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

Physica E 58 (2014) 84–87

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

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

H I G H L I G H T S







The effects of LFC and temperature on μ are remarkable for n o1012 cm À 2 or T $ 0.3TF.
The effects of LFC and temperature on τt =τs are nearly canceled in the ratio for IRS.
For CIS, the temperature effects on τt =τs are notable at T $ 0.3TF only for GH.
At low n, the dependence of the resistance ratio on LFC decreases when T increases.
With decreasing L, nMIT increases and becomes nearly independent of LFC and zi.

art ic l e i nf o

a b s t r a c t

Article history:
Received 22 September 2013
Received in revised form
19 November 2013
Accepted 25 November 2013
Available online 4 December 2013

We investigate the transport scattering time, the single-particle relaxation time and the magnetoresistance of a quasi-two-dimensional electron gas in a GaP/AlP/GaP quantum well at zero and finite
temperatures. We consider the interface-roughness and impurity scattering, and study the dependence
of the mobility, scattering time and magnetoresistance on the carrier density, temperature and local-field
correction. In the case of zero temperature and Hubbard local-field correction our results reduce to those
of Gold and Marty (Physica E 40 (2008) 2028; Phys. Rev. B 76 (2007) 165309). We also discuss the
possibility of a metal–insulator transition which might happen at low density.
& 2013 Elsevier B.V. All rights reserved.

Keywords:
AlP quantum well
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 LoLc ¼45 Å, and valley degeneracy
gv ¼2 for well width L4Lc [3]. Recently, Gold and Marty have
calculated the transport scattering time, single-particle relaxation
time and the magnetoresistance for GaP/AlP/GaP QW with LoLc [4].

In such thin QW, interface-roughness scattering (IRS) is the dominant
scattering mechanism [5]. The scattering mechanism, which is
responsible for limiting the mobility, can be determined by

n

Corresponding author. Fax: +848 38350096.
E-mail address: (N.Q. Khanh).

1386-9477/$ - see front matter & 2013 Elsevier B.V. All rights reserved.
/>
comparing experimental results with those of theoretical calculations
[6–8]. Recent measurements and calculations of transport properties
of a quasi-two-dimensional electron gas (Q2DEG) in a magnetic field
give additional tool for determining the main scattering mechanism
and scattering parameters [9–13]. To the author's knowledge, there is
no calculation of transport properties of the spin-polarized Q2DEG in
a GaP/AlP/GaP QW at finite temperatures. Therefore, in this paper, we
calculate the mobility, the scattering time and the magnetoresistance
of the 2DEG realized in AlP for IRS and charged impurity scattering
(CIS) at zero and finite temperature, taking into account exchange–
correlation effects. We also discuss the possibility of a metal–
insulator transition (MIT) which might happen at low density and
calculate the critical electron density nMIT for the MIT.
2. Theory
We consider a Q2DEG with parabolic dispersion determined by
the effective mass mn. We assume that the electron gas is in the xy


N.Q. Khanh, V.V. Tai / Physica E 58 (2014) 84–87

plane with infinite confinement for z o0 and z 4L. For 0r zrL,
the electron gas in theplowest
ffiffiffiffiffiffiffiffi subband is described by the wave
function ψ(0 r zrL)¼ 2=L sin(πz/L) [14].
When the in-plane magnetic field B is applied to the system,
the carrier densities n 7 for spin up/down are not equal [15–16]. At
T ¼0 we have
n7 ¼

n
ð1 7 BBs Þ;
2

n þ ¼ n;

B o Bs

n À ¼ 0;

B ZBs

ð1Þ

Here, n ¼ n þ þn À is the total density and Bs is the so-called
saturation field given by g μB Bs ¼ 2EF where g is the electron spin
g-factor, μB is the Bohr magneton and EF is the Fermi energy [15].
For T4 0, n 7 is determined using the Fermi distribution function
and given by [16]
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2
2x=t
2x=t
ð2 þ 2xÞ=t
n
n þ ¼ t ln 1 À e þ ðe 2 À 1Þ þ 4e
ð2Þ
2
nÀ ¼ nÀnþ
where x ¼ B=Bs and t ¼ T=T F with TF is the Fermi temperature. The
energy averaged transport relaxation time for the (7 ) components is given in the Boltzmann theory by [14–16]
R
7


dετðεÞε½ À ∂f ðεÞ=∂εŠ
τ7 ¼ R
ð3Þ
7
dε ε½ À ∂f ðεÞ=∂εŠ
where
Z

1
1
¼
τðkÞ 2πℏε

A ðqÞ ¼ 1 þ

2k

〈jUðqÞj2 〉
½ A ðqފ2

0

q2 dq
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi;
2
4k Àq2

ð4Þ

2πe2 1
F C ðqÞ½1 À GðqÞŠΠ ðq; T Þ;
AL q

ð5Þ

Π ðq; TÞ ¼ Π þ ðq; TÞ þ Π À ðq; TÞ;
Π 7 ðq; T Þ ¼

Π

β

Z

4

0
7 ðq; E F 7 Þ 

1
0

Π

ð6Þ
À

dμ=

Á

Π 07 q; μ=
À
Á;
2
cosh β=2 μ 7 À μ=
n

2

gv m 4
0
1À
7 ðqÞ ¼
2

2π ℏ

ð7Þ

3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2
2kF 7
1À
Θðq À2kF 7 Þ5;
q
ð8Þ

F C ðqÞ ¼

1
4π

2 þ a2 q2



3aq þ8π 2 32π 4 1 À e À aq
À 2 2
aq
a q 4π 2 þ a2 q2


ð9Þ

where ni is the 2D impurity density, zi is the distance between
remote impurities and 2DEG, and F CIS ðq; zi Þis the form factor for
the electron–impurity interaction given in Ref. [14].
The mobility of the nonpolarized and fully polarized 2DEG is
given by μ ¼ e〈τ〉=mn . The resistivity is defined by ρ ¼ 1=s where
s ¼ s þ þ s À is the total conductivity and s 7 is the conductivity of
the (7 ) spin subband given by s 7 ¼ n 7 e2 〈τ 7 〉=mn [15]. It was
shown that multiple-scattering effects can account for this MIT at
low electron density where interaction effects become inefficient
to screen the random potential created by the disorder [21–22].
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 [3,21–22]
D
 E
Z 1 U ðqÞ2 ÂΠ o ðqÞÃ2 qdq
1
A¼
ð12Þ
4π n2 0
½ A ðqފ2
For n4 nMIT, where Ao 1, the Q2DEG is in a metallic phase and for
n onMIT, where A41, the Q2DEG is in an insulating phase and the
mobility vanishes.

3. Numerical results
In this section, we present our numerical calculations for the
transport scattering time, the single-particle relaxation time and
the magnetoresistance of a Q2DEG in a GaP/AlP/GaP QW for the

case L oLc using the following parameters [3–4]: A L ¼ 9.8,
mn ¼0.3mo and mz ¼0.9mo, where mo is the free electron mass.
To treat the exchange–correlation effects we use the LFC which is
very important for low electron densities. In the Hubbard approximation, only exchange effects are taken into account and the LFC
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
has the form GH ðqÞ ¼ q=½g v g s q2 þ kF Š where gs is the spin degeneracy. We also use analytical expressions of the LFC (GGA) according to the numerical results obtained in Ref. [20] where both
exchange and correlation effects are taken into account.
3.1. Results for interface-roughness scattering
In Fig. 1, we show the mobility μ versus electron density n for
two different QW widths and temperatures. Two approximations
for the LFC, GH and GGA are used. It is seen that exchange and
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,
10

7

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 CIS the random potential has the form

2

2
2π e2 1
〈U CIS ðqÞ 〉 ¼ ni
F CIS ðq; zi Þ2

AL q

Δ=3Å; Λ=50Å; B/Bs=0
T/TF=0
T/TF=0.3

mobility μ(104cm2/Vs)

with f ðεÞ ¼ 1=f1 þ expðβ ½ε À μ 7 ðTފÞg, β ¼ ðkB TÞ À 1 , μ 7 ¼ ln ½ À 1
2
2
þexpðβEF 7 ފ=β , EF 7 ¼ ℏ2 k F 7 =ð2mn Þ and ε ¼ ℏ2 k =ð2 mn Þ. Here,
n
m is the effective mass in xy-plane, gv is the valley degeneracy, G
(q) is the local-field correction (LFC) describing the exchange–
correlation effects
[17–20],
A L is the background static dielectric

2
constant and 〈UðqÞ 〉 is the random potential which depends on
the scattering mechanism [14]. For IRS the random potential is
given by [14]

  n 2 
D
 E
m
π 4
2 2

U IRS ðqÞ2 ¼ 2 4π
ðεF ΔΛÞ2 e À q Λ =4
ð10Þ
2
mz
kF a
a

85

L=40Å

GH

1

GGA
GH

0.1
L=30Å

GGA

0.1

1

5

electron density n(1012cm-2)

ð11Þ

Fig. 1. Mobility μ versus electron density n for IRS for two different QW widths and
temperatures in two G(q) models.


86

N.Q. Khanh, V.V. Tai / Physica E 58 (2014) 84–87

6

10
GH
L = 40Å < LC

4

Δ = 3Å; Λ = 50Å; B/Bs=0
G=0

G=0(T/TF=0)
GH(T/TF=0)
GGA(T/TF=0)
GGA(T/TF=0.3)

ρ(Bs)/ρ(0)

GGA

τt/τs

2

T/TF=0
T/TF=0.3

1
1
(kFΛ)2/3

L = 40Å < LC
Δ = 3Å; Λ = 50Å

0.6
0.1

2/3
1

0.1

10

and hence reduces the mobility. The temperature effect is remarkable for T $ 0.3TF ( $2.75 K for n ¼1011 cm À 2). Note that the
mobility in Fig. 1 is shown on a log-plot. On a linear scale the
changes due to a finite LFC and temperature are much larger.
The ratio of the transport scattering time and the singleparticle relaxation time τt =τs versus electron density is shown in

Fig. 2 for L¼ 40 Å and different approximations for the LFC. We
observe that the dependence of τt =τs on the LFC and temperature
is very weak because the effects of both LFC and temperature are
nearly canceled in the ratio. The straight lines are the analytical
results for IRS given in Ref. [23]. Note that at high electron density
τt =τs % ðkF ΛÞ2 =3 and the ratio allows us to determine kF Λ, and for a
given density, the parameter Λ can be determined.
Results for the resistance ratio ρðBs Þ=ρðB ¼ 0Þ versus electron
density for L ¼40 Å and two temperatures in different approximations for the LFC are shown in Fig. 3. We see that the resistance
ratio depends strongly on approximations for the LFC. The resistivity of a fully 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. At low density, the Hubbard LFC GH
increases and the LFC GGA decreases the resistance ratio. This
behavior may stem from the dependence of the LFC on the spinpolarization. At low density, the dependence of the resistance ratio
on the LFC decreases when temperature increases and the temperature effect is remarkable for T $ 0.3TF.

Fig. 3. Resistance ratio ρðBs Þ=ρðB ¼ 0Þ versus electron density for IRS for L ¼ 40 Å
and two temperatures in different approximations for the LFC.

10
L=30Å; ni = n; B/Bs=0

zi = -L/2

T/TF = 0
T/TF = 0.3

mobility μ(103cm2/Vs)

Fig. 2. Ratio of the transport scattering time and the single-particle relaxation time
τt =τs versus electron density for IRS.

2

1

electron density n(1012cm-2)

electron density n(1012 cm-2)

GGA
GH

1
zi = 0
GH

GGA

zi = L/2 GH

0.1
GGA

0.1

1

10

electron density n(10 cm )
12

-2

Fig. 4. Mobility μ versus electron density n for CIS for two temperatures and three
values of the position of impurities zi in two approximations for the LFC.

10

L=30Å; ni = n; B/Bs=0
T/TF = 0
T/TF = 0.3

The mobility versus electron density for CIS, characterized by the
impurity density ni ¼n and the distance zi of the impurity layer from
the QW edge at z¼0, is shown in Fig. 4. We observe that, the LFC is
very important at low density. On the other hand, the dependence of
the mobility on zi is more pronounced at higher density. Again, the
temperature effects on the mobility are remarkable at T$ 0.3TF.
The ratio τt =τs versus electron density for CIS for two temperatures and three values of zi in two G(q) models is displayed in Fig. 5.
At low density, the finite LFC decreases τt =τs remarkably for zi ¼L/2
or G(q)¼GGA [3]. At high density, the ratio τt =τs for zi ¼ L/2 differs
strongly from that for zi ¼ ÀL/2. The temperature effects on τt =τs are
notable for T$ 0.3TF at low density only for the Hubbard LFC.
The resistance ratio ρðBs Þ=ρðB ¼ 0Þ versus electron density for
impurities with density ni ¼n and zi ¼L/2 is plotted in Fig. 6. We see

τt/τs

3.2. Results for charged impurity scattering

GGA

zi = -L/2

GH
GH

1

zi = L/2
GGA
0.1

1

10

electron density n (1012cm-2)
Fig. 5. Ratio τt =τs versus electron density for CIS for two temperatures and zi in two
approximations for the LFC.


N.Q. Khanh, V.V. Tai / Physica E 58 (2014) 84–87

nMIT is determined mainly by IRS (CIS) for small (large) QW width.
Therefore, when L approaches Lc, the nMIT for zi ¼ L/2 is larger than
that for zi ¼ À L/2 because the CIS is stronger for impurities inside

the QW in comparison with the case of impurities located outside
the QW.

L = 30 Å
zi = L/2

6

87

ni= n
G = 0 (T/TF = 0)

ρ(Bs)/ρ(0)

T/TF = 0

4

T/TF = 0.3

4. Conclusions

GH

2
GGA

0
0.1

1

10

electron density n (1012 cm-2)
Fig. 6. Resistance ratio ρðBs Þ=ρðB ¼ 0Þ versus electron density for CIS for L ¼30 Å and
two temperatures in different approximations for the LFC.

10

critical electron density nMIT(1012cm-2)

B/Bs= 0; T/TF= 0
Δ = 3Å;Λ = 50Å
n = ni = 1x1011cm-2
conducting phase
1

GGA

insulating phase

IRS
IRS+RIS (zi=L/2)

GH

Acknowledgement

IRS+RIS (zi=-L/2)

This research is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under Grant
number 103.02-2011.25.

0.1
10

20

We have calculated the mobility, the scattering time and
magnetoresistance ratio, and the critical electron density of the
2DEG realized in AlP for interface-roughness and impurity scattering. We find the remarkable effects of the LFC and temperature for
n o1012 cm À 2 and T $ 0.3TF. For IRS, the effects of both the LFC and
temperature on τt =τs are nearly canceled in the ratio. At low
density, the dependence of the resistance ratio ρðBs Þ=ρðB ¼ 0Þ on
the LFC decreases when temperature increases. For CIS, the
temperature effects on both τt =τs and the resistance ratio are
remarkable at T $ 0.3TF only for Hubbard approximation. The LFC
decreases the screening properties and hence increases the effective random potential and critical electron density nMIT. With the
decrease of the QW width, the interface-roughness scattering
becomes stronger, and the critical electron density increases and
becomes nearly independent of the approximations used for the
LFC. Near Lc, the critical density nMIT is determined mainly by CIS
and the nMIT for zi ¼L/2 is larger than that for zi ¼ À L/2. We hope
that our results can be used to obtain information about the
scattering mechanism and many-body effects in GaP/AlP/GaP QW
structures.

30

40

quantum well width L(Å)
Fig. 7. Critical electron density versus QW width for two values of zi and two
approximations for the LFC.

that the effects of LFC and temperature are notable at low density
only for Hubbard approximations.

References
[1]
[2]
[3]
[4]
[5]
[6]
[7]

3.3. Results for the metal–insulator transition
[8]

The critical electron density versus QW width for two approximations for the LFC is displayed in Fig. 7. We observe that the
critical electron density decreases with increasing QW width, and
depends considerably on the approximation for LFC. The LFC
decreases the screening properties and hence increases the effective random potential and critical electron density. With the
decrease of the QW width, the IRS becomes stronger and the
critical electron density increases. For Lo15 Å, the critical electron
density is high (nMIT 42 Â 1012 cm À 2) and becomes nearly independent of the approximation used for the LFC (because at

n 42 Â 1012 cm À 2 the exchange–correlation effect is small, see
Fig. 1). It is seen that, for scattering parameters used in Fig. 7, the

[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]

M.P. Semtsiv, et al., Phys. Rev. B 74 (2006) 041303. (R).
M.P. Semtsiv, et al., Appl. Phys. Lett. 89 (2007) 184102.
A. Gold, R. Marty, Phys. Rev. B 76 (2007) 165309.
A. Gold, R. Marty, Physica E 40 (2008) 2028.
A. Gold, Solid State Commun. 60 (1986) 53.
T. Ando, A.B. Fowler, F. Stern, Rev. Mod. Phys. 54 (1982) 437.
John H. Davies, The Physics of Low-Dimensional Semiconductors, An Introduction, Cambridge University Press, Cambridge, 1998.
Chihiro Hamaguchi, Basic Semiconductor Physics, Springer, Berlin, Heidelberg,
2001.
A.A. Shashkin, et al., Phys. Rev. B 73 (2006) 115420.
V.T. Dolgopolov, A. Gold, JETP Lett. 71 (2000) 27.

T.M. Lu, et al., Phys. Rev. B 78 (2008) 233309.
E.H. Hwang, S. Das Sarma, Phys. Rev. B 87 (2013) 075306.
Xiaoqing Zhou, et al., Phys. Rev. B 85 (2012) 041310. (R).
A. Gold, Phys. Rev. B 35 (1987) 723.
S. Das Sarma, E.H. Hwang, Phys. Rev. B 72 (2005) 205303.
Nguyen Quoc Khanh, Physica E 43 (2011) 1712.
M. Jonson, J. Phys. C 9 (1976) 3055.
A. Gold, L. Calmels, Phys. Rev. B 48 (1993) 11622.
A. Gold, Phys. Rev. B 50 (1994) 4297.
A. Gold, Z. Phys. B 103 (1997) 491.
A. Gold, Phys. Rev. Lett. 54 (1985) 1079.
A. Gold, W. Götze, Phys. Rev. B 33 (1986) 2495.
A. Gold, Phys. Rev. B 38 (1988) 10798.