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Diffuse Bragg Scattering in Corrugated Quantum WellsYisong ZHENG∗ and Tsuneya ANDO
Department of Physics, Tokyo Institute of Technology2–12–1 Ookayama, Meguro-ku, Tokyo 152-8551
A correlation function is derived for interface roughness in lateral superlattices realized incorrugated quantum wells. It is given by a sinusoidal function with an exponential decay in thedirection of the superlattice. Its validity is examined by a model in one dimension. The anisotropicconductivity is calculated by solving the Boltzmann transport equation exactly in the absence ofa magnetic field and in a self-consistent Born approximation in high magnetic fields.
Keywords: quantum wire, lateral superlattice, interface roughness, anisotropic mobility, Boltzmann equa-tion, self-consistent Born approximation
§1. Introduction
Recently, a new periodic quantum-wire array wasformed during growth of a GaAs/AlAs heterointerface onGaAs (775)B substrate by molecular beam epitaxy.1−5)
Under suitable growth condition, a zigzag heterointerfacecomes into being instead of a flat plane. It is consideredas an ideal structure for the study of transport proper-ties of the periodically modulated two-dimensional (2D)electron gas, since it possesses a good interface qualityand a period much shorter than conventional lateralsuperlattices. The purpose of this paper is to elucidateeffects of disorder in the corrugation on the transport inboth absence and presence of a magnetic field.
Transport experiments were started quite recentlyand high anisotropy of the electron mobility has beenobserved.5,6) Experiments show that the longitudinalmobility parallel to the quantum-wire direction is muchlarger than the transverse mobility. A possible origin ofanisotropic conductivity is the anisotropic effective mass.In fact the mass anisotropy is a dominant origin of theanisotropic conductivity, for example, in bulk Si or Ge,7)
and in Si metal-oxide-semiconductor structures at the(110) interface.8,9) In the present system, however, thecalculation gives an almost circular and isotropic Fermiline for the usual electron density and therefore the massanisotropy cannot give rise to considerable effects on theelectron mobility.10)
Anisotropic conductivity has been observed also insystems with an isotropic mass. A large anisotropy wasfound in the electron conductivity of an AlxGa1−xAs/GaAs heterostructure on a (001) substrate,11,12) in agrid-inserted quantum well with a thin AlAs layer de-posited in the well region,13) and in an AlAs/GaAs frac-tional layer superlattice.14) Anisotropic transport wasstudied for electrons in heterostructures with a periodicinterface corrugation on a (311)B GaAs substrate andfor holes on a (311)A.15−19) A periodic corrugation wasshown to be formed and a large anisotropic mobility wasobserved also on vicinal (111)B GaAs.20−22) Anisotropyin optical properties was observed in quantum-wire ar-rays23) and in grid-inserted quantum wells.24)
A mobility anisotropy originating from anisotropicinterface roughness11) and from nearly straight chain-like potential were studied theoretically.12) Anisotropictransport in systems with lateral superlattice potential
with randomness was discussed within a model corre-lation function assumed somewhat arbitrarily.25,26) ABoltzmann transport equation was solved exactly foranisotropic scatterers and their effects on anisotropictransport were discussed.27) Effects of anisotropy onweak-localization correction to the conductivity werealso discussed.28,29)
In this paper, we shall study anisotropic transport indense quantum-wire arrays made of corrugated quantumwells. In §2 a correlation function of the corrugationwith small fluctuations is obtained analytically. In §3 theconductivity tensor is calculated by solving a Boltzman-n transport equation and compared with approximateresults. In §4 the conductivity is calculated in highmagnetic fields. The results are summarized in §5. InAppendix A a model of corrugations with some disorderis assumed and the correlation function is calculatedexplicitly for the purpose of a demonstration of thevalidity of the analytic expression.
§2. Fluctuations of Corrugations
We shall consider a quantum-wire arrays on GaAs(775)B as described in Fig. 1. The bottom flat heteroin-terface is chosen as the xy plane (z=0), a quasi periodiccorrugation ∆(r) is in the x direction, and therefore, thequantum-wire direction is in the y direction. In an idealcase the corrugation is perfectly periodic and thereforecan be expanded into Fourier series:
∆(r) =∑
n
∆n exp(i2πnx
a
), (2.1)
where a is the period of the corrugation, ∆n is the Fouriercoefficients, and n is an integer. We have
∆n =
∆( 1
2πn
)2 a2
a1a2
[1−exp
(−i
2πna1
a
)](n �=0);
0 (n=0),(2.2)
where the origin x = 0 is chosen at a valley of thecorrugation and x = a1 at the peak or ridge position.As described in Fig. 1, a=a1+a2 is the period, d1 and d2
are the well thickness at the valley and the peak of thecorrugation, respectively, and ∆ = d2−d1. The typicalvalues of the parameters are ∆ = 1.2 nm, a = 12 nm,
Submitted to Journal of Physical Society of Japan
Page 2 Y. Zheng and T. Ando
a1 =4 nm, and a2 =8 nm.We consider the situation that the period a exhibits
a fluctuation whose amount is much smaller than theperiod itself, i.e., αj(y)/a � 1, where a+αj(y) denotesthe period of the jth cell at y. We assume further thatthere is approximately no correlation between αj(y) ofdifferent j and also that αj(y) varies slowly in the y
direction. Let ξj(y) be the x coordinate of the valleyof the jth cell for y. Then, we have
ξj(y) =j−1∑l=0
[a+αl(y)] + ξ0(y). (2.3)
This means that ξj(y) exhibits a fluctuation varying s-lowly in the y direction but the amount of the fluctuationitself is larger than a for j�1, i.e., 〈[ξj(y)−〈ξj(y)〉]2〉1/2 >
a, where 〈· · ·〉 denotes a sample average. The situationis illustrated schematically in Fig. 2.
In the jth period, we can expand the corrugationinto the Fourier series,
∆(r) =∑
n
∆n(r) exp(i2πn[x−ξ(r)]
a+α(r)
)
≈∑
n
∆n(r) exp[i2πn[x−ξ(r)]
a
(1−α(r)
a
)],
(2.4)where ξ(r)≡ξj(y), α(r)≡αj(y), and ∆n(r)≡∆n(j, y) isthe Fourier coefficient of the corrugation which variesslowly in the y direction. We should note that |x−ξ(r)|<∼ a, and therefore, we can safely neglect the fluctu-ation in the period, i.e., the term proportional to α(r)/a
inside the bracket.
∆(r) ≈∑
n
∆n(r) exp(i2πn[x−ξ(r)]
a
). (2.5)
Except for the term with n=0, the Fourier coefficient ∆n
has a nonzero average and its fluctuation is much smallerthan the average for small randomness. Therefore, wecan neglect the dependence of ∆n on the position forn �=0, i.e.,
∆n(r) = ∆n, (n �=0). (2.6)
It is clear that the average of the corrugation van-ishes, i.e., 〈∆(r)〉=0. The correlation function becomes
D(r−r′) ≡ 〈∆(r)∆(r′)〉 ≈⟨ ∑
n
∑n′
∆n(r)∆∗n′(r′)
× exp(i2πn[x−ξ(r)]
a
)exp
(− i
2πn′[x′−ξ(r′)]a
)⟩.
(2.7)There is essentially no correlation between ∆0(r) andξ(r′) and therefore the cross terms with n �= 0 andn′ = 0 or n′ �= 0 and n = 0 in the above equation allvanish identically. Further, ξ(r′) itself and its spatialvariation ξ(r)− ξ(r′) are essentially independent andtherefore after the average over ξ(r′) the terms withn �= n′ vanish identically. Consequently, the correlation
function becomes
D(r−r′) = 〈∆0(r)∆0(r′)〉
+∑n �=0
|∆n|2⟨
exp(i2πn[x−x′−(ξ(r)−ξ(r′))]
a
)⟩.
(2.8)For a slowly-varying fluctuation of the corrugation
in the y direction, we have
ξ(r)−ξ(r′) ≈ ξ(x, y′)−ξ(x′, y′) − (y−y′)∇yξ, (2.9)
where ∇yξ is the local tilt angle of the corrugation, asshown in Fig. 2. Let N be an integer giving the totalnumber of corrugations present between the point (x′, y′)and (x, y′). Then, we have
ξ(x, y′)−ξ(x′, y′) =
Na +N−1∑j=0
αj , (|x−x′|>∼ a)
0, (|x−x′|<∼ a)(2.10)
where a+αj is the period of the jth corrugation startingfrom x corresponding to j =0.
Let α be the fluctuation of the period, i.e.,
α =√〈α2
j 〉. (2.11)
In the limit of small α, we have to the lowest order inα/a
⟨exp
(i2πn
aαj
)⟩= exp
[− 1
2
(2πn
a
)2
α2]. (2.12)
Then, we have
⟨exp
[i2πn
a
N−1∑j=0
αj
]⟩= exp
[− 1
2
(2πn
a
)2
α2N]
≈ exp[− 1
2
(2πn
a
)2
α2 |x−x′|a
].
(2.13)The last approximate equality holds for |x−x′|>∼ a. Inthe case of weak fluctuations α/a�1, however, we have−(1/2)(2πn/a)2α2|x−x′|/a � 1 for |x−x′|<∼ a exceptfor extremely large n. Because terms with large n canusually be neglected, the above can be used for the wholerange of |x−x′|. Therefore, we have
⟨exp
(i2πn[ξ(r)−ξ(r′]
a
)⟩≈ exp
[− 1
2
(2πn
a
)2
α2 |x−x′|a
]
×⟨exp
[− i
2πn
a(y−y′)∇yξ
]⟩.
(2.14)The above derivation shows that the results so far arevalid for any distributions of αj as long as α/a�1.
We consider the situation that ∇yξ exhibits a Gaus-sian distribution and let
∇ξ =√
〈(∇yξ)2〉 (2.15)
be the fluctuation of the tilt angle. Then, we have⟨exp
[−i
2πn
a(y−y′)∇yξ
]⟩= exp
[−1
2
(2πn
a
)2
∇ξ2(y−y′)2].
(2.16)
Diffuse Bragg Scattering in Corrugated Quantum Wells Page 3
The correlation function becomes
D(r−r′) = 〈∆0(r)∆0(r′)〉
+∑n �=0
|∆n|2 exp[i2πn(x−x′)
a
]
× exp[− 1
2
(2πn
a
)2
α2 |x−x′|a
]
× exp[− 1
2
(2πn
a
)2
∇ξ2(y−y′)2].
(2.17)
Next, we consider 〈∆0(r)∆0(r′)〉 in the case thatthe average interface position of a corrugation periodfluctuates independently from each other. In this case,〈∆0(r)∆0(r′)〉 vanishes when r and r′ belong to differentcorrugations and becomes 〈∆2
0〉 when they belong tothe same corrugation. Further, there is essentially nocorrelation between ∇yξ and ∆0(y). Then, we have
〈∆0(r)∆0(r′)〉 = 〈∆0(y)∆0(y′)〉 〈g[x−x′−(y−y′)∇yξ]〉,(2.18)
with
g(x) =
1− |x|a
; (|x| ≤ a)
0. (|x| > a)(2.19)
In terms of the Fourier transform
g(x) =∫
dq
2πg(q) exp(iqx), (2.20)
with
g(q) =4a
(qa)2sin2 qa
2, (2.21)
we have
〈∆0(r)∆0(r′)〉 =∫
dq
2π〈∆0(y)∆0(y′)〉 g(q)
× 〈exp(iq[x−x′−(y−y′)∇yξ]
)〉.(2.22)
For the Gaussian distribution of ∇yξ, we have
〈∆0(r)∆0(r′)〉 =∫
dq
2π〈∆0(y)∆0(y′)〉 g(q)
× exp[iq(x−x′)] exp[− 1
2∇ξ2q2(y−y′)2
].
(2.23)
Assuming a conventional Gaussian distribution
〈∆0(y)∆0(y′)〉 = ∆20 exp
[− (y−y′)2
λ2
], (2.24)
with
∆0 =√〈∆0(y)2〉, (2.25)
we have the following final expression
〈∆0(r)∆0(r′)〉 = ∆20
∫dq
2πg(q) exp[iq(x−x′)]
× exp[− 1
2
(∇ξ2q2+
2λ2
)(y−y′)2
].
(2.26)It is straightforward to make extension toward the
case in which there is some correlation between the av-erage corrugations of neighboring periods. For example,
we can add a term with a conventional Gaussian form
(∆′0)
2 exp[− (x−x′)2
λ2x
− (y−y′)2
λ2y
], (2.27)
depending on the type of fluctuations of the corrugation,where obviously λx > a and λy > a. Effects of sucha term in the anisotropic transport have been studiedin previous works11,13,15,27) and therefore will not bediscussed in the following.
The full expression of the correlation function is
D(r−r′) = ∆20
∫dq
2πg(q) exp[iq(x−x′)]
× exp[− 1
2
(∇ξ2q2+
2λ2
)(y−y′)2
]
+∑n �=0
|∆n|2 exp[i2πn(x−x′)
a
]
× exp[− 1
2
(2πn
a
)2
α2 |x−x′|a
]
× exp[− 1
2
(2πn
a
)2
∇ξ2(y−y′)2].
(2.28)
The most important feature of the correlation functionis the exponential decay in the x direction and that thedecay rate is determined by the amount of fluctuations inthe period. The correlation function is characterized byfour parameters α, ∇ξ, ∆0, and λ. They are summarizedas follows:
α : fluctuations of the corrugation period∇ξ : fluctuations of the local quantum-wire direction∆0 : fluctuations of the height of the corrugation
averaged locallyλ : correlation length of the fluctuations of the
height of the corrugation averaged locally in thequantum-wire direction (the correlation lengthis of the order of a in the perpendicular direc-tion)
This expression is different from a simplified approx-imate expression
D(r−r′) = ∆(r)∆(r′) exp[− (x−x′)2
λ2x
− (y−y′)2
λ2y
]
=∑
n
|∆n|2 exp[i2πn(x−x′)
a
]
× exp[− (x−x′)2
λ2x
− (y−y′)2
λ2y
],
(2.29)assumed in refs. 25 and 26, where ∆(r)∆(r′) is the cor-relation function in the absence of randomness averagedover a period.
The Fourier transform D(q) is given by
D(r) =∫
dq
(2π)2D(q) exp(iq ·r). (2.30)
Page 4 Y. Zheng and T. Ando
Then, we have
D(q) = ∆20 g(qx)
√2π
( 2λ2
+∇ξ2q2x
)−1/2
× exp[− 1
2
( 2λ2
+∇ξ2q2x
)−1
q2y
]
+∑n �=0
|∆n|2(2πn
a
)2 α2
a
[(qx− 2πn
a
)2
+14
(2πn
a
)4 α4
a2
]−1
×√
2πa
2π|n|∇ξexp
[− 1
2
( a
2πn
)2 q2y
∇ξ2
].
(2.31)In a simple one-dimensional model, i.e., for y = y′, wehave
D(q) = 〈∆20〉 g(q) +
∑n �=0
|∆n|2(2πn
a
)2 α2
a
×[(
q− 2πn
a
)2
+14
(2πn
a
)4 α4
a2
]−1
,
(2.32)
by integrating out the above expression over qy.In the case of the present corrugation, the tilt angle
θ1 and θ2 are determined by the microfacets appearingduring the growth process. However, the height of thepeak and valley of the corrugation is likely to vary al-most at random between neighboring corrugations. Thesituation is illustrated in Fig. 3. For such a randomfluctuation of the bottom and peak heights ∆z=
√〈z2〉,we have
∆0(∆z)2 =∆2
4
[1 +
(a1
a
)2
+(a2
a
)2]∆z2
∆2,
α(∆z)2 = 2(a21−a1a2+a2
2)∆z2
∆2.
(2.33)
In Appendix A, we shall explicitly calculate thecorrelation function numerically within such a modelin one dimension. The analytic correlation functionobtained above turns out to reproduce the numericalresults quite well except that the amount of fluctuationsof the period can be larger than the above expression fora given ∆z =
√〈z2〉. Therefore, we shall increase thefluctuation by introducing a constant factor β such that
α = β α(∆z). (2.34)
In the following we shall use correlation functions withthese parameters for explicit numerical calculations ofthe conductivity.
Figure 4 shows some examples of the correlationfunction with β =1.3, λ/a=5, and ∇ξ=0.1 [∆z/∆=0.1in (a) and 0.2 in (b)]. An important feature is the pres-ence of a large Bragg peak at (±2π/a, 0) broadened inthe qx direction by a Lorentzian function correspondingto the exponential decay in the real space. In the qy
direction the correlation is essentially given by a Gaus-sian function, but its actual form varies as a functionof qx. By changing the four parameters we can actuallysimulate various cases from those with a Bragg peak at(±2π/a, 0) much higher than the peak at the origin tothose with negligible Bragg peaks. The addition of eq.(2.27) may increase the freedom considerably.
§3. Anisotropic ConductivityThe Boltzmann transport equation for the distribu-
tion function fk is given by
dk
dt· ∂fk
∂k= −
∫dk′
(2π)2[fk(1−fk′)−fk′(1−fk)]W (k′, k),
(3.1)with
W (k′, k) =2π
h〈|Vk′k|2〉δ(εk−εk′), (3.2)
where Vk′k is the matrix element of scattering potential.To the lowest order in the applied electric field E, wehave
fk = f0(εk) + gk,
dk
dt· ∂fk
∂k= (−e)E ·vk
∂f0
∂εk,
(3.3)
where vk is the velocity, f0(εk) is the Fermi distributionfunction, εk = h2k2/2m with m being the effective mass,and gk is proportional to E. The transport equation isrewritten as
eE ·vk
(− ∂f0
∂εk
)= −
∫dk′
(2π)2(gk−gk′)W (k′, k). (3.4)
In the present quantum-well system, the effectivepotential was shown to take the form as
V (r) = Feff ∆(r) (3.5)
with
Feff = −V0 |ζ(d)|2, (3.6)
where ζ(z) is the wavefunction of the ground state in anideal quantum well with width d=(d1+d2)/2 and V0 isthe height of the barrier potential.30,31,10) Note that
Feff =∂E0(d)
∂d, (3.7)
where E0(d) is the energy of the lowest subband in thequantum well with width d. In a single heterostructure,Feff is the effective field at the interface approximatelygiven by
Feff =4πe2
κ
(12n+ndepl
), (3.8)
where κ is the static dielectric constant, n is the electronconcentration per unit area, and ndepl is the concentra-tion of charges in the depletion layer.32−35)
It is sufficient to consider only the case that theelectric field is in two principal directions x and y,because the conductivity tensor in a field with an ar-bitrary direction can be obtained by a simple coordinatetransformation. Define gx
k and gyk as gk for the field in
the x and y directions, respectively. The correspondingconducitivity will be denoted by σxx and σyy. We shallfirst consider the case E = (E, 0). Define the relaxation
Diffuse Bragg Scattering in Corrugated Quantum Wells Page 5
time τk by
1τk
=∫
dk′
(2π)22π
h〈|Vk′k|2〉δ(εk−εk′), (3.9)
and hxk by
gxk = (−e)τkEhx
k
(− ∂f0
∂εk
). (3.10)
Then, the transport equation is written as
hxk = vx
k +∫
dk′
(2π)2τk′hx
k′2π
h〈|Vk′k|2〉δ(εk−εk′). (3.11)
There are various ways to solve this equation exactlyfor anisotropic scattering.36−38,11) In this paper, it issolved by a simple iteration starting with hx
k′ =vxk′ in the
right hand side. The number of iterations can becomequite large when the diffuse Bragg peak starts to makean appreciable contribution to the scattering, but thecalculation is quite straightforward and requires only ashort CPU time.
A conventional way to obtain an approximate solu-tion is to assume
hxk = τxvx
k/τ, (3.12)
with the average relaxation time τ defined by
1τ
=1
D(ε)
∫dk
(2π)2δ(ε−εk)
1τk
, (3.13)
where D(ε) is the density of states without spin andvalley degeneracy. This approximation is known to givea lower bound of the conductivity due to a variationalprinciple. Multiplying vx
k and taking the average overthe direction of k, we have
1τx
= 21
D(ε)
∫dk
(2π)2δ(ε−εk)
∫dk′
(2π)2
× v−2k [(vx
k)2−vxkvx
k′ ]2π
h〈|Vk′k|2〉δ(εk−εk′),
(3.14)
with vk = |vk|. Similar equations can be obtained forgy
k, hyk, and τy . In explicit numerical calculations the
conductivity will be measured in units of ne2τ0/m, whereτ0 is the average relaxation time in the limit of small kF
such that kF a/π�1.Figure 5 shows calculated conductivity in the x and
y direction as a function of the Fermi wave vector kF
obtained by an exact solution of the Boltzmann equa-tion. With the increase of kF the conductivity in the ydirection parallel to quantum wires increases drastically,while that in the perpendicular x direction exhibits muchsmaller increase. Such kF dependence is a result of thereduction in the backscattering intensity because of theslowly varying random potential. When kF becomesclose to π/a, diffuse Bragg scattering with wave vectoraround (±2π/a, 0) starts to play dominant roles. As aresult, the conductivity in the x direction drops downconsiderably. It is interesting that the conductivity inthe y direction exhibits also an appreciable decrease.
The figure contains the average relaxation time τdefined in eq. (3.13). The conductivity in the x directionis similar to the simplest expression σ = ne2τ/m while
that in the y direction is not. The figure contains also theresult obtained by the approximation discussed above.The approximation seems to work reasonably well for theconductivity in both directions. Note that the anisotropyis sensitive to the parameter ∇ξ (fluctuations in the tiltangle) and becomes larger with the decrease of ∇ξ.
Figure 6 shows some examples of the Boltzmann dis-tribution function obtained numerically as a function ofangle θ defined by kx =kF cos θ and ky =kF sin θ. WhenkF ∼ π/a, the distribution function deviates from thesimple sinusoidal function [see eq. (3.12), for example]expected for isotropic cases. The deviation is particularlyevident for gx
k. In fact, the maximum at θ =0 (and alsoat θ=π) for small kF turns into a dip when akF /2π >∼ 0.4and the dip splits into two when akF > 0.5, due to thediffuse Bragg scattering corresponding to ∆kx ∼ 2π/aand ∆ky ∼ 0 where ∆k is the momentum transfer ofscattering. This Bragg scattering is most effective whenakF /2π is slightly larger than 1/2, which is clear also inFig. 5. The Bragg scattering affects also the distributionfunction gy
k considerably, although a prominent peak(suppressed from a sinusoidal function) is present atθ = π/2 (also at θ = 3π/2). It is quite surprising thatthe simple approximation gives a conductivity not sodifferent from the exact result even if the distributionfunction is completely different. The reason presumablylies in the variational principle as mentioned above.
Experimentally, the well-width dependence of theelectron mobility and its anisotropy were measured incorrugated quantum wells with a = 12 nm.5) The low-temperature mobility exhibits a dependence on the wellwidth roughly proportional to F 2
eff in particular for sys-tems with small d where Feff is given by eq. (3.6). Thisis consistent with the fact that the dominant scatteringmechanism is the corrugation. However, the mobilityanisotropy seems to vary as a function of the well widthin a somewhat irregular manner, i.e., µy/µx = 5.4 ford=10 nm, 9.0 for d=7 nm, 71 for 5 nm, and 27 for d=4nm, where µx and µy are the mobilities in the x and ydirections, respectively.
We have akF /2π ≈ 0.2 for d = 4 nm, 0.25 for d = 5nm, 0.28 for d=7 nm, and 0.3 for d=10 nm. Therefore,the large anisotropy ratios 27 and 71 for d = 4 nmand 5 nm, respectively, are consistent with the resultshown in Fig. 5. However, the reduction of the ratiofor wider quantum wells cannot be explained, suggestingthat other more isotropic scattering mechanisms areimportant there. A quantitative comparison may bepossible if we assume other scattering mechanisms suchas ionized donors and interface roughness at the bottombarrier with proper screening effect. This requires preciseknowledge of scatterers and is left for a future study.
The dependence of the anisotropic mobility of theelectron concentration was calculated previously in amodel correlation function given by eq. (2.29) in grid-inserted quantum wells.26) The results showed a consid-erable reduction of the conductivity in the perpendiculardirection around akF /2π∼1/2. The qualitative behaviorof the reduction itself is essentially independent of details
Page 6 Y. Zheng and T. Ando
of the correlation function as long as it has considerableBragg peaks at (±2π/a, 0).
§4. High Magnetic Fields
In a high magnetic field with strength H , the broad-ening of the Landau level with index N (N =0, 1, . . .) isgiven in a self-consistent Born approximation39) by
Γ2N = 4
∫dq
(2π)2F 2
effD(q)JNN (q)2, (4.1)
where
JNN(q) = exp(− l2q2
4
)LN
( l2q2
2
), (4.2)
with l being the magnetic length defined by l2 = ch/eHand LN(x) being the Laguerre polynomial. The exten-sion of the general theory of quantum transport to thecase of anisotropic scatterers is straightforward.40) Thediagonal conductivity is given by
σxx =e2
π2h
∫dE
(− ∂f0
∂E
) ∑N
(ΓxxN
ΓN
)2[1−
(E−EN
ΓN
)2],
(4.3)where EN is the energy of Landau level N and
(ΓxxN )2 = 2
∫dq
(2π)2(qyl)2F 2
effD(q)JNN (q)2. (4.4)
The conductivity σyy is obtained by replacement of ΓxxN
by ΓyyN with
(ΓyyN )2 = 2
∫dq
(2π)2(qxl)2F 2
effD(q)JNN (q)2. (4.5)
The anisotropy of scattering does not have appreciableinfluence on the Hall conductivity σxy =−σyx≈−1/nec.
In the short-range case a/l�1, we can replace D(q)by D(0) and have
Γ2N = Γ2 ≡ 4
2πl2F 2
effD(0),
(ΓxxN )2 = (Γyy
N )2 =4
2πl2F 2
effD(0)(N +
12
),
(4.6)
which gives(Γxx
N
ΓN
)2
=(Γyy
N
ΓN
)2
= N +12. (4.7)
This gives a peak value of the diagonal conductivitydependent only on universal quantities apart from theLandau-level index N .41,39)
Figure 7 shows some examples of the peak value ofthe conductivity and broadening of the lowest Landaulevel as a function of the magnetic field (l/a)2. Thebroadening is essentially independent of the field. Thediagonal conductivity in the direction parallel to thequantum-wire direction increases or remains almost samewith the increase of the field, while that in the perpendic-ular direction decreases drastically with the field. Whenthe magnetic length becomes comparable to or largerthan the array period, the anisotropy becomes consid-erable. The reason is that the scattering correspondingto the momentum transfer (±2π/a, 0) mainly causes the
diffusion of the center coordinate in the direction parallelto the wire direction but not in the perpendicular direc-tion. This large difference in the diffusion constant is theorigin of the anisotropic conductivity in high magneticfields.
A large anisotropy in high magnetic fields was ob-served experimentally.6) For (a/l)2∼4 experiments showanisotropy σyy/σxx ∼ 30, which is consistent with thepresent result shown in Fig. 7. For a quantitative com-parison, effects of other scattering mechanisms such asdonors and their screening determined self-consistentlyshould also be considered.33,42,43) Further, experimentalstudy on the dependence on various parameters such asthe electron concentration and the quantum-well widthis highly expected.
§5. Summary and ConclusionIn summary, a correlation function has been derived
for interface roughness in lateral superlattices realized incorrugated quantum wells. It has been given by a sinu-soidal function with an exponential decay in the directionof the superlattice. Its validity has been examined by amodel in one dimension. The anisotropic conductivityhas been calculated by solving the Boltzmann transportequation exactly in the absence of a magnetic field andin a self-consistent Born approximation in high magneticfields. When other scattering mechanisms are neglected,corrugations with small fluctuations in the period and inthe local direction give rise to large anisotropy in bothpresence and absence of a magnetic field.
AcknowledgmentsThis work was supported in part by Grants-in-Aid
for Scientific Research and for COE (12CE2004 “Controlof Electrons by Quantum Dot Structures and Its Appli-cation to Advanced Electronics”) from Ministry of Edu-cation, Culture, Sports, Science and Technology Japan.Numerical calculations were performed in part using thefacilities of the Supercomputer Center, Institute for SolidState Physics, University of Tokyo.
Appendix A: One-Dimensional ModelsIn this appendix a correlation function is calculated
for a model one-dimensional corrugation explicitly. Weassume that the tilt angles of the corrugation, θ1 and θ2
in Fig. 1, are fixed because they are determined by thecrystal plane which forms most easily in the epitaxialgrowth process. A small randomness is introduced in a1
and a2 and therefore in the period a=a1+a2.One way to realize this is to assume that the height
and depth of the corrugation fluctuate at random with aGaussian distribution with width ∆z around the averagevalues and determine the form of a random corrugationaccordingly as illustrated in Fig. 3. It is possible tointroduce fluctuations in a1 and a2 directly. Therefore,we shall combine these cases by introducing a parameterp such that p represents the probability of the fluctuationbeing determined by fluctuation (a1/a)∆a in a1 and(a2/a)∆a in a2 and 1−p the probability of the fluctuationdetermined by that of the top and bottom height ofthe corrugation ∆z. Note that it is quite unrealistic
Diffuse Bragg Scattering in Corrugated Quantum Wells Page 7
that the x coordinate of the peak and valley positionsis distributed around its mean position.
Figure 8 shows some examples of the correlationfunction of the corrugated interface obtained in this mod-el. For p = 0, i.e., when the randomness is determinedby that of the peak and valley heights, ∆z, the largestpeak is at 2π/a followed by a small peak at 4π/a. Theintensity ratio of these Bragg peaks is well explained byeq. (2.32) with |∆n|2 given in eq. (2.2). The fundamentalBragg peak has a long tail in the long-wavelength regionwhere the correlation function is almost independent ofthe wave vector. With the increase of p, the randomnessin the average interface position becomes larger than thatspecified by ∆z and a peak starts to grow at q =0. Forp=1, in particular, D(0) becomes extremely large.
Figure 9 shows an example of actual corrugation forp = 0 and p = 1. When p = 1, the average height itselfhas a large fluctuation and also the correlation functiondecays very slowly with the distance. This behavioris responsible to the sharp peak at q = 0 mentionedabove. Actual systems are not likely to have such alarge fluctuations in the average height because of theepitaxial growth and therefore the results for large p areunrealistic. It is clear that the correlation for small p canbe expressed by the analytic expression (2.32) quite wellby an appropriate choice of the parameters.
Figure 10 shows the correlation function in the realspace together with exponential functions ±A exp(−κ|x−x′|) fitted to the peaks and dips of the oscillation. Thefitting works particularly well for small p, demonstratingthe exponential decay. The decay rate κ is slightly (by afactor of >∼1.3 for p = 0) larger than [2πα(∆z)/a]2/2agiven by eq. (2.32) with eq. (2.33). The correlationfunction for p=1 shows a completely different behaviorbecause of the reason mentioned above.
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Page 8 Y. Zheng and T. Ando
(1985) 1519.
Figure CaptionsFig. 1 A schematic illustration of a periodic quantum-
wire array consisting of a GaAs/AlAs heterostruc-ture. The quantum wires are along the y direction.
Fig. 2 A quantum-wire array with small fluctuationsin the period and the direction. Thick and thinsolid lines show valleys in arrays with and withoutrandomness, respectively. The x coordinate of thevalley position associated with r =(x, y) is denotedby ξ(r).
Fig. 3 A corrugation with fluctuations in the heightof the peak ∆jp and valleys ∆jv. These positionsdetermine the x positions of the peak and valleysfor fixed θ1 and θ2.
Fig. 4 Examples of the correlation function for ∇ξ =0.1, λ/a = 5, and β = 1.3. (a) ∆z/∆ = 0.1 and (b)0.2.
Fig. 5 Examples of calculated conductivity as a func-tion of akF /2π for ∇ξ = 0.1, λ/a = 5, and β = 1.3,where kF is the Fermi wave vector. (a) ∆z/∆=0.1and (b) 0.2. The solid lines represent the resultsobtained by the exact solution of the Boltzmanntransport equation, the dotted lines those by theapproximate (variational) solution, and the dashedline represents the relaxation time averaged over theequi-energy line. In this figure, τx and τy are thesame as σxx and σyy obtained approximately.
Fig. 6 Examples of the distribution functions, gx and
gy for the electric field perpendicular and parallelto the quantum-wire direction, respectively, as afunction of angle θ (kx =kF cos θ and ky =kF sin θ),obtained by the exact solution of the transportequation for ∇ξ = 0.1, λ/a = 5, and β = 1.3. (a)∆z/∆ = 0.1 and (b) 0.2. The distribution functiondeviates considerably from a simple sinusoidal func-tion in isotropic cases when akF /2π∼1/2.
Fig. 7 Examples of the peak values of the diago-nal conductivity and the broadening of the lowestLandau level obtained in the self-consistent Bornapproximation for ∇ξ = 0.1, λ/a = 5, and β = 1.3.(a) ∆z/∆=0.1 and (b) 0.2. The conductivity in thedirection perpendicular to quantum wires decreasesrapidly with the increase of the magnetic field, whilethat in the parallel direction exhibits a peak due todiffuse Bragg scattering in high magnetic fields.
Fig. 8 Examples of the Fourier transform of the correla-tion function of the corrugation in a one-dimensionalmodel. (a) ∆z/∆ = 0.1 and ∆a/a = 0.1. (b)∆z/∆=0.2 and ∆a/a=0.2.
Fig. 9 Some examples of actual corrugations in a one-dimensional model for ∆z/∆=0.1 and ∆a/a=0.1.
Fig. 10 Examples of the real-space correlation functionof the corrugation in a one-dimensional model for∆z/∆ = 0.1 and ∆a/a = 0.1 together with a fit ofthe envelope to a single exponential function. Theactual value of the curve for p = 1 should actuallybe multiplied by 10.
Diffuse Bragg Scattering in Corrugated Quantum Wells Page 9
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Page 10 Y. Zheng and T. Ando
0.0 0.5 1.0 1.5 2.0 2.50.0
0.5
1.0
1.5
Wave Vector (units of 2π/a)
Cor
rela
tion
Fun
ctio
n (u
nits
of ∆
2 a2 )
a1/a = 1/3a2/a = 2/3∆z/∆ = 0.10β = 1.3α/a = 0.106∆0/∆ = 0.062ξ = 0.10
λ/a = 5.0
0.30 0.20 0.10 0.05 0.00
aqy/2π
(a)
0.0 0.5 1.0 1.5 2.0 2.50.0
0.1
0.2
0.3
0.4
0.5
Wave Vector (units of 2π/a)
Cor
rela
tion
Fun
ctio
n (u
nits
of ∆
2 a2 )
a1/a = 1/3a2/a = 2/3∆z/∆ = 0.20β = 1.3α/a = 0.212∆0/∆ = 0.125ξ = 0.10
λ/a = 5.0
0.30 0.20 0.10 0.05 0.00
aqy/2π
(b)
Fig. 4 (a) Fig. 4 (b)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.710-1
100
101
102
103
104
Fermi Wave Vector (units of 2π/a)
Con
duct
ivity
(un
its o
f ne2
τ 0/m
)
10-1
100
101
102
103
104
Rel
axat
ion
Tim
e (u
nits
of τ
0)
a1/a = 1/3a2/a = 2/3
α/a = 0.10∆0/∆ = 0.06
ξ = 0.10λ/a = 5.0
Approximation
σyy
σxx
τy
τx
τ
(a)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.710-1
100
101
102
103
104
Fermi Wave Vector (units of 2π/a)
Con
duct
ivity
(un
its o
f ne2
τ 0/m
)
10-1
100
101
102
103
104
Rel
axat
ion
Tim
e (u
nits
of τ
0)
a1/a = 1/3a2/a = 2/3
α/a = 0.20∆0/∆ = 0.12
ξ = 0.10λ/a = 5.0
Approximation
σyy
σxx
τy
τx
τ
(b)
Fig. 5 (a) Fig. 5 (b)
Diffuse Bragg Scattering in Corrugated Quantum Wells Page 11
0.0 0.5 1.0-250
-200
-150
-100
-50
0
50
100
150
200
250
Angle (units of 2π)
Dis
trib
utio
n F
unct
ion:
gky
(uni
ts o
f -ev
τ 0E
)
-20
-15
-10
-5
0
5
10
15
20
Dis
trib
utio
n F
unct
ion:
gkx
(uni
ts o
f -ev
τ 0E
)
a1/a = 1/3a2/a = 2/3α/a = 0.10∆0/∆ = 0.06ξ = 0.10
λ/a = 5.0 gk
y
gkx
0.30 0.40 0.45 0.50 0.55
akF/2π
(a)
0.0 0.5 1.0-300
-200
-100
0
100
200
300
Angle (units of 2π)
Dis
trib
utio
n F
unct
ion:
gky
(uni
ts o
f -ev
τ 0E
)
-20
-15
-10
-5
0
5
10
15
20
Dis
trib
utio
n F
unct
ion:
gkx
(uni
ts o
f -ev
τ 0E
)
a1/a = 1/3a2/a = 2/3α/a = 0.20∆0/∆ = 0.12ξ = 0.10
λ/a = 5.0 gk
y
gkx
0.30 0.40 0.45 0.50 0.55
akF/2π
(b)
Fig. 6 (a) Fig. 6 (b)
0 5 10 1510-2
10-1
100
101
Magnetic Field: (a/l)2
Pea
k C
ondu
ctiv
ity (
units
of e
2 /2π
h)
10-2
10-1
100
101
Bro
aden
ing
(uni
ts o
f Γ)
a1/a = 1/3a2/a = 2/3
α/a = 0.1∆0/∆ = 0.06
ξ = 0.1λ/a = 5.0
σxx
σyy
Γ0
(a)
0 5 10 1510-2
10-1
100
101
Magnetic Field: (a/l)2
Pea
k C
ondu
ctiv
ity (
units
of e
2 /2π
h)
10-2
10-1
100
101
Bro
aden
ing
(uni
ts o
f Γ)
a1/a = 1/3a2/a = 2/3
α/a = 0.2∆0/∆ = 0.12
ξ = 0.1λ/a = 5.0
σxx
σyy
Γ0
(b)
Fig. 7 (a) Fig. 7 (b)
Page 12 Y. Zheng and T. Ando
0.0 0.5 1.0 1.5 2.0 2.50.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Wave Vector (units of 2π/a)
Cor
rela
tion
Fun
ctio
n (u
nits
of ∆
2 a)
a1/a = 1/3a2/a = 2/3∆z/∆ = 0.10∆a/a = 0.10
1.00 0.75 0.50 0.25 0.00 p
(a)
0.0 0.5 1.0 1.5 2.0 2.50.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Wave Vector (units of 2π/a)
Cor
rela
tion
Fun
ctio
n (u
nits
of ∆
2 a)
a1/a = 1/3a2/a = 2/3∆z/∆ = 0.20∆a/a = 0.20
1.00 0.75 0.50 0.25 0.00 p
(b)
Fig. 8 (a) Fig. 8 (b)
20 25 30-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Position (units of a)
Cor
ruga
tion
(uni
ts o
f ∆)
a1/a = 1/3a2/a = 2/3∆z/∆ = 0.10∆a/a = 0.10
p= 0.0 p= 1.0 Regular
0 5 10
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
Distance (units of a)
Cor
rela
tion
Fun
ctio
n (u
nits
of ∆
2 )
a1/a = 1/3a2/a = 2/3∆z/∆ = 0.10∆a/a = 0.10
(x10)
1.00 0.07 0.75 0.10 0.50 0.11 0.25 0.11 0.00 0.11
p α
Fig. 9 Fig. 10
