
ory28, the ESR spectrum is generally given by the dynam-
ical susceptibility function in the limit of zero-momentum
transfer
00
+ (q= 0;!> 0) = ImGR
+ (q= 0;!);(14)
where
GR
+ (0;!) = iZ1
0dtX
r;r0h[s+(r;t);s (r0;0)]iei!t
= iZ+1
0dth[S+(t);S (0)]iei!t; (15)
whereSmeans the ladder operator de ned with respect
to the direction ~ nHof the static eld, hi denotes the
quantum and ensemble average at the given temperature
T, and
S(t)X
rexp(iHt)s(r) exp( iHt) =X
rs(r;t);
(16)
whereHis the static Hamiltonian we consider (e.g.,
H=H0+Hz).
In the absence of SO coupling ( = 0),H0has the
exact SU(2) spin rotation symmetry which is broken
only \weakly"14by the Zeeman term HZ. As a gen-
eral principle of ESR, in this case, the ESR spectrum (if5
any) remains paramagnetic, namely a single -function
at!=H. As we will demonstrate later in Sec. III, in
the low-temperature limit, this paramagnetic ESR can
be attributed to the edge states of the SSH model. Once
the SO coupling is introduced ( 6= 0), the SU(2) sym-
metry is broken and we would expect a nontrivial ESR
lineshape. However, somewhat surprisingly, (as we will
show later), the ESR spectrum attributed to the edge
states remains a -function at !=Heven when 6= 0.
Thus, in order to investigate possible nontrivial e ects
of the SO coupling on ESR, we further consider the NNN
hoppings
H=X
j=1tcy
j+2exp[i
~ n
=2]cj+ h.c.; (17)
where tis the NNN electron hopping amplitude. The
angle
and~ n
are the SO turn angle and the axis for
the NNN hopping, respectively.
The Hamiltonian of the system to be considered is then
HESR=H0+HZ+ H: (18)
Once we include the NNN hoppings, the chiral symme-
try is broken and the edge states are not protected to be
at zero energy. Nevertheless, when the chiral symmetry
breaking perturbations are weak, we may still identify
\edge states" localized near the ends although they are
no longer at exact zero energy even when H= 0. Un-
der a magnetic eld H, contributions from these edge
states dominate the ESR in the low-energy limit. Now,
the ESR spectrum can be nontrivially modi ed by the
SO couplings and
. It is the main purpose of the
present paper to elucidate this e ect. In real materials,
the NNN hoppings might be small but they are generally
non-vanishing. Thus it is important to develop a theory
of ESR in the presence of the NNN hoppings, especially
because we can detect the NNN hoppings with ESR even
when they are small.
We will treat the NNN hopping H, which would be
smaller than the NN hopping H0in many experimental
realizations, as a perturbation. This also turns out to
be convenient for our theoretical analysis. We also as-
sume thatj j;j
j 1 since SO couplings are weak in
most of the realistic systems, and they will formulate a
perturbation expansion in t, , and
.
III. EDGE STATES OF H0ANDU(1)~S~ nHSYMMETRY
As we discussed earlier, ESR would be an ideal probe
to detect the edge state of the 1D topological insulator
and various perturbations. In this section, we discuss
and explicitly solve the edge states of the unperturbed
Hamiltonian H0to demonstrate the robustness of the
edge states against SO coupling. As a consequence, in the
modelH0only with NN hoppings, there is no nontrivial
change in the edge ESR spectrum.Since our Hamiltonian H0is bilinear in fermion op-
erators, we can focus on single-electron states and rep-
resent them with the ket notation. For a half-in nite
chain with sites j= 1;2;:::, wherej= 1 corresponds to
the end of the chain, we nd a single-electron eigenstate
jEdge;ilocalized near the edge in the \topological in-
sulator" phase 0>0. Here=1 represents the spin
component in the direction of the magnetic eld. Namely,
(H0+HZ)jEdge,i=E(0)
jEdge,i (19)
~S~ nHjEdge,i=
2jEdge,i; (20)
with the energy eigenvalues E(0)
= H=2 and~Sis the
total spin of the system. The wave function of the edge
states is exactly calculated as
hj;jEdge,0i=8
<
: 0p
N
1 0
1+0(j 1)=2
(j22N0+ 1 );
0 (otherwise) ;
(21)
where 2 N0is the set of non-negative even integers, and
Nis the normalization constant. The energy eigenvalue
of the edge state for H0is, independently of the spin
component, exactly zero, re
ecting its topological nature.
It is also remarkable that the edge state wavefunction
is independent of and~ n. This is a consequence of a
canonical \gauge transformation"29
~c2k+1=c2k+1;
~c2k= exp (i ~ n~ =2)c2k;(22)
which eliminates the SO coupling from H0. In this sense,
H0still has a hidden SU(2) symmetry30,31even though
the SO coupling breaks the apparent spin SU(2) sym-
metry. However, since the gauge transformation involves
the local rotation of spins, the uniform magnetic eld
HZgives rise to a staggered eld after the gauge trans-
formation. This staggered eld completely breaks the
SU(2) symmetry. This is similar to the situation in a
spin chain with a staggered Dzyaloshinskii-Moriya inter-
action13. Thus, following the general principle of ESR,
we would expect a nontrivial ESR spectrum in the pres-
ence of the staggered SO coupling as in H0.
Nevertheless, somewhat unexpectedly, we nd that the
edge ESR spectrum for the model H0remains the -
function(! H), as if there is no anisotropy at all.
This is due to the fact that the edge-state wavefunction
Eq. (21) is non-vanishing only on the even sites. Since
the gauge transformation can be de ned so that it only
acts on the even sites where the edge-state wavefunction
Eq. (21) vanishes, the edge state is completely insensi-
tive to the SO coupling. Therefore, the spectral shape of
the edge ESR remains unchanged by the staggered SO
coupling. In addition, since the edge wave functions are
eigenstates of the total spin component along ~ nHaccord-
ing to Eq. (20), the edge states have U(1)~S~ nHsymmetry
generated by ~S~ nH.
In fact, the robustness of the edge ESR spectrum is
valid for a wider class of models. The edge ESR only6
probes a transition between two states with opposite po-
larization of the spin, which form a Kramers pair in the
absence of the magnetic eld. Thus, at zero magnetic
eld, the time-reversal invariance of the model requires
these two states to be exactly degenerate. For a nite
magnetic eld, if the system still has U(1)~S~ nHsymme-
try of rotation about the magnetic eld axis, the two
states can be labelled by the eigenvalues of Sz=1=2,
and their energy splitting is exactly
!ESR=H: (23)
Thus, as far as the edge ESR involving only the Kramers
pair is concerned, the U(1)~S~ nHsymmetry is sucient to
protect the -function peak at !=H. We note that,
more generally, when more than two states contribute to
ESR, these symmetries are not sucient to protect the
single-peak ESR spectrum, as there can be transitions
between states not related by time reversal. In fact, this
would be the case for the absorption due to bulk excita-
tions which we do not discuss in this paper.
IV. PERTURBATION THEORY OF THE EDGE
ESR FREQUENCY SHIFT
As we have shown in the previous Section, even in the
presence of SO coupling, there is no frequency shift for
edge ESR in the NN hopping model H0. This is a con-
sequence of the chiral symmetry, which stems from the
bipartite nature of the NN hopping model.
As we will discuss below, the introduction of NNN hop-
pings breaks the chiral symmetry and generally causes a
nontrivial frequency shift of the edge ESR. In this Sec-
tion, we develop a perturbation theory of ESR for the
edge states, rst by regarding H0+HZas the unper-
turbed Hamiltonian and Has a perturbation.
In the presence of the perturbation with SO couplings,
the eigenstate of the Hamiltonian is generally no longer
an eigenstate of the spin component ~S~ nHas in Eq. (20).
Nevertheless, as long as the perturbation theory is valid,
two edge states can still be identi ed by using spin =
+ and . Namely, for the full Hamiltonian (18), we
can de ne the edge state labeled by =as the state
adiabatically connected to jEdge;ias H!0.
Here let us introduce new symbols E+( )andE(n)
+( )as
the energy eigenvalue of the almost spin up (down) edge
state in Eq. (18) and its n-th order correction in the
perturbation theory, respectively. With these symbols,
the ESR frequency is given by
!ESR=E E+=H+ !; (24)
where the ESR trivial peak position is given by E(0)
E(0)
+=H. The ESR frequency shift !, driven by the
perturbation H, is expanded in the perturbation the-
ory as
!= !(1)+ !(2)+:::; (25)where then-th order term is
!(n)=E(n)
E(n)
+; (26)
forn2N. In the following parts, we perturbatively solve
the single-electron problem to compute the eigen-energy
di erence of two edge states E+andE .
A. First order in t
The NNN hopping Hbreaks the chiral symmetry,
and thus it can change the edge ESR spectrum. In fact,
the energy of the edge states is already shifted in the rst
order of H. However, the energy shift is the same for
the two edge states with di erent spin polarizations. This
is a consequence of the time-reversal (TR) symmetry of
the SO coupling, as demonstrated below:
E(1)
=hEdge,jHjEdge,i
=h (Edge, )jH 1j (Edge, )i
=E(1)
= 2t1 0
1 +0cos
2
(27)
where is the TR operator and H 1= His
used. Therefore, the edge ESR spectrum remains un-
changed in the rst order of tas the frequency shift
vanishes in this order:
!(1)=E(1)
E(1)
+= 0: (28)
B. Second order in t
We can formally write down the second-order pertur-
bation correction
E(2)
=hEdge,jH(E(0)
H0) 1PHjEdge,i
where the projection operator P 1
jEdge,ihEdge,j. It is rather dicult to evaluate
this formula directly, since the intermediate states in the
perturbation term include the bulk eigenstates of H0in
which the hidden symmetry is generally broken. Assum-
ing thatj j;j
j 1 (i.e., the SO coupling is suciently
small), we can develop a perturbative expansion in
and
in addition to t. In this framework, we expand
Eas a Taylor series of ,
and t. The quantity
of interest is the energy splitting E E+, since it
corresponds to the ESR frequency.
The edge state energy splitting in the second order in
t, and up to the second order in ,
is required to take
the form
E(2)
E(2)
1
2Ht2
a(~ n
~ nH)2
2+b(~ n~ nH)2 2
+c(~ n~ nH)(~ n
~ nH)
g; (29)
based on the following symmetry considerations, and a,
bandcare constants to be determined.7
First of all, ( E E ) will not change under
(H;~ nH;)!( H; ~ nH; ); (30)
because this corresponds to a trivial rede nition of co-
ordinate system. Therefore, we attach the factor Hin
the r.h.s. of Eq. (29).
Next we notice that always appears with ~ n, and
with~ n
, which leads to further constraints as we will
see below. Let us consider the limiting case = 0 with
nonzero
. The splitting can only depend on the relative
angle between ~ n
and~ nH. Furthermore, if ~ n
k~ nH,
the hidden symmetry of the edge state implies that the
energy splitting is exactly given by the Zeeman energy
and there is no perturbative correction. Thus, for = 0,
the energy splitting can only depend on ( ~ n
~ nH)2
2up
toO(
2) since the energy split is a scalar and it must
be written in terms of inner and vector products of ~ nH,
~ nand~ n
. Then, similarly, if
= 0 with nonzero ,
the energy splitting can only depend on ( ~ n~ nH)2 2.
Finally, the O(
) term should be linear in ~ nand~ n
,
and it vanishes when ~ nk~ n
k~ nHbecause of the U(1)~S~ n
symmetry. These requirements uniquely determine the
form of (~ n~ nH)(~ n
~ nH). Thus, the symmetries
reduce the possible forms of the second-order corrections
to Eq. (29) with only the three parameters a,b, andc.
To obtain these parameters, we note that the expan-
sion in and
introduced above can be naturally done
by regarding the SSH model without SO coupling
~H0= +1X
j=1n
t
1 + ( 1)j0
cy
j+1cj+ h.c.o
HX
j=1cy
j(~ ~ nH)cj=2 (31)
as the unperturbed Hamiltonian, and
Hpert= H0+ H; (32)
as the perturbation, where
H0 +1X
j=1n
t
1 + ( 1)j0
cy
j+1
exp
( 1)ji ~ n~ =2
1
cj+ h.c.
(33)
and Has de ned in Eq. (17).
As the result of the perturbation calculation given in
the Appendix, we nd that the second-order term of fre-
quency shift is non-positive given in the form
!(2)= H
2X
m3;j3(~M+~N)y(~M+~N)0;(34)
where
~MX
m2;j2hm3;j3jH0
0jm2;j2ihm2;j2jH00jEdgei
m2Ej2Ej3
(~ n~ nH)
~Nhm3;j3jH0jEdgei
Ej3(~ n
~ nH)
: (35)Herejm;ji's are single-particle (bulk) energy eigenstates
of~H0in Eq. (31) where m=labels the positive or neg-
ative energy sector (i.e., band indices) and jlabels other
possible quantum numbers, which is not the wave vector
since we have the open-ended boundary condition. The
energyEjstands for the energy eigenvalue of j;ji.
The perturbation terms H0, H00, and H0
0are de-
ned as
H0X
j=1tcy
j+2cj h.c.; (36)
H00X
j=1tcy
j+2cj+ h.c.; (37)
H0
0 +1X
j=1n
t
1 + ( 1)j0
cy
j+1cj h.c.o
:(38)
We note that Eqs. (36) and (38) are anti-Hermitian.
After putting the de nitions of ~Mand~Ninto Eq. (34),
we nd the result consistent with the general form
Eq. (29) required by symmetries. The parameters are
then identi ed as
a=b=t
t(1 +0)2
; c = 2t
t(1 +0)2
:
The detailed derivation of a,bandcis given in Appendix.
The nal result can be given in a compact form as
!(2)= H
2 t
t(1 +0)( ~ n~ nH
~ n
~ nH) 2
;(39)
which is non-positive. Since the rst-order correction
vanishes as we have already seen, the second-order term
Eq. (39) gives the leading term for the frequency shift
of the edge ESR. It also implies that, although
is the
SO-coupling turn angle for the NNN hopping terms, it
is equally important in !as the NN SO coupling turn
angle even if the NNN hopping itself is small ( t t).
One of the most remarkable features of our result is
that, the shift (up to the second order in the perturba-
tion) vanishes when ~ n~ nH=
~ n
~ nH. This corre-
sponds to the zeros of curves in Fig. 4 where the direction
of the magnetic eld is in the plane spanned by ~ n= ^y
and~ n
= ^z, andis the angle between ~ nHand ^yin the
^y-^zplane, as shown in Fig. 5. Therefore, we predict two
directions of the magnetic eld for which the ESR shift
vanishes, when the magnetic eld direction ~ nHsweeps
the plane spanned by ~ nand~ n
.
V. NON-PERTURBATIVE CALCULATION OF
THE EDGE ESR SPECTRUM
Here, in order to see the validity of our perturba-
tion theory in the preceding section, let us re-compute
the edge ESR frequency with a more direct numerical
method. As already mentioned, when the magnetic eld8
FIG. 4. Angle dependence of the ESR frequency shift !.
We set the parameters t= 1:0, t=0= 0:2, andH=
0:05. Red and blue curves are obtained by the second-order
perturbation calculation in Sec. IV B. Square and circle points
are the results of direct numerical diagonalization for a system
(of 100 sites) with an open boundary condition. The de nition
of angleis indicated in Fig. 5. The zeros of !occur at
~ n~ nH=
~ n
~ nH.
H(and thus the ESR frequency !) is much smaller com-
pared to the bulk excitation gap 4 tj0j, we may ignore
the e ects of the bulk excitations in the ESR spectrum.
Then the edge ESR spectrum is of the -function form
00
+ edge(q= 0;!)/[! (E E+)]; (40)
whereE+( )is the energy eigenvalue of the \almost"
spin-up (spin-down) edge state. These energies can be
accurately computed by numerical diagonalization of a -
nite (but long) size full Hamiltonian with an open bound-
ary condition. Then we obtain the spectrum peak shift
!E E+ Hfrom the numerical results of E. In
the present work, we calculated Eusing a nite open
chain of 100 sites.
In Fig. 4, we compare the numerical results of the peak
shift with the analytical perturbation theory of !(2)
ESRin
Eq. (39) when the magnetic eld is in the plane spanned
by~ nand~ n
. The gure clearly shows that our perturba-
tion theory agrees with the numerical results quite well.
VI. CONCLUSIONS AND DISCUSSION
We have analyzed ESR of edge states in a general-
ized SSH model with staggered SO couplings and with
an open end. In this paper, we assume that the en-
ergy scales of the magnetic eld, the frequency, and the
temperature are suciently small compared to the bulk
gap. Then the ESR spectrum only consists of a single
-function spectrum corresponding to the transition be-
tween two spin states at the edge, but is expected to show
a nontrivial frequency shift in general as the SO coupling
FIG. 5. Geometric relation among some vectors in Fig. 4,
where the direction of magnetic eld is in the plane spanned
by~ n= ^yand~ n
= ^z. The parameter is de ned as the
angle between ~ nHand ^yin the ^y-^zplane.
breaks the SU(2) symmetry strongly under the applied
magnetic eld. Nevertheless, there is no ESR frequency
shift in the model with only NN hoppings, thanks to its
chiral symmetry.
The chiral symmetry is broken by NNN hoppings,
which should be generally present in any realistic materi-
als even if they are small. This NNN hoppings, together
with the SO coupling, can induce a nontrivial frequency
shift on the edge ESR. Thus we have developed a pertur-
bation theory of the frequency shift, regarding the NNN
hoppings and the SO couplings as perturbations. Our
main result, the ESR frequency shift up to second order
in the perturbation theory, is found in Eq. (29). It is
non-positive in this order. (The resonance eld shift for
a xed frequency, which is usually measured in experi-
ments, is always positive.) In the presence of the NNN
hoppings, the SO couplings in the NN hoppings, which
did not cause a frequency shift by themselves, also con-
tribute to the frequency shift. We nd an interesting
dependence of the ESR frequency shift on the direction
of the static magnetic eld, relative to the SO couplings
on the NN and the NNN hoppings. In particular, the
ESR frequency shift is predicted to vanish when the static
magnetic eld points to a certain direction on the plane
spanned by the two SO coupling axes (see Fig. 4). Fur-
thermore, we performed a direct estimate of the ESR
frequency shift by a numerical calculation of the edge
state spectrum, without relying on the perturbation the-
ory. The result agrees very well with the perturbation
theory, establishing its validity.
Our results indicate that, the chiral symmetry break-
ing by the NNN hoppings in the \SSH"-type topological
insulators in one dimension may be detected by ESR, in9
the presence of the SO couplings. If the NNN hoppings
are small, which would be the case in many realistic ma-
terials, it might be dicult to detect their e ects with
other experimental techniques. ESR has been success-
ful in detecting even very small magnetic anisotropies,
thanks to its high sensitivity and accuracy. We hope that
the present work will pave the way for a new application
of ESR in detecting (small) chiral symmetry breaking.
While we do not discuss any particular material in this
paper, let us discuss here the prospect of experimentally
observing the e ects we predict. The maximal frequency
shift is given by the order of
Ht
tmaxf ;
g2
: (41)
The ratio of NNN to NN hoppings, t=t, of course
strongly depends on each material. It is even possible
thatjtj=t 1, in which case the system may be re-
garded as two chains coupled weakly by zigzag hopping.
It should be however noted that our theory is valid only
whenjtj=tis suciently small. We still expect that
our theory works reasonably well for ( t=t)20:1. In
carbon-based systems, such as polyacetylene, the SO in-
teraction is known to be weak. For example, even with
the enhanced SO interaction due to a curvature32, ;
is of order of 10 5. This would give an ESR shift that
is too small to be observed in experiments. However,
the SO interaction is stronger in heavier atoms. In fact,
even in carbon-based systems, the SO interaction can be
signi cantly enhanced by heavy adatoms. For example,
placing Pb as adatoms can enhance ;
up to 0:1 or more
in graphene33. This would give the edge ESR shift cor-
responding to the g-shift up to the order of 10 3(1,000
ppm), which should be observable. In particular, even
if the absolute value of the shift is dicult to be deter-
mined, the angular dependence of the ESR shift would be
more evident in experiments. Furthermore, it is known
that an SO interaction generally becomes larger when
the electron system we consider is located in the vicinity
of an interface between two bulk systems or is under a
strong, static electric eld34{36. Therefore, if we set up an
SSH chain system under such an environment, it would
become easier to detect an ESR frequency shift due to a
strong SO coupling.
Throughout this paper, we have taken the NN SO cou-
pling in the model Eq. (1) to be staggered. However,
there are other possibilities. In particular, in a transla-
tionally symmetric system, the NN SO coupling is uni-
form. Although the only di erence is the signs in the
Hamiltonian, ESR spectra should be signi cantly di er-
ent between these two cases. This is clear if we consider
the limit of the zero NNN hopping. In the staggered SO
coupling case, there is no ESR frequency shift. This is
because the NN SO coupling can be gauged out by the
canonical transformation Eq. (22), without changing the
odd site amplitude and thus leaving the SSH edge state
wavefunction Eq. (21) unchanged. On the other hand,
in the uniform SO coupling case, gauging out the NNSO coupling a ects any wavefunction including the SSH
edge state wavefunction, resulting in the change of the
ESR spectrum. A similar di erence has been recognized
between the ESR spectum in the presence of a staggered
DM interaction13and that with a uniform DM interac-
tion along the chain29. The analysis of the edge ESR
spectrum in the presence of a uniform SO coupling is left
for future studies.
ACKNOWLEDGMENTS
M. S. was supported by Grant-in-Aid for Scienti c
Research on Innovative Area, \Nano Spin Conversion
Science" (Grant No.17H05174), and JSPS KAKENHI
Grants (No. 17K05513 and No. 15H02117), and M. O.
by KAKENHI Grant No. 15H02113. This work was also
supported in part by U.S. National Science Foundation
under Grant No. NSF PHY-1125915 through Kavli In-
stitute for Theoretical Physics, UC Santa Barbara where
a part of this work was performed by Y. Y. and M. O.
Appendix: Perturbation theory for the frequency
shift !
Here we explain how to determine the parameters a,b
andcin the general form of Eq. (29), based on a pertur-
bation theory. As we have discussed earlier, in order to
expand the edge-state energy eigenstates with respect to
the SO coupling parameters and
, which are assumed
to be small, we regard ~H0of Eq. (31) as the unperturbed
part, and ~Hpertof Eq. (32) as the perturbation. More-
over, we have already shown in Sections III and IV A that
the ESR shift vanishes exactly up to O(t). Therefore,
the leading terms with coecients a,bandcstem from
the second-order perturbation proportional to t2. Be-
low, we will determine three parameters a,bandcfrom
the perturbative expansion of ~Hpert.
1. Second order in Hpert
The shift of the edge energy eigenvalue in the second
order of Hpertis given by
E[2]
=X
m=;j;~=hEdge,jHpertjm;j; ~ihm;j; ~jHpertjEdge,i
H
2 Em;j;~
X
m=;jhEdgej(H0)yjm;jihm;jjH0jEdgei
E2
jH
jn
nHj2
2=4; (A.1)
where the bulk single-particle eigen-energy Em;j;~is
given by
Em;j;~mEj ~H=2 (A.2)10
withmEjthe energy eigenvalue of spinless single-particle
eigenstatesjm;ji, and we have used the facts that in the
unperturbed sector, the orbital and spin parts of single-
particle eigenstates can be decomposed, e.g. jm;j;i=
jm;jijisince ~H0commutes with the spin operator of
each site. Here we de ne E[n]
as the energy eigenvalue
of the edge state with spin in then-th order of pertur-
bation in Hpert. This is to be distinguished from E(n)
introduced in Eq. (26), where nrefers to the order in t.
In the second line of Eq. (A.1), we assume jEm;j;~j H
and perform the Taylor expansion of E[2]
with respect
toH. The matrixhm;jjH0jEdgeican be calculated
by using the anti-Hermitian operator H0of Eq. (36).
From Eq. (A.1), the ESR frequency shift driven by E(2)
is expressed as
E[2]
E[2]
+= jn
nHj2
2=2
X
m=;jhEdgej(H0)yjm;jihm;jjH0jEdgei
E2
jH:(A.3)
This indeed corresponds to the parameter a. To obtain
bandc, we should proceed to higher orders of Hpert.
2. Third order in Hpert
In the third order perturbation theory within t2, the
energy correction takes the form as
E[3]
=X
m;j;~hEdge,jHpertjm3;j3;~3ihm3;j3;~3jHpertjm2;j2;~2ihm2;j2;~2jHpertjEdge,i
( ~2)H
2 m2Ej2
( ~3)H
2 m3Ej3
+hEdge,jHpertjEdge,ijhEdge,jHpertjm3;j3;~3ij2
( ~3)H
2 m3Ej32
X
m;jRe"
hEdgejH00jm3;j3ihm3;j3jH0
0jm2;j2ihm2;j2jH0jEdgei
m3E2
j2Ej3H#
(~ n~ nH)(~ n
~ nH)
2;(A.4)
where H00is de ned by Eq. (37).
Therefore,
E[3]
E[3]
+
=X
m;j Re"
hEdgejH00jm3;j3ihm3;j3j(H0
0)yjm2;j2i
m3E2
j2Ej3
hm2;j2jH0jEdgeiH] (~ n~ nH)(~ n
~ nH)
:
(A.5)
We see that this correction term corresponds to the pa-
rametercin Eq. (29).
3. Fourth order in Hpert
In order to derive the leading term of the parameter b,
we have to calculate the fourth-order term. For conve-nience of the fourth-order calculation, we introduce the
abbreviated notation of the matrix elements:
Aqrhmr;jr;~rjAjmq;jq;~qi; (A.6)
EEqE[0]
Emq;jq;~q (A.7)
for any operator A, and denote the zeroth order edge
state by \E". In this notation, the fourth order edge-
energy correction up till ( t )2-order is given by
E[4]
=X
m;j;~HE4H4;3
pertH3;2
pertH2E
EE2EE3EE4 E[2]
HE42
(EE4)2 2HEEHE4H43
pertH3E
E2
E3EE4+
HEE2
HE42
(EE3)3
X
m;j;hEdgejH00jm2;j2ihm2;j2j(H0
0)yjm3;j3ihm3;j3jH0
0jm4;j4ihm4;j4jH00jEdgei
m2m4Ej2E2
j3Ej4Hj~ n~ nHj2 2
4:(A.8)11
Then we can arrive at
E[4]
E[4]
+
X
m;j;hEdgejH00jm2;j2ihm2;j2j(H0
0)yjm3;j3ihm3;j3jH0
0jm4;j4ihm4;j4jH00jEdgei
m2m4Ej2E2
j3Ej4Hj~ n~ nHj2 2
2:(A.9)
This correspond to the term coeciented by the param-
eterbin Eq. (29).4. Summation over the perturbation orders
Through some algebra, we nd that terms of the fth
and higher orders in Hpertdo not contribute to the order
ofO
t2 2
. Summing up Eqs. (28), (A.3), (A.5), and
(A.9), we arrive at the energy shift of the energy di er-
ence between down- and up-spin edge states in an elegant
form as in Eq. (34).
smartyao@issp.u-tokyo.ac.jp
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1809.10852v2.Spin_orbit_crossed_susceptibility_in_topological_Dirac_semimetals.pdf | arXiv:1809.10852v2 [cond-mat.mes-hall] 20 Feb 2019Spin-orbit crossed susceptibility in topological Dirac se mimetals
Yuya Ominato1, Shuta Tatsumi1, and Kentaro Nomura1,2
1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan and
2Center for Spintronics Research Network, Tohoku Universit y, Sendai 980-8577,Japan
(Dated: February 21, 2019)
We theoretically study the spin-orbit crossed susceptibil ity of topological Dirac semimetals. Be-
cause of strong spin-orbit coupling, theorbital motion of e lectrons is modulated byZeeman coupling,
which contributes to orbital magnetization. We find that the spin-orbit crossed susceptibility is pro-
portional totheseparation oftheDirac points anditis high lyanisotropic. The orbital magnetization
is induced only along the rotational symmetry axis. We also s tudy the conventional spin susceptibil-
ity. The spin susceptibility exhibits anisotropy and the sp in magnetization is induced only along the
perpendicular to the rotational symmetry axis in contrast t o the spin-orbit crossed susceptibility.
We quantitatively compare the two susceptibilities and find that they can be comparable.
I. INTRODUCTION
In the presence of an external magnetic field, magne-
tization is induced by both the orbital motion and spin
magnetic moment ofelectrons. When spin-orbit coupling
isnegligible, the magnetizationiscomposedoftheorbital
and spin magnetization, which are induced by the mini-
mal substitution, p→p+eA, and the Zeeman coupling,
respectively. Additionally, spin-orbit coupling gives rise
to the spin-orbitcrossedresponse, in whichthe spinmag-
netization is induced by the minimal substitution, and
the orbital magnetization is induced by the Zeeman cou-
pling. In the strongly spin-orbit coupled systems, the
spin-orbit crossed response can give comparable contri-
bution to the conventional spin and orbital magnetic re-
sponses.
Spin-orbit coupling plays a key role to realize a topo-
logical phase of matter, such as topological insulators [1]
and topological semimetals [2]. A natural question aris-
ing is what kind of the spin-orbit crossed response occurs
in the topological materials. Because of the topologically
nontrivial electronic structure and the existence of the
topological surface states, the topological materials ex-
hibit the spin-orbit crossed response as a topological re-
sponse [3–7]. The spin-orbit crossed response has been
investigated in several systems. In the literature the con-
nection between the spin-orbit crossed susceptibility and
the spin Hall conductivity was pointed out [3, 4]. In
recent theoretical work, the spin-orbit crossed response
has been investigated also in Rashba spin-orbit coupled
systems [8, 9].
The topological Dirac semimetal is one of the topo-
logical semimetals [10–15] and experimentally observed
in Na3Bi and Cd 3As2[16–18]. The topological Dirac
semimetals have an inverted band structure originating
from strong spin-orbit coupling. They are characterized
by a pair of Dirac points in the bulk and Fermi arcs on
the surface [10, 11]. The Dirac points are protected by
rotational symmetry along the axis perpendicular to the
(001) surface in the case of Na 3Bi and Cd 3As2[10, 11].
This is an important difference from the Dirac semimet-
als appearing at the phase boundary of topological in-
sulators and ordinary insulators [19–22], in which thereis no Fermi arc. A remarkable feature of the topological
Dirac semimetals is the conservation of the spin angular
momentum along the rotation axis within a low energy
approximation [23]. The topological Dirac semimetals
are regarded as layers of two-dimensional (2D) quantum
spin Hall insulators (QSHI) stacked in momentum space
and exhibit the intrinsic semi-quantized spin Hall effect.
The magnetic responses of the generic Dirac electrons
have been investigated in several theoretical papers. The
orbital susceptibility logarithmically diverges and ex-
hibits strong diamagnetism at the Dirac point [6, 24–26].
When spin-orbit coupling is not negligible, the spin sus-
ceptibility becomes finite even at the Dirac point where
the density of states vanishes [6, 27–29]. This is contrast
to the conventional Pauli paramagnetism and known as
the Van Vleck paramagnetism [29–32].
In this paper, we study the spin-orbit crossed suscep-
tibility of the topological Dirac semimetals. We find that
the spin-orbitcrossedsusceptibility isproportionalto the
separation of the Dirac points and independent of the
other microscopic parameters of the materials. We also
include the spin conservation breaking term which mixes
up and down spins [10, 11]. We confirm that the spin-
orbit crossedsusceptibility is approximatelyproportional
to the separation of the Dirac points even in the absence
of the spin conservation as long as the separation is suf-
ficiently small. We also calculate the spin susceptibility
and quantitatively compare the two susceptibilities. Us-
ing the material parameters for Na 3Bi and Cd 3As2, we
show that the contribution of the spin-orbit crossed sus-
ceptibilityisimportantinordertoappropriatelyestimate
the total susceptibility.
The paper is organized as follows. In Sec. II, we in-
troduce a model Hamiltonian and define the spin-orbit
crossed susceptibility. In Secs. III and IV, we calculate
the spin-orbit crossed susceptibility and the spin suscep-
tibility. In Secs. V and VI, the discussion and conclusion
are given.2
II. MODEL HAMILTONIAN
We consider a model Hamiltonian on the cubic lattice
Hk=HTDS+Hxy+HZeeman, (1)
which is composed of three terms. The first and sec-
ond terms describe the electronic states in the topolog-
ical Dirac semimetals, which reduces to the low energy
effective Hamiltonian around the Γ point [10–12, 14, 15].
The first term is given by
HTDS=εk+τxσztsin(kxa)−τytsin(kya)+τzmk,(2)
where
εk=C0−C1cos(kzc)−C2[cos(kxa)+cos(kya)],
mk=m0+m1cos(kzc)+m2[cos(kxa)+cos(kya)].
(3)
Pauli matrices σandτact on real and pseudo spin (or-
bital) degrees of freedom. aandcare the lattice con-
stants.t,C1, andC2are hopping parameters. C0gives
constant energy shift. m0,m1, andm2are related to
strength of spin-orbit coupling and lead band inversion.
There are Dirac points at (0 ,0,±kD),
kD=1
carccos/parenleftbigg
−m0+2m2
m1/parenrightbigg
. (4)
The separation of the Dirac points is tuned by chang-
ing the parameters, m0,m1, andm2. The first term,
HTDS, commutes with the spin operator σz, andHTDS
is regarded as the Bernevig-Hughes-Zhangmodel [12, 33]
extended to three-dimension. The second term is given
by
Hxy=τxσxγ[cos(kya)−cos(kxa)]sin(kzc)
+τxσyγsin(kxa)sin(kya)sin(kzc),(5)
which mixes up and down spins. When Hxyis expanded
around the Γ point, leading order terms are third or-
der terms, which are related to the rotational symmetry
alongtheaxisperpendiculartothe(001)surfaceinNa 3Bi
and Cd 3As2. In the currentsystem, this axiscorresponds
to thez-axis and we call it the rotational symmetry axis
in the following. γcorresponds to the coefficient of the
third order terms in the effective model [10, 11]. When γ
iszero, the z-componentofspin conserves. At finite γ, on
the otherhand, the z-componentofspin isnot conserved.
Aswementionedintheintroduction,theexternalmag-
netic field enters the Hamiltonian via the minimal substi-
tution,p→p+eA, and the Zeeman coupling. We for-
mally distinguish the magnetic field by the way it enters
the Hamiltonianinordertoextractthespin-orbitcrossed
response. BorbitandBspinrepresent the magnetic field
in the minimal substitution and in the Zeeman coupling
respectively. They are the same quantities so that we
have to set Borbit=Bspinat the end of the calculation.In the following, the subscripts α,β,γ,δ refer tox,y,z.
We define the orbital magnetization Morbit
αand the spin
magnetization Mspin
αas follows
Morbit
α=−1
V∂Ω
∂Borbitα, (6)
Mspin
α=−1
V∂Ω
∂Bspin
α, (7)
where Ω is the thermodynamic potential and Vis the
systemvolume. Thesequantitiesarewritten, up tolinear
order in BorbitandBspin, as
Morbit
α=χorbit
αβBorbit
β+χSO
αβBspin
β, (8)
Mspin
α=χspin
αβBspin
β+χSO
αβBorbit
β, (9)
where
χorbit
αβ=∂Morbit
α
∂Borbit
β, (10)
χspin
αβ=∂Mspin
α
∂Bspin
β, (11)
χSO
αβ=∂Morbit
α
∂Bspin
β=∂Mspin
α
∂Borbit
β. (12)
Spin-orbit coupling can give the spin-orbit crossed sus-
ceptibility χSO
αβ, in addition to the conventional spin and
orbital susceptibilities, χspin
αβandχorbit
αβ[6, 7].
In the rest of the paper, we focus on the Zeeman cou-
pling, which can induce both of the orbital and spin mag-
netization as we see in Eqs. (8) and (9). The Zeeman
coupling is given by
HZeeman=−µB
2/parenleftbigg
gsσ0
0gpσ/parenrightbigg
·Bspin,
=−g+µBτ0σ·Bspin−g−µBτzσ·Bspin,(13)
whereµBis the Bohr magneton and gs,gpcorrespond to
theg-factors of electrons in sandporbitals, respectively.
We define g+= (gs+gp)/4 andg−= (gs−gp)/4, so that
the Zeeman coupling contains two terms, the symmetric
termτ0σand the antisymmetric term τzσ[7, 34, 35].
III. SPIN-ORBIT CROSSED SUSCEPTIBILITY
A. Formulation
The orbital magnetization is calculated by the formula
[36–40],
Morbit
α=e
2/planckover2pi1/summationdisplay
n/integraldisplay
BZd3k
(2π)3fnkǫαβγ
×Im/angbracketleft∂βn,k|(εnk+Hk−2µ)|∂γn,k/angbracketright,(14)
wherefnk=/bracketleftbig
1+e(εnk−µ)/kBT/bracketrightbig−1is the Fermi distribu-
tion function, ∂α=∂
∂kα, and|n,k/angbracketrightis a eigenstate of Hk3
and its eigenenergy is εnk. The derivative of the eigen-
states|∂αn,k/angbracketrightis expanded as [39]
|∂αn,k/angbracketright=cn|n,k/angbracketright+/summationdisplay
m/negationslash=n/angbracketleftm,k|/planckover2pi1vα|n,k/angbracketright
εmk−εnk|m,k/angbracketright,(15)
where the velocity operator vαis given by vα=∂αHk//planckover2pi1
andcnis a pure imaginary number. Using Eq. (15), the
formula, Eq. (14), is written as
Morbit
α=e
2/planckover2pi1/summationdisplay
n/integraldisplay
BZd3k
(2π)3fnkǫαβγ
×Im/summationdisplay
m/negationslash=n/angbracketleftn,k|/planckover2pi1vβ|m,k/angbracketright/angbracketleftm,k|/planckover2pi1vγ|n,k/angbracketright
(εmk−εnk)2(εnk+εmk−2µ).
(16)
Weusetheaboveformulainnumericalcalculation. Using
the 2D orbital magnetization Morbit(2D)
z(kz) at fixed kz,
Morbit
zis expressed as
Morbit
z=/integraldisplayπ/c
−π/cdkz
2πMorbit(2D)
z(kz).(17)
The aboveexpressionis useful when wediscussnumerical
results for χSO
zz. We can relate χSO
αβto the Kubo formula
for the Hall conductivity,
σαβ=e2
/planckover2pi1/summationdisplay
n/integraldisplay
BZd3k
(2π)3fnkǫαβγ
×Im/summationdisplay
m/negationslash=n/angbracketleftn,k|/planckover2pi1vβ|m,k/angbracketright/angbracketleftm,k|/planckover2pi1vγ|n,k/angbracketright
(εmk−εnk)2.(18)
When the density of states at the Fermi level vanishes,
the intrinsic anomalous Hall conductivity is derived by
the Streda formula [3, 4, 41],
σαβ=−eǫαβγ∂Morbit
γ
∂µ,
=−eǫαβγ∂χSO
γδ
∂µBspin
δ. (19)
The topological Dirac semimetals possess time reversal
symmetry, so that the Hall conductivity is zero in the
absence of the magnetic field. On the other hand, in the
presence of the magnetic field, this formula suggests that
the anomalous Hall conductivity at the Dirac point be-
comes finite beside the ordinary Hall conductivity, if χSO
γδ
is not symmetric as a function of the Fermi energy εF.
In the following section, we only consider χSO
αα, because
χSO
αβ(α/negationslash=β) becomes zero from the view point of the
crystalline symmetry in Na 3Bi and Cd 3As2.
B. Numerical results
Numerically differentiating Eq. (16) with respect to
Bspin
α, we obtain χSO
αα. In Sec. III and IV, we omit εkin Eq. (2) for simplicity. This simplification does not
change essential results in the following calculations. In
Sec. V, we incorporate εkin order to compare the spin-
orbit crossed susceptibility and the spin susceptibility
quantitatively in Na 3Bi and Cd 3As2. Figure 1 shows
the spin-orbit crossed susceptibility χSO
zzatεF= 0 as
a function of the separation of the Dirac points kD. In
the present model, there are several parameters, such as
t,a,m 0,and so on. We systematically change them and
find which parameter affect the value of χSO
zz. Figure 1
(a), (b), and (c) show that χSO
zzincreases linearly with
kDand satisfy following relation,
χSO
zz=g+µB2e
hkD
π. (20)
χSO
zzis proportional to the separation of the Dirac points
kDand the coupling constant g+µB.
Eq. (20) is given by numerical calculation. This result
is understood as follows. χSO
zzis obtained as
χSO
zz=/integraldisplayπ/c
−π/cdkz
2πχSO
zz(2D)(kz), (21)
whereχSO
zz(2D)(kz) is the 2D spin-orbit crossed suscepti-
bility at fixed kz, which is defined in the same way as
Eq. (12). χSO
zz(2D)is quantized as 2 g+µBe/hin the 2D-
QSHI and vanishes in the ordinary insulators [4, 7]. The
topological Dirac semimetal is regarded as layers of the
2D-QSHI stacked in the momentum space and the spin
Chern number on the kx-kyplane with fixed kzbecomes
finite only between the Dirac points. As a result, we
obtain Eq. (20). The sign of χSO
zzdepends on the spin
Chern number on the kx-kyplane with fixed kzbetween
theDiracpoints. ThisisanalogoustotheanomalousHall
conductivity in the Weyl semimetals [2, 23, 42]. In Fig. 1
(d),χSO
zzincreases linearly at small kDbut deviates from
Eq. (20) for finite γ. This is because the z-component
of spin is not conserved in the presence of Hxy, Eq. (5),
and the above argument for 2D-QSHI is not applicable
to the present system. In the following calculation, we
setm0=−2m2,m1=m2,m1/t= 1 and c/a= 1.
Figure 2 shows χSO
ααatεF= 0 as a function of γ. At
γ= 0,χSO
zzis finite as we mentioned above. On the other
hand,χSO
xxandχSO
yyare zero. This means that the orbital
magnetization is induced only along z-axis, which is the
rotational symmetry axis. As a function of γ,χSO
zzis an
even function and χSO
xx(yy)is an odd function.
Figure 3 (a) shows χSO
zzaround the Dirac point as a
function of εF. Wheng−/g+= 0,χSO
zzisan evenfunction
around the Dirac point. At εF= 0,χSO
zzis independent
ofg−/g+as we see it in Fig. 1 (b). When g−/g+/negationslash= 0,
however, χSO
zzis asymmetric and the derivative of χSO
zzis
finite. This suggests that the Hall conductivity is finite
wheng−/g+/negationslash= 0. Calculating Eq. (18) numerically, We
confirm that the Hall conductivity is finite at εF= 0.
Figure 3 (b) shows σxyas a function g−/g+.σxylinearly
increases with g−/g+. The topological Dirac semimetal4
0.0 0.5
kD [ π/c]1.00.0 0.5 1.0(a)εF=0
0.0 0.5
kD [ π/c]1.00.0 0.5 1.0(c)
0.0 0.5
kD [ π/c]1.00.0 0.5 1.0(d)0.0 0.5
kD [ π/c]1.00.0 0.5 1.0(b)
m1/t=0.5
1.0
2.0 g-/g+=0.0
0.5
1.0
c/a=1.0
2.0
3.0 γ/t=0.0
0.5
1.0 χzz [2g+μBe/(ha)]SO
χzz [2g+μBe/(ha)]SO χzz [2g+μBe/(ha)]SO
χzz [2g+μBe/(ha)]SO
FIG. 1: The spin-orbit crossed susceptibility χSO
zzatεF= 0
as a function of kD. We set the parameters m1=m2,m1/t=
1,g−/g+= 1,c/a= 1,andγ= 0, if the parameters are
not indicated in each figure. The panels (a), (b), and (c)
show that χSO
zzis proportional to kD, which means that χSO
zz
reflects the topological property of the electronic structu re.
From these numerical results, we obtain analytical express ion
forχSO
zz, Eq. (20), which is independent of model parameters
except for kDandg+. The panel (d) show that Hxyreduces
χSO
zzbut it is negligible for sufficiently small kD.
is viewed as a time reversal pair of the Weyl semimetal
with up and down spin. Therefore, the Hall conductivity
completely cancel with each other. Even in the presence
ofg+Zeeman term (the symmetric term), the cancella-
tion is retained. In the presence of g−Zeeman term (the
antisymmetric term), on the other hand, the cancella-
tion is broken. This is because g−Zeeman term changes
the separation of the Dirac points and the direction of
the change is opposite for the up and down spin Weyl
semimetals. As a result, the Hall conductivity is finite in
g−/g+/negationslash= 0 and given by
σxy=2
πe2
hag−µBBspin
t. (22)
This expression is quantitatively consistent with the nu-
merical result in Fig. 3 (b).γ [t] 1.5
1.0
0.5
0.0
-0.5
0.0 1.0 0.5 -0.5 -1.0χzz
χyy χxx εF=0
SO
SO
SO χ [g+μB(2e/h)k D]SO
FIG. 2: The spin-orbit crossed susceptibility as a function of
γ. The solid black curve is χSO
zz, the blue dashed curve is
χSO
xx, and the red dashed curve is χSO
yy. We set the parameters
m0=−2m2,m1=m2,m1/t= 1,g−/g+= 1, and c/a= 1.
Breaking the conservation of σz, i.e., with the increase of γ,
χSO
zzis reduced, while χSO
xxandχSO
yybecome finite.
IV. SPIN SUSCEPTIBILITY
In this section, we calculate the spin susceptibility us-
ing the Kubo formula,
χspin
αα(q,εF) =1
V/summationdisplay
nmk−fnk+fmk−q
εnk−εmk−q
×µ2
B|/angbracketleftn,k|g+τ0σα+g−τzσα|m,k−q/angbracketright|2,
(23)
whereVis the system volume, fnkis the Fermi distribu-
tion function, εnkis energy of n-th band and |n,k/angbracketrightis a
Bloch state of the unperturbed Hamiltonian. Taking the
long wavelength limit |q| →0, we obtain
lim
|q|→0χspin
αα(q,εF) =χintra
αα(εF)+χinter
αα(εF),(24)
whereχintra
αα(εF) is an intraband contribution,
χintra
αα(εF) =1
V/summationdisplay
nk/parenleftbigg
−∂fnk
∂εnk/parenrightbigg
×µ2
B|/angbracketleftn,k|g+τ0σα+g−τzσα|n,k/angbracketright|2,(25)
andχinter
αα(εF) is an interband contribution,
χinter
αα(εF) =1
V/summationdisplay
n/negationslash=m,k−fnk+fmk
εnk−εmk
×µ2
B|/angbracketleftn,k|g+τ0σα+g−τzσα|m,k/angbracketright|2.(26)
At the zero temperature, only electronic states on the
Fermi surface contribute to χintra
αα. On the other hand, all
electronic states below the Fermi energy can contribute
toχinter
αα[29]. From the above expression, we see that
χinter
ααbecomes finite, when the matrix elements of the5
-0.02-0.010.000.02
0.01
0 2 1 -1 -2
g-/g +σxy [e2/(ha)] εF [t]-1.0-0.50.0
0.0 1.0 0.5 -0.5 -1.01.5
1.0
0.5
-1.5g-/g +=1 g-/g +=-1
g-/g +=0 (a)
(b)
εF=0
Bspin =0.01t/(g +μB)χzz [g+μB(2e/h)k D]SO
FIG.3: Thespin-orbitcrossed susceptibility χSO
zzasafunction
ofεFand the Hall conductivity as a function of g−/g+. We
set the parameters m0=−2m2,m1=m2,m1/t= 1,c/a= 1,
andγ= 0. At εF= 0, the value of χSO
zzis independent of
g−but itsεFdependence changes at finite g−. Consequently,
the Hall conductivity becomes finite in accordance with Eq.
(19).
spin magnetization operatorbetween the conduction and
valence bands is non-zero, i.e. the commutation relation
between the Hamiltonian and the spin magnetization op-
erator is non-zero. If the Hamiltonian and the spin mag-
netization operator commute,
/angbracketleftn,k|[Hk,g+τ0σα+g−τzσα]|m,k/angbracketright= 0,(27)
the interband matrix element satisfies
(εnk−εmk)/angbracketleftn,k|g+τ0σα+g−τzσα|m,k/angbracketright= 0.(28)
This equation means that there is no interband matrix
element and χinter
αα= 0, because εnk−εmk/negationslash= 0.
In the following, we set εF= 0, where the density of
states vanishes. Therefore, there is no intraband contri-
bution and we only consider the interband contribution.
We numerically calculate Eq. (26). Figure 4 shows the
spin susceptibility χspin
ααas a function of (a) γand (b)
g−/g+. In the following, we explain the qualitative be-
havior of χspin
ααusing the commutation relation between
the Hamiltonianand the spin magnetizationoperator. In
Fig. (4) (a), χspin
zzvanishes at γ= 0, because the Hamil-
tonian,HTDS, and the spin magnetization operator of
z-component, g+µBτ0σz, commute,
[HTDS,g+µBτ0σz] = 0. (29)For finite γ, on the other hand, χspin
zzincreases with |γ|.
This is because the commutation relation between Hxy
andg+µBτ0σzis non-zero,
[Hxy,g+µBτ0σz]/negationslash= 0, (30)
andχinter
zzgives finite contribution. χspin
xxandχspin
yyare
finite even in the absence of Hxy, i.e.γ= 0, because
HTDSandg+µBτ0σα(α=x,y) do not commute,
[HTDS,g+µBτ0σx]/negationslash= 0,
[HTDS,g+µBτ0σy]/negationslash= 0. (31)
Atγ= 0,χspin
xxis equal to χspin
yy. For finite γ, however,
they deviate from each other. This is because HTDSpos-
sessesfour-foldrotationalsymmetryalong z-axisbut Hxy
breaks the four-fold rotational symmetry. Figure (4) (b)
shows that χSO
zzbecomes finite when g−/g+/negationslash= 0. The
antisymmetric term, g−µBτzσz, andHTDSdo not com-
mute,
[HTDS,g−µBτzσz]/negationslash= 0. (32)
Consequently, χinter
zzgives finite contribution, though the
z-componentof spin is a good quantum number. The an-
tisymmetric term does not break the four-fold rotational
symmetry along z-axis, so that χspin
xxis equal to χspin
yyin
Fig. (4) (b).
The spin susceptibility χspin
ααis also anisotropic but
contrasts with the spin-orbit crossed susceptivity χSO
αα.
χspin
xxandχspin
yyare larger than χspin
zz, in contrast χSO
zzis
larger than χSO
xxandχSO
yy. Therefore, the angle depen-
dence measurement of magnetization will be useful to
separate the contribution from the each susceptibility.
V. DISCUSSION
In this section, we quantitatively compare the spin-
orbit crossed susceptibility χSO
zzand the spin susceptibil-
ityχspin
zzat the Dirac points as a function of g−/g+. In
the following calculation, we set the parameters to re-
produce the energy band structure around the Γ point
calculated by the first principle calculation for Cd 2As3
and Na 3Bi [10, 15]. The parameters are listed in the
table and we omit Hxy, i.e.γ= 0.
Figure 5 shows the two susceptibilities as a function of
g−/g+. We find that the two susceptibilities are approx-
imately written as
χspin
zz∼/parenleftbiggg−
g+/parenrightbigg2
, (33)
and
χSO
zz∼ −1
g+/parenleftbigg
χ0+g−
g+/parenrightbigg
, (34)
by numerical fitting. In the present parameters, χSO
zzis
negative and depends on g−/g+. The dependence on6
εF=0 χxx
χyy
χzz γ [t]
g-/g +εF=0
0.6
0.4
0.2
0.0
0.0 1.0 0.5 -0.5 -1.00.0 1.0 0.5 -0.5 -1.00.30.4
0.2
0.00.1
(b) (a)
χxx
χyy
χzz g-/g +=0
γ=0 spin
spin
spin
spin
spin
spinχαα [(g +μB/2) 2/(ta 3)]spin χαα [(g +μB/2) 2/(ta 3)]spin
FIG. 4: The spin susceptibility χspin
ααatεF= 0 as a function
of (a)γand (b) g−/g+. We set m0=−2m2,m1=m2,
m1/t= 1, and c/a= 1. At γ= 0 and g−/g+= 0,χspin
zz= 0
whileχspin
xx,χspin
yy>0. These behaviors are explained by the
commutation relation between the Hamiltonian and the spin
magnetization operators as discussed in the main text.
g−/g+originates from the existence of εk, which breaks
theparticle-holesymmetry. The g-factorsareexperimen-
tally estimated as gs= 18.6 for Cd 2As3[43] andg−= 20
for Na 3Bi [44]. Unfortunately, there is no experimental
datawhichdeterminesbothof gs,gporg+,g−. FromFig.
5, we see that χSO
zzcan dominate over χspin
zzifg−/g+≃0.
As far as we know, there is no experimental observation
of the magnetic susceptibility in these materials. We ex-
pect the experimental observation in near future and our
estimation of χSO
zzwill be useful to appropriately analyze
experimental data.
Material parameters
Cd3As2Na3Bi
C00.306[eV] -1.183[eV]
C10.033[eV] 0.188[eV]
C20.144[eV] -0.654[eV]
m00.376[eV] 1.754[eV]
m1-0.058[eV] -0.228[eV]
m2-0.169[eV] -0.806[eV]
t0.070[eV] 0.485[eV]
a12.64[˚A]5.07[˚A]
c25.43[˚A]9.66[˚A]
χ [(g +μB/2) 2/(ta 3)]0.030.04
0.02
0.000.01χ [(g +μB/2) 2/(ta 3)]
0.030.05
0.02
0.000.01
g-/g +0.0 0.4 0.2 -0.2 -0.40.04g+=5
g+=10
g+=15
g+=5
g+=10
g+=15 −χ zz SO −χ zz SO
χzz spinχzz spin
Na 3Bi Cd 2As 3
FIG. 5: The spin-orbit crossed susceptibility χSO
zzand the spin
susceptibility χspin
zzat the Dirac points as a function of g−/g+.
The dashed curve is χspin
zzand the solid lines are χSO
zz. The
upper (lower) panel shows Cd 2As3(Na3Bi). When g−/g+are
sufficiently small, χSO
zzbecomes comparable to χspin
zz.
VI. CONCLUSION
We theoretically study the spin-orbit crossed suscepti-
bility of topological Dirac semimetals. We find that the
spin-orbit crossed susceptibility along rotational symme-
try axis is proportional to the separation of the Dirac
points and is independent of the microscopic model pa-
rameters. This means that χSO
zzreflects topological prop-
erty of the electronic structure. The spin-orbit crossed
susceptibility is induced only along the rotational sym-
metryaxis. We alsocalculatethe spinsusceptibility. The
spin susceptibility is anisotropic and vanishingly small
along the rotational symmetry axis, in contrast to the
spin-orbit crossed susceptibility. The two susceptibilities
are quantitatively compared for material parameters of
Cd2As3and Na 3Bi. At the Dirac point, the orbital sus-
ceptibility logarithmically diverges and gives dominant
contribution to the total susceptibility. Off the Dirac
point, on the other hand, the orbital susceptibility de-
creases [6, 24, 25] and the contribution from the spin
susceptibility and the spin-orbit crossed susceptibility is
important for appropriate estimation of the total suscep-
tibility.7
ACNOWLEDGEMENT
This work was supported by JSPS KAKENHI Grant
Numbers JP15H05854 and JP17K05485, and JSTCREST Grant Number JPMJCR18T2.
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2003.12221v1.Generalized_magnetoelectronic_circuit_theory_and_spin_relaxation_at_interfaces_in_magnetic_multilayers.pdf | Generalized magnetoelectronic circuit theory and spin relaxation at interfaces in
magnetic multilayers
G. G. Baez Flores,1Alexey A. Kovalev,1M. van Schilfgaarde,2and K. D. Belashchenko1
1Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,
University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA
2Department of Physics, King’s College London, Strand, London WC2R 2LS, United Kingdom
(Dated: July 6, 2021)
Spin transport at metallic interfaces is an essential ingredient of various spintronic device con-
cepts, such as giant magnetoresistance, spin-transfer torque, and spin pumping. Spin-orbit coupling
plays an important role in many such devices. In particular, spin current is partially absorbed at
the interface due to spin-orbit coupling. We develop a general magnetoelectronic circuit theory and
generalize the concept of the spin mixing conductance, accounting for various mechanisms respon-
sible for spin-flip scattering. For the special case when exchange interactions dominate, we give a
simple expression for the spin mixing conductance in terms of the contributions responsible for spin
relaxation (i.e., spin memory loss), spin torque, and spin precession. The spin-memory loss param-
eteris related to spin-flip transmission and reflection probabilities. There is no straightforward
relation between spin torque and spin memory loss. We calculate the spin-flip scattering rates for
N|N, F|N, F|F interfaces using the Landauer-Büttiker method within the linear muffin-tin orbital
method and determine the values of using circuit theory.
I. INTRODUCTION
Spin-orbit coupling (SOC) plays an essential role at
metallicinterfaces, especiallyinthecontextofspintrans-
port related phenomena such as giant magnetoresistance
(GMR),1,2spin injection and spin accumulation,3spin
transfer torque,4spin pumping,5–7spin-orbit torque,8,9
spin Hall magnetoresistance (SMR),10and spin Seebeck
effect (SSE).11–13The concept of the spin mixing conduc-
tance,originallyintroducedwithinthemagnetoelectronic
circuit theory,14plays a very important role in describing
the spin transport at magnetic interfaces.15
Nevertheless, the spin mixing conductance in its orig-
inal form cannot account for various important con-
tributions associated with spin-flip processes,16–22cou-
pling to the lattice,23,24and other effects associated with
magnons.25–27One can generalize the concept of spin
mixing conductance by considering spin pumping in the
presence of spin-flip processes28or by considering the
magnetoelectronic circuit theory in the presence of spin-
flipscattering.29Sofarsuchgeneralizationswerenotable
to clarify the role of interfacial spin relaxation (usually
referred to as spin memory loss or spin loss) in processes
responsible for spin pumping and spin-transfer torque.
Recent progress in first-principles calculations of interfa-
cial spin loss29suggests that an approach fully account-
ing for spin-nonconserving processes can be developed.
Experimentally, a great deal of data is available on the
relationbetweenspin-orbitinteractionsandtheefficiency
of spin-orbit torque.30–33This data is often interpreted
intuitively in terms of the spin memory loss parameter,1
while lacking careful theoretical justification.
In this work, we develop the most general form of the
magnetoelectronic circuit theory and apply it to studies
of spin transport, concentrating on such phenomena as
spin-orbit torque and interfacial spin relaxation in mul-tilayers. We introduce a tensor form for the generalized
spin mixing conductance describing spin-nonconserving
processes, such as spin dephasing, spin memory loss, and
spin precession. We numerically calculate parts of the
spinmixingconductanceresponsibleforthespinmemory
loss in N|N, F|N, F|F interfaces in the presence of spin-
orbit interactions using the Landauer-Büttiker method
based on linear muffin-tin orbital (LMTO) method. We
show that the generalized spin mixing conductance can
be also used to describe spin-orbit torque when exchange
interactions dominate and the torque on the lattice can
be disregarded. Our results for the generalized spin mix-
ing conductance suggest that two distinct combinations
of scattering amplitudes are responsible for spin mem-
ory loss and torque, and in general there is no simple
connection between the two.
The paper is organized as follows. In Sec. II, we de-
velop a general formulation of the magnetoelectronic cir-
cuit theory in the presence of spin-flip scattering. In Sec.
III, we apply the magnetoelectronic circuit theory to cal-
culations of spin loss in (N 1N2)N, (N 1F2)N, or (F 1F2)N
multilayers connected to ferromagnetic leads. In Sec. IV,
we apply the magnetoelectronic circuit theory to spin-
orbit torque calculations. Computational details are de-
scribed in Sec. V, and the technicalities of the adiabatic
embedding approach are detailed in Sec. VI. Section VII
presents numerical results for the spin-flip transmission
and reflection rates and area-resistance products for N|N,
F|N, F|F interfaces. Section VIII concludes the paper.arXiv:2003.12221v1 [cond-mat.mes-hall] 27 Mar 20202
II. GENERALIZED CIRCUIT THEORY
A. Formalism
The magnetoelectronic circuit theory follows from
the boundary conditions linking pairs of nodes in a
circuit.14Here we consider the general case, allowing
spin-nonconserving scattering at interfaces between mag-
netic or non-magnetic metals due to the presence of spin-
orbit interaction or non-uniform magnetization. The
boundary condition at an interface between nodes 1 and
2, witharbitrarydistributionfunctions ^fa(a= 1;2labels
the node), is:
^I2=G0X
nmh
^t0
mn^f1(^t0
mn)y
M2^f2 ^rmn^f2(^rmn)yi
;
(1)
whereG0=e2=h,^rmnis the spin-dependent reflection
amplitude for electrons reflected from channel ninto
channelmin node 2, ^t0
mnis the spin-dependent transmis-
sion amplitude for electrons transmitted from channel n
in node 1 into channel min node 2, and the Hermitian
conjugate is taken only in spin space. Equation (1) can
be easily rewritten for the current ^I1in node 1. For a
ferromagnetic node, the spin accumulation is taken to be
parallel to its magnetization. The matrices ^rmnand^t0
mn
are generally off-diagonal in spin space.
It is customary to assume that the distribution func-
tions in the nodes, ^fa= ^0f0
a+^fs
a, are isotropic, i.e.,
independent of k. In this case Eq. (1) reduces to gener-
alized Kirchhoff relations:29
I0
2=Gcc
2f0+Gcs
2fs Gm
2fs
2;(2)
Is
2=Gsc
2f0+^Gss
2fs ^Gm
2fs
2;(3)
where f0=f0
1 f0
2andfs=fs
1 fs
2are interfacial
drops of charge and spin components of the distribution
function, and ^Ia= (^0I0
a+^Is
a)=2. The conductances in
Eq.(2-3)carryasubscript2emphasizingthattheygener-
ally differ from their counterparts describing the currents
in node 1; this subscript will be dropped where it doesn’t
lead to confusion. The conductances are related through
Gcs=Gsc Gt,^Gss=Gcc^0 ^Gt,Gm=Gt+Gr,
^Gm=^Gt+^Grto the following scalar, vector, and tensorquantities:
Gcc= 2G0X
mnT
mn; (4)
Gt
i= 4G0X
mni"ijkTjk
mn; (5)
Gr
i= 4G0X
mni"ijkRjk
mn; (6)
Gsc
i= 2G0X
mn(Ti0
mn+T0i
mn+i"ijkTjk
mn); (7)
Gt
ij= 2G0kl
ijX
mn(Tkl
mn+Tlk
mn+i"klp[T0p
mn Tp0
mn]);(8)
Gr
ij= 2G0kl
ijX
mn(Rkl
mn+Rlk
mn+i"klp[R0p
mn Rp0
mn]);
(9)
wherekl
ij=ijkl ikjl, Latin indices i;:::;ldenote
Cartesian coordinates and m,nthe conduction channels,
andrepeatedCartesianindicesaresummedoverhereand
below. In the above expressions, we defined the following
combinations of scattering matrix elements:
R
mn=Tr[(^rmn
^r
mn)(^
^)]=4;(10)
T
mn=Tr[(^t0
mn
^t0
mn)(^
^)]=4;(11)
where Greek indices can take values from 0 to 3.
In order to obtain the circuit theory equations (2) and
(3) from Eq. (1), we used the trace relations for Pauli
matrices, Tr (^i^j) = 2ij, Tr(^i^j^k) = 2i"ijk, and
Tr(^i^j^k^l) = 2(ijkl+iljk ikjl). The unitar-
ity condition gives the following identities:
X
mn^rmn^ry
mn+^t0
mn(^t0
mn)y=M2^0; (12)
X
mn^r0
mn(^r0
mn)y+^tmn(^tmn)y=M1^0;(13)
X
mn^rmn(^rmn)y+^tmn(^tmn)y=M2^0;(14)
X
mn^r0
mn(^r0
mn)y+^t0
mn(^t0
mn)y=M1^0;(15)
which relate the conductances defined for the two nodes
separated by the interface as Gcc
1=Gcc
2,Gcs
1=Gcs
2+
Gm
2, and Gcs
2=Gcs
1+Gm
1.
The interface conductances in the magnetoelectronic
circuit theory have to be renormalized by the Sharvin re-
sistancefortransparentOhmiccontacts34,35whichallows
comparison between ab initio studies and experiment.36
The circuit theory in Eqs. (2) and (3) can be general-
ized to account for the drift contributions in the nodes
by renormalizing the conductances Gcc,Gcs,Gsc,Gm,
^Gss, and ^Gm. This can be done by connecting nodes 1
and2to proper reservoirs with spin-dependent distribu-
tion functions ^fLand ^fRvia transparent contacts. The
currents in the nodes then become ^I1= 2G0^M1(^fL ^f1)
and^I2= 2G0^M2(^f2 ^fR), where ^M1(2)describethenum-
ber of channels (in general spin-dependent) in the nodes.3
Effectively, this leads to substitutions f"(#)
1!f"(#)
1+
I"(#)
1=(2G0M"(#)
1)andf"(#)
2!f"(#)
2 I"(#)
2=(2G0M"(#)
2)
in Eqs. (2) and (3).
Finally, we note that the conductance ^Gmdescribes
various spin-nonconserving processes, such as spin de-
phasing, spin loss, and spin precession. Therefore, it can
be interpreted as a tensor generalization of the spin mix-
ing conductance14,37,38to systems with spin-flip scatter-
ing. In the limiting case described in Ref. 28, our defi-
nition reduces to the generalized tensor expression sug-
gested there. However, our definition is more general as
it can account for processes corresponding to spin pre-
cession and spin memory loss. Spin-nonconserving pro-
cesses can also result in spin-charge conversion (i.e., spin
galvanic effect), which is described by GmandGcscon-
ductances. Furthermore, Gscdescribes the conversion of
charge imbalance into spin current (inverse spin galvanic
effect), and ^Gssis the tensor spin conductance.
B. Spin-conserving F|N interface
We now apply the generalized circuit theory to an
F|N interface. In the special case of a spin-conserving
interface, Eqs. (2) and (3) should be invariant under
SO(3)rotationsinspinspace, whichreproducesthespin-
conserving circuit theory:14,37,38
Gm= 0; (16)
Gcs=Gsc=Gscm; (17)
^Gss=Gccm
m; (18)
^Gm= 2G"#
r(^1 m
m) + 2G"#
im;(19)
where the tensor m
mimplements a projection onto
the magnetization direction, and G"#
randG"#
iare the
real and imaginary parts of the spin-mixing conductance
G"#=G0P
mn(nm r""
mnr##
mn t""
mnt##
mn).
C. General F|N interface
To understand further the structure of current re-
sponses, we expand the vector and tensor conductances
in powers of magnetization:
G
i=G (0)
i+G (1)
i;kmk+G (2)
i;klmkml+;(20)
G
ij=G (0)
ij+G (1)
ij;kmk+G (2)
ij;klmkml+;(21)
where stands forsc,cs,t,r, orm, stands forss,
t,r, orm, and the tensors G (0)
i,G (1)
i;k,G (2)
i;kl,G (0)
ij,
G (1)
ij;k,G (2)
ij;kl, etc. are invariant under the nonmagnetic
point group of the system.
The circuit theory substantially simplifies for axially
symmetric interfaces, which are common in polycrys-
talline heterostructures. Choosing the zaxis to be nor-
mal to the interface and applying the constraints corre-
sponding to the C1vsymmetry, we obtain the expansionof vector conductances Gsc,GscandGmto second order
inm:
~G =0
B@mxx (1)
1+mymzx (2)
1
myx (1)
1 mxmzx (2)
1
mzx (1)
21
CA;(22)
wherex (1)
1,x (1)
2, andx (2)
1are arbitrary coefficients.
For the tensor conductances ^Gssand ^Gmwe obtain
^G =0
B@x (0)
1 0 0
0x (0)
1 0
0 0x (0)
21
CA (23)
+0
B@0 mzx (1)
1myx (1)
2
mzx (1)
1 0 mxx (1)
2
myx (1)
3mxx (1)
3 01
CA (24)
+0
B@m2
xx (2)
1+m2
zx (2)
2mxmyx (2)
1mxmzx (2)
4
mxmyx (2)
1m2
yx (2)
1+m2
zx (2)
2mymzx (2)
4
mxmzx (2)
5 mymzx (2)
5m2
zx (2)
31
CA
(25)
wherex (0)
1,x (0)
2,x (1)
1,x (1)
2,x (1)
3,x (2)
1,x (2)
2,x (2)
3,
x (2)
4, andx (2)
5are arbitrary coefficients.
Theroleofspin-flipscatteringbecomesthemosttrans-
parent if both the magnetization and the spin accumu-
lation are either parallel or perpendicular to the inter-
face. In this case, the tensor and vector conductances
in Eqs. (2) and (3) can be simplified, and we arrive at
thefollowingrelationsforrelevantcomponentsassociated
with the in-plane and perpendicular directions:
Gcc=G0(T""+T##+T"#+T#"); (26)
Gsc=G0(T"" T##+T"# T#"); (27)
Gt= 2G0(T"# T#"); Gr= 2G0(R"# R#");(28)
Gt= 2G0(T"#+T#");Gr= 2G0(R"#+R#");(29)
alongwithGcs=Gsc Gt,Gss=Gcc Gt,Gm=Gt+Gr,
andGm=Gt+Gr. Of course, all quantities in these ex-
pressions are different for the in-plane and perpendicular
orientations of the magnetization; the corresponding in-
dex has been dropped to avoid clutter. The spin-resolved
dimensionless transmittances and reflectances
T0=X
mnt0
mn(t0
mn); (30)
R0=X
mnr0
mn(r0
mn)(31)
are defined in the reference frame with the spin quanti-
zation axis aligned with the magnetization.
Eqs. (26)-(29), together with Eqs. (2) and (3), are
also valid for axially symmetric F|F interfaces, as long as
the magnetizations of the two ferromagnets are collinear.
These expressions generalize the result given in Ref. 29
for axially symmetric N|N junctions to include F|N and
F|F interfaces.4
D. Relation to Valet-Fert theory
The Valet-Fert model39incorporates spin relaxation
in diffusive bulk regions but makes restrictive approx-
imations for the interfaces, treating them as transpar-
ent, spin-conserving, and prohibiting transverse spin
accumulation.2,16,40–43When spin relaxation at inter-
faces is of interest, the treatment based on the Valet-Fert
model is forced to replace the interfaces by fictitious bulk
regions,1,2which is restrictive even for N|N interfaces.29
Here we show how diffusive bulk regions can be incor-
porated in the generalized circuit theory. By introduc-
ing nodes near the interfaces and treating both interfaces
and bulk regions as junctions, the generalized Kirchhoff’s
rules2,16,40–43can be used to analyze entire devices with
spin relaxation in the diffusive bulk regions and arbitrary
spin-nonconserving scattering at interfaces.
The Valet-Fert model employs the following equations
to describe spin and charge diffusion in a normal metal:
@2
x(DfN
0) = 0; (32)
@2
@x2(DfN
s) =fN
s
N
sf; (33)
and in a ferromagnet:
@2
@x2(D"f"+D#f#) = 0; (34)
@2
@x2(D"f" D#f#) =f" f#
F
sf: (35)
Here fF
s=m(f" f#)=2is the spin accumulation in
the ferromagnet, and the spin-flip relaxation times N
sf=
(lN
sf)2=Dand F
sf= (lF
sf)2(1=D"+ 1=D#)=2are given in
terms of the spin-diffusion lengths lN
sf,lF
sfand diffusion
coefficients D,D. We now consider three basic circuit
elements.
1. Diffusive N region
For a diffusive N layer, the solution of Eqs. (32) and
(33) leads to a simplified version of Eqs. (2) and (3)
with vanishing vector conductances Gsc,Gsc,Gm, and
all tensor conductances reduced to scalars:
Gcc
N=2D
tN; (36)
Gss
N=Gcc
NN
sinhN; (37)
Gm
N=Gcc
NNtanhN
2; (38)
wheretNisthethicknessoftheNlayer, and N=tN=lN
sf.2. Diffusive F region
For a diffusive F layer with spin accumulation that is
parallel to the magnetization, the solution of Eqs. (34)
and (35) leads to vanishing Gmand the other conduc-
tances defined as follows:
Gcc
F= (D"+D#)=tF; (39)
Gsc
F=Gcs
F=m(D" D#)=tF; (40)
Gss
F=G
FF
sinhF+(Gsc
F)2
Gcc
F; (41)
Gm
F=G
FFtanhF
2; (42)
whereG
F= [(Gcc
F)2 (Gsc
F)2]=Gcc
Fis the effective con-
ductance and tFthe thickness of the F layer, and F=
tF=lF
sf.
3. Diffusive F|N junction
As a simple application, consider a composite junc-
tion consisting of F and N diffusive layers separated by
a transparent interface. Such an idealized junction can
be used to model an interface with spin-flip scattering
between F and N layers.2,16,40–43Combining the results
for F and N regions with boundary conditions, we find
Gm= 0and the following effective conductances:
Gcc= (1=Gcc
F+ 1=Gcc
N) 1; (43)
Gcs=Gsc=Gsc
F; (44)
^Gss=Gss
NGss
F
Gss
N+Gss
F+Gm
N+Gm
Fm
m; (45)
^Gm=Gss
N+Gm
N Gss
N(Gss
N+Gss
F)
Gss
N+Gss
F+Gm
N+Gm
Fm
m;(46)
where the conductances for the F and N layers should
be taken from the previous subsections. If spin-flip
scattering is negligible, we recover the known result:16
^Gm=Gss
N(1 m
m).
III. SPIN LOSS AT INTERFACES
The experimental data on interfacial spin relaxation
comes primarily from the measurements of magnetore-
sistance in (N 1N2)N, (N 1F2)N, or (F 1F2)Nmultilayers
connected to ferromagnetic leads,1,2whereNis the num-
ber of repetitions. The results have been reported1,2in
terms of the effective spin memory loss parameter Nor
Fobtained by treating the interface as a fictitious bulk
layer and fitting the data to the Valet-Fert model. Here
we relate the experimentally measured parameter Nor
Fto the generalized conductances appearing in Eqs. (2)
and (3). We assume that the interfaces are axially sym-
metricandthatthemagnetizationandspinaccumulation
are either parallel or perpendicular to the interface.5
A. N|N multilayer
We first consider a multilayer with repeated interfaces
between normal metals N 1and N 2. We would like to as-
sess the decay of spin current which may include the spin
relaxationbothatinterfacesandinthebulk. Tothisend,
we place nodes in both N 1and N 2layers and consider
the case of axially symmetric interfaces corresponding to
relations, Gsc=Gsc=Gm= 0. The relevant conduc-
tancesGcc,Gss,Gm
1, andGm
2account for the scattering
in the bulk and/or at the interfaces. Using Eq. (3), we
arrive at the following equations for the spin current in
some arbitrary node iin the superlattice:
Is
i=Gss(fs
i 1 fs
i) Gm
ifs
i; (47)
Is
i=Gss(fs
i fs
i+1) +Gm
ifs
i; (48)
which results in the recursive formula:
2Gm
i
Gssfs
i=fs
i 1 2fs
i+fs
i+1: (49)
This equation has analytical solutions:
fi
s=C1ei+C2e i; (50)
where the constants C1andC2are determined by the
boundary conditions. In the limit of weak spin-flip scat-
tering, we obtain the leading term for the decay rate:
2Gm
1+Gm
2
Gcc; (51)
where the constants C1andC2are defined by the bound-
ary conditions. Note that to the lowest order in the
spin-flip processes, only denominator in Eq. (51) needs
to be renormalized by the Sharvin resistance for trans-
parent Ohmic contacts, i.e., 1=~Gcc= 1=Gcc (1=M 1+
1=M 2)=(4G0). It is clear that the constant describes
how the spin current decays as we increase the num-
ber of layers in the superlattice. The conductances in
Eq. (51) may also include scattering in the bulk where
the total conductances can be calculated by concatenat-
ing the corresponding bulk and interface conductances
using Eqs. (2) and (3). When obtaining from experi-
mental data, one typically considers only interfacial con-
tributions in Eq. (51), while the bulk contributions are
simply removed.1This does not cause any problem when
spin-orbit interaction is weak as in this limit the total
Gmis a simple sum of contributions from interface and
bulk.
B. F|N and F|F multilayers
By considering F|N and F|F multilayers connected to
ferromagnetic leads one can also quantify spin relaxation
at magnetic interfaces.1In this case, a parameter de-
scribing the decay of spin current can also be relatedto the scattering matrix elements and to the general-
ized conductances in Eq. (2) and (3). We assume that
we have a superlattice with repeated interfaces between
normal (N 1) and ferromagnetic (F 2) layers. Normal can
be considered a special case of F in this section, equa-
tions derived below also apply to F|F multilayers with-
out any modifications. We would like to assess the decay
of spin current due to spin relaxation at interfaces and
in the bulk. We take nodes in F and N layers and con-
sider the case of axially symmetric interfaces. We also
assume collinear spin transport with the magnetization
being in-plane or perpendicular to interfaces. The gen-
eralized conductances may include scattering both in the
bulk and at the interfaces. Using Eqs. (2) and (3), we
arrive at the following equations for the spin and charge
currents in node i:
I0
i=Gcc(f0
i 1 f0
i) +Gcs
i 1(fs
i 1 fs
i) Gm
ifs
i;(52)
I0
i=Gcc(f0
i f0
i+1) +Gcs
i+1(fs
i fs
i+1) +Gm
ifs
i;(53)
Is
i=Gsc
i 1(f0
i 1 f0
i) +Gss(fs
i 1 fs
i) Gm
ifs
i;(54)
Is
i=Gsc
i+1(f0
i f0
i+1) +Gss(fs
i fs
i+1) +Gm
ifs
i;(55)
which results in the recursive formula:
2Gm
i=Gsc
i 1 2Gm
i=Gcc
Gss=Gsc
i 1 Gcs
i 1=Gccfs
i=fs
i 1 2fs
i+fs
i+1;(56)
Similar to non-magnetic case, the above equation has an-
alytical solutions:
fi
s=C1ei+C2e i: (57)
In the limit of weak spin-flip scattering, we obtain the
leading term for the decay rate:
2Gm
F+Gm
N
G; (58)
whereG= [(Gcc)2 (Gsc)2]=Gccis the effective con-
ductance of the scattering region. Note that to the low-
est order in the spin-flip processes, only denominator in
Eq. (58) needs to be renormalized by the Sharvin re-
sistance for transparent Ohmic contacts, i.e., 1=~G=
1=G (1=M"
1+ 1=M#
1+ 1=M"
2+ 1=M#
2)=(8G0). The
constantdescribes how the spin current decays as we
increase the number of layers in the multilayers. The
conductances in Eq. (58) may also include scattering in
the bulk. The bulk and interface conductances can be
concatenated using Eqs. (2) and (3).
IV. SPIN-ORBIT TORQUE
The discontinuity of spin-current at the interface fol-
lowing from the circuit theory in Eqs. (2) and (3) can
be used to calculate the total torque transferred to both
the magnetization and the lattice. In general, separating
these two contributions is not possible without consider-
ations beyond the circuit theory. When exchange inter-
actions dominate and the torque on the lattice can be6
disregarded, we can use the circuit theory to calculate
the spin torque on magnetization. Note that spin-flip
scattering and spin memory loss can still be present even
in the absence of the lattice torque, e.g., due to magnetic
disorder at the interface.
In the absence of angular momentum transfer to the
lattice, it is natural to assume axial symmetry with re-
spect to magnetization direction which results in simpli-
fications in Eqs. (22), (23), (24), and (25), i.e., x (2)
1= 0,
x (0)
1=x (0)
2,x (1)
1=x (1)
2=x (1)
3,x (2)
2= 0,x (2)
3=
x (2)
4=x (2)
5=x (2)
1. This leads to the following gener-
alization of Eq. (19) for the spin mixing conductance:
^Gm= 2G"#
r(^1 m
m) + 2Gm
km
m+ 2G"#
im;(59)
whereG"#
r=G0P
mnRe(nm r""
mnr##
mn t""
mnt##
mn)de-
scribes the absorption of transverse spin current and
Gm
k=G0(T"#+T#"+R"#+R#")the absorption of lon-
gitudinal spin current (i.e., spin memory loss); G"#
i=
G0P
mnIm(nm r""
mnr##
mn t""
mnt##
mn)describes the pre-
cession of spins. Even though the formal expressions for
G"#
randG"#
idid not change compared to Eq. (19), their
values can still be affected by the presence of spin-flip
scattering due to unitarity of the scattering matrix. The
effect of the unitarity constraint, however, does not have
a direct relation to the spin memory loss parameter .29
Using a typical spin-orbit torque geometry10and Eq.
(3), we can write a boundary condition determining the
torque:
2e2
~~ F=e(^1 m
m)js= (^1 m
m)^Gms;(60)
where sis the spin accumulation and ~ Fis the magneti-
zation torque. The spin current can be further calculated
from the diffusion equation:
r2s=s=l2
sf; (61)
and
js=
2e@zs+jSH^y; (62)
where the interface is orthogonal to zaxis andjSHis the
spin Hall current. We recover conventional antidamping
and field like torques:
~ F= (~jSH=2e)g"#
rtanh=2
1 + 2g"#
rcothm(m^y)(63)
+g"#
itanh=2
1 + 2g"#
icothm^y#
;
whereg"#
r(i)= (lsf=)G"#
r(i)andistheconductivityofthe
normal metal. The results of this section are inconsistent
with the notion that spin memory loss should directly af-
fect spin-orbit torque.30–33As can be seen from Eq. (59),
two separate parameters are responsible for spin mem-
ory loss and spin-orbit torque, and in general there is nodirect connection between the two. In the presence of
spin-orbit interactions, only the total torque acting on
the lattice and magnetization can be obtained from the
circuit theory. However, it seems that a similar conclu-
sion can be reached about the absence of direct relation
between spin memory loss and torque.
V. COMPUTATIONAL DETAILS AND
INTERFACE GEOMETRY
The transmittances and reflectances (30)-(31) were
calculated using the Landauer-Büttiker approach im-
plemented in the tight-binding linear muffin-tin orbital
(LMTO) method.44Spin-orbit coupling (SOC) was in-
troduced as a perturbation to the LMTO potential
parameters.44,45Local density approximation (LDA) was
used for exchange and correlation.46
We have considered a number of interfaces between
metals with the face-centered cubic lattice. The inter-
faceswereassumedtobeepitaxialwiththe(111)or(001)
crystallographic orientation. Lattice relaxations were ne-
glected, and the average lattice parameter for the two
lead metals was used for the given interface. The po-
larization of the spin current and the magnetization (in
F|N and F|F systems) were taken to be either parallel or
perpendicular to the interface.
Self-consistent charge and spin densities were obtained
using periodic supercells with at least 12 monolayers of
each metal. The surface Brillouin zone integration in
transport calculations was performed with a 512512
mesh for magnetic and 128128for non-magnetic sys-
tems.
We also studied the influence of interfacial intermix-
ing on spin-memory loss at Pt jPd and AujPd interfaces.
One layer on each side of the interface was intermixed
with the metal on the other side. The mixing concentra-
tions were varied from 11% to 50%. For example, an A|B
interface with 25% intermixing had two disordered lay-
ers with compositions A 0:75B0:25and A 0:25B0:75between
pure A and pure B leads. The transverse size of the su-
percell was 22for 25% and 50% intermixing and 33
for 11% intermixing. The conductances were averaged
over all possible configurations in the 22supercell and
over 18 randomly generated configurations in 33. In
addition, a model with long-range intermixing (LRI) was
considered where the transition from pure A to pure B
occurs over 8 intermixed monolayers with compositions
A8=9B1=9, A 7=9B2=9,..., A 1=9B8=9. This model was im-
plemented using 3 3 supercells.
VI. ADIABATIC EMBEDDING
In the Landauer-Büttiker approach, the active region
where scattering takes place is embedded between ideal
semi-infinite leads. In the circuit theory, the leads are
imagined to be built into the nodes of the circuit on7
both sides of the given interface. In order to define spin-
dependent scattering matrices with respect to the well-
defined spin bases, we turn off SOC in the leads.
Toavoidspuriousscatteringattheboundarieswiththe
SOC-free leads, we introduce “ramp-up” regions between
the interface and the leads, wherein the SOC is gradually
increased from zero at the edges of the active region to
its actual magnitude near the interface. Specifically, for
an atom at a distance xfrom the interface ( jxj>l0), the
SOC parameters are scaled by (L 2jxj)=(L 2l0), where
Lis the total length of the active region and l0the length
of the region on each side of the interface where SOC is
retained at full strength. In our calculations we set l0to
2 monolayers.
Because a slowly varying potential only allows scat-
tering with a correspondingly small momentum transfer,
such adiabatic embedding29allows a generic pure spin
state from the lead to evolve without scattering into the
bulk eigenstate of the metal before being scattered at the
interface.
In a non-magnetic metal, as explained in Ref. 29, adia-
batic embedding leads to strong reflection near the lines
on the Fermi surface where the group velocity is paral-
lel to the interface. Geometrically, when projected or-
thographically onto the plane of the interface, these lines
formtheboundariesoftheprojectedFermisurface. Elec-
trons with such wave vectors can backscatter from the
SOC ramp-up region both with and without a spin flip.
Thecontributionofthisbackscatteringtothespin-flipre-
flectance is an artefact of adiabatic embedding and needs
to be subtracted out.29In a magnetic lead such backscat-
teringconservesspinandis,therefore,inconsequentialfor
spin-memory loss calculations.
Adiabatic embedding can also produce strong scatter-
ing near the intersections of different sheets of the Fermi
surface, where an electron can scatter from one sheet to
another with a small momentum transfer. Such intersec-
tions do not exist in non-magnetic metals considered in
this paper (Cu, Ag, Au, Pd, Pt), but they are present in
all ferromagnetic transition metals. When the two inter-
secting sheets correspond to states of opposite spin, scat-
tering from one sheet to the other is a spin-flip process.
Depending on the signs of the normal (to the interface)
components v?of the group velocities at the intersection,
this scattering may or may not change the propagation
direction with respect to the interface and thereby show
up in spin-flip reflection or transmission. These two sit-
uations are illustrated in Fig. 1. If v?has opposite signs
on the two intersecting sheets [see Fig. 1(a-b)], then SOC
opens a gap at the avoided crossing, and incident elec-
trons with quasi-momenta close to the intersection are
fully reflected from the ramp-up region with a spin flip.
On the other hand, if v?has the same sign on the two
sheets [see Fig 1(c-d)], then, instead of backscattering,
there is a large probability of forward spin-flip scattering
as the electron passes through the ramp-up region.
Because we are interested in the spin-flip scattering
processes introduced by the interface, the contribution
FIG. 1. Crossing of the electronic bands in a ferromagnetic
lead near an intersection of two Fermi surface sheets of op-
posite spin. The parallel component of the quasi-momentum,
kk, is fixed. (a-b) and (c-d): Cases where the normal compo-
nent of the group velocity v?has the same or opposite sign
on the two sheets, resulting in resonant spin-flip reflection or
transmission, respectively. (a) and (c): no SOC; (b) and (d):
avoided crossings induced by SOC.
of spin-flip scattering due to the presence of the ramp-
up regions in the leads should be subtracted out. Un-
fortunately, this can only be done approximately. The
approach used for N 1|N2interfaces in Ref. 29 was to sub-
tract the spin-flip reflectances of auxiliary systems N 1|N1
and N 2|N2where the same lead material is used on both
sidesofanimaginaryinterfacewithadiabaticembedding.
This method is reasonable because the electrons incident
from one of the leads and backscattered by the ramp-up
region never reach the interface in the real N 1|N2system.
In an F|N system, the same is true for the backscattering
on Fermi sheet crossings in F [the case of Fig. 1(a-b)], but
not for the forward scattering [the case of Fig. 1(c-d)].
Nevertheless, as a simple approximation, we extend
the approach of Ref. 29 to the F|N interfaces, subtract-
ing both the spin-flip reflectances in auxiliary F|F and
N|N systems and the spin-flip transmittance in auxiliary
F|F.Likewise,foranF 1|F2interface,wesubtractbothre-
flectances and transmittances in F 1|F1and F 2|F2. Thus,
for any kind of interface, we define
T0
"#=T1j2
"# T1j1
"# T2j2
"#(64)
R0
a;"#=R1j2
a;"# Raja
"#; (65)
wherea=Lora=Rdenotes one of the leads, and the
primed quantities are used in Eq. (58). In the follow-
ing, we refer to this as the subtraction method, and the
parametercalculated in this way is denoted s.8
A.k-point filtering
A more fine-grained approach is to identify the loca-
tions in the surface Brillouin zone where spurious reflec-
tion or transmission occurs and filter out the contribu-
tions to spin-flip scattering probabilities from those lo-
cations. This filtering requires care, because some spin-
flip scattering processes near the Fermi surface crossings
are, in fact, physical, rather than merely being artefacts
of adiabatic embedding. This can be seen from Fig. 2,
which shows possible spin-flip scattering processes facili-
tated by the crossing of the Fermi sheets of opposite spin.
Figure 2(a) shows a spin-flip backscattering process in
the left lead, which can occur near a Fermi projection
boundary in a normal metal or near a Fermi crossing of
the type shown in Fig. 1(b). The processes shown in
Figs. 2(b) and 2(c) result from the forward scattering
near a Fermi crossing of the type shown in Fig. 1(d) in
the left lead, where the electron is then either transmit-
ted through or reflected from the interface, respectively.
Each process has a reciprocal version. The three pro-
cesses shown in Figs. 2(a-c) exist solely due to the pres-
ence of a ramp-up region, which provides the small mo-
mentum transfer needed to scatter from one Fermi sheet
to another.
In contrast, Figs. 2(d) and 2(e) show physical scatter-
ing processes. Here, the momentum of an electron inci-
dent from the left lead lies inside the spin-orbit gap of
the type shown in Fig. 1(b) in the right lead. As a re-
sult, the electron experiences a resonant spin-flip trans-
mission [Fig. 2(d)] or reflection [Fig. 2(e)] at the inter-
face. Resonant spin-flip transmission shown in Fig. 2(d)
is possible because an electron can scatter to a different
Fermi sheet with a large momentum transfer acquired
from the interface. Illustrations in Fig. 2(d-e) are highly
schematic because the wavefunction inside the spin-orbit
gap is evanescent in the right lead.
Letusfirstexaminethespin-flipscatteringprocessesin
systems without a physical interface, where all scattering
is due to adiabatic embedding alone. Spin-flip reflection
at the Fermi projection boundaries can be seen in Figs.
3(a) and 3(d) for adiabatically embedded Pt and Pd, re-
spectively, denoted in the figure caption as a fictitious
“interface” of a material with itself (e.g., Pd|Pd).29The
areas with strong spin-flip reflection are notably broader
in Pt, which has a larger spin-orbit constant compared
to Pd. Spin-flip reflection at Fermi crossings can be seen
in Figs. 4(a) and 4(b) for adiabatically embedded Ni and
Co, respectively. These two cases correspond to the dia-
gram in Fig. 2(a). Spin-flip transmission at Fermi cross-
ings in Ni and Co is seen, in turn, in Figs. 4(c) and 4(d);
thisistheprocessshowninFig.2(b)withoutthephysical
interface.
Nowconsiderphysicalinterfaces. Contourswithstrong
spin-flip reflection in, say, Fig. 3(d) for Pd|Pd are also
seen in Fig. 3(c) for electrons incident from the Pd lead
in Pt|Pd; the same comparison can be made for con-
tours with strong spin-flip reflection in, say, Fig. 4(a) forNi|Ni and 4(g) for Ni|Co. These processes correspond to
Fig. 1(a). Furthermore, the contours with strong spin-
flip transmission in Fig. 4(c) for Ni|Ni show up in both
Fig. 4(e) and 4(g) for spin-flip transmission and reflection
in Ni|Co, respectively. These processes correspond to
Fig. 2(b) and 2(c). The contours with resonant spin-flip
transmission in Co|Co [Fig. 4(d)] also show up in spin-
flip transmission for Ni|Co [Fig. 4(e)]; this corresponds
to Fig. 2(b) with the two leads interchanged.
All of the spin-flip scattering processes mentioned so
far and corresponding to Fig. 2(a-c) are artefacts of adia-
batic embedding and need to be filtered out in the calcu-
lation of the interfacial spin loss parameter. On the other
hand, the spin-flip transmission [Fig. 4(e)] and reflection
[Fig. 4(g)] functions for the Ni|Co interface also show the
spin-flip resonances of the types shown in Fig. 2(d-e).
Consider the spin-flip reflection function for electrons in-
cident from the Ni lead for the Ni|Co interface, which
is shown in Fig. 4(g). Apart from the resonant contours
appearing in Fig. 4(a) and 4(c) for spin-flip reflection and
transmission in Ni|Ni, there are also resonant contours in
Fig. 4(g) that correspond to the spin-flip reflection reso-
nances in Co|Co, which are seen in Fig. 4(b). The same
resonant contours appearing in Fig. 4(e) for the spin-flip
transmission in Ni|Co correspond to the process shown
in Fig. 2(d). These resonances correspond to the physical
process depicted in Fig. 2(e) and should notbe filtered
out in the calculation of the spin loss parameter.
This analysis shows that both artefacts of adiabatic
embedding [Fig. 2(a-c)] and physical resonant spin-flip
scattering processes [Fig. 2(d-e)] can be located in k-
spaceusingspin-fliptransmissionfunctionscalculatedfor
auxiliarysystems. Thus, asanalternativetothesubtrac-
tion method discussed above, the artefacts of adiabatic
embedding can be removed using k-point filtering.
For nonmagnetic (N 1|N2) interfaces, we first identify
thek-points where the spin-flip reflectance in an auxil-
iary system (N 1|N1or N 2|N2) exceeds a certain threshold
value, which is chosen so that the spin-flip reflectance in
the auxiliary system becomes less than 0:001G0if the
contributions from the identified k-points are excluded.
Then the contributions from those k-points are excluded
in the calculation of the spin-flip reflectance for electrons
incident from the corresponding lead. To ensure that
the artefacts are fully removed, the excluded regions are
slightly enlarged.
Ferromagnetic leads induce resonant scattering near
the crossings of the Fermi surfaces for opposite spins.
Processes of the types shown in Fig. 2(a-c) should be
filtered out, as explained above. We found that the spin-
flip reflectances and transmittances for all ferromagnetic
interfacesconsideredherearedominatedbyresonantpro-
cessesdepictedinFig.2(d-e)ratherthanbycontributions
from generic k-points. Indeed, the spin-loss parameters
obtained by excluding the processes of Fig. 2(a-c) or by
including only those in Fig. 2(d-e) are almost identical.
Figures 4(i-l) show the spin-flip scattering functions ob-
tained by starting from Figs. 4(e-h) and filtering out ev-9
FIG. 2. Spin-flip scattering mechanisms induced by a crossing of two Fermi sheets of opposite spin in an adiabatically embedded
interface with no disorder. Dashed vertical lines show the interface; the label Fspecifies that the given metal must be
ferromagnetic. Blue and red lines schematically show the trajectory of an electron before and after the spin flip. Crosses show
physical spin-flip scattering processes, while circles denote those that occurs solely due to adiabatic embedding.
FIG. 3.k-resolved spin-flip reflection functions for adiabatically embedded Pt|Pt, Pd|Pd, and Pt|Pd interfaces with and without
k-point filtering. (a) R#"in PtjPt; (b)RL#"in PtjPd; (c)RR#"in PtjPd; (d)R#"in PdjPd; (e)R#"in PtjPt, filtered; (f) RL#"
in PtjPd, filtered; (g) RR#"in PtjPd, filtered; (h) R#"in PdjPd, filtered.
erything other than the processes of Fig. 2(d-e). By per-
formingk-point filtering in this way we obtain a lower
bound on the spin-flip scattering functions and the spin-
loss parameter, ensuring that the artefacts of adiabatic
embedding are completely removed. The values flisted
in Table III were obtained in this way.
VII. RESULTS
A. Non-magnetic interfaces
Table I lists the area-resistance products ARand the
spin-loss parameters for nonmagnetic interfaces. The
subtraction and k-point filtering methods result in simi-
lar values of . For all material combinations, is quite
similar for (001) and (111) interfaces, suggesting that the
crystallographic structure of the interface does not havea strong effect on interfacial spin relaxation. In all cases,
the spin-loss parameter is slightly lower for the parallel
orientation of the spin accumulation relative to the inter-
face.
The calculated ARproducts and parameters are in
goodagreementwithexperimentalmeasurements1insys-
tems without Pd, but both are strongly overestimated
for (Au,Ag,Cu,Pd)|Pd interfaces. However, the results
for the Au|Pd (111) interface with the spin accumulation
parallel to the interface are in good agreement with re-
cent calculations of Gupta et al.47(AR= 0:81f
m2and
= 0:43) based on the analysis of the local spin currents
near the interface.
The large discrepancy in ARfor interfaces with Pd
suggests that the idealized interface model is inadequate
for these interfaces. Therefore, Pt jPd and AujPd with
interfacial intermixing were also constructed as described
in Section V. The results for intermixed interfaces are
listed in Table II. It is notable that intermixing increases10
FIG. 4.k-resolved spin-flip transmission and reflection functions for Ni|Ni, Co|Co, and Ni|Co, and an illustration of k-point
filtering. (a) R#"in NijNi; (b)R#"in CojCo; (c)T#"in NijNi; (d)T#"in Co|Co; (e) T#"in Ni|Co; (f) T"#in Ni|Co; (g) RL
#"in
NijCo; (h)RR
#"in NijCo; (i)T#"in NijCo, filtered; (j) T"#in NijCo, filtered; (k) RL
#"in NijCo, filtered; (l) RR
#"in NijCo, filtered.
theARproduct, while its values for ideal interfaces with
Pd are already too large compared with experimental
reports. The spin-loss parameter is also significantly
increased by intermixing, which moves it further away
from experimental data.
The disagreement with experiment in the values of AR
andforinterfaceswithPdislikelyduetothelackofun-
derstanding of the interfacial structure in the sputtered
multilayers, for which no structural characterization is
available, to out knowledge. It seems somewhat implau-
siblethattherealsputteredinterfacesaremuchlessresis-
tive compared to both ideal or intermixed interfaces con-
sidered here. It is possible that nominally bulk regions
in sputtered multilayers containing Pd are more disor-
dered and thereby have a higher resistivity and shorter
spin-diffusion length compared to pure Pd films. The fit-
ting procedure used to extract the ARandparameters
for the interface1would then ascribe this additional bulk
resistance and spin relaxation to the interfaces.B. Ferromagnetic interfaces
Table III lists the results for interfaces with one or two
ferromagnetic leads. The ARproducts for all interfaces
are in excellent agreement with experimental data.1The
values of the spin-loss parameter obtained using the sub-
traction method ( s) tend to be larger, by up to a factor
of2, comparedtothe k-pointfilteringmethod( f), which
isexpectedtobemoreaccurate. ForPt|Cotheresultsfor
ARandare in good agreement both with experiment
and with calculations using the discontinuity of the spin
current.47In other systems ARagrees very well with ex-
periment but is underestimated, which may be due to
the neglect of interfacial disorder and to the limitations
of the adiabatic embedding method.
VIII. CONCLUSIONS
We have developed a general formalism for analyzing
magnetoelectronic circuits with spin-nonconserving N|N,
F|N, or F|F interfaces between diffusive bulk regions. A
tensor generalization of the spin mixing conductance en-11
TABLE I. Area-resistance products AR(f
m2) and spin-
loss parameters obtained using the subtraction method ( s)
and the filtering method ( f) for nonmagnetic interfaces. M
denotes the orientation of the spin accumulation relative to
the interface.
NjNPlaneMARARexpsfexp
PtjPd001k0.42
0.140.030.600.57
0.130.08?0.44 0.710.65
111k0.28 0.410.36
?0.29 0.450.38
AujPd001k0.96
0.230.080.710.68
0.080.08?0.96 0.860.82
111k0.83 0.530.54
?0.87 0.730.69
AgjPd001k0.92
0.350.080.410.47
0.150.08?1.12 0.500.54
111k0.89 0.410.47
?0.92 0.500.55
CujPd001k0.81
0.450.0050.410.47
0.240.05?0.81 0.470.52
111k0.80 0.430.40
?0.81 0.530.48
CujAu001k0.13
0.150.0050.080.08
0.130.07?0.13 0.110.11
111k0.11 0.080.07
?0.12 0.110.10
CujPt001k0.90
0.750.051.000.87
0.90.1?0.89 1.070.9
111k0.75 0.880.72
?0.82 1.110.83
CujAg001k0.03
0.0450.0050.020.2
0?0.03 0.030.02
111k0.13 0.030.03
?0.13 0.040.04
codes all possible spin-nonconserving processes, such as
spin dephasing, spin loss, and spin precession. In the
special case when exchange interactions dominate, those
contributions can be clearly separated into terms respon-
sible for spin memory loss, spin-orbit torque, and spin
precession. Surprisingly, there is no direct relation be-
tween spin-orbit torque and spin memory loss; the two
effects are described by different combinations of scat-
tering amplitudes responsible for the absorption of the
transverse and longitudinal components of spin current
at the interface.
The spin relaxation (i.e., spin memory loss) param-
eterhas been numerically calculated using Eqs. (51)
and (58) for a number of N|N, F|N, and F|F interfaces.
First-principles calculations, aided by adiabatic embed-
ding, show reasonable agreement with experiment for
and the area-resistance products with the exception of
N|N interfaces including a Pd lead. For such interfacesTABLE II. Same as in Table I but for non-magnetic interfaces
with intermixing. The percentage indicates the composition
in the two intermixed layers. LRI refers to the long-range
intermixing model; see Section V for details.
NjN (mix %) PlaneMARARexpsfexp
PtjPd (11%) 111k0.29
0.140.030.450.38
0.130.08?0.30 0.560.40
PtjPd (25%) 111k0.32 0.520.46
?0.34 0.650.52
PtjPd (50%) 111k0.36 0.580.51
?0.38 0.720.57
PtjPd (LRI) 111k0.82 1.200.91
?0.85 1.340.96
AujPd (11%) 111k0.86
0.230.080.560.46
0.080.08?0.90 0.760.58
AujPd (25%) 111k0.96 0.600.58
?1.01 0.810.73
AujPd (50%) 111k0.95 0.600.58
?0.99 0.820.73
AujPd (LRI) 111k1.24 0.790.65
?1.29 0.980.76
TABLEIII.SameasinTableIbutforF|NandF|Finterfaces.
F(N)jFPlaneMAR"AR#ARARexpsfexp
CujCo001k0.292.060.59
0.51
0.050.220.12
0.33
0.05?0.312.050.59 0.240.14
111k0.361.540.48 0.180.11
?0.361.520.47 0.190.12
PtjCo001k0.464.671.28
0.85
0.121.120.91
0.9
0.4?0.444.601.26 1.170.96
111k1.701.360.76 0.810.72
?1.821.380.80 0.910.80
AgjCo001k0.401.870.57
0.56
0.060.330.21
0.33
0.1?0.431.840.57 0.380.29
111k0.221.580.45 0.200.12
?0.221.570.45 0.210.13
NijCo001k0.221.040.32
0.255
0.0250.320.15
0.35
0.05?0.241.020.32 0.340.16
111k0.210.730.23 0.270.17
?0.250.720.24 0.290.16
bothandARare strongly overestimated, which can
not be explained by short or long-range interfacial in-
termixing. The analysis of spin-flip scattering probabil-
ities for F|N and F|F interfaces suggests that interfacial
spin relaxation is dominated by electronic states near the
crossings of the Fermi surfaces for opposite spins in fer-
romagnets.
The generalized magnetoelectronic circuit theory pro-
vides a convenient framework for analyzing spin trans-
port in magnetic nanostructures with strong spin-orbit12
coupling at interfaces.
ACKNOWLEDGMENTS
A. K. is much indebted to Gerrit Bauer for stimulat-
ing discussions on circuit theory with spin-flip scatter-
ing. This work was supported by the National ScienceFoundation through Grant No. DMR-1609776 and the
Nebraska MRSEC, Grant No. DMR-1420645, as well as
by the DOE Early Career Award DE-SC0014189 (AK)
and the EPSRC CCP9 Flagship project, EP/M011631/1
(MvS). Computations were performed utilizing the Hol-
land Computing Center of the University of Nebraska,
which receives support from the Nebraska Research Ini-
tiative.
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1708.07247v2.Strong_influence_of_spin_orbit_coupling_on_magnetotransport_in_two_dimensional_hole_systems.pdf | arXiv:1708.07247v2 [cond-mat.mes-hall] 25 Aug 2017Strong influence of spin-orbit coupling on magnetotranspor t in two-dimensional hole
systems
Hong Liu, E. Marcellina, A. R. Hamilton and Dimitrie Culcer
School of Physics and Australian Research Council Centre of Excellence in Low-Energy Electronics Technologies,
UNSW Node, The University of New South Wales, Sydney 2052, Au stralia
(Dated: August 28, 2017)
With a view to electrical spin manipulation and quantum comp uting applications, recent signifi-
cant attention has been devoted to semiconductor hole syste ms, which have very strong spin-orbit
interactions. However, experimentally measuring, identi fying, and quantifying spin-orbit coupling
effects in transport, such as electrically-induced spin pol arizations and spin-Hall currents, are chal-
lenging. Here we show that the magnetotransport properties of two-dimensional (2D) hole systems
display strong signatures of the spin-orbit interaction. S pecifically, the low-magnetic field Hall co-
efficient and longitudinal conductivity contain a contribut ion that is second order in the spin-orbit
interaction coefficient and is non-linear in the carrier numb er density. We propose an appropriate
experimental setup to probe these spin-orbit dependent mag netotransport properties, which will
permit one to extract the spin-orbit coefficient directly fro m the magnetotransport.
Low-dimensional hole systems have attracted consid-
erable recent attention in the context of nanoelectron-
ics and quantum information [ 1–9]. They exhibit strong
spin-orbit coupling but a weak hyperfine interaction,
which allows fast, low-power electrical spin manipula-
tion [10,11] and potentially increased coherence times
[12–15] while their effective spin-3/2 is responsible for
physics inaccessible in electron systems [ 16–20]. Struc-
tures with strong spin-orbit interactions coupled to su-
perconductors may enable topological superconductivity
hosting Majorana bound states and non-Abelian particle
statistics relevant for topological quantum computation
[21–24]. In the past fabricating high-quality hole struc-
tures was challenging, but recent years have witnessed
extraordinary experimental progress [ 12,25–43].
A full quantitative understanding of spin-orbit cou-
pling mechanisms is vital for the realization of spin-
tronics devices and quantum computation architectures
[44,45]. At the same time experimental measurement
of spin-orbit parameters is difficult [ 46]. Spin-orbit con-
stants can be estimated from weak antilocalization [ 47–
50], Shubnikov-de Haas oscillations and spin precession
in large magnetic fields (up to 2 T) [ 51–53], and state-of-
the-art optical measurements [ 54,55]. Many techniques
yield only the ratio between the Rashba and Dresselhaus
terms or allow the determination of only one type of spin
splitting. Likewise,experimentallyquantifyingspin-orbit
induced effects, such as via spin-to-charge conversion or
vice versa, is difficult. For instance, current-induced spin
polarizations in spin-orbit coupled systems are small and
their relationship to theoretical estimates is ambiguous
[56–58], while spin-Hall currents [ 59] can only be identi-
fied via an edge spin accumulation [ 60–62].
Here we show that the spin-orbit interaction can have
a sizeable effect on low magnetic-field Hall transport in a
2D hole system, which is density-dependent and experi-
mentally visible. Our central result, shown in Fig. 1, is a(a)
1.2
1.1
1.0RH/R0
10 8 6 4 2 0
Fz(MV/m)p = 1 x 1011cm-2
p = 1.5 x 1011cm-2
p = 2 x 1011cm-2GaAs QW1.4
1.3
1.2
1.1
1.0RH/R0GaAs
InAs
InSbp = 2 x 1011cm-2
(b)
Figure 1. Spin-orbit correction to the Hall coefficient RHof
2D holes in various 15 nm quantum wells as a function of the
electric field Fzacross the well, where R0≡1
peis the bare
Hall coefficient. Panel shows results for (a) different quantu m
well materials at p= 1×1011cm−2and (b) GaAs quantum
wells at different densities.
correction to the low-field Hall coefficient
RH=1
pe/bracketleftbigg
1+/parenleftbigg64πm∗2α2
/planckover2pi14/parenrightbigg
p/bracketrightbigg
, (1)
whereαis the coefficient of the cubic Rashba spin-orbit
term, which arisesfromthe applicationofan electricfield
Fzacrossthequantumwell, m∗istheheavy-holeeffective
mass atα= 0,pis the hole density, and eis the elemen-2
tary charge. Note that here we have chosen the z−axis
as the quantization direction. In hole systems, where the
spin-orbit coupling can account for as much as 40% of
the Fermi energy [ 63], effects of second-order in the spin-
orbit strength can be sizable in charge transport. These
reflect spin-orbit corrections to the occupation probabili-
ties, densityofstates, andscatteringprobabilities, aswell
as the feedback of the current-induced spin polarization
on the charge current. Quantitative evaluation shows
that the spin-orbit corrections can reach more than 10%
in GaAs quantum wells, and are of the order ∼20−30%
in InAs and InSb quantum wells (Fig. 1a). The magni-
tude of the spin-orbit corrections also increase with den-
sity, which is consistent with the expectation that the
strength of spin-orbit interaction increases with density
(Fig.1b). It is worth noting that the correction due to
spin-orbit coupling has already taken into account the
fact that the spin-split subbands may have different hole
mobilities.
In the following we derive the formalism and show
how spin-orbit coupling can give rise to corrections in
the magnetotransport. We consider a 2D hole system in
the presence of a constant electric field Fand a perpen-
dicular magnetic field B=Bzˆz. The full Hamiltonian is
ˆH=ˆH0+ˆHE+ˆU+ˆHZ, where the band Hamiltonian ˆH0
is defined below in Eq. ( 2),ˆHE=−eF·ˆrrepresents the
interactionwith the externalelectricfield ofholes ˆristhe
position operator, and ˆUis the impurity potential, dis-
cussed below. The Zeeman term HZ= 3κµBσ·Bwith
κis a material-specific parameter [ 16],µBthe Bohr mag-
neton and σthe vector of Pauli spin matrices. Rashba
spin-orbit coupling is expected to dominate greatly over
the Dresselhaus term in 2D hole gases, even in materi-
als such as InSb in which the bulk Dresselhaus term is
very large [ 63]. With this in mind, the band Hamiltonian
used in our analysis in the absence of a magnetic field is
written as [ 64]
H0k=/planckover2pi12k2
2m∗+iα(k3
−σ+−k3
+σ+)≡/planckover2pi12k2
2m∗+σ·Ωk,(2)
wherem∗=m0
γ1+γ2, the Pauli matrix σ±=1
2(σx±iσy),
k±=kx±iky. ForB= 0 the eigenvalues of the band
Hamiltonian are εk±=/planckover2pi12k2/(2m∗)±αk3. In an exter-
nal magnetic field we replace kby the gauge-invariant
crystal momentum ˜k=k−eAwith the vector potential
A=1
2(−y,x,0). The magnetic field is assumed small
enough that Landau quantization can be neglected, in
other words ωcτp≪1, where ωc=eBz/m∗is the cy-
clotron frequency and τpthe momentum relaxation time.
To set up our transport formalism, in the spirit of
Ref. [65], we begin with a set of time-independent states
{ks}, where srepresents the twofold heavy-hole pseu-
dospin. We work in terms of the canonical momentum
/planckover2pi1k. The terms ˆH0,ˆHEandˆHZare diagonal in wave vec-
tor but off-diagonal in band index while for elastic scat-
tering in the first Born approximation Uss′
kk′=Ukk′δss′.
Without loss of generality, here we consider short-range
impurity scattering. The impurities are assumed uncor-related and the averageof ∝an}bracketle{tks|ˆU|k′s′∝an}bracketri}ht∝an}bracketle{tk′s′ˆU|ks∝an}bracketri}htoverim-
purityconfigurationsis( ni|¯Uk′k|2δss′)/V, whereniis the
impurity density, Vthe crystalvolume, and ¯Uk′kthe ma-
trix element of the potential of a single impurity.
The central quantity in our theory is the density oper-
ator ˆρ, which satisfies the quantum Liouville equation,
dˆρ
dt+i
/planckover2pi1[ˆH,ˆρ] = 0. (3)
The matrix elements of ˆ ρare ˆρkk′≡ˆρss′
kk′=∝an}bracketle{tks|ˆρ|k′s′∝an}bracketri}ht
with understanding that ˆ ρkk′is a matrix in heavy hole
subspace. The density matrix ρkk′is written as ρkk′=
fkδkk′+gkk′, wherefkis diagonal in wave vector, while
gkk′is off-diagonal in wave vector. The quantity of in-
terest in determining the charge current is fksince the
current operator is diagonal in wave vector. We there-
fore derive an effective equation for this quantity by first
breaking down the quantum Liouville equation into the
kinetic equations of fkandgkk′separately, and fkobeys
dfk
dt+i
/planckover2pi1[H0k+HZ,fk]+ˆJ(fk) =DE,k+DL,k,(4)
where the scattering term in the Born approximation
ˆJ(fk)=1
/planckover2pi12/integraldisplay∞
0dt′[ˆU,e−iH0t′
/planckover2pi1[ˆU,ˆf(t)]eiH0t′
/planckover2pi1]kk,(5)
and the driving terms
DE,k=−eE
/planckover2pi1·∂fk
∂k, (6a)
DL,k=1
2e
/planckover2pi1{ˆv×B,∂fk
∂k}, (6b)
stem from the applied electric field and Lorentz force re-
spectively [ 65]. In external electric and magnetic fields
one may decompose fk=f0k+fEk+fEBk, wheref0kis
theequilibriumdensitymatrix, fEkisacorrectiontofirst
orderin the electricfield (but atzeromagnetic field), and
fEBkis an additional correction that isfirst order in the
electricand magnetic fields. The equilibrium density ma-
trix is written as f0k= (1/2)[(fk++fk−)11+σ·ˆΩ(fk+−
fk−)], where ˆΩis a unit vector and Ωwas defined in
Eq. (2), andfk±represent the Fermi-Dirac distribution
functions corresponding to the two band energies εk±.
In linear response one may replace fk→f0kin Eq. (6a).
On the other hand it is trivial to check that the driving
termDL,kvanishes when the equilibrium density matrix
is substituted, so in Eq. ( 6b) one may replace fk→fEk.
Hence, in this work we perform a perturbation expansion
up to first order in the electric and magnetic fields, and
up to second orderin the spin-orbit interaction, retaining
termsuptoorder α2. ThedetailedsolutionofEq.( 4)and
the explicit evaluation of the scattering term Eq. ( 5) are
given in the Supplement. We briefly summarize the pro-
cedure here. Firstly, with f0kknown and only DE,kon
the right-hand side of Eq. ( 4), we obtain fEk. Next, with3
onlyDL,kon the right-hand side of Eq. ( 4), we obtain
fEBk. By taking the trace with current operator the lon-
gitudinal and transverse components of the current are
found as jx,y=eTr/bracketleftbig
ˆvx,yfk/bracketrightbig
, withvi= (1//planckover2pi1)∂H0k/∂k.
Finally, with σxxandσxythe longitudinal and Hall con-
ductivities respectively, the Hall coefficient appearing in
Eq. (1) is found through RH=σxy
Bz(σ2xx+σ2xy). For the
Hall conductivity on the other hand one needs fEBk. We
note that the topological Berrycurvature terms that give
contributions analogous to the anomalous Hall effect in
Rashba systems (with the magnetization replaced by the
magnetic field Bz) vanish identically when both the band
structure and the disorder terms are taken into account.
Table I. The maximal hole densities for which the current
theory is applicable for 15 nm-wide GaAs, InAs, and InSb
quantum wells. Densities in units of 1011cm−2.
GaAs InAs InSb
6.55 8.08 8.60
The limits of applicability of our approach are as fol-
lows. We assume that the magnetotransport considered
here occurs in the weak disorder regime, i.e. εFτp//planckover2pi1≫1,
whereεFis Fermi energy. Furthermore, we assume that
the scatteringdoes not changeappreciablywhen the gate
field is changed at low density [ 40], so the condition
εFτp//planckover2pi1≫1 is still valid when the gate field is changed.
We assume αk3
F/ǫkin≪1 where ǫkin=/planckover2pi12k2
F
2m∗is kinetic
energy, for example in Ref. [ 48], the spin-orbit-induced
splitting of the heavy hole sub-band at the Fermi level is
determined to be around 30% of the total Fermi energy.
In addition, Eq. ( 2) withαindependent of wave vector
is a result of the Schrieffer-Wolff transformation applied
to the Luttinger Hamiltonian, and its use requires the
Schrieffer-Wolff method to be applicable. Furthermore,
throughout this paper we consider cases where only the
HH1 band is occupied. We have calculated the exact
window of applicability of our theory in Table I.
Physically, the terms ∝α2entering the Hall coefficient
aretracedbacktoseveralmechanisms. Firstly, spin-orbit
coupling gives rise to corrections to: (i) the occupation
probabilities, through fk±; (ii) the band energies and
density of states, through dεk±/dk; and (iii) the scatter-
ing term, whichincludes intra-andinter-bandscattering,
as well as scattering between the charge and spin distri-
butions. Secondly, Rashba spin-orbit coupling gives rise
to a current-induced spin polarization [ 56], which is of
first order in α, and this in turn gives rise to a feedback
effect on the charge current, which is thenresponsible for
approximately a quarter of the overall spin-orbit contri-
bution to the Hall coefficient.
As a concrete example, a 2D hole system confined to
GaAs/AlGaAs heterostructures is particularly promising
since it has not only a very high mobility, but also a spin
splitting that hasbeen shownto be electricallytunable in
both square and triangular wells [ 66]. The spin splitting
can be tuned from large values to nearly zero in a square3500
3000
2500
2000
1500
1000
500
0α (meV nm3)
108 6 4 2 0
Fz (MV/m)360
340
320
300α (meV nm3)
10864
Fz (MV/m) GaAs
InAs
InSb
GaAs
Figure 2. The Rashba coefficient αof as a function of the
net perpendicular electric field Fzfor 15 nm GaAs, InAs,
and InSb quantum wells. The inset shows that αfor GaAs
decreases by ∼20% asFzis increased from 4 MV/m to 10
MV/m, due to the fact the well becomes quasi-triangular at
Fz/greaterorsimilar4 MV/m.
quantumwellwhosechargedistributioncanbecontrolled
from being asymmetric to symmetric via the application
of a surface-gate bias. Whereas thus far the theoretical
formalismhas been general, to makeconcrete experimen-
tal predictions we first specialize to a two-dimensional
hole gas (2DHG) in a 15 nm-wide GaAs quantum well
subjected to an electric field in the ˆ zdirection, so that
the symmetry ofthe quantum well can be tuned arbitrar-
ily. In the simplest approximation, taking into account
only the lowest heavy-hole and light-hole sub-bands, in a
2DHG the Rashba coefficient αmay be estimated as
α=3/planckover2pi14
m2
0∆Eγ2∝an}bracketle{tφL|φH∝an}bracketri}ht∝an}bracketle{tφH|(−id/dz)|φL∝an}bracketri}ht.(7)
where∆ Eis energysplitting ofthe lowestheavy-holeand
light-hole sub-bands and γ=γ2+γ3
2, andφH,L≡φH,L(z)
represents the orbitalcomponent of the heavy-hole and
light-hole wave functions respectively in the direction ˆz
perpendiculartotheinterface. Forasystemwith topand
back gates, where the electric field Fzacross the well can
be turned on or off, we use a modified infinite square well
wave function in which Fzis already encoded [ 67].
TheRashbacoefficient α, asafunctionof Fz, for15nm
hole quantum wells is shown Fig. 2. For GaAs, at low Fz
(Fz≪4 MV/m), the Rashba coefficient increases with
F, whichisinaccordancewiththetrendsreportedinRef.
[68]. AsFzis increased, αthen saturates, and, at larger
electricfields( Fz>4MV/m), thequantumwellbecomes
quasi-triangular and the Rashba coefficient αdecreases
with increasing electric field Fz. The decrease of αas
a function of Fzin quasi-triangular wells is consistent
with the experimental findings of Ref. [ 69]. Note that for
different materials, αsaturates at different values of Fz,4
(a)
(b)1.00
0.98
0.96
0.94
0.92
0.90
0.88σxx/σ0
10 8 6 4 2 0
Fz(MV/m)p = 1 x 1011cm-2
p = 1.5 x 1011cm-2
p = 2 x 1011cm-2GaAs QW1.00
0.95
0.90
0.85
0.80
0.75
0.70σxx/σ0GaAs
InAs
InSbp = 2 x 1011cm-2
Figure 3. Ratio of Drude conductivity at finite electric field s
to its zero electric field value, with the bare Drude conduc-
tivityσ0≡peµ, for (a) different quantum well materials at
p= 1×1011cm−2and (b) GaAs quantum wells at different
densities. Here, the well width is 15 nm.
and that the αis larger in materials with a higher atomic
number [ 63].
Giventhe dependence of α(Fig.2), andhence theHall
coefficient RH(Fig.1), onFz, we now outline how αcan
be deduced experimentally. Using a top- and backgated
quantum well, the quantum well is initially tuned to be
symmetric so that αwill be zero and the hole density
can be measured accurately. One subsequently increases
Fz, for example to ∼4 MV/m for the GaAs quantum
welldiscussedabove,whilst keepingthe densityconstant.
This in turn results in an appreciable increase in α, and
hence a large change in RHas a function of Fz.
The non-monotonic change in αas a function of Fz
likewiseaffectsthelongitudinalconductivity σxx(Fig.3),
which reads
σxx=σ0/bracketleftbigg
1−/parenleftbigg60πm∗2α2
/planckover2pi14/parenrightbigg
p/bracketrightbigg
. (8)
The spin-orbit corrections are larger in InAs and InSb
(Fig.3a) rather than GaAs. Furthermore, as the density
increases, σxxdecreases faster with Fz(Fig.3b). How-
ever, although the spin-orbit corrections to σxxhave a
similar functional form as and a similar magnitude to
the corrections to RH, it is difficult to single out the de-
pendence of σxxonαexperimentally. As the shape of
the wave functions changes with Fz, the spin-orbit in-dependent scattering properties are also altered, which
may then introduce a larger correction to σxxthan the
spin-orbitinduced corrections[ 70]. In fact, the spin-orbit
independent corrections can alter the carrier mobility by
∼20% even in electron quantum wells [ 40].
Various possibilities exist to extend the scope of the
calculations presented in this paper. Here we have re-
stricted ourselves, for the sake of gaining physical insight
and without loss of generality, to hole systems in which
the Schrieffer-Wolff approximation is applicable so that
αcan be approximated as constant. In a general 2D hole
systemα(k) is a function of wave vector, and decreases
withkatlargerwavevectors. Itsbehaviourisinprinciple
nottractableanalyticallythoughitcanstraightforwardly
be calculated numerically. The results we have found re-
main true in their general closed form for hole systems
at arbitrary densities provided αis replaced by α(k). An
alternative approach would be to start directly with the
4×4 Luttinger Hamiltonian and determine the charge
conductivity using a spin-3/2 model. However, calculat-
ing the conductivity as a function of Fzcan quickly be-
comeverycomplicated analytically, limiting the utility of
such an approach. Finally, the kinetic equation approach
wehavediscussedcanstraightforwardlybe generalizedto
arbitrary band structures in a way that makes it suitable
for fully numerical approaches relying on maximally lo-
calized Wannier functions [ 71].
It is worth mentioning how the corrections in the mag-
netotransport properties of 2D electrons will differ from
those of 2D holes. In 2D electrons, to lowest order the
spin-orbit coupling stems from k.pcoupling with the
topmost valence band, and the leading contribution to
spin-orbit interaction in 2D electrons is linear in k[16].
As a result, the spin-orbit dependent corrections to the
magnetotransport in 2D electrons will be much smaller
compared to 2D holes, and thus may not be detectable
within experimental resolution.
In summary, we have presented a quantum kinetic the-
ory of magneto-transport in 2D heavy-hole systems in
a weak perpendicular magnetic field and demonstrated
that the Hall coefficient, as well as the longitudinal con-
ductivity, display strong signatures of the spin-orbit in-
teraction. We have also shown that our theory provides
an excellent qualitative agreementto existing experimen-
tal trends for α, although to the best of our knowledge,
there has not been a demonstration of RHchanging as a
function of α. An appropriate experimental setup with
top and back gates can lead to a direct electrical mea-
surement of the Rashba spin-orbit constant via the Hall
coefficient.
ACKNOWLEDGMENTS
This research was supported by the Australian Re-
search Council Centre of Excellence in Future Low-
Energy Electronics Technologies (project CE170100039)
and funded by the Australian Government.5
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two-dimensional hole systems”
Hong Liu, E. Marcellina, A. R. Hamilton and Dimitrie Culcer1
1School of Physics and Australian Research Council Centre of Excellence in Low-Energy Electronics Technologies,
UNSW Node, The University of New South Wales, Sydney 2052, Au stralia
I. LUTTINGER HAMILTONIAN
We start from the bulk 4 ×4 Luttinger Hamiltonian [ 1]HL(k2,kz) describing holes in the uppermost valence band
with an effective spin J= 3/2. So the hole system with top and back gate in z-direction can be simplified as the
isotropic Luttinger-Kohn Hamiltonian plus a confining asymmetrical t riangular potential.
ˆH=HL(k2,kz)−eFzz, z > 0, (1)
whereFzis the gate electric field and Fz≥0. The 4 ×4 Luttinger Hamiltonian, which is expressed in the basis of Jz
eigenstates {|+3
2∝an}bracketri}ht,|−3
2∝an}bracketri}ht,|+1
2∝an}bracketri}ht,|−1
2∝an}bracketri}ht}, reads
HL(k2,kz)=
P+Q0L M
0P+Q M∗−L∗
L∗M P−Q0
M∗−L0P−Q
, (2)
where
P=µ
2γ1(k2+k2
z), Q=−µ
2γ2(2k2
z−k2),
L=−√
3µγ3k−kz, M=−√
3µ
2(γk2
−+δk2
+).(3)
withµ=/planckover2pi12
m0,γ1,γ2,γ3are the Luttinger parameters (Table I),γ=γ2+γ3
2,δ=γ2−γ3
2, andk2=/radicalig
k2x+k2y,k±=
kx±ikyandθ= arctanky
kx. To obtain the spectrum of our system, we use modified infinite squa re well wave functions
[3] for the heavy hole (HH) and light hole (LH) states
φv=sin/bracketleftbigπ
d/parenleftbig
z+d
2/parenrightbig/bracketrightbig
exp/bracketleftbig
−βv/parenleftbigz
d+1
2/parenrightbig/bracketrightbig
π/radicalig
e−βvdsinh(βv)
2π2βv+2β3v, (4)
wherev=h,ldenote the HH and LH states and dis the width of the quantum well. The eigenvalues of the heavy
hole and light hole as well as the corresponding kdependent expansion coefficients are then obtained by diagonalizing
the matrix ˜H, whose elements are given as
˜H=∝an}bracketle{tν|HL(k2,ˆkz)+V(z)|ν′∝an}bracketri}ht, (5)
where|ν∝an}bracketri}htdenotes the wave function Eq. ( 4) andˆkzstands for the operator −i∂
∂z. The two lowest eigenenergies of
the 4×4 matrix Eq. ( 5) correspond to the dispersion of the spin-split HH1 ±subbands. Usually, only the lowest
HH-subspace is taken into account at low hole densities. Accordingly , we perform a Schrieffer-Wolff transformation
on Eq.5to restrict our attention to the lowest HH subspace. Therefore, the effective Hamiltonian describing the two
dimensional hole gas is [ 4]
H0k=/planckover2pi12k2
2m∗+iα(k3
−σ+−k3
+σ+), (6)
Table I. Luttinger parameters used in this work [ 2].
GaAs InAs InSb
γ1 6.85 20.40 37.10
γ2 2.10 8.30 16.50
γ3 2.90 9.10 17.702
wherem∗≡m∗
hh=m0
γ1+γ2and the Pauli matrix σ±=1
2(σx±iσy). The eigenvalues of Eq. ( 6) areεk,±=ǫ0±αk3,
whereǫ0=/planckover2pi12k2
2m∗. The Rashba coefficient αis expressed as
α=3µ2
∆Eγ2∝an}bracketle{tφL|φH∝an}bracketri}ht∝an}bracketle{tφH|kz|φL∝an}bracketri}ht. (7)
where ∆ Eis energy splitting of heavy hole and light hole.
II. SCATTERING TERM
Thek-diagonal part of density matrix fkis a 2×2 Hermitian matrix, which is decomposed into fk=nk11+Sk,
wherenkrepresents the scalar part and 11 is the identity matrix into two dimensions. The component Skis written
purely in terms of the Pauli σmatrices Sk=1
2Sk·σ≡1
2Skiσi. With this notation, the scattering term is in turn
decomposed into
ˆJ(fk) =ni
/planckover2pi12/integraldisplayd2k′
(2π)2|Ukk′|2(nk−nk′) lim
η→0/integraldisplay∞
0dt′e−ηt′e−iH0k′t′//planckover2pi1eiH0kt′//planckover2pi1+H.c.
+ni
2/planckover2pi12/integraldisplayd2k′
(2π)2|Ukk′|2(Sk−Sk′)·lim
η→0/integraldisplay∞
0dt′e−ηt′e−iH0k′t′//planckover2pi1σeiH0kt′//planckover2pi1+H.c..(8)
We use perturbation theory solving the kinetic equation up to α2. In the process, we decompose the matrix Sk=
Sk/bardbl+Sk⊥and write those two parts as Sk/bardbl= (1/2)sk/bardblσk/bardblandSk⊥= (1/2)sk⊥σk⊥. The terms sk/bardblandsk⊥are
scalars and given by sk/bardbl=Sk·ˆΩkandsk⊥=Sk·ˆΘkwithˆΩk=−sin3θˆx+cos3θˆyandˆΘk=−cos3θˆx−sin3θˆy.
Withγ=θ′−θ, the scattering term becomes
ˆJ(n) =πni
2/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2(nk−nk′)·(1+ˆΩk′·ˆΩk)/bracketleftig
δ(ǫ+−ǫ′
+)+δ(ǫ−−ǫ′
−)/bracketrightig
+πni
2/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2(nk−nk′)·σ·(ˆΩk′+ˆΩk)/bracketleftig
δ(ǫ′
+−ǫ+)−δ(ǫ′
−−ǫ−)/bracketrightig
+πni
2/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2(nk−nk′)·(1−ˆΩk′·ˆΩk)/bracketleftig
δ(ǫ+−ǫ′
−)+δ(ǫ−−ǫ′
+)/bracketrightig
+πni
2/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2(nk−nk′)·σ·/bracketleftig
(ˆΩk−ˆΩk′)/bracketrightig/bracketleftig
δ(ǫ′
−−ǫ+)−δ(ǫ′
+−ǫ−)/bracketrightig
,(9)
and
ˆJ(S) =πni
4/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2(Sk−Sk′)·/bracketleftig
σ(1−ˆΩk·ˆΩk′)+(ˆΩk·σ)ˆΩk′+ˆΩk(ˆΩk′·σ)/bracketrightig/bracketleftig
δ(ǫ+−ǫ′
+)+δ(ǫ−−ǫ′
−)/bracketrightig
+πni
4/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2(Sk−Sk′)·(ˆΩk+ˆΩk′)/bracketleftig
δ(ǫ′
+−ǫ+)−δ(ǫ′
−−ǫ−)/bracketrightig
+πni
4/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2(Sk−Sk′)·/bracketleftig
σ(1+ˆΩk·ˆΩk′)−(ˆΩk·σ)ˆΩk′−ˆΩk(ˆΩk′·σ)/bracketrightig/bracketleftig
δ(ǫ+−ǫ′
−)+δ(ǫ−−ǫ′
+)/bracketrightig
+πni
4/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2(Sk−Sk′)·/bracketleftig
(ˆΩk−ˆΩk′)/bracketrightig/bracketleftig
δ(ǫ+−ǫ′
−)−δ(ǫ−−ǫ′
+)/bracketrightig
.
(10)
We now separate these terms according to the contributions from intra-band and inter-band scatterings
ˆJ(n) =πni
2/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2(nk−nk′)(1+cos3 γ)/bracketleftig
δ(ǫ+−ǫ′
+)+δ(ǫ−−ǫ′
−)/bracketrightig
+πni
2/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2(nk−nk′)(1−cos3γ)/bracketleftig
δ(ǫ+−ǫ′
−)+δ(ǫ−−ǫ′
+)/bracketrightig
,(11)3
ˆJ(S) =πni
4/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2/bracketleftig
(sk/bardbl−sk′/bardbl)(1+cos3 γ)σk/bardbl+(sk/bardbl−sk′/bardbl)sin3γσk⊥
+(sk⊥+sk′⊥)σk/bardblsin3γ+(sk⊥+sk′⊥)(1−cos3γ)σk⊥/bracketrightig/bracketleftig
δ(ǫ+−ǫ′
+)+δ(ǫ−−ǫ′
−)/bracketrightig
+πni
4/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2/bracketleftig
(sk/bardbl+sk′/bardbl)(1−cos3γ)σk/bardbl−(sk/bardbl+sk′/bardbl)sin3γσk⊥
−(sk⊥−sk′⊥)σk/bardblsin3γ+(sk⊥−sk′⊥)(1−cos3γ)σk⊥/bracketrightig/bracketleftig
δ(ǫ+−ǫ′
−)+δ(ǫ−−ǫ′
+)/bracketrightig
,(12)
and
ˆJS→n(S) =πni
4/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2/bracketleftig
(sk/bardbl−sk′/bardbl)(1+cos3 γ)+(sk⊥+sk′⊥)sin3γ/bracketrightig/bracketleftig
δ(ǫ′
+−ǫ+)−δ(ǫ′
−−ǫ−)/bracketrightig
=πni
4/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2/bracketleftig
(sk/bardbl+sk′/bardbl)(1−cos3γ)−(sk⊥−sk′⊥)sin3γ/bracketrightig/bracketleftig
δ(ǫ+−ǫ′
−)−δ(ǫ−−ǫ′
+)/bracketrightig
,(13)
ˆJn→S(n) =πni
2/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2(nk−nk′)/bracketleftig
σk/bardbl(1+cos3 γ)+σk⊥sin3γ/bracketrightig/bracketleftig
δ(ǫ′
+−ǫ+)−δ(ǫ′
−−ǫ−)/bracketrightig
+πni
2/planckover2pi1/integraldisplayd2k′
(2π)2|Ukk′|2(nk−nk′)/bracketleftig
σk/bardbl(1−cos3γ)−σk⊥sin3γ/bracketrightig/bracketleftig
δ(ǫ′
−−ǫ+)−δ(ǫ′
+−ǫ−)/bracketrightig
.(14)
We next decompose the kinetic equation as follows:
dnk
dt+ˆJn→n(nk) =Dkn,
dSk/bardbl
dt+P/bardblˆJS→S(Sk/bardbl) =Dk/bardbl,
dSk⊥
dt+i
/planckover2pi1/bracketleftbig
H0k,Sk⊥/bracketrightbig
=Dk⊥.(15)
Note that the projection operator P/bardblabove acts on a matrix Mas Tr(Mσk/bardbl), where Tr refers to the matrix (spin)
trace.
III. SOLUTION FOR THE LONGITUDINAL CONDUCTIVITY
Here we derive the longitudinal conductivity at zero magnetic field. E xpanding the δfunctions in Sec. IIup to
∝α2, we get the following
δ(ǫ+−ǫ′
+)≈δ(ǫ0−ǫ′
0)+α(k3−k′3)∂
∂ǫ0δ(ǫ0−ǫ′
0)+α2(k3−k′3)2
2∂2δ(ǫ0−ǫ′
0)
∂ǫ2
0
δ(ǫ−−ǫ′
−)≈δ(ǫ0−ǫ′
0)−α(k3−k′3)∂
∂ǫ0δ(ǫ0−ǫ′
0)+α2(k3−k′3)2
2∂2δ(ǫ0−ǫ′
0)
∂ǫ2
0
δ(ǫ+−ǫ′
−)≈δ(ǫ0−ǫ′
0)+α(k3+k′3)∂
∂ǫ0δ(ǫ0−ǫ′
0)+α2(k3+k′3)2
2∂2δ(ǫ0−ǫ′
0)
∂ǫ2
0
δ(ǫ−−ǫ′
+)≈δ(ǫ0−ǫ′
0)−α(k3+k′3)∂
∂ǫ0δ(ǫ0−ǫ′
0)+α2(k3+k′3)2
2∂2δ(ǫ0−ǫ′
0)
∂ǫ2
0.(16)
We now insert Eq. ( 16) into the electric driving term DE,kand scattering term ˆJ(fk). With ρ0k=f0++f0−
211 +
f0+−f0−
2σk/bardblandf0+,f0−equilibrium Fermi distribution function, the driving term DE,kbecomes,
DE,kn=−eE·ˆk
2/planckover2pi1(∂f0+
∂k+∂f0−
∂k)≈eE·ˆk
2/planckover2pi1/bracketleftig
2/planckover2pi12k
m∗δ(ǫ0−ǫF)+6α2k5∂δ(ǫ0−ǫF)
∂ǫ0/bracketrightig
,
DE,k/bardbl=−eE·ˆk
2/planckover2pi1(∂f0+
∂k−∂f0−
∂k)σk/bardbl≈eE·ˆk
2/planckover2pi1/bracketleftig
6αk2δ(ǫ0−ǫF)+2/planckover2pi12k
m∗αk3∂δ(ǫ0−ǫF)
∂ǫ0)/bracketrightig
.(17)4
Solving Eqs. ( 15), we obtain the density matrices
n(0)
Ek=τpeE·ˆk
/planckover2pi1/bracketleftig/planckover2pi12k
m∗δ(ǫ0−ǫF)/bracketrightig
, (18a)
S(1)
Ek/bardbl=τsαeE·ˆk
/planckover2pi1/bracketleftig/planckover2pi12k4
m∗∂δ(ǫ0−ǫF)
∂ǫ0+3k2δ(ǫ0−ǫF)/bracketrightig
σk/bardbl=s(1)
Ek/bardblσk/bardbl, (18b)
n(2)
Ek=τpα2/braceleftigeE·ˆk
/planckover2pi1/bracketleftig
3k5∂δ(ǫ0−ǫF)
∂ǫ0/bracketrightig
−3km∗2ni
4απ/planckover2pi15s(1)
Ek/bardblζ(γ)−n(0)
Ek6nim∗3
π/planckover2pi17k2ξ(γ)/bracerightig
. (18c)
whereǫF=/planckover2pi12k2
F
2m∗,τp=2π/planckover2pi13
m∗niξ(γ),τs=4π/planckover2pi13
m∗niβ(γ), and
ζ(γ)=/integraldisplay
dγ|Ukk′|2(cosγ−cos3γ), ξ(γ)=/integraldisplay
dγ|Ukk′|2(1−cosγ), β(γ)=/integraldisplay
dγ|Ukk′|2(1−cosγcos3γ).(19)
In the low temperature limit, the Thomas-Fermi wave-vector of a t wo-dimensional hole gas without spin-orbit
coupling is kTF=2
aB, withaB=/planckover2pi12ǫr
m∗e2. The screened Coulomb potential between plane waves is given by
|Ukk′|2=Z2e4
4ǫ2
0ǫ2r/parenleftbigg1
|k−k′|+kTF/parenrightbigg2
. (20)
With Eq. ( 20), we obtainζ(γ)
ξ(γ)≈2 andξ(γ)
β(γ)=1
3. Using the velocity operator
ˆvx=/planckover2pi1kx
m∗+α
/planckover2pi13k2[−sin2θσx+cos2θσy],ˆvy=/planckover2pi1ky
m∗+α
/planckover2pi13k2[−sin2θσy−cos2θσx], (21)
the longitudinal current is jx=eTr/bracketleftbig
ˆvxρEk/bracketrightbig
, where ρEk= (n(0)
Ek+n(2)
Ek)11 +S(1)
Ek/bardbl. Therefore, the longitudinal
conductivity with Rashba spin orbit coupling up to second order in αis
σxx=e2τp
2πm∗k2
F/bracketleftig
1−15
2/parenleftbiggαk3
F
ǫkin/parenrightbigg2/bracketrightig
, (22)
whereǫkin=/planckover2pi12k2
F
2m∗.
IV. SOLUTION FOR THE HALL COEFFICIENT
Now we consider the case of Bz>0. Firstly, we find that the Zeeman terms have no contribution to th e Hall
coefficient. With Eqs. ( 21), the Lorentz driving term DL,kbecomes
DL,k=1
2e
/planckover2pi1/braceleftig
ˆv×B,∂ρEk
∂k/bracerightig
=1
2eBz
/planckover2pi1/braceleftig/braceleftbig
ˆvy,∂ρEk
∂kx/bracerightbig
−/braceleftbig
ˆvx,∂ρEk
∂ky/bracerightbig/bracerightig
. (23)
We separate DL,kinto the scalar and spin parts with DL,k=DL,n+DL,S, and, switching from the rectangular
coordinates to polar coordinates with∂D
∂kx=∂D
∂kcosθ−∂D
∂θsinθ
k;∂D
∂ky=∂D
∂ksinθ+∂D
∂θcosθ
k, we obtain
DL,n=−eBz
m∗/bracketleftbig
n(0)
k+n(2)
k/bracketrightbig
(−sinθ)+eBz
/planckover2pi13αk
/planckover2pi1s(1)
k,/bardbl(−sinθ),
DL,S/bardbl=−/braceleftigeBz
m∗s(1)
k,/bardbl(−sinθ)+eBz
/planckover2pi13αk
/planckover2pi1/bracketleftbig
n(0)
k+n(2)
k/bracketrightbig
(−sinθ)/bracerightig
σk/bardbl,
DL,S⊥= cosθeBz
/planckover2pi13αk2
/planckover2pi1∂/bracketleftbig
n(0)
k+n(2)
k/bracketrightbig
∂kσk⊥,(24)5
withn(0)
Ek=n(0)
kcosθ,n(2)
Ek=n(2)
kcosθands(1)
Ek,/bardbl=s(1)
k,/bardblcosθ. Solving Eqs. ( 15), we obtain the following density
matrices in presence both electric and magnetic fields
nBz,k=−sinθτpeBz/braceleftign(0)
k+n(2)
k
m∗+3αk
/planckover2pi12s(1)
k,/bardbl/bracerightig
,
SBz,k/bardbl=−sinθτseBz/braceleftigs(1)
k,/bardbl
m∗+3αk
/planckover2pi12/bracketleftbig
n(0)
k+n(2)
k/bracketrightbig/bracerightig
σk/bardbl,
SBz,k⊥= cosθ3eBz
2/planckover2pi1k∂/bracketleftbig
n(0)
k+n(2)
k/bracketrightbig
∂kσz.(25)
The Hall current is jy=eTr/bracketleftbig
ˆvyρEB
k/bracketrightbig
, whereρEBz
k=nBz,k11 +SBz,k/bardbl+SBz,k⊥. The Hall coefficient, up to the
second order in α, is thus given as
RH=σxy
Bz(σ2xx+σ2xy)≈1
pe/bracketleftig
1+8/parenleftbiggαk3
F
ǫkin/parenrightbigg2/bracketrightig
, (26)
whereωc=eBz
m∗.
[1] J. M. Luttinger, Phys. Rev. 102, 1030 (1956) .
[2] R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole systems (Springer, Berlin, 2003).
[3] G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev. B 28, 3241 (1983) .
[4] R. Winkler, Phys. Rev. B 62, 4245 (2000) . |
2304.12632v1.Magnetization_Switching_in_van_der_Waals_Systems_by_Spin_Orbit_Torque.pdf | 1
Magnetization Switching in van der Waals Systems by Spin-Orbit Torque
Xin Lin1,2, Lijun Zhu1,2*
1. State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of
Sciences, Beijing 100083, China
2. College of Materials Science and Opto -Electronic Technology, University of Chinese Academy of Sciences, Beijing
100049, China
*ljzhu@semi.ac.cn
Abstract : Electrical switching of magnetization via spin -orbit torque (SOT) is of great potential in fast, dense, energy -
efficient nonvolatile magnetic memory and logic technologies. Recently, enormous efforts have been stimulated to
investigate switching of perpendicular magnetization in van der Waals systems that have unique, strong tunability and
spin-orbit coupling effect compared to conventional metals. In this review , we first give a brief, generalized introduction
to the spin -orbit torque and van der Waals materials. We will then discuss the recent advances in magnetization switching
by the spin current generated from van der Waals materials and summary the progress in the switching of Van der Waals
magnetization by the spin current .
1. Introduction
1.1 Spin -orbit torque
Spin-orbit torque s (SOT s) are a powerful tool to
manipulat e magnetization at the nanoscale for spintronic
devices, such as magnetic random access memory (MRAM)
and logic [1-5]. SOTs are exerted on a magnetization when
angular momentum is transferred from spin accumulation
or spin currents carried by a flow of electrons or magnons
(Fig. 1 ). A spin current with spin polarization vector σ, can
exert two types of SOTs on a magnetization M, i.e., a
damping -like (DL) torque [ τDL ~ M × (M × σ)] due to the
absorption of the spin current component transverse to M
and a field-like (FL) torque [ τFL ~ M × σ] due to the
reflection of the spin current with some spin rotation . In the
simple case of the spin-current generator /magnet bilayer,
the efficienc y of the damping -like SOT per unit bias
current density, 𝜉DL𝑗
, can be estimated as [6]
𝜉DL𝑗 ≈ Tint θSH τM-1/(τM-1+τso-1) (1)
where τM-1/(τM-1+τso-1) is the percentage of the spin
current relaxed via the spin-magnetization exchange
interaction (with spin relaxation rate τM-1) within the
magnetic layer and is less than 1 in presence of non -
negligible spin relaxation via the spin-orbit scattering
(with spin relaxation rate τM-1) [6], Tint is the interfacial
spin transparency which determines what fraction of the
spin current enters the magnet (less than 1 in presence
of spin backflow [7-11] and spin memory loss [12-15]),
and θSH is the charge -to-spin conversion efficiency of
the spin current generator (e.g., the spin Hall ratio in the
case of spin Hall materials [2-4]). The quantitative
understanding of the efficiency of the field -like torque ,
𝜉FL𝑗, remains an open question.
The same SOT physics can be expressed in terms of
effective SOT fields: a damping -like effective SOT field
(HDL) parallel to M × σ and a field -like effective SOT field
(HFL) parallel to σ. The magnitudes of the damping -like
and field -like SOT fields correlate to their SOT efficiencies
per unit bias current density via
HDL = (ℏ/2e) j𝜉DL𝑗Ms-1t -1 （2）
HFL = (ℏ/2e) j𝜉FL𝑗Ms-1t -1 （3） where e is the elementary charge, ℏ reduced Plank’s
constant, t the magnetic layer thickness, Ms the saturation
magnetization of the magnetic layer, and j the charge
current density in the spin current generating layer.
The damping -like SOT is technologically more
import ant because it can excite dynamics and switching of
magnetization (even for low currents for which HDL is
much less than the anisotropy field of a perpendicular
magnetization). Field -like SOTs by themselves can
destabilize magnets only if HFL is greater th an the
anisotropy field, but they can still strongly affect the
dynamics in combination with a damping -like SOT [16-18].
The g eneration of spin currents is central to the SOT
phenomena. The spin polarization vector σ can have
longitudinal, transverse, and perpendicular components,
i.e., σx, σy, and σz. Transversely polarized spin current can
be generated by a longitudinal charge current flow in either
magnetic or non -magnetic materials via a variety of
possible spin -orbit -coupling (SOC) effects. The latter
includes the bulk spin Hall effect (SHE) [19-22],
topological surface states [23,24], interfacial SOC effects
[16,25-28], orbit -spin conversion [29], the anomalous Hall
effect [30,31], the planar Hall effect [32,33], the magnetic
SHE [34-36], Dresselhaus effect [37], Dirac nodal lines
[38,39], etc. The bulk S HE has been widely observed in
thin-film heavy metal (HM) [40-42], Bi-Sb [43], Bi xTe1-x
[44], CoPt [45], FePt [46], Fe xTb1-x [47], and Co -Ni-B [48].
Fig. 1 Schematic of damping -like and field -like spin-orbit
torque s exerted on a magnetic layer by an incident spin
current .
2
Since the transverse spins, in principle, cannot switch
a uniform perpendicular magnetization without the
assistance of an in -plane longitudinal magnetic field either
via coherent rotation of macros pin or via domain wall
depinning [2-4,49], generation of perpendicular and
longitudinal spins [50-53] are of great interests. While
perpendi cular and longitudinal spins are not allowed in
nonmagnetic materials that are cubic crystal or
polycrystalline/amorphous, additional crystal or magnetic
symmetry breaking can be introduced to make
perpendicular and longitudinal spins permissive.
Generat ion of perpendicular spins has been argued
from low-symmetry crystals (e.g., WTe 2 [50], MoTe 2
[54,55], and CuPt [56]), non -collinear antiferromagnetic
crystals with magnetic asymmetry (e.g. , IrMn 3 [57],
Mn 3GaN [58,59], Mn 3Sn [34]), some collinear
antiferromagnets with spin conversions (e.g. , Mn 2Au
[60,61] and RuO 2 [62-64]), and also some magnetic
interfaces [65]. Longitudinal spins might be generated by
low-symmetry crystals ( e.g., MoTe 2 [54], (Ga, Mn)As [66],
NiMnSb [67], and Fe/GaAs [68]) or by a non -zero
perpendicular magnetization [69,70].
1.2 van der Waals materials
So far, the most widely used spin -source materials are
heavy metals with strong spin Hall effect (e.g., Pt with a
giant spin Hall conductivity of 1.6×106 (ℏ/2e) Ω-1 m-1 in the
clean limit) [71], while the 3 d ferromagnets (e.g., Co, Fe,
Ni81Fe19, CoFeB ) and ferrimagnets (e.g., FeTb, CoTb , and
GdFeCo) are well studied a s the spin -detect ors. However,
it is believed that the energy efficiency of SOT devices may
be improved by developing new materials and new
mechanisms that generate spin currents.
Van der Waals materials have attracted enormous
attention in the field of material science and condensed
matter physics since the discovery of single monolayer
graphene in ref. [72] and is increasingly investigated in the
field of spintronics . This is particularly due to the diversity
of materials, the flexibility of preparation (e.g., by
mechanical exfoliation), and strong tunability by interface
effects. In the field of spintronics, the van der Waals
materials are also interesting for intriguing SOC effects.
For example, the non-magnetic van der Waals materials of
transition metal dichalcogenides (TMDs) and topological
insulators (TIs) [73,74] exhibit a strong ability in
generating transversely polarized spin current via the spin
Hall effect and/or topological surface states [75-77], in
some cases also in generating currents of perpendicular and
longitudinal spins due to low crystal symmetry [50,51]. On
other hand, van der Waals materials that are magnetic are
also interesting for spin-orbitronics due to their highly
tunable magneti sm and low magnetization at small
thicknesses [78-83]. As indicated by Eq. (2) and Eq. (3),
the damping -like and field -like effective SOT fields
exerted on the magnetic layer by a given spin current scale
inversely with the thickness and magnetization of the
magnetic layer .
This review is intended to focus on highlighting t he
recent advances of magnetization switching in van der
Waals systems by spin -orbit torque , including switching of
conventional magnetic metals by spin current from non -
magnetic van der Waals TMDs and TIs and switching of
van der Waals magnets by incident spin currents . 2. Magnetization s witching by spin current from
Transition Metal Dichalcogenide semimetals
TMDs typically consist of transition metals (e.g., Mo,
W, Pt, Ta, Zr) and chalcogenide elements (e.g., S, Te, Se).
From point of view of SOT applications, it is most
interesting to develop the TMD semi -metals with relatively
high conductivity, strong spin -orbit coupling, high θSH, and
reduced structural symmetry. However, TMD
semiconductors (e.g., MoS 2 [84,85], WS 2 [86,87], WSe 2
[85]) are merely studied in spin -orbitronics due to their
very poor conductivity that is detrimental to the energy
TMD metal s (e.g., TaS 2 [88], NbSe 2 [89]) are highly
conductive but typically not efficien t in generating spin
currents. Therefore, b elow we mainly discuss the progress
in spin-orbit torque stud ies of TMD semimetals that are
interesting for SOT applications (e.g., WTe 2 [50,90,91],
PtTe 2 [92], MoTe 2 [51,93], and ZrTe 2 [94]).
2.1 Exfoliated Transition Metal Dichalcogenide
semimetals
The pioneering s tudies of TMD semimetal s in the
field of spintronics [50] mainly focused on the
characterization of damping -like and field -like SOTs of
transvers e, perpendicular, and longitudinal spins generated
in bilayers of mechanically exfoliated TMD semimetal s
and 3d ferromagnets (e.g., Ni81Fe19). These studies have
opened a new subject field that unitizes van der Waals
materials for the possible generation of transverse,
perpendicular, and longitudinal spin polarizations.
WTe 2 is a semimetal with a low inversion symmetry
along the a axis of the crystal (the space group Pmn21) [Fig.
2(a)] and the Weyl points at the crossing of the oblique
conduction and the valence bands only at low temperatures
(typically below 100 K [95]). In a WTe 2/ferromagnet
bilayer, the screw -axis and glide -plane symmetries of this
space group are broken at the interface, so that
WTe 2/ferromagnet bilayers have only one symmetry, a
mirror symmetry relative to the bc plane (depicted in Fig.
2(a)). There is no mirror symmetry in the ac plane, and
therefore no 180◦ rotational symmetry about the c axis
(perpendicular to the sample plane). MacNeill et al. [50]
first observe d in mechanically exfoliated WTe 2/Ni81Fe19
bilayers damping -like (≈ 8×103 (ℏ/2e) Ω-1 m-1) and field -
like spin -orbit torques of transverse spins at room
temperature as well as t he exotic damping -like SOT of
perpendicular spins (≈ 3.6×103 (ℏ/2e) Ω-1 m-1). The
damping -like SOT of perpendicular spins manifests as an
additional sin2 𝜑 term in the antisymmetric signal of spin -
torque ferromagnetic resonance ( ST-FMR )[Fig. 2(b)] and
was attributed to the symmetry breaking at the interface of
the WTe 2 crystal. The damping -like SOT of perpendicular
spins is found to maximize when current is applied along
the low-crystal -symmetry a axis and vanishes when current
is applied along the high-crystal -symmetry b axis [50].
This is in contrast to the torques of the transverse spins that
are independent of the c rystal orientation. The damping -
like SOT of perpendicular spins in WTe 2/Ni 81Fe19 was also
found to vary little with the WTe 2 thickness, which was
suggest ed as an indication that the spin current is mainly
generated near the interface of the WTe 2 [90,91,96]. Xie et
al. [52] reported that in -plane direct current along the a axis
of WTe 2 can induce partial switching of magnetization in
absence of an external magnetic field [Fig. 2(c)] and shift 3
of the anomalous Hall resistance loop in SrRuO 3/exfoliated
WTe 2 bilayers , which was speculated as an indication of
damping -like torque of perpendicular spins on the
perpendicular magnetization (macrospin ). However, the re
can be longitudinal and perpendicular Oersted field s due to
current spreading in the WTe 2 layer [90,91,96], which can
also induce “field -free” switching of perpendicular
magnetization and anomalous Hall loop shifts via adding to or subtracting from the domain wall depinning field
(coercivity) . The SOTs of the WTe 2/Ni 81Fe19 have also
been reported to switch the Ni81Fe19 layer with weak in -
plane magnetic anisotropy at a current density of ≈ 3×105
A cm-2 [91]. An in-plane current along the a axis of WTe 2
has also been reported to enable partial switching of
perpendicular magnetization in WTe 2/Fe2.78GeTe 2 without
an external magnetic field [53].
FIG. 2. (a) Crystal structure near the surface of WTe 2, displaying a mirror symmetry relative to the bc plane but not to the
ac plane. (b) Symmetric (VS) and antisymmetric (VA) components of ST -FMR signal for WTe 2 (5.5)/Py (6) device as a
function of the angle of the in-plane magnetic field [50]. Reprinted with permission from Mac Neill et al., Nat. Phys. 13,
300 (2017). (c) Current -induced magnetization switching of WTe 2(15)/SrRuO3 when the current is along the low -
symmetry a axis where the magnetization can be switched without an external magnetic field [52]. Reprinted with
permission from Xie et al., APL Mater. 9, 051114 (2021). (d) Structure of the MoTe 2 crystal in the monoclinic ( β or 1T′)
phase depicted in the a-c plane for which the mirror plane is within the page and the Mo chains run into the page. (e)
Symmetric and antisymmetric ST -FMR resonance components for the MoTe 2(0.7)/Py(6) device with a current applied
perpendicular to the MoTe 2 mirror plane as a function of the orientation of the in-plane magnetic field. (f) The
conductivities of damping -like torque of perpendicular spins (blue) and transverse spins (red) as a function of the MoTe 2
thickness for devices with current aligned perpendicular to the MoTe 2 mirror plane [51]. Reprinted with permission from
Stiehl et al ., Phys. Rev. B 100, 184402 (2019). (g) Crystal structure of the monoclinic 1T′ phase of MoTe 2. (h)
Antisymmetric ST -FMR components for MoTe 2 (83.1)/Py(6) as a function of the orientation of the in-plane magnetic field.
(i) MOKE images implying switching of Py by current [93]. Reprinted with permission from Liang et al., Adv. Mater. 32,
2002799 (2020).
4
FIG. 3. (a) Current -induced magnetization switching in sputter -deposited WTe 2(10)/CoTb(6)/Ta(2) Hall bar (Hx= ± 900
Oe) [97]. Reprinted with permission from Peng et al., ACS Appl. Mater. Interfaces 13, 15950 (2021). (b) Second harmonic
longitudinal resistance (𝑅2𝜔𝑥𝑥) of WTe x(5)/Mo(2)/CoFeB(1) measured as a function of pulse current amplitude Ipulse under
zero external field [98]. (c) Current -induced switching loops of WTe x(5)/Ti(2)/CoFeB (1.5) Hall bar under different in-
plane magnetic fields at 200 K [99]. Reprinted with permission from Xie et al., Appl. Phys. Lett. 118, 042401 (2021). (d)
Dependences on the WTe x thickness of damping -like SOT efficiency (𝜉DL𝑗) and the WTe x resistivity (𝜌𝑥𝑥) for WTe 2/CoFeB .
(e) Apparent spin Hall conductivity as a function of the longitudinal conductivity for WTe 2/CoFeB [98]. Data in (b), (d),
and (e) are r eprinted with permission from Li et al., Matter 4, 1639 (2021).
FIG. 4. (a) Schematic of the CVD growth process for PtTe 2. (b) High -resolution transmission electron microscopy image
of a 5 nm PtTe 2 thin film. (c) Current -induced magnetization switching in the PtTe 2(5)/Au(2.5) /CoTb (6) Hall bar under
different in -plane field s [92]. Reprinted with permission from Xu et al., Adv. Mater. 32, 2000513 (2020). (d) Spin torque
ferromagnetic resonance spectrum of a ZrTe 2/Py bilayer at room temperature. (e) Current -induced magnetization
switching in ZrTe 2(8 u.c.)/CrTe 2(3 u.c.) Hall bar under a 700 Oe in -plane field at 50 K [94]. Reprinted with permission
from Ou et al., Nat. Commun. 13, 2972 (2022).
β-MoTe 2 is a semimetal that retains inversion
symmetry in bulk but has a low-symmetry interface (the
group space is Pmn21 in bulk but Pm11 in few -layer
structures [Fig. 2(d)]). Stiehl et al. [51] observed damping -
like SOT of both transverse spins (≈ 8×103 (ℏ/2e) Ω-1 m-1)
and perpendicular spins (≈ 1×103 (ℏ/2e) Ω-1 m-1) in
mechanically exfoliated β-MoTe 2/Ni81Fe19 bilayers [Fig. 2(e)]. This torque of perpendicular spins is one -third strong
than that of WTe 2/Ni 81Fe19 and was attributed to
perpendicularly polarized spin current from the surface of
the low -symmetry β-MoTe 2 [Fig. 2(f)]. This appears to
suggest that the breaking of bulk inversion symmetry is not
an essential requirement for producing perpendicular spins.
However, 1T′ -MoTe 2 [Fig. 2(g)] was reported to generate
5
no damping -like SOT of perpendicular spins in contact
with Ni 81Fe19 [Fig. 2(h)][93]. Instead, 1T′ -MoTe 2 only
generates a nonzero damping -like SOT of transverse spins
that switches the in -plane magnetized Ni 81Fe19 layer at a
current density of 6.7×105 A cm-2 [Fig. 2(i)]. NbSe 2 with
resistivity anisotropy was reported to generate a
perpendicular Oersted field but no perpendicular or
longitudinal spins when interfaced with Ni 81Fe19 [89]. The
damping -like toque of transverse spins in mechanically
exfoliated NbSe 2/Ni81Fe19 is very weak and corresponds to
a spin Hall conductivity of ≈ 103 (ℏ/2e) Ω-1 m-1 [89].
Here it is important to note that, while the presence
of perpendicular spins has been widely concluded in the
literature from a sin2 φ-dependent contribution in
symmetric spin-torque ferromagnetic resonance signal of
in-plane magnetization ( φ is the angle of the external
magnetic field relative to the current), or a φ-independent
but field -dependent contribution in the second harmonic
Hall voltage of in -plane magnetization, or field -free
switching, none of the three characteristics can simply
“signify” the presence of a flow of perpendicular spins.
This is because non -uniform current effects that can
generally exist and generates out -of-plane Oersted field in
nominally uniform, symmetric Hall bars a nd ST -FMR
strips [90,91,96,100] also exhibit all three characteristics.
As demonstrated by Liu and Zhu [100], these
characteristics can be considerable especially when the
devices have strong current spreading, e.g., in presence of
non-symmetric electric contact s.
2.2 Large -area Transition Metal Dichalcogenides
So far, most TMD studies have been based on
mechanical exfoliation, which is unsuitable for the mass
production of spintronic applications. Recently, efforts
have been made in large -area growth of thin-film TMDs
towards the goal of SOT applications [97,98]. For example,
sputter -deposited WTe x has also developed to drive low -
current -density switching of CoTb ( jc ≈ 7.05×105 A cm-2
under in -plane assisting field of 900 Oe) [Fig. 3(a)] [97]
and in WTe x/Mo/CoFeB ( jc ≈ 7×106 A cm-2, under no in-
plane assisting field, Fig. 3(b))[99] and in WTe x/Ti/CoFeB
(jc ≈ 2.0×106 A cm-2 under in -plane assisting field of ± 30
Oe, Fig. 3(c))[98]. It has become a consensus that the spin -
orbit torque in these sputter -deposited WTe x/FM samples
arises from the bulk spin Hall effect of the WTe x [97,98].
As indicated in F igs. 3(d) and 3(e), the measured spin -orbit
torque efficiency increases but the apparent spin Hall
conductivity decreases as the resistivity increases in the
dirty limit [41] due to increasing layer thickness.
Large -area PtTe 2 films with relatively high electrical
conducti vity (≈ 3.3×106 Ω-1 m−1 at room temperature) and
spin Hall conductivity (2×105 ℏ/2e Ω −1m−1) have also been
reported by annealing a Pt thin film in tellurium vapor at ≈
460 °C [Figs. 4(a) and 4(b)][92]. PtTe 2 is predicted to be
a type -II Dirac semimetal with spin -polarized surface
states. However, there is no indication of the generation of
torques of out -of-plane spins. Partial switching of
magnetization by in -plane current has also been reported in
a PtTe 2(10)/Au(2.5)/CoTb(6) Hall bar ( jc ≈ 9.9×106 A cm−2
under in -plane assisting field of 2 kOe) [Fig. 4(c)].
Growth of ZrTe 2 by molecular beam epitaxy (MBE)
has also been reported. A ST -FMR study has measured a
small damping -like torque of transverse spins for MBE -grown ZrTe 2/Ni 81Fe19 bilayer at room temperature [Fig.
4(d)][94]. This is consistent with the theoretical prediction
that ZrTe 2 is a Dirac semimetal with massless Dirac
fermions in its band dispersion [101] but vanishing spin
Hall conductivity. Even so, a ZrTe 2(8 u.c.)/CrTe 2(3 u.c.)
bilayer has been reported to be partially switched at 50 K
by an in -plane current of 1.8×107 A cm-2 in density under
an in-plane assisting field of 700 Oe [Fig. 4(e)].
3. Magnetization s witching by spin current from Bi-
based topological insulators
Another kind of layered strong -SOC material is Bi -
based topological insulators [102-104]. As displayed in Fig.
5 (a) , TIs are insulating in the bulk but conducting at the
surface. The initial interest of TIs for spin-orbit torque
studies is the topological surface states ( Fig. 5 (b) ). In the
wavevector space, the spin and momentum of electrons are
one-to-one locked to each other at the Fermi level. With a
flow of charge current, the shift in the electron distribut ion
in the wavevector space induces non -equilibrium spin
accumulation (Fig. 5 ( c)).
Fig. 5. Topological surface states and spin -accumulation in
topological insulators. (a) Real -space picture of the
conducting surface states in an ideal topological insulator
[103]. Reprinted with permission from Han an d Liu, APL
Mater. 9, 060901 (2021). (b) Angle -resolved
photoemission spectrum that indicates the bulk and surface
bands of a six -quintuple -layer -thick Bi 2Se3 film [102].
Reprinted with permission from Zhang et al., Nat. Phys. 6,
584 (2010). Copyright 2010 Springer Nature Limited. (c)
Current -induced spin accumulation in a topological
insulator [104]. The arrows denote the directions of spin
magnetic moments, which are opposite to the
corresponding spin angular momenta. Reprinted with
permission from He et al., Nat. Mater. 21, 15 –23 (2 022).
3.1 MBE -grown and exfoliated Topological insulators
Topological insulators were first introduced in the
field of spin -orbit torque i n 2014 . From ST -FMR
measurement , Mellnik et al . measured a giant damping -
like spin-orbit torque efficiency (𝜉DL𝑗 = 3.5) at room
temperature in Bi 2Se3/Py bil ayers grown by MBE [Fig.
6(a)][23]. In the same year, Fan et al . reported from
harmonic Hall measurement a damping -like torque
efficiency of 4 25 and the spin–orbit torque switching in the
6
(Bi 0.5Sb0.5)2Te3/(Cr 0.08Bi0.54Sb0.38)2Te3 bilayers [24] at 1.9
K [Fig. 6(b)].
As shown in Fig. 7 , room -temperature magnetization
switching by spin current from TIs (e.g., Bi 2Se3, Bi 2Te3,
and BiSb ) has been demonstrated in Hall -bar samples
[99,105]. Han et al. [76,106] first reported magnetization
switching in Hall bars of Bi 2Se3(7.4)/Co 0.77Tb0.23(4.6)
bilayer ( Hx = 1000Oe, Hc ≈ 200Oe, Jc ≈ 2.8×106 A cm-2,
switching ratio=85%)[Fig. 7(a)]. The damping -like SOT
efficiency was determined to be 0.16 ± 0.02 for the
Bi2Se3/Co 0.77Tb0.23. Similar results have been also reported
by Wu et al. [77] in Bi 2Se3/Gd x(FeCo) 1−x Hall bars ( 𝜉DL𝑗=
0.13, Jc ≈ 2.2×106 A cm-2)[Fig. 7(b)]. These values of spin -
orbit torque efficiency are significantly low compared to
those from Bi 2Se3/Py samples, which may be understood
partly by the increased spin current relaxation via spin -
orbit scattering in the ferrimagnets [6]. Khang et al. have
reported a spin -orbit torque efficiency of 52 (as determined
from a coercivity change measurement) and resistivity of
400 μΩ cm for MBE -grown Bi 1-xSbx [107]. Switching of
MBE -grown fully epitaxial Mn 0.45Ga0.55/Bi0.9Sb0.1 has also
been demonstrated at a current density of 1.1×106Acm–2
(Hx = 3.5 kOe) in [Fig. 7(c)]. Non -epitaxial BiSb films (10
– 20 nm) grown by MBE were also reported to have a high
spin-orbit torque efficiency of up to 3.2 and to enable
magnetization switching at a current density of 2.2×106 A
cm-2. There have also been reports of SOT switching of
exfoliated van-der-Waals magnets at low temperatures,
such as in Fe 3GeTe 2 [108,109] and Cr 2Ge2Te6 [110,111], by
the spin current from TIs. Liu et al . [112] reported a
strongly temperature -dependent damping -like torque
efficiency of up to 70 from a field -dependent harmonic
Hall response measurement[Fig. 8(a)], and current
switching of MBE -grown Bi 2Te3/MnTe Hall bar at a
critical current density of down to 6.6×106 Acm–2 (Hx = ±
400 Oe, T = 90 K) [Fig. 8(b)].
FIG. 6. (a) Spin torque ferromagnetic resonance spectrum
for Bi 2Se3(8)/Py(16) bilayer at room temperature [23].
Reprinted with permission from Mellnik et al., Nature 511,
449 (2014). (b) Second harmonic Hall resistance for
(Bi 0.5Sb0.5)2Te3(3 QL)/(Cr 0.08Bi0.54Sb0.38)2Te3(6 QL) bilayer
as a function of the in -plane field angle for different applied
a.c. current [24]. Reprinted with permission from Fan et al.,
Nat. Mater. 13, 699 (2014).
3.2 Sputter -deposited Topological Insulators
Since exfoliation and molecular -beam epitaxy are less realistic methods for the preparation of large -area TI thin
films for practical SOT devices, s puttering has been
introduced to grow amorphous or polycrystalline
“topological insulators”. The first report of sputter -
deposited “topological insulators” is BixSe(1–x) [113] with
relatively high electrical conductivity ( 0.78×105 Ω-1m-1 for
4 nm thickness) . Such sputter -deposited Bi xSe(1–x) exhibits
a very high damping -like spin-torque efficiency of 18 and
enabled magnetization switching in a BixSe(1–
x)(4)/Ta(0.5)/CoFeB(0.6)/Gd(1.2)/CoFeB(1.1) at a low
current density of ≈ 4.3×105 A cm-2 [Fig. 8(c)]. Wu et al.
[105] also reported room -temperature witching of
Bi2Te3/Ti/CoFeB at a current density of 2 .4×106Acm–2 (Hx
= 100 Oe). In the Hall bar of sputter -deposited PMA
Bi2Te3(8)/CoTb(6) bilayer, current -induced magnetization
switching was reported at a low critical current density of
9.7×105 A cm-2 [Fig. 8(d)]. Sputter -deposited BiSb films
(10 nm) were reported to provide a spin -torque efficiency
of 1.2 and to drive switching of CoTb at 4×105 A/cm-2
[106].
3.3 Practical impact
As we have discussed above, some TIs and their
sputter -deposited counterparts are reported to have much
higher damping -like torque efficiency than heavy metals .
Meanwhile, the sputter -deposited TIs are typically several
times more resistive than the MBE -grown ones since
disordered films typically have stronger electron scattering
than crystalline films. However, for practical SOT
applications, the spin -source ma terials are required to have
low resistivity and large damping -like spin -orbit torque
efficiency . Despite their amazingly high damping -like
spin-orbit torque efficiency , most TIs are highly resistive (>
1×103 μΩ cm), much more resistive than ferromagnetic
metals in metallic spintronic devices (e.g., 110 μΩ cm for
CoFeB). Current shunting into the adjacent metallic layers
would be considerably more than that flow s within the
topological insulator layer, resulting in increases in the
total switching current a nd power consumption of devices.
3.4 Mechanism of the spin current generation
Despite the debate, t he two main mechanism s via
which the TIs and their disordered counterparts generate
spin current or spin accumulation are the spin Hall effect
and the surface states. As suggested by Khang et al. [82],
Chi et al. [43], Tian et al. [44], the bulk spin Hall effect is
the dominant source of the spin current for the spin -orbit
torque in Bi 0.9Sb0.1, Bi0.53Sb0.47, and BixTe1–x. As shown in
Figs. 9(a) and 9(b), in disordered Bi0.53Sb0.47 the apparent
spin Hall conductivity increases non -linearly with
increasing layer thickness, which is a typical spin diffusion
behavior and in good consistent with a bulk spin Hall effect
being the mechanism of the spin current generation. In
contrast, the surface states of the TIs have been suggested
to be the main spin current source in MBE -grown (Bi 1-
xSbx)2Te3 [105] and Bi 2Te3[112]. This suggestion is
consistent with the strong dependence of the damping -like
spin-orbit torque on the composition [105], the temperature
[112], and thus the location of the Fermi level relative to
the Dirac point [Figs. 9(c) - 9(e)][105,114]. In addition, DC
et al . suggested that the quantum confinement effect of
small grains should account for the high spin -torque
efficiency in the sputter -deposited Bi xSe(1–x) [113].
7
FIG. 7. Current -induced magnetization switching at room temperature in (a) Bi2Se3(7.4)/CoTb(4.6) bilayer ( the in-plane
magnetic field is 1000 Oe) [76], (b) Bi2Se3(6)/Gd x(FeCo) 1−x(15) bilayer ( an in -plane magnetic field is 1000 Oe) [77], and
(c) Mn 0.45Ga0.55(3)/Bi0.9Sb0.1(5) (3.5 kOe) [115]. Data in (a) is reprinted with permission from Han et al., Phys. Rev. Lett.
119, 077702 (2017). Data in (b) is reprinted with permission from Wu et al., Adv. Mater. 31, 1901681 (2019) ; Data in ( c)
is reprinted with permission from Khang et al., Nat. Mater. 17, 808 (2018).
FIG. 8. (a) Variation of the spin Hall ratio of Bi 2Te3 with temperature. ( b) Current -induced magnetization switching of
Bi2Te3(8)/MnTe(20) at 90 K under different in -plane magnetic field s [112]. Reprinted with permission from Liu et al.,
Appl. Phys. Lett. 118, 112406 (2021). (c) Bi xSe(1–x)(4)/Ta(0.5)/CoFeB(0.6)/Gd(1.2)/CoFeB(1.1) (the in-plane magnetic
field is 80 Oe) [113], reprinted with permission from DC et al., Nat. Mater. 17, 800 (2018) . (d) Current switching of
sputter -deposited Bi2Te3(8)/CoTb (6) under an in -plane magnetic field of 400 Oe at room temperature [116]. Reprinted
with permission from Zheng et al., Chin. Phys. B 29, 078505 (2020).
8
FIG. 9. (a) Scanning transmission electron microscopy image of the 0.5 Ta/[0.35 Bi|0.35 Sb] N/0.3 Bi/2 CoFeB/2 MgO/1
Ta structure with N = 8 . (b) thickness dependence of the apparent spin Hall conductiv ity σSH of Bi0.53Sb0.47 [43]. Reprinted
with permission from Chi et al., Sci. Adv. 6, eaay2324 (2020). (c) Fermi level , (d) Resistivity ρxx and 2D carrier density
|n2D| for of (Bi 1-xSbx)2Te3 with different Sb percentage s. (e) Switching current density |Jjc| and effective damping -like spin -
orbit torque field vs the Sb ratio of (Bi 1-xSbx)2Te3 [105]. Reprinted with permission from Wu et al., Phys. Rev. Lett. 123,
207205 (2019).
4. Magnetization s witching of Van der Waals magnet
4.1 Van der Waals magnet
The r ecent discovery of v an der Waals magnets (e.g.,
Cr2Ge2Te6 [82], CrI 3 [83], etc.) has attached remarkable
attention in the field of magnetism and spintronics. While
the origin of the long-range magnetic order is still under
debate, it has been suggested to have a close correlation
with the suppression of thermal fluctuations by magnetic
anisotropy. Note that in absence of magnetic anisotropy, no
long-range magnetic order is expected by the Mermin –
Wagner theorem [117] at finite temperature in a two-
dimensional system . Van-der-Waals magnets provide a
unique, highly tunable platform for spintronics. Most
strikingly, the properties of van der Waals FMs, such as
Curie temperature (TC) [78,80], coercivity [79,80], and
magnetic domain structure [81], can be tuned significantly
by a variety of techniques ( e.g., layer thickness, ionic liquid
gating [78], proton doping [79], strain [80,81], exchange
bias [118,119], interfacial proximity -effect [120], etc.). An
interesting example is CrI 3, whose magnetic ordering
depends on the number of layers and can be tuned by an
external magnetic field. As shown in Fig. 10(a) , the CrI 3 is
ferromagnetic at 1 monolayer thickness, antiferromagnetic
at 2 monolayer thickness , and ferromagnetic at 3
monolayer thickness. Ferromagnetic CrI 3 also shows a
relatively square perpendicular magnetization loop [83].
Following the long -range ordering of magnetic
lattices, magnetic materials can be grouped into
ferromagnets, ferrimagnets, and antiferromagnets . In
general, ferromagnets and ferrimagnets are considered
more friendly than antiferromagnet s to be integrated into
electric circuits because their magnetization states can be
electrically detected by anomal ous Hall effect or tunnel
magnetoresistance and efficiently switched by SOTs. In
contrast, electrical detection and switching of collinear
antiferromagnets [121-123] are generally much more
challenging [124], despite the recent discovery of magnetoresistance and anomalous Hall in non -collinear
antiferromagnets Mn 3Sn [34,125,126]. For this reason,
spin-torque switching of magnetization is mostly studied
and better understood in ferromagnetic and ferrimagnetic
systems than in antiferromagnets. Our discussion below
will be focused on van der Waals ferromagnet s [116-134].
The van-der-Waals magnet CrBr 3 (TC = 34
K)[127,128] (TC = 34 K), CrI 3 (TC = 45 K) [83], Cr 5Te8
[129] and VI 3 (TC = 60 K) [130] have perpendicular
magnetic anisotropy but low Curie temperature . So far,
room -temperature ferromagnetism and low-temperature
perpendicular magnetic anisotropy ha ve been reported for
van der Waals materials FeTe [131], Fe 4GeTe 2 (Fig.
10(b))[132], Fe 5GeTe 2 [133], CrTe [134], CrTe 2 (Fig.
10(c))[135-137], Cr 1+δTe2 [138], Cr 2Te3 [139], Cr 3Te4
[140]), CrSe [141], and Fe 3GaTe 2 (Fig. 10( d))[142]. In Fig.
11, we summar ize the representative results of the Curie
temperature and magnetization of relatively thin van der
Waals magnets (note that TC of van der Waals magnets is
strongly thickness dependent ). While FenGeTe 2 can have
good PMA at low temperatures and CrnTem and CrSe are
relatively stable in air, they lose square hysteresis loops at
room temperature [Fig. 10(b) and 10(c)]. The recently
discovered Fe 3GaTe 2 [142] is an outstanding van der Waals
ferromagnet that can have both a high Curie temperature
(TC ≈ 350 - 380 K) and large PMA energy density ( Ku ≈
4.8×105 J m-3) [Fig. 10(d)]. Search ing for Van der Waals
magnets with room -temperature ferromagnetism, strong
perpendicular magnetic anisotropy, a nd high stability in the
air at the same is expected to be an active topic in the field.
4.2 Magnetization s witching of v an der Waals
ferromagnets
Spin-orbit torque switching of v an der Waals
ferromagnets was first demonstrated in perpendicularly
magnetized Fe3GeTe 2/Pt bilayer s [108,109], where the spin
9
current generated by the SHE in the Pt exerts a damping -
like spin torque on the Fe 3GeTe 2 [Fig. 12(a)]. Interestingly ,
despite the small layer thicknesses and small magnetization
of the Fe 3GeTe 2, the Fe 3GeTe 2/Pt samples have a high
depinning field (coercivity) and strong perpendicular
magnetic anisotropy such that they typically require a large
current density of ~ 107 A cm-2 [108,109] as well as an in -
plane magnetic field [Fig. 12(a)]. As indicated by the
anomalous Hall resistance, the Fe 3GeTe 2 was also only
partially switched , with the switching ratio of 20%-30% in
[108] and 62% in [118], probably due to the non-
uniformity of the magnetic domains with the van der Waals
layer.
Cr2Ge2Te6 is another well -studied van der Waals
ferromagnet . Spin-orbit torque switching of Cr2Ge2Te6 has
been demonstrated in Ta/Cr 2Ge2Te6 bilayers (Curie
temperature < 65 K) at a low current density of 5×105 A cm-2 at 4 K, with an in -plane assisting magnetic field of 200
Oe [143]. Zhu et al . [110] reported SOT switching of
Cr2Ge2Te6/W with interface -enhanced Curie temperature
of up to 150 K [Fig. 12(b)].
Current -induced magnetization switching has also
been realized in all van der Waals heterojunction s. Nearly
full magnetization switching (88%) has been reported in
MBE -grown Cr2Ge2Te6/(Bi 1-xSbx)2Te3 bilayer s [144][Fig.
13(a)]. In the (Bi0.7Sb0.3)2Te3/Fe 3GeTe 2 bilayer [145], the
threshold current density for the magnetization switching
is 5. 8×106 A cm-2 at 100 K [Fig. 13(b)]. Field-free
switching magnetization has been reported in the
exfoliation -fabricated WTe 2/Fe 2.78GeTe 2 bilayers by a
current along the low symmetry axis (9.8×106 A cm-2, T
=170K) [53]. In the same bilayer structure, Shin et al. also
realized magnetization switching at a current density of
3.9×106 A cm-2 (T = 150 K, Hx = 300 Oe , Fig. 13(c)) [146].
FIG. 1 0. (a) Kerr rotation vs perpendicular magnetic field for monolayer (1L), bilayer (2L) , and trilayer (3L) CrI 3 flake
[83]. Reprinted with permission from Huang et al., Nature 546, 270 (2017). ( b) Hall conductivity hysteresis loop of a 11 -
mono layer -thick Fe 4GeTe 2 crystal at various temperature s [132]. Reprinted with permission from Seo et al., Sci. Adv. 6,
eaay8912 (2019). (c) Out-of-plane magnetization hysteresis loop of 7 monolayer CrTe 2 at different temperatures along
the out -of-plane direction [137]. Reprinted with permission from Zhang et al ., Nat. Commun. 12, 2492 (2021). ( d)
Anomalous Hall resistance hysteresis (of Fe3GaTe 2 with different thickness es at 3 K and 300 K [142]. Reprinted with
permission from Zhang et al., Nat. Commun. 13, 5067 (2022).
FIG. 1 1 Saturation magnetization Ms vs Curie temperature TC of representative van der Wa als ferromagnets .
10
FIG. 1 2. (a) Current -driven perpendicular magnetization switching for Fe 3GeTe 2(4)/Pt(6) bilayer under an in-plane
magnetic field of 50 0 Oe at 100 K [108]. Reprinted with permission from Wang et al., Sci. Adv. 5, eaaw8904 (2019). (b)
Current -driven perpendicular magnetization switching for Cr2Ge2Te6(10)/W(7) bilayer under in -plane magnetic fields of
± 1 kOe at 150 K [110]. Reprinted with permission from Zhu et al., Adv. Funct. Mater. 32, 2108953 (2022).
FIG. 1 3. Current induced switching of van der Waals magnets. (a) Normalized anomalous Hall resistance vs current
density for (Bi 1-xSbx)2Te3(6)/Cr 2Ge2Te6(t) (x = 0.5) with different Cr 2Ge2Te6 thicknesses under an in-plane magnetic field
of 1 kOe at 2 K [144]. Reprinted with permission from Mogi et al., Nat. Commun. 12, 1404 (2021). (b) Anomalous Hall
conductivity vs current density for (Bi 1-xSbx)2Te3(8)/Fe3GeTe 2(6) at different temperatures under an in-plane magnetic
field of 1 kOe [145]. Reprinted with permission from Fujimura et al., Appl. Phys. Lett. 119, 032402 (2021). (c) Hall
resistance vs current density for Fe 3GeTe 2(7.3)/WTe 2(12.6) under in -plane magnetic field Hx = 300 Oe . The Hall resistance
varies during three consecutive current scans due to temperature rise towards the Curie temperature [146]. Reprinted with
permission from Shin et al., Adv. Mater. 34, 2101730 (2022).
5. Simplifying models of switching current density
In this section, we provide a quantitative
understanding of the switching current densities in the van
der Waals system by considering the simplifying models.
the transverse spin damping -like SOT efficiency per unit
current density ( 𝜉𝐷𝐿𝑗 ) of a heterostructure with PMA
inversely cor relates to the critical switching current density
(jc) in the spin -current -generating layer via Eq. (4) in
macrospin limit [149,150] and via Eq. (5) in the domain
wall depinning limit [151-153], i.e., jc ≈ eμ0MstFM (Hk-√2|𝐻𝑥|)/ћ𝜉DL𝑗, (4)
jc = (4e/πћ) μ0MstFMHc/𝜉DL𝑗, (5)
where e is the elementary charge, ℏ is the reduced Planck
constant, μ0 is the permeability of vacuum, Hx is the applied
field along the current direction, and tFM, Ms, Hk, and Hc
are the thickness, the saturation magnetization, the
effective perpendicular anisotropy field, and the
perpendicular coercivity of the driven magnetic layer FM,
respectively.
11
However, r ecent experiments [154] on heavy
metal/m agnet bilayers have shown that neither Eq. (4) nor
Eq. (5) can provide a reliable prediction for the switching
current and 𝜉𝐷𝐿𝑗 and that there is no simple correlation
between 𝜉𝐷𝐿𝑗 and the critical switching current density of
realistic perpendicularly magnetized spin -current
generator/ferromagnet heterostructures. As shown in Table
I, the same is true for the van der Waals systems. The
macrospin analysis does not seem to apply to the switching
dynamics of micrometer -scale samples so that the values
of 𝜉𝐷𝐿𝑗 determined using the switching current density and
Eq. ( 4) can produce overestimates by up to hundreds of
times (𝜉𝐷𝐿,𝑚𝑎𝑐𝑟𝑜𝑗 and 𝜉𝐷𝐿,𝑚𝑎𝑐𝑟𝑜𝑗/𝜉𝐷𝐿𝑗 in Table I). A domain -
wall depinning analysis [ Eq. ( 5)] can either under - or over -estimated 𝜉𝐷𝐿𝑗 by up to tens of times (𝜉𝐷𝐿,𝐷𝑊𝑗 and
𝜉𝐷𝐿,𝐷𝑊𝑗/𝜉𝐷𝐿𝑗 in Table I). These observations consistently
suggest that the switching current or “switching efficiency”
of perpendicular heterostructures in the micrometer or sub -
micrometer scales cannot provide a quantitative estimation
of 𝜉𝐷𝐿𝑗.
While the underlying mechanis m of the failure of the
simplifying models remains an open question, it is obvious
that Joule heating during current switching of the resistive
or low Curie -temperature van der Waals systems can have
a rather significant influence on the apparent switching
current density. As shown in Fig. 13(c), the anomalous Hall
resistance hysteresis loop drifts for three consecutive
current scans because of Joule heating [146].
TABLE I. Comparison of spin -torque efficiencies determined from the harmonic response or ST -FMR (𝜉𝐷𝐿𝑗 ) and
magnetization switching ( 𝜉𝐷𝐿,𝐷𝑊𝑗, 𝜉𝐷𝐿,𝑚𝑎𝑐𝑟𝑜𝑗) of PMA samples, which is calculated from Eq. (5) and Eq. (4) using the
applied external magnetic field ( Hx), saturation magnetization ( Ms), the perpendicular coercivity ( Hc), and the effective
perpendicular anisotropy field ( Hk) of the driven magnetic layer , and the critical magnetization switching current density
(jc). The value of 𝜉𝐷𝐿,𝑚𝑎𝑐𝑟𝑜𝑗 is estimated to be negative for Te2(10)/CoTb(6) [97] and ZrTe 2(7.2) /CrTe 2(1.8) [94] because
according to th e original reports an in -plane field greater that the effective perpendicular anisotropy field was applied.
Sample Technique Ms
(emu cm-3) Hc
(Oe) Hk
(kOe) Hx
(Oe) jc
(MA cm-2) 𝜉𝐷𝐿𝑗 𝜉𝐷𝐿,𝐷𝑊𝑗 𝜉𝐷𝐿,𝑚𝑎𝑐𝑟𝑜𝑗 𝜉𝐷𝐿,𝐷𝑊𝑗/𝜉𝐷𝐿𝑗 𝜉𝐷𝐿,𝑚𝑎𝑐𝑟𝑜𝑗/𝜉𝐷𝐿𝑗
Te2(10)/CoTb(6) [97] sputtering 48 60 0.33 900 0.7 0.2 0.47 -5.8 2.4 -29
ZrTe 2(7.2) /CrTe 2(1.8) [94] MBE 100 -- 0.2 700 18 0.014 -- -0.12 -- -8.5
Bi2Se3(7.4)/CoTb(4.6) [76] MBE 280 300 -- 1000 2.8 0.16 2.7 -- 17 --
Bi2Se3(6)/Gd x(FeCo) 1−x(15) [77] MBE 46 160 0.35 1000 2.2 0.13 0.98 10 7.6 78
BixSe(1–x) (4)/Ta(0.5)/ CoFeB
(0.6)/Gd(1.2)/CoFeB(1.1) [113] sputtering 300 30 6 80 0.43 8.67 1.2 180 0.13 21
(BiSb) 2Te3(6QL)/Ti
(2)/CoFeB(1.5) [105] MBE 868 30 2.24 100 0.12 2.5 6.3 345 2.5 138
Bi2Te3/Ti(2) (6QL)/CoFeB(1.5)
[105] MBE 868 27 2.06 100 2.4 0.08 0.287 16 3.5 194
Bi2Te3(8)/MnTe(20) [112] MBE 175 100 ≈50 400 6.6 10 1.0 397 0.10 40
Bi0.9Sb0.1(5) /Mn 0.45Ga0.55(3)
[115] MBE 240 4500 10 3500 1.1 52 57 50 1.1 0.96
SnTe(6QL)/Ti(2)/CoFeB(1.5)
[105] MBE 868 53 2.18 100 1.5 1.41 0.91 27.5 0.65 20
Fe3GeTe 2(4)/Pt(6) [108] exfoliation 16 125 11 500 11.6 0.12 0.013 0.86 0.11 7.2
Fe3GeTe 2 (15)/Pt(5) [109] exfoliation 170 750 30 3000 20 0.14 1.8 50 13 355
6. Conclusion and outlook
We have reviewed recent advances in spin -orbit
torque and resultant magnetization switching in van der
Waals systems. Van der Waals materials provided unique
opportuni ties for spintronics because of their diversity of
materials, the flexibility of preparat ion (e.g., by mechanical
exfoliation), and strong tunability by interface effects.
Van der Waals TMDs such as WTe 2 also exhibit potential
as a spin current source of both transverse spins and exotic
perpendicular and lo ngitudinal spins . Bismuth -based
topological insulators and their sputter -deposited
disordered counterparts are shown to generate giant
damping -like SOT with the efficiency of up to 1-3 orders
of magnitudes greater than 5 d metals. Moreover, van der
Waals materials that are m agnetic are also interesting for
spin-orbitronics due to their highly tunable magnetism and
low magnetization at small thicknesses since the damping -like and field -like effective SOT fields exerted on the
magnetic layer by a given spin current scale invers ely with
the thickness and magnetization of the magnetic layer.
Efficient switching of several van der Waals magnet s by
spin current has been demonstrated .
Despite the se exciting progres s, essential challenges
have remained to overcome for spin-orbit torque switching
of van der Waals systems :
(i) While the generation of different spin components
has been demonstrated, the efficiencies are typically quite
low. It has remained unclear as to whether and how the
efficiency of generating exotic perpendicular and
longitudinal spins by the low -symmetry TMDs can be
improved significantly to be compelling for practical SOT
applications .
(ii) Some Bi -based t opological i nsulators and alloys
have both giant effective spin Hall ratio and resistivity at 12
the same time, the latter is undesirable for metallic SOT
devices that require energy efficiency, high endurance, and
low impedance. It would be interesting if new uniform,
stable spin -orbit materials can be developed to provide
damping -like SOT efficiency of greater than 1 but
substantially less resistive than the yet -know n topological
insulators.
(iii) So far, large -area growth of van der Waals
magnets that have high Cu rie temperature, strong
perpendicular magnetic anisotropy, and high stabilities
against heating and atmosphere at the same time has
remained a key obstacle that prevents van der Waals
magnets from applications in spintronic technologie s.
Breakthroughs in th e fabrication of such application -
friendly van der Waals magnets are in urgent need.
(iv) While spin -orbit torque switching of
magnetization has been demonstrated in Hall -bar devices
containing van der Waals magnets, TMDs, or/and
topological insulators, t he simplifying models of
macrospin rotation and domain wall depinning cannot
provide an accurate prediction for the switching current
density. So far, the quantitative understanding of the
switching current remains a fundamental problem .
(v) From the point of view of magnetic memory and
logic application, s witching of thermally stable nanodots of
van der Waals magnet s, such as the free layers of
nanopillars of magnetic tunnel junctions, by current pulse s
of picosecond and nanosecond durations. Efforts are a lso
required on the demonstration and optimization of t he key
performance , e.g., the endurance, the write error rates, the
retention, and the tunnel magnetoresistance .
Acknowledgments
This work was supported partly by the National Key
Research and Develop ment Program of China (Grant No.
2022YFA12 0400 4) and by the National Natural Science
Foundation of China (Grant No. 12274405 ).
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1606.08334v2.Nonreciprocal_Transverse_Photonic_Spin_and_Magnetization_Induced_Electromagnetic_Spin_Orbit_Coupling.pdf | 1
NONRECIPROCAL TRANSVERSE PHOTONIC SPIN AND MAGNETIZATION -INDUCED
ELECTROMAGNETIC SPIN -ORBIT COUPLING
Miguel Levy and Dolendra Karki
Physics Department, Michigan Technological University
Henes Center for Quantum Phenomena
ABSTRACT
A study of nonreciprocal transverse -spin a ngular -momentum -density shift s for evanescent waves
in magneto -optic waveguide media is presented. Their functional relation to electromagnetic spin-
and orbital -momenta is presented and analyzed. It is shown that the magneto -optic gyrotropy can
be re -interpreted as the nonreciprocal electromagnetic spin-density shift per unit energy flux , thus
providing an interesting alternative physical picture for the magneto -optic gyrotropy . The
transverse spin -density shift is fou nd to be thickness -dependent in slab optical waveguides. This
dependence is traceable t o the admixture of minority helicity component s in the transverse spin
angular momentum. It is also shown that the transverse spin is magnetically tunable . A formulation
of electromagnetic spin-orbit coupling in magneto -optic media is presented, and an alternative
source of spin -orbit coupling to non -paraxial optics vortices is proposed. It is shown that
magnetization -induced electromagnetic spin -orbit coupli ng is possible, and that it leads to spin to
orbit al angular momentum conversion in magneto -optic media evanescent waves .
INTRODUCTION
In 1939, F. J. Beli nfante introduced a spin momentum density expression for vector fields to
explain the spin of quantum part icles and symmetrize the energy -momentum tensor [ 1]. For
monochromatic electromagnetic waves in free -space , the corresponding time-averaged spin
momentum density reads
1
2BBps
, (Eq. 1)
and the spin angular momentum density is
* 1Im ( )2Bos E E
. (Eq. 2 )
is the optical frequency and
0 the permittivity of free -space [2].
This spin angular momentum , in its transverse electromagnetic form , has merited much
attention in recent years, as it can be studied in evanescent waves [ 3-7]. There are fundamental
and practical reasons for this.
Until recently, the quantum field theory of the electromagnetic field has lacked a description of
separate local conservation laws for the spin and orbital angular momentum -generating currents
[7]. Whether such spin -generating momenta , as opposed to the actual spin angular momenta they
induce, are indeed observable or merely ‘virtual’ is of fundamental interest . Moreover, if the
electromagnetic spin and orbital momenta are separable, the question arises as to whether there are
any photonic spin -orbit interaction effects . Bliokh and co -workers give a positive answer for non -
paraxial fields. [7] Using the conservation laws proposed by these authors, we show that it is
possible to magnetically induce electromagnetic spin -orbit coupling in magneto -optic media. 2
We know that the transverse optical spin is a physically meaningful quantity that can be
transferred to material particles [3-8]. This has potential ly appealing consequences for optical -
tweezer particle manipulation , or to locate and track nanoparticles with a high degree of te mporal
and spatial resolution [9 ]. Thus, developing means of control for the transverse optical spin is of
practical interest.
In this paper, w e address the latter question for both spin momenta
Bp
and angular momenta
Bs
. We show that the ir magnitude s and sense of circulation can be accessed and controlled in a
single structure , and propose a specific configuration to this end . Explicit expressions for these
physical quantities and for the spin -orbit coupling are presented. Moreover, we develop our
treatment for nonreciprocal slab optical waveguides , resulting in a different response upon time
reversal s.
We consider the behavior of evanescent waves in transverse -magnetic (TM) modes in magnetic
garnet claddings on silicon -on-insulator guides. This allows us to obtain explicit expressions for
the transverse Beli nfante spin momenta and angular momenta and to propose a means for
magnetically controlling these objects, with potential applicabilit y to nanoparticle manipulation.
MAGNETIC -GYROTROPY -DEPENDENT EVANESCENT WAVES
Consider a silicon -on-insulator slab waveguide with iron garnet top cladding , as in Fig. 1 . The
off-diagonal component s of the garnet’s dielectric permittivity tensor contr ol the structure’s
magneto -optic response. Infrared 1550nm wavelength light propagates in the slab, in the presence
of a magnetic field transverse to the dir ection of propagation .
The electromagnetic field-expressions in the top cladding for transverse magnetization (y-
direction) and monochromatic light propagating in the z -direction are,
()
()( , )e
( , )i z t
o
i z t
oE E x y
H H x y e
(Eq. 3)
Maxwell -Ampere’s and Faraday’s laws in ferrimagnetic media are
00
ˆ 0 0 0 0
00cc
o o c o c
ccig ig
EEH i Ettig ig
(Eq. 4)
ooHE i Ht
(Eq. 5)
The off -diagonal component of the dielectric permittivity tensor
ˆ is the gyrotropy parameter ,
parameterized by g.
We examine transverse -magnetic (TM) propagation in the slab. Vertical and transverse -
horizontal directions are x, and y, respectively, is the propagation constant, and the wave equation
in the iron garnet is given by 3
22
22
20y o c y
cgH k Hx , with
02k , for wavelength
[10, 11 ]. (Eq. 6)
Defining
2
eff c
cg
as an effective permittivity in the cover layer, we get:
,0effx
ycH H e x
(Top cladding ) (Eq. 7)
cos( ), x 0y f x cH H k x d
(Core ) (Eq. 8)
exp( ( )),y s sH H x d x d
, (Substrate) (Eq. 9)
where
22
eff o eff k
, (Eq. 10)
22
x o fkk
, (Eq. 11)
22
s o s k
(Eq. 12)
f
, and
s are the silicon -slab and substrate dielectric -permittivity constants, respectively, and d is
the slab thickness.
Solving for the electric -field components in the top cladding layer, we get,
22
0
22
0()
()c eff
zy
c
c eff
xy
cgE i Hg
gEHg
(Eq. 13 )
Notice that these two electric field components are
/2 out of phase, hence the polarization is
elliptical in the cover laye r, with optical spin transverse to the propagation direction. In addition,
the polarization evinces opposite helicities for counter -propagating beam s, as
/zxEE changes sign
upon propagation direction reversal.
This result already contains an important difference with reciprocal non-gyrotropic
formulations, where
//zxE E i , and
the decay constant in the top cladding. E quation 13
depends on the gyrotropy parameter g, both explicitly and implicitly through
, and is therefore
magnetically tunable, as we shall see below.
We emphasize that the magnitude and sign of the propagation constant
change upon
propagation direction reversal, and , separately, upon magnetization direction reversal. The
difference between forward and backward propagation constants is also gyrotropy dependent. This
nonreciprocal quality of magneto -optic waveguides is central to the proper functioning of certain
on-chip devices, such as Mach -Zehnder -based optical isolat ors [10, 11 ].
As pointed out before, Eq. 2 applies to free-space Maxwell electromagnetism. In a dielectric
medium, the momentum density expression must account for the electronic response to the optical 4
wave. Minkowski’s and Abraham’s formulations describe the canonical and the kinetic
electromagnetic momenta , respectively [12]. Here we w ill focus on Minkowski’s version ,
p D B
, as it is intimately linked to the generation of translations in the host medium, and hence
to optical phase shifts , of interest in nonreciprocal phenomena.
D
is the dis placement vector, and
B
the magnetic flux density.
Dual -symmetric versions of electromagnetic field theory in free space have been considered by
various authors [ 2, 7, 8, 13]. However, t he interaction of light and matter at the local level often
has an electric character. Dielectric probe particles will generally sense the electric part of the
electromagnetic momentum and spin densities [2, 7, 8, 13]. Hence, we treat the standard (electric -
biased) formulation of the electromagnetic spin and orbital angular momenta. In the presence of
dielectric media , such as iron garnets in the near -infrared range , the expression for spin angular
momentum become s
*
, Im ( )2o
BMs E E
. (Eq. 14)
The orbital momentum is
*Im ( ( ) )2Oop E E
, where (Eq. 15)
()x x y y z z X Y X Y X Y X Y
, and
is the relative dielectric permittivity of the medium [4,
a, D, E] .
In magneto -optic media, the dielectric permittivity
is
cg , depending on the helicity of the
propagating transverse circular polarization. This is usually a small correction to
c , as g is two -,
or three -, orders of magnitude smaller in iron garnets , in the near infrared range . For elliptical
spins, where one he licity component dominates, we account for the admixture level of the minority
component in
through a weighted average .
NONRECIPROCAL ELECTROMAGNETIC TRANSVERSE SPIN ANGULAR MOMENTUM
AND SPIN -ORBIT COUPLING
1. Transverse Spin Momentum and Angular M omentum Densities in Non -Reciprocal Media
In thi s section we present a formulation for the transverse -spin momentum and angular
momentum densities , as well as the orbi tal angular momentum density, induced by evanescent
fields in nonreciprocal magneto -optic media . The magnitude and tuning range of these objects in
terms of waveguide geometry and optical gyrotropy are expounded and discussed. We detail t heir
unequal response to given optical energy fluxe s in opposite propagation directions and to changes
in applied magnetic fields. And we apply the recently proposed Bliokh -Dressel -Nori
electromagnetic spin-orbit correction term to calculate the spin -orbit interaction for evanescent
waves in gyrotropic media [7].
Equation (13), together with Eq. (14) and Eq. (15) yield the following expressions for the
transv erse Belinfante -Minkowski spin angular momentum, spin momentum and the orbital
momentum densities in evanescent nonreciprocal electromagnetic waves, 5
2
, 3 2 2 2 2ˆc eff c eff
B M y
o c cggs H y
gg
(Eq. 16)
2
, 3 2 2 2 2ˆeff c eff c eff
B M y
o c cggp H z
gg
(Eq. 17)
22
2
3 2 2 2 2ˆ
2c eff c eff
Oy
o c cggp H z
gg
(Eq.18 )
And the ratio
, 2c eff c eff O
B M c eff c effgg p
s g g
(Eq. 19)
These expression s depend on the magneto -optic gyrotropy parameter g and the dielectric
permittivity of the waveguide core channel and of its cover layer under transverse magnetization.
They yield different values under magnetic field tuning, magnetization and beam propagation
direction reversals, and as a function of waveguide core thickness as discussed below . The
propagation constant
is gyrotropy -, propagation -direction -, and waveguide -core-thickness -
dependent, and this behavior strongly impacts the electromag netic spin and orbital momenta.
The time -averaged electromagnetic energy flux (Poynting’s vector) in the iron garnet layer is
2* 1
2 22
01ˆ Re( )2 ()c eff
y
cgS E H H z
g
. (Eq. 20)
Re-expressing the transverse Belinfante -Minkowski spin angular momentum and spin
momentum densities in terms of the energy flow
S
,
, 2 2 22ˆc eff
BM
cgs S y
g
(Eq. 21)
, 2 2 22eff c eff
BM
cgpS
g
(Eq. 22)
Figure 1 plots the nonreciprocal Belinfante -Minkowski transverse spin-angular -momentum -
density shift per unit energy flux, as a function of silicon slab thickness in an SOI slab waveguide
with Ce 1Y2Fe5O12 garnet top cladding . Calculations are performed for the same electromagnetic
energy flux in opposite propagation directions, at a wavel ength of 1550 nm,
0.0086 g . The
nonreciprocal shift is normalized to the average spin angular momentum, as follows, 6
,,2f c eff b c efffb
B M S
f c eff b c efffbgg
s
gg
. (Eq. 23)
Subscripts f and b stand for forward, and backward propagation , respectively. This expression
evinces a relatively stable value , close to 0.7% above 0 .3m thickness. What is the explanation for
this? It has to do with the ellipticity of the transverse polarization in the x -z plane. Above 0.3m,
the ellipticity ranges from 31.4° to 36.9°, where 45° corresponds to circular polarization. In other
words, the ellipticity stays fairly constants, with a moderately small admixture of the minority
circularly polarized component , ranging from 25% to 14%. Below 0.3m, the minority component
admixture increases precipitously, reaching 87% at 0.13m.
Magnetization reversals produce the same effect. Consider the nonreciprocal Belinfante -
Minkowski transverse spin -angular -momentum -density shif t, as a function of silicon slab
thickness. Figure 2 plots the normalized shift in Eq. 16 pre -factor,
2 2 2 2 2 2 2 2
,
2 2 2 2 2 21
2c eff c eff c eff c eff
c c c cgg
BM
c eff c eff c eff c eff
c c c cgg g g g
g g g g
s
g g g g
g g g
22
gg (Eq. 24)
We observe the same qualitative thickness dependence as in Fig. 1 , corresponding to the
moderate, and relatively stable, admixture of minority circularly -polarized component above
0.3m thickness. 7
Fig. 1. Normalized nonreciprocal Belinfante -Minkowski transverse spin -angular -momentum -
density shift per unit energy flux as a function of silicon slab thickness for
0.0086 g ,
corresponding to Ce 1Y2Fe5O12 garnet top cladding on SOI at
1.55 m wavelength . The inset
shows the slab waveguide structure. M stands for the magnetization in the garnet.
The magneto -optic gyrotropy of an iron garnet can be controlled through an app lied magnetic field.
These ferri magnetic materials evince a hysteretic response, suc h as the one displayed in Fig. 3
(inset) for 532nm wavelength in a sputter -deposited film. The target composition is
Bi1.5Y1.5Fe5.0O12. Shown here are actual experimental data extracted from Faraday rotation
measurements. Below saturation, the magneto -optic response exhibits an effective gyrotropy value
that can be tuned through th e applic ation of a magnetic field. These measurements correspond to
a 0.5m-thick film on a (100) -oriented terbium gallium garnet (Tb GG) substrate. The optical beam
is incident normal to the surface, and the hysteresis loop probes the degree of magnetization norma l
to the surface as a function of applied magnetic field. These data show that the electromagnetic
spin angular momentum can b e tuned below saturation and between opposite magnetization
directions.
Figure 3 also revea ls an interesting feature about the magneto -optic gyrotropy. The normalized
nonreciprocal Belinfante -Minkowski transverse spin -angular -momentum -density shift per unit
energy flux,
,,B M Ss
, linearly tracks the gyrotropy, and is of the same order of magnitude as g,
although thickness -dependent . Yet, as pointed out before, this thickness dependence reflects the
admixture of the minor helicity component in the spin ellipticity. At 0.4m, for example,
,, 0.0072B M Ss
when
0.0086 g . However, the major polarization helicity component
contribution to
,,B M Ss
is 84.4% at this thickness, translating into 0.00853 at 100%. At 0.25m,
8
,, 0.00655B M Ss
, and the major polarization helicity component contribution is 76.2%,
translating into 0.0086 at 100%. We, thus, re-interpret the magneto -optic al gyrotropy as the
normalized Belinfante -Minkowski spin -angular -momentum density shift per unit energy flux.
Fig. 2 . Normalized nonreciprocal Belinfante -Minkowski transverse spin -angular -momentum -
density pre-factor shift as a function of silicon slab thickness for
0.0086 g , corresponding to
Ce1Y2Fe5O12 garnet top cladding on SOI at
1.55 m wavelength.
9
Fig. 3. Normalized nonreciprocal Belinfante -Minkowski transverse spin -angular -momentum -
density shift per unit energy flux as a function of magneto -optical gyrotropy. Data correspond to
0.25m silicon -slab thickness with Ce1Y2Fe5O12 garnet top cladding ,
1.55 m wavelength. The
inset shows the gyrotropy versus magnetic field hysteresis loop of a magnetic garnet film at
532nm
, sputter -deposited using a Bi1.5Y1.5Fe5.0O12 target .
2. Magnetization -Induced Electro magnetic Spin -Orbit Coupling
Bliokh and co -authors have studied the electromagnetic spin -orbit coupling in non -paraxial optical
vortex beams [7, 13]. They find that there is a spin dependent term in the orbital angular
momentum expression that leads to spin -to-orbit angular momentum conversion. This
phenomenon occurs under tight focusing or the scattering of light [ 7, 13]. Here we consider an
alternativ e source of electromagnetic spin -orbit coupling, magnetization -induced coupling in
evanescent waves.
The time-averaged spin- and orbital -angular momenta co nservation laws put forth in [7] are
* * * * 1Im ( ) Im22o
t i j ij i j j i E E H E H E H E
, and (Eq. 25)
22* * * 11Im ( ) Im ( )2 2 4o
t j jkl l i k j i ijk oiE r E H r E H E H E
(Eq. 26)
Latin indices i, j,… take on values x, y, z and
ijk is the Levi -Civita symbol. Summation over
repeated indices is assumed.
The interesting term in these equations, responsible for spin -orbit coupling, is
*Im2o
jiHE
. Notice
that it appears with opposite signs in the above equations , signaling a transfer of angular
momentum from spin to orbital motion. As it stands, so far in our treatment , this term equals zero,
since the spin points in the y -direction and the electric -field components of the TM wave point in
the x -, and z -directions. A way to overcome this null coupling, and enable the angular momentum
transfer, is to partially rotate the applied magnetic field about the x -axis away from the y -direction ,
as in Fig. 4 . This action induces a Faraday rotation about the z -axis, generating a spin -orbit
coupling term in the angula r momentum conservation laws. A slight rotation or directional gradient
in the magnetization M will induce electromagnetic spin-orbit interaction in the magneto -optic
medium .
Maxwell -Ampere’s law acquires off -diagonal components
ig in the dielectric permittivity
tensor upon rotation of the magnetic moment in the iron garnet film away from the y -axis, as shown
in Eq. 27 [14]. Hence, non -zero electromagnetic field components
yE and
xH , and spin -orbit
coupling, are induced in the propagating wave. The spatial, non -intrinsic, component,
characteristic of orbital motion, emerges in the form of a z -dependence in the angular momentum, 10
embodied in the partial or total evanescence of the major circularly -polarized component as the
wave propagates along th e guide.
ˆ 0
0c
o o c
ci g ig
EH i g i Etig
(Eq. 27)
In what sense is there an angular momentum transfer from spin to orbital, in this case? As the
polarization rotates in the x -y plane due to the Faraday Effect, there will be a spatially -dependent
reduction in the circulating electric field component of the electromagnetic wave alo ng the
propagation -direction. This can be seen as a negative increase in circular polarization with z, i.e. ,
an orbital angular momentum in the opposite direction to the electromagn etic spin.
Fig. 4. Rotated magnetization M generates TM to TE waveguide mode coupling and
electromagnetic spin -orbit coupling.
Finally, we derive an explicit expression for the spin -orbit coupling term. The relevant term
appearing in the orbital angular momentum flux in the z -direction is
* 0Im2z z yHE
Eq. (28)
We assume t hat Faraday rotation induces the
,yzEH terms via TM to transverse -electric (TE)
mode conversion, where
0zyiHEx
, Eq. (29)
and
,0 sinTE TEx i z
y y FE E e e z
. Eq. (30)
,0yE
is the electric field amplitude corresponding to full TM to TE conversion,
F is the specific
Faraday rotation angle,
TE and
TE are the cover -layer decay constant and the propagation
constant for the TE mode, respectively. For simplicity, we assume no linear birefringence in the
wavegu ide, so
TE TM . Hence, the spin to orbital angular momentum coupling term is
22 * 0
,0 2Im sin 22 2TEx TE
z z y y F FH E E e z
Eq. (31)
11
Acknowledgment
The authors thank Ramy El -Ganainy for suggesting this problem and for useful discussion s. 12
References
1. F. J. Beli nfante, “On the current and the density of electric charge, the energy, the linear
momentum and the angular momentum of arbitrary fields,” Physica 7, 449 -474 (1940).
2. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in
evanescent waves,” Nature Communications 5, 4300 (2014).
3. M. Neuberger an co -workers, “Experimental demonstration of the geometric so in Hall
effect of light in highly focused vector beams,” Conference on Lasers and Electro -Optics,”
QW1E.A (OSA 2012).
4. P. Banzer a co -workers, “The photonic wheel demonstration of a state of light with purely
transverse momentum,” J. Eur. Opt. Soc. Rap. Publ. 8, 13032 (2013).
5. K. Y. Bliokh and F. Nori, “Transverse spin of a surface polariton,” Phys. Rev. A 85, 061801
(2012) .
6. K. –Y. Kim, I. –M. Lee, J. Kim, J. Jung, and B. Lee, “Time reversal and the spin angular
momentum of tra nsverse electric and transverse -magnetic surface modes,” Phys. Rev. A
86, 063805 (2012).
7. K. Y. Bliokh, J. Dressel and F. Nori, “Conservation of the s pin and orbital angular momenta
in electromagnetism,” New Journal of Physics 16, 093037 (2014).
8. A. Aiello, P. Banzer, M. Neugebauer and G. Leuchs, “From transverse angular momentum
to photonic wheels,” Nature Photonics 9, 789 -795 (2015).
9. M.E.J. Friese, J. Enger, H. Rubinsztein -Dunlop, and N. R. Heckenberg, “Optical angular -
momentum transfer to trapped absorbing particles, Phys. Rev. A 54 1593 -1596 (1996).
10. J. Fujita, M. Levy, R. M. Osgood, Jr. , L. Wilkens, and H. Dötsch, “Waveguide Optical
Isolator Based on Mach -Zehnder Interferometer,” Appl. Phys. Lett. 76, 2158 (2000).
11. B. J. Stadler and T. Mizumoto, “Integrated magneto -optical materials and isolators: a
review,” IEEE Photonics Journal 6, 0600215 (2014).
12. S. M. Barnett, “ Resolution of the Ab raham -Minkowski Dilemma ,” Phys. Rev. Lett. 104,
070401 (2010).
13. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Dual electromagnetism: helicity, spin,
momentum and angular momentum,” New Journal of Physics 15, 033026 (2013).
14. D. C. Hutchings, B. M. Holmes, C. Zhang, P. Dulal, A. D. Block, S. -Y. Sung, N. C. A.
Seaton, B. J. H. Stadler, “Quasi -phase -matched Faraday rotation in semiconductor
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|
1201.4842v2.Strong_Enhancement_of_Rashba_spin_orbit_coupling_with_increasing_anisotropy_in_the_Fock_Darwin_states_of_a_quantum_dot.pdf | arXiv:1201.4842v2 [cond-mat.mes-hall] 24 Jan 2012Strong Enhancement of Rashba spin-orbit coupling with incr easing anisotropy in the
Fock-Darwin states of a quantum dot
Siranush Avetisyan,1Pekka Pietil¨ ainen,2and Tapash Chakraborty‡1
1Department of Physics and Astronomy, University of Manitob a, Winnipeg, Canada R3T 2N2
2Department of Physics/Theoretical Physics, University of Oulu, Oulu FIN-90014, Finland
We have investigated the electronic properties of elliptic al quantum dots in a perpendicular ex-
ternal magnetic field, and in the presence of the Rashba spin- orbit interaction. Our work indicates
that the Fock-Darwin spectra display strong signature of Ra shba spin-orbit coupling even for a low
magnetic field, as the anisotropy of the quantumdot is increa sed. An explanation of this pronounced
effect with respect to the anisotropy is presented. The stron g spin-orbit coupling effect manifests
itself prominently in the corresponding dipole-allowed op tical transitions, and hence is susceptible
to direct experimental observation.
In recent years our interest in understanding the
unique effects of the spin-orbit interaction (SOI) in semi-
conductor nanostructures [2] has peaked, largely due to
the prospect of the possible realization of coherent spin
manipulation in spintronic devices [3], where the SOI is
destined to play a crucial role [4]. As the SOI couples
the orbital motion of the charge carriers with their spin
state, an all-electrical control of spin states in nanoscale
semiconductor devices could thus be a reality. In this
context the Rashba SOI [5] has received particular at-
tention, largelybecauseinatwo-dimensionalelectrongas
thestrengthoftheRashbaSOIhasalreadybeenshownto
be tuned by the application of an electric field [6]. While
the earlier studies were primarily in a two-dimensional
electron gas, the attention has now been focused on the
role of SOI in a single InAS quantum dot [7]. The quan-
tum dot (QD) [8], a system of few electrons confined in
the nanometer region has the main advantage that the
shape and size of the confinement can be externally con-
trolled, which provides an unique opportunity to study
the atomic-likepropertiesofthesesystems[8,9]. SOcou-
plinginquantumdotsgeneratesanisotropicspinsplitting
[10] which provides important information about the SO
coupling strength.
Extensivetheoreticalstudiesofthe influenceofRashba
SOI in circularly symmetric parabolic confinement have
already been reported earlier [11], where the SO cou-
pling wasfound to manifest itself mainly in multiple level
crossings and level repulsions. They were attributed to
an interplay between the Zeeman and the SOI present
in the system Hamiltonian. Those effects, in particular,
the level repulsions were however weak and as a result,
wouldrequireextraordinaryefforts todetect the strength
of SO coupling [12] in those systems. Here we show that,
by introducing anisotropy in the QD, i.e., by breaking
the circular symmetry of the dot, we can generate a ma-
jor enhancement of the Rashba SO coupling effects in a
quantum dot. As shown below, this can be observed di-
rectly in the Fock-Darwin states of a QD, and therefore
should be experimentally observable [8, 9]. We show be-
low that the Rashba SO coupling effects are manifestlystrongin an elliptical QD [13], which should providea di-
rect route to unambigiously determine (and control) the
SO coupling strength. It has been proposed recently that
the anisotropy of a quantum dot can also be tuned by an
in-plane magnetic field [14].
The Fock-Darwin energy levels in elliptical QDs sub-
jected to a magnetic field was first reported almost two
decadesago[13], whereit wasfound that the majoreffect
of anisotropy was to lift the degeneracies of the single-
particle spectrum [15]. The starting point of our present
study is the stationary Hamiltonian
HS=1
2m∗/parenleftBig
p−e
cAS/parenrightBig2
+Vconf(x,y)+HSO+Hz
=H0+HSO+Hz
where the confinement potential is chosen to be of the
form
Vconf=1
2m∗/parenleftbig
ω2
xx2+ω2
yy2/parenrightbig
,
HSO=α
/planckover2pi1/bracketleftbig
σ×/parenleftbig
p−e
cAS/parenrightbig/bracketrightbig
zis the Rashba SOI, and Hz
isthe Zeemancontribution. Here m∗is the effective mass
of the electron, σare the Pauli matrices, and we choose
the symmetric gauge vector potential AS=1
2(−y,x,0).
As in Ref. [13], we introduce the rotated coordinates and
momenta
x=q1cosχ−χ2p2sinχ,
y=q2cosχ−χ2p1sinχ,
px=p1cosχ+χ1q2sinχ,
py=p2cosχ+χ1q1sinχ,
where
χ1=−/bracketleftbig1
2/parenleftbig
Ω2
1+Ω2
2/parenrightbig/bracketrightbig1
2, χ2=χ−1
1,
tan2χ=m∗ωc/bracketleftbig
2/parenleftbig
Ω2
1+Ω2
2/parenrightbig/bracketrightbig1
2//parenleftbig
Ω2
1−Ω2
2/parenrightbig
,
Ω2
1,2=m∗2/parenleftbig
ω2
x,y+1
4ω2
c/parenrightbig
, ωc=eB/m∗c.
In terms of the rotated operators introduced above, the
Hamiltonian H0is diagonal [13]
H0=1
2m∗/summationdisplay
ν=1,2/bracketleftbig
β2
νp2
ν+γ2
νq2
ν/bracketrightbig
,2
/s48/s52/s56/s49/s50/s49/s54
/s32
/s32/s32/s69 /s32/s40/s109 /s101/s86 /s41
/s40/s97/s41
/s52/s56/s49/s50/s49/s54/s50/s48
/s32/s32
/s32/s40/s99/s41
/s48/s52/s56/s49/s50/s49/s54
/s32/s32/s69 /s32/s40/s109 /s101/s86 /s41
/s40/s98/s41
/s48 /s49 /s50 /s51 /s52/s52/s56/s49/s50/s49/s54/s50/s48
/s66/s32/s40/s84/s41
/s32/s32
/s66/s32/s40/s84/s41/s49 /s50/s51 /s52 /s48/s40/s100/s41
FIG. 1: Magnetic field dependence of the low-lying Fock-
Darwin energy levels of an elliptical dot without the Rashba
SO interaction ( α= 0). The results are for (a) ωx= 4 meV
andωy= 4.1 meV, (b) ωx= 4 meV and ωy= 6 meV, (c)
ωx= 4 meV and ωy= 8 meV, and (d) ωx= 4 meV and
ωy= 10 meV.
where
β2
1=Ω2
1+3Ω2
2+Ω2
3
2(Ω2
1+Ω2
2), γ2
1=1
4/parenleftbig
3Ω2
1+Ω2
2+Ω2
3/parenrightbig
,
β2
2=3Ω2
1+Ω2
2−Ω2
3
2(Ω2
1+Ω2
2), γ2
2=1
4/parenleftbig
Ω2
1+3Ω2
2−Ω2
3/parenrightbig
,
Ω2
3=/bracketleftBig/parenleftbig
Ω2
1−Ω2
2/parenrightbig2+2m∗2ω2
c/parenleftbig
Ω2
1+Ω2
2/parenrightbig/bracketrightBig1
2.
Since the operator H0is obviously equivalent to the
Hamiltonianoftwoindependentharmonicoscillators,the
states of the electron can be described by the state vec-
tors|n1,n2;sz/angbracketright. Here the oscillator quantum numbers
ni= 0,1,2,...correspond to the orbital motion and
sz=±1
2to the spin orientation of the electron.
The Rashba Hamiltonian, in terms of the rotated op-
erators is now written as,
/planckover2pi1
αHSO=σx(sinχχ1−cosχω0)q1
−σy(sinχχ1+cosχω0)q2
−σy(cosχ−sinχω0χ2)p1
+σx(cosχ+sinχω0χ2)p2,
whereω0=eB/2c. The effect of the SO coupling is
readily handled by resortingto the standard ladder oper-
ator formalism of harmonic oscillators and by diagonal-
izingHSOin the complete basis formed by the vectors
|n1,n2;sz/angbracketright.
The Fock-Darwin states in the absence of the Rashba
SOI (α= 0) are shown in Fig. 1, for ωx= 4 meV and/s48/s52/s56/s49/s50/s49/s54/s32
/s69 /s32/s40/s109 /s101/s86 /s41
/s32
/s32/s32
/s40/s97/s41
/s52/s56/s49/s50/s49/s54/s50/s48
/s32/s32/s32
/s50/s40/s100/s41
/s48/s52/s56/s49/s50/s49/s54
/s32/s32/s69 /s32/s40/s109 /s101/s86 /s41
/s40/s98/s41/s52/s56/s49/s50/s49/s54/s50/s48
/s66/s32/s40/s84/s41
/s66/s32/s40/s84/s41
/s32/s32
/s40/s99/s41
/s48 /s49 /s50 /s51 /s52/s48 /s49 /s51 /s52
FIG. 2: Same as in Fig. 1, but for α= 20.
ωy= 4.1,6,8,10 meV in (a)-(d) respectively. We have
considered the parameters of an InAs QD [11] through-
out, because in such a narrow-gap semiconductor sys-
tem, the dominant source of the SO interaction is the
structural inversion asymmetry [16], which leads to the
RashbaSO interaction. As expected, breakingof circular
symmetry in the dot results in lifting of degeneracies at
B= 0, that is otherwise present in a circular dot [13, 15].
In Fig. 1 (a), the QD is very close to being circularly
symmetric, and as a consequence, the splittings of the
zero-field levels are vanishingly small. As the anisotropy
of the QD is increased [(b) – (d)], splitting of the levels
becomes more appreciable.
As the SO term is linear in the position and momen-
tum operators it is also linear in the raising and lowering
ladder operators. It is also off-diagonal in the quantum
number sz. As a consequence, the SOI can mix only
states which differ in the spin orientation, and differ by 1
either in the quntum number n1or inn2but not in both.
In the case of rotationally symmetric confinements these
selection rules translate to the conservation of the total
angular momentum j=m+szin the planar motion of
the electron.
At the field B= 0 the ground states |0,0;±1
2/angbracketrightare two-
fold degenerate. Due to the selection rules, this degener-
acy cannot be lifted either by the eccentricity of the dot
or by the Rashba coupling. Many of the excited states,
such as |n1,n2;±1
2/angbracketrightretain their degeneracy no matter
how strong the SO coupling is or how eccentric the dot
is, as we can see in the Figs. 1-3. At the same time, many
other degeneracies are removed by squeezing or streching
the dot. At non-zero magnetic fields some of the cross-
ings of the energy spectra are turned to anti-crossings by
the Rashba term in the Hamiltonian. For example, the3
/s48/s52/s56/s49/s50/s49/s54/s69 /s32/s40/s109 /s101/s86 /s41
/s32/s32
/s32/s69 /s32/s40/s109 /s101/s86 /s41
/s40/s97/s41
/s48 /s49 /s50 /s51 /s52/s48/s52/s56/s49/s50/s49/s54
/s32/s32
/s40/s98/s41
/s48 /s49 /s50 /s51 /s52/s52/s56/s49/s50/s49/s54/s50/s48
/s32/s32
/s40/s100/s41/s52/s56/s49/s50/s49/s54/s50/s48
/s66/s32/s40/s84/s41/s66/s32/s40/s84/s41
/s32/s32
/s40/s99/s41
FIG. 3: Same as in Fig. 1, but for α= 40.
second and third excited states in Fig. 2 (a) – Fig. 2 (d)
are composed mainly of the states |0,0;1
2/angbracketrightand|1,0;−1
2/angbracketright
which are mixed by the HSOaroundB= 3T causing a
level repulsion. We can also see that the squeezing of the
dot enhances the SO coupling. This can be thought of as
a consequence of pushing some states out of the way, just
as in our example of the state |1,1;1
2/angbracketright. SOI mixes it with
the state |1,0;−1
2/angbracketrightcausing the latter state to shift down-
ward in energy thereby reducing the anti-crossing gap.
Squeezing the dot, however moves the state energetically
farther away from |1,0;−1
2/angbracketrightand so weakens this gap re-
duction effect. It is abundantly clear from the features
revealed in the energy spectra that for a combination of
strong anisotropy of the dot and higher values of the SO
coupling strength, large anti-crossing gaps would appear
even for relatively low magnetic fields.
The effects of anisotropy and spin-orbit interaction on
the energy spectra above are also reflected in the optical
absorption spectra. Let us turn our attention on the
absorption spectra for transitions from the ground state
to the excited states. For that purpose we subject the
dot to the radiation field
AR=A0ˆǫ/parenleftBig
ei(ω/c)ˆn·r−iωt+e−i(ω/c)ˆn·r+iωt/parenrightBig
,
whereˆǫ,ωandˆnare the polarization, frequency and
the direction of propagation of the incident light, respec-
tively. We let the radiation enter the dot along the direc-
tion perpendicular to the motion of the electron, that is
parallel to the z-axis. Due to the transversalitycondition
the polarization vector will then lie in the xy-plane.
As usual, we shall make two approximations. First we
assume the intensity of the field be so weak that only the
terms linear in ARhas to be taken into account. Then
the effect of the radiative magnetic field on the spin can/s48/s51/s54/s57/s69/s32/s40/s109/s101/s86/s41 /s69/s32/s40/s109/s101/s86/s41
/s32/s32/s69/s32/s40/s109/s101/s86/s41
/s40/s97/s41
/s48/s51/s54/s57/s49/s50
/s32/s32/s32
/s40/s98/s41
/s48 /s49 /s50/s48/s51/s54/s57/s49/s50
/s40/s99/s41/s48/s51/s54/s57
/s32/s32
/s40/s100/s41
/s48/s51/s54/s57/s49/s50
/s32/s32
/s40/s101/s41
/s48 /s49 /s50/s48/s51/s54/s57/s49/s50
/s66/s40/s84/s41
/s32/s32
/s66/s40/s84/s41/s40/s102/s41
FIG.4: Opticalabsorption(dipoleallowed) spectaofellip tical
QDs for various choice of parameters: (a) i α= 0,ωx= 4
meV,ωy= 6, (b) α= 20,ωx= 4 meV, ωy= 8 meV, and
(c)α= 40,ωx= 4,ωy= 6. The polarization of the incident
radiation is along the x-axis. The parameters for (d)-(f) are
the same, except that the incident radiation is polarized al ong
they-axis. The areas of the filled circles are proportional to
the calculated absorption cross-section.
be neglected as well. So we can simply replace in the
stationary Hamiltonian HSthe vector potential ASwith
the field A=AS+AR. Discarding terms higher than
linear order in ARleads to the total Hamiltonian
H=HS+HR,
where the radiative part HRis given by
HR=−e
mecAR·/parenleftBig
p−e
cAS/parenrightBig
−αe
/planckover2pi1c[σ×AR]z.
The radiative Hamiltonian, even in the presence of the
Rashba SO coupling can be expressed in the well-known
form
HR= ie
c/planckover2pi1AR·[x,HS],
xbeing the position operator in the xy-plane.
Our second approximation is the familiar dipole ap-
proximation. We assume that the amplitude of radiation
can be taken as constantwithin the quantum dot, so that
we are allowed to write the field as
AR≈A0ˆǫ/parenleftbig
e−iωt+eiωt/parenrightbig
.4
Since the transition energies expressed in terms of radia-
tion frequences are of the order of THz, the correspond-
ing wavelengths are much larger than the typical size of
a dot, thus justifying our approximation. Applying now
the Fermi Golden Rule leads to the dipole approximation
form
σabs(ω) = 4π2αfωni|/angbracketleftn|ˆǫ·x|i/angbracketright|2δ(ωni−ω)
of the absorption cross section for transitions from the
inital state |i/angbracketrightto the final state |n/angbracketright. Hereαfis the fine
structure constant and ωniis the frequency correspond-
ing to the transition energy /planckover2pi1ω.
The familiar dipole selection rules for oscillator states
dictate largely the features seen in Fig. 4. In the absence
of the SOI, these rules – the spin state is preserved and
eithern1orn2is changed by unity – completely deter-
mine the allowed two transitions/vextendsingle/vextendsingle0,0;−1
2/angbracketrightbig
→/vextendsingle/vextendsingle1,0;−1
2/angbracketrightbig
and/vextendsingle/vextendsingle0,0;−1
2/angbracketrightbig
→/vextendsingle/vextendsingle0,1;−1
2/angbracketrightbig
. In contrast to the case of
circular dots the absorption in the elliptical dot depends
strongly on the polarization. This is explained by noting
that the oscillator strengths
fni=2m∗ωni
/planckover2pi1|/angbracketleftn|ˆǫ·x|i/angbracketright|2.
actually probe the occupations of quantum states related
to oscillations in the direction of the polarization ˆǫ. In a
circular dot all oscillation directions are equally probable
at all energies implying that the oscillator strengths are
independent of the polarization and depend only slightly
on the transitionenergyvia ωni, and the final state quan-
tum numbers n1,2. When the dot is squeezed in the y-
direction,say,theoscillatorstatesrelatedtothe y-motion
are pushed up in energy. This means that the polariza-
tion being along x-axis most of the oscillator strength
comes from transitions to allowed states with lowest en-
ergies. Similarly, when the incident radiation is polarized
along the y-axis most of the contribution is due to the
transitions to the oscillator states pushed up in the en-
ergy. Inellipticaldotstheoscillatorstatesarenotpure x-
andy-oscillators but their superpositions. Therefore in
addition to the main absorption lines, other allowed final
states have also non-vanishing oscillator strength. Fur-
thermore, as one can see by looking at the phase space
rotation formulas the external magnetic field tends to ro-
tate directions of the oscillator motion causing a shift of
the oscillator strength from an allowed transition to an-
other. This is exactly what we see in Fig. 4 (a) and Fig. 4
(d).
Even in the presence of the SOI the two allowed final
oscillator states provide major contributions to the cor-
responding corrected states. Hence we still see two dom-
inant absorption lines. However, now many forbidden
transitions have become allowed. The lowest absorption
linecorrespondingtothetransitionbetweenZeemansplit
states with main components/vextendsingle/vextendsingle0,0;−1
2/angbracketrightbig
and/vextendsingle/vextendsingle0,0;1
2/angbracketrightbig
pro-
vides a typical example. The transition involves a spinflip and is therefore strongly forbidden without the SOI.
Because the SOI mixes the state/vextendsingle/vextendsingle1,0;1
2/angbracketrightbig
into the for-
mer one and the/vextendsingle/vextendsingle0,1;−1
2/angbracketrightbig
into the latter one, the tran-
sition becomes possible. The appearance of other new
lines can be explained by analogous arguments. There
are also additional features involving discontinuities and
anti-crossings in Fig. 4. A comparision with the energy
spectra indicates that these are the consequences of the
anti-crossings present in the energy spectra.
It is also readily verified that the oscillator strengths
satisfy the Thomas-Reiche-Kuhn sum rule [17]
/summationdisplay
nfni= 1.
In terms of the cross section this translates to the condi-
tion
/integraldisplay∞
−∞σabs(ω)dω=2π2/planckover2pi1αf
m∗.
The absorptions visible in Fig. 4 practically saturate the
sum rule, the saturation being, of course complete in the
absence of the SOI in panels (a) and (d). The largest
fraction (of the order of 1/10) of the cross section either
falling outside of the displayed energy scale or having too
low intensity to be discernible in our pictures is found at
the strongest Rashba coupling in the panels (c) and (f)
for large magnetic fields, as expected.
The results presented here clearly indicate that, the
anisotropyofaQDalonecausesliftingofthedegeneracies
oftheFock-DarwinlevelsatB=0,asreportedearlier[13].
However, for large SO coupling strengths α, the effects
of the Rashba SOI, mainly the level repulsions at finite
magnetic fields, are maginified rather significantly as one
introduces anisotropy in the QD. This is reflected also
in the corresponding dipole-allowed optical transitions
where the distinct anti-crossingbehavioris observedthat
is a direct manifestation of the anti-crossings in the en-
ergy spectra. This prominent effect of the Rashba SOI
predicted here could be confirmed experimentally in op-
tical spectroscopy and the Fock-darwin spectra of few-
electron QDs [9, 18, 19]. It would also provide a very
useful step to control the SO coupling in nanostructures,
en route to semiconductor spintronics [3].
The work was supported by the Canada Research
Chairs Program of the Government of Canada.
[‡] Electronic address: tapash@physics.umanitoba.ca
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Nano-Physics & Bio-Electronics: A New Odyssey , edited
by, T. Chakraborty, F. Peeters, and U. Sivan (Elsevier,
Amsterdam, 2002).
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[5] Y.A. Bychkov and E.I. Rashba, J. Phys. C 17, 6039
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sslin, D.C. Driscoll, and A.C. Gossard, ibid.103, 027201
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[7] H. Sanada, T. Sogawa, H. Gotoh, K. Onomitshu, M. Ko-
hda, J. Nitta, and P.V. Santos, Phys. Rev. Lett. 106,
216602 (2011); S. Takahashi, R.S. Deacon, K. Yoshida,
A. Oiwa, K. Shibata, K. Hirakawa, Y. Tokura, and S.
Tarucha, ibid.104, 246801 (2010).
[8] T. Chakraborty, Quantum Dots (North-Holland, Am-
sterdam, 1999); T. Chakraborty, Comments Condens.
Matter Phys. 16, 35 (1992); P.A. Maksym and T.
Chakraborty, Phys. Rev. Lett. 65, 108 (1990).
[9] D. Heitmann (Ed.), Quantum Materials (Springer, Hei-
delberg, 2010).
[10] J. K¨ onemann, R.J. Haug, D.K. Maude, V. Fal’ko, and
B.L. Altshuler, Phys. Rev. Lett. 94, 226404 (2005).
[11] T. Chakraborty and P. Pietil¨ ainen, Phys. Rev. Lett.
95, 136603 (2005); P. Pietil¨ ainen and T. Chakraborty,
Phys. Rev. B 73, 155315 (2006); T. Chakraborty and P.
Pietil¨ ainen, ibid.71, 113305 (2005); A. Manaselyan andT. Chakraborty, Europhys. Lett. 88, 17003 (2009); and
the references therein.
[12] H.-Y. Chen, V. Apalkov, andT. Chakraborty, Phys. Rev.
B75, 193303 (2007).
[13] A.V.MadhavandT. Chakraborty, Phys.Rev.B 49, 8163
(1994); See also, P.A. Maksym, Physica B 249-251 , 233
(1998).
[14] M.P. Nowak, B. Szafran, F.M. Peeters, B. Partoens, and
W.J. Pasek, Phys. Rev. B 83, 245324 (2011).
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Pinczuk, L.N. Pfeiffer, and K.W. West, Appl. Phys. Lett.
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S.M. Reimann, M. Koskinen, M. Manninen, Phys. Rev.
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chanics, second edition, (Addison-Wesley, New York,
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1505.04301v1.Dynamics_of_a_macroscopic_spin_qubit_in_spin_orbit_coupled_Bose_Einstein_condensates.pdf | arXiv:1505.04301v1 [quant-ph] 16 May 2015Dynamics of a macroscopic spin qubit in spin-orbit
coupled Bose-Einstein condensates
Sh Mardonov1,2,3, M Modugno4,5and E Ya Sherman1,4
1Department of Physical Chemistry, The University of the Basque C ountry, 48080
Bilbao, Spain
2The Samarkand Agriculture Institute, 140103 Samarkand, Uzbek istan
3The Samarkand State University, 140104 Samarkand, Uzbekistan
4IKERBASQUE Basque Foundation for Science, Bilbao, Spain
5Department of Theoretical Physics and History of Science, Univer sity of the Basque
Country UPV/EHU, 48080 Bilbao, Spain
E-mail:evgeny.sherman@ehu.eus
Abstract. We consider a macroscopic spin qubit based on spin-orbit coupled Bos e-
Einstein condensates, where, in addition to the spin-orbit coupling, spin dynamics
strongly depends on the interaction between particles. The evolut ion of the spin
for freely expanding, trapped, and externally driven condensate s is investigated. For
condensates oscillating at the frequency corresponding to the Ze eman splitting in the
synthetic magnetic field, the spin Rabi frequency does not depend on the interaction
between the atoms since it produces only internal forces and does not change the
total momentum. However, interactions and spin-orbit coupling br ing the system
into a mixed spin state, where the total spin is inside rather than on t he Bloch
sphere. This greatly extends the available spin space making it three -dimensional, but
imposes limitations on the reliable spin manipulation ofsuch a macroscop icqubit. The
spin dynamics can be modified by introducing suitable spin-dependent initial phases,
determined by the spin-orbit coupling, in the spinor wave function.
PACS numbers: 03.75.Mn, 67.85.-d
Keywords : Two-component Bose-Einstein condensate, spin-orbit coupling, spin
dynamics
Submitted to: J. Phys. B: At. Mol. Opt. Phys.Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 2
1. Introduction
The experimental realization of synthetic magnetic fields and spin-o rbit coupling (SOC)
[1, 2] in Bose-Einstein condensates (BECs) of pseudospin-1/2 par ticles has provided
novel opportunities for visualizing unconventional phenomena in qu antum condensed
matter [3, 4]. More recently, also ultracold Fermi gases with synthe tic SOC have been
produced and studied [5, 6]. These achievements have motivated an d intense activity,
and a rich variety of new phases and phenomena induced by the SOC h as been discussed
both theoretically and experimentally [7, 8, 9, 10, 11, 12, 13, 14, 15 , 16, 17, 18, 19, 20].
Recently, it has also been experimentally demonstrated [3, 21] the a bility of a reliable
measurement of coupled spin-coordinate motion.
One of the prospective applications of spin-orbit coupled Bose-Eins tein condensates
consists in the realization of macroscopic spin qubits [8]. A more detaile d analysis of
quantum computation based on a two-component BEC was propose d in [22]. The
gates for performing these operations can be produced by means of the SOC and of an
external synthetic magnetic field. Due to the SOC, a periodic mecha nical motion of
the condensate drives the spin dynamics and can cause spin-flip tra nsitions at the Rabi
frequency depending on the SOC strength. This technique, known in semiconductor
physics as the electric dipole spin resonance, is well suitable for the m anipulation of
qubits based on the spin of a single electron [23, 24, 25]. For the macr oscopic spin qubits
based onBose-Einstein condensate, the physics is different in at lea st two aspects. First,
a relative effect of the SOC compared to the kinetic energy can be mu ch stronger here
than in semiconductors. Second, the interaction between the bos ons can have a strong
effect on the entire spin dynamics.
Here we study how the spin evolution of a quasi one-dimensional Bos e-Einstein
condensate depends on the repulsive interaction between the par ticles and on the SOC
strength. The paper is organized as follows. In Section 2 we remind t he reader the
ground state properties of a quasi-one dimensional condensate a nd consider simple spin-
dipole oscillations. In Section 3 we analyze, by means of the Gross-Pit aevskii approach,
thedynamics of free, harmonically trapped, andmechanically driven macroscopic qubits
based on such a condensate. We assume that the periodic mechanic al driving resonates
with the Zeeman transition in the synthetic magnetic field and find diffe rent regimes of
the spin qubit operation in terms of the interaction between the ato ms, the driving
frequency and amplitude. We show that some control of the spin qu bit state can
be achieved by introducing phase factors, dependent on the SOC, in the spinor wave
function. Conclusions will be given in Section 4.
2. Ground state and spin-dipole oscillations
2.1. Ground state energy and wave function
Before analyzing the spin qubit dynamics, we remind the reader how t o obtain the
ground state of an interacting BEC. In particular, we consider a ha rmonically trappedDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 3
quasi one-dimensional condensate, tightly bounded in the transv erse directions. The
system can be described by the following effective Hamiltonian, where the interactions
between the atoms are taken into account in the Gross-Pitaevskii form:
/hatwideH0=/hatwidep2
2M+Mω2
0
2x2+g1|ψ(x)|2. (1)
Hereψ(x) is the condensate wave function, Mis the particle mass, ω0is the frequency
of the trap (with the corresponding oscillator length aho=/radicalbig
/planckover2pi1/Mω0), andg1= 2as/planckover2pi1ω⊥
is the effective one-dimensional interaction constant, with asbeing the scattering length
of interacting particles, and ω⊥≫ω0being the transverse confinement frequency. For
further calculations we put /planckover2pi1≡M≡1, and measure energy in units of ω0and length
in units of aho, respectively. All the effects of the interaction are determined by the
dimensionless parameter /tildewideg1N, where/tildewideg1= 2/tildewideas/tildewideω⊥, where/tildewideasis the scattering length in
the units of aho,/tildewideω⊥is the transverse confinement frequency in the units of ω0, andN
is the number of particles. In physical units, for a condensate of87Rb and an axial
trapping frequency ω0= 2π×10 Hz, the unit of time corresponds to 0 .016 s, the unit
of lengthahocorresponds to 3 .4µm, and the unit of speed ahoω0becomes 0.021 cm/s,
respectively. In addition, considering that as= 100aB,aBbeing the Bohr radius, in the
presence of a transverse confinement with frequency ω⊥= 2π×100 Hz the dimensionless
coupling constant /tildewideg1turns out to be of the order of 10−3.
In order to find the BEC ground state we minimize the total energy in a properly
truncated harmonic oscillator basis. To design the wave function we take the real sum
of even-order eigenfunctions
ψ0(x) =N1/2nmax/summationdisplay
n=0C2nϕ2n(x). (2)
Here
ϕ2n(x) =1/radicalbig
π1/2(2n)!22nH2n(x)exp/bracketleftbigg
−x2
2/bracketrightbigg
, (3)
whereH2n(x) are the Hermite polynomials, and the normalization is fixed by requirin g
that
nmax/summationdisplay
n=0C2
2n= 1. (4)
The coefficients C2nare determined by minimizing the total energy Etot, such that
Emin= min
C2n{Etot}, (5)
where
Etot=1
2/integraldisplay/bracketleftBig
(ψ′(x))2+x2ψ2(x)+/tildewideg1ψ4(x)/bracketrightBig
dx, (6)
and|C2nmax| ≪1.
Formulas (5) and (6) yield the ground state energy, while the width o f the
condensate is defined as:
wgs=/bracketleftbigg2
N/integraldisplay
x2|ψ0(x)|2dx/bracketrightbigg1/2
. (7)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 4
Figure 1. (Color online) Ground-state probability density of the condensate obtained
from (2)-(6) (blue solid line), compared with the Thomas-Fermi app roximation in (8)
(red dashed line) for /tildewideg1N= 40.
In the non interacting limit, /tildewideg1= 0,ψ0(x) is the ground state of the harmonic
oscillator (nmax= 0), that is a Gaussian function with wgs= 1. In the opposite, strong
coupling limit /tildewideg1N≫1, the exact wave function (2) is well reproduced (see Figure 1)
by the Thomas-Fermi expression
ψTF(x) =√
3
2√
N
w3/2
TF/parenleftbig
w2
TF−x2/parenrightbig1/2;|x| ≤wTF, (8)
wherewTF= (3/tildewideg1N/2)1/3.
In general, for a qualitative description of the ground state one ca n use instead of
the exact wave function (2), the Gaussian ansatz
ψG(x) =/parenleftbiggN
π1/2w/parenrightbigg1/2
exp/bracketleftbigg
−x2
2w2/bracketrightbigg
, (9)
where the width wis single variational parameter for the energy minimization. Then
the total energy (6) becomes:
Etot=N/bracketleftbigg1
4/parenleftbigg
w2+1
w2/parenrightbigg
+/tildewideg1N
2(2π)1/2w/bracketrightbigg
. (10)
The latter is minimized with respect to wby solving the equation
dEtot
dw=N/bracketleftbigg1
2/parenleftbigg
w−1
w3/parenrightbigg
−/tildewideg1N
2(2π)1/2w2/bracketrightbigg
= 0. (11)
Inthestrongcouplingregime, /tildewideg1N≫1, wehavew≫1sothat-toafirstapproximation
- the kinetic term ∝1/w3in (11) can be neglected, yielding the following value for the
width of the ground state:
/tildewidewG=/parenleftbigg/tildewideg1N√
2π/parenrightbigg1/3
. (12)
The first order correction can be obtained by writing w=/tildewidew+ǫ(ǫ≪1), so that from
(11) it follows:
w=/parenleftbigg/tildewideg1N√
2π/parenrightbigg1/3
+√
2π
3/tildewideg1N. (13)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 5
0 10 20 30 40012345
g/OverTilde
l1NEmin,wgs
Figure 2. (Color online) Ground state energy (black solid line) and condensate width
(red dashed line) vs. the interaction parameter /tildewideg1N.
By substituting (13) in (10) we obtain that the leading term in the gro und state energy
for/tildewideg1N≫1 is:
E[G]
min=3
4N/parenleftbigg/tildewideg1N√
2π/parenrightbigg2/3
. (14)
In figure 2 we plot the ground state energy and the condensate wid th as a function
of the interaction, as obtained numerically from (5) and (7), respe ctively. As expected,
in the strong coupling regime /tildewideg1N≫1 both quantities nicely follow the behavior (not
shown in the Figure) predicted both by the Gaussian approximation a nd by the exact
solution, namely Emin∝(/tildewideg1N)2/3andwgs∝(/tildewideg1N)1/3.
2.2. Simple spin-dipole oscillations
Let us now turn to the case of a condensate of pseudospin 1/2 ato ms. Here the system
is described by a two-component spinor wave function Ψ = [ ψ↑(x,t),ψ↓(x,t)]T, still
normalized to the total number of particles N.The interaction energy (third term in
the functional (6)) now acquires the form (see, e.g. [9])
Eint=1
2/tildewideg1/integraldisplay/bracketleftbig
|ψ↑(x,t)|2+|ψ↓(x,t)|2/bracketrightbig2dx, (15)
where, for simplicity and qualitative analysis, we neglect the depende nce of interatomic
interaction on the spin component ↑or↓and characterize all interactions by a single
constant/tildewideg1.
Here we consider spin dipole oscillations, induced by a given small initial s ymmetric
displacement of the two spin components ±ξ. For a qualitative understanding, we
assume a negligible spin-orbit coupling and a Gaussian form of the wave function
presented as
ΨG(x) =1√
2/bracketleftBigg
ψG(x−ξ)
ψG(x+ξ)/bracketrightBigg
, (16)
whereψGis given by (9), and ξ≪w. The corresponding energy is given by:
E=E[G]
min+Esh, (17)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 6
whereE[G]
minis defined by (14) and Eshis the shift-dependent contribution:
Esh=N
2ξ2/parenleftbigg
1−/tildewideg1N√
2πw3/parenrightbigg
. (18)
Then, it follows that the corresponding oscillation frequency is
ωsh=/radicalBigg
1−/tildewideg1N√
2πw3. (19)
For strong interaction ( /tildewideg1N≫1) by substituting (13) in (19) we obtain:
ωsh≈/parenleftBigg√
2π
/tildewideg1N/parenrightBigg2/3
. (20)
Therefore, for strong interaction the frequency of the spin dipo le oscillations falls as
(/tildewideg1N)−2/3, and this result is common for the Gaussian ansatz and for the exac t solution;
it will be useful in the following section.
3. Spin evolution and particles interaction
3.1. Hamiltonian, spin density matrix, and purity
To consider the evolution of the driven quasi one dimensional pseud ospin-1/2 SOC
condensate we begin with the effective Hamiltonian
/hatwideH=α/hatwideσz/hatwidep+/hatwidep2
2+∆
2/hatwideσx+1
2(x−d(t))2+/tildewideg1|Ψ|2. (21)
Hereαis the SOC constant (see [11] and [12] for comprehensive review on t he SOC in
coldatomicgases), /hatwideσxand/hatwideσzarethePauli matrices, ∆isthesynthetic Zeemansplitting,
andd(t) is the driven displacement of the harmonic trap center as can be ob tained by
a slow motion of the intersection region of laser beams trapping the c ondensate.
Thetwo-componentspinorwavefunctionΨisobtainedasasolutiono fthenonlinear
Schr¨ odinger equation
i∂Ψ
∂t=/hatwideHΨ. (22)
To describe spin evolution we introduce the reduced density matrix
ρ(t)≡ |Ψ∝an}bracketri}ht∝an}bracketle{tΨ|=/bracketleftBigg
ρ11(t)ρ12(t)
ρ21(t)ρ22(t)/bracketrightBigg
, (23)
where we trace out the x−dependence by calculating integrals
ρ11(t) =/integraldisplay
|ψ↑(x,t)|2dx, ρ22(t) =/integraldisplay
|ψ↓(x,t)|2dx,
ρ12(t) =/integraldisplay
ψ∗
↑(x,t)ψ↓(x,t)dx, ρ21(t) =ρ∗
12(t), (24)
and, as a result,
tr(ρ(t))≡ρ11(t)+ρ22(t) =N. (25)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 7
Figure 3. (Color online) (a) Separation of a freely expanding condensate in tw o spin-
up and spin-down components with opposite anomalous velocities. (b ) Oscillation of
the spin-up and spin-down components in the harmonic trap.
Although the |Ψ∝an}bracketri}htstate is pure, integration in (24) produces ρ(t) formally describing a
mixed state in the spin subspace. One can characterize the resultin g spin state purity
by a parameter Pdefined as
P=2
N2/parenleftbigg
tr/parenleftbig
ρ2/parenrightbig
−N2
2/parenrightbigg
, (26)
where 0≤P≤1,
tr/parenleftbig
ρ2/parenrightbig
=N2+2(|ρ12|2−ρ11ρ22), (27)
and we omitted the explicit t−dependence for brevity. The system is in the pure state
whenP= 1,that is tr(ρ2) =N2withρ11ρ22=|ρ12|2. In the fully mixed state, where
tr(ρ2) =N2/2, we have P= 0 with
ρ11=ρ22=N
2, ρ 12= 0. (28)
The spin components defined by ∝an}bracketle{t/hatwideσi∝an}bracketri}ht ≡tr(/hatwideσiρ)/Nbecome
∝an}bracketle{t/hatwideσx∝an}bracketri}ht=2
NRe(ρ12),∝an}bracketle{t/hatwideσy∝an}bracketri}ht=−2
NIm(ρ12),
∝an}bracketle{t/hatwideσz∝an}bracketri}ht=2
Nρ11−1, (29)
and the purity P=/summationtext3
i=1∝an}bracketle{t/hatwideσi∝an}bracketri}ht2, which allows one to match the value of Pand the length
of the spin vector inside the Bloch sphere. For a pure state/summationtext3
i=1∝an}bracketle{t/hatwideσi∝an}bracketri}ht2= 1, and the
total spin is on the Bloch sphere. Instead, for a fully mixed state/summationtext3
i=1∝an}bracketle{t/hatwideσi∝an}bracketri}ht2= 0, and
the spin null.
3.2. A simple condensate motion
Let us suppose that a condensate of interacting spin-orbit couple d particles is located
in a harmonic trap and characterized by an initial wave function
Ψ0(x,0) =1√
2ψin(x)/bracketleftBigg
1
1/bracketrightBigg
, (30)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 8
(a)
024680.00.20.40.60.81.0
tP/LParen1t/RParen1(b)
0246810120.00.20.40.60.81.0
t/LAngleBracket1Σ/Hat
x/LParen1t/RParen1/RAngleBracket1
Figure 4. (Color online) ( a) Purity and ( b) spin component as a function of time for
a condensate released from the trap, for α= 0.2. The lines correspond to /tildewideg1N= 0
(black solid line; for the purity cf. (32)), /tildewideg1N= 10 (red dashed line), and /tildewideg1N= 20
(blue dot-dashed line).
with the spin parallel to the x−axis.
The spin-orbit coupling modifies the commutator corresponding to t he velocity
operator by introducing the spin-dependent contribution as:
/hatwidev≡i/bracketleftbigg/hatwidep2
2+α/hatwideσz/hatwidep,/hatwidex/bracketrightbigg
=/hatwidep+α/hatwideσz. (31)
The effect of the spin-dependent anomalous velocity term on the co ndensate motion
was clearly observed experimentally in [3] as the spin-induced dipole os cillations and in
[21] as the Zitterbewegung . Since the initial spin in (30) is parallel to the x-axis, the
expectation value of the velocity vanishes, ∝an}bracketle{t/hatwidev∝an}bracketri}ht= 0.
Free and oscillating motion of the BEC is shown in figure 3(a) and figure 3(b),
respectively. When one switches off the trap, the condensate is se t free, and the two
spin components start to move apart and the condensate splits up , see figure 3(a).
Each spin-projected component broadens due to the Heisenberg momentum-coordinate
uncertainty and interaction. The former effect is characterized b y a rate proportional
to 1/wgs.At large/tildewideg1N,the width wgs∼(/tildewideg1N)1/3,and, as a result, the quantum
mechanical broadening rate decreases as ( /tildewideg1N)−1/3.At the same time, the repulsion
between the spin-polarized components accelerates the peak sep aration [26] and leads
to the asymptotic separation velocity ∼(/tildewideg1N)1/2. This acceleration by repulsion leads
to opposite time-dependent phase factors in ψ↑(x,t) andψ↓(x,t) in (24) and, therefore,
results in decreasing in |ρ12(t)|and in the purity. Thus, with the increase in the
interaction, the purity and the x−spin component asymptotically tend to zero faster,
as demonstrated in figure 4. For a noninteracting condensate with the initial Gaussian
wave function ψin∼exp(−x2/2w2) the purity can be written analytically as
P0(t) = exp/bracketleftBigg
−2/parenleftbiggαt
w/parenrightbigg2/bracketrightBigg
. (32)
In the presence of the trap (figure 3(b)), the anomalous velocity in (31) causes spin
components (spin-dipole) oscillations with a characteristic frequen cy of the oder of ωshDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 9
(a)
0102030405060700.00.20.40.60.81.0
tP/LParen1t/RParen1
(b)
0102030405060700.00.20.40.60.81.0
t/LAngleBracket1Σ/Hat
x/LParen1t/RParen1/RAngleBracket1
(c)
010203040506070/MinuΣ3/MinuΣ2/MinuΣ10123
t/LAngleBracket1xΣ/Hat
z/RAngleBracket1
Figure 5. (Color online) (a) Purity, (b) spin component, and (c) spin dipole mom ent
as a function of time for the system in the harmonic trap with α= 0.2,∆ = 0, d0= 0.
The different lines correspond to /tildewideg1N= 0 (black solid line), /tildewideg1N= 10 (red dashed
line),/tildewideg1N= 20 (blue dot-dashed line), and /tildewideg1N= 60 (green dotted line).
in (20). With the increase in the interatomic interaction, the freque ncyωshdecreases
and, therefore, the amplitude of the oscillations arising due to the a nomalous velocity
(∼α/ωsh) increases. As a result, the acceleration and separation of the sp in-projected
components increase, the off-diagonal components of the densit y matrix in (24) became
smaller, and the minimum in P(t) rapidly decreases to P(t)≪1 as shown by the exact
numerical results presented in figures 5(a) and (b) [27]. In figure 5 (c) we show the
corresponding evolution of spin density dipole moment
∝an}bracketle{tx/hatwideσz∝an}bracketri}ht=1
N/integraldisplay
Ψ†x/hatwideσzΨdx. (33)
Here the oscillation frequency is a factor of two larger than that of the spin density
oscillation.
3.3. Spin-qubit dynamics and the Rabi frequency
To manipulate the macroscopic spin qubit, the center of the trap is d riven harmonically
at the frequency corresponding to the Zeeman splitting ∆ as
d(t) =d0sin(t∆), (34)
whered0is an arbitrary amplitude and the corresponding spin rotation Rabi f requency
ΩRis defined as αd0∆.At ∆≪1,as will be considered here, for a noninteractingDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 10
(a)
0 50 100 150 2000.00.20.40.60.81.0
tP/LParen1t/RParen1(b)
050100 150 200/MinuΣ1.0/MinuΣ0.50.00.51.0
t/LAngleBracket1Σ/Hat
x/LParen1t/RParen1/RAngleBracket1
Figure 6. (Color online) ( a) Purity and ( b) spin component as a function of time for
a driven condensate with α= 0.1,∆ = 0.1, d0= 2. The lines correspond to /tildewideg1N= 0
(black solid line), /tildewideg1N= 10 (red dashed line), and /tildewideg1N= 20 (blue dot-dashed line).
condensate and a very weak spin-orbit coupling, the spin componen t∝an}bracketle{t/hatwideσx(t)∝an}bracketri}htis expected
to oscillate approximately as
∝an}bracketle{t/hatwideσx(t)∝an}bracketri}ht= cos(Ω Rt). (35)
The corresponding spin-flip time Tsfis
Tsf=π
ΩR. (36)
Figure 6 shows the time dependence of the purity and the spin of the condensate for
givenα,d0, and ∆ at different interatomic interactions. In figure 6(a) one can see that
the increase of /tildewideg1Nenhances the variation of the purity (cf. Fig 5(a)). This variation
prevents a precise manipulation of the spin-qubit state in the conde nsate [28]. It follows
from figure 6(b) that although increasing the interaction strongly modifies the spin
dynamics, it roughly conserves the spin-flip time Tsf= 50π, see (36). To demonstrate
the role of the SOC coupling strength αand interatomic interaction at nominally the
same Rabi frequency Ω R,we calculated the spin dynamics presented in Figure 7. By
comparing Figures 6 and 7(a),(b) we conclude that the increase in th e SOC, at the
same Rabi frequency, causes an increase in the variation of the pu rity and of the spin
component. These results show that to achieve a required Rabi fr equency and a reliable
control of the spin, it is better to increase the driving amplitude d0rather than the spin-
orbit couping α.The increase in the SOC strength can result in losing the spin state
purity and decreasing the spin length. Figure 7(c) shows the irregu lar spin evolution of
the condensate inside the Bloch sphere. In figure 7(a), for α= 0.2 and/tildewideg1N= 20,the
purity decreases almost to zero, placing the spin close to the cente r of the Bloch sphere,
as can be seen in figure 7(c). It follows from Figures 6(b) and 7(b) t hat in order to
protect pure macroscopic spin-qubit states, the Rabi frequenc y should be small. Then,
taking into account that the displacement of the spin-projected w ave packet is of the
order ofα(/tildewideg1N)2/3and the packet width is of the order of ( /tildewideg1N)1/3, we conclude that
forα/greaterorsimilar(/tildewideg1N)−1/3, the purity of the driven state tends to zero. As a result, the Rab i
frequency for the pure state evolution is strongly limited by the inte raction between the
particles and cannot greatly exceed d0∆/(/tildewideg1N)1/3.Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 11
(a)
0 50 100 150 2000.00.20.40.60.81.0
tP/LParen1t/RParen1(b)
050100 150 200/MinuΣ1.0/MinuΣ0.50.00.51.0
t/LAngleBracket1Σ/Hat
x/LParen1t/RParen1/RAngleBracket1
Figure 7. (Color online) ( a) Purity, ( b) spin component, and ( c) spatial trajectory of
the spin inside the Bloch sphere for the driven BEC with α= 0.2,∆ = 0.1, d0= 1
resulting in the same Rabi frequency as in Figure (6). In Figures ( a) and (b) the lines
correspond to: /tildewideg1N= 0 (black solid line), /tildewideg1N= 10 (red dashed line), and /tildewideg1N= 20
(blue dotted line). At /tildewideg1N= 0,the time dependence of ∝an}bracketle{tσx∝an}bracketri}htis rather accurately
described by cos(Ω Rt) formula, corresponding to a relatively small variation in the
purity, 1 −P(t)≪1.With the increase in /tildewideg1N,the purity variation increases and the
behavior of ∝an}bracketle{tσx∝an}bracketri}htdeviates stronger from the conventional cos(Ω Rt) dependence. ( c)
Here the interaction is fixed to /tildewideg1N= 20. The green and red vectors correspond to the
initial and final states of the spin, respectively. Here the final time is fixed to tfin=Tsf,
see (36).
In addition, it is interesting to note that for /tildewideg1N≫1,where the spin dipole
oscillates at the frequency of the order of ( /tildewideg1N)−2/3(as given by (20)), the perturbation
due to the trap motion is in the high-frequency limit already at ∆ ≥(/tildewideg1N)−2/3, having
a qualitative influence on the spin dynamics [29, 30, 31].
3.4. Phase factors due to spin-orbit coupling
The above results show that the spin-dependent velocity in (31), a long with the
interatomic repulsion, results in decreasing the spin state purity an d produces irregular
spin motion inside the Bloch sphere. To reduce the effect of these an omalous velocities
and to prevent the resulting fast separation (with the relative velo city of 2α) of the spin
components, we compensate them by introducing coordinate-dep endent phases (similar
to the Bragg factors) in the wave function [32]. To demonstrate th e effect of theseDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 12
(a)
0 50 100 150 2000.00.20.40.60.81.0
tP/LParen1t/RParen1(b)
050100 150 200/MinuΣ1.0/MinuΣ0.50.00.51.0
t/LAngleBracket1Σ/Hat
x/LParen1t/RParen1/RAngleBracket1
Figure 8. (Color online) ( a) Purity, ( b) spin component, and ( c) trajectory of the
spin inside the Bloch sphere for a driven BEC with initial phases as in (37 ) and
α= 0.2,∆ = 0.1, d0= 1.In Figures ( a) and (b) the black solid line is for /tildewideg1N= 0,
the red dashed line is for /tildewideg1N= 10, and the blue dotted line is for /tildewideg1N= 20. In Figure
(c) the interaction is /tildewideg1N= 20. The green and red vectors correspond to the initial
and final states of the spin, respectively ( tfin=Tsf). The initial spin state (a solid-line
circle with white filling) is inside the Bloch sphere since ∝an}bracketle{tσx(t= 0)∝an}bracketri}ht=/radicalbig
P(0), and
P(0)<1 due to the mixed character in the spin subspace of the state in (37 ).
phase factors, we construct the initial spinor Ψ α(x,0) by a coordinate-dependent SU(2)
rotation [33] of the state with ∝an}bracketle{tσx∝an}bracketri}ht= 1 in (30) as
Ψα(x,0) =e−iαx/hatwideσzΨ0(x,0) =ψin(x)√
2/bracketleftBigg
e−iαx
eiαx/bracketrightBigg
. (37)
The expectation value of the velocity (31) at each component ψin(x)exp(±iαx) is zero,
and, as a result, the α-induced separation of spin components vanishes, making, as can
be easily seen [33], the spinor (37) the stationary state of the spin- orbit coupled BEC
in the Gross-Pitaevskii approximation.
In terms of the spin density matrix (24), the state (37) is mixed. Fo r a Gaussian
condensate with the width w, we get the following expression for the purity at t= 0
P[G]
α(0) = exp/bracketleftbig
−2(αw)2/bracketrightbig
. (38)Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 13
Instead, in the case of a Thomas-Fermi wave function as in (8), in t he limitαwTF≫1
the initial purity behaves as
P[TF]
α(0)∼cos2(2αwTF)
(αwTF)4. (39)
Both cases are characterized by a rapid decrease as αwTFis increased [25].
In the absence of external driving, the spin components and purit y of (37) state
remain constant. With the driving, spinor components evolve with tim e and the
observables show evolution quantitatively different from that pres ented in Figure 7. In
figure 8 we show the analog of figure 7 for the initial state in (37), wit hψin(x) =ψ0(x)
given by (2)-(5). By comparing these Figures one can see that the inclusion of the
spin-dependent phases in (37) strongly reduces the oscillations in t hex−component of
the spin, making the spin trajectory more regular and decreasing t he variations in the
purityP(t) compared to the initial state without these phase factors.
A general effect of the interatomic interaction can be seen in both fi gures 7 and 8.
Namely, for smaller interactions /tildewideg1N, the destructive role of the interatomic repulsion
on the spin state purity is reduced and the spin dynamics becomes mo re regular. As a
result, at smaller /tildewideg1Nthe spin trajectory is located closer to the Bloch sphere.
4. Conclusions
We have considered the dynamics of freely expanding and harmonica lly driven
macroscopic spin qubits based on quasi one-dimensional spin-orbit coupled Bose-
Einstein condensates in a synthetic Zeeman field. The resulting evolu tion strongly
depends in a nontrivial way on the spin-orbit coupling and interaction between the
bosons. On one hand, spin-orbit coupling leads to the driven spin qub it dynamics. On
the other hand, it leads to a spin-dependent anomalous velocity cau sing spin splitting
of the initial wave packet and reducing the purity of the spin state b y decreasing the
off-diagonal components of the spin density matrix. This destruct ive influence of spin-
orbit coupling is enhanced by interatomic repulsion. The effects of th e repulsion can be
interpreted in terms of the increase in the spatial width of the cond ensate and the
corresponding decrease in the spin dipole oscillation frequency with t he interaction
strength. The joint influence of the repulsion and spin-orbit couplin g can spatially
separate and modify the spin components stronger than just the spin-orbit coupling
and result in stronger irregularities in the spin dynamics. The spin-flip Rabi frequency
remains, however, almost unchangedinthepresence oftheintera tomicinteractions since
they lead to only internal forces and do not change the condensat e momentum. As a
result, to preserve the evolution within a high-purity spin-qubit sta te, with the spin
being always close to the Bloch sphere, the spin-orbit coupling should be weak and, due
to this weakness, the spin-rotation Rabi frequency should be sma ll and spin rotation
should take a long time. The destructive effect of both the spin-orb it coupling and
interatomicrepulsiononthepurityofthespinstatecanbecontrolla blyandconsiderablyDynamics of a macroscopic spin qubit in spin-orbit coupled B EC 14
reduced, although not completely removed, by introducing spin-de pendent Bragg-like
phase factors in the initial spinor wave function.
Acknowledgments
This work was supported by the University of Basque Country UPV/ EHU under
program UFI 11/55, Spanish MEC (FIS2012-36673-C03-01 and FI S2012-36673-C03-
03), and Grupos Consolidados UPV/EHU del Gobierno Vasco (IT-47 2-10). S.M.
acknowledges EU-funded Erasmus Mundus Action 2 eASTANA, “evr oAsian Starter for
the Technical Academic Programme” (Agreement No. 2001-2571/ 001-001-EMA2).
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[24] Rashba E I and Efros Al L 2003 Phys. Rev. Lett. 91126405
[25] It is interesting to mention that an increase in spin-orbit coupling strength after a certain value
leads to a less efficient spin driving as shown with perturbation theory approach by Li R, You
J Q, Sun C P and Nori F 2013 Phys. Rev. Lett. 111086805
[26] This procedure models the von Neumann quantum spin measurem ent, see Sherman E Ya and
Sokolovski D 2014 New J. Phys. 16015013
[27] An analysis of the spin-dipole oscillations based on the sum rules wa s presented in [9]Dynamics of a macroscopic spin qubit in spin-orbit coupled B EC 15
[28] This low precision of the spin control can be seen as a general fe ature of the systems where
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V and Sherman E Ya 2012 Phys. Rev. B85125312
[29] A technique to study and engineer the high-frequency behavio r has been introduced in Bukov M,
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Salasnich L and Polls A 2014 Phys. Rev. A89063607
Citro R and Naddeo A 2014 arXiv:1405.5356
[33] Tokatly I V and Sherman E Ya 2010 Phys. Rev. B82161305 |
1203.2795v1.Impact_of_Dresselhaus_vs__Rashba_spin_orbit_coupling_on_the_Holstein_polaron.pdf | arXiv:1203.2795v1 [cond-mat.str-el] 13 Mar 2012Impact of Dresselhaus vs. Rashba spin-orbit coupling on the Holstein polaron
Zhou Li1, L. Covaci2, and F. Marsiglio1
1Department of Physics, University of Alberta, Edmonton, Al berta, Canada, T6G 2J1
2Departement Fysica, Universiteit Antwerpen, Groenenborg erlaan 171, B-2020 Antwerpen, Belgium
(Dated: November 6, 2018)
We utilize an exact variational numerical procedure to calc ulate the ground state properties of a
polaron in the presence of Rashba and linear Dresselhaus spi n-orbit coupling. We find that when
the linear Dresselhaus spin-orbit coupling approaches the Rashba spin-orbit coupling, the Van-Hove
singularity in the density of states will be shifted away fro m the bottom of the band and finally
disappear when the two spin-orbit couplings are tuned to be e qual. The effective mass will be
suppressed; the trend will become more significant for low ph onon frequency. The presence of two
dominant spin-orbit couplings will make it possible to tune the effective mass with more varied
observables.
I. INTRODUCTION
One of the end goals in condensed matter physics is to
achieve a sufficient understanding of materials fabrica-
tion and design so as to ‘tailor-engineer’ specific desired
properties into a material. Arguably pn-junctions long
ago represented some of the first steps in this direction;
nowadays, heterostructures1and mesoscopic geometries2
represent further progress towards this goal.
Inthefieldof spintronics , wherethespindegreeoffree-
dom is specifically exploited for potential applications,3,4
spin-orbit coupling5plays a critical role because con-
trol of spin will require coupling to the orbital mo-
tion. Spin orbit coupling, as described by Rashba6and
Dresselhaus,7is expected to be prominent in two dimen-
sional systems that lack inversion symmetry, including
surface states. These different kinds of coupling are in
principle independently controlled.8,9
The coexistence of Rashba and Dresselhaus spin-orbit
coupling has now been realized in both semiconductor
quantum wells4,9and more recently in a neutral atomic
Bose-Einstein condensate.10When the Rashba and (lin-
ear) Dresselhaus spin-orbit coupling strengths are tuned
to be equal, SU(2) symmetry is predicted to be recov-
ered and the persistent spin helix state will emerge.4,10,11
This symmetry is expected to be robust against spin-
independent scattering but is broken by the cubic Dres-
selhaus spin-orbit coupling and other spin-dependent
scattering which may be tuned to be negligible.4
While we focus on the spin-orbit interaction, other in-
teractionsarepresent. In particular, the electron-phonon
interaction will be present and may be strong in semi-
conductor heterostructures. Moreover, optical lattices12
with cold polar molecules may be able to realize a tune-
able Holstein model.13The primary purpose of this work
is to investigate the impact of electron-phonon coupling
(as modelled by the Holstein model14) on the proper-
ties of the spin-orbit coupled system. We will utilize a
tight-binding framework; previously it was noted that in
the presence of Rashba spin-orbit coupling the vicinity
of a van Hove singularity near the bottom of the elec-
tron band15–17had a significant impact on the polaronic
propertiesofanelectron; withadditional(linear)Dressel-haus spin-orbit coupling the van Hove singularity shifts
well away from the band bottom, as the two spin-orbit
couplings acquire equal strength. As we will illustrate
below, the presence of two separately tunable spin-orbit
couplings will result in significant controllability of the
electron effective mass.
II. MODEL AND METHODOLOGIES
We use a tight-binding model with dimensionless Hol-
stein electron-phonon coupling of strength g, and with
linear Rashba ( VR) and Dresselhaus ( VD) spin-orbit cou-
pling:
H=−t/summationdisplay
<i,j>,s =↑↓(c†
i,scj,s+c†
j,sci,s)
+i/summationdisplay
j,α,β(c†
j,αˆV1cj+ˆy,β−c†
j,αˆV2cj+ˆx,β−h.c.)
−gωE/summationdisplay
i,s=↑↓c†
i,sci,s(ai+a†
i)+ωE/summationdisplay
ia†
iai(1)
wherec†
i,s(ci,s) creates (annihilates) an electron at site
iwith spin index s, anda†
i(ai) creates (annihilates) a
phononatsite i. The operators ˆVj,j= 1,2arewritten in
terms of the spin-orbit coupling strengths and the Pauli
matrices as ˆV1=VRˆσx−VDˆσy, andˆV2=VRˆσy−VDˆσx,
The sum over iis over all sites in the lattice, whereas
< i,j > signifies that only nearest neighbour hopping
is included. Other parameters in the problem are the
phonon frequency, ωE, and the hopping parameter t,
which hereafter is set equal to unity.
Without the electron-phononinteractionthe electronic
structure is readily obtained by diagonalizing the Hamil-
tonian in momentum space. With the definitions
S1≡VRsin(ky)+VDsin(kx),
S2≡VRsin(kx)+VDsin(ky), (2)
we obtain the eigenvalues
εk,±=−2t[cos(kx)+cos(ky)]±2/radicalBig
S2
1+S2
2(3)2
-5-4-3-2-1 0 1 2 3 4
-3-2-1 0 1 2 3-3-2-1 0 1 2 3VR/t = 0.5 VD/t = 0.5
-5-4-3-2-1 0 1 2 3 4
-3-2-1 0 1 2 3-3-2-1 0 1 2 3VR/t = 0.8 VD/t = 0.2
-5-4-3-2-1 0 1 2 3 4
-3-2-1 0 1 2 3-3-2-1 0 1 2 3VR/t = 0.9 VD/t = 0.1
-5-4-3-2-1 0 1 2 3 4
-3-2-1 0 1 2 3-3-2-1 0 1 2 3VR/t = 0.99 VD/t = 0.01
FIG. 1. Contour plots for the bare energy bands with Rashba-
Dresselhaus spin-orbit coupling, for different values of VRand
VDwhile the sum is kept constant: VR+VD=tfor these
cases. (a) VR=VD= 0.5t, (b)VR= 0.8t,VD= 0.2t, (c)
VR= 0.9t,VD= 0.1t, and (d) VR= 0.99t,VD= 0.01t. Note
the clear progression from a two-fold degenerate ground sta te
to a four-fold degenerate one.and eigenvectors
Ψk±=1√
2/bracketleftBigg
c†
k↑±S1−iS2/radicalbig
S2
1+S2
2c†
k↓/bracketrightBigg
|0/angbracketright.(4)
The ground state energy is
E0=−4t/radicalbig
1+(VR+VD)2/(2t2). (5)
Without loss of generality we can consider only VR≥0
andVD≥0. Either Rashba and Dresselhaus spin-orbit
coupling independently behave in the same manner, and
give rise to a four-fold degenerate ground state with
wave vectors, ( kx,ky) = (±arctan(VR√
2t),±arctan(VR√
2t),
(VD= 0), and similarly for VD/negationslash= 0 and VR= 0. With
both couplings non-zero, however, the degeneracy be-
comes two-fold, with the ground state wave vectors,
(kx0,ky0) =±(k0,k0); where k0= tan−1(VR+VD√
2t).
(6)
It is clear that the sum of the coupling strengths replaces
the strength of either in these expressions, so that hence-
forth in most plots we will vary one of the spin-orbit
interaction strengths while maintaining their sum to be
fixed. Similarly, the effective mass, taken along the diag-
onal, is
mSO
m0=1/radicalbig
1+(VR+VD)2/(2t2), (7)
wherem0≡1/(2t) (lattice spacing, a≡1, and/planckover2pi1≡1)
is the bare mass in the absence of spin-orbit interaction,
andmSOis the effective mass due solely to the spin-orbit
interaction. As detailed in the Appendix, the effective
mass becomes isotropic when the Rashba and Dressel-
haus spin-orbit coupling strengths are equal.
The non-interacting electron density of states (DOS)
is defined for each band, as
Ds(ǫ) =/summationdisplay
kδ(ǫ−ǫks) (8)
withs=±1.
In Fig.2(a) we show the low energy DOS for various
values of the spin-orbit coupling strengths, VRandVD,
while keeping their sum constant; the low energy van
Hove singularity disappears for VR=VD. Note that only
D−(ǫ) is shown, as the upper band, with DOS D+(ǫ),
exists only at higher energies. Furthermore, informa-
tion concerning the upper band can always be obtained
through the symmetry
D+(ǫ) =D−(−ǫ). (9)
In Fig.2(b) we show the value of the density of states
at the bottom of the band vs. VD; as derived in the
Appendix, theDOSvalueattheminimumenergyisgiven
by
D−(E0) =1
2πt1/radicalBig
1+(VR+VD)2
2t2−(VR−VD)2
(VR+VD)2.(10)3
/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54
/s32/s86
/s82/s61/s48/s46/s57/s57/s44/s32/s86
/s68/s61/s48/s46/s48/s49
/s32/s86
/s82/s61/s48/s46/s57/s44/s32/s32/s32/s86
/s68/s61/s48/s46/s49
/s32/s86
/s82/s61/s48/s46/s56/s44/s32/s32/s32/s86
/s68/s61/s48/s46/s50
/s32/s86
/s82/s61/s48/s46/s53/s44/s32/s32/s32/s86
/s68/s61/s48/s46/s53/s68/s95/s40/s69/s41
/s69/s40/s97/s41
/s32
/s74/s117/s109/s112/s32/s111/s102/s32/s100/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s115/s116/s97/s116/s101/s115
/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54
/s32/s86
/s82/s61/s48/s44/s32/s86
/s68/s61/s49/s32/s111/s114/s32/s32/s86
/s68/s61/s48/s44/s32/s86
/s82/s61/s49
/s32/s86
/s82/s43/s86
/s68/s61/s48/s46/s52
/s32/s86
/s82/s43/s86
/s68/s61/s49/s68/s95/s40/s69
/s48/s41
/s86
/s82/s47/s40/s86
/s82/s43/s86
/s68/s41
/s32/s40/s98/s41
FIG. 2. (a)The non-interacting density of states D−(E)
near the bottom of the band for four values of the spin-
orbit coupling strengths: ( VR,VD)/t= (0.5,0.5) (dot-dashed
curve), (0 .8,0.2) (dotted curve), (0 .9,0.1) (dashed curve),
and (0.99,0.01) (solid curve). Note that for equal coupling
strengths there is no van Hove singularity at low energies.
(b) The value of the density of states at the bottom of the
band (ground state) as a function of VD(while the total cou-
pling strength, VR+VD, is held constant. The value of the
density of states achieves a minimum value when VR=VD.
ForVR= 0 orVD= 0 there is a discontinuity, caused by
the transition from a doubly degenerate ground state to a
four-fold degenerate ground state.Note that when the coupling strengths are equal, the
density of states has a minimum. Also note that when
one kind of spin-orbit coupling vanishes, e.g. VR= 0,
orVD= 0, there will be a discontinuity for the density
of states (the density of states jumps to twice its value).
This is caused by a transition from a doubly degenerate
groundstate to a four-fold degenerateground state. This
discontinuity will also appear for VD≃0 orVR≃0 near
the bottom of the band as can be seen from Fig.2(a) for
VR= 0.99,VD= 0.01.
III. RESULTS WITH THE
ELECTRON-PHONON INTERACTION
As the electron phonon interaction is turned on, the
ground state energy (effective mass) will decrease (in-
crease) due to polaron effects. To study the polaron
problem numerically, we adopt the variational method
outlined by Trugman and coworkers,18,19which is a con-
trolled numerical technique to determine polaron prop-
erties in the thermodynamic limit exactly. This method
was recently further developed20,21to study the polaron
problem near the adiabatic limit with Rashba spin-orbit
coupling.17This case was also studied in Ref. [16] using
the Momentum Average Approximation.22
In Fig. 3, we show the ground state energy and
the effective mass correction as a function of the elec-
tron phonon coupling λ≡2g2ωE/(4πt),20for various
spin-orbit coupling strengths, but with the sum fixed:
VR+VD=t. These are compared with the results
from the Rashba-Holstein model with VD= 0. Here the
phonon frequency is set to be ωE/t= 1.0, which is the
typical value used in Ref.[16], and for each value of VR,
the ground state energy is compared to the correspond-
ing result for λ= 0. The numerical results are compared
with results from the MA method and from Lang-Firsov
strong coupling theory23,24(see Appendix). In Fig. 3(a),
the ground state energycrossesoversmoothly (at around
λ≈0.8)fromthe delocalizedelectronregimetothe small
polaron regime. In the whole regime, the ground state
energy is shifted up slightly as the Dresselhaus spin-orbit
coupling, VD, is increased in lieu of the Rashba spin-
orbit coupling. We show results for VD≤VR, as the
complementary regime is completely symmetric. The
MA results agree very well with the exact results and
the Lang-Firsov strong coupling results agree well in the
λ≥1 regime. Similarly, weak coupling perturbation
theory17agrees with the exact results for λ≤1 (not
shown). Fig. 3(b) shows the effective mass as a func-
tion of coupling strength; it decreases slightly, for a given
value ofλ, by increasing VDin lieu of VR.
All these results are plotted as a function of the elec-
tron phonon coupling strength, λ, as defined above; this
definition requires the value of the electron density of
states at the bottom of the band, and we have elected to
use, for any value of spin-orbit coupling, the value 1 /(4πt
appropriate to nospin-orbit coupling. If the actual DOS4
/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s45/s53/s45/s52/s45/s51/s45/s50/s45/s49/s48
/s69/s47/s116/s61/s49/s46/s48
/s32/s76/s70/s44/s32 /s86
/s82/s47/s116/s61/s48/s46/s53
/s32/s76/s70/s44/s32 /s86
/s82/s47/s116/s61/s48/s46/s56
/s32/s76/s70/s44/s32 /s86
/s82/s47/s116/s61/s49/s46/s48/s86
/s82/s47/s116/s43/s86
/s68/s47/s116/s61/s49/s46/s48/s32/s69/s120/s97/s99/s116/s44/s32/s86
/s82/s47/s116/s61/s48/s46/s53
/s32/s69/s120/s97/s99/s116/s44/s32/s86
/s82/s47/s116/s61/s48/s46/s56
/s32/s69/s120/s97/s99/s116/s44/s32/s86
/s82/s47/s116/s61/s49/s46/s48
/s32/s77/s46/s65/s46/s32/s86
/s82/s47/s116/s61/s48/s46/s53
/s32/s77/s46/s65/s46/s32/s86
/s82/s47/s116/s61/s48/s46/s56
/s32/s77/s46/s65/s46/s32/s86
/s82/s47/s116/s61/s49/s46/s48/s40/s69
/s71/s83/s45/s69
/s48/s41/s47/s116
/s32/s40/s97/s41
/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48
/s86
/s82/s47/s116/s43/s86
/s68/s47/s116/s61/s49/s46/s48/s32/s77/s46/s65/s46/s32/s86
/s82/s47/s116/s61/s48/s46/s53
/s32/s77/s46/s65/s46/s32/s86
/s82/s47/s116/s61/s48/s46/s56
/s32/s77/s46/s65/s46/s32/s86
/s82/s47/s116/s61/s49/s46/s48
/s32/s32/s69/s120/s97/s99/s116/s44/s32/s86
/s82/s47/s116/s61/s48/s46/s53
/s32/s32/s69/s120/s97/s99/s116/s44/s32/s86
/s82/s47/s116/s61/s48/s46/s56
/s32/s32/s69/s120/s97/s99/s116/s44/s32/s86
/s82/s47/s116/s61/s49/s46/s48/s109/s42/s47/s109
/s83/s79
/s32/s69/s47/s116/s61/s49/s46/s48/s40/s98/s41
FIG. 3. (a) Ground state energy difference EGS−E0vs.λfor
VR/t= 0.5,0.8,1.0 andωE/t= 1.0 while the total coupling
strength is kept fixed: VR+VD=t. Exact numerical results
are compared with those from the Momentum Average (MA)
method. Agreement is excellent. Strong coupling results ar e
also plotted (in red) by utilizing the Lang-Firsov (LF) stro ng
coupling approximation. Agreement in the strong coupling
regime ( λ≥1) is excellent. (b) Effective mass m∗/mSOvs.
λ. MA results are plotted (symbols) with the exact numerical
results, and again, agreement is excellent. In both (a) and ( b)
the polaronic effects are minimized for VR=VD./s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49
/s86
/s82/s47/s116/s43/s86
/s68/s47/s116/s61/s49/s46/s48/s32/s77/s46/s65/s46
/s69/s47/s116/s61/s48/s46/s49
/s32/s77/s46/s65/s46
/s69/s47/s116/s61/s48/s46/s50
/s32/s77/s46/s65/s46
/s69/s47/s116/s61/s49/s46/s48
/s32/s69/s120/s97/s99/s116
/s69/s47/s116/s61/s48/s46/s49
/s32/s69/s120/s97/s99/s116
/s69/s47/s116/s61/s48/s46/s50
/s32/s69/s120/s97/s99/s116
/s69/s47/s116/s61/s49/s46/s48/s40/s69
/s71/s83/s45/s69
/s48/s41/s47/s116
/s86
/s68/s47/s116
/s32/s61/s48/s46/s51/s50/s40/s97/s41
/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s49/s46/s49/s48/s49/s46/s49/s53/s49/s46/s50/s48/s49/s46/s50/s53/s49/s46/s51/s48/s49/s46/s51/s53/s49/s46/s52/s48/s49/s46/s52/s53/s49/s46/s53/s48/s49/s46/s53/s53
/s32/s77/s46/s65/s46
/s69/s47/s116/s61/s48/s46/s49
/s32/s77/s46/s65/s46
/s69/s47/s116/s61/s48/s46/s50
/s32/s77/s46/s65/s46
/s69/s47/s116/s61/s49/s46/s48
/s32/s69/s120/s97/s99/s116
/s69/s47/s116/s61/s48/s46/s49
/s32/s69/s120/s97/s99/s116
/s69/s47/s116/s61/s48/s46/s50
/s32/s69/s120/s97/s99/s116/s44
/s69/s47/s116/s61/s49/s46/s48/s86
/s82/s47/s116/s43/s86
/s68/s47/s116/s61/s49/s46/s48/s109/s42/s47/s109
/s83/s79
/s86
/s68/s47/s116
/s32/s61/s48/s46/s51/s50/s40/s98/s41
FIG.4. (a)Groundstateenergy EGS−E0asafunctionofspin
orbit coupling VD/tforωE/t= 0.1,0.2,1.0 with weakelectron
phonon coupling, λ= 0.32, and moderate spin-orbit coupling,
VR+VD=t. (b) Effective mass m∗/mSOas a function of
spin orbit coupling VD/tfor the same parameters. MA results
are again compared with the exact numerical results, and are
reasonably accurate for these parameters.5
appropriate to the value of spin-orbit coupling were used
in the definition of λ, then the effective mass, for ex-
ample, would vary even more with varying VDvs.VR
(see Fig. 2(b)). Moreover, this variation would be more
pronounced for lower values of ωE.
In Fig. 4, we show results for the ground state energy
and effective mass for different values of the Einstein
phonon frequency, ωE; MA results are also shown for
comparison. In these plots the electron phonon coupling
strength is kept fixed and VDis varied while maintaining
the total spin-orbit coupling constant. The ground state
energy has a maximum when the two spin-orbit coupling
strengths, VDandVR, are tuned to be equal; similarly,
theeffectivemasshasaminimumwhenthetwoareequal.
As the phonon frequency is reduced the minimum in the
effective mass becomes more pronounced. The MA re-
sults track the exact results, and, as found previously,17
are slightly less accurate as the phonon frequency be-
comes much lower than the hopping matrix element, t.
IV. SUMMARY
Linear spin-orbit coupling can arise in two varieties;
taken on their own, they are essentially equivalent, and
their impact on a single electron, even in the presence of
electron phonon interactions, will be identical. However,with the ability to tune either coupling constant, in both
solid state and cold atom experiments, one can probe
the degree of Dresselhaus vs. Rashba spin-orbit coupling
throughtheimpactonpolaronicproperties. Theprimary
effect of this variation is the electron density of states,
where the van Hove singularity can be moved as a func-
tion of chemical potential (i.e. doping) through tuning of
the spin-orbit parameters. These conclusions are based
on exact methods (the so-called Trugman method), and
are not subject to approximations. These results have
been further corroborated and understood through the
Momentum Average approximation, and through weak
and strong coupling perturbation theory. The effect is
expected to be experimentally relevant since in typi-
cal materials with large spin-orbit couplings the phonon
frequency is small when compared to the bandwidth,
ωE/t≪1.
ACKNOWLEDGMENTS
This work was supported in part by the Natural
Sciences and Engineering Research Council of Canada
(NSERC), by ICORE (Alberta), by the Flemish Science
Foundation (FWO-Vl) and by the Canadian Institute for
Advanced Research (CIfAR).
1See, for example, Quantum Dot Heterostructures , byDieter
Bimberg, Marius Grundmann, and Nikolai N. Ledentsov
(John Wiley and Sons, Toronto, 1999).
2For example, Mesoscopic Systems , by Y. Murayama
(Wiley-VCH, Toronto, 2001).
3S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.
Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelka-
nova and D. M. Treger, Science 294, 1488, (2001).
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Shou-Cheng Zhang, S. Mack and D. D. Awschalom, Na-
ture,458, 610-613(2009).
5R. Winkler, Spin-Orbit Coupling Effects in Two-
Dimensional Electron and Hole Systems (Springer, Berlin,
2003).
6E.I. Rashba, Sov. Phys. Solid State 2, 1109 (1960).
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8S.K. Maiti, S. Sil, and A. Chakrabarti, arXiv:1109.5842v2.
9L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Sch¨ on, and
K. Ensslin, Nature Physics 3, 650 (2007).
10Y. J. Lin, K. Jimenez-Garcia and I. B. Spielman, Nature,
471, 83 (2011). See alsoT. Ozawa and G. Baym, Phys.
Rev. A85, 013612 (2012).
11B.A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev.
Lett.97, 236601 (2006).
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80, 885 (2008).
13F. Herrera and R.V. Krems, Phys. Rev. A 84, 051401(R),
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14T. Holstein, Ann. Phys. (New York) 8, 325 (1959).15E. Cappelluti, C. Grimaldi and F. Marsiglio, Phys. Rev.
Lett98, 167002 (2007); Phys. Rev. B 76, 085334 (2007).
See also, C. Grimaldi, E. Cappelluti, and F. Marsiglio,
Phys. Rev. Lett. 97, 066601 (2006); Phys. Rev. B 73,
081303(R) (2006).
16L. Covaci and M. Berciu, Phys. Rev. Lett 102, 186403
(2009).
17Zhou Li, L.Covaci, M. Berciu, D. Baillie and F. Marsiglio,
Phys. Rev. B 83, 195104, (2011).
18S.A. Trugman, in Applications of Statistical and Field The-
ory Methods to Condensed Matter , edited by D. Baeriswyl,
A.R. Bishop, and J. Carmelo (Plenum Press, New York,
1990).
19J. Bonˇ ca, S.A. Trugman, and I. Batist´ ıc, Phys. Rev.
B60,1633 (1999).
20Zhou Li, D. Baillie, C. Blois, and F. Marsiglio, Phys. Rev.
B81, 115114, (2010).
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B81, 165113 (2010).
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Appendix A: Density of States and effective mass
Expanding εk,−around the minimum energy E0,
by defining k′
x=kx±arctan(VR+VD√
2t),k′
y=ky±6
arctan(VR+VD√
2t),we have
εk,−=E0+˜t1/braceleftbig
k′2
x+k′2
y/bracerightbig
±˜t2k′
xk′
y,(A1)
where
˜t1=t/braceleftBigg
1+(VR+VD)2
2t2−(VR−VD)2
2(VR+VD)2/radicalbig
1+(VR+VD)2/(2t2)/bracerightBigg
,(A2)
and
˜t2=t/braceleftBigg(VR−VD)2
(VR+VD)2/radicalbig
1+(VR+VD)2/(2t2)/bracerightBigg
,(A3)
Note that, with generic spin-orbit coupling, the effective
mass is in general anisotropic, but when VD=VR, it
becomes isotropic.
To calculate the density of states at the bottom of the
band, from the definition, we have
D−(E0+δE) =1
4π2/integraldisplayπ
−πdkx/integraldisplayπ
−πdkyδ(E0+δE−εk,−),
(A4)
whereδEis a small amount of energy above the bottom
of the band, E0. Around the two energy minimum points
there are two small regions which will contribute to this
integral. We choose one of them (and then multiply our
resultbyafactorof2),thenusethedefinitionsof k′above
instead of k, and introduce a small cutoff kc, which is the
radius of a small circle around kmin. Thus the integral
becomes
D−(E0+δE) = 2×1
4π2/integraldisplaykc
0k′dk′/integraldisplayπ
−πdθ
δ/bracketleftbigg
δE−/braceleftbig˜t1+1
2˜t2sin2θ/bracerightbig
k′2/bracketrightbigg
=1
2πt1/radicalBig
1+(VR+VD)2
2t2−(VR−VD)2
(VR+VD)2(A5)
In the weak electron-phonon coupling regime, perturba-
tion theory can be applied to evaluate the effective mass;
the self energy to first order in λis given by
Σweak(ω+iδ) =πλtωE/summationdisplay
k,s=±1
ω+iδ−ωE−εk,s.(A6)Theeffectivemasscanbeobtainedthroughthederivative
of the self energy
m∗
weak
mSO= 1−∂
∂ωΣweak(ω+iδ)|ω=E0.(A7)
By inserting the expansion of εk,−around the minimum
energyE0into Eqn.[A6] and Eqn.[A7], we obtain the
effective mass near the adiabatic limit as
m∗
weak
mSO= 1+λ
21/radicalBig
1+(VR+VD)2
2t2−(VR−VD)2
(VR+VD)2.(A8)
The effective mass has a minimum for VR=VDwhile
VR+VDis a constant.
Appendix B: Strong coupling theory
To investigate the strong coupling regime of the
Rashba-Dresselhaus-Holstein model for a single polaron,
we use the Lang-Firsov2324unitary transformation H=
eSHe−S, whereS=g/summationtext
i,σni,σ(ai−a†
i). Following pro-
cedures similar to those in Ref. (17), we obtain the first
order perturbation correction to the energy as
E(1)
k±=e−g2εk±−g2ωE, (B1)
wheregis the band narrowing factor, as used in the Hol-
stein model. To find the second order correction to the
ground state energy, we proceed as in Ref. (17), and find
E(2)
k−=−4e−2g2t2+(VR)2+(VD)2
ωE
×/bracketleftbig
f(2g2)−f(g2)/bracketrightbig
−e−2g2f(g2)ǫ2
k−
ωE,(B2)
wheref(x)≡∞/summationtext
n=11
nxn
n!≈ex/x/bracketleftbig
1 + 1/x+ 2/x2+.../bracketrightbig
.
Thus the ground state energy, excluding exponentially
suppressed corrections, is
EGS=−2πtλ/parenleftbig
1+2t2+(VR)2+(VD)2
(2πtλ)2/parenrightbig
,(B3)
and there is a correction of order 1 /λ2compared to the
zeroth order result. Corrections in the dispersion enter
in strong coupling only with an exponential suppression.
The ground state energy predicted by strong coupling
theory has a maximum for VR=VDwhileVR+VDis a
constant. |
1606.05758v1.Spin_Transport_at_Interfaces_with_Spin_Orbit_Coupling__Formalism.pdf | Spin Transport at Interfaces with Spin-Orbit Coupling: Formalism
V. P. Amin1, 2,and M. D. Stiles2
1Maryland NanoCenter, University of Maryland, College Park, MD 20742
2Center for Nanoscale Science and Technology, National Institute
of Standards and Technology, Gaithersburg, Maryland 20899, USA
(Dated: June 21, 2016)
We generalize magnetoelectronic circuit theory to account for spin transfer to and from the atomic
lattice via interfacial spin-orbit coupling. This enables a proper treatment of spin transport at inter-
faces between a ferromagnet and a heavy-metal non-magnet. This generalized approach describes
spin transport in terms of drops in spin and charge accumulations across the interface (as in the
standard approach), but additionally includes the responses from in-plane electric elds and o sets
in spin accumulations. A key nding is that in-plane electric elds give rise to spin accumulations
and spin currents that can be polarized in any direction, generalizing the Rashba-Edelstein and
spin Hall e ects. The spin accumulations exert torques on the magnetization at the interface when
they are misaligned from the magnetization. The additional out-of-plane spin currents exert torques
via the spin-transfer mechanism on the ferromagnetic layer. To account for these phenomena we
also describe spin torques within the generalized circuit theory. The additional e ects included in
this generalized circuit theory suggest modi cations in the interpretations of experiments involving
spin orbit torques, spin pumping, spin memory loss, the Rashba-Edelstein e ect, and the spin Hall
magnetoresistance.
I. INTRODUCTION
The spin-orbit interaction couples the spin and mo-
mentum of carriers, leading to a variety of important
e ects in spintronic devices. It enables the conversion
between charge and spin currents [1, 2], allows for the
transfer of angular momentum between populations of
spins [3{9], couples charge transport and thermal trans-
port with magnetization orientation [10{17], and results
in magnetocrystalline anisotropy [18{20]. Many of these
e ects already facilitate technological applications. The
development of such applications can be assisted by both
predictive (yet complicated) rst-principles calculations
and clear phenomenological models, which would aid the
interpretation of experiments and help to predict device
behavior.
In multilayer systems, bulk spin-orbit coupling plays a
crucial role in spin transport but the role of interfacial
spin-orbit coupling remains largely unknown. This un-
certainty derives from the uncharacterized transfer of an-
gular momentum between carriers and the atomic lattice
while scattering from interfaces with spin-orbit coupling.
This transfer of angular momentum occurs because a car-
rier's spin is coupled via spin-orbit coupling to its orbital
moment, which is coupled via the Coulomb interaction to
the crystal lattice. Such interfaces behave as either a sink
or a source of spin polarization for carriers in a way that
does not yet have an accurate phenomenological descrip-
tion. In this paper we develop a formal generalization of
magnetoelectronic circuit theory to treat interfaces with
spin-orbit coupling. In a companion paper, we extract
the most important consequences of this generalization
vivek.amin@nist.govand show that they capture the dominant e ects found
in more complicated Boltzmann equation calculations.
To understand the impact of interfacial spin-orbit cou-
pling we consider a heavy metal/ferromagnet bilayer,
where in-plane currents generate torques on the magne-
tization through various mechanisms that involve spin-
orbit coupling [6, 7, 21{24]. For example, bulk spin-orbit
coupling converts charge currents in the heavy metal
into orthogonally-
owing spin currents, through a pro-
cess known as the spin Hall e ect [25{31]. Upon entering
the ferromagnetic layer these spin currents transfer angu-
lar momentum to the magnetization through spin trans-
fer torques [32{36]. Both the spin Hall e ect and spin-
transfer torques have been extensively studied, but addi-
tional sources contribute to the total spin torque. These
remaining contributions arise from interfacial spin-orbit
coupling, which enables carriers of the in-plane charge
current to develop a net spin polarization at the inter-
face [37{41]. In systems with broken inversion symmetry
(such as interfaces) the generation of such spin polariza-
tion is known as the Rashba-Edelstein e ect. This spin
polarization can exert a torque on any magnetization at
the interface via the exchange interaction [7, 42]. A re-
cent experiment suggests that this mechanism can induce
magnetization switching alone, without relying on the
bulk spin Hall e ect [24].
The spin torque driven by the Rashba-Edelstein e ect
is typically studied by con ning transport to the two-
dimensional interface. Semiclassical models can capture
the direct and inverse Rashba-Edelstein e ects [43{46] in
this scenario. However, such models are not realistic de-
scriptions of bilayers, in which carriers scatter both along
and across the interface. Since spin transport across the
interface is a ected by the transfer of angular momentum
to the atomic lattice, the resulting spin torques are modi-
ed in ways that two-dimensional models cannot capture.arXiv:1606.05758v1 [cond-mat.mes-hall] 18 Jun 20162
FIG. 1. (Color online) (a) A heavy metal/ferromagnet bilayer subject to an in-plane electric eld. The axes directly below the
bilayer is used to describe electron
ow, where the z-axis points normal to the interface plane. The other axes is used to describe
spin orientation, where the direction `points along the magnetization while the directions dandfspan the plane transverse to
`. (b) Depiction of the physics described by the spin mixing conductance. Spins incident from the heavy metal brie
y precess
around the magnetization when re
ecting o of the interface. The imaginary part of the spin mixing conductance describes the
extent of this precession. Interfacial spin-orbit coupling changes the e ective magnetic eld seen by carriers during this process
in a momentum-dependent way; this alters the precession axis for each carrier and thus modi es the spin mixing conductance.
(c) Depiction of the loss of spin polarization that carriers experience while crossing interfaces with spin-orbit coupling. Without
interfacial spin-orbit coupling, carriers retain the portion of their spin polarization aligned with the magnetization, but lose the
portion polarized transversely to the magnetization due to dephasing processes just within the ferromagnet. With interfacial
spin-orbit coupling, carriers trade angular momentum with the atomic lattice; this leads to changes in all components of the
spin polarization. This phenomenon, known as spin memory loss, a ects each component di erently. The panel illustrates only
the loss in spin polarization aligned with magnetization. (d) Depiction of interfacial spin-orbit scattering in the presence of
an in-plane electric eld. Interfacial spin-orbit coupling allows for spins aligned with the magnetization to become misaligned
upon re
ection and transmission. For the scattering potential discussed in Sec. IV, the spin of a single re
ected carrier cancels
the spin of a single carrier transmitted from the other side of the interface. However, a net cancellation of spin is prevented if
the total number of incoming carriers di ers between sides, as can happen in the presence of in-plane current
ow. This occurs
because an in-plane electric eld drives two di erent charge currents within each layer; this forces the number of carriers with
a given in-plane momentum to di er on each side of the interface. The scattered carriers then carry a net spin polarization
and a net spin current.
The various contributions to spin torques in bilayers re-
main dicult to distinguish experimentally [22, 23] in
part because of the lack of models that accurately cap-
ture interfacial spin-orbit coupling [42].
Interfacial spin-orbit coupling may play an impor-
tant role in other phenomena. Spin pumping is one
example; it describes the process in which a precess-
ing magnetization generates a spin current [8]. In
heavy metal/ferromagnet bilayers, the pumped spin cur-
rent
ows from the ferromagnet into the heavy metal,
where the inverse spin Hall e ect generates an orthog-
onal charge current [47{51]. However, because inter-
facial spin-orbit coupling transfers spin polarization to
the atomic lattice, it modi es the pumped spin current
as it
ows across the interface. This transfer of spin
polarization remains uncharacterized in many systems,
thus contributing to inconsistencies in the quantitative
interpretation of experiments [9, 52{54]. Another exam-
ple, known as the spin Hall magnetoresistance, describes
the magnetization-dependent in-plane resistance of heavy
metal/ferromagnet bilayers [55{61]. Currently this e ect
is attributed to magnetization-dependent scattering at
the interface, but may also contain a contribution frominterfacial spin-orbit scattering. The impact of interfa-
cial spin-orbit coupling on these e ects remains unclear
due to the absence of appropriate models with which to
analyze the data.
Magnetoelectronic circuit theory is the most frequently
used approach to model spin currents at the interface
between a non-magnet and a ferromagnet. It describes
spin transport in terms of four conductance parameters,
where drops in spin-dependent electrochemical poten-
tials across the interface play the role of traditional volt-
ages. However, the theory cannot describe interfaces with
spin-orbit coupling because it does not consider spin-
ip processes due to spin-orbit coupling at the interface.
Fig. 1(a) depicts a typical scattering process described by
one of these conductance parameters. Given its success
in describing spin transport in normal metal/ferromagnet
bilayers, generalizing magnetoelectronic circuit theory to
include interfacial spin-orbit coupling would make it a
valuable tool for describing heavy metal/ferromagnet bi-
layers.
To generalize magnetoelectronic circuit theory one
must consider all the ways that interfacial spin-orbit cou-
pling potentially a ects spin transport. One such e ect,3
known as spin memory loss, describes a loss of spin cur-
rent across interfaces due to spin-orbit coupling. We il-
lustrate a process that contributes to spin memory loss
in Fig. 1(b). This loss occurs when the atomic lattice
at the interface behaves as a sink of angular momen-
tum. Recent work [62] incorporates this behavior into a
theory for spin pumping, but descriptions of this e ect
date back to over a decade ago [63{66]. Thus generaliz-
ing magnetoelectronic circuit theory for interfaces with
spin-orbit coupling requires accounting for spin memory
loss. By incorporating spin-
ip processes at the inter-
face into magnetoelectronic circuit theory, one can treat
this aspect of the phenomenology of interfacial spin-orbit
coupling.
Another important consequence of interfacial spin-
orbit coupling is that in-plane electric elds can create
spin currents that
ow away from the interface. First
principles calculations of Pt/Py bilayers suggest that a
greatly enhanced spin Hall e ect occurs at the inter-
face (as compared to the bulk) that could generate such
spin currents [67]. This suggests that in-plane electric
elds (and not just drops in spin and charge accumula-
tions across the interface) must play a role in generaliza-
tions of magnetoelectronic circuit theory. It also suggests
that one cannot con ne transport to the two-dimensional
interface when describing the e ect of in-plane electric
elds. Instead, one must consider transport both along
and across the interface. Some of the consequences of this
three-dimensional picture have been investigated in mul-
tilayer systems containing an insulator [68, 69]. The only
semiclassical calculations of three-dimensional metallic
bilayers are based on the Boltzmann equation [42]. Like
spin memory loss, these spin currents must be included
in generalizations of magnetoelectronic circuit theory to
fully capture the e ect of interfacial spin-orbit coupling.
In the following we give a semiclassical picture of how
such spin currents arise, and how they exert magnetic
torques that are typically not considered in bilayers.
Fig. 1(c) depicts how spins aligned with the magneti-
zation scatter from an interface with spin-orbit coupling.
For the scattering potential discussed in Sec. IV, single
re
ected and transmitted spins cancel on each side of the
interface. However, the netcancellation of spin is avoided
if the number of incoming carriers di ers between sides.
In the simplest scenario, this occurs if the in-plane elec-
tric eld drives di erent currents within each layer, so
that the occupancy of carriers di ers on either side for
a given in-plane momentum. We nd that through this
mechanism, carriers subject to interfacial spin-orbit scat-
tering can carry a net spin current in addition to exhibit-
ing a net spin polarization. If the net spin polarization is
misaligned with the magnetization, it can exert a torque
on the magnetization at the interface. This describes the
contribution to the spin torque normally associated with
the Rashba-Edelstein e ect (discussed earlier). However,
the spin currents created by interfacial spin-orbit scatter-
ing can
ow away from the interface, and those that
ow
into the ferromagnet exert additional torques. Althoughthese spin currents generate torques via the spin-transfer
mechanism, they arise from interfacial spin-orbit scat-
tering instead of the spin Hall e ect. This mechanism,
which cannot be captured by con ning transport to the
two-dimensional interface, is not usually considered when
analyzing spin torques in bilayers. However, it can con-
tribute to the total spin torque in important ways. For
instance, it allows for spin torques generated by inter-
facial spin-orbit coupling to point in directions typically
associated with the spin Hall e ect. The spin polariza-
tion and
ow directions of these spin currents are not
required to be orthogonal to each other or the electric
eld, unlike the spin currents generated by the spin Hall
e ect in in nite bulk systems. More work is needed to
determine how this semiclassical description of interfacial
spin current generation compares with the rst principles
description of an enhanced interfacial spin Hall e ect [67].
In this paper, we generalize magnetoelectronic circuit
theory to include interfacial spin-orbit coupling. Not only
does interfacial spin-orbit coupling modify the conduc-
tance parameters introduced by magnetoelectronic cir-
cuit theory, it requires additional conductivity parame-
ters to capture the spin currents that arise from in-plane
electric elds and spin-orbit scattering. Furthermore, the
transfer of angular momentum between carriers and the
atomic lattice at the interface alters the spin torque that
carriers can exert on the magnetization; this introduces
additional parameters that are needed to distinguish spin
torques from spin currents. However, we nd that many
of the parameters in this generalized circuit theory may
be neglected when modeling spin-orbit torques in bilayer
systems, and that including the conductivity and spin
torque parameters is more important than modifying the
conductance parameters. As with magnetoelectronic cir-
cuit theory, we provide microscopic expressions for most
parameters.
In a companion paper, to highlight the utility of the
proposed theory, we produce an analytical model describ-
ing spin-orbit torques caused by the spin-Hall and inter-
facial Rashba-Edelstein e ects. We achieve this by solv-
ing the drift-di usion equations with this generalization
of magnetoelectronic circuit theory. In that paper, we
focus on only the parameters that describe the response
of in-plane electric elds, and neglect all other changes
to magnetoelectronic circuit theory. We show that this
simpli ed approach captures the most important e ects
found in Boltzmann equation calculations of a model sys-
tem. In this paper, we discuss the complete generaliza-
tion of magnetoelectronic circuit theory in the presence
of interfacial spin-orbit coupling.
In Sec. II of this paper we describe spin transport at in-
terfaces with and without interfacial spin-orbit coupling.
In Sec. III we motivate the derivation of all parame-
ters, leaving some details for appendices A and B. In
Sec. IV we perform a numerical analysis of each bound-
ary parameter for a scattering potential relevant to heavy
metal/ferromagnet bilayers. This analysis allows us to
determine which parameters matter the most in these4
systems. Finally, in Sec. V we discuss implications of our
theory on experiments involving spin orbit torque, spin
pumping, the Rashba-Edelstein e ect, and the spin Hall
magnetoresistance.
II. SPIN AND CHARGE TRANSPORT AT
INTERFACES
In the following we discuss the general phenomenology
of spin transport at interfaces with and without spin or-
bit coupling. We rst describe some conventional spin
transport models to build up to the proposed model, and
refrain from presenting explicit expressions of any param-
eters until later sections.
A. Collinear spin transport
In the absence of spin-
ip processes one often assigns
separate current densities for majority ( j") and minority
(j#) carriers, i.e.
j"=G""j#=G##: (1)
HereG"=#denotes the spin-dependent interfacial conduc-
tance, while "=#refers to the drop in quasichemical
potential for each carrier population across the interface.
We may then de ne charge ( c) and spin ( s) components
for the drop in quasichemical potential
c= "+ # (2)
s= " #; (3)
and for the current densities
jc=j"+j# (4)
js=j" j#: (5)
across the interface. Using the following modi ed con-
ductance parameters
G=1
2
G"G#
; (6)
we may rewrite Eq. (1) as
js
jc!
=
G+G
G G+!
s
c!
(7)
instead. In this case both spin and charge currents are
continuous across the interface.
B. Magnetoelectronic Circuit Theory
When describing spin orientation in bulk ferromagnetic
systems, the magnetization direction provides a naturalspin quantization axes. However, at the interface be-
tween a non-magnet and a ferromagnet, the net spin po-
larizations of each region need not align. To account
for this, one must consider spins in the non-magnet that
point in any direction. In the ferromagnet, spins are mis-
aligned with the magnetization near the interface but be-
come aligned in the bulk. This occurs because spins pre-
cess incoherently around the magnetization; eventually
the net spin polarization transverse to the magnetization
vanishes. In transition metal ferromagnets and their al-
loys, this dephasing happens over distances smaller than
the spin di usion length.
To describe electron
ow and spin orientation in non-
magnet/ferromagnet bilayers, we use two separate coor-
dinate systems. For electron
ow, we choose the x=y
plane to lie along the interface and the z-axis to point
perpendicular to it. The interface is located at the z-axis
origin, and z= 0 andz= 0+describe the regions just
within the non-magnet and ferromagnet respectively. To
describe spin orientation, we choose the direction `to
be along the magnetization ( ^`=^m) and the directions
dandfto be perpendicular to ^`. The damping-like
(d) and eld-like ( f) directions point along the vectors
^d/^m[^m( E^z)] and ^f/^m( E^z) re-
spectively. This provides a convenient coordinate system
for describing spin-orbit torques, because torques with a
damping-like component push the magnetization towards
the E^zdirection, while those with a eld-like com-
ponent force the magnetization to precess about E^z.
We rst de ne the spin and charge accumulations at
the interface ( ), where the index 2[d;f;`;c;`;c]
describes the type of accumulation. The rst four indices
denote the spin ( d,f,l) and charge ( c) accumulations in
the non-magnet at z= 0 . The last two indices describe
the spin (`) and charge ( c) accumulations in the fer-
romagnet at z= 0+. In the ferromagnet we omit spin
accumulations aligned transversely to the magnetization,
due to the dephasing processes discussed above. Note
that the charge and spin components of have units
of voltage. We then de ne the spin and charge current
densities
owing out-of-plane ( jz ) in an identical fash-
ion. The charge and spin components of jz have the
units of number current density [70]. We refer to as
the spin/charge index.
One may rede ne any tensor that contains spin/charge
indices in another basis when useful. For instance, we
may write the spin accumulations and spin current den-
sities with longitudinal spin polarization in terms of av-
erages and di erences across the interface:
`=1
2
` `
; `=1
2
`+`
; (8)
jz`=1
2
jz` jz`
; jz`=1
2
jz`+jz`
:(9)
We may de ne similar expressions for the charge accumu-
lations and charge current densities. As we shall see, this
basis ( 2[d;f;`;c;`;c]) provides a more physically
transparent representation of all quantities.5
In the absence of interfacial spin-orbit coupling, the
spin current polarized along the magnetization direction
remains conserved. However, the spin current with po-
larization transverse to the magnetization dissipates en-
tirely upon leaving the normal metal. The interface ab-
sorbs part of this spin current, while the remaining por-
tion quickly dissipates within the ferromagnet due to a
precession-induced dephasing of spins. The total loss
of spin current then results in a spin transfer torque.
Figure 2 depicts this process by use of solutions to the
drift-di usion equations. In this situation, one may show
[71, 72] that the spin and charge current densities at
z= 0become
jz =GMCT
(10)
for a conductance tensor GMCT
given by
GMCT=0
BBBBBBBB@d f lclc
d Re[G"#] Im[G"#] 0 0 0 0
f Im[G"#] Re[G"#] 0 0 0 0
l 0 0 G+G 0 0
c 0 0 G G+0 0
l 0 0 0 0 0 0
c 0 0 0 0 0 01
CCCCCCCCA:
(11)
This formalism|known as magnetoelectronic circuit
theory|disregards spin currents and accumulations in
the ferromagnet with polarization transverse to the mag-
netization (due to the precession-induced dephasing de-
scribed above). This amounts to assuming that the pro-
cesses occurring in the shaded regions of Fig. 2(b) hap-
pen entirely at the interface instead. While this restric-
tion helps to reduce the number of required parameters,
it need not apply to non-ferromagnetic systems or ex-
tremely thin ferromagnetic layers. Note that the rows
corresponding to average and discontinuous quantities
are switched from the columns corresponding to those
quantities. This is done to emphasize that drops in ac-
cumulations cause average currents in magnetoelectronic
circuit theory.
Equation (11) implies that spin populations polarized
transverse to the magnetization decouple from those po-
larized longitudinal to it. The charge and longitudinal
spin current densities still obey Eq. (7), whereas the
transverse (non-collinear) spin current densities experi-
ences a nite rotation in polarization about the magneti-
zation axis. Note that the spin mixing conductance G"#
governs the latter phenomenon. In general, one obtains
all parameters via integrals of the transmission and/or
re
ection amplitudes over the relevant Fermi surfaces.
FIG. 2. (Color online) Spin current densities plotted ver-
sus distance from the interface, calculated using the drift-
di usion equations. Panel (a) treats the case without in-
terfacial spin-orbit coupling using magnetoelectronic circuit
theory as boundary conditions, whereas panel (b) treats the
case with interfacial spin-orbit coupling by using Eq. (12) as
boundary conditions instead. Due to precession-induced de-
phasing,jzdandjzfdissipate entirely within the ferromag-
net some distance from the interface (denoted by the purple
dashed line). With no interfacial spin-orbit coupling, the spin
current density polarized along the magnetization ( jzl) is con-
served, while the spin current densities polarized transversely
(jzdandjzf) exhibit discontinuities at the interface. With
interfacial spin-orbit coupling, all spin currents are discon-
tinuous at the interface. Furthermore, interfacial spin-orbit
coupling introduces additional sources of spin current via the
conductivity i and torkivity
FM
tensors (when an in-plane
electric eld is present). These sources may oppose the spin
currents that develop in the bulk. For example, the inclusion
of interfacial spin-orbit coupling leads jzfto switch signs near
to the interface, as seen by comparing panels (a) and (b).
C. Spin transport with interfacial spin orbit
coupling
To generalize magnetoelectronic circuit theory, i.e.
Eq. (10), to account for interfacial spin orbit coupling
and in-plane electric elds, we introduce the following
expression for the spin and charge current densities at
the interface:
ji =Gi +i ~E: (12)
Here we use a scaled electric eld de ned by ~E E=e
so that the elements of the tensor i have units of con-
ductivity. Without loss of generality, we assume that the
electric eld points along the xaxis.
The explosion of new parameters (relative to magneto-6
electronic circuit theory) is an unfortunate consequence
of spin-
ip scattering at the interface. Like magnetoelec-
tronic circuit theory, one may express each parameter
as an integral of scattering amplitudes over the relevant
Fermi surfaces; to discover which parameters may be ne-
glected we numerically study these integrals in Sec. IV.
Here, we discuss the overarching implications of this
model. In particular, three new concepts emerge from
the above expression:
First of all, the current density ji now includes an
index describing its direction of
ow ( i2[x;y;z ]), which
was previously assumed to be out-of-plane. In this gen-
eralization, a buildup of spin and charge accumulation
at interfaces may lead to spin and charge currents that
ow both in-plane and out-of-plane. The treatment of
in-plane currents close to the interface requires not only
the evaluation of Eq. (12), but also an extension of the
drift-di usion equations themselves.
Secondly, Eq. (12) depends on values of the spin and
charge accumulations from each side of the interface,
rather than di erences in those values across the inter-
face. This suggests that currents result from both drops
in accumulations andnon-zero averages of spin accumu-
lation at the interface [73].
Finally, interfacial spin-orbit scattering results in a
conductivity tensor ( i ) that drives spin currents in the
presence of an in-plane electric eld. This feature rep-
resents the greatest conceptual departure from previous
theories describing spin transport and is motivated by
results from the Boltzmann equation. Figure 2 describes
how some of these properties alter solutions of the drift-
di usion equations, as compared with magnetoelectronic
circuit theory.
Without interfacial spin-orbit coupling the in-plane
conductance tensors ( Gx andGy ) vanish, implying
that accumulations do not create in-plane currents in this
scenario. The conductivity tensor vanishes as well. Spin
transport transverse to the magnetization still decouples
from that longitudinal to it, and magnetoelectronic cir-
cuit theory is recovered. In the presence of interfacial
spin-orbit coupling, none of the tensors elements intro-
duced in Eq. (12) necessarily vanish, and spin transport
in all polarization directions becomes coupled. However,
for the interfacial scattering potential studied in Sec. IV,
many parameters di er by orders of magnitude; thus cer-
tain parameters may be neglected on a situational basis.
D. Spin-orbit torques
Without interfacial spin-orbit coupling, spin and
charge accumulations at an interface create both a spin
polarization and spin currents. The spin polarization de-
velops atz= 0 and exerts a torque on any magnetization
at the interface via the exchange interaction. The spin
current that develops at z= 0+exerts an additional
torque by transferring angular momentum to the ferro-
magnetic region via dephasing processes. For simplicity,we assume that this spin current transfers all of its angu-
lar momentum to the magnetization rather than the bulk
atomic lattice. We do so under the assumption that the
dephasing processes within the ferromagnet diminish spin
currents faster than the spin di usive processes caused by
bulk spin-orbit coupling. All of the incident transverse
spin current is then lost at the interface ( z= 0) or in
the bulk of the ferromagnet ( z > 0), and carriers can
only exchange angular momentum with the magnetiza-
tion. Thus the spin current at z= 0 , which represents
the incident
ux of angular momentum on the magne-
tized part of the bilayer, equals the total spin torque on
the system. Furthermore, the spin torques at z= 0 and
z>0 add up to equal the spin current at z= 0 .
However, at interfaces with spin-orbit coupling, the
atomic lattice behaves as a reservoir that carriers may
transfer angular momentum to. In this scenario, carri-
ers exert spin torques on both the magnetization and the
lattice. We cannot compute spin torques solely from the
spin currents described by Eq. (12) if we are to account
for the losses to this additional reservoir of angular mo-
mentum. Thus, we introduce a separate expression for
the total spin torque on the bilayer:
= +
~E; (13)
Note that the index 2[d;f] describes the directions
transverse to the magnetization, since spin torques only
point in those directions. The tensor , known as the
torkance, describes contributions to the spin torque from
the buildup of spin and charge accumulation at an in-
terface. The tensor
, which we call the torkivity , cap-
tures the corresponding contributions from an external,
in-plane electric eld. The torkivity tensor originates
from interfacial spin-orbit scattering, much like the con-
ductivity tensor introduced earlier.
We may separate the total spin torque into two contri-
butions:
= mag
+ FM
(14)
=
mag
+
FM
: (15)
The rst tensors on the right hand side of Eqs. (14) and
(15) describe torques exerted by the spin polarization
atz= 0. The second tensors describe the spin torque
exerted in the bulk of the ferromagnet ( z > 0). Both
torques are exerted on the magnetization rather than on
the atomic lattice. Here we assume that the torque at
z > 0 equals the transverse spin current at z= 0+as
before. Thus, the spin torques exerted at z= 0 and
z > 0 are both included in the torkance and torkivity
tensors.
Without interfacial spin-orbit coupling, the torkivity
tensor vanishes and the torkance tensor becomes
identical to Gz . This indicates that the transverse spin
current at z= 0 equals the total spin torque, as ex-
pected. In the presence of interfacial spin-orbit coupling,
the lattice also receives angular momentum from carriers;7
in this case 6=Gz and
6= 0. Thus, by comput-
ing the tensors introduced in Eq. (13), one may calculate
spin-orbit torques such that the lattice torques are ac-
counted for. Furthermore, Eqs. (14) and (15) allow one
to separate the total spin torque into its interfacial and
bulk ferromagnet contributions.
III. DERIVATION OF BOUNDARY
PARAMETERS
Interfacial spin-orbit coupling causes both momen-
tum and spin-dependent scattering at interfaces. If
the incident distribution of carriers depends on mo-
mentum and/or spin, outgoing carriers may become
spin-polarized via interfacial spin-orbit scattering. This
gives rise to non-vanishing accumulations, currents, and
torques, which are related by Eqs. (12) and (13). We now
motivate these relationships, which can be expressed in
terms of scattering amplitudes. We do so by approx-
imating the non-equilibrium distribution function near
the interface.
We rst consider the total distribution function f (k),
which gives the momentum-dependent occupancy of car-
riers described by the spin/charge index . In equilib-
rium, this distribution function equals the Fermi-Dirac
distribution feq
(" k). Just out of equilibrium, f (k) is
perturbed as follows
f (k) = feq
(" k) +@feq
@" kg (k); (16)
whereg (k) denotes the non-equilibrium distribution
function. The equilibrium distribution functions vanish
for 2[d;f;` ] since the non-magnet exhibits no equi-
librium spin polarization. However, the non-equilibrium
distribution functions for all spin/charge indices are gen-
erally non-zero.
To obtain the tensors introduced in Eqs. (12) and (13),
we must evaluate g (k) near the interface. One could
evaluateg (k) by solving the spin-dependent Boltzmann
equation for the bilayer system. In this approach one cap-
tures spin transport both in the bulk and at the interface.
A more practical approach is to assume some generic
form forg near the interface that is physically plau-
sible. This approach yields boundary conditions suit-
able for simpler bulk models of spin transport such as
the drift-di usion equations. In the companion paper,
we show that solving the drift-di usion equations using
these boundary conditions produces quantitatively simi-
lar results to solving the Boltzmann equation.
For simplicity, we assume that spherical Fermi surfaces
describe carriers in both layers. Later we generalize this
formalism to describe non-trivial electronic structures.
In the non-magnet, all carriers belong to the same Fermi
surface. In the ferromagnet, majority ( ") and minor-
ity (#) carriers belong to di erent Fermi surfaces. Thus
we use the spin/charge basis 2[d;f;`;c;";#], since
in this model carriers belonging to those populationshave well-de ned Fermi surfaces and velocities. The ten-
sors derived in this section may be expressed in other
spin/charge bases by straightforward linear transforma-
tions.
To approximate g at the interface we use the following
expression:
gin
(kjj) = e
q +~EfE
(kjj)
; (17)
Equation (17) represents the portion of g incident on
the interface, where kjjdenotes the in-plane momentum
vector and eequals the elementary charge. The right
hand side of Eq. (17) describes two pieces of the incom-
ing distribution function; Fig. 3 depicts both pieces over
k-space for each side of the interface. The rst term
captures spin/charge currents incident on the interface.
They may arise, for example, from the bulk spin Hall
e ect or ferromagnetic leads. The quantities q denote
the isotropic spin/charge polarization of those currents.
The second term represents the anisotropic contribution
to the distribution function caused by an external electric
eld. We remind the reader that the scaled electric eld
~Epoints along the xaxis. The simplest approximation
forfE
(kjj) is to use the particular solution of the Boltz-
mann equation in the relaxation time approximation:
fE
(kjj) = evx (kjj)8
>>>>><
>>>>>:0 2[d;f;` ]
=c
" ="
# =#(18)
This term describes the in-plane charge current caused
by the external electric eld, but also describes an in-
plane spin current polarized opposite to the magnetiza-
tion in the ferromagnet. The momentum relaxation times
in the ferromagnet di er between majority ( ") and mi-
nority ( #) carriers. In the non-magnet, the momentum-
relaxation time ( ) is renormalized by bulk spin-
ip pro-
cesses (see appendix A).
The outgoing distribution function
gout
(kjj) =S (kjj)gin
(kjj); (19)
is speci ed by the incoming distribution function and the
unitary scattering coecients S , given by
S jvz (kjj)j
jvz (kjj)jS0
(kjj); (20)
where
S0
=8
>>>>>>><
>>>>>>>:1
2tr
ry r
; 2[d;f;`;c ]
1
2tr
ty t
2[d;f;`;c ]; 2[";#]
1
2tr
(t)y t
2[";#]; 2[d;f;`;c ]
1
2tr
(r)y r
; 2[";#]
(21)8
FIG. 3. (Color online) Non-equilibrium distribution functions g (k) in the presence of interfacial spin-orbit scattering, resulting
from an (a) incident spin and charge accumulation and an (b) in-plane external electric eld. The images depict g (k) on each
side of the interface plotted over k-space. The gray spheres represent the equilibrium Fermi surface. The colored surfaces
represent the non-equilibrium perturbation to the Fermi surface, given by the charge distribution gc(k) (not to scale). The
arrows denote the spin distribution g(k). The blue and red regions represent the wavevectors pointing incident and away
from the interface respectively. (a) Scenario in which the incident carriers exhibit a net spin and charge accumulation. The
spin-polarization of the outgoing carriers di ers from the incident carriers due to interfacial spin-orbit scattering. The total
spin/charge current density ( ji ) and the resulting spin torques ( ) are related to the total spin/charge accumulation ( ) by
the tensors Gi and respectively. (b) Scenario in which the incident carriers are subject to an in-plane electric eld. The
in-plane electric eld drives two di erent charge currents on each side of the interface, since each layer possesses a di erent bulk
conductivity. This shifts the occupancy of carriers (i.e. the charge distribution) di erently on each side of the interface. When
spin- unpolarized carriers scatter o of an interface with spin-orbit coupling they become spin-polarized. Because the occupancy
of incident carriers was asymmetrically perturbed at the interface, a net cancellation of spin is avoided in even the simplest
scattering model. The resulting spin/charge currents and spin torques are captured by the tensors i and
respectively.
Note that for a ferromagnetic layer, in-plane electric elds also create incident in-plane spin currents as well (suppressed for
clarity in this gure).
Here we de ne the Pauli vector such thatd=x,
f=y,`=z, and
c=
1 0
0 1!
; "=
1 0
0 0!
; #=
0 0
0 1!
:(22)
The coecient S0
gives the strength of scattering
for carriers with spin/charge index into those with
spin/charge index . The scattering coecients depend
on the 22 re
ection and transmission matrices for spins
pointing along the magnetization axis. In particular, the
matricesrandtdescribe re
ection and transmission
respectively into the ferromagnet. The matrices randt
describe re
ection and transmission into the non-magnet.
Note that the density of states and Fermi surface area
element di er between incoming and outgoing carriers.
Thus to conserve particle number one must include the
ratio of velocities within the scattering coecients, as
done in Eq. (20).
We obtain all non-equilibrium quantities near the in-
terface by integrating g over the relevant Fermi surfaces.
We note that the outgoing part of g includes the con-
sequences of interfacial scattering, since it depends on
the scattering coecients. For example, the interfacial
exchange interaction leads to spin-dependent scattering,
which is captured by the di erence in the diagonal ele-ments of the 22 re
ection and transmission matrices.
On the other hand, the interfacial spin-orbit interaction
introduces spin-
ip scattering, which is captured by the
o -diagonal elements within these matrices. Thus, to
describe the consequences of interfacial spin-orbit scat-
tering we must not limit the form of the re
ection and
transmission matrices as was often done in the past.
We write the current density ji for carriers with
spin/charge index
owing in direction i2[x;y;z ] as
follows:
ji =1
~(2)31
vF Z
FS d2kvi (k)g (k) (23)
Note that all integrals run over the Fermi surface cor-
responding to the population with spin/charge index .
The quantity vF denotes the Fermi velocity for that
population. To de ne the accumulations we follow
the example of magnetoelectronic circuit theory [71, 72]
and assume that the incoming currents behave as if they
originate from spin-dependent reservoirs. This implies
that the incoming polarization q approximately equals
the accumulation at the interface.
We have now discussed the requirements for deriv-
ing the conductance and conductivity tensors found in
Eq. (12). We obtain these tensors by plugging Eqs. (17)
and (19) into Eq. (23) and noting that q . In doing9
so we write the currents ji in terms of the accumulations
and the in-plane electric eld ~E. From the resulting
expressions one then obtains formulas for the conduc-
tance and conductivity tensors in terms of the interfacial
scattering coecients. We outline this remaining process
in appendix A. In appendix C we generalize those ex-
pressions for the case of non-trivial electronic structures,
which allows one to compute the conductance and con-
ductivity tensors for realistic systems.
Having discussed the currents that arise from interfa-
cial spin-orbit scattering, we now discuss the spin torques
caused by the same phenomenon. The transverse spin po-
larization at z= 0 exerts a torque on any magnetization
at the interface via the exchange interaction. The trans-
verse spin current at z= 0+exerts a torque by transfer-
ring angular momentum to the ferromagnet. The total
spin torque then equals the sum of these two torques. To
describe the spin torque at z= 0, we must compute the
spin polarization at the interface. To accomplish this we
de ne the following matrix
T =8
<
:1
2tr
(t)yt
2[d;f;`;c ]
1
2tr
tyt
2[";#](24)
which describes phase-coherent transmission from all
populations into transverse spin states at the interface.
We may then compute the ensemble average of spin den-
sityhsiatz= 0 as follows:
hsi=1
~(2)3X
1
vF Z
FS 2ind2kT (kjj)gin
(kjj):
(25)
The torque at z= 0 is then given by
mag
= 0+Z
0 dzJex
~
hsi^m
; (26)
whereJexequals the exchange energy at the interface.
We evaluate this integral over the region that describes
the interface, where the exchange interaction and strong
spin-orbit coupling overlap. Note that the cross prod-
uct
hsi^m
=0hs0iis evaluated by computing
Eq. (25).
To describe the spin torque at z= 0+, we introduce
an additional scattering matrix:
S =8
<
:1
2tr
(t)yt
2[d;f;`;c ]
1
2tr
(r)yr
2[";#](27)
This scattering matrix is used to calculate the transverse
spin current at z= 0+. Since this spin current rapidly de-
phases, it contributes entirely to the spin torque exerted
on the ferromagnet. Note that the currents discussedpreviously corresponded to carriers with well-de ned ve-
locities. However, transverse spin states in the ferromag-
net consist of linear combinations of majority and minor-
ity spin states. Since these spin states possess di erent
phase velocities, the velocities of transverse spin states
oscillate over position. These states also posses di erent
group velocities, and wave packets with transverse spin
travel with the average group velocity. The transverse
spin current at z= 0+then equals
FM
=1
~(2)3X
Z
2DBZdkjjvz(kjj)
vz (kjj)S (kjj)gin
(kjj);
(28)
where
vz(kjj)1
2
vz"(kjj) +vz#(kjj)
(29)
gives the average group velocity of carriers in the ferro-
magnet. Note that we write this integral over the max-
imal two-dimensional Brillouin zone common to all car-
riers (see appendices A and B). The total torque then
equals the sum of torques at the interface and in the bulk
ferromagnet:
= mag
+ FM
: (30)
As before we assume that the incoming polarizations
approximately equal the accumulations at the interface.
Thus we obtain mag
and FM
in terms of and ~E
by plugging Eqs. (17) and (19) into Eqs. (25), (26), and
(28). From the resulting expressions we may de ne the
torkance and torkivity tensors introduced in Eq. (13). In
appendix B we discuss this process, and in appendix C we
present generalized expressions for non-trivial electronic
structures.
We note that the conductance and conductivity tensors
describe the charge current and longitudinal spin current
in the ferromagnet, but not the transverse spin currents.
In the ferromagnet, the transverse spin currents dissipate
not far from the interface, while the charge current and
longitudinal spin current can propagate across the entire
layer. Thus the transverse spin currents in the ferromag-
net are best described as spin torques given by FM
; this
explains why we include them in the torkance and torkiv-
ity tensors instead of the conductance and conductivity
tensors. If we derive a similar formalism to describe a
non-magnetic bilayer, spin currents polarized in all di-
rections should be included in the conductance and con-
ductivity tensors. With no magnetism, no spin torques
are exerted at or near the interface and the torkance and
torkivity tensors are not meaningful.
IV. NUMERICAL ANALYSIS OF BOUNDARY
PARAMETERS
In the following we numerically analyze the boundary
parameters introduced in Eqs. (12) and (13) in the pres-
ence of an interfacial exchange interaction and spin-orbit10
FIG. 4. (Color online) Contour plots of various boundary parameters versus the interfacial exchange ( uex) and Rashba ( uR)
strengths. The magnetization points away from the electric eld 45oin-plane and 22 :5oout-of-plane. Note that the parameters
plotted in panels (a)-(c) describe the scattering processes illustrated in Figs. 1(a)-(c). (a) Plot of Gzdf, which generalizes
Im[G"#] in the presence of interfacial spin-orbit coupling. It describes a rotation of spin currents polarized transversely to the
magnetization. (b) Plot of Gzll, which contributes to spin memory loss longitudinal to the magnetization. It varies mostly
withuR, since interfacial spin-orbit coupling provides a sink for angular momentum. (c) Plot of
FM
d, which describes the
out-of-plane, damping-like spin current created by an in-plane electric eld and spin-orbit scattering. It exceeds its eld-like
counterpart (
FM
f); thus, the resulting spin current exerts a (mostly) damping-like spin torque upon entering the ferromagnet.
(d) An array of contour plots, with each plot shown over an identical range as those in (a)-(c). The plot in row and column
corresponds to the parameter Gz . From this one may visualize the coupling between spin/charge indices for this tensor,
shown across the parameter space of the scattering potential given by Eq. (31). The overall structure of Gz resembles that
of magnetoelectronic circuit theory, given by Eq. (11). The corresponding gures for (e) z , (f)
FM
, and (g)
mag
are also
shown.
scattering. We do so to provide intuition as to the rel-
ative strengths of each boundary parameter. We use a
scattering potential localized at the interface [42] that is
based on the Rashba model of spin orbit coupling
V(r) =~2kF
m(z)
u0+uex^m+uR(^k^z)
(31)
whereu0represents a spin-independent barrier, uexgov-
erns the interfacial exchange interaction, and uRdenotesthe Rashba interaction strength. Plane waves comprise
the scattering wavefunctions in both regions.
In Fig. 4 we plot various boundary parameters versus
the exchange interaction strength ( uex) and the Rashba
interaction strength ( uR). Figures 4(a)-(c) display indi-
vidual boundary parameters, while Figs. 4(d)-(g) display
multiple boundary parameters for a given tensor. The
plots in Figs. 4(d)-(g) are arranged as arrays to help visu-11
alize the coupling between spin/charge components. The
spin-orbit interaction misaligns the preferred direction of
spins from the magnetization axis. Thus, no two tensor
elements are identical, though many remain similar. As
expected, the coupling between the transverse spin com-
ponents and the charge and longitudinal spin components
does not vanish.
The conductance tensor Gz generalizes GMCT
in the
presence of interfacial spin-orbit coupling. Comparison
to Eq. (11) suggests that the parameters GzddandGzdf
represent the real and imaginary parts of a generalized
mixing conductance ( ~G"#). Each element of the conduc-
tance tensor experiences a similar perturbation due to
spin-orbit coupling. However, the tensor elements from
the 22 o -diagonal blocks in Fig. 4(d) either vanish or
remain two orders of magnitude smaller than those from
the diagonal blocks. This remains true even for values
ofuRapproaching the spin-independent barrier strength
u0. While these blocks are small for the simple model
treated here, they may become important for particular
realistic electronic structures. The fact that the elements
Gzc andGz cvanish for all ensures the conserva-
tion of charge current and guarantees no dependence on
an o set to the charge accumulations. Note that four
additional parameters vanish in the conductance tensor
shown in Fig. 4(d); this occurs because identical scatter-
ing wavefunctions were used for both sides of the inter-
face when computing the scattering coecients. These
parameters do not vanish in general.
The results shown in Fig. 4 were computed for a mag-
netization with out-of-plane components. In magneto-
electronic circuit theory, the parameters are independent
of the magnetization direction. With interfacial spin-
orbit coupling, this is no longer the case. In general all
of the parameters in Eqs. (12) and (13) depend on the
magnetization direction. However, we nd that this de-
pendence is weak for the model we consider here. For in-
plane magnetizations (not shown) the 2 2 o -diagonal
blocks vanish, but spin-orbit coupling still modi es the
diagonal blocks in the manner described above.
In the presence of interfacial spin-orbit coupling the
lattice also receives angular momentum from carriers.
This results in a loss of spin current across the inter-
face, or spin memory loss, which the elements Gzl
partly characterize. The computation of these parame-
ters for realistic electronic structures should help predict
spin memory loss in experimentally-relevant bilayers. In
particular, spin memory loss might play a crucial role
when measuring the spin Hall angle of heavy metals via
spin-pumping from an adjacent ferromagnet [9]. Here
Gzllprovides the strongest contribution to spin mem-
ory loss that is caused by accumulations, and approaches
the imaginary part of the generalized mixing conductance
in magnitude.
Until now, we have discussed the tensors that describe
how accumulations a ect transport. However, in-plane
electric elds and spin-orbit scattering create additional
currents that form near the interface. In particular, theconductivity parameters i describe the currents that
can propagate into either layer without signi cant de-
phasing. For instance, the element zldescribes an out-
of-plane longitudinal spin current driven by an in-plane
electric eld. The element zlthen gives the disconti-
nuity in this spin current across the interface. This dis-
continuity arises because of coupling to the lattice, and
thus contributes to spin memory loss.
Likewise, the torkivity tensors describe contributions
to the total spin torque that arise from in-plane electric
elds and spin-orbit scattering. This includes the torques
exerted by the spin polarization at z= 0 and by the
transverse spin currents at z= 0+. The tensors
mag
and
FM
describe these torques respectively. Since the
transverse spin currents at z= 0+quickly dephase in the
ferromagnet, we treat them as spin torques and do not
include them in the conductivity tensor.
To understand how the boundary parameters con-
tribute to spin-orbit torques, we note that
mag
f>
mag
d
over the swept parameter space. This implies that the
torque exerted at z= 0 is primarily eld-like, which
agrees with previous studies of interfacial Rashba spin or-
bit torques [42]. However, we also nd that
FM
d>
FM
f
for stronguR; in this case the resulting spin current exerts
a damping-like torque by
owing into the ferromagnet.
Both spin torque contributions result from the interfacial
Rashba interaction. This implies that interfacial spin-
orbit scattering provides a crucial mechanism to the cre-
ation of damping-like Rashba spin torques. In the com-
panion paper we support this claim by comparing spin-
orbit torques computed using both the drift-di usion and
Boltzmann equations.
V. OUTLOOK
In the previous section we demonstrated that only cer-
tain boundary parameters remain important when mod-
eling spin orbit torques. The interfacial conductivity and
torkivity parameters capture physics due to in-plane ex-
ternal electric elds. They depend on the di erence in
bulk conductivities, which are typically easier to estimate
than interfacial spin/charge accumulations. For this rea-
son, calculating the conductivity and torkivity tensors
for a realistically-modeled system should provide direct
insight into its spin transport behavior. In particular,
we showed that conductivity and torkivity parameters
strongly indicate the potential to produce damping-like
and eld-like torques. Further studies may yield signi -
cant insight into the underlying causes of these and other
phenomena for speci c material systems. Even so, treat-
ing the elements of these tensors as phenomenological
parameters should bene t the analysis of a variety of ex-
periments, which we discuss now.
(1) Spin pumping/memory loss | Spin pumping ex-
periments in Pt-based multilayers suggest that the mea-
sured interfacial spin current di ers from the actual spin
current in Pt, leading to inconsistent predictions of the12
spin Hall angle [9, 53]. Rojas-Sanchez et al. [9] ex-
plain this discrepancy in terms of spin memory loss while
Zhang et al. [53] attribute it to interface transparency.
The latter characterizes the actual spin current generated
at an interface when backscattering is accounted for; it
depends on G"#and does not require interfacial spin-
orbit coupling. Though further experimental evidence is
needed to resolve these claims, the elements of Gz char-
acterize both spin memory loss and transparency. Fig-
ure 4 implies that transparency depends on interfacial
spin-orbit coupling, while spin memory loss also depends
on the interfacial exchange interaction. Thus, the gen-
eralized boundary conditions introduced here unify these
two interpretations and allow for further investigation us-
ing a single theory.
(2) Rashba-Edelstein e ect | Sanchez et al. [41] mea-
sure the inverse Rashba-Edelstein e ect in an Ag/Bi in-
terface, in which interfacial spin orbit coupling converts
a pumped spin current into a charge current. The the-
oretical methods that describe this phenomena to date
[37, 44{46, 68] assume orthogonality between the direc-
tional and spin components of the spin current tensor.
However, the conductivity tensor introduced here is ro-
bust in general; this implies that interfacial spin-orbit
scattering converts charge currents into spin currents
with polarization and
ow directions not orthogonal to
the charge current. Onsager reciprocity implies that spin
currents should give rise to charge currents at the inter-
face that
ow in all directions as well. Thus, the con-
ductivity tensor describes a generalization of the direct
and inverse Rashba-Edelstein e ects as they pertain to
interfaces with spin-orbit coupling.
(3) Spin Hall magnetoresistance | The conductivity
tensor also leads to in-plane charge currents. These cur-
rents depend on magnetization direction via the scat-
tering amplitudes, and thus suggest a new contribution
to the spin Hall magnetoresistance based on the Rashba
e ect in addition to that from the spin Hall e ect. Pre-
liminary calculations of this mechanism suggest a mag-
netoresistance in Pt/Co of a few percent, which is com-
parable or greater than experimentally measured values
in various systems [58{61].
We expect that the most useful approach for inter-
preting experiments as above is to treat the new trans-
port parameters as tting parameters. In the future, this
approach can be checked by calculating the parameters
from rst principles [74, 75] as has been done for magne-
toelectronic circuit theory. This requires computing the
boundary parameters for realistic systems using the ex-
pressions given in appendix C. Such calculations would
provide a useful bridge between direct rst-principles cal-
culations of spin torques [76{79] and drift-di usion cal-
culations done to analyze experiments.
To conclude, we present a theory of spin transport at
interfaces with spin-orbit coupling. The theory describes
spin/charge transport in terms of resistive elements,
which ultimately describe measurable consequences of
interfacial spin-orbit scattering. In particular, the pro-posed conductivity and torkivity tensors model the phe-
nomenology of in-plane electric elds in the presence of
interfacial spin-orbit coupling, which was previously inac-
cessible to the drift-di usion equations. We calculate all
parameters in a simple model, but also provide general
expressions in the case of realistic electronic structure.
We found that elements of the conductivity and torkiv-
ity tensors are more important than the modi cations of
other transport parameters (such as the mixing conduc-
tance) in many experimentally-relevant phenomena, such
as spin orbit torque, spin pumping, the Rashba-Edelstein
e ect, and the spin Hall magnetoresistance.
ACKNOWLEDGMENTS
The authors thank Kyoung-Whan Kim, Paul Haney,
Guru Khalsa, Kyung-Jin Lee, and Hyun-Woo Lee for
useful conversations and Robert McMichael and Thomas
Silva for critical readings of the manuscript. VA ac-
knowledges support under the Cooperative Research
Agreement between the University of Maryland and the
National Institute of Standards and Technology, Cen-
ter for Nanoscale Science and Technology, Grant No.
70NANB10H193, through the University of Maryland.
Appendix A: Derivation of the conductance and
conductivity tensors
To derive the conductance and conductivity tensors
we must approximate the distribution function f (k)
at the interface. The distribution function gives the
momentum-dependent occupancy of carriers described
by the spin/charge index . Just out of equilibrium, it is
perturbed by the linearized non-equilibrium distribution
functiong (k), as seen in Eq. (16). In the following we
complete the derivation begun in Sec. III.
We write the portion of g (kjj) incident on the inter-
face as done in Eq. (17). The rst term on the right
hand side of Eq. (17) captures the spin and charge cur-
rents incident on the interface, while the second term
gives an anisotropic contribution caused by an external
electric eld. As discussed in Sec. III, the simplest ap-
proximation for g (kjj) is to use the particular solution
of the Boltzmann equation in the relaxation time approx-
imation, given by Eq. (18). The momentum relaxation
times that we use account for di ering majority ( ") and
minority ( #) relaxation times in the ferromagnet, and
are renormalized by bulk spin-
ip scattering in the non-
magnet:
( ) 1= ( mf) 1+ ( sf) 1: (A1)
We may better approximate Eq. (18) by forcing the dis-
tribution function to obey outer boundary conditions as
well. In the companion paper we present a more sophisti-
cated approximation for Eq. (18) that accomplishes this13
by using solutions to the homogeneous Boltzmann equa-
tion.
The outgoing distribution, given by Eq. (19), is speci-
ed by the incoming distribution and the scattering co-
ecientsS . The scattering coecients are given by
Eq. (20) and Eq. (21). Here we compute non-equilibrium
accumulations analogously to the currents de ned by
Eq. (23),
= 1
e1
AFS Z
FS d2kg (k); (A2)
where denotes the accumulation. Furthermore, AFS
gives the Fermi surface area while vF gives the Fermi
velocity. The quantities just de ned apply to the popula-
tion with spin/charge index . Likewise, all integrals are
evaluated over the Fermi surface that corresponds to the
spin/charge index . Note that we express the accumula-
tions in units of voltage and the current densities in units
of number current density. Using Eqs. (19) and (20) we
may rewrite these expressions as integrals over the max-
imal two-dimensional Brillouin zone common (2DBZ) to
all carriers
= c
eX
Z
2DBZdkjj1
vz (kjj)
+S
gin
(kjj)
(A3)
ji = cj
eX
Z
2DBZdkjjvi (kjj)
vz (kjj)
iz+S
gin
(kjj);
(A4)
where
cvF
AFS ; cj e
~(2)3: (A5)
Note that the velocities correspond to outgoing carriers.
The factor iz(1 2iz) accounts for the fact that
incoming and outgoing currents have the opposite sign
fori=zbut the same sign for i2[x;y]. By integrat-
ing over the maximal two-dimensional Brillouin zone we
encounter evanescent states, since kjjvectors not corre-
sponding to real Fermi surfaces have imaginary kzvalues.
Here we neglect the contributions to the currents and ac-
cumulations due to evanescent states. Such contributions
vanish very close to the interface.
We must now express the accumulations and currents
in terms of the incoming polarizations and the in-plane
electric eld. Plugging Eqs. (17) and (19) into Eqs. (A3)
and (A4), we obtain the following
=A q +a ~E (A6)
ji =Bi q +bi ~E (A7)
where the tensors that contract with the incident
spin/charge polarization are given by
A =cZ
2DBZdkjj1
vz
+S
(A8)
Bi =cjZ
2DBZdkjjvi
vz
iz+S
(A9)while the tensors that multiply the in-plane electric eld
become
a =cX
Z
2DBZdkjj1
vz
+S
fE
(A10)
bi =cjX
Z
2DBZdkjjvi
vz
iz+S
fE
(A11)
In the same spirit as magnetoelectronic circuit theory,
these tensors represent moments of the scattering coe-
cients weighted by velocities.
To determine exactly how the currents depend on the
accumulations, we solve for ji in terms of . Doing so
yields the following conductance and conductivity tensors
Gi =Bi
[A 1]
i =bi Gi a :
To further simplify these expressions, we follow the ex-
ample of magnetoelectronic circuit theory [71, 72] and
assume that the incoming spin-currents behave as if they
originate from spin-dependent reservoirs. This implies
that the incoming spin polarization q equals the qua-
sichemical potential at the interface. For this to be
true, we must nd that A / anda = 0 by in-
spection of Eq. (A6). These relationships hold if one
evaluates Eqs. (A8) and (A10) over the incoming portion
of the Fermi surface only. We nd that the contributions
from the outgoing portion of the Fermi surface cancel to
a good approximation, which suggests that:
Gi =Bi (A12)
i =bi : (A13)
The above equations give simpler expressions for the con-
ductance and conductivity tensors in terms of interfacial
scattering coecients.
Appendix B: Derivation of the torkance and
torkivity tensors
To describe the spin torque at z= 0, we must compute
the ensemble average of spin density hsiusing Eq. (25).
The resulting torque is given by Eq. (26). To describe the
spin torque at z= 0+, we must calculate the transverse
spin current in the ferromagnet using Eqs. (28) and (29).
We then express the spin torque in terms of the incoming
polarizations and the in-plane electric eld by plugging
Eqs. (17) and (19) into Eqs. (25), (26), and (28). In doing
so we obtain
=C q +c~E; (B1)
where
C =CFM
+Cmag
(B2)
c=cFM
+cmag
(B3)14
describes the separation of the spin torque into its bulk
ferromagnet and interface contributions. The tensors
that contract with the incident spin/charge polarization
are given by
CFM
=cjZ
2DBZdkjjvz
vz S ; (B4)
Cmag
= Jex
~cjX
0Z
2DBZdkjj1
vz 0T0 ; (B5)
while the tensors that multiply the in-plane electric eld
become
cFM
=cjX
Z
2DBZdkjjvz
vz S fE
: (B6)
cmag
= Jex
~cjX
Z
2DBZdkjj1
vz 0T0 fE
(B7)
where the velocity vz(kjj) corresponds to the outgoing
portion of the Fermi surface in the ferromagnet.
As we did for the currents, we solve for in terms of
. Doing so yields the following torkance and torkivity
tensors
=C
[A 1]
=c a :
The torkance tensor describes the contribution to the to-
tal spin torque that arises from the accumulations at the
interface. The torkivity tensor describes the subsequent
contribution from interfacial spin-orbit scattering when
driven by an in-plane electric eld. Following the argu-
ments made for Eq. (A12):
=C (B8)
=c: (B9)
As seen in the companion paper, this approximation pro-
duces good agreement with the interfacial charge cur-
rents, spin currents, and spin torques computed via the
Boltzmann equation.
Appendix C: Boundary Parameters for Realistic
Interfaces
To generalize the expressions from the previous section
to include electronic structure, we must consider the non-
equilibrium distribution function for all bands relevant to
transport:
fm (k) = feq
m ("m k) +@feq
m
@"m kgm (k): (C1)
Heremdescribes the spin-independent band number and
denotes the spin/charge index. If the case of a non-
magnet, for each spin-independent band there are two
degenerate states. Linear combinations of these statescan produce phase coherent spin states that point in any
direction. Thus, for the non-magnet, the spin/charge in-
dex should span 2[d;f;`;c ], where the `direction is
aligned with the magnetization in the neighboring ferro-
magnet for convenience. In the ferromagnet all bands are
non-degenerate, so each state possesses a di erent phase
velocity. As a result, linear combinations of these states
have spin expectation values that oscillate over position,
complicating the description presented above. There is
no natural pairing of non-degenerate spin states. How-
ever, if states are quantized along a particular axis, the
spin accumulations and spin currents with polarization
along that axis are well-de ned regardless of the choice
of pairing. Thus for each spin-independent band in the
ferromagnet, the spin/charge index spans the states de-
scribing majority and minority carriers, i.e. 2[";#].
We generalize the approximate distribution function
fE
m (kjj) caused by an external electric eld to allow for
a band dependence. We do so because the velocities now
depend on band number and the scattering times may as
well. However, we assume that the incoming polarization
q does not depend on band number; thus we treat in-
cident currents as if they originate from spin-dependent
(but not band-dependent) reservoirs. The momentum
relaxation times for each spin-independent band in the
non-magnet are renormalized using Eq. (A1).
To account for coherence between bands, we begin with
a more general expression for the ensemble average of the
outgoing current:
hhjout
i ii=1
~X
mnn0 Z
2DBZdkjjgin
m
jvmz j
tr
(sn0m)yJout
n0n;i snm
:
(C2)
Heresstands for re
ection or transmission, depending on
what region(s) incoming and outgoing carriers are from.
The indices mand correspond to incoming carriers,
whilen,n0, and describe the outgoing carriers. The
current operator Jout
n0n;i is given by
Jout
n0n;i =i~
2mZ
2DPCdrjj( n0k)y( @i !@i) nk
(C3)
where the integral runs over a two-dimensional slice of
the primitive cell (aligned parallel to the interface). The
22 matrix nkis de ned for outgoing modes in the
ferromagnet as
nk=eikjjrjj
u"
nk(r)eik"
nzz0
0u#
nk(r)eik#
nzz!
;(C4)
whereu"=#
nk(r) andk"=#
nzdenote the Bloch wavefunction
and out-of-plane wavevector for majority/minority car-
riers. Both are de ned at kjjon the Fermi surface cor-15
responding to band n. For outgoing modes in the non-
magnet, nksimpli es to:
nk=eikjjrjjeiknzzunk(r)I22: (C5)
The incoming current is de ned as follows
hhjin
i ii=1
~X
m Z
2DBZdkjjgin
m
jvmz jtr
Jin
m;i
(C6)
where
Jin
m;i =izi~
2mZ
2DPCdrjj( mk)y( @i !@i) mk
(C7)
gives the current operator for the incoming current. The
total current is then
hhji ii=hhjin
i ii+hhjout
i ii
=1
~X
mnn0 Z
2DBZdkjjgin
m
jvmz j
trh
izJin
m;i + (sn0m)yJout
n0n;i snm
i
(C8)
where the choice of scattering matrix depends on the in-
coming spin/charge index and outgoing spin/charge
index as follows:
snm=8
>>>>><
>>>>>:rnm ; 2[d;f;`;c ]
tnm 2[d;f;`;c ]; 2[";#]
t
nm 2[";#]; 2[d;f;`;c ]
r
nm ; 2[";#]
(C9)
By plugging in the generalizations of Eqs. (17) and (18)
into Eq. (C8), we nd that Eq. (A9) generalizes to the
following
Bi = e
~X
mnn0Z
2DBZdkjj1
jvmz j
trh
izJin
m;i + (sn0m)yJout
n0n;i snm
i
;
(C10)
while Eq. (A11) now becomes:
bi = e
~X
mnn0 Z
2DBZdkjjfE
m
jvmz j
trh
izJin
m;i + (sn0m)yJout
n0n;i snm
i
:
(C11)Assuming as before that the incoming spin-currents be-
have as if they originate from spin-dependent reservoirs
( q ), we have:
Gi =Bi (C12)
i =bi : (C13)
Thus, Eqs. (C10) and (C11) generalize the conductance
and conductivity tensors respectively to include non-
trivial electronic structure.
The transverse spin current that develops in the ferro-
magnet atz= 0+may be obtained by using similar ex-
pressions. The tensor CFM
, originally given by Eq. (B4),
now becomes
CFM
= e
~X
mnn0Z
2DBZdkjj1
jvmz j
8
<
:tr
(t
n0m)yJout
n0n;zt
nm
2[d;f;`;c ]
tr
(r
n0m)yJout
n0n;zr
nm
2[";#]:
(C14)
Likewise, the tensor cFM
, rst described by Eq. (B6),
generalizes to the following:
cFM
= e
~X
mnn0 Z
2DBZdkjjfE
m
jvmz j
8
<
:tr
(t
n0m)yJout
n0n;zt
nm
2[d;f;`;c ]
tr
(r
n0m)yJout
n0n;zr
nm
2[";#]:
(C15)
Evaluating the trace in Eq. (C15) gives the ensemble av-
erage of velocity for the transverse spin states in the fer-
romagnet. Here we do not assume that the velocity of
these states equals the average velocity of majority and
minority carriers. However, for the simple model dis-
cussed in the previous section, one can show that the
current operator Jout
n0n;zsimpli es to the following:
Jout
n0n;z!Jout
z/1
2
vz"+vz#
(C16)
In this scenario, Eqs. (C14) and (C15) reduce to
Eqs. (B4) and (B6) as expected. This justi es the use
of the average velocity to describe transverse spin states
in the simple model. For q we have:
FM
=CFM
(C17)
FM
=cFM
: (C18)
Thus we have generalized the torkance and torkivity
tensors that describe bulk ferromagnet torques for non-
trivial electronic structures.
For realistic systems, the interface should be modeled
over a few atomic layers so that an exchange potential16
Parameter Value
E ective mixing conductance
Re[~G"#]GzddorGzff
Im[~G"#]GzdforGzfd
Spin current due to interfacial spin-orbit scattering
jE
d(0 )zd~E
jE
f(0 )zf~E
Spin torque on the lattice at the interface
latt
d
zd
d~E
latt
f
zf
f~E
TABLE I. Table of phenomenological parameters relevant to
the drift-di usion model of spin-orbit torque developed in the
companion paper, chosen by the numerical study performed
in Sec.IV. All other boundary parameters are discarded in
that model. As can be seen in section IIIA of the companion
paper, the rst four parameters govern the total spin torque
thickness dependence, while the last two parameters describe
the spin torque's zero-thickness intercept. Note that here all
boundary parameters obey the sign convention that positive
currents
ow towards from the ferromagnet.
and spin-orbit coupling may simultaneously exist. If
these atomic layers make up the scattering region used
to obtain the scattering coecients, then the expressions
presented here describe the currents on either side of the
interface as intended. However, in order to describe the
interfacial torque, the tensors Cmag
andcmag
must be
written as sums of the layer-resolved torques within the
interfacial scattering region. We save the generalization
of Eqs. (B5) and (B7) for future work, since in this paper
we treat the interface as a plane rather than a region of
nite thickness.
Appendix D: Boundary parameters relevant to
bilayer spin-orbit torques
In Sec. IV, we numerically analyze each boundary pa-
rameter for an interfacial scattering potential that in-
cludes the exchange interaction and spin-orbit coupling.
We nd that many parameters di er by several orders
of magnitude. In the companion paper, we use this in-
formation to derive an analytical drift-di usion model of
spin-orbit torques in heavy metal/ferromagnet bilayers.
In the following we discuss the minimal set of parameterscrucial to that solution.
Table I includes six parameters important to the inter-
face of heavy metal/ferromagnet bilayers. Along with the
spin di usion length ( lsf), the bulk conductivity ( NM
bulk),
and the spin Hall current density ( jsH
d) in the non-
magnet, they describe all of the phenomenological pa-
rameters used by the analytical drift-di usion model in
the companion paper. The rst two parameters are the
real and imaginary parts of the spin mixing conductance.
The generalized version of these parameters may be ex-
tracted from the conductance tensor Gz . Numerical
studies show that these parameters depend weakly on
magnetization direction. In the companion paper, the
ungeneralized spin mixing conductance is used. The pa-
rametersjE
d(0 ) andjE
f(0 ) denote the interfacial spin
currents just within the non-magnet that arise due to
in-plane electric elds and spin-orbit scattering. In anal-
ogy to the bulk spin Hall current, these parameters act
as sources of spin current for the drift-di usion equa-
tions. Thus, in the absence of jE
d(0 ),jE
f(0 ), andjsH
d,
all bulk currents and accumulations vanish. In addition
to the spin mixing conductance, these parameters deter-
mine the non-magnet thickness-dependence of spin-orbit
torques. The nal two parameters give the approximate
loss of angular momentum to the interface. They equal
the damping-like and eld-like spin-orbit torques in the
limit of vanishing non-magnet thickness. They are de-
rived by subtracting the interfacial torque from the loss
in out-of-plane spin current density across the interface.
Our numerical analysis suggests that spin and charge
accumulations cause negligible di erences in these two
quantities. Thus, we assume that latt
dand latt
fstem
primarily from spin-orbit scattering at the interface. The
treatment of the lattice torque presented in the compan-
ion paper begins from this assumption.
The model introduced in the companion paper general-
izes the drift-di usion model used in Ref. [42] to include
interfacial spin-orbit e ects. Only two additional phe-
nomenological parameters ( jE
d(0 ) andjE
f(0 )) are re-
quired to capture the non-magnet thickness dependence,
while an additional two parameters ( latt
dand latt
f) de-
scribe the corresponding zero-thickness intercept. Table I
provides formulas for these phenomenological parameters
in terms of the boundary parameters contained within
Eqs. (12) and (13). We note that in magnetoelectronic
circuit theory, the conductance parameters are given by
sums of interfacial scattering coecients over the avail-
able scattering states. All of the boundary parameters
introduced here possess a similar form, as discussed in
appendices A, B, and C.
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1002.0441v2.Spin_resolved_scattering_through_spin_orbit_nanostructures_in_graphene.pdf | arXiv:1002.0441v2 [cond-mat.mes-hall] 26 Apr 2010Spin-resolved scattering through spin-orbit nanostructu res in graphene
D. Bercioux1,2,∗and A. De Martino3,†
1Freiburg Institute for Advanced Studies, Albert-Ludwigs- Universit¨ at, D-79104 Freiburg, Germany
2Physikalisches Institut, Albert-Ludwigs-Universit¨ at, D-79104 Freiburg, Germany
3Institut f¨ ur Theoretische Physik, Universit¨ at zu K¨ oln, Z¨ ulpicher Straße 77, D-50937 K¨ oln, Germany
(Dated: November 10, 2018)
We address the problem of spin-resolved scattering through spin-orbit nanostructures in graphene,
i.e., regions of inhomogeneous spin-orbit coupling on the nanom eter scale. We discuss the phe-
nomenon of spin-double refraction and its consequences on t he spin polarization. Specifically, we
study the transmission properties of a single and a double in terface between a normal region and
a region with finite spin-orbit coupling, and analyze the pol arization properties of these systems.
Moreover, for the case of a single interface, we determine th e spectrum of edge states localized at
the boundary between the two regions and study their propert ies.
PACS numbers: 72.80.Vp, 73.23.Ad, 72.25.-b, 72.25.Mk, 71. 70.Ej
I. INTRODUCTION
Graphene1,2— a singlelayerofcarbonatoms arranged
in a honeycomb lattice — has attracted huge atten-
tion in the physics community because of many unusual
electronic, thermal and nanomechanical properties.3,4In
graphene the Fermi surface, at the charge neutrality
point, reduces to two isolated points, the two inequiv-
alent corners KandK′of the hexagonal Brillouin zone
of the honeycomb lattice. In their vicinity the charge
carriers form a gas of chiral massless quasiparticles with
a characteristic conical spectrum. The low-energy dy-
namics is governed by the Dirac-Weyl (DW) equation5,6
in which the role of speed of light is played by the elec-
tron Fermi velocity. The chiral nature of the quasipar-
ticles and their linear spectrum lead to remarkable con-
sequences for a variety of electronic properties as weak
localization, shot noise, Andreev reflection, and many
others. Also the behavior in a perpendicular magnetic
field discloses new physics. Graphene exhibits a zero-
energy Landau level, whose existence gives rise to an un-
conventional half-integer quantum Hall effect, one of the
peculiar hallmarks of the DW physics.
Driven by the prospects of using this material in spin-
tronic applications,7,8the study of spin transport is one
of the most active field in graphene research.9–14Sev-
eral experiments have recently demonstrated spin injec-
tion, spin-valve effect, and spin-coherent transport in
graphene, with spin relaxation length of the order of
few micrometers.10,14In this context a crucial role is
playedbythe spin-orbitinteraction. Ingraphenesymme-
tries allow for two kinds of spin-orbit coupling (SOC).15
Theintrinsic SOC originates from carbon intra-atomic
SOC. It opens a gap in the energy spectrum and con-
verts graphene into a topological insulator with a quan-
tized spin-Hall effect.15This term has been estimated
to be rather weak in clean flat graphene.16–19Theex-
trinsicRashba-like SOC originates instead from inter-
actions with the substrate, presence of a perpendic-
ular external electric field, or curvature of graphene
membrane.16–18,20This term is believed to be responsiblefor spin polarization21and spin relaxation22,23physics in
graphene. Optical-conductivitymeasurementscouldpro-
vide a way to determine the respective strength of both
SOCs.24
In this article we address the problem of ballistic spin-
dependent scattering in the presence of inhomogeneous
spin-orbit couplings. Our main motivation stems from
a recent experiment that reported a large enhancement
of Rashba SOC splitting in single-layer graphene grown
on Ni(111) intercalated with a Au monolayer.25Further
experimental results show that the intercalation of Au
atoms between graphene and the Ni substrate is essen-
tial in order to observe sizable Rashba effect.26,27The
preparation technique of Ref. 25seems to provide a sys-
tem with properties very close to those of freestanding
graphene in spite of the fact that graphene is grown on
a solid substrate. The presence of the substrate does not
seem to fundamentally alter the electronic properties ob-
served in suspended systems, i.e., the existence of Dirac
points at the Fermi energy and the gapless conical dis-
persion in their vicinity.
These results suggest that a certain degree of control
on the SOC can be achieved by appropriate substrate
engineering, with variations of the SOC strength on sub-
micrometer scales, without spoiling the relativistic gap-
less nature of quasiparticles. This could pave the way for
the realization of spin-optics devices for spin filtration
and spin control for DW fermions in graphene. An opti-
mal design would require a detailed understanding of the
spin-resolved ballistic scattering through such spin-orbit
nanostructures , which is the aim of this paper.
The problem of spin transport through nanostructures
with inhomogeneous SOC has already been thoroughly
studied in the case of two-dimensional electron gas in
semiconductor heterostructures with Rashba SOC.28–30
Here the Rashba SOC31— arising from the inversion
asymmetry of the confinement potential — couples the
electron momentum to the spin degree of freedom and
thereby lifts the spin degeneracy. In this case, a region
with finite SOC between two normal regions has prop-
erties similar to biaxial crystals: an electron wave inci-2
N region SO region
ky
kxky
kxφE k+k-
ξ+ξ-
Figure 1: (Color online) Illustration of the kinematics of t he
scattering at a N-SO interface in graphene. The circles rep-
resent constant energy contours.
dent from the normal region splits at the interface and
the two resulting waves propagate in the SO region with
different Fermi velocities and momenta.28This effect —
analogousto the opticaldouble-refraction— producesan
interference pattern when the electron waves emerge in
the second normal region. Moreover, electrons that are
injected in an spin unpolarized state emerge from the SO
region in a partially polarized state.
Herewe shallfocus on the twosimplest examplesofSO
nanostructuresingraphene: (i)asingleinterfacebetween
two regions with different strengths of SOC; (ii) a SOC
barrier, or double interface, i.e., a region of finite SOC
in between two regions with vanishing SOC.
Our analysis shows — in analogy to the case of 2DEG
— that the ballistic propagation of carriers is governed
bythespin-doublerefraction. Wefind thatthescattering
properties of the structure strongly depend on the injec-
tion angle. As a consequence, an initially unpolarized
DW quasiparticle emerges from the SOC barrier with a
finite spin polarization. In analogy to the edge states in
the quantum spin-Hall effect,15we also consider the pos-
sibility of edge states localized at the interface between
regions with and without SOC.
This paper is organized as follows. In Sec. IIwe intro-
duce the model and the transfer matrix formalism used
in the rest of the paper. In Sec. IIIwe discuss the scat-
tering problem at a single interface and the spectrum of
edge states. In Sec. IVwe address the case of a dou-
ble interface — a SOC barrier — and the final Sec. Vis
devoted to the discussion of results and conclusions.
II. MODEL AND FORMALISM
We consider a clean graphene sheet in the xy-
plane with SOCs15,16,21,32,33inhomogeneous along the
x-direction. We shall restrict ourselves to a single-
particle picture and neglect electron-electron interaction
effects. The length scale over which the SOCs vary is as-
sumed to be much largerthan graphene’slattice constant
(a= 0.246 nm) but much smaller than the typical Fermiwavelength of quasiparticles λF. Since close to the Dirac
pointsλF∼1/|E|, at low energy Ethis approximation
is justified. This assumption ensures that we can use the
continuum DW description, in which the two valleys are
not coupled. Yet close to a Dirac point we can approxi-
mate the variation of SOCs as a sharp change. Focusing
on a single valley, the single-particle Hamiltonian reads
H=vFσ·p+HSO, (1)
HSO=λ(x)
2(σ×s)z+∆(x)σzsz, (2)
wherevF≈106m/s is the Fermi velocity in graphene.
In the following we set /planckover2pi1=vF= 1. The vector of Pauli
matrices σ= (σx,σy) [resp.s= (sx,sy)] acts in sublat-
tice space [resp. spin space]. The term HSOcontains the
extrinsic or Rashba SOC of strength λand the intrinsic
SOC of strength ∆. While experimentally the Rashba
SOC can be enhanced by appropriate optimization of
the substrate up to values of the order of 14 meV,25the
intrinsic SOC seems at least two orders of magnitude
smaller. Yet, the limit of large intrinsic SOC is of con-
siderable interest since in this regime graphene becomes
a topological insulator.15Thus in this paper we shall
not restrict ourself to the experimentally relevant regime
λ≫∆ but consider also the complementary regime.
The wave function Ψ is expressed as
ΨT= (ΨA↑,ΨB↑,ΨA↓,ΨB↓),
where the superscriptTdenotes transposition. Spectrum
and eigenspinors of the Hamiltonian ( 1) with uniform
SOCs are briefly reviewed in Appendix A. The spec-
trum consists of four branches Eα,ǫ(k) labelled by the
two quantum numbers ǫ=±1 andα=±1. Here, the
first distinguishes particle and hole branches, the second
gives the sign of the expectation value of the spin pro-
jection along the in-plane direction perpendicular to the
propagation direction k. The spectrum strongly depends
on the ratio
η=∆
λ. (3)
Forη >1/2 a gap separates particle and hole branches.
The gap closes at η= 1/2 and forη <1/2 one particle
branch and one hole branch are degenerate at k= 0 (see
Fig.8in App.A).
We now briefly summarize the transfer matrix ap-
proach employed in this paper to solve the DW scat-
tering problem.35–38We assume translational invariance
in they-direction, thus the scattering problem for the
Hamiltonian( 1)reducestoaneffectivelyone-dimensional
(1D) one. The wave function factorizes as Ψ( x,y) =
eikyyχ(x), wherekyis the conserved y-component of the
momentum, which parameterizes the eigenfunctions of
the Hamiltonian of given energy E.
For simplicity we consider piecewise constant profiles
of SOCs, and solve the DW equation in each region of
constant couplings. Then we introduce the x-dependent3
0 !/4 !/2 0 !/4 !/2
Incident angle !Refraction angles "#"-"+
0 !/4 !/2 !-!+(a) (b)
Figure 2: (Color online) Refraction angles as function of th e
incidence angle for fixed energy and fixed SOCs. Panel (a):
E= 5,λ= 0.5, ∆ = 2; panel (b): E= 5,λ= 2, ∆ = 0 .5.
4×4 matrix Ω( x), whose columns are given by the com-
ponents of the independent, normalized eigenspinor of
the 1D DW Hamiltonian at fixed energy.39Due to the
continuity of the wave function at each interface between
regions of different SOC, the wave function on the left
of the interface can be expressed in terms of the wave
function on the right via the transfer matrix
M=/bracketleftbig
Ω(x−
0)/bracketrightbig−1Ω(x+
0), (4)
wherex0is the position of the interface and x±
0=x0±δ
with infinitesimal positive δ. The condition det M= 1
guarantees conservation of the probability current across
theinterface. Thegeneralizationtothecaseofasequence
ofNinterfaces at positions xi,i= 1,...,N, is straight-
forward since the transfer matrices relative to individual
interfaces combine via matrix multiplication:
M=N/productdisplay
i=1/bracketleftbig
Ω(x−
i)/bracketrightbig−1Ω(x+
i). (5)
From the transfer matrix it is straightforward to deter-
mine transmission and reflection matrices, which encode
all the relevant information on the scattering properties.
III. THE N-SO INTERFACE
First we concentrate on the elastic scattering problem
at the interface separating a normal region N ( x <0),
where SOCs vanish, and a SO region ( x >0), where
SOCs are finite and uniform.
We consider a quasiparticle of energy E, withEas-
sumed positive for definiteness and outside the gap pos-
sibly opened by SOCs. This quasiparticle incident from
the normal region is characterized by the y-component
of the momentum, or equivalently, the incidence angle φ
measured with respect to the normal at the interface, seeFig.1. Conservation of kyimplies that
kN
y=Esinφ=Eαsinξα=kSO
y (6a)
kN
x=Ecosφ (6b)
kSO
xα=Eαcosξα (6c)
whereα=±1 andEα=/radicalbig
(E−∆)(E+∆−αλ). The
refraction angles ξαare fixed by momentum conservation
along the interface ( 6a) and read
ξα= arcsin/parenleftbiggE
Eαsinφ/parenrightbigg
. (7)
Figure1illustrates the refraction process at the N-
SO interface. The incident wave function, assumed to
have fixed spin projection in the z-direction, in the SO
region splits in a superposition of eigenstates of the
SOCs Hamiltonian corresponding to states in the differ-
ent branches of the spectrum. These eigenstates prop-
agate along two distinct directions characterized by the
anglesξα, whose difference depends on SOC and is an in-
creasing function of the incidence angle, see Fig. 2. The
anglesξαcoincide only for normal incidence or for λ= 0.
Equation ( 7) implies that there exists a critical angle
for each of the two modes given by
˜φα= arcsin/parenleftbiggEα
E/parenrightbigg
. (8)
Forφlarger than both critical angles ˜φα, the quasipar-
ticle is fully reflected, since there are no available trans-
mission channels in the SO region. For φin between the
two critical angles the quasiparticletransmits only in one
channel.40
After this qualitative discussion of the kinematics, we
now present the exact solution of the scattering problem.
In the N region x <0 a normalized scattering state of
energyE >0, incident from the left on the interface with
incidence angle φand spin projection sis given by
χN(x) =[δ↑,s|↑/angb∇acket∇ight+δ↓,s|↓/angb∇acket∇ight]/parenleftbigg1
eiφ/parenrightbiggeikxx
/radicalbig2vx
F
+[r↑s|↑/angb∇acket∇ight+r↓s|↓/angb∇acket∇ight]/parenleftbigg1
−e−iφ/parenrightbigge−ikxx
/radicalbig2vx
F,(9)
wherekx≡kN
x(cf. Eq. 6b). Here the index s=↑,↓
specifies the spin projection of the incoming quasipar-
ticle with |↑/angb∇acket∇ightand|↓/angb∇acket∇ighteigenstates of szandδi,jis the
Kronecker delta. The velocity vx
F= cosφis included to
ensure proper normalization of the scattering state. The
complex coefficients rs′sare reflection probability ampli-
tudes for a quasiparticle with spin sto be reflected with
spins′. The associated matrix Ω N(x) reads
ΩN(x) =1/radicalbig2vx
F
eikxxe−ikxx0 0
ei(kxx+φ)−e−i(kxx+φ)0 0
0 0 eikxxe−ikxx
0 0 ei(kxx+φ)−e−i(kxx+φ)
.4
00.1 0.2 0.3 0.4 0.5
-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (a)
00.1 0.2 0.3 0.4 0.5
-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (b)
00.1 0.2 0.3 0.4 0.5 0.6
-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (c)
Figure 3: (Color online) Angular dependence of the transmis -
sion probabilities T+↑(blue dashed line) and T−↑(red solid
line) at energy E= 2.5. The SOC are fixed as follows: (a)
λ= 0.1 and ∆ = 0, (b) λ= 0 and ∆ = 0 .1, and (c) λ= 0.5
and ∆ = 0 .1.
Similarly the wave function in the SO region ( x>0) can
be expressed in general form as
χSO(x) =1/radicalbigvx
++/bracketleftbig
t+ψ++(x)+r+¯ψ++(x)/bracketrightbig
+1/radicalbigvx
−+/bracketleftbig
t−ψ−+(x)+r−¯ψ−+(x)/bracketrightbig
(10)
wheret±(resp.r±) are complex amplitudes for right-
moving (resp. left-moving) states. The coefficient tα
represents the transmission amplitude into mode α. The
wave functions ψα+and the Fermi velocities vx
α+in the
SO region are obtained from the expressions given in
App.Awith the replacement kx→kSO
xα, where for no-
tational simplicity the label SO will be understood. The
wave functions ¯ψα+are in turn obtained from ψα+by
replacingkxα→ −kxα. The matrix Ω SO(x) then reads
ΩSO(x) = (11)
e−iξ+−θ+
2−eiξ+−θ+
2e−iξ−−θ−
2−eiξ−−θ−
2
eθ+
2 eθ+
2 eθ−
2 eθ−
2
ieθ+
2 ieθ+
2 −ieθ−
2−ieθ−
2
ieiξ+−θ+
2−ie−iξ+−θ+
2−ieiξ−−θ−
2ie−iξ−−θ−
2
N+eikx++x0 0 0
0N+e−ikx+x0 0
0 0 N−eikx−x0
0 0 0 N−e−ikx−x
where in the second matrix the normalization factors are
defined as Nα= 1/(2/radicalbig
vα+sinhθα).
According to Eq. ( 4) the transfer matrix for the sin-
gle interface problem is given by the matrix product
M= [ΩN(0−)]−1ΩSO(0+). FromMwe obtain the trans-
mission and the reflection probabilities for a spin-up orspin-down incident quasiparticle:
T+s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleM33δs,↑+M13δs,↓
M13M31−M11M33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
Υ+(φ), (12)
T−s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleM31δs,↑+M11δs,↓
M13M31−M11M33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
Υ−(φ), (13)
R↑s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleM31M23−M33M21
M13M31−M11M33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
δs,↑
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleM13M21−M11M23
M13M31−M11M33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
δs,↓,(14)
R↓s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleM31M43−M33M41
M13M31−M11M33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
δs,↑
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleM13M41−M11M43
M13M31−M11M33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
δs,↓,(15)
where Υ α(φ) =θ(˜φα−φ)θ(˜φα+φ) withθ(x) the Heav-
iside step function. Here, Tαsis the probability for an
incident quasiparticle with spin projection sto be trans-
mittedinmode αintheSOregion. Ofcourse,probability
current conservation enforces T+s+T−s+R↑s+R↓s= 1.
Figures3(a)–(c) show the angular dependence of the
transmission probabilities for an incident spin-up quasi-
particle into the (+) and ( −) modes of the SO region for
different values of the SOCs. Panel (a) refers to the case
of vanishing intrinsic SOC (∆ = 0). Here the (+) and
the (−) energy bands are separated by a SOC-induced
splitting ∆Eext=λ. Therefore at fixed energy the two
propagating modes in the SO region have two different
momenta, which gives rise to the two different critical
angles (cf. Eq. ( 8) with ∆ = 0). Panel (b) refers to the
caseλ= 0, where the SOC opens a gap ∆ Eint= 2∆
between the particle- and the hole-branches, however the
(+)/(−)-modes remain degenerate. Therefore at fixed
energy these modes have the same momentum and, as a
consequence, the same critical angles (cf. Eq. ( 8) for
λ= 0 and ∆ /negationslash= 0). When both SOCs are finite —
the situation illustrated in panel (c) — the transmission
probabilitiesexhibit morestructure. Forincidenceangles
smaller than ˜φ+no particular differences with the cases
of panels (a) and (b) are visible. When the (+) mode
is closed, an increase (resp. decrease) of the ( −) mode
transmission is observed for positive (resp. negative) an-
gles, before the transmission drops to zero for incidence
angles approaching ˜φ−. The asymmetry between posi-
tive and negative angles is reversed if the spin state of
the incident quasiparticle is reversed.
These symmetryproperties can be rationalizedby con-
sidering the operator of mirror symmetry through the
x-axes.41This consists of the transformation y→ −y
and at the same time the inversion of the spin and the
pseudo-spin states. It reads
Sy= (σx⊗sy)Ry, (16)
whereRytransforms y→ −y. The operator Sycom-
mutes with the total Hamiltonian of the system [ Sy,H0+5
HSO] = 0, therefore allows for a common basis of eigen-
states. For the scattering states in the SO region ( 10)
we haveSyχ+(ξ+) =χ+(ξ+) andSyχ−(ξ−) =−χ−(ξ−).
Instead, it induces the following transformation on the
scattering states ( 9) in the normal region: Syχs(φ) =
iχ−s(−φ). By comparing the original scattering matrix
with the Sy-transformed one we find that
Tα,s(φ) =Tα,−s(−φ) (17)
withα=±ands=↑,↓, which is indeed the symmetry
observedintheplots. Theasymmetryofthetransmission
coefficients occurs only when both SOCs are finite.
A. Edge states at the interface
In addition to scatteringsolutions ofthe DW equation,
it is interesting to study the possibility that edge states
exist at the N-SO interface, which propagate alongthe
interface but decay exponentially away from it. The in-
terestinthesetypesofsolutionsisconnectedtothestudy
of topological insulators. It has been shown — first by
Kane and Mele15— that a zig-zag graphene nanoribbon
with intrinsic SOC supports dissipationless edge states
within the SOC gap. In fact, similar states are always
expected to exist at the interface between a topologically
trivial and a topologically non-trivial insulator. In our
case, the latter is represented by graphene with intrinsic
SOC. Of course SOC-free graphene is not an insulator,
howeverit is topologicallytrivial and edgestate solutions
do arise for |ky|>|E|. WhenEis within the gap in the
SO region the corresponding mode is evanescent along
thexdirection on both sides of the interface. Note that
012345012345
kyE(ky)
Figure 4: (Color online). Energy dispersion of the edge stat e
at the N-SO interface as a function of the momentum along
the interface kyfor different values of SOCs. Solution of the
transcendental equation is allowed only for |ky|>|E|(white
area). In all three cases shown η >1/2: ∆ = 1 and λ= 0.4
(lower-red line), ∆ = 1 .5 andλ= 0.7 (middle-blue line), and
∆ = 2 and λ= 0.9 (upper-green line).the edge state we find is different from the one discussed
in Refs.15,32where zig-zag or hard-wall boundary con-
ditions at the edge of the SOC region were imposed.
The wave function on the N side then reads
χN(x) =/parenleftbigg1
−iq+iky
E/parenrightbigg
(A|↓/angb∇acket∇ight+B|↑/angb∇acket∇ight)eqx(18)
withq=/radicalbig
|ky|2−E2. In the SO region the wave func-
tion can be written as
χSO(x) =C
(−q++ky)
i(E−∆)
E−∆
i(q++ky)
e−q+x+D
(q−−ky)
−i(E−∆)
E−∆
(q−+iky)
e−q−x
withqα=/radicalig
k2y−(E−∆)(E+∆−αλ). The continu-
ity of the wave function at the N-SO interface leads to
a linear system of equations for the amplitudes AtoD.
The matrix of coefficients must have a vanishing deter-
minant for a non-trivial solution to exist. This condition
provides a transcendental equation for the energy of pos-
sible edge states, whose solutions are illustrated in Fig. 4
for different values of the intrinsic and extrinsic SOCs.
The condition |ky|>|E|implies that solutions only exist
outside the shadowed area. In addition, they are allowed
onlyin the caseSOCsopena gapinthe energyspectrum,
which occurs when η>1/2 (see App. Aand Eq. ( 3)). As
can be seen in Fig. 4the result is quite insensitive to the
precise value of the extrinsic SOC.
Edge states exist only for values of the momentum
along the interface larger than the intrinsic SOC, i.e.,
ky>kmin
y= ∆. The apparent breaking of time-reversal
invariance (the dispersion is not even in ky) is due to the
fact that we are considering a single-valley theory. The
full two-valley SOC Hamiltonian is invariant under time-
reversal symmetry, that interchanges the valley quantum
number. This invariance implies that solutions for neg-
ative values of kycan be obtained by considering the
Dirac-Weyl Hamiltonian relative to the other valley. The
twocounter-propagatingedge states live then at opposite
valleysand haveoppositespinstateandrealizeapeculiar
1D electronic system.
As mentioned in the Introduction, the intrinsic SOC
is estimated to be much smaller than the extrinsic one,
therefore in a realistic situation one would not expect the
opening of a significant energy gap and the presence of
edge states. It would be interesting to explore the pos-
sibility to artificially enhance the intrinsic SOC, thereby
realizing the condition for the occurrence of edge states.
IV. THE N-SO-N INTERFACE
The analysis of the scattering problem on a N-SO in-
terface of the previous section can be straightforwardly
generalized to the case of a double N-SO-N interface (SO
barrier). Here the transmission matrix Dis given by6
00.2 0.4 0.6 0.8 1
-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (a)
00.2 0.4 0.6 0.8 1
-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (b)
00.2 0.4 0.6 0.8 1
-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (c)
Figure 5: (Color online). Panel (a): Angular plots for T↑↑as
a function of the injection angle for E= 2, ∆ = 1 and λ= 0.
The three lines correspond to different distance between the
interfaces: d=π/2 (dashed black), d=π(dotted red), and
d= 2π(solid blue). The spin-precession length is ℓSO=π.
Whenλ= 0 the transmission probability in the spin state
opposed to the injected spin is always zero. Panel (b) and
(c): angular plots of T↑↑(solid-blue) and T↓↑(dashed red) as
a function of the injection angle for E= 2,λ= 1 and ∆ = 0.
The distance between the two interfaces is d=πin panel
(a) and d= 2πin panel (b). The spin-precession length is
ℓSO= 2π.
Eq. (5) withN= 2. The transmission and the reflection
probabilities in the case of a spin-up or -down incident
quasiparticle read
T↑s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleD33δs,↑+D13δs,↓
D13D31−D11D33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
, (19)
T↓s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleD31δs,↑+D11δs,↓
D13D31−D11D33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
, (20)
R↑s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleD31D23−D33D21
D13D31−D11D33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
δs,↑
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleD13D21−D11D23
D13D31−D11D33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
δs,↓,(21)
R↓s=/vextendsingle/vextendsingle/vextendsingle/vextendsingleD31D43−D33D41
D13D31−D11D33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
δs,↑
+/vextendsingle/vextendsingle/vextendsingle/vextendsingleD13D41−D11D43
D13D31−D11D33/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
δs,↓.(22)
Inthiscasethereisanadditionalparameterwhichcon-
trols the scattering properties of the structure, namely
the widthdof the SO region. In order to compare this
length to a characteristic length scale of the system, we
introduce the spin-precession length defined as
ℓSO= 2π/planckover2pi1vF
λ+2∆. (23)
The intrinsic SOC alone cannot induce a spin preces-
sion on the carriers injected into the SO barrier — an
injected spin state, say up, is obviously never converted
into a spin-down state. Figure 5(a) shows the angular00.2 0.4 0.6 0.8 1
-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (a)
00.2 0.4 0.6 0.8 1
-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (b)
00.2 0.4 0.6 0.8 1
-90 -75 -60 -45 -30 -15 015 30 45 60 75 90 (c)
Figure 6: (Color online). Angular plot of T↑↑(solid-blue)
andT↓↑(dashed-red) as a function of the injection angle for
E= 2,λ= 1 and (a) ∆ = λ/4, (b) ∆ = λ/2, and ∆ = λ. The
distance between the two interfaces is kept fixed to d=ℓSO.
dependence of the transmission in the case of injection of
spin-up. The behavior of the transmission as a function
of the injection angle depends sensitively on the width d
compared to the spin-precession length. For small width
d < ℓSO(dashed line) the transmission is a smooth de-
creasing function of φand stays finite also for φlarger
than the critical angle. In the case d≥ℓSO(dotted- and
solid-lines) instead the transmission probability exhibits
a resonant behavior and drops to zero as soon as the
injection angle equals the critical angle.
When only the extrinsic SOC is finite, the transmis-
sion behavior changes drastically. Two different critical
angles appear — the biggest coincides usually with π/2.
The extrinsic SOC induces spin precession because of the
coupling between the pseudo- and the real-spin. This is
illustrated in Fig. 5(b)-(c). In Panel (b) we consider the
case of spin-up injection with d=ℓSO/2. At normal inci-
dence the transmission is entirely in the spin-down chan-
nel (dashed line). Moving away from normal incidence,
the transmission in the spin-up channel (solid line) in-
creases from zero and, after the first critical angle, the
transmissions in spin-up and spin-down channels tend to
coincide. In panel (c) the width of the barrier is set to
d=ℓSO. Here, the width of the SO region permits to an
injected carrier at normal incidence to perform a com-
plete precession of its spin state — the transmission is in
the spin-up channel. For finite injection angles the spin-
down transmission (dashed line) also becomes finite. For
φ/lessorsimilar˜φ+the transmission in the spin-up channel is almost
fully suppressed while that in the spin-down channel is
large. Finally, for φ >˜φ+the two transmission coeffi-
cients do not show appreciable difference.
In the case where both extrinsic and intrinsic SOC
are finite, the transmission probability exhibits a richer
structure. We focus again on the case of injection of
spin-up quasiparticles. Moreover we fix the width of the
SO regionsothat it is alwaysequalto the spin-precession
lengthd=ℓSO. Fig.6illustrates the transmission proba-
bilitiesTs↑forthree values ofthe ratio∆ /λ= 1/4,1/2,1.
Notice that from panel (a) to (c) the width of SO region7
00.2 0.4 0.6 0.8 1-90 -75 -60 -45 -30 -15 015
30
45
60
75
90 (a)
-90 -45 0 45 90 -0.4 -0.2 00.2 0.4
ϕ Injection angle Pz Polarization (b)
Figure 7: (Color online). Panel (a): total transmission T
as a function of the injection angle for E= 2,d= 2πand
several values of SOCs: λ= 1 and ∆ = 0 (blue-solid line),
λ= 0 and ∆ = 0 .5 (red-dotted line), λ= 1 and ∆ = λ/4
(yellow-dashed line), ∆ = λ/2 (orange-dashed-dotted line),
andλ= ∆ (black-dotted-dotted-dashed line). Panel (b): z-
component of the spin polarization Pzas a function on the
injection angle for E= 2 and d= 2πand the following values
of the SOCs: λ= 1, ∆ = 0 and λ= 0, ∆ = 1 (same black-
dashed line), λ= 1 and ∆ = λ/4 (red-dotted), ∆ = λ/2
(blue-dotted-dashed line), and ∆ = λ(green-solid line).
decreases.
The symmetry properties of the transmission func-
tion can be rationalized by using the symmetry opera-
tion (16). Proceeding in a similar manner as in the case
of the single interface, for the SO barrier we find the
following symmetry relations
Ts,s(φ) =Ts,s(−φ), (24a)
Ts,−s(φ) =T−s,s(−φ), (24b)
which are confirmed by the explicit calculations.
So far we have considered the injection of a pure spin
state — the injected carrier was either in a spin-up state
or a spin-down state. Following Ref. 30we now address
the transmission of an unpolarized statistical mixture of
spin-up and spin-down carriers. This will characterize
the spin-filtering properties of the SO barrier. In the
injection N region, an unpolarized statistical mixture of
spins is defined by the density matrix
ρin=1
2|χ↑/angb∇acket∇ight/angb∇acketleftχ↑|+1
2|χ↓/angb∇acket∇ight/angb∇acketleftχ↓|, (25)
where|χs/angb∇acket∇ight ≡ |s/angb∇acket∇ight ⊗ |σ/angb∇acket∇ightwith|σ/angb∇acket∇ight= (1/√
2)(1,eiφ) cor-
responds to a pure spin state. When traveling throughthe SO region, the injected spin-unpolarized state is sub-
jected to spin-precession. The density matrix in the out-
put N region can be expressed in terms of the transmis-
sion functions ( 19) as
ρout=1
2T↑|ζ↑/angb∇acket∇ight/angb∇acketleftζ↑|+1
2T↓|ζ↓/angb∇acket∇ight/angb∇acketleftζ↓|,(26)
where the coefficients Ts=T↑s+T↓sare the total trans-
missions for fixed injection state. The spinor part is de-
fined as
|ζs/angb∇acket∇ight=1√Ts/parenleftbigg
t↑s
t↓s/parenrightbigg
⊗|σ/angb∇acket∇ight, (27)
wherets′,sare the transmission amplitudes for incoming
(resp. outgoing) spin s(resp.s′). The output density
matrix is used to define the total transmission
T=T↑+T↓
2(28)
andthe expectation valueofthe zcomponent ofthe spin-
polarization
Pz=1
2(T↑↑+T↑↓−T↓↑−T↓↓).(29)
In Fig.7we report the total transmission (panel (a))
and thez-component of the spin-polarization (panel (b))
as a function of the injection angle for fixed energy and
width of the SO region. We observe that for an un-
polarized injected state the transmission probability is
an even function of the injection angle T(φ) =T(−φ).
Moreover, for injection angles larger than the first criti-
cal angleφ >˜φ+, the transmission has an upper bound
atT= 1/2. On the contrary Pzis an odd function of
the injection angle Pz(φ) =−Pz(−φ). It is zero when
at least one SOC is zero. When both SOC parameters
are finite Pzis finite and reaches the largest values for
φ>˜φ+. The maxima in this case increase as a function
of the intrinsic SOC.
To experimentally observe this polarization effect the
measurement should not involve an average over the an-
gleφ, which, otherwise — due to the antisymmetry of Pz
— would wash out the effect. To achieve this, one could
use,e.g., magnetic barriers,37,42which are known to act
as wave vector filters.
V. CONCLUSIONS
In this paper we have studied the spin-resolved trans-
missionthroughSOnanostructuresin graphene, i.e., sys-
tems where the strength of SOCs — both intrinsic and
extrinsic — is spatially modulated. We have considered
the case ofan interface separatinga normal regionfrom a
SO region, and a barrier geometry with a region of finite
SOC sandwiched between two normal regions. We have
shown that — because of the lift of spin degeneracy due
to the SOCs — the scatteringat the single interface gives8
rise to spin-double refraction: a carrier injected from the
normal region propagates into the SO region along two
different directions as a superposition of the two avail-
able channels. The transmission into each of the two
channels depends sensitively on the injection angle and
on the values of SOC parameters. In the case of a SO
barrier, this result can be used to select preferential di-
rections along which the spin polarization of an initially
unpolarized carrier is strongly enhanced.
We have also analyzed the edge states occurring in the
single interface problem in an appropriate range of pa-
rameters. These states exist when the SOCs open a gap
in the energy spectrum and correspond to the gapless
edge states supported by the boundary of topological in-
sulators.
A natural follow-up to this work would be the detailed
analysis of transport properties of such SO nanostruc-
tures. From our results for the transmission probabil-
ities, spin-resolved conductance and noise could easily
be calculated by means of the Landauer-B¨ uttiker formal-
ism. Moreoverweplantostudyothergeometries,as, e.g.,
nanostructureswith aperiodic modulation ofSOCs. The
effects of various types of impurities on the properties
discussed here is yet another interesting issue to address.
We hope that our work will stimulate further theo-
retical and experimental investigations on spin transport
properties in graphene nanostructures.
Acknowledgments
We gratefully acknowledge helpful discussions with
L. Dell’Anna, R. Egger, H. Grabert, M. Grifoni, W.
H¨ ausler, V. M. Ramaglia, P. Recher and D. F. Urban.
The work of DB is supported by the Excellence Initia-
tive of the German Federal and State Governments. The
work of ADM is supported by the SFB/TR 12 of the
DFG.
Appendix A: Graphene with uniform spin-orbit
interactions.
In this appendix we briefly review the basic proper-
ties of DW fermions in graphene with homogeneous SO
interactions.21The energy eigenstates are plane waves
ψ∼Φ(k)eik·rwith Φ a four-componentspinorand eigen-
values given by ( vF=/planckover2pi1= 1)
Eα,ǫ(k) =αλ
2+ǫ/radicaligg
k2x+k2y+/parenleftbigg
∆−αλ
2/parenrightbigg2
,(A1)
whereα=±andǫ=±. The energy dispersion as a
function of kxat fixedky= 0 is illustrated in Fig. 8for
several values of ∆ and λ. The index ǫ=±specifies the
particle/holebranchesofthe spectrum. The eigenspinorsΦα,ǫ(k) read
ΦT
α,ǫ(k) =1
2√coshθα× (A2)
(e−iφ−ǫθα/2,ǫeǫθα/2,iαǫeǫθα/2,iαeiφ−ǫθα/2),
whereTdenotes transposition and
sinhθα=αλ/2−∆
k, (A3)
eiφ=kx+iky
k, (A4)
withk=/radicalig
k2x+k2y. The spin operator components are
expressed as Sj=1
2sj⊗σ0. Their expectation values in
the eigenstate Φ α,ǫread
/angb∇acketleftSx/angb∇acket∇ight=−ǫαsinφ
2coshθα, (A5a)
/angb∇acketleftSy/angb∇acket∇ight=ǫαcosφ
2coshθα, (A5b)
/angb∇acketleftSz/angb∇acket∇ight= 0, (A5c)
which shows that the product ǫαcoincides with the sign
of the expectation value of the spin projection along the
inplanedirectionperpendiculartothedirectionofpropa-
gation. For vanishing extrinsic SOC, the eigenstates Φ α,ǫ
reduce to linear combinations of eigenstates of Sz.
Similarly, the expectation value of the pseudo-spin op-
(b)
-4 -2 024
Momentum (d)-4 -2 024Energy (a)
-4 -2 024
Momentum -4 -2 024Energy (c)
Figure 8: Spectrum of the DW Hamiltonian with intrinsic
and Rashba SOC as a function of kxforky= 0 . Panel (a):
dashed lines refer to ∆ = 0 .5 andλ= 0; solid and dotted lines
refer to ∆ = 0 and λ= 1. Panel (b): ∆ = 0 .4 andλ= 1.
Panel (c): ∆ = 0 .5 andλ= 1. Panel (d): ∆ = 0 .8 andλ= 1.9
eratorσis given by
/angb∇acketleftσx/angb∇acket∇ight=ǫcosφ
coshθα, (A6a)
/angb∇acketleftσy/angb∇acket∇ight=ǫsinφ
coshθα. (A6b)
Since the SOCs in graphene do not depend on momen-
tum, thevelocityoperatorstillcoincideswiththepseudo-
spin operator: v=˙r= i[H,r] =σ. Thus the velocityexpectationvalueinthestateΦ α,ǫisgivenbyEqs. ( A6a–
A6b). Alternatively, it can be obtained from the energy
dispersion as
vα,ǫ=∇kEα,ǫ=ǫk/radicalig
k2+/parenleftbig
∆−αλ
2/parenrightbig2.(A7)
The groupvelocity is then independent of the modulus of
the wave vector if either the SOCs vanish or ∆ = αλ/2.
∗Electronic address: dario.bercioux@frias.uni-freiburg.de
†Electronic address: ademarti@thp.uni-koeln.de
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1509.06118v1.Nature_of_Valance_Band_Splitting_on_Multilayer_MoS2.pdf | 1 Nature of Valance Band Splitting on M ultilayer MoS 2
Xiaofeng Fana, *, W.T. Zhenga, and David J. Singha,b, †
a. College of Materials Science and Engineering, Jilin University, Changchun 1300 12, China
b. Department of Physics and Astronomy, University of Missouri, Columbia, Missouri
65211 -7010, USA
*E-mail: xffan@jlu.edu.cn ; † E-mail: singhdj@ missouri.edu
Abstract
Understanding the origin of splitting of valance band is important since it govern s the
unique spin and valley ph ysics in few -layer MoS 2. With first principle methods, we explore
the effects of spin -orbit coupling and layer ’s coupling on few -layer MoS 2. It is found that
intra-layer spin -orbit coupling has a major contribution to t he splitting of valance band at K.
In double -layer MoS 2, the layer ’s coupling result s in the widen ing of energy gap of splitted
states induced by intra -layer spin -orbit coupling. The valance band splitting of bulk MoS 2
in K can follow this model. We also f ind the effect of inter -layer spin -orbit coupling in
triple -layer MoS 2. In addition , the inter -layer spin -orbit coupling is found to become to be
stronger under the pressure and results in the decrease of main energy gap in the splitting
valance bands at K. .
Introduction
A new class of 2D materials, the single -layer and/or few -layer of hexagonal transition
metal dichalcogenides (h-TMDs) , have attracted broad attentions due to the extraordinary
physic al properties and promising applications in electric and optoelectronic devices1-5. As
the prototypical 2D materials, single -layer h -TMDs are direct band gap semiconductors
with spin -splitting at valance band maximum which is much different from graphene6-9.
This promises a chance to manipulate the spin degree of freedom and valley
polarization10-12. In addition , with extreme dimensional confinement , tightly -bound
excitons and strong electron -electron interactions due to weak screening, h -TMDs have
been ideal low -dimensional compounds to explore many interesting quantum phenomena11,
13-16, such as spin- and valley - Hall effects and superconductivity17-19. There are also a lot
of fascinating optical properties in single -layer h -TMDs, such as the strong band gap
photoluminescence at edge5, surface sensitive luminescence20, 21 and strain -controlled
optical band gap22-25, and so on.
Among the se h-TMDs, MoS 2 is a representative . Bulk MoS 2 is a layered compound
stacked with the weak van der Waals interaction26. Due to the highly anisotropic
mechanic al property , it is used in dry lubrication . It has also made the interest due to the
special catalytic activity from its edge27. In each layer of MoS 2, there are three atomic
layers with a center layer of Mo around S layers in both sides. The state s near band gap are 2 well-known to be mainly from the d -orbitals of Mo28. There is a priori proposal that the
layer ’s coupling is possible to have a very weak effect to the states near band gap. Bulk
MoS 2 is an indirect -gap semiconductor with a band gap of 1.29 eV. However, following the
reduction of layers to sinlge -layer, there is a transition between indirect band gap and direct
gap3. The single -layer MoS 2 is found to have a direct band gap of about 1.8 eV5, 29.
Therefore, the layer’s coupling has a strong effect to the states near band gap with recent
reports30. Especially, the states of valance band top a t point (VB-) and cond uction band
bottom along Λ (CB-Λ) is much sensitive to the layer ’s coupling (LC) . Compared with the
states of VB - and CB - Λ, the LC effects on the states of valance band top and conduction
band bottom at K point (VB -K, CB -K) are very weak. Therefore, there remains an open
question about the origin of the splitting at valance band of K point which govern s the
unique spin and valley ph ysics. In the single -layer limit, the splitting can be attributed
entirely to spin -orbit coupling (SOC). In bulk limit, it is considered to be a result of
combination of SOC and LC. However, there is disagreement about the relative strength of
both mechanisms31-37.
In the work, we explore the effect of SOC on few -layer MoS 2 with th e rule of LC by
first principle methods in details. We analyze the splitting of states at VB -, VB -K CB -Λ,
and CB -K and explore the change of splitting by following the increase of distance of both
layers for double -layer MoS 2. It is found that intra-layer SOC (intra -SOC) has a major
contribution to the splitting at VB -K, while LC can open effectively the degeneracy of
states at VB -K. With the analysis of charge distribution in real space, the double -
degeneracy of states at the valance band maximum of K po int, which isn't broken due to
the inter-layer inverse symmetry for both layers which result s in the forbidding of
inter-layer SOC, are mainly from the spin -up state of first -layer and spin -down state of
second layer. For triple -layer MoS 2, the LC with int er-layer SOC due to the absence of
inter-layer inverse symmetry in t hree-layer system makes the splitting complicated. The
intra-layer SOC results in two main bands splitting, while in each main band, the
triple -degeneracy is broken mainly due to the inter -layer SOC. With the pressure, it is
found in double -layer MoS 2 that the double -degeneracy of states in each main band isn ’t
broken when the splitting of both main bands is increased due to the strengthening of LC.
For triple -layer Mo S2 under large pressure, the splitting of triple -degeneracy in each main
band is very obvious.
Computational Method
The present calculations are performed within density functional theory using
accurate frozen -core full -potential projector augmented -wave (P AW) pseudopotentials , as
implemented in the VASP code38-40. The generalized gradient approximation (GGA) with
the parametrization of Perdew -Burke -Ernzerhof (PBE) and with added van der Waals
corrections is used41. The k-space integrals and the plane -wave basis sets are chosen to
ensure that the total energy is converged at the 1 meV/atom level. A kinetic energy cutoff of
500 eV for the plane wave expansion is found to be sufficient . The effect of dispersion
interaction is included by the empirical correction scheme of Grimme (DFT +D/PBE)42.
This approach has been successful in describing layered structures43, 44.
The lattice constants a and c of bulk MoS 2 are 3.191 Å and 12.374 Å which is similar 3 to that from the experiments (3.160 Å and 12.295 Å). For the different layered MoS 2, the
supercells are constructed with a vacuum space of 20 Å along z direction. The Brillouin
zones are sampled with the Γ -centered k -point grid of 18 181. With the state -of-the-art
method of adding the stress to stress tensor in V ASP code39, 40, the structure of bulk MoS 2
is optimized under a specified hydrostatic pressure of 15GPa . With these structural
parameters from bulk MoS 2, the double - and triple -layer MoS 2 structures under the
pressures are constructed. The electronic properties can analyzed with and/or without
spin-orbit coupling to explore the band splitting near band gap. The calculated band gap of
single -layer MoS 2 without the consideration of spin -orbit interaction is 1.66 eV and less
than the experimental report of about 1.8 eV . Obviously , the band gap from PBE is
underestimated as in common in usual density functional calculations . Though the band
gap is underestimated b y PBE, the band structure near Fermi level doesn ’t have obvious
difference from that from other many body method s.
Results and discussion
The structure of single -layer MoS 2 has the hexagonal symmetry with space group
P-6m2. The six sulfur atoms near each Mo atom form a trigonal prismatic structure with
the mirror symmetry in c direction. Obviously, the reversal symmetry is absent, and the
intra-SOC in the band structure becomes to be free. An obvious band splitting on the
valance band maximum around K (K’) point has been observed and is contributed to the
SOC. In addition , the SOC also results in the band splitting on conduction band minimum
around Λ point, while the splitting at VB - and CB -K is not opened. The states of VB-
and CB -K are contribut ed mostly from d z2 orbital of Mo and the effect of the spin-orbit
effect is very weak . At the same time, the states of VB-K and CB -Λ are mainly from d x2
-y2
and d xy orbitals of Mo and the spin -orbit effect on Mo can be revealed in the case of the
absence of reversal symmetry . The band splitting at VB -K (149 meV) is larger than that at
CB-Λ (about 79 meV) . It may be that the distribution of change (or wave function) at
CB-Λ around Mo atoms in xy plane is more localized than that at VB -K30. We also notice
that the charge distribution of spin -up state is much different from that of spin -down state
at VB -K. The spin -down state around Mo is more localized than the spin -up state.
For double -layer MoS 2, the interaction between two layers become s important to the
states near Fermi level. One of much evident effect s is the direct band gap (K-K) of
single -layer becomes to be indirect band gap (-K) due to the uplift of state at VB -. It can
be ascribe d to large band splitting (0.618 eV) at VB -. It is also observed that the band
splitting at the conduction band bottom around is about 0.352 eV . However, without the
consideration of SOC effect , the band splitting at VB -K is just 73.8 meV . The difference of
LC’s strength of different states at VB-, VB -K and CB - is ascribed to the charge
distribution near sulfur atoms. The large contribution of charge on sulfur atoms for the
states at VB - makes the LC to become easier30. The weak LC at VB-K may make the
SOC important. In order to explore the rules of SOC and LC in double -layer MoS 2, we
calculat ed the change of band structures by following the change of distance between both
layers with and without the consideration of spin -orbit effect. As shown in Fig. 1, the band
splitting including that at VB -, VB -K and CB - approach es zero quickly following the
increase of distance, especially that of VB -K, if the spin -orbit effect is not considered. With 4 SOC, the band splitting at VB-K and CB - converge towards some constants (about 149
meV and 79 meV), while the band splitting at VB - approaches zero. Obviously, there is
no spin -orbit effect at VB-. With the large distance between both layers, the effect of LC
can be ignored and the splitting is from SOC.
It has been well-known that the spin -up and spin -down states at VB -K’ are reversed
by compared with that at VB -K in single -layer MoS 2. For double -layer MoS 2, both
splitting bands at VB -K are two -degeneracy . As shown in Fig. 2c, the upper band of both
bands is composed by the spin -up state of first layer and spin -down state of second layer
(~|1, ~|2). The low er band is with the spin -down state of first layer and spin-up state of
second layer (~|1, ~|2). Obviously, the energies of spin-up and spin -down of second
layer at VB -K are reversed by compared with that of first layer. Since there is reversal
symmetry for double -layer system, the energ ies of states |1 and |2 with same energy
cannot be split due to the absence of inter-layer SOC (inter -SOC). Therefore, we can
understand the splitting at VB -K based on the intra -SOC and LC with the theoretical model
shown in Fig. 2a. Because of the splitting of intra -SOC, the energies of |1 and |2 are
very different . This reduces largely the coupling of both states due to layer ’s interaction.
From the band splitting value (166 meV) of double layer at VB -K, the increased splitting
from LC effect is about 17 meV and is much less than that (73.8 meV) from LC without
the consideration of spin -orbit effect. For the spin -down channel , the mechanism of band
splitting is same to that of spin -up channel. Therefore , the contribution of intra -SOC (149
meV) to the band splitting at VB -K is much larger than that of LC. The same mechanism
about the splitting at VB -K (shown in Fig. 2a) can be used for bulk MoS 2. The contribution
of LC is increased to about 59 meV since the band splitting at VB -K is about 0.208 eV . If
no considering the spin -orbit effect, the band splitting due to LC is abou t 145.7 meV which
much similar to the value from SOC(149 meV) in single -layer. This may be the reason that
there is disagreement about the relative strength of both effects in bulk limit. Based on the
model mentioned above and analysis, the intra -SOC effec t is the main mechanism for the
splitting at VB -K in bulk limit .
For triple -layer MoS 2, the band splitting near band gap is complicated , since there are
three states from three layers which are coupling with each other and hybridized with
possible int er-SOC . For the states at VB -, there is no SOC effect and the three degenerate
states will be splitting due to LC. It is found that the two splitting values ( 1 and 2 in
Fig. 3a) which control the relative energy difference of three states after the hybridization
are 0.293 eV and 0.502 eV , respectively . The much different value of both splitting implies
that there is strong coupling between first layer and third layer since both splitting value s
should be equivalent if the nearest -neighbor interaction is just considered for the three
degenerate states . Without the consideration of spin -orbit effect, the splitting values at
CB- (1 and 2) are 0.241 eV and 0.225 eV and that at VB -K (K1 and K2) are 49
meV and 55 meV , respectively . Based on the nearest -neighbor LC strength (73.8/2 meV) at
VB-K from double layer, the LC strength between first-layer and third layer of triple layer
is about 2 meV at VB -K and may be ignored. Therefore, we propose a coupling model
based on the in tra-SOC a nd nearest -neighbor LC, as shown in Fig. 3c. With this model, the
spin-up and spin -down bands of each layer are splitted by the intra -SOC. Then the LC will
perturb these states for each spin channel . For example , the spin -up states are composed 5 with two degenerate upper states (|1 , |3) and one lower state (|2 ) in Fig. 3c and LC
will result in the splitting of two degenerate upper states with the increase of energy gap
between |2 and |3. With the spin -up and spin -down channels together after LC, there
should two main bands and each main band is composed with two degenerate states and
one single states. The LC doesn't change the energy gap between the main bands. It is
found that the energy gap ’SOC is 148.7 meV and similar to the splitting from i ntra-SOC
(SOC =149 meV). However, it is interesting that the two degenerate states in each main
band, such as the upper states ~|1 and ~|2 and the lower states ~|2 and ~|3, are
splitted, as shown in the inset of Fig. 3c. In addition , it is found that the splitting values are
so large (such as, 11.3 meV between ~|1 and ~|2) that the contribution of LC between
first layer and third layer is not enough. We propose that the splitting of degenerate states
in each main band is from the inter-SOC.
While the small pressure doesn ’t induce the obvious splitting from intra -SOC in
single -layer MoS 2, it is possible there is strong effect to the splitting from inter -SOC in
triple -layer MoS 2. For double -layer MoS 2, the inter -SOC is forbidden and the de generate
state in each main band isn ’t opened and the energy gap between both main bands is
increased with the strengthening of LC under the pressure in Fig. 4a. In triple -layer, the
strengthening of LC under pressure should have the obvious effect, such as the increase of
energy gap between ~|1 and ~|3 in Fig. 4c. Besides the enhanced LC effect, an
apparent observation is the energy gap between main bands ’SOC has been decreased to
135.8 meV under 15 GPa in Fig. 4b. This should be the typical evidence for the inter -SOC.
Conclusions
We study the band splitting at valance band maximum of multi -layer MoS 2 by first
principle methods in details. We propose a model based on the int ra-layer spin -orbit
coupling to solve t he valance band splitting at K point of multi -layer MoS 2 and bulk MoS 2
with the perturbation of layer ’s coupling and inter -layer spin -orbit coupling. It is also found
that the direct interaction between second near -neighbor layers is weak at VB -K. While the
inter-layer spin -orbit coupling is forbidden in double -layer MoS 2, this effect appear s in
triple -layer MoS 2. Especially, under the pressure, the inter -layer spin -orbit coupling is
raised with the decrease of energy gap between main bands from intr a-layer spin -orbital
coupling.
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9 Fig. 1.
Fig.1 Band structure of double -layer MoS 2 calculated without spin -orbit coupling (a) and with spin -orbit
coupling, and the changes of conduction band splitting ( ) at point and valance band splitting at
point () and K point (K) following the distance between two layers of double -layer MoS 2, calculated
without spin -orbit coupling (c) and with spin -orbit coupling (d). Note the red circle s in Fig. 1c and d
represent the data from the equilibrium (or stable) state (ES) and the dot and dash dot lines presen t the
conduction band splitting ( ) at point and valance band splitting ( K) at K point of single -layer
MoS 2, respectively.
10 Fig.2
Fig. 2 Schematic of valance band splitting of valance band maximum at K point due to the spin -orbit
coupling in the each layer (intra -SOC) and layer ’s coupling (LC) in band structure of double -layer MoS 2
(a), schematic structure of double -layer MoS 2 (b), the isosurface of band -decomposed charge density of
four states at valance band maximum of K point including the states ~|1 , ~|2, ~|1 and ~|2
shown in Fig. 2a after considering the effects of intra -SOC and LC.
11 Fig. 3
Fig. 3 Band structure of tri ple-layer MoS 2 calculated without spin -orbit coupling (a) and with spin -orbit
coupling (b), S chematic of valance band splitting of valance band maximum at K point due to the
spin-orbit (SOC) and layer ’s coupling (LC) in band structure of triple -layer MoS 2 (c). Note that in the
inset of Fig. 3c, the band structure is plotted with two directions K and KM and the lengths for
K and KM are the 1/10 of total lengths in the two directions, respectively.
12 Fig. 4
Fig. 4 Band structure s of double -layer MoS 2 (a) and triple -layer MoS 2 (b) under the pressure of 15 GPa
calculated with spin -orbit coupling, and valance bands of triple -layer MoS 2 under 15 GPa near K Point
and the related s chematic about band splitting due to the spin -orbit (SOC) and layer ’s couplin g (LC).
|
0805.1028v1.Frustration_and_entanglement_in_the__t__2g___spin__orbital_model_on_a_triangular_lattice__valence__bond_and_generalized_liquid_states.pdf | arXiv:0805.1028v1 [cond-mat.str-el] 7 May 2008Frustration and entanglement in the t2gspin–orbital model on a triangular lattice:
valence–bond and generalized liquid states
Bruce Normand
D´ epartement de Physique, Universit´ e de Fribourg, CH–170 0 Fribourg, Switzerland
Theoretische Physik, ETH–H¨ onggerberg, CH–8093 Z¨ urich, Switzerland
Andrzej M. Ole´ s
Marian Smoluchowski Institute of Physics, Jagellonian Uni versity, Reymonta 4, PL–30059 Krak´ ow, Poland
Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany
(Dated: November 1, 2018)
We consider the spin–orbital model for a magnetic system wit h singly occupied but triply degen-
eratet2gorbitals coupled into a planar, triangular lattice, as woul d be exemplified by NaTiO 2. We
investigate the ground states of the model for interactions which interpolate between the limits of
pure superexchange and purely direct exchange interaction s. By considering ordered and dimerized
states at the mean–field level, and by interpreting the resul ts from exact diagonalization calculations
on selected finite systems, we demonstrate that orbital inte ractions are always frustrated, and that
orbital correlations are dictated by the spin state, manife sting an intrinsic entanglement of these
degrees of freedom. In the absence of Hund coupling, the grou nd state changes from a highly res-
onating, dimer–based, symmetry–restored spin and orbital liquid phase, to one based on completely
static, spin–singlet valence bonds. The generic propertie s of frustration and entanglement survive
even when spins and orbitals are nominally decoupled in the f erromagnetic phases stabilized by a
strong Hund coupling. By considering the same model on other lattices, we discuss the extent to
which frustration is attributable separately to geometry a nd to interaction effects.
PACS numbers: 71.10.Fd, 74.25.Ha, 74.72.-h, 75.30.Et
I. INTRODUCTION
Frustration in magnetic systems may be of geometri-
cal origin, or may arise due to competing exchange in-
teractions, or indeed both.1For quantum spins, frustra-
tion acts to enhance the effects of quantum fluctuations,
leading to a number of different types of magnetically
disordered state, among which some of the more familiar
are static and resonating valence–bond (VB) phases. A
further form of solution in systems with frustrated spin
interactionsis the emergenceof novelorderedstates from
ahighlydegeneratemanifoldofdisorderedstates,andthe
mechanism for their stabilizationhas become known sim-
ply as “order–by–disorder”.1,2Many materials are now
known whose physical properties could be understood
only by employing microscopic models with frustrated
spin interactions in which some of these theoretical con-
cepts operate.
A different and still richer situation occurs in the class
of transition–metal oxides or fluorides with partly filled
3dorbitals and near–degeneracy of active orbital degrees
of freedom. In undoped systems, large Coulomb interac-
tions on the transition–metal ions localize the electrons,
and the low–energy physics is that of a Mott (or charge–
transfer3) insulator. Their magnetic properties are de-
scribed by superexchange spin–orbital models, derived
directly from the real electronic structure and contain-
ing linearly independent but strongly coupled spin and
orbital operators.4Such models emerge from the charge
excitations which involve various multiplet states,5,6in
which ferromagnetic (FM) and antiferromagnetic (AF)interactions, as well the tendencies towards ferro–orbital
(FO) and alternating orbital (AO) order, compete with
eachother. This leadsto a profound, intrinsic frustration
of spin–orbital exchange interactions, which occurs even
in case of only nearest–neighbor interactions for lattices
with unfrustrated geometry, such as the square and cu-
bic lattices.7The underlying physics is formulated in the
Goodenough–Kanamorirules,8which imply that the two
types of order are complementary in typical situations:
AO order favorsa FM state while FO order coexists with
AF spin order. Only recently have exceptions to these
rules been noticed,9and the search for such exceptions,
and thus for more complex types of spin–orbital order or
disorder, have become the topic of much active research.
A case study for frustration in coupled spin–orbital
systems is provided by the one–dimensional (1D) SU(4)
model.10One expects a priori no frustration in one di-
mension and with only nearest–neighbor interactions.
However, spin and orbital interactions, the latter for-
mulated in terms of pseudospin operators, appear on a
completelysymmetricalfooting foreverybond, and favor
respectively AF and AO ordering tendencies, which com-
pete with each other. In fact a low–energy but magnet-
ically disordered spin state also frustrates the analogous
pseudospin–disordered state, and conversely. This com-
petition results in strong, combined spin–orbital quan-
tum fluctuations which make it impossible to separate
the two subsystems, and it is necessary to treat explicitly
entangled spin–pseudospin states.9,11While in one sense
this may be considered as a textbook example of frustra-
tion and entanglement, the symmetry of the entangled2
sectorsissohighthatjointspin–pseudospinoperatorsare
asfundamentalasthe separatespinandpseudospinoper-
ators, forming parts of a larger group of elementary (and
disentangled) generators. The fact that the 1D SU(4)
modelisexactlysolvablealsoresultsinfundamentalsym-
metriesbetween the intersitecorrelationfunctions forthe
spin and orbital (and spin–orbital) sectors.12We return
below to a more detailed discussion of entanglement and
its consequences. Although indicative of the rich under-
lying physics (indeed, unconventional behavior has been
identified for the SU(4) Hamiltonian on the triangular
lattice,10,13)theimplicationsofthismodelareratherlim-
ited because it does not correspond to the structure of
superexchange interactions in real correlated materials.
Realistic superexchange models for perovskite
transition–metal oxides with orbital degrees of freedom
have been known for more than three decades,5,6but
the intrinsic frustrating effects of spin–orbital interac-
tions have been investigated only in recent years.7,14
A primary reason for this delay was the complexity
of the models and the related quantum phenomena,
which require advanced theoretical methods beyond a
straightforward mean–field theory. The structure of
spin–orbital superexchangeinvolves interactionsbetween
SU(2)–symmetric spins {/vectorSi,/vectorSj}on two nearest–neighbor
transition–metal ions {i,j}, each coupled to orbital op-
erators{/vectorTi,/vectorTj}which obey only much lower symmetry
(at most cubic for a cubic lattice), and its general form
is4
HJ=J/summationdisplay
/angbracketleftij/angbracketright/bardblγ/braceleftBig
ˆJ(γ)
ij/parenleftBig
/vectorSi·/vectorSj/parenrightBig
+ˆK(γ)
ij/bracerightBig
.(1.1)
The energy scale Jis determined (Sec. II) by the in-
teraction terms and effective hopping matrix elements
between pairs of directional egorbitals [(ddσ) element]
ort2gorbitals [(ddπ) element] The orbital operators ˆJ(γ)
ij
andˆK(γ)
ijspecifytheorbitalsoneachbond /an}bracketle{tij/an}bracketri}ht /bardblγ,which
participate in dn
idn
j⇀↽dn+1
idn−1
jvirtual excitations, and
thus have the symmetry of the lattice. The form of the
orbital operators depends on the valence n, on the type
(egort2g) of the orbitals and, crucially, on the bond di-
rectionin realspace.15It is clearfrom Eq.(1.1) that indi-
vidualtermsintheHamiltonian HJcanbeminimized for
particularly chosen spin and orbital configurations,4but
in general the structure of the orbital operators ensures
a competition between the different bonds.
This directional nature is the microscopic origin of the
intrinsic frustration mentioned above, which is present
even in the absence of geometrical frustration. Both
spin andorbitalinteractionsarefrustrated, makinglong–
range order more difficult to realize in either sector, and
enhancing the effects of quantum fluctuations. Quite
generally, because insufficient (potential) energy is avail-
able from spin or orbital order, instead the system is
driven to gain (kinetic) energy from resonance processes,
promoting phases with short–range dynamical correla-
tions and leading naturally to spin and/or orbital dis-order. Disordering tendencies are particularly strong in
highlysymmetricsystems, whichforcrystallinematerials
means cubic and hexagonal structures. Among possible
magnetically disordered phases for spin systems, tenden-
cies towards dimer formation are common in the regime
of predominantly AF spin interactions, and new phases
with VB correlations occur. This type of physics was
discussed first for egorbitals on the cubic lattice,7and,
in the context of BaVS 3, for one version of the problem
oft2gorbitals on a triangular lattice.16The same generic
behaviorhassincebeenfoundfor t2gorbitalsonthecubic
lattice,17eg–orbitalsystemsonthetriangularlattice,18,19
and fort2gorbitals in the pyrochlore geometry.20,21By
analogy with spin liquids, the orbital–liquid phase1has
been introduced for systems with both eg7,22andt2g14,23
orbital degrees of freedom. The orbital liquid is a phase
in which strongorbital fluctuations restorethe symmetry
of the orbital sector, in the sense that the instantaneous
orbital state of any site is pure, but the time average
is a uniform occupation of all available orbital states.
We note that in the discussion of orbital liquids in t2g
systems,14,23it was argued that the spin sector would be
ordered. To date little is known concerning the behavior
oforbitalcorrelationsin an orbitalliquid, the possible in-
stabilities of the orbital liquid towards dimerized or VB
phases, or its interplay with lattice degrees of freedom.
Onepossiblemechanismforthe formationofanorbital
liquid state is the positional resonanceof VBs. There has
been considerable recent discussion of spin–orbital mod-
els in the continuing search for a realistic system real-
izing such a resonating VB (RVB) state,19including in
a number of the references cited in the previous para-
graph. While the RVB state was first proposed for the
S=1
2Heisenberg model on a triangular lattice,24exten-
sive analysis of spin–only models has not yet revealed
a convincing candidate system, although the nearest–
neighbor dimer basis has been shown to deliver a very
good description of the low–energy sector for the S=1
2
Heisenberg model on a kagome lattice.25To date, the
only rigorous proof for RVB states has been obtained in
rather idealized quantum dimer models (QDMs),26most
notably on the triangular lattice.27The insight gained
from this type of study can, however, be used19to for-
mulate some qualitative criteria for the emergence of an
RVB ground state. These combine energetic and topo-
logical requirements, both of which are essential: the
energetics of the system must establish a proclivity for
dimer formation, a high quasi–degeneracy of basis states
in the candidate ground manifold, and additional energy
gains from dimer resonance; exact degeneracy between
topological sectors (determined by a non–local order pa-
rameter related to winding of wave functions around the
system) is a prerequisite to remove the competing possi-
bility of a “solid” phase with dimer, plaquette or other
“crystalline” order.28
We comment here that the “problem” of frustration,
and the resulting highly degenerate manifolds of states
whichmaypromoteresonancephenomena,isoftensolved3
Na-ionO-ion
Ti-ionYZ
XYZX
(a) (b)
FIG. 1: (Color online) Structure of the transition–metal ox ide
with edge–sharing octahedra realized for NaTiO 2: (a) frag-
ment of crystal structure, with Ti and Na ions shown respec-
tively by black and green (grey) circles separated by O ions
(open circles); (b) titanium /an}bracketle{t111/an}bracketri}htplane with adjacent oxy-
gen layers, showing each Ti3+ion coordinated by six oxygen
atoms (open circles). The directions of the Ti–Ti bonds are
labeled as XY,YZ, andZX, corresponding to the plane
spanned by the connecting Ti–O bonds. This figure is repro-
duced from Ref. 33, where it served to explain the structure
of LiNiO 2.
by interactions with the lattice. Lattice deformations act
to lift degeneracies and to stabilize particular patterns
of spin and orbital order, the most familiar situation
being that in colossal–magnetoresistance manganites.29
The samephysicsis alsodominantin anumberofspinels,
where electron–lattice interactions are responsible both
for the Verwey transition in magnetite30and fort2gor-
bital order below it, as well as for inducing the Peierls
state in CuIr 2S4and MgTi 2O4.31Similar phenomena are
also expected31to play a role in NaTiO 2. Here, however,
we will not introduce a coupling to phonon degrees of
freedom, and focus only on purely electronic interactions
whose frustration is not quenched by the lattice.
The spin–orbital interactions on a triangular lattice
are particularly intriguing. This lattice occurs for edge–
sharing MO 6octahedra in structures such as NaNiO 2or
LiNiO 2, where the consecutive /an}bracketle{t111/an}bracketri}htplanes of Ni3+ions
are well separated. These two eg–electron systems be-
havequitedifferently: while NaNiO 2undergoesacooper-
ativeJahn–Tellerstructuraltransitionfollowedbyamag-
netic transition at low temperatures ( TN= 20 K), both
transitions are absent in LiNiO 2.32Possible reasons for
this remarkable contrast were discussed in Ref. 33, where
the authors noted in particular that realistic spin–orbital
superexchange neither has an SU(2) ⊗SU(2) structure,18
nor can it ever be reduced only to the consideration of
FM spin terms.34These studies showed in addition that
LiNiO 2is not a spin–orbital liquid, and that the rea-
sons for the observed disordered state are subtle, as spins
and orbitals are thought likely to order in a strictly two–
dimensional (2D) spin–orbital model.33
Thepossibilitiesofferedforexoticphasesinthistypeof
model and geometry motivate the investigation of a real-
istic spin–orbital model with active t2gorbitals, focusing
first on 3d1electronic configurations. The threefold de-generacyofthe orbitalsismaintained, although, asnoted
above, this condition may be hard to maintain in real
materials at low temperatures. A material which should
exemplify this system is NaTiO 2(Fig. 1), which is com-
posed of Ti3+ions int1
2gconfiguration, but has to date
had rather limited experimental35,36and theoretical37
attention. Considerably more familiar is the set of tri-
angular cobaltates best known for superconductivity in
NaxCoO2: here the Co4+ions havet5
2gconfiguration and
are expected to be analogous to the d1case by particle–
hole symmetry. The effects of doping have recently been
removed by the synthesis of the insulating end–member
CoO2.38Another system for which the same spin–orbital
model could be applied is Sr 2VO4, where the V4+ions
occupy the sites of a square lattice.39
The model with hopping processes of pure superex-
changetypewasconsideredinthecontextofdopedcobal-
tates by Koshibae and Maekawa.40These authors noted
that, like the cubic system, two t2gorbitals are active
for each bond direction in the triangular lattice, but that
thesuperexchangeinteractionsareverydifferentfromthe
cubic case because the effective hopping interchanges the
activeorbitals. Here we focus onlyon insulating systems,
whose entire low–energy physics is described by a spin–
orbital model. In addition to superexchange processes
mediated by the oxygen ions, on the triangular lattice
it is possible to have direct–exchange interactions, which
result from charge excitations due to direct d−dhop-
ping between those t2gorbitals which do not participate
in the superexchange. The ratio of these two types of
interaction ( α, defined in Sec. II) is a key parameter of
the model. Further, in transition–metal ions4the coef-
ficients of the different microscopic processes depend on
the Hund exchange JHarising from the multiplet struc-
ture of the excited intermediate d2state,41and we intro-
duce
η=JH
U, (1.2)
as the second parameter of the model. The aim of this
investigation is to establish the general properties of the
phase diagram in the ( α,η) plane.
We conclude our introductory remarks by returning to
the question of entanglement. In the analysis to follow
wewillshowthat thepresenceofconflictingorderingten-
dencies driven by different components of the frustrated
intersite interactions can be related to the entanglement
of spin and orbital interactions. By “entanglement” we
mean that the correlations in the ground state involve
simultaneous fluctuations of the spin and orbital com-
ponents of the wave function which cannot be factor-
ized. We will introduce an intersite spin–orbital corre-
lation function to identify and quantify this type of en-
tanglement in different regimes of the phase diagram. It
has been shown9that such spin–orbital entanglement is
present in cubic titanates or vanadates for small values
of the Hund exchange η. Here we will find entanglement
to be a generic feature of the model for all exchange in-4
teractions, even in the absence of dimer resonance, and
that only the FM regime at sufficiently high η, which
is fully factorizable, provides a counterpoint where the
entanglement vanishes.
The paper is organized as follows. In Sec. II we derive
the spin–orbitalmodel for magneticions with the d1elec-
tronicconfiguration(Ti3+orV4+)onatriangularlattice.
The derivation proceeds from the degenerate Hubbard
model, and the resulting Hamiltonian contains both su-
perexchange and direct exchange interactions. We begin
our analysis of the model, which covers the full range
of physical parameters, in Sec. III by considering pat-
terns of long–ranged spin and orbital order representa-
tive of all competitive possibilities. These states compete
with magnetically or orbitally disordered phases domi-
nated by VB correlations on the bonds, which are in-
vestigated in Sec. IV. The analysis suggests strongly
that all long–range order is indeed destabilized by quan-
tum fluctuations, leading over much of the phase dia-
gram to liquid phases based on fluctuating dimers, with
spin correlations of only the shortest range. In Sec. V
we present the results of exact diagonalization calcula-
tions performed for small clusters with three, four, and
six bonds, which reinforce these conclusions and provide
detailed information about the local physical processes
leading to the dominance of resonating dimer phases. In
each of Secs. III, IV, and V, we conclude with a short
summary of the primary results, and the reader who is
more interested in an overview, rather than in detailed
energetic comparisons and actual correlation functions
for the different phases, may wish to read only these.
Some insight into the competition and collaboration be-
tween frustration effects of different origin can be ob-
tainedbyvaryingthegeometryofthesystem,andSec.VI
discusses the properties of the model on related lattices.
A discussion and concluding summary are presented in
Sec. VII.
II. SPIN–ORBITAL MODEL
A. Hubbard model for t2gelectrons
We consider the spin–orbital model on the triangular
lattice which follows from the degenerate Hubbard–like
model fort2gelectrons. It contains the electron kinetic
energy and electronic interactions for transition–metal
ions arranged on the /an}bracketle{t111/an}bracketri}htplanes of a compound with
localcubic symmetryandwith the d1ionicconfiguration,
and as such is applicable to Ti3+or V4+[Fig. 1(a)]. The
kinetic energy is given by
Ht=−/summationdisplay
/angbracketleftij/angbracketright/bardblγ,µν,σt(γ)
µν/parenleftBig
d†
iµσdjνσ+d†
jνσdiµσ/parenrightBig
,(2.1)
whered†
iµσare creation operators for an electron with
spinσ=↑,↓andorbital“color” µat sitei, and the sum is
/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0
/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
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/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1
(a)
c2
(c)3
23
1 1(bc) (ca)
(ab)ba
(b)
FIG. 2: (Color online) (a) Schematic representation of the
hopping processes in Eq. (2.1) which contribute to magnetic
interactions on a representative bond /an}bracketle{tij/an}bracketri}htalong the c–axis
in the triangular lattice. The t2gorbitals are represented by
different colors (greyscale intensities). Superexchange p ro-
cesses involve O 2 pzorbitals (violet), and couple pairs of a
andborbitals (red, green) with effective hopping elements t,
interchanging their orbital color. Direct exchange couple sc
orbitals (blue) with hopping strength t′. (b) Pairs of t2gor-
bitals active in superexchange and (c) single orbitals acti ve in
direct exchange; horizontal bonds correspond to the situat ion
depicted in panel (a).
made over all the bonds /an}bracketle{tij/an}bracketri}ht/bardblγspanning the three direc-
tions,γ=a,b,c, of the triangular lattice. This notation
isadoptedfromthesituationencounteredinacubicarray
of magnetic ions, where only two of the three t2gorbitals
areactiveonanyonebond /an}bracketle{tij/an}bracketri}ht, andcontribute t(γ)
µνtothe
kineticenergy, whilethe third liesinthe planeperpendic-
ular to the γaxis and thus hopping processes involving
the 2pπoxygen orbitals is forbidden by symmetry.42,43
We introduce the labels a≡yz,b≡xz, andc≡xyalso
for the three orbital colors, and in the figures to follow
their respective spectral colors will be red, green, and
blue.
For the triangular lattice formed by the ions on the
/an}bracketle{t111/an}bracketri}htplanes of transition–metal oxides (Fig. 1) it is also
thecasethatonlytwo t2gorbitalsparticipatein(superex-
change)hoppingprocessesviatheoxygensites. However,
unlike the cubic lattice, where the orbital color is con-
served, here any one active orbital color is exchanged for
the other one [Fig. 2(a)]. Using the same convention,
that each direction in the triangular lattice is labeled by
its inactive orbital color44γ=a,b,c, the hopping ele-5
ments for a bond oriented (for example) along the c–axis
in Eq. (2.1) are t(c)
ab=t(c)
ba=t, whilet(c)
aa=t(c)
bb= 0. In
addition, and also in contrastto the cubic system, for the
triangular geometry a direct hopping from one corbital
tothe other, i.e.withoutinvolvingtheoxygenorbitals, is
also permitted on this bond (Fig. 2), and this element is
denoted by t′=t(c)
cc. We will also refer to these hopping
processes as off–diagonal and diagonal. We stress that
while the lattice structure of magnetic ions is triangular,
the system under consideration retains local cubic sym-
metry of the metal–oxygen octahedra, which is crucial
to ensure that the degeneracy of the three t2gorbitals is
preserved.
The electron–electroninteractionsaredescribedby the
on–site terms45
Hint=U/summationdisplay
iµniµ↑niµ↓+/parenleftbigg
U−5
2JH/parenrightbigg/summationdisplay
i,µ<ν,σσ′niµσniνσ′
−2JH/summationdisplay
i,µ<ν/vectorSiµ·/vectorSiν+JH/summationdisplay
i,µ/negationslash=νd†
iµ↑d†
iµ↓diν↓diν↑,(2.2)
whereUandJHrepresent respectively the intraorbital
Coulomb and on–site Hund exchange interactions. Each
pair of orbitals {µ,ν}is included only once in the inter-
action terms. The Hamiltonian (2.2) describes rigorously
the multiplet structureof d2ions within the t2gsubspace,
and is rotationally invariant in the orbital space.45
When the Coulomb interaction is large compared with
the hopping elements ( U≫t,t′), the system is a Mott
insulator with one delectron per site in the t2gorbitals,
whence the local constraint in the strongly correlated
regime is
nia+nib+nic= 1, (2.3)
whereniγ=niγ↑+niγ↓. The operators act in the re-
stricted space niγ= 0,1. The low–energy Hamiltonian
may be obtained by second–order perturbation theory,
andconsistsofasuperpositionoftermswhichfollowfrom
virtuald1
id1
j⇀↽d2
id0
jexcitations. Because each hopping
process may be of either off–diagonal ( t) [Fig. 2(b)] or
diagonal (t′) type [Fig. 2(c)], the Hamiltonian consists
of several contributions which are proportional to three
coupling constants,
Js=4t2
U, Jd=4t′2
U, Jm=4tt′
U.(2.4)
These represent in turn the superexchange term, the di-
rect exchange term, and mixed interactions which arise
from one diagonal and one off–diagonal hopping process.
We chooseto parameterizethe Hamiltonian by the sin-
gle variable
α= sin2θ, (2.5)
with
tanθ=t′
t, (2.6)which gives Js=Jcos2θ,Jm=Jsinθcosθ, andJd=
Jsin2θ;Jis the energy unit, which specifies respectively
the superexchange ( J=Js) and direct–exchange ( J=
Jd) constants in the two limits α= 0 andα= 1. The
Hamiltonian
H=J/braceleftBig
(1−α)Hs+/radicalbig
(1−α)αHm+αHd/bracerightBig
(2.7)
consists of three terms which follow from the processes
described by the exchange elements in Eqs. (2.4), each of
which contains contributions from both high– and low–
spin excitations.
B. Superexchange
Superexchange contributions to Hcan be expressed in
the form
Hs=1
2/summationdisplay
/angbracketleftij/angbracketright/bardblγ/braceleftBig
r1/parenleftBig
/vectorSi·/vectorSj+3
4/parenrightBig/bracketleftBig
A(γ)
ij+1
2(niγ+njγ)−1/bracketrightBig
+r2/parenleftBig
/vectorSi·/vectorSj−1
4/parenrightBig/bracketleftBig
A(γ)
ij−1
2(niγ+njγ)+1/bracketrightBig
−2
3(r2−r3)/parenleftBig
/vectorSi·/vectorSj−1
4/parenrightBig
B(γ)
ij/bracerightBig
, (2.8)
where one recognizes a structure similar to that for su-
perexchange in cubic vanadates,4,14with separation into
aspinprojectionoperatoronthetripletstate,( /vectorSi·/vectorSj+3
4),
and an operator ( /vectorSi·/vectorSj−1
4) which is finite only for
low–spin excitations. These operators are accompanied
by coefficients ( r1,r2,r3) which depend on the Hund ex-
change parameter (1.2), and are given from the multiplet
structure of d2ions41by
r1=1
1−3η, r2=1
1−η, r3=1
1+2η.(2.9)
TheCoulombandHundexchangeelementsdeducedfrom
the spectroscopic data of Zaanen and Sawatzky46are
U= 4.35 eV and JH= 0.59 eV, giving a realistic value
ofη≃0.136 for Ti2+ions. For V2+one finds46U= 4.98
eV andJH= 0.64 eV, whence η≃0.13, and the values
for V3+ions are expected to be very similar. Finally,
for Co3+ions,47U= 6.4 eV andJH= 0.84 eV, giving
againη≃0.13. The value η= 0.13 therefore appears to
be quite representative for transition–metal oxides with
partly filled t2gorbitals, whereas somewhat larger values
have been found for systems with active egorbitals due
to a stronger Hund exchange.4
The orbital operators AijandBijin Eq. (2.8) depend
on the bond direction γand involve two active orbital
colors,
A(γ)
ij=/parenleftBig
T+
iγT+
jγ+T−
iγT−
jγ/parenrightBig
−2Tz
iγTz
jγ+1
2n(γ)
in(γ)
j,(2.10)
B(γ)
ij=/parenleftBig
T+
iγT−
jγ+T−
iγT+
jγ/parenrightBig
−2Tz
iγTz
jγ+1
2n(γ)
in(γ)
j.(2.11)6
For illustration, in the case γ=c(/an}bracketle{tij/an}bracketri}ht /bardblc), the orbitals
aandbat siteiare interchanged (off–diagonal hopping)
at sitej, and the electron number operator is n(γ)
i=
nia+nib. The quantity niγin Eq. (2.8) is the number
operator for electrons on the site in orbitals inactive for
hopping on bond γ,niγ= 1−n(γ)
i, ornicin this example.
For a single bond, the orbital operators in Eq. (2.10)
may be written in a very suggestive form by performing
a local transformation in which the active orbitals are
exchanged on one bond site, specifically |a/an}bracketri}ht → |b/an}bracketri}htand
|b/an}bracketri}ht → |a/an}bracketri}hton bondγ=c.40Then
A(γ)
ij= 2/parenleftBig
/vectorTiγ·/vectorTjγ+1
4n(γ)
in(γ)
j/parenrightBig
, (2.12)
B(γ)
ij= 2/parenleftBig
/vectorTiγ×/vectorTjγ+1
4n(γ)
in(γ)
j/parenrightBig
,(2.13)
where the scalar product in Aijis the conventional
expression for pseudospin–1/2 variables, and the cross
product in Bijis defined as
/vectorTiγ×/vectorTjγ=1
2(T+
iγT+
jγ+T−
iγT−
jγ)+Tz
iγTz
jγ.(2.14)
Equations (2.8) and (2.12) make it clear that for a sin-
gle superexchange bond, the minimal energy is obtained
either by forming an orbital singlet, in which case the op-
timal spin state is a triplet, or by forming a spin singlet,
in which case the preferred orbital state is a triplet; we
refer to these bond wavefunctions respectively as (os/st)
and (ss/ot). The two states are degenerate for η= 0,
while for finite Hund exchange
E(os/st)=−Jr1, (2.15)
E(ss/ot)=−1
3J(2r2+r3), (2.16)
and the (os/st) state is favored. This propensity for sin-
glet formation in the α= 0 limit will drive much of the
physics to be analyzed in what follows.
Becauseoftheoff–diagonalnatureofthehoppingterm,
in the original electronic basis (before the local transfor-
mation) the orbital singlet is the state
|ψos/an}bracketri}ht=1√
2(|aa/an}bracketri}ht−|bb/an}bracketri}ht), (2.17)
while the orbital triplet states are
|ψot+/an}bracketri}ht=|ab/an}bracketri}ht, (2.18)
|ψot0/an}bracketri}ht=1√
2(|aa/an}bracketri}ht+|bb/an}bracketri}ht), (2.19)
|ψot−/an}bracketri}ht=|ba/an}bracketri}ht. (2.20)
The locally transformed basis then gives a clear analogy
which can be used for single bonds and dimer phases in
combination with all of the understanding gained for the
Heisenberg model. However, we stress here that the local
transformationfailsforsystemswithmorethan1bondinthe absence of static dimer formation. This arises due to
frustration, and can be shown explicitly in numerical cal-
culations, but we will not enter into this point in more
detail here. However, we take the liberty of retaining
the notation of the local transformation, particularly in
Sec. IV when considering dimers. Because the transfor-
mation interchanges the definitions of FO and AO con-
figurations, we will state clearly in each section the basis
in which the notation is chosen.
C. Direct Exchange
The direct exchange part is obtained by considering
virtual excitations of active γorbitals on a bond /an}bracketle{tij/an}bracketri}ht /bardblγ,
which yield
Hd=1
4/summationdisplay
/angbracketleftij/angbracketright/bardblγ/braceleftBig/bracketleftBig
−r1/parenleftBig
/vectorSi·/vectorSj+3
4/parenrightBig
+r2/parenleftBig
/vectorSi·/vectorSj−1
4/parenrightBig/bracketrightBig
×/bracketleftBig
niγ(1−njγ)+(1−niγ)njγ/bracketrightBig
+1
3(2r2+r3)/parenleftBig
/vectorSi·/vectorSj−1
4/parenrightBig
4niγnjγ/bracerightBig
.(2.21)
Here there are no orbital operators, but only number
operators which select electrons of color γon bonds ori-
entedalongthe γ–axis. Whenonlyonlyoneactiveorbital
is occupied [ niγ(1−njγ)], this electron can gain energy
−1
4Jfrom virtual hopping at η= 0, a number which
has only a weak dependence on the bond spin state at
η>0. When both active orbitals are occupied ( niγnjγ),
placing the two electrons in a spin singlet yields the far
lower bond energy −J, and thus again one may expect
muchofthe discussionto followtocenterondimer–based
states of the extended system. Again the triplet d2spin
excitation corresponds to the lowest energy, ( U−3JH),
and only the lower two excitations involve spin singlets
which could minimize the bond energy. The structure
of these terms is the same as in the 1D egspin–orbital
model,48or the case of the spinel MgTi 2O4.20A simpli-
fied model for the triangular–lattice model in this limit,
using a lowest–order expansion in ηfor the spin but not
for orbital interactions, was introduced in Ref. 49.
D. Mixed Exchange
Finally, the twodifferent types ofhopping channelmay
also contribute to two–step, virtual d1
id1
j⇀↽d2
id0
jexci-
tations with one off–diagonal ( t) and one diagonal ( t′)
process. The occupied orbitals are changed at both sites
(Fig. 2), and as for the superexchange term the result-
ing effective interaction may be expressed in terms of or-
bital fluctuation operators. To avoid a more general but
complicated notation, we write this term only for c–axis
bonds,
H(c)
m=−1
4/summationdisplay
/angbracketleftij/angbracketright/bardblc/bracketleftBig
r1/parenleftBig
/vectorSi·/vectorSj+3
4/parenrightBig
−r2/parenleftBig
/vectorSi·/vectorSj−1
4/parenrightBig/bracketrightBig7
×/parenleftBig
T+
iaT+
jb+T−
ibT−
ja+T+
ibT+
ja+T−
iaT−
jb/parenrightBig
,(2.22)
where the orbital operators are
T+
ia=b†
ici, T+
ib=c†
iai,
T−
ia=c†
ibi, T−
ib=a†
ici. (2.23)
These definitions are selected to correspond to the ↑–
pseudospin components of both operators being |bi/an}bracketri}htfor
Tz
iaand|ci/an}bracketri}htforTz
ib. The form of the H(a)
mandH(b)
mterms
is obtained from Eq. (2.22) by a cyclic permutation of
the orbital indices. By inspection, this type of term is
finite only for bonds whose sites are occupied by linearsuperpositions of different orbital colors, and creates no
strong preference for the spin configuration at small η.
E. Limit of vanishing Hund exchange
In the subsequent sections we will give extensive con-
sideration to the model of Eq. (2.7) at η= 0. In this
special case the multiplet structure collapses (spin sin-
glet and triplet excitations are degenerate), one finds a
singlechargeexcitationofenergy U, andtheHamiltonian
reduces to the form
H(η= 0) =J/summationdisplay
/angbracketleftij/angbracketright/bardblγ/braceleftBig
(1−α)/bracketleftBig
2/parenleftBig
/vectorSi·/vectorSj+1
4/parenrightBig/parenleftBig
/vectorTiγ·/vectorTjγ+1
4n(γ)
in(γ)
j/parenrightBig
+1
2(niγ+njγ)−1/bracketrightBig
+α/bracketleftBig/parenleftBig
/vectorSi·/vectorSj−1
4/parenrightBig
niγnjγ−1
4/parenleftBig
niγ(1−njγ)+(1−niγ)njγ/parenrightBig/bracketrightBig
−1
4/radicalbig
α(1−α)/parenleftBig
T+
i¯γT+
j˜γ+T−
i˜γT−
j¯γ+T+
i˜γT+
j¯γ+T−
i¯γT−
j˜γ/parenrightBig/bracerightBig
, (2.24)
which depends only on the ratio of superexchange to di-
rect exchange (0 ≤α≤1). The first line of Eq. (2.24)
makes explicit the fact that the spin and orbital sectors
are completely equivalent and symmetrical at α= 0, at
least at the level of a single bond. However, we will
showthatthisequivalenceisbrokenwhenmorebondsare
considered, and no higher symmetry emerges because of
the color changes involved for different bond directions,
which change the SU(2) orbital subsector. The second
line of Eq. (2.24) emphasizes the importance of bond oc-
cupation and singlet formation at α= 1 (Sec. IIC).
In the third line of Eq. (2.24), the labels ¯ γ/ne}ationslash= ˜γre-
fer to the two mixed orbital operators on each bond
[Eq. (2.23)]. Orbital fluctuations are the only processes
contributing to the mixed terms in this limit, where the
spin state of the bond has no effect. We draw the at-
tention of the reader to the fact that for the parameter
choiceα= 0.5, anelectronofanycoloratanysitehasthe
same matrix element to hop in any direction. However,
because of the different color changes involved in these
processes, again the spin–orbital Hamiltonian does not
exhibit ahigher symmetry at this point, a result reflected
in the different operator structures of superexchange and
direct exchange components.
III. LONG–RANGE–ORDERED STATES
In this Section we study possible ordered or partially
ordered states for the Hamiltonian of Eq. (2.7). As ex-
plained in Sec. II, the parameters of the problem arethe ratio of the direct and superexchange interactions,
α(2.5), and the strength of the Hund exchange interac-
tion,η(1.2). Regarding the latter, we will discuss briefly
the transition to ferromagnetic (FM) spin order for in-
creasingηin this framework.
The first necessary step in any analysis of such an
interacting system is to establish the energies of differ-
ent (magnetically and orbitally) ordered states. The
high connectivity of the triangular–lattice system sug-
gests that ordered states will dominate, and claims of
more exotic ground states are justifiable only when these
are shown to be uncompetitive. The calculations in this
Section will be performed for static orbital and spin con-
figurations, with the virtualprocessesresponsiblefor(su-
per)exchange as the only fluctuations. In the language
of the discussion in Sec. I, fully ordered states gain only
potential energy at the cost of sacrificing the kinetic (res-
onance) energy from fluctuation processes, which we will
show in Secs. IV and V is of crucial importance here.
A. Possible orbital configurations
The results to follow will be obtained by first fixing
the orbital configuration, either on every site or on par-
ticular bonds, and then computing the spin interaction
and optimizing the spin state accordingly. While this is
equivalent to the converse, the procedure is more trans-
parent and offers more insight into the candidate phases.
We limit the number of states to ordered phases with
small unit cells, and the orbital states to be considered8
(a) (b)
(c) (d)
(e) (f)
FIG. 3: (Color online) Schematic representation of possibl e
orbital states with a single color on each site of the triangu -
lar lattice: (a) one–color state; (b) and (c) two inequivale nt
two–color states; (d) three–sublattice three–color state ; (e)
and (f) two inequivalent three–color states. The latter two
configurations are degenerate with similar states where the
lines of occupied aandborbitals repeat rather than being
staggered along the direction perpendicular to the lines of
occupied corbitals. The three–sublattice state (3d) is nonde-
generate ( d= 1), states (3a), (3b), and (3e) have degeneracy
d= 3, and states (3c) and (3f) have degeneracy d= 6.
are enumerated in this subsection. For clarity we adopt
the conventionofFig. 2(c) that horizontal( c) bonds have
diagonal (direct exchange) hopping of corbitals, which
areshowninblue, andoff–diagonal(superexchange)hop-
pingprocessesfor aandborbitals[Fig.2(b)], respectively
red and green; up–slanting ( a) bonds have diagonal hop-
ping foraorbitals and off–diagonal hopping between b
andcorbitals; down–slanting ( b) bonds have diagonal
hopping for borbitals and off–diagonal hopping between
aandcorbitals. All Hamiltonians and energies are func-
tions ofαandη, as given by Eqs. (2.7), (2.8), (2.21),
and (2.22). To minimize additional notation, they will
be quoted in this and in the next section as functions of
the single argument α, with implicit η–dependence con-
tained in the parameters ( r1,r2,r3). The orbital bond
indexγwill also be suppressed here and in Sec. IV.
We continue to refer to the orbital type as a “color”,
and begin by listing symmetry–inequivalent states where(a)
(b) (c)
(d) (e)
FIG. 4: (Color online) Schematic representation of possibl e
orbital configurations with superpositions of (a) two orbit als
in a two–color state, (b) three orbitals, (c) two orbitals wi th
equal net weight, and (d) and (e) two orbitals with differing
net weights of all three orbitals. State (a) has degeneracy
d= 3, states (b) and (c) have d= 1, and the degeneracies of
states (d) and (e) are d= 6 and d= 3.
each site has a unique color. If the same orbital is occu-
pied at every site [Fig. 3(a)], the three states with a,b,
orcorbitals occupied are physically equivalent (degener-
acy isd= 3). When lines of the same occupied orbitals
alternate along the perpendicular direction there are two
basicpossibilities, whichareshowninFigs.3(b) and3(c).
These two–color states differ in their numbers of active
superexchange or direct–exchange bonds, which depend
on how the monocolored lines are oriented relative to the
active hopping direction(s) of the orbital color. There
is only one three–color configuration with equal occupa-
tions, which is shown in Fig. 3(d).
Turning to orbital states with unequal occupations,
motivated by the tendency of Hto favor dimer forma-
tion in certain limits we extend our considerations to the
possibility of a four–site unit cell [Figs. 3(e) and 3(f)].
More elaborate three–color unit cells are not considered.
In this case the same state is obtained when the fourth
site is occupied by electrons whose orbital color is any of
the other three. Again this state, which breaksrotational
symmetry, differs depending on its orientation relative to
the active hopping axes.9
States involving a superposition of either two or three
orbitals at each site can be expected to allow a signif-
icantly greater variety of hopping processes. When ei-
ther two or three orbital states are partially occupied
at each site (we stress that the condition of Eq. (2.3)
is always obeyed rigorously), one finds the two uniform
states represented in Figs. 4(a) and 4(b). These denote
the symmetric wavefunctions |ψ2/an}bracketri}ht= (|φa/an}bracketri}ht+|φb/an}bracketri}ht)/√
2
and|ψ3/an}bracketri}ht= (|φa/an}bracketri}ht+|φb/an}bracketri}ht+|φc/an}bracketri}ht)/√
3 at every site, where
|φγ/an}bracketri}ht=γ†|0/an}bracketri}ht. The remaining states shown in Fig. 4 in-
volveonlytwoorbitalspersite, but with allthreeorbitals
partly occupied in the lattice. The average electron den-
sitypersiteandperorbitalis1 /3in thestateofFig.4(c),
while in Figs. 4(d) and 4(e) it is nc=1
2,na=nb=1
4.
The latter two states are neither unique nor (for general
interactions) equivalent to each other, and represent two
classes of states with respective degeneracies 3 and 6.
B. Ordered–state energies: superexchange
Before analyzing the different possible ordered states
for any of the model parameters, we stress that the spin
interactions on a given bond depend strongly on the or-
bital occupation of that bond. We begin with the pure
superexchange model Hs(2.8), meaning α= 0, for which
the question of spin and orbital singlets was addressed
in Sec. IIB. Here the spin and orbital scalar products
/an}bracketle{t/vectorSi·/vectorSj/an}bracketri}htand/an}bracketle{t/vectorTi·/vectorTj/an}bracketri}htmay take only values consistent
with long–range order throughout the system and thus
vary between −1/4 and +1/4.
For a bond on which both electrons occupy active or-
bitals, one has the possibility of either FO or AO states.
For the FO state, /an}bracketle{t/vectorTi·/vectorTj/an}bracketri}ht= 1/4 =/an}bracketle{t/vectorTi×/vectorTj/an}bracketri}htand
/an}bracketle{tAij/an}bracketri}ht=/an}bracketle{tBij/an}bracketri}ht= 1, whence the terms of Hscan be sepa-
rated into the physically transparent form
H(FO)
1(0) =1
2Jr1/parenleftbigg1
2/an}bracketle{tniγ+njγ/an}bracketri}ht/parenrightbigg/parenleftbigg
/vectorSi·/vectorSj+3
4/parenrightbigg
= 0,
H(FO)
2(0) =1
2Jr2/parenleftbigg
2−1
2/an}bracketle{tniγ+njγ/an}bracketri}ht/parenrightbigg/parenleftbigg
/vectorSi·/vectorSj−1
4/parenrightbigg
=Jr2/parenleftbigg
/vectorSi·/vectorSj−1
4/parenrightbigg
, (3.1)
H(FO)
3=1
3J(r3−r2)/parenleftbigg
/vectorSi·/vectorSj−1
4/parenrightbigg
,
specifying a net spin interaction which, because niγ= 0,
must be AF if any hopping processes are to occur. In
the AO case, /an}bracketle{t/vectorTi·/vectorTj/an}bracketri}ht=−1/4 =/an}bracketle{t/vectorTi×/vectorTj/an}bracketri}htand/an}bracketle{tAij/an}bracketri}ht=
/an}bracketle{tBij/an}bracketri}ht= 0, giving
H(AO)
1(0) =−1
2Jr1/parenleftbigg
/vectorSi·/vectorSj+3
4/parenrightbigg
,
H(AO)
2(0) =1
2Jr2/parenleftbigg
/vectorSi·/vectorSj−1
4/parenrightbigg
,(3.2)
H(AO)
3(0) = 0,and the spin interaction is constant at η= 0, with only a
weak FM preference emerging at finite η. We remind the
reader here that the designations FO and AO continue
to be based on the conventional notation22obtained by a
local transformation on one bond site, and in the basis of
the original orbitals correspond respectively to opposite
active orbitals and to equal active orbitals. Cases where
only one orbital is active on a bond are by definition AO,
but do contribute a finite spin interaction
H1
1(0) =−1
4Jr1/parenleftbigg
/vectorSi·/vectorSj+3
4/parenrightbigg
,
H1
2(0) =1
4Jr2/parenleftbigg
/vectorSi·/vectorSj−1
4/parenrightbigg
, (3.3)
H1
3(0) = 0,
which again has only a weak FM tendency at η >0.
Clearly, when neither electron may hop, the bond does
not contribute a finite energy.
We begin with the uniform, one–color orbital state of
Fig. 3(a), meaning that all bonds are AO by the defini-
tion of the previous paragraph. In two directions both
electrons are active, while in the third none are. The
energy per bond is
E(3a)
FM(0) =−1
3Jr1. (3.4)
and the spin configuration is FM. However, an antifer-
romagnetic (AF) spin configuration on the square lattice
defined by the active hopping directions has energy
E(3a)
AF(0) =−1
6J(r1+r2), (3.5)
from which one observes that all spin states are degen-
erate atη= 0. The ordered spin state spin is then FM
for any finite η. We note in passing that the energy per
bond for a square lattice would have the significantly
lower value −1
2Jfor the same Hsconvention, by which
is meant the presence of the constants +3
4and−1
4in
Eq. (2.8). This result is a direct reflection of the geo-
metrical frustration of the triangular lattice, an issue to
which we return in Sec. VI.
The state of Fig. 3(b) involves one set of (alternating)
AO lines with two active orbitals and two sets of (AO)
lines each with one active orbital. All sets of lines favor
FM order at finite η, with
E(3b)
FM(0) =−1
3Jr1. (3.6)
Here the square–lattice state which becomes degenerate
atη= 0, with
E(3b)
AF(0) =−1
6J(r1+r2), (3.7)
is more accurately described as one with two lines of AF
spins andone ofFM spins [Fig. 5(a)], and will be denoted
henceforth as AFF.10
(a) (b)
FIG. 5: (Color online) Spin configurations minimizing the
total energy of the superexchange Hamiltonian Hs(α= 0)
for given fixed patterns of orbital order: (a) AFF state for
the orbital ordering pattern of Fig. 3(c), showing how the FM
lineisselected bythedirection(here b)givingzerofrustration;
(b) 60–120◦ordered spin configuration minimizing the total
energy for the orbital ordering pattern of Fig. 3(d).
The state of Fig. 3(c) involves one set of FO lines with
twoactiveorbitals,onesetoflineswithoneactiveorbital,
one half set of AO lines with two active orbitals and one
half set of inactive lines. The two–active FO lines will
favor AF order, while the AO and the one–active lines
will favor FM order only at η>0, giving
E(3c)
AFF(0) =−1
72J(9r1+11r2+4r3) (3.8)
from the AFF configuration, but with 2 equivalent di-
rections for the FM line. At η= 0 the energy is again
−1
3J. BothE(3b)
AF(0) andE(3c)
AFF(0) can be regarded as the
energy of an unfrustrated system, in the sense that the
spin order enforced in any one direction by the orbital
configuration at no time denies the system the ability to
adopt the energy–minimizing configuration in other di-
rections. However, at finite ηthe configurations shown
in Figs. 3(b) and 3(c) will be penalized relative to the
uniform (AO) order of Fig. 3(a) due to the presence of
AF bonds.
We insert here an important observation: the orbital
state of Fig. 3(c) also admits the formation of 1D AF
Heisenberg spin chains on the FO ( b–axis) lines. The
energy per bond of such a state includes constant inter-
chain contributions which are independent of the spin
state (/an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht= 0) on these bonds. Of these interchain
bonds, 1/4 are FO with two active orbitals and 1/2 have
one active orbital. One finds
E(3c)
1D(0) =−1
9Jln2 (2r2+r3)−1
24J(3r1+r2),(3.9)
which gives E(3c)
1D(0) =−0.3977Jatη= 0. This energy
is significantly lower than that of an ordered magnetic
state, a result showing that the kinetic energy gained
from resonance processes on the chains is far more signif-
icantthan minimalpotential energygainobtainablefrom
an ordering of the magnetic moments on the active inter-
chain bonds which are active, and thus provides strong
evidence in favorof the hypothesis that any orderedstatewill “melt” to a quantum disordered one in this system.
We will return to this issue below.
For the two–color superposition [Fig. 4(a)], one set
of bonds always has two active orbitals, but with equal
probability of being FO or AO, while the other two sets
of bonds have a 1/4 probability of having two active or-
bitals, which are FO, or a 1/2 probability of having one
active orbital (and a 1/4 probability of having none).
Under these circumstances, the net system Hamiltonian
can be expressedby summing overall the possible orbital
states, although this is not necessarily a useful exercise
when the spin state may not be isotropic. By insert-
ing the three most obvious ordered spin states, FM, AF
(meaning here the AF state of the triangular lattice with
120◦bond angles and /an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht=−1
8) and AFF, the can-
didate energies are
E(4a)
FM(0) =−1
6Jr1, (3.10)
E(4a)
AF(0) =E(4a)
AFF(0) =−1
48J(5r1+7r2+2r3).
The coincidence for the results for the AF and AFF or-
dered states in this case is an accidental degeneracy. The
final energy E(4a)
AF(F)=−7
24Jatη= 0 shows that both
states are compromises, and it is not possible to put all
bonds in their optimal spin state simultaneously. This
arises because of the presence of two–active FO compo-
nents in all three lattice directions, and will emerge as a
quite generic feature of superposition states, albeit not
one without exceptions.
In general there is no compelling reason (given by H
for any value of α) to expect that two–color superposi-
tions of this type may be favorable. While the 120◦state
ofatriangular–latticeantiferromagnetisonecompromise
withinaspaceofSU(2)operators,thistypeofsymmetry–
breaking is not relevant within the orbital sector, where
there are three colors and the two–color subsector of ac-
tive orbitals in the α= 0 limit changes as a function of
the bond orientation.
In the equally weighted three–color state [Fig. 3(d)],
all bonds are FO and it is easy to show that 1/3 of them
(arranged as isolated triangles) have two active orbitals
while the other 2/3 have one active orbital. The two–
active bonds favor AF order while the one–active bonds
have only a weak preference for FM order at finite η.
In this case the problem becomes frustrated and is best
resolved by a kind of AF state on the triangular lattice
where the strong triangles have 120◦angles and alternat-
ing triangles have spins either all pointing in or all point-
ing out [Fig. 5(b)]; then 2/3 of the intertriangle bonds
have 60◦angles while the other 1/3 have 120◦angles.
The energy of this state is
E(3d)(0) =−1
144J(19r1+17r2+6r3),(3.11)
andE(3d)(0) =−7
24Jatη= 0, a value again inferior to
the optimal energy due to the manifest spin frustration.11
In the state of Fig. 3(e), the only AO bonds (1/6 of the
total) contain inactive orbitals. Of the remaining bonds,
3/6 have two active FO orbitals (in all three directions)
and 2/6 have one active orbital. Once again the system
is composed of strongly coupled triangles, but this time
in a square array and with strong coupling in their basal
direction by one set of two–active FO bonds. Possible
competitive spin–ordered states would be AF or AFF,
with energies
E(3e)
AF(0) =−1
96J(5r1+15r2+6r3),(3.12)
E(3e)
AFF(0) =−1
288J(15r1+35r2+16r3).
The lowest energy is obtained for 120◦AF order, with
the frustrated value E(3e)
AF(0) =−13
48Jforη= 0.
For the state in Fig. 3(f) the FO bonds (1/6) and only
1/6 of the AO bonds have two active orbitals, while the
other 2/3 of the bonds have one active orbital. In this
case
E(3f)
FM(0) =−1
4Jr1,
E(3f)
AF(0) =−1
96J(15r1+13r2+2r3),(3.13)
E(3f)
AFF(0) =−1
192J(36r1+17r2+5r3),
leading again to an AF spin state. At η= 0 one has
E(3f)
AF(0) =−5
16J,i.e.relatively weaker frustration.
Turning now to three–color superpositions, the “uni-
form” orbital state [Fig. 4(b)] is one in which on every
bond there is a probability 2/9 of having two active FO
orbitals, 2/9 for two active AO orbitals, 4/9 of one active
orbital and 1/9 of no active orbitals. The appropriately
weighted bond interaction strengths may be summed to
give the net interaction, which for the three spin states
considered results in the energies
E(4b)
FM(0) =−2
9Jr1,
E(4b)
AF(0) =−1
36J(5r1+5r2+r3),(3.14)
E(4b)
AFF(0) =−1
81J(12r1+10r2+2r3),
andthusthe AFstateislowest, with thevalue E(4b)
AF(0) =
−11
36Jatη= 0. While this orbital configuration does not
attain the minimal energy of −1
3J, it is a close competi-
tor: although it involves every bond, the fractional prob-
abilities of each being in a two–active state mean that
it cannot maximize individual bond contributions. How-
ever, we will see in Sec. IIID that state (4b) lies lowest
over much of the phase diagram (0 <α<1) as a result
of the contributions from mixed terms.
For states with unequal site occupations, in Fig. 4(c)
one has a situation where on 1/3 of the bonds (arranged
in separate triangles) there is a 1/4 probability of twoactive FO orbitals and a 1/2 probability of one active
orbital, while on the remaining 2/3 of the bonds there is
a 1/4 probability of two active AO orbitals, 1/4 of two
active FO orbitals and 1/2 of having one active orbital.
On computing the net energies for the three standard
spin configurations, one obtains
E(4c)
FM(0) =−5
24Jr1,
E(4c)
AF(0) =−1
192J(25r1+25r2+8r3),(3.15)
E(4c)
AFF(0) =−1
216J(30r1+25r2+8r3),
wherethe AF statewith E(4c)
AF(0) =−29
96Jisthe lowestat
η= 0. However, this state is also manifestly frustrated.
In the unequally weighted state of Fig. 4(d), the prob-
lemisbestconsideredonceagainaslinesofdifferentbond
types. Here 1/6 of the lines have two active orbitals (1/2
FO and 1/2 AO), 1/6 of the lines have probability 1/4
of two active orbitals (AO) and 1/2 of one active orbital,
1/3 of the lines have probability 1/4 of two active FO or-
bitals, 1/4oftwoactiveAOorbitalsand1/2ofoneactive
orbital, and the remaining 1/3 of the lines have probabil-
ity 1/4 of two active orbitals (FO) and 1/2 of one active
orbital. The ordered spin states yield the energies
E(4d)
FM(0) =−5
24Jr1,
E(4d)
AF(0) =−1
192J(25r1+27r2+6r3),(3.16)
E(4d)
AFF(0) =−1
144J(21r1+17r2+4r3),
whenceitisagaintheAFstate, withasmalldegreeofun-
relieved frustration in its energy E(4d)
AF(0) =−29
96J, which
lies lowest at η= 0.
Finally, the state of Fig. 4(e) has the orbital pattern of
Fig. 4(d) rotated in such a way that the number of active
orbitals in different bond directions is changed. Now 1/3
of the bonds have probabilities 1/4 of two active orbitals
(AO) and 1/2 of one active orbital, while the remaining
2/3 have probabilities 1/4 of two active orbitals (FO),
1/4 of two active orbitals (AO) and 1/2 of one active
orbital. The ordered–state energies are
E(4e)
FM(0) =−1
4Jr1,
E(4e)
AF(0) =−1
96J(15r1+13r2+2r3),(3.17)
E(4e)
AFF(0) =−1
36J(6r1+5r2+r3),
of which the AFF states lies lowest at η= 0, achieving
the unfrustrated value E(4e)
AFF(0) =−1
3J. That it is pos-
sible to obtain this energy in an orbital superposition is
because of the absence of FO bond contributions in one
direction, which can then be chosen to be FM.
The results of this section and the conclusions one may
draw from them are summarized in Subsec. IIIE below.12
C. Ordered–state energies: direct exchange
In the limit of only direct exchange, the analysis is
somewhat simpler. The Hamiltonian is Hdof Eq. (2.21),
and in this case a particle on any site is active in only
one direction, which leads to the immediate observation
thatin astaticorbitalconfigurationit isneverpossibleto
have, on average,activeexchangeprocessesonmorethan
2/3 of the bonds. For simplicity we repeat the Hamilto-
nianforthe twocasesofAO orderbetweensites, inwhich
case by definition at most one of the orbitals is active,
and FO order between sites, which is restricted to the
case where neighboring sites have the same orbital color
and the correct bond orientation. We stress that in this
subsection the definitions FO and AO are entirely con-
ventional, as the local transformation of Sec. IIB is not
relevant at α= 1, and thus the designation FO implies
orbitals of the same color, and AO orbitals of different
colors. One obtains the expressions
H(AO)(1) =1
4J/bracketleftbigg
−r1/parenleftbigg
/vectorSi·/vectorSj+3
4/parenrightbigg
+r2/parenleftbigg
/vectorSi·/vectorSj−1
4/parenrightbigg/bracketrightbigg
,(3.18)
H(FO)(1) =1
3J(2r2+r3)/parenleftbigg
/vectorSi·/vectorSj−1
4/parenrightbigg
,(3.19)
which in the η= 0 limit reduce to the forms
H(AO)(1) =−1
4J, (3.20)
H(FO)(1) =J/parenleftbigg
/vectorSi·/vectorSj−1
4/parenrightbigg
.(3.21)
It is clear (Sec. II) that for a single bond, the most fa-
vorablestate is aspin singlet, which would contribute en-
ergy−J, but at the possible expense of placing all of the
neighboring bonds in suboptimal states. The very strong
preference for such singlet bonds means that any mean–
field study of the minimal energy is incomplete without
the consideration of dimerized (or valence–bond) states
(Sec. IV). The analysis of this section can be considered
as elucidating the optimal energies to be gained from
long–ranged magnetic and orbital order on these bonds,
where the optimal energy of any one is −1
2J. Also as
noted in Sec. II, any active AO bond gains an exchange
energy (−1
4J) simply because it does not prevent one of
the two particles from performing virtual hopping pro-
cesses, and this we term “avoided blocking”. In the limit
of zero Hund exchange, these will give a highly degener-
ate manifold of all possible spin states, from which FM
states are selected at finite η.
We beginagainwith one–colorstateofFig. 3(a), which
we denote henceforth as (3a). Only one set of lattice
bonds has finite interactions, all FO, and therefore the
system behaves as a set of AF Heisenberg spin chains
with energy per bond
E(3a)
AF1D(1) =−1
9Jln2 (2r2+r3),(3.22)whenceE(3a)
AF1D(1) =−0.2310Jatη= 0.
In state (3b), the FO lines do not correspond to active
hopping directions. The remaining two directions then
form an AO square lattice with
E(3b)
FM(1) =−1
6Jr1. (3.23)
This can be called a “pure avoided–blocking” energy.
The spins are unpolarized at η= 0, where all bond spin
states are equivalent, but any finite ηwill select FM or-
der (hence the notation). We will see in the remainder of
this section that E=−1
6Jis the optimal energy obtain-
able by a 2D ordered state in the direct–exchange limit
(α= 1), where the net energy is generically higher than
atα= 0quitesimplybecausetherearehalfasmanyhop-
ping channels. Thus the “melting” of such ordered states
into quasi–1D states becomes clear from the outset, and
can be understood due to the very low connectivity of
the active hopping network on the triangular lattice.
In state (3c), one of the FO lines is active, and forms
AF Heisenberg spin chains. Electrons in the other FO
line areactiveonly in across–chaindirection, wheretheir
bonds areAO,and gainavoided–blockingenergy, whence
E(3c)
AF(1) =−1
12J(2ln2+1) = −0.1988J(3.24)
atη= 0. As in the preceding subsection, the coherent
state of each Heisenberg chain is not altered by the pres-
ence of additional electrons from other chains executing
virtual hopping processes onto empty orbitals of individ-
ual sites. The spin chains remain uncorrelated and only
quasi–long–range–ordereduntil a finite value of η, where
FM spin polarization and a long–range–orderedstate are
favored.
In the two–color superposition (4a), 1/3 of the bonds
are inactive, while on the other 2/3 one has probabil-
ity 1/4 of two active electrons (FO), 1/2 of one active
(AO) and 1/4 of two inactive electrons. In this case, one
obtains an effective square lattice on which an AF spin
configuration is favored by the FO processes, with
E(4a)
AF(1) =−1
72J(3r1+7r2+2r3),(3.25)
so againE(4a)
AF(1) =−1
6Jatη= 0.
The uniform three–color state (3d) maximizes AO
bonds, but 1/3 of the bonds on the lattice remain in-
active. Thus
E(3d)
FM(1) =−1
6Jr1, (3.26)
and Hund exchange will select the FM spin state.
The three–color state (3e) has FO lines oriented in
their active direction and will, as in state (3c), form
Heisenberg chains linked by bonds with AO order. While
the geometry of the interchain coupling can differ de-
pending on the orbital alignment in the inactive chains,
it does not create a frustrated spin configuration and13
the net energy is E(3e)
AF(1) =E(3c)
AF(1). The state (3f)
has only inactive FO lines and so gains only avoided–
blocking energy, from 2/3 of the bonds in the system,
whenceE(3f)
FM(1) =E(3d)
FM(1).
In the uniform three–color superposition (4b), every
bond has probability 1/9 of containing two active elec-
trons (FO), 4/9 of one active electron and 4/9 of remain-
ing inactive. For the three different ordered spin config-
urations considered in Subsec. IIIB the energies are
E(4b)
FM(1) =−1
9Jr1,
E(4b)
AF(1) =−1
72J(5r1+5r2+r3),(3.27)
E(4b)
AFF(1) =−1
81J(6r1+5r2+r3),
and one finds the energy E(4b)
AF(1) =−11
72Jfor the 1200
AF state at η= 0.
The three–color state (4c) is one in which 1/3 of the
bonds (arranged on isolated triangles) have probability
1/4 of being in a state with two active electrons and 1/2
of containing one active electron, while on the other 2/3
of the bonds there is simply a 1/2 probability of one
active orbital. The respective energies are
E(4c)
FM(1) =−1
8Jr1,
E(4c)
AF(1) =−1
192J(15r1+13r2+2r3),(3.28)
E(4c)
AFF(1) =−1
216J(18r1+13r2+2r3).
Atη= 0, the energy E(4c)
FM(1) =−5
32Jis minimized
by a 120◦state on the triangles, which are also isolated
magnetically in this limit. Finite values of ηresult in
FM interactions between the triangles, and a frustrated
problem in the spin sectorwhich by inspection is resolved
in favor of a net FM configuration only at large η(η >
0.23).
Finally, the three–color states (4d) and (4e) yield two
possibilities in the α= 1 limit, namely where one of the
minority colors is aligned with its active direction and
where neither is. In the former case,
E(4d)
FM(1) =−5
48Jr1,
E(4d)
AF(1) =−1
384J(25r1+27r2+6r3),(3.29)
E(4d)
AFF(1) =−1
96J(7r1+7r2+2r3),
and the lowest energy E(4d)
AFF(1) =−1
6Jatη= 0 is given
by the directionally anisotropic AFF spin configuration.
This is because 1/2 of the lines, in two of the three di-
rections, have some AF preference from their 1/4 prob-
ability of containing two active orbitals, while the third
direction has no preference at η= 0, and in any case
favors FM spins at η >0. In the latter case, the onlyAF tendencies arise along lines in a single direction, but
avoided–blocking energy is sufficient to exclude the pos-
sibility of a Heisenberg chain state. Here
E(4e)
FM(1) =−1
8Jr1,
E(4e)
AF(1) =−1
192J(15r1+13r2+2r3),(3.30)
E(4e)
AFF(1) =−1
72J(6r1+5r2+r3),
whenceE(4e)
AFF(1) =−1
6Jatη= 0, in fact with two de-
generate possibilities for the orientation of the FM line.
D. Ordered–state energies: α= 0.5
Toillustratethepropertiesofthemodelinthepresence
of finite direct and superexchange contributions, i.e.at
intermediate values of α, we consider the point α= 0.5.
As shown in Sec. II, there is no special symmetry at this
point, because the contributions from diagonal and off–
diagonal hopping remain intrinsically different. States
with long–ranged orbital (and spin) order at α= 0.5
are mostly very easy to characterize, because all virtual
processes, of both types, allowed by the given configura-
tion are able to contribute in full to the net energy. For
the many of the states considered in this section, the en-
ergetic calculation for α= 0.5 is merely an exercise in
adding the α= 0 andα= 1 results with equal weight.
Exceptions occur for superposition states gaining energy
from processes contained in Hm[Eq. (2.22)], and are in
fact decisive here. Because these terms involve explicitly
a finite density of orbitals of all three colors on the bond
in question, with the active diagonalcolor representedon
both sites, only for states (4b), (4c), and (4d), but not
(4e) [Figs. 4(b–e)], will it be necessary to consider this
contribution.
For state (3a), in two directions both electrons are
active by off–diagonal hopping, while in the third both
may hop diagonally. Diagonal hopping favorsan AF spin
configuration, while the off–diagonalhopping bonds have
only a weak preference (by Hund exchange) for FM or-
der. The ordered–state spin solution is then a doubly
degenerate AFF state with energy per bond
E(3a)(0.5) =−1
72J(9r1+7r2+2r3),(3.31)
givingE(3a)(0.5) =−1
4Jatη= 0. We remind the
reader that the prefactor of the superexchange and di-
rect exchange contributions is only half as large as in
Subsecs. IIIB and IIIC [Eq. (2.7)], so the overall effect
of additional hopping processes in this state is in fact
an unfrustrated energy summation. We also comment
that, exactly at η= 0, there is no obvious preference
for any magnetic order between the diagonal–hopping
chains. Only at unrealistically large values of ηwould14
the system sacrifice this diagonal–hopping energy to es-
tablish a square–lattice FM state. At finite η, the one–
colororbitalstate representsa compromisebetween com-
peting spin states preferred by the two types of hopping
contribution.
State (3b) has no diagonal–hopping chains, and these
processes therefore enforce only a weak preference for a
FMsquarelattice. Becausetheoff–diagonalhoppingpro-
cesses also favor FM order at finite η(Subsec. IIIB), the
two types of contribution cooperate and one obtains
E(3b)(0.5) =−1
4Jr1. (3.32)
State (3c) contains one half set of diagonal–hopping
chains, which fall along one of the directions which in
the spin state favored by the off–diagonal hopping pro-
cesses could be FM or AF; this degeneracy will therefore
be broken. The other half set of chains will gain only
avoided–blocking energy from diagonal processes, which
will take place in the FM direction and thus cause no
frustration even at finite η. One obtains
E(3c)(0.5) =−1
144J(3r1+7r2+2r3),(3.33)
and thusE(3c)(0.5) =−1
4Jatη= 0 from this AFF
configuration. The additive contributions from superex-
change and direct exchange remove the possibility that
Heisenberg–chain states in either of the directions fa-
vored separately by off–diagonal (Sec. IIIB) or diagonal
(Sec. IIIC) hopping could result in an overall lowering of
energy.
As in Subsec. IIIC, in the two–colorsuperposition (4a)
the diagonal hopping processes are optimized by an AFF
spin configuration. Although this is one of the degener-
ate states minimizing the off–diagonal Hamiltonian, the
directions of the FM lines do not match. Insertion of the
four possible spin states yields
E(4a)
FM(0.5) =−1
8Jr1,
E(4a)
AF(0.5) =−1
96J(8r1+10r2+3r3),
E(4a)
AFF(0)(0.5) =−1
72J(6r1+7r2+r3),(3.34)
E(4a)
AFF(1)(0.5) =−1
144J(12r1+14r2+3r3),
whence the lowest final energy is E(4a)
AF(0.5) =−7
32Jat
η= 0. As noted in the previous sections for this spin
configuration, the optimal energy for all bonds is not
attainable within the off–diagonal hopping sector, and
the addition of the (small) diagonal–hopping contribu-
tion causes little overall change.
The equally weighted three–color state (3d) has no
lines of diagonal–hopping bonds, and in fact these con-
tribute only avoided–blocking energy on the bonds be-
tween the strong triangles defined by the off–diagonalproblem, adding to the weak propensity for FM intertri-
angle bonds arising only from the Hund exchange. The
diagonal processes can be taken only to alter this energy,
and not to promote any tendency towards an alteration
of the spin state, whose energy is then
E(3d)(0.5) =−1
144J(19r1+11r2+3r3),(3.35)
withE(3d)(0.5) =−11
48Jatη= 0.
State (3e) is already frustrated in the off–diagonal sec-
tor, and diagonal–hoppingprocessescontributeprimarily
on otherwise inactive bonds without changing the frus-
tration conditions. For the two candidate spin configu-
rations
E(3e)
AF(0.5) =−1
96J(5r1+12r2+4r3),
E(3e)
AFF(0.5) =−1
192J(11r1+19r2+8r3),(3.36)
a competition won by the 120◦AF–ordered state with
E(3e)
AF(0.5) =−7
32Jatη= 0.
State (3f) lacks active lines of diagonal–hopping pro-
cesses, and thus the avoided–blocking energy may be
added simply to the results for the off–diagonal sector,
giving
E(3f)
FM(0.5) =−5
24Jr1,
E(3f)
AF(0.5) =−1
192J(25r1+19r2+2r3),(3.37)
E(3f)
AFF(0.5) =−5
384J(12r1+5r2+r3),
or a minimum of E(3f)
AF(0.5) =−23
96Jatη= 0.
In the uniform three–color superposition (4b), on ev-
ery bond there is a probability 4/9 of having only off–
diagonal hopping processes, 2/9 for 2 active FO orbitals
and 2/9 for two active AO orbitals, a probability 1/9 of
having only diagonal hopping processes, and a probabil-
ity 4/9 of other processes. These last include the contri-
butions fromoneactivediagonaloroff–diagonalelectron,
and mixed processes contained in the Hamiltonian Hm
(2.22); none of these three possibilities favors any given
bond spin configuration other than a FM orientation at
finiteη. The net energy contributions are
E(4b)
FM(0.5) =−2
9Jr1,
E(4b)
AF(0.5) =−1
144J(20r1+18r2+3r3),(3.38)
E(4b)
AFF(0.5) =−1
54J(8r1+6r2+r3),
and thus the AF state is lowest, with E(4b)
AF(0.5) =−41
144J
atη= 0. While this energy differs from that for the AFF
spin configuration by only1
144J, its crucial property is
that it lies below the value −1
4Jobtained by direct sum-
mation of the superexchangeand direct–exchangecontri-
butions.15
For this orbital configuration, all three spin states gain
a net energy of −1
18Jatη= 0 from mixed processes,
and these are sufficient, as we shall see, to reduce the
otherwisepartiallyfrustratedordered–stateenergytothe
global minimum for this value of α. By a small extension
of the calculation, the energy of the 1200AF spin state
may be deduced at η= 0 for all values of α, and is given
by
E(4b)
AF(α) =−1
72J/parenleftbig
22−11α+8/radicalbig
α(1−α)/parenrightbig
.(3.39)
Comparison with the value obtained by direct summa-
tion,E=−1
6(2−α), revealsthat state(4b) isthe lowest–
lying fully spin and orbitally ordered configuration in the
region0.063<α<0.983. Thatthis statedominatesover
the majorityofthe phasediagramis a directconsequence
of its ability to gain energy from mixed processes.
The non–uniform three–colorstate (4c) also presents a
delicate competition between spin configurations of very
similar energies. From the preceding subsections, it is
clearthat inthis casediagonalandoff–diagonalprocesses
favor different ground states, while there will also be a
mixed contribution from 1/3 of the bonds. The energies
of the three standard spin configurations are
E(4c)
FM(0.5) =−1
12Jr1,
E(4c)
AF(0.5) =−1
384J(45r1+41r2+10r3),(3.40)
E(4c)
AFF(0.5) =−1
432J(54r1+41r2+10r3),
where the AF state, obtaining E(4c)
AF(0.5) =−1
4Jis the
lowest atη= 0.
Finally, in the three–color states (4d) and (4e), which
are composed of lines of two–colorsites, this delicate bal-
ance between different spin configurations persists. For
configuration (4d), an AFF state with the same orien-
tation of the FM line (along the b–axis) is both favored
by diagonal hopping processes and competitive for off–
diagonal processes. With inclusion of a small contribu-
tion due to mixed processes, the three orderedspin states
have energies
E(4d)
FM(0.5) =−51
288Jr1,
E(4d)
AF(0.5) =−1
768J(85r1+84r2+18r3),(3.41)
E(4d)
AFF(0.5) =−1
576J(71r1+59r2+14r3),
from which the AFF state minimizes the energy at η= 0
withE(4d)
AFF(0.5) =−1
4J.
For state (4e), which has no mixed contribution, the
orientationsofthe FM lines in the optimal AFF states do
not match, and it is necessary, as above, to consider both
possibilities when performing a full comparison. These
four ordered spin states yield the energies
E(4e)
FM(0.5) =−3
16Jr1,E(4e)
AF(0.5) =−1
128J(15r1+13r2+2r3),
E(4e)
AFF(0)(0.5) =−1
144J(18r1+13r2+2r3),
E(4e)
AFF(1)(0.5) =−1
48J(6r1+4r2+r3),(3.42)
among which the AF state in fact lies lowest at η= 0,
achievingthe weaklyfrustratedvalue E(4e)
AF(0.5) =−15
64J.
E. Summary
Here we summarize the results of this section in a con-
cise form. For the superexchange model ( α= 0), a con-
siderable number of 2D ordered orbital and spin states
exist which return the energy −1
3Jatη= 0. This de-
generacy is lifted at any finite Hund exchange in favor
of orbital states [(3a), (3b)] permitting a fully FM spin
alignment. Most other orbital configurations introduce a
frustration in the spin sector at small η, while some offer
the possibility of a change of ground–state spin configu-
ration at finite η, wherer1exceeds the r2andr3contri-
butions and begins to favor states with more FM bonds.
However, the value E=−1
3Jper bond remains a
rather poor minimum for a system as highly connected
as the triangular lattice, even if, as in the superexchange
limit, active hopping channels exist only in two of the
three lattice directions for each orbital color. Indeed,
the limitations of the available ordering (potential) en-
ergy are clearly visible from the fact that a significantly
lower overall energy is attained in systems which aban-
don spin order in favor of the resonance (kinetic) en-
ergy gains available in one lattice direction. The result
E(3c)
1D(0) =−0.3977Jis the single most important ob-
tained in this section, and in a sense obviates all of the
considerations made here for fully ordered states, man-
dating the full consideration of 2D magnetically and or-
bitally disordered phases.
In the study of ordered states, it becomes clear that
the Hund exchange acts to favor FM spin alignments at
highη. Because the “low–spin” states of minimal energy
are in fact stabilized by quantum corrections due to AF
spin fluctuations, the lowest energies at η= 0 are never
obtained for FM states, and therefore increasing ηdrives
a phase transition between states of differing spin and
orbital order. We show in Fig. 6 the transitions from
quasi–1D AF–correlated states at low η, for bothα= 0
andα= 1, to FM states of fixed orbital and spin order
(3b). The transitions occur at the values ηc(0) = 0.085
andηc(1) = 0.097, indicating that FM ordered states
may well compete in the physical parameter regime. We
note again that the energies in the superexchange limit
are lower by approximately a factor of two compared to
the direct–exchange limit simply because of the number
of available hopping channels.
Wenotealsothatthereisneverasituationinwhichthe
spin Hamiltonian becomes that of a Heisenberg model on16
0 0.05 0.1 0.15 0.2
η−0.8−0.6−0.4−0.2EAF/J, EFM/J
FIG. 6: (Color online) Minimum energies per bond obtained
for orbitally ordered phases, showing a transition as a func -
tion of Hund exchange ηfrom quasi–1D, AF–correlated to
FM ordered spin states. For the superexchange Hamiltonian
H∫of Sec. II ( α= 0), the transition is from the quasi–1D
spin state on orbital configuration (3c) [black, dashed line
from Eq. (3.9)] to the one–color orbital state (3a) [red, sol id
line from Eq. (3.4)]. For the direct–exchange Hamiltonian
Hd(α= 1), the transition is from the purely 1d spin state
on the one–color orbital state (3a) [green, dot–dashed line
from Eq. (3.22)] to the two–colour, avoided–blocking state
(3b) [blue, dotted line from Eq. (3.23)]. The transitions to
FM order as obtained from the mean–field considerations of
this section are marked by arrows.
a triangular lattice. This demonstrates again the inher-
entfrustrationintroducedbytheorbitalsector. However,
the fact that the ordered–state energy can never be low-
ered to the value EHAF=−3
8J, which might be expected
for a two–active FO situation on every bond, far less the
value−1
2Jwhich could be achieved if it were possible to
optimize every bond in some ordered configuration, can
be taken as a qualitative reflection of the fact that on
the triangular lattice the orbital degeneracy “enhances”
rather than relieves the (geometrical) frustration of su-
perexchange interactions (Sec. VI).
The limit of direct exchange ( α= 1) is found to be
quite different: the very strong tendency to favor spin
singlet states, and the inherent one–dimensionalityof the
model in this limit (one active hopping direction per or-
bital color), combine to yield no competitive states with
long–ranged magnetic order. Their optimal energy is
very poor because of the restricted number of hopping
channels, and coincides with the (“avoided–blocking”)
value for the model with only AO bonds, E=−1
6J.
Thus these states form part of a manifold with very
high degeneracy. However, even at this level it is clear
that more energy, meaning kinetic (from resonance pro-
cesses) rather than potential, may be gained by forming
quasi–1D Heisenberg–chain states with little or no inter-
chain coupling and only quasi–long–ranged magnetic or-
der. Studies of orbital configurations permitting dimer-
ized states are clearly required (Sec. IV). Finite Hund
exchange acts to favor ordered FM configurations, whichwill take over from chain–like states at sufficiently high
values ofη(Fig. 6).
Finally, orderedstatesofthemixedmodelshowanum-
ber of compromises. At α= 0.5, where the coefficients of
superexchange and direct–exchange are equal, some con-
figurations are able to return the unfrustrated sum of the
optimal states in each sector when considered separately,
namely−1
4J. However, superposition states, which are
not optimal in either limit, can redeem enough energy
from mixed processes to surpass this value, and in fact
the maximally superposed configuration (4b) is found
to minimize the energy over the bulk of the phase di-
agram. Still, the net energy of such states remains small
compared to expectations for a highly connected state
with three available hopping channels per orbital color.
Because of the directional mismatch between the diago-
nal and off–diagonal hopping sectors, no quasi–1D states
with only chain–like correlations are able to lower the
ordered–state energy in the intermediate regime.
IV. DIMER STATES
As shown in Sec. II, the spin–orbital model on a sin-
gle bond favors spin or orbital dimer formation in the
superexchange limit, and spin dimer formation in the
direct–exchange limit. The physical mechanism respon-
sible for this behavior is, as always, the fluctuation en-
ergy gain from the highly symmetric singlet state. On
the basis of this result, combined with our failure to find
any stable, energetically competitive states with long–
ranged spin and orbital order in either limit of the model
(Sec. III), we proceed to examine states based on dimers.
Given the high connectivity of the triangular lattice,
dimer–based states are not expected a priorito be capa-
ble of attaining lower energies than ordered ones, and
if found to be true it would be a consequence of the
high frustration, which as noted in Sec. I has its ori-
gin in both the interactions and the geometry. Here we
consider static dimer coverings of the lattice, and com-
pute the energies they gain due to inter–singlet corre-
lations. The tendency towards the formation of singlet
dimerstates will be supportedby the numericalresultsin
Sec. V, which will also address the question of resonant
dimer states.
A. Superexchange model
Motivated by the fact that the spin and orbital sectors
inHs(2.8) are not symmetrical, we proceed with a sim-
ple decoupling of spin and orbital operators. Extensive
research on spin–orbital models has shown that this pro-
cedure is unlikely to capture the majority of the physical
processes contributing to the final energy, particularly
in the vicinity of highly symmetric points of the general
Hamiltonian. The results to follow are therefore to be
treated as a preliminary guide, and a basis from which17
to consider a more accurate calculation of the missing
energetic contributions. We remind the reader that the
notation FO and AO used in this subsection is again that
obtained by performing a local transformation on one
site of every dimer. As noted in Sec. IIB, this procedure
is valid for the discussion of states based on individual
dimerized bonds, where it represents merely a notational
convenience. For FO configurations, which in the origi-
nal basis have different orbital colors, one might in prin-
ciple expect that, because of the color degeneracy, there
should be more ways to realize these without frustration
than thereareto realizeAF spinconfigurations; however,
because of the directional dependence of the hopping, we
will find that this is not necessarily the case (below).
Thebasicpremiseofthespin–orbitaldecouplingisthat
if the spin (orbital) degrees of freedom on a dimer bond
form a singlet state, their expectation value /an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht
(/an}bracketle{t/vectorTi·/vectorTj/an}bracketri}ht) on the neighboring interdimer bonds will be
precisely zero. The optimal orbital (spin) state of the
interdimer bond may then be deduced from the effec-
tive bond Hamiltonian obtained by decoupling. Because
Hsdepends on the number of electrons on the sites of
a given bond which are in active orbitals, and this num-
ber is well defined only for the dimer bonds, the effective
Hamiltonian will be obtained by averaging over all occu-
pation probabilities. In contrast to the pure Heisenberg
spin Hamiltonian, here the interdimer bonds contribute
with finite energies, and the dimer distribution must be
optimized. A systematic optimization will not be per-
formed in this section, where we consider only represen-
tative dimer coverings giving the semi–quantitative level
ofinsightrequiredasapreludetoaddingdimerresonance
processes (Sec. V).
On the triangular lattice there are three essentially
different types of interdimer bond, which are shown in
Fig. 7). For a “linear” configuration [Fig. 7(a)], the num-
ber of electrons in active orbitals on the interdimer bond
is two; for the 8 possible configurations where one dimer
bond is aligned with the interdimer bond under consider-
ation [Fig. 7(b)], the number is one on the corresponding
site and one or zero with equal probability on the other;
for the 14 remaining configurations where neither dimer
bond is aligned with the interdimer bond [Fig. 7(c)], the
number is one or zero for both sites. The number of elec-
trons in active orbitals is then two for type (7a), two or
one, each with probability 1/2, for type (7b), and two,
one or zero with probabilities 1/4, 1/2, and 1/4 for type
(7c).
The effective interdimer interactions for each type of
bond can be deduced in a manner similar to the treat-
ment of the previous section. Considering first the situa-
tion for a bond of type (7a) with (os/st) dimers, setting
/an}bracketle{t/vectorTi·/vectorTj/an}bracketri}ht= 0 yields one high–spin and two low–spin terms
which contribute
H(os,7a)
1(0) =−1
4Jr1/parenleftBig
/vectorSi·/vectorSj+3
4/parenrightBig
,
H(os,7a)
2(0) =3
4Jr2/parenleftBig
/vectorSi·/vectorSj−1
4/parenrightBig
, (4.1)(a) (b)
(c)
FIG. 7: (Color online) Types of interdimer bond differing in
effective interaction due to dimer coordination: (a) “linea r”,
(b) “semi–linear”, (c) “non–linear”.
H(os,7a)
3(0) =1
6J(r3−r2)/parenleftBig
/vectorSi·/vectorSj−1
4/parenrightBig
.
ClearlyH(os,7a)
1favors FM (high–spin) interdimer spin
configurations with coefficient1
4, whileH(os,7a)
2and
H(os,7a)
3favor AF (low–spin) configurations with coeffi-
cient3
8(both atη= 0). Because r1exceedsr2andr3
when Hund exchangeis finite, one expects a criticalvalue
ofηwhere FM configurations will be favored. Simple al-
gebraic manipulations using all three terms suggest that
this value, which should be relevant for a linear chain of
(os/st)dimers, is ηc=1
8. In the limit η→0, the effective
bond Hamiltonian simplifies to
H(os,7a)
eff(0) =1
2J/parenleftbigg
/vectorSi·/vectorSj−3
4/parenrightbigg
.(4.2)
For a bond of type (7a) with (ss/ot) dimers, setting
/an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht= 0 on the interdimer bond yields
H(ss,7a)
1(0) =3
4Jr1/parenleftbigg
/vectorTi·/vectorTj−1
4/parenrightbigg
,
H(ss,7a)
2(0) =−1
4Jr2/parenleftbigg
/vectorTi·/vectorTj+3
4/parenrightbigg
, (4.3)
H(ss,7a)
3(0) =−1
6J(r3−r2)/parenleftbigg
/vectorTi×/vectorTj+1
4/parenrightbigg
.
HereH(ss,7a)
1favorsAO configurations with coefficient3
8,
whileH(ss,7a)
2andH(ss,7a)
3both favor FO configurations
with coefficient1
4(atη= 0). Over the relevant range of
Hund exchangecoupling, 0 <η<1/3, there is no change
in sign and AO configurations are always favored. The
effective bond Hamiltonian for η→0 is
H(ss,7a)
eff(0) =1
2J/parenleftbigg
/vectorTi·/vectorTj−3
4/parenrightbigg
.(4.4)
For bonds of type (7b), when only one electron occu-
pies an active orbital the corresponding decoupled inter-
dimer bond Hamiltonians are, for (os/st) dimers,
H(os,1)
1(0) =−1
4Jr1/parenleftBig
/vectorSi·/vectorSj+3
4/parenrightBig
,18
H(os,1)
2(0) =1
4Jr2/parenleftBig
/vectorSi·/vectorSj−1
4/parenrightBig
,(4.5)
H(os,1)
3(0) = 0.
The final interdimer interaction is obtained by averag-
ing over these expressions and those (4.1) for two active
orbitalsper bond, and takes the rathercumbersome form
H(os,7b)
eff(0) =1
12J(r3+5r2−3r1)/vectorSi·/vectorSj
−1
48J(9r1+5r2+r3),(4.6)
which reduces in the limit η→0 to
H(os,7b)
eff(0) =1
4J/parenleftbigg
/vectorSi·/vectorSj−5
4/parenrightbigg
.(4.7)
For (ss/ot) dimers, the situation cannot be formulated
analogously, because if only one electron on the bond
is active, the orbital state of the other electron has no
influence on the hopping process, i.e./vectorTi·/vectorTjis not a
meaningful quantity. The resulting expressions lead then
to
H(ss,7b)
eff(0) =1
8J(3r1−r2)/vectorTi·/vectorTj−1
12J(r3−r2)/vectorTi×/vectorTj
−1
48J(9r1+5r2+r3), (4.8)
which has the η→0 limit
H(ss,7b)
eff(0) =1
4J/parenleftbigg
/vectorTi·/vectorTj−5
4/parenrightbigg
.(4.9)
Finally, forabondoftype(7c), thereisnocontribution
from interdimer bond states with no electrons in active
orbitals, so the above results [(4.1, 4.5) and (4.3, 4.8)]
are already sufficient to perform the necessary averaging.
With (os/st) dimers
H(os,7c)
eff(0) =1
48J(2r3+13r2−9r1)/vectorSi·/vectorSj
−1
192J(27r1+13r2+2r3),(4.10)
which reduces in the limit η→0 to
H(os,7c)
eff(0) =1
8J/parenleftbigg
/vectorSi·/vectorSj−7
4/parenrightbigg
,(4.11)
while for (ss/ot) dimers,
H(ss,7c)
eff(0) =1
16J(3r1−r2)/vectorTi·/vectorTj−1
24J(r3−r2)/vectorTi×/vectorTj
−1
192J(27r1+13r2+2r3),(4.12)
which in the η→0 limit gives
H(ss,7c)
eff(0) =1
8J/parenleftbigg
/vectorTi·/vectorTj−7
4/parenrightbigg
.(4.13)These results have clear implications for the nearest–
neighbor correlations in an extended system. By inspec-
tion, systems composed of either type of dimer would
favor AF (spin) and AO interdimer bonds, to the extent
allowed by frustration, and “linear” [type (7a)] bonds
over “semi–linear” [type (7b)] bonds over “non–linear”
[type (7c)] bond types in Fig. 7, to the extent allowed
by geometry. Discussion of this type of state requires in
principletheconsiderationofallpossibledimercoverings,
but will be restricted here to a small number of periodic
arrays which illustrate much of the essential physics of
extended dimer systems within this model.
We begin by considering the periodic covering of
Fig. 8(a), a fully linear conformation (of ground–state
degeneracy 12) whose interdimer bond types (Table I)
maximize the possible number of bonds of type (7a).
The counterpoint shown in Fig. 8(b) consists of pairs
of dimer bonds with alternating orientations in two of
the three lattice directions, and constitutes the simplest
configuration minimizing (to zero) the number of type–
(7a) interdimer bonds. The coverings in Figs. 8(c) and
(d) have the same property. These configurations exem-
plify a quite general result, that any dimer covering in
which there are no linear configurations [type (7a)] of
any pair of dimers will have 1/3 type–(7b) bonds, and
thus the remaining 1/2 of the bonds must be of type
(7c). The coverings shown in Figs. 8(a) and (b, c, d)
represent the limiting cases on numbers of each type of
bond, in that any random dimer covering will have val-
ues between these. Indeed, it is straightforward to argue
that, in changes of position of any set of dimers within
a covering, the creation of any two bonds of type (7b)
will destroy one of type (7a) and one of type (7c), and
conversely.
Havingestablishedthiseffectivesumrule, weturnnext
to the energies of the dimer configurations. First, for
both types of dimer [(os/st) and (ss/ot)], all states with
equal numbers of each bond type are degenerate, sub-
ject to equal solutions of the frustration problem. Next,
if frustration is neglected, it is clear from Eqs. (4.2,4.4),
(4.7,4.9), and (4.11,4.13), that the AF and AO energy
values for the three bond types (obtained by substitut-
ing−1
4for/vectorSi·/vectorSjand/vectorTi·/vectorTj) are respectively −1
2J,−3
8J
and−1
4J, which, when taken together with the sum rule,
suggest a very large degeneracy of dimer covering ener-
gies.
Returning to the question of frustration, a covering of
minimal energy is one which both minimizes the number
TABLE I: Occurrence probabilities for bonds of each type for
four simple periodic dimer coverings of the triangular latt ice.
configuration dimer bond (7a) bond (7b) bond (7c)
Fig. 8(a)1
61
602
3
Fig. 8(b)1
601
31
2
Fig. 8(c)1
601
31
2
Fig. 8(d)1
601
31
219
(a) (b)
(c) (d)
FIG. 8: (Color online) Periodic dimer coverings on the trian -
gular lattice, each representative of a class of coverings: (a)
linear; (b) plaquette; (c) 12–site unit cell; (d) “zig–zag” .
of FM or FO bonds, and ensures that they fall on bonds
of type (7c); both criteria are equally important. For
the dimer covering (8a), with maximal aligned bonds,
it is possible by using the spin (for (os/st) dimers) or
orbital (for (ss/ot) dimers) configuration represented by
the arrows in Fig. 9(a) to make the number of frustrated
(FM/FO) interdimer bonds equal to 1/6 of the total.
Bearing in mind that the 1/6 of bonds covered by dimers
are also FM/FO, and that at least 1/3 of bonds on the
triangular lattice must be frustrated for collinear spins,
this number is an absolute minimum. [Here we do not
considerthe possibilityof non–collinearorderof the non–
singlet degree of freedom.] Further, for this configuration
one observes that all of the FM/FO bonds already fall on
bondsoftype(7c), providinganoptimalcasewith energy
Edim(0) =−J/parenleftbigg1
6+1
6·1
2+1
2·1
4+1
6·3
16/parenrightbigg
=−13
32J (4.14)
atη= 0. This value constitutes a basic bound which
demonstrates that a simple, static dimer covering has
lowerenergythan any long–range–orderedspin ororbital
state discussed in Sec. III in this limit ( α= 0) of the
model.
It remains to establish the degeneracy of the ground–
state manifold of such coverings, and we provide only
a qualitative discussion using further examples. If al-
ternate four–site (dimer pair) clusters in Fig. 8(a) are
rotated to give the covering of Fig. 8(b), the minimal
frustration is spoiled: by analogy with Fig. 9, it is easy
to show that, if only 1/6 of the bonds are to be frus-(a) (b)
FIG. 9: (Color online) Spin or orbital configurations (black
arrows) within (a) linear and (b) zig–zag orbital– or spin–
singlet dimer coverings of the triangular lattice. The numb er
of frustrated interdimer bonds is reduced to 1/6 of the total ,
and all are of type (7c). This figure emphasizes that for the
spin–orbital model, dimer singlet formation does not exhau st
the available degrees of freedom.
trated, then they are of type (7b), and otherwise 1/3
of the bonds are frustrated if all are to be of type (7c).
On the periodic 12–site cluster [Fig. 8(c)], one may place
three four–site clusters in each of the possible orienta-
tions, which as above removes all bonds of type (7a) and
maximizes those of type (7b). Within this cluster it is
possible to have only four frustrated interdimer bonds
out of 18, while between the clusters there is again an
arrangement of the spin or orbital arrows ( cf.Fig. 9)
with only six FM or FO bonds out of 24, for a net total
of 1/6 frustrated interdimer bonds, of which half are of
type (7b). The covering of Fig. 8(d) represents an exten-
sion ofthe procedureof enlargingunit cells and removing
four–site plaquettes, which demonstrates that it remains
possiblein the limit ofno type–(7a) bonds to reduce frus-
tration to 1/6 of the bonds, and to bonds of type (7c)
[Fig. 9(b)], whence the energy of the covering is again
Edim(0) =−13
32(4.14). Thus it is safe to conclude that,
for the static–dimer problem, the ground–state manifold
forα= 0 consists of a significant number of degener-
ate coverings. We do not pursue these considerations
further because of degeneracy lifting by dimer resonance
processes, and because the energetic differences between
staticdimerconfigurationsarelikelytobedwarfedbythe
contributions from dimer resonance, the topic to which
we turn in Sec. V.
B. Direct exchange model
The very strong preference for bond spin singlets (the
factor of 4 in Eq. (2.21)] suggests that dimer states will
also be competitive in this limit, even though only 1/6 of
the bonds may redeem an energy of −J. Following the
considerations and terminology of the previous subsec-
tion, we note (i) that /an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht= 0 on interdimer bonds
and (ii) that in this case, interdimer bonds have energy
−1
4Jatη= 0 for types (7a) and (7b), and 0 for type
(7c). Because any state with a maximal number (1/6) of20
type–(7a) bonds must have only bonds of type (7c) for
the other 2/3 [states (8a)], such a state is manifestly less
favorable at α= 1 than those of type (8b)–(8d), where
there are no aligned pairs of dimers. In this latter case,
the full calculation gives
Edim(1) =−1
144J(9r1+19r2+8r3),(4.15)
andEdim(1) =−1
4Jforη= 0. This energy does now ex-
ceed that availablefrom the formationof Heisenbergspin
chains in one of the three lattice directions (Sec. IIIC),
which gave the value E1DAF(1) =−0.231J.
At the level of these calculations, the manifold of de-
generate states with this energy is very large, and its
counting is a problem which will not be undertaken here.
We will show in Sec. V that, precisely in this limit, no
dimerresonanceprocessesoccurandthestaticdimercov-
erings do already constitute a basis for the description of
the ground state. The question of fluctuations leading to
the selection of a particular linear combination of these
states which is of lowest energy, i.e.of a type of order–
by–disorder mechanism, is addressed in Ref. 49.
At finite values of the Hund exchange, this type of
statewillcomeintocompetitionwiththesimpleavoided–
blocking states which gain, with a FM spin state, an
energy
EFM(1) =−1
6Jr1, (4.16)
as 2/3 of the bonds contribute with an energy of −1
4Jr1.
The critical value of ηrequired to drive the transition
from the low–spin dimerized state to the FM state is
found to be
ηc= 0.1589. (4.17)
C. Mixed model
Because both of the endpoints, α= 0 andα= 1, favor
dimerized states over states of long–ranged order, it is
natural to expect that a dimer state will provide a lower
energy also at α= 0.5. However, we remind the reader
that there are no intermediate dimer bases, and caution
there is no strong reason to expect one or other of the
limiting dimer states to be favored close to α= 0.5. By
inspection, the energy of an α>0 state can be obtained
by direct addition of the diagonal interdimer bond con-
tributions in an (ss/ot) or (os/st) dimer state, which is
established by pure off–diagonalhopping, because no site
occupancies arise which allow mixed processes. For the
same reason, no interdimer terms impede a calculation
of the energy of an α <1 state by summing the off–
diagonal interdimer bond contributions in a spin–singlet
dimer state stabilized by purely diagonal processes. We
will not analyze the static dimer solutions for the inter-
mediate regime in great detail, and provide only a crudeestimate of the α= 0.5 energy by averagingoverboth re-
sults at the limits of their applicability. We will make no
attempt here to exclude other forms of disordered state
atα= 0.5, and return to this question in Sec. V.
For each type of bond it is straightforward to compute
the energy gained from interdimer hopping processes of
the type not constituting the dimer state, and the re-
sults are shown in Table II. The first four lines give the
energies per bond from diagonal hopping processes oc-
curring on the bonds of the different α= 0 dimer states,
and conversely for the final two lines. It is clear that the
occupations of type (7a) bonds preclude any hopping of
the opposite type. For α= 0 dimer configurations, the
interdimer diagonal hopping on (7b) bonds is always of
avoided–blocking type, while on (7c) bonds a blocking
can occur, and like the other terms is evaluated using
/an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht. Forα= 1, off–diagonal hopping on the inter-
dimer bonds is evaluated with /an}bracketle{t/vectorSi·/vectorSj/an}bracketri}ht= 0 between the
spinsinglets: allprocesseson(7b)bondsarethoseforone
active orbital; complications arise only for (7c) bonds,
where an interdimer bond between parallel dimers has
two active AO orbitals, while one between dimers which
are not parallel has two active FO orbitals.
Atα= 0.5, the energy of an (os/st) or (ss/ot) dimer
state augmented by diagonal hopping processes is mini-
mized by states (8a) and (8d): the interdimer bond con-
tributions of all coverings in Fig. 8 are equal, despite the
differenttypecounts, soonlythe α= 0energyisdecisive.
Atη= 0,
E(8a)
o(0.5) =−1
2/parenleftbigg13
32+2
3·1
4/parenrightbigg
J=−55
192J,(4.18)
E(8d)
o(0.5) =−1
2/parenleftbigg13
32+1
3·1
8+1
2·1
4/parenrightbigg
J
=−55
192J. (4.19)
The energy of a spin–singlet dimer state augmented by
off–diagonal hopping is minimal in states (8b) and, cu-
riously, (8a): although the latter has explicitly a worse
ground–state energy than the other states shown, the
effect of the additional hopping is strong, not least be-
cause all interdimer type–(8c) bonds arebetween parallel
TABLE II: Additional interdimer bond energies at α= 0.5
due respectively to (i) diagonal hopping occurring in a stat e
(designated by α= 0) stabilized by off–diagonal processes
and (ii) off–diagonal hopping in a state ( α= 1) stabilized by
diagonal processes.
bond (7a) (7b) (7c)
α= 0, (os/st), AF 0 −1
16(r1+r2)−1
8(r1+r2)
α= 0, (os/st), FM 0 −1
8r1 −1
4r1
α= 0, (ss/ot), AO 0 −1
32(3r1+r2)−1
16(3r1+r2)
α= 0, (ss/ot), FO 0 −1
32(3r1+r2)−1
16(3r1+r2)
α= 1,/bardbldimers 0 −1
16(3r1+r2)−1
8(3r1+r2)
α= 1, non– /bardbldimers 0 −1
16(3r1+r2)−1
12(2r2+r3)21
dimers. Thus at η= 0,
E(8a)
d(0.5) =−1
2/parenleftbigg5
24+1
2·2
3/parenrightbigg
J=−13
48J,
E(8b)
d(0.5) =−1
2/parenleftbigg1
4+1
3·1
4+1
2·2
3·1
2+1
2·1
3·1
4/parenrightbigg
J
=−13
48J. (4.20)
Despite the fact that these are two completely differ-
ent expansions, it is worth noting that the two sets of
numbers are rather similar, which occurs because the
significantly inferior energy of the α= 1 ground state
is compensated by the significantly greater interdimer
bond energies available from off–diagonal hopping pro-
cesses. However, this result also implies that no special
combinations of diagonal and off–diagonal dimers can be
expected to yield additional interdimer energies beyond
this value.
Takingthe covering(8a)asrepresentativeofthelowest
available energy, but bearing in mind that many other
states lie very close to this value, an average over the two
approaches yields
E(8a)
dim(0.5) =−107
384J (4.21)
atη= 0. This number is no longer lower than the value
obtained in Sec. IIID for fully ordered states gaining en-
ergy from mixed processes, raising the possibility that
non–dimer–based phases may be competitive in the in-
termediate regime, where neither of the limiting types
of dimer state alone is expected to be particularly suit-
able. However, we will not investigate this question more
systematicallyhere, and cautionthat the approximations
made both in Sec. IIID and here make it difficult to draw
a definitive conclusion.
D. Summary
The results of this section make it clear that static
dimer states, while showing the same energetic trend,
are considerably more favorable than any long–range–
ordered states (Sec. III) over most of the phase diagram.
As a function of α, the dimer energy increases mono-
tonically from −13
32Jto−1
4J, and both end–point values
also lie below the results obtained for quasi–1D spin–
disordered states in Sec. III. We stress that the results of
this section are provisional in the sense that we have not
performed a systematic exploration of all possible dimer
coverings, but rather have focused on a small number
of examples illustrative of the limiting cases in terms of
interdimer bond types. More importantly, we have con-
sidered only static dimer coverings with effective inter-
dimer interactions: the kinetic energy contributions due
to dimer resonance processes for all values of α <1 are
missing in this type of calculation. For this reason, wehave also refrained from investigating higher–order pro-
cesses, which may select particular dimer states from a
manifold of static coverings degenerate at the level of the
current considerations. Gaining some insight into the
magnitude and effects of resonance contributions is the
subject of the following section.
V. EXACT DIAGONALIZATION
A. Clusters and correlation functions
In this Section we present results obtained for small
systems by full exact diagonalization (ED). Because each
site has two spin and three orbital states, the dimension
of the Hilbert space increases with cluster size as 6N,
whereNis the number of sites. As a consequence, we
focus here only on systems with N= 2, 3, and 4 sites:
all three clusters can be considered as two–, three– or
four–site segments of an extended triangular lattice, con-
nected with periodic boundary conditions. For the single
bond and triangle this only alters the bond energies by
a factor of two, a rescaling not performed here, but for
the four–site system it is easy to see that the intercluster
bonds ensure that the system connectivity is tetrahedral.
We will also compare some of the single–bond and tetra-
hedron results with those for a four–site chain. Other ac-
cessible cluster sizes ( N= 5 and 6) yield awkwardshapes
which disguise the intrinsic system properties. Indeed we
will emphasize throughout this Section those features of
our very small clusters which can be taken to be generic,
and those which are shape–specific.
Given the clear tendency to dimerization illustrated in
Secs. III and IV, it is to be expected that spin correla-
tion lengths in all regimes of αare very small. To the
extent that the behavior of the model for any parameter
setis drivenby localphysics, the clusterresultsshouldbe
highly instructivefor such trends asdimer formation, rel-
ative roles of diagonal and off–diagonal hopping, dimer
resonance processes, lifting of degeneracies both in the
orbital sector and between states of (os/st) and (ss/ot)
dimers, and the importance of joint spin–orbital corre-
lations. However, generic features of extended systems
which cannot be accessed in small clusters are those con-
cerning questions of high system degeneracy and subtle
selection effects favoring specific states.
We will compute and discuss the cluster energies, de-
generacies, site occupations, bond hopping probabili-
ties in diagonal and off–diagonal channels (discussed in
Sec. VC), and the spin, orbital, and spin–orbital (four–
operator) correlation functions. All of these quantities
will be calculated for representative values of αandη
coveringthefullphasediagram,andeachcontainsimpor-
tant information of direct relevance to the local physics
properties listed in the previous paragraph. Although
the systems we study are perforce rather small, we will
showthat onemayrecognizein them anumber ofgeneral
trends valid also in the thermodynamic limit.22
We introduce here the three correlation functions,
which for a bond /an}bracketle{tij/an}bracketri}htoriented along axis γare given
respectively by
Sij≡1
d/summationdisplay
n/angbracketleftbig
n/vextendsingle/vextendsingle/vectorSi·/vectorSj/vextendsingle/vextendsinglen/angbracketrightbig
, (5.1)
Tij≡1
d/summationdisplay
n/angbracketleftbig
n/vextendsingle/vextendsingle/vectorTiγ·/vectorTjγ/vextendsingle/vextendsinglen/angbracketrightbig
, (5.2)
Cij≡1
d/summationdisplay
n/angbracketleftbig
n/vextendsingle/vextendsingle(/vectorSi·/vectorSj)(/vectorTiγ·/vectorTjγ)/vextendsingle/vextendsinglen/angbracketrightbig
−1
d2/summationdisplay
n/angbracketleftbig
n/vextendsingle/vextendsingle/vectorSi·/vectorSj/vextendsingle/vextendsinglen/angbracketrightbig/summationdisplay
m/angbracketleftbig
m/vextendsingle/vextendsingle/vectorTiγ·/vectorTjγ/vextendsingle/vextendsinglem/angbracketrightbig
,(5.3)
wheredis the degeneracy of the ground state. The
definitions of the spin ( Sij) and orbital ( Tij) correla-
tion functions are standard, and we have included ex-
plicitly all of the quantum states {|n/an}bracketri}ht}which belong to
the ground–state manifold. The correlation function Cij
(5.3) contains information about spin–orbital entangle-
ment, as defined in Sec. I: it represents the difference
between the average over the complete spin–orbital op-
erators and the product of the averagesover the spin and
orbital parts taken separately. It is formulated in such
a way that Cij= 0 means the mean–field decoupling of
spin and orbital operators on every bonds is exact, and
bothsubsystemsmaybetreatedindependentlyfromeach
other. Such exact factorizability is found9in the high–
spin states at large η; its breaking, and hence the need
to handle coupled spin and orbital correlations in a sig-
nificantly more sophisticated manner, is what is meant
by “entanglement” in this context.
B. Single bond
We consider first a single bond oriented along the c–
axis (Fig. 10). In the superexchange limit the active or-
bitals areaandb, while for direct exchange only the cor-
bitalscontributeinEq.(2.7). AsdiscussedinSec.IVA, a
single bond gives energy −Jin the superexchange model
(α= 0) [Fig. 10(a)], where the ground state has degener-
acyd= 6 atη= 0, from the two triply degenerate wave
functions (ss/ot) and (os/st). At finite η, the latter is
favored as it permits a greater energy gain from excita-
tions to the lowest triplet state in the d2configuration
[Eqs. (2.15) and (2.16)].
Although orbital fluctuations which appear in the
mixed exchange terms in Eq. (2.22) may in principle
contribute at α >0, one finds that the wave function
remains precisely that for α= 0,i.e.(ss/ot) degenerate
with (os/st), all the way to α= 0.5. Thus for the param-
eter choice specified in Sec. II, the ground–state energy
increases to a maximum of E0=−0.5Jhere [Fig. 10(a)].
The degeneracy d= 6 is retained throughout the regime
α<0.5, and only at α= 0.5 do several additional states
join the manifold, causing the degeneracy to increase to
d= 15. For the entire regime α∈(0.5,1], the ground0 0.2 0.4 0.6 0.8 1
α−0.8−0.6−0.4−0.20.0Sij , Tij , Cij0 0.2 0.4 0.6 0.8 1−1.0−0.8−0.6−0.4−0.20.0En/J
{a,b} orbitals c orbitals(ss/ot)
(st/os)(ss/cc)(a)
(b)
resonating static13
18
86
6
FIG. 10: (Color online) Evolution of the properties of a sing le
bondγ≡cas a function of αatη= 0: (a) energy spectrum
(solid lines) with degeneracies as shown; (b) spin ( Sij, filled
circles), orbital ( Tij, empty circles), and spin–orbital ( Cij,×)
correlations: Sij=Tij=Cij=−0.25 forα <0.5, while
Tij=Cij= 0 for α >0.5. The ground–state energy E0is
−Jfor both the superexchange ( α= 0) and direct–exchange
(α= 1) limits, and its increase between these is a result of the
scaling convention. The transition between the two regimes
occurs by a level crossing at α= 0.5. For α <0.5, the
two types of dimer wave function [(ss/ot) and (os/st)] are
degenerate ( d= 6) for resonating orbital configurations {ab},
while at α >0.5, the nondegenerate spin singlet is supported
by occupation of corbitals at both sites [(ss/cc)].
state is a static orbitalconfigurationwith corbitals occu-
pied at both sites to support the spin singlet, and d= 1.
The evolution of the spectrum with αdemonstrates not
only that superexchange and direct exchange are physi-
cally distinct, unable to contribute at the same time, but
that the two limiting wave functions are extremely ro-
bust, their stability quenching all mixed fluctuations for
a single bond. In this situation it is not the ground–state
energy but the higher first excitation energy which re-
veals the additional quantum mechanical degrees of free-
dom active at α= 0 compared to α= 1 [Fig. 10(a)].
The spin, orbital, and composite spin–orbital corre-
lation functions defined in Eqs. (5.1)–(5.3) give more
insight into the nature of the single–bond correlations.
The degeneracy of wavefunctions (ss/ot) and (os/st) for
0≤α <0.5 leads to equal spin and orbital correlation
functions, as shown in Fig. 10(b), and averaging over the
different states gives Sij=Tij=−1
4. As a singlet for
onequantityismatchedbyatripletfortheother, the two23
sectors are strongly correlated, and indeed Cij=−1
4, in-
dicating an entangled ground state. However, a consider-
ablymoredetailedanalysisispossible. Eachofthe sixin-
dividual states {|n/an}bracketri}ht}within the ground manifold has the
expectation value/angbracketleftbig
n/vextendsingle/vextendsingle(/vectorSi·/vectorSj)(/vectorTiγ·/vectorTjγ)/vextendsingle/vextendsinglen/angbracketrightbig
=−3
16, which
we assert is the minimum possible when the spin and
pseudospin are the quantum numbers of only two elec-
trons. It is clear that if the operator in Cijis evaluated
for any one of these states alone, the result is zero. En-
tanglement arisesmathematically because of the product
ofaveragesinthesecondtermofEq.(5.3), andphysically
because the ground state is a resonant superposition of
a number of degenerate states. We emphasize that the
resulting value, Cij=−1
4, is the minimum obtainable
in this type of model, reflecting the maximum possible
entanglement. We will show in Sec. VE that this value
is also reproduced for the Hamiltonian of Eq. (2.7) on
a linear four–site cluster, whose geometry ensures that
the system is at the SU(4) point of the 1D SU(2) ⊗SU(2)
model.9
By contrast, for α >0.5 those states favored by su-
perexchangebecomeexcited, andthespin–singletground
state hasSij=−3
4. The orbital configuration is charac-
terized by /an}bracketle{tnicnjc/an}bracketri}ht= 1, a rigid order which quenches all
orbital fluctuations (indeed, the orbital pseudospin vari-
ables/vectorTiγare zero). Thus the spin and orbital parts are
trivially decoupled, giving Cij= 0. Finally, at the tran-
sition point α= 0.5, averaging over all 15 degenerate
states yields Sij=−0.15,Tij=−0.10, andCij=−0.09.
In summary, the very strong tendency to dimer forma-
tion in the two limits α= 0 andα= 1 precludes any
contribution from mixed terms on a single bond, lead-
ing to a very simple interpretation of the ground–state
properties for all parameters.
C. Triangular cluster
We turn next to the triangle, which has one bond in
each of the lattice directions a,b, andc. Unlike the case
of the single bond, here the spin–orbital interactions are
strongly frustrated, in a manner deeper than and quali-
tatively different from the Heisenberg spin Hamiltonian.
Not only can interactions on all three bonds not be sat-
isfied at the same time, but also the actual form of these
interactions changes as a function of the occupied or-
bitals. The triangle is sufficient to prove (numerically
and analytically) the inequivalence in general of the orig-
inal model and the model after local transformation, for
frustration reasons discussed in Sec. IIB.
Webeginwiththeobservationthattheresultstofollow
are interpreted most directly in terms of resonant dimer
states on the triangle. This fact is potentially surprising,
given that the number of sites is odd and dimer forma-
tion must always exclude one of them, but emphasizes
the strong tendencies to dimer formation in all param-
eter regimes of the model. For their interpretation we
use a VB ansatz where it is assumed that one bond isoccupied by an optimal dimer state, minimizing its en-
ergy, and the final state of the system is determined by
the contributions of the other two bonds. This ansatz is
perforce only static, and breaks the symmetry at a crude
level, but enables one to understand clearly the effects of
the resonance processes captured by the numerical stud-
ies in restoring symmetries and lowering the total energy.
Considering first the VB ansatz for the superexchange
model, the energy −Jmay be gained only on a single
bond, in one of two ways. For the bond spin state to
be a singlet ( S= 0, (ss/ot) wave function), two differ-
ent active orbitals are occupied at both sites in one of
the orbital triplet states. The other two bonds lower the
total energy when the third site has an electron of the
third orbital color, each gaining an energy of −0.25Jdue
to the orbital interactions in Eq. (2.8). The energy of
the triangle is then EVB(0) =−0.5Jper bond, and the
cluster has a low–spin ( S=1
2) ground state with degen-
eracyd= 6 from the combination of the orbital triplet
and the spin state of the third electron. We stress that
the location ( a,b, orcbond) of the spin singlet does not
contribute to the degeneracybecause the three VB states
are mixed within the ground state by the contributing
off–dimer hopping processes. The same considerations
applied to an (os/st) dimer on one of the bonds of the
triangle shows that there is no color and spin state of
the third electron which allows both non–dimer bonds
to gain the energy −0.25Jsimultaneously, so the cluster
has a higher energy of −5
12Jper bond. Thus the VB
ansatz illustrates a lifting of the degeneracy between the
two types singlet state, the physical origin of which lies
in the permitted off–dimerfluctuation processes,and this
will be borne out in the calculations below. However, the
net spin state of the cluster has little effect on the esti-
mated energy of the (os/st) case, and its high–spin ver-
sion (S=3
2) will be a strong candidate for the ground
state at higher values of η. In the direct–exchange limit
(α= 1), the VB ansatz for spin singlets again returns an
energyEVB(1) =−5
12J, alsobecauseonlyonenon–dimer
bond cancontribute. Herethe off–dimerprocessesarere-
stricted to the third electron, which has arbitrary color
and spin, and cannot mix the three VB states, whence
the degeneracy is d= 12.
With this framework in mind, we turn to a description
of the numerical calculations at all values of α, beginning
withthemostimportantresults: at α= 0thedegeneracy
isd= 6, and hence VB resonance is confirmed, yielding
an energy very much lower than the static estimate, at
E0=−0.75Jper bond [Fig. 11(a)]. Thus strong orbital
dynamics and positional resonance effects operate in the
ground–state manifold. These break the (ss/ot)/(os/st)
symmetry, but act to restore other symmetries broken in
the VB ansatz. At α= 1, the energy and degeneracy
from the VB ansatz are exact, showing that the orbital
sector is classical andintroduces no resonance effects.
Figure 11(a) shows the complete spectrum of the tri-
angular cluster for all ratios of superexchange to di-
rect exchange, and in the absence of Hund coupling.24
0.0 0.2 0.4 0.6 0.8 1.0−0.80−0.60−0.40−0.200.000.20En/J
0.0 0.2 0.4 0.6 0.8 1.0−0.3−0.2−0.10.00.1Sij , Tij , Cij
0.0 0.2 0.4 0.6 0.8 1.0α0.00.20.40.6n1γ
(ss/ot) (ss/cc) orbitalsfluctuating{a,b} orbitals c orbitals(b)(a)
resonating static488
(c)
FIG. 11: (Color online) (a) Energy spectrum per bond for
a triangular cluster as a function of αforη= 0. Ground–
state degeneracies are as indicated, with d= 6 at α= 0
andd= 12 atα= 1. The arrows mark two transitions in the
nature of the (low–spin) ground state, which are further cha r-
acterized in panels (b) and (c). (b) Spin ( Sij, filled circles),
orbital (Tij, empty circles), and spin–orbital ( Cij,×) correla-
tion functions on the cbond. (c) Average electron densities in
thet2gorbitals at site 1 [Figs. 2(b,c)], showing n1b(solid line)
andn1a=n1c(dashed). The orbital labels are shown for a c
bond. All three panels show clearly a superexchange regime
forα <0.32, a direct–exchange regime for α >0.69, and an
intermediate regime (0 .32< α <0.69). A full description is
presented in the text.
Frustration of spin–orbital interactions is manifest in
rather dense energy spectra away from the symmetric
points, and in a ground–state energy per bond signifi-
cantly higher than the minimal value −J. Atα= 0
the spectrum is rather broad, with a significant number
of states of relatively low degeneracy due to the strong
fluctuations and consequent mixing of VB states in this
regime. However, even in this case the ground state is
well separated from the first excited state. As empha-
sized above, the ground–state energy, E0(0) =−0.75J,
is quite remarkable, demonstrating a very strong energy
gain from dimer resonance processes. By contrast, thevalueE0(1) =−5
12Jper bond found at α= 1 is exactly
equal to that deduced from the VB ansatz, demonstrat-
ing that this wave function is exact. Here the excited
states have high degeneracies, mostly of orbital origin,
and thus the spectrum shows wide gaps between these
manifolds of states; this effect is more clearly visible in
Fig. 12(c). The degeneracies shown in Fig. 11(a) are dis-
cussed below. In the intermediate regime, many of the
degeneracies at the end–points are lifted, leading to a
very dense spectrum. The two transitions at α= 0.32
andα= 0.69 appear as clear level–crossings: the inter-
mediate ground state is a highly excited state in both
of the limits ( α= 0,1), reinforcing the physical picture
of a very different type of wave function dominated by
orbital fluctuations and, as we discuss next, with little
overt dimer character.
Thecorrelationfunctionsforanyonebondofthetrian-
gle are shown in Fig. 11(a). That Sijis constant for all α
can be understood in the dimer ansatz by averaging over
the three configurations with one (ss/ot) or (ss) bond
and one ’decoupled’ spin on the third site, which gives
Sij=−1
4everywhere. The orbital and spin–orbital cor-
relation functions show a continuous evolution accompa-
nied by discontinuous changes at two transitions, where
the nature of the ground state is altered. The orbital
correlation function Tij=−1
12atα= 0 may be under-
stood as an average over the orbital triplet (+1
4) and the
two non–dimer bonds (each −1
4). Whenαincreases, this
value is weakened by orbital fluctuations, and undergoes
atransitionat α= 0.32toaregimewhereorbitalfluctua-
tions dominate, and Tijis close to zero. Above α= 0.69,
Tijbecomes positive, and approaches +1
12asα→1,
indicating that the wave function changes to the static–
dimer limit. While /vectorTicvanishes on the cbond here, the
cluster average has a finite value due to the contribution
Tij=1
4from the active non–singlet bond.
The spin–orbital correlation function Cijalso marks
clearly the three different regimes of α. Whenα<0.32,
Cijhasasignificantnegativevalue[Fig.11(b)]whosepri-
mary contributions are given by the four–operator com-
ponent/an}bracketle{t(/vectorSi·/vectorSj)(/vectorTiγ·/vectorTjγ)/an}bracketri}ht. Bycontrast, Cijis closetozero
in the intermediate regime, increasing again to positive
values forα>0.69. For all α>0.32,Cijcan be shown
to be dominated by the term −SijTijin Eq. (5.3), while
the four–operator contribution is small, and vanishes as
α→1. Thus entanglement, defined as the lack of fac-
torizability of the spin and orbital sectors, can be finite
even for vanishing joint spin–orbital dynamics.
Further valuable information is contained in the or-
bital occupancies at individual sites [Fig. 11(b)], which
show clearly the three different regimes. Although there
is always on average one electron of each orbital color
on the cluster, these are not equally distributed, as each
site participates only in two bonds and the symmetry is
broken. A representative site, labelled 1 in Figs. 2(b,c)]
has onlyaandcbonds, and hence the electron density
in theborbital is expected to differ from the other two.
The values nb= 2/3 andna=nc= 1/6 found in the25
regimeα<0.32 is understood readily as following from
a 1/3 average occupation of ( ab) and (bc) orbital triplet
states on the candabonds, respectively, and of an ( ac)
orbitaltriplet state on the bbond, which ensuresthat the
electron at site 1 is in orbital b[Fig. 2(b)]. By contrast,
in the regime α >0.69, only the two static orbital con-
figurations ( cc) and (bb) on thecandbbonds contribute,
andna=nc=1
2, whilenb= 0; when the system is in the
third possible spin–singlet state, with a ( bb) orbital state
on thebbond, the third electron is either aorc. Between
these two regimes(0 .32<α<0.69) is an extended phase
with equal averageoccupancy of all three orbitals at each
site, a potentially surprising result given the broken site
symmetry of the cluster. While this may be interpreted
as a restoration of the symmetry of the orbital sector by
strong orbital fluctuations, including those due to terms
inHm(2.22), it does not imply a higher symmetry of the
strongly frustrated interactions at α= 0.5.
ThespectraasafunctionofHundcoupling ηareshown
in Fig. 12 for the α= 0 andα= 1 limits, and at α= 0.5
to represent the intermediate regime. The lifting of de-
generacies as a function of ηis a generic feature. States
of higher spin are identifiable by their stronger depen-
dence onη, and in all three panels a transition is vis-
ible from a low–spin to a high–spin state. At α= 0
[Fig. 12(a)], the large low– ηgap to the next excited state
resultsinthetransitionoccurringattheratherhighvalue
ofηc= 0.158. This can be taken as a further indication
of the exceptional stability of the resonance–stabilized
ground state in the low–spin sector. The degeneracy
d= 12 of the high–spin state is discussed below.
The transition to the high–spin state at α= 1 also
occurs at a high critical value, ηc= 0.169 [Fig. 12(a)],
due in this case quite simply to the lack of competition
for the strong singlet states on individual bonds. Only
in the intermediate regime, 0 .32< α <0.69, where we
have shown already that the orbital state is quite differ-
ent from that in either limit [Fig. 11], is the transition
to the high–spin state much more sensitive to η. The or-
bitalfluctuations inthis phaseoccurboth in the low–spin
and the high–spin channel, making these very similar in
energy, and the transition occurs for α= 0.5 at only
ηc= 0.033 [Fig. 12(b)]. As expected from the α= 0
limit, where fluctuations are also strong, the characteris-
tic features of this energy spectrum are low degeneracy
and a semicontinuous nature. The location of the high–
spin transition as a function of αmay be used to draw a
phase diagram for the triangular cluster, which has the
rather symmetric form shown in Fig. 13.
Yet more information complementary to that in the
energy spectra and correlation functions can be obtained
by considering the average “occupation correlations” for
a bond/an}bracketle{tij/an}bracketri}ht /bardblγ,
P=/an}bracketle{tniγnjγ/an}bracketri}ht, (5.4)
Q=/an}bracketle{tniγ(1−njγ)/an}bracketri}ht+/an}bracketle{t(1−niγ)njγ/an}bracketri}ht,(5.5)
R=/an}bracketle{t(1−niγ)(1−njγ)/an}bracketri}ht. (5.6)
These probabilities ( P+Q+R= 1) reflect directly the0 0.05 0.1 0.15 0.2−1.0−0.6−0.20.2En/J0 0.05 0.1 0.15 0.2−1.2−0.8−0.40.0En/J
0 0.05 0.1 0.15 0.2
η−0.6−0.4−0.20.0En/J12
6
128(a)
(b)
(c)44
FIG.12: (Color online) Energyspectrafor atriangular clus ter
as a function of Hund exchange η. Energies are quoted per
bond, and shown for: (a) α= 0, (b) α= 0.5, and (c) α= 1.
The arrows indicate transitions at ηcfrom the low–spin ( S=
1/2) to the high–spin ( S= 3/2) ground state. The numbers
in all panels give degeneracies for the two lowest states for
η < ηcandη > ηc, respectively.
nature of the resonance processes contributing to the en-
ergy of the cluster states, in that they show the relative
importance of diagonal and off–diagonal hopping in the
ground states, and the evolution of these contributions
withαandη. We do not present these quantities in de-
tail here, but only summarize the overall picture of the
ground state whose understanding they help elucidate.
For this summary we return to the VB framework,
which accounts for many of the basic properties illus-
trated in the numerical results presented above. Con-
sidering first the low–spin states ( η= 0), atα= 0 the
ground state is given by one (ss/ot) dimer resonating
around the three bonds of the cluster; the third site has
the third color, its hopping gives a largevalue of Q= 1/3
(R= 2/3 from the pure superexchange channel) and its26
0 0.2 0.4 0.6 0.8 1
α0.000.040.080.120.160.20η
S=1/2S=3/2
FIG. 13: (Color online) Phase diagram of the triangular clus -
ter in the plane ( α,η). The spin states below and above the
transition line ηc(α) are respectively spin doublet ( S= 1/2)
and spin quartet ( S= 3/2).
spin an addition twofold degeneracy ( d= 3×2 = 6);
the orbital occupation of the (ss/ot) dimer is responsible
for the net 1/6:1/6:2/3 occupation distribution. When
α >0 the state remains essentially one with a resonat-
ing spin singlet, large Qand dominant R, but the orbital
triplet degeneracy is lifted to 2+1 and the ground–state
degeneracy to d= 2×2. All quantities, including P,Q,
andR, undergo discontinuous changes at α≃0.32, and
in this regime there is no longer strong evidence for an
interpretation in terms of resonating spin singlets: large
Q≃2/3 and the equal site occupations suggest the dom-
inance of mixed hopping processes which are not con-
sistent with either mechanism of singlet formation. The
retention of fourfold degeneracy across this transition is
largely accidental, and stems from twofold spin and or-
bital contributions. Only for α >0.69 is a spin–singlet
description once again valid: here Pbecomes significant,
astheresonatingsingletisstabilizedbydiagonalhopping
where the orbital has the bond color. The third site now
has one of two possible colors, its hopping keeps Qlarge,
and its spin yields another twofold degeneracy, as do the
orbital states, whence the net degeneracy is d= 2×2×2.
Only atα= 1 does the spin singlet become static, while
the third site still has either of the other colors, yield-
ing the symmetric result P= 1/9,Q=R= 4/9, and
degeneracy d= 12.
A similar description is possible in the high–spin states
atη >ηc. Atα= 0 the (os/st) dimer is rendered static
by the fact that hopping to the third site is now excluded
if it has the third color, and so instead this site takes one
of the singlet colors, a twofold degree of freedom which,
however, does not allow singlet motion; as a consequence
the orbital occupation is uniform (1/3:1/3:1/3), the hop-
pingprocessesincludecontributionsinthediagonalchan-
nel (P= 1/6,Q= 1/3,R= 1/2) and the degeneracy
isd= 3×4 = 12. For α >0 the orbital singlet mayagain resonate, but the third site retains one of the sin-
glet colors, orbital degeneracy is broken and d= 4. Once
again strong mixed processes dominate the intermedi-
ate regime, in which the spin state is not an important
determining factor. Above α= 0.69 the critical value
ηcrequired to overcome spin singlet formation becomes
largeagain,andthehigh–spinstateisonewhereavoided–
blocking processes (large Q) dominate, while broken or-
bital degeneracy keeps d= 4. Finally, at α= 1 one
obtains a pure avoided–blocking state with orbital con-
figurations acborcbafor the sites (1 ,2,3) of Fig. 2(c),
and consequent degeneracy d= 4×2 = 8. Thus it is
clear that the high– ηregion is also one yielding interest-
ing orbital models with nontrivial ground states, some
including orbital singlet states.
D. Tetrahedral cluster
Asinthecaseofthetriangularlattice,interpretationof
thenumericalresultsforthetetrahedralcluster(four–site
plaquette of the triangular lattice) is aided by considera-
tion of the VB ansatz in the two limits of superexchange
and direct–exchange interactions. The tetrahedral clus-
ter can accommodate exactly two dimers, with all inter-
dimer bonds of type (7c), and may thus be expected to
favor dimer–based states by simple geometry. However,
because the considerations and comparisons of this sub-
sectionaregivenonlyforthis singleclustertype, anybias
of this sort would not invalidate the results and trends
discussed here.
Because of the different forms and symmetries of the
spin and orbital sectors, there is no possibility of elemen-
tary spin–orbital operators, or of a ground–state wave
function which is a net singlet of a higher symmetry
group. The state with two orbital singlets on one pair
of bonds, two spin singlets on a second pair and pure
interdimer bonds on the third pair does exist, but is not
competitive: the energy cost forremovingthe orbital sin-
glets from the spin state maximizing their energy is by
no means compensated by the energy gain from having
two spin singlet bonds in an orbital state which also does
not maximize their energy. This result may be taken as
a further indication for the stability of dimers only in
the forms (os/st) or (ss/ot) in this model, and states of
sharedorbitalandspinsingletsarenotconsideredfurther
here. We return to this point in the following subsection,
in the context of the four–site chain.
We discuss only the energies of the VB wave functions
atη= 0. The minimal values obtainable for /an}bracketle{t/vectorSi·/vectorSj/an}bracketri}htand
/an}bracketle{t/vectorTiγ·/vectorTjγ/an}bracketri}htonthe interdimerbondsis −1/4,corresponding
to the AF/AO order. Thus at α= 0 the energy per bond
is
Eos/st(0) =Ess/ot(0) =−1
2J, (5.7)
with the degeneracy of the (ss/ot) and (os/st) wave func-
tionsrestoredasforthe singlebond. In the limit ofdirect27
exchange, the VB wave function consists of spin singlets
with twoactiveorbitalsofthe bond. Thegeometryofthe
cluster precludes these orbitals from being active on any
of the interdimer bonds, as a result of which the energy
per bond at η= 0 is
E(1) =−1
3J, (5.8)
and the ground state has degeneracy d= 3.
The most important results for the tetrahedron, which
we discuss in detail in the remainder of the subsection,
are the following. At α= 0, the exact ground state
energy isE0=−0.5833J: while not as large as in the
case of the triangle (Sec. VC), the resonance energy
contribution is very significant also for an even number
of cluster sites. The degeneracy of the numerical ground
state,d= 6, has its origin in only one of the (ss/ot) or
(os/st) wavefunctions (below), demonstratingagainthat
thereisnosensein whichthe quantumfluctuationsinthe
spinandorbitalsectorsaresymmetrical,andthattheVB
ansatz is capturing the essence of the local physics only
at a very crude level. At α= 1, as also for the triangular
cluster, the numerical results confirm not only the energy
given by the VB ansatz but every detail (degeneracies,
occupations, correlations) of this state.
We begin the systematic presentation of results by dis-
cussing the energy spectra at η= 0 [Fig. 14(a)]. As soon
as the degeneracies of the superexchange limit ( α= 0)
are broken, the spectrum becomes very dense, and re-
mains so across almost the complete phase diagram until
a level–crossing at αc= 0.92. The ground–state energy
for all intermediate values of αinterpolates smoothly to-
wards the transition, showing an initial decrease not ob-
served in the triangle: for the tetrahedron, mixed hop-
ping terms make a significant contribution, leading to an
overall energy minimum around α= 0.15. The domi-
nance of these terms is indicated by both the extremely
high value of αcand the steepness of the low– αcurve
where the transition to the static VB phase is finally
reached.
The bond correlation functions shown in Fig. 14(b)
illustrate the effects of corrections to the VB ansatz.
The spin correlations always have the constant value
Sij=−1
4, which is the most important indication of the
breakingofsymmetrybetween(ss/ot)and(os/st)sectors
at lowα: this value is an average over the spin–singlet
result−3/4 (on two bonds) and four bonds with value 0,
and thus it is clear that (ss/ot) dimers afford more res-
onance energy. However, the proximity of (os/st) states
suggests that a low value of ηc, the critical Hund cou-
pling for the transition to the high–spin state, is to be
expected (below).
The orbital correlations average to zero at α= 0, a
non–trivial result whose origin lies in the breaking of
nine–fold degeneracy within the orbital sector, and re-
main close to this value until the transition at αc. It is
worth noting here that Tij= 0 implies a higher frustra-
tion in the orbital sector than would be obtained in the0.0 0.2 0.4 0.6 0.8 1.0
α−0.3−0.2−0.10.00.1Sij , Tij , Cij0.0 0.2 0.4 0.6 0.8 1.0−0.6−0.5−0.4−0.3−0.2En/J
(a)
(b)2
1
FIG. 14: (Color online) (a) Energy spectrum per bond for
a tetrahedral cluster as a function of αforη= 0. Ground–
state degeneracies are as indicated, with d= 6 at α= 0
andd= 150 at α= 1. The arrow marks a transition in the
nature of the (low–spin) ground state. (b) Spin ( Sij, filled
circles), orbital ( Tij, empty circles), and spin–orbital ( Cij,×)
correlation functions on the cbond of the tetrahedral cluster
as functions of αforη= 0.
spin sector for an (os/st) state ( Sij=−1
12), which is due
to the complex direction–dependence of the orbital de-
grees of freedom. This phase is maintained across much
of the phase diagram, with only small changes to the cor-
relation functions, the negative value of Tijreflecting an
easing of orbital frustration. The lack of a phase transi-
tion throughout the region in which mixed processes are
also important suggests that a dimer–based schematic
picture of the ground state remains appropriate for the
four–site system, with only quantitative evolution as a
function of αuntilαc= 0.92. Atα= 1, the result
Tij=−1
6is the consequence of c–orbital operators on
the interdimer aandbbonds.
Significant spin–orbital correlations, Cij≃ −0.1 at
α= 0 [Fig. 14(b)], are found to be due exclusively to
the four–operator term at low α. While these negative
contributions drop steadily through most of the regime
α < α c, signifying a gradual decoupling of orbitals and
spins as the static limit ( α= 1) is approached, near αc
the negative value of Cijis again enhanced by the con-
tribution −SijTijdue to the interdimer bonds. Thus,
as for the triangle (Sec. VC), the entanglement is finite,28
0 0.05 0.1 0.15 0.2−1.0−0.8−0.6−0.4−0.20.0En/J
0 0.05 0.1 0.15 0.2
η−0.4−0.3−0.2−0.10.0En/J0 0.05 0.1 0.15 0.2−1.0−0.8−0.6−0.4−0.20.00.2En/J(a)
(b)
(c)
FIG. 15: (Color online) Energy spectra for a tetrahedral clu s-
ter as a function of Hundexchange η. Energies are quoted per
bond, and shown for: (a) α= 0, (b) α= 0.5, and (c) α= 1.
The arrows indicate transitions from the low–spin ( S= 0) to
the high–spin ( S= 2) ground state.
complete factorization is not possible, and a finite value
Cij=−1
24is found even at α= 1. We note here that on
the tetrahedron there is little information in the orbital
occupations, which are constant ( nγ=1
3) over the entire
phase diagram, demonstrating only the symmetry of this
cluster geometry, and are therefore not shown.
ThespectraasafunctionofHundcoupling ηareshown
for the three parameter choices α= 0, 0.5, and 1 in
Fig. 15. Once again, the spectra become very dense away
fromη= 0. Atα= 0 [Fig. 15(a)] high–spin states are
found also in the low–energy sector, as a consequence of
the near–degeneracy of (ss/ot) and (os/st) states, and
the high–spin transition occurs at a very low value of ηc
[Fig. 15(a)]. The direct–exchange limit is both qualita-
tively and quantitatively different, because the quantum
fluctuations and the corresponding energy gains are lim-
ited to the spin sector, making the low–spin states con-0 0.2 0.4 0.6 0.8 1
α0.000.040.080.120.160.20η
S=0S=2
FIG.16: (Color online)Phase diagram of thetetrahedral clu s-
ter in the plane ( α,η). As for the triangular cluster, the spin
states below and above the line ηc(α) are respectively singlet
(S= 0) and quintet ( S= 2), with no intermediate triplet
phase.
siderably more stable and giving ηc= 0.175 [Fig. 15(c)].
The spin excitation gap decreasesgraduallywith increas-
ingη, but until just below ηc, for all values of α, the spin
excitation is to S= 1 states. However, these triplet
states are never the ground state in the entire regime
ofη, a single transition always occurring directly into
anS= 2 state. In the intermediate regime represented
byα= 0.5, the energy spectrum is so dense that indi-
vidual states are difficult to follow (a more systematic
analysis of the spectra in different subspaces of Szis not
presented here). The high–spin transition occurs at the
relatively high value ηc= 0.136, due mainly to the large
energygains in the low–spinsector from mixed exchange.
Further evidence for the importance of the orbital exci-
tations in Hm(2.22) can be found in the broadening of
the spectrum which leads to the occurrence of quantum
states with weakly positive energies: for both superex-
change and direct–exchange processes, the Hamiltonians
are constructed as products of projection operators with
negative coefficients, so positive energies are excluded.
The low– to high–spin transition points at all values of
αcan be collected to give the full phase diagram of the
tetrahedron shown in Fig. 16. As shown above, in the
superexchange limit the high–spin state lies very close
to the low–spin ground state, and the transition to an
S= 2 spin quintet occurs at ηc= 0.017. We comment
here that this high–spin state is in no sense classical or
trivial, being based on orbital singlets which are stabi-
lized by strong orbital fluctuations, and emphasize again
that the high–spin sector also contains a manifold of rich
problems in orbital physics, which we will not consider
further here. The near degeneracy of (ss/ot) and (os/st)
states is further lifted in the presence of the mixed terms
inHm, raisingηcto values on the order of 0.12 acrossthe
bulk of the phase diagram. For no choice of parameters29
is a spin triplet state found at intermediate values of η.
The reentrant behavior close to α= 0.5 is an indication
oftheimportanceofmixedtermsinstabilizingalow–spin
state, the tetrahedral geometry providing one of the few
examples we have found of anything other than a direct
competition, andhenceaninterpolation,betweenthetwo
limiting cases. The rapid upturn in the limit of α→1
reflects the anomalous stability of the static VB states
in the direct–exchange limit. The very strong asymme-
try of the transition line in Fig. 16 contrasts sharply with
the near–symmetryabout α= 0.5 observed for the trian-
gle (Fig. 13), and shows directly the differences between
those features of the phase diagram which are universal
and those which are effects of even or odd cluster sizes in
a dimer–based system.
We close our discussion of the tetrahedral cluster with
a brief discussion of degeneracies and summary of the
picture provided by the VB ansatz with additional res-
onance. For the orbital occupation correlations and de-
generacies, we begin with the low–spin sector ( η= 0).
Atα= 0 one has two (ss/ot) VBs resonating around the
6 bonds of the cluster, a state characterized by P= 1/6,
Q= 1/3, andR= 1/2; however, a mixing of the orbital
triplet states lowers the degeneracy from 9 to d= 6.
Forα >0 the state is the same, with slow evolution of
P <1/6,Q >1/2, andR >1/3, but now mixed hop-
ping terms break all orbital degeneracies, giving d= 1.
Only when α>0.92 is the ground state more accurately
characterized as one based on spin singlets of the bond
color, with significant values of Pand the restoration of
anorbitaldegeneracy d= 2. Asα→1, thediagonalhop-
ping component is strengthened ( P→1/3) as the pair of
bond–colored spin singlets resonates, until at α= 1 they
become static and the degeneracy is d= 3.
For the high–spin states in the regime η>ηc, atα= 0
one has two resonating (os/st) VBs, with the hopping
channels unchanged and only the spin degeneracy d= 5.
This state is not altered qualitatively for any α <0.92,
a transition value independent of η. For 0.92< α <
1, orbital correlations are strongly suppressed and the
state is characterized by hopping processes largely of the
avoided–blocking type (one active orbital, Qdominant),
still withd= 5. Finally, α= 1 represents the limit of a
pure avoided–blocking state ( P= 0,Q= 2/3,R= 1/3),
where the degeneracy jumps to 150, a number which can
be understood as 5 (spin degeneracy) ×[6 (number of
two–colorstates with no bonds requiring spin singlets) +
24 (number of three–color states with no bonds requiring
spin singlets)].
E. Four–site chain
As a fourth and final case, we present results from
a linear four–site cluster. While not directly relevant
to the study of the triangular lattice, this system offers
further valuable insight into the intrinsic physics of the
spin–orbital model. The cluster is oriented along the c–0 0.2 0.4 0.6 0.8 1
α−0.6−0.4−0.20.0Sij , Tij , Cij0 0.2 0.4 0.6 0.8 1−1.0−0.8−0.6−0.4−0.20.0En/J
{a,b} orbitals c orbitals(ss/os) (ss/cc)(a)
(b)
resonating static1 1h
FIG. 17: (Color online) Evolution of the properties of the
four–site chain as a function of αatη= 0: (a) energy spec-
trum and (b) spin ( Sij, filled circles), orbital ( Tij, empty cir-
cles), and spin–orbital ( Cij,×) correlation functions. Both
panels show a transition occurring at a level crossing at
α= 4/7. In panel (a), the labels show a nondegenerate
ground state ( d= 1) in both regimes, which has predimi-
nantly spin singlet character at α >0.571, but both spin
and orbital singlet components at α <0.571. In panel (b),
Sij=Tij=Cij=−1
4forα <0.571 due to a resonating ( ab)
orbital configuration, while Tij=Cij= 0 forα >0.571 as a
consequence of the static corbital configuration.
axis with periodic boundary conditions. As for the single
bond (Sec. VB), only the aandborbitals contribute at
α= 0, where indeed one finds average electron densities
per sitenia=nib=1
2, andnic= 0. Likewise, at α= 1
only thecorbitals are occupied, with nic= 1, a result
dictated by the spin singlet correlations, which are fully
developed only for complete orbital occupation.
The energy per bond for the four–site chain in the
superexchange limit is again −J, as for a single bond
[Fig. 17(a)]: somewhat surprisingly, the bonds do not
”disturb” each other, and joint spin–orbital fluctuations
extend over the entire chain. However, in contrast to
a single bond, this behavior is due to only one quan-
tum state, the SU(4) singlet. In this geometry, only
one SU(2) orbital subsector is selected, and the result-
ingSU(2) ⊗SU(2) systemislocatedpreciselyattheSU(4)
point ofthe Hamiltonian.50Thus, exactly asin the SU(4)
chain, all spin, orbital and spin–orbital correlation func-
tions are equal, Sij=Tij=Cij=−0.25, as shown
in Fig. 17(b). For SijandTij, this result may be under-30
stoodasan averageoverequalprobabilitiesofsinglet and
triplet states on each bond. In more detail, the condition
set on the correlation functions by SU(4) symmetry12is
4
3/an}bracketle{t(/vectorSi·/vectorSj)(/vectorTic·/vectorTjc)/an}bracketri}ht=Sij=Tij, an equality also obeyed
by the single bond (Sec. VB). The product of Sijand
Tijin its definition ensures the identity for Cij. The
unique ground state is nevertheless a linear superposi-
tion of states expressed in the spin and orbital bases,
and has not only finite but maximal entanglement. This
state persists, with a perfectly linear α–dependence, all
the way to α= 1, but ceases to be the ground state
atα=4
7[Fig. 17(a)], where there is a level–crossing
with theα= 1 ground state (also perfectly linear). This
latter state has a completely different, fluctuation–free
orbital configuration, with pure c–orbital occupation at
every site, and gains energy solely in the direct–exchange
channel. The spins and orbitals are decoupled, Tijand
Cijvanish, and the spin state has Sij=−0.50: this re-
sult can be understood as an equal average over bond
states with /vectorSi·/vectorSj=−3
4and−1
4, and matches that ob-
tained for the four–site AF Heisenberg model with a res-
onating VB (RVB) ground state.2The energy at α= 1,
E0=−0.75J[Fig. 17(a)], is given directly by including
the constant term, −1
4Jper bond, in the definition of the
Hamiltonian (2.21).
The results for the linear four–site cluster demonstrate
again the competition between superexchange and direct
exchange. The orbital fluctuations arising due to the
mixed exchange term, Hm(2.22), are responsible for re-
moving the high degeneracies of the eigenenergies in the
limitsα= 0 andα= 1 [Fig. 17(a)]. In fact the spectrum
of the excited states is quasi–continuous in the regime
aroundα= 0.5, but has a finite spin and orbital gap ev-
erywhereother than the quantum critical point at α=4
7.
These chain results raise a further possibility for the
spontaneous formation at α= 0 of a 1D state not dis-
cussed in Sec. III. A set of (for example) c–axis chains,
with onlyaandborbitalsoccupied in the pseudospinsec-
tor, would createexactly the 1D SU(4) model, and would
therefore redeem an energy E=−3
4Jper bond from the
formation of linear, four–site spin–orbital singlets. The
energy of the triangular lattice would receive a further,
constant contribution from the cross–chain bonds, which
wascalculatedin Eq.(3.9) forgeneral η, andhence would
be given at η= 0 by
ESU(4)
1D(0) =−1
3·3
4J−1
6J=−5
12J. (5.9)
This energy represents a new minimum compared with
all of the results in Sec. III. That it was obtained from
a melting of both spin and orbital order confirms the
conclusion that ordered phases are inherently unstable
in this class of model, being unable to provide sufficient
energy to compete with the kinetic energy gains avail-
able through resonance processes. That its value is now
lower than that obtained for a static, 2D dimer covering
(Sec. IV) is not of any quantitative significance, given
the results of Sec. V confirming the importance of the0.0 0.2 0.4 0.6 0.8 1.0
α−0.8−0.6−0.4−0.20.0E0/J
FIG. 18: (Color online) Ground–state energy per bond as a
functionof α, obtainedwith η= 0for atriangular clusterwith
3 bonds (blue, dashed line), and a tetrahedral cluster with 6
bonds (red, solid line). For comparison, the energies obtai ned
from the VB ansatz in the limiting cases α= 0 and α=
1 are shown for the triangular cluster (blue, diamonds) and
tetrahedral cluster (red, yellow–filled, open circles); at α= 0
both VB energies are the same, while at α= 1 they match
the exact solutions. Green, upward–pointing triangles sho w
the static–dimer results of Sec. IV for the extended system,
and the black, dot–dashed line the lowest energy per bond
obtained for fully spin and orbitally ordered phases in Sec. III.
The violet, downward–pointing triangle shows the energy of
the orbitally ordered but spin–disordered Heisenberg–cha in
state at α= 0 [Eq. (3.9)] and the open, yellow–filled square
that of the analogous state at α= 1 [Eq. (3.22)], while the
cross shows the energy of the spin– and orbitally disordered ,
SU(4)–chain state [Eq. (5.9)].
positional resonance of dimers.
F. Summary
To summarize, we have shown in this section the re-
sults of exact numerical diagonalization calculations per-
formed on small clusters. Detailed analysis of ground–
state energies, degeneracies, site occupancies and a num-
ber of correlation functions can be used to extract valu-
able information about the local physics of the model
across the full regime of parameters. Essentially all of
the quantities considered show strong local correlations
and the dominance of quantum fluctuations of the short-
est range, with ready explanations in terms of resonating
dimer states.
We draw particular attention to the extremely low
ground–state energy of the triangular cluster, which
shows large gains from dimer resonance. The tetrahedral
cluster also has a very significant resonance contribution,
although more of its ground–state energy is captured at31
the level of a static dimer model. Such a VB ansatz
provides the essential framework for the understanding
of all the results obtained, even for systems with odd
site numbers. The energies and their evolution with α
contain some quantitative contrasts between even– and
odd–site systems, allowing further insight concerning the
range over which the qualitative features of the cluster
results extend.
Focusing in detail upon these energies, Fig. 18 sum-
marizes the exact diagonalization results at zero Hund
coupling, and provides a comparison not only with the
VB ansatz, but with all of the other results obtained
in Secs. III–V. From bottom to top are shown: the ex-
act cluster energies including all physical processes; the
clusterVB ansatz, showingthe importance ofdimer reso-
nance energy; the static VB ansatz for extended systems,
suggestingbycomparisonwithclusterstheeffectsofreso-
nance; the energies of “melted” states with 1D spin (and
orbital) correlations; the optimal energy of states with
full, long–ranged spin and orbital order.
Returning to the cluster results, their degeneracies can
be understood precisely, and demonstrate the restora-
tion of various symmetries due to resonance processes.
We provide a complete explanation for all the correla-
tion functions computed, and use these to quantify the
entanglement as a function of α,ηand the system size.
There is a high–spin transition as a function of ηfor all
values ofα, which sets the basic phase diagram and es-
tablishes a new set of disentangled orbital models at high
η.
The extrapolation of the cluster results to states of
extended systems, some approximations for which are
shown in Fig. 18, is not straightforward, and cannot be
expected to include any information relevant to subtle
selection effects within highly degenerate manifolds of
states. However, with the exception of the static–dimer
regime around α= 1, our calculations suggest that noth-
ing subtle is happening in this model over the bulk of
the phase diagram, where the physics is driven by large
energetic contributions from strong, local resonance pro-
cesses.
VI. RHOMBIC, HONEYCOMB, AND KAGOME
LATTICES
In Sec. I we alluded to the question of different sources
offrustrationincomplexsystemssuchasthe spin–orbital
model of Eq. (2.7). More specifically, this refers to the
relative effects of pure geometrical frustration, as under-
stood for AF spin interactions, and ofinteraction frustra-
tion of the type which can arise in spin–orbital models
even on bipartite lattices.7Because the interaction frus-
tration depends in a complex manner on system geom-
etry, no simple separation of these contributions exists.
In this section we alter the lattice geometry to obtain
somequalitativeresultswithabearingonthisseparation,
by considering the same spin–orbital model on the three(a) (b)
(c)
FIG. 19: (Color online) (a) Rhombic lattice, showing a two–
color orbitally ordered state. (b) Honeycomb lattice, show ing
a one–color orbitally ordered state. (c) Kagome lattice, sh ow-
ing a three–color orbitally ordered state.
simple lattice geometries which can be obtained from the
triangular lattice by the removal of active bonds or sites.
The geometries we discuss are rhombic, obtained by
removing all bonds in one of the three triangular lat-
tice directions [Fig. 19(a)], honeycomb, or hexagonal, ob-
tained by removing every third lattice site [Fig. 19(b)],
and kagome, obtained by removing every fourth lattice
site in a 2 ×2 pattern [Fig. 19(c)]. Simple geometrical
frustration is removed in the rhombic and honeycomb
cases, but for Heisenberg spin interactions the kagome
geometry is generally recognized (from the ground–state
degeneracy of both classical and quantum problems) to
be even more frustrated than the triangular lattice. We
consider only the α= 0 andα= 1 limits of the model,
andη= 0. We discuss the results for long–range–ordered
states (Sec. III) and for static dimer states (Sec. IV) for
all three lattice geometries. Here we do not enter into nu-
merical calculations on small clusters, and comment only
on those systems for which exact diagonalization may be
expected to yield valuable information not accessible by
analytical considerations.
A. Rhombic lattice
While the connectivity of this geometry is precisely
that of the square lattice, we refer to it here as rhombic
to emphasize the importance of the bond angles of the32
(a) (b)
FIG. 20: (Color online) Rhombic lattice with (a) columnar
and (b) plaquette dimer coverings.
chemical structure in maintaining the degeneracy of the
t2gorbitalsandindeterminingthenatureoftheexchange
interactions. It is worth noting that the spin–orbital
model (2.7) on this lattice may be realized in Sr 2VO4
(below). In the absence of geometrical frustration, the
spin problem created by imposing any fixed orbital con-
figuration selected from Sec. III (Figs. 3 and 4) is gen-
erally rather easy to solve. Further, at η= 0 both FM
and AF, and by extension AFF, spin states have equal
energies, leading to a high spin degeneracy.
Following Sec. III, the α= 0 energies for the majority
of the orbitally ordered states of Fig. 3 are
Erh
lro(0) =−1
2J (6.1)
per bond at η= 0 for a number of possible spin con-
figurations, whose degeneracy is lifted (in favor of FM
lines or planes) at finite η. Indeed, the only exceptions
to this rule occur for the three–color state [Fig. 3(d)]
and for orientations of the other states which preclude
hopping in one of the two lattice directions, whose tri-
angular symmetry properties are broken by the missing
bond. As noted in Sec. III, for superpositions is it the
exception rather than the rule for all hopping processes
to be maximized, but on the rhombic lattice this is pos-
sible for the states in Fig. 4(a) and some orientations of
those in Figs. 4(d) and 4(e).
Forα= 1, the energy limit even on the triangular lat-
tice wasset ratherby the number ofactive bonds than by
the problem of minimizing their frustration. Similar to
theα= 0 case, all states where the active hopping direc-
tion is one of the two lattice directions, plus in this case
state (3d), can redeem the maximum energy available,
Erh
lro(1) =−1
4J (6.2)
atη= 0, which is simply the avoided–blocking energy,
for a large number of possible spin configurations. Finite
Hund exchange favors FM spin states.
Turning to dimerized states, the calculation of the en-
ergy of any given dimer covering proceeds as in Sec. IV,
namely by counting for each the respective numbers of
bonds of types (7a), (7b), and (7c) [Fig. 7]. For the
rhombic lattice, lack of geometrical frustration meansthat all interdimer bonds can be chosen to be AF/AO.
The two most regular dimer coveringsof the rhombic lat-
tice with small unit cells may be designated as “colum-
nar” [Fig. 20(a)] and “plaquette” [Fig. 20(b)]. In both
cases, 1/4 of the bonds are the dimers, and by inspection
1/4 of the interdimer bonds in the columnar state are of
type (7a), while the remainder are (7c); by contrast, the
plaquette state has no type–(7a) bonds, 1 /2 type–(7b)
bonds, and the remainder are of type (7c). For α= 0,
the energies are
Erh
dc(0) =−1
4J−1
4·1
2J−1
2·1
4J=−1
2J,
Erh
dp(0) =−1
4J−1
2·3
8J−1
4·1
4J=−1
2J(6.3)
atη= 0, both for (ss/ot) and for (os/st) dimers. The de-
generacyofthese two limiting cases, in the sense of maxi-
malandminimalnumbersoftypes–(7a)and–(7b)bonds,
suggests a degeneracy of all dimer coverings at this level
of analytical sophistication. Further, all of these dimer
coverings are degenerate with all of the unfrustrated or-
dered states at η= 0. The selection of a true ground
state from this large manifold of static states (order–
by–disorder) would hinge on higher–order processes, but
these considerations are likely to be rendered irrelevant
by dimer resonance (Sec. V).
For the spin–singlet dimer states at α= 1 one finds
Erh
dc(1) =−1
4J−1
4·1
4J−1
2·0J=−5
16J,
Erh
dp(1) =−1
4J−1
2·1
4J−1
4·0J=−3
8J,(6.4)
atη= 0, and thus that, as for the triangular lattice, the
energy is minimized by dimer configurations excluding
linear interdimer bonds. This remains a large manifold
ofdimer coverings,whose energyis manifestly lowerthan
any of the possible orbitally ordered states in this limit
of the model, and within which order–by–disorder is ex-
pected to operate (Sec. V).49
The considerations of this subsection, extended to fi-
nite values of η, may be relevant in the understanding of
experimentalresultsforSr 2VO4. ThesesuggestweakFM
order,51accompanied by an AO order52which could be
interpreted as arising from the formation of dimer pairs.
When the oxygen octahedra distort, the threefold degen-
eracy of the t2gorbitals is lifted, to give a model con-
taining only two degenerate orbitals, dyzanddxz. This
leads to a situation with Ising–like superexchange inter-
actionsand quasi–1Dholepropagationin aneffective t–J
model.53
B. Honeycomb lattice
The situation for the honeycomb lattice is very simi-
lar to that for the rhombic case. Again the absence of
geometrical frustration makes it possible to obtain the33
(a)
(b)
FIG. 21: (Color online) Honeycomb lattice with (a) columnar
and (b) three–way dimer coverings.
minimal energy for a number of orbital orderings, with a
high spin degeneracy at η= 0. For pure superexchange
interactions, once again
Eh
lro(0) =−1
2J (6.5)
per bond, while in the direct–exchange limit
Eh
lro(1) =−1
4J, (6.6)
both atη= 0, for the same physical reasons as above.
For dimer states, on the honeycomb lattice all inter-
dimer bonds are by definition of type (7c), and again can
be made AF/AO because frustration is absent, so the
energies of all dimer coverings are de facto identical. By
way of demonstration, the two simplest regular configu-
rations, which we label “columnar”and “three–way”,are
shown in Fig. 21, and, from the fact that now 1 /3 of the
bonds contain dimers, their energies are
Eh
dc(0) =−1
3J−2
3·1
4J=−1
2J,
Eh
d3(0) =−1
3J−2
3·1
4J=−1
2J,(6.7)
per bond at α= 0 =η. Thus static dimer states are
again degenerate with unfrustrated ordered states in thesuperexchangelimit, anddetailedconsiderationofkinetic
processes would be required to deduce the lowest total
energy. In this context, the dimer coverings shown in
Fig. 21 exemplify two limits about which little kinetic
energy can be gained from resonance (Fig. 21(a), where
large numbers of dimers must be involved in any given
process)and in which kinetic energy gainsfrom processes
involving short loops [the three dimers around 2/3 of the
hexagons, Fig. 21(b)] are maximized.
Atα= 1, only the dimer energy is redeemed, and this
on 1/3 of the bonds, so
Eh
d(1) =−1
3J (6.8)
atη= 0 for a large manifold of coverings. This energy
is once again significantly better than any of the possi-
ble ordered states, a result which can be ascribed to the
low connectivity. That the ground state of the extended
system in this limit for both the rhombic and honeycomb
lattices involves a selection from a large number of nearly
degeneratestates suggeststhat numerical calculationson
small clusters would not be helpful in resolving detailed
questions about its nature. The same model for the hon-
eycomb geometry in the α= 1 limit has been discussed
for theS= 1 compound Li 2RuO3,54where the authors
invoked the lattice coupling, in the form of a structural
dimerization driven by the formation of spin singlets, to
select the true ground state.
C. Kagome lattice
The kagome lattice occupies something of a special
place among frustrated spin systems1as one of the most
highly degenerate and intractable problems in existence,
for both classical and quantum spins, and even with
only nearest–neighbor Heisenberg interactions. Inter-
est in this geometry has been maintained by the dis-
covery of a number of kagome spin systems, and has
risen sharply with the recent synthesis of a true S= 1/2
kagome material, ZnCu 3(OH)6Cl2.55Preliminary local–
probe experiments56,57show a state of no magnetic or-
der and no apparent spin gap, whose low–energy spin
excitations have been interpreted58as evidence for an
exotic spin–liquid phase. Both experimentally and the-
oretically, kagome systems of higher spins ( S= 3/2 and
5/2) are found to have flat bands of magnetic excita-
tions, reflecting the very high degeneracy of the spin
sector.59While no kagome materials are yet known with
both spin and orbital degrees of freedom, Maekawa and
coworkers40,44have considered the itinerant electron sys-
tem on the triangular lattice for α= 0 (actually for the
motion of holes in Na xCoO2), demonstrating that the
combination of orbital, hopping selection, and geometry
leads to any one hole being excluded from every fourth
site, and thus moving on a system of four interpenetrat-
ing kagome lattices.34
(a) (b)
FIG. 22: (Color online) Kagome lattice with unequally
weighted two–color states oriented (a) with and (b) against
the lattice direction corresponding to the majority orbita l
color.
Considering first the energies per bond for states of
long–ranged spin and orbital order, in a number of cases
the values for the kagome lattice are identical to those of
the triangular lattice. This is easy to show by inspection
for the one–color state (3a), and for the superposition
states (4a), (4b), and (4c), where bonds of all types are
removed in equal number. However, for the less symmet-
rical orbital color configurations a more detailed analysis
of the type performed in Sec. III is required, and yields
provocative results. The two simple possibilities for or-
dered two–color states with a single color per site are
shown in Fig. 22, and differ only in the orientation of
the continuous lines (the majority color) relative to the
active orbitals. These can be considered as the kagome–
lattice analogs of states (3b) and (3c), as well as of (3e)
and (3f).
When the lines of c–orbitalsarealignedwith the c–axis
[Fig. 22(a)], this direction is inactive at α= 0, and only
the other two directions contribute, one with two active
FO orbitals, mandating an AF spin state to give energy
−1
2Jper bond, and the other with energy −1
4Jand no
strong spin preference, whence
E(k3b)(0) =−1
4J (6.9)
atη= 0 for sets of unfrustrated AF chains. By contrast,
when the lines of c–orbitals fall along the b–direction
[Fig. 22(b)], the α= 0 problem contains one FO and
one AO line each with two active orbitals, and one line
with one active orbital. Only the first requires AF spin
alignment, while the other two lines are not frustrating,
with the result that an energy
E(k3c)(0) =−5
12J (6.10)
can be obtained. This value is lower than that on the
triangular lattice, showing that for the class of models
under consideration, where not all hopping channels are
activein alldirections, asystem oflowerconnectivitycan
lead to frustration relief even when its geometry remains
purely that of connected triangles.(a) (b)
FIG. 23: (Color online) Kagome lattice with two different,
equally weighted three–color states: (a) two–color lines o ri-
ented such that only one superexchange channel, plus the di-
rect exchange channel, is active on every bond. (b) two–colo r
lines oriented such that all superexchange channels are act ive,
but no direct exchange channels.
With this result in mind, we consider again the possi-
bilities offered by different three–color states, specifically
those shown in Fig. 23. With reference to the superex-
change problem, the state in Fig. 19(c), which by anal-
ogy with (3d) we denote as (k3d), contains only a small
numberofremnanttrianglesand isolatedbonds still with
two active orbitals. However, the state (k3d1), shown in
Fig. 23(a) is that which ensures that no such bonds re-
main, and every single bond of the lattice has one active
superexchange channel. The state (k3d2) in Fig. 23(b)
is that in which every single bond of the lattice has two
active (FO) superexchange channels: this possibility can
be realized for the kagome geometry, at the cost of creat-
ing a frustrated magnetic problem requiring a 120◦spin
state to minimize the energy,
E(k3d)(0) =−5
16J, (6.11)
E(k3d1)(0) =−1
4J, (6.12)
E(k3d2)(0) =−3
8J. (6.13)
Thus one finds that lower energies than the value −1
3J
per bond, which was the lower bound for fully (or-
bitally and spin–)ordered states on the triangular lattice,
are again possible for three–color ordered states. How-
ever, the residual spin frustration means that the lowest
ordered–stateenergyonthekagomelatticeisgivenbythe
unfrustrated, two–color AFF state, E(k3c)(0) =−5
12J.
We present briefly the energies of the same states at
α= 1,whereonlyamaximumofonehoppingchannelper
bond can be active, and as noted abovethis is generallya
stricter energetic limit than any frustration constraints.
The results at η= 0 are
E(k3b)(1) =−1
4J (6.14)
for an AFF state gaining most of its energy from the35
c–axis chains, and
E(k3c)(1) =−1
12J (6.15)
due to the dearth of active orbitals in this orientation.
Similarly, by counting active orbitals in the three–color
states,
E(k3d)(1) =−1
6J, (6.16)
E(k3d1)(1) =−1
4J, (6.17)
E(k3d2)(1) = 0, (6.18)
and it is the state of Fig. 23(a) which achieves the un-
frustrated value −1
4Jby permitting one active hopping
channel on every bond of the kagome lattice.
We will not discuss the orbital superposition states
which are the analogs of (4d) and (4e), noting only
that these present again two different possibilities on the
kagome lattice, depending on the orientation of the ma-
jority lines. Even with the frustration relief offered by
this geometry for the type of model under consideration,
superposition states contain too many hopping channels
for all to be satisfied simultaneously, and it is not possi-
ble to equal the energy values found respectively for the
configurations in Figs. 23(a) and (b) at α= 1 andα= 0.
It remains to consider dimer states on the kagome lat-
tice, as these have been of equal or lower energy for ev-
ery case analyzed so far. The set of nearest–neighbor
dimer coverings of the kagome lattice is large, and for
theS= 1/2 Heisenberg model in this geometry the spin
singlet manifold has been proposed as the basis for an
RVB description.25Two dimer coverings degenerate at
the level of the current treatment are shown in Fig. 24.
Dimer coverings of the kagome lattice have the prop-
erty that 3 /4 of the triangles contain one dimer. In
this case, the other bonds of the triangle are interdimer
bonds, one of which is of type (7b) while the other is of
type (7c). The other 1/4 of the triangles, known60as
“defect triangles”, have no dimers, and their three bonds
are either all of type (7b), with probability 1/4, or one
each of types (7a), (7b), and (7c), with probability 3/4.
The frustration of the system is contained in the problem
of minimizing the number of FM/FO interdimer bonds;
this exercise is complex and no solution is known, so only
an upper bound will be estimated here.
The bonds of a defect triangle connect three differ-
ent dimers, and so one (or all three) must be FM/FO. A
hexagonofthekagomelatticewithnodimersonitsbonds
issurroundedbysixnon–defectivetriangles,onewithone
dimer by one defective neighbor, with two dimers two,
and a hexagon with three dimers shares its non–dimer
bonds with three defect triangles. Hexagons with odd
dimer numbers must create a FM/FO bond between at
least one pair of dimers, and it is reasonable to place this
bond on the defect triangle(s) where an energy cost is al-
ready incurred. We note immediately that the cost of re-
versing the type–(7a) bond,1
4J(Sec. IVA), exceeds that(a)
(b)
FIG. 24: (Color online) Kagome lattice with two different
dimer coverings, (a) and (b). In both examples, only two of
the twelve triangles shown explicitly on the cluster are “de fec-
tive”(contain nodimer), butthereader maynotice thatmany
of the next twelve triangles adjoining the boundary must als o
be so.
of reversingboth interdimer bonds of a non–defective tri-
angle, which is1
8J+1
16J. As a consequence, we take this
cost, which is equal to that of reversing both a non–
defective triangle and the weakest bond of the defect tri-
angle, to be an upper bound on the effect of frustration.
The net energy of a dimer state for α= 0 =ηis then
estimated to be
Ekd(0) =−3
4·1
3J−3
4·1
3J/parenleftbigg3
8+1
4/parenrightbigg
−1
4J/bracketleftbigg1
4/parenleftbigg2
3·3
8+1
3·1
4/parenrightbigg
+3
4·1
3/parenleftbigg1
4+3
8+1
4/parenrightbigg/bracketrightbigg
=−209
384J≃ −13
24J. (6.19)
This is a very large number for the kagome lattice, ex-
ceeding even the value −1
2Jper bond (which, however, is
of no special significance here). Thus we find that dimer
states in this type of model are strongly favored, gaining
a very much higher energy than even the best ordered
states. Qualitatively, the dimer energy shares with the
ordered–state energy the feature that it is considerably36
betterthananythingobtainableforthetriangularlattice.
This implies that the reduced connectivity of the lattice
geometry for a model where the orbital degeneracy pro-
vides a number of mutually exclusive hopping channels
makesit easier to find states where every remainingbond
can support a favorable hopping process without strong
frustration.
Applying all of the above geometrical considerations
to the direct–exchange model ( α= 1), where there is no
frustration problem between the spin singlets, one finds
Ekd(1) =−3
4·1
3J−3
4·1
3J/bracketleftbigg1
4+0/bracketrightbigg
−1
4·1
3J/bracketleftbigg1
4·3
4+1
4·1
2/bracketrightbigg
=−21
64J (6.20)
atη= 0. Once again this energy is significantly lower
than the value Edim(1) =−1
4Jobtained for the triangu-
lar lattice in Eq. (4.15), demonstrating that the multi-
channel spin–orbital model of the type considered here is
less frustrated in the kagome geometry.
Wecommentinclosingthatthe dimerenergieswehave
estimatedareonlythoseofstaticVBconfigurations,and,
away from α= 1, the possibility remains of a signifi-
cant resonance energy gain from quantum fluctuations
between these states ( cf.Sec. V). Numerical calculations
on small clusters of sufficient size (here at least 6 sites for
a unit cell) would be helpful in this frustrated case.
To summarize this section, the spin–orbital model on
bipartite lattices appears to present competing ordered
and dimerized states with the prospect of high degenera-
cies. Among “frustrated” systems (in the sense of being
non–bipartite), the kagome lattice provides an example
where geometrical and orbital frustration effects cancel
partially, affording favorable dimerized solutions. Thus,
while it is possible to ascribe some of the frustration ef-
fects we have studied in the triangular lattice to a purely
geometrical origin, for more complex models it is in gen-
eralnecessaryto extend the concept of “geometricalfrus-
tration” beyond that applicable to pure spin systems.
VII. DISCUSSION AND SUMMARY
We have considered a spin–orbital model representa-
tive of a strongly interacting 3 d1electron system with
the cubic structural symmetry of edge–sharing metal–
oxygen octahedra, conditions which lead to a triangular
lattice of magnetic interactions between sites with un-
broken, threefold orbital degeneracy. We have elucidated
the qualitativephasediagram,whichturns outto be very
rich, in the physicalparameterspace presented by the ra-
tio (α) of superexchange to direct–exchange interactions
and the Hund exchange ( η).
Despite the strong changes in the fundamental nature
of the model Hamiltonian as a function of αandη, anumber of generic features persist throughout the phase
diagram. With the exception ofthe ferromagneticphases
at highη, which effectively suppressesquantum spin fluc-
tuations (below), there is no long–rangedmagnetic or or-
bital order anywhere within the entire parameter regime.
This shows a profound degree of frustration whose origin
lies both in the geometry and in the properties of the
spin–orbital coupling; a qualitative evaluation of these
respective contributions is discussed below.
All of the phases ofthe model show a strong preference
for the formation of dimers. This can be demonstrated
in a simple, static valence–bond (VB) ansatz, and is re-
inforced by the results of numerical calculations. The
staticansatzis alreadyanexactdescriptionofthe direct–
exchange limit, α= 1, and gives the best analytic frame-
work for understanding the properties of much of the re-
mainder of the phase diagram. The most striking single
numerical result is the prevalence of VB states even on a
triangular cluster, and the underlying feature reinforced
by all of the calculations is the very large additional “ki-
netic” contribution to the ground–state energy arising
from the resonance of VBs due to quantum fluctuations.
It is this resonance which drives symmetry restoration in
some or all of the spin, orbital, and translational sectors
over large regions of the parameter space. The sole ex-
ception to dimerization is found at high ηand around
α= 1, where the only mechanism for virtual hopping is
the adoption of orbital configurations which permit one
orbital to be active (“avoided blocking”).
The “most exotic” region of the phase diagram is that
at smallαandη, and this we have assigned tentatively
as an orbital liquid. In this regime, quantum fluctua-
tions are at their strongest and most symmetrical, and
every indication obtained from energetic considerations
of extended systems, and from microscopic calculations
of a range of local quantities on small clusters, suggests
a highly resonant, symmetry–restored phase. While this
orbital liquid is in all probability (again from the same
indicators) based on resonating dimers, an issue we dis-
cuss in full below, we cannot exclude fully the possibility
of a type of one–dimensional physics: short, fluctuating
segments of frustration–decoupled spin or orbital chains,
whose character persists despite the high site coordina-
tion. It should be stressed here that the point ( α,η) =
(0,0)isnotinanysenseaparentphaseforexoticstatesin
the restofthe phasediagram: mixed and direct exchange
processes are qualitatively different elements, which in-
troduce different classes of frustrated model at finite α.
While the matter is somewhat semantic, we comment
only that one cannot argue for the point α= 0.5 being
“more exotic” than α= 0 despite having the maximal
number of equally weighted hopping channels, because it
does not possess any additional symmetries which man-
date qualitative changes to the general picture. In this
sense, the limit α= 1 serves as a valuable fixed point
which is understood completely, and yet is still domi-
nated by the purely quantum mechanical concept of sin-
glet formation.37
One indicatorwhich can be employed to quantify “how
exotic” a phase may be is the entanglement of spin and
orbital degrees of freedom. We define entanglement as
the deviation of the spin and orbital sectors from the
factorized limit in which their fluctuations can be treated
separately. We compute a spin–orbital correlation func-
tion and use it to measure entanglement, finding that
this is significant over the whole phase diagram. Qual-
itatively, entanglement is maximal around the superex-
change limit, which is dominated by dimers where sin-
glet formation forces the other sector to adopt a local
triplet state. However, for particular clusters and dimer
configurations, the high symmetry may allow less entan-
gled possibilities to intervene exactly at α= 0. The
direct–exchange limit, α= 1, provides additional in-
sight into the entanglement definition: the four–operator
spin–orbital correlation function vanishes, reflecting the
clear decoupling of the two sets of degrees of freedom at
this point, but the finite product of separate spin and
orbital correlation functions violates the factorizability
condition.
This preponderance of evidence for quantum states
based on robust, strongly resonating dimers implies fur-
ther that the (spin and orbital) liquid phase is gapped.
Such a state would have only short–ranged correlation
functions. However, these gapped states are part of a
low–energy manifold, and for the extended system we
have shown that this consists quite generally of large
numbers of (nearly) degenerate states. The availability
of arbitrary dimer rearrangements at no energy cost has
been suggested to be sufficient for the deconfinement of
elementary S= 1/2 (and by analogy T= 1/2) excita-
tions with fractional statistics.61However, the spinons
(orbitons) are massive in such a model, in contrast to
the properties of algebraic liquid phases.62
Alow–spintohigh–spintransition,occurringasafunc-
tion ofη, is present for all values of α. The quantitative
estimation of ηcin the extended system remains a prob-
lem for a more sophisticated analysis. At the qualitative
level, largeηcan be considered to suppressquantum spin
fluctuations by promoting parallel–spin (ferromagnetic)
intermediate states on the magnetic ions. However, even
when this sector is quenched, the orbital degrees of free-
dom remain frustrated, and contain non–trivialproblems
in orbital dynamics. In the superexchange (low– α, high–
η) region, frustration is resolved by the formation of or-
bital singlet (spin triplet) dimers, whose resonance min-
imizes the ground–state energy. The frustration in the
direct–exchange (high– α, high–η) region is resolved by
avoided–blocking orbital configurations, and order–by–
disorder effects are responsible for the selection of the
true ground state from a degenerate manifold of possi-
bilities; this is the only part of the phase diagram not
displaying dimer physics. Thus the ferromagnetic orbital
models in both limits exhibit a behavior quite different
from that of systems with only S= 1/2 spin degrees of
freedom on the triangular lattice.
We have commented on both geometry and spin–orbital interactions as the origin of frustration in the
models under consideration. However, a statement such
as “on the triangular lattice, geometrical frustration en-
hances interaction frustration for spin–orbital models”
must be qualified carefully. We have obtained anecdotal
evidence concerning such an assertion in Sec. VI by con-
sideringotherlatticegeometries,andfind thatindeed the
same model on an unfrustrated geometry appears capa-
ble of supporting ordered states; however, the interplay
of the two effects is far from direct, as the kagome lattice
presents a case where dimer formation acts to reduce the
net frustation. Quite generally, spin–orbital models con-
tain in principle more channels which can be used for re-
lievingfrustration,buttheexactnatureofthecouplingof
spin and orbital sectors may result in the opposite effect.
Specificdatacharacterizingmutualfrustrationcanbeob-
tained from the spin and orbital correlations computed
on small clusters: as shown in Sec. V, for the triangular
lattice there are indeed regimes where, for example, the
effective orbital interactions enforced by the spin sector
make the orbital sector more frustrated (higher Tij) than
would be the analogous pure spin problem (measured by
Sij), and conversely.
We comment briefly on other approaches which might
be employed to obtain more insight into the states of the
extended system, with a view to establishing more defini-
tively the nature and properties of the candidate orbital
liquid phase. More advanced numerical techniques could
be used to analyze larger unit cells, but while Lanczos
diagonalization, contractor renormalization63or other
truncation schemes might afford access to systems two,
or even four, times larger, it seems unlikely that these
clusters could provide the qualitatively different type of
data required to resolve the questions left outstanding
in Sec. V. An alternative, but still non–perturbative and
predominantly unbiased, approach would be the use of
variational wave functions, either formulated generally
or in the more specific projected wave function technique
which leads to different types of flux phase.64,65Adapt-
ing this type of treatment to the coupled spin and orbital
sectorswithout undue approximationremainsa technical
challenge.
Within the realm of effective models which could be
obtained by simplification of the ground–state manifold,
we cite only the possibility motivated by the current re-
sults of constructing dimer models based on (ss/ot) and
(os/st) dimers. Dimer models26are in general highly
simplified, and there is no systematic procedure for their
derivation from a realistic Hamiltonian, but they are
thought to capture the essential physics of certain classes
of dimerized systems. Because QDM Hamiltonians pro-
vide exact solutions, and in some cases genuine exam-
ples of exotica long sought in spin systems, including the
RVB phase and deconfined spinon excitations, they rep-
resent a valuable intermediate step in understanding how
such phenomena may emerge in real systems. Here we
have found (i) a very strong tendency to dimer forma-
tion, (ii) a large semi–classical degeneracy of basis states38
formed from these dimers, and (iii) that resonance pro-
cesses even at the four–site plaquette scale providea very
significant energetic contribution. From the final obser-
vation alone, a minimal QDM, meaning only exchange of
parallel dimers of all three directions and on all possible
plaquette units, would already be expected to contain
the most significant corrections to the VB energy. At
this point we emphasize that, because of the change of
SU(2) orbitalsectorwith lattice direction, our2Dmodels
are not close to the SU(4) point where four–site plaque-
tte formation, and hence very probably a crystallization,
would be expected.13From the results of Secs. IV and
V, a rather more likely phase of the QDM would be one
with complete plaquette resonance through all three col-
ors, and without breaking of translational symmetry.
Rigorous proof of a liquid phase, such as that repre-
sented by an RVB state, is more complex, and as noted
in Sec. I it requires satisfying both energetic and topo-
logical criteria. Following the prescription in Ref. 19,
three conditions must be obeyed: (i) a propensity for
dimer formation, (ii) a highly degenerate manifold of ba-
sis states from which the RVB ground state may be con-
structed, and (iii) a mapping of the system to a liquid
phase of a QDM. Criteria (i) and (ii) match closely the
labels in the previous paragraph, and both dimer forma-
tion and high degeneracy have been demonstrated ex-
tensively here. The energetic part of criterion (iii) also
appears to be obeyed here: static dimers have an en-
ergy (V), and allowing their location and orientation
to change gains more ( t). The regime V/t <1 of the
triangular–lattice QDM is the RVB phase demonstrated
in Ref. 27, whose properties include short–range corre-
lation functions and gapped, deconfined spinons. This
mapping also contains the criterion of togological degen-
eracy, andcouldinprinciplebe partiallycircumventedby
a direct demonstration. However, no suitable numericalstudies are available of non–simply connected systems,
and so here we can present only plausibility arguments
based on the high degeneracy and spatial topology of the
dimer systems analyzed in Secs. IV and V. It is safe to
concludethatthe threefold–degenerate t2gorbitalsystem
on the triangular lattice is one of best candidates yet for
a true spin–orbital RVB phase.
In closing, spin–orbital models have become a frontier
of intense current interest for both experimental and the-
oretical studies of novel magnetic and electronic states
emerging as a consequence of intrinsic frustration. Our
model has close parallels to, and yet crucial differences
from, similar studies of manganites (cubic systems of eg
orbitals), LiNiO 2(triangular, eg), YTiO 3and CaVO 3
(cubic,t2g), and many other transition–metal oxides, ap-
pearing in some respects to be the most frustrated yet
discussed. One of its key properties, arising from the ex-
treme(geometricalandinteraction–driven)frustration,is
that ordered states become entirely uncompetitive com-
pared to the resonance energy gained by maximizing
quantum (spin and orbital) fluctuations. In the orbital
sector, the restoration of symmetry by orbital fluctua-
tions makes the model a strong candidate to display an
orbital liquid phase. Because this liquid is based on ro-
bustdimerstates, the mechanismforitsformationisvery
likely to be spin–orbital RVB physics.
Acknowledgments
We thank G. Khaliullin and K. Penc for helpful
discussions, and J. Chaloupka for technical assistance.
A. M. Ole´ s acknowledges support by the Foundation for
Polish Science (FNP) and by the Polish Ministry of Sci-
enceandEducationunderProjectNo.N20206832/1481.
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1604.04623v3.Determining_the_spin_orbit_coupling_via_spin_polarized_spectroscopy_of_magnetic_impurities.pdf | arXiv:1604.04623v3 [cond-mat.mes-hall] 15 Oct 2016Determining the spin-orbit coupling via spin-polarized sp ectroscopy of magnetic
impurities
V. Kaladzhyan,1,2,∗P. Simon,2and C. Bena1,2
1Institut de Physique Th´ eorique, CEA/Saclay, Orme des Meri siers, 91190 Gif-sur-Yvette Cedex, France
2Laboratoire de Physique des Solides, CNRS, Univ. Paris-Sud ,
Universit´ e Paris-Saclay, 91405 Orsay Cedex, France
(Dated: October 1, 2018)
We study the spin-resolved spectral properties of the impur ity states associated to the presence of
magnetic impurities in two-dimensional, as well as one-dim ensional systems with Rashba spin-orbit
coupling. We focus on Shiba bound states in superconducting materials, as well as on impurity
states in metallic systems. Using a combination of a numeric al T-matrix approximation and a
direct analytical calculation of the bound state wave funct ion, we compute the local density of
states (LDOS) together with its Fourier transform (FT). We fi nd that the FT of the spin-polarized
LDOS, a quantity accessible via spin-polarized STM, allows to accurately extract the strength of the
spin-orbit coupling. Also we confirm that the presence of mag netic impurities is strictly necessary
for such measurement, and that non-spin-polarized experim ents cannot have access to the value of
the spin-orbit coupling.
I. INTRODUCTION
The electronic bands of materials that lack an inver-
sion center are split by the spin-orbit (SO) coupling. A
strong SO coupling implies that the spin of the electron
is tied to to the direction of its momentum. Materials
with strong SO coupling have been receiving a consider-
able attention in the past decade partly because SO is
playing an important role for the discovery of new topo-
logical classes of materials.1,2Two-dimensional topolog-
ical insulators, first predicted in graphene,3have been
discovered in HgTe/CdTe heterostructures4following a
theoretical prediction by Bernevig et al..5They are char-
acterized by one-dimensional helical edge states where
the spin is locked to the direction of propagation due
to the strong SO coupling. Similar features occur for
the surface states of 3D topological insulators which also
haveastrongbulk SO coupling.1The spin-to-momentum
locking was directly observed by angle-resolved photoe-
mission spectroscopy (ARPES) experiments.6,7
Topological superconductors share many properties
with topological insulators. They possess exotic edge
statescalled Majoranafermions, particleswhich aretheir
own antiparticles.1Topological superconductivity can be
either induced by the proximity with a standard s-wave
superconductor or be intrinsic. In the former case, Ma-
jorana states have thus been proposed to form in one-
dimensional8,9and two-dimensional semiconductors10,11
withstrongSOcouplingwhenproximitizedwithas-wave
superconductor, and in the presence of a Zeeman field.
Following this strategy, many experiments have reported
signatures of Majorana fermions through transport spec-
troscopy in one dimensional topological wires.12–16How-
ever, there are presently only a few material candidates
such as strontium ruthenate,17certain heavy fermion
superconductors,18or some doped topological insulators
such as Cu xBi2Se3,19that may host intrinsic topological
superconductivity.Although SO coupling has been playing an essential
role in the discovery of new topological materials, it is
also of crucial importance in the physics of spin Hall
effect,20in spintronics21and quantum (spin) computa-
tion since it allows to electrically detect and manipulate
spin currents in confined nanostructures (see Ref. 22 for
a recent review).
Based on the prominent role played by SO in the
past decades, it is thus of great interest to be able
to evaluate the SO coupling value in a given mate-
rial accurately, though in general this is a very dif-
ficult task. Inferences can be made from ARPES
measurements;23–25in particular spin-polarized ARPES
measurements have been used to evaluate the SO cou-
pling in variousmaterials.26–32Other possibilities involve
magneto-transport measurements in confined nanostruc-
tures: this technique has been used to measure the
SO coupling in clean carbon nanotubes33or in InAs
nanowires.34
Here we propose a method to measure the magnitude
of the SO coupling directly using spin-polarized scan-
ning tunnelling microscopy (STM),35and the Fourier
transform (FT) of the local density of states (LDOS)
near magnetic impurities (FT-STM). The FT-STM tech-
nique has been used in the past in metals, where it
helped in mapping the band structure and the shape
and the properties of the Fermi surface,36–43as well
as in extracting information about the spin properties
of the quasiparticles.44More spectacularly, it was used
successfully in high-temperature SCs to map with high
resolution the particular d-wave structure of the Fermi
surface, as well as to investigate the properties of the
pseudogap.45–47
In this paper, on one hand, we calculate the Fourier
transform of the spin-polarized local density of states
(SP LDOS) of the so-called Shiba bound state48–51as-
sociated with a magnetic impurity in a superconductor.
Shiba bound states have been measured experimentally
by STM52–54and it has actually been shown that the ex-2
tentoftheShibawavefunction canreachtensofnanome-
ters in 2D superconductors, which allows one to mea-
sure the spatial dependence of the LDOS of such states
withhighresolution.55Weconsiderbothone-dimensional
and two-dimensional superconductors with SO coupling.
While two-dimensional systems such as e.g. Sr 2RuO4,17
or NbSe 255,56become superconducting when brought at
lowtemperature, one-dimensionalwiressuchasInAs and
InSb are not superconducting at low temperature. In or-
der to see the formation of Shiba states one would need
to proximitize them by a SC substrate. The formation of
Shiba states in such systems,57,58as well as in p-wave
superconductors,59,60has been recently touched upon,
but the effect of the SO coupling on the FT of the SP
LDOS in the presence of magnetic impurities has not
previously been analyzed.
On the other hand we focus on the effects of the spin-
orbit coupling on the impurity states of a classical mag-
netic impurity in one-dimensional and two-dimensional
metallic systems such as Pb61and Bi, as well as InAs
and InSb semiconducting wires that can be also mod-
eled as metals in the energy range that we consider. We
should note that for these systems no bound state forms
at a specific energy, but the impurity is affecting equally
the entire energy spectrum.
By studying the two classesofsystems described above
we show that the SO coupling can directly be read-off
from the FT features of the SP LDOS in the vicinity
of the magnetic impurity. We note that such a signa-
ture appearsonly formagneticimpurities, and onlywhen
thesystemisinvestigatedusingspin-polarizedSTMmea-
surements, the non-spin-polarized measurements do not
provide information on the SO, as it has also been previ-
ously noted.62The main difference between the SC and
metallic systems, beyond the existence of a bound state
in the former case, is that the spin-polarized Friedel os-
cillations around the impurity haveadditional features in
theSCphase, themostimportantonebeingtheexistence
of oscillations with a wavelength exactly equal to the SO
coupling length scale; such oscillations are not present in
the metallic phase. Another difference is the broadening
of the FT features in the superconducting phase com-
pared to the non-SC phase in which the sole broadening
is due to the quasiparticle lifetime.
We focus on Rashba SO coupling as assumed to be
the most relevant for the systems considered, but we
have checked that our conclusion holds for other types
of SO. To obtain the SP LDOS we use a T-matrix
approximation,43,63,64and we present both numerical
and analytical results which allow us to obtain a full un-
derstandingofthe observedfeatures, ofthe splittings due
to the SO, as well as of the spin-polarization of the im-
purity states and of the symmetry of the FT features.
In Sec. II, we present the general model for two-
dimensional and one-dimensional cases and the basics of
the T-matrix technique. In Sec. III we show our results
for the SP LDOS, calculated both numerically and an-
alytically, for 2D systems, both in the SC and metallicphase. Sec. IV is devoted to SP LDOS of impurity in
one-dimensional systems. Our Conclusions are presented
in section V. Details of the analytical calculations are
given in the Appendices.
II. MODEL
We consider an s-wave superconductor with a SC
paring ∆ s, and Rashba SO coupling λ, for which
the Hamiltonian, written in the Nambu basis Ψ p=
(ψ↑p,ψ↓p,ψ†
↓−p,−ψ†
↑−p)T, is given by:
H0=/parenleftbigg
ξpσ0∆sσ0
∆sσ0−ξpσ0./parenrightbigg
+HSO. (1)
The energy spectrum is ξp≡p2
2m−εF, whereεFis the
Fermi energy. The operator ψ†
σpcreates a particle of spin
σ=↑,↓of momentum p≡(px,py) in 2D and p≡pxin
1D. Below we set /planckover2pi1to unity. The system is considered to
lay in the (x,y) plane in 2D case, whereas in 1D case we
setpyto zero in the expressions above, and we consider
a system lying along the x-axis. The metallic limit is
recovered by setting ∆ s= 0. The Rashba Hamiltonian
can be written as
HSO=λ(pyσx−pxσy)⊗τz, (2)
in 2D and simply as HSO=λpxσy⊗τzin 1D. We
have introduced σandτ, the Pauli matrices acting re-
spectively in the spin and the particle-hole subspaces.
The unperturbed retarded Green’s function can be ob-
tained from the above Hamiltonian via G0(E,p) =
[(E+iδ)I4−H0(p)]−1, whereδis the inverse quasipar-
ticle lifetime.
In what follows we study what happens when a sin-
gle localized impurity is introduced in this system. We
consider magnetic impurities of spin J= (Jx,Jy,Jz) de-
scribed by the following Hamiltonian:
Himp=J·σ⊗τ0·δ(r)≡V·δ(r),(3)
whereJis the magnetic strength. We only consider here
classical impurities oriented either along the z-axis,J=
(0,0,Jz), or along the x-axis,J= (Jx,0,0). This is
justifiedprovidedtheKondotemperatureismuchsmaller
than the superconducting gap.51
To find the impurity states in the model described
above we use the T-matrix approximation described in
[51, 63, and 64] and [43]. We also neglect the renormal-
ization of the superconducting gap because it is mainly
local51,65and therefore only introduces minor effects for
our purposes. Since the impurity is localized, the T-
matrix is given by:
T(E) =/bracketleftbigg
1−V/integraldisplayd2p
(2π)2G0(E,p)/bracketrightbigg−1
V.(4)3
The real-space dependence of the non-polarized,
δρ(r,E), and SP LDOS, Sˆn(r,E), with ˆn=x,y,z, can
be found as
Sx(r,E) =−1
πℑ[∆G12+∆G21],
Sy(r,E) =−1
πℜ[∆G12−∆G21],
Sz(r,E) =−1
πℑ[∆G11−∆G22],
δρ(r,E) =−1
πℑ[∆G11+∆G22],
with
∆G(E,r)≡G0(E,−r)T(E)G0(E,r),
where ∆Gijdenotes the ij-th component of the matrix
∆G, andG0(E,r) is the unperturbed retarded Green’s
function in real space, given by the Fourier transform
G0(E,r) =/integraldisplaydp
(2π)2G0(E,p)eipr. (5)
The FT of the SP LDOS components in momentum
space,Sˆn(p,E) =/integraltextdrSˆn(r,E)e−ipr, with ˆn=x,y,z,
as well as the FT of the non-polarized LDOS, δρ(p,E) =/integraltext
drδρ(r,E)e−iprare given by
Sx(p,E) =i
2π/integraldisplaydq
(2π)2[˜g12(E,q,p)+ ˜g21(E,q,p)],(6)
Sy(p,E) =1
2π/integraldisplaydq
(2π)2[g21(E,q,p)−g12(E,q,p)],(7)
Sz(p,E) =i
2π/integraldisplaydq
(2π)2[˜g11(E,q,p)−˜g22(E,q,p)],(8)
δρ(p,E) =i
2π/integraldisplaydq
(2π)2[˜g11(E,q,p)+ ˜g22(E,q,p)],(9)
wheredq≡dqxdqy,
g(E,q,p) =G0(E,q)T(E)G0(E,p+q)
+G∗
0(E,p+q)T∗(E)G∗
0(E,q),
˜g(E,q,p) =G0(E,q)T(E)G0(E,p+q)
−G∗
0(E,p+q)T∗(E)G∗
0(E,q),
andgij, ˜gijdenote the corresponding components of the
matricesgand ˜g. Note that while the non-polarized and
the SP LDOS are of course real functions when evalu-
ated in position space, their Fourier transforms need not
be. Sometimes we get either or both real and imaginary
components for the FT, depending on their correspond-
ing symmetries. In the figureswe shallindicate eachtime
if we plot the real or the imaginarycomponent of the FT.
To obtain the FT of the non-polarized and the SP
LDOS, we first evaluate the momentum integrals in
Eqs. (4-9) numerically. For this we use a square lattice
version of the Hamiltonians (1) and (2), where we take
the tight-binding spectrum Ξ p≡µ−2t(cospx+cospy)
with chemical potential µand hopping parameter t. We
set the lattice constant to unity. It is also worth noting
thatallthe numericalintegrationsareperformedoverthe
first Brillouin zone and that we use dimensionless units
by settingt= 1.Alternatively, as detailed in the appendices, we find
the exact form for the non-polarized and SP LDOS in
the continuum limit by performing the integrals in the
FTofthe Green’sfunctions analytically. Moreover,when
considering the SC systems, the energies Eof the Shiba
statestogetherwith the correspondingeigenstatesforthe
ShibawavefunctionsΦattheorigincanbeobtainedfrom
the corresponding eigenvalue equation66
[I4−VG0(E,r=0)]Φ(0) = 0. (10)
The spatial dependence of the Shiba state wave function
is determined using
Φ(r) =G0(E,r)VΦ(0). (11)
The real-space Green’s function is obtained simply by a
Fourier transformof the unperturbed Green’s function in
momentum space, G0(E,p). The non-polarized and the
SP LDOS are given by
ρ(E,r) = Φ†(r)/parenleftbigg
0 0
0σ0/parenrightbigg
Φ(r), (12)
and
S(E,r) = Φ†(r)/parenleftbigg
0 0
0σ/parenrightbigg
Φ(r), (13)
where we take into account only the hole components of
the wave function, and not the electron ones. This is
because the physical observables are related to only one
of the two components, for example in a STM measure-
ment one injects an electron at a given energy and thus
have access to the allowed number of electronic states,
not to both the electronic and hole states simultaneously.
The Bogoliubov-de Gennes Hamiltonian contains the so-
called particle-hole redundancy, and the electron and the
holecomponentscanbesimplyrecoveredfromeachother
by overall changes of sign, and/or changing the sign of
the energy. Belowwecompute only the hole components,
but there would have been no qualitative differenced had
we computed the electron component.
III. RESULTS FOR TWO DIMENSIONAL
SYSTEMS
A. Real and momentum space dependence of the
2D Shiba bound states
For a 2D superconductorwith SO coupling in the pres-
ence of a magnetic impurity one expects the formation
of a single pair of Shiba states.57,58The energies of the
particle-hole symmetric Shiba states67are given by (in-
dependent of the direction of the impurity):
E1,¯1=±1−α2
1+α2∆s,
whereα=πνJandν=m
2π. (See Appendix A for details
of how the energies of the Shiba states are calculated.)4
Up to the critical value αc= 1 these energies are ordered
the following way: E1>E¯1. As soon as α>α c, energy
levelsE1andE¯1exchange places, making the order the
following:E¯1> E1. This corresponds to a change of
the ground state parity.51,68,69Forα≫1 the subgap
states approach the gap edge and eventually merge with
the continuum. For the type of impurities considered
here, there is no dependence of these energies on the SO
couplingin thelow-energyapproximation,thoughaweak
dependence is introduced when one takes into account
the non-linear form of the spectrum. The dependence
of energy of the Shiba states on the impurity strength
Jis depicted in Fig. 1 where we plot the total spin of
the impurity state S(p= 0) as a function of energy and
impurity strength. Note that the two opposite-energy
Shiba states have opposite spins.
FIG. 1. (Color) The averaged SP LDOS induced by an impu-
rity as a function of the impurity strength for an in-plane
magnetic impurity. The dashed line shows the supercon-
ducting gap. A similar result is obtained when the impu-
rity spin is perpendicular to the plane. Note that the two
Shiba states with opposite energies have opposite spin. We
sett= 1,µ= 3,δ= 0.01,λ= 0.5,∆s= 0.2.
We are interested in studying the spatial structure of
the Shiba states in the presence of magnetic impurities
oriented both perpendicular to the plane, and in plane.
Thiscanbedonebothinrealspaceandmomentumspace
bycalculatingthe Fouriertransformofthe spin-polarized
LDOSusingtheT-matrixtechniquedetailedintheprevi-
ous section. We focus on the positive-energy Shiba state,
noting that its negative energy counterpart exhibits a
qualitativelysimilarbehavior. InFig. 2weshowthereal-
space dependence of the non-polarized and SP LDOS.
Each of the panels corresponds to the interference pat-
terns originating from different types of scattering. Note
that the spin-orbit value cannot be accurately extracted
from these type of measures, since the system contains
oscillationswith manydifferentsuperposingwavevectors.
ToovercomethisproblemwefocusontheFTofthesefea-
tures, as it is oftentimes done in spatially resolved STM
experiments, which allow for a more accurate separation
of the different wavevectors.36–43Thus in Fig. 3 we focus
on the FT of the SP LDOS for two types of impuritieswith spin oriented along zandxaxes respectively.
z-impurity x-impurity
Jz(Jx=Jy= 0) Jx(Jy=Jz= 0)
FIG. 2. (Color) The real-space dependence of the non-
polarized as well as of the SP LDOS components for the
positive energy Shiba state, for a magnetic impurity with
Jz= 2 (left column), and Jx= 2 (right column). We take
t= 1,µ= 3,δ= 0.01,λ= 0.5,∆s= 0.2.
Note that the SO introducesnon-zerospin components
in the directions different from that of the impurity spin.
These components exhibit either two-fold or four-fold
symmetric patterns. Also the SO is affecting strongly
the spin component parallelto the impurity, in particular
when the impurity is in-plane, in which case the struc-
ture of the SP LDOS around the impurity is no longer
radially symmetric. However, as can be seen in the bot-
tom panel of Fig. 3, the non-spin-polarized LDOS is5
not affected by the presence of SO, preserving a radially
symmetric shape quasi-identical to that obtained in the
absence of SO. Thus the SO coupling can be measured
onlyviathe spin-polarizedcomponentsofthe LDOS, and
not the non-polarized LDOS.
These results, which are obtained using a numerical
integration of the T-matrix equations, are also supported
by analytical calculations which help to understand the
fine structure of the FT of the SP LDOS (see Appendices
for details). These calculations yield for the SP LDOS
generated by a magnetic impurity perpendicular to the
plane
Sx(r) = +J2
z/parenleftbigg
1+1
α2/parenrightbigge−2psr
rcosφr×
×/braceleftigg/summationdisplay
σσν2
σ
pσ
Fcos(2pσ
Fr−θ)+2ν2v2
F
v2pFsinpλr/bracerightigg
,
Sy(r) = +J2
z/parenleftbigg
1+1
α2/parenrightbigge−2psr
rsinφr×
×/braceleftigg/summationdisplay
σσν2
σ
pσ
Fcos(2pσ
Fr−θ)+2ν2v2
F
v2pFsinpλr/bracerightigg
,
Sz(r) =−J2
z/parenleftbigg
1+1
α2/parenrightbigge−2psr
r×
×/braceleftigg/summationdisplay
σν2
σ
pσ
Fsin(2pσ
Fr−θ)−2ν2v2
F
v2pFcospλr/bracerightigg
,
ρ(r) = +J2
z/parenleftbigg
1+1
α2/parenrightbigge−2psr
r×
×/braceleftbigg
2ν2
mv+2ν2v2
F
v2pFsin(2mvr−θ)/bracerightbigg
,(14)
with
tanθ=/braceleftigg
2α
1−α2,ifα/ne}ationslash= 1
+∞,ifα= 1. (15)
We have introduced eiφr=x+iy
r, and
pσ
F=−σmλ+mv, (16)
pλ= 2mλ, (17)
ps=/radicalig
∆2s−E2
1/v, (18)
withv=/radicalbig
v2
F+λ2, andvF=/radicalbig
2εF/m. Herepσ
F, pλ
andpsare the different momenta which can be read off
from the SP LDOS. For an in-plane magnetic impuritywe have
Ss
x(r) =−J2
x/parenleftbigg
1+1
α2/parenrightbigg/braceleftbigg/summationdisplay
σν2
σ
pσ
F[1+sin(2pσ
Fr−2β)],
+2ν2v2
F
v2pF[cospλr+sin(2mvr−2β)]/bracerightbigge−2psr
r
Sa
x(r) = +J2
x/parenleftbigg
1+1
α2/parenrightbigg/braceleftbigg/summationdisplay
σν2
σ
pσ
F[1−sin(2pσ
Fr−2β)]
−2ν2v2
F
v2pF[cospλr+sin(2mvr−2β)]/bracerightbigge−2psr
rcos2φr,
Sy(r) = +J2
x/parenleftbigg
1+1
α2/parenrightbigg/braceleftbigg/summationdisplay
σν2
σ
pσ
F[1−sin(2pσ
Fr−2β)]
−2ν2v2
F
v2pF[cospλr−sin(2mvr−θ)]/bracerightbigge−2psr
rsin2φr,
Sz(r) =−J2
x/parenleftbigg
1+1
α2/parenrightbigg/braceleftbigg
2/summationdisplay
σσν2
σ
pσ
Fcos(2pσ
Fr−θ)
+4ν2v2
F
v2pFsinpλr/bracerightbigge−2psr
rcosφr,
ρ(r) = +J2
x/parenleftbigg
1+1
α2/parenrightbigg/braceleftbigg
4ν2
mv+4ν2v2
F
v2pF×
×sin(2mvr−θ)/bracerightbigge−2psr
r, (19)
with tanβ=α.
TheSxcomponent is the sum of symmetric part a Ss
x
and an asymmetric part Sa
x. Note that the features ob-
served in the FT of the SP LDOS plots are well captured
by the analytical calculations. In particular we note that
the oscillations in the SP LDOS are dominated by the
following four wavevectors:
2p±
F, p+
F+p−
F= 2mv,andp−
F−p+
F=pλ≡2mλ,
which should give rise in the FT to high-intensity fea-
tures at these wavevectors (the red arrows in Fig. 3).
Indeed, we note in the numerical results for the FT of
the SP LDOS the existence of four rings, correspond-
ing to 2p±
F,p+
F+p−
F= 2mvandp−
F−p+
F=pλ, hav-
ing the proper two-fold or fold-fold symmetries, consis-
tent with the cos /sinφrand cos/sin2φrdependence of
the SP LDOS obtained analytically. For example, in the
xcomponent of the SP LDOS induced by an ximpu-
rity, the 2p+
F, 2p−
Fandpλrings have a maximum along x
and a minimum along y, while the 2 mvring has a sym-
metry corresponding to a rotation by 90 degrees. The
ycomponent of the FT of the SP LDOS has a four-
fold symmetry in which we can again identify the same
wavevectors,while the Szcomponent has a two-foldsym-
metry, and the 2 mvvector is absent. Similarly, for the
Sxand theSycomponents of the SP LDOS induced by
azimpurity (these components should be zero in the
absence of the SO coupling) only the 2 p±
Fandpλwave
vectors are present, with similar symmetries, while the
Szcomponent is symmetric. Note also the central peak6
z-impurity x-impurity
Jz(Jx=Jy= 0) Jx(Jy=Jz= 0)
FIG. 3. (Color) The FT of the non-polarized as well as of the
SP LDOS components for the positive energy Shiba state as a
function of momentum, for a magnetic impurity with Jz= 2
(left column), and Jx= 2 (right column). We take t= 1,µ=
3,δ= 0.01,λ= 0.5,∆s= 0.2. For a z-impurity we depict
the real part of the FT for δρand forSz, and the imaginary
part for and SxandSy, whereas for an x-impurity we take
the imaginary part only for the Szcomponent. Black two-
headed arrows correspond to the value of 2 pλ≡4mλ(see the
analytical results) and thus allow to extract the SO couplin g
constant directly from these strong features in momentum
space. The other arrows correspond to the other important
wavevectors that can be observed in these FTs, as identified
with the help of the analytical results.
atpx=py= 0 which is due to the terms independent of
FIG. 4. (Color) The FT of various SP LDOS component for a
Shiba state as a function of the SO coupling λand ofpy(for
px= 0 - vertical cut). We take t= 1,µ= 3,δ= 0.01,∆s=
0.2,Jz= 2.
position in the SP LDOS.
The most important observation is that all the compo-
nents of the FT of the SP LDOS exhibit a strong feature
at wave vector pλ. Thus an experimental observation of
this feature via spin-polarized STM would allow one to
read-off directly the value of the SO coupling. The spin
orbit can also be read-off from the distance between the
2p+and 2p−peaks, though the intensity of these fea-
tures is not as strong. This appears clearly in Fig. 4, in
which we plot a horizontal cut though two of the FT –
SP LDOS above as a function of the SO coupling λ.
Note that the only wave vector present in the non-
polarized LDOS is 2 mv, which has only a very weak de-
pendence on λfor not too large values of the SO with re-
spect to the Fermi velocity, thus it is quasi-impossible to
determine the SO coupling from a measurement without
spin resolution. Note also the typical two-dimensional
1/rdecay of the Friedel oscillations is overlapping in this
case with an exponential decay with wave vector ps.
B. Comparison to the metallic phase
Asimilaranalysiscanbe performedforimpurity states
forming in the vicinity of a magnetic impurity in a metal-
lic system. Here the classical magnetic impurity does not
lead to any localized bound states at a specific energy,
and the intensity of the impurity contribution is roughly
independent of energy.
Thus in Fig. 5 we plot the FT of the impurity con-
tribution to the LDOS and SP LDOS at a fixed energy
E= 0.1. We note that we have similar features to those
observed in the SC regime, with the main differences be-
ing that the long-wavelengthcentral features are now ab-
sent, andthat the FT peaksaremuch sharperthan in the
SC regime. This behavior can be explained from the an-
alytical expressions of the non-polarized and SP LDOS,
whose derivation is presented in Appendix B. The results7
z-impurity x-impurity
Jz(Jx=Jy= 0) Jx(Jy=Jz= 0)
FIG. 5. (Color) The FT of the impurity contributions to the
non-polarized and SP LDOS for an energy E= 0.1 and for
a magnetic impurity with Jz= 2 (left column), and Jx= 2
(right column). We take the inverse quasiparticle lifetime
δ= 0.03 and we set t= 1,µ= 3,λ= 0.5,∆s= 0. For
az-impurity we depict the real part of the FT for δρand
forSz, and the imaginary part for SxandSy, whereas for
anx-impurity we take the imaginary part only for the Sz
component. UnlikeintheSCcase, thestrongpeaksappearing
in the center and at pλare absent here. The arrows denote
the wavevectors of the observed features as identified from t he
analytical calculations.are presented below for an out-of-plane spin impurity:
Sx(r)∼J
1+α2cosφr
r/summationdisplay
σσν2
σ
pσsin2pσr,
Sy(r)∼J
1+α2sinφr
r/summationdisplay
σσν2
σ
pσsin2pσr,
Sz(r)∼ −J
1+α22
r/summationdisplay
σν2
σ
pσcos2pσr,
ρ(r)∼ −J
1+α24αν2v2
F
v21/radicalbig
p2
F+2mE+E2/v2×
×sinpεr
r, (20)
while for an xdirected impurity (in-plane):
Sx(r)∼ −J
1+α2/braceleftbigg
2ν2v2
F
v21−cos2φr
rcospεr/radicalbig
p2
F+2mE+E2/v2
+/summationdisplay
σν2
σ
pσ1+cos2φr
rcos2pσr,
Sy(r)∼ −J
1+α2sin2φr
r/bracketleftbigg
−2ν2v2
F
v2cospεr/radicalbig
p2
F+2mE+E2/v2
+/summationdisplay
σν2
σ
pσcos2pσr/bracketrightbigg
,
Sz(r)∼ −J
1+α2cosφr
r/summationdisplay
σσν2
σ
pσsin2pσr,
ρ(r)∼ −J
1+α2·α
r·4ν2v2
F
v2sinpεr/radicalbig
p2
F+2mE+E2/v2,(21)
withpF=mvF,pσ=pσ
F+E/v/ne}ationslash= 0,pε≡p++p−=
2(mv+E/v) andνσ=ν/bracketleftbig
1−σλ
v/bracketrightbig
.
Note that these expressions are very similar to those
obtained in the SC regime, except that the wave vec-
tors of the oscillations now do not include pλ. However,
this could still be read-off experimentally from the differ-
ence between p−andp+. Another important difference
between the SC and non-SC regimes is the presence of
the exponentially decaying term in the expressions de-
scribing the LDOS dependence for the Shiba states in
the SC regime. The Shiba states have an exponential
decay for distances larger than the superconducting co-
herence length, while the impurity states in the non-SC
regime only decay algebraically as 1 /r. In the Fourier
space this is translated into a much larger broadening of
the features corresponding to the Shiba states in the SC
regime with respect to that of the features corresponding
to the impurity contributions in metals. The width of
the peaks in the latter is solely controlled by the inverse
quasiparticle lifetime δand is generally quite small.
Note also that in both regimes one needs to use the
spin-polarized LDOS and magnetic impurities to be able
to extract the value of the SO coupling, while the non-
polarized LDOS is not sensitive to this wavevector. Last
but notleast, asdescribedin Appendix B,theLDOS per-
turbations induced by a non-magnetic impurity do not8
show any direct signature of the SO coupling (the only
contributing wavevector is 2 mvin the metallic regime,
while in the SC regime no Shiba state form for a non-
magnetic impurity), thus the only manner to have access
to the SO coupling is via spin-polarized STM in the pres-
ence of magnetic impurities.
IV. ONE-DIMENSIONAL SYSTEMS
While in one-dimensional systems superconductivity is
not intrinsic, a superconducting gap can be opened via
proximitizing them with a superconducting substrate.
For such systems it is thus particularly interesting to
study the FT of the SP LDOS for both the supercon-
ducting and non-superconducting regimes, as both these
regimes can be achieved experimentally at low tempera-
ture for the same materials.
WeconsidertheHamiltoniangivenbyEqs.(1-3), where
we setpy→0, and we perform a T-matrix analysis sim-
ilar to that described in the previous section for both
the SC and non-SC phases, for different directions of the
magnetic impurity. The wire is considered to be oriented
along thexdirection, and the SO coupling is oriented
alongy.8,9We thus expect a similar and more exotic be-
havior for impurities directed along xandz, and a more
classical behavior for impurities with the spin parallel to
the direction of the SO, thus oriented along y.
The energies and wave functions of the Shiba states
can be found using the same procedure as for the two-
dimensional systems (see Appendix C). This yields for
the energies of the states:
E1,¯1=±1−α2
1+α2∆s,whereα=J/v.
The FT of the positive energy state as a function of
momentum and the SO coupling is presented in Fig. 6
for a SC (left column) and non-SC state (right column),
for an impurity directed along z. For this situation the
spin of the Shiba state has two non-zero components,
one parallel to the wire, and one parallel to the impurity
spin, and these two components are depicted in Fig. 6.
Note that, similar to the two-dimensional case, there is
a split of the FT features increasing linearly with the SO
coupling strength. Also note that in the non-SC phase
the central feature, whose wave vector is given by pλ,
is absent, and that the FT features are broadened in
the SC regime with respect to the non-SC one. Also,
same as in the two-dimensional case, the SO affects the
spin-polarized components but almost do not change the
non-polarized LDOS, as it can be seen in Fig. 6 where
it appears that the non-polarized LDOS FT features do
not evolve with the SO coupling.
These results are confirmed by analytical calculations.
Below we give the spin components and the LDOS in
the SC state for an impurity directed along zobtained
analytically (see Appendix C), for the positive energy
Shiba state:SC case Non-SC case
FIG. 6. (Color) The FT of various SP LDOS component
for a Shiba state (left column), and for an impurity state at
E= 0.1 (right column), as a function of the SO coupling λ
and of momentum p, for an impurity perpendicular to the
wire and directed along z. We set t= 1,µ= 1. We take
∆s= 0.2,Jz= 4,δ= 0.01 in the SC case and ∆ s= 0,Jz=
2,δ= 0.05 in the non-SC case.
Sx(x) =1+α2
4[2sinpλx+sin(2mv|x|+pλx−2θ)
−sin(2mv|x|−pλx−2θ)]·e−2ω|x|/v
Sy(x) = 0
Sz(x) =−1+α2
4[2cospλx+cos(2mv|x|+pλx−2θ)
+cos(2mv|x|−pλx−2θ)]·e−2ω|x|/v
ρ(x) =1+α2
2[1+cos(2mv|x|−2θ)]·e−2ω|x|/v(22)
where tanθ=α. We also present the FT of the SP
LDOSforthenon-SCphasefortheimpuritycontribution9
corresponding to the energy E(see Appendix D):
Sx(x) = +α
1+α2·1
πv[cos(pε|x|−pλx)−cos(pε|x|+pλx)]
Sy(x) = 0
Sz(x) = +α
1+α2·1
πv[sin(pε|x|−pλx)+sin(pε|x|+pλx)]
ρ(x) =−2α2
1+α2·1
πvcospεx
As before, in the expressions above pε= 2(mv+
E/v),pλ= 2mλ.
Indeed these calculations confirm our observations, in
the SC state the dominantwavevectorsare2 p±
F= 2mv±
pλ, 2mvandpλ, while in the non-SC phase only pǫ±pλ,
and 2mv.
Similar results are obtained if the impurity is oriented
alongx, with the only difference that the xandzcompo-
nents will be interchanged, up to on overall sign change
(see Appendices C and D). For impurities parallel to y,
and thus to the SO vector, we expect the SP LDOS to
be less exotic, and indeed in this case the only non-zero
component of the impurity SP LDOS is Sy. In the SC
regime we thus find
Sx(x) = 0
Sy(x) =−(1+α2)[1+cos(2mv|x|−2θ)]·e−2ω|x|/v
Sz(x) = 0
ρ(x) = +(1+α2)[1+cos(2mv|x|−2θ)]·e−2ω|x|/v
while in the non-SC regime we have
Sx(x) = 0
Sy(x) = +2α
1+α2·1
πvsinpε|x|
Sz(x) = 0
ρ(x) =−2α2
1+α2·1
πvcospεx
We see that Syexhibits features only at the 2 mvand
correspondingly at the pǫwave vectors, same as the non-
polarized LDOS, thus not allowing for the detection of
the SO coupling.
For intermediate directions of the impurity spin, all
three components will be present, with the xandzex-
hibiting all the wave vectors, while the ycomponent
solely the 2 mv, and with relative intensities given by the
relative components of the impurity spin.
Thus, we conclude that, same as in the 2D case, the
SO can be measured using spin-polarized STM and mag-
netic impurities; moreover, in the 1D case one needs to
consider impurities that have a non-zero component per-
pendicular to the direction of the SO.
V. CONCLUSIONS
We have analyzed the formation of Shiba states and
impuritystatesin1Dand2Dsuperconductingandmetal-lic systems with Rashba SO coupling. In particular we
have studied the Fourier transform of the local density of
states of Shiba states in SCs and of the impurity states
in metals, both non-polarized and spin-polarized. We
have shown that the spin-polarized density of states con-
tains information that allows one to extract experimen-
tally the strength of the SO coupling. In particular the
features observed in the FT of the SP LDOS split with
a magnitude proportional to the SO coupling strength.
Moreover, the Friedel oscillations in the SP LDOS in the
SC regime show a combination of wavelengths, out of
which the SO length can be read off directly and non-
ambiguously. We note that these signatures are only vis-
ible in the spin-polarized quantities and in the presence
of magnetic impurities. For non-spin-polarized measure-
ments, no such splitting is present and the wave vectors
observed in the FT of the SP LDOS basically do not
depend on the SO coupling. When comparing the re-
sults for the SC Shiba states to the impurity contribu-
tion in the metallic state and we find a few interesting
differences, such as a broadening of the FT features cor-
responding to a spatial exponential decay of the Shiba
states compared to the non-SC case. Moreover, the FT
of the SP LDOS in the SC regime exhibits extra fea-
tures with a wavelengthequal to the SO length which are
not present in the non-SC phase. It would be interest-
ing to generalize our results to more realistic calculations
which may include some specific lattice characteristics,
more realistic material-dependent tight-binding parame-
ters for the band structure and the SO coupling values.
However, we should note that our results have a fully
general characteristic, independent of the band structure
or other material characteristics, and that the features
in the FT of the non-polarized LDOS will correspond to
split features in the spin-polarized LDOS, and thus the
spin-orbit can be measured unequivocally from the split
obtained from the comparison between the non-polarized
and spin-polarized measurements. We have checked that
up to a rotation in the spin space our results hold also
for other types of SO coupling such as Dresselhaus.
According to our knowledge, the FT-STM is a well-
established experimental technique which does not deal
with large systematic errors.36–43The experimental data
presented e.g. in Ref. 43 shows that the resolution in
the Fourier space (momentum space) reaches 0 .05˚A−1,
whereas a typical value of spin-orbit coupling wave vec-
torpλ∼0.15˚A−1(see e.g. Ref. 22), and thus it is suf-
ficient to resolve the features originating from the spin-
orbit coupling. Moreover, we would like to point that
the exponent e−2psrdefines in the real space how far the
impurity-induced states areextended, and it manifests in
the momentum space as the widening of the ring-like fea-
tures appearing at particular momenta. The condition of
resolving the spin-orbit is thus 2 ps< pλ, otherwise the
widening is large enough to blur the spin-orbit feature.
This condition can be rewritten in a more explicit way,10
namely
1/radicalbig
1+(λ/vF)2·α
1+α2·∆s
εF<λ
vF
For any realistic parameters the first two factors on the
left side are of the order of unity, and ∆ s/εF∼10−3
for superconductors. However, for realistic values of the
spin-orbit coupling λ, this inequality holds and therefore
there should not be any technical problem with resolving
those features.
Our results can be tested using for example materials
such as Pb, Bi, NbSe 2or InAs and InSb wires, which
are known to have a strong SO coupling, using spin-
polarized STM which is nowadays becoming more andmore available.35
While finalizing this manuscript we became aware of
a recent work70focusing on issues similar to some of the
subjects (in particular the real space Friedel oscillations
in the metallic regime) addressed in our work.
ACKNOWLEDGMENTS
This work is supported by the ERC Starting Indepen-
dent Researcher Grant NANOGRAPHENE 256965. PS
would like to acknowledge interesting discussions with T.
Cren and financial support from the French Agence Na-
tionale de la Recherche through the contract Mistral.
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Appendix A: Analytical calculation of the Shiba states wave functions for a 2D system
Wecancalculateanalyticallythenon-polarizedandtheSPLDOSforth eShibastatesexploitingthemodeldescribed
by the Hamiltonians in Eqs. (1-3). All the integrations below are perf ormed using a linearization around the Fermi
energy. The energies of the Shiba states can be found by solving th e corresponding eigenvalue equation66
[I4−VG0(E,r=0)]Φ(0) = 0 (A1)
whereG0(E,r) is the retarded Green’s function in real space obtained by a Fourie r transform from the retarded
Green’s function in momentum space G0(E,p) = [(E+iδ)I4−H0(p)]−1, whereδis the inverse quasiparticle lifetime.
In all the calculations below we take the limit of δ→+0, and we specify + i0 only in the cases when it affects the
results. The wave functions of the Shiba states at r= 0 are given by the eigenfunctions obtained from the equation
above. Their spatial dependence is determined using
Φ(r) =G0(E,r)VΦ(0) (A2)12
Consequently, the non-polarized and the SP LDOS are given by
ρ(E,r) = Φ†(r)/parenleftbigg
0 0
0σ0/parenrightbigg
Φ(r), (A3)
S(E,r) = Φ†(r)/parenleftbigg
0 0
0σ/parenrightbigg
Φ(r). (A4)
Thus, in order to find the energies and the wave functions corresp onding to the Shiba states we need to find the
real-space Green’s function. This is obtained simply by a Fourier tran sform of the unperturbed Green’s function in
momentum space, G0(E,p). We start by writing down the unperturbed Green’s function in mom entum space, which
is given by G0(E,p) =1
2/summationtext
σ=±Gσ
0(E,p), where
Gσ
0(E,p) =−1
ξ2σ+ω2/parenleftbigg
1iσe−iφp
−iσeiφp1/parenrightbigg
⊗/parenleftbigg
E+ξσ∆s
∆sE−ξσ/parenrightbigg
, (A5)
whereω=/radicalbig
∆2s−E2, ξσ=ξp+σλp. Toobtainitsreal-spacedependence oneneedstoperformthe Fo uriertransform:
Gσ
0(E,r) =/integraldisplaydp
(2π)2Gσ
0(E,p)eipr
We will have four types of integrals:
Xσ
0(r) =−/integraldisplaydp
(2π)2eipr
ξ2σ+ω2(A6)
Xσ
1(r) =−/integraldisplaydp
(2π)2ξσeipr
ξ2σ+ω2(A7)
Xσ
2(s,r) =−/integraldisplaydp
(2π)2−isσeisφpeipr
ξ2σ+ω2(A8)
Xσ
3(s,r) =−/integraldisplaydp
(2π)2−isσeisφpξσeipr
ξ2σ+ω2(A9)
Since the spectrum is split by SO coupling, there will be two Fermi mome nta which can be found the following way:
p2
2m+σλp−εF= 0, pσ
F=−σλ+/radicalbig
λ2+2εF/m
1/m
Forp>0 we linearize the spectrum around the Fermi momenta, thus:
ξσ≈/parenleftbiggpσ
F
m+σλ/parenrightbigg
(p−pσ
F) =/radicalbig
λ2+2εF/m(p−pσ
F)≡v(p−pσ
F),
thereforep=pF+ξσ/v, wherev=/radicalbig
v2
F+λ2. We rewrite:
dp
(2π)2=m
2π/bracketleftbigg
1−σλ
v/bracketrightbigg
dξσdφ
2π=νσdξσdφ
2π,
whereνσ=ν/bracketleftbig
1−σλ
v/bracketrightbig
, withν=m/2π. Due to the symmetry all the integrals are zero at r=0except for the first
one, namely,
Xσ
0(0) =−νσπ
ω. (A10)
All the coordinate dependences can be calculated using the formalis m introduced in Ref. 60. Finally we get:
Xσ
0(r) =−2νσ·1
ω·ℑK0[−i(1+iΩσ)pσ
Fr] (A11)
Xσ
1(r) =−2νσ·ℜK0[−i(1+iΩσ)pσ
Fr] (A12)
Xσ
2(s,r) = 2sσνσ·1
ω·eisφr·ℜK1[−i(1+iΩσ)pσ
Fr] (A13)
Xσ
3(s,r) =−2sσνσ·eisφr·ℑK1[−i(1+iΩσ)pσ
Fr], (A14)13
where Ω σ=ω/pσ
Fvdefines the inverse superconducting decay length, and pS=ω/v. Therefore, the Green’s function
can be written as
Gσ
0(E,r) =
EXσ
0(r)+Xσ
1(r)EXσ
2(−,r)+Xσ
3(−,r) ∆ sXσ
0(r) ∆ sXσ
2(−,r)
EXσ
2(+,r)+Xσ
3(+,r)EXσ
0(r)+Xσ
1(r) ∆ sXσ
2(+,r) ∆ sXσ
0(r)
∆sXσ
0(r) ∆ sXσ
2(−,r)EXσ
0(r)−Xσ
1(r)EXσ
2(−,r)−Xσ
3(−,r)
∆sXσ
2(+,r) ∆ sXσ
0(r)EXσ
2(+,r)−Xσ
3(+,r)EXσ
0(r)−Xσ
1(r)
.
(A15)
Thus we have:
G0(E,r=0) =−πν/radicalbig
∆2s−E2/parenleftbigg
Eσ0∆sσ0
∆sσ0Eσ0/parenrightbigg
. (A16)
1. z-impurity
The coordinate dependence of the eigenfunctions is given by
Φ¯1(r) = +Jz
2/summationdisplay
σ=±
(E¯1−∆s)Xσ
0(r)+Xσ
1(r)
(E¯1−∆s)Xσ
2(+,r)+Xσ
3(+,r)
−(E¯1−∆s)Xσ
0(r)+Xσ
1(r)
−(E¯1−∆s)Xσ
2(+,r)+Xσ
3(+,r)
,Φ1(r) =−Jz
2/summationdisplay
σ=±
(E1+∆s)Xσ
2(−,r)+Xσ
3(−,r)
(E1+∆s)Xσ
0(r)+Xσ
1(r)
(E1+∆s)Xσ
2(−,r)−Xσ
3(−,r)
(E1+∆s)Xσ
0(r)−Xσ
1(r)
.
(A17)
Using these expressions we can compute the asymptotic behavior o f the non-polarized and SP LDOS in coordinate
space for the state with positive energy (thus we omit index 1 below) :
Sx(r) = +J2
z/parenleftbigg
1+1
α2/parenrightbigg/braceleftigg/summationdisplay
σσν2
σcos(2pσ
Fr−θ)
pσ
F+2ν2v2
F
v2·sinpλr
pF/bracerightigg
·e−2psr
rcosφr (A18)
Sy(r) = +J2
z/parenleftbigg
1+1
α2/parenrightbigg/braceleftigg/summationdisplay
σσν2
σcos(2pσ
Fr−θ)
pσ
F+2ν2v2
F
v2·sinpλr
pF/bracerightigg
·e−2psr
rsinφr (A19)
Sz(r) =−J2
z/parenleftbigg
1+1
α2/parenrightbigg/braceleftigg/summationdisplay
σν2
σsin(2pσ
Fr−θ)
pσ
F−2ν2v2
F
v2·cospλr
pF/bracerightigg
·e−2psr
r(A20)
ρ(r) = +J2
z/parenleftbigg
1+1
α2/parenrightbigg/braceleftbigg
2ν2
mv+2ν2v2
F
v2·sin(2mvr−θ)
pF/bracerightbigg
·e−2psr
r(A21)
with tanθ=/braceleftigg
2α
1−α2,ifα/ne}ationslash= 1
+∞,ifα= 1,andpλ= 2mλ. Performing the Fourier transforms of these expressions we can
obtain information about the main features and symmetries that we observe in momentum space:
Sx(p) = +2πiJ2
z/parenleftbigg
1+1
α2/parenrightbigg
cosφp+∞/integraldisplay
0drJ1(pr)/braceleftigg/summationdisplay
σσν2
σcos(2pσ
Fr−θ)
pσ
F+2ν2v2
F
v2·sinpλr
pF/bracerightigg
·e−2psr(A22)
Sy(p) = +2πiJ2
z/parenleftbigg
1+1
α2/parenrightbigg
sinφp+∞/integraldisplay
0drJ1(pr)/braceleftigg/summationdisplay
σσν2
σcos(2pσ
Fr−θ)
pσ
F+2ν2v2
F
v2·sinpλr
pF/bracerightigg
·e−2psr(A23)
Sz(p) =−2πJ2
z/parenleftbigg
1+1
α2/parenrightbigg+∞/integraldisplay
0drJ0(pr)/braceleftigg/summationdisplay
σν2
σsin(2pσ
Fr−θ)
pσ
F−2ν2v2
F
v2·cospλr
pF/bracerightigg
·e−2psr(A24)
ρ(p) = +2πJ2
z/parenleftbigg
1+1
α2/parenrightbigg+∞/integraldisplay
0drJ0(pr)/braceleftbigg
2ν2
mv+2ν2v2
F
v2·sin(2mvr−θ)
pF/bracerightbigg
·e−2psr(A25)14
2. x-impurity
The coordinate dependence of the eigenfunctions is given by
Φ¯1(r) = +Jx
2/summationdisplay
σ=±
+(E¯1−∆s)[Xσ
0(r)+Xσ
2(−,r)]+Xσ
1(r)+Xσ
3(−,r)
+(E¯1−∆s)[Xσ
0(r)+Xσ
2(+,r)]+Xσ
1(r)+Xσ
3(+,r)
−(E¯1−∆s)[Xσ
0(r)+Xσ
2(−,r)]+Xσ
1(r)+Xσ
3(−,r)
−(E¯1−∆s)[Xσ
0(r)+Xσ
2(+,r)]+Xσ
1(r)+Xσ
3(+,r)
, (A26)
Φ1(r) =−Jx
2/summationdisplay
σ=±
+(E1+∆s)[Xσ
0(r)−Xσ
2(−,r)]+Xσ
1(r)−Xσ
3(−,r)
−(E1+∆s)[Xσ
0(r)−Xσ
2(+,r)]−Xσ
1(r)+Xσ
3(+,r)
+(E1+∆s)[Xσ
0(r)−Xσ
2(−,r)]−Xσ
1(r)+Xσ
3(−,r)
−(E1+∆s)[Xσ
0(r)−Xσ
2(+,r)]+Xσ
1(r)−Xσ
3(+,r)
. (A27)
For the positive energy state we compute the asymptotic behavior of the non-polarized and SP LDOS in coordinate
space. We write Sx(r) =Ss
x(r)+Sa
x(r):
Ss
x(r) =−J2
x/parenleftbigg
1+1
α2/parenrightbigg/braceleftigg/summationdisplay
σν2
σ1+sin(2pσ
Fr−2β)
pσ
F+2ν2v2
F
v2·cospλr+sin(2mvr−2β)
pF/bracerightigg
·e−2psr
r(A28)
Sa
x(r) = +J2
x/parenleftbigg
1+1
α2/parenrightbigg/braceleftigg/summationdisplay
σν2
σ1−sin(2pσ
Fr−2β)
pσ
F−2ν2v2
F
v2·cospλr+sin(2mvr−2β)
pF/bracerightigg
·e−2psr
rcos2φr(A29)
Sy(r) = +J2
x/parenleftbigg
1+1
α2/parenrightbigg/braceleftigg/summationdisplay
σν2
σ1−sin(2pσ
Fr−2β)
pσ
F−2ν2v2
F
v2·cospλr−sin(2mvr−θ)
pF/bracerightigg
·e−2psr
rsin2φr(A30)
Sz(r) =−J2
x/parenleftbigg
1+1
α2/parenrightbigg/braceleftigg
2/summationdisplay
σσν2
σcos(2pσ
Fr−θ)
pσ
F+4ν2v2
F
v2·sinpλr
pF/bracerightigg
·e−2psr
rcosφr (A31)
ρ(r) = +J2
x/parenleftbigg
1+1
α2/parenrightbigg/braceleftbigg
4ν2
mv+4ν2v2
F
v2·sin(2mvr−θ)
pF/bracerightbigg
·e−2psr
r(A32)
with tanβ=α. Sameasbefore, performingthe Fouriertransformsoftheseex pressionsallowsus to obtaininformation
about the most important features and symmetries we observe in m omentum space:
Ss
x(p) =−2πJ2
x/parenleftbigg
1+1
α2/parenrightbigg+∞/integraldisplay
0drJ0(pr)/braceleftigg/summationdisplay
σν2
σ1+sin(2pσ
Fr−2β)
pσ
F+ (A33)
+2ν2v2
F
v2·cospλr+sin(2mvr−2β)
pF/bracerightbigg
·e−2psr(A34)
Sa
x(p) =−2πJ2
x/parenleftbigg
1+1
α2/parenrightbigg
cos2φp+∞/integraldisplay
0drJ2(pr)/braceleftigg/summationdisplay
σν2
σ1−sin(2pσ
Fr−2β)
pσ
F− (A35)
−2ν2v2
F
v2·cospλr+sin(2mvr−2β)
pF/bracerightbigg
·e−2psr(A36)
Sy(p) =−2πJ2
x/parenleftbigg
1+1
α2/parenrightbigg
sin2φp+∞/integraldisplay
0drJ2(pr)/braceleftigg/summationdisplay
σν2
σ1−sin(2pσ
Fr−2β)
pσ
F− (A37)
−2ν2v2
F
v2·cospλr−sin(2mvr−θ)
pF/bracerightbigg
·e−2psr(A38)
Sz(p) =−2πiJ2
x/parenleftbigg
1+1
α2/parenrightbigg
cosφp+∞/integraldisplay
0drJ1(pr)/braceleftigg
2/summationdisplay
σσν2
σcos(2pσ
Fr−θ)
pσ
F+4ν2v2
F
v2·sinpλr
pF/bracerightigg
·e−2psr(A39)
ρ(p) = +2πJ2
x/parenleftbigg
1+1
α2/parenrightbigg+∞/integraldisplay
0drJ0(pr)/braceleftbigg
4ν2
mv+4ν2v2
F
v2·sin(2mvr−θ)
pF/bracerightbigg
·e−2psr(A40)15
Appendix B: The SPDOS for a 2D metallic system in the presence of a magnetic impurity
The low-energy Hamiltonian can be written as
H0=ξpσ0+λ(pyσx−pxσy) =/parenleftbigg
ξpiλp−
−iλp+ξp/parenrightbigg
, (B1)
whereξp=p2
2m−εF. The corresponding spectrum is given by E=ξp±λp. The retarded Green’s function reads
G0(E,p) =1
(E−ξp+i0)2−λ2p2/parenleftbigg
E−ξp+i0iλp−
−iλp+E−ξp+i0/parenrightbigg
(B2)
To compute the eigenvalues for a single localized impurity we calculate
G0(E,r=0) =/integraldisplaydp
(2π)2E−ξp+i0
(E−ξp+i0)2−λ2p2/parenleftbigg
1 0
0 1/parenrightbigg
=1
2/summationdisplay
σ/integraldisplaydp
(2π)21
E−ξσ+i0/parenleftbigg
1 0
0 1/parenrightbigg
,
whereξσ=ξp+σλp. Forp>0 we linearize the spectrum around Fermi momenta, thus:
ξσ≈/parenleftbiggpσ
F
m+σλ/parenrightbigg
(p−pσ
F) =/radicalbig
λ2+2εF/m(p−pσ
F)≡v(p−pσ
F),
withpσ
F=m[−σλ+v], and thus we rewrite:
dp
(2π)2=m
2π/bracketleftbigg
1−σλ
v/bracketrightbigg
dξσdφ
2π=νσdξσdφ
2π,
whereνσ=ν/bracketleftbig
1−σλ
v/bracketrightbig
, withν=m/2π. Thus we get:
/integraldisplaydp
(2π)21
E−ξσ+i0=νσ/integraldisplay
dξσ1
E−ξσ+i0=−iπνσ,
and therefore:
G0(E,r=0) =1
2/summationdisplay
σ(−iπνσ)/parenleftbigg
1 0
0 1/parenrightbigg
=−iπν/parenleftbigg
1 0
0 1/parenrightbigg
(B3)
Since there is no energy dependence, there will be no impurity-induc ed states. To find the coordinate dependence of
the Green’s function we calculate:
Xσ
0(r) =/integraldisplaydp
(2π)2eipr
E−ξσ+i0(B4)
Xσ
1(s,r) =/integraldisplaydp
(2π)2−iseisφpeipr
E−ξσ+i0(B5)
Below we use the Sokhotsky formula:
1
x+i0=P1
x−iπδ(x)
Xσ
0(r) =/integraldisplaydp
(2π)2eipr
E−ξσ+i0=νσ/integraldisplay
dξσ/integraldisplaydφp
2πeiprcos(φp−φr)
E−ξσ+i0=νσ/integraldisplay
dξσJ0[(pσ
F+ξσ/v)r]
E−ξσ+i0=
=νσ/braceleftbigg
P/integraldisplay
dξσJ0[(pσ
F+ξσ/v)r]
E−ξσ−iπ/integraldisplay
dξσδ(E−ξσ)J0[(pσ
F+ξσ/v)r]/bracerightbigg
=♠
We calculate separately the first integral:
P/integraldisplay
dξσJ0[(pσ
F+ξσ/v)r]
E−ξσ=2
π+∞/integraldisplay
1du√
u2−1P/integraldisplay
dξσsin[(pσ
F+ξσ/v)r]
E−ξσ=
=2
πℑ+∞/integraldisplay
1du√
u2−1P/integraldisplay
dξσei(pσ
F+ξσ/v)r
E−ξσ=2
πℑ+∞/integraldisplay
1du√
u2−1eipσru·P/integraldisplay
dxe−ir
vx
x=♣16
P/integraldisplay
dxe−ir
vx
x=P/integraldisplaycosr
vx
xdx−iP/integraldisplaysinr
vx
xdx= 0−iπ=−iπ
Therefore:
♣=−2ℑ+∞/integraldisplay
1ieipσru
√
u2−1du=−2+∞/integraldisplay
1cospσru√
u2−1du=πY0(pσr), pσ/ne}ationslash= 0
♠=πνσ[Y0(pσr)−iJ0(pσr)].
The second integral is
Xσ
1(s,r) =/integraldisplaydp
(2π)2−iseisφpeipr
E−ξσ+i0=νσ/integraldisplay
dξσ/integraldisplaydφp
2π−iseisφpeiprcos(φp−φr)
E−ξσ+i0=seisφrνσ/integraldisplay
dξσJ1[(pσ
F+ξσ/v)r]
E−ξσ+i0=
=seisφr·νσ/braceleftbigg
P/integraldisplay
dξσJ1[(pσ
F+ξσ/v)r]
E−ξσ−iπ/integraldisplay
dξσδ(E−ξσ)J1[(pσ
F+ξσ/v)r]/bracerightbigg
=♥
We calculate separately the first integral:
P/integraldisplay
dξσJ1[(pσ
F+ξσ/v)r]
E−ξσ=P/integraldisplay
dxJ1[(pσ−x/v)r]
x=−∂
∂rP/integraldisplay
dxJ0[(pσ−x/v)r]
x(pσ−x/v)=
=−∂
∂rP/integraldisplay
dyJ0[(pσ−y)r]
y(pσ−y)=−∂
∂(pσr)/bracketleftbigg
P/integraldisplay
dyJ0[(pσ−y)r]
y+P/integraldisplay
dyJ0[(pσ−y)r]
pσ−y/bracketrightbigg
=
=−∂
∂(pσr)2
πℑ+∞/integraldisplay
1du√
u2−1/bracketleftbigg
P/integraldisplayei(pσ−y)ru
ydy+P/integraldisplayei(pσ−y)ru
pσ−ydy/bracketrightbigg
=−2∂
∂(pσr)ℑ+∞/integraldisplay
1idu√
u2−1/bracketleftbig
1−eipσru/bracketrightbig
=
=−2+∞/integraldisplay
1usinpσru√
u2−1du= 2∂
∂(pσr)+∞/integraldisplay
1cospσru√
u2−1du=−π∂
∂(pσr)Y0(pσr) =πY1(pσr), pσ/ne}ationslash= 0
Therefore:
♥=πνσ[Y1(pσr)−iJ1(pσr)].
Finally:
Xσ
0(r) =πνσ[Y0(pσr)−iJ0(pσr)] (B6)
Xσ
1(s,r) =seisφr/braceleftig
πνσ[Y1(pσr)−iJ1(pσr)]/bracerightig
≡seisφr˜Xσ
1(r), (B7)
wherepσ=pσ
F+E/v/ne}ationslash= 0. Thus the Green’s function for r/ne}ationslash=0can be written as:
G0(E,r) =1
2/summationdisplay
σ/parenleftbigg
Xσ
0(r)−σe−iφr˜Xσ
1(r)
σeiφr˜Xσ
1(r)Xσ
0(r)/parenrightbigg
(B8)
Below we compute the T-matrix for different types of impurities. Imp urity potentials take the following forms:
Vsc=U/parenleftbigg
1 0
0 1/parenrightbigg
, Vz=Jz/parenleftbigg
1 0
0−1/parenrightbigg
, Vx=Jx/parenleftbigg
0 1
1 0/parenrightbigg
(B9)
The corresponding T-matrices are
Tsc=U
1+iπνU/parenleftbigg
1 0
0 1/parenrightbigg
, Tz=/parenleftbiggJ
1+iπνJ0
0−J
1−iπνJ/parenrightbigg
, T x=J
1+π2ν2J2/parenleftbigg
−iπνJ 1
1−iπνJ/parenrightbigg
(B10)
For each type of impurity we can compute the SP and non-polarized L DOS using
∆G(E,r) =G0(E,−r)T(E)G0(E,r) (B11)17
Sx(E,r) =−1
π[ℑ∆G12+ℑ∆G21] (B12)
Sy(E,r) =−1
π[ℜ∆G12−ℜ∆G21] (B13)
Sz(E,r) =−1
π[ℑ∆G11−ℑ∆G22] (B14)
∆ρ(E,r) =−1
π[ℑ∆G11+ℑ∆G22] (B15)
Asymptotic expansions of Bessel functions
Since the integrals are expressed in terms of Neumann function and Bessel function of the first kind, we give their
asymptotic behavior for x→+∞:
J0(x)∼+/radicalbigg
2
πxcos/parenleftig
x−π
4/parenrightig
, J 1(x)∼ −/radicalbigg
2
πxcos/parenleftig
x+π
4/parenrightig
Y0(x)∼ −/radicalbigg
2
πxcos/parenleftig
x+π
4/parenrightig
, Y 1(x)∼ −/radicalbigg
2
πxcos/parenleftig
x−π
4/parenrightig
Fourier transforms in 2D
F[f(r)] = 2π+∞/integraldisplay
0rJ0(pr)f(r)dr (B16)
F[cosφrf(r)] = 2πicosφp·+∞/integraldisplay
0rJ1(pr)f(r)dr,F[sinφrf(r)] = 2πisinφp·+∞/integraldisplay
0rJ1(pr)f(r)dr(B17)
F[cos2φrf(r)] =−2πcos2φp+∞/integraldisplay
0rJ2(pr)f(r)dr,F[sin2φrf(r)] =−2πsin2φp+∞/integraldisplay
0rJ2(pr)f(r)dr(B18)
1. z-impurity
We denote α=πνJand write the asymptotic expansions of the non-polarized and SP LD OS components in
coordinate space:
Sx(r)∼J
1+α2cosφr
r/summationdisplay
σσν2
σ
pσsin2pσr (B19)
Sy(r)∼J
1+α2sinφr
r/summationdisplay
σσν2
σ
pσsin2pσr (B20)
Sz(r)∼ −J
1+α22
r/summationdisplay
σν2
σ
pσcos2pσr (B21)
ρ(r)∼ −J
1+α24αν2v2
F
v21/radicalbig
p2
F+2mE+E2/v2·sinpεr
r, (B22)18
wherepε= 2(mv+E/v). and we get for pσ>0:
Sx(p)∼+J
1+α2·2πicosφp+∞/integraldisplay
0drJ1(pr)/summationdisplay
σσν2
σ
pσsin2pσr (B23)
Sy(p)∼+J
1+α2·2πisinφp+∞/integraldisplay
0drJ1(pr)/summationdisplay
σσν2
σ
pσsin2pσr (B24)
Sz(p)∼ −J
1+α2·4π+∞/integraldisplay
0drJ0(pr)/summationdisplay
σν2
σ
pσcos2pσr (B25)
ρ(p)∼ −J
1+α2·8παν2v2
F
v21/radicalbig
p2
F+2mE+E2/v2+∞/integraldisplay
0drJ0(pr)sinpεr (B26)
2. x-impurity
Sx(r)∼ −J
1+α21
r/braceleftigg
2ν2v2
F
v2cospεr/radicalbig
p2
F+2mE+E2/v2+/summationdisplay
σν2
σ
pσcos2pσr+ (B27)
+cos2φr/bracketleftigg
−2ν2v2
F
v2cospεr/radicalbig
p2
F+2mE+E2/v2+/summationdisplay
σν2
σ
pσcos2pσr/bracketrightigg/bracerightigg
(B28)
Sy(r)∼ −J
1+α2sin2φr
r/bracketleftigg
−2ν2v2
F
v2cospεr/radicalbig
p2
F+2mE+E2/v2+/summationdisplay
σν2
σ
pσcos2pσr/bracketrightigg
(B29)
Sz(r)∼ −J
1+α2cosφr
r/summationdisplay
σσν2
σ
pσsin2pσr (B30)
ρ(r)∼ −J
1+α2·α
r·4ν2v2
F
v2sinpεr/radicalbig
p2
F+2mE+E2/v2(B31)
With the corresponding Fourier transforms:
Sx(p) =Ssym
x(p)+Sasym
x(p) =−J
1+α2·2π+∞/integraldisplay
0drJ0(pr)/bracketleftigg
2ν2v2
F
v2cospεr/radicalbig
p2
F+2mE+E2/v2+/summationdisplay
σν2
σ
pσcos2pσr/bracketrightigg
−(B32)
−J
1+α2·2πcos2φp+∞/integraldisplay
0drJ2(pr)/bracketleftigg
2ν2v2
F
v2cospεr/radicalbig
p2
F+2mE+E2/v2−/summationdisplay
σν2
σ
pσcos2pσr/bracketrightigg
(B33)
Sy(p) =−J
1+α2·2πsin2φp+∞/integraldisplay
0drJ2(pr)/bracketleftigg
2ν2v2
F
v2cospεr/radicalbig
p2
F+2mE+E2/v2−/summationdisplay
σν2
σ
pσcos2pσr/bracketrightigg
(B34)
Sz(p)∼ −J
1+α2·2πicosφp+∞/integraldisplay
0drJ1(pr)/summationdisplay
σσν2
σ
pσsin2pσr (B35)
ρ(p)∼ −J
1+α2·8παν2v2
F
v21/radicalbig
p2
F+2mE+E2/v2+∞/integraldisplay
0drJ0(pr)sinpεr (B36)19
Appendix C: Analytical calculation of the Shiba states wave functions for a 1D system
The unperturbed Green’s function in momentum space is G0(E,p) =1
2/summationtext
σ=±Gσ
0(E,p), where
Gσ
0(E,p) =−1
ξ2σ+∆2s−E2/parenleftbigg
1iσ
−iσ1/parenrightbigg
⊗/parenleftbigg
E+ξσ∆s
∆sE−ξσ/parenrightbigg
, (C1)
whereξσ=ξp+σλp. To get the coordinate value one needs to perform the Fourier tra nsform:
Gσ
0(E,x) =/integraldisplaydp
2πGσ
0(E,p)eipx
We will have two types of integrals:
Xσ
0(x) =−/integraldisplaydp
2πeipx
ξ2σ+ω2, (C2)
Xσ
1(x) =−/integraldisplaydp
2πξσeipx
ξ2σ+ω2, (C3)
whereω2= ∆2
s−E2. Since the spectrum is split by SO coupling, there will be two Fermi mom enta which can be
found the following way:
p2
2m+σλp−εF= 0, pσ
F=−σλ+/radicalbig
λ2+2εF/m
1/m≡m[−σλ+v]
Forp>0 we linearize the spectrum around Fermi momenta, thus:
ξσ≈/parenleftbiggpσ
F
m+σλ/parenrightbigg
(p−pσ
F) =/radicalbig
λ2+2εF/m(p−pσ
F)≡v(p−pσ
F),
thereforep=pσ
F+ξσ/vand we get:
Xσ
0(x) =−/integraldisplaydp
2πeipx
ξ2σ+ω2=−
+∞/integraldisplay
0dp
2πeipx
ξ2σ+ω2++∞/integraldisplay
0dp
2πe−ipx
ξ2
−σ+ω2
=♣
+∞/integraldisplay
0dp
2πeipx
ξ2σ+ω2≈1
2πveipσ
Fx/integraldisplay
dξσeiξσx/v
ξ2σ+ω2=1
2vωeipσ
Fxe−ω|x|/v
+∞/integraldisplay
0dp
2πe−ipx
ξ2
−σ+ω2≈1
2πve−ip−σ
Fx/integraldisplay
dξ−σe−iξ−σx/v
ξ2
−σ+ω2=1
2vωe−ip−σ
Fxe−ω|x|/v
♣=−1
2vω/bracketleftig
eim[−σλ+v]x+e−im[σλ+v]x/bracketrightig
e−ω|x|/v=−1
v·1
ωcosmvxe−iσmλxe−ω|x|/v
Xσ
1(x) =−/integraldisplaydp
2πξσeipx
ξ2σ+ω2=−
+∞/integraldisplay
0dp
2πξσeipx
ξ2σ+ω2++∞/integraldisplay
0dp
2πξ−σe−ipx
ξ2
−σ+ω2
=♠
+∞/integraldisplay
0dp
2πξσeipx
ξ2σ+ω2≈1
2πveipσ
Fx/integraldisplay
dξσξσeiξσx/v
ξ2σ+ω2=i
2vsgnxeipσ
Fxe−ω|x|/v
+∞/integraldisplay
0dp
2πξ−σe−ipx
ξ2
−σ+ω2≈1
2πve−ip−σ
Fx/integraldisplay
dξ−σξ−σe−iξ−σx/v
ξ2
−σ+ω2=−i
2vsgnxe−ip−σ
Fxe−ω|x|/v20
♠=−i
2vsgnx/bracketleftig
eim[−σλ+v]x−e−im[σλ+v]x/bracketrightig
e−ω|x|/v=1
v·sinmv|x|e−iσmλxe−ω|x|/v
Finally:
Xσ
0(x) =−1
v·1
ωcosmvxe−iσmλxe−ω|x|/v(C4)
Xσ
1(x) = +1
v·sinmv|x|e−iσmλxe−ω|x|/v(C5)
and
G0(E,x) =1
2/summationdisplay
σ=±/parenleftbigg
1iσ
−iσ1/parenrightbigg
⊗/parenleftbigg
EXσ
0(x)+Xσ
1(x) ∆ sXσ
0(x)
∆sXσ
0(x)EXσ
0(x)−Xσ
1(x)/parenrightbigg
(C6)
G0(ǫ,x= 0) =−1
v1√
1−ǫ2/parenleftbigg
ǫσ0σ0
σ0ǫσ0/parenrightbigg
,whereǫ=E
∆s(C7)
The eigenvalues and eigenfunctions at r=0can be obtained using Eq. (10) The energy levels are
E1,¯1=±1−α2
1+α2∆s,whereα=J/v. (C8)
In case of an impurity along the z-axis the corresponding eigenvectors are
Φ¯1(0) =/parenleftbig1 0−1 0/parenrightbigT,Φ1(0) =/parenleftbig0 1 0 1/parenrightbigT(C9)
and in case of an impurity along the x-axis:
Φ¯1(0) =/parenleftbig1 1−1−1/parenrightbigT,Φ1(0) =/parenleftbig1−1 1−1/parenrightbigT. (C10)
1. z-impurity
Φ¯1(x) = +Jz
2/summationdisplay
σ
+(E¯1−∆s)Xσ
0(x)+Xσ
1(x)
−iσ[(E¯1−∆s)Xσ
0(x)+Xσ
1(x)]
−(E¯1−∆s)Xσ
0(x)+Xσ
1(x)
+iσ[(E¯1−∆s)Xσ
0(x)−Xσ
1(x)]
,Φ1(x) =−Jz
2/summationdisplay
σ
+iσ[(E1+∆s)Xσ
0(x)+Xσ
1(x)]
(E1+∆s)Xσ
0(x)+Xσ
1(x)
+iσ[(E1+∆s)Xσ
0(x)−Xσ
1(x)]
(E1+∆s)Xσ
0(x)−Xσ
1(x)
.
(C11)
Using these expressions we can compute the non-polarized and SP L DOS in both coordinate and momentum space
for the positive energy state (omitting the index 1):
Sx(x) =1+α2
4[2sinpλx+sin(2mv|x|+pλx−2θ)−sin(2mv|x|−pλx−2θ)]·e−2ω|x|/v(C12)
Sy(x) = 0 (C13)
Sz(x) =−1+α2
4[2cospλx+cos(2mv|x|+pλx−2θ)+cos(2mv|x|−pλx−2θ)]·e−2ω|x|/v(C14)
ρ(x) =1+α2
2[1+cos(2mv|x|−2θ)]·e−2ω|x|/v(C15)21
where tanθ=α. We perform the Fourier transform to get the momentum space be havior, exploiting the following
’standard’ integrals:
/integraldisplay
e−2ω|x|/ve−ipxdx= 22ω/v
p2+(2ω/v)2(C16)
/integraldisplay
cospλx·e−2ω|x|/ve−ipxdx=2ω
v/bracketleftbigg1
(p+pλ)2+(2ω/v)2+1
(p−pλ)2+(2ω/v)2/bracketrightbigg
(C17)
/integraldisplay
sinpλx·e−2ω|x|/ve−ipxdx=i2ω
v/bracketleftbigg1
(p+pλ)2+(2ω/v)2−1
(p−pλ)2+(2ω/v)2/bracketrightbigg
(C18)
/integraldisplay
sin2mv|x|·e−2ω|x|/ve−ipxdx=p+2mv
(p+2mv)2+(2ω/v)2−p−2mv
(p−2mv)2+(2ω/v)2(C19)
We rewrite these expressions using p±
F, thus we get:
/integraldisplay
cospλx·e−2ω|x|/ve−ipxdx=2ω
v/braceleftigg
1
/bracketleftbig
p+(p−
F−p+
F)/bracketrightbig2+(2ω/v)2+1
(/bracketleftbig
p−(p−
F−p+
F)/bracketrightbig2+(2ω/v)2/bracerightigg
(C20)
/integraldisplay
sinpλx·e−2ω|x|/ve−ipxdx=i2ω
v/braceleftigg
1
/bracketleftbig
p+(p−
F−p+
F)/bracketrightbig2+(2ω/v)2−1
/bracketleftbig
p−(p−
F−p+
F)/bracketrightbig2+(2ω/v)2/bracerightigg
(C21)
/integraldisplay
sin2mv|x|·e−2ω|x|/ve−ipxdx=p+(p−
F+p+
F)
/bracketleftbig
p+(p−
F+p+
F)/bracketrightbig2+(2ω/v)2−p−(p−
F+p+
F)
/bracketleftbig
p−(p−
F+p+
F)/bracketrightbig2+(2ω/v)2(C22)
For the last two integrals we introduce symbols/summationtext
p′and/tildewider/summationtext
p′(wide tilde signify that we take the difference, not sum),
wherep′∈ {p−pλ,p+pλ}. Thus we have
/integraldisplay
cos(2mv|x|−2θ)cospλx·e−2ω|x|/ve−ipxdx= (C23)
=1
2/summationdisplay
p′/braceleftbigg1−α2
1+α2·2ω
v/bracketleftbigg1
(p′+2mv)2+(2ω/v)2+1
(p′−2mv)2+(2ω/v)2/bracketrightbigg
+ (C24)
+2α
1+α2·/bracketleftbiggp′+2mv
(p′+2mv)2+(2ω/v)2+p′−2mv
(p′−2mv)2+(2ω/v)2/bracketrightbigg/bracerightbigg
(C25)
/integraldisplay
cos(2mv|x|−2θ)sinpλx·e−2ω|x|/ve−ipxdx= (C26)
=1
2i/tildewidest/summationdisplay
p′/braceleftbigg1−α2
1+α2·2ω
v/bracketleftbigg1
(p′+2mv)2+(2ω/v)2+1
(p′−2mv)2+(2ω/v)2/bracketrightbigg
+ (C27)
+2α
1+α2·/bracketleftbiggp′+2mv
(p′+2mv)2+(2ω/v)2+p′−2mv
(p′−2mv)2+(2ω/v)2/bracketrightbigg/bracerightbigg
(C28)
We rewrite these expressions using p±
F, thus we get:
/integraldisplay
cos(2mv|x|−2θ)cospλx·e−2ω|x|/ve−ipxdx= (C29)
=1−α2
1+α2·ω
v/bracketleftbigg1
(p+2p+
F)2+(2ω/v)2+1
(p−2p−
F)2+(2ω/v)2/bracketrightbigg
+
+α
1+α2·/bracketleftbiggp+2p+
F
(p+2p+
F)2+(2ω/v)2+p−2p−
F
(p−2p−
F)2+(2ω/v)2/bracketrightbigg
+
+1−α2
1+α2·ω
v/bracketleftbigg1
(p+2p−
F)2+(2ω/v)2+1
(p−2p+
F)2+(2ω/v)2/bracketrightbigg
+
+α
1+α2·/bracketleftbiggp+2p−
F
(p+2p−
F)2+(2ω/v)2+p−2p+
F
(p−2p+
F)2+(2ω/v)2/bracketrightbigg22
/integraldisplay
cos(2mv|x|−2θ)sinpλx·e−2ω|x|/ve−ipxdx= (C30)
=1
i/braceleftbigg1−α2
1+α2·ω
v/bracketleftbigg1
(p+2p+
F)2+(2ω/v)2+1
(p−2p−
F)2+(2ω/v)2/bracketrightbigg
+
+α
1+α2·/bracketleftbiggp+2p+
F
(p+2p+
F)2+(2ω/v)2+p−2p−
F
(p−2p−
F)2+(2ω/v)2/bracketrightbigg/bracerightbigg
−
−1
i/braceleftbigg1−α2
1+α2·ω
v/bracketleftbigg1
(p+2p−
F)2+(2ω/v)2+1
(p−2p+
F)2+(2ω/v)2/bracketrightbigg
+
+α
1+α2·/bracketleftbiggp+2p−
F
(p+2p−
F)2+(2ω/v)2+p−2p+
F
(p−2p+
F)2+(2ω/v)2/bracketrightbigg/bracerightbigg
Using the formula cos2γ= (1+cos2 γ)/2 we can write the momentum space expressions for the non-polariz ed and
SP LDOS components:
Sx(p) =i(1+α2)ω
v/braceleftigg
1
/bracketleftbig
p+(p−
F−p+
F)/bracketrightbig2+(2ω/v)2−1
/bracketleftbig
p−(p−
F−p+
F)/bracketrightbig2+(2ω/v)2/bracerightigg
+ (C31)
+1
i/braceleftbigg1−α2
2·ω
v/bracketleftbigg1
(p+2p+
F)2+(2ω/v)2+1
(p−2p−
F)2+(2ω/v)2/bracketrightbigg
+
+α
2·/bracketleftbiggp+2p+
F
(p+2p+
F)2+(2ω/v)2+p−2p−
F
(p−2p−
F)2+(2ω/v)2/bracketrightbigg/bracerightbigg
−
−1
i/braceleftbigg1−α2
2·ω
v/bracketleftbigg1
(p+2p−
F)2+(2ω/v)2+1
(p−2p+
F)2+(2ω/v)2/bracketrightbigg
+
+α
2·/bracketleftbiggp+2p−
F
(p+2p−
F)2+(2ω/v)2+p−2p+
F
(p−2p+
F)2+(2ω/v)2/bracketrightbigg/bracerightbigg
Sz(p) =−(1+α2)ω
v/braceleftigg
1
/bracketleftbig
p+(p−
F−p+
F)/bracketrightbig2+(2ω/v)2+1
/bracketleftbig
p−(p−
F−p+
F)/bracketrightbig2+(2ω/v)2/bracerightigg
− (C32)
−1−α2
2·ω
v/bracketleftbigg1
(p+2p+
F)2+(2ω/v)2+1
(p−2p−
F)2+(2ω/v)2/bracketrightbigg
−
−α
2·/bracketleftbiggp+2p+
F
(p+2p+
F)2+(2ω/v)2−p−2p−
F
(p−2p−
F)2+(2ω/v)2/bracketrightbigg
−
−1−α2
2·ω
v/bracketleftbigg1
(p+2p−
F)2+(2ω/v)2+1
(p−2p+
F)2+(2ω/v)2/bracketrightbigg
−
−α
2·/bracketleftbiggp+2p−
F
(p+2p−
F)2+(2ω/v)2+p−2p+
F
(p−2p+
F)2+(2ω/v)2/bracketrightbigg
ρ(p) = (1+α2)/braceleftigg
2ω/v
p2+(2ω/v)2+/bracketleftigg
ω/v
/bracketleftbig
p+(p−
F+p+
F)/bracketrightbig2+(2ω/v)2+ω/v
(/bracketleftbig
p−(p−
F+p+
F)/bracketrightbig2+(2ω/v)2/bracketrightigg/bracerightigg
+ (C33)
+α/braceleftigg
p+(p−
F+p+
F)
/bracketleftbig
p+(p−
F+p+
F)/bracketrightbig2+(2ω/v)2−p−(p−
F+p+
F)
/bracketleftbig
p−(p−
F+p+
F)/bracketrightbig2+(2ω/v)2/bracerightigg23
2. x-impurity
Φ¯1(x) = +Jx
2/summationdisplay
σ
(1+iσ)[(E¯1−∆s)Xσ
0(x)+Xσ
1(x)]
(1−iσ)[(E¯1−∆s)Xσ
0(x)+Xσ
1(x)]
−(1+iσ)[(E¯1−∆s)Xσ
0(x)−Xσ
1(x)]
−(1−iσ)[(E¯1−∆s)Xσ
0(x)−Xσ
1(x)]
,Φ1(x) = +Jx
2/summationdisplay
σ
−(1−iσ)[(E1+∆s)Xσ
0(x)+Xσ
1(x)]
(1+iσ)[(E1+∆s)Xσ
0(x)+Xσ
1(x)]
−(1−iσ)[(E1+∆s)Xσ
0(x)−Xσ
1(x)]
(1+iσ)[(E1+∆s)Xσ
0(x)−Xσ
1(x)]
.
(C34)
Using these expressions we can compute the non-polarized and SP L DOS in both coordinate and momentum space.
We perform the calculation for the positive-energy state, and we fi nd, omitting index 1:
Sx(x) =−1+α2
2[2cospλx+cos(2mv|x|+pλx−2θ)+cos(2mv|x|−pλx−2θ)]·e−2ω|x|/v(C35)
Sy(x) = 0 (C36)
Sz(x) =−1+α2
2[2sinpλx+sin(2mv|x|+pλx−2θ)−sin(2mv|x|−pλx−2θ)]·e−2ω|x|/v(C37)
ρ(x) = (1+α2)[1+cos(2mv|x|−2θ)]·e−2ω|x|/v(C38)
where tanθ=α. Momentum space dependence can be derived from the z-impurity expressions since everything
coincides up to coefficients.
3. y-impurity
Φ¯1(x) = +Jy
2/summationdisplay
σ
(1−σ)[(E¯1−∆s)Xσ
0(x)+Xσ
1(x)]
i(1−σ)[(E¯1−∆s)Xσ
0(x)+Xσ
1(x)]
−(1−σ)[(E¯1−∆s)Xσ
0(x)−Xσ
1(x)]
−i(1−σ)[(E¯1−∆s)Xσ
0(x)−Xσ
1(x)]
,Φ1(x) = +Jy
2/summationdisplay
σ
−(1+σ)[(E1+∆s)Xσ
0(x)+Xσ
1(x)]
i(1+σ)[(E1+∆s)Xσ
0(x)+Xσ
1(x)]
−(1+σ)[(E1+∆s)Xσ
0(x)−Xσ
1(x)]
i(1+σ)[(E1+∆s)Xσ
0(x)−Xσ
1(x)]
,
(C39)
after summation over σ:
Φ¯1(x) = +Jy
+/bracketleftbig
(E¯1−∆s)X−
0(x)+X−
1(x)/bracketrightbig
i/bracketleftbig
(E¯1−∆s)X−
0(x)+X−
1(x)/bracketrightbig
−/bracketleftbig
(E¯1−∆s)X−
0(x)−X−
1(x)/bracketrightbig
−i/bracketleftbig
(E¯1−∆s)X−
0(x)−X−
1(x)/bracketrightbig
,Φ1(x) = +Jy
−/bracketleftbig
(E1+∆s)X+
0(x)+X+
1(x)/bracketrightbig
i/bracketleftbig
(E1+∆s)X+
0(x)+X+
1(x)/bracketrightbig
−/bracketleftbig
(E1+∆s)X+
0(x)−X+
1(x)/bracketrightbig
i/bracketleftbig
(E1+∆s)X+
0(x)−X+
1(x)/bracketrightbig
.(C40)
Using these expressions we can compute the non-polarized and SP L DOS in coordinate space
Sx(x) = 0, (C41)
Sy(x) =−(1+α2)[1+cos(2mv|x|−2θ)]·e−2ω|x|/v, (C42)
Sz(x) = 0, (C43)
ρ(x) = +(1+α2)[1+cos(2mv|x|−2θ)]·e−2ω|x|/v. (C44)
Appendix D: The SPDOS for a non-superconducting one-dimens ional system in the presence of a magnetic
impurity
The low-energy Hamiltonian in the non-SC regime can be written as
H0=ξpσ0+λ(pyσx−pxσy) =/parenleftbigg
ξpiλp
−iλp ξ p/parenrightbigg
(D1)24
whereξp=p2
2m−εF. The corresponding spectrum is given by E=ξp±λpand the retarded Green’s function reads
G0(E,p) =1
(E−ξp+i0)2−λ2p2/parenleftbigg
E−ξp+i0iλp
−iλp E −ξp+i0/parenrightbigg
. (D2)
To compute the eigenvalues for a single localized impurity we calculate
G0(E,x= 0) =/integraldisplaydp
2πE−ξp+i0
(E−ξp+i0)2−λ2p2/parenleftbigg
1 0
0 1/parenrightbigg
=1
2/summationdisplay
σ/integraldisplaydp
2π1
E−ξσ+i0/parenleftbigg
1 0
0 1/parenrightbigg
, (D3)
whereξσ=ξp+σλp. Forp>0 we linearize the spectrum around the Fermi momenta, thus:
ξσ≈/parenleftbiggpσ
F
m+σλ/parenrightbigg
(p−pσ
F) =/radicalbig
λ2+2εF/m(p−pσ
F)≡v(p−pσ
F),
wherepσ
F=m[−σλ+v], and thus we get:
/integraldisplaydp
2π1
E−ξσ+i0≈1
2πv/bracketleftbigg/integraldisplaydξσ
E−ξσ+i0+/integraldisplaydξ−σ
E−ξ−σ+i0/bracketrightbigg
=−i
v
This leads to:
G0(E,x= 0) =1
2/summationdisplay
σ/parenleftbigg
−i
v/parenrightbigg/parenleftbigg
1 0
0 1/parenrightbigg
=−i
v/parenleftbigg
1 0
0 1/parenrightbigg
(D4)
Since there is no energy dependence, there will be no impurity-induc ed states. The Green’s function coordinate
dependence is given by the following expression:
G0(E,x) =1
2/summationdisplay
σ/integraldisplaydp
2πeipx
E−ξσ+i0/parenleftbigg
1iσ
−iσ1/parenrightbigg
(D5)
To find the coordinate dependence of the Green’s function we calcu late:
Xσ
0(x) =/integraldisplaydp
2πeipx
E−ξσ+i0(D6)
Integral calculation
Below we use the Sokhotsky formula1
x+i0=P1
x−iπδ(x):
Xσ
0(x) =/integraldisplaydp
2πeipx
E−ξσ+i0=1
2πv/bracketleftbigg
eipσ
Fx/integraldisplay
dξσeiξσx/v
E−ξσ+i0+e−ip−σ
Fx/integraldisplay
dξ−σe−iξ−σx/v
E−ξ−σ+i0/bracketrightbigg
We compute explicitly only one of the integrals in the brackets since th e other one can be computed in the similar
fashion:
/integraldisplay
dξσeiξσx/v
E−ξσ+i0=P/integraldisplay
dξσeiξσx/v
E−ξσ−iπ/integraldisplay
dξσδ(E−ξσ)eiξσx/v=−iπ(1+sgnx)eiEx/v
Finally we have:
Xσ
0(x) =−i
vexp/bracketleftbigg
i/parenleftbigg
mv+E
v/parenrightbigg
|x|/bracketrightbigg
e−iσmλx, (D7)
and the Green’s function can be written as:
G0(E,x) =1
2/summationdisplay
σ/parenleftbigg
1iσ
−iσ1/parenrightbigg
Xσ
0(x). (D8)25
Below we compute the T-matrix for different types of impurities. Imp urity potentials take the following forms:
Vsc=U/parenleftbigg
1 0
0 1/parenrightbigg
, Vz=Jz/parenleftbigg
1 0
0−1/parenrightbigg
, Vx=Jx/parenleftbigg
0 1
1 0/parenrightbigg
(D9)
The corresponding T-matrices are
Tsc=U
1+iU/v/parenleftbigg
1 0
0 1/parenrightbigg
, Tz=/parenleftiggJ
1+iJ/v0
0−J
1−iJ/v/parenrightigg
, T x=J
1+J2/v2/parenleftbigg
−iJ/v1
1−iJ/v/parenrightbigg
(D10)
For each type of impurity we can compute the non-polarized and SP L DOS using Eq. (B11) and Eqs. (B15) where
we replace rbyx. By taking the Fourier transforms of the expressions above we ge t the the momentum space
dependence. Below we denote α=J/v.
1. z-impurity
Sx(x) = +α
1+α2·1
πv[cos(pε|x|−pλx)−cos(pε|x|+pλx)] (D11)
Sy(x) = 0 (D12)
Sz(x) = +α
1+α2·1
πv[sin(pε|x|−pλx)+sin(pε|x|+pλx)] (D13)
ρ(x) =−2α2
1+α2·1
πvcospεx (D14)
where we denote pε= 2(mv+E/v),pλ= 2mλ. After taking the Fourier transform we get:
Sx(p) = +α
1+α2·i
πv/bracketleftbigg1
p+pε+pλ−1
p+pε−pλ−1
p−pε+pλ+1
p−pε−pλ/bracketrightbigg
(D15)
Sy(p) = 0 (D16)
Sz(p) = +α
1+α2·1
πv/bracketleftbigg1
p+pε+pλ+1
p+pε−pλ−1
p−pε+pλ−1
p−pε−pλ/bracketrightbigg
(D17)
ρ(p) =−2α2
1+α2·1
v[δ(p−pε)+δ(p+pε)] (D18)
2. x-impurity
Sx(x) = +α
1+α2·1
πv[sin(pε|x|−pλx)+sin(pε|x|+pλx)] (D19)
Sy(x) = 0 (D20)
Sz(x) =−α
1+α2·1
πv[cos(pε|x|−pλx)−cos(pε|x|+pλx)] (D21)
ρ(x) =−2α2
1+α2·1
πvcospεx (D22)
We do not give the Fourier transform for these expressions since t hey coincide with the ones for a z-impurity if we
exchangeSzandSxand change the overall sign.26
3. y-impurity
Sx(x) =Sz(x) = 0 (D23)
Sy(x) = +2α
1+α2·1
πvsinpε|x| (D24)
ρ(x) =−2α2
1+α2·1
πvcospεx (D25)
The corresponding Fourier transform is:
Sy(p) =2α
1+α2·1
πv/bracketleftbigg1
p+pε−1
p−pε/bracketrightbigg
(D26) |
1403.4728v1.Spin_orbit_coupling_effects_on_spin_dependent_inelastic_electronic_lifetimes_in_ferromagnets.pdf | arXiv:1403.4728v1 [cond-mat.mtrl-sci] 19 Mar 2014Spin-Orbit Coupling Effects on Spin Dependent Inelastic Ele ctronic Lifetimes in
Ferromagnets
Steffen Kaltenborn and Hans Christian Schneider∗
Physics Department and Research Center OPTIMAS,
University of Kaiserslautern, 67663 Kaiserslautern, Germ any
(Dated: July 4, 2021)
For the 3d ferromagnets iron, cobalt and nickel we compute th e spin-dependentinelastic electronic
lifetimes due to carrier-carrier Coulomb interaction incl uding spin-orbit coupling. We find that
the spin-dependent density-of-states at the Fermi energy d oes not, in general, determine the spin
dependence of the lifetimes because of the effective spin-fli p transitions allowed by the spin mixing.
The majority and minority electron lifetimes computed incl uding spin-orbit coupling for these three
3-d ferromagnets do not differ by more than a factor of 2, and ag ree with experimental results.
PACS numbers: 71.70.Ej,75.76.+j,75.78.-n,85.75.-d
I. INTRODUCTION
The theoretical and experimental characterization of
spin dynamics in ferromagnetic materials due to the in-
teraction with short optical pulses has become an impor-
tant part of research in magnetism.1–6In this connec-
tion, spin-dependent hot-electron transport processes in
metallic heterostructures have received enormous inter-
est in the past few years.7In particular, superdiffusive-
transport theory has played an increasingly important
role in the quantitative interpretationof experimental re-
sults.4,6,8Superdiffusive transport-theory, which was in-
troduced and comprehensively described in Refs. 9and
10, uses spin- and energy-dependent electron lifetimes
as input,10and its quantitative results for hot-electron
transportonultrashorttimescalesinferromagneticmate-
rials rely heavily, to the best of our knowledge, on the re-
lation between majority and minority electrons for these
materials.
The spin-dependent lifetimes that are used for hot-
electrontransport,bothinferromagnetsandnormalmet-
als, are the so-called “inelastic lifetimes.” These state
(orenergy)dependentlifetimesresultfromout-scattering
processes due to the Coulomb interaction between an ex-
cited electron and the inhomogeneous electron gas in the
system. These lifetimes can be measured by tracking op-
tically excited electrons using spin- and time-resolved 2-
photon photoemission (2PPE)11,12and can be calculated
asthebroadeningoftheelectronicspectralfunctionusing
many-body Green function techniques.13,14The problem
oftheaccuratedeterminationoftheselifetimes hasfueled
method development on the experimental and theoreti-
cal side,15but has always suffered from the presence of
interactions (electron-phonon, surface effects) that can-
not be clearly identified in experiment and are difficult
to include in calculations. Qualitative agreement was
reached for the spin-integrated lifetimes in simple met-
als and iron,16but even advanced quasiparticle calcula-
tions including many-body T-matrix contributions, have
yielded a ratio between majority and minority lifetimes,
which is in qualitative disagreement with experiment forsome ferromagnets. A particularly important material
in recent studies has been nickel,4,6,10for which the the-
oretical ratio comes out between 6 and 8,16while the
experimental result17is 2. Recent experimental results
point toward a similar disagreement for cobalt.12
In light normal metals and ferromagnets spin-orbit
coupling generally leads to very small corrections to
the single-particle energies, i.e., the band structure , but
it changes the single-particle states qualitatively by in-
troducing a state-dependent spin mixing. With spin-
orbit coupling, the average spin of an electron can be
changed in transitions due to any spin-diagonal interac-
tion, in particular by electron-phonon momentum scat-
tering.18–21This isalsotrue forthetwo-particleCoulomb
interaction,22,23as long as one monitors only the average
spin of one of the scattering particles, as is done in life-
time measurements by 2-photon photoemission experi-
ments. While this spin mixing due to spin-orbit cou-
pling has recently been included in lifetime calculations
for lead,24it was not included in DFT codes used for ex-
isting lifetime calculations for 3d-ferromagnets and alu-
minum,16,25,26whose results are nowadays widely used.
This paper presents results for electron lifetimes in
metals and spin-dependent lifetimes in ferromagnets in-
cluding spin-orbit coupling . We show that spin-orbit cou-
pling can be important for electron lifetimes in metals
in general. Moreover, the ratio between the calculated
majority and minority lifetimes is, for the first time, in
agreement with experiment.11,12,17We believe that our
calculated electronic lifetimes should be used as an accu-
rate input for calculationsof spin-dependent hot-electron
dynamics in ferromagnets.
II. SPIN-DEPENDENT ELECTRON AND HOLE
LIFETIMES IN CO AND NI
We first discuss briefly our theoretical approach to cal-
culate the lifetimes. We start from the dynamical and
wave-vector dependent dielectric function ε(/vector q,ω) in the
random phase approximation (RPA).13,14,25,26Our ap-
proach, cf. Ref. 27, evaluates the wave-vector summa-2
tions inε(/vector q,ω) without introducing an additional broad-
ening of the energy-conserving δfunction. This proce-
dure removes a parameter whose influence on the calcu-
lation for small qis not easy to control and which would
otherwise need to be separately tested over the whole
energy range.
The/vectork- and band-resolved electronic scattering rates,
i.e., the inverse lifetimes, γν
/vectork= (τν
/vectork)−1, are calculated
using the expression25,26
γν
/vectork=2
¯h/summationdisplay
µ/vector q∆q3
(2π)3Vq/vextendsingle/vextendsingleBµν
/vectork/vector q/vextendsingle/vextendsingle2fµ
/vectork+/vector qℑε(/vector q,∆E)
|ε(/vector q,∆E)|2.(1)
Here, the band indices aredenoted by µandν, and/vectorkand
/vector qdenote wave-vectors in the first Brillouin zone (1. BZ).
The energies ǫµ
/vectork, occupation numbers fµ
/vectorkand overlap ma-
trix elements Bµν
/vectork/vector q=/angbracketleftψµ
/vectork+/vector q|ei/vector q·/vector r|ψν
/vectork/angbracketrightare extracted from
the ELK DFT (density functional theory) code,28which
employs a full-potential linearized augmented plane wave
(FP-LAPW) basis. Last, Vq=e2/(ε0q2) denotes the
FouriertransformedCoulombpotentialand∆ E=ǫµ
/vectork+/vector q−
ǫν
/vectorkis the energy difference between initial and final state.
For negative ∆ E, the distribution function has to be re-
placed by −(1−fµ
/vectork+/vector q). By using the overlap matrix el-
ements as defined above we neglect corrections due to
local field effects. In the language of many-body Green
functions, this corresponds to an on-shell G0W0calcula-
tion,16,26where the screened Coulomb interaction ( W0)
is obtained from the full RPA dielectric function. The
/vectork- and band-dependent wave-functions that result from
the DFT calculations including spin-orbit coupling are of
the form |ψµ
/vectork/angbracketright=aµ
/vectork|↑/angbracketright+bµ
/vectork|↓/angbracketright,18where|σ/angbracketrightare spinors
identified by the spin projection σ=↑,↓along the mag-
netization direction. According to whether |aµ
/vectork|2or|bµ
/vectork|2
is larger, we relabel each eigenstate by its dominant spin
contribution σ, so that we obtain spin-dependent life-
times,τσ
/vectork. Our choice of quantization axis is such that
σ=↑denotes majority carriers states and σ=↓minor-
ity carrier states. Due to the existence of several bands
(partially with different symmetries) in the energy range
of interest and the anisotropy of the DFT bands ǫ(ν)(/vectork),
several lifetimes τν
/vectorkcan be associated with the same spin
and energy. When we plot these spin and energy depen-
dent lifetimes τσ(E) in the following, in particularFigs. 1
and2, this leads to a scatter of τσ(E) values.
Figures1and2displaythecalculatedenergy-andspin-
resolved carrier lifetimes τσ(E) around the Fermi energy
for cobalt and nickel. The spread of lifetimes at the same
energy, which was mentioned above, can serve as an indi-
cation for the possible range of results for measurements
of energy resolved lifetimes. These “raw data” are im-
portant for the interpretation of the theoretical results
because they already show two important points. First,
we checked that there is no good Fermi-liquid type fit to
these lifetimes. Second, even if one fits the lifetimes in
a restricted energy range by a smooth τ(E) curve, this−2−1 012051015202530τCo(fs)
E−EF(eV)
FIG. 1. (Color online) Energy-resolved majority ( τ↑, blue +)
and minority ( τ↓, red◦) carrier lifetimes for cobalt. There are
in general several different lifetime points at the same ener gy
(see text). We used 173/vectork-points in the full BZ.
−2−1 012020406080
E−EF(eV)τNi(fs)
FIG. 2. (Color online) Same as Fig. 1for nickel.
ignoresthe spread of lifetimes, which can be quite sizable
as shown in Figs. 1and2. We believe that such a spread
of electronic lifetimes, in particular in the range around
1eV above the Fermi energy should be important for the
interpretation of photoemission experiments in this en-
ergy range, and when these results are used as input in
hot-electron transport calculations.
Figure1shows the energy- and spin-resolved lifetimes
in cobalt. In addition to the longer lifetimes close to the
Fermi energy, hole lifetimes in excess of 5fs occur at the
top of some d-bands around −1.5,−1.2, and−1eV. For
electronic states with energies above 0.5eV longer life-
timesoccuratsome /vectork-points. Therearealso kstateswith
apronouncedspin-asymmetryinthelifetimes(seediscus-
sion below). Another important property of cobalt is the
existence of two different conduction bands, which inter-
sect the Fermi surface with different slope. This leads to3
tworather well-defined lifetime curves, both for electrons
and holes. This can be best seen between −0.6 and 0eV,
where the two curves are shifted by about 0.2eV.
The calculated lifetimes in nickel, see Fig. 2, do
not show a pronounced influence of d-bands and/or
anisotropy below the Fermi energy as in cobalt, which is
due to the smaller number of bands in the vicinity of the
Fermi energy. However, there is a clear spin-dependence
of electronic lifetimes, which is most pronounced around
0.4eV, but persists almost up to 2eV.
III. SPIN ASYMMETRY OF ELECTRON
LIFETIMES IN FE, CO, AND NI
In the following, we will mainly be concerned with life-
times above 0.3eV above the Fermi energy, which is the
interesting energy range for the interpretation of photoe-
mission experiments and hot-electron transport calcula-
tions, because close to the Fermi energy the influence
of phonons is expected to become more pronounced and
leadtosignificantlyshorterlifetimes thanthosepredicted
by a calculation that includes only the Coulomb interac-
tion. To facilitate comparison with experiment we aver-
age the lifetimes in each spin channel in bins of 100meV
and denote the result by ¯ τ(E). The standard deviation
of the averaging process then yields “error bars” on the
¯τ(E) values. Note that this procedure does not corre-
spond to a “random k” approximation.
Figure3displays the averagedelectron lifetimes deter-
mined from the data shown in Figs. 1and2. As insets
we have included the ratio of majority and minority life-
times,τ↑/τ↓, together with experimental data11,12,17for
iron, cobalt and nickel. Figure 3(a) shows that there is
only a veryweakspin dependence for iron, and the agree-
ment of the ratio τ↑/τ↓with experiment17and recent in-
vestigations15,16,29is quite good, but there is a slight dis-
agreement with earlier, semiempirical studies.30,31How-
ever, even an increase of the ratio around 0 .5eV in the
experiment17is well reproduced in our results.
The averaged lifetimes of cobalt, which are shown in
Fig.3(b), agree quite well with the experimental life-
times,11,12but the large error bars extend to a much
wider energy range than in iron. This can be traced
back to the scatterof lifetimes in Fig. 1. The correspond-
ing figure for iron (not shown) exhibits a much smaller
scatter. The ratio of majority and minority electron life-
times, see inset in Fig. 3(b), is around 1 below 0.5eV
and increases to τ↑/τ↓≃2 for larger energies, a trend
that agrees extremely well with measurements.11,12,17To
put this result into perspective we note that the experi-
mental data in Ref. 17were compared with a theoretical
model based on the random kapproximation.32If the
random-kinteraction matrix elements are taken to be
spin and energy independent, the majority and minor-
ity relaxation times are determined by double convolu-
tions over the spin-dependent density-of-states (DOS).16
It was found that the experimental results were not in050100150200¯τFe(fs)
01020¯τCo(fs)
00.511.522.530204060
E−EF(eV)¯τNi(fs)
0.30.50.70.91.10.511.52
E−EF(eV)τ↑
Fe/τ↓
Fe
0.30.50.70.91.1123
E−EF(eV)τ↑
Co/τ↓
Co
0.30.50.70.91.1123
E−EF(eV)τ↑
Ni/τ↓
Ni(c)(a)
(b)
FIG. 3. (Color online) Energetically averaged majority (bl ue
up triangles) and minority (red down triangles) lifetimes f or
(a) Fe, (b) Co and (c) Ni. The error bars denote the standard
deviation obtained from the scatter of the lifetimes as show n
in Figs.1and2. The insets show the calculated ratio of ma-
jority and minority electrons (“ ◦”),τ↑/τ↓, in comparison to
experimental data, where the “ •” (“×”) correspond to values
extracted from Ref. 11and17(12).
agreement with the ratio of the DOS at the Fermi en-
ergy, which led the authors of Ref. 17to speculate that
the matrix elements for parallel and antiparallel spins
should be different due to the Pauli exclusion principle.
In our calculations, the effective spin-dependence of the
matrix elements is caused exclusively by the spin-mixing
due to spin-orbit coupling, but the effect is the same: It
makes the ratio of the lifetimes different from the spin-
dependent DOS at the Fermi energy.
In Fig.3(c) we turn to nickel. Here, as in the case
of iron, the average lifetimes are slightly larger than
the measured ones17(not shown), but due to the small
anisotropy in the band structure, the lifetimes in nickel
show the smallest error bars and thus an extremely well-
defined spin dependence. Only our calculated majority
electron lifetimes are similar to earlier ab-initio evalua-
tions,15,16,29but there is an important discrepancy in the
ratioτ↑/τ↓: The inset of Fig. 3(c) shows a ratio of about
τ↑/τ↓≃2, which is independent of energy above 0.4eV.4
012345020406080100
E−EF(eV)τAl(fs)
FIG. 4. (Color online) Calculated energy-resolved electro nic
lifetimes for aluminum (blue “ ◦”) in comparison to earlier in-
vestigations without spin-orbit interaction. The black sq uares
(stars) correspond to some data extracted from Ref. 25(33).
There are in general several different lifetime points at the
same energy (see text). We used 173/vectork-points in the full BZ.
This results compares extremely well with experiment,
and should be contrasted with the calculated result of
Ref.16forτ↑/τ↓≃8 around 0.5eV. These GW calcu-
lations (even with a T-matrix approach) gave very simi-
lar results to those of the random kapproximation16in
the energy range above 0.5eV. This indicates that the
resulting spin asymmetry τ↑/τ↓≃6–8 is solely deter-
mined by the spin-dependent DOS Dσ(E). Indeed, one
hasD↑(EF)/D↓(EF)≃8. With the inclusion of spin-
orbit coupling, which gives rise to effective spin-flip tran-
sitions, the spin asymmetry is no longer determined by
the spin-dependent DOS alone. This interpretation is
again supported by Ref. 17where a strongly enhanced
spin-flip matrix element had to be introduced by hand to
improve the agreement between a random- kcalculation
and experiment.
To conclude the discussion of the ferromagnets, we
comment on the spin-integrated lifetimes which can be
obtained from the spin-dependent lifetimes, but are not
shown here. Compared with experimental lifetimes of
Ref.17we generally find an agreement for energies above
0.5eV that is on par with earlier calculations.15–17,29For
energies below 0.5eV where the error bars on the av-
eraged lifetimes are largest, the calculated lifetimes are
larger than the measured ones, but in this energy range
a good agreement with experiments cannot be expected
because of scattering processes, which appear as elastic
due to the energy resolution of the photoemission exper-
iments.IV. INFLUENCE OF SPIN-ORBIT COUPLING
ON ELECTRON LIFETIMES IN AL
To underscore the importance of spin-orbit coupling
for lifetime calculations, we also briefly discuss our calcu-
lated results for electronic lifetimes in aluminum in com-
parison with earlier investigations25,33without spin-orbit
effects. Fig. 4shows that smaller electronic lifetimes re-
sult for aluminum when spin-orbit coupling is included.
In particular in the energy range between 1 and 3eV
the lifetimes differ by almost a factor of two. Thus the
inclusion of spin-orbit coupling improves the agreement
withexperiment(see,forinstance, Ref. 34), whichwasal-
ready quite good for the existing calculations.33It is con-
ceivable that the use of more sophisticated many-body
techniques, such as the inclusion of vertex corrections
or using a T-matrix approach,33might lead to further
improvements. As in the case of the ferromagnets, the
electronic band structure is practically unchanged by the
inclusion of spin-orbit coupling, but the rather large ef-
fect of the spin-orbit coupling on spin relaxation in alu-
minum through spin hot-spots has already been demon-
strated.18Another argument for the importance of the
spin-orbit coupling is that the spin-mixing allows transi-
tion between the Kramersdegeneratebands. These tran-
sitionsbetweenKramersdegeneratebandsmayhaveare-
markable influence even on electron-gas properties that
are usually assumed to be spin-independent, such as the
intraband plasma frequency.27
V. CONCLUSION
In conclusion, we presented ab-initio results for spin-
dependent electronic lifetimes in ferromagnets and alu-
minum including spin-orbit coupling. We found that
the electronic lifetimes in iron exhibit no visible spin de-
pendence in the range of −2 up to 3eV in agreement
with earlier results, whereas the ratio τ↑/τ↓between
majority and minority lifetimes does not exceed 2 for
cobalt and nickel. Our results agree well with experi-
mental data, but differ from earlier calculations, which
found thatτ↑/τ↓was essentially determined by the spin-
dependent density-of-states. We showed that, by allow-
ing for effectively spin-changing transitions as contribu-
tions to the lifetime, spin-orbit coupling is the essential
ingredient that can make the spin asymmetry of the elec-
troniclifetimes much smallerthan the spin-asymmetryof
the density-of-states. Inclusion of our calculated spin de-
pendentlifetimesintransportcalculationsshouldmakeit
possible to more accurately characterize the influence of
spin-dependent hot-electron transport on magnetization
dynamics.
ACKNOWLEDGMENTS
We are grateful for a CPU-time grant from the J¨ ulich
Supercomputer Centre (JSC). We acknowledge helpful
discussions with M. Aeschlimann, M. Cinchetti, and S.
Mathias.5
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0710.2866v1.Intersubband_spin_orbit_coupling_and_spin_splitting_in_symmetric_quantum_wells.pdf | arXiv:0710.2866v1 [cond-mat.mtrl-sci] 15 Oct 2007Intersubband spin-orbit coupling and spin splitting in sym metric quantum wells
F. V. Kyrychenko and C. A. Ullrich
Department of Physics and Astronomy, University of Missour i, Columbia, Missouri, 65211, USA
I. D’Amico
Department of Physics, University of York, York YO10 5DD, Un ited Kingdom
In semiconductors with inversion asymmetry, spin-orbit co upling gives rise to the well-known
Dresselhaus and Rashba effects. If one considers quantum wel ls with two or more conduction sub-
bands, an additional, intersubband-induced spin-orbit te rm appears whose strength is comparable
to the Rashba coupling, and which remains finite for symmetri c structures. We show that the con-
duction band spin splitting due to this intersubband spin-o rbit coupling term is negligible for typical
III-V quantum wells.
PACS numbers: 73.50.-h, 73.40.-c, 73.20.Mf, 73.21.-b
Keywords: spintronics, spin Coulomb drag, spin-orbit coup ling, quantum wells
I. INTRODUCTION
Research in nanoscience is crucial for its technologi-
cal implications and for the fundamental exploration of
the quantum properties of nanostructures such as quan-
tum wells, wires and dots. Of particular interest is the
study of spin dynamics, which hopes to revolutionizetra-
ditional electronics using the spin properties of the carri-
ers (spintronics) [1]. In this context, the theoretical pre-
diction [2] and experimental confirmation [3] of the spin-
Coulomb drag (SCD) effect was of great importance, as
this effect results in the natural decay ofspin current and
intrinsic dissipation in AC-spintronic circuits [4]. Due
to Coulomb interactions between spin-up and spin-down
electrons, theupanddowncomponentsofthetotallinear
momentum are not separately conserved. This momen-
tum exchange between the two populations represents
an intrinsic source of friction for spin currents, known as
spin-transresistivity [5].
In [4] we demonstrated that the SCD produces an in-
trinsic linewidth in spin-dependent optical excitations,
which can be as big as a fraction of a meV for intersub-
band spin plasmons in parabolic semiconductor quan-
tum wells (QWs). This intrinsic linewidth would be
ideal to experimentally verify the behavior of the spin-
transresistivity in the frequency domain.
In our proposed experiment, we suggested to use sym-
metricparabolic QWs to avoid an undesired splitting of
the spin plasmons due to Rashba spin-orbit (SO) cou-
pling. We based our discussion on earlier work [6], in
which collective intersubband spin excitations in QWs
weredescribedinthepresenceofDresselhausandRashba
SO interaction terms [7, 8] for strictly two-dimensional
(2D) systems [9]. In symmetric QWs, the Rashba term
vanishes and only bulk inversion asymmetry (Dressel-
haus) interaction is present.
However, as shown recently by Bernardes et al. [10],
the Rashba SO coupling gives finite contributions even
for symmetric structures, if treated in higher order per-
turbation theory. As a consequence, for QWs with morethan one subband, there appears an additional intersub-
band SO interaction, whose magnitude can become com-
parable to that of 2D Dresselhaus and Rashba interac-
tions. This interaction gives rise to a nonzero spin-Hall
conductivity and renormalizes the bulk mass by ∼5% in
InSb double QWs [10]. This raises the question whether
this effect must be accounted for when extracting the
SCD from intersubband spin plasmon linewidths [4].
In this paper we are going to show that while intersub-
band SO interaction may manifest itself in some special
cases, as for example in the double well analyzed in Ref.
[10], it has little to no effect on spin splitting and spin
mixing in QWs once the 2D Dresselhaus and/or Rashba
terms are taken into account.
In Sec. II we present the general formalism of calcu-
lating conduction band states in quantum structures in-
cluding both 2D and intersubband SO interaction. In
Sec. III we consider the specific case of symmetric single-
well quantum structures, and in Sec. IV we present re-
sults for a parabolic model QW. Sec. V gives a brief
summary.
II. GENERAL FORMALISM
We consider conduction electrons in a QW described
by the Hamiltonian
ˆH=ˆH0+ˆHso, (1)
whereˆH0is spin independent and ˆHsois the SO inter-
action projected on the conduction band. For simplic-
ity we will consider only spin off-diagonal (spin-mixing)
terms in ˆHso. The eigenfunctions associated with ˆH0
alone can be obtained by solving a single-particle equa-
tion of the Schr¨ odinger-Poisson or Kohn-Sham type, re-
sulting in spin-independent subband envelope functions
ψi(z,k/bardbl) and energy eigenvalues εi, whereiis the sub-
band index and zis the direction of quantum confine-
ment.2
Let us now consider the two lowest conduction sub-
bands of the QW. In the basis of the first two subband
spinors|ψ1↑/angbracketright,|ψ1↓/angbracketright,|ψ2↑/angbracketright,|ψ2↓/angbracketright, the Schr¨ odinger
equation with the full Hamiltonian (1) has the form
ε1α10β
α∗
1ε1β′0
0β′∗ε2α2
β∗0α∗
2ε2
A=εA, (2)
where
α1=/angbracketleftψ1↑ |ˆHso|ψ1↓/angbracketright
α2=/angbracketleftψ2↑ |ˆHso|ψ2↓/angbracketright,
β=/angbracketleftψ1↑ |ˆHso|ψ2↓/angbracketright,
β′=/angbracketleftψ1↓ |ˆHso|ψ2↑/angbracketright. (3)
To remove the off-diagonal terms mixing the ↑,↓states
within the same subband, we apply the unitary transfor-
mationB=U·Awith
U=1√
2
1−α1
|α1|0 0
1α1
|α1|0 0
0 0 1 −α2
|α2|
0 0 1α2
|α2|
. (4)
Equation (2) then transforms into
ε1−|α1|0 −γ1γ2
0ε1+|α1| −γ2γ1
−γ∗
1−γ∗
2ε2−|α2|0
γ∗
2γ∗
1 0ε2+|α2|
B=εB,
(5)
where the off-diagonal matrix elements
γ1,2=1
2/bracketleftig
βα∗
2
|α2|±β′α1
|α1|/bracketrightig
(6)
connect the first and second subbands. We treat these
contributions to the conduction band Hamiltonian per-
turbatively to second order, and obtain the following so-
lutions of Eq. (5):
ε±
1=ε1±|α1|
+|γ1|2
ε1±|α1|−ε2∓|α2|+|γ2|2
ε1±|α1|−ε2±|α2|,
ε±
2=ε2±|α2|
+|γ1|2
ε2±|α2|−ε1∓|α1|+|γ2|2
ε2±|α2|−ε1±|α1|
and
B−
1=
1
0
−γ∗
1
ε1−|α1|−ε2+|α2|
γ∗
2
ε1−|α1|−ε2−|α2|
, (7)B+
1=
0
1
−γ∗
2
ε1+|α1|−ε2+|α2|
γ∗
1
ε1+|α1|−ε2−|α2|
, (8)
B−
2=
−γ1
ε2−|α2|−ε1+|α1|
−γ2
ε2−|α2|−ε1−|α1|
1
0
, (9)
B+
2=
γ2
ε2+|α2|−ε1+|α1|γ1
ε2+|α2|−ε1−|α1|
0
1
.(10)
The eigenvectors B±
iare normalized up to first order in
the off-diagonal perturbation.
In the absence of intrasubband (2D) terms, α1=α2=
0, theintersubband SO interactiongives rise to spin mix-
ing without lifting the spin degeneracy( ε+
i=ε−
i); it only
causes a spin-independent shift of the subband energies.
By contrast, if an intrasubband interaction is present (or
ifspindegeneracyislifted byothermeans, e.g.,byamag-
netic field), the spin splitting is affected. For the lowest
subband it is given by ε+
1−ε−
1= ∆ε1, where
∆ε1= 2|α1|+2|γ1|2 |α2|−|α1|
(ε2−ε1)2−(|α2|−|α1|)2
−2|γ2|2 |α2|+|α1|
(ε2−ε1)2−(|α2|+|α1|)2.(11)
To proceed further we need the explicit form of the SO
Hamiltonian ˆHso.
III. RASHBA AND DRESSELHAUS SO
INTERACTION IN SYMMETRIC QWS
By folding down the 14 ×14k·pHamiltonian for a
QW grown in [001] direction in a zinc-blende crystal to
a 2×2 conduction band problem [11], one obtains an
effective SO Hamiltonian in the conduction band:
ˆHso≈/parenleftbigg
0hso
h∗
so0/parenrightbigg
, (12)
where
hso=R(z)k−−iλk+∂2
∂z2−iλ
4(k2
−−k2
+)k−,(13)
with
λ= 4√
2
3PQP′/parenleftbigg1
(E∆−ε)(E′v−ε)−1
(Ev−ε)(E′
∆−ε)/parenrightbigg3
and
R(z) =√
2
3P2/bracketleftbigg∂
∂z/parenleftbigg1
Ev−ε−1
E∆−ε/parenrightbigg/bracketrightbigg
+√
2
3P′2/bracketleftbigg∂
∂z/parenleftbigg1
E′v−ε−1
E′
∆−ε/parenrightbigg/bracketrightbigg
.(14)
Here,k±=1√
2(kx±iky),εis the electron energy, Ev(z)
andE∆(z) are the position-dependent Γ 8and Γ7valence
bandedges, and P=−i/planckover2pi1
m/angbracketleftS|ˆpx|X/angbracketright=/radicalig
Ep/planckover2pi12
2misthe mo-
mentum matrix element. Primed quantities correspond
to the higher lying Γ 8−Γ7conduction band and Qis
the momentum matrix element between the valenceband
and the higher conduction band. Along with the Rashba
and linear Dresselhaus terms in Eq. (12) we keep the cu-
bic Dresselhaus term as well. During the derivation we
assumed that the variation of the band edges is small
compared with the energy gaps in the material.
In symmetric structures, due to parity conservation
theintrasubband SO interaction contains only the Dres-
selhaus contribution,
α1=−λ
4√
2k3sin(2ϕ)e−iϕ+D11√
2kei(ϕ+π
2),(15)
α2=−λ
4√
2k3sin(2ϕ)e−iϕ+D22√
2kei(ϕ+π
2),(16)
and theintersubband SO interaction(between the lowest
two subbands) involves only the Rashba term
β=β′∗=R12√
2ke−iϕ, (17)
whereϕis the polar angle of the in-plane vector k/bardblmea-
sured from the [100] direction, and k=|k/bardbl|. Further-
more,
Dii=−λ/angbracketleftbigg
ψi(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2
∂z2/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi(z)/angbracketrightbigg
(18)
and
R12=/angbracketleftψ1(z)|R(z)|ψ2(z)/angbracketright. (19)
The quantity R12corresponds to the coupling parameter
ηderived in Ref. [10] using an 8-band k·pmodel.
For smallkthe linear term in Eqs. (15)-(16) dominates
and we can approximate
α1
|α1|≈α2
|α2|≈ei(ϕ+π
2). (20)
Then,
γ1=1√
2R12kcos/parenleftig
2ϕ+π
2/parenrightig
(21)
γ2=−i√
2R12ksin/parenleftig
2ϕ+π
2/parenrightig
, (22)and the ground state spin splitting follows from Eq. (11)
as
∆ε1≈2|α1|−R2
12D11√
2(ε2−ε1)2k3−R2
12D22√
2(ε2−ε1)2k3cos(4ϕ).
(23)
Theintersubband interaction results thus in an addi-
tional spin splitting proportional to k3.
Next, we expand the spin splitting that is induced by
theintrasubband SO interaction. Up to order k3we ob-
tain
|α1| ≈D11√
2k+λ
8√
2k3−λ
8√
2k3cos(4ϕ),(24)
which givesthe final expressionforthe subband splitting:
∆ε1=√
2D11k+/parenleftbiggλ
4−R2
12D11
(ε2−ε1)2/parenrightbiggk3
√
2
−/parenleftbiggλ
4+R2
12D22
(ε2−ε1)2/parenrightbiggk3
√
2cos(4ϕ).(25)
One finds that the intersubband SO interaction produces
an additional spin splitting of the same symmetry as the
intrasubband cubic Dresselhaus term. We will now es-
timate the magnitude of this additional contribution for
GaAs parabolic QWs.
IV. SUBBAND SPIN SPLITTING IN
PARABOLIC WELLS
Let us consider a parabolic QW with conduction band
confining potential
V(z) =1
2Kz2, (26)
resulting in the noninteracting energy spectrum
εj=/radicalbigg
/planckover2pi12K
m∗/parenleftbigg
j−1
2/parenrightbigg
, j = 1,2,...(27)
The first and second subband envelope functions are
ψ1(z) =4/radicalbigg
2ξ
πe−ξz2, (28)
ψ2(z) =4/radicalbigg
32ξ3
πze−ξz2, (29)
whereξ=/radicalbig
m∗K/4/planckover2pi12. Straightforward calculations
give
/angbracketleftbigg
ψ1/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2
∂z2/vextendsingle/vextendsingle/vextendsingle/vextendsingleψ1/angbracketrightbigg
=−ξ, (30)
/angbracketleftbigg
ψ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2
∂z2/vextendsingle/vextendsingle/vextendsingle/vextendsingleψ2/angbracketrightbigg
=−3ξ, (31)
/angbracketleftψ1|z|ψ2/angbracketright=−1
2√ξ. (32)4
For our parabolic well, the positional dependence of the
valence band edge (the valence band potential) is
Ev=−1
4Kz2,
corresponding to a valence band offset VBO=0.33. For
GaAs parameters ( Eg= 1.42 eV, ∆ = 0 .34 eV,Ep=
22 eV) Eq. (14) gives R(z)≈ −/parenleftbig∂Ev
∂z/parenrightbig
7˚A2. Using Eqs.
(18), (19) and (30)–(32) we then get
R12=−(7˚A2)K
4√ξ, D 11=λξ, D 22= 3λξ,
and
R2
12D22
(ε2−ε1)2=
∆ε
/planckover2pi12
2m∗˚A2
2
147
64λ∼10−6λ,
form∗= 0.065m0and ∆ε=ε2−ε1= 40 meV. The con-
tribution of the intersubband SO interaction to the spin
splitting of the lowest conduction subband is six orders
of magnitude weaker than that of the cubic Dresselhaus
intrasubband termand thus canbe completely neglected.
The spin mixing induced by the intersubband SO in-
teraction can be estimated from Eq. (7):
/vextendsingle/vextendsingle/vextendsingle/vextendsingleγ2
ε2−ε1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
≈R2
12k2
2(∆ε)2=49
32
∆ε
/planckover2pi12
2m∗˚A2
k2˚A2∼10−7,
fork= 0.01˚A−1. This is seven orders of magnitude
weaker than the spin mixing induced by intrasubband
SO interaction and also can be completely neglected.
SimilarresultswereobtainedforGaAssymmetricrect-
angular QWs.V. CONCLUSIONS
In this paper, we have considered the effects of SO
coupling on the conduction subband states in symmetric
QWs. Our work was motivated by Ref. [10], which dis-
cussed a SO coupling effect specific to QWs with more
than one subband and showed that it can affect the elec-
tronic and spin transport properties in some systems.
We found that although the magnitude of this in-
tersubband SO interaction can be comparable to that
of the 2D Dresselhaus and Rashba terms, its effect on
the spin splitting and spin mixing of conduction band
states is several orders of magnitude weaker since it con-
nects states with different energies. This is due to the
fact that the spin splitting and spin mixing of conduc-
tion band states are renormalized by the intersubband
energy difference.
Therefore, ifone considerssystem with non-degenerate
subbands, one can completely neglect the intersubband
SO interaction compared to the usual 2D Dresselhaus
and Rashba terms. These findings provide an a posteri-
orijustification for the approach used to calculate sub-
band splittings and spin plasmon dispersions carried out
in Ref. [6]. This opens the way for a comprehensive the-
ory of collective intersubband excitations in QWs in the
presence of SCD and SO coupling.
Acknowledgments
This work was supported by DOE Grant No.
DE-FG02-05ER46213, the Nuffield Foundation Grant
NAL/01070/G, and by the Research Fund 10024601 of
the Department of Physics of the University of York.
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1805.00047v1.Superconducting_tunneling_spectroscopy_of_spin_orbit_coupling_and_orbital_depairing_in_Nb_SrTiO__3_.pdf | Superconducting tunneling spectroscopy of spin-orbit coupling
and orbital depairing in Nb:SrTiO 3
Adrian G. Swartz,1, 2, 3,Alfred K. C. Cheung,4Hyeok Yoon,1, 2, 3Zhuoyu
Chen,1, 2, 3Yasuyuki Hikita,2Srinivas Raghu,2, 4and Harold Y. Hwang1, 2, 3
1Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA
2Stanford Institute for Materials and Energy Sciences,
SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
3Department of Applied Physics, Stanford University, Stanford, California 94305, USA
4Department of Physics, Stanford University, Stanford, California 94305, USA
(Dated: May 2, 2018)
We have examined the intrinsic spin-orbit coupling (SOC) and orbital depairing in thin lms of
Nb-doped SrTiO 3by superconducting tunneling spectroscopy. The orbital depairing is geometrically
suppressed in the two-dimensional limit, enabling a quantitative evaluation of the Fermi level spin-
orbit scattering using Maki's theory. The response of the superconducting gap under in-plane
magnetic elds demonstrates short spin-orbit scattering times so1:1 ps. Analysis of the orbital
depairing indicates that the heavy electron band contributes signi cantly to pairing. These results
suggest that the intrinsic spin-orbit scattering time in SrTiO 3is comparable to those associated
with Rashba e ects in SrTiO 3interfacial conducting layers and can be considered signi cant in all
forms of superconductivity in SrTiO 3.
The relativistic spin-orbit interaction is fundamental
in the solid state, connecting the conduction electron
spin to the atomic, electronic, orbital, and structural
symmetry properties of the material [1]. SrTiO 3is an
oxide semiconductor with highly mobile t2gconduction
electrons and exhibits superconductivity at the lowest
known carrier density of any material [2{4]. The rele-
vance of the intrinsic spin-orbit coupling (SOC) for su-
perconductivity in the bulk material remains an open
question: the atomic SOC produces a relatively small
splitting (29 meV [2]) of the t2gbands, butmight be an
important energy scale considering the small supercon-
ducting gap in SrTiO 3. Moreover, SrTiO 3is the host
material for unconventional two-dimensional (2D) super-
conductors such as FeSe/SrTiO 3[5],-doped SrTiO 3[3],
and LaAlO 3/SrTiO 3[6]. Spin-orbit coupling in SrTiO 3
interfacial accumulation layers has been extensively stud-
ied both experimentally and theoretically [7{13]. In these
systems, Rashba SOC has been suggested to give rise to
many of the unusual normal- and superconducting-state
properties due to the broken inversion symmetry and the
highly asymmetric con nement potential. Understand-
ing the competition between the intrinsic and Rashba
coupling scales is critical to understanding the spin-orbit
textures and superconducting phases in both bulk and
2D systems.
The spin-orbit coupling strength can be quantitatively
extracted from superconducting tunneling spectra of thin
lms in large parallel magnetic elds [14{16]. In a con-
ventionals-wave superconductor, a magnetic eld acts
in two ways on the conduction electrons: by inducing
cyclotron orbits and via the electron magnetic moment
(spin). Both of these e ects lead to the breaking of
Cooper pairs once their energy scale competes with the
condensation energy. For thin lms in the 2D limit, theorbital depairing can be geometrically suppressed, lead-
ing to highly anisotropic upper-critical elds with large
in-planeHc2;k. In the absence of spin-orbit coupling, spin
is a good quantum number and Hc2;kis determined by
the Pauli paramagnetic limit ( HP= 0=p
2B, where
0is the superconducting gap at T= 0 andBis the
Bohr magneton) [14, 17, 18]. The application of an in-
plane magnetic eld splits the spin-up and spin-down
superconducting quasiparticle density of states (DOS)
through the Zeeman e ect (Fig. 1 left panel) [14]. In-
creasing the spin-orbit coupling leads to a mixing of the
spin-up and spin-down states and lifts Hc2;kabove the
Pauli limit [14, 19, 20]. If the spin-orbit scattering rate is
very fast (h= so>0, where sois the normal-state spin-
orbit scattering time), then the superconducting DOS
does not exhibit measurable Zeeman splitting (Fig. 1
right panel). Fitting the tunneling spectra using Maki's
theory [21{23] enables a quantitative extraction of both
the orbital depairing parameter ( o) and sofrom the
tunneling spectra. This approach, pioneered by Tedrow
and Meservey, has been used extensively to explore de-
pairing mechanisms of conventional elemental supercon-
ductors [14{16, 22, 23].
Here we examine spin-orbit coupling and orbital de-
pairing in thin lms of Nb-doped SrTiO 3(NSTO) using
tunneling spectroscopy. Recently, we have developed an
approach for realizing high-quality tunneling junctions
for bulk NSTO with eV resolution of the superconduct-
ing gap [24, 25]. By carefully engineering the band align-
ments using polar tunneling barriers, the interfacial car-
rier density probed by tunneling corresponds to the nom-
inal density of dopants. We study the tunneling conduc-
tance (di=dv ) of NSTO lms in the 2D limit ( d < GL,
wheredis the lm thickness and GLis the Ginzburg-
Landau coherence length). We nd a single supercon-arXiv:1805.00047v1 [cond-mat.supr-con] 30 Apr 20182
-4-2024E/Δ2.52.01.51.00.50.0ρ↑,↓ /ρ0 -4-2024E/Δ
!∥!#SrTiO3(001)Nb:SrTiO3LaAlO3AgdΩΩa)
b)
SrTiLaAlO(SrO)0(TiO2)0(LaO)+(AlO2)-(AlO2)-(LaO)+
%=0(=0.1+,-∆/=0.6%=6(=0.1+,-∆/=0.61↑+1↓
FIG. 1. (Color online) a) Schematic of the tunneling junc-
tion device structure and atomic stacking of the oxide het-
erostructure. b) Expected e ect of Zeeman splitting on the
spin-dependent DOS for two cases: zero spin-orbit coupling
(b= 0) (left panel) and large spin-orbit coupling ( b= 6) (right
panel). The dimensionless SOC parameter b= h=(3 so0) re-
ects the strength of the SOC relative to the gap energy scale.
Dashed blue (dashed grey) and solid red (solid grey) curves
represent the spin-up and spin-down DOS, respectively, while
the solid black curve gives the total DOS from "+#(shifted
upwards by 1 for clarity). The spectra were calculated using
Maki's theory (Eq. (2)) at T= 0 K, lifetime broadening
parameter= 0:1, and magnetic eld BH=0= 0:6.
ducting gap which closes at the superconducting transi-
tion temperature ( Tc). We extract Hc2;kfrom the tun-
neling spectra and nd that it greatly exceeds the Pauli
limit. Under in-plane applied elds, Zeeman splitting is
not observed and an apparent single gap persists at all
elds until closing completely near 1.6 T, indicating that
the spin-orbit coupling scale ( h= so) is larger than 0.
We analyze the data using Maki's theory [21{23] and ex-
amine the relative contributions from orbital depairing
and spin-orbit scattering. Due to the heavy mixing of
the spin states, Maki's theory provides an upper-bound
for the spin-orbit scattering time of so1:1 ps and spin
di usion length s32 nm.
We fabricated tunneling junctions consisting of super-
conducting NSTO thin lms of thickness d= 18 nm, with
a 2 unit cell (u.c.) epitaxial LaAlO 3tunneling barrier,
and Ag counter electrodes as described elsewhere [24, 25].
NSTO with 1 at.% Nb-doping was homoepitaxially de-
posited on undoped SrTiO 3(001) by pulsed-laser deposi-
tion [26]. Films grown by this technique exhibit full car-
6050403020100 Δ (µeV)0.60.40.20.0T (K)1.00.80.60.40.20.0R/RN0.60.50.40.30.20.10.0di/dv (mS)-400-2000200400V (µV)!"#= 0 T!"#= 0 Ta)b)FIG. 2. (Color online) Tunneling spectroscopy and resis-
tivity in zero eld. a) Tunneling conductance ( di=dv ) of 18
nm thick Nb-doped SrTiO 3thin lm measured at the base
temperature of the dilution refrigerator. b) Superconducting
gap amplitude () (open circles, left axis) compared to the
normalized resistance (solid blue (grey) line, right axis). The
superconducting gap closes at T= 3155 mK, which is very
close to the resistive transition temperature Tc= 330 mK
de ned as 50% of the normal state resistivity at T= 0:6 K.
rier activation and bulk-like electron mobility. The polar
LaAlO 3tunnel barrier plays a crucial role in enabling
access to the electronic structure of NSTO in the 2D su-
perconducting limit. The LaAlO 3layer provides an in-
terfacial electric dipole which shifts the band alignments
between the Ag electrode and semiconducting SrTiO 3by
0:5 eV/u.c. [27{29]. Aligning the Fermi-level between
the two electrodes signi cantly reduces the Schottky bar-
rier and eliminates the long depletion length which pro-
hibits direct tunneling.
First, we report the zero- eld superconducting behav-
ior of the sample. Figure 2a shows di=dv measured at
base temperature ( T= 20 mK) and 0H= 0 T exhibit-
ing a single superconducting gap (). Although we ob-
serve high-energy coupling to longitudinal-optic phonon
modes (not shown) as reported recently [24], we do not
nd other strong-coupling renormalizations (i.e. McMil-
lan and Rowell [30]) in the tunneling spectra. The su-
perconducting gap is well t by the Bardeen-Cooper-
Schrie er (BCS) equation for the density of states with
0= 471eV. Due to the nite resolution of the
measurement and thermal broadening, the minimum of
the superconducting gap is nite. Here, the gap mini-
mum is two-orders of magnitude smaller than the normal
state conductance, demonstrating the dominance of elas-
tic tunneling and the high quality of the junction, even
in the 2D limit. The superconducting gap closes near
Tc= 330 mK as measured by four-point resistivity (Fig.
2b). Importantly, we do not observe a pseudogap as was
recently observed in LaAlO 3/SrTiO 3[31], indicating the
pseudogap is speci c to the LAO/STO interface and not
a generic feature in the 2D limit.
We now turn to the magnetic- eld response of the su-
perconducting gap. Figure 3a shows the superconducting
gap at several characteristic values of applied magnetic3
eld (left panel: H?, right panel: Hk). Figure 3b dis-
plays the zero-bias conductance (gap minimum) normal-
ized to the normal-state zero-bias conductance for both
eld orientations. We nd a large anisotropy between
Hc2;?andHc2;kwith a ratio Hc2;?/Hc2;k= 0.052. We
extract the Ginzburg-Landau superconducting coherence
lengthGL=p
0=(2Hc2;?) = 62 nm > d, con rming
the superconducting state is in the 2D regime. SrTiO 3
is a type-II superconductor with large London penetra-
tion depth compared to GLandd, and the quenching
of superconductivity due to an out-of-plane eld can be
attributed to the formation of vortices. For elds applied
in-plane, the large size of a vortex core is energetically un-
favorable to form in the 2D limit and the orbital depair-
ing is dramatically suppressed leading to enhanced Hc2;k.
We nd that the superconducting gap exhibits large Hc2;k
far in excess of the Pauli limit ( HP= 0=p
2B= 0.574
T), and in agreement with a study of upper-critical elds
from resistivity measurements in -doped SrTiO 3quan-
tum wells [20]. Here, we can examine the spin-dependent
response of the superconducting gap spectra to extract
the relevant contributions to orbital and spin depairing
mechanisms.
The superconducting DOS has been given by Maki's
theory, which takes into account orbital depairing, Zee-
man splitting of the spin states, and SOC [15, 21]. The
spin-dependent DOS is given by,
";#=0
2sgn(E)Re0
@uq
u2
11
A; (1)
where0is the normal-state DOS and uare de ned by,
u=EBH
0+uq
1 u2
+b0
@u uq
1 u2
1
A;(2)
for whichEis the energy relative to the Fermi level ( EF),
b= h=(3 so0) is a dimensionless quantity representing
the strength of the spin-orbit scattering relative to 0,
andrepresents spin-independent lifetime corrections.
Maki's equation (Eq. (2)) reduces to the BCS DOS in
the limit of vanishing andb. The quantity BHrepre-
sents the Zeeman splitting of the spin-dependent states
and observation of this splitting in the experimental data
depends on the strength of b(see Fig. 1). The parame-
ter= i+ oH2
kincludes eld-independent broadening
( i) and o=De2d2=(6h0) is the standard orbital de-
pairing for a thin lm in a parallel magnetic eld ( Dis
the di usion coecient) [14, 15, 21]. We follow the nu-
merical approach of Worledge and Geballe in applying
Eq. (2) to the tunneling data [21{23].
We now focus on the spectra shown in Fig. 3a (right
panel) for in-plane applied elds. The magnetic elds ex-
plored here ( BHk=0<2) are large enough to observe
Zeeman splitting in the weak spin-orbit limit ( b<1) [32].
1.21.00.80.60.40.20.0(σ / σN)|V=0
2.01.51.00.50.0µ0H (T)0.60.50.40.30.20.10.0di/dv (mS)-400-2000200400V (µV) 0.11 T 0.09 T 0.08 T 0.07 T 0.06 T 0.05 T 0.04 T 0.03 T 0.02 T 0.01 T 0 T-400-2000200400V (µV) 2 T 1.6 T 1.5 T 1.4 T 1.2 T 1.0 T 0.8 T 0.5 T 0.3 T 0 T!"#,%!"#,∥'(!∥='(!%=
!)a)
b)FIG. 3. (Color online) Tunneling spectroscopy of the super-
conducting gap under applied magnetic eld. a) Raw di=dv
data for several values of magnetic elds applied out-of plane
(0H?, left panel) and in-plane ( 0Hk, right panel). b) Zero-
bias conductivity ( =di=dv ) of the gap minimum normalized
to the normal-state conductance ( N) for both eld orienta-
tions. The out-of plane ( Hc2;?) and in-plane ( Hc2;k) upper
critical elds are indicated. The vertical dashed blue line in-
dicates the Pauli paramagnetic limiting eld ( HP).
However, for all measured magnetic elds, the data does
not exhibit a clear signature of Zeeman splitting indicat-
ing strong spin scattering relative to the superconducting
gap (compare Fig. 1 with Fig. 3a right panel) and con-
sistent with the violation of the Pauli-limit. While the
spin-orbit parameter bis eld-independent, the e ect of
Zeeman splitting in combination with rapid spin mixing
is to produce an e ective broadening of the total DOS
("+#, see Fig. 1) following an H2dependence [15].
Therefore since both orbital depairing and the large SOC
produce quasiparticle broadening under an applied eld,
it is a useful exercise to rst consider a reduced version of
Maki's theory which ignores the spin-degree of freedom
in the problem, such that,
u!u=E
0+0up
1 u2; (3)
which in zero- eld is equivalent to the Dynes formulation
were the phenomenological Dynes quasiparticle broaden-
ing parameter is given by = 00[33]. We rst t the
data of Fig. 3a right panel using Eq. (3) where 0is4
the only free parameter. The results for 0are shown in
Fig. 4a as a function of H2
kand are well described by
0= i+H2with i= 0.056 and = 0.4 T 2. The
small intrinsic quasiparticle broadening ( i) gives =
2eV and identical to our previous report in the bulk
limit [24]. The extracted value re
ects the total contri-
bution to eld-induced broadening from both spin-orbit
coupling and orbital depairing.
To quantify the spin-orbit and orbital depairing con-
tributions, we apply Maki's full theory (Eq. (2)) to
the set of tunneling data between 300 and 700 mT
(0:3< BH=0<0:86) including the spin-dependent
density of states, spin-orbit parameter b, and depairing
parameter= i+ oH2. The only free parameters are b
and owhich must both be singly valued at all elds. We
nd that the best ts are statistically equivalent for b>4
(with varying 0) [32], indicating short spin-orbit scat-
tering times so<1:1 ps. In this regime ( b>4), 0and
bare correlated. This can be understood as a competi-
tion between the spin-orbit induced e ective broadening
and orbital depairing. For instance, in the limit b!1 ,
the broadening from SOC vanishes and orbital depairing
must asymptotically approach to account for the exper-
imentally observed broadening. Fig. 4b shows a charac-
teristic best t for 0Hk= 0.5 T (BH=0= 0.61) with
b= 6 and 0= 0:11. Additionally, an upper-bound on
the spin di usion length is given by s=q
3
4Dtr so<32
nm [34], where Dtr=v2
F tr=30:0012 m2/s is the trans-
port di usion coecient. Here we have estimated the
Fermi velocity vFin a single-band approximation with
e ective mass m= 1.24m0[3] and Fermi level EF=
61 meV [2, 24]. We have used the Drude scattering time
tr=me=ewheree= 300 cm2/Vs is the experimen-
tally measured electron mobility.
The contribution from orbital depairing in the tunnel-
ing data provides additional information on the super-
conducting phase. The best ts from Maki's theory in
the rangeb >4 correspond to 0.016 T 2< o, for
which oincreases commensurately with b. Thus, even
though spin-orbit and orbital depairing cannot be quan-
ti ed independently, there are clear experimental limits
on o. We can compare the experimental owith the ex-
pected orbital contribution from normal-state transport
parameters with o=Dtre2d2=(6h0)2 T 2, which is
far in excess of the measured total broadening of = 0:4
T 2. This apparent discrepancy can be resolved by con-
sidering the multi-band nature of bulk SrTiO 3with three
occupiedt2gorbitals comprised of two light- and one
heavy-electron bands [2, 10]. Normal-state transport co-
ecients are dominated by the highly mobile light elec-
trons, but these carriers only make-up a fraction of the
total DOS, whereas the lowest lying heavy band com-
prises the majority of the electrons at EF[2, 3, 10, 35]. In
other words, the experimental data cannot be explained
by solely considering highly mobile, light electrons in
1.61.20.80.40.0σ / σN-400-2000200400V (µV)0.80.60.40.20.0ζ´1.61.20.80.40.0(µ0H||)2 (T)a)b)
!"#∥= 0.5 T%=6("=0.11FIG. 4. (Color online) Maki analysis of the superconducting
gap spectra under in-plane magnetic elds. a) Total quasi-
particle broadening 0(black dots) determined by tting the
tunneling data of Fig. 3a right panel using Eq. (3). The
total broadening exhibits a dependence on the square of the
applied magnetic eld and the solid line represents a t to
0= i+H2. b) Normalized di=dv data (solid black line)
measured at 0Hk= 0.5 T and theoretical t (dashed red
(grey) line) using Maki's full theory as expressed in Eq. (2)
withb= 6, i= 0.056, o= 0.11, and 0= 47eV.
forming the superconducting phase. We can transpose
the orbital depairing extracted from the superconduct-
ing tunneling data to DSCrepresenting the di usion co-
ecient for electrons which contribute to pairing. We
nd 0:110 4m2/s< D SC<2:310 4m2/s, which
agrees very well with a simplistic estimate of the di u-
sion constant for the heavy electron band with m6m0
[36] and momentum scattering time he100 fs, giving
Dhe110 4m2/s. Therefore, the robustness of super-
conductivity at high magnetic elds is consistent with the
established bulk band structure for which the heavy elec-
tron band dominates the total DOS and results in weak
orbital depairing. We note that the importance of the
heavy bands for superconductivity has been suggested in
LaAlO 3/SrTiO 3[37, 38]
The spin-orbit scattering times observed here are com-
parable to the momentum scattering time ( tr= so0:1)
and signi cantly shorter than those suggested theoreti-
cally in a single band limit [39]. Additionally, we can
expect that Rashba and Dresselhaus elds are minimal
in the current sample structure under investigation [32].
Therefore, the rapid spin mixing near the Fermi level
can be understood in the context of the multiband elec-
tronic structure of bulk SrTiO 3with hybridized orbital
character arising from the tetragonal crystal eld split-
ting and the intrinsic atomic spin-orbit interaction [2].
This picture is analogous to p-type Si where short spin
relaxation times are characteristic despite the modest
SOC [40, 41]. The spin-orbit scattering explored here
re
ects the electrons with the largest contribution to
the density of states and the superconducting conden-
sate, which in bulk SrTiO 3is the heavy electron band.
This is in contrast to transport experiments exploring5
spin-orbit coupling in the normal state (i.e. weak (anti-
)localization, Subnikov de Haas oscillations) which are
most sensitive to the highly mobile subset of carriers
[3, 35]. Therefore, careful analysis of the sub-band struc-
ture and orbital character in con ned SrTiO 3-based het-
erostructures (e.g. LaAlO 3/SrTiO 3) is critical to un-
derstanding the spin-orbit properties of the normal and
superconducting phases. Regardless, it is interesting to
note that the scattering times found here ( so1 ps),
are in the ballpark of the vast majority of experimen-
tal ndings in LaAlO 3/SrTiO 3[7, 8], suggesting that the
spin-orbit scattering at the Fermi level arising from the
intrinsic atomic spin-orbit interaction contributes at least
on equal footing with Rashba e ects.
In conclusion, we have performed tunneling experi-
ments on the dilute superconductor SrTiO 3doped with
1 at.% Nb in the 2D superconducting limit. These re-
sults were enabled by precisely designing the tunneling
junction with epitaxial dipole tunnel barriers, which shift
band alignments and facilitates high-resolution tunneling
spectroscopy. The data indicates a single superconduct-
ing gap which closes at Tc. By geometrically suppressing
the orbital depairing, we show that the large intrinsic
SOC can be observed directly in the tunneling spectra
by the violation of the Pauli-limit and the absence of
Zeeman splitting. Surprisingly short spin-orbit scatter-
ing times of order 1 ps were obtained. Examination of
the orbital depairing parameter indicates that the heavy
electron band, which is dicult to explore in transport
experiments, plays an important role in the formation of
the superconducting phase.
We thank M. E. Flatt e for useful discussions. This
work was supported by the Department of Energy,
Oce of Basic Energy Sciences, Division of Mate-
rials Sciences and Engineering, under Contract No.
DE-AC02-76SF00515; and the Gordon and Betty
Moore Foundation's EPiQS Initiative through Grant
GBMF4415 (dilution fridge measurements).
aswartz@stanford.edu
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1803.03549v1.Spin_vorticity_coupling_in_viscous_electron_fluids.pdf | arXiv:1803.03549v1 [cond-mat.mes-hall] 9 Mar 2018Spin-vorticity coupling in viscous electron fluids
Ruben J. Doornenbal,1Marco Polini,2and Rembert A. Duine1,3
1Institute for Theoretical Physics, Utrecht University,
Leuvenlaan 4, 3584 CE Utrecht, The Netherlands
2Istituto Italiano di Tecnologia, Graphene Labs, Via Morego 30, I-16163 Genova, Italy
3Department of Applied Physics, Eindhoven University of Tec hnology,
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
(Dated: November 7, 2021)
We consider spin-vorticity coupling—the generation of spi n polarization by vorticity—in viscous
two-dimensional electron systems with spin-orbit couplin g. We first derive hydrodynamic equations
for spin and momentum densities in which their mutual coupli ng is determined by the rotational
viscosity. We then calculate the rotational viscosity micr oscopically in the limits of weak and strong
spin-orbit coupling. We provide estimates that show that th e spin-orbit coupling achieved in recent
experiments is strong enough for the spin-vorticity coupli ng to be observed. On the one hand, this
coupling provides a way to image viscous electron flows by ima ging spin densities. On the other
hand, we show that the spin polarization generated by spin-v orticity coupling in the hydrodynamic
regime can, in principle, be much larger than that generated , e.g. by the spin Hall effect, in the
diffusive regime.
PACS numbers: 85.75.-d, 75.30.Ds, 04.70.Dy
Introduction. —The field of spintronics is concerned
with electric control of spin currents [1]. For the de-
scription of experimentally relevant systems it has, until
very recently, been sufficient to consider their coupled
spin-charge dynamics in the diffusive regime where the
time scale for electron momentum scattering is fast com-
pared to other time scales. The celebrated Valet-Fert
theory for electron spin transport in magnetic multilay-
ers [2] and the Dyakonov-Perel drift-diffusion theory for
spin generation by the spin Hall effect [3], for example,
fall within this paradigm.
Very recent experimental developments have brought
about solid-state systems, such as ultra-clean encapsu-
lated graphene, in which the momentum scattering time
can be much longer than the time scale for electron-
electron interactions [4–7]. In this so-called hydrody-
namic regime, the electron momentum needs to be in-
cluded as a hydrodynamic variable and the viscosity of
the electron system cannot be neglected [8–17]. The
finite electron viscosity leads to several physical conse-
quences, such as a negative nonlocal resistance [4] and
super-ballistic transport through point contacts [7, 18].
These developments have spurred on a great deal of re-
search, including proposals for measuring the Hall vis-
cosity [19–21] and connections to strong-coupling predic-
tions from string theory [22].
In a seemingly unrelated development, spin-
hydrodynamic generation, i.e. the generation of voltages
from vorticity, was recently experimentally observed in
liquid Hg [23]. Spin-hydrodynamic generation is believed
to be a consequence of spin-vorticity coupling. Phe-
nomenological theories of spin-vorticity coupling were
developed early on [24] and have been applied to fluids
consisting of particles with internal angular momentum
such as ferrofluids [25], molecular nanofluids [26], andnematic liquid crystals [27]. In these phenomenological
theories, the coupling between orbital angular momen-
tum, i.e. vorticity of the fluid, and internal angular
is governed by a dissipative coefficient, the so-called
“rotational viscosity”. This type of viscosity has been
estimated microscopically for classical systems (see e.g.
[27]) and Hg [23], but not for viscous electrons in a
crystal.
Motivated by the recent realization of solid-state sys-
tems hosting viscous electron fluids, we develop in this
Letter the theory for spin-vorticity coupling in such sys-
tems. Wederivethephenomenologicalequationsdescrib-
ing coupled spin and momentum diffusion, and compute
the rotational viscosity microscopically. We apply our
theory to viscous electron flow through a point contact
and show that the spin densities generated hydrodynam-
ically can be much larger than the ones that are gen-
erated by the spin Hall effect in the diffusive transport
regime. Our results may therefore stimulate experimen-
tal research towards novel ways of spin detection and
generation.
Phenomenology. —We consider two-dimensional (2D)
electron systems with approximate translation invari-
ance and approximate rotation invariance around the
axis perpendicular to the plane (chosen to be the ˆz-
direction). The conserved quantities of this system are
energy, charge, linear momentum in the plane and an-
gular momentum in the ˆz-direction. For brevity, we do
not consider energy conservation explicitly and focus on
momentum and angular momentum conservation. In the
following, we follow the discussion of Ref. [24] and gen-
eralize it to include spin diffusion and lack of Galilean
invariance. The momentum density is denoted by p(r,t)
and is a 2D vector p= (px,py) in the ˆx-ˆy-plane with
r= (x,y) = (rx,ry). The total angular momentum den-2
sity in the ˆz-direction is the sum of orbital angular mo-
mentum density ǫαβrαpβand spin density s(r,t) (in the
ˆz-direction). Here, ǫαβis the 2D Levi-Civita tensor and
summation over repeated indices α,β,γ,δ ∈ {x,y}is im-
plied. We denote with vthe conjugate variable to the
momentum density, i.e., the velocity, whereas the spin
chemical potential, commonly referred to as spin accu-
mulation,µsis the conjugate variable to the spin density.
Conservation of linear momentum yields
∂pα(r,t)
∂t=−∂Παβ(r,t)
∂rβ, (1)
with Π αβ(r,t) the stress tensor. Conservation of angular
momentum in the z-direction is expressed as
∂[ǫαβrαpβ(r,t)+s(r,t)]
∂t=−∂jJ
α(r,t)
∂rα,(2)
withjJ
α(r,t) theα-th component of the angular momen-
tum current and in the above equations the summation
is overboth αandβ. The equation for the spin density is
found by subtracting the cross-product of rwith Eq. (1)
from Eq. (2) and yields
∂s(r,t)
∂t=−∂js
α(r,t)
∂rα−2Πa(r,t),(3)
with Πa(r,t) =ǫαβΠβα(r,t)/2 the antisymmetricpart of
the stress tensor and js
α(r,t) =jJ
α(r,t)−ǫβγrβΠγα(r,t)
the spin current.
Anonzerovelocityandspindensityincreasetheenergy
of the system. By symmetry, a nonzero velocity leads to
a contribution ρkinv2/2 to the energy density. This ex-
pression defines the kinetic mass density ρkin, such that
p(r,t) =ρkinv(r,t)[28]. Forthecasethatisofinterestto
us, i.e., 2D electrons with spin-orbit coupling, the kinetic
mass density is not equal to the average mass density ρ
because spin-orbit coupling breaks Galilean invariance.
Likewise, a nonzero spin density contributes χsµ2
s/2 to
the energy density, where χsis the static spin suscepti-
bility, so that s(r,t) =/planckover2pi1χsµs(r,t). These terms in the
energy density lead to contributions to the entropy pro-
duction from which relations between the fluxes (the spin
current and antisymmetric part of the pressure tensor)
and the forces (spin accumulation and velocity) are de-
rived phenomenologically. In terms of µs(r,t) andv(r,t)
we havefor the antisymmetric part of the pressuretensor
that [24]
Πa(r,t) =−ηr[ω(r,t)−2µs(r,t)//planckover2pi1],(4)
withω(r,t) =ǫαβ∂vβ(r,t)/∂rαthe vorticity and ηrthe
rotational viscosity . The aboveexpression showsthat an-
gular momentum is transferred, by spin-orbit coupling,
between orbital and spin degrees of freedom until the an-
tisymmetric part of the pressure tensor is zero. For the
spin current we have that js
α(r,t) =−σs∂µs(r,t)/∂rα=−Ds∂s(r,t)/∂rαwhich defines the spin diffusion con-
stantDsand spin conductivity σs, which obey the Ein-
stein relation σs=/planckover2pi1Dsχs. Note that we are omitting an
advective contribution ∼vαsto the spin current as we
restrict ourselves to the linear-responseregime. Inserting
these results for the fluxes into Eq. (3) and using Eq. (1)
leads to
∂s(r,t)
∂t=Ds∇2s(r,t)
+2ηr/bracketleftbigg
ω(r,t)−2s(r,t)
/planckover2pi12χs/bracketrightbigg
−s(r,t)
τsr;
ρkin∂vα(r,t)
∂t=−eρEα
m+νρkin∇2vα(r,t)
+ηrǫαβ∂
∂rβ/bracketleftbigg
ω(r,t)−2s(r,t)
/planckover2pi12χs/bracketrightbigg
−ρkinvα(r,t)
τmr.(5)
In the above we have assumed the linear-response regime
and introduced the kinematic viscosity νusing that the
symmetric part of the stress tensor is given by Π αβ=
νρkin∂vα/∂rβ. Furthermore, we have added spin and
momentum relaxation terms, parameterized by the phe-
nomenological time scales τsrandτmr, respectively. We
have also included an electric field E(the electron has
charge−e).
Eqs. (5) are the main phenomenological equations for
spin density and velocity. The term proportional to ηrin
the first equation describes generation of spin accumula-
tion in response to vorticity, e.g., spin-vorticity coupling.
In the steady state the hydrodynamic equations are
characterized by three length scales. The first is a length
scalethatresultsfromthespin-vorticitycouplingequalto
ℓsv=/radicalbig
Ds/planckover2pi12χs/(2ηr), which is the characteristic length
over which the orbital and spin angular momentum equi-
librate. Furthermore, we have the spin diffusion length
ℓsr=√Dsτsrthat determines the length scales for relax-
ation of spin due to impurities, and the momentum dif-
fusion length ℓmr=√ντmr. The most interesting regime,
whichoccursinthe limitofstrongspin-orbitcouplingrel-
ative to momentum and spin relaxation, is the one where
ℓsvis the shortest length scale. In this case the spin den-
sity locally follows the vorticity, which is determined by
the electron flow.
Application. —We consider electron flow through a
point contact (PC) [7, 18] driven by a voltage V. Tak-
ingτmr,τsr→ ∞we have from Ref. [18] for the velocity
distribution at the PC that
vy(x) =−πρeV
4mνρkin/radicalbigg/parenleftBigw
2/parenrightBig2
−x2, (6)
wheretheflowisinthe y-directionand wisthePCwidth.
From Eq. (5), in the limit ℓsv≪wthe steady-state spin
density generated at the PC by spin-vorticity coupling in
the hydrodynamic regime is then
s(x)
/planckover2pi12χsjc=−m
πewρ4x/radicalbig
(w/2)2−x2, (7)3
wherejc=−eρ/integraltext
dxvy(x)/(mw) is the average current
density.
Let us compare Eq. (7) with the spin density gener-
ated by the spin Hall effect in the diffusive limit. In
the latter case, the spin accumulation is determined
by∂2µs/∂x2=µs/ℓ2
sr, which follows from Eqs. (5) in
the limitℓsr≪ℓsv, together with the expression js
y=
−σs∂µx/∂x+θSH/planckover2pi1jc
y/(2e) for the spin current. Here
jc
y=σeEyis the diffusive charge current through the
PC, withσe=e2ρ2τmr/(m2ρkin) the electrical conduc-
tivity andθSHthe spin Hall angle. Using the boundary
conditionsjs(−w/2) =js(w/2) = 0, we find for the spin
density in the diffusive limit that
sdiff(x)
/planckover2pi12χsjcy=θSHℓsr
2eσssech/parenleftbiggw
2ℓsr/parenrightbigg
sinh/parenleftbiggx
ℓsr/parenrightbigg
.(8)
A crucial difference is thus that for diffusive spin trans-
port and when w≫ℓsr, the spin density is only nonzero
within a distance ∼ℓsraway from the edges of the PC,
while when w≫ℓsvand in the hydrodynamic limit, the
spin density [see Eq. (7)] is nonzero everywhere (except
atx= 0 where it vanishes by symmetry).
In both hydrodynamic and diffusive limits, the max-
imum spin density occurs at the edges. In the hy-
drodynamic limit the spin density formally diverges as
|x| →w/2, since the vorticity that results from the ve-
locity in Eq. (6) diverges in the same limit. This diver-
gence is, however, unphysical, as there will be a micro-
scopic length scale ℓedgeover which the velocity goes to
zero near the edge of the sample, resulting in a maxi-
mum spin density of |s(±w/2)|/(/planckover2pi12χsjc)∼m/(eρℓedge)
near the edges of the sample. We expect the lat-
ter to be much larger than the maximum spin density
|sdiff(±w/2)/(/planckover2pi12χsjc)| ∼m2θSHℓsr/(e/planckover2pi1ρτmr) generated
by the spin Hall effect in the diffusive regime (where we
estimatedσs∼/planckover2pi1ρτmr/m2), because /planckover2pi1τmr/(mθSHℓsr)∼
ℓmr/(θSHkFℓsr) is expected to be much larger than the
microscopic length scale ℓedge. Here,kFis the Fermi
wave number.
Microscopic theory. —We proceed by calculating the
rotational viscosity microscopically. This is most easily
achieved [29] by noting that even when spin relaxation
due to impurities is absent ( τsr→ ∞), the spin-vorticity
coupling opens a channel for spin relaxation, with rate
4ηr//planckover2pi12χs, which microscopically stems from the com-
bined effect of spin-orbit coupling and electron-electron
interactions. Hence, ηrcan be extracted from the re-
tarded spin-spin response function (for spin in the ˆz-
direction) at zero wave vector, denoted by χ(+)
s(ω), when
this response function is computed for a clean system
with spin-orbit coupling and interactions. From Eqs. (5)
we find that for v=0this response function has the
form
χ(+)
s(ω) =χs
1−iω/planckover2pi12χs/(4ηr). (9)Hence, we have that
1
ηr=−/parenleftbigg2
/planckover2pi1χs/parenrightbigg2
lim
ω→0Im[χ(+)
s(ω)]
ω.(10)
As a representative example, we compute the rota-
tionalviscosityusingstandardlinear-responsetechniques
for a 2D electron gas with Rashba spin-orbit coupling,
which has the following Hamiltonian [30]:
ˆH=/integraldisplay
dr/summationdisplay
σ∈{↑,↓}ˆψ†
σ(r)/bracketleftbigg
−/planckover2pi12∇2
2m+λ/planckover2pi1ˆz·/parenleftbigg∇
i×τ/parenrightbigg/bracketrightbigg
ˆψσ(r),
(11)
whereˆψσ(r) [ˆψ†
σ(r)] is an electron annihilation [creation]
operator and τis a vector of Pauli matrices. The unit
vector in the ˆz-direction is denoted by ˆz. The con-
stantλparametrizes the strength of spin-orbit interac-
tions. The spin density operator in imaginary time τis
ˆs(r,τ) =/planckover2pi1[ˆψ†
↑(r,τ)ˆψ↑(r,τ)−ˆψ†
↓(r,τ)ˆψ↓(r,τ)]/2, where
the dependence on τof the electron creation and annihi-
lationoperatorsindicates theircorrespondingHeisenberg
evolution in imaginary time. We have for the imaginary-
time spin-spin response function
χs(iωn) =1
/planckover2pi1/integraldisplay
dr/integraldisplay/planckover2pi1β
0dτ∝an}b∇acketle{tˆs(r,τ)ˆs(r,0)∝an}b∇acket∇i}ht0eiωnτ,(12)
whereiωn= 2πn/(/planckover2pi1β) is a bosonic Matsubara frequency
withβ= 1/(kBT) the inverse thermal energy, and the
expectation value ∝an}b∇acketle{t···∝an}b∇acket∇i}ht0is taken at equilibrium. Neglect-
ing vertex corrections due to interactions, this is worked
out to yield
χs(iωn) =−1
4/planckover2pi1V/summationdisplay
k/summationdisplay
δ/ne}ationslash=δ′/integraldisplay
d/planckover2pi1ωd/planckover2pi1ω′Aδ(k,ω)Aδ′(k,ω′)
×/bracketleftbiggN(/planckover2pi1ω)−N(/planckover2pi1ω′)
ω−ω′+iωn/bracketrightbigg
, (13)
withN(/planckover2pi1ω) =/bracketleftbig
eβ(/planckover2pi1ω−µ)+1/bracketrightbig−1the Fermi-Dirac distri-
bution function at chemical potential µ. The spectral
functionsAδ(k,ω) are labeled by the Rashba spin-orbit-
split band index δ=±. We incorporateelectron-electron
interactions into the spectral function by taking them
equal to Lorentzians broadened by the electron collision
timeτee[this corresponds to dressing bare propagator
lines in the spin bubble in Eq. (12) by self-energy inser-
tions], i.e.,
Aδ(k,ω) =/planckover2pi1
2πτee1
[/planckover2pi1ω−/planckover2pi1ωδ(k)]2+/parenleftBig
/planckover2pi1
2τee/parenrightBig2,(14)
where/planckover2pi1ωδ(k) =/planckover2pi12k2/2m+δ/planckover2pi1λkis the Rashba band dis-
persion. Inserting Eq. (14) into Eq. (13) and performing
a Wick rotation iωn→ω+i0+yields
ηr=4π2/planckover2pi14χ2
s
mτee/bracketleftBigg
2π+8/parenleftbigµτee
/planckover2pi1/parenrightbig
1+4/parenleftbigµτee
/planckover2pi1/parenrightbig2+4tan−1/parenleftbigg2µτee
/planckover2pi1/parenrightbigg/bracketrightBigg
,
(15)4
where we took λ→0. In the limit µτee//planckover2pi1≫1, we have
ηr=π/planckover2pi14χ2
s/(mτee).
Since we have neglected vertex corrections, the result
in Eq. (15) does not vanish in the λ→0 limit and is
strictly speaking only valid when spin-orbit coupling is
so strong that the spin-vorticity coupling is limited by
electron-electron interactions, i.e., when λkFτee≫1. In
the opposite limit, where the bottleneck for spin relax-
ation is the spin-orbit coupling, we perform a Fermi’s
Golden Rule calculation to determine the decay rate of
a spin polarization to second order in the strength of the
spin-orbit interactions. This gives at low temperatures
that
ηr=−π/planckover2pi1
8/integraldisplaydk
(2π)2A2(k,µ)(λ/planckover2pi1k)2,(16)
whereA(k,µ) is the spectral function obtained from
Eq. (14) by replacing /planckover2pi1ωδ(k)→/planckover2pi12k2/2m. Carrying out
the remaining integral gives
ηr=mλ2
2/planckover2pi1/bracketleftbigg
1+π/parenleftBigµτee
/planckover2pi1/parenrightBig
+2/parenleftBigµτee
/planckover2pi1/parenrightBig
tan−1/parenleftbigg2µτee
/planckover2pi1/parenrightbigg/bracketrightbigg
,
(17)
which indeed vanishes as λ→0. Whenµτee//planckover2pi1≫1, we
have that /planckover2pi1ηr∼(λkF)(λkFτee), showing the dependence
on the small parameter λkFτee≪1 explicitly. Inter-
estingly, since the kinematic viscosity ν∝τee, we have
that the rotational viscosity ηr∝1/νin the limit of
strongspin-orbit coupling and ηr∝νin the limit ofweak
spin-orbit coupling, with a maximum rotational viscosity
whenλkFτee∼1.
Estimates. —Next, we estimate the spin-vorticity cou-
pling for graphene with proximity-induced spin-orbit
coupling. We take λ/planckover2pi1kFto be on the order of 1 meV
[32]. Furthermore, we take τee∼100 fs [4]. We thus
have thatλ/planckover2pi1kFis about one order of magnitude smaller
than/planckover2pi1/τeeand use the weak spin-orbit coupling expres-
sion in Eq. (16). Evaluating Eq. (16) for a linear disper-
sion/planckover2pi1vFk, wherevF∼106m/s is the graphene Fermi
velocity, we find that
ηr∼(λ/planckover2pi1kF)2
/planckover2pi1v2
F/parenleftBigµτee
/planckover2pi1/parenrightBig
, (18)
usingµτee≫/planckover2pi1. We estimate the corresponding inverse
time scale as
ηr
/planckover2pi12χs∼(λ/planckover2pi1kF)2
/planckover2pi13χsv2
F/parenleftBigµτee
/planckover2pi1/parenrightBig
∼100 GHz,(19)
where we took µτee//planckover2pi1∼10, and estimated the spin sus-
ceptibility as χs∼D(µ), with the density of states at the
Fermi level D(µ)∼√ne/(/planckover2pi1vF), and the electron number
densityne∼1012cm−2[4].
To estimate the corresponding length scale ℓsv, we as-
sume that spin diffusion is in the hydrodynamic regime
determined by electron-electron interactions that lead tospin drag [33]. We then have for the spin diffusion con-
stant thatDs∼/planckover2pi1ρτee/(m2χs). The spin-vorticity length
scale is then ℓsv∼vF/planckover2pi1/radicalbig
τeeχs/ηr∼1µm. This is
the same order of magnitude as the momentum relax-
ation length scale ℓmr[4], so that the rotational viscosity
appears to be high enough to lead to observable spin-
vorticity coupling. Moreover, the limit where ℓsv< ℓmr
seems to be within experimental reach. Note that in the
regime of weak spin-orbit coupling we have for the spin
relaxation the Dyakonov-Perel result that 1 /τsr∝τmr
[36], which yields that in the hydrodynamic regime we
haveℓsr∼ℓsv/radicalbig
τee/τmr≫ℓsv.
A simple interpretation of the spin-vorticity coupling
is that the electron spins are polarized by an effective
magnetic field /planckover2pi1ω(r,t)/µB, withµBthe Bohr magneton,
in the frame that rotates with the electron flow vorticity.
We estimate the vorticity ω∼v/ℓmrusingℓmr∼0.1-
1µm, and a drift velocity of v∼100 m/s [4], which
yields a substantial effective magnetic field of 1-10 mT.
Discussion and conclusions. —We have developed the
theory for spin-vorticity coupling in viscous electron flu-
ids, both phenomenologically and microscopically, and
we have estimated that the proximity-induced spin-orbit
coupling in graphene is large enough for observable ef-
fects. As an example, we predict alargespin polarization
induced by spin-hydrodynamic generation in a PC. This
large spin density may e.g. be observed optically [37] or
via nitrogen-vacancy centre magnetometry [34, 35]. The
imaged spin density would provide a fingerprint of the
vorticity of the electron flow.
An interesting direction for future research is general-
ization of the phenomenological and microscopic deriva-
tion to other spin-orbit couplings, including, in particu-
lar, also the effects of violation of translational and ro-
tational invariance beyond the phenomenological relax-
ation terms that we included here. One example would
be that of Weyl semi-metals that naturally have size-
able spin-orbit coupling and have also been reported to
be able to reach the hydrodynamic regime [38]. Other
candidates are bismuthene [39] and stanene [40] that
combine strong spin-orbit coupling with high mobility.
Further interesting directions of research include incor-
porating effects of a magnetic field and computation of
the rotational viscosity in the regime where spin-orbit
interactions and electron-electron interactions are com-
parable in magnitude. In this regime, the crossover from
weak-to-strong spin-orbit coupling takes place, whereas
inclusion of momentum-relaxing scattering would lead to
a crossover from the spin-vorticity coupling to the spin
Hall effect.
Acknowledgements. —We thank Denis Bandurin, Eu-
gene Chudnovsky, and Harold Zandvliet for useful com-
ments. R.D. is member of the D-ITP consortium, a
program of the Netherlands Organisation for Scientific
Research (NWO) that is funded by the Dutch Min-
istry of Education, Culture and Science (OCW). This5
work is in part funded by the Stichting voor Funda-
menteel Onderzoek der Materie (FOM) and the Euro-
pean Research Council (ERC). M.P. is supported by
the European Union’s Horizon 2020 research and inno-
vation programme under grant agreement No. 696656
“GrapheneCore1”.
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1402.5817v2.Low_Energy_Effective_Hamiltonian_for_Giant_Gap_Quantum_Spin_Hall_Insulators_in_Honeycomb_X_Hydride_Halide__X_N_Bi__Monolayers.pdf | Low-Energy Effective Hamiltonian for Giant-Gap Quantum Spin Hall Insulators in
Honeycomb X-Hydride/Halide ( X= N-Bi) Monolayers
Cheng-Cheng Liu,1Shan Guan,1Zhigang Song,2Shengyuan A. Yang,3Jinbo Yang,2,4and Yugui Yao1,
1School of Physics, Beijing Institute of Technology, Beijing 100081, China
2State Key Laboratory for Mesoscopic Physics, and School of Physics, Peking University, Beijing 100871, China
3Engineering Product Development, Singapore University of Technology and Design, Singapore 138682, Singapore
4Collaborative Innovation Center of Quantum Matter, Beijing, China
Usingthetight-bindingmethodincombinationwithfirst-principlescalculations,wesystematically
derive a low-energy effective Hilbert subspace and Hamiltonian with spin-orbit coupling for two-
dimensional hydrogenated and halogenated group-V monolayers. These materials are proposed to
be giant-gap quantum spin Hall insulators with record huge bulk band gaps opened by the spin-orbit
coupling at the Dirac points, e.g., from 0.74 to 1.08 eV in Bi X(X= H, F, Cl, and Br) monolayers.
We find that the low-energy Hilbert subspace mainly consists of pxandpyorbitals from the group-V
elements, and the giant first-order effective intrinsic spin-orbit coupling is from the on-site spin-orbit
interaction. These features are quite distinct from those of group-IV monolayers such as graphene
and silicene. There, the relevant orbital is pzand the effective intrinsic spin-orbit coupling is from
the next-nearest-neighbor spin-orbit interaction processes. These systems represent the first real 2D
honeycomb lattice materials in which the low-energy physics is associated with pxandpyorbitals.
A spinful lattice Hamiltonian with an on-site spin-orbit coupling term is also derived, which could
facilitate further investigations of these intriguing topological materials.
PACS numbers: 73.43.-f, 73.22.-f, 71.70.Ej, 85.75.-d
I. INTRODUCTION
Recent years have witnessed great interest in two-
dimensional (2D) layered materials with honeycomb lat-
tice structures. Especially, the 2D group-IV honey-
comb lattice materials, such as successively fabricated
graphene,1,2and silicene,3,4have attracted considerable
attention both theoretically and experimentally due to
their low-energy Dirac fermion behavior and promising
applications in electronics. Recently, we have discovered
stable 2D hydrogenated and halogenated group-V hon-
eycomb lattices via first-principles (FP) calculations.5
Their structures are similar to that of a hydrogenated
silicene (silicane), as shown in Fig. 1(a). In the absence
of spin-orbit coupling (SOC), the band structures show
linear energy crossing at the Fermi level around Kand
K0points of the hexagonal Brillouin zone. It is quite
unusual that the low-energy bands of these materials are
ofpxandpyorbital character. Previous studies in the
context of cold atoms systems have shown that pxand
pyorbital character could lead to various charge and or-
bital ordered states as well as topological effects.6,7Our
proposed materials, being the first real condensed mat-
ter systems in which the low-energy physics is associated
withpxandpyorbitals, are therefore expected to exhibit
rich and interesting physical phenomena.
The quantum spin Hall (QSH) insulator state has gen-
erated great interest in condensed matter physics and
materialscienceduetoitsscientificimportanceasanovel
quantum state and its potential technological applica-
tions ranging from spintronics to topological quantum
computation.8–10This novel electronic state is gaped in
the bulk and conducts charge and spin in gapless edge
states without dissipation protected by time-reversal
(b)
ΓΚ
Μ
(a)FIG. 1. (Color online). (a) The lattice geometry for 2D
X-hydride/halide ( X= N-Bi) monolayer from the side view
(top) and top view (bottom). Note that two sets of sublat-
tice in the honeycomb group V element Xare not coplanar
(a buckled structure). The monolayer is alternatively hydro-
genated or halogenated from both sides. (b) The first Bril-
louin zone of 2D X-hydride/halide monolayer and the points
of high symmetry.
symmetry. The concept of QSH effect was first proposed
by Kane and Mele in graphene in which SOC opens a
nontrivial band gap at the Dirac points.11,12Subsequent
works, however, showed that the SOC for graphene is
tiny, hence the effect is difficult to be detected experi-
mentally.13–15So far, QSH effect has only been demon-
strated in HgTe/CdTe quantum wells,16,17and experi-
mental evidence for helical edge modes has been pre-
sented for inverted InAs-GaSb quantum wells.18–20Nev-
ertheless, these existing systems more or less have serious
limitationsliketoxicity,difficultyinprocessing,andsmall
bulk gap opened by SOC. Therefore, an easy and envi-arXiv:1402.5817v2 [cond-mat.mtrl-sci] 22 Sep 2014ronmental friendly realization of a QSH insulator is much
desired. Extensive effort has been devoted to the search
for new QSH insulators with large SOC gap.21–28For
instance, new layered honeycomb lattice type materials
such as silicene, germanene24or stanene25, and chemi-
cally modified stanene27have been proposed. Ultrathin
Bi(111) films have drawn attention as a candidate QSH
insulator, whose 2D topological properties have been re-
ported.29An approach to design a large-gap QSH state
on a semiconductor surface by a substrate orbital filter-
ing process was also proposed.30However, desirable QSH
insulators preferably with huge bulk gaps are still rare.
A sizable bulk band gap in QSH insulators is essential for
realizing many exotic phenomena and for fabricating new
quantum devices that can operate at room temperature.
Using FP method, we have recently demonstrated that
the QSH effect can be realized in the 2D hydrogenated
and halogenated group-V honeycomb monolayers family,
with a huge gap opened at the Dirac points due to SOC.5
Although the low-energy spectrum of these materials is
similar to the 2D group-IV honeycomb monolayers such
as graphene and silicene, the low-energy Hilbert space
changes from the pzorbital to orbitals mainly consisting
ofpxandpyfrom the group-V atoms (N-Bi). More-
over, the nature of the effective SOC differs between
the two systems. Motivated by the fundamental inter-
est associated with the QSH effect and huge SOC gaps
in these novel 2D materials, we develop a low-energy ef-
fective model Hamiltonian that captures their essential
physics. In addition, we propose a minimal four-band
latticeHamiltonianwiththeon-siteSOCtermusingonly
thepxandpyorbitals.
From the symmetry analysis, the next-nearest-
neighbor (NNN) intrinsic Rashba SOC should exist in
these systems due to the low-buckled structure, similar
to the case of silicene.25However, as we shall see, the
dominant effect is from the much larger first-order SOC
of on-site origin. Therefore, in the following discussion,
we shall focus on the first-order on-site SOC and neglect
the higher-order effects. This point will be further dis-
cussed later in this paper.
The paper is organized as follows. In Sec. II, we de-
rivestepbystepthelow-energyeffectiveHilbertsubspace
and Hamiltonian for honeycomb X-hydride ( X= N-Bi)
monolayers, and also investigate in detail the effective
SOC.SectionIIIpresentsthederivationofthelow-energy
effectivemodelfor X-halide( X=N-Bi, halide=F-I)hon-
eycomb monolayers. In Sec. IV, a simple spinful lattice
Hamiltonian for the honeycomb X-hydride/halide mono-
layers family is constructed. We conclude in Sec. V with
a brief discussion of the effective SOC and present a sum-
mary of our results.II. LOW-ENERGY EFFECTIVE HAMILTONIAN
FOR HONEYCOMB XH(X= N-BI)
MONOLAYERS
A. Low-energy Hilbert subspace and effective
Hamiltonian without SOC
As is shown in Fig. 1(a), there are two distinct sites
A and B in the unit cell of X-hydride ( X= N-Bi)
honeycomb lattice with full hydrogenation from both
sides of the 2D Xhoneycomb sheet. The primitive
lattice vectors are chosen as ~ a1=a(1=2;p
3=2)and
~ a2=a( 1=2;p
3=2), whereais the lattice constant.
We consider the outer shell orbitals of textitX ( X=
N-Bi), namely s,px,py,pz, and also the sorbital of
H in the modeling. Therefore, in the representation
fjpA
yi;jpA
xi;jpA
zi;jsA
Hi;jsAi;jpB
yi;jpB
xi;jpB
zi;jsB
Hi;jsBig
(for simplicity, the Dirac ket symbol is omitted in the
following), the Hamiltonian (without SOC) at Kpoint
with the nearest-neighbor hopping considered in the
Slater-Koster formalism31reads
H0=HAA
0HAB
0
HABy
0HBB
0
; (1)
with
HAA
0=2
666640 0 0 0 0
0 0 0 0 0
0 0 0 VH
sp 0
0 0 VH
sp HVH
ss
0 0 0 VH
ss 3
77775;(2)
HAB
0=2
66664 V0
1 iV0
10 0V0
2
iV0
1V0
10 0 iV0
2
0 0 0 0 0
0 0 0 0 0
V0
2iV0
20 0 03
77775;(3)
HBB
0=2
666640 0 0 0 0
0 0 0 0 0
0 0 0VH
sp 0
0 0VH
sp HVH
ss
0 0 0VH
ss 3
77775; (4)
whereVH
sp(VH
ss) is the hopping between the pz(s)
orbital from Xatom and the sorbital from H, and
V0
1(3=4) (Vpp Vpp)andV0
2(3=2)VspwithVpp,
Vsp, andVppbeing the standard Slater-Koster hopping
parameters. andHare on-site energies for sorbitals
of atom Xand of atom H, respectively. The on-site en-
ergies forporbitals are taken to be zero.
To diagonalize the Hamiltonian, we first perform the
2(c)(a) (b)
(d)FIG. 2. (Color online). (a)(b) The partial band structure projection for NH and NF without SOC, respectively. Symbol size
is proportional to the population in the corresponding states. The Fermi level is indicated by the dotted line. (c)(d) Band
structures for BiH and BiF without (black dash lines) and with (red solid lines) SOC. The four band structures are obtained
from the first-principles methods implemented in the VASP package32using projector augmented wave pseudo-potential, and
the exchange-correlation is treated by PAW-GGA. The Fermi level is indicated by the solid line.
following unitary transformation:
'A
1= 1p
2
pA
x+ipA
y
=jpA
+i;
'B
2=1p
2
pB
x ipB
y
=jpB
i;
'3=1p
2
1p
2
pA
x ipA
y
1p
2
pB
x+ipB
y
;
'4=1p
21p
2
pA
x ipA
y
1p
2
pB
x+ipB
y
:(5)
In the basisf'A
1;sB;sB
H;pB
z;'B
2;sA;sA
H;pA
z;'3;'4g,
the Hamiltonian can be written as a block-diagonal form
with three decoupled blocks H ,H , andH
:
H0 !H1=Uy
1H0U1; (6)U1=2
666666666666664 ip
20i
2 i
20 0 0 0 0 0
1p
20 1
21
20 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0
0 ip
2 i
2 i
20 0 0 0 0 0
01p
2 1
2 1
20 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 13
777777777777775;(7)
H1=H H H
; (8)
with
H =2
6640iV20 0
iV2VH
ss 0
0VH
ssHVH
sp
0 0VH
sp 03
775; (9)
3H =2
6640 iV2 0 0
iV2VH
ss 0
0VH
ss H VH
sp
0 0 VH
sp 03
775;(10)
H
=diagfV1; V1g; (11)
whereV1= 2V0
1andV2=p
2V0
2.
The eigenvectors for the first diagonal block H can
be easily obtained as
j"ii=1
Ni2
66641
i"i
V2
i"2
i "i V2
2
V2VHss
iVH
sp
"i"2
i "i V2
2
V2VHss3
7775;(12)
where"iandNi(i= 1;2;3;4)are the correspond-
ing eigenvalues and normalization factors, respectively.
Therefore, upon performing the unitary transforma-
tionf 1; 2; 3; 4g=f'A
1;sB;sB
H;pB
zgU withU =
fj"iigi=1;2;3;4fu
jig, the above upper-left 44block
H is diagonalized.
For the second diagonal block H , its eigenvalues are
denoted as"4+i(i= 1;2;3;4), and it can be easily shown
that"4+i="i, where"iare eigenvalues of H . This is
consistent with FP results, i.e., there are four two-fold
degeneracy points at Kpoint as shown in Fig. 2(a). The
eigenvectors of H are given by
j"ii=1
Ni2
66641
i"i
V2
i"2
i "i V2
2
V2VH
ss
iVH
sp
"i"2
i "i V2
2
V2VHss3
7775;(13)
where"iandNi(i= 5;6;7;8)are the corresponding
eigenvalues and normalization factors. Similar to the
case ofH , upon performing the unitary transforma-
tionf 5; 6; 7; 8g=f'B
2;sA;sA
H;pA
zgU withU =
fj"i+4igi=1;2;3;4fu
jig, the block H is diagonalized.
The third block H
is already diagonal
with eigenvalues fV1; V1gand eigenvectors
f'3;'4g f 9; 10g. Therefore, in the new
basisf 1; 2; 3; 4; 5; 6; 7; 8; 9; 10g
'A
1;sB;sB
H;pB
z;'B
2;sA;sA
H;pA
z;'3;'4
U2, where
U2u u I22, the total Hamiltonian (1) takes
a fully diagonlized form. The whole diagonalization
process can be summarized as follows:
f 1; 2; 3; 4; 5; 6; 7; 8; 9; 10g
=
pA
y;pA
x,pA
z;sA
H;sA;pB
y;pB
x;pB
z;sB
H;sB
U;(14)
where
U=U1U2; (15)
H0 !H0
0=UyH0U; (16)H0
0=diagf"1;"2;"3;"4;"5;"6;"7;"8;V1; V1g:(17)
From the band components projection as shown in
Fig. 2(a), in the vicinity of the Dirac points (around
Fermilevel), themaincomponentsofthebandcomefrom
thepxandpyorbitals of group-V element textitX mixed
with a small amount of sorbital of textitX. Compared
with the expressions of the eigenstates obtained above,
we find that the orbital features agree with that of j"1i
andj"5iifwetaketheireigenenergiesastheFermienergy.
Therefore the corresponding states 1and 5constitute
thelow-energyHilbertsubspace. Inthefollowing, wewill
give the explicit forms of the low-energy states 1and 5
as well as their eigenvalues.
Note that, in the above 44H , the scale of the 22
non-diagonal block H 12is smaller than the difference of
the typical eigenvalues between the upper 22diagonal
blockH 11and the lower 22diagonal block H 22.
Hence, through the downfolding procedure33, we could
obtain the low-energy effective Hamiltonian as
Heff
11=H 11+H 12(" H 22) 1H 21:(18)
Up to the second order, one obtains
"1=1
2
0+q
02+ 4V2
2
; (19)
with
0= +"VH
ss2
"2 H" VHsp2;
"=1
2
+q
2+ 4V2
2
:(20)
Consequently, we can obtain the explicit expressions of
j"1i
u
j1
j=1;4and 1. In a similar way, the explicit
expressions ofj"5ifu
j1gj=1;4and 5can also be ob-
tained. So far, we have obtained the eigenvalues "1="5
[Eqs. (19) and (20)] and the corresponding low-energy
Hilbert subspace consisting of 1and 5,
1=u
11'A
1+u
21sB+u
31sB
H+u
41pB
z;
5=u
11'B
2 u
21sA u
31sA
H+u
41pA
z:(21)
The above coefficients fu
j1gj=1;4are given in Eq. (12).
Further simplification could be made in order to cap-
ture the main physics. We can omit the second-order
correction for the eigenvalues and the first-order correc-
tion for the eigenvectors, i.e., the terms (u
31sB
H+u
41pB
z)
for 1and( u
31sA
H+u
41pA
z)for 5, and only keep the
zeroth-order eigenvectors and eigenvalues,
1=u
11'A
1+u
21sB;
5=u
11'B
2 u
21sA;
"1="=1
2
+q
2+ 4V2
2
:(22)
4This approximation is justified by our FP calculations,
namely in the vicinity of the Fermi level, px,py, and
sorbitals overwhelmingly dominate over the sHandpz
orbitals in the band components.
In the Hamiltonian (17), one can take the Fermi en-
ergyEF="1="5as energy zero point. Hence, states
1and 5, which constitute the low-energy Hilbert sub-
space, take the following explicit forms:
1=u
11
1p
2
pA
x+ipA
y
+u
21sB;
5=u
111p
2
pB
x ipB
y
u
21sA;(23)
with
u
11=
+q
2+ 18V2sp
r
22+ 36V2sp 2q
2+ 18V2sp;
u
21= 3p
2iVspr
22+ 36V2sp 2q
2+ 18V2sp:
Since we are interested in the low-energy physics near
theDiracpoint, weperformthesmall ~kexpansionaround
Kby~k!~k+Kand keep the terms that are first order
in~k. We find that
HK=
0vFk
vFk+ 0
; (24)
withvFbeing the Fermi velocity
vF=p
3a
21
2ju
11j2(Vpp Vpp) +ju
21j2Vss
;
(25)
and
k=kxiky:
Either following similar procedures, or using the in-
version symmetry (or time-reversal symmetry ) of the
system, we can easily obtain the low-energy Hilbert sub-
space and the low-energy effective Hamiltonian around
theK0point. Finally, we can summarize the basis for
the low-energy Hilbert subspace as
1=u
11
1p
2
pA
x+i zpA
y
+u
21 zsB;
5=u
111p
2
pB
x i zpB
y
u
21 zsA;(26)
and the low-energy effective Hamiltonian without SOC
reads
H =vF(kxx+ zkyy); (27)
where Pauli matrices denote the orbital basis degree
of freedom, and z=1labels the two valleys Kand
K0. Note that under the space inversion operation P=
x xand the time-reversal operation T= x^K(^Kis
the complex conjugation operator), the above low-energy
effective Hamiltonian [Eq. (27)] is invariant.B. Low-energy effective Hamiltonian involving
SOC
The SOC can be written as
Hso=0^L^s=0
2L+s +L s+
2+Lzsz
;(28)
wheres=sxisyandL=LxiLydenote the ladder
operators for the spin and orbital angular momenta, re-
spectively. Here ^s= (~=2)~ s, and in the following we shall
take ~= 1.0is the magnitude of atomic SOC. Because
of the presence of pxandpyorbital component in the
low-energy Hilbert subspace [Eq. (26)] f 1; 5g
f";#g,
an on-site effective SOC is generated with
Hso=so zzsz; (29)
where
so=1
2ju
11j20
=1
22
41 9V2
sp
2 q
2+ 18V2sp+ 18V2sp3
50:
(30)
Again we stress that in the honeycomb textitX-hydride
monolayers the dominant intrinsic effective SOC is on-
site rather than from the NNN hopping processes as in
the original Kane-Mele model.
Consequently, from the above Hamiltonian (27) and
(29), we obtain the generic low-energy effective Hamil-
tonian around the Dirac points acting on the low-energy
Hilbert subspace:
Heff=H +Hso=vF(kxx+ zkyy) +so zzsz;
(31)
where the analytical expressions for Fermi velocity vF
and magnitude of intrinsic effective SOC soare given in
Eqs. (25) and (30), whose explicit values are presented
in Table I via FP calculations. Again we note that the
above spinful low-energy effective Hamiltonian is invari-
ant under both the space-inversion symmetry operation
and time-reversal symmetry operation with T=isy x^K.
The two model parameters vFandsocan be obtained
by fitting the band dispersions of the FP results. Their
values are listed in Table I.
III. LOW-ENERGY EFFECTIVE
HAMILTONIAN FOR HONEYCOMB
TEXTITX-HALIDE ( X= N-BI) MONOLAYERS
A. Low-energy Hilbert subspace and effective
Hamiltonian without SOC
For the textitX-halide ( X= N-Bi) systems, the outer
shellorbitalsofXlabeledas Xs,Xpx,Xpy,Xpz, andthe
5TABLE I. Values of Fermi velocity vFand magnitude of in-
trinsic SOC sofor textitX-hydride honeycomb monolayers
obtained from FP calculations. Note that so=Eg=2, with
Egthe gap opened by SOC at the Dirac point.
system vF
105m=s
so(eV)
NH 6.8 6:710 3
PH 8.3 1810 3
AsH 8.7 9710 3
SbH 8.6 0.21
BiH 8.9 0.62
outer shell orbitals of halogen labeled as Hs,Hpx,Hpy,
Hpzwith(H=F-I)aretakenintoaccountinthefollowing
derivation. As is shown in Fig. 1(a), there are also two
distinct sites A and B in the honeycomb lattice unit cell
of textitX-halide with full halogenation from both sides
ofthe2DtextitXhoneycombsheet. Intherepresentation
fXpA
y,XpA
x,XpA
z,HpA
z,HpA
y,HpA
x,HsA,XsA,XpB
y,
XpB
x,XpB
z,HpB
z,HpB
y,HpB
x,HsB,XsBgand at theK
point, the total Hamiltonian with the nearest-neighbor
hopping considered in the Slater-Koster formalism reads
Hha
0=
hAA
0hAB
0
hAB
0yhBB
0!
; (32)
with
hAA
0=
2
66666666666640 0 0 0 Vha
pp 0 0 0
0 0 0 0 0 Vha
pp 0 0
0 0 0 Vha
pp 0 0 Vha
sp 0
0 0Vha
pp ha
p 0 0 0 Vha
sp
Vha
pp 0 0 0 ha
p 0 0 0
0Vha
pp 0 0 0 ha
p 0 0
0 0 Vha
sp 0 0 0 ha
sVha
ss
0 0 0 Vha
sp 0 0Vha
ss 3
7777777777775;
(33)
hAB
0=2
6666666666664 V0
1 iV0
10 0 0 0 0 V0
2
iV0
1V0
10 0 0 0 0 iV0
2
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
V0
2iV0
20 0 0 0 0 03
7777777777775;(34)hBB
0=
2
66666666666640 0 0 0 Vha
pp 0 0 0
0 0 0 0 0 Vha
pp 0 0
0 0 0 Vha
pp 0 0Vha
sp 0
0 0Vha
pp ha
p 0 0 0 Vha
sp
Vha
pp 0 0 0 ha
p 0 0 0
0Vha
pp 0 0 0 ha
p 0 0
0 0Vha
sp 0 0 0 ha
sVha
ss
0 0 0 Vha
sp 0 0Vha
ss 3
7777777777775;
(35)
where ha
pis the on site energy for the porbitals of the
halogen atom, (ha
s) is the on site energy for the sor-
bital of textitX (halogen) atom, the on site energies for p
orbitals of textitX atoms are taken to be zero. Vha
pp(Vha
pp
)isthehoppingbetweenthe pzorbitalfromtextitXatom
and thepzorbital from halogen atom in the "shoulder by
shoulder" ("head to tail") type. Vha
spis the hopping be-
tween thepz(s) orbital from textitX atom and the s(pz)
orbital from halogen atom. Vha
ssis the hopping between
thesorbital from textitX atom and the sorbital from
halogen atom. The parameters V0
1andV0
2take the same
expressions as in Sec.II A.
Firstly, we perform the unitary transformation as in
Eq. (5), as well as the following unitary transformation
H'A
1= 1p
2
HpA
x+iHpA
y
H'B
2=1p
2
HpB
x iHpB
y
H'A
3= 1p
2
HpA
x iHpA
y
H'B
4= 1p
2
HpB
x+iHpB
y: (36)
In the new basis fX'A
1,XsB,H'A
1,HsB,XpB
z,HpB
z,
X'B
2,XsA,H'B
2,HsA,XpA
z,HpA
z,X'3,X'4,H'A
3,
H'B
4g=fXpA
y,XpA
x,XpA
z,HpA
z,HpA
y,HpA
x,HsA,
XsA,XpB
y,XpB
x,XpB
z,HpB
z,HpB
y,HpB
x,HsB,XsBg
Uha
1, we could rewrite the Hamiltonian in the following
block-diagonalformwiththreedecoupleddiagonalblocks
Hha
1=Hha
1; Hha
1; Hha
1;
; (37)
Hha
1; =2
6666666640iV2Vha
pp 0 0 0
iV2 0 Vha
ss 0 Vha
sp
Vha
pp 0 ha
p 0 0 0
0Vha
ss 0 ha
sVha
sp 0
0 0 0 Vha
sp 0Vha
pp
0 Vha
sp 0 0Vha
pp ha
p3
777777775;
(38)
6Hha
1; =2
6666666640 iV2Vha
pp 0 0 0
iV2 0 Vha
ss 0Vha
sp
Vha
pp 0 ha
p 0 0 0
0Vha
ss 0 ha
s Vha
sp 0
0 0 0 Vha
sp 0Vha
pp
0Vha
sp 0 0 Vha
pp ha
p3
777777775;
(39)
Hha
1;
=2
6666664V1 0Vha
ppp
2Vha
ppp
2
0 V1 Vha
ppp
2Vha
ppp
2
Vha
ppp
2 Vha
ppp
2ha
p 0
Vha
ppp
2Vha
ppp
20 ha
p3
7777775:(40)
For the first diagonal block Hha
1; , in the presentation
fX'A
1;XsB;H'A
1;HsB;XpB
z;HpB
zgits eigenvectors can
be written as
j"ha
ii=1
Nha
i
2
666666666641
i
C
Vha
pp
"ha
i hap
i[Vha
pp(Vha2
sp+Vha
ppVha
ss) "ha
iVha
ss("ha
i ha
p)]
DC
iVha
sp[ha
sVha
pp ha
pVha
ss "ha
i(Vha
pp Vha
ss)]
DC
iVha
sp[Vha2
sp+Vha
ssVha
pp "ha
i("ha
i ha
s)]
DC3
77777777775;(41)
with
D
"ha
i
"ha
i ha
s
Vha2
pp "ha
i
"ha
i ha
p
+
"ha
i ha
p
Vha2
sp;
(42)
and
CV2
"ha
i ha
p
Vha2pp "ha
i
"ha
i hap: (43)
Here,"ha
iandNha
i(i= 1;2;;6)are the corresponding
eigenvalues and the normalization factors, respectively.
Therefore, by the unitary transformation
ha
1; ha
2; ha
3; ha
4; ha
5; ha
6
=
X'A
1;XsB;H'A
1;HsB;XpB
z;HpB
z
U ;(44)
withU =fj"ha
iigi=1;2;;6fu
jig, the above 66block
Hha
1; is diagonalized.
From our FP calculations [Fig. 2(b)], the main compo-
nents of the band around the Dirac points and the Fermi
level come from the XpxandXpyorbitals, mixed with
a small amount of the HpxandHpyorbitals as well as
Xsorbital. The orbital features are identical with the
eigenvectors of "ha
1. When we take its eigenvalue as theFermi energy EF. Following similar procedures as in the
previous section, we can obtain the eigenvalues up to the
second-order correction and the eigenvectors up to the
first-order correction with
"ha
1=1
20
@0+vuut02+ 4V2
2 20Vha2pp
" hap+Vha4pp
" hap21
A;
(45)
where
0=
"ha02
1
Vha2
ss+Vha2
sp
"ha0
1
ha
pVha2
ss+ ha
sVha2
sp
D
"ha0
i +
+
Vha2
sp+Vha
ssVha
pp
D
"ha0
i;
(46)
"ha0
1=1
20
@ +vuut2+ 4V2
2 2Vha2pp
" hap+Vha4pp
" hap21
A;
(47)
and
"=1
2
+q
2+ 4V2
2
: (48)
Uptothispoint, wehavefoundthelow-energyeigenvalue
"ha
1and the corresponding basis ha
1. Again, in order to
capture the essential physics, we simply the above ex-
pressions by taking only the zeroth-order terms. So in
the following, we take "ha
1="ha0
1and omit the correc-
tion withfHsB;XpB
z;HpB
zgfor the eigenvector fj"ha
1ig.
Consequently, the eigenvector has the following form in
the basisfX'A
1;XsB;H'A
1g
j"ha
1i=1
nha
12
6641
iV2
"ha0
1
Vha
pp
"ha0
1 hap3
7752
64uha
11
uha
21
uha
313
75;(49)
withnha
1being a normalization constant, and the eigen-
value"ha0
1is given in Eqs. (47) and (48).
The eigenvalues of the second diagonal block Hha
1; are
denoted as "ha
6+i(i= 1;2;;6), and one finds that
"ha
6+i="ha
i(i= 1;2;;6), where"ha
iare eigenvalues of
Hha
1; . Through similar procedures, the low-energy eigen-
vectorfj"ha
7ighas the following simple form in the basis
fX'B
2;XsA;H'B
2g:
j"ha
7i=1
nha
12
6641
iV2
"ha0
1
Vha
pp
"ha0
1 hap3
775=2
64uha
11
uha
21
uha
313
75:(50)
7The third diagonal block Hha
1;
are of high energy hence
is not of interest here.
Fromtheaboveanalysis, thelow-energystates ha
1and
ha
7constitute the low-energy Hilbert subspace. They
have the following explicit forms:
ha
1=uha
11
1p
2
XpA
x+iXpA
y
+uha
21XsB
+uha
31
1p
2
HpA
x+iHpA
y
;
ha
7=uha
111p
2
XpB
x iXpB
y
uha
21XsA
+uha
311p
2
HpB
x iHpB
y
:(51)
Againweperformthesmall ~kexpansionintheabovelow-
energy Hilbert subspace around Kpoint by~k!~k+K
and keep the first-order terms in ~k,
HK=
0vFk
vFk+ 0!
; (52)
withvFthe Fermi velocity
vF=p
3a
21
2juha
11j2(Vpp Vpp) +juha
21j2Vss
:
(53)
Note that for the textitX-halide systems, juha
11j2is
much larger than juha
21j2andjuha
31j2. Either follow-
ing similar procedures, or via the inversion symmetry
(or time-reversal symmetry ), one can obtain the low-
energy Hilbert subspace and and the low-energy effective
Hamiltonian around the K0point. Finally the basis for
low-energy Hilbert subspace can be summarized as
ha
1=uha
11
1p
2
XpA
x+i zXpA
y
+uha
21 zXsB
+uha
31
1p
2
HpA
x+i zHpA
y
;
ha
7=uha
111p
2
XpB
x i zXpB
y
uha
21 zXsA
+uha
311p
2
HpB
x i zHpB
y
:(54)
and the low-energy effective Hamiltonian without SOC
reads
H =vF(kxx+ zkyy); (55)
where Pauli matrices denote the orbital basis degree of
freedom, and zlabels the two valleys KandK0. Note
thatunderthespacereversaloperation P=x xandthe
time-reversal operation T= x^K, the above low-energy
effective Hamiltonian Eq. (55) is also invariant.B. Low-energy effective Hamiltonian involving
SOC
In a similar way as in Sec. II B, we obtain an on-
site SOC in the spinful low-energy Hilbert subspace
f 1; 7g
f";#g,
Hso=so zzsz; (56)
so=1
2juha
11j2X
0+1
2juha
31j2ha
0;(57)
whereuha
11anduha
31are given in Eq. (49), and X
0(ha
0)
is the magnitude of atomic SOC of pnictogen (halogen).
It should be noted that due to the presence of major px
andpyorbital components, the first-order on-site effec-
tive SOC also dominates in the textitX-halide systems.
Equation (49) explains the tendency that the soin-
creases with the atomic number of halogen for the same
pnictogen element, as shown in Table II.
From Eqs. (55) and (56), we obtain the generic low-
energyeffectiveHamiltonianaroundtheDiracpointsact-
ing on the low-energy Hilbert subspace f 1; 7g
f";#g
Heff=H +Hso=vF(kxx+ zkyy) +so zzsz;
(58)
where Fermi velocity vFand magnitude of intrinsic effec-
tive SOCsoare given in Eqs. (53) and (57), and their
values are listed in Table II. One notes that this Hamil-
tonian is also invariant under both the space-inversion
symmetry and time-reversal symmetry with T=isy x^K.
The two model parameters vFandsofor halides ob-
tained by fitting the band dispersions of the FP results
are listed in Table II.
IV. A SIMPLE SPINFUL LATTICE
HAMILTONIAN FOR THE HONEYCOMB
TEXTITX-HYDRIDE/HALIDE ( X= N-BI)
MONOLAYERS FAMILY
For the purpose of studying the topological proper-
ties of the honeycomb textitX-hydride/halide ( X= N-
Bi) monolayers family, as well as their edge states, it is
convenient to work with a lattice Hamiltonian via lat-
tice regularization of the low-energy continuum models
(Eq. (31) and Eq. (58)). Taking into account the main
physics involving pxandpyorbitals, we construct the fol-
lowing spinful lattice Hamiltonian for the 2D honeycomb
textitX-hydride/halide ( X= N-Bi) monolayers
H=X
hi;ji; ; =px;pyt
ijcy
i cj
+X
i; ; =px;py;;0=";#
;0cy
i ci 0sz
;0;(59)
wherehi;jimeansiandjsites are nearest neighbors,
and are the orbital indices. The first term is the
hoppingtermandthesecondoneistheon-siteSOCterm.
8TABLE II. Values of two model parameters vFandsofor honeycomb textitX-halide ( X= N-Bi) monolayers obtained from
FP calculations. Note that so=Eg=2, withEgthe gap opened by SOC at the Dirac point.
system vF
105m=s
so(eV) system vF
105m=s
so(eV)
NF 5.5 8:510 3NBr 4.2 1910 3
PF 7.2 1310 3PBr 8.0 1710 3
AsF 7.3 8010 3AsBr 8.2 9810 3
SbF 6.6 0.16 SbBr 7.7 0.20
BiF 7.2 0.55 BiBr 7.3 0.65
NCl 4.3 9:710 3NI 3.8 2810 3
PCl 7.8 1710 3PI 8.1 1910 3
AsCl 8.0 9510 3AsI 9.1 0:10
SbCl 7.3 0.19 SbI 7.7 0.21
BiCl 6.9 0.56 BiI 7.7 0.65
After Fourier transformation of the above lattice
Hamiltonian, its energy spectrum over the entire Bril-
louin zone can be obtained. Since here spin is good quan-
tum number, we can divide the model Hamiltonian into
two sectors for spin up and spin down separately. For
each sector, the corresponding model Hamiltonian reads
H"(k) =2
66640 i0
2hAB
xx(k)hAB
xy(k)
0hAB
xy(k)hAB
yy(k)
0 i0
2y03
7775;(60)
H#(k) =2
66640i0
2hAB
xx(k)hAB
xy(k)
0hAB
xy(k)hAB
yy(k)
0i0
2y03
7775;(61)
where
hAB
xx(k)1
2(3Vpp+Vpp) coskx
2
exp
iky
2p
3
+Vppexp
ikyp
3
;
hAB
xy(k)ip
3
2(Vpp Vpp) sinkx
2
exp
iky
2p
3
;
and
hAB
yy(k)1
2(Vpp+ 3Vpp) coskx
2
exp
iky
2p
3
+Vppexp
ikyp
3
:
For simplicity, we choose the lattice constant a= 1. The
on-site energies for porbitals are taken to be zero. Near
theKandK0points, the above model Hamiltonian re-
duces to the low-energy effective Hamiltonian [Eq. (31)
and (58)] with vF=p
3a
4(Vpp Vpp)andso=0=2.
-4-3-2-101234
Energy(eV)Κ
ΜΓ Γ FIG. 3. (Color online). A comparison of the band structures
for monolayer SbH calculated using FP and TB methods with
SOC . The dashed green curve is the FP result. The solid red
and blue curves are the TB model results. The red curve is
with the NN hopping only, while the blue curve also includes
the NNN hopping terms. For the NN case, the parameters
are taken as Vpp= 1:68eV,Vpp= 0:60eV. For the NNN
case, the parameters are taken as Vpp= 1:69eV,Vpp=
0:62eV,VNNN
pp = 0eV,VNNN
pp = 0:23eV. For both cases,
so= 0:21eV. The superscript NNN means the next-nearest-
neighbor hoping. The Fermi level is set to zero.
Taking SbH as an example, we compare the results
from FP calculations and from the lattice models. As
shown in Fig. 3, there is a good agreement between the
two results around the Kpoint. The fitting away from
Kpoint can be improved by including hopping terms
between far neighbors. In Fig. 3, we also show the result
with NNN hopping, for which a fairly good agreement
with the FP low-energy bands over the whole Brillouin
zone can be achieved.
9V. DISCUSSION AND SUMMARY
Wehaveobtainedthelow-energyeffectiveHamiltonian
for the textitX-hydride and textitX-halide ( X= N-Bi)
family of materials, which is analogous to the Kane-Mele
model proposed for the QSH effect in graphene.11The
important difference is that in Kane-Mele model the ef-
fective SOC is of second-order NNN type, which is much
weaker than the on-site SOC in our systems. The SOC
term in our Hamiltonian opens a large nontrivial gap
at the Dirac points. From KtoK0the mass term
changes sign for each spin species and the band is in-
verted. As a result, the QSH effect can be realized in the
textitX-hydride and textitX-halide ( X= N-Bi) mono-
layers. Some of these materials, such as BiH/BiF, have
record huge SOC gap with magnitude around 1 eV, far
higher than the room-temperature energy scale, hence
making their detection much easier.
On the experimental side, the buckling honeycomb
Bi(111) monolayer and film have been manufactured via
molecular beam epitaxy (MBE).23,29,34On the other
hand, chemical functionalization of 2D materials is a
powerful tool to create new materials with desirable
features, such as modifying graphene into graphane,
graphone, and fluorinated graphene via H and F, re-
spectively.35Therefore, it is very promising that Bi-
Hydride/Halidemonolayer, thehugegapQSHinsulators,
may be synthesized by chemical reaction in the solvents
or by the exposure of the Bi (111) monolayer and film
to the atomic or molecular gases. It is noted that even
though one side (full passivation) instead of both sides
(alternatingpassivation)ofBi(111)bilayersispassivated,
the band structure is almost unchanged and the topol-
ogy properties remain nontrivial. This will provide more
freedom to realize these kinds of materials.
It is known that the low-energy Hilbert space for
graphene consists of the pzorbital from carbon atoms. In
that system, the SOC term from NNN second-order pro-
cesses is vanishingly small, and the on-site SOC as well
as the nearest neighbor SOC are forbidden by symme-
try constraint. In contrast, for the honeycomb textitX-
hydride/halide monolayers, pxandpyorbitals from the
group V elements constitute the low-energy Hilbert sub-
space. In fact, this represents the first class of materials
for which the Dirac fermion physics is associated with
pxandpyorbitals. Because of this, the effective on-site
SOC can has nonzero matrix elements and results in the
huge SOC gap at the Dirac points.
The leading-order effective SOC processes in the
textitX-hydride and textitX-halide systems, silicene, and
graphene are schematically shown in Fig. 4. As shown
in Figs. 44(a) and 4(b), the representative leading-order
effective SOCprocesses aroundthe Kpoint inthe honey-
comb textitX-hydride and textitX-halide monolayers are
jpA
+"iso !jpA
+"i;jpA
+#i so !jpA
+#i;
jpB
"i so !jpB
"i;jpB
#iso !jpB
#i;(62)wheresorepresents the atomic spin-orbit interaction
strength, which is given in Eq. (30) for textitX-hydride
systems and Eq. (57) for textitX-halide systems. In a
Hilbert subspace consisting of pxandpyorbitals, such
effective SOC arises in the first-order on-site processes,
which leads to its huge magnitude.
As for silicene, which has a low-buckled structure, the
typical leading-order SOC is from the (first-order) NNN
processes,25as shown in Fig. 4(c),
jpA
z"iV !jpB
"i 0
2 !jpB
"iV !jpA
z"i;
jpA
z#iV !jpB
#i0
2 !jpB
#iV !jpA
z#i;
jpB
z"iV !jpA
+"i0
2 !jpA
+"iV !jpB
z"i;
jpB
z#iV !jpA
+#i 0
2 !jpA
+#iV !jpB
z#i;(63)
whereVis the nearest-neighbor direct hopping ampli-
tudeand0representstheatomicintrinsicSOCstrength.
The whole process can be divided into three steps. For
example, we consider the pA
zorbital. Firstly, due to the
low-buckled structure, pA
zcouples topB
. Carriers in pA
z
orbital then hop to the nearest neighbor pB
orbital. Sec-
ondly, the atomic intrinsic SOC shifts the energy of the
spin up and spin down carriers by 0
2. In the third step,
carriersin the pB
orbitalhop toanothernearest-neighbor
pA
zorbital, making the resulting effective SOC an NNN
process and of first order in 0.
As for graphene, around Dirac point, the leading-order
effective SOC is from (second-order) NNN effective SOC
process, as shown in Fig. 4(d):
jpA
z"i0=p
2 ! jpA
+#iV !jsB
#iV !jpA
+#i0=p
2 ! jpA
z"i;
jpB
z#i0=p
2 ! jpB
"iV !jsA
"iV !jpB
"i0=p
2 ! jpB
z#i:(64)
During the whole NNN hopping process, the atomic SOC
appears twice, making the effective SOC second order in
0and hence much weaker.
In summary, using the TB method and the FP calcu-
lation, we have derived the low-energy effective Hilbert
subspace and Hamiltonian for the honeycomb textitX-
hydride/halide monolayers materials. These 2D group-V
honeycomb lattice materials have the same low-energy
effective Hamiltonian due to their same D3dpoint group
symmetry and the same D3small group at the Kand
K0points. The low-energy model contains two key pa-
rametersvFandso. We have obtained their analytic
expressions and also their numerical values by fitting the
FP calculations. Moreover, we have found that the low-
energy Hilbert subspace consists of pxandpyorbitals
from the group-V elements, which is a key reason for the
huge SOC gap. This feature is distinct from the group-
IV honeycomb lattice monolayers such as silicene and
graphene. Finally, we construct a spinful lattice Hamil-
tonian for these materials. Our results will be useful for
further investigations of this intriguing class of materials.
10ππ
πσ
σsoc
ππ
πσσsocsocσ
σsoc
σσσ
σ
σsocσ
AB
AB
BB
A A(a) (b)
(c) (d)FIG. 4. (Color online). The leading-order effective SOC pro-
cesses in textitX-hydride or textitX-halide ( X= N-Bi), sil-
icene and graphene. (a) and (b) Sketches of the huge effective
on-site SOC in textitX-hydride systems and textitX-halide
systems. (c) Sketch of the effective SOC from NNN hopping
processes caused by the buckling in silicene. (d) Sketch of the
second-order effective SOC from NNN hopping processes in
graphene.ACKNOWLEDGMENTS
This work was supported by the MOST Project
of China (Nos. 2014CB920903, 2010CB833104, and
2011CBA00100), the National Natural Science Foun-
dation of China (Grant Nos. 11225418, 51171001,
and 11174337), SUTD-SRG-EPD2013062, and the Spe-
cialized Research Fund for the Doctoral Program of
HigherEducationofChina(GrantNo. 20121101110046).
Cheng-Cheng Liu was supported Excellent young schol-
ars Research Fund of Beijing Institute of Technology
(Grant No. 2014CX04028).
Note added Recently, we notice another relevant
work36discussing effective models of a honeycomb lat-
tice withpxandpyorbitals.
ygyao@bit.edu.cn
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12 |
2206.05041v1.Quantum_heat_engine_based_on_a_spin_orbit_and_Zeeman_coupled_Bose_Einstein_condensate.pdf | Quantum heat engine based on a spin-orbit and Zeeman-coupled Bose-Einstein condensate
Jing Li,1,E. Ya Sherman,2, 3, 4and Andreas Ruschhaupt1
1Department of Physics, University College Cork, T12 H6T1 Cork, Ireland
2Departamento de Qu´ ımica-F´ ısica, UPV /EHU, Apartado 644, 48080 Bilbao, Spain
3IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain
4EHU Quantum Center, University of the Basque Country UPV /EHU
We explore the potential of a spin-orbit coupled Bose-Einstein condensate for thermodynamic cycles. For
this purpose we propose a quantum heat engine based on a condensate with spin-orbit and Zeeman coupling
as a working medium. The cooling and heating are simulated by contacts of the condensate with an external
magnetized media and demagnetized media. We examine the condensate ground state energy and its dependence
on the strength of the synthetic spin-orbit and Zeeman couplings and interatomic interaction. Then we study
the eciency of the proposed engine. The cycle has a critical value of spin-orbit coupling related to the engine
maximum e ciency.
Introduction Quantum cycles are of much importance both
for fundamental research and for applications in quantum-
based technologies[1, 2]. Quantum heat engines have been
demonstrated in recent on several quantum platforms, such as
trapped ions [3, 4], quantum dots [5] and optomechanical os-
cillators [6–9]. Well-developed techniques for experimental
control make Bose-Einstein condensates (BECs) [10] a suit-
able system for a quantum working medium of a thermal ma-
chine [11–13].
Recently, a quantum Otto cycle was experimentally realized
using a large quasi-spin system with individual cesium (Cs)
atoms immersed in a quantum heat bath made of ultracold ru-
bidium (Rb) atoms [14, 15]. Several spin heat engines have
been theoretically and experimentally implemented using a
single-spin qubit [16], ultracold atoms [17], single molecule
[18], a nuclear magnetic resonance setup [19] and a single-
electron spin coupled to a harmonic oscillator flywheel [20].
These examples have motivated our exploration of the spin-
orbit coupled BEC considered in this paper.
Spin-orbit coupling (SOC) links a particle’s spin to its
motion, and artificially introduces charge-like physics into
bosonic neutral atoms [21]. The experimental generation
[22–25] of SOC is usually accompanied by a Zeeman field,
which breaks various symmetries of the underlying system
and induces interesting quantum phenomena, e.g. topological
transport[26]. In addition, in the spin-orbit coupled BEC sys-
tem, more studies on moving solitons [27–29], vortices [30],
stripe phase [31] and dipole oscillations [32] have been re-
ported.
In this paper, we propose a BEC with SOC as a working
medium in a quantum Stirling cycle. The classic Stirling cy-
cle is made of two isothermal branches, connected by two
isochore branches. The BEC is characterized by SOC, Zee-
man splitting, a self-interaction, and is located in a quasi-
one-dimensional vessel with a moving piston that changes the
length of the vessel. The external ”cooling” and ”heating”
reservoirs are modelled by the interaction of the spin-1 /2 BEC
with an external magnetized and demagnetized medias. The
expansion and compression works depend on the SOC and
Corresponding author: jli@ucc.ieZeeman coupling. A main goal is to examine the condensate
ground state energy and its dependence on the strength of the
synthetic spin-orbit, Zeeman couplings, interatomic interac-
tion and length of the vessel. For the semiquantitative analy-
sis, perturbation theory is applied to understand the e ects of
SOC and Zeeman splitting. We further analyze several impor-
tant parameters and investigate how they a ect the e ciency
of the cycle, e.g. the critical SOC strength for di erent self-
interactions.
Model of the heat engine: Working medium We consider a
quasi-one dimensional BEC, extended along the x axis and
tightly confined in the orthogonal directions. The mean-field
energy functional of the system is then given by E=R+1
1"dx
with spin-independent self-interaction of the Manakov’s sym-
metry [33]:
"= yH0 +g
2(j "j2+j #j2)2; (1)
where ( "; #)T(here T stands for transposition) and
the wavefunctions "and #are related to the two pseudo-
spin components. The parameter grepresents the strength of
the atomic interaction which can be tuned by atomic s wave
scattering length using Feshbach resonance [34, 35] with g>
0,g<0, and g=0 giving the repulsive, attractive, and no
atomic interaction, respectively. The Hamiltonian H0in Eq.
(1) of the spin-1 /2 BEC, trapped in an external potential V(x),
is given by
H0=ˆp2
2mˆ0+
~ˆpˆx+~
2ˆz+V(x); (2)
with ˆ p= i~@xbeing the momentum operator in the longi-
tudinal direction, ˆ x;zbeing the Pauli matrices, and ˆ 0being
the identity matrix. Here is the SOC constant and is the
Zeeman field. We choose a convenient length unit , an energy
unit~2=(m2) and a time unit m2=~and express the following
equations in the corresponding dimensionless variables. The
coupled Gross-Pitaevskii equations are now given by
i@
@t "=
1
2@2
@x2+
2+g n(x)+V(x)!
" i @
@x #;(3)
i@
@t #=
1
2@2
@x2
2+g n(x)+V(x)!
# i @
@x ";(4)arXiv:2206.05041v1 [cond-mat.quant-gas] 10 Jun 20222
a a aP
=0
>0A
B
CD
magnetization
source
magnetization source
B C demagnetization source
D Ademagnetization
source
1 2(a)
(b)
FIG. 1. (a) The schematic diagram of the quantum Stirling cycle
based on the Zeeman and SOC. (b) Visualization of the demagne-
tization (left) and magnetization (right) processes with the external
sources; the blue dots represent the BEC atoms and the orange dots
represent the external source.
where the density is given by n(x)=j "j2+j #j2. We fix the
norm N=R1
1n(x)dx=1.
We consider a hard-wall potential V(x) of half width a:
V(x)=0;(jxja); V(x)=1(jxj>a):(5)
This potential is analogous to a piston in a thermodynamic
cycle and it allows one to define the work of the quantum cy-
cle. The ground state of the BEC then depends on the half
width a, the detuning , the interactions gand the SOC , i.e.
;g(a;), and the corresponding total ground state energy of
the BEC is then denoted as E ;g(a;). We define also the pres-
sure P ;g(a;) as a measure of the energy E ;g(a;) stored per
total length 2 a:
P ;g(a;) @E ;g(a;)
2@a: (6)
In the special case of = 0 and for the spin-independent
self-interaction proportional to n(x), the energy [36, 37]
is given by E ;g(a;0)=E0;g(a;0) 2=2 resulting in
independent pressure P ;g(a;0). Notice that at both
nonzero and, the system is characterized by a magne-
tostriction in the form M ;g(a;)=@P ;g(a;)=@:
Model of the heat engine: Quantum Stirling cycle We con-
sider a quantum Stirling cycle keeping the interaction gand
the SOC fixed during the whole process. The key idea is that
the external ”cooling” and ”heating” reservoirs are modelled
by the interaction of the spin 1 /2 BEC with an external mag-
netized media (see Fig. 1(b), right) resp. demagnetized media
(see Fig. 1(b), left). This external, (de)magnetized source
leads to a random magnetic field in the condensate and be-
cause of the Zeeman-e ect this corresponds to a detuning ofthe condensate to with some probability density distribution
p(). We assume that this external source brings the system to
a stationary state with the condensate described by a density
operator
ˆ=Z
p()j ;g(a;)ih ;g(a;)jd: (7)
The probability density distribution of the demagnetizing
source pdm() is centered around hidmR
pdm()d =0
while the one of the magnetizing source pm() is centered
around a positive value him>0. As an increase in de-
creases the BEC energy [10] by an dependent amount, the
demagnetization source plays the role of a “hot thermal bath”
here and the magnetization source plays the role of a “cold
thermal bath”. In general there could exist a stationary exter-
nal magnetic field leading to an additional detuning during the
cycle. We neglect this possibility in the following in order to
simplify the notation.
The realization of the Stirling cycle is described by a four-
stroke protocol, illustrated in Fig. 1(a). We start at point
Awith the BEC being in contact with the demagnetiza-
tion source, leading to an e ective detuning centered around
hidm=0. The potential is of half width a1. The BEC state is
given by Eq. 7 with p()=pA()pdm().
Quantum “isothermal” expansion stroke A !B:Dur-
ing this stroke, the working medium stays in contact with
the external demagnetization source while the potential ex-
pands adiabatically from a1toa2without excitation in the
BEC. The probability density distribution p() stays con-
stant during this ”isothermal” stroke, i.e. we have pA()=
pB()=pdm() (e ective detuning centered around hidm=
0). The average work done during this “isothermal” ex-
pansion stroke can be then calculated as [38] hWei=R
pdm()
E ;g(a1;) E ;g(a2;)
d.
Quantum isochore cooling stroke B !C:The contact
with the demagnetization source is switched o and the work-
ing medium is brought into contact with the magnetization
source while keeping a2constant. The probability distribu-
tion p() is changed to pC()pm(), this corresponds
to a ”cooling” (as the total energy of the BEC is lowered).
The average heat exchange in this stroke can be calculated as
hQci=R
(pm() pdm())E ;g(;a2)d.
Quantum “isothermal” compression stroke C ! D:
During this stroke, the working medium stays in contact
with the external magnetization source while the BEC com-
presses adiabatically from potential half width a2toa1with-
out excitation in the BEC. The probability density distribu-
tion p() remains constant during this ”isothermal” stroke,
i.e. we have pD()=pC()=pm() leading to an ef-
fective detuning centered around him>0. The average
work done during this “isothermal” compression is hWci=R
pm()
E ;g(a2;) E ;g(a1;)
d.
Quantum isochore heating stroke D !A:The contact
with the magntetization source is switched o and the work-
ing medium is brought again into contact with the demagne-
tization source while keeping a1constant. The probability
distribution p() is changed back to pA()=pdm(), this
corresponds to a ”heating” (as the total energy of the BEC is3
increased). The average heat exchange in this stroke can be
calculated ashQhi=R
(pdm() pm())E(;a1)d.
To study this quantum cycle, it is important to examine and
understand the dependence of the BEC ground-state energy on
the di erent parameters. This will be done in the following.
Perturbation theory for the ground state energy The com-
plex BEC system used in the thermodynamic cycle does not
have an exact analytical solution. However, we can obtain
analytical insight by considering perturbation theory of the
ground state energy E ;0(a;) of the non-selfinteracting BEC
(i.e. g=0) at small (and nonzero ), as well as at small
(and nonzero ).
In the case of small then 1=a, the Hamiltonian in
Eq. (1) can be written as H0=H0;0+H0
0whereH0=ˆp2=2+
ˆz=2+V(x) and the perturbation term being H0
0= ˆpˆx.
The eigenstate basis of H0;0is given by (0)
n;#(x)=0; n(x)T,
(0)
n;"(x)= n(x);0T, where n(x) are the eigenstates of the
potential in Eq.(5). The first-order correction to the energy
vanishes and the second-order correction becomes:
(0)
2= X
n>1jh (0)
n;"(x)jH0
0j (0)
0;#(x)ij2
(n2 1)2=(8a2)+ : (8)
Thus, the total ground state energy E ;0(a;) of the system up
to second order in is given by
E ;0(a;)2
8a2
2 2 2
4a2+2 2
8a42 (a;) cot (a;)
2;(9)
where (a;)p
2 8a2:We can simplify Eq. (9) by
approximating the expression up to first order in :
E ;0(a;)2
8a2
2 2
2+2 6
32 a
`sr!2
: (10)
The first three terms on the right-hand side of Eq. (10) corre-
spond to kinetic energy, Zeeman energy (at =0) and SOC
energy (at =0). Here we introduced the spin rotation length
`sr1= with a=`sr 1.
Alternatively, in the case of large then >1=aand small
detuning , the Hamiltonian can be written as H0=H0;1+H0
1
whereH0=ˆp2=2+ ˆpˆx+V(x);and the perturbation term
H0
1= ˆz=2:The unperturbedH0;1has pairs of degenerate
eigenstates (0)
aand (0)
bwith the energy E ;0(a;0):
(0)
a(x)= n(x)e i x
1
1!
; (0)
b(x)= n(x)ei x
1
1!
:(11)
Based on the perturbation theory for degenerate states and tak-
ing into account that the diagonal matrix elements of the per-
turbation, h (0)
ijzj (0)
ii=2=0, we obtain at a=`sr 1 the
ground state energy in the form:
E ;0(a;)2
8a2 2
2 2
4`sr=a (2a=`sr)2 2 sin 2a
`sr! :(12)
When we look at the corresponding pressure following from
Eq. (12), we can calculate approximately the pressure di er-
encePbetween the points BandCin the cycle (at a2, see Fig.
1.56 1.58 1.60 1.62 1.64-0.04-0.020.000.02
1.0 1.2 1.4 1.6 1.8 2.00.00.20.40.60.81.01.2FIG. 2. Pressure P ;0(a2;) versus potential half width afor the cases
of = 0; =1:6 (solid black, essentially, -independent), =
1; =1 (dashed blue) and = 1; =1:6 (dot-dashed red). (Inset)
The pressure di erence between points CandB,P=P ;0(a2;1)
P ;0(a2;0).
0.5 1 1.5 21.41.61.82.2.2
FIG. 3. Critical c(g;) versus detuning for di erent nonlineari-
ties: attractive g= 1 (black solid line), non-interaction g=0 (blue
dashed line), and repulsive g=1 (dot-dashed red line).
1). The di erencePjumps from negative to positive at cer-
tain widths where 2 a2=`sr(n+1)or (n+1)=(2a2) with
n=1;2;3;4;:::. In addition, there is always an between two
consecutive “jump points” where Pbecomes zero. We will
denote the first corresponding value of ;where the change of
Pfor negative to positive occurs, as the critical c(g;).
Energy and pressure We examine now the exact numerical
values of energy and pressure where we fix a1=1 and a2=2.
The corresponding pressure is illustrated in Fig. 2 for a non-
interacting BEC ( g=0). The shown pressure P ;0(a2;0) for
=0 does not depend on the strength of SOC as discussed
above. We can also see that the pressure P ;0(a2;1) is approx-
imately equal to the pressure P ;0(a2;0) at a2;providing
crossing of the red and dotted lines; this corresponds then to a
critical c(0;1)1:6. The corresponding di erence in pres-
surePis shown in detail in the inset; it can be seen that P
changes from negative to positive at c(0;1) as one expects it
from the perturbation theory above.
In Fig. 3, the relations between the critical c(g;) and de-
tuning for di erent nonlinearities gare plotted. From the
perturbation theory for g=0 and for small , one expects a
value of c(g;)=a21:57. The figure shows that the ex-4
act cis increasing with increasing for all cases of g. There
is a competition between SOC and Zeeman field, therefore, a
larger detuning requires automatically a larger (and there-
fore a larger c) to have an e ect. We also see that cis larger
(smaller) for attractive g= 1 (repulsive g=1) for all . The
heuristic reason is that there is a (kind of) compression (ex-
pansion) of the wavefunction for g<0 (g>0) and, therefore,
a weaker (stronger) e ect of SOC. This requires heuristically
a larger (smaller) (and therefore c) to show an e ect.
Work, heat and e ciency of the engine Here we are mainly
interested in the properties of the cycle originating from the
BEC and not in the details of the (de)magnetization source.
Therefore, we assume that the probability density distribu-
tions pdmresp. pmare strongly peaked around hidm=0
resp.him= 0>0 such that we approximate pdm()=()
and pm()=( 0) (whereis the Dirac distribution).
In this case, the black-solid line and the blue-dashed line
in Fig. 2 present an example of the expansion and com-
pression strokes of the cycle shown in the schematic Fig.
1. The work done during the “isothermal” expansion pro-
cess in Fig. 1,hWei, is then given by the energy di er-
ences:hWei=E ;g(a1;0) E ;g(a2;0). The cooling heat
exchange from BtoChQcithrough contact with the mag-
netization source, becomes hQci=E ;g(a2;0) E ;g(a2;0):
The workhWcidone during the compression stroke is then
hWci=E ;g(a2;0) E ;g(a1;0). The heat in the last stroke
can be calculated by hQhi=E ;g(a1;0) E ;g(a1;0). The
total work then becomes:
A=hWci+hWei=I
ABCDP ;g(a;0)da: (13)
For small 0,
A= 0Z2a2
2a1M ;g(a;0!0)da: (14)
As defined above, at = c(g;0), the pressures at a2for
= 0 and 0>0 approximately coincide. If > c(g;0);
the pressure-dependencies on afor = 0 and 0>0 cross
at a certain half width eawith a1<ea<a2. In that case, the
work done at the interval ( ea;a2) provides a negative contri-
bution while the contribution of the interval ( a1;ea) can still
increase. In the following, we restrict our analysis to the case
c(g;0) while we expect a maximum of the total work
close to c(g;0).
The e ciency of each quantum cycle is now defined as
=A
hQhi: (15)
At small 1=(2a2) we may approximate the e ciency of
the quantum cycle in terms of 0as
2666642
22
00BBBB@1
a2
1 1
a2
21CCCCA+2
43
0377775 2; (16)
where the coe cientis
= (a1;0)
a4
1cot (a1;0)
2 (a2;0)
a4
2cot (a2;0)
2:(17)
0.0 0.5 1.0 1.50.00.20.40.60.81.0
0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0(a)
(b)
FIG. 4. E ciencyversus with 0=0:5 (solid black), 0=
1:0 (solid blue) and 0=2:0 (solid red); the dotted vertical lines
denote the critical SOC strength c(g;0). (a) g=0; results based
on perturbation theory in Eq. (16) (blue, red and black dashed lines);
the dashed pink line is given by Eq. (18) for 0!0. (b) g= 1.
In the limit of 0!0, the e ciencysimplifies to
=2
2 6
32
a2
2 a2
1
2: (18)
It is worth noticing that Eq. (18) has two limits with respect to
the value of a2:Let us define cas the e ciency at the critical
c. First, Eq. (18) is applicable only at a2<; thus, limiting
the criticalcto the values of the order of 0.1. Secondly, for
g<0 the value of a2is limited to 2 =jgj[39], thuscis limited
correspondingly. (Note that Eq. (18) is not directly applicable
tog,0 BEC).
Figure 4 shows that the e ciencygrows as increases.
The approximate e ciency in Eq. (16) is a quadratic function
of , and this is in good agreement with the numerical results
in Fig. 4(a) for the case g=0. In the limit of 0!0,
the eciency 2, see Eq. (18). This limit case is also
shown by the dashed pink line in Fig. 4(a). As one expects
a maximum of the total work close to c(g;0), one expects
also that the e ciency reaches the maximum at close to
c(g;0). The e ciencycat a critical cwith respect to 0
is shown in Fig. 5. The e ciency decreases with increasing
. This corresponds to Eq. (16) when = c=a2for all
three cases of g(see Fig. (3)).
Discussion and conclusions Here we return to the physical
units and discuss the possibility of experimental realization of
the present Stirling cycle. In the one-dimensional realization5
0.5 1 1.5 20.50.60.70.80.9
FIG. 5. E ciencycat c(g;0) versus 0. Nonlinearities: attrac-
tiveg= 1 (black solid), non-interaction g=0 (blue dashed), and
repulsive g=1 (dot-dashed red). Values of c(g;0) are the same as
those in Fig. 3.
considered above, with the physical unit of length , the re-
sulting dimensionless coupling constant gcan be estimated as
2Naat=sp;where spis the condensate cross-section, physi-
cally corresponding to the piston cross-section. Here aatis the
interatomic scattering length (typically of the order of 10 aB,
where aBis the Bohr radius) dependent on the Feshbach res-
onance realization, and N 103is the total number of atoms
in the condensate. A reasonable for optical setups is of the
order of 10 m. Thus, the choice of a1;a2of the order of 10
m allows one to achieve dimensionless and0of the order
of unity [25], and thus explore the operational regimes of the
Stirling cycle up to the critical values.
In summary, we have explored the potential of a spin-orbit coupled Bose-Einstein condensate in a thermodynamic
Stirling-like cycle. It takes advantage of both the non-
commuting synthetic spin-orbit and Zeeman-like contribu-
tions. The ”cooling” and ”heating” is assumed to originate
by interaction with external magnetization and demagnetiza-
tion media. We have examined the ground-state energy of
the condensate and how the corresponding pressure depends
on the di erent parameters of the system. We have studied
the eciency of the corresponding engine in the dependence
on the strength of these spin-related couplings. The cycle is
characterized by a critical spin-orbit coupling, corresponding,
essentially, to the maximum e ciency. The dependence of
the eciency on the spin-dependent coupling and nonlinear
self-interaction paves the way to applications of these cycles.
While we have concentrated here on e ects originating from
the BEC, it will be interesting to study the details of the e ects
of the external magnetization and demagnetization sources in
the future.
ACKNOWLEDGMENTS
We are grateful to C. Whitty and D. Rea for commenting
on the manuscript. J.L. and A.R. acknowledge support from
the Science Foundation Ireland Frontiers for the Future Re-
search Grant “Shortcut-Enhanced Quantum Thermodynam-
ics” No.19 /FFP/6951. The work of E.S. is financially sup-
ported through the Grant PGC2018-101355-B-I00 funded by
MCIN /AEI/10.13039 /501100011033 and by ERDF “A way
of making Europe”, and by the Basque Government through
Grant No. IT986-16.
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0810.5405v1.Berry_Phase_in_a_Single_Quantum_Dot_with_Spin_Orbit_Interaction.pdf | arXiv:0810.5405v1 [cond-mat.mes-hall] 30 Oct 2008APS/123-QED
Berry Phase in a Single Quantum Dot with Spin-Orbit Interact ion
Huan Wang∗, Ka-Di Zhu†
Department of Physics, Shanghai Jiao Tong University, Shan ghai 200240, People’s Republic of China
(Dated: November 3, 2018)
Berry phase in a single quantum dot with Rashba spin-orbit co upling is investigated theoretically.
Berry phases as functions of magnetic field strength, dot siz e, spin-orbit coupling and photon-spin
coupling constants are evaluated. It is shown that the Berry phase will alter dramatically from 0
to 2πas the magnetic field strength increases. The threshold of ma gnetic field depends on the dot
size and the spin-orbit coupling constant.
PACS numbers: 71.70.Ej, 03.65.Vf
I. INTRODUCTION
Due to its important role in encoding information, the
phase of wavefunction attracts a lot of interest in infor-
mation science. Those properties can also be used in fu-
ture quantum information and quantum computer. Thus
selectingone kindofphaseswhich canbe manipulated by
quantumeffectisveryimportant. Berryphaseisbelieved
to be a promising candidate. As a quantum mechanical
system evolvescyclically in time such that it return to its
initial physical state, its wavefunction can acquire a ge-
ometric phase factor in addition to the familiar dynamic
phase[1, 2]. If the cyclic change of the system is adia-
batic, this additional factor is known as Berry’s phase[3],
and is, in contrast to dynamic phase, independent of en-
ergy and time. Fuentes-Guridi et al.[4] calculated the
Berry phase of a particle in a magnetic field in consid-
eration of the quantum nature of the light field. Yi et
al.[5] studied the Berry phase in a composite system and
showed how the Berry phases depend on the coupling be-
tween the two subsystems. In a recent paper, San-Jose
et al.[6] have described the effect of geometric phases in-
duced by either classical or quantum electric fields acting
on single electron spins in quantum dots. Wang and Zhu
[7] have investigated the voltage-controlled Berry phases
in two vertically coupled InGaAs/GaAs quantum dots.
Most recently, observations of Berry phases in solid state
materials are reported[8, 9, 10]. Leek et al.[10] demon-
strated the controlled Berry phase in a superconducting
qubit which manipulates the qubit geometrically using
microwave radiation and observes the phase in an inter-
ference experiment.
Spin-relatedeffectshavepotentialapplicationsinsemi-
conductor devices and in quantum computation. Rashba
et al.[11] have described the orbital mechanisms of
electron-spin manipulation by an electric field. Sonin[12]
has demonstrated that an equilibrium spin current in a
2D electron gas with Rashba spin-orbit interaction can
result in a mechanical torque on a substrate near an edge
of the medium. Serebrennikov[13] considered that the
∗Email: wanghuan2626@sjtu.edu.cn
†Email: zhukadi@sjtu.edu.cncoherent transport properties of a charge carrier. The
transportation will cause a spin precession in zero mag-
netic fields and can be described in purely geometric
terms as a consequence of the corresponding holonomy.
The spin-orbit interaction in semiconductor heter-
structures is increasingly coming to be seen as a tool
whichcan manipulate electronicspin states[14, 15]. Two
basic mechanisms of the spin-orbit coupling of 2D elec-
trons are directly related to the symmetry properties of
QDs. They stem from the structure inversion asymme-
try mechanism described by the Rashba term[11, 16] and
the bulk inversion asymmetry mechanism described by
the Dresselhaus term[17]. Recently, Debald and Emary
[18] have investigated a spin-orbit driven Rabi oscilla-
tion in a single quantum dot with Rashba spin-orbit cou-
pling. However, the influence of spin-orbit interaction
on Berry phase in a single quantum dot is still lacking.
In the present paper we will give a detail study on the
Berry phase evolution of a single quantum dot with spin-
orbit interaction in a time-dependent quantized electro-
magnetic environment. We will borrow quantum optics
method to investigate the impact of the spin-orbit inter-
action and spin-photon interaction on Berry phase.
The paper is organized as follows. In Sec.II, we give
the model Hamiltonian including both spin-orbit inter-
action and spin-photon interaction and calculated Berry
phases as functions of magnetic field strength, dot size,
spin-orbit coupling and photon-spin coupling constants.
In Sec.III, we draw the figures of the Berry phase as a
function of magnetic field strength and some discussions
are given. The final conclusion is presented in Sec.IV.
II. THEORY
We consider a simple two-dimensional quantum dot
with parabolic lateral confinement potential in a perpen-
dicular magnetic field Bwhich points along zdirection.
Then the electron system can be described by the Hamil-
tonian [18],
Hs=(p+e
cA)2
2m∗+m∗
2ω2
0(x2+y2)+1
2gµBBσz,(1)
wherepis the linear momentum operatorof the electron,
A(r) =B
2(−y,x,0)isthevectorpotentialinthesymmet-2
ric gauge,ω0is the characteristic confinement frequency,
andσ= (σx,σy,σz) is the vector Pauli matrices. m∗
is the effective mass of the electron and gits gyromag-
netic factor. µBis the Bohr magneton. In the second
quantized notation, Eq.(1) becomes
Hs= (a+
xax+a+
yay+1)/planckover2pi1/tildewideω+/planckover2pi1ωc
2i(a+
xay−axa+
y)
+1
2gµBBσz,(2)
whereωc=eB
m∗cand/tildewideω2=ω2
0+ω2
c
4. If we set
a+=1√
2(ax−iay),a−=1√
2(ax+iay),(3)
Then, the Hamiltonian (2) can be written as
Hs=n+/planckover2pi1ω++n−/planckover2pi1ω−+1
2gµBBσz, (4)
whereω±=/tildewideω±ωc/2,n+=a+
+a+andn−=a+
−a−. In
what follows we include the spin-orbit interaction which
is described as Rashba Hamiltonian in this system [11]
Hso=−α
/planckover2pi1[(p+e
cA)×σ]z, (5)
whereαis the spin-orbit coupling constant which can be
controlledbygatevoltageinexperiment. Onsubstituting
Eq.(3) into Eq.(5) and then
Hso=α
/tildewidel[γ+(σ+a++σ−a+
+)−γ−(σ−a−+σ+a+
−)],(6)
whereγ±= 1±1
2(/tildewidel/lB)2,/tildewidel= (/planckover2pi1/m∗/tildewideω)1
2andlB=
(/planckover2pi1/m∗ωc)1
2.
The Hamiltonians of photons and the coupling to the
electron spin can be written as follows:
Hp=/planckover2pi1ωpb+b, (7)
Hp−s=gc(σ++σ−)(b†+b), (8)
whereb+(b) andωpare the creation (annihilation) op-
erator and energy of the photons, respectively. gcis the
spin-photon coupling constant. Hence we obtain the to-
tal Hamiltonian of the electron and photons:
H=Hs+Hso+Hp+Hp−s
=/planckover2pi1ω+a+
+a++/planckover2pi1ω−a+
−a−+1
2gµBBσz
+α
/tildewidel[γ+(σ+a++σ−a+
+)−γ−(σ−a−+σ+a+
−)]
+/planckover2pi1ωpb+b+gc(σ++σ−)(b†+b).(9)
Performing a unitary rotation of the spin such that σz→
−σzandσ±→ −σ∓, we arrive at the Hamiltonian
H=/planckover2pi1ω+a+
+a++/planckover2pi1ω−a+
−a−−1
2gµBBσz
+α
/tildewidel[γ−(σ+a−+σ−a+
−)−γ+(σ−a++σ+a+
+)]
+/planckover2pi1ωpb+b−gc(σ++σ−)(b†+b).(10)We now derive an approximation form of this Hamilto-
nian by borrowing the observation from quantum optics
that the terms preceded by γ+in Eq.(10) are counterro-
tating, and thus negligible under the rotating-wave ap-
proximation when the spin-orbit coupling is small com-
pared to the confinement [18]. The last term in Eq.(10)
treats in the conventional rotaing-waveapproximation of
quantum optics.
H=/planckover2pi1ω+a+
+a++/planckover2pi1ω−a+
−a−+1
2|g|µBBσz
+λ(σ+a−+σ−a+
−)+/planckover2pi1ωpb+b−gc(σ+b+σ−b+),(11)
whereλ=αγ−//tildewidel. Sincegis negative in InGaAs, we
choose the absolute value |g|ofg. It is obvious that the
ω+mode is decoupled from the rest of the system, giving
H=/planckover2pi1ω+n++HJCwhere
HJC=/planckover2pi1ω−a+
−a−+1
2|g|µBBσz+/planckover2pi1ωpb+b
+λ(σ+a−+σ−a+
−)−gc(σ+b+σ−b+).(12)
This is the well-known two mode Jaynes-Cummings
model of quantum optics. In general this Hamiltonian
can not be solved exactly except ωp=ω−. In what fol-
lows, for the sake of analytical simplicity, we consider
ωp=ω−which we can use a frequency-controllable laser
and a special circuit to satisfy this condition in real ex-
periments.
In order to solve the above Hamiltonian, we define the
normal-mode operators:
A=e1a−+e2b, (13)
K=e2a−−e1b, (14)
where
e1=λ/radicalbig
λ2+g2c,e2=−gc/radicalbig
λ2+g2c,(15)
withe2
1+e2
2= 1. The new operators satisfy the commu-
tation relations[19]
[A,A†] = 1,[NA,A] =−A,[NA,A†] =A†,
[K,K†] = 1,[NK,K] =−K,[NK,K†] =K†,
[A,K] = 0,[A,K†] = 0,[NA,NK] = 0,(16)
whereNA=A†A(NK=K†K) is the number opera-
tor related to the normal-mode operator A(K). Intro-
ducing the number-sum operator S=NA+NKand
the number-difference D=NA−NK, we can verify
that the Hamiltonian (12) transforms into the follow-
ing Hamiltonian:(i) S=na+nbis a conserved quan-
tity (na=a†
−a−andnb=b†b); (ii) the operator D can
be written in terms of the generators {Q+,Q−,Q}of the
SU(2) Lie algebra,
D= 2(e2
1−e2
2)Q0+2e1e2(Q++Q−),(17)3
whereQ−=a−b†,Q+=a†
−b, andQ0=1
2(a†
−a−−b†b),
with [Q−,Q+] =−2Q0and [Q0,Q±] =±Q±;(iii) the
commutation relation between the operators S and D is
null, i.e., [S,D] = 0; and consequently, (iv) the Hamilto-
nianHJCsimplifies to HJC=H0+V, where
H0=/planckover2pi1ωp(S+1
2σz),
V=1
2δσz+λA(σ−A++σ+A),(18)
with [H0,V] = 0.λA=/radicalbig
λ2+g2cis an effective coupling
constantand δ=ωp−|g|µBB//planckover2pi1. The aboveHamiltonian
can be solved exactly. The eigenstates of this Hamilto-
nian are given by
|Ψ(n,±)/an}bracketri}ht=cosθ(n,±)|n,↑/an}bracketri}ht+sinθ(n,±)|n+1,↓/an}bracketri}ht,(19)
tanθ(n,±)= (δ±∆n)/2λA/radicalbig
(n+1),(20)
where ∆ n=/radicalbig
δ2+4λ2
A(n+1) and | ↑>(| ↓>) is the
spin-up (down) state.
According to Ref.[4], since only the quasi-mode Ais
coupled with the spin of the electron, so the phase shift
operatorU(ϕ) =e−iϕA†Ais introduced. Applied adi-
abatically to the Hamiltonian (18), the phase shift op-
erator alters the state of the field and gives rise to the
following eigenstates:
|ψ(n,±)>=e−inϕcosθ(n,±)|n,↑>+
e−i(n+1)ϕsinθ(n,±)|n+1,↓>.(21)
Changingϕslowly from 0 to 2 π, the Berry phase is cal-
culated as Γ l=i/integraltext2π
0l/an}bracketle{tψ|∂
∂ϕ|ψ/an}bracketri}htldϕwhich is given by
Γl= 2π[sinθ(n,l)]2. (22)
This Berry phase is composed of two parts. One is in-
duced by spin-orbit interaction, the other is induced by
quantized light. Therefore if we can measure the total
Berry phase and either part of two Berry phase, we will
measure the other part of Berry phase.
III. NUMERICAL RESULTS
For the illustration of the numerical results, we choose
the typical parametersof the InGaAs: g=−4,m∗/me=
0.05 (meis the mass of free electron). The dot size is
defined by l0=/radicalbig
/planckover2pi1/m∗ω0. Figure 1 depicts the Berry
phases Γ +as a function of the magnetic field strength
for three spin-orbit couplings. In Figure 1, we can find
that all the Berry phases change almost from 0 to 2 πas
the magnetic field strength varies from 20 mTto 50mT.
When other parametersare fixed, the spin-orbit coupling
constantchangesas α= 0.4×10−12eVm, 0.8×10−12eVm
and 1.2×10−12eVm, the Berry phases Γ +will have a
FIG. 1: The Berry phase Γ +as a function of magnetic field
strength Bwith three spin-orbit coupling constants ( α=
0.4×10−12eVm, 0.8×10−12eVmand 1.2×10−12eVm). The
other parameters used are g=−4,m∗/me= 0.05,gc=
0.01meV,l0= 80nm, andn= 0.
FIG. 2: The Berry phases of Γ +and Γ −as a function of
magnetic field strength B. The parameters used are α=
0.4×10−12eVm,g=−4,m∗/me= 0.05,gc= 0.01meV,
l0= 80nm, andn= 0.
slight movement in the figure. When B <20mTand
B >50mT, the Berry phase changes gradually, while
when 20mT <B < 50mT, the Berry phase changes dra-
matically. As the coupling constant increases, the Berry
phase changes from sharply to slowly. The Sh¨ ordinger
equationhastwodifferenteigenenergieswhen n= 0. The
two eigenenergies will give two different Berry phases.
Figure 2 illustrates these two Berry phases. In Figure
2, when the others parameter are fixed, one of the Berry
phasechangesfrom0to2 π, whiletheotherchangesfrom
2πto 0 as the magnetic field strength varies from 20 mT
to 50mT. Two Berry phases have an intersecting point
at approximatively B= 33mT, which is corresponding
to the resonant point.4
FIG. 3: The Berry phases Γ +as a function of Bwith three
three different dot sizes ( l0= 70nm,80nm,90nm). The pa-
rameters used are α= 0.4×10−12eVm,g=−4,m/me=
0.05,gc= 0.01meV, andn= 0.
FIG. 4: The Berry phases Γ +as a function of B
with three different light coupling constants ( gc=
0.01meV,0.02meV,0.03meV). The parameters used are α=
0.4×10−12eVm,g=−4,m/me= 0.05,l0= 80nm, and
n= 0.
Figure 3 shows the effect of the dot size on the Berry
phase. When dot size becomes large from 70nm to 90nm,
although all three Berry phases change from 0 to 2 π, the
threshold points of the magnetic field have a large move-
ment. When the dot size is 70nm, the Berry phase will
change dramatically at approximately 40mT, while the
dot sizes are 80nm and 90nm, the turning points are ap-
proximately at 30mT and 20mT, respectively. This im-
plies that the bigger the dot, the smaller the threshold of
the magnetic field strength. Figure 4 illustrates the in-
fluence of spin-photon coupling constant on Berry phase.
As the coupling constant becomes large, the Berry phase
becomes less drastic as shown in Figure 4.
In a recent paper, Giuliano et al.[20] have designed an
experimental arrangement, which is capacitively coupledthe dot to one arm of a double-path electron interferome-
ter. The phase carried by the transported electrons may
be influenced by the dot. The dot’s phase gives raise to
an interference term in the total conductance across the
ring. More recently, Leek et al.[10] have measured Berry
phase in a Ramsey fringe interference experiment. Our
experimentalsetupproposedhereisanalogouswith these
two arrangements as shown in Figure 5. A beam light is
split into two beams, one of the beams passes through
the dot, and interferes with the other one. Accurate con-
trol of the light field for dot is achieved through phase
and amplitude modulation of laser radiation coupled to
the dot. We choose a special designed electric circuit
to ensure the magnetic and laser vary synchronistically.
Through detecting the interfered light, we can measure
the Berry phase.
FIG. 5: A sketch of a possible experimental setup to detect
the Berry phase. a, b and c are three mirror, d is a beam
splitter.
IV. CONCLUSIONS
In conclusion, we have theoretically investigated the
Berry phase in a single quantum dot in the presence of
Rashba spin-orbit interaction. Berry phases as functions
of magnetic field strength, dot size, spin-orbit coupling
and photon-spin coupling constants are evaluated. It
is shown that for a given quantum dot, the spin-orbit
coupling constant and photon-spin coupling constant the
Berry phase will alter dramatically from 0 to 2 πas the
magnetic field strength increases. The threshold of mag-
neticfieldisdependent onthe Rashbaspin-orbitcoupling
constant, spin-photon coupling constantand the dot size.
Wealsoproposeapracticablemethod todetectthe Berry
phase in such a quantum dot system. Finally, we hope
that our predictions in the present work can be testified
by experiments in the near future.5
Acknowledgments
This work has been supported in part by National
Natural Science Foundation of China (No.10774101) andthe National MinistryofEducationProgramforTraining
PhD.
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2307.12177v2.Spin_orbit_coupling_induced_phase_separation_in_trapped_Bose_gases.pdf | Spin-orbit-coupling-induced phase separation in trapped Bose gases
Zhiqian Gui,1Zhenming Zhang,2Jin Su,3Hao Lyu,4and Yongping Zhang1,∗
1Department of Physics, Shanghai University, Shanghai 200444, China
2CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
3Department of Basic Medicine, Changzhi Medical College, Changzhi 046000, China
4Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
In a trapped spin-1 /2 Bose-Einstein condensate with miscible interactions, a two-dimensional spin-
orbit coupling can introduce an unconventional spatial separation between the two components. We
reveal the physical mechanism of such a spin-orbit-coupling-induced phase separation. Detailed
features of the phase separation are identified in a trapped Bose-Einstein condensate. We further
analyze differences of phase separation in Rashba and anisotropic spin-orbit-coupled Bose gases. An
adiabatic splitting dynamics is proposed as an application of the phase separation.
I. INTRODUCTION
Phase separation is a generic phenomenon from clas-
sical physics to quantum physics, for example, the
oil-water separation and spin Hall effect [1]. Two-
component atomic Bose-Einstein condensates (BECs)
provide a tunable platform for the investigations of phase
separation [2–8]. The two components can be real-
ized by using different atomic species or same species
with different electric hyperfine states. Such a sys-
tem features intra- and inter-component interactions.
When the inter-component interactions dominate over
the intra-component interactions, two components pre-
fer to be phase-separated in order to minimize the inter-
component interactions [9, 10]. The interactions for
phase separation are called immiscible. The immiscibility
of two-component BECs are completely tunable in exper-
iments. Phase separation effect induces rich physics in
quantum gases, such as the formation of vector solitons
and vortex-soliton structures, coherent spin dynamics,
and pattern formations [11–19].
In a two-component BEC, an artificial spin-orbit cou-
pling can be synthesized between different hyperfine
states via Raman lasers [20–23]. Such a Raman-induced
spin-orbit coupling is one dimensional. Rashba spin-
orbit coupling, which is two dimensional, has also been
experimentally realized in BECs [24, 25]. The imple-
mentation of spin-orbit coupling in BECs gives rise to
exotic quantum phases and rich superfluid properties,
which opens an avenue for simulating topological mat-
ters and exploring superfluid dynamics [25–32]. In a
Raman-type spin-orbit-coupled BEC, a stripe phase [33]
can exist in miscible interactions [26, 28]. In contrast,
for a Rashba spin-orbit-coupled BEC, the stripe phase
may appear in the immiscible regime [34]. Very inter-
estingly, Refs. [35, 36] have numerically found a spatially
phase-separated ground state in a Rashba-coupled and
harmonically trapped BEC with miscible interactions.
Such a ground state in the miscible regime is unexpected
∗yongping11@t.shu.edu.cnfor a usual two-component BEC without spin-orbit cou-
pling. Reference [35] identifies that the exotic phase sep-
aration satisfies a combined symmetry of parity and a
spin flip. The existence of this state is attributed by
Refs. [36, 37] to a spin-dependent force. The force is in-
trinsic in the presence of Rashba spin-orbit coupling and
drives the two components moving in opposite directions.
The force concept provides an intuitive picture for the un-
expected phase separation. However, its weakness is ob-
vious. The force is proportional to the square of Rashba
spin-orbit coupling strength. Therefore, a large strength
is expected to generate a larger spatial separation. In
contrast, numerical results show that the separation de-
creases with an increasing strength [36, 37]. So far, the
physical origin of the unconventional phase separation in
the miscible regime is yet to be addressed. It calls for an
unambiguous interpretation, since the phase separation
has already found a broad application in other excited
states. In Refs. [38, 39], a spin-orbit-coupled bright soli-
ton is found to be spatially separated in center-of-mass
between the two components. Dynamics of the separa-
tion in bright solitons is analyzed by varying spin-orbit
coupling strength [40]. A spin-orbit-coupled single-vortex
state, in which each component carries a singly quan-
tized vortex, shows spatial separation between two com-
ponents, and the separation is inversely proportional to
spin-orbit coupling strength [41]. Recently, dynamics of
the separation is triggered by a sudden quench of spin-
orbit coupling strength in a trapped BEC [42]. All men-
tioned separations occur in the miscible regime, causing
a counterintuitive expectation.
In this paper, we provide the physical mechanism for
the unconventionally spin-orbit-coupling-induced phase
separation. Eigenstates of a two-dimensional spin-orbit
coupling have a momentum-dependent relative phase
φ(⃗k) between the two components. Closely around a
fixed momentum ⃗k0, the relative phase may present a
linear dependence φ(⃗k)∝(⃗k−⃗k0)·⃗ r0with a constant
⃗ r0. The linear dependence is a momentum kick to move
two components relatively. The superposition of these
eigenstates distributing around ⃗k0constitutes a spatially
separated wave packet. The separation, whose amplitude
can be calculated, is a completely single-particle effect ofarXiv:2307.12177v2 [cond-mat.quant-gas] 12 Oct 20232
spin-orbit coupling. A weakly trapped BEC with two-
dimensional spin-orbit coupling is a perfect platform to
simulate the phase separation. The miscible interactions
force atoms to condense at a certain spin-orbit-coupled
momentum state with a momentum-dependent relative
phase. Meanwhile, weak traps broaden the condensed
momentum so that the condensation occupies momen-
tum states in a narrow regime, which give rise to the
linear dependence of the relative phase. We numerically
identify detailed features of spin-orbit-coupling-induced
phase separation in a trapped BEC with miscible inter-
actions by analyzing its ground states. The phase sepa-
ration matches with the single-particle prediction when
spin-orbit coupling strength dominates. We also com-
pare the separation differences between Rashba and an
anisotropic spin-orbit coupling. Finally, as an applica-
tion of the phase separation, we propose an adiabatic
splitting dynamics.
The paper is organized as follows. In Sec. II, the phys-
ical mechanism of spin-orbit-coupling-induced phase sep-
aration is unveiled. The separation amplitudes are pre-
dicted. From the mechanism, we know that the sepa-
ration is a single-particle effect. In Sec. III, we identify
separated features of ground states in a trapped BEC
with Rashba spin-orbit coupling by the imaginary-time
evolution method and the variational method. In Sec. IV,
we reveal the effect of the anisotropy of spin-orbit cou-
pling on the phase separation. In Sec. V, we propose an
adiabatic dynamics to dynamically split two components
basing on the phase separation. For the completeness
of our discussion, immiscible-interaction-induced phase
separation is shown in Sec. VI. The conclusion follows in
Sec. VII.
II. SPIN-ORBIT-COUPLING-INDUCED PHASE
SEPARATION
Rashba spin-orbit-coupling-induced phase separation
is a completely single-particle effect. We reveal the phys-
ical origin of such phase separation. The Rashba spin-
orbit-coupled Hamiltonian is
HSOC=p2
x+p2
y
2+λ(pxσy−pyσx), (1)
where pxandpyare the momenta along the xandydirec-
tions respectively, λis the spin-orbit coupling strength,
andσx,yare spin-1/2 Pauli matrices. The eigenenergy of
the Hamiltonian has two bands. The lower band is
E=k2
x+k2
y
2−λq
k2x+k2y, (2)
with associated eigenstates being
Φ =1√
2eikxx+ikyy
eiφ
2
e−iφ
2
. (3)
Since the Hamiltonian possesses continuously transla-
tional symmetry, the eigenstates are plane waves withkx,ybeing the quasimomenta along the xand ydi-
rections, respectively. The outstanding feature is that
Rashba spin-orbit coupling generates a relative phase φ
between the two components, which satisfies
tan(φ) =kx
ky. (4)
It is noted that ( kx, ky) = (0 ,0) is a singularity, closely
around which the relative phase cannot be defined.
Therefore, the eigenstate in Eq. (3) works beyond the
regime around the singularity.
We construct a wave packet by superposing these
eigenstates,
Ψ =Z∞
−∞dkxdkyG(k−¯k)Φ, (5)
with the superposition coefficient Gbeing a momentum-
dependent localized function centering around ¯k. For a
straightforward illustration, we take a Gaussian distribu-
tion as an example,
G(k−¯k) =1
2πp
∆x∆ye−(kx−¯kx)2
2∆x−(ky−¯ky)2
2∆y.(6)
The Gaussian distributed superposition coefficient is cen-
tered at ¯k= (¯kx,¯ky) with the packet widthsp
∆x,yalong
xandydirections. If the widths are narrow, the superpo-
sition mainly happens around ¯k. Therefore, we analyze
the eigenstates around ¯k, and the relative phase becomes
φ(k)≈φ(¯k) + (kx−¯kx)∂φ
∂kx ¯k+ (ky−¯ky)∂φ
∂ky ¯k,(7)
which is linearly dependent on the momenta kx,y. This
is true since expanding any continuous function around
a certain parameter point leads to dominant linear-
dependence. Such linear dependence in Eq. (7) induced a
momentum kick, generating the relative motion between
the two components. After substituting the Gaussian
distribution in Eq. (6) and φin Eq. (7) into Eq. (5) and
performing integration, we get the wave packet,
Ψ =1√
2ei¯kxx+i¯kyy
×
e−∆x
2h
x+1
2∂φ(¯k)
∂kxi2−∆y
2h
y+1
2∂φ(¯k)
∂kyi2+iφ(¯k)
2
e−∆x
2h
x−1
2∂φ(¯k)
∂kxi2−∆y
2h
y−1
2∂φ(¯k)
∂kyi2−iφ(¯k)
2
.(8)
The outstanding feature of the resultant wave packet is
that the two components have a relative position dis-
placement. The displacements along the xandydirec-
tions are
∂φ(k)
∂kx ¯k=¯ky
¯k2x+¯k2y,∂φ(k)
∂ky ¯k=−¯kx
¯k2x+¯k2y. (9)
The nonzero displacements give rise to a phase separation
between two components. From the construction of the3
phase-separated wave packets, we can see that the origin
of the phase separation is the existence of the momentum-
dependent relative phase in eigenstates and the occupa-
tion of these eigenstates confined in a narrow momentum
regime.
Rashba spin-orbit coupled BEC is an ideal platform to
generate such a phase-separated state. The lower band
in Eq. (2) has infinite energy minima which locate at
the quasimomenta satisfying k2
x+k2
y=λ2; therefore,
kx=λcos(θ) and ky=λsin(θ) with θbeing an angle.
The interacting atoms spontaneously choose one of en-
ergy minima to condense and form a BEC [33, 35, 36].
This means that θis spontaneously chosen to be a value
¯θ. In real atomic BEC experiments, traps are inevitable.
A weak harmonic trap naturally broadens the BEC mo-
mentum giving rise to a Gaussian distribution centered
at (¯kx,¯ky) =λ(cos( ¯θ),sin(¯θ)). Furthermore, the broad-
ening is narrow so that Eq. (7) is satisfied. Conse-
quently, the Rashba-coupled BEC presents as a phase
separated state in Eq. (8) with ∂φ(k)/∂kx|¯k= sin( ¯θ)/λ
and∂φ(k)|/∂ky|¯k=−cos(¯θ)/λ. The position displace-
ment is inversely proportional to the spin-orbit coupling
strength λ, which clearly indicates Rashba spin-orbit-
coupling-induced phase separation. When the strength
goes to zero ( λ≈0), the momentum ( ¯kx,¯ky)≈(0,0)
becomes a singularity so that the eigenstate in Eq. (3)
is not physical. Without the spin-orbit coupling, the
BEC becomes the conventional one, and there is no phase
separation between two components. If the strength
is enhanced gradually from zero, the position displace-
ments should continuously increase from zero to catch up
with the predicted value (sin( ¯θ)/λ,−cos(¯θ)/λ). When
the strength λis large enough, the position displace-
ments decrease towards zero again since they are in-
versely proportional to λ. In this case, the plane-wave
phase ( ¯kxx+¯kyy) dominates, while the relative phase
in the eigenstates [Eq. (3)] is independent of λ, i.e.,
tan(φ) = ¯kx/¯ky= cot( ¯θ). Consequently, the effect of
the relative phase is obliterated by the plane-wave phase,
and phase separation disappears.
According to the above mechanism of the phase sepa-
ration, if there is no weak trap to broaden the condensed
momentum, the spin-orbit-coupled BEC can not present
the position displacement. This is why a spatially ho-
mogeneous BEC with spin-orbit coupling does not show
phase separation as studied in most literature. Never-
theless, in order to broaden the condensed momentum
without traps, we may consider spatially localized exci-
tation states, such as bright solitons and vortices. These
spatially self-trapped states naturally broadens the con-
densed momentum. Therefore, the resultant phase sep-
aration between two components in Rashba spin-orbit-
coupled bright solitons and quantum vortices, which have
been numerically revealed in Refs. [38, 39, 41], can be
understood by a generalization of our mechanism. Inter-
estingly, the position displacement of quantum vortex is
inversely proportional to the spin-orbit coupling strength
as uncovered numerically in [41], can be explained unam-biguously.
We emphasize that the spin-orbit-coupling-induced
phase separation only works for a two-dimensional spin-
orbit coupling. For an one-dimensional spin-orbit cou-
pling, i.e., the Raman-induced one, the single-particle
Hamiltonian is H′=p2
x/2+λpxσz+Ωσxwith Ω being the
Rabi frequency due to Raman lasers [20, 23]. The lower
energy band of this system is E=k2
x/2−p
λ2k2x+ Ω2
with eigenstates being Φ = eikxx(−sin(Θ) ,cos(Θ))T.
Here, tan(Θ) = Ω /(λkx), and Tis the transpose oper-
ator. It is noted that there is no momentum-dependent
relative phase in the eigenstates. Therefore, the Raman-
induced spin-orbit coupling, in principle, can not gener-
ate the phase separation.
In above, we have revealed the physical mecha-
nism of Rashba spin-orbit-coupling-induced phase sep-
aration. We demonstrate that a weakly trapped spin-
orbit-coupled BEC satisfies requirements of the mech-
anism. Quantum phase in trapped spin-orbit-coupled
BECs may be phase separated states. The spin-orbit-
coupling-induced phase separation is a single-particle ef-
fect. The role of nonlinearity in the BEC is to sponta-
neously choose one energy minimum for condensation. In
the following, we study ground states of a trapped spin-
orbit-coupled BEC and identify features of spin-orbit-
coupling-induced phase separation.
III. RASHBA
SPIN-ORBIT-COUPLING-INDUCED PHASE
SEPARATION IN TRAPPED BECS
We consider a quasi-two-dimensional spin-1/2 BEC
with Rashba spin-orbit coupling. The trap frequency ωz
along the zdirection is assumed to be very large so that
the dynamics is completely frozen into the ground state of
thez-directional harmonic trap. Such the strong trap can
be implemented by an optical lattice in the zdirection
in experiments. After integrating the atomic state along
zdirection, we are left with a quasi-two-dimensional sys-
tem. Rashba spin-orbit coupling can be artificially im-
plemented by an optical Raman lattice [24], generating
the Hamiltonian HSOC shown in Eq. (1). The spin-
orbit-coupled BEC is described by the following Gross-
Pitaevskii (GP) equation,
i∂Ψ
∂t= (HSOC+V+Hint) Ψ. (10)
with Ψ = (Ψ 1,Ψ2)Tbeing the two-component wave
function. The harmonic trap in the x−yplane is
V=1
2ω2(x2+y2) with ωthe dimensionless trap fre-
quency. Hintdenotes nonlinear interactions,
Hint=
g|Ψ1|2+g12|Ψ2|20
0 g12|Ψ1|2+g|Ψ2|2
,(11)
The GP equation is dimensionless, and the units of
length, time, momentum and energy are chosen as lz=4
- 808y / lz( a )
- 202lzky/s47 /s1115( c )
- 8 0 8- 808
x / lz( b )
- 2 0 2- 202
lzkx/s47 /s1115( d )
FIG. 1. Ground state of a trapped Rashba spin-orbit-coupled
BEC with miscible interactions. (a) and (b) Density distri-
butions |Ψ1|2and|Ψ2|2in the coordinate space. (c) and (d)
Density distributions |Ψ1|2and|Ψ2|2in the momentum space.
The parameters are ω/ωz= 0.1,lzλ/ℏ= 0.2,g= 12 and
g12= 8.
p
ℏ/(mωz), 1/ωz,ℏ/lzandℏωz, respectively. With the
units, the inter-and intra-component interaction coeffi-
cients become g=Na√
8π/lzandg12=Na12√
8π/lz.
Here, Nis the atom number, and aand a12are
corresponding s-wave scattering lengths, respectively.
The wave functions satisfy the normalization condition,R
dxdy(|Ψ1|2+|Ψ2|2) = 1. In numerical calculations, ex-
perimentally accessible parameters are used. The typical
trap frequency is ωz= 2π×200 Hz, leading to the unites
of length and time lz= 0.76µmand 1 /ωz= 0.8msre-
spectively. a∼100a0with a0being the Bohr radius,
andN∼300, lead to g∼10. The spin-orbit coupling
strength can be changed by tuning parameters of Raman
lasers in experiments [24].
When g > g 12, the interactions are miscible. We
first study ground states of the system in this regime
by performing the imaginary-time evolution of the GP
equation. The evolution is numerically implemented
by the split-step Fourier method. The window of two-
dimensional space is chosen as ( x, y)∈[−6π,6π] and is
discretized into a 256 ×256 grid.
A typical result is shown in Fig. 1. As expected from
the prediction in the previous section, the ground state
is phase-separated. The two components are spatially
separated along the xdirection, as shown by Figs. 1(a)
and 1(b). The ground state spontaneously chooses ¯θ=
−π/2 so that the atoms condense at ( ¯kx,¯ky) = (0 ,−λ),
which can be seen from the momentum-space density dis-
tributions in Figs. 1(c) and 1(d). In this case, according
to Eq. (9), the position displacement occurs along the x
direction, and the first component shifts by 1 /(2λ) on the
right side and the second component shifts oppositely by
1/(2λ) on the left side.
In the presence of interactions, it is impossible to con-
struct analytical wave function of ground state from the
procedure demonstrated in the previous section. Never-
theless, the single-particle wave functions in Eq. (8) and
-101-
2-100
1 2 0.0580.0950
1 2 0.0560.063x/lz(a)
lz k'y//s295(b)Δ'x/lzl
z /s108/ /s295(c)Component 1C
omponent 2Δ
'y/lzl
z /s108/ /s295(d)FIG. 2. Rashba spin-orbit-coupling-induced phase sepa-
ration in a trapped BEC. The parameters are ω/ωz= 0.1,
g= 12 and g12= 8. (a) The center of mass for the two com-
ponents along the xdirection as a function of the spin-orbit
coupling strength λ. The solid lines are from the variational
method, and the circles are obtained by the imaginary-time
evolution of the GP equation. The dashed lines are ±1/(2λ)
predicted from the single-particle model. The red (blue) color
represents the first (second) component. (b) The condensed
momentum ¯k′
yas a function of λ. The red solid line and blue
circles are obtained by the imaginary-time evolution and the
variational method, respectively. The variational parameters
∆′
x,yare shown in (c) and (d).
phase-separated results shown in Fig. 1 stimulate us to
use a trial wave function to study the phase separation
by the variational method [37]. The trial wave function
is assumed to be
Ψ(x, y) =(∆′
x∆′
y)1
4
√
2πei¯k′
yy
e−∆′
x
2(x−δx)2−∆′
y
2y2
e−∆′
x
2(x+δx)2−∆′
y
2y2
.(12)
Here, we have assumed that the atoms spontaneously
condenses at (0 ,¯k′
y) in momentum space and therefore
the phase separation only happens along the xdirection
with the relative position displacement 2 δx. 1/p∆′x,y
characterize the widths of the wave packet along the x, y
directions. The unknown parameters ¯k′
y, δx,∆′
x,yare to
be determined by minimizing the energy functional,
E=Z
dxdyΨ∗(HSOC+V)Ψ
+Z
dxdyhg
2(|Ψ1|4+|Ψ2|4) +g12|Ψ1|2|Ψ2|2i
,
(13)
Substituting the trial wave function into the energy func-5
4
6
8
1
0
-
1
.
0
-
0
.
8
0
.
8
1
.
0
0
.
1
0
.
2
0
.
3
-
0
.
7
0
.
0
0
.
7
x
/
l
z
g
1
2
(
a
)
x
/
l
z
/s119
/
/s119
z
(
b
)
FIG. 3. The center of mass of two components for a non-
dominant lzλ/ℏ= 0.5. The solid lines are obtained by the
variational method and the circles are from the imaginary-
time evolution of the GP equation. The red and blue colors
represent the first and second components, respectively. (a)
The center-of-mass as a function of the inter-component in-
teraction coefficient g12.ω/ωz= 0.1 and g= 12. (b) The
center-of-mass as a function of the trap frequency ω.g= 12
andg12= 8.
tional Eleads to
E=¯k′2
y
2+∆′
x+ ∆′
y
4
1 +ω2
∆′x∆′y
+1
2ω2δ2
x
+λ(∆′
xδx−¯k′
y)e−∆′
xδ2
x+p∆′x∆′y
8π
g+g12e−2∆′
xδ2
x
.
(14)
By minimizing the energy functional with respect to the
unknown parameters, ∂E/∂X = 0 ( X=¯k′
y, δx,∆′
x,y), we
obtain all information of the trial wave function. The
phase separation can be characterized by the center of
mass of each component,
¯r1,2=Z
r|Ψ1,2(r)|2dr, (15)
withr= (x, y). In Fig. 2(a), the solid lines show
¯x1,2=±δxcalculated from the variational method, while
the results obtained by the imaginary-time evolution of
the GP equation are demonstrated by the circles. We
find that the results from the two calculation methods
agree very well. Without spin-orbit coupling ( λ= 0),
the conventional BEC has ¯ x1,2= 0 and condensates at
¯k′
y= 0, as shown in Fig. 2(b). With the growth of λ,
¯k′
yalways increases linearly [see Fig. 2(b)]. The displace-
ment ¯ xfirst increases drastically to a maximum value and
then declines to the predicted ±1/(2λ) obtained by the
single-particle model [see the dashed lines in Fig. 2(a)].
The dependence of the displacement on λexactly follows
the expectation in the previous section. In the dramatic
increase regime for ¯ x, the variational parameters ∆′
x,y
also change dramatically [see Figs. 2(c) and 2(d)].
Rashba spin-orbit coupling introduces an intrinsic
force,
F=dp
dt=−
[r, HSOC], HSOC
= 2λ2(p×ez)σz, (16)withezbeing the unit vector along the zdirection and
pthe atom momentum. The force originates from spin-
orbit-coupling-induced anomalous velocity [43–45]. Con-
sidering the ground states shown in Fig. 2, the force op-
erator in momentum space becomes Fx= 2λ2¯k′
yσzand
Fy= 0. The two components feel opposite force Fxalong
thexdirection. Ground states must compensate the in-
trinsic force to reach equilibrium. It can be implemented
by displacing two component opposite to the force. Since
¯k′
y<0 in the case shown in Fig. 2, the first component is
displaced towards to the right side and the second to the
left side. The force concept has been used in Refs. [36, 37]
to explain the phase separation. Since the force is pro-
portional to λ2, it seems that a large displacement would
be induced for a large λ. However, as shown in Fig. 2(a),
the dependence of the displacement on λdoes not fol-
low the force. We can see that the intrinsic force cannot
explain the phase separation in the large λregime.
Figure 2(a) shows that the separation follows the
single-particle prediction ±1/(2λ) when λdominates.
When λis weak, the displacement also depends on other
parameters, such as nonlinear coefficients and the har-
monic trap. In Fig. 3(a), we plot the displacement ¯ xas
a function of the inter-component interaction coefficient
g12for a non-dominant λ. The displacement slightly
rises with an increasing g12, and it reaches the maxi-
mum when g12=g. If g12> g, the interactions be-
come immiscible, leading to ground states different from
the trial wave function in Eq. (12). The dependence
of the displacement on the trap frequency is shown in
Fig. 3(b). We find that the displacement decreases as
the trap frequency increases. This is because the dis-
placement requires more kinetic energy in a tight trap.
It is notice that there is a slight mismatching between
the results from the variational method (the solid line)
and the imaginary-time evolution (the circles) in Fig. 3.
The origin of such the mismatching is that the Gaus-
sian profile in the trial wave function in Eq. (12) cannot
exactly describe the imaginary-time-evolution-generated
wave function as shown in Fig. 1.
IV. THE ANISOTROPIC
SPIN-ORBIT-COUPLING-INDUCED PHASE
SEPARATION IN TRAPPED BECS
Rashba-spin-orbit-coupling-induced phase separation
has been analyzed in the previous section. In the two-
dimensional spin-orbit-coupled BEC experiment [24], the
spin-orbit coupling strengths are tunable, which leads
to an anisotropic coupling. It has been revealed that
the anisotropic spin-orbit coupling has a great impact
on ground states of a spatially homogeneous BEC [46].
In this section, we study anisotropic-spin-orbit-coupling-
induced phase separation. The single-particle Hamilto-
nian of the anisotropic spin-orbit coupling is
H′
SOC=p2
x+p2
y
2+λ1pxσy−λ2pyσx, (17)6
FIG. 4. The anisotropic spin-orbit-coupling-induced phase separation in a trapped BEC. The parameters are ω/ωz= 0.1,
g= 12, and g12= 8. (a1) The lower band of H′
SOCin Eq. (17) with lzλ1/ℏ= 0.3 and lzλ2/ℏ= 0.6. (a2) and (a4) show
corresponding ground-state density distributions of the first component |Ψ1|2in coordinate and momentum spaces, respectively.
(a3) and (a5) are for the second component |Ψ2|2. (b1)-(b5) are same as (a1)-(a5) but with lzλ1/ℏ= 0.6 and lzλ2/ℏ= 0.3. In
(a2), (a3), (b2), and (b3), white stars represent the center of wave packets predicted by the single-particle model.
with the anisotropic strengths λ1̸=λ2. The lower band
ofH′
SOCis
E=k2
x+k2
y
2−q
λ2
1k2x+λ2
2k2y. (18)
with the associated eigenstates being the same as Eq. (3)
but having the different relative phase which can be writ-
ten as
tan(φ) =λ1kx
λ2ky. (19)
According to the mechanism of the spin-orbit-coupling-
induced phase separation, the anisotropic coupling can
generate position displacements related to the derivatives
of the relative phase. The displacements along the xand
ydirections are
∂φ(k)
∂kx ¯k=λ1λ2¯ky
λ2
1¯k2x+λ2
2¯k2y,
∂φ(k)
∂ky ¯k=−λ1λ2¯kx
λ2
1¯k2x+λ2
2¯k2y. (20)Here, ¯k= (¯kx,¯ky) is the momentum at which the atoms
condense. The lowest energy minima of the lower band
depend on the anisotropy. When λ1< λ 2, the two
minima locate at ( ¯kx,¯ky) = (0 ,±λ2) [see Fig. 4(a1)].
They locate at ( ¯kx,¯ky) = (±λ1,0) when λ1> λ 2[see
Fig. 4(b1)]. With the miscible interactions, the BEC
spontaneously chooses one of these two minima to con-
dense. The ground state that spontaneously condenses
at (¯kx,¯ky) = (0 ,−λ2) for λ1< λ 2is demonstrated
in Figs. 4(a2)-(a5). We obtain ground states by the
imaginary-time evolution of the GP equation with H′
SOC.
From the single-particle prediction in Eq. (20), the phase
separation of this ground state happens only along the
xdirection, and the center-of-mass of the first compo-
nent is λ1/(2λ2
2) and that of the second component is
−λ1/(2λ2
2) [see the white stars in Figs. 4(a2) and 4(a3)].
Density distributions shown in Figs. 4(a2) and 4(a3)
clearly indicate the phase separation following the pre-
dictions. The ground state that spontaneously condenses
at (¯kx,¯ky) = ( −λ1,0) for λ1> λ 2is demonstrated
in Figs. 4(b2)-(b5). The single-particle mechanism in7
0.00 .51 .01 .52 .0-0.40.00.4l
z /s1082//s295x/lz-
0.40.00.4y
/lz
FIG. 5. Anisotropic-spin-orbit-coupling-induced phase sepa-
ration as a function of λ2with a fixed lzλ1/ℏ= 1. Circles are
for the first component and crosses are for the second compo-
nent. The blue (red) color represents separation along the x
(y) direction. Other parameters are ω/ωz= 0.1,g= 12 and
g12= 8.
Eq. (20) predicts that for this ground state the separa-
tion happens along the ydirection and the center-of-mass
are∓λ2/(2λ2
1) for two components [see the white stars in
Figs. 4(b2) and 4(b3)]. The results from the imaginary-
time evolution shown in Figs. 4(b2) and 4(b3) match with
the single-particle predictions.
These analyses have shown that the center of mass
of each component strongly depends on the ratio of
the spin-orbit coupling strengths. To reveal the depen-
dence of phase separation on λ2/λ1, we calculate ground
states with a fixed λ1and a changeable λ2by using the
imaginary-time evolution. The results are summarized in
Fig. 5, where the circles (crosses) represent the center of
mass for the first (second) component. For λ2< λ1= 1,
the phase separation occurs along the ydirection and
|¯y|increases with the increase of λ2[see red circles and
crosses in Fig. 5], while ¯ xis zero [see blue circles and
crosses in Fig. 5]. When λ2= 0, the spin-orbit coupling
becomes one-dimensional, there is no phase separation
due to the absence of the relative phase. The results
change for λ2> λ 1= 1 and the phase separation along
thexdirection is observed. In this case, the separation
decreases with λ2increasing. For a very large λ2, the
separation disappears since the spin-orbit coupling effec-
tively turns to be one-dimensional. The results in Fig. 5
demonstrate that the maximum separation happens for
λ1=λ2which is Rashba spin-orbit coupling. This is also
expected from the single-particle prediction in Eq. (20).
V. ADIABATIC SPLITTING DYNAMICS
We have shown that ground states of a trapped BEC
with two-dimensional spin-orbit coupling and miscible in-teractions are phase-separated. As an important appli-
cation, we study adiabatic dynamics of the phase separa-
tion. As pointed out by previous works, a linear coupling
between two component favors miscibility regardless of
interactions [47, 48]. Therefore, a miscible-to-immiscible
transition may occur by decreasing the coupling. The
adiabatic dynamics is stimulated by slowly switching off
the linear coupling. Theoretically, the process is de-
scribed by the time-dependent GP equation,
i∂Ψ
∂t= [HSOC+ Ω(t)σx+V+Hint] Ψ. (21)
Here, Ω( t)σxrepresents the linear coupling between the
two components, and can be experimentally achieved by
using a radio-frequency coupling [6]. The time-dependent
Rabi frequency is
Ω(t) = Ω 0(1−t/τq), (22)
with Ω 0being the initial value of the linear coupling and
τqis the quench duration. At t= 0, the presence of Ω 0
greatly suppresses the ground-state phase separation. We
obtain ground state by the imaginary-time evolution of
Eq. (21) with Ω( t) = Ω 0. A typical ground state is shown
in insets (a) and (b) of Fig. 6, and the separation be-
tween two components is not obvious. Using this ground
state as initial state, we evolve the time-dependent GP
equation. The center-of-mass ¯ xfor two components is
recorded during the time evolution in Fig. 6. By de-
creasing the linear coupling adiabatically, the separation
between two component gradually increases. When it
is completely switched off, i.e., t=τq, the separation
04 08 01 20-0.6-0.30.00.30.6x/lz/s119
z t
-808-808x
/lzy/lz
-808-808x
/lzy/lz
-808-808x
/lzy/lz
-808-808x
/lzy/lz(
a
)
(
b
)
(
c
)
(
d
)
FIG. 6. Adiabatic splitting dynamics of a trapped BEC with
Rashba spin-orbit coupling by slowly switching off the linear
coupling. The parameters are ω/ωz= 0.1,g= 12, g12= 8,
Ω0/ωz= 3, and ωzτq= 150. The red (blue) dots represent
the center-of-mass of the first (second) component. Insets
(a,b) [(c,d)] are density distributions of the first and second
components at t= 0 [t=τq], respectively.8
FIG. 7. Immiscible-interaction-induced phase-separated ground states in a trapped BEC with Rashba spin-orbit coupling.
The parameters are g= 4, g12= 8 and ω/ωz= 0.1. (a1)-(a4) When lzλ/ℏ= 0.5 the ground state is a half-quantum vortex
state. (a1) [a(3)] and (a2) [(a4)] are the coordinate (momentum) space density distributions of the first and second components
respectively. (b1)-(b4) When lzλ/ℏ= 1.5 the ground state is a stripe state. (b1) [b(3)] and (b2) [(b4)] are the coordinate
[momentum] space density distributions of the first and second components respectively.
is maximized [see corresponding density distributions in
insets (c) and (d)]. The two components can realize a
dynamically spatial splitting, which move along opposite
directions. Such adiabatic splitting dynamics are remi-
niscent of a kind of “atomic spin Hall effect” [49].
VI. IMMISCIBLE INTERACTIONS INDUCED
PHASE SEPARATION
In all above, the interactions are miscible ( g > g 12),
which support atoms to condense at a particular momen-
tum state. On the other hand, immiscible interactions
(g < g 12) prefer a spatial separation between two com-
ponents in order to minimize the inter-component inter-
actions proportional to g12. In the presence of spin-orbit
coupling, the immiscible-interaction-induced phase sepa-
ration presents interesting features [35, 36, 50]. In Fig. 7,
we show two different kinds of immiscible-interaction-
induced phase separated ground states with different val-
ues of spin-orbit coupling strength λ. For λ= 0.5, the
ground state obtained by the imaginary-time evolution
is a half-quantum vortex state which has been first re-
vealed in Refs. [35, 50]. The first component distribution
has a Gaussian shape [see Figs. 7(a1) and 7(a3)], and
the second component is a vortex with a winding num-
berw= 1 [see Figs. 7(a2) and 7(a4)]. The first compo-
nent is filled in the density dip of the second one, form-
ing a spatial separation along the radial direction. For
λ= 1.5, the ground state becomes a stripe state which
has been first revealed in Refs. [33, 34]. The ground state
condenses simultaneously at two different momenta [see
Figs. 7(b3) and 7(b4)]. Such momentum occupation gen-
erates spatially periodic modulations in density distribu-
tions [Figs. 7(b1) and 7(b2)]. Meanwhile, stripes of the
two components are spatially separated.We emphasize that phase separations induced by spin-
orbit coupling and immiscible interaction have different
physical origins. The spin-orbit-coupling-induced phase
separation only works for a two-dimensional spin-orbit
coupling. However, phase separations have been also
studied for a BEC with a one-dimensional spin-orbit cou-
pling, the mechanism of which is different. In the pi-
oneered spin-orbit-coupled experiment, experimentalists
observed a spatial separation between two dressed states
with a Raman-induced spin-orbit coupling [20]. The
spin-orbit coupling generates two energy minima, whose
occupations can be considered as two dressed states. In
the dressed state space, atomic interactions turn to be
immiscible between two dressed states in the presence
of the Rabi frequency. The phase separation happens
in the dressed state space due to immiscibility. In ad-
dition, Ref. [47] reveals the existence of phase separa-
tion in a spin-1 BEC with the Raman-induced spin-orbit
coupling. The single-particle Hamiltonian of the system
isH= (px+λ′Fz)2/2 + Ω′Fx+ϵF2
z. Here, Fx,y,z are
the spin-1 Pauli matrices, λ′is the spin-orbit coupling
strength, Ω′is the Rabi frequency, and ϵis the quadratic
Zeeman shift. The spinor interactions include density-
density part with the coefficient c0and spin-spin part
with the coefficient c2. In particular, a very negative
quadratic Zeeman shift ϵ=−λ2/2 was considered. With
such a large negative ϵ, the occupation in the second
component can be eliminated. The spinor only occu-
pies the first and third components. Interestingly, the
spinor interactions between the first and third compo-
nents are immiscible for a negative spin-spin interaction
(c2<0). Different phase-separated states between the
first and third component are due to immiscible interac-
tions [47] .9
VII. CONCLUSION
In summary, we have revealed the physical mechanism
of spin-orbit-coupling-induced phase separation. The
mechanism, which is very different from the conventional
immiscible-interaction-induced separation, is a complete
single-particle effect of spin-orbit coupling. We have ana-
lyzed separation features in a trapped BEC with Rashba
spin-orbit coupling and miscible interactions and studied
the effects of the anisotropy of spin-orbit coupling on theseparation. All features can be explained by the single-
particle mechanism. As an interesting application of the
phase separation, we propose an adiabatic dynamics that
can dynamically split two components spatially.
ACKNOWLEDGMENTS
This work was supported by National Natural Sci-
ence Foundation of China with Grants No.12374247 and
11974235. H.L. acknowledges support from Okinawa In-
stitute of Science and Technology Graduate University.
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1601.06935v2.Double_Quantum_Spin_Vortices_in_SU_3__Spin_Orbit_Coupled_Bose_Gases.pdf | arXiv:1601.06935v2 [cond-mat.quant-gas] 24 Sep 2016Double-quantum spin vortices in SU(3) spin-orbit coupled B ose gases
Wei Han,1,2Xiao-Fei Zhang,1Shu-Wei Song,3Hiroki Saito,4Wei Zhang∗,5Wu-Ming Liu†,2and Shou-Gang Zhang‡1
1Key Laboratory of Time and Frequency Primary Standards,
National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China
2Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
3State Key Laboratory Breeding Base of Dielectrics Engineer ing,
Harbin University of Science and Technology, Harbin 150080 , China
4Department of Engineering Science, University of Electro- Communications, Tokyo 182-8585, Japan
5Department of Physics, Renmin University of China, Beijing 100872, China
We show that double-quantum spin vortices, which are charac terized by doubly quantized circu-
lating spin currents and unmagnetized filled cores, can exis t in the ground states of SU(3) spin-orbit
coupled Bose gases. It is found that the SU(3) spin-orbit cou pling and spin-exchange interaction
play important roles in determining the ground-state phase diagram. In the case of effective fer-
romagnetic spin interaction, the SU(3) spin-orbit couplin g induces a three-fold degeneracy to the
magnetized ground state, while in the antiferromagnetic sp in interaction case, the SU(3) spin-orbit
coupling breaks the ordinary phase rule of spinor Bose gases , and allows the spontaneous emergence
of double-quantum spin vortices. This exotic topological d efect is in stark contrast to the singly
quantized spin vortices observed in existing experiments, and can be readily observed by the current
magnetization-sensitive phase-contrast imaging techniq ue.
PACS numbers: 03.75.Lm, 03.75.Mn, 67.85.Bc, 67.85.Fg
I. INTRODUCTION
The recent experimental realization of synthetic spin-
orbit (SO) coupling in ultracold quantum gases [1–10] is
considered as an important breakthrough, as it provides
new possibilities for ultracold quantum gases to be used
as quantum simulation platforms, and paves a new route
towards exploring novel states of matter and quantum
phenomena [11–19]. It has been found that the SO cou-
pling can not only stabilize various topological defects,
such as half-quantum vortex, skyrmion, composite soli-
ton and chiral domain wall, contributing to the design
and exploration of new functional materials [20–23], but
alsoleadtoentirelynew quantum phases, suchasmagne-
tized phase and stripe phase [24–26], providing support
for the study of novel quantum dynamical phase transi-
tions [27, 28] and exotic supersolid phases [29–31].
All the intriguing features mentioned above are based
on the characteristics that the SO coupling (either of the
NIST[1], Rashba[24]orWeyl[32]types)makestheinter-
nal states coupled to their momenta via the SU(2) Pauli
matrices. However, if the (pseudo)spin degree of freedom
involves more than two states, the SU(2) spin matrices
cannot describe completely all the couplings among the
internal states. For example, a direct transition between
the states |1∝angbracketrightand|−1∝angbracketrightis missing in a three-component
system [24, 33]. From this sense, an SU(3) SO coupling
with the spin operator spanned by the Gell-Mann matri-
∗wzhangl@ruc.edu.cn
†wliu@iphy.ac.cn
‡szhang@ntsc.ac.cnces is more effective in describing the internal couplings
among three-component atoms [33, 34]. The SU(3) SO
coupled system has no analogue in ordinary condensed
matter systems, hence may lead to new quantum phases
and topological defects.
In this article, we show that a new type of topological
defects, double-quantum spin vortices, can exist in the
ground states of SU(3) SO coupled Bose-Einstein con-
densates (BECs). It is found that the SU(3) SO coupling
leads to two distinct ground-state phases, a magnetized
phase or a lattice phase, depending on the spin-exchange
interaction being ferromagnetic or antiferromagnetic. In
the magnetized phase, the SU(3) SO coupling leads to a
groundstatewiththree-folddegeneracy,in starkcontrast
to the SU(2) case where the degeneracy is two, thus may
offer new insights into quantum dynamical phase transi-
tions [27]. In the lattice phase, the SU(3) SO coupling
breaks the ordinary phase requirement 2w 0= w1+w−1
for ordinary spinor BECs, where w iis the winding num-
ber of thei-th spin component [35–37], and induces three
types of exotic vortices with cores filled by different mag-
netizations. The interlaced arrangementof these vortices
leads to the spontaneous formation of multiply quantized
spin vortices with winding number 2. This new type of
topological defects can be observed in experiments us-
ing magnetization-sensitive phase-contrast imaging tech-
nique.
II. SU(3) SPIN-ORBIT COUPLING
We consider the F= 1 spinor BECs with SU(3)
SO coupling. Using the mean-field approximation, the
Hamiltonian can be written in the Gross-Pitaevskii form2
as
H=/integraldisplay
dr/bracketleftbigg
Ψ†/parenleftbigg
−/planckover2pi12∇2
2m+Vso/parenrightbigg
Ψ+c0
2n2+c2
2|F|2/bracketrightbigg
,(1)
where the order parameter Ψ= [Ψ1(r),Ψ0(r),Ψ−1(r)]⊤
is normalized with the total particle num-
berN=/integraltextdrΨ†Ψ. The particle density is
n=/summationtext
m=1,0,−1Ψ∗
m(r)Ψm(r), and the spin density
vectorF= (Fx,Fy,Fz) is defined by Fν(r) =Ψ†fνΨ
withf= (fx,fy,fz) being the vector of the spin-1 ma-
trices given in the irreducible representation [35, 38–40].
TheSOcouplingtermischosenas Vso=κλ·p, whereκis
the spin-orbit coupling strength, p= (px,py) represents
2D momentum, and λ= (λx,λy) is expressed in terms
ofλx=λ(1)+λ(4)+λ(6)andλy=λ(2)−λ(5)+λ(7),
withλ(i)(i= 1,...8) being the Gell-Mann matrices, i.e.,
the generators of the SU(3) group [41]. Note that the
SU(3) SO coupling term in the Hamiltonian involves all
the pairwise couplings between the three states. This is
distinct from the previously discussed SU(2) SO coupling
in spinor BECs, where the states Ψ 1(r) and Ψ −1(r)
are coupled indirectly [24, 42, 43]. The parameters c0
andc2describe the strengths of density-density and
spin-exchange interactions, respectively.
The Hamiltonian with SU(3) SO coupling can be re-
alized using a similar method of Raman dressing as in
the SU(2) case [1, 9, 44]. As shown in Fig. 1(a), three
laser beams with different polarizations and frequencies,
intersecting at an angle of 2 π/3, are used for the Ra-
man coupling. Each of the three Raman lasers dresses
one hyperfine spin state from the F= 1 manifold ( |F=
1,mF= 1∝angbracketright,|F= 1,mF= 0∝angbracketrightand|F= 1,mF=−1∝angbracketright) to
the excited state |e∝angbracketright[See Fig. 1(b)]. When the standard
rotatingwaveapproximationisusedandtheexcitedstate
is adiabatically eliminated due to far detuning, one can
obtain the effective Hamiltonian in Eq. (1), as discussed
in Appendix A.
By diagonalizing the kinetic energy and SO coupling
terms, we can obtain the single-particleenergyspectrum,
which can provide useful information about the ground
stateofBosecondensates. FortheSU(2)case,itisknown
that the single-particle spectrum with the NIST type SO
coupling acquires either a single or two minima, depend-
ing on the strength of the Raman coupling [1], while for
the case of Rashba type there exist an infinite number
of minima locating on a continuous ring in momentum
space [45]. For the SU(3) SO coupling discussed here, we
find that there are in general three discrete minima re-
siding on the vertices of an equilateral triangle [See Figs.
1(c)-1(d)]. This unique property of the energy band im-
plies the possibilityofa three-folddegeneratemany-body
magnetized state [27] or a topologically nontrivial lattice
state, depending on the choices among the three minima
made by the many-body interactions.
FIG. 1: (Color online) Scheme for creating SU(3) spin-orbit
coupling in spinor BECs. (a) Laser geometry. Three laser
beams with different frequencies and polarizations, inters ect-
ing at an angle of 2 π/3, illuminate the cloud of atoms. (b)
Leveldiagram. EachofthethreeRamanlasers dresses onehy-
perfine Zeeman level from |F= 1,mF= 1/angbracketright,|F= 1,mF= 0/angbracketright
and|F= 1,mF=−1/angbracketrightof the87Rb 5S1/2,F= 1 ground state.
δ1,δ2andδ3correspond to the detuning in the Raman tran-
sitions. (c) Triple-well dispersion relation. The SU(3) sp in-
orbit coupling induces three discrete minima of the single-
particle energy band on the vertices of an equilateral trian gle
in thekx-kyplane. (d) Projection of the first energy band on
a 2D plane. Units with /planckover2pi1=m= 1 are used for simplicity.
III. PHASE DIAGRAM
Next, we discuss the phase diagram of the many-body
ground states. For the case of SU(2) SO coupling, it is
shown that two many-body ground states, magnetized
state and stripe state, can be stabilized in a homoge-
neous system [7, 24, 26]. Although the Rashba SO cou-
pling provides infinite degenerate minima in the single-
particle spectrum, a many-body ground state condensed
in one or two points in momentum space is always ener-
getically favorable due to the presence of spin-exchange
interaction [24]. As a result, a lattice state with the con-
densates occupying three or more momentum points for
SU(2) SO coupling is unstable, unless a strong harmonic
trap is introduced [21, 42, 46].
For the present case of SU(3) SO coupling, we first
analytically calculate the possible ground states using
a variational approach with a trial wave function Ψ=
α1Ψ1+α2Ψ2+α3Ψ3, where3
Ψ1=1√
3
1
1
1
e−i2κx, (2a)
Ψ2=1√
3
e−iπ
3
eiπ
3
eiπ
eiκ(x−√
3y),(2b)
Ψ3=1√
3
eiπ
3
e−iπ
3
eiπ
eiκ(x+√
3y), (2c)
correspond to the many-body states with all particles
condensing on one of the three minima of the single-
particlespectrum, and αi=1,2,3areexpansioncoefficients.
Substituting Eqs. (2a)-(2c) into the interaction energy
functional
E=/integraldisplay
dr/parenleftigc0
2n2+c2
2|F|2/parenrightig
, (3)
one obtains
E
N=/parenleftbiggc0
2+4c2
9/parenrightbigg
¯n−7c2
9¯n/summationdisplay
i/negationslash=j|αi|2|αj|2,(4)
where ¯n=|α1|2+|α2|2+|α3|2is the mean particle den-
sity. By minimizing the interaction energy with respect
to the variation of |αi|2, one finds that the spin-exchange
interaction plays an important role in determining the
phase diagram.
Whenc2>0, it favors |α1|2=|α2|2=|α3|2= ¯n/3,
indicating that the ground state is a triangular lattice
phasewithanequallyweightedsuperpositionofthe three
single-particle minima. On the other hand, as c2<0,
the system prefers a state with either |α1|2=¯n,|α2|2=
|α3|2= 0, or |α2|2= ¯n,|α1|2=|α3|2= 0, or |α3|2=
¯n,|α1|2=|α2|2= 0, indicating that the ground state
occupies one single minimum in the momentum space,
and corresponds to a three-fold degenerate magnetized
phase.
Note that the variational wave function Eqs. (2a)-(2c)
is a good starting point as the SO coupling is strong
enough to dominate the chemical potential. For the case
with weak SO coupling, one must rely on numerical sim-
ulations to determine the many-body ground state. In
such a situation, we find a stripe phase with two minima
in momentum space occupied for c2≫κ2, which will be
discussed latter.
The many-body ground states can be numerically ob-
tained by minimizing the energy functional associated
with the Hamiltonian Eq. (1) via the imaginarytime evo-
lution method. It is found that the numerical results are
consistent with the analytical analysis discussed above
for rather weak interaction with c2/lessorsimilarκ2. Figure 2 il-
lustrates the two possible ground states of spinor BECs
with SU(3) SO coupling. When c2>0, the three compo-
nents are immiscible and arranged as an interlaced tri-
angular lattice with the spatial translational symmetry
FIG. 2: (Color online) Two distinct phases present in SU(3)
spin-orbit coupled BECs. (a)-(d) The topologically nontri vial
lattice phase for antiferromagnetic spin interaction ( c2>0)
with (a) the density and phase of Ψ 1represented by heights
and colors, (b) the phase within one unit cell showing the
positions of vortices (white circles) and antivortices (bl ack
circles), (c) the corresponding momentum distributions, a nd
(d) the structural schematic drawing of the phase separatio n.
(e)-(f) The three-fold degenerate magnetized phase for fer ro-
magnetic spin interaction ( c2<0) with (e) the density and
phase distributions of Ψ 1and (f) the corresponding momen-
tum distributions.
spontaneously broken [See Figs. 2(a)-2(d)]. This lattice
is topologically nontrivial and embedded by vortices and
antivortices as shown in Fig. 2(b). From this result,
we conclude that a lattice phase can be stabilized in a
uniform SU(3) SO coupled BEC, which is in clear con-
trast to the SU(2) case where a strong harmonic trap is
required[21,42,46]. Moredetailsonthestructureofvor-
tices aswell astheir unique spin configurationswill be in-
vestigated later. On the other hand, as c2<0, the three
components are miscible, and the system forms a magne-
tized phase with the spatial transitional symmetry pre-
served but the time-reversal symmetry broken [See Figs.
2(e)-2(f)]. This magnetized phase occupies one of the
three minima of the single-particle spectrum by sponta-
neous symmetry breaking, hence is three-fold degenerate
instead of doubly degenerate in the SU(2) case [26, 27].
For strong antiferromagnetic spin interaction with
c2≫κ2, however, a stripe phase is identified with two
of three minima occupied in the momentum space. We
take the states with two or three minima occupied in
the momentum space as trial wave functions, and per-
form imaginary time evolution to find their respective
optimized ground state energy. A typical set of results
are summarized in Figure 3(a), showing the energy com-
parison with different values of interatomic interactions.
Obviously, one finds that the stripe phase will has lower
energy than the lattice phase when the interatomic in-
teraction exceeds a critical value. Due to the finite mo-
mentum in vertical direction of the stripe [See Fig. 3(d)],
both the spatial translational and time-reversal symme-
tries are broken [See Figs. 3(b)-3(c)]. This is distinct4
FIG. 3: (color online) (a) Energy comparison between the
lattice and stripe phases. The energy difference ∆ Ebetween
the numerical simulation and the variational calculation a re
shown by solid (lattice state) and dashed (stripe state) lin es.
(b)-(d) The ground-state density, phase and momentum dis-
tributions of the stripe phase with the parameters c2= 20κ2
andc0= 10c2.
from the stripe phase induced by SU(2) SO coupling,
where the time-reversal symmetry is preserved [24].
IV. PHASE REQUIREMENT
The vortex configuration of spinor BECs depends on
the phase relation between the three components. We
next discuss the influence of SO coupling on the phase
requirement of the vortex configuration. We first assume
that the spinor order parameter of a vortex in the polar
coordinate ( r,θ) can be described as
ψj(r,θ) =φjeiwjθ+αj, (5)
wherej= 0,±1 andφj≥0.
A. Without spin-orbit coupling
In the absence of SO coupling, the phase-dependent
terms in the Hamiltonian are
Hphase=Ephase
kin+Ephase
int
=−1
2/integraldisplay
Ψ∗1
r2∂2
∂θ2Ψdr+2c2/integraldisplay
ℜ(ψ∗
1ψ∗
−1ψ2
0)dr,(6)
where the first term results from the kinetic energy and
the second from the spin-exchange interaction. Substi-tuting Eq. (5) into (6), one obtains
Ephase
kin=/summationdisplay
j=1,0,−1w2
j/integraldisplayπφ2
j
rdr, (7)
Ephase
int= 2c2/integraldisplay
φ1φ−1φ2
0rdr
/integraldisplay
cos[(w 1−2w0+w−1)θ+(α1−2α0+α−1)]dθ.(8)
It is easy to read from Eq. (7) that the system favors
small winding numbers energetically. Moreover, from
Eq. (8) the energy minimization requires the winding
number and phase satisfy the following relations
w1−2w0+w−1= 0, (9a)
α1−2α0+α−1=nπ, (9b)
wherenis odd forc2>0 and even for c2<0. The phase
requirementofEq. (9a)indicatesthatthefollowingtypes
of winding combination, such as ∝angbracketleft±1,×,0∝angbracketright,∝angbracketleft0,×,±1∝angbracketright,
∝angbracketleft±1,0,∓1∝angbracketright,∝angbracketleft±1,±1,±1∝angbracketright,∝angbracketleft±2,±1,0∝angbracketrightand∝angbracketleft0,±1,±2∝angbracketrightare
allowed in a spinor BEC, where the symbol “ ×” denotes
the absence of the Ψ 0component.
B. With SU(2) spin-orbit coupling
For the case of SU(2) SO coupling, we take the Rashba
type as an example, and write the Hamiltonian as
Esoc=/integraldisplay
κψ†
0−i∂x−∂y0
−i∂x+∂y0−i∂x−∂y
0−i∂x+∂y0
ψdr,(10)
whereψ= [ψ1,ψ0,ψ−1]⊤. Substituting Eq. (5)into(10),
one can obtain
Esoc=/integraldisplay
drdθ/bracketleftig
(φ0r∂rφ1−w1φ0φ1)ei[(w1−w0+1)θ+(α1−α0−π
2)]
−(φ1r∂rφ0+w0φ1φ0)e−i[(w1−w0+1)θ+(α1−α0−π
2)]
+(φ0r∂rφ−1+w−1φ0φ−1)ei[(w−1−w0−1)θ+(α−1−α0−π
2)]
−(φ−1r∂rφ0−w0φ−1φ0)e−i[(w−1−w0−1)θ+(α−1−α0−π
2)]/bracketrightig
.
(11)
In order to minimize the SO coupling energy, it is pre-
ferred that
w1−w0+1 = 0, (12a)
w−1−w0−1 = 0, (12b)
α1−α0−π
2=mπ, (12c)
α−1−α0−π
2=nπ. (12d)
Then the SO coupling energy is rewritten as
Esoc= 2π/integraldisplay
[φ0r∂rφ1−φ1r∂rφ0−(w1+w0)φ0φ1]drcosmπ
+2π/integraldisplay
[φ0r∂rφ−1−φ−1r∂rφ0+(w−1+w0)φ0φ−1]drcosnπ,
(13)5
wheremandnare odd or even, which can be deter-
mined by minimizing the energy expressed in Eq. (13).
It is found that the SU(2) SO coupling does not vio-
late the ordinary requirement on the winding combina-
tion in Eq. (9a), but introduces further requirements in
Eqs.(12a)-(12b). Asaresult, while ∝angbracketleft−1,0,1∝angbracketright,∝angbracketleft−2,−1,0∝angbracketright
and∝angbracketleft0,1,2∝angbracketrightare still allowed, some winding combinations
such as ∝angbracketleft±1,±1,±1∝angbracketright,∝angbracketleft±1,×,0∝angbracketright,∝angbracketleft0,×,±1∝angbracketright,∝angbracketleft1,0,−1∝angbracketright,
∝angbracketleft2,1,0∝angbracketrightand∝angbracketleft0,−1,−2∝angbracketrightare forbidden. Obviously, one
can see that the SO coupling break the chiral symmetry,
thus may lead to chiral spin textures.
C. With SU(3) spin-orbit coupling
For the case of SU(3) SO coupling, the effective Hamil-
tonian can be written as
Esoc=/integraldisplay
κψ†
0−i∂x−∂y−i∂x+∂y
−i∂x+∂y0−i∂x−∂y
−i∂x−∂y−i∂x+∂y0
ψdr.(14)
Substituting Eq. (5) into (14), we get
Esoc=/integraldisplay
drdθ/bracketleftig
(φ0r∂rφ1−w1φ0φ1)ei[(w1−w0+1)θ+(α1−α0−π
2)]
−(φ1r∂rφ0+w0φ1φ0)e−i[(w1−w0+1)θ+(α1−α0−π
2)]
+(φ0r∂rφ−1+w−1φ0φ−1)ei[(w−1−w0−1)θ+(α−1−α0−π
2)]
−(φ−1r∂rφ0−w0φ−1φ0)e−i[(w−1−w0−1)θ+(α−1−α0−π
2)]
+(φ−1r∂rφ1+w1φ−1φ1)ei[(w1−w−1−1)θ+(α1−α−1−π
2)]
−(φ1r∂rφ−1−w−1φ1φ−1)e−i[(w1−w−1−1)θ+(α1−α−1−π
2)]/bracketrightig
.
(15)
By minimizing the SO coupling energy, one obtains the
following relations
w1−w0+1 = 0, (16a)
w−1−w0−1 = 0, (16b)
w1−w−1−1 = 0, (16c)
α1−α0−π
2=mπ, (16d)
α−1−α0−π
2=nπ, (16e)
α1−α−1−π
2=lπ. (16f)
Then the SO coupling energy can be rewritten as
Esoc=2π/integraldisplay
[φ0r∂rφ1−φ1r∂rφ0−(w1+w0)φ0φ1]drcosmπ
+2π/integraldisplay
[φ0r∂rφ−1−φ−1r∂rφ0+(w−1+w0)φ0φ−1]drcosnπ
+2π/integraldisplay
[φ−1r∂rφ1−φ1r∂rφ−1+(w1+w−1)φ−1φ1]drcoslπ,
(17)wherem,nandlare odd or even, which can be
determined from Eq. (17). However, the three winding
requirements Eqs. (16a)-(16c) can not be satisfied simul-
taneously. Thus the SU(3) SO coupling may choose two
out of the three winding requirements for the following
three cases:
Case I:
w1−w0+1 = 0, (18a)
w−1−w0−1 = 0, (18b)
α1−α0−π
2=mπ, (18c)
α−1−α0−π
2=nπ. (18d)
Case II:
w1−w0+1 = 0, (19a)
w1−w−1−1 = 0, (19b)
α1−α0−π
2=mπ, (19c)
α1−α−1−π
2=lπ. (19d)
Case III:
w−1−w0−1 = 0, (20a)
w1−w−1−1 = 0, (20b)
α−1−α0−π
2=nπ, (20c)
α1−α−1−π
2=lπ. (20d)
For case I, the winding combination ∝angbracketleft−1,0,1∝angbracketrightis allowed,
while∝angbracketleft1,0,−1∝angbracketrightis not allowed, indicating the chiral sym-
metry is broken. For case II and case III, one can find
that the SU(3) SO coupling breaks the ordinary require-
ment on the winding combination in Eq. (9a), thus new
winding combinations, such as ∝angbracketleft0,1,−1∝angbracketrightand∝angbracketleft1,−1,0∝angbracketright,
are possible.
V. VORTEX CONFIGURATIONS
The vortex configurations of spinor BECs can be clas-
sified according to the combination of winding numbers
and the magnetization of vortex core [35–37]. For ex-
ample, a Mermin-Ho vortex has winding combination
∝angbracketleft±2,±1,0∝angbracketrightwith a ferromagnetic core, where the plus
and minus signs represent different chirality of the vor-
tices [47], and the expression of ∝angbracketleftw1,w0,w−1∝angbracketrightindicates
that the components of Ψ 1, Ψ0and Ψ −1in the wave
function acquire winding numbers of w 1, w0and w −1,
respectively. Using this notation, a polar-core vortex has
winding combination ∝angbracketleft±1,0,∓1∝angbracketrightwith an antiferromag-
netic core, and a half-quantum vortex has winding com-
bination ∝angbracketleft±1,×,0∝angbracketrightwith a ferromagnetic core, where the
symbol “ ×” denotes the absence of the Ψ 0component.6
FIG. 4: (Color online) Vortex configurations in antiferro-
magnetic spinor BECs with SU(3) spin-orbit coupling. (a)
Vortex arrangement among the three components of the con-
densates. One can identify three types of vortices, includ-
ing a polar-core vortex with winding combination /angbracketleft−1,0,1/angbracketright
(blue line) and two ferromagnetic-core vortices with wind-
ing combinations /angbracketleft1,−1,0/angbracketright(green line) and /angbracketleft0,1,−1/angbracketright(red
line). (b)-(d) Spherical-harmonic representation of the t hree
types of vortices. The surface plots of |Φ(θ,φ)|2for (b) the
polar-core vortex /angbracketleft−1,0,1/angbracketright, (c) the ferromagnetic-core vor-
tex/angbracketleft1,−1,0/angbracketright, and (d) the ferromagnetic-core vortex /angbracketleft0,1,−1/angbracketright
are shown with the colors representing the phase of Φ(θ,φ).
Here,Φ(θ,φ) =/summationtext1
m=−1Y1m(θ,φ)ΨmandY1mis the rank-1
spherical-harmonic function.
In the lattice phase induced by the SU(3) SO cou-
pling with antiferromagnetic spin interaction, there ex-
ists three types of vortices: one is a polar-core vortex
with winding combination ∝angbracketleft−1,0,1∝angbracketright, and the other two
are ferromagnetic-core vortices with winding combina-
tions∝angbracketleft1,−1,0∝angbracketrightand∝angbracketleft0,1,−1∝angbracketright[See Fig. 4(a)]. However,
the vortex configurations with opposite chirality of each
type, such as ∝angbracketleft1,0,−1∝angbracketright,∝angbracketleft−1,1,0∝angbracketrightand∝angbracketleft0,−1,1∝angbracketright, are not
allowed, because the chiral symmetry is intrincically bro-
kenin SU(3) SO coupledsystems, asdiscussed inSec. IV.
Surprisingly, one finds that the two types of
ferromagnetic-core vortices ∝angbracketleft1,−1,0∝angbracketrightand∝angbracketleft0,1,−1∝angbracketrightvio-
late the conventional phase requirement 2w 0= w1+w−1
for ordinary spinor BECs [35–37]. This can be under-
stood by noting that the relative phase among different
wave function components are no longer uniquely deter-
mined by the spin-exchange interaction but also affected
by the SU(3) SO coupling, as qualitatively explained in
Sec. IV. Thus, the interlaced arrangement of the three
types of vortices forms a new class of vortexlattice which
has no analogue in systems without SO coupling.
The configurations of the three types of vortices in-
duced by the SU(3) SO coupling with antiferromagnetic
interaction are essentially different from those usually
observed in ferromagnetic spinor BECs, as can be il-
lustrated by the spherical-harmonic representation [35].
From Figs. 4(b)-4(d) one can find that for the polar-corevortex,theantiferromagneticorderparametervariescon-
tinuously everywhere, while for the ferromagnetic-core
vortex,themagneticorderparameteracquiresasingular-
ity at the vortex core. In contrast, in the ordinary ferro-
magnetic spinor BECs, the ferromagnetic order parame-
ter varies continuously everywhere for the ferromagnetic-
corevortex,buthasasingularityatthecoreforthepolar-
core vortex [35].
VI. DOUBLE-QUANTUM SPIN VORTICES
Spin vortex is a complex topological defect resulting
from symmetry breaking, and is characterized by zero
net mass current and quantized spin current around an
unmagnetized core [35, 38, 48–51]. It is not only different
from the magnetic vortex found in magnetic thin films
[52–54], but alsofromthe 2Dskyrmion[55,56] dueto the
existence of singularity in the spin textures [57]. Single-
quantum spin vortex with the spin current showing one
quantum ofcirculation has been experimentally observed
in ferromagnetic spinor BECs [58]. Multi-quantum spin
vortices with l(l≥2) quanta circulating spin current,
however, are considered to be topologically unstable and
have not been discovered yet [35].
A particularly important finding of our present work
is that the polar-core vortex in the lattice phase has a
spin current with two quanta of circulation around the
unmagnetized core, hence can be identified as a double-
quantum spin vortex. Figure 5 presents the transverse
magnetization F+=Fx+iFy, longitudinal magnetiza-
tionFz, and amplitude of the total magnetization |F|
in the lattice phase, which are experimentally observ-
able by magnetization-sensitive phase-contrast imaging
technique [59]. From these results, one can find two dis-
tinct types of topological defects, double-quantum spin
vortex (DSV) and half skyrmion (HS) [60, 61], which
correspond to the polar-core vortex with winding com-
binations ∝angbracketleft−1,0,1∝angbracketrightand the ferromagnetic-core vortex
with winding combinations ∝angbracketleft1,−1,0∝angbracketrightor∝angbracketleft0,1,−1∝angbracketright, re-
spectively. In particular, for the double-quantum spin
vortex, the core is unmagnetized and the orientation of
the magnetization along a closed path surrounding the
core acquires a rotation of 4 π. This finding indicates
that a regular lattice of multi-quantum spin vortices can
emerge spontaneously in antiferromagnetic spinor BECs
with SU(3) SO coupling. By exploring the effect of a
small but finite temperature, we confirm that the double-
quantum spin vortices are robust against thermal fluctu-
ations and hence are observable in experiments, as dis-
cussed in Appendix B.
The emergence of spin current with two quanta of cir-
culation can be analytically understood by expanding
the wave function obtained by the variational methods
around the center of a double-quantum spin vortex. We
suppose that the wave function of the lattice phase is7
FIG. 5: (Color online) Double-quantum spin vortex in an-
tiferromagnetic spinor BECs with SU(3) spin-orbit couplin g.
(a) Spatial maps of the transverse magnetization with color s
indicating the magnetization orientation. (b) Longitudin al
magnetization. (c) Amplitude of the total magnetization |F|.
Two kinds of topological defects, double-quantum spin vor-
tex (DSV) and half skyrmion (HS) are marked by big and
small circles, respectively. The transverse magnetizatio n ori-
entation arg F+along a closed path (indicated by big circles)
surrounding the unmagnetized core shows a net winding of
4π, revealing the presence of a double-quantum spin vortex.
written as
ψ=1
3
1
1
1
e−i2κx+1
3
e−iπ
3
eiπ
3
eiπ
eiκ(x−√
3y)+1
3
eiπ
3
e−iπ
3
eiπ
eiκ(x+√
3y).
(21)
Then one can expand ψaround the center of a vortex
with winding number ∝angbracketleft−1,0,1∝angbracketright, e.g., at the location of
(x,y) = (0,π/(3√
3κ)). Substituting x=ǫcosθandy=
π/(3√
3κ)+ǫsinθintoψand expanding with respect to
the infinitesimal ǫ, we obtain
ψ=
−iκe−iθǫ−1
2κ2ei2θǫ2
1−κ2ǫ2
−iκeiθǫ−1
2κ2e−i2θǫ2
+O/parenleftbig
ǫ3/parenrightbig
.(22)
Notice that the second-order terms with e±i2θhave
no essential influence on the phases, thus the wind-
ing number for each component can still be represented
as∝angbracketleft−1,0,1∝angbracketright[See Figs. 6(a)-6(c)]. However, since the
first-order terms are canceled out when calculating the
transverse magnetization F+=√
2[ψ∗
1ψ0+ψ∗
0ψ−1], the
second-order terms play a dominant role, leading to the
emergence of spin current with two quanta of circulation
around an unmagnetized core
F+∝ǫ2e−i2θ, (23)
as illustrated in Fig. 6(d).
FIG. 6: (color online) (a)-(c) Phases of the polar-core vort ex
described by the wave function in Eq. (22), displaying the
winding combination /angbracketleft−1,0,1/angbracketright. (d) Direction of the trans-
verse magnetization, indicating the emergence of spin curr ent
with two quanta of circulation.
VII. CONCLUSION
To summarize, we have mapped out the ground-state
phase diagram of SU(3) spin-orbit coupled Bose-Einstein
condensates. Several novel phases are discovered includ-
ing a three-fold degenerate magnetized phase, a vortex
lattice phase, as well as a stripe phase with time-reversal
symmetry broken. We also investigate the influence of
SU(3) spin-orbit coupling on the phase requirement of
thevortexconfiguration,anddemonstratethattheSU(3)
spin-orbit coupling breaks the ordinary phase rule of
spinor Bose-Einstein condensates, and allows the sponta-
neous emergence of stable double-quantum spin vortices.
As a new member in the family of topological defects,
double quantum spin vortex has never been discovered
in any other systems. Our work deepen the understand-
ing of spin-orbit phenomena, and will attract extensive
interest of scientists in the cold atom community.
ACKNOWLEDGMENTS
This work was supported by NKRDP under grants
Nos. 2016YFA0301500, 2012CB821305; NSFC under
grants Nos. 61227902, 61378017, 11274009, 11434011,
11434015, 11447178, 11522436, 11547126; NKBRSFC
under grant No. 2012CB821305; SKLQOQOD un-
der grant No. KF201403; SPRPCAS under grant
Nos. XDB01020300, XDB21030300; JSPS KAKENHI8
grant No. 26400414 and MEXT KAKENHI grant No.
25103007. W. H. and X.-F. Z. contributed equally to
this work.
APPENDIX A: DERIVING THE EFFECTIVE
HAMILTONIAN
We consider spinor Bose-Einstein condensates (BECs)
illuminated by three Raman laser beams, which couple
two of the three hyperfine spin components respectively,
as illustrated in Figs. 1(a)-1(b) of the main text. The
internal dynamics of a single particle under this scheme
can be described by the Hamiltonian
H=3/summationdisplay
j=1/parenleftbigg/planckover2pi12k2
2m+εj/parenrightbigg
|j∝angbracketright∝angbracketleftj|+n/summationdisplay
l=1El|l∝angbracketright∝angbracketleftl|
+3/summationdisplay
j=1n/summationdisplay
l=1/bracketleftig
Ωjei(Kj·r+ωjt)Mlj|l∝angbracketright∝angbracketleftj∝angbracketright+h.c./bracketrightig
,(A1)
where/planckover2pi1kis the momentum of the particles, and εjand
Elare the energies of the ground and excited states, re-
spectively. In the atom-light coupling term, Kjandωj
are the wave vectors and frequencies of the three Raman
lasers with Ω jthe corresponding Rabi frequencies, and
Mljis the matrix element of the dipole transition. One
can see that this Hamiltonian is similar to that used in
the scheme for creating 2D spin-orbit (SO) coupling in
ultracold Fermi gases [9], thus can be readily realized in
Bose gases. Taking the standard rotating wave approxi-
mation to get rid of the time dependence of the Hamilto-
nian, and adiabatically eliminating the excited states for
far detuning, the Hamiltonian can be rewritten as
H=
/planckover2pi12(k+K1)2
2m+δ1Ω12 Ω13
Ω21/planckover2pi12(k+K2)2
2m+δ2Ω23
Ω31 Ω32/planckover2pi12(k+K3)2
2m+δ3
,(A2)
whereδ1,δ2andδ3arethetwo-photondetunings, andthe
realparametersΩ jj′= Ωj′jdescribe the Raman coupling
strength between hyperfine ground states |j∝angbracketrightand|j′∝angbracketright,
which can be expressed as [9, 62]
Ωjj′=−/radicalbigIjIj′
/planckover2pi12cǫ0/summationdisplay
m′∝angbracketleftj′|erq|m′∝angbracketright∝angbracketleftm′|erq|j∝angbracketright
∆.(A3)
Here,Ijis the intensity of each Raman laser, and ∆
denotes the one-photon detuning. Other parameters c,
ǫ0andein Eq. (A3) are the speed of light, permittiv-
ity of vacuum and elementary charge, respectively. In
Eq. (A3),q=x,y,zis an index labeling the components
ofrin the spherical basis, and |m′∝angbracketrightdescribes the middle
excited hyperfine spin state in the Raman process. For
simplicity, we assume Ω = Ω 12= Ω13= Ω23, which can
always be satisfied by adjusting the system parameters,
such as the laser intensity.
Introducing a unitary transformation
U=1√
3
1 1 1
−e−iπ
3−eiπ
31
−eiπ
3−e−iπ
31
(A4)and a time-dependent unitary transformation U(t) =
ei/parenleftbigg
/planckover2pi12K2
0
2m+δ2−Ω/parenrightbigg
t, the effective Hamiltonian becomes
H=
/planckover2pi12k2
2m+δ1−δ20 0
0/planckover2pi12k2
2m0
0 0/planckover2pi12k2
2m+δ3−δ2+3Ω
+Vso,(A5)
where the laser vectors K1=−K0ˆ ey,K2=√
3K0
2ˆ ex+
K0
2ˆ eyandK3=−√
3K0
2ˆ ex+K0
2ˆ eyare defined with K0=
2mκ//planckover2pi1. The spin-dependent uniform potential induced
by the Raman detuning δiand Raman coupling strength
Ω can be eliminated by applying a Zeeman field, leading
to
H=
/planckover2pi12k2
2m+ǫ10 0
0/planckover2pi12k2
2m0
0 0/planckover2pi12k2
2m+ǫ2
+Vso,(A6)
whereǫ1=δ1−δ2+∆1+∆2andǫ2=δ3−δ2−∆1+∆2+
3Ω with ∆ 1and ∆ 2denoting the linear and quadratic
Zeeman energy respectively. By tuning the detuning, the
Zeeman energy and the Raman coupling strength, one
can reach the regime ∆ 1=δ3−δ1+3Ω
2and ∆ 2=δ2−
δ1+δ3+3Ω
2which satisfying ǫ1=ǫ2= 0. Then we have
H=/planckover2pi12k2
2m+Vso, (A7)
which is the single-particle Hamiltonian with SU(3) SO
coupling considered in the main text.
APPENDIX B: STABILITY OF THE
DOUBLE-QUANTUM SPIN VORTEX STATES
In order to verify the stability of the phases discovered
in this manuscript, we have explored the effects of a
small but finite temperature, and concluded that the
double-quantum spin vortex states are robust against
the thermal fluctuations. In particular, we considered
a random fluctuation ∆ φin the real-time evolution
of the Gross-Pitaevskii equation, which causes an
energy fluctuation about ∆ E= 0.03EgwithEgthe
ground-state energy. An estimation shows that this
level of fluctuation corresponds to the energy scale
kBTwithT∼300 nK, which is higher enough for a
usual system of Bose-Einstein condensates in realistic
experiments. According to numerical simulations, we
find that the structure of the double-quantum spin
vortex state is stable under this level of fluctuation in
tens of millisenconds [See Fig. 7], suggesting that this
phase is indeed observable in experiments.9
FIG. 7: (color online) Stable double-quantum
spin vortex under a random fluctuation. The
images are taken at t= 20msin the real-time
evolution, with thermal fluctuation in the en-
ergy scale of kBTwithT∼300 nK. (a) Spa-
tial maps of the transverse magnetization with
colors indicating the magnetization orientation.
(b) Longitudinal magnetization. (c) Amplitude
of the total magnetization |F|. It is shown that
the double-quantum spin vortices are topologi-
cally stable under external fluctuations with a
fairly long lifetime of tens of ms.
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1208.2923v2.Chaos_driven_dynamics_in_spin_orbit_coupled_atomic_gases.pdf | Chaos-driven dynamics in spin-orbit coupled atomic gases
Jonas Larson,1, 2,Brandon M. Anderson,3and Alexander Altland2
1Department of Physics, Stockholm University, Se-106 91 Stockholm, Sweden
2Institut f ur Theoretische Physik, Universit at zu K oln, K oln, De-50937, Germany
3Joint Quantum Institute, national Institute of Standards and technology
and the University of Maryland, Gaithersburg, Maryland 20899-8410, USA
(Dated: May 24, 2022)
The dynamics, appearing after a quantum quench, of a trapped, spin-orbit coupled, dilute atomic
gas is studied. The characteristics of the evolution is greatly in
uenced by the symmetries of the
system, and we especially compare evolution for an isotropic Rashba coupling and for an anisotropic
spin-orbit coupling. As we make the spin-orbit coupling anisotropic, we break the rotational sym-
metry and the underlying classical model becomes chaotic; the quantum dynamics is a ected ac-
cordingly. Within experimentally relevant time-scales and parameters, the system thermalizes in a
quantum sense. The corresponding equilibration time is found to agree with the Ehrenfest time,
i.e. we numerically verify a log(h 1) scaling. Upon thermalization, we nd the equilibrated
distributions show examples of quantum scars distinguished by accumulation of atomic density for
certain energies. At shorter time-scales we discuss non-adiabatic e ects deriving from the spin-orbit
coupled induced Dirac point. In the vicinity of the Dirac point, spin
uctuations are large and, even
at short times, a semi-classical analysis fails.
PACS numbers: 03.75.Kk, 03.75.Mn
I. INTRODUCTION
The physics of ultracold atomic gases has greatly ad-
vanced in recent years [1]. The high control of system pa-
rameters, together with the isolation of the system from
its environment, have made it possible to use such se-
tups to simulate various theoretical models of condensed
matter physics [1, 2]. Of signi cance in many condensed
matter models is the response to external magnetic elds.
Since atoms are neutral, there is no direct way to imple-
ment a Lorentz force in these systems. Early experiments
created a synthetic magnetic eld via rotation [3]. While
simple theoretically, these methods are impractical for
certain setups, and they are limited to weak, uniform
elds. The rst experimental demonstration of laser-
induced synthetic magnetic elds for neutral atoms [4],
on the other hand, paves the way for an avenue of new
situations to be studied in a versatile manner [5{7]. Ow-
ing to numerous fundamental applications in the con-
densed matter community [8, 9], maybe the most im-
portant direction appears when the laser elds induce
a synthetic spin-orbit (SO) coupling. Indeed, a certain
kind of SO-coupling for neutral atoms has already been
demonstrated [10], and it is expected that more general
SO-couplings will be attainable within the very near fu-
ture [11, 12].
While SO-couplings can in principle bear identical
forms in condensed matter and cold atom models, there is
an inevitable di erence, often overlooked, between these
two systems. The presence of a con ning potential for the
atomic gas can qualitatively change the physics [1, 3],
jolarson@fysik.su.seand has only recently been addressed [13{17]. Fur-
thermore, most of these studies are concerned with
ground/stationary state properties of the system [13{
15], while few works discuss dynamics or non-equilibrium
physics. Notwithstanding, the experimental isolation of
these systems suggests that they are well suited for stud-
ies of closed quantum dynamics [18].
Historically, some of the nest experiments regard-
ing dynamics of closed quantum systems have been per-
formed in quantum optics [19, 20]. An early example
proved quantization of the electromagnetic eld by mak-
ing explicit use of quantum revivals [21]. Such quantum
recurrences, in general connected to integrability or small
system sizes, are now well understood. The situation be-
comes more complex for non-integrable systems [18] or
systems with a large number of degrees-of-freedom [22].
One particularly interesting question is whether any ini-
tial state relaxes to an asymptotic state, and if so, what
are then the properties of this \equilibrated" state and
the mechanism behind the equilibration. Both these
questions have inspired numerous publications during
the last decade, both theoretical [23, 24] as well as ex-
perimental [25{27]. A rule of thumb is that in order
for a closed quantum system to thermalize , i.e. all ex-
pectation values can be obtained from a microcanonical
state, its underlying classical Hamiltonian should be non-
integrable [18]. While true in most cases studied so far,
exceptions to this hypothesis has been found [28]. More-
over, the behavior near the transition from regular to
chaotic dynamics, classically explained by Kolmogorov-
Arnold-Moser theory [29], is not well understood for a
quantum system [30]. It is therefore desirable to study
a system where these two regimes can be explored by
tuning an external parameter, and for which the exper-
imental methods in terms of preparation and detectionarXiv:1208.2923v2 [quant-ph] 28 Jan 20132
are already well developed.
Motivated by the above arguments, in this paper we
consider dynamics of a trapped SO-coupled cold dilute
atomic gas. The SO-coupling is assumed tunable from
isotropic (Rashba-like) to anisotropic, and hence the sys-
tem can be tuned between regular and chaotic. Note
that even though this crossover is generated by a change
in the form of the SO-coupling, the con ning trap causes
the system to become non-integrable. We distinguish
between short and long time evolution, where by \long
time" we mean times similar to the Ehrenfest time. In
fact, the corresponding time-scale for the thermalization
is found to agree with the Ehrenfest time, and thereby
scale as log( h 1)=whereis the maximum Lyaponov
exponent. This scaling for the thermalization has been
conjectured in Ref. [31], but was not numerically ver-
i ed in these works. At shorter times when the wave
packet remains localized, we especially study the rapid
changes in the spin as the wave packet evolves in the
vicinity of the Dirac point (DP). For energies below the
DP (E < 0), we utilize an adiabatic model derived in the
Born-Oppenheimer approximation (BOA) [32]. Aside
from some special initial states, we encounter thermal-
ization in all cases. These exceptions correspond to
states evolving within a regular \island" in the otherwise
chaotic sea. Among the thermalized states, the equi-
librated distributions are found to show quantum scars
originating from periodic orbits of the underlying classi-
cal model. The experimental relevance of all our theoret-
ical predictions are discussed and put in a state-of-the-art
experimental perspective.
The paper is outlined as follows. The following sec-
tion introduces the system Hamiltonian and discusses
its symmetries. Section II B derives the adiabatic model
by imposing the BOA. A semi-classical analysis, demon-
strating classical chaos for anisotropic SO-couplings, is
presented in Sec. III. The following section considers the
full quantum model at short times, Sec. IV A, and long
times, Sec. IV B. Section IV C contains a discussion re-
garding experimental relevance of our results. Finally,
Sec. V gives some concluding remarks.
II. SPIN-ORBIT COUPLED COLD ATOMS
A. Model spin-orbit Hamiltonian
Several proposals exist for implementing spin-orbit
couplings in cold atoms [33{35]. In general, these syn-
thetic spin-orbit elds are generated through the appli-
cation of optical and Zeeman elds to produce a set of
dressed states that are well separated energetically from
the remaining dressed states [5]. We denote these states
as pseudo-spin, but emphasize that there is no connection
to real space rotations. Spatial variation of the dressed
states will couple the pseudo-spin to the orbital motion
of the atom. An atom prepared in a pseudo-spin state
will therefore see an e ective Hamiltonian, provided theatom is suciently cold.
For a speci c con guration of optical elds, one can
induce the e ective Hamiltonian [35]
^HSO=^p2
2m+1
2m!2r2+vx^px^x+vy^py^y;(1)
where ^p= (^px;^py) is the momentum operator, ^r= (^x;^y)
is the position operator, mis the mass of the atom, and
!the frequency of a harmonic trap. The operator ^ i
is thei-th Pauli matrix in pseudo-spin space, and the
velocitiesvicouple pseudo-spin to an e ective momen-
tum dependent Zeeman eld, B(p) = (vxpx;vypy). This
momentum-dependent Zeeman eld can simulate any
combination of the Rashba [38] and Dresselhaus [39] SO-
couplings experienced in semiconductor quantum wells
and systems alike.
In the absence of a trap, != 0, the spectrum of (1) is
E(px;py) =1
2m
p2
x+p2
y
+q
(vxpx)2+ (vypy)2(2)
with the corresponding eigenfunctions
j ;pi=eim(vxx+vyy)j'i (3)
where
j'i=1p
2
e i'=2j"i ei'=2j#i
; (4)
is a spinor with helicity =1 and'=
arctan(vypy=vxpx). These states have well de ned mo-
mentum, but have no velocity since h_ri=hrpHi= 0,
provided the optical elds are maintained. Note further
that the eigenstates are parametrically dependent on px
andpy.
We remark that for an isotropic SO-coupling, vx=vy,
the Hamiltonian (1) is equivalent to the dual E"Jahn-
Teller model, frequently appearing in chemical/molecular
physics and condensed matter theories [37]. With a
simple unitary rotation of the Pauli matrices, the SO-
coupling attains the more familiar Rashba form [38] (or
equivalently Dresselhaus form [39]). For vx6=vy, i.e.
when the SO-coupling is anisotropic, the model becomes
the dualE( x+ y) Jahn-Teller model [37]. In par-
ticular, the ^ z-projection of total angular momentum,
^Jz=^Lz+^z
2, is a constant-of-motion for the isotropic
but not for the anisotropic model. More precisely, break-
ing of the SO isotropy implies a reduction in symmetry
fromU(1) toZ2.
Throughout we will use dimensionless parameters
where the oscillator energy Eo= h!sets the energy-
scale,l=p
h=m! the length-scale, and the characteristic
time is =! 1. We note that for typical experimental
setups,!10 100 Hz and m(v2
x+v2
y)=h1 10 kHz.
Moreover, in what follows we will refer to pseudo-spin
simply as spin. When necessary, we introduce a param-
eterhserving as a dimensionless Planck's constant, i.e.
hh. In this way, hcontrols the strength of Planck's con-
stant and by varying it we can explore how the dynamics
depends on h.3
B. Adiabatic model
The large ratio of the SO energy to trapping energy,
typicallymv2=h!10 1000, suggests that a BOA [32]
will be valid for experimental implementations. The sep-
aration of timescales of the spin and orbital degrees of
freedom implies that in some regimes we can factorize
the wavefunction as the product of spin and orbital wave-
functions. A spin initially aligned with the adiabatic
momentum-dependent magnetic eld B(p) will remain
locked to that eld at future times, provided the center
of mass motion avoids the DP. We then solve for the spin
wavefunction at an instantaneous orbital con guration
and use this answer to nd an adiabatic potential for the
orbital motion. This is in analogy with the traditional
BOA, where the electronic and nuclear wavefunctions are
approximated as a product, and the electron degrees of
freedom instantaneously adjust to the adiabatic potential
given by the nuclear degrees of freedom.
In our BOA, we have chosen the adiabatic states [32]
for the orbital motion to be the spin-helicity states, given
by (4). If we project the Hamiltonian into the basis j'i,
we arrive at the adiabatic potential
^H()
ad=^x2
2+^y2
2+^p2
x
2+^p2
y
2+q
v2x^p2x+v2y^p2y:(5)
The trap thus takes the role of kinetic energy and (5) can
be pictured as a particle in a (dual) adiabatic potential
V(^px;^py) =^p2
x
2+^p2
y
2+q
v2x^p2x+v2y^p2y: (6)
shown in Fig. 1 for both the isotropic (a) and anisotropic
(b) cases. We have neglected non-adiabatic corrections
arising from the vector potential and the Born-Huang
term [40]. For example, an additional scalar potential
Vnad(px;py)(vxvy)2(p2
x+p2
y)
v2xp2x+v2yp2y2: (7)
will emerge from the action of the SO-coupling on the
spinorj'i. This term is order Vnadh'jr2
pj'i1=p2.
There will also be an additional vector potential term
A1=p. The non-adiabatic corrections diverge near the
DP, but then fall o rapidly at nite p. The adiabatic
approximation, i.e. BOA, will be valid if the particle
avoidsp= 0. We will show later that this condition is
met if the particle is in the lower band, = 1, and has
energyE < 0.
Imposing the BOA, any state propagating on the lower
adiabatic potential will be denoted ( px;py;t), and it is
understood that
(px;py;t) = (px;py;t)j' i: (8)
The real space wave function ( x;y;t ) is given as
usual from the Fourier transform of (px;py;t).
The time-evolution follows from (px;py;t) =exp
i^H( )
adt
(px;py;0). It is also clear that the
state (px;py;t) determines the spin orientation which
is inherent in the ket-vector j'ii. More explicitly, the
time-evolved Bloch vector
R(t) = (Rx(t);Ry(t);Rz(t))(h^xi;h^yi;h^zi) (9)
takes the form
Rx(t) =Z
dpxdpyj (px;py;t)j2cos(');
Ry(t) =Z
dpxdpyj (px;py;t)j2sin('); (10)
Rz(t) = 0
in the BOA, and it is remembered that the parameter '
depends on pxandpy. Note that the Bloch vector pre-
cesses in the equatorial spin xy-plane. If the wave packet
(px;py;t) is sharply localized, a crude approximation
for the Bloch vector is given by
Rx(t) =vxpx(t)q
(vxpx(t))2+ (vypy(t))2; (11)
Ry(t) =vypy(t)q
(vxpx(t))2+ (vypy(t))2; (12)
Rz(t) = 0; (13)
where p (t) =R
dpxdpyj(px;py;t)j2p with =x; y.
III. CLASSICAL DYNAMICS
Quantum chaos is often de ned by having an under-
lying chaotic classical model. For the full model (1),
the spin degrees-of-freedom cannot be eliminated in a
straightforward manner in the vicinity of the Dirac point
and as a consequence it is not a priori clear what the
underlying classical model would be in this regime. On
the other hand, in the BOA, the adiabatic Hamiltonian
^H( )
adcan serve as our classical model Hamiltonian. Still,
it should be noted that we assume h^H( )
adi 0, such that
the spectrum contains a suciently large number of en-
ergies below E= 0. Furthermore, we point out that jus-
ti cation of the BOA does not necessarily imply approval
of a semi-classical approximation which depends on the
system energy and the actual shape of the dual poten-
tialV (px;py). Nevertheless, as we will demonstrate in
the following, for the chosen parameters, the agreement
is indeed very good.
The classical equations-of-motion of the Hamiltonian4
V ( )p+V ( )p-V ( )p+
V ( )p-py
pypxpx(a)
(b)
FIG. 1. Adiabatic potentials of the isotropic (a) and
anisotropic (b) SO-coupled models. In both gures, the E= 0
plane is the one including the DP at px=py= 0. A nec-
essary, but not sucient, condition for the validity of the
BOA is that E < 0. In (a), the lower adiabatic potential
V (px;py) has the characteristic sombrero shape. By con-
sidering an anisotropic SO-coupling, the rotational symme-
try is broken and V (px;py) possesses two global minima at
(px;py) = (0;vy).
−20 0 20−20−1001020
x−20 0 20−50050
xpx(a) (b)
FIG. 2. Two examples of classical trajectories (( x(t);Px(t))
for regular (a) and chaotic (b) dynamics. In (a), typical for
regular motion the trajectories evolve upon a tori. Contrary,
in (b) the trajectory is much more irregular which is char-
acteristic for the chaotic evolution. The regular motion is
calculated for the SO-coupling strengths vx=vy= 30, and
the chaotic motion with vx= 20 andvy= 30. In both cases,
the energy is E= 192.
FIG. 3. Poincar e sections of the Rashba SO-coupled adiabatic
model (5) for the intersections y= 0 (a) and py= 0 (b).
The initial energy is E= 192, the SO-coupling strengths
vx=vy= 30, and the number of simulated semi-classical
trajectories 18.
^H( )
adare
_x=px v2
xpxq
v2xp2x+v2yp2y; (14)
_px= x; (15)
_y=py v2
ypyq
v2xp2x+v2yp2y; (16)
_py= y: (17)
For the Rashba SO-coupling, vx=vy=v, there is one
unstable x point ( px;py) = (0;0) and a seam of sta-
ble x points p2
x+p2
y=v2, see Fig. 1 (a). For the
anisotropic case, vy> vx, there are three unstable x
points, (px;py) = (0;0) and (px;py) = (vx;0), while
there are two stable x points ( px;py) = (0;vy), see
Fig. 1 (b).
The classical energy E(x;px;y;py) =p2
x=2 +p2
y=2 +
x2=2 +y2=2 q
v2xp2x+v2yp2ydetermines a hypersurface
in phase space for any given energy E(x;px;y;py) =E0.
The semi-classical trajectories ( x(t);px(t);y(t);py(t)) live
on this surface. For the integrable case, vx=vy,
these surfaces form di erent tori characteristic for quasi-
periodic motion. As the rotational symmetry is slightly
broken,vx6=vy, the tori deforms and the motion loses
its quasi-periodic structure [29]. This is the generic
crossover from regular to chaotic classical dynamics. As
an example of this generic behavior, we show in Fig. 2
two randomly sampled trajectories in the xpx-plane for
regular (a) and chaotic (b) evolution. For all results of
this section, we solve the set of coupled di erential equa-
tions (14) using the Runge-Kutta (4,5) algorithm mod-
i ed by Gear's method , suitable for sti equations. We
have also numerically veri ed our results employing dif-
ferent algorithms [41]. As will be discussed further below,
even in the chaotic regime, periodic orbits may persist
and will greatly a ect the dynamics, both at a classical
and a quantum level [42]. Such orbits are not, however,
visible from Fig. 2.
The semi-classical behavior of classical dynamical sys-
tems is favorable visualized using Poincar e sections [43].5
Corresponding sections for the system (14)-(17) are de-
picted in Figs. 3 and 4. In the rst gure we display the
Poincar e sections in the xpxplane for the intersections
determined by y= 0 (a) or py= 0 (b) of the isotropic
model with the SO-coupling amplitudes vx=vy= 30.
The initial energy is taken as E= 192, well below
the DP, consistent with the BOA. In (b), the section
de ned bypy= 0, the evolution results in ellipses in the
Poincar e section, characteristic of quasi periodic motion.
The structure of the Poincar e section for y= 0 (a) is
somewhat more complex. This can be understood from
the sombrero shape of the adiabatic potential V (px;py);
for givenx=x0,px=p0
x,y= 0, and energy E0, there
are four possible values of py, and this multiplicity of
possiblepy's allow the \curves" in Fig. 3 (a) to cross.
It should be noted that any single curve does not cross
itself. Furthermore, by adding the pyvalues to Fig. 3
we have veri ed that neither of the corresponding three
dimensional curves cross.
Figure 4 presents two examples for anisotropic SO-
couplings, both with vx= 20 and vy= 30. The
quasi-periodic evolution is lost and the dynamics become
mixed, with regions of both chaos and regular dynamics.
The same conclusions were found in Ref. [44] where a re-
lated Jahn-Teller model was studied. The two lower plots
consider the same energies as in Fig. 3, i.e. E= 192,
while for (a) and (b) E= 88. Expectedly, the higher
energy increases the accessible volume of phase space.
For both energies we nd islands free from chaotic tra-jectories. As will be demonstrated in the next section,
within these islands the evolution is regular and the sys-
tem does not thermalize. The plots also demonstrate
clear structures also appearing in the chaotic regimes of
the Poincar e sections in which the density of solutions
changes.
IV. QUANTUM DYNAMICS
The idea of this section is to analyze how the corre-
sponding quantum evolution is a ected by whether the
classical dynamics is regular or chaotic. Of particular im-
portance is the long time evolution in which the system
state may or may not equilibrate. However, we study
also the short time dynamics arising for a localized wave
packet traversing the Dirac point. In this regime, clearly
the classical results of the previous section does not hold.
To study the system beyond the classical approxima-
tion, we solve the time-dependent Schr odinger equation,
represented by the Hamiltonians (1) or (5), to obtain
the corresponding wave function ( x;y;t ) at time t.
Note that for the full model (1), the wave function con-
tains the spin degree-of-freedom ( x;y;t ) = "(x;y;t )j"
i+ #(x;y;t )j#i. The non-equilibrium initial state ap-
pears after a quench in the center of the trap. We prepare
the system in a quasi-ground state for a shifted trap, and
att= 0 suddenly move the trap center to xs=ys= 0,
V(x;y) =(x xs)2
2+(y ys)2
2;
xs6= 0 and=orys6= 0;t<0;
xs=ys= 0; t0:(18)
By \quasi-ground state" in an anisotropic SO-coupled
system, we consider an initial state predominantly pop-
ulated in one of the two minima of the adiabatic po-
tentialV (px;py). This seems experimentally reasonable
where small
uctuations will favor one of the two min-
ima. For the isotropic case, the phase of ( px;py;t= 0)
is taken randomly in agreement with symmetry breaking.
Given the evolved states ( x;y;t ), we are interested in
the Bloch vector (10) or its components, and the distri-
butionsj(px;py;t)j2andj (x;y;t )j2.
The numerical calculation is performed employing the
split-operator method [45] which relies on factorizing, for
short times t, the time-evolution operator into a spatial
and a momentum part. For small SO-couplings vxand
vy, the method is relatively fast, while as vxand/orvyare
increased the time-steps tmust be considerably reduced
and the necessary computational power rises rapidly. In
addition, for large vxandvy, the grid sizes of position
and momentum space must be increased, which also in-
creases the computation time. Thus, we will limit the
analysis to SO-couplings vx; vy30. Furthermore, we
have found by convergence tests that the full model (1)requires much smaller time-steps tthan the adiabatic
one (5), and most of our simulations will therefore be re-
stricted to energies E < 0 for which the BOA is justi ed.
The full quantum simulations are complemented by the
semi-classical truncated Wigner approximation (TWA),
which has turned out very ecient in order to re-
produce quantum dynamics [46]. The TWA considers
a set ofNdi erent initial values ( xi;yi;pxi;pyi) ran-
domly drawn from the distributions j (x;y;0)j2and
j(px;py;0)j2. These are then propagated according to
the classical equations-of-motion (14). The propagated
set (xi(t);yi(t);pxi(t);pyi(t)) gives the semi-classical dis-
tributions, from which expectation values can be evalu-
ated.
A. Short time dynamics
Before investigating the prospects of thermalization,
we rst consider short time dynamics, by which we mean
time-scales where the wave packet remains localized. In6
FIG. 4. Poincar e sections of the anisotropic SO-coupled adia-
batic model (5) for y= 0 (a) and (c), and for py= 0 (b)
and (d). The initial energies are E= 88 (a) and (b),
andE= 192 (c) and (d), and the SO-coupling strengths
vx= 20 andvy= 30 for both cases. The corresponding maxi-
mum Lyaponov exponents have been derived to 0:12 and
= 0:090 respectively. The number of semi-classical trajec-
tories is the same as for Fig. 3, namely 18.
this respect, it is tempting to think of the dynamics as
semi-classical. However, in the vicinity of the the DP any
classical description would fail. Equivalently, the spin
degrees-of-freedom will show large
uctuations which are
dicult to capture classically. The short time dynamics
is consequently most interesting for situations with ener-
giesE > 0 where both the semi-classical approximation
and the BOA break down, implying that the simulation
0 5 10 15 20 25 30−101
tRα−101Rα (a)
(b)FIG. 5. Bloch vector components Rx(dashed lines) and Ry
(solid lines). For the upper plot (a), the trap has been dis-
placed in th y-direction,xs= 0 andys= 28, while in the lower
plot (b) the displace direction is the perpendicular, xs= 28
andys= 0. In both gures, vx= 10 andvy= 15, and the
average energy E280.
is performed using the full model Hamiltonian (1). For
these energies, the wave packet can traverse the DP and
population transfer between the two adiabatic potentials
V(^px;^py) typically occurs. It is known that such non-
adiabatic transitions can play important roles for the dy-
namics, and that the actual transition probabilities be-
tween the two potentials may be extremely sensitive to
small
uctuations in the state [31, 48]. In this subsection
we especially address such non-adiabatic e ects.
There are indeed several relevant time-scales in the dy-
namics: (i) The spin precession time Tspgives the typical
time for spin evolution and is proportional to the e ective
magnetic eldjB(p)j, (ii) the classical oscillation period
Tcl= 2, and (iii) the thermalization time Tth, which
estimates the time it takes for the system to thermalize,
i.e. when expectation values become approximately time
independent. Normally, the magnitudes of these times
follow the list above (in growing order), except in the
vicinity of the DP where TspTclor evenTsp Tcl
very close to the DP. While the rst two are well de ned,
de ning the last one is non-trivial. We can say that ( i)
and (ii) characterizes short time-time scales, and ( iii)
long time-scales. As will be numerically demonstrated,
the thermalization time turns out to scale as log( h 1)=,
wherehis the e ective dimensionless Planck's constant
andthe maximum Lyaponov exponent. This suggests
that the thermalization time agrees with the Ehrenfest
time
TE= log(V=h)=; (19)
withVthe e ective occupied phase space volume. TE
is also the typical time-scale where semi-classical (TWA)
expectation values no longer agree with quantum expec-
tation values, which can be seen as a breakdown of Ehren-
fest's theorem [49].
From the form of the non-adiabatic coupling (7), it fol-
lows that transitions between the adiabatic states (4) are
restricted to the vicinity of the DP. These non-adiabatic
transitions are manifested as rapid changes in the Bloch7
vector (10). In Fig. 6 we present two examples of the
Bloch vector evolution (in both examples Rz(t)0). In
Fig. 6 (a), the trap has been shifted in the y-direction.
For short times, the shift of the trap induces a build-up of
momentum in the opposite y-direction as a consequence
of the Ehrenfest theorem. This adds with the non-zero
y-component of momentum before the quench. The av-
erage momentum in the x-direction remains zero and as
a consequence Rx(t)0, see Eq. (11).
These dynamics change qualitatively if the trap is
shifted in the x-direction instead of the y-direction. For
suciently large shifts of xs, the wave packet will set o
along the adiabatic potentials and encircle the DP. The
spin dynamics should therefore not display the same type
of \jumps" that appear when the wave packet traverses
the DP. Moreover, since the average momentum in the
x-direction is in general non-zero, Rx(t) will also be non-
zero. The results are demonstrated in Fig. 6 (b). Com-
pared to the rst example in (a), the wave packet does
not spend much time near the DP so the wave packet
delocalization occurs more slowly. To a large extent the
evolution is driven by harmonicity, in contrast to the ex-
ample of Fig. 6 (a) where the anharmonicity of the Born-
Huang term, and the non-adiabatic transitions near the
DP, push the system away from semi-classical evolution.
The gure demonstrates how the dynamics can depend
on the initial conditions, in both (a) and (b), E280 but
the wave packet broadening starts earlier in (a) than in
(b). This type of state-dependence has been discussed in
Ref. [47]; generically there is a period tswhere the width
of the wave packet stays nearly constant, followed by a
rapid broadening. The time-scale tsdepends strongly
on the initial conditions, while the proceeding evolution
aftertsseems pretty generic for chaotic systems.
B. Long time dynamics; thermalization
Whenever we consider an anisotropic SO-coupling,
vx6=vy, from the Figs. 3 and 4 it is clear how the
adiabatic classical model becomes chaotic. Beyond the
adiabatic model, it has been shown [50] that the full
anisotropic model, i.e., E( x+ y) Jahn-Teller model,
is chaotic in the sense of level repulsion [51] of eigenen-
ergies. For the isotropic E"Jahn-Teller model, on
the other hand, the level repulsion e ect is not as evi-
dent, however a weak repulsion also in this model signals
emergence of quantum chaos [52].
The goal of this subsection is to study the long time
dynamics of the system; speci cally if equilibration oc-
curs, and if so, does the equilibrated state mimic a ther-
mal state. A distinguishing property of thermal states is,
for example ergodicity, i.e., the distributions j (x;y;t )j2
andj(px;py;t)j2spread out over their accessible energy
shells. Moreover, for a thermally equilibrated state, the
distributions show seemingly irregular interference struc-
tures on scales of the order of the Planck cells, which
normally become even ner in the Wigner quasi distri-
FIG. 6. (Color online) Distributions j (x;y;t f)j2((a) and
(c)) andj(px;py;tf)j2((b) and (d)) at tf= 400 for the
Rashba SO-coupled model. At time t= 0, the trap is sud-
denly displaced from x0=y0= 16 tox0=y0= 0. The
initial ground state is then quenched into a localized excited
state. The upper two plots (a) and (b) display the results
from full quantum simulations of the adiabatic model (5),
while the lower plots (c) and (d) show the corresponding
semi-classical TWA distributions. The average semi-classical
energy E 192 with a standard deviation E22. The
dimensionless SO-coupling strengths vx=vy= 30.
bution [53{55]. Non-thermalized states, on the contrary,
typically leave much more regular traces of quantum in-
terference in their distributions. While such often sym-
metrical structures are absent for thermalized states, we
will demonstrate that thermalized distributions may still
show clear density
uctuations on scales larger than the
Planck cells. These are examples of quantum scars and
they are remnants of classical periodic orbits [42].
We begin by considering the adiabatic isotropic model
withvx=vy= 30, and trap shifts xs=ys= 16. Af-
ter a quench of the trap position, the initial energy is
E=h^H( )
adi 192. This energy corresponds to the
energy of the Poincar e section presented in Fig. 3. The
resulting distributions are shown in Fig. 7 (a) and (b)
after a propagation time tf= 400 . The nal time tfap-
proximates 60 classical oscillations. Both the real space
densityj (x;y;t )j2and momentum density j(px;py;t)j2
reveal clear interference patterns as anticipated. The DP
at the origin ( px;py) = (0;0) repels the wave function
forming a \hole." The lack of zero momentum states in-
duces a mass
ow in real space and a similar \hole" in
its distribution. The classically energetically accessible8
FIG. 7. (Color online) Same as Fig. 6 but for the anisotropic
SO-coupled model with vx= 20 andvy= 30. The largely
populated regions are so called quantum scars and derive from
properties of the underlying classical model, i.e. they are not
outcomes of some coherent quantum mechanism.
regions are given by
x2+y22Emax+v2
y;
p2
x+p2
y 2q
v2xp2x+v2yp2y2Emax;(20)
whereEmaxis the maximum energy component notice-
ably populated by the state.
The quantum results are compared with the TWA dis-
tributions displayed in the lower plots (c) and (d) of the
same Fig. 7. The same kind of ring-shape is obtained,
and the concentration in density appears at the same lo-
cations for both the quantum and classical simulations.
Expectedly, the quantum interference taking place within
the wave packet is not captured by the TWA. This fol-
lows since single semi-classical trajectories are treated
independently, i.e. added incoherently, while a quantum
wave packet must be considered as one entity. For a TWA
approach of the full isotropic E"Jahn-Teller model (1)
we refer to Ref. [56].
The situation is drastically changed when we break
the rotational U(1) symmetry by assuming vx6=vy. The
result for low initial energy is depicted in Fig. 7 (a) and
(b). The energy is comparable to the potential barrier
separating the two minima in the adiabatic potentials,
and as a consequence, the wave packet is predominantly
localized in the left minima. The density modulations
seems now much more irregular in comparison to Fig. 6.
In the seemingly random density distribution, some clear
density maxima emerge, both in momentum as well as
in real space. These density accumulations derive from
periodic orbitals of the underlying classical model and
FIG. 8. (Color online) Same as Fig. 7 but for an initial energy
E > 0. The dimensionless SO-couplings vx= 14 andvy= 21,
while the shifts xs=ys= 16 giving an average energy E=
h^HSOi36:5.
0100200300∆x(t)
0 10 20 30 40 50 60 70 800100200300
t∆x(t+δ)(a)
(b)h
FIG. 9. Examples of the phase space area x(t) for di erent
h-values (h= 1;2;3; :::; 10). The upper plot (a) gives x(t)
without shifting the time, while for the lower one (b) time has
been shifted by = log(h)=. The arrow indicates increasing
h-values. It is clear how the spread in x(t) between di erent
hvalues is suppressed when we shift the time. The trap shifts
xs=ys= 19 resulting in an energy E 88. The maximum
Lyaponov exponent = 0:18.
are termed quantum scars [42, 57, 58]. The appearance
of scars is an example of the classically chaotic model
leaving a trace in its quantum counterpart. The scars
are also captured in the semi-classical TWA, shown in
Fig. 7 (c) and (d), supporting their classical origin.
When we shift the trap for larger values on xsand
ys, the energy is increased and at some point the BOA
breaks down. An example, obtained from integrating the
full model (1), is presented in Fig. 8. For these higher
energies there are no signs of quantum scars. As for the
situation of Fig. 7, the spread of the wave packet and the
irregular interference patterns indicates thermalization.
This far we have demonstrated thermalization for the
anisotropic SO coupled model, but not discussed corre-
sponding time-scales. One related question is how the
evolution of various expectation values scale with h(di-
mensionless Planck constant). It has been argued that
the Ehrenfest time, Eq. (19), can be a measure of the
thermalization time [31]. We will now explore how the
phase space area (t) = p ( =x; y), where9
−7 −6.5 −6 −5.5 −5 −4.5 −4−0.02−0.0100.010.020.030.04
x|ψ(x,0)|
FIG. 10. (Color online) Sections of j (x;y= 0)jfor di erent
values on the dimensionless Planck's constant h:h= 1 (black
solid line), h= 2 (blue dotted line), and h= 3 (red dashed
line). The nal time tf= 80,xs=ys= 16, and vx= 14
andvy= 20. As a comparison between classical and quan-
tum results, we also include the TWA results as a green solid
line, calculated for h= 1. The green line has been shifted
downward with 0.02 for clarity.
and p are the variances of ^ and ^p respectively,
evolves for di erent values of h. Since x(t) and y(t)
behave similarly we focus only on x(t). For thermaliza-
tion, x(t)y(t) is an e ective measure of the covered
phase space volume, and for large times tit should more
or less approach the accessible phase space volume as the
distribution spreads over the whole energy shell. We have
chosen to study x(t) since it
uctuates relatively little
before reaching its asymptotic value. In Fig. 9 (a) we dis-
play x(t) for 10 di erent values on hranging from h= 1
toh= 10. The arrow in the plot shows the direction of
increasingh's. As is seen, by increasing hthe wave packet
broadening starts earlier and the state equilibrates faster.
If the Ehrenfest time TEsets the typical time scale in the
process, by shifting the time with = log(h)=we should
recover a \clustering" of the curves. This is indeed ver-
i ed in Fig. 9 (b) where the curves have been shifted in
time by. The corresponding Lyaponov exponent has
been optimized in order to minimize the spread in the
curves. The obtained value = 0:18 is somewhat larger
than the numerically calculated one = 0:12 but still of
the same order. The picture also makes clear that the
wave packet broadening kicks in after some time tsas
anticipated above.
The route to thermalization can typically be di-
vided into; ( i) a classical drift, and ( ii) quantum dif-
fusion [31]. The role of the quantum di usion for ther-
malization was analyzed in Ref. [31], where it was found
to \smoothen" the phase space distributions preventing
sub-Planck structures. For the classical drift there is no
lower bound on the neness of density structures that can
form, and characteristic for classical chaotic dynamics is
that ever ner formations build-up as a result of the typ-
ical \stretching-and-folding" mechanism. However, in aquantum chaotic system, when the structures reach the
Planck cell regime, the quantum pressure becomes too
strong and the quantum di usion then prevents any fur-
ther structures to form. Thus, Planck's constant sets
a lower bound for the
uctuations in the distributions.
This quantum smoothening is demonstrated in Fig. 10,
where we plot a section of j (x;y= 0)jfor di erent values
on the scaled dimensionless Planck's constant h(= 1;2;3
for black, blue, and red lines respectively). The e ect is
clearly seen in the gure. A similar pattern is found
(not shown) also for the momentum distributions. For
the classical system, corresponding to h= 0, there is no
lower limit on how ne the structures can be. We indicate
this by also plotting the TWA results in the same gure
as a green line (note that the green line has been shifted
downward in order to separate it from the quantum re-
sults). The number of trajectories used for the gure is
250 000, and if we would like to produce ner structures
(by propagating the system for longer times) we would
need many more trajectories and the simulation would
rapidly become very time consuming.
Related to the above discussion a note on quantum
phase space distributions is in order. It is well known
that sub-Planck structures are common in the Wigner
distribution [53]. This is not contradicting any quan-
tum uncertainty relation. After all, the Wigner distribu-
tion is not a proper probability distribution, despite the
fact that its marginal distributions reproduce the cor-
rect real and momentum space probability distributions.
The Husimi Q-function, while not possessing the proper
marginal distributions, is positive de nite and lacking
singularities, and it is indeed found that the Q-function
does not support sub-Planck structures [60].
We nish this subsection by analyzing the dynamics in
the islands of the Poincar e sections of Fig. 4 where the
classical theory predicts regular evolution. From Fig. 4
(c) we have that for px20 andxy0 the evolu-
tion should be regular. We can achieve such a situation
by using the quench-shifts xs= 20 andys= 0. As for
the examples above, we propagate the state for a time
tf= 400, and the resulting distributions are given in
Fig. 11. The striking di erence with Figs. 7 and 8 is evi-
dent; no irregular structure is apparent, but clear regular
interference patterns are. We have veri ed that the in-
terference structure prevails also after doubling the time,
tf= 800.
C. Proposed experimental realization
Much of the above dynamics can be observed in a sys-
tem of cold atoms with synthetic SO-coupling, for exam-
ple, a system of87Rb with a synthetic eld induced by
the 4-level scheme [33]. In this system, the recoil energy
Er=mv2h50 kHz. The synthetic eld limits the
lifetime of the experiment to tl1s[4, 10]. To push
the experiment into the long time regime, we will use
a trapping frequency of !=2= 30 Hz. These parame-10
FIG. 11. (Color online) Same as Fig. 7 but for the shifts
xs= 20 andys= 0. For the given dimensionless parameters,
the initial state is such that its dynamics should be regular
according to the corresponding Poincar e section, Fig. 4. The
energy E 250.
ters will give a dimensionless value of vy=q
Er
h!11,
withvxtunable between 0 and 11. The large trapping
frequency will provide a sucient number of oscillations
for thermalization to occur. We could consider values of
vy30 by decreasing the trapping frequency to 10Hz,
but then the lifetime of the system may be at the boarder
for thermalization.
The condensate can be adiabatically loaded to one of
the two states at the bottom of the momentum-space
potential, de ned by p=mvy^y. The quench can then
be preformed by shifting the minimum of the real-space
trapping potential. We then let the system evolve until
we reach either the thermalization time, or the lifetime
of the experiment. The momentum distribution can be
measured with a destructive time-of-
ight (TOF) mea-
surement [4, 10], which should reveal thermalization as
well as signatures of quantum scars. Repeated experi-
mental measurements allow for time-resolved calculation
of expectation values. Similarly, the quantum spin jumps
near the DP, as discussed in Sec. IV A, can be observed
using a spin-resolved TOF measurement.
As a nal remark, for a weakly interacting gas we
work near a Feschbach resonance [61]. However, for re-
alistic parameters [62], we estimate a scattering length
as310 9m,N5105atoms, and a transverse
harmonic trapping frequency !z100 Hz. For these
parameters, the characteristic scale of the non-linearity
ish1kHz, which is smaller than the recoil energy
above, suggesting the non-linear term will play only a
minor role. We have numerically veri ed that the results
do not change qualitatively by solving the corresponding
non-linear Gross-Pitaevskii equation. Indeed, we nd the
deviations with a non-linearity are not large enough to
be seen by eye.
V. CONCLUSIONS
In this paper we studied dynamics, deriving from a
quantum quench, in anisotropic SO-coupled cold gases,
focusing primary on aspects arising from the fact thatthe underlying classical model is chaotic. The evolution
of the initially localized wave packet on its way to equili-
bration has been analyzed, and we have shown how a clas-
sical period of limited spreading is followed by a collapse
regime dominated by rapid spreading. After the collapse
period, the wave packet is maximally delocalized, but
still possesses quantum interference structures. At the
Ehrenfest time, the state has approximately equilibrated
as is seen in the decay of expectation values, as well as
seemingly irregular density
uctuations both in real and
momentum space. We showed that the ne structure of
these
uctuations are limited by the quantum di usion,
and thereby the size of the Planck's constant h. For the
isotropic model, after the collapse no thermalization is
found, as is expected from the integrability of the under-
lying classical model.
For smaller energies, when the wave packet predom-
inantly populates one of the dual potential wells, ther-
malization is again seen. Here, however, an additional
phenomenon appears in terms of quantum scars. These
density enhancements emerge along classically periodic
orbits. They are classical in nature and long lived. Quan-
tum scars have also been studied in di erent cold atom
settings; atoms in an optical lattice and con ned in an
anisotropic harmonic trap [58]. The results on thermal-
ization presented in this work is most likely also applica-
ble to the set-up of Ref. [58]. We also demonstrated that
for certain ne tuned initial states, the dynamics stays
regular even in the anisotropic model. In the classical
picture, these solutions correspond to the ones belong-
ing to regular islands in the otherwise chaotic Poincar e
sections.
We argue that the present system is ideal for studies
of quantum chaos and quantum thermalization for nu-
merous reasons. The system parameters can be tuned
externally by adjusting the wavelength of the lasers in-
ducing the SO-coupling, and as we discussed in Sec. IV C
the SO dominated regime is reachable in current exper-
iments. Moreover, both state preparation and detection
are relatively easily performed in these setups. Equally
important, the system is well isolated from any environ-
ment and coherent dynamics can be established up to
hundreds of oscillations which is well beyond the themral-
ization time. The energy of the state is simply controlled
by the trap displacement, and it should for example be
possible to give the system small energies such that the
atoms reside mainly in one potential well where quantum
scars develop.
We nish by pointing out that the present model is also
di erent from most earlier studies on quantum thermal-
ization [18, 24] in the sense that the dynamics is essen-
tially \single-particle" and not arising from many-body
physics. Related to this, we have numerically veri ed
that adding a non-linear term gj (x;y;t )j2to the Hamil-
tonian does not change our results qualitatively for mod-
erate realistic interaction strengths g. In order to enter
into the regime where interaction starts to a ect the re-
sults, one would need a condensate with a large number11
of atoms (millions of atoms) or alternatively externally
tune the scattering length via the method of Feshback
resonances.
ACKNOWLEDGMENTS
The authors thank Ian Spielman for helpful comments.
SFB/TR 12 is acknowledged for nancial support. JLacknowledges Vetenskapsr adet (VR), DAAD (Deutscher
Akademischer Austausch Dienst), and the Royal Re-
search Council Sweden (KVA) for nancial help. BA
acknowledges the sponsorship of the US Department of
Commerce, National Institute of Standards and Technol-
ogy, and was supported by the National Science Founda-
tion under Physics Frontiers Center Grant PHY-0822671
and by the ARO under the DARPA OLE program.
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2207.06347v1.Giant_orbital_Hall_effect_and_orbital_to_spin_conversion_in_3d__5d__and_4f_metallic_heterostructures.pdf | PHYSICAL REVIEW RESEARCH 4, 033037 (2022)
Giant orbital Hall effect and orbital-to-spin conversion in 3 d,5d,a n d4 fmetallic heterostructures
Giacomo Sala*and Pietro Gambardella†
Department of Materials, ETH Zurich, 8093 Zurich, Switzerland
(Received 4 April 2022; revised 21 June 2022; accepted 23 June 2022; published 13 July 2022)
The orbital Hall effect provides an alternative means to the spin Hall effect to convert a charge current into
a flow of angular momentum. Recently, compelling signatures of orbital Hall effects have been identified in 3 d
transition metals. Here, we report a systematic study of the generation, transmission, and conversion of orbitalcurrents in heterostructures comprising 3 d,5d,a n d4 fmetals. We show that the orbital Hall conductivity of
Cr reaches giant values of the order of 5 ×10
5[¯h
2e]/Omega1−1m−1and that Pt presents a strong orbital Hall effect in
addition to the spin Hall effect. Measurements performed as a function of thickness of nonmagnetic Cr, Mn, andPt layers and ferromagnetic Co and Ni layers reveal how the orbital and spin currents compete or assist eachother in determining the spin-orbit torques acting on the magnetic layer. We further show how this interplaycan be drastically modulated by introducing 4 fspacers between the nonmagnetic and magnetic layers. Gd and
Tb act as very efficient orbital-to-spin current converters, boosting the spin-orbit torques generated by Cr by afactor of 4 and reversing the sign of the torques generated by Pt. To interpret our results, we present a generalizeddrift-diffusion model that includes both spin and orbital Hall effects and describes their interconversion mediatedby spin-orbit coupling.
DOI: 10.1103/PhysRevResearch.4.033037
I. INTRODUCTION
The interconversion of charge and spin currents underpins
a variety of phenomena and applications in spintronics, in-cluding spin-orbit torques, spin pumping, the excitation ofmagnons, and the tuning of magnetic damping [ 1,2]. The spin
Hall effect (SHE) mediates this interconversion through thecombination of intrinsic and extrinsic scattering processes,all of which require sizable spin-orbit coupling [ 3]. Recent
theoretical work has shown that the intrinsic SHE is accompa-nied by a complementary process involving the orbital angularmomentum, the so-called orbital Hall effect (OHE), whichconsists in the flow of orbital momentum perpendicular tothe charge current [ 4–10]. According to theoretical calcula-
tions, the OHE is more common and fundamental than theSHE because it does not require spin-orbit coupling and canthus occur in a wider range of materials. The intrinsic SHEthen emerges as a by-product of the OHE resulting from theorbital-to-spin conversion in materials with nonzero spin-orbitcoupling. In this case, the spin Hall conductivity has the samesign as the product between the orbital conductivity and theexpectation value of spin-orbit coupling: σ
S∼σL/angbracketleftL·S/angbracketright.T h e
OHE was first predicted in 4 dand 5 dtransition elements
[11,12] and recently in light metals [ 4] and their interfaces
*giacomo.sala@mat.ethz.ch
†pietro.gambardella@mat.ethz.ch
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.[13] as well as in two-dimensional (2D) materials [ 14,15].
The theoretical orbital Hall conductivity of light elements iscomparable to or even larger than the spin Hall conductivityof Ta, W, and Pt, which provide a strong SHE [ 4]. The OHE
is thus intrinsically more efficient than the SHE, and orbitalcurrents are expected to contribute to magnetotransport effectssuch as the anisotropic, spin Hall, and unidirectional mag-netoresistance as well as spin-orbit torques [ 6,16–20]. The
ubiquity and strength of the OHE, besides making it funda-mentally interesting, broaden the material palette available forspintronic applications and provide an additional handle tooptimize the efficiency of spin-orbit torques. Yet, differentlyfrom spins, nonequilibrium orbital currents do not coupledirectly to the magnetization of magnetic materials and cantorque magnetic moments only indirectly through spin-orbitcoupling [ 19–21]. Optimizing the orbital-to-spin conversion
is thus a prerequisite for taking advantage of large orbitalcurrents.
The prediction of the OHE in light elements has triggered
intense research on current-induced orbital effects. Recentexperiments have identified signatures of the OHE in ma-terials with low [ 18,19] and high [ 20] spin-orbit coupling
and revealed its contribution to spin-orbit torques [ 16,18],
whose strength can be tuned by improving the orbital-to-spinconversion ratio [ 19,22]. However, experimental values of the
orbital Hall conductivity are smaller than theoretical estimates[16,18,19,23], and a systematic investigation of orbital effects
as a function of the type and thickness of nonmagnetic, ferro-magnetic, and spacer layers is still missing.
Here, we present a comprehensive study of the interplay of
the OHE and SHE in structures combining different light andheavy nonmagnetic metals (NM =Cr, Mn, Pt), ferromagnets
(FM=Co, Ni), and rare-earth spacers ( X=G d ,T b ) .W e
2643-1564/2022/4(3)/033037(14) 033037-1 Published by the American Physical SocietyGIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)
(a) (b)
(a)LS
MTSTLSJCxz
L SE
MTSTLSLSJC E
NM
FML
S
FIG. 1. (a) The spin Hall effect and orbital Hall effect induced
by an electric field Ein a nonmagnet (NM) produce spin ( TS)a n d
orbital ( TLS) torques on the magnetization Mof an adjacent ferro-
magnet (FM). The strength of the torques depends on the intensityof the spin and orbital currents and on the spin-orbit coupling of the
ferromagnet. The schematic shows the direction of the induced spin
(S, blue dots and crosses) and orbital ( L, red circling arrows) angular
momenta when the spin and orbital Hall conductivities σ
S,L>0.
(b) The insertion of a spacer layer may increase the orbital torque
relative to the spin torque by converting the orbital current (red) intoa spin current (blue) prior to their injection into the ferromagnet.
provide evidence of the OHE in Pt and Mn and report giant
values of the orbital Hall conductivity in Cr, which extrapolateto the theoretical limit of 10
6[¯h
2e](/Omega1m)−1in Cr films thicker
than the orbital diffusion length [ 4], which we estimate to
be/greaterorsimilar20 nm. Because of the simultaneous presence of strong
OHE and SHE in Pt and Cr, we argue that experimental resultsare best described by a combined spin-orbital conductivityrather than by separating the two effects. We show that theinterplay between orbital and spin currents can be tailoredby varying the thickness of the ferromagnetic layer as wellas by inserting a Gd or Tb conversion layer between thenonmagnet and the ferromagnet. Rare-earth spacers do notgenerate significant spin-orbit torques by themselves, but theyenhance the torque efficiency up to four times when Cr isthe source of spin and orbital currents and reverse the signof the torques generated by Pt. The latter effect is attributedto the OHE overcoming the SHE in Pt. Finally, we present aphenomenological extension of the spin drift-diffusion modelthat includes orbital effects and the conversion between spinand orbital moments, which accounts for both the thicknessdependence and sign change of the spin-orbit torques gen-erated by the interplay of OHE and SHE in NM/FM andNM/ X/FM heterostructures.
II. BACKGROUND
According to the theory of the OHE, an electric field ap-
plied along the xdirection in a material with orbital texture
inkspace induces interband mixing that results in electron
states with finite orbital angular momentum [ 5–7]. Electrons
occupying these nonequilibrium states carry the angular mo-mentum as they travel in real space. Therefore, although thetotal orbital momentum vanishes, a nonzero orbital current isproduced along the z(y) direction with orbital polarization
parallel to ±y(±z), similar to the SHE [see Fig. 1(a)]. Thelatter occurs concomitantly with the OHE when the nonmag-
net has nonzero spin-orbit coupling /angbracketleftL·S/angbracketright
NM. The primary
spin current injected into the adjacent ferromagnet exerts adirect torque on the local magnetization (spin torque). Or-bitals, instead, act indirectly through the spin-orbit couplingof the ferromagnet that converts the orbital current into asecondary spin current. We refer to the torque generated bythis secondary spin current as orbital torque. The (indepen-dence) dependence of the (spin) orbital torque on /angbracketleftL·S/angbracketright
FM
is the key difference between SHE and OHE. In the SHE
scenario, the angular momentum is entirely generated in thenonmagnet, and the ferromagnet behaves almost as a passivelayer since it only contributes to the properties of the NM/FMinterface. In contrast, the OHE in a NM/FM bilayer dependson both the interfacial and bulk properties of the ferromagnet,which is directly involved in the torque generation. Sincethe orbital conductivity is typically large [ ≈10
5(/Omega1m)−1][4]
but the spin-orbit coupling of 3 dferromagnets is relatively
weak [ 24], the orbital torque efficiency in NM/FM bilayers
is finite but small. Alternatively, the orbital torque may beenhanced by realizing most of the orbital-to-spin conversionin a spacer layer sandwiched between the nonmagnet and theferromagnet [Fig. 1(b)]. The effectiveness of this approach
depends on the conversion efficiency of the spacer, its spinand orbital diffusion lengths, and the quality of the additionalinterfaces, as discussed later.
Here, we summarize fundamental theoretical predictions
and experimental confirmations of the OHE. We list ap-proaches to distinguish orbital and spin effects by meansof torque measurements in heterostructures with differentelements, thickness, and stacking order. Furthermore, we es-tablish a parallel between known spin-transport effects andpossible orbital counterparts that have not been observed yetbut could contribute to answering open questions about orbitaltransport.
(i) Large orbital Hall conductivities have been predicted in
several 3 d,4d, and 5 dtransition elements [ 4,11,12] and 2D
materials [ 9,10]. Experimental evidence is so far limited to Cr
[19,25], Cu [ 20,21,26], Zr [ 18], and Ta [ 20]. Recent experi-
ments on V [ 23,27] can also be reinterpreted in light of the
OHE. The coexistence of the OHE and the SHE, especially inheavy metals, makes it difficult to distinguish the two effects.
(ii) The spin and orbital torques are expected to add
constructively (destructively) when /angbracketleftL·S/angbracketright
NM·/angbracketleftL·S/angbracketrightFM>0
(<0). This competition can be tailored by properly choosing
the ferromagnet, as recently observed in Refs. [ 19,20].
(iii) In a NM/FM bilayer, the orbital Hall efficiency should
depend on the thickness of both the nonmagnet ( tNM) and
the ferromagnet ( tFM). In contrast, the spin Hall efficiency is
nominally independent of the latter and results in an inversedependence of the spin torque on t
FM[1]. The dependence
of the orbital Hall efficiency on tNMhas been addressed in
Ref. [ 19], but the role of tFMis still unknown.
(iv) The spin diffusion in transition metals with strong
SHE is typically limited to a few nanometers [ 28]. Al-
though recent measurements suggest longer orbital dif-fusion lengths [ 19,29], the length scale of the orbital
diffusion and its conversion into spins remain to be es-tablished. These quantities and the nature of the mech-anisms underlying the orbital scattering may be ad-
033037-2GIANT ORBITAL HALL EFFECT AND ORBITAL-TO-SPIN … PHYSICAL REVIEW RESEARCH 4, 033037 (2022)
dressed by torque measurements in thick nonmagnetic
films and by nonlocal transport measurements, which couldalso verify the existence of the inverse OHE.
(v) Spacer layers between the nonmagnet and the ferro-
magnet can alter spin torques in several ways, namely, byintroducing an additional interface with different spin scat-tering properties, by suppressing the spin backflow, and bymodifying the spin memory loss [ 1,30–33]. Such effects are
expected to influence also the orbital torque. In addition,spacers can either increase or decrease the orbital torquedepending on the sign of their orbital and spin Hall conduc-tivities, and spin-orbit coupling, which converts orbitals intospins and vice versa. Pt spacers have been shown to increasethe orbital torques in light metal systems [ 19,21]; however, Pt
is also a well-known SHE material. A systematic investigationof the enhancement or suppression of spin and orbital currentsin materials with different combinations of orbital and spinconductivities is required.
(vi) The spin diffusion in multilayer structures is usu-
ally modeled by semiclassical drift-diffusion equations thataccount for, e.g., spin backflow at interfaces, the spin-orbittorque dependence on the thickness of the nonmagnet, and thespin Hall and unidirectional magnetoresistance [ 34–39]. The
model has not been extended yet to the OHE, which requiresthe inclusion of the spin-orbital interconversion mediated byspin-orbit coupling.
(vii) The orbital transmission at the NM/FM interface is
more sensitive to the interface quality than spins and, hence,to growth conditions and stacking order [ 7,18]. It is an open
question whether the transmission can be described by a singleparameter equivalent to the spin-mixing conductance, whichwe dub orbital mixing conductance.
(viii) The SHE generates dampinglike and fieldlike spin-
orbit torques of comparable strength [ 1,40]. So far, no
theoretical or experimental work has determined with cer-tainty the relative magnitude of the two components ofthe orbital torque. Assessing their strength may help us tounderstand the mechanism of accumulation, transfer, and con-version of orbitals.
(ix) The OHE has been attributed to an intrinsic scattering
mechanism in elements with orbital texture. The analogy withthe SHE [ 41,42] suggests that also extrinsic processes may
contribute to the generation of orbital currents. Measuring theorbital Hall efficiency as a function of the element resistivitymay reveal extrinsic orbital effects.
(x) The transmission and absorption of spins and orbitals
at the interface with an insulating ferromagnet [ 43], e.g.,
yttrium iron garnet (YIG), may be fundamentally differentsince the latter do not interact with the magnetization. Earlyexperiments reported spin pumping effects in YIG/light metalbilayers, but they were interpreted in terms of the inverse SHE[44].
(xi) The generation and accumulation of orbitals at the
NM/FM interface can modulate the longitudinal resistance bythe combination of direct and inverse OHE, as recently foundin Ref. [ 22]. Compared with the spin Hall magnetoresistance
[35], such orbital Hall magnetoresistance may have a different
dependence on the type of ferromagnet, its thickness, and thethickness of the nonmagnet.(xii) Orbital accumulation might also give rise to a unidi-
rectional magnetoresistance, in analogy to the unidirectionalspin Hall magnetoresistance [ 45]. The underlying mechanism,
however, would be intrinsically different since orbitals wouldnot directly alter the magnon population, whereas orbital-dependent scattering might contribute to the conductivityin addition to spin-dependent scattering [ 46]. On the other
hand, the injection into the ferromagnet of electrons withfinite orbital momentum may induce an additional source oflongitudinal magnetoresistance analogous to the anisotropicmagnetoresistance [ 17].
Orbital effects are thus rich and intertwined with spin
transport, allowing for additional means to tune the spin-orbittorque efficiency as well as to understand the transport ofangular momentum in thin-film heterostructures. In the fol-lowing, we address points (i)–(vii) listed above. We providecomprehensive evidence for the occurrence of giant OHEsin 3dand 5 dtransition metals, reveal the interplay of the
OHE and SHE in ferromagnets of variable thickness withand without spacer layers, and establish a phenomenologicalframework to analyze and efficiently exploit the interplay ofspin and orbital currents in metallic heterostructures.
III. EXPERIMENTS
We studied NM/FM and NM/ X/FM multilayers where NM
=Cr, Mn, or Pt, FM =Co or Ni, and X=Gd or Tb.
The samples were grown by magnetron sputtering on a SiNsubstrate, capped with either Ti(2) or Ru(3.5) (thicknessesin nanometers), and patterned in Hall-bar devices by opticallithography and lift-off. All samples have in-plane magneti-zation. Current-induced spin-orbit torques were quantified bythe harmonic Hall voltage method [ 40] using angle-scan mea-
surements [ 51]. We detected the first- and second-harmonic
Hall voltage while applying an alternate current with 10 Hzfrequency and rotating a constant magnetic field in the easyplane of the magnetization [ xyplane; see Fig. 2(a)]. The
harmonic signals were measured as a function of currentamplitude and field strength [Figs. 2(b) and2(c)]. The second-
harmonic resistance depends on the field angle φasR
2ω
xy=
/Theta1Tcosφ+/Phi1T(2 cos3φ−cosφ). Here, /Theta1Tis the sum of the
dampinglike spin-orbit field BDLand contribution from the
thermal gradient along z, and /Phi1Tdepends on the fieldlike
spin-orbit field BFLand the Oersted field BOe. Thus the anal-
ysis of R2ω
xymeasured at different magnetic fields allows for
the separation of torques and thermal effects, yielding themagnitude of the spin-orbit fields for a given electric field[51]. These spin-orbit fields exert spin and orbital torques
on the magnetization T
DL=MsBDLm×(p×m) and TFL=
MsBFLm×p, where pis the net spin polarization direction,
mis the magnetization vector, and Msis the saturation mag-
netization. In the following, we consider uniquely BDLsince,
apart from Ni/Cr and Co/Pt samples, BFLwas too small to
distinguish from the Oersted field. The difficult detection ofB
FLin our samples originates from the very small planar Hall
coefficient (of the order of 1 m /Omega1) to which /Phi1Tis propor-
tional. To compare samples with different elements, thickness,and stacking order, we converted B
DLinto a spin-orbital
033037-3GIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)
(b) (c)JCzmϕ Vxy(a)B
xy
0.0 0.5 1.0 1.5 2.0-2.2-2.02.02.22.4 Co/Cr(12)
Ni/Cr(12)ΘT/RAHE-3 (10 )
1/Beff (1/T)0 60 120 180 240 300 3600.00.10.2 180 mT
1350 mTRxy2ω (mΩ)
ϕ (° )
FIG. 2. (a) Schematic of the harmonic Hall effect measurements.
An alternating current Jcflows along the Hall bar and generates
transverse first- and second-harmonic Hall signals that depend on
the angle φrelative to xof a magnetic field Bof constant ampli-
tude. (b) Representative second-harmonic Hall resistance measuredin Co(2) /Cr(12) during the rotation of Bin the xyplane. The solid
lines are fits to the function R
2ω
xy=/Theta1Tcosφ+/Phi1T(2 cos3φ−cosφ)
(see text). (c) Dependence of /Theta1T(dampinglike field BDL+ther-
mal signal) normalized to the anomalous Hall resistance ( RAHE)
on the effective field given by the sum of the applied magnetic
field and demagnetizing field. Data are shown for Co(2) /Cr(12) and
Ni(4)/Cr(12). The slope of the linear fit (solid lines) is proportional
toBDL, while its intercept with the vertical axis corresponds to the
thermal contribution, which is field independent and can be easilydistinguished.
conductivity according to the formula
ξLS=2e
¯hMstFMBDL
E, (1)
where eis the electron charge, ¯ his Planck’s constant, tFMis
the thickness of the ferromagnet, and E=ρJcis the applied
electric field ( ρis the longitudinal resistivity, and Jcis the
current density) [ 1,52]. The normalization to the applied elec-
tric field avoids the ambiguities intrinsic to the calculation ofthe current density in a heterostructure. Since in our samplesthe ferromagnet lies below the nonmagnet, we invert the signof the measured B
DLto follow the convention that Pt has
positive spin Hall conductivity. In the literature, Eq. ( 1)i s
usually referred to as the spin Hall conductivity orspin-orbit
torque efficiency , which is related to the effective spin Hall
angle of the NM layer by θLS=ρξLS(Appendix A). Here,
we point out that, when the SHE and OHE are consideredtogether, both spin and orbital currents influence ξ
LSand their
individual quantitative contributions cannot be disentangledbecause the spin-orbit torques depend on the total nonequi-librium spin angular momentum in the ferromagnet (primaryspins+converted spins) but not on the orbital component.
This reasoning implies the impossibility of determining sep-arately the spin and orbital Hall conductivities of a materialby measuring nonequilibrium effects on an adjacent ferro-magnet, even for transparent interfaces. Thus we call ξ
LSthe
spin-orbital conductivity and θLSthe spin-orbital Hall angle.
However, spin and orbital effects can still be distinguished(b)σ > 0S σ > 0LL S
σ < 0S σ > 0LL S0.10.2 Ni/Cr
Ni/Mn
02468 1 0 1 2 1 4 1 6-2-10
Co/Cr
Co/MnξLS (105 Ω-1m-1)
t (nm)NMNM02468 1 0 1 201234
Co/Pt
Ni/PtξLS (105 Ω-1m-1)
t (nm)(a)
FIG. 3. (a) Spin-orbital conductivity as a function of the thick-
ness of the Pt layer in Co(2) /Pt(tNM) and in Ni(4)/Pt(5). The solid
line is a fit to the drift-diffusion equation [Eq. ( 2)]. The sign of
the spin and orbital Hall conductivities in the nonmagnet is indi-
cated and color-coded in the schematic representing the generation,transmission, and conversion of orbital (red) and positive (blue) or
negative (white) spin currents. (b) The same as (a) in FM /Cr(t
NM)
and FM /Mn(tNM), where FM =Co(2) or Ni(4).
at a qualitative level, as discussed in the following. We also
note that a finite OHE could explain, at least in part, the largevariability of the spin-orbit torque efficiency found in sampleswith different ferromagnets, thicknesses, stacking order, andpreparation conditions [ 1].
IV . OHE in Cr, Mn, and Pt
A. Dependence of ξLSon the thickness of the NM layer
Figure 3compares ξLSmeasured in FM/NM bilayers,
where FM is an in-plane magnetized Co(2) or Ni(4) layer andNM is a Cr, Mn, or Pt layer of variable thickness t
NM.W e
find that the two 3 dlight metals generate sizable spin-orbit
torques, similar to previous measurements in materials withweak spin-orbit coupling such as V , Cr, and Zr [ 16,23,44,53].
The torques are remarkably strong in Cr-based samples, forwhich ξ
LSreaches values similar to those for Co/Pt. To the best
033037-4GIANT ORBITAL HALL EFFECT AND ORBITAL-TO-SPIN … PHYSICAL REVIEW RESEARCH 4, 033037 (2022)
TABLE I. Sign of the spin-orbit coupling /angbracketleftL·S/angbracketrightand orbital
and spin Hall conductivity σL,Sin selected transition metals (see
Refs. [ 4,11,12,47–49]). A positive orbital (spin) Hall conductivity
means that a charge current along +xinduces an orbital current (spin
current) along +zwith orbital (spin) angular momentum along −y
[50].
Cr Mn Co Ni Gd Tb Pt
/angbracketleftL·S/angbracketright –– ++ –– +
σL ++ + +
σS –– ++ –– +
of our knowledge, this is the highest torque efficiency reported
in the literature for a FM/NM bilayer made of light elements.However, the dependence of ξ
LSon the type of ferromagnet
and on tNMis very different in Cr and Mn with respect to Pt.
Co/Pt(tNM) and Ni/Pt(5) have torque efficiencies of com-
parable magnitude and identical sign. In contrast, when Cror Mn is used, ξ
LSchanges sign when Co is replaced with
Ni [see Fig. 3(b)]. A comparison between Fig. 3and Table I
indicates that Co/Cr and Co/Mn behave as expected withinthe framework of the SHE, namely, the sign of the torques isopposite to Co/Pt because the spin Hall conductivity σ
Shas
opposite sign in Cr and Mn relative to Pt. The same argu-ment, however, cannot explain the positive sign of ξ
LSin the
Ni-based samples since the direction of the spin polarizationinduced by the SHE is fixed and determined by /angbracketleftL·S/angbracketright
NM.
The sign change can be accounted for only by consideringthe OHE and the opposite sign of the spin and orbital Hallconductivities of Cr and Mn. In this case, the negative ξ
LS
measured in Co/Cr and Co/Mn indicates that in these samples
the spin torque overwhelms the orbital torque. The positivespin-orbital conductivity found with Ni shows instead that theorbital-to-spin conversion in this ferromagnet is so efficientas to make the orbital torque stronger than the spin torque[19]./angbracketleftL·S/angbracketright
FMis indeed predicted to be larger in Ni than in
Co and positive [ 7,20,54]; thus a larger amount of the orbital
current can be converted into a spin current of opposite signto the primary spin current generated by Cr or Mn. Thereforethe torques exerted on Co are mostly generated outside theferromagnet thanks to the orbital-to-spin conversion occurringin the nonmagnet. In contrast, the torques on Ni result fromthe orbital-to-spin conversion inside the ferromagnet.
The variation of ξ
LSwith the thickness tNMof Cr and
Mn is also different from the thickness dependence of thetorque efficiency in heavy elements [ 1,52]. In Co/Pt( t
NM),ξLS
saturates at about 9 nm [Fig. 3(a)]. The fit to the drift-diffusion
equation
ξLS(t)=σLS/bracketleftbigg
1−sech/parenleftbiggt
λ/parenrightbigg/bracketrightbigg
(2)
yields a diffusion length λ=2.2 nm and an intrinsic spin-
orbital Hall conductivity σLS=3.5×105[¯h
2e](/Omega1m)−1.T h i s
value, which agrees with previous works [ 1,3,41,52,55], as-
sumes a transparent interface and is thus an underestimationof the intrinsic spin-orbital Hall conductivity of Pt. In Cr andMn,ξ
LSincreases with tNMand does not saturate, even at
tNM=15 nm. The intrinsic spin-orbital Hall conductivity ofCr is thus significantly larger than ξLSreported in Fig. 3(b).
Indeed, fitting ξLSin Co/Cr(tNM) with λfixed in the range 15–
25 nm yields 5 ×105<|σLS|<12×105(/Omega1m)−1, in good
agreement with the predicted giant orbital Hall conductivityof Cr [ 4].
The trend of ξ
LS(tNM) hints at two alternatives. The first
possibility is that the spin ( λS) and orbital ( λL) diffusion
lengths of Cr and Mn are larger than the typical spin diffusionlength of heavy elements. For example, λ
Sis found to be
about 13 and 11 nm in Cr and Mn, respectively, in Ref. [ 44],
whereas λS=1.8 nm and λL=6.1 nm in Cr according to
Ref. [ 19]. Alternatively, we argue that it suffices to have a
large orbital diffusion length and a nonzero /angbracketleftL·S/angbracketrightNMfor spins
to accumulate over long distances, even if the spin diffusionlength in the nonmagnet is short (see Sec. VI). Spin torque
measurements cannot distinguish between the two possibili-ties. Nonetheless, the trends in Fig. 3suggest the possibility
to increase the spin-orbital conductivity in FM/Cr sampleswith large t
NMup to and beyond the maximal efficiency of
Co/Pt. This possibility has gone unnoticed so far because thinnonmagnetic films ( t
NM≈5 nm) are typically considered in
torque measurements.
A very long orbital diffusion length in Mn may also ex-
plain why ξLSis smaller in Mn than in Cr at any thickness
and independently of the ferromagnet. This result contrastswith theoretical calculations that predict large and comparableorbital conductivities in Cr and Mn [ 4] but agrees with the spin
pumping measurements of Ref. [ 44]. We notice that Ref. [ 4]
considered the bcc structure to calculate the orbital conduc-tivity of Mn, but different crystalline phases can compete andcoexist in Mn thin films [ 56]. This difference may account for
the small experimental value of ξ
LS. Alternatively, the small
spin-orbital conductivity may be determined by a differentquality of the FM/Cr and FM/Mn interfaces, to which theorbital current is very sensitive [ 7,18], possibly because of
Co and Mn intermixing [ 57]. Owing to the larger resistivity
of Mn compared with Cr, however, we note that the effectivespin-orbital Hall angle of Co(2)/Mn(9) is θ
LS=−0.03, which
is comparable to θLS=−0.05 of Co(2)/Cr(9) (Appendix A).
We also notice that interfacial effects (interfacial torques,
spin memory loss, and spin transparency) can influence thestrength of the torques and hence the spin-orbital conductivity,as shown by the different ξ
LSmeasured in Co/Pt and Ni/Pt
samples [ 58]. However, interfacial effects cannot explain our
results, namely, the sign change of ξLSwith the ferromagnet
and its monotonic increase with tNM, because they should be
independent of the thickness of the nonmagnet and becomenegligible in thick films.
Overall, these measurements provide strong evidence of
the OHE and orbital torques in Cr and Mn, in agreement withtheoretical predictions and previous studies of Cr-based sam-ples [ 4,19,25]. Additionally, they show that the spin-orbital
diffusion length is much longer in light elements than in Pt,a difference that could be exploited to boost the effectivespin-orbital conductivity beyond the limit of FM/Pt samples.
B. Dependence of ξLSon the thickness of the FM layer
Theoretical calculations of the spin and orbital transfer at
the FM/NM interface predict a different dependence of the
033037-5GIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)
spin and orbital torque on the thickness tFMof the ferromagnet
[59]. The former is dominant when tFMis small, whereas the
orbital torque can be comparable to or larger than the spintorque in thick ferromagnets. As a consequence, the totaltorque may change sign when t
FMincreases if /angbracketleftL·S/angbracketrightNM·
/angbracketleftL·S/angbracketrightFM<0, as, for instance, in the case of Co/Cr. To test
this possibility, we measured the torque on the magnetizationof Co( t
FM)/Cr(9) and Co( tFM)/Pt(5) as a function of tFM.
Figures 4(a) and4(b) show the torque per unit electric field
calculated as T=MsBDL
E. The sign of the torque is opposite
in the two sets of samples and does not change in the exploredthickness range. This result might suggest that in Co( t
FM)/Cr
the orbital torque is always negligible compared with thespin torque. However, a careful analysis indicates a differentscenario. After taking into account the dead magnetic layer(0.5 and 0.3 nm in the samples with Cr and Pt, respectively;see Appendix B), we tentatively fit the dependence of Ton the
ferromagnet thickness to ∼1/t
FM. This scaling should reflect
the inverse proportionality of the torque amplitude to the mag-netic volume when the current-induced angular momentumis generated outside the ferromagnet. In this case, the spinHall conductivity is constant and solely determined by thecharge-to-spin conversion efficiency of the nonmagnet [ 1],
and Eq. ( 1) yields
T=¯h
2eξLS
tFM. (3)
Equation ( 3) captures well the variation of the torque only
fortFM<1–2 nm, in both Co( tFM)/Cr and Co( tFM)/Pt. The
discrepancy at large thicknesses suggests the presence of atorque mechanism additional to the spin current injection fromthe nonmagnetic layer. This possibility is corroborated by thethickness dependence of the spin-orbital conductivity, whichis different in the two series of samples. In Co( t
FM)/Pt,|ξLS|
is approximately constant up to 3 nm and increases at larger
tFMby about 20% [see Fig. 4(c)]. In Co( tFM)/Cr, instead,
|ξLS|initially increases as the ferromagnet becomes thicker,
possibly due to the formation of a continuous Co/Cr interface;then it decreases starting from t
FM=1 nm and drops by more
than 50% at tFM=3 nm relative to the maximum. Beyond this
thickness, it remains approximately unchanged. The distinctthickness dependence in Co( t
FM)/Cr and Co( tFM)/Pt cannot be
ascribed to strain [ 60] since Co is grown on an amorphous
substrate. In addition, strain-induced effects should be similarin the two sample series. Moreover, it cannot be attributed to avariation of the interface quality. Since the latter is expected toimprove as Co becomes thicker, the spin-orbital conductivityshould increase or remain approximately constant for t
FM>
1 nm. Furthermore, we exclude that the measured trend de-pends on uncertainties in the saturation magnetization due toproximity effects since ξ
LSdepends on the areal magnetization
[see Eq. ( 1)], which is free from ambiguity (see Appendix B).
Finally, we rule out self-torques due to the SHE inside the Colayer [ 49] since control measurements in Co(7)/Ti(3) do not
give evidence of torques within the experimental resolution.
Alternatively, we propose that the decrease of the spin-
orbital conductivity with t
FMin Co/Cr results from the
competition between spin and orbital torques in the ferromag-net. As sketched in Figs. 4(d) and4(e), the spin and orbital
currents J
SandJLdecay inside the ferromagnet on a lengthzJL
JSJLS
zJL
JSJLS
tFM tFM|ξ |LS(a) (b)
(c)
(d) (e)
|ξ |LS02468 1 0 1 2-1.6-0.4-0.200.2
Co/Cr Co/PtT (ΩTm-2)
tFM (nm)
0123456789 1 0 1 1 1 201234|ξLS| (105 Ω-1m-1)
t (nm)FM02468 1 0 1 2-0.04-0.0200.020.04T (ΩTm-2)
tFM (nm)
FIG. 4. (a) Dependence of the spin-orbit torque normalized to the
applied electric field on tFMin Co( tFM)/Cr(9) and Co( tFM)/Pt(5). The
solid lines are fits to1
tFM. (b) Enlarged view of (a). (c) Dependence of
|ξLS|ontFMin the two sample series. (d) and (e) Schematics showing
qualitatively the interplay of the spin JSand orbital JLcurrents, which
are injected into the ferromagnet from the interface with the nonmag-
netic metal and decay with the distance z. Part of the orbital moments
is converted into spin moments and generates a spin current JLSwith
the same (opposite) polarization as the primary spin current in Pt/Co
(Cr/Co). JSyields the spin torque, and JLSyields the orbital torque.
The spin-orbital conductivity ξLSis constant when the orbital-to-spin
conversion is negligible (dashed line). It increases with tFMwhen JS
andJLSadd up and decreases when JSandJLScompete (solid line).
scale determined by the respective dephasing lengths. In the
absence of orbital-to-spin conversion, the spin-orbital conduc-tivity, which depends on the absorption of the injected spincurrent ξ
LS∼JS(0)−JS(tFM), increases rapidly with tFMand
033037-6GIANT ORBITAL HALL EFFECT AND ORBITAL-TO-SPIN … PHYSICAL REVIEW RESEARCH 4, 033037 (2022)
remains constant afterwards because spin dephasing occurs
within a few atomic layers from the interface [ 61,62]. On the
other hand, if we assume that the orbital current is transmittedover a distance longer than its spin counterpart [ 59] and that
part of it is also converted into a spin current J
LS, then ξLS∼
JS(0)−JS(tFM)±JLS(tFM) can increase or decrease with tFM
depending on the relative sign of JSandJLS, i.e., on the prod-
uct/angbracketleftL·S/angbracketrightNM·/angbracketleftL·S/angbracketrightFM. Since the latter is positive (negative)
in Co( tFM)/Pt [Co( tFM)/Cr], our qualitative model envisages
an increase (decrease) of the spin-orbital conductivity with
tFM, in agreement with our measurements and the thickness
dependence predicted in Ref. [ 59].
The dependence of both T andξLSontFMshows that
Co, rather than being a passive layer subject to an externallygenerated spin current, participates in the overall generationof spin-orbit torques. The active role of the ferromagnet inval-idates the assumption on which Eq. ( 3) rests and explains the
deviation of the torque measured at large thicknesses from the1/t
FMdependence. Interestingly, these measurements point to
a non-negligible OHE in Pt, in accordance with the measure-ments discussed next.
V . ORBITAL-TO-SPIN CONVERSION IN A SPACER LAYER
The results presented in Sec. IVshow that the spin-orbital
conductivity of a light metal can be maximized by a properchoice of the ferromagnet and its thickness. There is, however,a limitation from both a practical and theoretical point of view.According to Hund’s third rule, light metals have oppositespin-orbit coupling relative to ferromagnetic Fe, Co, and Ni;thusξ
LScannot be maximized in such bilayers. As proposed
in Sec. II, this optimization may be possible, instead, if the
orbital current is converted into the spin current prior to the in-jection into the ferromagnetic layer [Fig. 1(b)]. This approach
requires materials with high spin-orbit coupling between thelight metal and the ferromagnet [ 21]. Although the additional
layer can itself be a source of spin current, we show in thefollowing how thickness-dependent measurements reveal theunderlying orbital-to-spin conversion and indicate the optimalconversion conditions.
Figure 5shows the spin-orbital conductivity measured in
Co(2)/X(t
X)/Cr(9) and Co(2) /X(tX)/Pt(5) as a function of
the rare-earth thickness, where Xis either Gd or Tb. We find
a drastic change of the magnitude and sign of the torquesupon increasing t
X. As the rare-earth layer becomes thicker
in Co/X(tX)/Cr,|ξLS|first increases, reaching its maximum
magnitude at about t=3 nm, and then decreases [notice the
negative sign of ξLSin Fig. 5(a)]. At this thickness, |ξLS|
of Co/ X(3)/Cr is three to four times larger than in Co/Cr
and is thus comparable to or larger than the highest spin-orbital conductivity of Co/Pt [cf. Figs. 3(a) and 5(a)]. In
Co/X(t
X)/Pt, instead, ξLSdecreases rapidly with tX, changes
sign at 2 nm, and saturates thereafter. This variation, which issimilar in samples containing Gd and Tb, is in direct contrastwith the widespread assumption that the positive spin Hallconductivity of Pt determines the sign and magnitude of thedampinglike spin-orbit torque in Pt heterostructures.
Indeed, our findings cannot be attributed to the sole SHE
in the nonmagnetic layer, nor can they be attributed to thespin-orbit torques generated by the rare-earth layer, which,(b)σ < 0S σ > 0LL S
L S
σ > 0S σ > 0L(a)
0123456-5-4-3-2-10
Co/Gd/Cr
Co/Tb/CrξLS (105 Ω-1m-1)
t (nm)X
02468 1 0-3-2-10123 Co/Gd/Pt
Co/Tb/PtξLS (105 Ω-1m-1)
tX (nm)
FIG. 5. (a) Dependence of the spin-orbital conductivity on
the thickness of the rare-earth spacer in Co(2) /Gd(tX)/Cr(9) and
Co(2)/Tb(tX)/Cr(9). The schematic depicts the conversion of the
orbital current into a spin current. Since the spin and orbital Hallconductivities of Cr are opposite and the spin-orbit coupling of
Gd and Tb is negative, the primary and converted spin currents
have the same sign. (b) The same as (a) in Co(2) /Gd(t
X)/Pt(5) and
Co(2)/Tb(tX)/Pt(5). In this case, the primary (blue) and converted
(white) spin currents have opposite sign because Pt has positive
spin and orbital Hall conductivities and Gd and Tb have negativespin-orbit coupling.
although present, are too small to explain the sizable change
ofξLSin the trilayers with respect to the Co/Cr and Co/Pt
bilayers (see control measurements of Co/Tb and Co/Gd in theConclusions). Moreover, samples with inverted position of Gdand Tb with respect to the Co layer present spin-orbital Hallconductivities similar to the samples without the spacer, whichindicates that the rare-earth layer is not the dominant source ofspin-orbit torques (see the Conclusions). Instead, the results inFig. 5can be rationalized by considering the combination of
OHE, SHE, and orbital-to-spin conversion in the spacer. Thenet spin current transferred from Cr or Pt to Co depends onthe transmission at the interface, the spin and orbital diffusionin the rare-earth layer, and its orbital-to-spin conversion effi-ciency. Whereas the first two effects always diminish the spincurrent reaching the ferromagnet, the orbital-to-spin conver-sion enhances it when /angbracketleftL·S/angbracketright
NM·/angbracketleftL·S/angbracketrightX>0 and weakens
it when /angbracketleftL·S/angbracketrightNM·/angbracketleftL·S/angbracketrightX<0. This is the case for samples
containing Cr and Pt, respectively (see Table I). The length
scale over which the effect takes place is determined by thecombination of the spin and orbital diffusion lengths of Gdand Tb. When the spacer is thin relative to these two lengths,the orbital-to-spin conversion supplies the spin current withmore spins than those lost by scattering. On the other hand,
033037-7GIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)
spin-flip events become dominant at large thicknesses and
decrease the transmitted spin current. The spin-orbital con-ductivity saturates then to a finite value determined by theSHE of the rare-earth layer, as indicated by the similar ξ
LS
measured in samples with either Cr or Pt and thick spacers
(tX/greaterorequalslant6n m ) .
These findings highlight the importance of achieving ef-
ficient orbital-to-spin conversion. This can be pursued bysandwiching a rare-earth spacer of optimal thickness betweenthe ferromagnet and the nonmagnet because rare-earth met-als are effective enhancers of the conversion but not strongsources of spin-orbit toques [ 63]. Remarkably, our results also
provide evidence of a strong OHE in Pt.
VI. GENERALIZED DRIFT-DIFFUSION MODEL
OF ORBITAL AND SPIN CURRENTS
To shed light on the interplay between spin and orbital
currents, we developed a 1D model that takes into accountthe generation and diffusion of both spin and orbital angu-lar momenta as well as their interconversion mediated byspin-orbit coupling. We consider first a single nonmagneticlayer where an electric field Eapplied along xinduces the
SHE and OHE. Let μ=μ
S,Lbe the spin or orbital chemical
potential and Jμ=JS,Lbe the corresponding spin or orbital
current along zwith spin and orbital polarization along y.
The generation, drift, and diffusion of spins and orbitals aregoverned by [ 34–39]
d
2μ
dz2=μ
λ2μ, (4)
Jμ=−σ
2edμ
dz+σHE, (5)
where λμis the diffusion length, σis the longitudinal electri-
cal conductivity, and σHis the spin or orbital conductivity, i.e.,
the off-diagonal element of the conductivity tensor. Solvingthese equations yields μ=Ae
z/λμ+Be−z/λμ, with the coeffi-
cients AandBobtained by imposing the boundary condition
thatJμvanishes at the edges of the nonmagnet. In this form,
however, the equations of the spin and orbital components areindependent and cannot account for the orbital-to-spin andspin-to-orbital conversion mediated by spin-orbit coupling.To capture this process, we add a phenomenological termto Eq. ( 4) for the spin (orbital) chemical potential that is
proportional to its orbital (spin) counterpart, i.e.,
d
2μS
dz2=μS
λ2
S±μL
λ2
LS, (6)
d2μL
dz2=μL
λ2
L±μS
λ2
LS, (7)
JS=−σ
2edμS
dz+σSE, (8)
JL=−σ
2edμL
dz+σLE, (9)
where the +(−) sign corresponds to negative (positive) spin-
orbit coupling. Physically, this additional term represents theconversion between spins and orbitals at a rate proportional tothe respective chemical potential. Thus, even when the SHE
is negligible, a finite spin imbalance is produced in responseto the orbital accumulation. The parameter controlling thisprocess is the coupling length λ
LS, which is a measure of both
the efficiency and length scale over which the conversion takesplace.
We remark that Eqs. ( 6)–(9) are phenomenological and
based on the hypothesis that spin and orbital transport canbe described on an equal footing. They assume implicitly thepossibility of defining spin and orbital potentials and currentseven if the spin and orbital angular momenta are not conservedin the presence of spin-orbit coupling and the crystal field[64]. In this regard, we notice that the spin diffusion model
has found widespread use in the quantitative analysis of spin-orbit torques [ 1,52,65], spin Hall magnetoresistance [ 35], and
surface spin accumulation [ 55] despite the nonconservation
of spin angular momentum. Moreover, there is a fundamentaldifference between spin and orbital transport that makes theapproximations underlying the orbital drift-diffusion modelless critical. Contrary to intuition, the crystal field does notquench the nonequilibrium orbital moment as efficiently as it
suppresses the equilibrium orbital moment. This is because
the orbital moment is carried by a relatively narrow subsetof conduction electron states, namely, its transport is medi-ated by “hot spots” in kspace. Since the orbital degeneracy
of the hot spots is in general protected against the crystalfield splitting, the orbital momentum can be transported overlonger distances than its spin counterpart [ 5,59]. This orbital
transport mechanism has no spin equivalent and is supportedby the experimental evidence that orbital diffusion lengths innonmagnets and dephasing lengths in ferromagnets are signif-icantly longer than the corresponding spin lengths, as shownin this paper and in Refs. [ 19,29]. Further theoretical work
is required to ascertain the limits of our spin-orbital modeland determine how to capture analytically the spin-orbitalinterconversion. However, our model is consistent with theBoltzmann approach proposed in Ref. [ 66] and also repro-
duces the experimental results, as explained in the following.
To solve the coupled equations ( 6) and ( 7), we substitute
the former into the latter and obtain
d
4μS
dz4−/parenleftbigg1
λ2
S+1
λ2
L/parenrightbiggd2μS
dz2+/parenleftbigg1
λ2
Sλ2L−1
λ4
LS/parenrightbigg
μS=0.(10)
The solution to Eq. ( 10) reads
μS(z)=Aez/λ1+Be−z/λ1+Cez/λ2+De−z/λ2, (11)
where
1
λ2
1,2=1
2⎡
⎣1
λ2
S+1
λ2
L±/radicalBigg/parenleftbigg1
λ2
S−1
λ2
L/parenrightbigg2
+4
λ4
LS⎤
⎦ (12)
are the combined spin-orbital diffusion lengths that result
from the coupling of the spin and orbital degrees of freedomintroduced by λ
LS. Equation ( 11) is the generalization of the
standard diffusion of spins valid in the absence of spin-orbitalinterconversion. Two additional exponentials appear becauseof the coupling between LandS. For the same reason, the
spin-orbital diffusion lengths λ
1,2are a combination of the
spin, orbital, and coupling lengths. The same formal solutionas Eqs. ( 11) and ( 12) holds for the orbital chemical potential
033037-8GIANT ORBITAL HALL EFFECT AND ORBITAL-TO-SPIN … PHYSICAL REVIEW RESEARCH 4, 033037 (2022)
because our model treats μSandμLon an equal footing.
However, the eight unknown coefficients in Eq. ( 11) (four
forμSand four for μL) are in general different between μS
andμL. They are found by imposing that the spin and orbital
currents vanish at the edges of the nonmagnet and that the pairof solutions for μ
SandμL[Eq. ( 11)] satisfies Eqs. ( 6) and ( 7)
at any z. Then, we find that the spin chemical potential at the
surface of the nonmagnet increases with the thickness tNMas
μS(tNM)=2eλ1/parenleftBiggσS∓σL
λ2
LSγ2
1−γ2
γ1/parenrightBigg
E
σtanh/parenleftbiggtNM
2λ1/parenrightbigg
+2eλ2/parenleftBiggσS∓σL
λ2
LSγ1
1−γ1
γ2/parenrightBigg
E
σtanh/parenleftbiggtNM
2λ2/parenrightbigg
, (13)
where γi=1
λ2
i−1
λ2S. Equation ( 13) captures the interplay be-
tween the SHE and OHE, which reinforce or weaken each
other depending on the sign of the spin-orbit coupling and onλ
LS. In comparison, in the absence of coupling between Sand
L,E q .( 13) would read
μS(tNM)=2eλSσS
σEtanh/parenleftbiggtNM
2λS/parenrightbigg
, (14)
consistent with the standard spin drift-diffusion model. We
note that Eqs. ( 11) and ( 13) are valid under the condition
λLS>√λSλLbecause for smaller values of λLSthe solution
to Eq. ( 10) is a linear combination of complex exponential
functions, i.e., μSandμLhave an oscillatory dependence on z.
Similar oscillations have been predicted in Ref. [ 9]. However,
we argue that complex solutions to Eq. ( 12) are incompati-
ble with experimental results since an oscillatory dependenceof spin-orbit torques or spin Hall magnetoresistance on thethickness of the nonmagnetic layer has never been observed.The condition λ
LS>√λSλLalso implies that the conversion
between spins and orbitals cannot occur on a length scaleshorter than the shortest distance over which either spins ororbitals diffuse. At the same time, it shows that the conver-sion is always less efficient than the intrinsic spin and orbitalrelaxation.
We apply our model to study the interplay of nonequi-
librium spins and orbitals induced by the SHE and OHEin two exemplary situations. First, we take a single non-magnetic layer with negative spin-orbit coupling, e.g., Cr.Figure 6shows the spin and orbital chemical potentials in
three different conditions. In Fig. 6(a), the OHE is turned
off (σ
L=0), and the SHE is active [ σS=−105(/Omega1m)−1]. In
Figs. 6(b)and6(c), the situation is opposite, namely, σL=105
(/Omega1m)−1andσS=0. In all cases, orbitals (spins) accumulate
at the interfaces even if the OHE (SHE) is set to zero. Since/angbracketleftL·S/angbracketright
NM<0, the two chemical potentials are of opposite
sign. When λLSdecreases, the spin accumulation resulting
from the orbital conversion increases approximately as λ−2
LS
[Fig. 6(b) and Eq. ( 13)]. Interestingly, we find that even if
λSis small, spins accumulate on a long distance because the
spin-orbital diffusion lengths λ1,2are dominated by λL[cf.
Figs. 6(b) and6(c)].
AsλLSdecreases and the spin-orbit conversion becomes
more efficient, both the spin accumulation and the orbitalaccumulation increase at the sample edges. This effect might(a) (b)
σ= 0L
σ= 0S (c)
-30030
0 5 10 15 20-0.80.00.8L (μeV ) S (μeV)
z (nm) λ = 5 n mL λ = λ = 2 nmLSσ= 0S
λ = λ = 2 nmLS
λ = 2 nmS-0.20.00.2
05 1 0 1 5 2 0-10010L (μeV) S (μeV)
z (nm)-10010
05 1 0 1 5 2 0-202 λLS= 10 nm
λLS= 3 nmL (μeV)
λLS= 10 nm
λLS= 3 nmS (μeV)
z (nm)
FIG. 6. (a) Orbital and spin chemical potentials in a single non-
magnetic layer with tNM=20 nm, σS=−105(/Omega1m)−1,σL=0,
λS=λL=2n m ,a n d λLS=10 nm. (b) The same as (a) with σS=0,
σL=105(/Omega1m)−1,λS=λL=2n m ,a n d λLS=3 or 10 nm. (c) The
same as (a) with σS=0,σL=105(/Omega1m)−1,λS=2n m , λL=5n m ,
andλLS=10 nm. In all cases, the resistivity of the NM layer was
set to ρ=56×10−8/Omega1m as measured for Cr, and an electric field
E=5×104V/m was considered.
seem counterintuitive, because spin-orbit coupling usually
induces dissipation of angular momentum. In our model, how-ever, the dissipation of SandLis included in the parameters
λ
SandλL, respectively, whereas λLSdescribes the nondissipa-
tive exchange of angular momentum between the orbital andspin reservoirs. Thus λ
LSeffectively increases the spatial ex-
tent of orbital and spin accumulation. Formally, this happensbecause one of the two spin-orbital diffusion lengths λ
1,2in-
creases while the other changes weakly when λLSis reduced.
As a consequence, more spins and orbitals can accumulate atthe sample edges. This result is similar to the model withoutspin-orbit coupling, which predicts an increase in μ
Swith the
spin diffusion length: μS(tNM/greatermuchλS)∼λS[see Eq. ( 14)].
Next, we consider a trilayer structure representative of
the samples Co(2) /X(tX)/Cr(9) and Co(2) /X(tX)/Pt(5). We
model the spatial variations of μSandμLby four equations of
the same type as Eq. ( 11), two for the rare-earth layer and
two for Cr (or Pt). We assume that μS,μL,JS, and JLare
continuous at the X/NM interface ( z=tX) and impose the
constraint that JSandJLrelate to μSandμL, respectively,
033037-9GIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)
(a) (b)
-1.5-1.0-0.50.0
01234560510JS (109 A/m2) JL (109 A/m2)
tX (nm)012
02468 1 0048JS (109 A/m2) JL (109 A/m2)
tX (nm)σ > 0S σ > 0L
Pt
X
CotXCr
X
CotXσ < 0S σ > 0L
FIG. 7. (a) Calculated spin and orbital currents at the FM/ X
interface as a function of tXin Co(2) /X(tX)/Pt(5). (b) The same as
(a) for Co(2) /X(tX)/Cr(9). The orbital and spin Hall conductivities
of Pt and Cr are indicated above the graphs. The parameters usedto calculate the orbital and spin currents can be found in Table II
(Appendix C).
through the mixing conductance GS,L:
JX
S(tX)=GX
S
eμS(tX), (15)
JX
L(tX)=GX
L
eμL(tX). (16)
In doing so, we introduce the orbital equivalent of the spin
mixing conductance, which is expected to depend on thespin-orbit coupling of the ferromagnet and to influence thestrength of the orbital torque. Thus, in our model, G
Ltakes
into account the additional orbital-to-spin conversion occur-ring in the ferromagnet or at the interface. Furthermore, weonly consider the real part of G
S,Lsince the fieldlike torque
is small in our samples. Finally, we assume a finite SHE inboth the nonmagnetic and rare-earth layers, but smaller inthe latter, whereas the OHE is present only in the nonmagnet(see Appendix Cfor a list of the parameters). We set σ
S>0
in Pt, σS<0 in Cr and in the spacer, and σL>0 in both
Cr and Pt. The spin-orbit coupling is assumed positive in Ptand negative in Cr and in the rare-earth layer. With thesereasonable assumptions, we can reproduce qualitatively theresults of Fig. 5, namely, the enhancement of the spin-orbital
conductivity upon insertion of a rare-earth spacer betweenCo and Cr and the sign change of the torques when thesame layer is sandwiched between Co and Pt. Figure 7shows
the calculated spin and orbital currents, to which spin-orbittorques are proportional, that reach the FM/ Xinterface as a
function of the rare-earth thickness t
X. In both the case of
Cr and the case of Pt the orbital current decreases monoton-ically as t
Xincreases because of the orbital diffusion away
from the X/NM interface. In contrast, the spin current variesdifferently with tXdepending on whether Cr or Pt is chosen
because the primary spin current and the current obtainedupon orbital-to-spin conversion in the rare-earth element havethe same sign with Cr and have opposite sign with Pt. Thuscalculations based on a generalized drift-diffusion model con-firm the interpretation of the data in Fig. 5, which cannot
be explained without the inclusion of the OHE. We believethat a better quantitative agreement with the measurementscould be obtained by including additional effects that we havedisregarded, namely, the interfacial resistance ( μ
SandμLnot
continuous), the interfacial spin and orbital scattering ( JSand
JLnot continuous), and the thickness dependence of the spacer
resistivity and, possibly, of the diffusion lengths. The modelcould be extended to account for the orbital conversion in theferromagnet, which is hidden here behind the orbital mixingconductance. Finally, it may be employed to investigate othertransport effects such as the spin Hall magnetoresistance andits orbital counterpart.
VII. CONCLUSIONS
Our measurements of spin-orbit torques in FM/NM and
FM/X/NM multilayers with light and heavy metals provide
comprehensive evidence for strong OHE effects in 3 dand 5 d
metals and establish a systematic framework to analyze andefficiently exploit the interplay of spin and orbital currents.Owing to the entanglement of the orbital and spin degreesof freedom in materials with finite spin-orbit coupling, thisinterplay is best described by combined spin-orbital conduc-tivity ( ξ
LS) and diffusion length ( λLS) parameters rather than
by considering the OHE and SHE as two separate effects.The experimental values of ξ
LSfor the different systems and
control samples are summarized in Fig. 8. Corresponding
values of the spin-orbital Hall angle θLSare reported in Ap-
pendix A. We found strong spin-orbit torques produced by the
light elements Cr and Mn, whose sign depends on the adja-cent ferromagnet, in contrast with torques generated by theSHE. The spin-orbital conductivity increases with the thick-ness of the light metal layer without indications of saturation.This trend is compatible with spin-orbital diffusion lengthsλ
LS/greaterorsimilar20 nm in these elements and extrapolates to a giant
intrinsic spin-orbital conductivity as predicted by theory [ 4].
Because of the competition between spin and orbital torques,the spin-orbital conductivity varies with the thickness of theferromagnet in a monotonic or nonmonotonic way dependingon the relative sign of /angbracketleftL·S/angbracketright
NMand/angbracketleftL·S/angbracketrightFM. Furthermore,
we show that the interplay between spin and orbital torquescan be drastically enhanced by inserting a 4 fspacer layer
between the nonmagnet and the ferromagnet. As summarizedin Fig. 8, the inclusion of a Tb (Gd) spacer results in a fourfold
(threefold) increase of the torques generated by Cr and Mnthat cannot be attributed to spin currents generated by therare-earth element. Instead, the enhancement results from theconversion of the orbital current into a secondary spin currentof the same sign as the primary spin current. The orbital-to-spin conversion has a striking effect in Pt, when the primaryspin current generated by the SHE and the secondary spincurrents generated by the OHE interfere destructively. Thiseffect results in the reversal of the spin-orbit torque generatedby Pt when the orbital-to-spin conversion rate is stronger than
033037-10GIANT ORBITAL HALL EFFECT AND ORBITAL-TO-SPIN … PHYSICAL REVIEW RESEARCH 4, 033037 (2022)
FM/Cr(9)
Co(2)/Gd(3)/Cr(9)
Co(2)/Tb(3)/Cr(9)
Co(2)/Gd(4)
Co(2)/Tb(4)
Gd(3)/Co(2)/Cr(9)-4-3-2-101(a) (b)
(c)23ξLS (105 Ω-1m-1)
CoNi
Cr
FM/Mn(9)
Co(2)/Gd(3)/Mn(9)
Gd(3)/Co(2)/Mn(9)CoNi
Mn
FM/Pt(5)
Co(2)/Gd(3)/Pt(5)
Co(2)/Tb(3)/Pt(5)
Gd(4)/Co(2)/Pt(5)
Tb(4)/Co(2)/Pt(5)NiCo
Pt
FIG. 8. Comparison of the effective spin-orbital conductivity ξLSmeasured in NM/FM and NM/ X/FM layers where (a) NM =Cr,
(b) NM =Mn, and (c) NM =Pt; FM =Co,Ni,X=Gd,Tb. The thickness of each layer is indicated in nanometers in parentheses. The
results of control experiments on X/FM and NM/FM/ Xlayers are also shown.
the primary spin current. These findings indicate the presence
of a strong OHE and SHE in Cr, Mn, and Pt and highlightthe importance of orbital-to-spin conversion phenomena indifferent types of heterostructures. The largest ξ
LS=−4.3×
105(/Omega1m)−1andθLS≈0.25 are found in Co(2)/Gd(3) /Cr(9)
and Co(2)/Tb(3) /Cr(9) layers. Both of these parameters are
larger compared with Co/Pt and previous measurements, in-dicating that optimization of the thickness of 3 dmetal layers
and the insertion of 4 fspacers lead to giant spin-orbital Hall
effects and ensuing spin-orbit torques. The fits of ξ
LSas a
function of thickness indicate that the spin-orbital conductiv-ity of Cr saturates at values of the order of 10
6(/Omega1m)−1,i n
agreement with theoretical estimates [ 4]. Finally, we propose
an extended drift-diffusion model that treats the orbital andspin moment on an equal footing and includes the orbital-to-
FM/Cr(9)
Co(2)/Gd(3)/Cr(9)
Co(2)/Tb(3)/Cr(9)Co(2)/Gd(4)
Co(2)/Tb(4)
Gd(3)/Co(2)/Cr(9)-0.3-0.2-0.10.00.10.2θH
CoNi
Cr
FM/Mn(9)
Co(2)/Gd(3)/Mn(9)
Gd(3)/Co(2)/Mn(9)MnCoNi
FM/Pt(5)
Co(2)/Gd(3)/Pt(5)
Co(2)/Tb(3)/Pt(5)
Gd(4)/Co(2)/Pt(5)
Tb(4)/Co(2)/Pt(5)Co
Ni
Pt075150225300ρ (μΩ cm) Co
Ni
FIG. 9. Resistivity (top) and effective spin-orbital Hall angle
(bottom) of the samples in Fig. 8in the main text.spin conversion mediated by spin-orbit coupling. The model
explains both the monotonic and nonmonotonic behavior ofξ
LSobserved in the FM/NM and FM/ X/NM multilayers as
a function of thickness and spin-orbit coupling of the con-stituent layers. It also shows how the spatial profiles of theorbital and spin accumulation are determined by the combinedspin-orbital diffusion lengths and spin and orbital mixing con-ductances. Overall, our results provide a useful framework tomaximize the orbital-to-spin conversion efficiency, interpretexperimental results, and address open fundamental questionsabout orbital transport.
(a) (b)
01234560.00.51.01.52.02.5
Co/Gd( tX)/Cr
Co/Tb( tX)/CrMs*t (mA )
tX (nm)02468 1 00.00.40.81.21.62.02.4
Co/Gd( tX)/Pt
Co/Tb( tX)/PtMs*t (mA)
tX (nm)
02468 1 0 1 20481216
Co(tFM)/Cr
Co(tFM)/PtMs*t (mA)
tFM (nm)(c)
FIG. 10. (a) Dependence of the areal magnetization on the
thickness of the rare-earth layer in Co(2) /X(tX)/Cr(9) samples.
(b) The same as (a) in Co(2) /X(tX)/Pt(5) samples. (c) Dependence
of the areal magnetization on the thickness of the ferromagnet in
Co(tFM)/Cr(9) and Co( tFM)/Pt(5) samples.
033037-11GIACOMO SALA AND PIETRO GAMBARDELLA PHYSICAL REVIEW RESEARCH 4, 033037 (2022)
TABLE II. Parameters used in the drift-diffusion model to calculate the spin and orbital currents in a FM/ X/NM trilayer, where NM is
either Cr or Pt. λS,Lis the spin or orbital diffusion length, λLSis the spin-orbital conversion length, σS,Lis the spin or orbital Hall conductivity,
α=±1 is the sign of the spin-orbit coupling, GS,Lis the spin or orbital mixing conductance, and ρis the electrical resistivity. The thickness
of the Cr (Pt) layer was 9 (5) nm. An electric field E=5×104V/m was considered in both cases.
λNM
LλXLλNMSλXSλNMLSλXLS σNM
L σX
L σNM
S σX
S GL GS ρNM ρX
(nm) (nm) (nm) (nm) (nm) (nm) [( /Omega1m)−1][ (/Omega1m)−1][ (/Omega1m)−1][ (/Omega1m)−1]αNMαX[(/Omega1m2)−1][ (/Omega1m2)−1](/Omega1m) ( /Omega1m)
C r 8262 2 0 2 . 5 8 .2×1050 −0.7×105−0.15×105−1−13×1014101456×10−8115×10−8
P t 12222 2 . 5 8 .8×10503 .5×105−0.15×105+1−13×1014101433×10−8115×10−8
ACKNOWLEDGMENT
We acknowledge the support of the Swiss National Science
Foundation (Grant No. 200020_200465).
APPENDIX A: EFFECTIVE SPIN-ORBITAL HALL ANGLE
Figure 9shows the effective spin-orbital Hall angle of
the samples presented in Fig. 8in the main text. The Hall
angle was calculated according to θLS=ξLSρ, where ρis the
resistivity of the entire stack. However, we refrain from esti-mating the resistivity of the individual layers and comparingquantitatively θ
LSof the NM layers alone in this way, because
the resistivity of the heterostructures depends strongly on in-terfaces and the thickness of all layers. Similarly to ξ
LS,w e
interpret θLSas a parameter that describes the simultaneous
occurrence of the OHE, the SHE, and orbital-to-spin conver-sion. The values reported in Figs. 8and9are measured in
samples with the thickness specified in the axis labels.
APPENDIX B: SATURATION MAGNETIZATION
Figure 10shows the surface saturation magnetization
of samples belonging to the series Co(2) /X(tX)/Cr(9),
Co(2)/X(tX)/Pt(5), Co( tFM)/Cr(9), and Co( tFM)/Pt(5) as a
function of the corresponding thickness. The magnetiza-tion was measured by superconducting quantum interferencedevice (SQUID) magnetometry on blanket films grown si-multaneously to the measured devices. The measurementyields the magnetic moment of the sample, which, after nor-malization to the sample area, defines the areal saturationmagnetization M
stFM. This parameter is to be preferred over
the volume saturation magnetization, since the latter dependson the thickness of the ferromagnetically active material. Thisis in turn difficult to define with certainty in the studied sam-ples because of interdiffusion at interfaces, proximity effects,and possible ferrimagnetic coupling. Such a complexity, how-ever, does not impinge on the calculation of the spin-orbitalconductivity because the quantity appearing in Eq. ( 1)i st h e
areal saturation magnetization M
stFM, not the volume magne-
tization.
Figures 10(a) and 10(b) show that the areal magnetiza-
tion decreases upon increasing the thickness of either Gdor Tb in both Cr- and Pt-based samples. We attribute thisreduction to the antiferromagnetic interaction between Co and
the rare-earth layer. We note that the ferrimagnetic couplingcannot explain our results, namely, the trends presented inFig. 5. First, the torque efficiency rises by a factor of 3–4
when the thickness of the rare earth t
Xincreases from 0 to
3 nm, while the areal magnetization decreases only by 20%.Second, the areal magnetization decreases monotonically with
t
X, while the trends in Fig. 5are not monotonic with respect
to the thickness. For example, the spin-orbital conductivity ofCo(2)/X(t
X)/Pt(5) saturates in the limit of large tX, whereas
the magnetization does not.
The areal magnetization of Co( tX)/Cr(9) and Co( tX)/Pt(5)
samples increases linearly with tX, as expected. The linear fits
yield dead layers of about 0.5 and 0.3 nm in the two series, re-spectively. The dead layer is likely located at the substrate/Cointerface and is probably thinner in the Co( t
FM)/Pt(5) series
because of proximity effects with Pt. These values have beentaken into account in the torque calculation in Fig. 4.
Finally, the saturation magnetization of Co(2) /NM( t
NM)
and Ni(4) /NM( tNM) was found to be independent of tNM,
except for Co(2) /Pt(tNM), where the areal magnetization in-
creases by 7% from tPt=1n mt o tPt=12 nm (not shown).
APPENDIX C: PARAMETERS OF THE
DRIFT-DIFFUSION MODEL
Table IIlists all the parameters used for the calculation of
the orbital and spin currents in the Co(2) /X(tX)/Cr(9) and
Co(2)/X(tX)/Pt(5) samples (Fig. 7). Some of them have been
measured (spin diffusion length of Pt; spin Hall conductivityof Cr and Pt from Co/Cr and Co/Pt samples, respectively;and resistivity). Others have been chosen in accordance withthe literature (spin mixing conductance, sign of the spin-orbitcoupling, orbital conductivity). The remaining parameters,mostly involving orbitals and the spin-orbital interconver-sion, are not available in the literature and have been chosensuch that the calculations agree qualitatively with the mea-surements. From this perspective, the extended drift-diffusionmodel can be used to estimate the order of magnitude ofthe unknown parameters. For instance, the spin and orbitaldiffusion length and the spin-orbital coupling length of therare-earth layer must be of the order of a few nanometers atmost for the model to reproduce the measurements in Fig. 5.
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033037-14 |
1211.2055v1.Metal_insulator_transition_in_three_band_Hubbard_model_with_strong_spin_orbit_interaction.pdf | arXiv:1211.2055v1 [cond-mat.str-el] 9 Nov 2012Metal-insulator transition in three-band Hubbard model wi th strong spin-orbit
interaction
Liang Du and Xi Dai
Beijing National Laboratory for Condensed Matter Physics,
and Institute of Physics, Chinese Academy of Sciences, Beij ing 100190, China
Li Huang
Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, China and
Science and Technology on Surface Physics and Chemistry Lab oratory,
P.O. Box 718-35, Mianyang 621907, Sichuan, China
(Dated: June 19, 2018)
Recent investigations suggest that both spin-orbit coupli ng and electron correlation play very
crucial roles in the 5 dtransition metal oxides. By using the generalized Gutzwill er variational
methodanddynamicalmean-fieldtheorywith thehybridizati onexpansioncontinuoustime quantum
Monte Carlo as impurity solver, the three-band Hubbard mode l with full Hund’s rule coupling and
spin-orbit interaction terms, which contains the essentia l physics of partially filled t2gsub-shell of
5dmaterials, is studied systematically. The calculated phas e diagram of this model exhibits three
distinct phase regions, including metal, band insulator an d Mott insulator respectively. We find that
the spin-orbit coupling term intends to greatly enhance the tendency of the Mott insulator phase.
Furthermore, the influence of the electron-electron intera ction on the effective strength of spin-orbit
coupling in the metallic phase is studied in detail. We concl ude that the electron correlation effect
on the effective spin-orbit coupling is far beyond the mean-fi eld treatment even in the intermediate
coupling region.
I. INTRODUCTION
The Mott metal-insulator transition (MIT) induced by
electron-electron correlation has attracted intensive re-
searchactivities in the past several decades1–4. Although
the main features of Mott transition have already been
captured by single-band Hubbard model, most of Mott
transition in realistic materials have multi-orbital nature
and should be described by multi-band Hubbard model.
Unlike the situation in single-band case, where the Mott
transition is completely driven by the local Coulomb in-
teraction U, the Mott transition in multi-band case is
affected by not only Coulomb interaction but also crys-
tal field splitting and Hund’s rule coupling among dif-
ferent orbitals5–7. The interplay between Hund’s rule
coupling and crystal field splitting generates lots of inter-
esting phenomena in the multi-band Hubbard model, for
examples, orbital selective Mott transition, high-spin to
low-spin transition and orbital ordering. Therefore, most
ofthe intriguingphysicsin 3 dor4dtransitionmetalcom-
pounds can be well described bythe multi-band Hubbard
model with both Hund’s rule coupling and crystal field
splitting.
In the present paper, we would like to concentrate our
attention on the Mott physics in another group of inter-
esting compounds, the 5 dtransition metal compounds,
where spin-orbit coupling (SOC), the new physical in-
gredient in Mott physics, plays an important role. Com-
pared to 3 dorbitals, the 5 dorbitals are much more ex-
tended and the correlation effects are not expected to be
important here. While as firstly indicated in reference8,
the correlation effects can be greatly enhanced by SOC,which is commonly strong in 5 dmaterials. The first well
studied 5 dMott insulator with strong SOC is Sr 2IrO4,
wheretheSOCsplitsthe t2gbandsinto(upper) jeff= 1/2
doublet and (lower) jeff= 3/2 quartet bands and greatly
suppresses their bandwidths8–12. Since there are totally
five electrons in its 5 dorbitals, the jeff= 1/2 bands are
half filled and the jeff= 3/2 bands are fully occupied,
which makes the system being effectively a jeff= 1/2
single-band Hubbard model with reduced bandwidth.
Therefore the checkerboard anti-ferromagnetic ground
state of Sr 2IrO4can be well described by the single-band
Hubbard model with half filling.
Here, we will focus on the 5 dmaterials with four elec-
trons in the t2gsub-shell. These materials include the
newly discovered BaOsO 3, CaOsO 3and NaIrO 3etc13.
All these materials share one important common feature:
in low temperature, these materials are insulators with-
out magnetic long-range order. The origin of the insu-
lator behavior can be due to two possible reasons, the
strong enough Coulomb interaction and SOC. We will
have Mott insulator in the former and band insulator in
the latter case respectively. Therefore it is interesting
to study the features of metal-insulator transition in a
generict2gsystem occupied by four electrons with both
Coulomb interaction and SOC.
In the present paper, we study the t2gHubbard model
withSOCandfourelectronsfillingbyusingrotationalin-
variantGutzwillerapproximation(RIGA)anddynamical
mean-field theory combined with the hybridization ex-
pansion continuous time quantum Monte Carlo (DMFT
+ CTQMC) respectively. The paramagnetic U−ζphase
diagram is derived carefully. Further, the interplay be-2
tween SOC ζand Coulomb interaction Uis analyzed in
detail. We will mainly focus on the following two key is-
sues: (i) How does the SOC affect the boundary of Mott
transitions in this three-band model? (ii) How does the
Coulomb interaction modify the effective SOC strength?
Thispaperisorganizedasfollows. InSec. II,thethree-
band model is specified, and the generalized multi-band
Gutzwiller variational wave function is introduced. In
Sec. IIIA, the calculated results, including U−ζphase
diagram, quasi-particle weight and charge distribution,
for the three-band model are presented. The effect of
Coulomb interaction on SOC is analyzed in Sec. IIIB.
Finally we make conclusions in section IV.
II. MODEL AND METHOD
The three-band Hubbard model with full Hund’s rule
coupling and SOC terms is defined by the Hamiltonian:
H=−/summationdisplay
ij,aσtijd†
i,aσdj,aσ+/summationdisplay
iHi
loc=Hkin+Hloc,(1)
whereσdenotes electronic spin, and arepresents the
threet2gorbitals with a= 1,2,3 corresponding to
dyz,dzx,dxyorbitals respectively. The first term de-
scribes the hopping process of electrons between spin-
orbital state “ aσ” on different lattice sites iandj. Local
Hamiltonian terms Hi
loc=Hi
u+Hi
soccontain Coulomb
interaction Hi
uand SOC Hi
soc(In the following, the site
index is suppressed for sake of simplicity).
Hu=U/summationdisplay
ana↑na↓+U′/summationdisplay
a<b,σσ′naσnbσ′−Jz/summationdisplay
a<b,σnaσnbσ
−Jxy/summationdisplay
a<b/parenleftig
d†
a↑da↓d†
b↓db↑+d†
a↑d†
a↓db↑db↓+h.c./parenrightig
,(2)
Hsoc=/summationdisplay
aσ/summationdisplay
bσ′ζ/angbracketleftaσ|lxsx+lysy+lzsz|bσ′/angbracketrightd†
aσdbσ′,(3)
whereU(U′) denotes the intra-orbital (inter-orbital)
Coulomb interaction, Jzterm describes the longitudinal
part of the Hund’s coupling. While the other two Jxy
terms describe the spin-flip and pair-hopping process re-
spectively. ζis SOC strength, l(s) is orbital (spin) an-
gular momentum operator. Here we assume the studied
system experiences approximately cubic symmetry ( Oh
symmetry), in which two parameters U′andJxyfollow
the constraints U′=U−2JandJxy=Jz=J. Here we
chooseJ/U= 0.25 for the systems studied in this paper
unless otherwise noted. This lattice model is solved in
the framework of RIGA14–17and DMFT(CTQMC)18,19
methods respectively, which are both exact in the limit
of infinite spacial dimensions20,21. In this work, a semi-
elliptic bare density of states ρ(ǫ) = (2/πD)/radicalbig
1−(ǫ/D)2
is adopted, which corresponds to Bethe lattice with infi-
nite connectivity. In the present study, the energy unitis set to be half bandwidth D= 1 and all orbitals are
assumed to have equal bandwidth.
Next, we will briefly introduce the recently developed
RIGA method16. The generalized Gutzwiller trial wave
function |ΨG/angbracketrightcan be constructed by acting a many-
particle projection operator Pon the uncorrelated wave
function |Ψ0/angbracketright22–24,
|ΨG/angbracketright=P|Ψ0/angbracketright, (4)
with
P=/productdisplay
iPi=/productdisplay
i/summationdisplay
ΓΓ′λΓΓ′|Γ/angbracketrightii/angbracketleftΓ′|. (5)
|Ψ0/angbracketrightis normalized uncorrelated wave function in which
Wick’s theorem holds. |Γ/angbracketrightiare atomic eigenstates on
siteiandλΓΓ′are Gutzwiller variational parameters. In
our work, |Γ/angbracketrightiare eigenstates of atomic Hamiltonian Hu,
each|Γ/angbracketrightiis labeled by good quantum number N,J,Jz,
whereNis total number of electrons, Jis total angular
momentum, Jzis projection of total angular momen-
tum along zdirection. The non-diagonal elements of the
previously defined variational parameter matrix λΓΓ′are
assumed to be finite only for state |Γ/angbracketright,|Γ′/angbracketrightbelonging to
the sameatomic multiplet, i.e, with the same three quan-
tum labels15. In the following, we assume the local Fock
terms are absent in |Ψ0/angbracketright,
/angbracketleftΨ0|c†
iαciβ|Ψ0/angbracketright=δαβ/angbracketleftΨ0|c†
iαciα|Ψ0/angbracketright=δαβn0
iα.(6)
For general case, a local unitary transformation matrix
Ais needed to transform the original diα-basis into the
so-called natural cim-basis16, i.e,dα=/summationtext
mAαmcm. In
the original single particle basis ( dyz↑,dyz↓,dzx↑,dzx↓,
dxy↑,dxy↓), SOC term is expressed as :
Hsoc=−ζ
2
0 0−i0 0 1
0 0 0 i−1 0
i0 0 0 0 −i
0−i0 0−i0
0−1 0i0 0
1 0i0 0 0
.(7)
Then the transformation matrix Ais as follows:
A=1√
6
−√
3 0 1 0 0 −√
2
0−1 0√
3−√
2 0
−i√
3 0−i0 0 i√
2
0−i0−i√
3−i√
2 0
0 2 0 0 −√
2 0
0 0 2 0 0√
2
,(8)
where the t2gorbitals have been treated as a system with
leff= 1.
In the natural single particle basis, SOC matrix is
transformed into:
Hsoc=
−ζ/2 0 0 0 0 0
0−ζ/2 0 0 0 0
0 0 −ζ/2 0 0 0
0 0 0 −ζ/2 0 0
0 0 0 0 ζ0
0 0 0 0 0 ζ
.(9)3
Meanwhile the Coulomb interaction term is transformed
as:
/summationdisplay
αβδγUαβδγd†
αd†
βdδdγ=/summationdisplay
mnkl˜Umnklc†
mc†
nckcl,(10)
with
˜Umnkl=/summationdisplay
αβδγUαβδγA†
mαA†
nβAδkAγl.(11)
Fixn0in each orbital
initial guess of renormalization factor R
construct Gutzwiller effective one-particle Hamiltonian,
ˆHeff
0=/summationtext
i/negationslash=j/summationtext
αβ/summationtext
γδtαβ
ijR†
αγˆc†
iγˆcjδRδβ+/summationtext
iαηαˆc†
iαˆciα
and solve the Hamiltonian to get ∂E/∂R.
∂E
∂λΓΓ′=/summationdisplay
δβ/parenleftigg
∂Ekin
∂Rδβ∂Rδβ
∂λΓΓ′+∂Ekin
∂R†
βδ∂R†
βδ
∂λΓΓ′/parenrightigg
+∂Eloc
∂λΓΓ′+/summationdisplay
αηα∂n′
α
∂λΓΓ′= 0
solve the equations to get renormalization factor R
check ifRis
self-consistent.
calculate quantities:
total energy, occupation ...YesNo
FIG. 1. Flowchart of the RIGA self-consistent loop to mini-
mize total energy E(n0) with respect to |Ψ0/angbracketrightandλΓΓ′.
In this paper, we define expectation value with uncor-
related wave function:
O0=/angbracketleftΨ0|ˆO|Ψ0/angbracketright, (12)
while expectation value with Gutzwiller wave function is
defined as:
O=OG=/angbracketleftΨG|ˆO|ΨG/angbracketright. (13)
During the minimization process, two following con-
straints are forced,
/angbracketleftΨ0|P†P|Ψ0/angbracketright= 1, (14)
and
/angbracketleftΨ0|P†Pniα|Ψ0/angbracketright=/angbracketleftΨ0|niα|Ψ0/angbracketright. (15)
In the present paper, the second constraint is satisfied
in the following way. We first calculate the total energy
of the trial wave function with both the left-hand andright-hand side of the above equation equaling to some
desired occupation number n0
α. Then we minimize the
energy respect to n0
αat the last step.
The remaining task is to minimize the variational
ground energy E=Ekin+Elocwith respect to λΓΓ′
and|Ψ0/angbracketright, and fulfill the previous two constraints. Here,
Ekin=/angbracketleftΨG|Hkin|ΨG/angbracketright=/summationdisplay
ij/summationdisplay
γδ˜tγδ
ij/angbracketleftΨ0|c†
iγcjδ|Ψ0/angbracketright,
(16)
and
Eloc=/angbracketleftΨG|Hloc|ΨG/angbracketright= Tr(φ†Hlocφ),(17)
with˜t,Randφdefined as:
˜tγδ
ij=/summationdisplay
αβtαβ
ijR†
αγRδβ (18)
R†
αγ=Tr/parenleftbig
φ†c†
αφcγ/parenrightbig
/radicalig
n0γ(1−n0γ), (19)
φII′=/angbracketleftI|P|I′/angbracketright/radicalbig
/angbracketleftΨ0|I′/angbracketright/angbracketleftI′|Ψ0/angbracketright, (20)
where|I/angbracketright(|I′/angbracketright) stands for a many-body Fock state and
n0
γ=/angbracketleftΨ0|nγ|Ψ0/angbracketright.
The flowchart of RIGA method is shown in Fig.1. For
fixedn0
αin each orbital, minimizing Ewith respect to
|Ψ0/angbracketrightandλΓΓ′can be divided into two steps in each iter-
ative process. Firstly, fix Gutzwiller variational parame-
tersλΓΓ′and find optimal uncorrelated wave function by
solving effective single particle Hamiltonian,
Heff
0=/summationdisplay
i/negationslash=j/summationdisplay
γδ˜tγδ
ijc†
iγcjδ+/summationdisplay
iαηαc†
iαciα,(21)
where Lagrange parameters ηαare used to minimize the
variational energy fulfilling Gutzwiller constraints. Sec-
ondly, we fix the uncorrelated wave function, and opti-
mize the variational energy with respect to Gutzwiller
variational parameters λΓΓ′,
∂E
∂λΓΓ′=/summationdisplay
δβ/parenleftigg
∂Ekin
∂Rδβ∂Rδβ
∂λΓΓ′+∂Ekin
∂R†
βδ∂R†
βδ
∂λΓΓ′/parenrightigg
+∂Eloc
∂λΓΓ′+/summationdisplay
αηα∂n′
α
∂λΓΓ′= 0, (22)
wheren′
α=/angbracketleftΨ0|P†Pnα|Ψ0/angbracketright. In this way, λΓΓ′and|Ψ0/angbracketright
are self-consistently determined.
For the fix n0
αalgorithm, we need to scan the n0
αto
get the global ground state of the studied system. In
this paper, because SOC will split the t2gorbitals into
two fold jeff= 1/2 and four fold jeff= 3/2 states, we can
introduce an alternative variable δn0to determine n0
αfor4
each orbital. The occupation polarization δn0is defined
as:
δn0=n0
3/2−n0
1/2, (23)
in which n0
3/2andn0
1/2stand for the average occupa-
tion number of lower ( jeff= 3/2) and upper ( jeff= 1/2)
orbitals respectively. Since total electron number of
the system is fixed to be 4 n0
3/2+ 2n0
1/2= 4, we have
0≤δn0≤1.δn0(n0) corresponding to ground state is
denoted by δn0
g(n0
g).
In the present paper, we also use DMFT+CTQMC
method to crosscheck our results derived by RIGA. For
DMFT+CTQMC method, the system temperature is set
to beT= 0.025 (corresponding to inverse temperature
β= 40). In each DMFT iteration, typically 4 ×108
QMC samplings have been performed to reach sufficient
numerical accuracy25.
III. RESULTS AND DISCUSSION
A.U-ζphase diagram
In this subsection, we mainly focus on phase diagram
for the three-band model proposed in Eq.(1). The ob-
tainedU−ζphase diagrams with J/U= 0.25 are shown
in Fig.2. The upper panel shows the phase diagram cal-
culated by zero temperature RIGA method, while the
calculated results by DMFT+CTQMC method at finite
temperature is shown in the lower panel. The results ob-
tained by two different methods are consistent with each
other quite well except that DMFT+CTQMC can not
distinguish between band insulator and Mott insulator,
which will be explained later. Apparently, there exists
three different phasesin U−ζplane: metallic state in the
lower left corner, band insulator in the lower right region
and Mott insulator in the upper right region. The gen-
eral shape of the phase diagram can be easily understood
by considering two limiting cases: (i) For ζ= 0, one has
a degenerate three-band Hubbard model populated by 4
electrons per site. The model will undergo an interaction
driven Mott transition at critical Uc/D∼11.0 with each
band filled by 4 /3 electrons. (ii) For non-interacting case
(U= 0.0), the model is exactly soluble. The three bands
are degenerate and filled by 4 /3 electrons at ζ=0.0. Fi-
niteζwill split the three degenerate bands into a (lower)
jeff= 3/2quadrupletand(upper) jeff= 1/2doubletwith
energy separation being 1 .5ζ. Increasing ζwill pump
electrons from upper to lower orbitals until the upper
bands are completely empty and the system undergoes a
metal to band insulator transition, which is expected at
ζ/D= 1.33.
In order to clarify the way we determine the metal,
band insulator, and Mott insulator phases by RIGA
method, in Fig.3 we plot the total energy and quasipar-
ticle weight as a function of δn0defined in the previous
section , where the SOC strength is fixed at ζ/D= 0.7,
U/D
ζ/D 0 2 4 6 8 10 12
0 0.2 0.4 0.6 0.8 1 1.2 1.4MetalBand InsulatorMott InsulatorU/D
ζ/D 0 2 4 6 8 10 12
0 0.2 0.4 0.6 0.8 1 1.2 1.4MetalInsulator
FIG. 2. (Color online) Phase diagram of three-band Hub-
bard model with full Hund’s coupling terms in the plane of
Coulomb interaction U(J/U= 0.25) and spin-orbit coupling
ζ. Upper panel: The phase diagram is calculated by RIGA at
zero temperature. Lower panel: The phase diagram is calcu-
lated by DMFT+CTQMC with finite temperature T= 0.025.
and from top to bottom the Coulomb interaction is
U/D= 1.0,3.0,6.0. The ground state of the system
is the state with the lowest energy respect to δn0. The
typical solution for the metal phase is shown in Fig.3a,
where the energy minimum occurs at 0 < δn0
g<1.0 cor-
responding to the case that all orbitals being partially
occupied. While for a band insulator, as shown in Fig.3b
as a typical situation, the energy minimum happens at
δn0
g= 1.0 corresponding to the case that the jeff= 3/2
orbitals are fully occupied and jeff= 1/2 orbitals are
empty, and more over the quasiparticle weight Zkeeps
finite when δn0approching unit. And finally the situ-
ation of a Mott insulator is illustrated in Fig.3c, where
the quasiparticle weight Zvanishes at some critical δn0,
above which the system is in Mott insulator phase and
can no longer be described by the Gutzwiller variational
method.
While in the DMFT+CTQMC calculations, the phase
boundary between metal and insulator is identified by
measuring the imaginary-time Green function at τ=
β/226,27. SinceG(β/2) can be viewed as a representa-5
1.601.802.00
(a) 0.91.0
(d)
7.407.607.80E
(b) 0.50.60.7
Z
(e)
16.116.216.3
0.0 0.2 0.4 0.6 0.8
δn0(c)
0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.3
δn0(f)U=1 U=1,j=3/2
U=1,j=1/2
U=3 U=3,j=3/2
U=3,j=1/2
U=6 U=6,j=3/2
U=6,j=1/2
FIG. 3. (Color online) Total energy E(δn0) and quasipar-
ticle weight Z(δn0) as a function of occupation polarization
δn0=n0
3/2−n0
1/2for different values of interaction strength
U/D= 1,3,6 (J/U= 0.25) at fixed ζ/D= 0.7 and zero
temperature, where n0
3/2is average occupation number of
jeff= 3/2 quadruplet, n0
1/2is average occupation number of
jeff= 1/2 doublet.
tion of the integrated spectral weight within a few kBT
ofEF, so it can be used as an important criterion to
judge whether metal-insulator transition occurs. The
corresponding results for SOC strength ζ/D= 0.5 are
shown in Fig.4. Clearly seen in this figure, the critical Uc
is about 3.5, and both the jeff= 3/2 andjeff= 1/2
orbitals undergo metal-insulator transitions simultane-
ously. Since there is chemical potential ambiguity in
the insulator phase, it is difficult to further distinguish
Mott insulator from band insulator by DMFT+CTQMC
method. Therefore we only calculatethe phase boundary
between the metal and insulator phase, which is in good
agreement with the results obtained by RIGA.
Now, we come back to discussed the phase diagram
obtained by RIGA and DMFT. When both the Coulomb
interaction Uand SOC ζare finite, the phase diagram
looks a bit complicated. By considering different values
ofSOC, we divide the phase diagramverticallyinto three
regions.
Firstly, for 0 .00< ζ/D < 0.24, with increasing
Coulomb interaction U, our calculation by RIGA pre-
dicts a transition from metal to Mott insulator. The
transition is characterizedby the vanishing of quasiparti-
cle weight as discussed previously. The critical Coulomb
interaction Ucdecreasesdrasticallywith the increment of
SOC, the effect of SOC tends to enhance the Mott MIT
greatly. The DMFT results show very similar behavior
in this region as shown in lower panel of Fig.2, except
that the Ucobtained by DMFT has weaker dependence
on the strength of SOC compared to that of RIGA. From
the view point of DMFT, the suppression of the metallic
phase by SOC can be explained quite clearly. For the
effective impurity model in DMFT, the metallic phase
corresponds to a solution when the local moments on the0.20.40.60.81.0
1 2 3 4 5 6 7 8 9 10|G(β/2)|
U/Dζ=0.5j=3/2
j=1/2
FIG. 4. (Color online) The imaginary-time Green function
atτ=β/2 as a function of Coulomb interaction strength U.
The SOC strength ζis chosen to be 0.5 as a illustration. The
calculation is done by DMFT+CTQMC method at β= 40.
In this figure the normalized quantities by G(β/2) atU/D=
1.0 are shown and the arrows correspond to phase transition
points.
impurity site are fully screened by the electrons in heat-
bath through the Kondo like effect. With the SOC, there
is an additional channel to screen the local spin moment
otherthanthe Kondoeffect, whichleadstothe formation
of spin-orbital singlets. This additional screening chan-
nel, which is completely local, will thus compete with the
Kondo effect and suppress the metallic solution. There
is no net local moment left in this type of Mott phase,
and the ground state is simply a product state of local
spin-orbital singlets on each site.
For 0.24< ζ/D < 1.33, two successive phase transi-
tions are observed with the increment of U. The transi-
tion from metallic to band insulator phases occurs firstly,
and followed by another transition to the Mott insula-
tor phase. In the intermediate Uregion, the effective
band width of system is reduced by the correlation ef-
fects, whichdrivesthe systemintoabandinsulatorphase
with relative small band splitting induced by SOC. Fur-
ther increasing interaction strength Uwill push the sys-
tem to the Mott limit. Although this process is believed
to be a crossover rather than a sharp phase transition,
our RIGA calculation provides a mean field description
for these two different insulators, where in the band in-
sulator phase the interaction effects only renormalize the
effective band structure and do not suppress the coher-
ent motion of the electrons entirely. Similar behavior can
also be obtained by DMFT method, where the quasipar-
ticle weight determined by DMFT selfenergy keeps finite
for the band insulator phase and vanishes for the Mott
insulator phase.
At last, for ζ/D >1.33 region, the orbitals are fully
polarized with electrons fully occupied jeff= 3/2 bands
and fully empty jeff= 1/2 bands at U= 0.0, indicating
that the system is in the band insulator state already6
in the non-interacting case. Similar band insulator to
Mott insulator transition will be induced with further
increment of Uin the RIGA description, as discussed
before.
0.00.20.40.60.8Zζ=0.1, j=3/2
ζ=0.5, j=3/2
ζ=1.5, j=3/2
0.00.20.40.60.8
0 1 2 3 4 5 6 7Z
U/Dζ=0.1, j=1/2
ζ=0.5, j=1/2
ζ=1.5, j=1/2
FIG. 5. (Color online) Quasiparticle renormalization fact ors
Zof the lower orbitals ( jeff= 3/2 quadruplet) and upper
orbitals ( jeff= 1/2 doublet) as function of Coulomb inter-
actionU(J/U= 0.25) for different values of SOC ( ζ/D=
0.1,0.5,1.5). The dashed lines label the critical Ufor transi-
tion to Mott state. The results are obtained by zero temper-
ature RIGA method.
0.650.700.750.800.850.900.951.00
0 1 2 3 4n(jeff= 3/2)
U/Dζ=0.1, RIGA
ζ=0.1, DMFT
ζ=0.5, RIGA
ζ=0.5, DMFT
ζ=1.5, RIGA
ζ=1.5, DMFT
FIG. 6. (Color online) Occupation number of the lower
orbitals ( jeff= 3/2 quadruplet) with increasing Coulomb
U(J/U= 0.25) for selected SOC ( ζ/D= 0.1,0.5,1.5). Both
thecalculated results byRIGAandDMFT(CTQMC) are pre-
sented.
For several typical SOC parameters ( ζ/D=
0.1,0.5,1.5) in the three regions defined above, we study
the evolutions of quasiparticle weight and band specific
occupancy with Coulomb interaction. The quasiparti-
cle weight for selected SOC with increasing Uis plotted
in Fig.5. The upper (lower) panel shows the quasiparti-
cle weight for jeff= 3/2 (1/2) orbitals. Note the quasi-
particle weight in RIGA is defined as the eigenvalues of
the Hermite matrix R†R. For both ζ/D= 0.1 and 1.5,1.71.92.1/angbracketleftL2/angbracketright(a)
1.01.41.8/angbracketleftS2/angbracketright(b)
0.00.51.01.5
0 0 .5 1 1 .5 2 2 .5 3 3 .5 4/angbracketleftJ2/angbracketright
U/D(c)DMFT
RIGA
DMFT
RIGA
DMFT
RIGA
FIG. 7. (Color online) Expectation value of orbital angular
momentum /angbracketleftL2/angbracketright, spin angular momentum /angbracketleftS2/angbracketright, and total an-
gular momentum /angbracketleftJ2/angbracketrightas function of Coulomb interaction U
with fixed spin-orbit coupling strength ζ/D= 0.7. It is de-
rived by RIGA at zero temperature and DMFT+CTQMC at
β= 40 respectively.
the quasiparticle weights decrease from 1 to 0 monoton-
ically with the increasing interaction strength UandJ
until the transition to Mott insulator phase. While for
ζ/D= 0.5, there exists a kink at U/D= 2.7 in the lower
panel (jeff= 1/2), which corresponds to the transition
from metal to band insulating state. For transition to
Mott insulating state, quasiparticle weights for all the
orbitals reach zero simultaneously, with Uc/D= 6.7 for
ζ/D= 0.1,Uc/D= 4.5 for ζ/D= 0.5, and Uc/D= 4.0
forζ/D= 1.5.
The occupation number of the (lower) jeff= 3/2 or-
bitals as a function of on-site Coulomb interaction U
is ploted in Fig.6 for three typical SOC strength. For
ζ/D= 0.1, to some extent, the occupation behavior is
similar to ζ= 0 case, in which the occupation number
is only slightly changed by the interaction. The situa-
tion is quite different for ζ/D= 0.5, where the occupa-
tion of the jeff= 3/2 orbital increases with interaction
at the beginning and decreases slightly after the tran-
sition to the band insulator phase. The non-monotonic
behavior here is mainly due to the competition between
the repulsive interaction Uand Hund’s rule coupling J.
The effect of Uwill always enhance the splitting of the
local orbitals to reduce the repulsive interaction among
these orbitals. While the Hund’s rule coupling intents
to distribute the electrons more evenly among different
orbitals. For ζ/D= 1.5 case, occupation number in the
two subsets is fully polarized at U= 0 and the effect of
Hund’s coupling term will reduce the occupation of the
jeff= 3/2 orbital monotonically.
At last, the expectation value of the total orbital angu-
lar momentum /angbracketleftL2/angbracketright, spin angular momentum /angbracketleftS2/angbracketrightand
total angular momentum /angbracketleftJ2/angbracketrightas a function of Coulomb
interaction Uare plotted in Fig.7, where ζ/Dis fixed
to 0.70. In the non-interacting case, all the three ex-7
pectation values can be calculated exactly and they will
approach the atomic limit with the increment of inter-
actionUandJ. In the atomic limit the SOC strength
is much weaker than the Hund’s coupling J, the ground
state can be well described by the LScoupling scheme,
where the four electrons will first form a state with total
orbital angular momentum L= 1 and total spin momen-
tumS= 1,andthenformaspin-orbitalsingletstatewith
total angularmomentum J= 0. From Fig.7, we can find
that the system approaches the spin-orbital singlet quite
rapidly after the transition to the band insulator phase.
0.81.01.21.41.61.82.02.22.42.62.8
0 0.5 1 1.5 2 2.5ζeff/ζ
U/Dζ=0.1, RIGA
ζ=0.1, HFA
ζ=0.5, RIGA
ζ=0.5, HFA
FIG. 8. (Color online) Evolution of effective SOC strength
with increasing Coulomb U(J/U= 0.25) for selected SOC
(ζ/D= 0.1,0.5), A comparison of results derived by RIGA
and HFA are presented.
B. Effective spin-orbit coupling
In the multi-orbital system, the interaction effects will
mainly cause two consequences for the metallic phases:
(1) It will introduce renormalizationfactor for the energy
bands; (2) It will modify the local energy level for each
orbital which splits the bands. For the present model,
the second effect will modify the effective SOC, which
is another very important problem for the spin orbital
coupled correlation system. Within the Gutzwiller vari-
ational scheme used in the present paper, the effective
SOC can be defined as:
ζeff=−1
2∂Eint(δn0)
∂δn0−1
2∂Esoc(δn0)
∂δn0,(24)
whereEintandEsocare the ground state expectation
values of interaction and SOC terms in the Hamilto-
nian respectively. Note the second term is different from
the bare SOC ζ0unlessnαis a good quantum num-
ber. If the interaction energy is treated by Hartree Fock
mean field approximation (HFA), the above equation
givesζeff=−∂EHF
int(δn0)/(2∂δn0) +ζ0, which will al-
ways greatly enhance the spin-orbital splitting with the
increasing Uas found by some works based on LDA+Umethod28,29. In this section, we compare the results ob-
tained by RIGA and HFA. As shown in Fig.8, the effec-
tive SOC obtained by HFA increases quite rapidly with
the interaction UandJ. While the results obtained
by RIGA show very different behavior. For weak SOC
strength, i.e. ζ/D= 0.1, the effective SOC obtained
by RIGA increases first then decrease. This interesting
non-monotonicbehaviorreflectsthecompetitionbetween
the repulsive interaction U, which intends to increase the
occupation difference for jeff= 1/2 andjeff= 3/2 or-
bitals, and the Hund’s rule coupling J, which intents to
decrease the occupation difference. While for relatively
strong SOC strength, the effective SOC increases with
interaction U(andJ) monotonically all the way to the
phase boundary indicating the repulsive interaction U
plays a dominate role here. But compared to HFA, the
enhancement of effective SOC induced by the interaction
is much weaker even for the latter case. This is mainly
due to the reduction of the high energy local configura-
tions in the Gutzwiller variational wave function com-
pared to the Hatree Fock wave function, which greatly
reduces the interaction energy and its derivative to the
orbital occupation.
IV. CONCLUDING REMARKS
The Mott MIT in three-band Hubbard model with full
Hund’s rule coupling and SOC is studied in detail using
RIGA and DMFT+CTQMC methods. First, we propose
the phase diagram with the strength of electron-electron
interaction and SOC. Three different phases have been
found in the U−ζplane, which are metal, band insula-
tor and Mott insulator phases. For 0 .00< ζ/D < 0.24,
increasing Coulomb interaction will induce a MIT tran-
sition from metal to Mott insulator. For 0 .24< ζ/D <
1.33,effect of Uwill causetwosuccessivetransitions, first
frommetaltobandinsulator, thentoMottinsulator. For
ζ/D >1.33, a transition from band insulator to Mott in-
sulatorisobserved. Fromthephasediagram,wefindthat
the critical interaction strength Ucis strongly reduced by
the presence of SOC, which leads to the conclusion that
the SOC will greatly enhance the strong correlation ef-
fects in these systems. Secondly, we have studied the ef-
fect of electron-electron interaction on the effective SOC.
Our conclusion is that the enhancement of effective SOC
found in HFA is strongly suppressed once we go beyond
the mean field approximationand include the fluctuation
effects by RIGA or DMFT methods.
ACKNOWLEDGMENT
We acknowledge valuable discussions with profes-
sor Y.B. Kim and professor K. Yamaura, and finan-
cial support from the National Science Foundation of
China and that from the 973 program under Contract
No.2007CB925000 and No.2011CBA00108. The DMFT8
+ CTQMC calculations have been performed on the SHENTENG7000 at Supercomputing Center of Chinese
Academy of Sciences (SCCAS).
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1110.6798v1.Spin_Orbit_Coupled_Quantum_Gases.pdf | November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
Spin-Orbit Coupled Quantum Gases
Hui Zhai
Institute for Advanced Study, Tsinghua University, Beijing, 100084, China
In this review we will discuss the experimental and theoretical progresses in studying
spin-orbit coupled degenerate atomic gases during the last two years. We shall rst review
a series of pioneering experiments in generating synthetic gauge potentials and spin-orbit
coupling in atomic gases by engineering atom-light interaction. Realization of spin-orbit
coupled quantum gases opens a new avenue in cold atom physics, and also brings out a
lot of new physical problems. In particular, the interplay between spin-orbit coupling and
inter-atomic interaction leads to many intriguing phenomena. Here, by reviewing recent
theoretical studies of both interacting bosons and fermions with isotropic Rashba spin-
orbit coupling, the key message delivered here is that spin-orbit coupling can enhance
the interaction e ects, and make the interaction e ects much more dramatic even in the
weakly interacting regime.
Keywords : Cold Atoms, Synthetic Gauge Potential, Spin-Orbit Coupling, Super
uidity,
Feshbach Resonance
Many interesting phenomena in condensed matter physics occur when electrons
are placed in an electric or magnetic eld, or possess strong spin-orbit (SO) cou-
pling. However, in the cold atom systems, neutral atoms neither possess gauge
coupling to electromagnetic elds nor have SO coupling. Recently, by controlling
atom-light interaction, one can generate an arti cial external abelian or non-abelian
gauge potential coupled to neutral atoms. The basic principle is based on the Berry
phase e ect1and its non-abelian generalization2. An important application of this
scheme is creating an e ective SO coupling in degenerate atomic gases. Since 2009,
Spielman's group in NIST has successfully implemented this principle and gener-
ated both synthetic uniform gauge eld6, magnetic eld7, electric eld8and SO
coupling9. We shall discuss the experimental progresses along this line in Sec. 1.
The e ects of SO coupling in electronic systems have been extensively discussed
in condensed matter physics before and are also important topics nowadays. One of
the most famous example is recently discovered topological insulators3;4;5. In this
review, we try to convey the message that SO coupling in degenerate atomic gases
will bring out new physics which have not been considered before, mainly due to
the interplay between SO coupling and the unique properties of atomic gases. For
bosonic atoms, SO coupled interacting bosons is a system never explored in physics
before. For fermionic atoms, since a lot of intriguing physics have been revealed
during the last ten years by utilizing Feshbach resonance technique to achieve in-
teraction as strong as Fermi energy, the interplay between resonance physics and
SO coupling is de nitely a subject of great interests.
1arXiv:1110.6798v1 [cond-mat.quant-gas] 31 Oct 2011November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
2Hui Zhai
A key point we want to emphasize in this review is that a nearly isotropic
SO coupling will dramatically enhance the e ects of inter-particle interactions, so
that the interaction e ects are not weak even in the regime where the interaction
strength itself is small . This is because an isotropic Rashba SO coupling or a nearly
isotropic SO coupling signi cantly changes the low-energy states of single particle
Hamiltonian, as we shall discuss in Sec 2. In Sec. 3, we discuss many-body system
of bosons. Because the single particle ground state has large degeneracy, it is the
inter-particle interaction that selects out a unique many-body ground state and
determines its low-energy
uctuations. In Sec. 4, we discuss many-body system of
fermions. Because the low-energy density-of-state (DOS) is largely enhanced, the
interaction e ects become much more profound, in particular for weak attractions.
A brief summary and future perspective are given in Sec. 5.
1. Synthetic Gauge Potentials and Spin-Orbit Coupling
In 2009, Spielman's group in NIST rst realized a uniform vector potential in Bose-
Einstein condensate (BEC) of87Rb6. In this experiment, two counter propagating
Raman laser beams couple jF;m Fi=j1; 1ilevel of87Rb toj1;0ilevel, and couple
j1;0ilevel toj1;1ilevel, as shown in Fig. 1(a) and (b), which can be described by
the Hamiltonian
H=0
B@k2
x
2m+1
2ei2k0x0
2e i2k0xk2
x
2m
2ei2k0x
0
2e i2k0xk2
x
2m 21
CA (1)
wherek0= 2=,is the wave length of two lasers. 2 k0is therefore the momentum
transfer during the two-photon processes. 1= 1+!+2, and2= 1+! 2,
where 1denotes the linear Zeeman energy, !denotes the frequency di erence of
two Raman lasers, and 2is the quadratic Zeeman energy.
Applying a unitary transformation to wave function = U , where
U=0
@e i2k0x0 0
0 1 0
0 0ei2k0x1
A (2)
and the e ective Hamiltonian becomes
He =UHUy=0
B@(kx+2k0)2
2m+1
20
2k2
x
2m
2
0
2(kx 2k0)2
2m 21
CA: (3)
When both 1and2are large, the single particle energy dispersion of He is shown
as Fig. 1(c), which displays a single energy minimum at nite kx. In this regime the
low energy physics is dominated by a single dressed state described by1
2m(kx Ax)2,
whereAxis a constant. This leads to a uniform vector gauge eld.
In a follow up experiment, Spielman's group applied a Zeeman eld gradient
along ^ydirection to this system7. In this case, Axbecomes a function of yinsteadNovember 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
Spin-Orbit Coupled Quantum Gases 3
/Minus3/Minus2/Minus1123k/Slash1k00.51.01.5E-101(b)BEC
(a)(c)(d)/Minus3/Minus2/Minus1123k/Slash1k0/Minus224E
Fig. 1. (a) A schematic of NIST experiment, in which two counter propagating Raman beams are
applied to87Rb BEC. (b) A schematic of how three F= 1 levels are coupled by Raman beams. (c)
Dispersion in the regime of uniform vector potential. (d) Dispersion in the regime of non-abelian
gauge eld.
of a constant. It gives rise to a non-zero synthetic magnetic eld Bz= @yAx6= 0.
They observed a critical magnetic eld above which many vortices are generated in
the BEC7. In another experiment8, they made Axtime dependent which gives rise
to a non-zero electric led Ex= @tAx6= 0. They observed collective oscillation of
BEC after a pulse of electric eld8.
By tuning the Zeeman energy and the laser frequency, one can also reach the
regime where 1+!2, and thus20, while122is still large. In 2011,
Spielman's group rst reached this regime and showed that a SO coupling can be
generated9. As shown in Fig. 1(d), in this regime the low-energy physics contains
two energy minima which are dominated by j1; 1iandj1;0istates, respectively.
Therefore we can deduce the low-energy e ective Hamiltonian by keeping both
j1; 1iandj1;0i, and rewrite the Hamiltonian as
H= k2
x
2m+h
2
2ei2k0x
2e i2k0xk2
x
2m h
2!
(4)
whereh=2. Similarly, by applying a unitary transformation to the wave function
with
U=e ik0x0
0eik0x
(5)
one reaches an e ective Hamiltonian that describes SO coupling
HSO=UHUy=
(kx+k0)2
2m+h
2
2
2(kx k0)2
2m h
2!
=1
2m(kx+k0z)2+
2x+h
2z:
(6)November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
4Hui Zhai
In fact, upon a pseudo-spin rotation x! zandz!x, the above Hamiltonian
is equivalent to
HSO=1
2m(kx+k0x)2
2z+h
2x; (7)
where the rst term can be viewed as an equal weight of Rashba ( kxx+kyy)
and Dresselhaus ( kxx kyy) SO couplinga. The Hamiltonian of Eq. 7 can also
be viewed as a Hamiltonian with synthetic non-abelian gauge eld, since the vector
potentialAx= k0zdoes not commute with the scale potential =
2x+h
2z.
Later on, Campbell et al. discussed how to generalize the NIST scheme to create
SO coupling in both ^ xand ^ydirection, and nearly isotropic Rashba SO coupling10.
Xu and You recently introduce a dynamics generalization of the NIST scheme11.
Sau et al. discussed an explicit method to create Rashba SO coupling in fermionic
40K system at nite magnetic eld, where a magnetic Feshbach resonance is avail-
able12. Beside the NIST scheme, there are also other theoretical proposals that use
-type or tripod system to generate synthetic magnetic eld13;14;15;16, non-abelian
gauge eld with a monopole17and SO coupling18;19;20;21. However, for those pro-
posals using dark states13;14;15;16;20;21, collisional stability is a concern since there
is always one eigen-state whose energy is below the dark state manifold, and multi-
particle collision can lead to decay out of the dark state manifold. A recent review
paper by Dalibard et al. has discussed di erent proposals in detail22.
2. Single Particle Properties with Rashba Spin-Orbit Coupling
In the rest part of this review we will consider SO coupling in both ^ xand ^ydirections,
whose Hamiltonian is given by
H0=1
2m
(kx xx)2+ (ky yy)2+k2
z
(8)
and in particular, we will consider the most symmetric Rashba case x=y=
>0, where the physics is most interesting. In this case, the Hamiltonian can be
rewritten as
H0=1
2m
k2
? 2k?+2+k2
z
(9)
Obviously, spin is no longer a good quantum number. However \helicity" is a good
quantum number. \Helicity" means that the spin direction is either parallel or
anti-parallel to the in-plane momentum direction. For these two helicity branches,
their dispersion are given by
k=1
2m(k2
?2k?+2+k2
z) (10)
aIn many literature, Rashba SO coupling denotes kxy kyx, while Dresselhaus SO coupling
denoteskxy+kyx. They are equivalent to the notations used in this paper by a pseudo-spin
rotationx! yandy!x.November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
Spin-Orbit Coupled Quantum Gases 5
(a)(b)/Minus2/Minus10120.00.51.01.52.0
k/UpTee/Slash1ΚE/Slash1ER012340.000.050.100.150.20E/Slash1ERDOS/Slash1/LParen1Κm/RParen1
Fig. 2. Energy dispersion with kz= 0 (a) and density-of-state (b) for Rashba spin-orbit coupled
single particle Hamiltonian. In (a), \helicity" is \ + " for red solid line and is \ " for blue dashed
line.ER=2=(2m) is introduced as energy unit.
where k?= (kx;ky) andk?=q
k2x+k2y. This Hamiltonian displays a symmetry
of simultaneously rotation of spin and momentum along ^ zdirection.
As shown in Fig. 2(a), helicity plus branch has lower energy. The single par-
ticle energy minimum has nite in-plane momentum k?=, and all the single
particle states with same k?=andkz= 0 but di erent azimuthal angle are
degenerate ground states. The single particle DOS also has non-trival feature. In a
three-dimensional system without SO coupling, DOS vanishes aspat low ener-
gies. However, in this system, as shown in Fig. 2(b), the low-energy DOS becomes
a constant when <E R, similar to a conventional two-dimensional system.
Without SO coupling, the single particle Hamiltonian has a unique ground state
atk= 0. At zero-temperature, bosons are all condensed at k= 0 state. If the inter-
action is weak, interaction e ect is perturbative which only creates nite quantum
depletion to the zero-momentum condensate. Also, because the vanishing DOS at
low energies, two-body bound state appears only when the attractive interaction is
above a threshold. Far below the threshold, when the attractive interaction is weak,
the strength of fermion pairing and the fermion super
uid transition temperature
are both exponentially small. That is to say, without SO coupling, not surprisingly,
the interaction e ect is weak in the regime where the interaction strength is small.
With SO coupling, the e ects of interaction are signi cantly enhanced even when
its strength is still weak, because SO coupling signi cantly changes the single par-
ticle behaviors as discussed above. First, because the single particle ground state
is not unique now, the ground state of a boson condensate is also not unique if
there is no interactions. That is to say, it is the interaction that selects out a unique
ground state among many possibilities. In this sense, the e ect of interaction is
non-perturbative even for very weak interactions. Secondly, because the DOS is a
constant at low energies, a two-body bound state appears for any weak attractions.
Therefore, for weak attractive interactions, both pairing gap and super
uid transi-
tion temperature are largely enhanced by SO coupling. In particular, the super
uid
transition temperature can reach the same order as the Fermi temperature even forNovember 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
6Hui Zhai
weak attractions. These points will be discussed in the next two sections in more
detail.
On the other hand, we shall always keep in mind that in real experiment, one can
never achieve a perfect isotropic Rashba SO coupling. Therefore, in our following
discussions, we always rst obtain interesting results in the isotropic Rashba limit,
and then discuss how robust the results are if there is a little anisotropy, namely, a
slight di erence between xandy.
3. Spin-Orbit Coupled Bose Gases
We will rst consider (pseudo-)spin-1 =2 bosons. We shall rst consider the mean-
eld ground stateband then discuss the e ects of quantum and thermal
uctuations
on top of mean- eld saddle points. Let us rst consider the most simpli ed form of
interactions
^Hint=Z
d3r
gn2
1(r) +gn2
2(r) + 2g12n1(r)n2(r)
(11)
By minimizing Gross-Pitaevskii energy functional, Wang et al. found that the mean-
eld ground state has two di erent phases depending on the sign of g12 g23. When
g12<g, the system is in the \plane wave phase", where all bosons are condensed into
a single plane wave state, and the direction of plane wave is spontaneously chosen
in thexyplane. For instance, if the plane wave momentum is along ^ xdirection, the
condensate wave function is given by
=r
2eix1
1
: (12)
The density of each component is uniform, but their phases modulate from zero
to 2periodically, as found from numerical solution of Gross-Pitaevskii equation
and shown in Fig. 3(a). This state spontaneously breaks time-reversal, rotational
symmetry and the U(1) symmetry of super
uid phase. When g12>g, all bosons are
condensed into a superposition of two plane wave states with opposite momentums,
whose condensate wave function is given by
=p
2
eix1
1
+e ix1
1
=pcos(x)
isin(x)
(13)
In this phase, the spin density n1 n2=cos(2x) which has a periodic modulation
in space, as shown in Fig. 3(b), and therefore is named as \stripe super
uid". The
direction of the stripe is also spontaneously chosen in the xyplane. Here, without
loss of generality, we choose it along ^ xdirection. In this state, the high density regime
of one component coincides with the low-density regime of the other component, so
that the inter-component repulsive interaction is maximumly avoided. That is the
bThere are also proposals of non-mean- eld fragmented state, like N00N state, as ground state of
SO coupled bosons24, however, the conventional wisdom is that such a state is very fragile when
external perturbations are present.November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
Spin-Orbit Coupled Quantum Gases 7
(b)
(a)
Fig. 3. (a) Phase of condensate wave function in the \plane wave phase"; (b) Spin density in the
\stripe super
uid" phase.
reason why the spin stripe state is favored when g12is larger than g. In addition to
theU(1) super
uid phase, this state also breaks the rotational symmetry (but keeps
the re
ection symmetry), and the translational symmetry along the stripe direction.
Hence, symmetry wise, it can also been called \smectic super
uid". In contrast to
the \plane wave phase", this state does not break time-reversal symmetry.
In principle, the condensate wave function can be any superposition of single
particle ground states as
=X
'c'ei(cos'x+sin'y)1
ei'
(14)
One may wonder why only a single state or a superposition of a pair of states
are favored by interactions. In fact, if there are more than a pair of single particle
states in the superposition, the condensate wave function will exhibit interesting
structure, such as various types of skyrmion lattices. And if all the degenerate
states enter the condensate wave function with equal weight and speci ed relative
phases, condensate will exhibit interesting structure of half vortices, as rst proposed
by Stanescu, Anderson, Galitski24and Wu, Mondragon-Shem25. However, such a
state is not energetically favorable in spin-1 =2 case for a uniform (or nearly uniform)
system. This is because that the interaction part can be rewritten as
^Hint=Z
d3rg+g12
2n2(r) +g g12
2S2
z(r)
; (15)
wheren(r) =n1(r) +n2(r) andSz=n1(r) n2(r). TheS2
z-term can be satis ed
by either choosing the \plane wave phase" or the \stripe phase", while the n2-
term with positive coecient always favors a uniform density. One can easily show
that, if there are more than a pair of states in the superposition, the condensate
density will always have non-uniform modulation, which causes energy of n2-term.
Similar situation has also been found for spin-1 Hamiltonian26;23. However, there
are several situations where skyrmion lattices or half vortices are found as ground
state. When a strong harmonic con nement potential V(r) =m!r2
?=2 is applied
to the system, condensate density is no longer uniform because of the trapping
potential and the requirement from n2-term becomes less restrictive. In addition,November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
8Hui Zhai
one notes in the limit of zero interaction, the ground state in a harmonic trap can
be solved exactly and it is a half-vortex state31;32;54. Recently, three groups have
found that if one largely increases the con nement potential so that a=p
~=m! is
comparable to 1 =, or reduces the interaction energy to be comparable to ~!, the
ground state will evolve continuously to skyrmion lattice phases, and nally to half
vortex phases31;32;54. Besides, Xu et al. and Kawakami et al. found in spin-2 case,
because of an addition interaction term (which favors cyclic phase in absence of SO
coupling29;30), there exists a parameter regime where various types of skyrmion
lattice phases are ground state even for a uniform system.
Back to the discussion of a nearly uniform spin-1 =2 systemc, the \plane wave
phase" and the \stripe phase" are in fact two very robust mean- eld states. In
real situation, the interactions between two pseudo-spin states have much more
complicated form than the simpli ed form of Eq. (11). Both Yip and Zhang et al.
considered a speci c type of complicated interaction form using a concrete real-
ization of Rashba SO coupling, and they found the ground state is still either the
\plane wave phase" or the \stripe phase"34;80. And also, the SO coupling is always
not perfectly isotropic, say, x>y, then the Hamiltonian itself does not have rota-
tional symmetry anymore. The single particle energy has two minima at ( ;0;0)
instead of a continuous circle of degenerate states. At mean- eld level, the e ect of
anisotropic SO coupling is to pin the direction of plane wave or stripe into certain
direction (^xdirection for this case). The NIST experimental situation discussed in
Sec. 1 corresponds to the case y= 0 and also with a Zeeman eld. As shown by
Ho and Zhang36, and also in the experimental paper9, the phase diagram of this
system also only contains such two phases.
To go beyond the mean- eld description, three di erent approaches have been
tried so far. The rst is the e ective eld theory approach37, which treats Gaussian
uctuations of low-energy modes. The second is Bogoliubov approach41;42;43, which
more focuses on the gapless phonon excitations. And the third is the renormalization
approach38;39;40, which discusses how scattering vertices are renormalized by high
order processes. These di erent approaches address beyond-mean- eld e ects from
di erent perspectives, and the results are consistent with each other where they
overlap.
Taking e ective eld theory approach as an example, for the \stripe" phase, the
super
uid phase and the relative phase ubetween two momentum components
are two low-lying modes:
'ST=p
2ei
ei(x+u)1
1
+e i(x+u)1
1
: (16)
In fact, the relative phase udescribes the phonon mode of stripe order. The dy-
cFor typical experimental parameters, the BEC is in the nearly uniform regime rather than strong
harmonic con nement regime.November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
Spin-Orbit Coupled Quantum Gases 9
namics ofandu elds are governed by an e ective energy function37
HST
e =
2m"
(@x)2+(@y)2
2+ (@xu)2+
@2
yu2
42#
; (17)
where >1 is a constant. For the \plane wave" phase, the superfuid phase is the
only low energy mode
'PW=pei(x+)1
1
(18)
and its e ective energy is derived as37
HPW
e =
2m
(@x)2+1
42(@2
y)2
: (19)
One notes that in the \stripe" phase, the quadratic term ( @yu)2is absent in Eq.
(17), and in the \plane wave" phase, the quadratic term ( @y)2is absent in Eq.
(19). This is in fact a manifestation of rotational symmetry in this system. Similar
e ective theory has also been found for \FFLO" state in fermion super
uid44;46.
Such an energy function is also the classical energy of smectic liquid crystal.
In two dimensions, the nite temperature phase transition is driven by prolif-
eration of topological defects. For usual XYmodel, both the energy of topological
vortex and its entropy logarithmically depend on system size. Thus, only above
a critical temperature, entropy wins energy and the topological defects proliferate
which drives system into normal phase. However, in the \stripe" phase, because the
absence of ( @yu)2term, the energy for a topological defect of u, i.e. a dislocation in
the stripe, no longer logarithmically depends on system size. Hence it will lose to en-
tropy at any nite temperature, and the proliferation of these dislocations will melt
the stripe order. This restores translational symmetry and drives the system into a
boson paired super
uid phase37. Such a boson pairing state can also be predicted
by looking at pairing instability of normal state with renormalized interactions38.
By considering the renormalization e ects of scattering vertex from high energy
states, Gopalakrishnan et al.38and Ozawa, Baym39;40found that the scattering
amplitudes between two states with opposite momentum become smaller and even
vanishing, which makes the \stripe phase" more favorable at the low-density limit,
and meanwhile leads to more signi cant the
uctuation of stripe order38. For same
reason, in the \plane wave" phase, the super
uid phase will immediately disorder
at nite temperature and the system becomes normal.
In addition, the e ective theory Eq.(17) and Eq. (19) also imply that there is a
Goldstone mode which has linear dispersion along ^ xdirection (direction of stripe
or plane wave momentum), and quadratic dispersion along ^ ydirection (direction
perpendicular to the direction of stripe or plane wave momentum). Same results
have been reached by Bogoliubov calculation41;42;43. Similar behaviors of Goldstone
modes also exist in \FFLO" phase of fermion super
uid44;45;46.November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
10Hui Zhai
When the SO coupling is anisotropic ( x6=y), there is no rotational symmetry
in the Hamiltonian. Therefore, one will have ( @yu)2in the stripe phase, and ( @y)2
in the plane wave phase. However, the coecients of those terms are propositional
to 1 (y=x)2. In the regime y=xis very close to unity, the energy of a disloca-
tion is much smaller than the energy of a vortex or a half vortex. Hence, the system
will undergo two Kosterlitz-Thouless phase transitions. At a lower critical tempera-
ture, dislocations proliferate and the system becomes paired super
uid. Then, at a
higher critical temperature, vortices proliferate and the system becomes normal. A
completed phase diagram in term of interaction parameter, SO coupling anisotropy
and temperature is given by Jian and Zhai37.
For SO coupled bosons, many questions remain open. Recently several works
begin to address the questions about the super
uid critical velocity42, vortices in
presence of external rotation47;48;49, the e ects of dipolar interactions50, the collec-
tive modes51, super
uid to Mott insulator transition in a lattice52, the interplay
between magnetic eld and SO coupling53, and the dynamical e ects nearby Dirac
point due to SO coupling54;55;56. The research along this direction will de nitely
reveal more interesting physics and stimulate more interesting experiments.
4. Spin-Orbit Coupled Fermi Gases across a Feshbach Resonance
Even for a non-interacting system, the thermodynamic behavior of a Fermi gas is
dramatically changed by a strong SO coupling because of the change of the low-
energy DOS57. In this section, we focus on Fermi gas with attractive interaction,
and in particular, across a Feshbach resonance. The interaction part is modeled as
Hint=gZ
d3r y
"(r) y
#(r) #(r) "(r) (20)
where
1
g=m
4~2as X
k1
~2k2=m: (21)
Here we relate the bare interaction gtos-wave scattering length via Eq. (21), which
is the same as the scheme widely used in free space without SO coupling. This is
based on the assumption that the interaction Hamiltonian is not changed by SO
coupling, and the ashere should be understood as scattering length in free space.d
Using this interaction Hamiltonian, Vyasanakere and Shenoy rst studied the
two-body problem across a Feshbach resonance59. Because the low-energy DOS is
now a constant, an arbitrary weak attractive interaction will give rise to a bound
state. Similar situations are two-body problem in two-dimension and Cooper prob-
lem in three-dimension in absence of SO coupling, where DOS are also constants.
dThis is equivalent to assume that the s-wave pseudo-potential is still a valid approximation
for a short-range realistic potential in presence of SO coupling. In fact, the validity of such an
approximation is not quite obvious, and it has only been examined recently by Cui58.November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
Spin-Orbit Coupled Quantum Gases 11
Fig. 4. (a) Gap size as a function of =kFfor di erent values of 1 =(kFas). (b) Size of Cooper
pair inxyplanel?and along ^zdirectionlzas a function of =kFat resonance with as=1; (c)
Super
uid transition temperature Tc=TFas a function of 1 =(kFas) for di erent values of =kF.
Reprinted from arXiv: 1105.2250 (Phys. Rev. Lett. to be published).
The binding energy can be easily calculated by looking at poles of T-matrix59;61
or by reducing the two-body Schr odinger equation to a self-consistent equation62.
The two-body properties at the BCS side and at resonance regime are signi cantly
changed by SO coupling. At weakly interacting BCS side with small negative as,
the binding energy behaves as
Eb ~22
2m4
e2e 2
jasj: (22)
where a large binding energy can always been reached by increasing the strength
of SO coupling. At resonance with as=1, 1=is the only length scale in the
two-body problem and therefore one has a universal result
Eb= 0:88~22
2m: (23)
While for the BEC side with small positive as, at leading order Ebis still given by
~2=(2ma2
s) and is not a ected by SO coupling. Moreover, because of SO coupling,
the two-body wave function has both singlet and triplet components. For a two-
body bound state with zero center-of-mass momentum, the wave function behaves
as59
= s(r)j"# #"i +
a(r)j""i + a(r)j##i (24)
where s(r) and a(r) are symmetric and anti-symmetric functions, respectively.
Furthermore, by looking at binding energy with nite center-of-mass momentum,
one can determine the e ective mass of molecules (two-body bound state). Hu et al.
61and Yu and Zhai62found that at the BCS limit, the e ective mass of molecule
nally saturates to 4 m, and at resonance, the e ective mass is a universal number
of 2:40m. In the BEC limit, the e ective mass saturates to 2 mas conventional case.
Generalizing conventional BEC-BCS crossover mean- eld theory to the case
with SO coupling and equal population hn"i=hn#i, one can show that the system
remains gapped for all kFas, although there are triplet p-wave components60;61;62;63,November 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
12Hui Zhai
and the pair wave function obtained from the mean- eld theory nally approaches
the wave function of two-body bound state (i.e. molecule wave function)60;62. Hence,
it is still a crossover as 1 =(kFas) changes from negative to positive. However, on the
other hand, the change of low-energy DOS and the presence of two-body bound
state at the BCS side and resonance regime will signi cantly change the properties
of crossover60;61;62;64. For example, as shown in Fig. 4(a), the pairing gap at the
BCS side is dramatically enhanced when =kFis comparable or larger than unitye.
For such a strong SO coupling, the DOS at Fermi energy becomes a constant, and
is much larger than the DOS in absence of SO coupling. That is the reason why
the pairing e ects become much dramatic even for same interaction strength. For
another example, in absence of SO coupling, Fermi energy is the only energy scale at
resonance, and therefore the size of Cooper pair is a universal constant times 1 =kF.
SO coupling introduces another scale at resonance, which is 1 =. For large , as
the pairing gap approaches two-body binding energy, the size of Cooper pairs also
approaches 1 =. Fig. 4(b) shows that the behavior of lundergoes a crossover from
1=kFto 1=as=kFincreases. This plot also shows that the size of Cooper pair in
thexyplane is di erent from the size along ^ zdirection, namely, the Cooper pairs
are anisotropic.61;62. In addition, one can also show that the super
uid transition
temperature at the BCS side can also be enhanced a lot by SO coupling. For large
enough SO coupling, it eventually approaches the BEC temperature of molecules
with mass 4 m62;64, which is a sizable fraction of Fermi temperature. The critical
temperature across resonance is rst estimated by Yu and Zhai62as shown in Fig.
4(c).
If SO coupling is slightly anisotropic, DOS at very low-energy will nally vanish.
However, there is still a large energy window .~22=(2m) where the DOS is greatly
enhanced by SO coupling. Hence, pairing gap will still be enhanced as long as the
density of fermions is not extremely low. Moreover, although it is no longer true
that arbitrary small attraction can cause a bound state, the critical value for the
appearance of a bound state will move from unitary point as=1to the BCS side
with negative as59. Once the bound state is present, it will in
uence the universal
behavior of pair size at resonance and super
uid critical temperature as discussed
above.
For the imbalanced case, the phase diagram becomes more richer. Several groups
have studied the phase diagram in presence of a Zeeman eld65;66;67;68;69and
in various other circumstance70;71;72;73;74;75;76;77;78;79;80 f. They have shown that,
eIn cold atom system, SO coupling is generated by atom-light coupling, and therefore is on the
order of the inverse of the laser wave length. And since the laser wave length and the inter-particle
distance are comparable (between 0:1mand1m) in atomic gases, the strength of SO
coupling in cold atom systems can naturally reach the regime =kF1.
fIskin and Subasi studied SO coupled Fermi gas with mass imbalance72. They use mixture of
di erent species as motivation of this study. We caution that the current way of generating SO
coupling is based on light coupling of di erent internal state of atoms, which can not generate SO
coupling if di erent internal states are di erent atomic speciesNovember 1, 2011 0:15 WSPC/INSTRUCTION FILE Review_SOC
Spin-Orbit Coupled Quantum Gases 13
instead of a crossover, there are phase transitions between topological and non-
topological phases65;66;67;68. They have also discussed how SO coupling in
uences
the competition between a uniform super
uid and a phase separation66;67.
5. Summary and Future Developments
SO coupled quantum gases with interactions are new systems in cold atom physics.
Moreover, SO coupled bosonic system has never been thought in physics before.
Currently our understanding of this system is still very limited, and many questions
remain open. However, even from our limited experience with this new system, one
can already get a feeling that this system has many unusual behaviors. This gives
a lot of opportunities for theorists and experimentalists.
Acknowledgements
I would like to thank Chao-Ming Jian and Zeng-Qing Yu for collaboration on this
subject, and thank Xiaoling Cui for helpful discussions. This work is supported by
Tsinghua University Initiative Scienti c Research Program, NSFC under Grant No.
11004118 and No. 11174176, NKBRSFC under Grant No. 2011CB921500.
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2010.01970v1.Detection_of_the_Orbital_Hall_Effect_by_the_Orbital_Spin_Conversion.pdf | Detection of the Orbital Hall E ect by the Orbital-Spin Conversion
Jiewen Xiao,1Yizhou Liu,1and Binghai Yan1,
1Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel
(Dated: October 6, 2020)
The intrinsic orbital Hall e ect (OHE), the orbital counterpart of the spin Hall e ect, was pre-
dicted and studied theoretically for more than one decade, yet to be observed in experiments. Here
we propose a strategy to convert the orbital current in OHE to the spin current via the spin-orbit
coupling from the contact. Furthermore, we nd that OHE can induce large nonreciprocal magne-
toresistance when employing the magnetic contact. Both the generated spin current and the orbital
Hall magnetoresistance can be applied to probe the OHE in experiments and design orbitronic
devices.
I. INTRODUCTION
The intrinsic orbital Hall e ect (OHE), where an elec-
tric eld induces a transverse orbital current, was pro-
posed by the Zhang group [1] soon after the prediction
of the intrinsic spin Hall e ect (SHE) [2, 3]. The SHE
was soon observed [4, 5], later applied for the spintronic
devices[6, and references therein], and also led to the sem-
inal discovery of the quantum SHE, i.e., the 2D topolog-
ical insulator [7, 8]. Di erent from the SHE, the OHE
does not rely on the spin-orbit coupling (SOC), and thus,
it was predicted to exist in many materials [1, 9{18] with
either weak or strong SOC, for example, in metals Al,
Cu, Au, and Pt.
In an OHE device, the transverse orbital current leads
to the orbital accumulation at transverse edges, similar
to the spin accumulation in a SHE device. Zhang et al
[1] proposed to measure the edge orbital accumulation by
the Kerr e ect. Recently, Ref. 19 predicted the orbital
torque generated by the orbital current. However, the
OHE is yet to be detected in experiments until today.
The detection of the orbital is rather challenging, because
the orbital is highly non-conserved compared to the spin,
especially at the device boundary.
A very recent work by us proposed [20] that the lon-
gitudinal current through DNA-type chiral materials is
orbital-polarized, and contacting DNA to a large-SOC
material can transform the orbital current into the spin
current. Thus, we are inspired to conceive a similar way
to detect the transverse OHE by converting the orbital
to the spin by the SOC proximity.
In this article, we propose two ways to probe the OHE,
where the strong SOC from the contact transforms the
orbitronic problem to the spintronic measurement. One
way is to generate spin current or spin polarization from
the transverse orbital current by connecting the edge to a
third lead with the strong interfacial SOC. Then the edge
spin polarization and spin current is promising to be mea-
sured by the Kerr e ect [4] and the inverse SHE [21{23],
respectively. The other way is to introduce a third mag-
netic lead and measure the magnetoresistance. We call
binghai.yan@weizmann.ac.il
FIG. 1. Illustration of the orbital-spin conversion and
the orbital Hall magnetoresistance (OHME). (a) The
orbital Hall e ect and the spin polarization/current genera-
tion. Opposite orbitals (red and blue circular arrows) from
the left lead de
ect into opposite boundaries. The red and
blue backgrounds represent the orbital accumulation at two
sides. Because of the SOC region (yellow) at one side, the
orbital current is converted into the spin current (indicated
by black arrows). (b) The two-terminal (2T) OHMR. The
third lead is magnetized but open. (c) The three-terminal
(3T) OHMR. The third lead is magnetized and conducts cur-
rent. The 2T/3T conductance between di erent leads relies
on the magnetization sensitively. The thickness of grey curves
represent the relative magnitude of the conductance.
it the orbital Hall magnetoresistance (OHMR), similar
to the spin Hall magnetoresistance [24, 25]. In our pro-
posal, the OHE refers to orbitals that resemble atomic-
like orbitals, which naturally couple to the spin via the
atomic SOC. We rst demonstrate detection principles in
a lattice model by transport calculations. Then we incor-
porate these principles into the metal copper, which has
negligible SOC and avoids the co-existence of the SHE,
as a typical example of realistic materials. In the copper
based device, we demonstrate the resultant spin polariza-
tion/current and very large OHMR (0 :31:3 %), which
are measurable by present experiment techniques.
II. RESULTS AND DISCUSSIONS
A. Methods and General Scenario
To detect the OHE, we introduce an extra contact with
the strong SOC on the boundary of the OHE material,arXiv:2010.01970v1 [cond-mat.mtrl-sci] 5 Oct 20202
as shown in Figure 1. This device can act for both two-
terminal (2T) and three-terminal (3T) measurements (or
more terminals). In theoretical calculations, we com-
pletely exclude SOC from all leads so that we can well
de ne the spin current. We also remove SOC in the OHE
material, the device regime in the center, to avoid the ex-
istence of SHE. Only nite atomic SOC is placed in the
interfacial region (highlighted by yellow in Figure 1) be-
tween the OHE and the third lead.
We rst prove the principle by a simple square-lattice
model that hosts OHE. As shown in the inset of Figure
2(a), a tight-binding spinless model is constructed, withthree orbitals s,pxandpyassigned to each site. Under
the above basis, the atomic orbital angular momentum
operator ^Lzis written as
^Lz= h2
40 0 0
0 0 i
0i03
5 (1)
And three eigenstates p(pxipy)=p
2;scorrespond
to eigenvalues Lz=1;0, respectively. After consider-
ing the nearest neighboring hopping, the Hamiltonian is
written as
H(kx;ky) =0
@Es+ 2tscoskxa+ 2tscoskya 2itspsinkxa 2itspsinkya
2itspsinkxa E px+ 2tpcoskxa+ 2tpcoskya 0
2itspsinkya 0 Epy+ 2tpcoskxa+ 2tpcoskya1
A
(2)
whereEs,EpxandEpyare onsite energies of s,pxandpy
orbitals.ts,tp,tp,tspare electron hopping integrals
betweensorbitals,type oriented porbitals,type
orientedporbitals, and sandporbitals, respectively. In
the following calculations, their values are speci ed as
Es= 1:3,Epx=Epy= 1:9,ts= 0:3,tp= 0:6,
tp= 0:3, andtsp= 0:5, in the unit of eV. To realize
the OHE, it requires the inter-orbital hopping to induce
the transverse Lzcurrent. Since pxandpyorbital are
orthogonal under the square lattice geometry, the inter-
orbital hopping tspbecomes the critical parameter that
controls the existence of the OHE. Then we introduce the
atomic SOC on the boundary to demonstrate the OHE
detection by soc^Sz^Lz, where ^Szis the spin operator.
We estimate the OHE conductivity ( OH) with the
orbital Berry curvature in the Kubo formula [26, 27],
OH=e
hX
nZd3k
(2)3fnk
Lz
n(k) (3)
Lz
n(k) = 2h2X
m6=nIm[hunkjjLzyjumkihumkj^vxjunki
(Enk Emk)2]
(4)
where
Lzn(k) is the \orbital" Berry curvature for the
nthband with Bloch state junkiand energy eigenvalue
Enk.fnkis the Fermi-Dirac distribution function. vx
is thexcomponent of the band velocity operator while
jLzyis the orbital current operator in the ydirection, de-
ned asjLzy= (^Lz^vy+ ^vy^Lz)=2. Therefore, the above
formula indicates that the interband perturbation in-
duces the orbital Berry curvature, further reiterating
the importance of inter-orbital hopping. We also note
that, the orbital Berry curvature is even under the time-
reversal symmetry or the spatial inversion symmetry,
Lzn(k) =
Lzn( k).
For the device schematically presented in Figure 1, wecalculated the conductance by the Landauer-B uttiker for-
mula [28] with the scattering matrix from lead ito lead
j,
Gi!j=e2
hX
n2j;m2ijSnmj2; (5)
whereSmnis the scattering matrix element from the m-
th eigenstate in lead ito thentheigenstate in lead j.
In all three leads ( i;j= 1;2;3), spin (Sz="#) is a con-
served quantity because of the lack of SOC. We turn
o the inter-orbital hopping in leads so that Lzis also
conserved, i.e., Lzcommutes with the Hamiltonian (See
Supplementary Materials). Therefore, with the spin and
orbital conserved leads, we can specify the conductance
in eachSzandLzchannel, and de ne the orbital- and
spin-polarized conductance as:
Gij
Sz=Gi!j" Gj!j#(6)
Gij
Lz=Gi!j+ Gi!j ; (7)
whereGij
Sz(Lz)is the conductance from lead ito the
Sz(Lz) channel of lead j.Gi!j0is omitted here since
Lz= 0 contributes no polarization. We performed the
conductance calculations with the quantum transport
package Kwant [29].
As illustrated in Figure 1(a), electrons with the op-
posite orbital angular momentum de
ect into transverse
directions in the OHE region, resulting in the transverse
orbital current. Therefore, orbital accumulates at two
sides, and the orbital polarization emerges. To detect the
orbital polarization, atomic SOC is added at one side, as
highlighted by yellow in Figure 1(a). After electron de-
ecting into the SOC region, the right-handed orbital
(red circular arrows) is converted to the up spin polar-
ization. If a third lead is further attached, the SOC re-
gion converts the orbital current into the spin current.3
If the third lead exhibits magnetization along z(Mz)
(Figure 1(b) and 1(c)), inversely, the OHE induces the
OHMR, relying on whether Mzis parallel or anti-parallel
to the generated spin polarization. In the 2T measure-
ment (Figure 1(b)), the conductance from lead 1 to lead 2
(G1!2) changes when the Mzdirection is reversed. And
the changing direction of G1!2depends sensitively on
the size of the device, due to the complex orbital accu-
mulation and re
ection with an open lead. While for its
spin counterpart, the SHE-induced magnetoresistance is
commonly measured in a 2T setup [24, 25]. In the 3T
device (Figure 1(c)), the situation is simpler since the
transverse orbital current can
ow into the third lead.
IfMzand spin polarization is parallel (anti-parallel), the
transverse orbital current matches (mismatches) the lead
magnetization, resulting in the high (low) G1!3and low
(high)G1!2accordingly. We point out that the 3T mea-
surement is usually more favorable than 2T, since the 3T
device avoids the 2T reciprocity constrain [30] and the
conductance change [ G=G(Mz) G( Mz)] is also
relatively larger in the third lead, as discussed in the fol-
lowing.
B. Spin Polarization and Spin Current Generated
by the OHE
The band structure weighted by the orbital Berry cur-
vature for the square lattice is plotted in Figure 2(a).
The highest band corresponds to the sorbital dispersion,
while two lower bands are dominated by porbitals. The
orbital Berry Curvature concentrates near the point,
Mpoint and - Mline in the Brillouin zone, where band
hybridization is strong. After integrating
Lzin the Bril-
louin zone, the orbital Hall conductivity is derived and
presented in Figure 2(a). It shows that, due to the inter-
orbital hopping tsp, states below ( porbitals) and above ( s
orbital) Fermi level both exhibits signi cant OH. How-
ever, iftspis turned o so that Lzis conserved, both
Lz
andOHvanishes.
Based on the square lattice with nite tsp, the 2T de-
vice is constructed, as shown in Figure 2(b). Without
SOC at two sides, the orbital density distribution is plot-
ted in Figure 2(c), which shows that opposite orbitals
accumulate and polarize at two boundaries. With SOC
turned on, spin density appears and largely concentrates
on the local SOC atoms, which is promising to be de-
tected by the Kerr e ect [4]. Since SOC couples the p+
(p ) orbital to the"(#) spin and forms the jjm=3
2i
(jjm= 3
2i) state, the spin density near the SOC region
largely follows the orbital density pattern: positive at
the upper side and negative at the lower side. To ver-
ify that the spin polarization is directly induced by the
OHE rather than SOC, we turned o the OHE by setting
tsp= 0 eV and preserve the SOC at the interface. The
supplementary Figure S2 shows that both the orbital and
spin polarization disappear.
On the basis of 2T device, a third lead is attached to
FIG. 2. Orbital-spin conversion in the two terminal
(2T) and three terminal (3T) device. (a) Band struc-
ture of the square lattice with tsp= 0:5 eV (left) and the
orbital Hall conductivity with tsp= 0:5 eV andtsp= 0:0 eV
(right). In the inset, the tight binding model of the square
lattice is presented. (b) 2T and 3T detection devices, where
larger spheres at two sides represent SOC regions. The yel-
low spheres at left, right and upper sides represent leads. (c)
Orbital and spin density distribution in the 2T setup, at the
energy level of 0.2 eV. (d) Total, orbital and spin conductance
from lead 1 to lead 3 with (left) and without (right) SOC.
the SOC side to form a 3T device, as shown in Figure
2(b). Therefore, rather than the orbital accumulation,
the orbital current will
ow into the third lead and gen-
erate the spin current. Figure 2(d) shows that the orbital
current from lead 1 to lead 3 ( G13
Lz) exists with and with-
out SOC at the interface. For instance, for states above
Fermi level, Lz= +1 states are more easily transported
into lead 3 than Lz= 1 states, and thus polarizes the
lead, being consistent with the positive orbital polariza-
tion at the upper side in Figure 2(c). On the other hand,
for the spin conductance, it only appears when turning on
SOC, and the energy dependence of G13
Szlargely follows
the orbital conductance, further demonstrating the spin
generation process from the orbital. If we increase the
SOC strength, G13
Szincreases accordingly, because of the
higher orbital-spin conversion eciency (see Figure S3).
We also test orbital non-conserved leads with nonzero
tsp, whose spin conductance remains the similar feature
(see Figure S4).
C. Orbital Hall Magnetoresistance
As discussed above, the current injected into lead 3 is
spin-polarized. When lead 3 is magnetized along the z
axis, we expect the existence of magnetization-dependent
conductance, i.e. G13(Mz)6=G13( Mz). From the cur-4
FIG. 3. Orbital Magnetoresistance with magnetic leads. (a) 2T and 3T detection devices with the exchange eld Mz
in the open and conducting lead 3. Larger spheres represent the interfacial SOC region. (b) Total conductance from lead 1 to
lead 2 in Mz eld in the 2T device, with the dephasing term set to 0.001. In the inset, the dephasing dependent G12for
the peak inside the circle is presented. (c) Total conductance from lead 1 to lead 2 in Mz eld in the 3T device. (d) Total,
(e) Spin and (f) Orbital conductance from lead 1 to lead 3 in Mz eld in the 3T device. In all these calculations, SOC and
Mzis set to 0.2 eV and 0.4 eV, respectively.
rent conservation [30], we deduce the relation,
G13= G12(8)
where GijGij(Mz) Gij( Mz). To demonstrate
this,Mzis introduced to lead 3 as an exchange eld to
the spin, as shown in the 3T setup in Figure 3(a). Results
in Figure 3(c) and 3(d) indicate that G12and G13
can reach several percentage of the total conductance at
some energies. We also con rm that G12and G13are
proportional to the exchange eld strength (see Figure
S5).
To understand the orbital induced magnetoresistance,
the spin and orbital conductance from lead 1 to lead 3 are
calculated. As shown in Figure 3(e), G13
Szalmost changes
its sign when
ipping Mzin lead 3, as expected. And G13
Sz
now inversely a ects G13
Lzbecause of the interfacial SOC.
When further comparing Figure 3(d) and 3(f), we found
that the change of the magnitude of G13
Lzis proportional
to the change of total conductance G13. Therefore,
it veri es the scenario in Figure 1(c): when the orbital
matches the spin in magnetic leads, G13
Lzand thusG13is
higher while G12is accordingly lower. Thus, it indicates
the essential role of the orbital in connecting charge and
spin in the transport.However, the 2T results exhibit qualitatively di erent
features from the 3T results. According to the reciprocity
relation [30], the 2T conductance obeys G12(Mz) =
G12( Mz). Only when the current conservation is bro-
ken, one may obtain the 2T magentoresistance. There-
fore, we introduce a dephasing term ito leak electrons
into virtual leads [31] to release the above constrain. As
shown in the inset of Figure 3(b), the G12is zero at
= 0, rst increases quickly and soon decreases as fur-
ther increasing . In the large limit, the system is to-
tally out of coherence and thus, the conductance cannot
remember the spin and orbital information. We note that
the dephasing exists ubiquitously in experiments due to
the dissipative scattering for example by electron-phonon
interaction and impurities.
For the same Mz, the 2T G12(Figure (3b)) roughly
exhibits the opposite sign compared to the 3T G12(Fig-
ure (3c)) in the energy window investigated. Unlike that
G13follows the change of G13
Lz(Figure (3f)), the change
direction of G12depends on the geometry of the 2T de-
vice (see Figure S6). The magnitude of the 2T G12
also depends sensitively on the value of . Its peak value
(Figure (3b)), with around 0.001, is comparable with
the 3T value in the same parameter regime. However,5
the 3T conductance avoids the strict constrain of the 2T
reciprocity, and the existence of 3T OHMR does not rely
on the dephasing. Furthermore, in the 3T setup, the
magnetoresistance ratio G13=G13in the third lead is
also larger than G12=G12, because of the lower total
conductance of G13. Therefore, we propose that the 3T
setup may be more advantageous to detect the OHE.
D. Realistic Material Cu
Based on the simple square lattice model, we demon-
strate two main phenomena, the OHE-induced spin po-
larization / spin current current assisted by the atomic
SOC on the boundary and the existence of OHMR. We
further examine them in a realistic material Cu. This
light noble metal is predicted to exhibit the strong OHE.
As shown in Figure 4(a), the 2T (without lead 3) and
3T devices are composed of Cu (without SOC) in both
the scattering region and leads, and the heavy metal Au
(with SOC) at two boundaries. We adopted the tight-
binding method to describe the Cu and leads, where 9
atomic orbitals ( s,px,py,pz,dz2,dx2 y2,dxy,dyz,dzx)
are assigned to each site. The nearest-neighboring and
the second-nearest-neighboring hoppings are considered
with the Slater-Koster type parameters from Ref. 32. For
the heavy metal Au at two sides, the SOC strength is
set to 0.37 eV as suggested by Ref. 32. With the tight-
binding approach, the rst-principles band structure of
Cu is reproduced (see Figure S7).
As shown in Figure 4(b), the orbital Berry curvature
concentrates on the dorbital region ( 4 eV 2 eV)
due to the orbital hybridization, consistent with previ-
ous works [12, 15]. After integrating
Lz, Figure 4(b)
shows that the orbital Hall conductivity is around 6000
(h=e)(
cm) 1in thedorbital region, even larger than
the spin Hall conductivity of Pt. Near the Fermi level, the
orbital Hall conductivity is determined by the s-orbital
derived bands and reduces to around 1000 ( h=e)(
cm) 1.
For the 2T device, the orbital and spin density at Fermi
level are plotted in Figure 4(c). The orbital polarization
exists at two sides as a consequence of the OHE. With the
heavy metal Au attached, spin polarization is generated,
which concentrates on Au atoms and follows the orbital
density pattern. To con rm that the spin polarization
is induced by the OHE, we arti cially turn o the inter-
orbital hopping in Cu to eliminate the OHE, but still
keep the SOC in the Au region. Result show that both
the orbital and spin polarization disappear (see Figure
S8), in accordance with our prediction.
For the 3T device, we add a third Cu lead to one SOC
side and calculate the spin conductance from lead 1 to
lead 3 (G13
Sz). As shown in Figure 4(d), the generatedspin conductance displays an energy-dependence similar
to the bulk OH. Near the Fermi level, the spin polar-
ization rate can reach 4 %, and it is even around 20%
in thedorbital region. Therefore, a sizable spin current
can also be generated from the OHE by adding an inter-
facial SOC layer. Similarly, when arti cially switching
o the OHE of Cu but keeping the Au part, the spin cur-
rent disappears, eliminating the contribution of the SHE
brought by the thin Au layer (see Figure S9).
We also studied the OHMR by applying an exchange
eldMzin the lead 3. We choose Mz= 0:95 eV ac-
cording to the approximate spin splitting in the tran-
sition metal Co (see Figure S10). As shown in Figure
4(e) and 4(f), the 3T OHMR is rather large, where we
nd G12=G120:3% and G13=G131:3% at the
Fermi level. In experiment, the SHE magentoresistance is
around 0:050:5% (see Ref. 33 for example). Therefore,
the sizable OHMR in copper can be fairly measurable by
present experimental techniques. We should point out
that similar e ects can be generalized to other OHE ma-
terials like Li and Al [15].
III. SUMMARY
In summary, we have proposed the OHE detec-
tion strategies by converting the orbital to spin by
the interfacial SOC, and inducing the strong spin cur-
rent/polarization. Inversely, the OHE can also generate
the large nonreciprocal magnetoresistance when employ-
ing the magnetic contact. We point out that, compared
to the two-terminal one, the three-terminal OHMR does
not require the dephasing term , and may be more ad-
vantageous to detect the OHE. Using the device setup
based on the metal Cu, we demonstrate that the gener-
ated spin polarization and OHMR are strong enough to
be measured in the present experimental condition. Our
work will pave a way to realize the OHE in experiment,
and further design orbitronic or even orbitothermal de-
vices for future applications.
IV. ACKNOWLEDGEMENT
We honor the memory of Prof. Shoucheng Zhang. This
article follows his earlier works on the intrinsic orbital
Hall e ect and spin Hall e ect. B.Y. acknowledges the
nancial support by the Willner Family Leadership Insti-
tute for the Weizmann Institute of Science, the Benoziyo
Endowment Fund for the Advancement of Science, Ruth
and Herman Albert Scholars Program for New Scientists,
and the European Research Council (ERC) under the
European Union's Horizon 2020 research and innovation
programme (Grant No. 815869, NonlinearTopo).
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FIG. 4. The orbital-spin conversion and orbital magnetoresistance in Cu. (a) 2T/3T device based on real materials,
where the scattering region and leads are treated as Cu and the SOC region is treated as Au. (b) The band structure of Cu,
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2206.00784v2.Substrate_Effects_on_Spin_Relaxation_in_Two_Dimensional_Dirac_Materials_with_Strong_Spin_Orbit_Coupling.pdf | Substrate E ects on Spin Relaxation in Two-Dimensional Dirac Materials with Strong
Spin-Orbit Coupling
Junqing Xu1,and Yuan Ping1,y
1Department of Chemistry and Biochemistry, University of California, Santa Cruz, CA 95064, USA
(Dated: December 6, 2022)
Understanding substrate e ects on spin dynamics and relaxation in two-dimensional (2D) mate-
rials is of key importance for spintronics and quantum information applications. However, the key
factors that determine the substrate e ect on spin relaxation, in particular for materials with strong
spin-orbit coupling, have not been well understood. Here we performed rst-principles real-time
density-matrix dynamics simulations with spin-orbit coupling (SOC) and quantum descriptions of
electron-phonon and electron-impurity scattering for the spin lifetimes of supported/free-standing
germanene, a prototypical strong SOC 2D Dirac material. We show that the e ects of di er-
ent substrates on spin lifetime ( s) can surprisingly di er by two orders of magnitude. We nd
that substrate e ects on sare closely related to substrate-induced modi cations of the SOC- eld
anisotropy, which changes the spin-
ip scattering matrix elements. We propose a new electronic
quantity, named spin-
ip angle "#, to characterize spin relaxation caused by intervalley spin-
ip
scattering. We nd that the spin relaxation rate is approximately proportional to the averaged value
of sin2
"#=2
, which can be used as a guiding parameter of controlling spin relaxation.
INTRODUCTION
Since the long spin di usion length ( ls) in large-area
graphene was rst reported by Tombros et al.[1], sig-
ni cant advances have been made in the eld of spin-
tronics, which has the potential to realize low-power
electronics by utilizing spin as the information car-
rier. Various 2D materials have shown promising spin-
tronic properties[2], e.g., long lsat room temperatures
in graphene[3] and ultrathin black phosphorus[4], spin-
valley locking (SVL) and ultralong spin lifetime sat
low temperatures in transition metal dichalcogenides
(TMDs)[5] and germanene[6], and persistent spin helix
in 2D hybrid perovskites[7].
Understanding spin relaxation and transport mecha-
nism in materials is of key importance for spintronics
and spin-based quantum information technologies. One
critical metric for ideal materials in such applications is
spin lifetime ( s), often required to be suciently long
for stable detection and manipulation of spin. In 2D-
material-based spintronic devices, the materials are usu-
ally supported on a substrate. Therefore, for the design
of those devices, it is crucial to understand substrate ef-
fects on spin relaxation. In past work, the substrate ef-
fects were mostly studied for weak SOC Dirac materials
like graphene[8{12]. How substrates a ect strong SOC
Dirac materials like germanene is unknown. In partic-
ular, the spin relaxation mechanism between weak and
strong SOC Dirac materials was shown to be drastically
di erent. [6] Therefore, careful investigations are required
to unveil the distinct substrate e ects on these two types
of materials.
Here we focus on the dangling-bond-free insulating
jxu153@ucsc.edu
yyuanping@ucsc.edusubstrates, which interact weakly with the material thus
preserve its main physical properties. Insulating sub-
strates can a ect spin dynamics and relaxation in sev-
eral aspects: (i) They may induce strong SOC elds, so
called internal magnetic elds Binby breaking inversion
symmetry[9] or through proximity e ects[10]. For ex-
ample, the hexagonal boron nitride substrate can induce
Rashba-like elds on graphene and dramatically accel-
erate its spin relaxation and enhance the anisotropy of
sbetween in-plane and out-of-plane directions[8]. (ii)
Substrates may introduce additional impurities [11, 12]
or reduce impurities/defects in material layers, e.g.,
by encapsulation[13]. In consequence, substrates may
change the electron-impurity (e-i) scattering strength,
which a ects spin relaxation through SOC. (iii) Ther-
mal vibrations of substrate atoms can introduce addi-
tional spin-phonon scattering by interacting with spins
of materials[9].
Previously most theoretical studies of substrate e ects
on spin relaxation were done based on model Hamil-
tonian and simpli ed spin relaxation models[9, 11, 12].
While those models provide rich mechanistic insights,
they are lack of predictive power and quantitative ac-
curacy, compared to rst-principles theory. On the other
hand, most rst-principles studies only simulated the
band structures and spin polarizations/textures of the
heterostructures[14{16], which are not adequate for un-
derstanding spin relaxation. Recently, with our newly-
developed rst-principles density-matrix (FPDM) dy-
namics approach, we studied the hBN substrate e ect on
spin relaxation of graphene, a weak SOC Dirac material.
We found a dominant D'yakonov-Perel' (DP) mechanism
and nontrivial modi cation of SOC elds and electron-
phonon coupling by substrates[8]. However, strong SOC
Dirac materials can have a di erent spin relaxation mech-
anism - Elliott-Yafet (EY) mechanism[17], with only
spin-
ip transition and no spin precession, unlike the DParXiv:2206.00784v2 [cond-mat.mes-hall] 4 Dec 20222
mechanism. How substrates a ect spin relaxation of ma-
terials dominated by EY mechanism is the key question
here. Furthermore, how such e ects vary among di er-
ent substrates is another outstanding question for guiding
experimental design of interfaces.
In our recent study, we have predicted that mono-
layer germanene (ML-Ge) is a promising material for
spin-valleytronic applications, due to its excellent prop-
erties including spin-valley locking, long sandls, and
highly tunable spin properties by varying gates and ex-
ternal elds[6]. As discussed in Ref. 6, ML-Ge has strong
intrinsic SOC unlike graphene and silicene. Under an
out-of-plane electric eld (in consequence broken inver-
sion symmetry), a strong out-of-plane internal magnetic
eld forms, which may lead to mostly EY spin relax-
ation [6]. Therefore, predicting sof supported ML-Ge
is important for future applications and our understand-
ing of substrate e ects on strong SOC materials. Here,
we examine the substrate e ects on spin relaxation in
ML-Ge through FPDM simulations, with self-consistent
SOC and quantum descriptions of e-ph and e-i scatter-
ing processes[6, 8, 18{20]. We study free-standing ML-
Ge and ML-Ge supported by four di erent insulating
substrates - germanane (GeH), silicane (SiH), GaTe and
InSe. The choice of substrates is based on similar lat-
tice constants to ML-Ge, preservation of Dirac Cones,
and experimental synthesis accessibility[21, 22]. We will
rst show how electronic structures and sof ML-Ge
are changed by di erent substrates - while sof ML-
Ge on GeH and SiH are similar to free-standing ML-
Ge, the GaTe and InSe substrates strongly reduce sof
ML-Ge due to stronger interlayer interactions. We then
discuss what quantities are responsible for the disparate
substrate e ects on spin relaxation, which eventually an-
swered the outstanding questions we raised earlier.
RESULTS AND DISCUSSIONS
Substrate e ects on electronic structure and spin
texture
We begin with comparing band structures and spin
textures of free-standing and supported ML-Ge in Fig. 1,
which are essential for understanding spin relaxation
mechanisms. Since one of the most important e ects of
a substrate is to induce an out-of-plane electric eld Ez
on the material layer, we also study ML-Ge under a con-
stantEzas a reference. The choice of the Ezis based on
reproducing a similar band splitting to the one in ML-Ge
with substrates. The band structure of ML-Ge is similar
to graphene with two Dirac cones at KandK0 K,
but a larger band gap of 23 meV. At Ez= 0, due to
time-reversal and inversion symmetries of ML-Ge, every
two bands form a Kramers degenerate pair[17]. A nite
Ezor a substrate breaks the inversion symmetry and in-
duces a strong out-of-plane internal B eld Bin(Eq. 21),
which splits the Kramers pairs into spin-up and spin-down bands[6]. Interestingly, we nd that band struc-
tures of ML-Ge-SiH (Fig. 1c) and ML-Ge-GeH (Fig. S4)
are quite similar to free-standing ML-Ge under Ez=-7
V/nm (ML-Ge@-7V/nm, Fig. 1b), which indicates that
the impact of the SiH/GeH substrate on band structure
andBinmay be similar to a nite Ez(see Fig. S4). This
similarity is frequently assumed in model Hamiltonian
studies[9, 11]. On the other hand, the band structures of
ML-Ge-InSe (Fig. 1d) and ML-Ge-GaTe (Fig. S4) have
more di erences from the free-standing one under Ez,
with larger band gaps, smaller band curvatures at Dirac
Cones, and larger electron-hole asymmetry of band split-
tings. This implies that the impact of the InSe/GaTe
substrates can not be approximated by applying an Ez
to the free-standing ML-Ge, unlike SiH/GeH substrates.
We further examine the spin expectation value vectors
Sexpof substrate-supported ML-Ge. Sexpis parallel to
Binby de nition (Eq. 21). Sexp
Sexp
x;Sexp
y;Sexp
z
withSexp
ibeing spin expectation value along direction i
and is the diagonal element of spin matrix siin Bloch
basis. Importantly, from Fig. 1e and 1f, although Sexpof
ML-Ge on substrates are highly polarized along z(out-of-
plane) direction, the in-plane components of Sexpof ML-
Ge-InSe (and ML-Ge-GaTe) are much more pronounced
than ML-Ge-SiH (and ML-Ge-GeH). Such di erences are
crucial to the out-of-plane spin relaxation as discussed in
a later subsection.
Spin lifetimes of germanene on substrates and spin
relaxation mechanism
We then perform our rst-principles density-matrix
calculation [6, 18{20] at proposed interfaces, and examine
the role of electron-phonon coupling in spin relaxation of
ML-Ge at di erent substrates. Throughout this paper,
we focus on out-of-plane sof ML-Ge systems, since their
in-plane sis too short and less interesting. We com-
pare out-of-plane sdue to e-ph scattering between the
free-standing ML-Ge (with/without an electric eld) and
ML-Ge on di erent substrates in Fig. 2a. Here we show
electron sfor most ML-Ge/substrate systems as intrin-
sic semiconductors, except hole sfor the ML-Ge-InSe
interface. This choice is because electron sare mostly
longer than hole sat lowTexcept for the one at the
ML-Ge-InSe interface; longer lifetime is often more ad-
vantageous for spintronics applications. From Fig. 2, we
nd that sof ML-Ge under Ez= 0 and -7 V/nm are
at the same order of magnitude for a wide range of tem-
peratures. The di erences are only considerable at low
T, e.g, by 3-4 times at 20 K. On the other hand, sof
supported ML-Ge are very sensitive to the speci c sub-
strates. While sof ML-Ge-GeH and ML-Ge-SiH have
the same order of magnitude as the free-standing ML-
Ge, in particular very close between ML-Ge-GeH and
ML-Ge@-7 V/nm, sof ML-Ge-GaTe and ML-Ge-InSe
are shorter by at least 1-2 orders of magnitude in the
whole temperature range. This separates the substrates3
FIG. 1. Band structures and spin textures around the Dirac cones of ML-Ge systems with and without substrates. (a)-(d) show
band structures of ML-Ge under Ez= 0 and under -7 V/nm and ML-Ge on silicane (SiH) and on InSe substrates respectively.
(e) and (f) show spin textures in the kx-kyplane and 3D plots of the spin vectors Sexp
k1on the circlej !kj= 0:005 bohr 1of
the band at the band edge around Kof ML-Ge on SiH and InSe substrates respectively. Sexp
Sexp
x;Sexp
y;Sexp
z
withSexp
i
being spin expectation value along direction iand is the diagonal element of spin matrix siin Bloch basis. The red and blue
bands correspond to spin-up and spin-down states. Due to time-reversal symmetry, band structures around another Dirac cone
atK0= Kare the same except that the spin-up and spin-down bands are reversed. The grey, white, blue, pink and green
balls correspond to Ge, H, Si, In and Se atoms, respectively. Band structures of ML-Ge on germanane (GeH) and GaTe are
shown in Fig. S4 in the Supporting Information, and are similar to those of ML-Ge on SiH and InSe substrates, respectively.
In subplots (e) and (f), the color scales Sexp
zand the arrow length scales the vector length of in-plane spin expectation value.
into two categories, i.e. with a weak e ect (ML-Ge-GeH
and ML-Ge-SiH) and a strong e ect (ML-Ge-GaTe and
ML-Ge-InSe).
We further investigate the role of electron-impurity (e-
i) scattering in spin relaxation under di erent substrates,
by introducing defects in the material layer. We consider
a common type of impurity - single neutral Ge vacancy,
whose formation energy was found relatively low in previ-
ous theoretical studies[23, 24]. From Fig. 2b, we can see
that sof all ve systems decrease with impurity density
ni. Since carrier scattering rates 1
p(carrier lifetime p)
increases (decrease) with ni, we then obtain sdecreases
with p's decrease, an evidence of EY spin relaxation
mechanism. Moreover, we nd that sis sensitive to the
type of the substrate with all values of ni, and for each of
four substrates, sis reduced by a similar amount with
di erentni, from low density limit (109cm 2, where e-
ph scattering dominates) to relatively high density (1012
cm 2, where e-i scattering becomes more important).
Since the bands near the Fermi energy are composed
of the Dirac cone electrons around KandK0valleys in
ML-Ge, spin relaxation process arises from intervalleyand intravalley e-ph scatterings. We then examine rel-
ative intervalley spin relaxation contribution (see its
de nition in the Fig. 2 caption) in Fig. 2c. being close
to 1 or 0 corresponds to intervalley or intravalley scatter-
ing being dominant in spin relaxation. becomes close
to 1 below 70 K for electrons of ML-Ge-SiH, and below
120 K for holes of ML-Ge-InSe. This indicates that at
lowTonly intervalley scattering processes are relevant
to spin relaxation in ML-Ge on substrates. This is a re-
sult of spin-valley locking (SVL), i.e. large SOC-induced
band splittings lock up or down spin with a particular K
or K' valley [6]. According to Fig. 1 and 2c, the SVL
transition temperature ( TSVL; below which the propor-
tion of intervalley spin relaxation rate is close to 1)
seems approximately proportional to SOC splitting en-
ergy SOC, e.g. for electrons (CBM) of ML-Ge-GaTe and
ML-Ge-SiH, and for holes (VBM) of ML-Ge-InSe, SOC
are15,24 and 40 meV respectively, while TSVLare
50, 70 and 120 K respectively. As SOCcan be tuned by
Ezand the substrate, TSVLcan be tuned simultaneously.
Under SVL condition, spin or valley lifetime tends to be
exceptionally long, which is ideal for spin-/valley-tronic4
FIG. 2. The out-of-plane spin lifetime sof intrinsic free-standing and substrate-supported ML-Ge. (a) sof ML-Ge under
Ez= 0, -7 V/nm and substrate-supported ML-Ge as a function of temperature without impurities. Here we show electron s
for intrinsic ML-Ge systems except that hole sis shown for ML-Ge-InSe, since electron sare longer than hole sat lowT
except ML-Ge-InSe. (b) sas a function of impurity density niat 50 K. The impurities are neutral ML-Ge vacancy with 50% at
higher positions and 50% at lower ones of a Ge layer. The dashed vertical line corresponds to the impurity density where e-ph
and e-i scatterings contribute equally to spin relaxation ( ni;s). And e-ph (e-i) scattering is more dominant if ni<(>)ni;s. (c)
The proportion of intervalley spin relaxation contribution of (electrons of) ML-Ge-SiH and (holes of) ML-Ge-InSe without
impurities. is de ned as =( inter
s;z) 1
( inters;z) 1+w( intras;z) 1, where inter
s;z and intra
s;z are intervalley and intravalley spin lifetimes,
corresponding to scattering processes between KandK0valleys and within a single KorK0valley, respectively. being close
to 1 or 0 corresponds to dominant intervalley or intravalley spin relaxation, respectively. wis a weight factor related to what
percentage of total Szcan be relaxed out by intravalley scattering itself. wbeing close to 0 and 1 correspond to the cases that
intravalley scattering can only relax a small part (0) and most of excess spin (1) respectively. In Supporting Information Sec.
SII, we give more details about de nition of w. (d) Electron and hole sat 20 K of ML-Ge without impurities on hydrogen-
terminated multilayer Si, labeled as Si nH withnbeing number of Si layers. Si nH is silicane if n= 1, and hydrogen-terminated
Silicon (111) surface if n=1.
applications.
Additionally, the studied substrates here are mono-
layer, while practically multilayers or bulk are more com-
mon, thus it is necessary to understand how schanges
with the number of substrate layers. In Fig. 2d, we
show sat 20 K of ML-Ge on hydrogen-terminated mul-tilayer Si, ML-Ge-Si nH, withnbeing number of Si layer.
SinH becomes hydrogen-terminated Silicon (111) surface
ifn=1. We nd that sare changed by only 30%-40%
by increasing nfrom 1 to 3 and kept unchanged after
n3. For generality of our conclusion, we also test
the layer dependence of a di erent substrate. We found5
the sof ML-Ge on bilayer InSe ( n= 2) is changed by
8% compared to monolayer InSe at 20 K, even smaller
change than the one at Si nH substrates. Given the dis-
parate properties of these two substrates, we conclude
using a monolayer is a reasonable choice for simulating
the substrate e ects on sin this work.
The correlation of electronic structure and phonon
properties to spin relaxation at di erent substrates
We next analyze in detail the relevant physical quan-
tities, and determine the key factors responsible for sub-
strate e ects on spin relaxation. We focus on results
under lowTas spin relaxation properties are superior at
lowerT(the realization of SVL and longer s).
First, to have a qualitative understanding of the
material-substrate interaction strength, we show charge
density distribution at the cross-section of interfaces in
Fig. 3a-d. It seems that four substrates can be catego-
rized into two groups: group A contains GeH and SiH
with lower charge density distribution in the bonding re-
gions (pointed by the arrows); group B contains GaTe
and InSe with higher charge density distribution in the
bonding regions. In Fig. S5, we investigate the charge
density change e(de ned by the charge density dif-
ference between interfaces and individual components).
Consistent with Fig. 3, we nd that efor GaTe and
InSe substrates overall has larger magnitude than the
one for GeH and SiH substrates. Therefore the material-
substrate interactions of group B seem stronger than
those of group A. Intuitively, we may expect that the
stronger the interaction, the stronger the substrate e ect
is. The FPDM simulations in Fig. 2a-b indeed show that
the substrate e ects of group B being stronger than those
of group A on s, consistent with the above intuition.
Next we examine electronic quantities closely related
to spin-
ip scattering responsible to EY spin relaxation.
Qualitatively, for a state k1, its spin-
ip scattering rate
1
s(k1) is proportional to the number of its pair states
k2allowing spin-
ip transitions between them. The num-
ber of pair states is approximately proportional to den-
sity of states (DOS) around the energy of k1. Moreover,
for EY mechanism, it is commonly assumed that spin
relaxation rate is proportional to the degree of mixture
of spin-up and spin-down states (along the zdirection
here), so called \spin-mixing" parameter[17] b2
z(see its
de nition in Sec. SII), i.e., 1
s/
b2
z
, where
b2
z
is the
statistically averaged spin mixing parameter as de ned in
Ref. 6. Therefore, we show DOS, energy-resolved spin-
mixingb2
z(") and
b2
z
as a function of temperature in
Fig. 3e-g.
We nd that in Fig. 3e DOS of ML-Ge-GeH and ML-
Ge-SiH are quite close to that of ML-Ge@-7V/nm, while
DOS of ML-Ge-GaTe and ML-Ge-InSe are 50%-100%
higher around the band edge. Such DOS di erences
are qualitatively explained by the staggered potentials of
ML-Ge-GaTe and ML-Ge-InSe being greater than thoseof ML-Ge-GeH and ML-Ge-SiH according to the model
Hamiltonian proposed in Ref. 25. In Fig. 3f-g, b2
zof ML-
Ge-GeH and ML-Ge-SiH are found similar to ML-Ge@-
7 V/nm, and not sensitive to energy and temperature.
On the contrast, for ML-Ge-GaTe and ML-Ge-InSe, their
b2
z(") and
b2
z
increase rapidly with energy and temper-
ature. Speci cally, we can see at 300 K,
b2
z
of ML-Ge-
GaTe and ML-Ge-InSe are about 4-20 times of the one
of ML-Ge-GeH and ML-Ge-SiH in Fig. 3g. Thus the one
order of magnitude di erence of sbetween group A (ML-
Ge-GeH and ML-Ge-SiH) and group B (ML-Ge-GaTe
and ML-Ge-InSe) substrates at 300 K can be largely ex-
plained by the substrate-induced changes of DOS and
b2
z
. On the other hand, at low T, e.g., at 50 K,
b2
z
of ML-Ge-GaTe and ML-Ge-InSe are only about 1.5 and
2.5 times of the ones of ML-Ge-GeH and ML-Ge-SiH,
and DOS are only tens of percent higher. However, there
is still 1-2 order of magnitude di erence of sbetween
di erent substrates. Therefore, the substrate e ects on
scan not be fully explained by the changes of
b2
z
and
DOS, in particular at relatively low temperature.
We then examine if substrate-induced modi cations of
phonon can explain the changes of spin relaxation at dif-
ferent substrates, especially at low T. We emphasize that
at lowT, since spin relaxation is fully determined by
intervalley processes (Fig. 2c), the related phonons are
mostly close to wavevector K. From Fig. 4, we nd that
the most important phonon mode for spin relaxation at
lowThas several similar features: (i) It contributes to
more than 60% of spin relaxation (see Fig 4a). (ii) Its
energy is around 7 meV in the table of Fig. 4a. (iii)
Its vibration is
exural-like, i.e., atoms mostly vibrate
along the out-of-plane direction as shown in Fig. 4b-
d. Moreover, for this mode, the substrate atoms have
negligible thermal vibration amplitude compared to the
one of the materials atoms. This is also con rmed in
the layer-projected phonon dispersion of ML-Ge-InSe in
Fig. 4e. The purple box highlights the critical phonon
mode around K, with most contribution from the mate-
rial layer. (iv) The critical phonon mode does not couple
with the substrate strongly, since its vibration frequency
does not change much when substrate atoms are xed
(by comparing Fig. 4e with f). We thus conclude that
the substrate-induced modi cations of phonons and ther-
mal vibrations of substrate atoms seem not important for
spin relaxation at low T(e.g. below 20 K).
Therefore, neither the simple electronic quantities
b2
and DOS nor the phonon properties can explain the sub-
strate e ects on spin relaxation at low T.
The determining factors of spin relaxation derived
from spin-
ip matrix elements
On the other hand, with a simpli ed picture of spin-
ip transition by the Fermi's Golden Rule, the scattering
rate is proportional to the modulus square of the scat-
tering matrix elements. For a further mechanistic un-6
FIG. 3. Charge density, density of states (DOS), and spin mixing parameters of free-standing and substrate-supported ML-Ge.
Cross-section views of charge density at interfaces of ML-Ge on (a) GeH, (b) SiH, (c) GaTe, and (d) InSe. The Ge layers
are above the substrate layers. The unit of charge density is e=bohr3. Charge densities in the regions pointed out by black
arrows show signi cant di erences among di erent systems. (e) DOS and (f) energy-solved spin-mixing parameter along zaxis
b2
z(") of ML-Ge under Ez=-7 V/nm and on di erent substrates. "edgeis the band edge energy at the valence band maximum
or conduction band minimum. The step or sudden jump in the DOS curve corresponds to the edge energy of the second
conduction/valence band or the SOC-induced splitting energy at K. (g) The temperature-dependent e ective spin-mixing
parameter
b2
z
of various ML-Ge systems.
derstanding, we turn to examine the modulus square of
the spin-
ip matrix elements, and compare their qual-
itative trend with our FPDM simulations. Note that
most matrix elements are irrelevant to spin relaxation
and we need to pick the \more relevant" ones, by de ning
a statistically-averaged function. Therefore, we propose
an e ective band-edge-averaged spin-
ip matrix element
jeg"#j2(Eq. 8). Here the spin-
ip matrix element can be
for general scattering processes; in the following we focus
on e-ph process for simplicity. We also propose a so-called
scattering density of states DSin Eq. 9, which measures
the density of spin-
ip transitions and can be roughly re-
garded as a weighted-averaged value of the usual DOS.
Based on the generalized Fermi's golden rule, we approx-
imately have 1
s/jeg"#j2DSfor EY spin relaxation (see
the discussions above Eq. 11 in \Methods" section).
As shown in Fig. 5a, 1
sis almost linearly propor-
tional tojeg"#j2DSat 20 K. As the variation of DSamong
ML-Ge on di erent substrates is at most three times (see
Fig. 3e and Fig. S6), which is much weaker than the
large variation of 1
s, this indicates that the substrate-
induced change of sis mostly due to the substrate-
induced change of spin-
ip matrix elements. Although
jeg"#j2was often considered approximately proportionalto
b2
, resulting in 1
s/
b2
, our results in Fig. 3
in the earlier section indicate that such simple approx-
imation is not applicable here, especially inadequate of
explaining substrate dependence of sat lowT.
To nd out the reason why jeg"#j2for di erent sub-
strates are so di erent, we rst examine the averaged
spin-
ip wavefunction overlap jo"#j2(with the reciprocal
lattice vector G= 0), closely related to jeg"#j2(Eq. 18
and Eq. 17). From Fig. 5b, 1
sandjo"#j2have the same
trend, which implies jeg"#j2andjo"#j2may have the same
trend. However, in general, the G6=0elements ofjo"#j2
may be important as well, which can not be unambigu-
ously evaluated here. (See detailed discussions in the
subsection \Spin-
ip e-ph and overlap matrix element"
in the \Methods" section).
To have deeper intuitive understanding, we then pro-
pose an important electronic quantity for intervalley
spin-
ip scattering - the spin-
ip angle "#between two
electronic states. For two states ( k1;n1) and (k2;n2) with
opposite spin directions, "#is the angle between Sexp
k1n1
andSexp
k2n2or equivalently the angle between Bin
k1and
Bin
k2.
The motivation of examining "#is that: Suppose two
wavevectors k1andk2= k1are in two opposite valleys7
(a)
Substrate !K(meV) Contribution
Ge@-7V/nm 7.7 78%
Ge-GeH 6.9 70%
Ge-SiH 7.1 64%
Ge-GaTe 6.4 90%
Ge-InSe 7.2 99%
FIG. 4. (a) The phonon energy at wavevector Kof the mode
that contributes the most to spin relaxation, and the per-
centage of its contribution for various systems at 20 K. We
consider momentum transfer K, as spin relaxation is fully de-
termined by intervalley processes between KandK0valleys.
(b), (c) and (d) Typical vibrations of atoms in 3 3 supercells
of (b) ML-Ge@-7 V/nm, (c) ML-Ge-SiH, and (d) ML-Ge-
InSe of the most important phonon mode at Karound 7 meV
(shown in (a)). The red arrows represent displacement. The
atomic displacements smaller than 10% of the strongest are
not shown. (e) The layer-projected phonon dispersion of ML-
Ge-InSe within 12 meV. The red and blue colors correspond
to the phonon displacements mostly contributed from the ma-
terial (red) and substrate layer (blue) respectively. The green
color means the contribution to the phonon displacements
from the material and substrate layers are similar. The pur-
ple boxes highlight the two most important phonon modes
aroundKfor spin relaxation.(f) Phonon dispersion of ML-
Ge-InSe within 12 meV with substrate atoms (InSe) being
xed at equilibrium structure and only Ge atoms are allowed
to vibrate.
Qand -Qrespectively and there is a pair of bands, which
are originally Kramers degenerate but splitted by Bin.
Due to time-reversal symmetry, we have Bin
k1= Bin
k2,
which means the two states at the same band natk1
andk2have opposite spins and "#between them is
zero. Therefore, the matrix element of operator bAbe-
tween states ( k1;n) and (k2;n) -Ak1n;k2nis a spin-
ipone and we name it as A"#
k1k2. According to Ref. 26,
with time-reversal symmetry, A"#
k1k2is exactly zero. In
general, for another wavevector k3within valley - Qbut
not k1,A"#
k1k3is usually non-zero. One critical quan-
tity that determines the intervalley spin-
ip matrix ele-
mentA"#
k1k3for a band within the pair introduced above
is"#
k1k3. Based on time-independent perturbation theory,
we can prove that A"# between two states is approxi-
mately proportional to sin
"#=2 . The derivation is
given in subsection \Spin-
ip angle "#for intervalley
spin relaxation" in \Methods" section.
As shown in Fig. 5c, 1
sof ML-Ge on di erent
substrates at 20 K is almost linearly proportional to
sin2("#=2)DS, where sin2("#=2) is the statistically-
averaged modulus square of sin
"#=2
. This indicates
that the relation jeg"#j2/sin2("#=2) is nearly perfectly
satis ed at low T, where intervalley processes dominate
spin relaxation. We additionally show the relations be-
tween 1
sandjeg"#j2DS,jo"#j2DSandsin2("#=2)DSat
300 K in Fig. S7. Here the trend of 1
sis still approxi-
mately captured by the trends of jeg"#j2DS,jo"#j2DSand
sin2("#=2)DS, although not perfectly linear as at low T.
Since"#is de ned by Sexpat di erent states, sis
highly correlated with Sexpand more speci cally with
the anisotropy of Sexp(equivalent to the anisotropy of
Bin). Qualitatively, the larger anisotropy of Sexpleads to
smaller"#and longer salong the high-spin-polarization
direction. This nding may be applicable to spin re-
laxation in other materials whenever intervalley spin-
ip
scattering dominates or spin-valley locking exists, e.g., in
TMDs[5], Stanene[27], 2D hybrid perovskites with persis-
tent spin helix[7], etc.
At the end, we brie
y discuss the substrate e ects
on in-plane spin relaxation ( s;x), whereas only out-of-
plane spin relaxation was discussed earlier. From Table
SI, we nd that s;xof ML-Ge@-7V/nm and supported
ML-Ge are signi cantly (e.g., two orders of magnitude)
shorter than free-standing ML-Ge, but the di erences
between s;xof ML-Ge on di erent substrates are rela-
tively small (within 50%). This is because: With a non-
zeroEzor a substrate, the inversion symmetry broken
induces strong out-of-plane internal magnetic eld Bin
z
(>100 Tesla), so that the excited in-plane spins will pre-
cess rapidly about Bin
z. The spin precession signi cantly
a ects spin decay and the main spin decay mechanism
becomes DP or free induction decay mechanism[28] in-
stead of EY mechanism. For both DP and free induc-
tion decay mechanisms[20, 28], s;xdecreases with the
uctuation amplitude (among di erent k-points) of the
Bincomponents perpendicular to the xdirection. As
the
uctuation amplitude of Bin
zof ML-Ge@-7V/nm and
supported ML-Ge is large (Table SI; much greater than
the one ofBin
y), their s;xcan be much shorter than the
value of ML-Ge at zero electric eld when EY mechanism
dominates. Moreover, since the
uctuation amplitude of
Bin
zof ML-Ge on di erent substrates has the same or-8
FIG. 5. The relation between 1
sand the averaged modulus square of spin-
ip e-ph matrix elements jeg"#j2, of spin-
ip overlap
matrix elementsjo"#j2andsin2("#=2) multiplied by the scattering density of states DSat 20 K. See the de nition of jeg"#j2,
jo"#j2andDSin Eq. 8, 19 and 9 respectively. "#is the spin-
ip angle between two electronic states. For two states ( k;n) and
(k0;n0) with opposite spin directions, "#is the angle between Sexp
knandSexp
k0n0.sin2("#=2) is de ned in Eq. 24. The variation
ofDSamong di erent substrates is at most three times, much weaker than the variations of 1
sand other quantities shown
here.
der of magnitude (Table SI), s;xof ML-Ge on di erent
substrates are similar.
CONCLUSIONS
In this paper, we systematically investigate how spin
relaxation of strong SOC Dirac materials is a ected by
di erent insulating substrates, using germanene as a pro-
totypical example. Through FPDM simulations of sof
free-standing and substrate supported ML-Ge, we show
that substrate e ects on scan di er orders of magni-
tude among di erent substrates. Speci cally, sof ML-
Ge-GeH and ML-Ge-SiH have the same order of mag-
nitude as free-standing ML-Ge, but sof ML-Ge-GaTe
and ML-Ge-InSe are signi cantly shortened by 1-2 orders
with temperature increasing from 20 K to 300 K.
Although simple electronic quantities including charge
densities, DOS and spin mixing
b2
z
qualitatively ex-
plain the much shorter lifetime of ML-Ge-GaTe/InSe
compared to ML-Ge-GeH/SiH in the relatively high T
range, we nd they cannot explain the large variations
of samong substrates at low T(i.e. tens of K). We
point out that spin relaxation in ML-Ge and its inter-
faces at low Tis dominated by intervalley scattering pro-
cesses. However, the substrate-induced modi cations of
phonons and thermal vibrations of substrates seem to be
not important. Instead, the substrate-induced changes
of the anisotropy of Sexpor the spin-
ip angles "#which
changes the spin-
ip matrix elements, are much more cru-
cial."#is at the rst time proposed in this article to the
best of our knowledge, and is found to be a useful elec-
tronic quantity for predicting trends of spin relaxation
when intervalley spin-
ip scattering dominates.Our theoretical study showcases the systematic inves-
tigations of the critical factors determining the spin re-
laxation in 2D Dirac materials. More importantly we
pointed out the sharp distinction of substrate e ects on
strong SOC materials to the e ects on weak SOC ones,
providing valuable insights and guidelines for optimizing
spin relaxation in materials synthesis and control.
METHODS
First-Principles Density-Matrix Dynamics for Spin
Relaxation
We solve the quantum master equation of density ma-
trix(t) as the following:[19]
d12(t)
dt= [He;(t)]12+
0
BBB@1
2P
3458
<
:[I (t)]13P32;4545(t)
[I (t)]45P
45;1332(t)9
=
;
+H:C:1
CCCA;
(1)
Eq. 1 is expressed in the Schr odinger picture, where the
rst and second terms on the right side of the equa-
tion relate to the coherent dynamics, which can lead
to Larmor precession, and scattering processes respec-
tively. The rst term is unimportant for out-of-plane
spin relaxation in ML-Ge systems, since Larmor preces-
sion is highly suppressed for the excited spins along the
out-of-plane or zdirection due to high spin polarization
alongzdirection. The scattering processes induce spin9
relaxation via the SOC. Heis the electronic Hamiltonian.
[H;] =H H. H.C. is Hermitian conjugate. The
subindex, e.g., \1" is the combined index of k-point and
band.P=Pe ph+Pe iis the generalized scattering-
rate matrix considering e-ph and e-i scattering processes.
For the e-ph scattering[19],
Pe ph
1234 =X
qAq
13Aq;
24; (2)
Aq
13=r
2
~gq
12q
G(1 2!q)q
n
q;(3)
whereqandare phonon wavevector and mode, gq
is the e-ph matrix element, resulting from the absorp-
tion ( ) or emission (+) of a phonon, computed with
self-consistent SOC from rst-principles,[29] n
q=nq+
0:50:5 in terms of phonon Bose factors nq, andG
represents an energy conserving -function broadened to
a Gaussian of width .
For electron-impurity scattering[19],
Pe i
1234=Ai
13Ai;
24; (4)
Ai
13=r
2
~gi
13q
G(1 3)p
niVcell; (5)
whereniandVcellare impurity density and unit cell vol-
ume, respectively. giis the e-i matrix element computed
by the supercell method and is discussed in the next sub-
section.
Starting from an initial density matrix (t0) prepared
with a net spin, we evolve (t) through Eq. 1 for a long
enough time, typically from hundreds of ps to a few s.
We then obtain spin observable S(t) from(t) (Eq. S1)
and extract spin lifetime sfromS(t) using Eq. S2.
Computational details
The ground-state electronic structure, phonons, as well
as electron-phonon and electron-impurity (e-i) matrix
elements are rstly calculated using density functional
theory (DFT) with relatively coarse kandqmeshes in
the DFT plane-wave code JDFTx[30]. Since all sub-
strates have hexagonal structures and their lattice con-
stants are close to germanene's, the heterostructures
are built simply from unit cells of two systems. The
lattice mismatch values are within 1% for GeH, GaTe
and InSe substrates but about 3.5% for the SiH sub-
strate. All heterostructures use the lattice constant 4.025
A of free-standing ML-Ge relaxed with Perdew-Burke-
Ernzerhof exchange-correlation functional[31]. The in-
ternal geometries are fully relaxed using the DFT+D3
method for van der Waals dispersion corrections[32].
We use Optimized Norm-Conserving Vanderbilt (ONCV)
pseudopotentials[33] with self-consistent spin-orbit cou-
pling throughout, which we nd converged at a ki-
netic energy cuto of 44, 64, 64, 72 and 66 Ry forfree-standing ML-Ge, ML-Ge-GeH, ML-Ge-SiH, ML-Ge-
GaTe and ML-Ge-InSe respectively. The DFT calcula-
tions use 2424kmeshes. The phonon calculations em-
ploy 33 supercells through nite di erence calculations.
We have checked the supercell size convergence and found
that using 66 supercells lead to very similar results of
phonon dispersions and spin lifetimes. For all systems,
the Coulomb truncation technique[34] is employed to ac-
celerate convergence with vacuum sizes. The vacuum
sizes are 20 bohr (additional to the thickness of the het-
erostructures) for all heterostructures and are found large
enough to converge the nal results of spin lifetimes. The
electric eld along the non-periodic direction is applied
as a ramp potential.
For the e-i scattering, we assume impurity density is
suciently low and the average distance between neigh-
boring impurities is suciently long so that the interac-
tions between impurities are negligible, i.e. at the dilute
limit. The e-i matrix gibetween state ( k;n) and (k0;n0)
isgi
kn;k0n0=hknjVi V0jk0n0i, whereViis the poten-
tial of the impurity system and V0is the potential of the
pristine system. Viis computed with SOC using a large
supercell including a neutral impurity that simulates the
dilute limit where impurity and its periodic replica do
not interact. To speed up the supercell convergence, we
used the potential alignment method developed in Ref.
35. We use 55 supercells, which have shown reasonable
convergence (a few percent error of the spin lifetime).
We then transform all quantities from plane wave basis
to maximally localized Wannier function basis[36], and
interpolate them[29, 37{41] to substantially ner k and
q meshes. The ne kandqmeshes are 384384 and
576576 for simulations at 300 K and 100 K respectively
and are ner at lower temperature, e.g., 1440 1440 and
24002400 for simulations at 50 K and 20 K respectively.
The real-time dynamics simulations are done with our
own developed DMD code interfaced to JDFTx. The
energy-conservation smearing parameter is chosen to
be comparable or smaller than kBTfor each calculation,
e.g., 10 meV, 5 meV, 3.3 meV and 1.3 meV at 300 K,
100 K, 50 K and 20 K respectively.
Analysis of Elliot-Yafet spin lifetime
In order to analyze the results from real-time rst-
principles density-matrix dynamics (FPDM), we com-
pare them with simpli ed mechanistic models as dis-
cussed below. According to Ref. [18], if a solid-state
system is close to equilibrium (but not at equilibrium)
and its spin relaxation is dominated by EY mechanism,
its spin lifetime sdue to the e-ph scattering satis es (for
simplicity the band indices are dropped)10
1
s/N 2
k
X
kq8
<
:jg"#;q
k;k qj2nqfk q(1 fk)
(k k q !q)9
=
;;(6)
=N 1
kX
kfk(1 fk); (7)
wherefis Fermi-Dirac function. !qandnqare
phonon energy and occupation of phonon mode at
wavevector q.g"#is the spin-
ip e-ph matrix element
between two electronic states of opposite spins. We will
further discuss g"#in the next subsection.
According to Eq. 6 and 7, 1
sis proportional to jg"#
qj2
and also the density of the spin-
ip transitions. Therefore
we propose a temperature ( T) and chemical potential
(F;c) dependent e ective modulus square of the spin-
ip e-ph matrix element jeg"#j2and a scattering density
of statesDSas
jeg"#j2=P
kqwk;k qP
jg"#;q
k;k qj2nqP
kqwk;k q; (8)
DS=N 2
kP
kqwk;k q
N 1
kP
kfk(1 fk); (9)
wk;k q=fk q(1 fk)(k k q !c); (10)
where!cis the characteristic phonon energy speci ed
below, and w k;k qis the weight function. The matrix
element modulus square is weighted by nqaccording to
Eq. 6 and 7. This rules out high-frequency phonons at
lowTwhich are not excited. !cis chosen as 7 meV
at 20 K based on our analysis of phonon-mode-resolved
contribution to spin relaxation. w k;k qselects transitions
between states separated by !cand around the band edge
orF;c, which are \more relevant" transitions to spin
relaxation.
DScan be regarded as an e ective density of spin-
ip e-ph transitions satisfying energy conservation be-
tween one state and its pairs. When !c= 0, we
haveDS=R
d
df
d
D2()=R
d
df
d
D() with
D() density of electronic states (DOS). So DScan
be roughly regarded as a weighted-averaged DOS with
weight
df
d
D().
Withjeg"#j2andDS, we have the approximate relation
for spin relaxation rate,
1
s/jeg"#j2DS: (11)
Spin-
ip e-ph and overlap matrix element
In the mechanistic model of Eq. 6 in the last section,
the spin-
ip e-ph matrix element between two electronic
states of opposite spins at wavevectors kandk qof
phonon mode reads[29]g"#;q
kk q=D
u"(#)
k qvKS u#(")
k qE
; (12)
qvKS=s
~
2!qX
e ;q@ qvKS
pm; (13)
@ qvKS=X
leiqRl@VKS
@ jr Rl; (14)
VKS=V+~
4m2c2rrVp; (15)
whereu"(#)
kis the periodic part of the Bloch wavefunc-
tion of a spin-up (spin-down) state at wavevector k.is
the index of ion in the unit cell. is the index of a di-
rection. Rlis a lattice vector. Vis the spin-independent
part of the potential. pis the momentum operator. is
the Pauli operator.
From Eqs. 12-15, g"#can be separated into two parts,
g"#=gE+gY; (16)
wheregEandgYcorrespond to the spin-independent
and spin-dependent parts of VKSrespectively, called El-
liot and Yafet terms of the spin-
ip scattering matrix
elements respectively.[28]
Generally speaking, both the Elliot and Yafet terms
are important; for the current systems swith and with-
out Yafet term have the same order of magnitude. For
example, sof ML-Ge-GeH and ML-Ge-SiH without the
Yafet term are about 100% and 70% of swith the Yafet
term at 20 K. Therefore, for qualitative discussion of s
of ML-Ge on di erent substrates (the quantitative calcu-
lations of sare performed by FPDM introduced earlier),
it is reasonable to focus on the Elliot term gEand avoid
the more complicated Yafet term gY.
De neVE
qas the spin-independent part of qvKS,
so thatgE=D
u"(#)
k VE
q u#(")
k qE
. Expanding VE
qas
P
GeVE
q(G)eiGr, we have
gE=X
GeVE
q(G)o"#
kk q(G); (17)
o"#
kk q(G) =D
u"(#)
k eiGr u#(")
k qE
; (18)
whereo"#
kk q(G) isG-dependent spin-
ip overlap func-
tion. Without loss of generality, we suppose the rst
Brillouin zone is centered at .
Therefore,gEis not only determined by the long-range
component of o"#
kk q(G), i.e.,o"#
kk q(G= 0) but also the
G6= 0 components. But nevertheless, it is helpful to
investigate o"#
kk q(G= 0) and similar to Eq. 8, we pro-
pose an e ective modulus square of the spin-
ip overlap
matrix elementjo"#j2,11
jo"#j2=P
kqwk;k qP
jo"#
k;k q(G= 0)j2
P
kqwk;k q: (19)
Internal magnetic eld
Suppose originally a system has time-reversal and in-
version symmetries, so that every two bands form a
Kramers degenerate pair. Suppose the k-dependent spin
matrix vectors in Bloch basis of the Kramers degenerate
pairs are s0
kwiths(sx;sy;sz). The inversion symme-
try broken, possibly due to applying an electric eld or
a substrate, induces k-dependent Hamiltonian terms
HISB
k=BgeBin
ks0
k; (20)
whereBgeis the electron spin gyromagnetic ratio.
Bin
kis the SOC eld and called internal magnetic elds.
Binsplits the degenerate pair and polarizes the spin along
its direction. The de nition of Bin
kis
Bin
k2SOC
kSexp
k=(Bge); (21)
where Sexp
Sexp
x;Sexp
y;Sexp
z
withSexp
ibeing spin
expectation value along direction iand is the diagonal
element ofsi. SOCis the band splitting energy by SOC.
Spin-
ip angle "#for intervalley spin relaxation
Suppose (i) the inversion symmetry broken induces Bin
k
(Eq. 21) for a Kramers degenerate pair; (i) there are two
valleys centered at wavevectors Qand Qand (iii) there
are two wavevectors k1andk2nearQand Qrespec-
tively. Due to time-reversal symmetry, the directions of
Bin
k1andBin
k2are almost opposite.
De ne the spin-
ip angle "#
k1k2as the angle between
Bin
k1andBin
k2, which is also the angle between Sexp
k1
andSexp
k2. We will prove that for a general operator bA,
A"#
k1k2 2
sin2
"#
k1k2=2 A##
k1k2 2
; (22)
whereA"#
k1k2andA##
k1k2are the spin-
ip and spin-
conserving matrix elements between k1andk2respec-
tively.
The derivation uses the rst-order perturbation theory
and has three steps:
Step 1: The 22 matrix of operator bAbetween k1and
k2of two Kramers degenerate bands is A0
k1k2. According
to Ref. 26, with time-reversal symmetry, the spin-
ip
matrix element of the same band between kand kis
exactly zero, therefore, the spin-
ip matrix elements ofA0
k1k2are zero at lowest order as k1+k20, i.e.,A0;"#
k1k2
A0;#"
k1k20.
Step 2: The inversion symmetry broken induces Bin
k
and the perturbed Hamiltonian HISB
k(Eq. 20). The
new eigenvectors Ukare obtained based on the rst-order
perturbation theory.
Step 3: The new matrix is Ak1k2=Uy
k1A0
k1k2Uk2. Thus
the spin-
ip matrix elements A"#
k1k2with the inversion
symmetry broken are obtained.
We present the detailed derivation in SI Sec. III.
From Eq. 22, for the intervalley e-ph matrix elements
of ML-Ge systems, we have
g"#
k1k2 2
sin2
"#
k1k2=2 g##
k1k2 2
: (23)
As g"#
k1k2 2
largely determines sof ML-Ge systems,
the di erences of sof ML-Ge on di erent substrates
should be mainly due to the di erence of sin2
"#
k1k2=2
.
For the intervalley overlap matrix elements, we should
have o"#
k1k2 2
sin2
"#
k1k2=2 o##
k1k2 2
. Since o##
k1k2 2
is of order 1, o"#
k1k2 2
is expected proportional to
sin2
"#
k1k2=2
and have the same order of magnitude as
sin2
"#
k1k2=2
.
Finally, similar to Eq. 8, we propose an e ective mod-
ulus square of sin2
"#
k1k2=2
,
sin2("#=2) =P
kqwk;k qsin2
"#
k;k q=2
P
kqwk;k q: (24)
DATA AVAILABILITY
The data that support the ndings of this study are
available upon request to the corresponding author.
CODE AVAILABILITY
The codes that were used in this study are available
upon request to the corresponding author.
ACKNOWLEDGEMENTS
We thank Ravishankar Sundararaman for helpful dis-
cussions. This work is supported by the Air Force Of-
ce of Scienti c Research under AFOSR Award No.
FA9550-YR-1-XYZQ and National Science Foundation
under grant No. DMR-1956015. This research used
resources of the Center for Functional Nanomaterials,12
which is a US DOE Oce of Science Facility, and the
Scienti c Data and Computing center, a component of
the Computational Science Initiative, at Brookhaven Na-
tional Laboratory under Contract No. DE-SC0012704,
the lux supercomputer at UC Santa Cruz, funded by NSF
MRI grant AST 1828315, the National Energy Research
Scienti c Computing Center (NERSC) a U.S. Depart-
ment of Energy Oce of Science User Facility operated
under Contract No. DE-AC02-05CH11231, and the Ex-
treme Science and Engineering Discovery Environment
(XSEDE) which is supported by National Science Foun-
dation Grant No. ACI-1548562 [42].AUTHOR CONTRIBUTIONS
J.X. performed the rst-principles calculations. J.X.
and Y.P. analyzed the results. J.X. and Y.P. designed all
aspects of the study. J.X. and Y.P. wrote the manuscript.
ADDITIONAL INFORMATION
Supplementary Information accompanies the pa-
per on the npj Computational Materials website.
Competing interests: The authors declare no com-
peting interests.
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1908.00927v2.Two_dimensional_orbital_Hall_insulators.pdf | Two-dimensional orbital Hall insulators
Luis M. Canonico,1Tarik P. Cysne,2,3Tatiana G. Rappoport,2,4and R. B. Muniz1
1Instituto de Física, Universidade Federal Fluminense, 24210-346 Niterói RJ, Brazil
2Instituto de Física, Universidade Federal do Rio de Janeiro,
Caixa Postal 68528, 21941-972 Rio de Janeiro RJ, Brazil
3Departamento de Física, Universidade Federal de São Carlos,
Rod. Washington Luís, km 235 - SP-310, 13565-905 São Carlos, SP, Brazil
4Department of Physics and Center of Physics, University of Minho, 4710-057, Braga, Portugal
(Dated: March 4, 2020)
Detailed analyses of the spin and orbital conductivities are performed for different topological
phases of certain classes of two-dimensional (2D) multiorbital materials. Our calculations show
the existence of orbital-Hall effect (OHE) in topological insulators, with values that exceed those
obtained for the spin-Hall effect (SHE). Notably, we have found non-topological insulating phases
that exhibit OHE in the absence of SHE. We demonstrate that the OHE in these systems is deeply
linked to exotic momentum-space orbital textures that are triggered by an intrinsic Dresselhaus-
type of interaction that arises from a combination of orbital attributes and lattice symmetry. Our
resultsstronglyindicatesthatotherclassesofsystemswithnon-trivialorbitaltexturesand/ororbital
magnetism may also exhibit large OHE even in their normal insulating phases.
The OHE, similarly to the SHE, refers to the creation
of a transverse flow of orbital angular momentum that is
induced by a longitudinally applied electric field [1]. It
has been explored mostly in three dimensional metallic
systems, where it can be quite strong [2–5]. For systems
in which the spin-orbit coupling (SOC) is sizeable, the
orbital and spin angular momentum degrees of freedom
are coupled, establishing an interrelationship between
charge, spin, and orbital angular momentum excitations.
However, the OHE does not necessarily require SOC, it
can be associated to the presence of orbital textures [5]
and be especially significant in various materials.
Chiral orbital textures in the reciprocal space have
been discussed in connection with orbital magnetism at
the surface of spmetals [6], photonic graphene [7] and
also in topological insulators with strong SOC. More re-
cently they were observed in chiral borophene [8], single-
layer transition metal dichalcogenides [9] and tin tel-
luride monolayers for photocurrent generation [10]. Or-
bital magnetism is enhanced in surfaces [11], indicating
that orbital effects can be crucial in 2D materials, which
can also be evidenced by the observation of orbital tex-
tures in van der Waals materials. Still, OHE remains
mostly unexplored in 2D materials [12, 13].
Here, we investigate the role of orbital textures for the
OHE displayed by multi-orbital 2D materials. We pre-
dict the appearance of rather large OHE in these systems
both in their metallic and insulating phases. The orbital
Hall currents can be considerably larger than the spin
Hall ones, and be present even in the absence of SHE.
Their use as information carriers widens the development
possibilities of novel spin-orbitronic devices.
In our analyses, we consider a minimal tight-binding
(TB) model Hamiltonian that involves only two orbitals
(pxandpy) per atom in a honeycomb lattice [14, 15]:H=X
hijiX
st
ijpy
ispjs+X
is
i+I`z
z
ss
py
ispis;
(1)
whereiandjdenote the honeycomb lattice sites posi-
tioned at~Riand~Rj, respectively. The symbol hijiindi-
cates that the sum is restricted to the nearest neighbour
(n.n) sites only. The operator py
iscreates an electron of
spinsin the atomic orbitals p=p=1p
2(pxipy)
centred at ~Ri. Here,s=";#labels the two electronic
spin states, and iis the atomic energy at site i, which
may symbolise a staggered on-site potential that takes
valuesi=VAB, when site i belongs to the A and B
sub-lattices of the honeycomb arrangement, respectively.
The transfer integrals t
ijbetween the porbitals centred
on n.n atoms are parametrised according to the standard
Slater-Koster TB formalism [16]. They depend on the
direction cosines of the n.n. interatomic directions, and
may be approximately expressed as linear combinations
of two other integrals ( VppandVpp) involving the p
andporbitals, where andrefer to the usual compo-
nents of the angular momentum around these axes.
Since our model does not include the orbital pz, it is
restricted to a sector of the `= 1angular momentum
vector space spanned only by the eigenstates of `z p
associated with m`=1. Within this sector it is useful
to introduce a pseudo angular momentum SU(2)-algebra
where the Pauli matrices act on p
. In this case, there
is a one-to-one correspondence between the representa-
tions of the Cartesian components of the orbital angular
momentum operators in this basis and the usual Pauli
matrices, and `zis not conserved (details are given Sec.
I of the supplementary material - SM). The last term in
Eq.1 describes the intrinsic atomic SOC.
This simple model describes relatively well the low-arXiv:1908.00927v2 [cond-mat.mes-hall] 2 Mar 20202
Figure 1: (a) Band structure calculations along some symme-
try lines in the 2D BZ for Vpp= 0,Vpp=1 eV, andI= 0.
The blue line represents the results for VAB= 0:0, and the
red line for VAB= 0:8. (b) Orbital Hall conductivities cal-
culated for the same sets of parameters. The insets show the
in-plane contribution to the orbital angular momentum tex-
tures calculated in the neighbourhoods the (left inset) and
K(right inset) symmetry points of the 2D Brillouin zone, for
VAB= 0:0. The left and right inset textures are associated
with the lower flat and dispersive bands, respectively.
energy electronic properties of novel group V based 2D
materials [15, 17, 18]. Its topological characteristics were
previously investigated in the context of optical lattices,
and it has been verified that it exhibits a rich topological
phase diagram, which includes quantum spin-Hall insu-
lator (QSHI) phases [17, 19–22].
Following Ref. 21 we shall assume, for simplicity, that
Vpp= 0andVpp= 1eV. Our focus is on three dis-
tinct phases that manifest themselves depending on the
parameters specified in Eq. (1). In the absence of SOC
and sub-lattice resolved potentials, the electronic band
structure consists of four gapless bulk energy bands, two
of which form Dirac cones at the KandK0symmetry
points of the 2D first Brillouin zone (BZ), whereas the
other two are flat. Each flat band is tangent to one of
the dispersive bands at the point, as Fig. 1 (a) illus-
trates.
Our results for the orbital Hall conductivities ( z
OH),
calculated as functions of energy by means of the Kubo
formula [23], with the orbital current defined as J`z
y=
1
2f`z;vyg, are shown in Fig. 1 (b) for VAB= 0:0(blue
line), and for VAB= 0:8(red line). Details of these
calculations are given in Sec. II of SM. Here we notice a
strong orbital Hall conductivity, which peaks at energies
close to where the flat bands touch the dispersive bands
at . ForVAB6= 0, the electronic structure develops
an energy gap around E= 0that eliminates the original
Dirac cones in the vicinities of KandK0. The flat bands,
however, remain tangent to the dispersive bands at , as
shown in Fig. 1 (a), and the large OHE in this case also
occurs for energies close to where they touch each other.
The insets of Fig. 1 (b) depict the in-plane contribu-
tion to the orbital angular momentum textures, calcu-
lated on a circle around the (left inset) and K(right
inset) symmetry points of the 2D first BZ. They are both
computed for VAB= 0. The colours of the arrows em-
phasise their in-plane azimuthal angles. At the point,the texture displays a dipole-field like structure, whereas
in the vicinity of the Kpoints it is identical to the spin-
texture produced by the Dresselhaus SOC in zinc blende
lattice systems [24]. Here, the texture is not caused by
SOC, but results only from the orbital features and crys-
talline symmetry, as we shall subsequently show.
In the presence of SOC, three energy gaps open: one
originating from the K(K0)points, and the other two at
, while the flat bands acquire a slight energy disper-
sion - see Fig. SI of the SM. When the relative values
ofIandVABvary, this model exhibits a rich topolog-
ical phase diagram [21]. We shall focus on three phases
that display distinct topological gap features. They are
classified by sets of spin Chern numbers (i,j,k,l) associ-
ated with the four "-spin bands, namely A1 (1,-1,1,-1),
B1 (1,0,0,-1), and B2 (0,1,-1,0), according to the nota-
tion of Ref. 21. We remind that the spin Chern numbers
for the#-spin sector have opposite signs. This codifica-
tion clearly indicates that when the system is in the A1
phase the two lateral energy gaps are topological, but the
central one is not. The reverse occurs in the B2 phase,
where only the central energy gap is topological. Last
but not least, all the three energy gaps are topological
in the B1 phase. This is explicitly verified in the left
panels of Fig. 2, which show the spin Hall conductivi-
ties (z
SH) (red curves) calculated as functions of energy
for three different sets of parameters that simulate sys-
tems in each of these phases. In the absence of sublattice
asymmetry ( VAB= 0) and forI= 0:2, the system as-
sumes the B1 phase and becomes a QSHI within all the
three energy gaps, as the quantised plateaux of z
SHin
Fig. 2 (a) show. For I= 1:1andVAB= 0:8, the system
is in the B2 phase, which exhibits a quantised spin Hall
conductivity plateau in the central energy gap, and two
non topological side gaps within which it behaves as an
ordinary insulator, displaying no QSHE, as Fig. 2 (c) il-
lustrates. For I= 0:2andVAB= 0:8, the system takes
on the A1 phase, where it becomes a QSHI for energies
within the lateral energy gaps, but behaves as a conven-
tional insulator inside the central gap, as portrayed in
Fig. 2 (e).
The corresponding orbital Hall conductivities ( z
OH)
calculated for the three phases (blue curves) are also de-
picted in the left panels of Fig. 2, together with the
respective densities of states (grey lines) represented in
arbitrary units. We notice that within the lateral gaps,
z
OHexhibits plateaux with much higher intensities than
those of the SHE. However, in contrast with the latter,
the OHE is not quantised. Its plateaux heights depend
uponIandVAB, increasinginmodulusasthegapwidth
reduces, thoughlimitedbytheOHEvaluefor I!0(see
Sec. IV of SM). A remarkable result illustrated in Fig.
2 (c) is the existence of finite OHE within the two (non-
topological) side energy gaps of phase B2, where the sys-
tem becomes an ordinary insulator with no QSHE. This
is particularly interesting because there are no electronic3
edge states crossing these energy gaps (see Sec. V of the
SM), and raises the question on how the orbital Hall cur-
rent propagates through the system in this case. It is
also noticeable that the OHE is an odd function of the
Fermi energy ( EF)and vanishes in the central energy
gaps for all three phases. This is due to symmetry lim-
itations of this simplified model, which we shall address
subsequently.
Figure 2: Spin Hall conductivity (red), and orbital Hall con-
ductivity (blue), together with the density of states (grey),
calculated as functions of energy for: (a) I= 0:2and
VAB= 0- B1 phase; (c) I= 1:1, andVAB= 0:8- B2
phase, and (e) I= 0:2, andVAB= 0:8- A1 phase. The
densities of states are depicted in arbitrary units. Panels (b),
(d) and (f) show the associated orbital textures, calculated
for the lower"-spin band, with the same sets of parameters,
respectively. The density plots illustrate their corresponding
h`zipolarisations.
It is also important to examine how disorder affects
the transport properties of these systems. To simulate
it we consider on-site potentials iwith values randomly
distributed within [-W/2, W/2], where W is the disorder
strength. We calculate the spin- and orbital-Hall conduc-
tivities for different values of W using Chebyshev poly-
nomial expansions and the Kubo-Bastin formula, which
are efficiently implemented in the open-source software
KITE. Similarly to what we have previously found forthe SHE [22], the orbital Hall plateaux are robust to rel-
atively strong Anderson disorder. Details of these calcu-
lations are described in sections VI, VII and VIII of the
SM.
We know that OHE is linked to orbital textures in re-
ciprocal space [5], and to establish this relationship we
have calculated these textures for the "-spin lowest en-
ergybandintheentire2DfirstBZ.Theresultsareshown
in panels (b), (d), and (f) for systems in the B1, B2, and
A1 phases, respectively. The orbital characters for all "-
spin eigenstates are depicted in Fig. SV of the SM. It is
worth noticing that when either IorVABare different
from zero, the orbital textures display finite out-of-plane
components for each spin direction. However, due to
time reversal symmetry the `zorbital polarisations for
inverted spin directions are opposite, and consequently
the total`zpolarisation vanishes. The structure of the
in-plane texture, nevertheless, remains the same for both
spin components, which means that the in-plane orbital
texture survives. It is also noteworthy that both the low-
est two energy bands as well as the upper ones display
opposite in-plane orbital textures for this simple model.
Consequently, the OHE vanishes at the onset of the cen-
tral energy gap, where the accumulated in-plane orbital
texture of the occupied states becomes zero. The absence
of electronic states within an energy gap leads to a con-
stant value for z
OH[2, 25, 26] in its range, which justifies
the lack of OHE in the central energy gap found for the
three phases.
Contour curves are also shown for certain values of
EFranging from the bottom of the energy band to the
beginningofthelowestenergygap. Inallphases, wenote
that close to the point, where the lowest energy band
value is minimum, there is virtually no in-plane orbital
angular momentum texture, and the OHE is very small.
AsEFincreases the in-plane orbital texture builds up,
assuming a dipole-field like configuration. Eventually,
whenEFapproaches the onset of the first energy gap, it
develops a Dresselhaus-like arrangement near the K and
K’ points, with opposite signs in each valley.
In order to uncover the raison d’etre of these exotic or-
bital textures that promote OHE in this systems we de-
rive an effective theory near the Dirac points KandK0.
Around them, the orbital angular momentum texture is
perfectly captured by a linear approximation in the crys-
talline momentum, whereas it requires a fourth-order ex-
pansion near the point. Our effective Hamiltonian Heff
can be expressed in terms of SU(2)
SU(2)orbital and
sub-lattice algebras, and written as: Heff=H0+HAB+
HSOC+H`. HereH0= ~vF(kxx+ kyy)istheusual
Dirac Hamiltonian, with Fermi velocity vF=ap
3
2~,ade-
notes the lattice constant, and =1for theKandK0
valleys, respectively. HSOC=sI`zrepresents the SOC,
wheres=1for"and#spin electrons, respectively.
HAB=VABzis the sub-lattice resolved potential. H`4
breaksthedegeneracybetween `zeigenstatesandisgiven
by:
H`= ~vF
4 (k+`+ +k ` ) p
3~vF
2a(`xx+ `yy);
(2)
where =x+i y, = ,` ( =x;y) are the
orbital angular momentum matrices in the corresponding
Hilbert space, k=kxiky, and`=`xi`y.
As shown in the section X of the SM, in the absence
ofH`each valley presents two degenerated Dirac cones.
The first term in the right hand side of Eq. (2) alters
the Fermi velocity of the Dirac cones and leads to an in-
plane orbital texture profile similar to the one portrayed
around the point. The second term, however, pro-
duces a Dresselhaus-like splitting in the Dirac cones and
is primarily responsible for the orbital angular momen-
tum texture found in our TB calculations. Our effective
theory confirms that the exotic in-plane texture exhib-
ited by these 2D systems is an intrinsic property that
arises solely from the px-pyorbital characteristics and
crystalline symmetries.
Figure 3: (a) Energy band spectrum calculated for the simple
model with second n.n. hopping integrals V(2)
pp= 0:2. Here
we keepVpp= 0,Vpp= 1,I= 0:2andVAB= 0:8. (b)
SH (red line) and OH (blue line) conductivities calculated as
functions of energy. The grey line depicts the density of states
(DOS) in arbitrary units. The inset shows a closeup of the
central energygaphighlightingthenon-zero valueoftheOHE
within this energy range.
We shall now address the absence of OHE in the cen-
tral energy gap as results from our calculations. This
limitation actually comes from a combination of electron-
hole and parity symmetries, which lead to energy levels
that are symmetric with respect to the zero energy for
this simple model[21]. One way of breaking it is by in-
troducing second n.n. hopping integrals, as Fig. 3 (a)
illustrates. Here, just as a proof of concept, we kept
Vpp= 0, and choose the second n.n. hopping integrals
Vpp2= 0:2. In this case only the central energy gap
survives, and within it the system assumes an ordinary
insulating phase. Fig. 3 (b) clearly shows that the SHE
vanishes in this energy range, whereas the OHE is fi-
nite. Here, the in-plane orbital texture associated with
Figure 4: SH (red line) and OH (blue line) conductivities
calculated as functions of energies for flat bismuthene: (a)
without sublattice asymmetry ( VAB= 0) and (b) with VAB=
0:87. The insets highlight the non-zero values of the OHE
within the corresponding central energy-gap ranges.The grey
line depicts the DOS in arbitrary units.
the second lowest energy band no longer cancels the con-
tribution from the first band. Thus, the OHE does not
vanish at the onset of the central energy gap and keeps
its non-zero value constant within it. This result, al-
though relatively small in this particular case, unequiv-
ocally shows that it is possible to obtain a finite OHE
for a non-topological insulating phase, as we previously
found for the lateral energy gaps of the B2 phase. Hav-
ing shown that this effect happens for our simple-model
system, it is instructive to inquire into the possibility of
observing it in a real system. A candidate is the recently
synthesised flat bismuthene grown on SiC, whose low en-
ergy electronic properties are reasonably well described
by an effective TB model Hamiltonian that includes only
two orbitals ( pxandpy) per atom [14, 15, 17, 18]. It
is a real solid state system, typical of a promising class
of 2D materials based on the group group VA elements
that exhibit relatively large energy gaps. In fact, a very
good TB fit of both the valence and conduction bands
of flat bismuthene can be obtained with the inclusion of
second n.n. hopping integrals, as Fig. SX of the SM
illustrates. Results for the associated SHE and OHE cal-
culated as functions of EFfor planar bismuthene em-
ploying a Chebyshev polynomial expansion method are
shown in Fig.4. We clearly see in this case that the spec-
traarenotsymmetricwithrespecttothezeroenergyand
the right-hand side gap disappear. Results for VAB= 0,
depicted in Fig.4 (a), show that the remaining gaps are
topological, displaying a quantised SHE, and significant
OHE. For sufficiently large sublattice asymmetry, how-
ever, the central gap ceases to be topological, exhibiting
noSHE,asFig.4(b)illustrates. Notwithstanding, theor-
bital Hall conductivity is appreciable within this energy
range. This validates our original prediction that pure
orbital angular momentum currents can be triggered by
a longitudinally applied electric field in some normal in-
sulators.
In summary, we have performed detailed analyses of
the spin and orbital Hall conductivities for a class of 2D5
systems, relating the corresponding OHE, SHE and or-
bital textures. Our calculations show the existence of
OHE in topological insulators, with values that exceed
those obtained for the SHE. Remarkably, we also obtain
OHE for normal insulating phases where the SHE is ab-
sent and no edge states cross their energy gaps. We show
that the OHE in these systems is associated with exotic
momentum-space orbital textures that are caused by an
intrinsic Dresselhaus-type of interaction. This is rather
generalandshowthatcertain2Dinsulatingmaterialscan
generateorbitalangularmomentumcurrentsthatmaybe
useful for developing novel spin-orbitronic devices.
We acknowledge CNPq/Brazil, FAPERJ/Brazil
and INCT Nanocarbono for financial support, and
NACAD/UFRJ for providing high-performance com-
puting facilities. TGR acknowledges COMPETE2020,
PORTUGAL2020, FEDER and the Portuguese Foun-
dation for Science and Technology (FCT) through
project POCI-01- 0145-FEDER-028114. TPC acknowl-
edges São Paulo Research Foundation (FAPESP) grant
2019/17345-7.
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Supplementary material for “Spin and Charge Transport of Multi-Orbital Quantum
Spin Hall Insulators”
HAMILTONIAN IN RECIPROCAL SPACE AND ORBITAL ANGULAR MOMENTUM OPERATORS
DEFINITION
The Hamiltonian given by Eq. (1) of the main text can be rewritten in the reciprocal space. Using the basis
f p+;A
; p ;A
; p+;B
; p ;B
g, the hopping term reads,
~H0(~k) =0
BBB@0 0 ~A(~k)~D(~k)
0 0 ~C(~k)~B(~k)
~A(~k)~C(~k) 0 0
~D(~k)~B(~k) 0 01
CCCA; (S1)
where
2~A(~k) =txx(~k) +tyy(~k) +i
txy(~k) tyx(~k)
; (S2)
2~B(~k) =txx(~k) +tyy(~k) +i
tyx(~k) txy(~k)
; (S3)
2~C(~k) =txx(~k) tyy(~k) +i
txy(~k) +tyx(~k)
; (S4)
2~D(~k) =txx(~k) tyy(~k) i
txy(~k) +tyx(~k)
; (S5)
The transfer integrals t (~k)are Fourier transforms of Slater-Koster coefficients in the honeycomb lattice,
t ; (~k) =2X
m=0
p ;A ^V p ;B
(m)e i~k~ am; (S6)
where,mruns over the three nearest-neighbours of a site in sublattice A, that are located in sublattice B, and ~ amis
the vector connecting the atom at sublattice A with its mth neighbor at sublattice B. The Slater-Koster integrals are
given by
t (~k) =n2
(m)Vpp+ (1 n2
(m))Vpp; (S7)
t = n (m)n (m)
Vpp Vpp
; (S8)
withn ; (m)being the direction cosine connecting the site of sublattice Awithm-th first neighbor on sublat-
ticeB. In all results presented in this work, we set Vpp= 0, unless it is mentioned. This condition can be
relaxed without changing any of the main conclusions of our work. In this basis, the SOC term is diagonal,
Hs
soc=sIdiag(1; 1;1; 1)and the sublattice potential is given by HAB=VABdiag(1;1; 1; 1).
As mentioned on the main text, due the absence of the pzorbital, the electronic states are restricted to the subspace
associated with m`=1only, hence the angular momentum operators can be redefined in terms of a SU(2) algebra
as:
lz= p+
p+ p
p ;
lx= p+
p + p
p+ ;
ly=i p
p+ p+
p
(S9)
KUBO FORMULA FOR LINEAR RESPONSE CONDUCTIVITY
Inthemaintext,wecomputethespinandorbitalHallconductivityfordifferenttopologicalphasesotheHamiltonian
of Eq. (1) in the main text. For the cases of the pristine system, we used Kubo formalism to compute both OHE and2
SHE. In this formalism, the spin Hall (SH) and orbital Hall (OH) -polarized response, in ^ydirection, to an electric
field applied in the ^xdirection is given by,
OH(SH)=e
~X
n6=mX
s=";#Z
B:Z:d2k
(2)2(fm~k fn~k)
X
n;m;~k;s; (S10)
X
n;m;~k;s=~2Im"
s
n;~k jXy(~k) s
m;~k
s
m;~k vx(~k) s
n;~k
(Es
n;~k Es
m;~k+i0+)2#
(S11)
Were
OH(SH)istheorbitalHall(SpinHall)DCconductivitywithpolarizationin -direction,
X
n;m;~k;sistherelated
gauge-invariant Berry curvature. In Eq. S11, Es
n(m);~kandj s
n(m);~k
are eigenvalues and eigenvectors of Hamiltonian
of Eq. (S1), for n(m)Bloch band, with n;m = 1;::;4(in crescent order of energy), and s=";#spin-sector. Velocity
operators are defined by, vx(y)(~k) =@H(~k)=@kx(y), whereH(~k)is the tight-binding Hamiltonian in reciprocal space.
The current operator in ^ydirection is defined by jXy(~k) =
Xvy(~k) +vy(~k)X
=2, whereX=^`(^s)for OH (SH)
conductivities polarized in direction.
As it was discussed in the main text, this model presents a non-vanishing z
OH, even in absence of SOC, in contrast
to the spin Hall ( z
SH) response which depends on the presence of SOC or exchange interaction. Added to this, in the
presence of an exchange term, it was shown that the model presents a non-vanishing x
OHassociated with in-plane
polarized orbital Hall effect. It is important to mention that equations S10 and S11 are valid only in the clean limit
and do not take into account the effect of disorder. However, as was briefly pointed in the main text, the effect of
disorder should not affect our results for insulating phases (Fermi energy inside an electronic gap) due to the absence
of the Fermi surface, responsible to generates the leading-order contribution in the computation of vertex corrections
[26]. We confirm the robustness of our results against Anderson disorder using a real-space computation method
which is discussed in the next sections of the SM.
ANALYSIS OF THE BAND STRUCTURE
We have examined the orbital Hall conductivity properties of three distinct topological phases displayed by the
Hamiltonian Hdefined by Eq. (1) in the main text. They are labelled as B1, A1, and B2 phases, according to the
classifications used in Ref. 21. Figure SI shows the "-spin electron energy bands for the system in these three phases.
The spin-#bands can be deduced by applying a time-reversal symmetry operation on H. Panel (a) illustrates the band
structure of the B1 phase, calculated for I= 0:2Vpp, andVAB= 0. We notice that the SOC causes three energy
gaps to open, one originating from the K(K0) points, and the other two at , while the flat bands acquire a slight
energy dispersion. Panel (b) shows the energy bands for the system in the A1 phase, calculated with I= 0:2Vpp, and
VAB= 0:8Vpp. The sub-lattice potential affects each valley differently, as expected, because it breaks the degeneracy
between eigenvalues at the KandK0symmetry points. By examining the opposite spin polarisation one finds that
this phase exhibits a strong spin-valley locking, as discussed in Refs. 18, 21, 22. Panel (c) displays the energy bands
for the system in the B2 phase, calculated with I= 1:1VppandVAB= 0:8Vpp. In this case, Iis comparable but
slightly larger than VAB, and we note that they lead to effects that are similar to those exhibited panel (b), including
a strong spin-valley locking with valley polarisation stronger than in the previous case due to the relatively large
values ofIandVAB.
EVOLUTION OF THE ORBITAL HALL EFFECT PLATEAUX
In the main text, it was mentioned that the height of the orbital Hall plateaux within the lateral gaps depends upon
the SOC coupling constant and the sub-lattice resolved potential. To demonstrate this, we show in Figure SII results
for the spin and orbital Hall conductivities calculated for different sets of parameters for the B1, A1, and B2 phases.
The results depicted in each panel of Figure SII are obtained for a fixed value of VABand two different values of I
that are represented in the left and right columns, respectively. In panel (a) we show the conductivities calculated
forVAB=0;I= 0:2VppandI= 1:0Vpp, which correspond to situations in which the system is in the B1 phase. It
is clear that the height of the OHE plateau decreases as the SOC increases. In fact, the height of the plateau scales3
Figure SI:"-spin electron energy bands calculated as functions of wave vectors along some symmetry directions in the 2D
Brillouin zone for three distinct topological phases: (a) B1 with I= 0:2VppandVAB= 0. (b) A1 with I= 0:2Vppand
VAB= 0:8Vpp(c) B2 with I= 1:1VppandVAB= 0:8Vpp.
with the size of the lateral gap, being close to the maximum value of the metallic limit for very small gaps. The same
trend is observed in the other two phases, in contrast with the heights of the spin Hall plateaux that remain the same
in all cases .
ZIGZAG NANO-RIBBONS SPECTRA
Thorough the main text we analysed the spin and orbital Hall effects for the three distinct topological phases
B1;A1andB2. Tofurthersubstantiateourfindingsofthenon-zeroorbitalHallconductivityinthetriviallyinsulating
phases, we analysed the energy spectrum of a zigzag nano-ribbon in our system for the three distinct phases. Figure
SIII shows the spectra for each of the phases studied in the main text. As expected, the number of pairs of edge
states corresponds with the index Z2of each of these phases. Panel (a) shows the energy bands corresponding to the
phaseB1, here the most interesting features are the pairs of edge channels that cross the gap and the fully symmetric
spectrum for both spin polarizations. Panel (b) displays the spectrum of the A1phase, here we can see the strong
spin-valley locking that results from the inversion symmetry breaking produced by staggered sub-lattice potential.
Interestingly here we can observe the absence of edge states traversing the central gap. Finally panel (c) shows the
band structure of a ribbon in the phase B2. Here we can see how due the strong spin-orbit coupling and staggered
sub-lattice potential the edge states in both the lateral gaps do not cross the gap while the edge estates of the central
gap are crossing again. The results are fully consistent with the spin Cher number characterization of these phases
done in Ref [21]. The results of panel (c) are the most striking ones, because they indicate that differently from
the spin Hall conductivity, the orbital Hall effect plateau does not require electronic conducting channels to have a
constant non-quantized value.
latex onecolumn undefined
CHEBYSHEV POLYNOMIAL EXPANSION
To study the transport and spectral properties of the honeycomb lattice with px pyorbitals we used the
Chebyshev polynomial expansion. In this numerical method, the Green and spectral functions are accurately
expanded in terms of Chebyshev polynomial of first kind of the Hamiltonian matrices[28, 29]. This set of polynomials
are commonly chosen due their unique convergence properties, their relation with the Fourier transform and their
convenient recurrence relations that allows the iterative construction of higher order polynomials[28, 29]. In recent
years this method has gained much attention in the study of the transport properties of 2D systems[30–34]. Because
of its high scalability, It was used to study of topological phase transitions induced by disorder[22], and more recently,
in the analysis of the electronic properties of graphene encapsulated between two twisted hBN structures[35]. The
method requires a rescaling of the Hamiltonian and it spectrum to make them fit into the interval ( 1;1)where4
Figure SII: Spin Hall conductivity z
SH(red) and orbital Hall conductivity z
OH(blue)calculated for: (a) VAB= 0and
I= 0:2Vpp(left) andI= 1:0Vpp(right). (b) VAB= 0:8VppandI= 0:2Vpp(left) andI= 0:5Vpp(right). (c)
VAB= 0:8Vpp,I= 1:1Vpp(left) andI= 1:5Vpp(right)
the Chebyshev polynomials are defined and consequently the convergence of the method is assured. This scaling is
achieved by means of the transformations ~H= (H b)=aand ~E= (E b)=awherea(ET EB)=(2 )and
b(ET+EB)=2. In the later ETandEBrepresents the top and bottom limits of the spectrum, respectively, and
is a small cut-off parameter introduced to avoid numerical instabilities.
With this later conditions fulfilled, the Chebyshev polynomial expansion of the density operator considering N
polynomials can be written as:5
Figure SIII: Zigzag Nano-ribbons spectra for phases (a) B1withI= 0:2VppandVAB= 0, (b) A1 with I= 0:2Vppand
VAB= 0:8Vpp, and (c) B2 with I= 1:1VppandVAB= 0:8Vpp.
~E
=1
p
1 ~E2N 1X
m=0gmmTm
~E
; (S12)
wheregmis a kernel introduced to control the Gibbs oscillations produced by the sudden truncation of the series
expansion[28, 29]. The coefficients are calculated with m=hTrTm
~H
i, in whichh:::irepresents the average
over different disorder configurations. The calculation of the density operator of a given system is reduced to the
computation of the trace of a matrix. To further decrease the computational cost of the calculation of quantities such
as the density operator, instead of calculating the full trace of the polynomial matrices[29], we simply approximate
the expansion coefficient mas
m1
RhRX
r=1h rjTm~~H
j rii (S13)
In the laterj rirepresent a set of random vectors which are defined as j ri=D 1=2PD
i=1ei ijii. Herefjiigi=1;:::;D
denotes the original basis set, in which orbitals and spins on the lattice sites are treated equivalently, Drepresents
the dimension of the Hamiltonian matrix, and iis the phase of each of the state vectors that comprise each of the
random vectors. Ris the number of random vectors used in the trace estimation and the convergence of the later
goes as 1=p
DR.
CHEBYSHEV POLYNOMIAL EXPANSION OF KUBO FORMULA
To compute the spin and orbital conductivities of disordered systems, we employed the efficient algorithm developed
by J. García et. al.[36, 37], which is based in the Chebyshev expansion of the Kubo-Bastin formula[38]:
(;T) =i~
Z+1
1dEf(E;;T)
Trhj (E H)j dG+
dE j dG
dEj (E H)i; (S14)6
in which
represents the area of the 2Dsample,f(E;;T)is the Fermi-Dirac distribution for the energy E,
chemical potential and temperature T.G+(G )symbolise the advanced(retarded) one electron Green function.
As it can be seen from (S14) the Kubo-Bastin formula is expressed as a current-current correlation function. Then,
to adapt this formula to calculate the spin hall conductivity z
SH, we define j as the current-density operator like
j jx=ie
~[x;H]andj as the spin current-density as j js
y=1
2fz;vygwherezis the usual Pauli’s matrix and
vyis they-Component of the velocity operator. For the computation of the orbital Hall conductivity, again z
OHwe
define the current operator j asj jx=ie
~[x;H]and we write j as the orbital current density operator, which
is defined like j js
y=1
2f`z;vygwhere`zis thez. It is noteworthy to mention that for the spin hall conductivity
calculations we used the open-source code from the KITE project[39].
NUMERICAL SIMULATION OF THE DISORDERED CASE
It is instructive to investigate how disorder affects the OHE in these two-dimensional systems and more specifically,
how it modifies the plateaux in the orbital Hall conductivity that, as discussed before, is not dominated by conducting
edge states. For this purpose we include in our Hamiltonian an on-site Anderson disorder term iwhose values are
randomly picked from an uniform distribution that goes from
W
2;W
2
, in whichWrepresents the Anderson disorder
strength and then proceeded with the aforementioned Chebyshev polynomial expansions to compute the density of
states (DOS), and the transverse components of the spin and orbital conductivity tensors. In these calculations we
have considered systems of 8256256orbitals, Chebyshev polynomials up to the order M= 1280and we averaged
overR= 150random vectors. It is noteworthy to mention that due the large number lattice sites that we are
considering, we restricted ourselves to only one disorder realization, this is based on the assumption that almost every
possible configuration is contained on our system due it large size.7
Figure SIV: Spin (red) and orbital (blue) Hall conductivities calculated as functions of energy for: (a) I= 0:2VppVAB= 0,
(b)I= 0:2VppVAB= 0:8Vpp, and (c)I= 1:1VppVAB= 0:8Vppin the presence of disorder. The left, central and right
panels show the results obtained in the relatively weak ( W= 0:05Vpp), intermediate ( W= 0:2Vpp) and strong ( W= 0:4Vpp)
disorder regimes, respectively. The grey lines represent the density of states calculated for the same set of parameters.
Figure SIV shows the spin and orbital Hall conductivities calculated for both weak ( W= 0:05Vpp), intermediate
(W= 0:2Vpp) and strong ( W= 0:4Vpp) disorder. Similarly to what was previously observed for the SHE [34], the
orbital Hall plateau remains present, even for a relatively strong disorder that closes the lateral gaps. Our preliminary
results indicate that the orbital Hall effect in two-dimensional insulators is robust against Anderson disorder.
ORBITAL TEXTURE ANALYSIS
In contrast with the SHE, our calculations show that the OHE is not quantised, and occurs even in the absence of
metallic edge states. In order to explore the origin of the OHE in this model system, we investigated the characteristics
ofitsorbitalangularmomentumin reciprocalspace withinthe2DfirstBZ.Tothisend, wecomputetheorbitaltexture
in reciprocal space defined as,
~Ls
n;~k=
`x s
n;~k^x+
`y s
n;~k^y+
`z s
n;~k^z; (S15)8
Where,
`x;y;z s
n;~k=
s
n;~k `x;y;z s
n;~k
is the expected value of angular-momentum operator in reciprocal space for
states of Bloch band nand spin sector s. To study the orbital texture and how it affects the OHE, we separate the
in-plane textures (
`x;y s
n;~k), which are represented by arrows, see Fig. SV and panels of Fig.2 of main text, and
out-of-plane textures (
`z s
n;~k), which we represent as a color plot (dark blue color for
`z
1and dark red color for
`z
1). Following the semi-classical argument of Ref. [40], it is possible to show that z
OHis a consequence of the
existence of non-trivial in-plane orbital texture (
`x;y s
n;~k). Some features of the function z
OH(Ef)can be understood
from these textures, as we briefly mentioned in the main text, and now we detailed here.
Figure SV: Orbital character of the "-spin eigenstates of H[Eq. S1 or Eq. (1) of the main text] calculated for: (a) I= 0:2Vpp,
andVAB= 0; (b)I= 0:2Vpp, andVAB= 0:8Vpp; (c)I= 1:1Vpp, andVAB= 0:8Vpp.
Figure 2 of the main text displays both the in-plane and the out-of-plane orbital polarisations of the lowest "-spin
energy band for the B1, A1 and B2 phases. Results for the #-spin bands can be easily obtained by time-reversal
symmetry operation. In Figure SV we complement our analysis by showing the orbital textures of the four "-spin
energy bands for each one of the three phases. The orbital projections depicted in panel (a) were calculated for
I= 0:2VppandVAB= 0, and correspond to the case in which the system assumes the B1 phase. Clearly, the
in-plane orbital textures of the first and second energy bands are opposite to each other, and the same happens to the
third and fourth bands, which leads z
OH(Ef)to be an odd function of Fermi energy, and consequently, the absence
of OHE in the central gap. As it was shown in Fig. 3 of the main text, if we include second neighbors hopping
in the tight-binding Hamiltonian, the particle-hole symmetry around the central gap is broken, and the cancelation
of in-plane orbital texture is lost, leading to the appearance of a central plateau in the orbital Hall conductivity.
It is also noteworthy that h`zi"
n;~kfor the second and third bands are opposite, as well as around the K(K0) and
symmetry points. Conversely, the first and fourth bands respectively exhibit h`zi"
n;~k1in the vicinities of the 9
point, but virtually vanishing values around KandK0. Panel (b) displays the orbital projections of the eigenstates
corresponding to the A1 phase, calculated for I= 0:2VppandVAB= 0:8Vpp. One of the main eye-catching
characteristics of this phase is the opposed out-of-plane orbital polarisations around the K0andKpoints, which is
a manifestation of the orbital-valley locking produced by VAB. Similarly to phase B1, the out-of-plane polarisations
of the first and second "-spin energy bands are opposed to the fourth and third ones, respectively. In addition, the
in-plane orbital angular momentum polarisations for this phase exhibit the same configuration as those obtained for
the B1 phase,which means that, also in this phase, sigma is an odd function of Fermi energy, with no central plateau.
However, due to the orbital-valley locking, the corresponding absolute values are smaller, which explains the different
curve derivative of the OHE in the phase A1 when compared with the OHE of the phase B1. Finally, panel (c) shows
the orbital character of the system, calculated for I= 1:1VppandVAB= 0:8Vpp, when it is in the B2 phase.
In this case we find that h`zi"
n;~k 1for the lowest energy band, which goes along with a substantial reduction of
the in-plane texture. Similarly to the previous cases, h`zi"
n;~kfor the lowest and highest energy bands are inverted.
However, there a noticeable change in h`zi"
n;~kin comparison with the results obtained for the A1 phase, which is
accompanied by a relatively strong orbital-valley locking produced by the combined action of the large values of I
andVAB.
LOW-ENERGY APPROXIMATION
As discussed in the main text, our effective Hamiltonian in the vicinity of the K=K0point can be expressed in
terms ofSU(2)
SU(2)orbital and sub-lattice algebras. Expanding the matrix of Eq. S1 near valleys K= 4=3a
andK0= 4=3a, we obtain, up to first order in electronic momentum, the effective theory
Heff= ~vF(kxx+ kyy) +sI`z+VABz+H`: (S16)
Here,vF=ap
3
2~Vpprepresents the Fermi velocity, and ais the lattice constant; =1for theKandK0valleys,
respectively, and s=1for"and#spin electrons, respectively. The last term H`breaks the degeneracy between `z
eigenstates and can be separated in two contributions:
H`=H`k+HD;whereH`k= ~vF
4 (k+`+ +k ` )andHD= p
3~vF
2a(`xx+ `yy):(S17)
=x+i y, = ,` ( =x;y) represent the orbital angular momentum matrices in the corresponding Hilbert
space,k=kxiky, and`=`xi`y.
Figure SVI shows a comparison between the energy band spectra obtained by our tight-binding (blue dashed lines)
and effective models (red solid lines) calculations in the vicinities of KandK0. In the left column we notice for the
three phases that our effective linear model describes rather well the two inner energy bands, but fails to properly
do so for the two outer ones. This can be corrected with the inclusion of quadratic terms in our approximation, as
illustrated in the right column of Figure SVI. It is noteworthy that the orbital texture near KandK0are very well
described by our effective model. Nevertheless, to reproduce the orbital texture in the vicinity of , it is necessary to
perform an even higher-order expansion up to 4th order.
To provide insight on how H`affects the energy spectrum and orbital textures of this model, we examine the
corresponding contributions of each term in Eq. S17. For simplicity, we consider only one spin sector. In this case,
the energy spectrum of H0= ~vF(kxx+ kyy)consists of two degenerate Dirac cones that are associated with
the two eigenstates of the angular momentum pseudo-spinor. Similarly to what occurs in graphene, the inclusion of
HAB=VABzopens an energy gap in the spectrum, while HSOC=sI`zacts as an orbital exchange interaction,
shifting upwards (downwards) the Dirac cone associated with the `zeigenvalue +1(-1). To understand how H`
modifies the energy spectrum, we introduce a multiplicative factor that regulates its overall intensity and inspect the
energy band structure of H0+H`for two different values of in the following situations: (i) H`K6= 0;HD= 0, (ii)
H`K= 0;HD6= 0, and (iii)H`K6= 0;HD6= 0. The results for the energy bands calculated as functions of kxfor
ky= 0are exhibited in Figure SVII. In panels (a) and (d) we note that H`Klifts the orbital degeneracy of the two
Dirac cones for kx6= 0, by differently renormalising their corresponding Fermi velocities. Panels (b) and (d) show
howHDaffects the energy bands. HDdoes not depend upon the wave vector ~k, and has the same functional form10
Figure SVI: Comparison between the tight-binding energy band calculations (blue dashed lines) with the eigenvalues of our
effective Hamiltonians in the vicinities of KandK0(red solid lines). The eigenvalues obtained with the linear and quadratic
order expansions are depicted in the left and right panels, respectively. The results are for: (a) I= 0:2VppVAB= 0, (b)
I= 0:2VppVAB= 0:8Vpp, and (c)I= 1:1VppVAB= 0:8Vpp.
of a Dresselhaus SOC for Dirac Fermions. It may be regarded as equivalent to a Dresselhaus SOC for orbital states.
As expected, HDleads to a Dresselhaus-like band splitting, without opening a gap at E= 0. In panels (c) and (f)
of Figure SVII, we clearly see the formation of a single Dirac cone and the two outer bands when both H`Kand
HDare present. It is worth recalling that to reproduce the flat-bands, it is necessary to consider high-order terms in
k. Similarly to what is observed in quantum anomalous Hall insulators, the gap opening at E= 0is a consequence
of the interplay between the orbital equivalent of a SOC and an exchange interaction. There is, however, a rich
phenomenology involving the contributions of the distinct terms in Eq. S16 that arises when is varied, but this goes
beyond the scope of the present discussion.
Finally, we examine the role of H`andHDin the orbital texture of this model system. Figure SVIII shows the
orbital textures calculated for: (a) HD6= 0andH`K= 0; (b) forHD= 0andH`K6=, and (c) for the effective complete
Hamiltonian without SOC and VAB. By comparing the three panels, it is clear that the orbital texture of our effective
model is basically governed by the Dresselhaus-like coupling associated with the orbital angular momentum spinor,
which reproduces rather well the in-plane texture of our tight-binding calculations near K.11
Figure SVII: Energy bands calculated as functions of kx(forky= 0) by means of our effective theory around the Ksymmetry
point. Panels (a), (b) and (c) depict the results obtained for = 0:3in the cases: H`K6= 0;HD= 0;H`K= 0;HD6= 0and
H`K6= 0;HD6= 0, respectively. Panels (d), (e), and (f) show the results calculated for the same cases, but with = 1:0.
Figure SVIII: Comparison between of the in-plane texture profile for: (a) HD6= 0andH`K= 0; (b)HD= 0andH`K6= 0,
and (c)HD6= 0andH`K6= 0.
SECOND NEAREST NEIGHBOURS AND ORBITAL TEXTURE ANALYSIS
As mentioned in the main text, the absence of OHE plateau in the central electronic spectrum gap of px-py-model of
Eq. (1) is a consequence of the combination of the particle-hole and parity symmetries of spectrum which translates in
cancellation of in-plane orbital texture at half-filling. To understand better the consequences of the breaking of these
symmetries, we introduce a toy-model of the px-pyHamiltonian, where we have included second nearest-neighbours
hopping. This model is described by,
H=X
hijiX
st
ijpy
ispjs+X
hhijiiX
st
ijpy
ispjs+X
isipy
ispis+X
ishz
spy
ispis; (S18)
here as before, iandjrepresents the honeycomb lattice sites whose position is given by ~Riand~Rj, respectively. The12
symbolshijiandhhijiiindicates that the summations are restricted to the nearest and second nearest neighbour
sites respectively. The operator py
iscreates an electron of spin sin the atomic orbitals p=p=1p
2(pxipy)
centred at ~Ri. Here,s=";#labels the two electronic spin states, and, now, iis the atomic energy at site i, which
encodes the effect of the combination of a sublattice potential VAB, and the on-site energy of porbitals"p. This terms
take values i="pVAB, when site i belongs to the A and B sub-lattices of the honeycomb arrangement, respectively.
Figure SIX: Comparison between the orbital (spin- ") texture profiles of the px-pywith only nearest neighbours (a), and the
orbital texture of the same model when second nearest neighbours are considered (b). Left: Orbital Texture profile for the
deepest energy band. Center: Orbital Texture profile for the second lowest energy band. Right: addition of the orbital textures
in Left and Right panels with the in-plane component scaled by a factor 5.
As it was shown in figure 3 of the main text, the principal effect of the particle-hole and parity symmetries breaking,
a consequence of the inclusion of the second nearest neighbours, is the appearance of an orbital Hall conductivity
plateau in the central gap. In order to uncover the connection between the appearance of this plateau and the orbital
textures, we analyse the texture profiles of the two deepest energy bands for two different cases of this model. In
panel (a) of the figure SIX are shown the orbital textures of the two deepest energy bands(left and central panels) of
the simple model that does not include second nearest neighbours [Eq. (1) in the main text] and their summation
(right panel) in which the in-plane components of the texture are in a larger scale to make easier the analysis of
their details. The in-plane component of orbital textures in left and central panels present the dipole configuration
around the point and the anti-vortices in the KandK0points, and the out-of-plane component appears due to the
inversion symmetry breaking produced by the inter-lattice potential. At the right panel of the figure (a), we show that
the addition of the orbital textures of the left and central panels results in a zero net in-plane orbital texture. Once
that orbital Hall conductivity ( z
OH) appears as a consequence of dynamics of in-plane orbital texture, in presence of
an external electric field, this explains the absence of OHE in the central gap of the simplified px-pymodel of Eq. (1)
in the main text. Now, in panel (b) of the figure SIX, we consider the orbital texture of the Hamiltonian with the
inclusion of second nearest-neighbours hopping (see. Eq. S18). Again, the left and the central panels of the figure
show the orbital textures of two deepest bands and the right panel shows the sum of these two textures, with the
in-plane component multiplied by a scaling factor to facilitate its visualization. To maintain the resemblance between13
the aforementioned case and this new case, we set the same Slater-Koster parameters that we used for the phase A1
of the simplified model with the addition of "p= 0:3andVpp2= 0:2. With these parameters, as it was shown
in figure 3 (a) of the main text the energy bands of this modified model are not particle-hole symmetric, an effect
caused exclusively by the inclusion of second nearest-neighbours. We note in Fig. SIX (b) that the overall features of
in-plane orbital-texture of two deepest bands (left and central panels) are not qualitatively modified, i.e., still present
a dipole-like texture near -point and anti-vortices textures at valleys. However, as can be seen in Fig. SIX (b), right
panel, the exact cancellation of the in-plane texture of two deepest bands is lost, causing the existence of a net in-
plane orbital texture which produces an OHE in the central gap of the spectrum shown in figure 3 (b) of the main text.
Once shown by means of the simply px pymodel that the orbital Hall effect is present in systems where the
particle-hole and parity symmetries are absent, we now focus on a real material. For this purpose, we have chosen
the flat bismuthene grown on SiC as a candidate for the observation of OHE in the central plateau. The observation
of orbital-insulator phase in the central gap of bismuthene should be easier in the experimental point of view, once it
corresponds to neutrality situation. In the past, this system has been studied by means of the aforementioned minimal
px-pytight-binding Hamiltonian [14, 22, 41]. However, we have noticed that by including second nearest-neighbours
in the tight-binding Hamiltonian used in Ref. 22 the electronic structure is better reproduced. In the bismuthene/SiC
heterostructure, the break of inversion symmetry induces a small Rashba SOC,
HR= 2iRX
hi;jiX
spy
is[^z(~ ^eij)]sspjs+H:c: (S19)
where~ symbolises the Pauli vector, ^eijdenotes the unit vector along the n.n. inter-site direction of ~Rj ~Ri,Ris the
Rashba SOC constant, and sdesignates the opposite spin direction specified by s. We will consider this term only in
the fitting of tight-binding Hamiltonian to DFT spectrum and, we neglect it in transport calculations presented here.
The reasons is that the non-conserving spin character of this coupling complicates the analysis of orbital texture and
its typical small value does not alter the main conclusions of our discussion, as it was checked by us. In figure SX is
shown a direct comparison of the DFT energy band structure obtained in Ref. 14, with and without the inclusion of
Rashba spin-orbit coupling. From this figure, it is rapidly noticeable the agreement between the DFT energy bands
and the tight-binding model in describing the top of the valence band and the bottom of the conduction bands and
the indirect gap in the point. In the table SI are shown the two-centre integral parameters used in the description
of this model. Once shown the agreement of energy band of the complete model, we are going to restrict ourselves to
the situation in which the Rashba SOC is neglected and the system is subject to a staggered potential VAB= 0:87.
The first of these constraints is to avoid complications in the analysis due to possible contributions to the orbital
texture by the Rashba SOC, which does not conserve spin as a good quantum number, and the second one is to leave
the system in a topological phase similar to the phase A1of thepx pymodel with only nearest neighbours. This
will allow us to focus on the analysis of the Orbital texture and its connection with the orbital Hall conductivity.
Table SI: Second Nearest-neighbour two-centre energy integrals, and spin orbit coupling constants (all in eV) for the Bi/SiC.
Two-centre integrals Intrinsic SOC Rashba SOC On-site energy
Vpp= +1:51522I= 0:435R= 0:032p= 0:279865
Vpp= 0:575788
Vpp2= 0:18
Vpp2= 0:00658
In figure SXI is displayed the orbital textures of the two deepest energy bands of bismuthene grown over SiC in
the phaseA1and without Rashba SOC. The principal difference that is noticeable when one looks at this figure is
the change of the out-of-plane orbital texture of the spin- "sector, with respect to textures of A1-phase in previous
cases, produced by the change of sign of the spin-orbit coupling. The in-plane components of orbital texture for the
model with parameters of bismuthene shown in Fig. SXI left and centre panels, do not present noticeable differences
from those of Fig. SIX (b) of the model with second nearest neighbours. Again, as can be seen from the right panel
of Fig. SXI, there is a non-zero total in-plane orbital texture when we add up the textures of two deepest bands
of left and central panels what again explain the existence of OHE in the central gap, as shown in Figure 4 of the
main text. This suggests the recent synthesized flat bismuthene as a realistic platform to observe the orbital Hall
insulator phase. The central plateau will persist by the inclusion of the Rashba term [Eq. (S19)] on Hamiltonian,14
Figure SX: Comparison between the DFT energy bands (blue doted line) and the tight-binding model with second nearest
neighbours bands (red solid line) for: (a) R= 0:032eV and (b) R= 0.
once the spectrum keeps the particle-hole asymmetry. When the Rashba term is included, we cannot separate the
textures by spin sectors because it breaks the sz-symmetry. So the previous analysis of sum of orbital texture must
be done summing the four lowest energy bands to obtain total texture related to the central plateau. But the main
conclusions are the same and we do not present this analysis here.
Figure SXI: Orbital Textures profile of spin- "sector for the two lowest energy bands of bismuthene over SiC in which the
Rashba SOC is neglected and VAB= 0:87eV. Left: Orbital texture profile of the lowest energy band. Center: Orbital texture
profile of the second lowest energy band. Right: Addition of the later texture profiles. The resultant in-plane orbital texture
are scaled by a factor 5to facilitate the visualization |