DSpace at VNU: Transport properties of the two-dimensional electron gas in wide AlP quantum wells including temperature and correlation effects

Physica E 67 (2015) 84–88

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Physica E
journal homepage: www.elsevier.com/locate/physe

Transport properties of the two-dimensional electron gas in wide AlP
quantum wells including temperature and correlation effects
Vo Van Tai, Nguyen Quoc Khanh n
Department of Theoretical Physics, National University in Ho Chi Minh City, 227-Nguyen Van Cu Street, 5th District, Ho Chi Minh, Vietnam

H I G H L I G H T S

 The difference between the results of GH and GGA model is remarkable.
 The temperature effects are notable at very low temperature T $ 0.3TF.
 The correlation effects increase the critical density for a MIT considerably.

art ic l e i nf o

a b s t r a c t

Article history:
Received 15 August 2014
Received in revised form
5 November 2014
Accepted 24 November 2014
Available online 26 November 2014

We investigate the mobility, magnetoresistance and scattering time of a quasi-two-dimensional electron
gas in a GaP/AlP/GaP quantum well of width L4Lc ¼45.7 Å at zero and finite temperatures. We consider

the interface-roughness and impurity scattering, and study the dependence of the mobility, the resistance and scattering time ratio on the carrier density and quantum well width for different values of
the impurity position and temperature using different approximations for the local-field correction. In
the case of zero temperature and Hubbard local-field correction our results reduce to those of Gold and
Marty (Phys. Rev. B. 76 (2007) 165309) [3]. We also study the correlation and multiple scattering effects
on the total mobility and the critical density for a metal–insulator transition.
& 2014 Elsevier B.V. All rights reserved.

Keywords:
AlP quantum wells
Magnetoresistance
Scattering time
Temperature effect

1. Introduction
GaP/AlP/GaP quantum well (QW) structures, where the electron gas is located in the AlP, have been studied recently at low
temperatures via cyclotron resonance, quantum Hall effect,
Shubnikov de Haas oscillations [1] and intersubband spectroscopy
[2]. In this structure, due to biaxial strain in the AlP and confinement effects in the quantum well of width L, the electron gas has
valley degeneracy gv ¼1 for well width L oLc ¼45.7 Å, and valley
degeneracy gv ¼2 for well width L4 Lc [3,4]. Recently, we have
calculated the mobility, scattering time and magnetoresistance for
a GaP/AlP/GaP QW with L oLc including the temperature and exchange-correlations effects [5]. In this paper, we present our calculation for the case of wide GaP/AlP/GaP QW with L 4Lc. We
consider interface-roughness and randomly distributed charged
impurities as source of disorder. We investigate the dependence of
the mobility, the resistance and scattering time ratio on the carrier
density and QW width for different values of the impurity position
n

Corresponding author. Fax: þ84 8 38350096.
E-mail address: (N.Q. Khanh).

/>1386-9477/& 2014 Elsevier B.V. All rights reserved.

and temperature. We also study the correlation and multiple
scattering effects (MSE) [6] on the total mobility and the critical
density for a metal–insulator transition (MIT) [6,7].

2. Theory
We assume that the electron gas (EG), with parabolic dispersion determined by the effective mass mn, is in the xy plane with
infinite confinement for z o0 and z 4L. For 0 rz rL, the EG in the
lowest subband is described by the wave function ψ
(0 rzr L) ¼ 2/L sin(πz/L) [5,8].
When the in-plane magnetic field B is applied to the system,
the carrier densities n± for spin up/down are not equal [5,9,10]. At
T¼ 0 we have

n± =

n⎛
B⎞
⎜1 ± ⎟, B < Bs
Bs ⎠
2⎝

n+ = n, n− = 0,

B ≥ Bs

(1)

Here n = n+ + n− is the total density and Bs is the so-called saturation field given by gμB Bs = 2EF where g is the electron spin g-


V.V. Tai, N.Q. Khanh / Physica E 67 (2015) 84–88

factor, μB is the Bohr magneton and EF is the Fermi energy. For
T 40, n± is determined using the Fermi distribution function and
given by [5,9]

1 − e2x / t +
n
t ln
2
n− = n − n+

n+ =

(e2x / t − 1)2 + 4e (2 + 2x)/ t
2
(2)

where x = B /Bs and t = T /TF with TF is the Fermi temperature. The
energy averaged transport relaxation time for the (7 ) components are given in the Boltzmann theory by [5,8,9]

τ±

∫ dε τ (ε) ε [ − (∂f ± (ε)/∂ε)]
=
∫ dε ε [ − (∂f ± (ε)/∂ε)]

(3)

where

1
1
=
2π ℏε
τ (k)

∈ (q) = 1 +

∫0

2k

U (q)

2

q2dq

[∈ (q)]2

4k 2 − q 2

,
(4)

2π e 2 1

FC (q)[1 − G (q)] Π (q, T ),
∈L q

(5)

Π (q, T ) = Π+ (q, T ) + Π − (q, T )

Π± (q, T ) =

β
4

∫0

∞

dμ ′

Π±0 (q, μ′)

c osh2 (β/2)

Π±0 (q, EF ± ) ≡ Π±0 (q) =

FC (q) =

4π 2

(6)

(μ± − μ′)

⎡
g v m⁎ ⎢
1−
2π ℏ2 ⎢⎣

,
(7)

⎤
⎛ 2k F ± ⎞2
⎥
±
Θ
1−⎜
(
q
2
k
)
−
⎟
F ⎥,
⎝ q ⎠
⎦

⎛
1
8π 2

32π 4 1 − e−aq ⎞
⎜3aq +
⎟,
−
2
2
aq
a2q2 4π 2 + a2q2 ⎠
+aq ⎝

(8)

(9)

with β = (kB T )−1, ε = ℏ2k 2/(2m⁎) and ∈L denotes the backg
round static dielectric constant. Here kF ± = (4πn± /gv )1/2,

EF ± = ℏ2k 2F ±/(2m⁎), μ± = ln [ − 1 + exp (βEF ± )]/β , and Π± (q , T ) is
the 2D Fermi wave vector, Fermi energy, chemical potential, Fermi
distribution function and polarizability for the up/down spin state,
respectively. G(q) is the local-field correction (LFC) describing the
exchange-correlation effects [8,11] and

U (q)

2

is the random

potential which depends on the scattering mechanism [8]. For

interface-roughness scattering (IRS) the random potential is given
by [8]

UIRS (q)

2

⎛ 4π ⎞ ⎛ m⁎ ⎞2 ⎛ π ⎞4
2 2
= 2⎜ ⎟⎜
⎟ ⎜
⎟ (ε F ΔΛ)2e−q Λ /4
⎝ a2 ⎠ ⎝ m z ⎠ ⎝ k F a ⎠

85

where σ = σ+ + σ − is the total conductivity and σ± is the conductivity of the (7 ) spin subband given by σ± = n± e2 τ± /m⁎ [9]. It
was shown that multiple-scattering effects can account for the MIT
at low electron density where interaction effects become inefficient to screen the random potential created by the disorder
[6,7]. The MIT is described by parameter A, which depends on the
random potential, the screening function including the LFC and
the compressibility of the electron gas, and is given by [3,6,7]

1
A=
4πn2

∫0

∞

U (q)

2

[Π o (q)]2 qdq

[∈ (q)]2

.

(12)

For n4 nMIT, where A o1, the 2DEG is in a metallic phase and
for n onMIT, where A 41, the 2DEG is in an insulating phase and
the mobility vanishes.

3. Numerical results
For the case L 4Lc we use the following parameters [1–3,12]:
∈L ¼9.8, gv ¼ 2, mn ¼ 0.52mo and mz ¼0.3mo, where mo is the free
electron mass. The LFC is very important at low electron densities.
In the Hubbard approximation, only exchange effects are taken
into account and the LFC has the form GH (q) = q /[gv gs q2 + kF2 ]
where gs is the spin degeneracy [3–5]. We also use analytical expressions of the LFC (GGA) according to the numerical results obtained in Ref. [11] where both exchange and correlation effects are
taken into account.
In Fig. 1, we show the mobility μ versus electron density n for a
QW of width L ¼60 Å for IRS with Δ ¼3 Å and Λ ¼ 50 Å for different
temperatures in two G(q) models. It is seen that correlation effects
are very important for no 1012 cm À 2 and the mobility depends
strongly on the approximation for LFC. The LFC reduces the

screening, increases the effective scattering potential, and hence
reduces the mobility. The temperature effect is remarkable for
T $ 0.3TF ( $ 1.6 K for n ¼1011 cm À 2 ). We have chosen Δ ¼3 Å and
Λ ¼50 Å because, using these values, Gold and Marty [4] have
calculated the mobility for thin AlP QW of width L¼40 Å and
obtained good agreement with experimental results. Furthermore,
although there is not very much known about the parameters Δ
and Λ, Δ values around 3 Å and Λ values between 60 and 10 Å
seem to be most realistic [4,13]. Our results can be helpful for
experimenters in determining the interface-roughness parameters
Δ and Λ for GaP/AlP/GaP QW structures. The minimum in the

(10)

where Δ represents the average height of the roughness perpendicular to the 2DEG, Λ represents the correlation length parameter
of the roughness in the plane of the 2DEG and mz is the effective
mass perpendicular to the xy-plane.
For remote charged impurity scattering (CIS) the random potential has the form

UCIS (q)

2

⎛ 2πe2 1 ⎞2
⎟ FCIS (q, zi )2
= Ni ⎜
⎝ ∈L q ⎠

(11)

where Ni is 2D impurity density, zi is the distance between remote
impurities and 2DEG, and FCIS (q , zi ) is the form factor for the
electron–impurity interaction given in Ref. [8].
The mobility of the nonpolarized and fully polarized 2DEG is
given by μ = e < τ > /m⁎. The resistivity is defined by ρ = 1/σ

Fig. 1. Mobility μ versus electron density n for a QW of width L ¼60 Å for IRS with
Δ ¼3 Å and Λ ¼50 Å for different temperatures in two G(q) models.


86

V.V. Tai, N.Q. Khanh / Physica E 67 (2015) 84–88

Fig. 2. Resistance ratio ρ (Bs )/ρ (B = 0) versus electron density for a QW of width
L ¼ 60 Å for IRS with Δ ¼ 3 Å and Λ ¼ 50 Å at different temperatures in different
approximations for the LFC.

mobility versus carrier density appeared in Fig. 1 has been explained by Gold [8] using an approximation derived in [14]. Due to
the factor e−q

2Λ2 /4

in

UIRS (q)

2

, the low temperature mobility

μIRS ∼ 1/n at low density (kFΛ«1) and μIRS ∼ n3/2 at high density (kFΛ»1). The crossover of the μIRS ∼ 1/n (decreasing function
of n ) to the μIRS ∼ n3/2 (increasing function of n) behavior is at
n ¼gv/(2πΛ2), where the minimum of the mobility is reached.
In Fig. 2 we show the resistance ratio ρ (Bs )/ρ (B = 0) versus
electron density for a QW of width L ¼60 Å for IRS with Δ ¼3 Å
and Λ ¼50 Å at different temperatures in different approximations
for the LFC. We see that the resitivity of a polarized 2DEG limited
by IRS is higher compared to that of the nonpolarized case. This
effect is due to spin-splitting in the parallel magnetic field leading
to reduced screening in a spin-polarized electron gas. The LFC
increases considerably the resistance ratio especially in the case of
GGA. In the Hubbard approximation, the effects of LFC are nearly
canceled by the temperature effect at T $ 0.3TF for
n 45 Â 1011 cm À 2. Recall that for L¼40 Å o Lc, at low density, the
Hubbard LFC GH increases and the LFC GGA decreases the resistance

a

ratio [5]. We note that in the case of IRS, the height parameter Δ
cancels out for the magnetoresistance ratio ρ (Bs )/ρ (B = 0). Without LFC, the limiting behavior for small density is
ρ (Bs )/ρ (B = 0) ¼8 [15]. When many-body effects described by a
LFC are included, using the approximation derived in [14] for the
scattering time, it can be shown that the magnetoresistance ratio
might increase remarkably at low density [16]. These features of
the magnetoresistance ratio found in Refs. [15,16] are seen also in
Fig. 2.
The ratio of the transport scattering time and the single-particle scattering time τt/τs can be used to determine microscopic
parameters of disorder such as Λ and zi [3]. In Fig. 3, we show the
ratio τt/τs versus electron density for a QW of width L ¼60 Å for IRS

and CIS in different G(q) models. In the case of IRS, only small
effects due to the LFC GH are seen. In the case of CIS, Ni is canceled
out in the ratio τt/τs and the results for zi ¼L/2 differs strongly from
that for zi ¼ À L/2 at high density. For low density, the difference
between the results of GH and GGA model is notable for both
IRS and CIS. Note that the ratio τt/τs in Fig. 3 is shown on a log-plot.
On a linear scale the changes due to a finite LFC are much
larger.
The mobility versus electron density for a QW of width
L¼100 Å for CIS with the impurity concentration Ni ¼n at two
temperatures for different values of the distance zi of the impurity
layer from the QW edge at z ¼0 is shown in Fig. 4. We use two G
(q) models and observe that the correlation effects are very important at low density. The dependence of the mobility on zi is
more pronounced at high density. The temperature effects on the
mobility are remarkable at T $ 0.3TF. We note that, as in the case of
IRS, the minimum in the mobility μCIS versus carrier density for
T¼ 0 appeared in Fig. 4 can be explained using Gold's arguments
given in [8].
In Fig. 5 we show the mobility μ versus QW width L for CIS with
Ni ¼n ¼1012 cm À 2 at two temperatures for three values of the
impurity position zi in two G(q) models. For zi ¼L/2, we see a very
weak QW width dependence of the mobility. For zi r 0, the mobility increases with increasing well width because the distance
between the electron gas and the impurity layer increases with
increasing well width. The temperature effects and the differences
between the results of two G(q) models are notable for the wide
range of QW width.

b

Fig. 3. Ratio τt/τs versus electron density for a QW of width L ¼60 Å in different G(q) models (a) for IRS with Δ ¼ 3 Å and Λ¼ 50 Å and (b) for CIS for two values of the impurity

position zi.


V.V. Tai, N.Q. Khanh / Physica E 67 (2015) 84–88

Fig. 4. Mobility μ versus electron density for a QW of width L ¼ 100 Å for CIS with
Ni ¼n at two temperatures for different values of impurity position zi in two G
(q) models.

Fig. 5. Mobility μ versus QW width L for CIS at two temperatures for three values of
the impurity position zi in two G(q) models.

a

87

Fig. 7. Total mobility versus electron density n for a QW of width L ¼ 60 Å for IRS
with Δ ¼ 3 Å and Λ ¼ 50 Å and CIS with zi ¼ À L/2 and Ni ¼ n in two G(q) models.

The resistance ratio ρ (Bs )/ρ (B = 0) versus QW width L for CIS
with Ni ¼n at two temperatures for different values of electron
density n in two G(q) models is plotted in Fig. 6. We see that the
difference between the results of two G(q) models is very large for
impurities both inside (zi ¼L/2 ) and outside (zi ¼ À L/2 ) the QW.
The temperature effects in both GH and GGA model are notable at
very low temperature T $ 0.1TF ( $ 0.53 K for n ¼1011 cm À 2 ) for the
entire range of QW width considered here.
The total mobility versus electron density n for a QW of width
L¼ 60 Å for IRS with Δ ¼3 Å and Λ ¼ 50 Å and CIS with zi ¼ ÀL/2
and Ni ¼n in two G(q) models is displayed in Fig. 7. We observe

that the correlation and multiple-scattering effects are remarkable
at low densities. The LFC decreases the screening properties and
hence increases the effective random potential and critical electron density. The MSE are very strong for the case of GGA when
correlation effects are taken into account. At high densities
(n 45 Â 1012 cm À 2) the MSE are negligible and the total mobility
becomes nearly independent of the approximation used for LFC.
Note that the low-temperature MIT is caused by quantum fluctuations and its behavior in the critical region is very complex.

b

Fig. 6. Resistance ratio ρ (Bs )/ρ (B = 0) versus QW width L for CIS with Ni ¼ n at two temperatures for different values of electron density n in two G(q) models (a) for zi ¼ À L/2
and (b) for zi ¼L/2.


88

V.V. Tai, N.Q. Khanh / Physica E 67 (2015) 84–88

Therefore, an adequate description of the MIT, especially in the
vicinity of the transition point, requires more sophisticated theory
[17–19].

Science and Technology Development (NAFOSTED) under Grant
no. 103.01-2014.01.

References
4. Conclusion
In summary, we have calculated the mobility, scattering time
and magnetoresistance of the 2DEG in a GaP/AlP/GaP QW of width
L4 Lc ¼45.7 Å for interface-roughness and impurity scattering. We

find that the difference between the results of GH and GGA model is
remarkable for n o1012 cm À 2. The temperature effects are notable
at very low temperature T $ 0.3TF ( $ 1.6 K for n ¼1011 cm À 2 ). The
MSE lead to a MIT at low density and the correlation effects increase the critical density considerably. Our results can be used to
obtain information about the scattering mechanism and manybody effects in GaP/AlP/GaP QW structures.

Acknowledgment
This research is funded by Vietnam National Foundation for

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