Solution of electric-field-driven tight-binding lattice in contact with fermion reservoir

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by  Jong E. Han

Solution of electric-field-driven tight-binding lattice in contact with fermion reservoirs
Jong E. Han
Department of Physics, State University of New York at Buffalo, Buffalo, New York 14260, USA
(Dated: October 24, 2012)
Electronsintight-bindinglatticedrivenbyDCelectricfielddissipatetheirenergythroughon-site
fermionic thermostats. Due to the translational invariance in the transport direction, the problem
can be block-diagonalized. We solve this time-dependent quadratic problem and demonstrate that
the problem has an oscillatory steady-state. The steady-state occupation number shows that the
Fermisurfacedisappearsforanydampingfromthethermostatsandanyfiniteelectricfield. Despite
thelackofmomentumscattering,theconductivitytakesthesameformasthesemi-classicalOhmic
expression from the relaxation-time approximation.
2
PACSnumbers: 71.27.+a,71.10.Fd,71.45.Gm
1
0
2
I. INTRODUCTION surface at small fields and a recovery of the Bloch oscil-

t lation in the high-field limit.
c
O Nonequilibriumphenomenainlatticeisoneoftheold- However, it is unclear so far which of the approx-
imations, such as single-band approximation without
estandmostfundamentalproblemsinsolidstatephysics.
3 Landau-Zener tunneling or the nature of on-site inter-
In conventional solids, the acceleration due to external
2
action, are responsible for the rather peculiar long-time
field is relatively small compared to electronic energy,

states. One of the goals of this paper is that we provide
] and various scattering mechanisms make the transport
el diffusive enough so that the small field approximation exact solutions to one of the dissipation models with on-
sitefermionthermostatsandgiveanalyticunderstanding
- has often been used. The quantum Boltzmann method
tr has been applied effectively1,2 and linear response limit of the problem, and guide possible future modeling.
s Duetothenatureoftheone-bodyreservoirs,theprob-
has been widely used in the literature. However, recent
.
at progress in nano-devices and optical lattice systems has lem can be solved exactly (see Fig. 1). With identical
reservoirs on each site, the Hamiltonian can be block-
m made rigorous high-field formalism necessary to under-
diagonalized according to the wave-vector of electrons in
standtheirnon-perturbativeeffectssuchastheBlochos-
-
d cillation. In such regime, understanding the interplay of thetransportdirection. Theblock-diagonalHamiltonian
n non-perturbative field-effect and the many-body physics canthenbeexactlysolvedbyatime-dependentperturba-
o tiontheory18,19 usingthenonequilibriumGreenfunction
has emerged as one of the most pressing problems in
c theory. The calculation of the occupation number sup-
nano-science.
[
Combining the nonequilibrium and quantum many- ports the numerical observations that the Fermi surface

1 body effects is an extremely challenging task. Much ef- of the unperturbed equilibrium is completely lost with
v effectively infinite temperature, in contrast to the con-
fort has been exerted towards understanding strong cor-
7 ventional picture of the displaced Fermi sea under finite
relation physics in quantum dot physics, especially the
9
nonequilibrium Kondo problem. Analytical theories3–5 electric field. Despite the lack of momentum scattering,
2
theDCelectriccurrentofthismodelrecoversthefamiliar
6 and many numerical methods have been proposed along
. thetime-dependent6–8,andsteady-statesimulations10,11. semi-classical Boltzmann equation result20.
0
Insuchsystemswithlocalizedinteractingregion,theim-
1
2 portant question of energy dissipation could have been
II. MODEL
1 side-stepped, and the existence of steady-state has not
: been a major issue.
v
We study a quadratic model of a one-dimensional s-
i In the past few years, non-perturbative inclusion of
X orbitaltight-bindingmodelconnectedtofermionicreser-
electric-field and many-body effects has been one of the
voirs (see Fig. 1) under a uniform electric field E. The
r importantissuesinthefield. Theoriesforlatticenonequi-
a librium have been formulated12,13, mostly based on the effect of the electric field is absorbed in the temporal
gaugeasthePeierlsphaseϕ(t)=eEattothehoppingin-
dynamical mean-field theory (DMFT) for an s-orbital
tight-binding (TB) lattice with on-site interaction14–17. tegral12 γ. The time-dependent Hamiltonian then reads
Although a long-held belief in solid-state transport has
been that, under a finite electric-field, the Fermi sea is Hˆ(t) = −γ (eiϕ(t)d†i+1di+H.c.)+ (cid:15)αc†iαciα
perturbatively shifted by a drift velocity, many calcula- (cid:88)i (cid:88)iα
tions performed under the DMFT framework have sug- −g (c†iαdi+H.c.), (1)
gested that the system approaches a steady-state with iα
(cid:88)
infinitely hot electron gas even for small field. With in-
clusion of proper dissipation mechanism, correct models with d†i as the (spinless) electron operator on the tight-
should expect the Boltzmann picture of displaced Fermi binding chain on site i, c†iα with the reservoir fermion
2
E It is important to note that the k-dependence enters the
problem as k+ϕ(t) for t > 0. This problem is simply a
� di di+1 resonant level model18 where the level is modulated si-
� main chain nusoidallyfort>0. Consideringthesetwoobservations,
g we can infer that different k only adds different phase to
k+ϕ(t) and steady-state properties such as the average
occupationnumbern (t)= d (t)c(t) areexpectedtobe
k †
(cid:104) (cid:105)
k-independent. Thisshowsthattheon-sitefermionther-
s
ain mostat problem completely obliterates the Fermi surface
ch at any non-zero field and, even in the small damping
r
oi limit, and does not restore the conventional picture of
v
r shifted Fermi surface due to an external field.
e
s
e
r
III. SOLUTION FOR OCCUPATION NUMBER
AND CURRENT
H(t) F=IG. 1�: �One-di(meei⌦nstidon†i+al1tdigih+t-bHind.cin.)g+lattice ✏o↵f co†ir↵bictai↵l di
underanelectricfieldE. Eachlatticesiteisconnectedtoan
identicalferXmiionicbathof{c }withthecoXni↵tinuumindexα The time-dependent Hamiltonian (4) can be exactly
iα
along t�hegreserv(ocir†i↵cdhaiin+dHire.cct.io)nw. ith ⌦ = eEa. smoelvthedodb19y. theWneonwerqiuteilibthrieumHaKmeilldtyosnhianGraesenHˆfu(ntc)tio=n
k
Xi↵ Hˆ +Vˆ(t) with the time-independent unperturbed part
0
The electric fisetaldtesiscoanbnseoctrebdedtoathseasittiemiew-idtehptehnedceonnttinpuhuamseinfdaecxtor Htˆo0 = Hˆk(0) and the time-dependent perturbation as
the electron hαtohpeapleoxinnpglgicei(attcechomnrpneseoecrrtvaivoliitrgyacouhfagtiehn)e..reHseerrevowirechdaoinnso,tbustpeecaicfhy Vˆ(t)=Hˆk(t)−Hˆk(0),
chains are assumed to have an identical dispersion re- Vˆ(t)= 2γ[cos(k+ϕ(t)) cos(k)]d d v(t)d d. (5)
† †
lation (cid:15) . Notice that the electric field is applied only − − ≡
α
on the tight-binding chain {d†i}. The coupling between Whentheperturbationisone-bodyondiscretestatesthe
the TB site and the reservoir is given by the identical lesser and greater part of the self-energy is zero, and the
tunneling parameter g. The Peierls phase ϕ(t) is given lesser d-Green function G< is expressed only in terms of
as the transient term, symbolically written as19
0, for t<0 G< =[I+GrV]G<[I+VGa] (6)
ϕ(t)= (2) 0
Ωt, for t>0
(cid:26) andtheretardedGreenfunctionGr isgivenbytheusual
Ω = eEa is the Bloch-oscillation frequency due to the
Dyson’s equation
electric field.
We note that the whole system has discrete trans- Gr =Gr+GrVGr, (7)
0 0
lational symmetry in the transport direction and the
Hamiltonian is readily block-diagonalized with respect wherethematrixproductmeansconvolution-integralsin
time.
to the wave-vector k as d†k = Nd−1 jeikRjd†j and First with the retarded functions, the non-interacting
c†kα = Nd−1 jeikRjc†jα (with(cid:113)lattice(cid:80)sites Rj = aj limit has the time-translational symmetry and
and the(cid:113)numbe(cid:80)r of sites along the TB chain Nd), Gr(t t) = iθ(t t) ∞ d(cid:15) Γ/π e i(cid:15)(t t(cid:48))
0 − (cid:48) − − (cid:48) (cid:15)2+Γ2 − −
Hˆ(t) = k (cid:34)−2γcos(k+ϕ(t))d†kdk+ α (cid:15)αc†kαckα = −iθ(t−t(cid:48))e(cid:90)−−i∞(cid:15)d(k)(t−t(cid:48))−Γ|t−t(cid:48)|, (8)
(cid:88) (cid:88)
whereweweuseaflat-bandDOSforthereservoirinthe
−g α (c†kαdk+H.c.)(cid:35). (3) iπngfi2nNit(e0-)baannddlitmhietdweinthsitthyeohfysbtraidteizsaotfiotnhberofeardmenioinngbΓat=h
(cid:88)
N(0) = δ((cid:15) ). Writing Gr(t,t) = Gr(t t)gr(t,t),
Here (cid:15)d(k) = 2γcos(k) is the tight-binding dispersion α α (cid:48) 0 − (cid:48) (cid:48)
− Eq. (7) becomes
at zero E-field. Then each k-sector can be treated and
(cid:80)
solved separately. So from now on, we suppress the k- t
subscriptuntilnecessarywiththefollowingHamiltonian, gr(t,t)=1 i dsv(s)gr(s,t), (9)
(cid:48) (cid:48)
−
(cid:90)t(cid:48)
Hˆ (t) = 2γcos(k+ϕ(t))d d+ (cid:15) c c
k − † α †α α which can be solved as
α
(cid:88)
t
−g (c†αd+H.c.). (4) gr(t,t(cid:48))=exp −i v(s)ds , (10)
(cid:88)α (cid:20) (cid:90)t(cid:48) (cid:21)
3
[ ,t] for easier analytic treatment. After an integral-
−∞
by-parts and some straightforward steps, we have
n (t) Γ 0
k nk(t) = dω (15)
v (t) π ×
1 k (cid:90)−∞
0 2
dse i(ω+iΓ)s i(2γ/Ω)sin(k+Ω(t+s)) .
− −
0.5 (cid:12)(cid:12)(cid:90)−∞ (cid:12)(cid:12)
(cid:12) (cid:12)
An identity(cid:12)for Bessel functions Jn(x) (cid:12)
0
∞
eixcosθ = inJ (x)einθ (16)
n
m=
-0.5 (cid:88)−∞
0 20 40 can be used to perform the integrals as
time (g -1)
n (t) = Γ Jn(2Ωγ)Jm(2Ωγ)ei(m−n)(k+Ωt)
k
FIG. 2: Expectation value of occupation number n (t) of π (m n)Ω+2iΓ ×
k nm − −
wave-vector k and the current J (t) for γ = 1, Ω = 0.5, (cid:88)
k 1 m2Ω2+Γ2
Γ=0.1 and at k =π/2+0.1. After the initial time of Γ−1, log +iχ (17)
transient behavior dies out and the expectation values reach 2 n2Ω2+Γ2 mn
(cid:20) (cid:21)
steady oscillation.
with
mΩ nΩ
and finally we have for the full retarded Green function χmn =π+tan−1 +tan−1 . (18)
Γ Γ
t
Gr(t,t(cid:48))= iθ(t t(cid:48))e−i(cid:15)d(k)(t−t(cid:48))−Γ|t−t(cid:48)|exp i v(s)ds .As pointed out earlier, it is important to note that the
− − (cid:20)− (cid:90)t(cid:48) (cid:21) k and t-dependence is only in the exponential factor as
(11) ei(m n)(k+Ωt). Therefore, taking a time-average over the
−
For the same-time argument for G< we have the period 2πΩ 1 gives k-independent results, which have
−
Dyson’s equation been confirmed in numerical calculations. For instance,
the DC limit of n (t) only involves the terms of m = n
G<(t,t)=G<(t,t) k
0 in Eq. (17), and we have
t
+ Gr(t,s)v(s)G<(s,t)+G<(t,s)v(s)Ga(s,t) ds
0 0 1 ∞ 2γ 2 1
(cid:90)0 n¯ = J = . (19)
t(cid:2) t (cid:3) k 2 m Ω 2
+ Gr(t,s)v(s)G<0(s,s(cid:48))v(s(cid:48))Ga(s(cid:48),t)dsds(cid:48).(12) m(cid:88)=−∞(cid:20) (cid:18) (cid:19)(cid:21)
(cid:90)0 (cid:90)0 For electronic models without thermostats, it has been
WesettheinitiallesserGreenfunctionwiththehalf-filled argued that the effective electron temperature quickly
reservoir at zero temperature as becomes infinity14 from k-independent occupation num-
ber n¯ . The same conclusion applies to this model with
0 Γ/π k
G<(t,t)=i dω e iω(t t(cid:48)). (13) on-site fermionic thermostats.
0 (cid:48) (cid:90)−∞ (ω−(cid:15)d(k))2+Γ2 − − Now we turn to the calculation of electric current,
Aftersomestraightforwardsteps,theoccupationnumber ∂(cid:15) (k+Ωt)
d
for the wave-vector k, nk(t)=−iG<(t,t), becomes Jk(t)= ∂k nk(t)=2γsin(k+Ωt)nk(t). (20)
0 Γ/π Due to the sine-function, the DC current has contribu-
n (t) = dω (14)
k (ω (cid:15) (k))2+Γ2 × tions only from m n = 1 in Eq. (17). After some
(cid:90)−∞ − d manipulations, we h−ave ±
t 2
1 i dsv(s)ei(ω−(cid:15)d(k)+iΓ)(t−s)−i(cid:82)stv(s(cid:48))ds(cid:48) .
− 2γΓ 2γ 2γ
(cid:12) (cid:90)0 (cid:12) J¯ = J J
Fig. 2 sho(cid:12)(cid:12)ws the above n (t) numerically evaluated fo(cid:12)(cid:12)r k π(Ω2+4Γ2) m m(cid:18)Ω(cid:19) m−1(cid:18)Ω(cid:19)×
(cid:12) k (cid:12) (cid:88)
γ = 1, Ω = 0.5, Γ = 0.1 and at k = π/2+0.1. Due to m2Ω2+Γ2
Γlog +Ωχ . (21)
theexponentialfactore−Γ(t−s), theintegralconvergesto (m 1)2Ω2+Γ2 m,m−1
a steady-state oscillation state after time t Γ 1 and (cid:20) − (cid:21)
−
≈
the transient behavior dies out. Therefore, for long-time As in the case for n (t), the DC limit of J (t) becomes
k k
behavior, thetime-integralrange[0,t]canbechangedto independent of k. The total current is shown in Figs. 3
4
1 0.35
0.9 "dc.dat"
0.4 � 0.3
/ 0.8
G = 0.1 E 0.7 0.25
G = 0.05 ,
d 0.6
0.3 el 0.2
fi 0.5
nt c- 0.4 0.15
e i
r r
C cur0.2 lect 00..23 0.1
D e 0.05
0.1
0.1 0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
damping,�/�
0
0 0.2 0.4 0.6 0.8 1
E-field
FIG.4: ContourplotofDCcurrentasafunctionofdamping
and electric field.
FIG.3: DCcurrentasafunctionofthedampingΓandelec-
tric field Ω = eEa. For small field, the current has a linear
IV. CONCLUSIONS
dependenceontheE-fieldshowinganOhm’slaw-likebehav-
ior. AsE increases,theBlochoscillationbehaviortakesover
and the DC current decreases. The dashed lines are the sim- Calculationsonelectrontransportwithfermionicther-
plified expression, Eq. (22). mostat confirm salient features of numerical results,
such as uniform occupation number for all k-vectors in
the Brillouin zone and the Ohmic-like limit of electric
and 4. Similar plot has been obtained in the interact- current. In particular, the simplified electric current,
ing model from numerical calculation of Hubbard model Eq. (22), recovered the semi-classical Boltzmann trans-
connected to fermion bath16. portresult. However,thenatureoftheoscillatorysteady-
state is quite different from the conventional solid-state
Itisinstructivetosimplifytheaboveexpressioninthe
transportpicture. Inthepresenceofastaticelectricfield,
limitofΩ,Γ γ wheretheDCcurrentisreducedtothe
(cid:28) thismodelpredictsthattheFermisurfacecompletelydis-
expression
appearsforanyfinitedampingΓandelectricfieldE after
the time-evolution of the interval ∆t (cid:38) max(Ω 1,Γ 1).
4γΓΩ − −
J¯ . (22) Theapplicabilityofthismodelmaybejustifiedforstud-
≈ π(Ω2+4Γ2) iesonshort-timedynamicsanddissipation. However,us-
ing this model to study the strongly correlated nonequi-
Detailed derivation is provided in Appendix. This ap- libriumsteady-staterequiresmuchcautionsincetheuni-
proximate expression is shown as dashed lines in Fig. 3. form on-site coupling to reservoirs does not seem to pre-
Despite that the formula was derived for Ω,Γ γ, it serve the Fermi surface.
(cid:28)
shows remarkable accuracy to the DC current for a wide
Wemayspeculatewhatcouldbetheminimalelements
range of Γ and E.
for a realistic model which may bridge the conventional
It is also interesting to note that a similar formula picture of Fermi surface shift and the non-perturbative
has been derived for a super-lattice system with Ohmic field-effect, hence eventually accessing the nonequilib-
scatteringwithinthesemi-classicalBoltzmanntransport rium strongly correlated steady-state. One may consider
equation20. Although the current has the same depen- momentum scattering (i) in phononic Ohmic bath (ii) in
dence on the damping and the electric field, it should be fermionic bath with site-disorder and (iii) Landau-Zener
emphasizedthatthetwomodelshavequitedifferentscat- tunneling in multi-band dispersion.
tering mechanism where in the Boltzmann approach20
the momentum relaxation is explicitly built-in while in
our case the lattice wave-vector scattering does not hap-
pen and a very different Fermi surface structure results. V. ACKNOWLEDGEMENTS
In the low-field limit, the current (22) recovers the form
of the Drude conductivity per electron,
Author is grateful for helpful discussions with Kwon
Park and Woo-Ram Lee. This work has been supported
J¯ γΩ Eτ, (23) bytheNationalScienceFoundationwiththeGrantnum-
≈ πΓ ∝ m ber DMR-0907150. Author also thanks the Asia-Pacific
∗
Center for Theoretical Physics at Postech University
with γ 1/m and Γ 1/τ. where part of the work has been completed.
∗
∼ ∼
5
Appendix A: Derivation of current at small field Using the asymptotic expression21 for x/Ω,γ/Ω ,
→∞
For both Γ,Ω γ, we expand Eq. (21) to the leading
(cid:28)
order of m as,
Ω/2γ (x <2γ)
J¯ 2γΓ J 2γ J 2γ JΩx(2Ωγ) 2 ∼(cid:40)0π√1−(x/2γ)2 (|x|>2γ) ,
k ≈ π(Ω2+4Γ2) m Ω m−1 Ω × (cid:2) (cid:3) | |
m (cid:18) (cid:19) (cid:18) (cid:19)
(cid:88)
2mΓΩ2 mΩ
+2Ωtan 1 . (A1)
−
m2Ω2+Γ2 Γ
(cid:20) (cid:21) the integral simplifies to
Rearranging the summation and using the identity
x[J (x)+J (x)]=2mJ (x), we write
m 1 m+1 m
−
2γ 1 x2Γ x
J¯k ≈ π(Ω22γ+Γ4Γ2) m Jm[Jm−1+Jm+1]× (cid:90)−2γ π 4γ2−x2 (cid:18)x2+Γ2 +xtan−1 Γ(cid:19)dx.
(cid:88)
mΓΩ2 mΩ (cid:112)
+Ωtan 1 (A2)
−
m2Ω2+Γ2 Γ
(cid:20) (cid:21)
In the limit Γ γ, the second term in the parenthesis
2ΓΩ mΓΩ mΩ
= π(Ω2+4Γ2) mΩJm2 m2Ω2+Γ2 +tan−1 Γ .dominates and(cid:28)we have
m (cid:18) (cid:19)
(cid:88)
ForΩ γ, wedefinex=mΩintheregimem=x/Ω
(cid:28) (cid:29)
1, the summation becomes 2ΓΩ 2γ xdx 4γΓΩ
J¯ = .
∞ xJΩx(2Ωγ)2 x2x+ΓΓ2 +tan−1 Γx dΩx. ≈ π(Ω2+4Γ2)(cid:90)0 (cid:112)4γ2−x2 π(Ω2+4Γ2)(A3)
(cid:90)−∞ (cid:18) (cid:19)
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