Crossover from bias-induced to field-induced breakdowns in one-dimensional band and Mott insulators attached to electrodes

1.5 MB·English

by  Yasuhiro Tanaka

Crossover from bias-induced to field-induced breakdowns in one-dimensional band
and Mott insulators attached to electrodes
Yasuhiro Tanaka∗ and Kenji Yonemitsu
Institute for Molecular Science, Okazaki 444-8585, Japan
Department of Functional Molecular Science, Graduate University for Advanced Studies, Okazaki 444-8585, Japan and
JST, CREST, Sanbancho, Chiyoda-ku, Tokyo 102-0075, Japan
(Dated: January 13, 2011)
Nonequilibrium states induced by an applied bias voltage (V) and the corresponding current-
voltage characteristics of one-dimensional models describing band and Mott insulators are investi-
1 gated theoretically by using nonequilibrium Green’s functions. We attach the models to metallic
1 electrodeswhoseeffectsareincorporatedintotheself-energy. Modulationoftheelectrondensityand
0 thescalarpotentialcomingfromtheadditionallong-rangeinteractionarecalculatedself-consistently
2 within the Hartree approximation. For both models of band and Mott insulators with length L ,
C
thebiasvoltageinducesabreakdownoftheinsulatingstate,whosethreshold showsacrossoverde-
n
a pendingon LC. It is determined basically by the bias Vth ∼∆ for LC smaller than the correlation
J lengthξ=W/∆whereW denotesthebandwidthand∆theenergygap. ForsystemswithLC ≫ξ,
2 thethresholdisgovernedbytheelectricfield,Vth/LC,whichisconsistentwithaLandau-Zener-type
1 breakdown, Vth/LC ∝∆2/W. We demonstrate that the spatial dependence of the scalar potential
iscruciallyimportantforthiscrossoverbyshowingthecasewithoutthescalar potential,wherethe
] breakdown occurs at Vth ∼∆ regardless of thelength LC.
l
e
PACSnumbers: 77.22.Jp,73.40.Rw,71.10.Fd,72.20.Ht
-
r
t
s
. I. INTRODUCTION portant results is that the dielectric breakdown of Mott
t
a insulators is interpreted as a many-body counterpart of
m the Landau-Zener (LZ) breakdown.23,24 This is known
Nonlinear conduction in correlated electron systems
- to describe the breakdown of band insulators where the
d such as one-dimensional Mott insulators1,2 and two-
one-particle picture holds. The threshold is given as
n dimensional charge-ordered materials3–8 has been of
E ∝∆2 with ∆ being an energygap. A mean-fieldap-
o great interest in the past few decades. They offer in- th
proach to electric-field-induced insulator-to-metal tran-
c
triguingsubjects ofnonequilibriumphysics incondensed
[ sitions by using Keldysh Green’s functions has been re-
matter and possibility for novel functions of electronic
ported in Ref. 19.
1 devices. For example, in a typical quasi-one-dimensional
v Mott insulator, Sr CuO ,1 a dielectric breakdown has The latter approach is to consider an insulator at-
2 3
3 been observed experimentally by applying a strong elec- tached to electrodes. Interface structures must be ex-
8
tric field. A dielectric breakdown has been reported plicitly takeninto accountin realelectric devices. Along
2
also in an organic spin-Peierls insulator, K-TCNQ2 thisline,Okamotoinvestigatedthecurrent-voltage(I-V)
2
. [TCNQ=tetracyanoquinodimethane]. For another or- characteristics of heterostructures that consist of Mott-
1
ganic compound, (BEDT-TTF)(F TCNQ) [BEDT- insulatorlayerssandwichedbymetallic leadsby combin-
0 2
TTF=bis(ethylenedithio)tetrathiafulvalene], which is a ing the dynamical mean-field theory with the Keldysh
1
1 quasi-one-dimensional Mott insulator, metal-insulator- Green’s function technique.20,21 Ajisaka et al. studied
: semiconductor field-effect transistor device structures theI-V characteristicsofanelectron-phononsystemcou-
v have been reported,9 where its field-effect characteristics pled with two reservoirs by a field-theoretical method.22
i
X are different from those of band insulators.10–12 In the studies listed above, Okamoto discussed the I-V
characteristics in the framework of the LZ breakdown,
r So far, theoretical investigations on nonlinear conduc-
a whileAjisakaet al. proposedadifferentmechanismwith
tion forinteracting electronsystems thatare initially in-
the threshold bias voltage V ∼ ∆. These results seem
sulating in their equilibrium states have been done by th
to be inconsistent with each other.
several authors.10–22 In general, these studies are classi-
fied into two approaches depending on whether an ex- When we consider a nanostructure in which some ma-
ternal force is written as an electric field13,14,16–19 or a terial is attached to left and right metallic electrodes,
bias voltage.10–12,15,20–22 The former approachis to con- thebiasvoltageV appliedtothematerialisdescribedas
sider electron systems without electrodes. The electric V =µ −µ with µ and µ being the chemicalpoten-
L R L R
field is usually applied with open boundary condition,14 tialsoftheleftandrightelectrodes,respectively. Herewe
or equivalently with periodic boundary condition by us- assumethatthework-functiondifferenceattheinterfaces
ing a time-dependent magnetic flux.13,16 In one dimen- is absent for simplicity. In this case, one might expect
sion, Oka and Aoki studied the Hubbard model under that the current flows when some energy levels of the
a strong electric field by the time-dependent density- material appear in the region between µ and µ , indi-
L R
matrix-renormalization-groupmethod.14Oneoftheirim- catingthethresholdgovernedbytheappliedbiasvoltage
2
V ∼∆. In fact, for the transport in a field-effect tran-
th
sistorwithasmallchannel,suchanexplanationhasbeen
used frequently.25 However, for a bulk insulator with an
energy gap ∆, this picture does not hold and should be
FIG. 1: (Color online) Schematic view of the model. A one-
replaced by the LZ mechanism where the threshold is
determined by the electric field, E ∝ ∆2. This con- dimensionalbandorMottinsulatorinthecentralpartiscon-
th nectedbyleftandrightelectrodes. Solid(open)symbolsrep-
sideration poses us a question about which parameter
resent the sites in thecentral part (electrodes).
determines the mechanism of the breakdown. In partic-
ular, we address the condition for the realization of the
LZ breakdown in a structure with electrodes. It is also
half filling:
important to know the relation between the approaches
using the structure with electrodes and those which do
LC−1
notincludethemexplicitly. Wepointoutthatthesizeof H = − [t+(−1)i−1δt)](c† c +h.c.)
aninsulatoraswellasthe potentialdistributioninsideit iσ i+1σ
Xi=1 Xσ
determine the nature of the breakdown.
1 1
In this paper, we study one-dimensional band and + U (ni↑− )(ni↓− )
2 2
Mott insulators attached to two electrodes (see Fig. Xi
1), using the nonequilibrium Green’s function approach + V (n −1)(n −1), (1)
ij i j
that has previously been used to discuss the suppres- X
hhijii
sion of rectification at metal-Mott-insulator interfaces.12
This approach more naturally describes nonequilibrium
where c† (c ) denotes the creation (annihilation) oper-
steady states than the approach based on the time- iσ iσ
ator for an electron with spin σ at the ith site, n =
dependent Schr¨odinger equation because a current os- iσ
c† c , and n = n +n . L is the total number of
cillation is inevitable in the latter owing to finite-size iσ iσ i i↑ i↓ C
sites in the central part. The parameter t denotes the
effects. The present method can be easily applied to
transferintegral,δt itsmodulation,andU the on-sitein-
higher-dimensional systems. Preliminary results for the
teraction. We use t as a unit of energy. For δt = 0 and
I-V characteristics of two-dimensional charge-ordered
U >0, the first and second terms in Eq. (1) become the
systems are reported in Ref. 26.
one-dimensionalrepulsive Hubbard model, whereas they
We show that the applied bias voltage V induces a
describeabandinsulatorforδt6=0andU =0. Thelong-
breakdownofbandandMottinsulatorsatzerotempera-
range Coulomb interaction term with V = V /|i− j|
ture. Forbothinsulators,thethresholdshowsacrossover ij p
is introduced because it is responsible for the potential
as a function of the size of the insulating regionLC. For modulation near the metal-insulator interfaces (i >∼ 1
is.yes.,tetmhescwhiatrhacLtCerissmticaldleercatyhalenntghthe coofrtrheelawtiaovnelefunngctthioξn, and i<∼LC). The Vij term is treated by the Hartree ap-
proximation,whichis equivalent to the introductionof a
intheinsulator,thebreakdowntakesplacewhenthebias
scalarpotentialthat satisfies the Poissonequation. Here
V exceeds the energy gap ∆. For L ≫ξ, it is governed
C hhijiimeansthesummationoverpairsoftheithandjth
by the electric field, V/L , which is consistent with the
C sites with i 6= j in the central part (1 ≤ i,j ≤ L ). We
LZtunnelingmechanism. Whetherthechargegapispro- C
briefly review our formulation12 below to treat steady
duced by the band structure or by the electron-electron
states with a finite voltage.
interaction is irrelevant to the crossover phenomenon.
Forthemetallicelectrodes,weconsidernoninteracting
Wewillfocusonthespatialmodulationofthewavefunc-
electrons. The effects of the left and right (α = L,R)
tions. For the crossoverbehaviorandthe deformationof
electrodes on the central part are then described by the
the wave functions, the spatial dependence of the scalar
retarded self-energies.27–29 For simplicity, we take the
potentialinsidethe bandorMottinsulatoris important.
wide-bandlimitsothattheself-energiesareindependent
This is demonstrated in the Appendix by showing the
of energy. In the present case, they become diagonal
casewithoutthescalarpotential,wherethebias-induced
matrices29
transitionoccursatV ∼∆regardlessofthe lengthL .
th C
i i
(Σr) =− (Γ ) =− γ δ δ , (2)
α ij 2 α ij 2 α iiα jiα
II. MODEL AND FORMULATION
where δ and δ are the Kronecker delta and i = 1
iiα jiα L
(i =L )denotesthesiteconnectedwiththeleft(right)
R C
We consider a one-dimensional insulator that is at- electrode. We consider the case of γ =γ . Within the
L R
tachedtosemi-infinitemetallicelectrodesontheleftand Hartree-Fock approximation for the on-site term in Eq.
rightsidesasshowninFig. 1. Theinsulatorisreferredto (1), the retarded Green’s function for spin σ is written
asthecentralpart,anddescribedbytheHubbardmodel as
for a Mott insulator and a tight-binding model with al-
ternating transfer integrals for a band insulator, both at [Gr(ǫ)−1] =ǫδ −(Hr ) , (3)
σ ij ij HFσ ij
3
with InordertoobtainδG<α(ǫ),wefirstdecomposethelesser
σ
self-energy in the wide-band limit29 as in Eq. (9):
(Hr ) =(H ) + (Σr) . (4)
HFσ ij HFσ ij α ij Σ<(ǫ) = i(Γ f (ǫ)+Γ f (ǫ))
α=XL,R σ L L R R
= Σ<eq(ǫ)+ δΣ<α(ǫ), (12)
Theoff-diagonalelementsof(H ) comefromthefirst σ σ
HFσ ij Xα
termofEq. (1)andthe diagonalelements arewrittenas
with
(H ) =ψ +Uhn −1/2i, (5)
HFσ ii i iσ¯ Σ<eq(ǫ)=i(Γ +Γ )f (ǫ), (13)
σ L R C
with σ¯ =−σ. Here, the scalar potential ψi is defined by and
the Hartree approximation to the long-range Coulomb
interaction as δΣ<σα(ǫ)=iΓα[fα(ǫ)−fC(ǫ)], (14)
where f (ǫ) = θ(µ −ǫ). Then, we employ the Keldysh
ψ = V (hn i−1)+ai+b, (6) α α
i ij j equation
Xj6=i
δG<α(ǫ)=Gr(ǫ)δΣ<α(ǫ)Ga(ǫ), (15)
σ σ σ σ
with a and b being constants. These constants are so
determined that ψ satisfies the boundary conditions: where Gaσ(ǫ) is the Hermitian conjugate of Grσ(ǫ). The
expressions for neq and δnα, with which the numerical
iσ iσ
V for i=1 calculationsarecarriedout,areobtainedbysubstituting
ψ = 2 (7)
i (cid:26)−V2 for i=LC, Eq. (8) into Eqs. (10) and (11). The results are
for the bias voltage V. When V is positive, the left neq = Re[uσ(i)]2 1 tan−1 2(µC −ǫσm) + 1 , (16)
electrodehasa higherpotentialfor the electronsandthe iσ Xm m hπ γmσ 2i
current flows from left to right.12 Here we assume that
and
the work-function difference is absent at the interfaces.
By diagonalizing the complex symmetric matrix HHrFσ, δnα = γα ∞ dǫ|[Gr(ǫ)] |2[f (ǫ)−f (ǫ)]
the retarded Green’s function is obtained as iσ 2π Z σ iiα α C
−∞
[Gr(ǫ)] = uσm(i)uσm(j), (8) = γα Im uσm(i)uσm(iα)uσn∗(i)unσ∗(iα)
σ ij Xm ǫ−Emσ 2π Xn,mn h ǫσm−ǫσn−iγmσ/2−iγnσ/2 io
2(µ −ǫσ) 2(µ −ǫσ)
where Eσ ≡ ǫσ − iγσ/2 (ǫσ and γσ are real) is the × tan−1 α m −tan−1 C m
m m m m m γσ γσ
eigenvalue of Hr and uσ(i) is the correspondingright h m m
HFσ m 2(µ −ǫσ) 2(µ −ǫσ)
eigenvector. + tan−1 α n −tan−1 C n
The electron density is calculated by decomposing it γnσ γnσ i
into the “equilibrium” and “nonequilibrium” parts12 as uσ(i)uσ(i )uσ∗(i)uσ∗(i )
+ Re m m α n n α
ǫσ −ǫσ−iγσ/2−iγσ/2
hn i=neq+ δnα. (9) h m n m n i
iσ iσ iσ 1 (µ −ǫσ)2+(γσ/2)2
Xα × ln α m m
2 (µ −ǫσ)2+(γσ/2)2
h C m m
The“equilibrium”partisdefinedbyintegratingthelocal 1 (µ −ǫσ)2+(γσ/2)2
density of states as − ln α n n . (17)
2 (µ −ǫσ)2+(γσ/2)2
C n n i
1 ∞
neq ≡− dǫ[Gr(ǫ)] f (ǫ), (10) In the above equations, we recover the electron density
iσ π Z σ ii C in the equilibrium state without the electrodes if γσ =
−∞ m
0, since the bracket in Eq. (16) is reduced to the step
wherefC(ǫ)=θ(µC−ǫ)istheFermidistributionfunction function and δnα =0.
iσ
with the chemical potential µC at the midpoint of the The currentfromthe leftelectrode isexpressedby the
right and left chemical potentials, µC = (µR +µL)/2. “nonequilibrium” part of the density as12,30
Since µ = V/2 and µ = −V/2, we have µ = 0.
L R C ∞ dǫ
For V > 0, where the left chemical potential is higher J = Tr[Γ Gr(ǫ)Γ Ga(ǫ)][f (ǫ)−f (ǫ)]
than the right, δnL (δnR) is interpreted as the inflow Z 2π L σ R σ L R
iσ iσ −∞ Xσ
(outflow). γ γ ∞
The “nonequilibrium” part of the density δnα is ob- = L R dǫ |[Gr(ǫ)] |2[f (ǫ)−f (ǫ)]
iσ 2π Z σ iLiR L R
tained from the “nonequilibrium” part of the lesser −∞ Xσ
Green’s function: = γ δnL −γ δnR , (18)
R iRσ L iLσ
∞ Xσ Xσ
δnα ≡ dǫ[δG<α(ǫ)] . (11)
iσ Z σ ii where we set e=¯h=1.
−∞
4
0.05 1.3
(a)
1.2
0.04
1.1
0.03
1
0.2
0.02 0.9 0.1
0
0.8 -0.1
0.01
-0.2
0 50 100 150 200
0.7
0 50 100 150 200
0
0 0.4 0.8 1.2 1.6 i
0.2
(b)
FIG.2: (Coloronline)I-V characteristics ofone-dimensional
0.1
band insulators for δt = 0.025 and δt = 0.05. The other
parameters are L = 200, U = 0, V = 0.1, and γ = γ =
C p L R
0.1. Resultsforδt=0arealsoshownwherewesetLC =200, 0
U =0, V =0, and γ =γ =0.1.
p L R
-0.1
III. RESULTS
-0.2
0 50 100 150 200
In this section, we showthe results ofI-V characteris-
i
tics,chargedensities,andthespatialdependenceofwave
functionsforbandandMottinsulators. Forbothmodels,
abreakdownofthe insulatingstate takesplacewhenthe FIG. 3: (Color online) (a) Electron density, hn i, “equilib-
i
biasV becomes sufficientlylarge. The thresholdshowsa rium”part,neq,and(b)“nonequilibrium”partsδnRandδnL
i i i
crossoverbehaviorasa functionofthe sizeofthe central for δt = 0.025, L = 200, U = 0, V = 0.1, γ = γ = 0.1,
C p L R
part L , which indicates the mechanism of the break- and V =0.3. The scalar potential ψ isshown in theinset of
C i
down changes according to L . The profile of the scalar (a).
C
potential ψ has crucial importance on the way of the
i
breakdown. This is demonstrated in the Appendix by
showing that, if ψ is artificially set to zero for all i, the
i tron density hn i = hn i+hn i and its “equilibrium”
i i↑ i↓
crossoverphenomenon disappears. part neq = neq + neq for V = 0.3 are shown. A 2k
i i↑ i↓ F
oscillation in the charge distribution is induced by the
boundaries.12Foralli,hn iisalmostunityasintheequi-
i
A. Band insulators librium case (hn i = neq = 1 for V = 0). The electron
i i
density hn i is basically unchanged by the bias voltage
i
Figure 2 shows the I-V characteristics for band insu- V when J is small. The scalar potential ψi has a lin-
lators with δt = 0.025 and 0.05. The other parameters ear dependence on i as shown in the inset of Fig. 3(a).
are L =200, U =0, V =0.1, and γ =γ =0.1. For This is because the long-range interaction term in Eq.
C p L R
comparison, we show the results for the regular trans- (6) is small for hnii ∼ 1, so that ψi is determined only
fer integrals (δt = 0) with L = 200, U = 0, V = 0, bytheboundaryconditions. The“equilibrium”partneq,
C p i
and γ = γ = 0.1. For δt = 0, the current J becomes on the other hand, deviates from unity near the left and
L R
nonzero for V 6=0 since the centralpartis metallic. The right electrodes, where the deviation is canceled by the
I-V curve has stepwise structures owing to the finite- “nonequilibrium” parts δnα, as shown in Fig. 3(b). The
i
size effect. For V = 0 and δt > 0, the central part is a quantity δnL have nonnegative values for all i because
i
band insulator with the energy gap ∆ = 4δt. Because electrons come in from the left electrode. Although δnL
i
of the gap, J is suppressed near V =0. The I-V curves is large near the left electrode, it decays as i increases.
for finite δt show more complex structures than that for On the other hand, δnR have nonpositive values for all i
i
δt = 0. Apart from the fine structures, J increases al- becauseelectronsgoouttotherightelectrode. Notethat
most linearly for large V, which indicates a breakdown δnRi =−δnLLC+1−i for γL =γR, no work-function differ-
of the band insulator. ences,andathalffilling.12 ThebehaviorsofδnL andδnR
i i
For finite voltages applied, the charge distributions in indicate that electrons and holes hardly penetrate into
resistive and conductive states for δt = 0.025 are shown the central part and the resistive state is maintained.
in Figs. 3 and 4, respectively. In Fig. 3(a), the elec- The current hardly flows through the central part, be-
5
1.3
(a)
0.04
1.2
1.1
0.03
1
0.6
0.4 0.02
0.9 0.2
0
-0.2
0.8 -0.4 0.01
-0.6
0 50 100 150 200
0.7
0 50 100 150 200
0
i 0 0.4 0.8 1.2 1.6
0.2
(b)
FIG.5: (Coloronline)I-V characteristicsofone-dimensional
0.1
bandinsulatorsforseveralvaluesofL withδt=0.025, U =
C
0, V = 0.1, and γ = γ = 0.1. The solid lines show the
p L R
0 function J =aVe−Vth/V which fits to theresults.
1 0.01
-0.1
0.8 0.008
-0.2
0 50 100 150 200
0.6 0.006
i
0.4 0.004
FIG. 4: (Color online) (a) Electron density, hn i, “equilib-
i
rium”part,neq,and(b)“nonequilibrium”partsδnR andδnL 0.2 0.002
i i i
for δt = 0.025, L = 200, U = 0, V = 0.1, γ = γ = 0.1,
C p L R
and V =1.0. Thescalar potentialψi is shown in theinset of 0 0
(a). 0 100 200 300 400
cause J in Eq. (18) is determined by the difference be-
tweenthedensitymodulationbytheleftelectrodeatthe FIG.6: (Coloronline)Dependenceofthethresholdbiasvolt-
right boundary, δnL , and that by the right electrode at age Vth and thethreshold electric field Eth on thesize of the
iR central part L . The other parameters are the same as in
the left boundary, δnR. Both terms δnL and δnR are C
iL iR iL Fig. 5. The error bars in thefittingare also shown.
vanishingly small, as shown in Fig. 3(b).
The charge distribution for V = 1.0, where the sys-
tem is conductive,is qualitatively different from that for and go out to the right electrode. A finite current flows
V = 0.3 as shown in Fig. 4. The spatial dependences through the central part: δnL and δnR are finite and
of neiq and hnii are nearly the same. They increase al- have the opposite signs. iR iL
most linearlyfrom left to rightexcept in the vicinities of Next, we discuss the breakdown mechanism of band
the electrodes,wheresomeoscillatorystructure appears. insulators. InFig. 5,weshowtheI-V curvesfordifferent
The distributionsofneiq andhniiareunderstoodbythat sizes of the central parts LC with δt = 0.025. To the
of ψi shown in the inset of Fig. 4(a). The electron den- numerical results, the function,
sity is higher (lower) on the right (left) half where ψ
i
is low (high). This behavior is caused by the electrons J =aVe−Vth/V, (19)
that move through the system in the conductive phase.
The profile of ψ shows almost a linear dependence on is well fitted, where a and V are parameters. This
i th
i although a small deviation from the linearity near the expression originates from the LZ tunneling mechanism
electrodes is visible in contrast to the resistive phase, throughwhichtheinsulatorbreaksdownwiththethresh-
which is because the charge redistribution (hn i 6= 1) is old voltage V .14 For V < V , the current J is ex-
i th th
easier in the conductive phase. As for the “nonequilib- ponentially suppressed due to the energy gap, while it
rium” parts of the density shown in Fig. 4(b), δnL have increases linearly for V > V . When the central part
i th
positive values for all i, while δnR are negative for all is large, the fitting works well as shown in Fig. 5, so
i
i because the electrons come in from the left electrode that the breakdown is consistent with the LZ tunneling
6
picture, although there exist fine structures in the I-V -4
1.2 10
characteristics which come from the discreteness of the
energy spectrum of the central part. As LC decreases, 1 V=0.6
the structure becomes more prominent. For L = 50,
C
forexample,adeviationfromthefittingcurveduetothe 0.8 ba
stepwisestructurebecomeslarge,whichindicatestheLZ V=0.4
mechanism is no longer applicable to small-LC systems. 0.6 c
Figure 6 shows V determined by fitting Eq. (19) to
th
the data for each LC, together with the corresponding 0.4 V=0.2
electric field E ≡ V /L . For large L , V is pro-
th th C C th
portional to LC, so that Eth becomes a constant. In 0.2
band insulators, the LZ breakdown is known to be in- V=0
duced by the applied electric field.23,24 Since the one- 0
particlepicture holds inbandinsulators,this breakdown -2 -1 0 1 2
can be analyzed as a usual interband tunneling problem
and the threshold electric field becomes E ∝∆2/W.31
th
The breakdown occurs when the energy gain by displac-
ing an electron with charge −e in an electric field E by FIG. 7: (Color online) |uσ(i )uσ(i )|2 with σ =↑ plotted
m R m L
the distance ξ = W/∆, eEξ, overcomes the energy gap against ǫ↑ for V = 0, 0.2, 0.4, and 0.6 in the case of δt =
m
∆. Here, W ≃ 4 is the bandwidth and ξ ∼ 40. We 0.025, L = 400, U = 0, V = 0.1, and γ = γ = 0.1. For
C p L R
haveobtainedthe thresholdE =0.0022,which is com- V =0.2,0.4,and0.6,thelinesareshiftedupwardby0.3,0.6,
th
parable with the value obtained by the LZ formula,23,24 and 0.9, respectively.
(∆/2)2/v = 0.00125 with v = 2. In short, the threshold
is governed by the electric field.
When the central part is small, the fitting to the I- well as the spatial modulation of the wave functions as
V curve becomes worse because the finite-size effect be- discussedbelow. ForsmallV,ψ hasalineardependence
i
comes severe. The LZ mechanism is not suitable for un- onithroughoutthecentralpartbecausetheelectronsare
derstanding this breakdown. In this case, another mech- localized, hn i ≃ 1, so that the effect of the long-range
i
anism, in which the threshold is determined by the bias interaction on ψ is small. When the system is conduc-
i
voltage,is more appropriatefor the following reason. As tive, the charge redistribution occurs near the interfaces
L decreases,iteventuallybecomessmallerthanthecor- where a small deviation from the linearity is seen in ψ .
C i
relation length ξ. The tunneling occurs when the en- This charge redistribution weakens the electric field on
ergy gain by displacing an electron by the distance L , the sites away from the interfaces in the central part.
C
eEL , overcomes the energy gap ∆. This indicates that For comparison,we show in the Appendix the I-V char-
C
the mechanism of the breakdown continuously changes acteristicsthatareobtainedbyartificiallysettingψ =0
i
aroundL ∼ξ as a function ofL . When V exceeds ∆, for all i. This corresponds to a hypothetical case where
C C
some energy levels of the central part come in between a sufficiently large charge redistribution occurs near the
µ and µ . For L < ξ, the wave functions do not interfaces. There is no electric field in the central part:
L R C
fully decay in the system: the electron injected from the a voltage drop occurs only at the interfaces. In this ex-
left electrodewithenergyhigherthan∆/2canreachthe tremecase,weobtainthethresholdbiasvoltageV ∼∆
th
right electrode through these levels so that the current regardless of the length L , which is in contrast to the
C
flows. This can be clearly seen in Fig. 5 for L = 50 LZ-typebehaviorforlargeL inFig. 6. Suchasituation
C C
where the gap is ∆ ∼ 0.2 due to the finite-size effect. never occurs in our calculations with ψ and for realis-
i
The I-V curve shows an abrupt increase at V ∼ ∆ be- tic parameters. As we will discuss in the next section
cause µ exceeds the lowest unoccupied energy level of andalsoin the Appendix, the effectof the spatialprofile
L
the centralpart. Eachstepwiseincreaseinthe I-V char- of ψ on the breakdown mechanism of Mott insulators
i
acteristics corresponds to the increase in the number of is basically the same as in the case of band insulators.
energy levels located between µ and µ . Thus, the model without ψ is inappropriate for realistic
L R i
We have numerically confirmed that the results are insulators.
qualitatively unchanged even if the long-range Coulomb In discussing the breakdown for L ≫ ξ, the spatial
C
interactionstrengthVpandthesystem-electrodecoupling dependences of the wave functions uσm(i) are crucial as
strengthγL(=γR)arevaried. Thus,thethresholdshows explained above. In Fig. 7, we show |u↑m(iR)u↑m(iL)|2
a crossover as a function of LC. When LC > ξ, the LZ- as a function of the real part of the one-particle energy
type breakdownoccurs and the thresholdis governedby ǫ↑ for several values of V in the case of L = 400.
m C
the electric field. For LC < ξ, on the other hand, the The behavior of this quantity for σ =↓ is the same. It
current flows when V exceeds the energy gap ∆. shows whether a given one-particle state contributes to
It is noted that the spatial dependence of ψ is impor- the current J. Note that J is obtained by integrating
i
tantfortherealizationofthefield-inducedbreakdownas |[Gr(ǫ)] |2 over µ < ǫ < µ . Since ǫ − ǫσ ap-
σ iLiR R L m
7
pears in the denominator for [Gr(ǫ)] [Eq. (8)], the 0.015
σ ij
one-particle state m with finite |uσ(i )uσ(i )|2 in the (a)
m R m L
interval −V/2 = µR < ǫσm < µL = V/2 gives a large 0.01
contribution to J. This quantity directly shows whether
the one-particle state m is localized or delocalized be- 0.005
cause it comes from the product of the amplitudes of
the wave function at the two interfaces i and i . If
L R 0
|uσ(i )uσ(i )|2 is large, the state has finite amplitudes
m R m L (b)
atbothsidesofthe centralpart,sothatitisdelocalized.
If |uσ(i )uσ(i )|2 is small, on the other hand, the state
m R m L 0.005
has a small amplitude at either of the interfaces.
For V = 0, |u↑ (i )u↑ (i )|2 shows two bands that
m R m L
correspond to the conduction and valence bands in the
0
band insulator. Because of the energy gap, no state
(c)
exists in the region −∆/2 < ǫσ < ∆/2 for V = 0
m
so that the current does not flow at least for V < ∆.
When V = 0.2, several states appear in the region 0.005
−∆/2<ǫ↑ <∆/2, correspondingto the leakage of one-
m
particle states from the electrodes to the central part.
However, these states do not contribute to the current 0
because their |u↑ (i )u↑ (i )|2 are vanishingly small as 0 100 200 300 400
m R m L i
shown in Fig. 7. Note that the line for each V 6= 0
is shifted upward by 3V. For V = 0.4, the number of
2
states around ǫ↑ = 0 with small |u↑ (i )u↑ (i )|2 in- FIG. 8: (Color online) Spatial dependence of one-particle
m m R m L
creases. The energy range where these localized states states|u↑m(i)|2for(a)m=201,(b)m=191,and(c)m=175
appear becomes wider as V increases. Consequently,the inthecaseofV =0.4. Theotherparametersarethesameas
in Fig. 7. The corresponding ǫ↑ are indicated by the arrows
delocalized states that contribute to the current depart m
in Fig. 7.
from the region µ < ǫ < µ . Therefore, the current
R L
does not flow even if V barely exceeds the gap. As we
showintheAppendix,thelocalizedstatesdonotappear
|u↑ (i)|2 on the (i,m) plane for V = 0.4 in Fig. 9. For
idfewpeensdeetnψcei =of0ψfoirntaollaic.cIotuinstctroucoibatlationttahkeemthoedusplaattiioanl Vm= 0.4, the states 190 <∼ m <∼ 200 are localized near
i
the left electrode, and the states below are delocalized.
of the wave functions.
At the bottom of the lower band, the states are local-
Figure 8 shows the spatial dependences of the squares
ized near the right electrode because of the low scalar
of the absolute values of the wave functions |u↑ (i)|2 for
m potential near the right electrode.
V = 0.4 and several m whose ǫ↑ are located at the po-
m
sitions indicated by the arrows in Fig. 7. Here m is so
labeled that ǫσ <ǫσ for 1≤m<m′ ≤L =400. Fig-
m m′ C
ure 8(a) shows the one-particle state in the lower band B. Mott insulators
with ǫ↑ = 0.0348 (m = 201) which is inside the gap for
m
V =0. This state is localized on the left half of the cen- In this section, we consider the case where the cen-
tralpart. Thereasonisasfollows. Thescalarpotentialis tral part is described by the Hubbard model. Figure
high(low) nearthe left (right) electrode. Within eachof 10 shows the I-V characteristics for δt = 0, L = 200,
C
theconductionandvalencebands,thestatewhoseweight U = 1.5, V = 0.1, and γ = γ = 0.1. When V = 0,
p L R
islargeneartheleft(right)electrodehasahigher(lower) thesystemisanantiferromagneticinsulatorowingtothe
energy than others. In the present case, the indexes m Hartree-Fock approximation. The energy gap ∆ is then
forthevalencebandarem=206,203,201,199,...,3,2, 0.25. InthepreviousstudiesontheI-V characteristicsof
1,while thosefortheconductionbandarem=195,198, metal-Mott-insulatorinterfaces,10,11 the currentwascal-
200, 202,..., 398, 399, 400. As m is lowered, the wave culated by solving the time-dependent Scho¨dinger equa-
functionoftheone-particlestateisgenerallyextendedto tion. It is argued that the results obtained by the time-
a wider region and its largestamplitude is shifted to the dependent Hartree-Fock approximation for the electron-
right, as shown in Fig. 8(b) for the case of ǫ↑ =−0.157 electron interaction are consistent with those obtained
m
(m=191). Asmisloweredfurther,e.g.,forǫ↑m =−0.400 byexactmany-electronwavefunctionsonsmallsystems.
(m =175) in Fig. 8(c), the one-particle state is delocal- For example, the suppression of rectification at metal-
izedtoreachtherightelectrode. Then,itswavefunction Mott-insulatorinterfacesisdescribedbybothmethods.11
has large amplitudes near both electrodes. Althoughthepresenttime-independentHartree-Fockap-
In order to overview the behaviors of the one-particle proximation is worse, we expect the present approach
states, we show the contour map of one-particle states captures the essential features of nonequilibrium steady
8
1.3
(a)
1.2
400
1.1
0.06
1
0.05 0.3
300
0.04 0.2
0.9 0.1
0.03 0
0.02 0.8 -0.1
m 200 -0.2
0.01 -0.3
0 50 100 150 200
0 0.7
0 50 100 150 200
100 i
0.2
(b)
0
0 100 200 300 400 0.1
i
0
-0.1
FIG. 9: (Color online) Contour map of one-particle states
|u↑ (i)|2 in (i,m) plane for V = 0.4. The other parameters
m
are thesame as in Fig. 7.
-0.2
0 50 100 150 200
0.05 i
0.04
FIG. 11: (Color online) (a) Electron density, hn i, “equilib-
i
rium”part,neq,and(b)“nonequilibrium”partsδnRandδnL
i i i
0.03 for δt=0, L =200, U =1.5, V =0.1, γ =γ =0.1, and
C p L R
V =0.5. Thescalar potential ψ is shown in theinset of (a).
i
0.02
0.01 cays as i increases (decreases) [Fig. 11(b)]. Electrons
and holes do not penetrate into the central part so that
0 the current does not flow.
0 0.4 0.8 1.2 1.6 For V = 1.0 > V , the spatial dependences of neq
th i
and hn i are shown in Fig. 12(a). They increase almost
i
linearly from left to right. This reflects the profile of
the scalar potential ψ that is higher (lower) on the left
FIG.10: (Coloronline)I-V characteristicsofone-dimensional i
(right) half. The “nonequilibrium” parts of the densi-
Hubbardmodel forδt=0, L =200, U =1.5, V =0.1, and
C p ties, δnL and δnR are extended over the whole system
γL=γR =0.1. i i
with small spatial dependence [Fig. 12(b)],12 which is in
contrast to the resistive phase. Since δnL and δnR are
iR iL
statesunder the bias voltage. As we increaseV, the cur- finite with opposite signs, a finite current flows through
rentbeginstoflowatV ∼0.8. Thebreakdownbecomes the central part.
th
a first-order transition due to the Hartree-Fock approxi- InFig. 13, weshowthe thresholdbiasvoltageV and
th
mation,whichisincontrasttothecaseofbandinsulators the corresponding electric field E = V /L for the
th th C
in the previous section. first-ordertransitionas afunction ofL . When the cen-
C
The charge distributions in resistive and conductive tral part is small, i.e., LC <∼50, Vth is almost a constant
phases at finite V are shown in Figs. 11 and 12, respec- near the energy gap ∆ ≃ 0.25. For small L , the elec-
C
tively. Theiroverallfeatures aresimilarto thoseinband tron injected from the left electrode with energy higher
insulators. For all i and V < V , the electron density than∆/2 canreachthe rightelectrode since the correla-
th
hn iisalmostunity,whichisbasicallythe sameasinthe tion length ξ = W/∆ ≃ 16 is comparable to L . Thus,
i C
equilibrium case (hn i = neq = 1 for V = 0). The scalar the threshold for L <ξ is determined by the bias volt-
i i C
potentialψ hasalineardependenceoniasshowninthe age. For L ≫ξ, on the other hand, V is proportional
i C th
inset of Fig. 11(a). As for the “nonequilibrium” parts, to L , so that the threshold is governed by the electric
C
δnL (δnR) is large near the left (right) electrode and de- field. In recent theoretical studies,14 Oka and Aoki have
i i
9
1.3 showninFig. 11(a),whichmeansthattheelectronsfeel
(a) a uniform electric field in the central part. Therefore,
1.2
our model describes the LZ breakdown as in the open
1.1 Hubbard chain as long as the electrodes do not affect
the nature of the breakdown for L ≫ ξ. In fact, the
C
1 0.6 threshold Eth is about 0.0033, which is comparable to
0.4 the LZ value,14 (∆/2)2/v = 0.0078, with v = 2. Thus,
0.9 0.2
0 the thresholdshowsa crossoveras afunctionofL asin
C
-0.2
0.8 -0.4 the case of band insulators.
-0.6
0 50 100 150 200 The LZ breakdown is explained as before by compar-
0.7
0 50 100 150 200 ing the charge gap ∆ and the work which is done by the
electric field on an electron moving over the correlation
i
length ξ. If the work eEξ exceeds ∆, the electron in
0.2 the lower band may go over to the upper band so that
(b) the current flows. According to the results by Oka and
Aoki,14 this consideration is applicable to Mott insula-
0.1
tors where the correlation effects are important. Thus,
we expect that the results obtained by the Hartree-Fock
0 approximation are qualitatively unchanged even if we
takeaccountof the electroncorrelation. Itis wellknown
-0.1 that the Hartree-Fock theory overestimates the charge
gap ∆. It predicts the antiferromagnetic spin ordering
which is actually destroyed if quantum fluctuations are
-0.2
appropriately taken into account. However, the overes-
0 50 100 150 200
timated ∆ will alter the threshold only quantitatively.
i
We also expect that the spin ordering does not essen-
tiallyaffectthebreakdownitselfsinceonlythechargede-
FIG. 12: (Color online) (a) Electron density, hn i, “equilib- greesoffreedomarerelevanttothemechanism. Wenote
i
rium”part,neq,and(b)“nonequilibrium”partsδnR andδnL thatthetime-dependentdensity-matrix-renormalization-
i i i
for δt=0, LC =200, U =1.5, Vp =0.1, γL =γR =0.1, and groupmethod has been applied to a Mott insulator with
V =1.0. The scalar potential ψi is shown in the inset of (a). electrodes very recently,32 and that the I-V characteris-
ticshavebeenconsistentlyexplainedbytheLZtunneling
1.6 0.016 mechanism. However,itisalsoshownthatphysicalquan-
titiessuchasthespinstructurefactorandthedoubleoc-
cupancy do notreacha stationarystate inthe accessible
1.2 0.012
time window. Therefore,a directdescriptionofnonequi-
libriumsteadystatesasinthepresentstudyisimportant
0.8 0.008 to deepen our understanding of the breakdown.
Since the crossoverbehavioris obtainedforbothband
and Mott insulators, the electron-electron interaction is
0.4 0.004
notresponsible for the phenomenon. We emphasize that
the spatial profile of ψ is important for the realization
i
0 0 of the LZ breakdown. In the Appendix, this is demon-
0 100 200 300 400
strated for the Mott insulator by showing the I-V char-
acteristicsthatareobtainedbyartificiallysettingψ =0
i
for all i. In this case, no electric field exists in the cen-
tral part so that the modification of the wave functions
FIG. 13: (Color online) Dependence of the threshold bias
does not occur. The I-V curves do not show any L
voltage Vth and the threshold electric field Eth on the size of C
dependence apart from the fine structures coming from
the central part L . The other parameters are the same as
C
the discreteness of the energy spectrum. We obtain the
in Fig. 10.
threshold bias voltage V ∼ ∆ regardless of the length
th
L as in band insulators.
C
proposedthat the LZ breakdownoccurs also in Mott in- In Fig. 14, we show |u↑ (i )u↑ (i )|2 as a function
m R m L
sulatorsby applyingthe time-dependent density-matrix- of the real part of the one-particle energy ǫ↑ for sev-
m
renormalization-group method to the one-dimensional eral values of V, where each line for V 6= 0 is shifted
Hubbard modelunder anelectric fieldwith openbound- upward by V. The behavior of this quantity for σ =↓
ary condition. In our calculations, the scalar potential is the same. For V = 0, no state exists in the re-
ψ is linearly increasing with i in the resistive phase as gion −∆/2 < ǫσ < ∆/2 since the energy gap opens.
i m
10
c
-4 (a)
2 10 0.04
V=0
V=0.3
1.6
V=0.6 0.02
V=0.9
1.2
a 0
(b)
0.8
0.01
0.4
b
0
-2 -1 0 1 2 0
(c)
0.01
FIG. 14: (Color online) |uσ(i )uσ(i )|2 with σ =↑ plotted
m R m L
against ǫ↑ for V =0, 0.3, 0.6, and 0.9 in the case of δt=0,
m
L = 200, U = 1.5, V = 0.1, and γ = γ = 0.1. For
C p L R 0
V =0.3,0.6,and0.9,thelinesareshiftedupwardby0.3,0.6, 0 50 100 150 200
and 0.9, respectively. i
FIG. 15: (Color online) Spatial dependence of one-particle
|u↑m(iR)u↑m(iL)|2 shows two bands that correspond to states |u↑m(i)|2 for (a) m=100, (b) m=90, and (c) m=85
the upper and lower Hubbard bands. When V = 0.3, inthecaseofV =0.6. Theotherparametersarethesameas
inFig. 14. Thecorrespondingǫ↑ areindicatedbythearrows
one-particle states leak from the electrodes to the cen- m
in Fig. 14.
tral part so that several states appear in the region
−∆/2 < ǫ↑ < ∆/2. As in band insulators, these states
m
do not have any contributions to the current because
their |u↑m(iR)u↑m(iL)|2 are vanishingly small. As we in- resistive phase are similar to those in band insulators.
crease V further, e.g., V = 0.6, the states with van-
ishingly small |u↑ (i )u↑ (i )|2 appear in a wider range
m R m L
around ǫ↑ = 0. The number of these localized states
m
also increases. The appearance of the localized states
IV. SUMMARY
keeps the central part resistive until V/L reaches the
C
threshold electric field E even if V > ∆ holds. When
th
the system is conductive (V >Vth), the upper and lower We have investigated the I-V characteristics of the
Hubbard bands are merged into a single metallic band. one-dimensional band and Mott insulators attached to
In this case, all the states around ǫ↑m ∼0 are delocalized electrodes. Atightbindingmodelwithalternatingtrans-
and contribute to the current. fer integrals for the band insulator and the Hubbard
Figure 15 shows |u↑ (i)|2 as a function of i for V = model for the Mott insulator are studied by using the
m
0.6. Here m is chosen at 100, 90, and 85, whose ǫ↑ nonequilibrium Green’s function method. The applied
m
are located at the positions indicated by the arrows in bias voltage induces a breakdown of the insulating state
Fig. 14. In Fig. 15(a), we show the wave function with to convert into a conductive state for both models. The
ǫ↑ =−0.00395 (m =100), which is localized on the left threshold shows a crossover as a function of the size L
m C
half of the central part. This state belongs to the lower of the insulators. For LC <∼ ξ = W/∆, the breakdown
Hubbard band. Since the scalar potential is high (low) occurs at V ∼ ∆ so that the threshold is governed by
th
near the left (right) electrode, the state is located near the bias voltage. For L ≫ ξ, the electric field deter-
C
the top of the lower Hubbard band. The one-particle mines the threshold, V /L ∝ ∆2/W, which is con-
th C
states in the band gradually lose their localized nature sistent with the LZ breakdown reported previously.13,14
as m is lowered. This can be seen in Fig. 15(b) for Since the crossover is obtained for both band and Mott
the case of ǫ↑ = −0.361 (m = 90), where its largest insulators,theelectron-electroninteractionisnotrespon-
m
amplitudeisshiftedtotherightcomparedtothatofm= sibleforthephenomenon. Theprofileofthescalarpoten-
100. Figure 15(c) shows the one-particle state for ǫ↑ = tial ψ , whichis linearly increasingwith i in the resistive
m i
−0.497 (m = 85), which is completely delocalized. Its phase so that the electrons in the central part feel an
wavefunctionhaslargeamplitudesnearbothelectrodes. almostuniformelectricfield,isimportantfortherealiza-
The spatial dependences of the one-particle states in the tion of the LZ breakdown and the crossoverbehavior.