PHYSICAL REVIE% 8 VOLUME 48, NUMBER 15 15 OCTOBER 1993-I

Anderson localization in an anisotropic model

Qian-Jin Chu and Zhao-Qing Zhang*Institute ofPhysics, Academia Sinica, Beijing, China

(Received 14 June 1993)

The phase diagram of the Anderson-localization problem in an anisotropic model which has bothtwo-dimensional (2D) and 3D scattering is studied using a diagrammatic method. It is found that themobility edge follows the relation 0=a exp( —b/8' ) in the anisotropy limit, where 8 is the anisotropyparameter which interpolates a system of two-dimensionally coupled pure 1D chains, each having adifferent site energy, and a system of 3D isotropic randomness. 8', is the critical randomness, and a and

b are two Fermi-energy-dependent constants. This behavior is different from the results of a 1D-to-3Dcrossover model studied previously. In that model, there exists a nonzero critical anisotropy 0, below

which all states are localized. Physical reasons are given to explain this difference.

I. INTRODUCTION

Much has been known about the properties of Ander-son localization transition of independent electrons in anisotropically random potential. It is now widely accept-ed that the marginal spatial dimension for localization istwo and all states are localized in one and two dimen-sions. However, an important assumption underlyingthese investigations is that the correlation length of theimpurity is finite and the medium is macroscopicallyhomogeneous. The long-range correlations between twoimpurities can alter the localization behaviors in one andtwo dimensions. ' A fractal type impurity can also leadto different transport behaviors in two dimensions.

Recently, an anisotropically random model has beenproposed. ' In this model, the site energy c in the An-derson Hamiltonian

H=g eaa+t pa a&,a (nn)

consists of two kinds of randomness, i.e.,

E =87),, +(1—8)y, , (2)

a,a are the creation and annihilation operators, respec-tively, a,P are indices for sites on a three-dimensional(3D) lattice, and t is the nearest-neighbor hopping matrix.g,, and y, are random variables with the same Hat distri-bution of width 8', i.e.,

lxl ~ W/2P(y. ) =P(n;. ) =P(x)= '0,th (3)

y, depends on the z coordinate only and i depends on thex-y coordinates. 0 is an anisotropy parameter, varyingbetween 0 and 1 which interpolates the randomness be-tween a 1D layered (8=0) and 3D isotropic (8=1) sys-tem. For any 8(1, the existence of y, in Eq. (2) intro-duces infinite range correlations among the site energiesin any given layer.

Both diagrammatic and numerical calculations ' haveshown that there exists a critical value of 0, below whichall states are localized. The existence of a nonzero 0, can

be understood as follows. In the 8=0 limit, the systembecomes a randomly layered medium and the localizationlength g, along the layering direction is proportional to

A small 3D randomness introduces an "inelastic"scattering length l;„, which is proportional to(1—8)/(OW ). This length measures the distance that a3D randomness will take to scatter an electron awayfrom the layering direction to the x-y plane. If 6 is smallsuch that l;„, is larger than g„ the electrons cannot feelthe 3D scattering and remain localized. The critical 0,is, thus, the value of 0 at which the above lengths becomecomparable and the system loses its 1D character.

Here, we consider a 2D-to-3D crossover model. In thismodel, the site energy c is of the following form:

E =Op,,+(1—8)y;, (4)

where y; depends on the coordinates x-y only and hasthe same distribution as given in Eq. (3). In the 8=0 lim-it, Eq. (4) gives a system which consists of two-dimensionally coupled pure 1D chains, each having adifferent site energy. This system crosses over to isotro-pic random 3D behaviors as 0 is increased to 1. In thiscase, the infinite range correlations exist among the siteenergies of any given 1D chain with fixed x-y coordi-nates. Since two is the marginal dimension and the local-ization length perpendicular to the 1D chain grows ex-ponentially as

g~~~ exp(a/W' ) in the small W limit, it is

likely that a small 3D scattering can drive the system intoan extended state. Thus, one would expect a verydifFerent mobility edge behavior in the small 0 regime.

The purpose of this work is to study the phase diagramof this 2D-to-3D crossover model described in Eqs. (1),(3), and (4) using a diagrammatic theory developed inRefs. 5 and 6. Despite the approximations involved inthe diagrammatic theory, the method, nevertheless,yields the phase diagrams which are in good agreementwith those obtained by the numerical transfer-matrixmethod.

Since the formulation developed in this work is parallelto that described in detail in Ref. 6, we will write downonly the main steps in Sec. II, which lead to the final mo-bility edge equation. The numerical results as well assome discussions are given in Sec. III.

0163-1829/93/48(15)/10761(5)/$06. 00 48 10 761 1993 The American Physical Society

10 762 QIAN-JIN CHU AND ZHAO-QING ZHANG

II. FORMULATION

Using the momentum representation, the Hamiltonianof Eq. (1) can be recast into the form

where

E+ =E+-CO

2(12a)

H= pe(p)a a + g v(q)a + aP

with

(5)and

u (p)= Be(p)J (12b)

and

e(p)=t g e' '~

5

v(q)= —g E, ea

(7)

where 5 is the vector pointing from a site to one of itsnearest neighbors, R is the coordinate of the site a, andX is the total number of sites in the cubic lattice. Usingthe self-consistent Born approximation (SCBA), the aver-aged single-particle Green's function (GF), R, and itsself-energy g, can be derived from Eqs. (3)—(7) as

Z

R;+=&G;+),= 1

E —e(p) —X& (E)

(8)

and

X—(E)= g & v(q)v( —q) ),R~ +q(E)q

WO 1

12N ~ E —e(p') —X—,(E)P Pz

W'(1 —())' 1

12Nii, E —e(p +pti X~ (E)

Here, 6+ and 6 are the retarded and advanced GF's,respectively, E denotes the Fermi energy, Nii is the num-ber of sites in an x-y plane, "z" and "II" denote com-ponents of a vector in the z and x-y directions, respective-ly, and & ), denotes the configurational averaging. Thefactors W 8 /12 and W (1—8) /12 represent thescattering strengths from 3D and 2D randomness in Eq.(4), respectively. The "p," dependence of the self-energyarises from the scattering of the 2D random potentialwhich conserves the momentum in the z direction.

The conductivity o (co,E) in the j direction is given bythe Kubo formula

Equation (11) can be expanded into diagrams which con-tain the averaged single-particle GF. Only two classes ofdiagrams will be evaluated: ladder diagrams and maxi-mally crossed diagrams (MCD's). The ladder diagrams,which are shown in Fig. 1, yield the bare conductivityo . , and the MCD's, which represents the coherent-backscattering e6'ect, give the localization corrections.The summation of the ladder diagrams yields

I u, (p)+n (p)]b,R (E)o (E)= 2~N~ ~r (E)

12e u, (E)~8' 0

o, (E)=.(13)

(14)

with

u, (E)= g u, (p)bRp(E),2vrp E N

(E)=R —(E)—R (E

b,X (E)=X (E)—&~ (E),

p(E)= g ERp(E)P

(15)

(17)

(18)

Pq

' (p')—

Pi

Equation (14) gives the desired property that. a, (E)diverges as 0 in the anisotropy limit. The vertexcorrection 5P""'" '(co, E) from the MCD can be ex-pressed as

gyRA(~ E) gyRA(l)(~ E)+gyRA(2)(~ E)

g+yR A ( )(nE ) +JJ

where ( n ) stands for the nth-order summation of MCD's.For the z component, we have

oi (co,E)= [P. "(co,E)+Pii"(co,E))

with

yiRiA( AR)( E) y ( ) (p )

1

P~P

vj (P)

Pq PT7/ir II

Pq P

X

(a) (b) (c) (d) (e) (f)

X +I

)

x « PIG+' '(E )lp')

x&p'IG '+'(E —)lp)), ,

FIG. 1. Each vertex in the ladder diagrams (bold solid line)consists of two types of scattering. The dotted line denotes the2D scattering with a strength 8 (1—0) /12. 3D isotropicscattering, denoted by a dashed line, has the strength 8 0 /12.The solid line denotes the averaged Green's function R.

48 ANDERSON LOCALIZATION IN AN ANISOTROPIC MODEL 10 763

5P„"'"'=2 g v, (p)u, (p')RA(n)

7T p p P1'P2' ' ' ' 'P2n

R+(E+ )R (E )U' '(p, p&, co, E)

X U (p»pz, co, E)U (p2, p3, co, E)R& (E—+ )Rz (E )

X U' '(p2„„p2„,o2, E)U' '(p2„, p', co, E)Rp~(E+ )R p (E ), (20)

while for the x-y component we have

5$;;"'"'(ro,E)=2 g v;(p))u;(p„+))R+(E+)R (E )U' '(p), p2, co, E)Rp~(E+)R (E )

P1' " 'Pn+1

X U™(p„,p„+„co,E)R* (E+ )R p (E ) . (21)

U' ' represents the summation of ladder diagrams con-taining only the 2D scattering. U' ' represents the sum-mation of MCD's containing both 2D and 3D scatteringas shown in Fig. 2. Following the procedure described inAppendixes 8 and C of Ref. 6, the evaluations of U' '

and U' ' yield

i (1—8) b,R~ (E)U' (p&, p2, 0+,E)=5 +5~1'1 2 ~ lz'1 2z 22rp(E)82N

II

(22)

and

S (E)

(1 —8)'S (E)D~~

(E,P 1, )k

8 b,X (E)»z

with

p(E) W8. '

6

z z= —j~+D (E)k +Do(E)k

b X (E) b X~ (E)+iiS (E—)—F (p&„kll'co, E) b X (E

(24)

(25)

(26)

U(M)( E) W (1—8)12N

II

8' 012N

and

D E 1

4 (E)hX Nii»z

X $(k, co,E),where k=p&+pz and, for small co and k,

(23)X g [v'(pl)+u (p~~)]b, R (E),

(27)

where D~~

and D, are the bare diffusion constants, which are related to the bare conductivity of Eqs. (13) and (14) by theEinstein relation cr =2e pD.J)

. Finally, the vertex corrections of Eq. (19) can be obtained by substituting Eqs. (22) and

(23) into Eqs. (20) and (21). Combining the bare conductivity cr (E) of Eqs. (13).and (14) with the vertex corrections ofEqs. (19)—(21), we obtain the conductivity o~~(co, E). The diffusion constants have the forms

D, (co,E)=D, (E)—mS (E)p(E)N 1 —8

D, (co,E)—i~+D (E)k +Do(E)k

(2&)

1(((o2,E)—D

(((E)—

2

Dii(co,E)

ico+D~~ (E)kl +D, (E—)k,

1 D(((a),E)p(E)N, '+D „(E)[k (E)+ k „]— (29)

10 764 QIAN-JIN CHU AND ZHAO-QING ZHANG 48

U( ) (p p&, ~, E) =

Pq Pp

,X. ——

Pq Pg

(E) 2r (E)8D'„(E)(1—8)

1/2

(31)

where

r(E)= X~ b, X~ (E)P,

[W8+W(1 —8) ]12(30)

FIG. 2. The MCD's that give rise to U' '(p&, p~, co,E). Thebold solid line consists of both 2D and 3D scattering. The hor-izontal line denotes the averaged Green's function R.

In Eq. (29), there are two coherent-backscattering correc-tions to D~~. The contributions from both 2D and 3Dscatterings give an anisotropic diffusion pole. The contri-bution from mainly 2D scattering has a 2D diffusion polewith an "inelastic" scattering length ko ' which measuresthe length that electrons can stay in a 2D plane before be-ing scattered off by a 3D scattering. Since this pure 2Dcoherent-backscattering effect is prohibited for an elec-tron moving in the z direction, we have ignored an un-physical positive 2D coherent-backscattering correctionterm in Eq. (28).

For a self-consistent theory, the bare diffusion con-stants Dll and D, in Eqs. (28) and (29) should be replacedby D~~ and D„respectively. The mobility edge equationis then obtained by asking both Dll(0+, E) and D, (0+,E)equal to zero. From that we find

21 (E*) 1

AS(E*)p(E*)N k 2r(E*)/S(E*)[Dll E*)gll E ~kll+D,'(E")k, (32)

with

g (E)= 1 1

~p(E)Nll lk k2(E)+k2. (33)

vll(E)= g [v„(p)+vy'(p)]bRp(E) .4~p z N

(34)

The summations over k and kll in Eqs. (32) and (33) canbe replaced by integrals. The upper cutoff of these in-tegrals is inversely proportional to the mean free path,i e kll=xo/ill and k;=xo/I, . ill and l, are related tothe diffusion constant by l =3D (E)/vl (E) andI, = 3D, /u, (E), where u, (E) is given by Eq. (15) and ull isdefined as

—b0=a expWC

(35)

where a and b are two constants that depend on the Fer-mi energy. In order to check this relation, we have solvedEq. (32) numerically for the case E/t =0 at very small

gives an exponential dependent localization length in thex-y plane, i.e., gll~(1/W )exp(b/W'). A small 3Dscattering produces an inelastic scattering length l;„„which is inversely proportional to ko(E) of Eq. (31). Atransition to the extended state can occur if 0 is such thatI;„, becomes comparable to gll. Since ko(E) ~ W 8 atsmall 0, the mobility edge should have the form

The constant xo is of order unity and chosen to be 0.42 sothat Eq. (32) recovers the known critical value ofW, /t = 16.2 in the isotropic limit 8= 1 at E/t =0.

III. RESULTS AND DISCUSSIONS

Based on Eq. (32), the mobility edge W, /t is calculatednumerically for a fixed value of the Fermi energy. Theresult for E/t =0 is shown in Fig. 3. In the isotropic lim-it, similar to the case of the 1D-to-3D crossover model ofRef. 6, the value of W, (8 ) /t increases as 8 decreases from1, due to the reduction of 3D scattering strength. As 0approaches the anisotropy regime, 2D scattering becomesdominant and turns the mobility edge curve downward.However, unlike the model of Ref. 6, the value of W, /t isalways finite except at 0=0. The nonexistence of a finitecritical value of 0, in this model can be understood in thefollowing way. At 0=0 and small W a 2D scattering

30

25

O20

15

1 0

00 0.2 0.4 0.8

FIG. 3. Value of W, /t plotted as a function of 0 for E/t =0.

48 ANDERSON LOCALIZATION IN AN ANISOTROPIC MODEL 10 765

0.3

C

0.2

0. 1

0-14 -12 -10 - 8I n (0)

— 6

FIG. 4. Mobility edge is plotted in the small 0 limit, as 8'vs ln(0) for (a) E/t =0, (b) E/t =2, and (e) E/t =4.

values of 8 and plot 1/W', vs ln(8) in Fig. 4(a). A linearcurve in Fig. 4(a) demonstrates that the mobility edgeindeed obeys the relation of Eq. (35). The slope of thecurve could indicate the value of —1/b. We have alsocalculated the mobility edge for the cases of E/t =2 and4. The overall phase diagrams are close to the one shown

in Fig. 3. In the small 8 limit, the mobility edge againfollows the relation of Eq. (35) as can be seen in Figs. 4(b)and 4(c) for E/t =2 and 4, respectively. However, theabsolute value of the slope of these curves, 1/b, increasesas energy moves towards the band edge. This is con-sistent with the fact that the value of b decreases as theband edge is approached.

In conclusion, we have introduced an anisotropicallyrandom model. By changing the anisotropy parameter,the system crosses over from 2D-to-3D randomness. Theself-consistent diagrammatic theory of Vollhardt andWolfe, which has been generalized to treat a 1D-to-3Dcrossover model previously, is used in this model to ob-tain the localization phase diagram. In the anisotropylimit, unlike the 1D-to-3D crossover model, we do notfind a nonzero critical anisotropy below which all statesare localized. Instead, it is found that the mobility edgefollows a 8=a exp( b/W—, ) relation due to the nature ofexponential dependence of localization length in the x-ydirection.

ACKNOWLEDGMENTS

This work was partially supported by Grant No.LWTZ-1298. The authors thank P. Sheng for useful dis-cussions.

*Also at Exxon Research and Engineering Company, Annan-dale, NJ 08801.

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