Hidden Symmetries of Electronic Transport in a Disordered One-Dimensional Lattice

0.23 MB·English

by  Wonkee Kim

Hidden Symmetries of Electronic Transport in a Disordered One-Dimensional Lattice
Wonkee Kim1, L. Covaci1, and F. Marsiglio1−3
1Department of Physics, University of Alberta, Edmonton, Alberta, Canada, T6G 2J1
2DPMC, Universit´e de Gen`eve, 24 Quai Ernest-Ansermet, CH-1211 Gen`eve 4, Switzerland,
3National Institute for Nanotechnology, National Research Council of Canada, Edmonton, Alberta, Canada, T6G 2V4
(Dated: February 6, 2008)
6
Correlated, or extended, impurities play an important role in the transport properties of dirty
0
metals. Here, we examine, in the framework of a tight-binding lattice, the transmission of a single
0
electron through an array of correlated impurities. In particular we show that particles transmit
2
through an impurity array in identical fashion, regardless of the direction of transversal. The

n demonstration of this fact is straightforward in the continuum limit, but requires a detailed proof
a forthediscretelattice. Wealsobrieflydemonstrateanddiscussthetimeevolutionofthesescattering
J states, to delineate regions (in time and space) where the aforementioned symmetry is violated.

5
1 PACSnumbers: 72.10.Bg,72.10.Fk

]
r I. INTRODUCTION Fig.2 ( Kim et al )
e
h
1
t
o In a one-dimensional lattice, correlated impurities or
. defectsmaygiverisetoextendedelectronicstatesforpar- 1
t
a ticularenergylevels.1,2,3Variousdisorderedsystemswith 0.8
m
correlation have been investigated, such as the random
- dimer,1,3 the random trimer,4 non-symmetric dimers,5
d the randomtrimer-dimer,6 the Thue-Morse lattice,7 and 0.6
n the random polymer chain.8 A built-in internal struc- 2T| 50
o |
c ture of the impurity configurationis important1,3 evenif 0.4
[ aninternalsymmetryisnotnecessary.5 Inthe contextof 500

electronictransport,thetransmissionresonanceforacer-
1 0.2
tain energy value implies the disordered system behaves
v
3 like an ordered lattice for the corresponding electronic
2 state,whichisextendedthroughouttheentirelattice. In 0
-2 -1 0 1 2
3 this respect, it is possible to understand the increase in E
1 the conductivity for a conducting polymer2.
0
One may, in fact, envision many other impurity con- FIG.1: (Coloronline)Transmission probabilityasafunction
6
figurations that give rise to extended states for various of E for randomly distributed 50 impurity dimers(blue) and
0
energy values. For example, let us consider a pair of im- 500 dimers (green) in comparison with a single dimer (red).
/
t purities with a potential V with respect to the lattice Theinnerspacingofapairnd =3andthepotentialV =1/2
a
m potential (taken to be zero). We assume that the spac- while the critical value V0 = 2/3. The transmission proba-
bility is calculated as an ensemble average over 50 impurity
ing between the two impurities is always n a, where a
- d configurations.
d is the lattice constant and n is a positive integer. This
d
n is then a ‘dimer’, whose length can be any value. If we
o adopt a tight-binding model, so that energies are given
c
: by E = −2t0cos(ka), where t0 is the nearest-neighbor case,considerationofone ‘dimer’helps us to understand
v hopping amplitude, then one can obtain the transmis- electronic transport in a lattice with many ‘dimers’, fol-
Xi sion probability |T|2 for this state. It is lowing the analysis of Ref.1. In Fig. 1, we show |T|2
as a function of E for randomly distributed 50 impurity
r sin2(k)
a |T|2 = . (1) dimers(blue)and500dimers(green)incomparisonwith
2
sin2(k)+V2 cos(knd)+ V2 sisni(nk(nk)d) a single dimer (red). The inner spacing of a pair nd =3
h i and the potential V = 1/2. Note that in this instance
where t and a have been set to unity for simplicity. V < V = 2/3. The transmission probability is calcu-
0 c
Inspection of Eq. (1) reveals a critical strength of the lated to be an ensemble average over 50 impurity con-
impurity potential V /t = ±2/n for repulsive and at- figurations. Remarkably, the transmission remains unity
c 0 d
tractive interactions,respectively, whichdetermines how at the three energies for which a single ‘dimer’ has unit
manystatesareextended,i.e. haveunit transmission. A transmission, and the ’width’ of the resonance narrows
graphicalconstruction readily shows that, for V >2/n , as the number of impurities increases.
d
therearen −1extendedstateswhileforV ≤2/n ,there Thisspecificexampleisgivenbywayofintroductionto
d d
aren extendedstates. Asinthenearest-neighbordimer more general considerations of impurity potentials on a
d
2
lattice. Inthispaperwewillshowthereexisthiddensym- where the hopping amplitude t will be set to be unity,
0
metries underlying the physics of electronic transport in c+ creates an electron at a site i, and I represents an
i
aone-dimensionallatticewithanarbitraryimpuritycon- impurity configuration. The Schr¨odinger equation is
figuration, using the transfer matrix formalism. We will H|ψi=E|ψi with |ψi= ψ c+|0i; this becomes
j j j
also demonstrate the time evolution of a wave packet by P
directdiagonalizationoftheHamiltoniansincethetrans- −[ψ +ψ ]+U ψ =Eψj (3)
j+1 j−1 j j
fermatrixformalismdoesnotproducethe dynamicsofa
wavefunction. While the dynamics canbe insightful, we where ψj is an amplitude to find an electron at site j.
arealsohopefulthatpresent-daytechnologycanspatially In order to use the transfer matrix formalism we write
and temporally resolve some of the interesting dynamics Eq. (3) in a matrix form as follows:
that result from our (and future) work.
ψ U −E −1 ψ ψ
Perhaps the most general correlated impurity config- j+1 = i j ≡M j .
uration is the following. Suppose there are N impuri- (cid:18) ψj (cid:19) (cid:18) 1 0 (cid:19)(cid:18)ψj−1 (cid:19) j(cid:18)ψj−1 (cid:19)
ties A A ···A embedded at the sites 1 throughN in a (4)
1 2 N
host lattice. The corresponding impurity potentials are Note that Mi is a unimodular matrix, i.e. det(Mi) = 1.
given by (U1, U2, ···, UN), where Ui (i = 1,···,N) ThewavefunctionsψL (fori<1)andψR (fori>N)are
can be positive, zero, or negative. Our main result is ψL =eikxi+Re−ikxi andψR =T eikxi,wherexi =a·i.
thatthe transmissionamplitudeT forthis impuritycon- Usingthe transfermatrixformalism,onecanexpressthe
figuration is identical to T for its reverse configuration; coefficients R and T in terms of k, Ui, and E as follows:
namely, T(U , U , ···, U ) = T(U , U , ···, U ).
1 2 N N N−1 1
T 1+R
Giventhisthenobviouslytheequalityholdsforthetrans- =P , (5)
mission |T|2 and reflection |R|2 probabilities. In other (cid:18)iT (cid:19) (cid:18)i(1−R)(cid:19)
words, |T|2 and |R|2 are independent of the direction of
the incoming wave. This symmetry is striking because where P = S−1MS with S = cos(k) sin(k) , and
the impurity configuration is assumed arbitrary without (cid:18) 1 0 (cid:19)
possessing any parity. For example, it has been shown M = MNMN−1···M1. It is misleading to express M =
that the chains of ABBABAAB··· and BAABABBA··· ΠNi=1Mi because Mi and Mj (i 6= j) do not commute
contribute identically for electronic transport7. This with each other. Solving Eq. (5), one can obtain10
is a simple example of the general symmetry we de-
2i
scribe. Note that an exchange of any two impurity T = (6)
i(P +P )+P −P
potentials does not show this symmetry; for example, 11 22 12 21
T(U1, U2, U3, U4)6=T(U2, U1, U3, U4)9. R = P12+P21−i(P11−P22) . (7)
We should emphasize now that this symmetry does i(P +P )+P −P
11 22 12 21
not hold in the impurity region (as perhaps one would
expect), but this can only be determined by solv- Consequently, the transmission probability |T|2 =
ing the time-dependent Schr¨odinger equation (see be- 4 tr(PP˜)+2 and the reflection probability |R|2 =
low). We also explain another symmetry associated (cid:14)h i
with a particular momentum k = π/(2a), where a tr(PP˜)−2 tr(PP˜)+2 , where tr(P) means the
is the lattice constant: |T(U1, U2, ···, UN)|2 = thrace of P,ia(cid:14)ndh P˜ is the itranspose of P. Note that
|T(−U , −U , ···, −U )|2. This equality indicates a
1 2 N Eq. (1) is readily obtained with this formalism by us-
potential barrier or well produces identical transmission ing M′ =M Mnd−1M , where M =M (U =V) and
and reflection probabilities for the electronic state with V 0 V V 1 1
M = M (U = 0), and n is the inner spacing of the
k = π/(2a). This is unrelated to the other remarkable 0 1 1 d
impurity pair.
featureofwavepacketsonaone-dimensionallatticewith
One of the symmetries we introduced earlier is
nearest-neighbor hopping and k = π/(2a): they do not
T(U , U ,···, U ) = T(U , U ,···, U ). In or-
diffuse asa functionoftime, becausethey lackgroupve- 1 2 N N N−1 1
der to show this equality, let us introduce Q= S−1M′S
locity dispersion. Indeed, one can show that no matter
where M′ = M M ···M . Notice the order of the ma-
what the range of the hopping is, there will be a wave 1 2 N
trix multiplication because M and M (i 6= j) do not
vectorforwhichthewavepacketretainsitsinitialwidth. i j
commute witheachother. Since the desiredequalityim-
plies that T[P] = T[Q], we need to show P +P =
11 22
Q +Q andP −P =Q −Q . Ontheotherhand,
11 22 12 21 12 21
II. FORMALISM
from the definitions of P = S−1MS and Q = S−1M′S,
this equality in turn implies that M = M′ , M =
11 11 22
We start with a tight-binding Hamiltonian for a one- M′ , and M −M =M′ −M′ . Note, however,that
22 12 21 12 21
dimensional lattice with N impurities: in general M 6=M′, as we will show later.
For simplicity, we define U = U − E. We also in-
i i
H =−t c+c +c+ c + U c+c (2) 1 0
0 i i+1 i+1 i i i i troduce 2×2 matrices α and β such as α =
Xi (cid:2) (cid:3) Xi∈I (cid:18)0 0(cid:19)
3
0 −1 M =M M ···M , we obtain
and β = to have M = U α+β. Note that N N−1 1
(cid:18)1 0 (cid:19) i i
α2 = α, β2 = −1, αβ +βα = β, and αβα = 0. Since
M = (U α+β)(U α+β)···(U α+β)
N N−1 1
N N
= βN + (U U ···U )βN−jnαβjn−jn−1−1α···αβj2−j1−1αβj1−1 , (8)
j1 j2 jn
Xn {Xjn}
where {j } means j ,j ,···,j = 1,2,,···,N with j < j < ··· < j . Since αβα = 0 and β2 = −1, non-vanishing
n 1 2 n 1 2 n
terms should have j −j −1=even, where l=2,3,···n for a given n. Therefore, we obtain
l l−1
N N
M =βN + (Uj1Uj2···Ujn)(−1)21(jn−j1−n+1)βN−jnαβj1−1 , (9)
Xn {{Xjn}}
where{{j }}means{j }withj −j −1=even. Since As we mentioned, the equality of the transmission am-
n n l l−1
1 0 0 −1 0 0 plitude, T[P] = T[Q], is associated with some particu-
α = , αβ = , βα = , and
(cid:18)0 0(cid:19) (cid:18)0 0 (cid:19) (cid:18)1 0(cid:19) lar relations among the components of the two matrices
0 0 M =M1M2···MN andM′ =MNMN−1···M1. Follow-
βαβ = (cid:18)0 −1(cid:19), we use these as the basis matrices to ing the same way as for M, we expand M′ in terms of
expand M: Ui (i=1,2,···,N), α, and β to obtain
c −c
M =c α+c αβ+c βα+c βαβ = 1 2 . (10)
1 2 3 4 (cid:18)c3 −c4 (cid:19)
M′ = (U α+β)(U α+β)···(U α+β)
1 2 N
N N
= βN + (U U ···U )βj1−1αβj2−j1−1α···αβjn−jn−1−1αβN−jn . (11)
j1 j2 jn
Xn {Xjn}
It is also true for M′ that non-vanishing terms have j −j −1=even because αβα=0. Therefore,
l l−1
N N
M′ =βN + (Uj1Uj2···Ujn)(−1)12(jn−j1−n+1)βj1−1αβN−jn . (12)
Xn {{Xjn}}
Expanding M′, again, in terms of the basis matrices, we To obtain i) c of M, we should have j −1= even and
1 1
know N −j = even. Similarly, ii) for c , j −1 = odd and
n 2 1
N −j = even, iii) for c , j −1 = even and N −j =
c′ −c′ n 3 1 n
M′ =c′1α+c′2αβ+c′3βα+c′4βαβ =(cid:18)c′1 −c′2 (cid:19) . (13) odd, and iv) for c4, j1−1= odd and N −jn = odd. In
3 4 terms of U , we obtain
i
ComparingEqs.(10)and(13),theequalitiesM =M′ ,
11 11
M =M′ , andM −M =M′ −M′ areequivalent N N
to2t2he equ2a2lities c11=2 c′1, c241= c′4,12and c221+c3 = c′2+c′3. c1 = (Uj1Uj2···Ujn)(−1)21(N−n)
Since βN does not depend on Ui and βN = (−1)N/2 or Xn {{Xjn}}1
(−1)(N−1)/2β for even (or odd) N, we can ignore βN for N N
the purposeofshowingtheseequalities,or,alternatively, c2 = (Uj1Uj2···Ujn)(−1)21(N−n−1)
we can redefine M as M −βN and M′ and M′ −βN. Xn {{Xjn}}2
4
N N We alsofoundanothersymmetry associatedwithelec-
c3 = (Uj1Uj2···Ujn)(−1)21(N−n−1) tronictransportinaone-dimensionallattice; inthis case
Xn {{Xjn}}3 thereisno applicabilityinthe continuumlimit. Suppose
c4 = N N (Uj1Uj2···Ujn)(−1)21(N−n−2) k = π/2; then S = (cid:18)01 10(cid:19), and S = S˜ = S−1. Note
Xn {{Xjn}}4 thatforasymmetric(2×2)matrixX suchthatX =X˜,
(wehveerne, {e{vjenn}).}1Smimeailnasrly{,{j{n{}j}}w}ith=(j{1{j−}1},Nwi−thj(no)dd=, MusicsoantsiisdfieerstMr(iPXP˜M˜):i =M˜iXMi with Ui →−Ui. Now let
n 2 n
even),{{j }} ={{j }}with(even,odd),and{{j }} =
n 3 n n 4
{{j }} with (odd, odd). tr PP˜ = tr M ···M M˜ ···M˜
n N 1 1 N
For c′, c′, c′, and c′ of Eq. (13), we should have (cid:16) (cid:17) (cid:16) (cid:17)
1 2 3 4
(j1−1,N−jn)=(even, even),(even, odd), (odd, even), = tr M˜N···M˜1M1···MN
and (odd, odd), respectively. Note that we need (even, (cid:16) (cid:17)U→−U
odd) for c′ while (odd, even) for c . On the other hand, = tr Q˜Q =tr QQ˜ (15)
2 2
we need (odd, even) for c′ while (even, odd) for c . (cid:16) (cid:17)−U (cid:16) (cid:17)−U
3 3
Therefore, we obtain This symmetry indicates that for k =π/2, |T|2 and |R|2
N N depend only on |Ui|. If E =−2cos(k), then Ui = Ui. In
c′1 = (Uj1Uj2···Ujn)(−1)12(N−n) this instance, |T(U1,···,UN)|2 = |T(−UN,···,−U1)|2.
Xn {{Xjn}}1 Since |T(U1,···,UN)|2 = |T(UN,···,U1)|2 from a gen-
eralderivation,|T(U ,···,U )|2 =|T(−U ,···,−U )|2.
N N 1 N 1 N
c′2 = (Uj1Uj2···Ujn)(−1)12(N−n−1) Consequently, a potential barrier and a potential well
giveidenticaltransmissionandreflectionprobabilitiesfor
Xn {{Xjn}}3
k =π/2 and E =−2cos(k).
N N
c′3 = (Uj1Uj2···Ujn)(−1)21(N−n−1)
Xn {{Xjn}}2 III. TIME EVOLUTION OF A WAVE PACKET
N N
c′4 = (Uj1Uj2···Ujn)(−1)21(N−n−2) So far we have been using the transfer matrix for-
Xn {{Xjn}}4 malism. Since, however, the transfer matrix formalism
Consequently, we have c = c′, c = c′, and c +c = is based on the time-independent Schr¨odinger equation,
1 1 4 4 2 3
c′ + c′; in other words, T[P] = T[Q], or equivalently, we cannot see the time evolution of a wave function,
2 3
T(U , U , ···, U ) = T(U , U , ···, U ) In fact, which may indicate the significance of the symmetries
1 2 N N N−1 1
M = −M′ and M = −M′ . Note, however, that in wehaveshown. Letusconsiderawavepacketwithaver-
12 21 21 12
general M 6=M′. agepositionx andaveragemomentumk attime t=0:
0 0
One may anticipate a similar symmetry in the con- |Ψ(0)i= ϕ(x ,0)c+|0i, where
i i i
tinuum limit.11 Consider a time-dependent Scho¨dinger P
etoqnuiaatnionH:(xi∂)tψin(ctl,uxd)es=aHn (ixm)pψu(tr,itxy),pwotheenrteiatlhVe (Hxa)mfoilr- ϕ(xi,0)= (2πα12)1/4eik0(xi−x0)e−41(xi−x0)2/α2 . (16)
|x| ≤ l. The wave function for |x| > l can be rep-
resented as ψ(|x| > l) = eikx+Re−ikx Θ(−x − l) + We introduced the initial uncertainty α associated with
TeikxΘ(x − l), where Θ(x(cid:2)) is the step f(cid:3)unction. Let position. We wish to propagate (in real time) this wave
packet towards the potentials. To do this we first diago-
us introduce another time-dependent Scho¨dinger equa-
tion: i∂ ψ′(t,x) = H(−x)ψ′(t,x), where ψ(|x| > l) = nalizetheHamiltonian,whichisan(L×L)matrix,where
t
eikx+R′ e−ikx Θ(−x − l) + T′ eikxΘ(x − l). Invok- Listhetotalnumberoflatticesitesunderconsideration.
Wethusobtaintheeigenstates|niandthecorresponding
i(cid:2)ng both space i(cid:3)nversion and time reversal, one can see
that ψ(t,−x) and ψ∗(−t,−x) satisfy the same equation eigenvalues ǫn such that H|ni = ǫn|ni. An eigenstate is
as ψ′(t,x) does. Equating a ψ(t,−x)+a ψ∗(−t,−x) = a column vector with L components which describes the
1 2
ψ′(t,x) for |x|>l leads to T′ =T and R′T∗+R∗T =0. probability of finding an electron at a particular site in
the lattice. With eigenstates in hand, the time evolution
ThefirstrelationcorrespondstoT[P]=T[Q]inalattice.
of the wave packet is given by
The second relation, however, does not hold in a lattice.
Instead, we found in a lattice
L
T∗[P](R[Q]−R[P]) |Ψ(t)i= |nihn|Ψ(0)ie−iǫnt . (17)
=e2ik . (14)
T[P](R∗[Q]−R∗[P]) nX=1
Nevertheless, we can still define R[Q] = eiδR[P]. The As time goes on, the wave packet initially at x moves
0
phase δ is determined by eiδ = −e2ik R∗T/(RT∗) in a to the potential region and scatters off the potential. A
lattice while eiδ =−R∗T/(RT∗) in the continuum limit. part of the wave packet is reflected and the other part
5
Fig.2 ( Kim et al ) depending on the impurity configurations. In particular,
the scattering completely differentiates [I] from [II], as
indicated by the very differently behaved red and blue
curves in the scattering region. Nonetheless the proba-
bility that emerges (either in transmission or reflection)
is identical for both, in agreement with the symmetry
we just proved. Note that the transmitted wave packets
are identical in all other respects as well whereas the re-
flectedwavepacketshaveidenticalshape, butarephase-
shifted with respect to one another. In fact, one can
show that the phase shift originates from the phase δ in
R[Q] = eiδ R[P]. The shift measured by the difference
between the two reflected wave packets for [I] and [II] at
their half width is determined by ∂δ(k)/∂k .
k0
IV. CONCLUSIONS (cid:12)
(cid:12)
100 200 300 400 500
FIG. 2: Time evolution of a wave packet in two impurity We have determined the transmission and reflection
configurations [I] (red solid) and [II] (blue dashed curve). characteristics for various impurity configurations on a
Each configuration has 5 impurities: for [I] the potential is onedimensionallattice. Inparticularweprovedthatthe
(−0.5, 0, 0.5, 1.5, 1) while for [II] its reverse holds. transmission of a particle through an array of impuri-
ties is independent of the direction of travel. This the-
orem may help understand, among other things, weak
localization, where time-reversed paths play an impor-
is transmitted. Mathematically we can define the re-
tant role. Some important differences arise because of
flection and the transmission as |R|2 = |ϕ(x ,t)|2
left i thelattice: first,itbecomesclearthatthegroupvelocity
and |T|2 = |ϕ(x ,t)|2, respectivePly as t → ∞,
right i is most important (in the continuum limit the energy is
where left(Pright) means the summation includes only usually emphasized). Secondly, other symmetries exist
the leftP(right) side of the impurity region. on a lattice for particular wavevectors. Naturally these
Fig. 2 shows the time evolution of a wave packet symmetries do not hold in the actual scattering region.
impinging on 5 impurities located at sites 300 to 304. While we have no specific proposals at present, we hope
The parameters of the wave packet are given as follows: this work motivates experimentalists to look for these
x0 =150, α=20, and k0 =π/3. Note we choose α=20 violations in the vicinity of particular impurity configu-
so that the size of the wave packet is much greater than rations. Furtherworkisinprogresswithtrimerimpurity
the size of the impurity region. This means that we can configurations,moregeneralbandstructures,andhigher
consider the wave packet as a plane wave in this case. dimensionality.
In fact, the numerical results of the transmissionand re-
flection probabilities are in close agreement with those
obtained through the transfer matrix analysis. The im- This work was supported in part by the Natural
purity configuration is determine by 5 impurity poten- Sciences and Engineering Research Council of Canada
tials: (U , U , U , U , U ). For [I] (red solid), we (NSERC), by ICORE (Alberta), and by the Canadian
1 2 3 4 5
have (−0.5, 0, 0.5, 1.5, 1) while for [II] (blue dashed Institute for Advanced Research (CIAR). FM is appre-
curve), we have the reverse order. As clearly shown in ciativeofthehospitalityoftheDepartmentofCondensed
Fig. 2, the time evolution of the wave packet is different Matter Physics at the University of Geneva.
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