Table Of ContentImage-potential-induced spin-orbit interaction in one-dimensional electron systems
Yasha Gindikin and Vladimir A. Sablikov
Kotel’nikov Institute of Radio Engineering and Electronics,
Russian Academy of Sciences, Fryazino, Moscow District, 141190, Russia
We study the spin-orbit interaction effects in a one-dimensional electron system that result from
the image charges in a nearby metallic gate. The nontrivial property of the image-potential-induced
spin-orbit interaction (iSOI) is that it directly depends on the electron density because of which a
positivefeedbackarisesbetweentheelectrondensityandtheiSOImagnitude. Asaresult,thesystem
becomes unstable against the density fluctuations under certain conditions. In addition, the iSOI
contributes to the electron-electron interaction giving rise to strong changes in electron correlations
andcollectiveexcitationspectra. Wetracetheevolutionofthespectrumofthecollectiveexcitations
7 and their spin-charge structures with the change in the iSOI parameter. One out of two collective
1 modes softens as the iSOI amplitude grows to become unstable at its critical value. Interestingly,
0 this mode evolves from a pure spin excitation to a pure charge one. At the critical point its velocity
2 turns to zero together with the charge stiffness.
n
a
y
J The Rashba spin-orbit interaction (RSOI) in low-
Gate
5 dimensionalsystemsarisesbecauseofastructureinversion
a
2 asymmetry, which results from an external electric field
] actingonelectronsinadditiontothecrystallinefield. The 2a Wire
l RSOI plays a central role in such areas as the generation,
e manipulation and detection of spin, topological states, 0 x
- d
r Majoranafermions,low-dimensionalmaterialswithDirac-
t
s type spectra and even cold-atom systems (for a recent
.
t review see Ref. [1]).
a
m The RSOI is described by the Rashba Hamiltonian [2]
Figure 1. The schematic of a one-dimensional electron system
- H =α(E×k)σ, (1) with image charges induced on a gate. The arrows show the
d RSOI
electricfieldsactingonelectronsfromtheirownimagecharges
n
o where E is an external electric field, which is usually and from the images of neighboring electrons.
c considered as a given value. By tuning the field E, one
[ can gain control over the RSOI parameter α = αE.
R
This is important for the spin manipulation by electrical in a dramatic transformation of the ground state. The
2
mechanism is as follows. An electron density fluctuation
v means.
7 In the present paper we consider a principally different induces an additional image charge and hence increases
8 situation where the structure symmetry is broken by a an electric field component normal to the gate surface.
9 This enhances the iSOI parameter α and consequently
metallic gate placed in close proximity to the electronic R
1
lowers the electron energy within the fluctuation region,
system and coupled to it by the Coulomb forces. This
0
attracting there electrons from adjacent regions or reser-
. situation is close to the experiments where the electron
1 voirs. Thus the density fluctuation once appeared starts
system under investigation is placed directly on a con-
0
to grow.
7 ductive gate [3]. In this case the RSOI can arise even
Let us begin with a qualitative description of the pro-
1 without any potential applied to the gate thanks to the
: image charges electric field as shown in Fig. 1. This field cess. To be specific, consider a single-mode quantum wire
v
parallel to a metallic gate, separated by a distance of a/2
i is strong enough in the vicinity of the interface. One may
X from the latter. Let us determine the electron density in
therefore expect strong effects due to the image-potential-
r induced spin-orbit interaction (iSOI). The presence of the the wire for the case of a fixed chemical potential µ. For
a
now,werestrictourselvestoamean-fieldtheory,assuming
iSOIrecentlywasconfirmedbyseveralexperimentswhere
the electron density n to be uniformly distributed.
the spin-orbit splitting was observed in the surface elec-
The single-electron state energy reads as
tron states formed by the image potential on the Au(001)
surface [4] and at the graphene/Ir(111) interface [5]. The (cid:126)2
values of α measured in these experiments agree well ε = [(k+sk )2−k2 ]+vn, (2)
R ks 2m so so
with the calculations performed by McLaughlan et al. [6].
A novel and fascinating property of the iSOI is that where k is the longitudinal wave vector and s = ±1
α depends on the electron density. This dependence is the spin index. The Coulomb interaction energy is
R
creates an efficient mechanism for density fluctuations v= 2e2 ln(a/d) with d being the quantum wire diameter
(cid:15)
to grow, which under certain circumstances can result and (cid:15) as the dielectric constant. The iSOI wave vector
2
is kso = αRm/(cid:126)2. The iSOI parameter αR = αE⊥ is n/n0
proportional to the normal component of the electric 6
field where the SOI constant α does not depend on the
field. It is important that the field is determined by the
electron density E⊥ = 2ne/(cid:15)a. Whence it follows that 4
kso =2enαm/(cid:126)2(cid:15)a. Theequationfortheelectrondensity n-
isfoundbysummingovertheoccupiedstates. Takinginto
account that there are two values of the Fermi momenta
for each spin direction, k(s) = −sk ±(k2 +2m(µ− 2
F so so
vn)/(cid:126)2)1/2, weobtainanequationtodeterminenatzero
n
temperature, +
α*
2(cid:115)(cid:18)2αme(cid:19)2 2m 0 1 αc*
n= n2+ (µ−vn). (3)
π (cid:126)2(cid:15)a (cid:126)2 Figure 2. The electron density dependence on the iSOI pa-
rameter for v∗ =2.
Its solutions are
√
−v∗± 1−α∗2+v∗2 In what follows we study the spectra of collective exci-
n (α∗)=n , (4)
± 0 1−α∗2 tations in a one-dimensional (1D) electron system below
a threshold of a possible instability to find out the condi-
where n = √8mµ/π(cid:126), v∗ = v(cid:112)2mµ−1/π(cid:126), and α∗ = tions under which the stability of the excitations could
0
4αme/π(cid:126)2(cid:15)a is a dimensionless iSOI parameter. be lost.
The electron density exhibits an S-type dependence on An important aspect of the iSOI is a nontrivial modi-
α∗ asseenfromFig.2. AtweakiSOIα∗ <1,thesolution fication of the electron-electron (e-e) interaction Hamil-
is unique. In the range of 1<α∗ <α∗ there appear two tonian. The image charges not only screen the Coulomb
c
solutions, the stability of which should be examined. At interaction to make it dipole-like, but also create a new
α∗ > α∗ the solution is at all absent within the simple spin-dependent component of the e-e interaction. This
c
model considered. The critical iSOI magnitude is given effect should be manifested in a qualitative change in the
by correlation functions. To the best of our knowledge, the
properties of the correlated electron state and its collec-
(cid:112)
α∗ = 1+v∗2. (5) tive excitations were not investigated in literature in such
c
circumstances.
Suchbehaviorofn(α∗)indicatesapossibleinstabilityof Our model Hamiltonian reads as
theelectronsystematsufficientlystrongiSOIα∗ ∈(1,α∗)
c H =H +H +H . (6)
andatendencyforaradicaltransformationoftheelectron kin e−e iSOI
state at α∗ > α∗, which may lead to the emergence of
c The first term is the kinetic energy H =
spatially inhomogeneous structures or a new correlated (2m)−1(cid:80) (cid:82) dxψ+(x)p2ψ (x),whereψ (x)standskifnorthe
state. Nontrivial effects are expected already when α∗ is s s x s s
electron field operator and p stands for momentum.
oftheorderofunity. Ourestimatesshowthatsuchvalues x
The operator of the e-e interaction energy is
of α∗ can be attained in materials with a strong spin-
orbitinteraction[1]. PresentlythetunableRSOIwiththe 1 (cid:88)(cid:90)
parameter as large as αR ∼4×10−10 eV m is attained in He−e = 2 dx1dx2ψs+1(x1)ψs+2(x2) (7)
suchmaterialsasBi2Se3 inquantumwellsinthepresence s1s2
of the electric field of the order of 3×105 V/cm [1, 7]. ×U(x1−x2)ψs2(x2)ψs1(x1).
Using thesedata one canestimate the distance a between
the electron system and the gate at which α∗ ∼ 1. For Here U(x) = √ e2 − √ e2 is the e-e interaction
x2+d2 x2+a2
m = 0.1m and (cid:15) ∼ 10 we estimate a ∼ 40 Å, which is potential screened by the image charges. Its Fourier
e
realizable in modern heterostructures. transform U = (cid:82) dxU(x)e−iqx is a table integral [8],
q
Mechanisms stabilizing the electron system at strong equal to U =2e2(K (qd)−K (qa)), with K being the
q 0 0 0
iSOI and the nature of the emerging electron state con- modified Bessel function [9].
stitute a challenging problem that deserves a separate The iSOI Hamiltonian can be formulated on the basis
study. Apossiblemechanismshouldincludetheprocesses of the standard form (1) taking into account that the
leading to an essential rearrangement of the density of electric field is produced by all the charges in the system.
states, such as the population of the higher transverse Using Eq. (1) in the case of the iSOI is supported by
sub-bands in the quantum wire and the formation of a calculations carried out in Ref. [6] within the relativistic
new correlated state. multiple-scattering methods.
3
The iSOI Hamiltonian reads as
α(cid:88)1
H = (E (x )p +p E (x ))σ , (8)
iSOI (cid:126) 2 y i xi xi y i zi
i
where σ is the Pauli matrix of the ith electron and
zi
E (x ) is the y component of the electric field acting on
y i
the electron. This field contains two principally different
contributions that come from external charges and the
images of all electrons in the system. We emphasize
that the iSOI can not be described by a single-particle
Hamiltonian as opposed to RSOI described in Refs. [10–
18] by a fixed parameter α .
R
The two-particle contribution is the total field of other
electron images acting on a given electron,
Eee(x )=(cid:88)E(x −x ), (9) Figure 3. The square of the frequency ω−2 of collective ex-
y i i j citations as a function of wave vector and iSOI amplitude.
j(cid:54)=i Additionally, a plane ω2 = 0 is shown. The frequency is
−
normalized at ω =v k . The system parameters are taken
0 F F
where E(x −x ) = −ea(cid:0)(x −x )2+a2(cid:1)−32. A corre- as follows: kFaB =1.27, d=0.078aB, a=0.39aB, nion =n0.
i j i j
sponding collective contribution to the Hamiltonian (8)
5 5
equals
Hee = α (cid:88)(cid:90) dx dx ψ+(x )ψ+(x ) (10) 4 q/k0F.1 4
iSOI 2(cid:126) 1 2 s1 1 s2 2 0.6
×{E(x −xs1s)2S +S E(x −x )}ψ (x )ψ (x ), ω- 3 1.1 3ξ
1 2 12 12 1 2 s2 2 s1 1 qvF 1.6 -
2 2
with S =(p σ +p σ )/2.
12 x1 z1 x2 z2
The Hamiltonian (10) together with Eq. (7) forms a 1 1
modified Hamiltonian of the e-e interaction that contains
a spin-dependent component appearing because of iSOI. 0 0.02 0.05 0.08 0.11 0.14 0
A single-particle contribution, coming from the image α
of the positive background charge n in the wire (and
ion
the charge in the gate, should there be any) as well as Figure 4. The spin-charge separation parameter (solid line)
the field of the electron’s own image E(0) equals E0y = and normalized phase velocity (dashed line) for the ω− collec-
E(0) − n E , where E is the q = 0 component of tive mode as a function of iSOI amplitude. The same system
ion 0 0
parameters as in Fig. 3.
the Fourier-transform E = −2e|q|K (|q|a) of the field
q 1
E(x) [8]. This leads to a single-particle contribution to
the Hamiltonian (8),
component of spin s, wave-vector q, and frequency ω are
α(cid:88)(cid:90) shown to satisfy the following system of linear equations:
H0 = dxψ+(x)E0p σ ψ (x). (11)
iSOI (cid:126) s y x z s
s n(s)(cid:18)χ−1−U + mα2Eq(2F +n E )−sω2mαEq(cid:19)
qω qω q (cid:126)2 0 0 q (cid:126)q
Below we investigate a linear response of the system
definedbytheHamiltonian(6)–(11)toanexternalpertur- +n(−s)(cid:18)−U + mα2Eq(2F +n E )(cid:19)=ϕ(s). (12)
bation of the form H =(cid:80) (cid:82) dxψ+(x)ϕ(s)(x,t)ψ (x). qω q (cid:126)2 0 0 q qω
ext s s s
The calculations are based on two independent meth-
ods, viz. the random phase approximation (RPA) and The mean electric field F0 = E(0)+(n0 −nion)E0 as
bosonization. Both approaches yield compatible results. compared to E0 contains additionally the contribution
y
The calculations are performed for a 1D system of length from the mean electron density. By χqω we denote the
L with fixed mean electron density n . The periodic Lindhard susceptibility,
0
boundary conditions are imposed, and the limit L→∞
considered. χ = m ln(q−2kF)2−(2m(cid:126)ωq+i0)2 , (13)
RPA calculations are based on the equation of motion qω 2π(cid:126)2q (q+2k )2−(2mω+i0)2
F (cid:126)q
for the Wigner function derived in the Appendix. The
Fourier components n(s) of electron density with the z where k =πn /2.
qω F 0
4
Setting the determinant of (12) to zero, we obtain
the dispersion equation for both branches of collective
excitations,
(cid:18)ω (cid:19)2 (cid:16) (cid:17)
± =1+ U˜ −α˜2F˜ E˜ (14)
qv q 0 q
F
(cid:114)
(cid:16) (cid:17)2
± U˜ −α˜2F˜ E˜ +α˜2E˜2.
q 0 q q
2αn U
Dimensionless amplitudes are α˜ = 0, U˜ = q ,
πea q π(cid:126)v
B F
F E (cid:126)k
F˜ = 0, and E˜ = q with v = F and a =
0 en2 q en F m B
0 0
(cid:126)2/me2.
Of most interest is branch ω since it has an unusual
−
dependence on the wave-vector q and the iSOI parameter Figure 5. The square of the frequency ω2 of collective ex-
+
α˜. This dependence is demonstrated in Fig. 3 in the case citations as a function of wave vector and iSOI amplitude.
where the distance a is small enough. The frequency of Additionally, a plane ω2 = 0 is shown. The same system
+
thismodeanditsvelocitydecreasewithincreasingα˜. The parameters as in Fig. 3.
frequency squared ω2(q) turns to zero at some condition,
−
(cid:113) On the contrary, at α˜ = 0 another branch ω corre-
1+2U˜ +
α˜ =α˜0 ≡ q , (15) sponds to purely charge excitations, which transform into
q (cid:113)
E˜2+2F˜ E˜ purely spin ones as α˜ increases. Their spectrum is shown
q 0 q
in Fig. 5. Upon the increase in α˜ their velocity v (q) al-
+
andevenbecomesnegativeintheregionofα˜ >α˜0,where waysremainspositive. Thestiffnessofthespinsubsystem
q
the excitations become unstable. It is worth noting that does not turn to zero.
upon the increase in α˜ the excitations start losing their Letuscomparethecritical iSOIvalueα˜00 fromEq.(15)
stability in the long-wave region where also the largest at which the long-wave collective excitations start losing
frequency increment appears in the instability regime. their stability with αc∗ of Eq. (5), corresponding to the
Spin-dependent interactions break the spin-charge sep- instability of the ground state of a system with a fixed
aration between the branches ω of collective excitations. chemical potential. For the case when the system is
±
It is interesting to investigate how the spin-charge struc- sufficiently close to the gate√n0a (cid:28) 1, we obtain for
ture of the excitations evolves as α˜ is increased. From dimensionaliSOIvaluesα˜00 ∝ n0aαc∗,whichmeansthat
Eq. (12) we determine the spin-charge separation param- the collective excitations instability develops first. In the
eter ξ± for both branches of excitations, opposite limiting case of n0a(cid:29)1, they are of the same
order of magnitude.
(cid:12)
n+ +n− (cid:12) 1 (cid:18)ω qv (cid:19) The bosonization [19] treatment of the problem leads
ξ = qω qω(cid:12) = ± − F . (16)
± n+qω−n−qω(cid:12)(cid:12) α˜E˜q qvF ω± to similar results. The presentation is simplified greatly
ω± in the absence of the mean electric field F . Then the
0
Atα˜ =0,theparameterξ =0,whichmeansthatbranch eigenstates of the kinetic energy can be chosen as the
−
ω corresponds to a purely spin excitation (n+ =−n− ) basis functions. Linearizing their spectrum, we introduce
w−ith dispersion law ω = v q. However, aqsωα˜ → αq˜ω0, the bosons a+(q) = ( 2π )1/2 (cid:80) θ(rq)ρ (q), where the
− F q s L|q| rs
the frequency ω (q)→0 and the parameter ξ →∞ as r=±
− − normal ordered density of fermions with spin projection
showninFig.4. Consequently, nearthethresholdα˜ =α˜q0 s on branch r is ρ (q) = (cid:80) : c+(p+q)c (p) :. The
the collective excitation ω (q) turns into a purely charge rs p rs rs
− quadratic part of a bosonized Hamiltonian (6) is
excitation (n+ =n− ).
qω qω
The system stiffness κ = −limq→0χ−nn1(q,0) with a H = (cid:126)vF (cid:88) q(cid:104)a+(rq)a (rq)(2+U˜ +rsα˜E˜ )
charge susceptibility χ (q,ω)=(n+ +n− )/ϕ deter- 2 s s q q
nn qω qω qω
q>0
mined from Eq. (12) equals r,s=±
κ =π(cid:126)v (1+2U˜ )(cid:34)1−(cid:18) α˜ (cid:19)2(cid:35) . (17) + 21(a+s(rq)a+−s(−rq)+H.c.)(U˜q+rsα˜E˜q) (18)
F 0 α˜00 + 1(a+(rq)a+(−rq)+a+(rq)a (rq)+H.c.))U˜ (cid:21) .
2 s s s −s q
The stiffness turns to zero at α˜ =α˜0. This points at the
0
instability of the charge subsystem. This is the most pro- We diagonalize it by the Bogoliubov-Tyablikov transfor-
nounced manifestation of the iSOI in the e-e correlations. mation [20]. For this purpose, matrices defining com-
5
mutators [H,a+]=(cid:80) a+A +a B for each boson a rather general for a wide class of 1D, quasi-1D, and 2D
k i i ik i ik k
from (18) are constructed. Then the squares of the ele- systems in materials with strong spin-orbit interaction.
mentary excitations frequencies are just the eigenvalues The instability leads to the formation of a new correlated
of matrix (A−B)(A+B). They are state that needs to be investigated further.
ThisworkwaspartiallysupportedbytheRussianFoun-
(cid:18)ω± (cid:19)2 =1+U˜ ±(cid:113)U˜2+α˜2E˜2 , (19) dation for Basic Research (Grant No 17-02-00309) and
qv q q q the Russian Academy of Sciences.
F
which coincides with (14) at F =0.
0
In conclusion, we have shown that the Coulomb in- APPENDIX
teraction of 1D electrons with the image charges in the
nearby metallic gate has a spin-dependent component
Here we derive Eq. (12) of the main text. Define single-
caused by the Rashba spin-orbit interaction. This iSOI
and two-particle Klimontovich operators [21] as
can strongly affect both the ground state of the system
and the collective excitations. The main effect is an insta- 1 (cid:90) η η
fˆ(s)(x,p,t)= dηeipηψ+(x+ ,t)ψ (x− ,t) (20)
bility which occurs as the iSOI parameter is large enough. 2π s 2 s 2
Our estimations have shown that the critical conditions
areattainableinrealisticsystems. Thiseffectseemstobe and
1 (cid:90) η η η η
fˆ(s1,s2)(x ,p ,x ,p ,t)= dη dη ei(p1η1+p2η2)ψ+(x + 1,t)ψ+(x + 2,t)ψ (x − 2,t)ψ (x − 1,t).
1 1 2 2 (2π)2 1 2 s1 1 2 s2 2 2 s2 2 2 s1 1 2
(21)
Theaveragevaluesf(s)(x,p,t)andf(s1s2)(x1,p1,x2,p2,t) Bycommutingfˆ(s)(x,p,t)withH+Hextandtakingthe
of these operators w.r.t. the ground state are just the average, the equation of motion for the WDF is obtained,
Wigner distribution functions (WDFs), which allow one
to find the observables of interest. Thus, the electron
density is expressed as
(cid:90)
n(s)(x,t)= dpf(s)(x,p,t). (22)
i(cid:126)2p 1 (cid:90) (cid:16) η η (cid:17)
i(cid:126)∂ f(s)(x,p,t)=− ∂ f(s)(x,p,t)+ dηdp ei(p−p1)ηf(s)(x,p ,t) ϕ (x− ,t)−ϕ (x+ ,t) (23)
t m x 2π 1 1 s 2 s 2
1 (cid:88)(cid:90) (cid:16) η η (cid:17)
+ dξdηdp dp ei(p−p1)ηf(s,ς)(x,p ,ξ,p ,t) U(x−ξ− )−U(x−ξ+ )
2π 1 2 1 2 2 2
ς
iαs(cid:90) (cid:16) η η (cid:17)
−iαsE0∂ f(s)(x,p,t)− dξdηdp dp ei(p−p1)ηf(s,−s)(x,p ,ξ,p ,t) E(cid:48)(x−ξ− )+E(cid:48)(x−ξ+ )
y x 2π 1 2 1 2 2 2
iα (cid:88) (cid:90) (cid:18)1 (cid:19) η
− ς dξdηdp dp ei(p−p1)η ∂ f(s,ς)(x,p ,ξ,p ,t)+ip f(s,ς)(x,p ,ξ,p ,t) E(x−ξ− )
2π 1 2 2 ξ 1 2 2 1 2 2
ς
iα (cid:88) (cid:90) (cid:18)1 (cid:19) η
− s dξdηdp dp ei(p−p1)η ∂ f(s,ς)(x,p ,ξ,p ,t)+ip f(s,ς)(x,p ,ξ,p ,t) E(x−ξ− )
2π 1 2 2 x 1 2 1 1 2 2
ς
iα (cid:88) (cid:90) (cid:18)1 (cid:19) η
− ς dξdηdp dp ei(p−p1)η ∂ f(s,ς)(x,p ,ξ,p ,t)−ip f(s,ς)(x,p ,ξ,p ,t) E(x−ξ+ )
2π 1 2 2 ξ 1 2 2 1 2 2
ς
iα (cid:88) (cid:90) (cid:18)1 (cid:19) η
− s dξdηdp dp ei(p−p1)η ∂ f(s,ς)(x,p ,ξ,p ,t)−ip f(s,ς)(x,p ,ξ,p ,t) E(x−ξ+ ).
2π 1 2 2 x 1 2 1 1 2 2
ς
This is the first equation in the Bogoliubov-Born-Green- RPA by factorizing the two-particle WDF [23],
Kirkwood-Yvon hierarchy [22]. We truncate it using the
f(s1,s2)(x ,p ,x ,p ,t)=f(s1)(x ,p ,t)f(s2)(x ,p ,t).
1 1 2 2 1 1 2 2
(24)
6
Thisdefinesthewaythepaircorrelationsaretakenintoac- isobtainedbysettingthedeterminantof (32)tozero,are
count. Introduce the deviation f(s)(x,p,t) of f(s)(x,p,t) the spin-charge separated common plasmons and spinons.
1
from its equilibrium value f(s)(p) as a result of the exter- Their velocity does not depend on SOI.
0
The terms (30) and (31) reflect the collective elec-
nal perturbation H ,
ext
tron contribution to iSOI. Whereas the structure of the
f(s)(x,p,t)=f(s)(x,p,t)−f(s)(p). (25) term(30)resemblestheCoulombcontribution(28),there
1 0
also appears a qualitatively new integral term (31). Inte-
Theequationofmotionforf(s)(x,p,t), linearizedw.r.t. grate the equation of motion w.r.t. p to get
1
H , in Fourier representation reads as
ext
(−(cid:126)ω+αqsF )n(s) =
0 qω
−(cid:126)ωf1(s)(q,p,ω)=−(cid:126)2mpqf1(s)(q,p,ω) (26) − (cid:126)m2q (cid:90) κf1(s)(q,κ,ω)dκ−αqsEqn20 (cid:88)n(qςω). (35)
q q
+ϕ(s)(f(s)(p+ )−f(s)(p− )) (27) ς
qω 0 2 0 2
+U (f(s)(p+ q)−f(s)(p− q))(cid:88)n(ς) (28) Substitute the integral term from Eq. (35) to Eq. (31),
q 0 2 0 2 qω express f(s)(q,p,ω), and integrate the latter w.r.t. p to
ς 1
obtain the Eq. (12) of the main text.
−αqsF f(s)(q,p,ω) (29)
0 1
+αsE p(f(s)(p+ q)−f(s)(p− q))(cid:88)n(ς) (30)
q 0 2 0 2 qω
ς
+αE (f(s)(p+ q)−f(s)(p− q))(cid:88)ς(cid:90) κf(ς)(q,κ,ω)dκ. [1] A.Manchon,H.C.Koo,J.Nitta,S.M.Frolov, andR.A.
q 0 2 0 2 1 Duine, Nature materials 14, 871 (2015).
ς
[2] R. Winkler, Spin-orbit Coupling Effects in Two-
(31)
Dimensional Electron and Hole Systems (Springer, 2003).
[3] A. Bachtold, P. Hadley, T. Nakanishi, and C. Dekker,
The terms (26)–(28) reflect the contribution of kinetic
Science 294, 1317 (2001).
energy, external potential and Coulomb e-e interaction.
[4] T. Nakazawa, N. Takagi, M. Kawai, H. Ishida, and
Theterm(29)reflectsthepartofiSOIduetothemean
R. Arafune, Phys. Rev. B 94, 115412 (2016).
electric field F0 =E(0)+(n0−nion)E0. Let us discuss [5] S. Tognolini, S. Achilli, L. Longetti, E. Fava, C. Mariani,
the effects of the mean field in some more detail. RPA M. I. Trioni, and S. Pagliara, Phys. Rev. Lett. 115,
assumes that the single-particle states, the distribution 046801 (2015).
over which is given by f(s)(p), are formed by a single- [6] J. R. McLaughlan, E. M. Llewellyn-Samuel, and
0 S. Crampin, Journal of Physics: Condensed Matter 16,
particle part of the Hamiltonian, the mean electric field
6841 (2004).
included. For a system with a fixed particle number,
[7] P. D. C. King, R. C. Hatch, M. Bianchi, R. Ovsyannikov,
this sets the Fermi momenta for a spin direction s to C.Lupulescu,G.Landolt,B.Slomski,J.H.Dil,D.Guan,
be k(s) = −sk ±k , where k = αmF /(cid:126)2 and k J. L. Mi, E. D. L. Rienks, J. Fink, A. Lindblad, S. Svens-
F so F so 0 F
stands for πn /2. Restricting the equation of motion son, S. Bao, G. Balakrishnan, B. B. Iversen, J. Oster-
0
walder, W. Eberhardt, F. Baumberger, and P. Hofmann,
to include just terms (26)–(29), we easily can find the
Phys. Rev. Lett. 107, 096802 (2011).
electrondensityforthecaseofiSOIexclusivelyduetothe
[8] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals,
mean field. For this purpose express the f1(s)(x,p,t) and series, and products (Academic press, 2014).
integrate over p to obtain the equations for the density, [9] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W.
Clark, NIST Handbook of Mathematical Functions (Cam-
(cid:88)
n(s) =ϕ(s)χ(s)+U χ(s) n(ς). (32) bridge University Press, 2010).
qω qω qω q qω qω
[10] A. V. Moroz, K. V. Samokhin, and C. H. W. Barnes,
ς
Phys. Rev. B 62, 16900 (2000).
Here the Lindhard susceptibility [11] W. Häusler, Phys. Rev. B 63, 121310 (2001).
[12] E. N. Bulgakov and A. F. Sadreev, Phys. Rev. B 66,
(cid:90) f(s)(κ+ q)−f(s)(κ− q) 075331 (2002).
χ(s) = dκ 0 2 0 2
qω −(cid:126)(ω+i0)+ (cid:126)2κq +αqsF [13] M. Governale and U. Zülicke, Phys. Rev. B 66, 073311
m 0 (2002).
= m ln(q−2kF)2−(2m(cid:126)ωq+i0)2 (33) [[1145]] SJ..SDuenb,aSld.aGnadngBa.dKhraarmaiearh,,Pahnyds.OR.evA..BSt7a1ry,k1h15,3P2h2y(s2.0R0e5v)..
2π(cid:126)2q (q+2k )2−(2mω+i0)2
F (cid:126)q Lett. 98, 126408 (2007).
[16] B.Braunecker,G.I.Japaridze,J.Klinovaja, andD.Loss,
turns out to be independent of spin s and of the mean-
Phys. Rev. B 82, 045127 (2010).
field F0. Hence, the collective excitations, the dispersion [17] O. A. Tretiakov, K. S. Tikhonov, and V. L. Pokrovsky,
relation of which Phys. Rev. B 88, 125143 (2013).
[18] C. Sun and V. L. Pokrovsky, Phys. Rev. B 91, 161305
χ−1(χ−1−2U )=0 (34) (2015).
qω qω q
7
[19] F. D. M. Haldane, Journal of Physics C: Solid State [21] I. L. Klimontovich, Soviet Physics JETP 6, 753 (1958).
Physics 14, 2585 (1981). [22] M. Bonitz, Quantum kinetic theory (Springer, 2016).
[20] J. Avery, Creation and annihilation operators (McGraw- [23] T. Hasegawa and M. Shimizu, Journal of the Physical
Hill Companies, 1976). Society of Japan 38, 965 (1975).
