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Theory of weak localization in a lateral superlattice
Superlattices
and Microstructures,
THEORY
199
Vol. 11. No. 2, 7992
OF WEAK
LOCALIZATION
IN A LATERAL
SUPERLATTICE
W. Szott, NanoFAB
C. Jedrzejek,’ and W. P. Kirk Center, Department of Physics Texas ABM University College Station, Texas 77843 (Received
19 May 1991)
A theory of weak localization for the parallel and transverse conductivity in lateral semiconductor superlattices is presented. Based on our previous studies of planar superlattices, we assume that there exists coherent transport between quan turn wires. The theory explicitly accounts for the superlattice periodic structure. Small collisional broadening calculations are performed as quantum corrections to Boltzmann transport. Similar to planar superlattices, weak localization corrections to the conductivity depend on the width of the superlattice miniband. This theory allows for (i) direct determination of the effective interwire electron coupling by measurement of the parallel conductivity, and (ii) computation of the electron dephasing time (such as the electron-electron scattering time) for this quantum structure
1. Introduction The theory of weak localization (WL) of electrons in a single quantum wire was formally developed by Altshuler and Aronov.’ For a wire of finite thickness the parameters of the theory (magnetic time 7~ and diffusion coefficient D) depend on the width of a wire (see, Beenakker and van Houten’). Takagaki et aL3 showed experimentally the applicability of 1-D WL theory for 100 nm wires. Here we develop a WL theory for lateral superlattices of wires which parallels our work on 3-D superlattices. 4-6 For decoupled wires this theory reduces to the case of very-small-width (one mode) single wires. In this paper we are primarily concerned with the weak localization aspect of superlattice transport properties in the low collision limit. The aim of this paper is, t,herefore, to provide a theoretical model to interpret. experiments that involve weak localization effects in lateral superlattices. The model avoids the restriction of a uniform impurity distribution and takes into account electron wave function modulation in a superlattice. We employ, however, the simplified assumption that scattering from a single impurity is isotropic. Upon impurity averaging the effective interaction (and the lifetime) becomes anisotropic in a form that goes beyond the Bhatt, Wijlfle, and Ramakrishnan assump tion,’ vir. the scattering amplitude depends on the momentum difference of the final and initial state of the electron. Formally, this work generalizes the previously
t Permanent address: Department of Physics, ellonian University, Cracow, Poland.
0749-6036/92/020199+06$02.00/0
Jag
obtained results for three-dimensional superlattices to the case of two-dimensional superlattices. As for the case of three-dimensional superlattices one of the main results we find is that the Cooperon for superlattices still has the diffusive form (that is the dominant contribution to conductivity comes from backscattering). We consider this an important result. However, in contrast to distinguish
to the Bhatt
et al.’ result, we have
between the diffusion coefficient
fiZ that
enters the classical expression for conductivity, o 0: fi,,, and the effective diffusion constant D, appearing in the Cooperon. We find that D, consists of two parts. The first part depends on the width of the miniband. The other contribution has a hopping-like character and is mostly governed by the electron wave function modulation in a superlattice. Very recently, a two dimensional electron gas experiencing an atomically precise, lateral Kronig-Penney potential was fabricated by Pfeiffer et a1.s on the cleaved edge of a GaAs/AlGaAs system. In this system carriers are capable of coherently crossing 200 interfaces (nanometer-size barriers).g Therefore, it is now possible to investigate coherent coupling effects of quantum wires by comparing the theory presented here with experimental results. 2. Tight-binding
Model
for a Superlattice
A. Hamiltonian We assume a structure (with geometry shown in Fig. 1) in which the miniband is described by a tightbinding model. The total Hamiltonian consists of three parts:
0 1992 Academic Press Limited
Superlattices
200
f(z - qa)
Pz(‘i
nia energy profile for relevant
H = Ho f H, + He,. Ho is the kinetic electrons
energy of the in-wire motion
(1) of free
f(z - (fj +M
)
\
I
Fig. 1. Superlattice potential spatial coordinates.
Vol. 11, No. 2, 1992
and Microstructures,
/
(q +$a
(q+l>a
Fig. 2. Schematic representation of relevant distribution functions in a superlattice. Solid line represents the bottom of the conduction band (potential profile). Dashed lines depict the impurity distribution, and the wave function f(z - nia), respectively. nio is the center of a potential well and (ni + t)o is the center of the barrier.
with H, is an inter-wire
tunneling,
v,(k,,p,)
=
&(k.-P’)nyf++f_e-‘k’a)*
x (,f+ +
and H,, is responsible for electron-impurity scattering. In these formulas I;, is the parallel momentum in the wires, n is the wire number, cc(IC=) is the dispersion relation for in-wire motion, and t,,, is the inter-wire tunneling amplitude. The transfer between the wirrs is restricted to nearest neighbors. Since we assume the single scattering approximation, we consider only one impurity so that at the end of the calculations all appropriate quantities are simply multiplied by the number of impurities. We assume that the basis functions
The following notation position)
are products of Wannier functions localized on the superlattice wires in the z-direction with period a, with the free waves in parallel planes. These functions constitute a complete and orthogonal set; however, they are not the eigenfunctions of H, + Ht. The eigenfunctions of Ho + Ht, which are used to construct H,,, are the Bloch functions that are expanded in terms of the basis functions (4). The scattering potential from an impurity at (X,, 2;) is assumed to be the 6 function in the configuration space
where k, = E,
V,(Z, 2) = V&(x - X*)6(2 - 2;).
(7)
f-e-‘p=a) is used (where 2, is the impurity
f(Z, - na) = { p ; where n,n < Z, < (n, + l)a.
fp,h,“,,s;:: nz+’ Here f+ = f(zht$e + z,),
where z, = Z, - (n, + $)u and Iz,I < 4. Also
F = 0, 1, . . . . N - 1. A schematic
sentation of the relevant tice is shown in Fig. 2. B.
Impurity
distributions
repre-
in the superlat-
Averaging
We assume that the total distribution function of impurities factors into independent distributions in the parallel and t directions:
(5)
Consequently, the interaction part of the Hamiltonian in the tight-binding approximation is given by
where pL is a uniform r-distribution, p,(X,) = coast., and pz is a periodic z-distribution, pz( Z, +na) = pl(Zi), which preserves the symmetry of the superlattice. The functions F(X,, Z,) that we average with respect to the impurity distribution are bilinear forms of
Superlattices
and Microstructures,
the scattering potential, Eq. (7), according to the following equation
Vol. 1 I, No. 2, 1992 and they factorize
(10) where F,(X,)
= e’qrx8, Fr(n,)
O,l, . . . . N - 1, and F~(z*) = The average value of F is
I
wrth P(z,) = Np(:a
$0
a1 =<
jt
>& + <
a2 =< fzfa3 =< fif?
Fz(f+(zt), f-(2,)).
dZzdXzp(X,, Zz)F(Xi,
= L,o4,
where the a,‘s are determined
= e’qZnSa,with qZ = $$,
E=
r
201
2,)
3.
(11)
a/2 _-a,2 Xz,)&(z,)&,
J
A. Scattering
+ 2%) being the z-distribution
Transport
(17)
Properties
Time
nor-
Ps
B(P, e,,) = %V2
J
~li’(p.q,cl,p)Go(~,~“l~ (18) Green’s function, are: E,(Q,) = s2m, , and
is the unperturbed
and the dispersion
:v
relations
E,(Q,) = w(l - cosq,a). The scattering life-time
@ IV Pz_
f” >&is=2 < f; >b,,
>p,=< f3f+ >pz, >pz
where GO(qre,) t +
= j?=(z,))
The self energy, C(p, E,). in the Born approximation is
malized per wire. We are now in a position to calculate the impurity averaged scattering probability W illustrated in Fig. 3 PI +
by (P*(-z,)
1 __ = -2l?nC(p, T(P)
4 +P4
Fig. 3. A double scattering amplitude, Ws, from a single impurity (8) with a point like potential V. Averaging over the impurity distribution (consistent with the superlattice structure) restores momentum conservation, p1 + pz = pa + pd.
T(P) for EF > 210 reatls
1 to+) = $1
+ bcosp,o),
119)
where (20) and
I+. = N, < V,V, >,=, =
b_ b2(0)
~ti(Pl,P2,Ps,P4), (2T)2
6 P,+P?>Ps+P4
where the average impurity
density,
= W*(s, U, u) + Here
Here g(eF) n, = A,
(13)
Wz(s,
O<_h
is the density of states
per spin.
u, u).
u = pl, - pzr, v = psr - pd,, and
where K is the complete elliptic integral of the first kind. The average scattering time ‘e has the following form Te = (T.(q))Fermi surface = ~
M/,(s,u,u)
= lb, + bzcosy][bl
and W2(~,u,v)=b;cos-cos--.
(14)
+bz~~+>
‘5
A
ua
2)l7
2
2
(15)
Coefficients b; are functionals of the impurity distributions and functions of the total momentum s,
b,(s) =
d a,
bz(s) =
4a2 cos Ja b,(S) ’
(21)
and
= WP1*,Pz*,P3z.P4*)
WP,,PZ,P3.P4)
b,(O)
( 12)
+ 2a3 cos Sd
(16)
(23)
Note that one obtains the suppressed scattering results for b = 1 or P(Z) = 6(z). This is solely due to the pointlike scattering potential. B. Weak ductivity
Localization
Corrections
to the Con-
Weak localization corrections are determined by the diffusive term in the Cooperon, C (see Fig. 4). The Cooperon structure, as a function of external momenta, is
Superlattices
202
Fig. 4. Pictorial
representation
and Microstructures,
of the Cooperon. In the quasi-particle
+s-p
Vol. 11, No. 2, 1992
approximation
(ef
+s-q lim s ---t o.&
+ o+
X,JS,Gz,WP)
1 0
= i fl
Only Cii has a diffusive character ;s+p
>> 3) l=j=l, ifj. (=j=2.
for small s and ‘“‘I
;stq
Fig. 5. The Bethe-Salpeter equation for the two-particle scattering amplitude (Cooperon) in the maximally crossed diagram approximation.
The
diffusion
constant
in
the
r-direction
is
D, = (a:~)~~ = t~r,~~~(l - b*)-‘/’ for r:,,~ = consf on the Fermi surface (EF > 2~). The diffusion constant in the t-direction is more complicated, i.e. it consists of coherent and incoherent, (hopping) parts. D, = D,,, + D;,h. where
where by definition Ct,(s,P,s,%I,~e) Here U,(s,p)=
Gth
b,(s,)+ b~(sr)cospza, and Uz(s,p)=
b3(Sz) cospzc. We can write Bethe-Salpeter C,, as shown in Fig. 5 ) = n,L’W,(s.
C’3(S,P,%~,,,w’P
+ wV2~,(s, P) x Gii
(32)
= ~*)(S,E,,IJoU~(s,P)I;I(s,q)
J
d2r
-Ut(s, (‘*)*
+ r,c” +wt)
C
(BS)
equations
for
p)li,(s,p)d,, r)G(; - r, F,)
(25)
+ 1
< (ff
fif? >pz + f2J2 >& 1
D,_u = -.
2r
BS equation
From Eqs. (28) and (30) it follows that part of the Cooperon is
for c,,
(33)
which is the relative probability for inter-wirr, scattcring. There is an analogy of Dz,h to one-dimensional hoppmg diffusion. viz.
u2
Ckl(s,r,q,t,,dg).
k=i.L
and the corresponding
<
P Tl.“il
pn,,+ Prl,“il =
(34) the diffusive
(‘-6) i~-,k(S,En.U?P)~kJ(S,F,,.wP), x (1 + bcosp,a)(l + bcosq,a) Ia/ + Dzs; + D,sz,
k=I.Z
where
(35) Finally, the following result for the weak localization corrections to conductivity is obtained:
x G(i
+ r,
E,
+
WP)U~S, r)U,(s, r), (27)
with the following solutions
SuPertattices and Microstructures,
D D*,WL
=
Here, rpphis a dephasing the standard
‘CUz,WL D,
time which was introduced
replacement
by
Ju() -+ T;,
Also! au,~ has the following 1. scaling relation
203
Vol. 11, No. 2, 1992
properties:
tion cannot be relaxed without breaking the discrete translational symmetry of the whole periodic structure. Small perturbations of the periodic distribution are acceptable and will affect the total elastic scattering time in a form similar to that which describes surface scattering. Finally, a complete expression for the Cooperon in the magnetic field contains oscillatory terms which reflect the commensurability of the magnetic length and superlattice period. Such an effect was observed by Paris et ~1.‘~ and was interpreted as periodic replicas of the standard weak localization effect. This interesting effect requires more detailed, numerical studieq.
(37) 4. Conclusions 2. when D, ---* 0, CT,,\~‘Lapproaches coupled wires)
1-D limit
(de-
(38)
3. when D, is large (* > l), c~,~‘L acqtiires the standard form for 2-D anisotropic systems, i.e. e2 U,,WL
=
-py-&
D, dD,
In rpTph + const.
(39)
Upon neglecting the oscillating part of the Cooperon, the weak localization corrections to magnetoconductivity becomes huw~(B) z ow~(B)-uw~(O) with
Au,,wL(B)
=
&
Au,,~,
D
= ~Au=,,~, D,
where Bph is the characteristic cesses:
field for dephasing
A new tight-binding model of quantum corrections to the conductivity of a lateral semiconducting superlattice is proposed. The effects of electron wave function modulation and spatial distribution of impurities were studied. The main results of this work can be summarized as follows: 1. The superlattice structure causes an effective anisotropic elastic scattering time for isotropic scatterers. 2. Quantum corrections to conductivity is suppressrcl by a hopping-like diffusion. 3. Scaling relations are satisfied by the anisotropic conductivity tensor for a model that does not belong to the Bhatt et aL7 class of anisotropic models (anisotropic mass tensor, anisotropic scatterers). Acknowledgement - This material is based in part on work supported by the National Science Foundation (DMR 8800359). One of the authors (C.J.) acknowledges partial travel support from the Polish Research Council (grant # 2 0401 91 01).
(40) References
pro-
B. L. Al’tshuler and A. G. Aronov, Pis’ma Eksp. Teor. Fiz. 33, 515 (1981) [JETP Lett. 499 (1981)]. At this point we summarize with a few comments about the crucial assumptions made in the above model. We employed a &like scattering potential (commonly used in weak localization theories) which allowed us to obtain the final results in an analytical form. But we caution the reader that this form may be a rather rough approximation for more realistic potentials such as those associated with weakly screened charged impurities. As shown in our previous work,‘j the effect of a nonuniform distribution of long-range scattering centers is, at least qualitatively, equivalent to the effect caused by a more uniform distribution of point-like impurities. Thus, the scattering distribution may be better taken into account by introducing an effective impurity distribution instead of a geometrical distribution. Another important assumption concerns the periodicity of the impurity distribution. This assump-
Zh. 33,
C.W. Beenakker and H. van Houten, Phys. Rev. B 38, 3232 (1988). Y. Takagaki, K. Ishibashi, S. Ishida, S. Takaoka, K. Game, K. Murase, and S. Namba. Jap. J. Appl. Phys. 28, 645 (1989). W. Szott, C. Jedrzejek, Rev. B 40, 1790 (1989). W. Szott, Rev. Lett. W. Szott, Rev. B (in
and W. P. Kirk,
Phys.
C. Jedrzejek, and W. P. Kirk, 63, 1980 (1989). C. Jedrzejek, and W. P. Kirk. press).
Phys. Phys.
R. N. Bhatt, P. WGlfle, and T. V. Ramakrishnan. Phys. Rev. B 32, 569 (1985). L. N. Pfeiffer, K. W. West, H. L. StGrmer, J. P. Eisenstein, K. W. Baldwin, D. Gershoni, and 3. Spector, Appl. Phys. Lett. 56, 1697 (1990).
204 ’ H. L. Stiirmer, L. N. Pfeiffer, K. W. West, and K. W. Baldwin, m Proceedings of the International Symposium Nanoslructures and Mesoscopic Systerns, (eds. W. P. Kirk and M. A. Reed, Academic Press, 1991), p. 51.
Superlattices
and Microstructures,
Vol. 7 7, No. 2, 1992
lo E. Paris, J. Ma, A. bl. Kriman, D. K. Ferry, and E. Barbier, in Proceedings of the International Symposium Nanostructures and Mesoscopic S~~stems~ (eds. W. P. Kirk and M. A. Reed, Academic Press, 1991), p. 311.
and Microstructures,
THEORY
199
Vol. 11. No. 2, 7992
OF WEAK
LOCALIZATION
IN A LATERAL
SUPERLATTICE
W. Szott, NanoFAB
C. Jedrzejek,’ and W. P. Kirk Center, Department of Physics Texas ABM University College Station, Texas 77843 (Received
19 May 1991)
A theory of weak localization for the parallel and transverse conductivity in lateral semiconductor superlattices is presented. Based on our previous studies of planar superlattices, we assume that there exists coherent transport between quan turn wires. The theory explicitly accounts for the superlattice periodic structure. Small collisional broadening calculations are performed as quantum corrections to Boltzmann transport. Similar to planar superlattices, weak localization corrections to the conductivity depend on the width of the superlattice miniband. This theory allows for (i) direct determination of the effective interwire electron coupling by measurement of the parallel conductivity, and (ii) computation of the electron dephasing time (such as the electron-electron scattering time) for this quantum structure
1. Introduction The theory of weak localization (WL) of electrons in a single quantum wire was formally developed by Altshuler and Aronov.’ For a wire of finite thickness the parameters of the theory (magnetic time 7~ and diffusion coefficient D) depend on the width of a wire (see, Beenakker and van Houten’). Takagaki et aL3 showed experimentally the applicability of 1-D WL theory for 100 nm wires. Here we develop a WL theory for lateral superlattices of wires which parallels our work on 3-D superlattices. 4-6 For decoupled wires this theory reduces to the case of very-small-width (one mode) single wires. In this paper we are primarily concerned with the weak localization aspect of superlattice transport properties in the low collision limit. The aim of this paper is, t,herefore, to provide a theoretical model to interpret. experiments that involve weak localization effects in lateral superlattices. The model avoids the restriction of a uniform impurity distribution and takes into account electron wave function modulation in a superlattice. We employ, however, the simplified assumption that scattering from a single impurity is isotropic. Upon impurity averaging the effective interaction (and the lifetime) becomes anisotropic in a form that goes beyond the Bhatt, Wijlfle, and Ramakrishnan assump tion,’ vir. the scattering amplitude depends on the momentum difference of the final and initial state of the electron. Formally, this work generalizes the previously
t Permanent address: Department of Physics, ellonian University, Cracow, Poland.
0749-6036/92/020199+06$02.00/0
Jag
obtained results for three-dimensional superlattices to the case of two-dimensional superlattices. As for the case of three-dimensional superlattices one of the main results we find is that the Cooperon for superlattices still has the diffusive form (that is the dominant contribution to conductivity comes from backscattering). We consider this an important result. However, in contrast to distinguish
to the Bhatt
et al.’ result, we have
between the diffusion coefficient
fiZ that
enters the classical expression for conductivity, o 0: fi,,, and the effective diffusion constant D, appearing in the Cooperon. We find that D, consists of two parts. The first part depends on the width of the miniband. The other contribution has a hopping-like character and is mostly governed by the electron wave function modulation in a superlattice. Very recently, a two dimensional electron gas experiencing an atomically precise, lateral Kronig-Penney potential was fabricated by Pfeiffer et a1.s on the cleaved edge of a GaAs/AlGaAs system. In this system carriers are capable of coherently crossing 200 interfaces (nanometer-size barriers).g Therefore, it is now possible to investigate coherent coupling effects of quantum wires by comparing the theory presented here with experimental results. 2. Tight-binding
Model
for a Superlattice
A. Hamiltonian We assume a structure (with geometry shown in Fig. 1) in which the miniband is described by a tightbinding model. The total Hamiltonian consists of three parts:
0 1992 Academic Press Limited
Superlattices
200
f(z - qa)
Pz(‘i
nia energy profile for relevant
H = Ho f H, + He,. Ho is the kinetic electrons
energy of the in-wire motion
(1) of free
f(z - (fj +M
)
\
I
Fig. 1. Superlattice potential spatial coordinates.
Vol. 11, No. 2, 1992
and Microstructures,
/
(q +$a
(q+l>a
Fig. 2. Schematic representation of relevant distribution functions in a superlattice. Solid line represents the bottom of the conduction band (potential profile). Dashed lines depict the impurity distribution, and the wave function f(z - nia), respectively. nio is the center of a potential well and (ni + t)o is the center of the barrier.
with H, is an inter-wire
tunneling,
v,(k,,p,)
=
&(k.-P’)nyf++f_e-‘k’a)*
x (,f+ +
and H,, is responsible for electron-impurity scattering. In these formulas I;, is the parallel momentum in the wires, n is the wire number, cc(IC=) is the dispersion relation for in-wire motion, and t,,, is the inter-wire tunneling amplitude. The transfer between the wirrs is restricted to nearest neighbors. Since we assume the single scattering approximation, we consider only one impurity so that at the end of the calculations all appropriate quantities are simply multiplied by the number of impurities. We assume that the basis functions
The following notation position)
are products of Wannier functions localized on the superlattice wires in the z-direction with period a, with the free waves in parallel planes. These functions constitute a complete and orthogonal set; however, they are not the eigenfunctions of H, + Ht. The eigenfunctions of Ho + Ht, which are used to construct H,,, are the Bloch functions that are expanded in terms of the basis functions (4). The scattering potential from an impurity at (X,, 2;) is assumed to be the 6 function in the configuration space
where k, = E,
V,(Z, 2) = V&(x - X*)6(2 - 2;).
(7)
f-e-‘p=a) is used (where 2, is the impurity
f(Z, - na) = { p ; where n,n < Z, < (n, + l)a.
fp,h,“,,s;:: nz+’ Here f+ = f(zht$e + z,),
where z, = Z, - (n, + $)u and Iz,I < 4. Also
F = 0, 1, . . . . N - 1. A schematic
sentation of the relevant tice is shown in Fig. 2. B.
Impurity
distributions
repre-
in the superlat-
Averaging
We assume that the total distribution function of impurities factors into independent distributions in the parallel and t directions:
(5)
Consequently, the interaction part of the Hamiltonian in the tight-binding approximation is given by
where pL is a uniform r-distribution, p,(X,) = coast., and pz is a periodic z-distribution, pz( Z, +na) = pl(Zi), which preserves the symmetry of the superlattice. The functions F(X,, Z,) that we average with respect to the impurity distribution are bilinear forms of
Superlattices
and Microstructures,
the scattering potential, Eq. (7), according to the following equation
Vol. 1 I, No. 2, 1992 and they factorize
(10) where F,(X,)
= e’qrx8, Fr(n,)
O,l, . . . . N - 1, and F~(z*) = The average value of F is
I
wrth P(z,) = Np(:a
$0
a1 =<
jt
>& + <
a2 =< fzfa3 =< fif?
Fz(f+(zt), f-(2,)).
dZzdXzp(X,, Zz)F(Xi,
= L,o4,
where the a,‘s are determined
= e’qZnSa,with qZ = $$,
E=
201
2,)
3.
(11)
a/2 _-a,2 Xz,)&(z,)&,
J
A. Scattering
+ 2%) being the z-distribution
Transport
(17)
Properties
Time
nor-
Ps
B(P, e,,) = %V2
J
~li’(p.q,cl,p)Go(~,~“l~ (18) Green’s function, are: E,(Q,) = s2m, , and
is the unperturbed
and the dispersion
:v
relations
E,(Q,) = w(l - cosq,a). The scattering life-time
@ IV Pz_
f” >&is=2 < f; >b,,
>p,=< f3f+ >pz, >pz
where GO(qre,) t +
= j?=(z,))
The self energy, C(p, E,). in the Born approximation is
malized per wire. We are now in a position to calculate the impurity averaged scattering probability W illustrated in Fig. 3 PI +
by (P*(-z,)
1 __ = -2l?nC(p, T(P)
4 +P4
Fig. 3. A double scattering amplitude, Ws, from a single impurity (8) with a point like potential V. Averaging over the impurity distribution (consistent with the superlattice structure) restores momentum conservation, p1 + pz = pa + pd.
T(P) for EF > 210 reatls
1 to+) = $1
+ bcosp,o),
119)
where (20) and
I+. = N, < V,V, >,=, =
b_ b2(0)
~ti(Pl,P2,Ps,P4), (2T)2
6 P,+P?>Ps+P4
where the average impurity
density,
= W*(s, U, u) + Here
Here g(eF) n, = A,
(13)
Wz(s,
O<_h
is the density of states
per spin.
u, u).
u = pl, - pzr, v = psr - pd,, and
where K is the complete elliptic integral of the first kind. The average scattering time ‘e has the following form Te = (T.(q))Fermi surface = ~
M/,(s,u,u)
= lb, + bzcosy][bl
and W2(~,u,v)=b;cos-cos--.
(14)
+bz~~+>
‘5
A
ua
2)l7
2
2
(15)
Coefficients b; are functionals of the impurity distributions and functions of the total momentum s,
b,(s) =
d a,
bz(s) =
4a2 cos Ja b,(S) ’
(21)
and
= WP1*,Pz*,P3z.P4*)
WP,,PZ,P3.P4)
b,(O)
( 12)
+ 2a3 cos Sd
(16)
(23)
Note that one obtains the suppressed scattering results for b = 1 or P(Z) = 6(z). This is solely due to the pointlike scattering potential. B. Weak ductivity
Localization
Corrections
to the Con-
Weak localization corrections are determined by the diffusive term in the Cooperon, C (see Fig. 4). The Cooperon structure, as a function of external momenta, is
Superlattices
202
Fig. 4. Pictorial
representation
and Microstructures,
of the Cooperon. In the quasi-particle
+s-p
Vol. 11, No. 2, 1992
approximation
(ef
+s-q lim s ---t o.&
+ o+
X,JS,Gz,WP)
1 0
= i fl
Only Cii has a diffusive character ;s+p
>> 3) l=j=l, ifj. (=j=2.
for small s and ‘“‘I
;stq
Fig. 5. The Bethe-Salpeter equation for the two-particle scattering amplitude (Cooperon) in the maximally crossed diagram approximation.
The
diffusion
constant
in
the
r-direction
is
D, = (a:~)~~ = t~r,~~~(l - b*)-‘/’ for r:,,~ = consf on the Fermi surface (EF > 2~). The diffusion constant in the t-direction is more complicated, i.e. it consists of coherent and incoherent, (hopping) parts. D, = D,,, + D;,h. where
where by definition Ct,(s,P,s,%I,~e) Here U,(s,p)=
Gth
b,(s,)+ b~(sr)cospza, and Uz(s,p)=
b3(Sz) cospzc. We can write Bethe-Salpeter C,, as shown in Fig. 5 ) = n,L’W,(s.
C’3(S,P,%~,,,w’P
+ wV2~,(s, P) x Gii
(32)
= ~*)(S,E,,IJoU~(s,P)I;I(s,q)
J
d2r
-Ut(s, (‘*)*
+ r,c” +wt)
C
(BS)
equations
for
p)li,(s,p)d,, r)G(; - r, F,)
(25)
+ 1
< (ff
fif? >pz + f2J2 >& 1
D,_u = -.
2r
BS equation
From Eqs. (28) and (30) it follows that part of the Cooperon is
for c,,
(33)
which is the relative probability for inter-wirr, scattcring. There is an analogy of Dz,h to one-dimensional hoppmg diffusion. viz.
u2
Ckl(s,r,q,t,,dg).
k=i.L
and the corresponding
<
P Tl.“il
pn,,+ Prl,“il =
(34) the diffusive
(‘-6) i~-,k(S,En.U?P)~kJ(S,F,,.wP), x (1 + bcosp,a)(l + bcosq,a) Ia/ + Dzs; + D,sz,
k=I.Z
where
(35) Finally, the following result for the weak localization corrections to conductivity is obtained:
x G(i
+ r,
E,
+
WP)U~S, r)U,(s, r), (27)
with the following solutions
SuPertattices and Microstructures,
D D*,WL
=
Here, rpphis a dephasing the standard
‘CUz,WL D,
time which was introduced
replacement
by
Ju() -+ T;,
Also! au,~ has the following 1. scaling relation
203
Vol. 11, No. 2, 1992
properties:
tion cannot be relaxed without breaking the discrete translational symmetry of the whole periodic structure. Small perturbations of the periodic distribution are acceptable and will affect the total elastic scattering time in a form similar to that which describes surface scattering. Finally, a complete expression for the Cooperon in the magnetic field contains oscillatory terms which reflect the commensurability of the magnetic length and superlattice period. Such an effect was observed by Paris et ~1.‘~ and was interpreted as periodic replicas of the standard weak localization effect. This interesting effect requires more detailed, numerical studieq.
(37) 4. Conclusions 2. when D, ---* 0, CT,,\~‘Lapproaches coupled wires)
1-D limit
(de-
(38)
3. when D, is large (* > l), c~,~‘L acqtiires the standard form for 2-D anisotropic systems, i.e. e2 U,,WL
=
-py-&
D, dD,
In rpTph + const.
(39)
Upon neglecting the oscillating part of the Cooperon, the weak localization corrections to magnetoconductivity becomes huw~(B) z ow~(B)-uw~(O) with
Au,,wL(B)
=
&
Au,,~,
D
= ~Au=,,~, D,
where Bph is the characteristic cesses:
field for dephasing
A new tight-binding model of quantum corrections to the conductivity of a lateral semiconducting superlattice is proposed. The effects of electron wave function modulation and spatial distribution of impurities were studied. The main results of this work can be summarized as follows: 1. The superlattice structure causes an effective anisotropic elastic scattering time for isotropic scatterers. 2. Quantum corrections to conductivity is suppressrcl by a hopping-like diffusion. 3. Scaling relations are satisfied by the anisotropic conductivity tensor for a model that does not belong to the Bhatt et aL7 class of anisotropic models (anisotropic mass tensor, anisotropic scatterers). Acknowledgement - This material is based in part on work supported by the National Science Foundation (DMR 8800359). One of the authors (C.J.) acknowledges partial travel support from the Polish Research Council (grant # 2 0401 91 01).
(40) References
pro-
B. L. Al’tshuler and A. G. Aronov, Pis’ma Eksp. Teor. Fiz. 33, 515 (1981) [JETP Lett. 499 (1981)]. At this point we summarize with a few comments about the crucial assumptions made in the above model. We employed a &like scattering potential (commonly used in weak localization theories) which allowed us to obtain the final results in an analytical form. But we caution the reader that this form may be a rather rough approximation for more realistic potentials such as those associated with weakly screened charged impurities. As shown in our previous work,‘j the effect of a nonuniform distribution of long-range scattering centers is, at least qualitatively, equivalent to the effect caused by a more uniform distribution of point-like impurities. Thus, the scattering distribution may be better taken into account by introducing an effective impurity distribution instead of a geometrical distribution. Another important assumption concerns the periodicity of the impurity distribution. This assump-
Zh. 33,
C.W. Beenakker and H. van Houten, Phys. Rev. B 38, 3232 (1988). Y. Takagaki, K. Ishibashi, S. Ishida, S. Takaoka, K. Game, K. Murase, and S. Namba. Jap. J. Appl. Phys. 28, 645 (1989). W. Szott, C. Jedrzejek, Rev. B 40, 1790 (1989). W. Szott, Rev. Lett. W. Szott, Rev. B (in
and W. P. Kirk,
Phys.
C. Jedrzejek, and W. P. Kirk, 63, 1980 (1989). C. Jedrzejek, and W. P. Kirk. press).
Phys. Phys.
R. N. Bhatt, P. WGlfle, and T. V. Ramakrishnan. Phys. Rev. B 32, 569 (1985). L. N. Pfeiffer, K. W. West, H. L. StGrmer, J. P. Eisenstein, K. W. Baldwin, D. Gershoni, and 3. Spector, Appl. Phys. Lett. 56, 1697 (1990).
204 ’ H. L. Stiirmer, L. N. Pfeiffer, K. W. West, and K. W. Baldwin, m Proceedings of the International Symposium Nanoslructures and Mesoscopic Systerns, (eds. W. P. Kirk and M. A. Reed, Academic Press, 1991), p. 51.
Superlattices
and Microstructures,
Vol. 7 7, No. 2, 1992
lo E. Paris, J. Ma, A. bl. Kriman, D. K. Ferry, and E. Barbier, in Proceedings of the International Symposium Nanostructures and Mesoscopic S~~stems~ (eds. W. P. Kirk and M. A. Reed, Academic Press, 1991), p. 311.