DSpace at VNU: Transport properties of a spin-polarized quasi-two-dimensional electron gas in an InP In1-xGaxAs InP quantum well including temperature effects

Physica E 43 (2011) 1712–1716

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

Transport properties of a spin-polarized quasi-two-dimensional electron gas
in an InP/In1 À xGaxAs/InP quantum well including temperature effects
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 City, Vietnam

a r t i c l e i n f o

a b s t r a c t

Article history:
Received 29 March 2011
Received in revised form
18 May 2011
Accepted 25 May 2011
Available online 2 June 2011

We investigate the mobility and resistivity of a quasi-two-dimensional electron gas in an InP/In1 À xGaxAs/
InP quantum well at arbitrary temperatures and spin polarizations caused by an applied in-plane
magnetic field. We consider the carrier density, impurity concentration and layer thickness parameters
such that the ionized impurity and alloy disorder scattering are the main mechanisms. We investigate the
dependence of the mobility and resistivity on the carrier density, layer thickness and magnetic field.
& 2011 Elsevier B.V. All rights reserved.

1. Introduction

The transport properties of a quasi-two-dimensional electron gas
(Q2DEG) in the lattice matched InP/In0.53Ga0.47As/InP quantum well
(QW) have been studied by several authors [1–5]. It is an attractive
system for high-speed electronic device applications due to the
negligible concentration of DX centers and discolations on the InP
donor layers [1]. The scattering mechanism, which is responsible for
limiting the mobility, can be determined by comparing experimental results with those of theoretical calculations [6–9]. Recent
measurements and calculations of transport properties of a 2DEG
in a magnetic field give additional tool for determining the main
scattering mechanism [10–14]. To the author’s knowledge, there is
no calculation of transport properties of the spin-polarized 2DEG in
an InP/In1À xGaxAs/InP quantum well at finite temperatures. Therefore, we decide to investigate here in this paper the magnetic field
and temperature effects on the mobility and resistivity of a 2DEG in
an InP/In1À xGaxAs/InP quantum well.
2. Theory
We consider a single InP/In1 À xGaxAs/InP QW of width a with
infinite confinement. We assume that the electrons are free to
move in x–y plane with the effective mass mn and confined in the
z-direction. We neglect the subband structure and include only
the lowest subband in our calculation. The wave function for the
z-direction is given via [6]
rffiffiffi
pz
2
cðzÞ ¼
sin
, 0 oz o a
ð1Þ
a
a

and is zero for all other z.
n

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doi:10.1016/j.physe.2011.05.028

When the in-plane magnetic field is applied to the system, the
carrier densities n 7 for spin up/down are not equal. At T¼0 we
have [11]
8


< n 7 ¼ n 17 B , B o Bs
2
Bs
ð2Þ
: n þ ¼ n, n- ¼ 0, B Z Bs
Here n¼n þ þn À is the total density and Bs is the so-called
saturation field given by gmBBs ¼2EF where g is the electron spin
g-factor and mB is the Bohr magnetron. For T40, n 7 is determined using the Fermi distribution function and given by [11,15]
8
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
2
2b=t
ð2 þ 2bÞ=t
<

n þ ¼ n2 t ln1Àe2b=t þ ðe À1 2Þ þ 4e
ð3Þ
: nÀ ¼ nÀn þ
where b¼ B/Bs and t ¼T/TF with TF is the Fermi temperature. The
energy averaged transport relaxation time for the ( 7) components are given in the Boltzmann theory by [7,11]
R
detðeÞe½À@f 7 ðeÞ=@eŠ
ð4Þ
/t 7S ¼ R
dee½À@f 7 ðeÞ=@eŠ
Here t(e) is the energy dependent relaxation time, and f 7 (e) is
the Fermi distribution function
f 7 ðeÞ ¼

1
1þ eb½e-m 7 ðTފ

ð5Þ

where b ¼(kBT) À 1 and

m7 ¼

1

b

Â
Ã
ln À1 þ expðbEF 7 Þ

ð6Þ

is the chemical potential for the up/down spin state (with the
Fermi energy EF 7 ). The energy dependent relaxation time t(e)
depends on the scattering mechanism and given by [7–9]
Z 2k
2
/9UðqÞ9 S q2 dq
1
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼
ð7Þ
tðkÞ 2p_e 0
½ A ðqފ2
4k2 Àq2


N.Q. Khanh / Physica E 43 (2011) 1712–1716

where e ¼ _2 k2 =ð2mn Þ, U(q) is the random potential for wave
number q and [16–18]
A ðqÞ ¼ 1 þ

2pe2 1
FC ðqÞ½1ÀGðqފPðq,TÞ
AL q

ð8Þ

is the finite wave vector dielectric screening function. Here G(q) is
the local field correction (LFC), FC(q) is the Coulomb form
factor arising from the subband wave functions c(z), AL is the
background static dielectric constant and P(q,T) is the 2D
irreducible finite-temperature polarizability function given by
P(q,T) ¼ P þ (q,T)þ P À (q,T) with P 7 (q,T) are the polarizabilities
of the polarized up/down spin states. At finite temperature we
have [11,19]
Z
P07 ðq, m0 Þ
b 1 0
ð9Þ
P 7 ðq,TÞ ¼
dm
2
4 0
cosh b2 ðm 7 Àm0 Þ
where
3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2
m 4
2kF 7
0
0
7
P 7 ðq,EF 7 Þ  P 7 ðqÞ ¼ 2 1À 1À
YðqÀ2kF Þ5

q
p_
n

ð10Þ

À1

and for our infinite quantum well model, we have [6]


1
8p2 32p4 1ÀeÀaq
À
3aq
þ
FC ðqÞ ¼
aq
a2 q2 4p2 þ a2 q2
4p2 þa2 q2
We will use the Hubbard approximation GH ðqÞ ¼

1
gs

ð12Þ

q
pffiffiffiffiffiffiffiffiffiffiffi
for the

2
2
q þ kF

LFC [20] where gs is the spin degeneracy. For the unpolarized
electron gas, we apply gs ¼2 and for the fully polarized electron
gas, we use gs ¼ 1. In this paper we will consider four scattering
mechanisms: surface-roughness (S), alloy disorder (A), remote
(R) and homogenous background (B) doping. The random potentials for these scattering mechanisms are given as follows [4,6]

  n 2 
4p
m
p 4
2 2
2
/9US ðqÞ9 S ¼ 2 2
ðeF DLÞ2 eÀq L =4
ð13Þ
mz
kF a
a

2

/9UR ðqÞ9 S ¼ ni

FR ðq,zi Þ ¼



 

A3
3
ðdVÞ2
2
4a



In this section, we present our numerical calculations for
the mobility and resistivity of a Q2DEG in an InP/In1 À xGaxAs/InP
QW using the following parameters [4]: NB ¼1016 cm À 3,
˚ L ¼50 A,
˚ eL ¼13.3, x ¼0.47, dV ¼0.6 eV,
ni ¼1011 cm À 2, D ¼1 A,
A¼5.9 A˚ and mn ¼mz ¼0.041mo, where mo is the vacuum mass
of the electron.

The mobility of the unpolarized and fully polarized 2DEG is
given by m ¼e/mn o t 4. The mobility m limited by different
scattering mechanisms versus electron density n at T¼0, B¼0
for the well width a¼100 A˚ (thin lines) and a ¼150 A˚ (thick lines)
is plotted in Fig. 1. It is seen from the figure that the contribution
of surface-roughness scattering to the mobility can be neglected
for a $ 150 A˚ and no1012 cm À 2. We note that our results are
similar to those given earlier by Gold [4].
In Fig. 2 we show the mobility limited by the alloy disorder,
remote and background impurity scattering versus electron

density n at T¼0 for B ¼0 (thin lines) and B¼Bs (thick lines) and
˚ We observe that the alloy disorder scattering depends
a¼150 A.
strongly on the magnetic field at low densities. This dependence
stems from the dependence of the screening function on the spinpolarization caused by the magnetic field. At higher densities
n $ 1012 cm À 2 the alloy disorder scattering shows a weak dependence on the magnetic field and becomes the main scattering
mechanism in mobility limitation. For a comparison, we now
discuss the scattering mechanisms in an AlxGa1 À xAs/GaAs/AlxGa1 À xAs QW. In the case of large aluminum concentration, the
band edge discontinuity increases leading to increasing confinement of the electron wave function in the GaAs layer and
correspondingly decreasing degree of wave function penetration

ð14Þ

2
2pe2 1
FR ðq,zi Þ2
AL q

8p2
1
aq 4p2 þ a2 q2
8
1 qzi
Àaq
Þ, zi o 0
>
>
2 e ð1Àe
>
>

<
À Á
2 2
1 Àqz
1 ÀqðaÀzi Þ
þ a2pq2 sin2 pazi ,
 1À 2 e i À 2 e
>
>
> 1 Àqðzi ÀaÞ
>
ð1ÀeÀaq Þ, zi 4 a
: 2e

3. Numerical results

107

ð15Þ

106
μ (cm2/Vs)

2

/9UA ðqÞ9 S ¼ xð1ÀxÞ

z-direction, L is the correlation length parameter of the roughness in the xy direction, A3 is the alloy unit cell, dV is the spatial
average of the fluctuating alloy potential over the alloy unit cell, ni
is the 2D impurity density, zi is the distance between remote

impurities and 2DEG and NB is the density of homogenous
background impurities.

3.1. The mobility

2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
with kF 7 ¼ 4pn 7 is the 2D Fermi wave vector for the spin
up/down carriers. The Coulomb form factor is given by
Z þ1
Z þ1
0
2
2
dz9cðzÞ9
dz0 9cðz0 Þ9 eÀq9zÀz 9
ð11Þ
FC ðqÞ ¼
À1

1713

0 rzi ra

105
NB = 1016cm-3
ni = 1011cm-2, zi = -a/2
δV = 0.6 eV, x = 0.47
Λ = 50Å, Δ = 1Å

ð16Þ

2
2pe2 1
2
/9UB ðqÞ9 S ¼ NB a
FB ðqÞ
AL q
FB ðqÞ ¼


2
1
4p2
6 Àaq
6
e
þ 2 2 ðeÀaq À1Þ
aq 4p2 þ a2 q2
aq
a q

!
2aq 3a3 q3 8ð1ÀeÀaq Þ
þ
þ
À
4p2 þ a2 q2
p2

8p4

104
ð17Þ

103
1010

1011

1012

B
R
A
S
1013

-2

n (cm )
ð18Þ

In above expressions mz is the mass perpendicular to the
interface, D is the average height of the roughness in the

Fig. 1. Mobility m versus electron density n at T¼ 0 and B¼ 0. The lines refer to the
mobility limited by: surface-roughness (S), alloy disorder (A), remote (R) and
background (B) impurity scattering for the well width a ¼100 A˚ (thin lines) and
a¼ 150 A˚ (thick lines).

1714

N.Q. Khanh / Physica E 43 (2011) 1712–1716

5

107
16

NB = 10 cm
ni = 1011cm-2, zi = -a/2
δV = 0.6 eV, x = 0.47

105
B
R
A

104

103
1010

NB = 1016cm-3
ni = 1011cm-2, zi = -a/2
δV = 0.6 eV, x = 0.47

4

ρ (Bs)/ρ (B = 0)

μ (cm2/Vs)

106

-3

B
R
A

3

2

1

1011
n (cm-2)

0
1010

1012

Fig. 2. Mobility m limited by alloy disorder, remote and background impurity
scattering versus electron density n at T¼ 0 for B¼0 (thin lines) and B ¼Bs (thick
˚
lines) and a¼ 150 A.

1011
n (cm-2)

1012

Fig. 4. Resistivity ratio r(Bs)/r(B¼ 0) versus electron density n at T¼0 for
˚ The thin and thick lines correspond to the cases of G(q) ¼0 and
a¼ 150 A.
G(q) ¼ GH(q), respectively.

107
2000

105

103
1010

1011

ni = 1011cm-2, zi = -a/2

1200

δV = 0.6 eV, x = 0.47

B
R
A

800

B
R
A

104

n = 1011 cm-2
NB = 1016cm-3

1600
ρ (Ω)

μ (cm2/Vs)

106

NB = 1016cm-3
ni = 1011cm-2, zi = -a/2
δV = 0.6 eV, x = 0.47

400
100

1012

n (cm-2)
Fig. 3. Mobility m limited by alloy disorder, remote and background impurity

˚ The thin and
scattering versus electron density n at T¼0 and B¼ 0 for a¼150 A.
thick lines correspond to the cases of G(q) ¼ GH(q) and G(q) ¼0, respectively.

120

140

160

180

200

a (Å)
Fig. 5. Resistivity r due to alloy disorder, remote and background impurity
scattering versus the well width at T¼ 0 for B ¼0 and B¼ Bs (thick lines).

of the ( 7) spin subband given by
into the AlxGa1 À xAs barrier layer. Thus, our infinite confining
potential well model is reasonable and the alloy disorder scattering can be neglected [5]. Furthermore, interfaces extremely flat
are obtainable by the state-of-art molecular-beam epitaxy technology and interface roughness scattering is still excluded from
our calculations [21]. We have also found that, for scattering
parameters used in this paper, the mobility limited by ionized
impurities is about two times lower than that in an InP/In1 À x
GaxAs/InP QW due to the higher electron effective mass in GaAs
(mn ¼0.067mo).
We now discuss the effect of the LFC G(q) appeared in the
screening function (8) on the mobility. We use the Hubbard
approximation GH(q) for the LFC. The results for T ¼0, B ¼0 and

a ¼150 A˚ plotted in Fig. 3 indicate that the effect of the LFC is
remarkable at low densities.
3.2. The resistivity
The resistivity of the polarized 2DEG is given by r ¼ 1/s where
s ¼ s þ þ s À is the total conductivity and s 7 is the conductivity

s7 ¼

n 7 e2 /t 7 S
mn

ð19Þ

Results for the resistivity ratio r(Bs)/r(B¼0) versus electron
density n at T¼0 for a ¼150 A˚ are shown in Fig. 4. We observe
again that the effect of the LFC is remarkable at low densities. We
note that our results are similar to those given in earlier works
[20,22] for other structures.
The dependence of the resistivity on the well width at T¼0 for
two cases of B¼ 0 and B ¼Bs is depicted in Fig. 5. It is seen that the
resistivity shows a weak dependence on the well width for
homogeneous background doping. In the case of remote doping
and alloy disorder scattering the resistivity decreases with
increase in the well width.
In Fig. 6 we plot the temperature dependence of the resistivity
˚ As seen from the figure, the resistivity due to the
for a ¼150 A.
alloy disorder scattering shows a weak dependence on
temperature.
The temperature dependences of the resistivity for a ¼150 A˚

in two cases of B¼0 and B ¼Bs are plotted in Figs. 7 and 8,


N.Q. Khanh / Physica E 43 (2011) 1712–1716

1600

n = 1011 cm-2
NB = 1016cm-3
ni = 1011cm-2, zi = -a/2
δV = 0.6 eV, x = 0.47

B
R
A

1400

B/Bc = 1

B
R
A

NB = 1016cm-3
1011cm-2,

1000

ρ (Ω)

ρ (Ω)

1200

105

1715

800

10

ni =
zi = -a/2
δV = 0.6 eV, x = 0.47

4

103

600
400
200
0.2

0.4

0.6

102

0.8

4

8

T/TF
Fig. 6. Resistivity r due to alloy disorder, remote and background impurity
scattering as a function of the temperature for a¼ 150 A˚ in two cases of B¼0
and B¼ Bs (thick lines).

105
B=0
NB = 1016 cm-3

ρ (Ω)

104

ni = 1011cm-2, zi = -a/2
δV = 0.6 eV, x = 0.47

B
R
A

103

102

4

8

12

16

20
24
T (K)

28

32

36

40

Fig. 7. Resistivity r due to alloy disorder, remote and background impurity
scattering as a function of the temperature for a¼150 A˚ and B¼0 in two cases
of n¼ 1010 cm À 2 and n¼ 1011 cm À 2 (thick lines).

respectively. We observe that at high temperatures the resistivity
shows a weak temperature dependence.

12

16

20
24
T (K)

28

32

36

40

Fig. 8. Resistivity r due to alloy disorder, remote and background impurity
scattering as a function of the temperature for a ¼150 A˚ and B¼ Bs in two cases
of n¼ 1010 cm À 2 and n¼ 1011 cm À 2 (thick lines).

Hubbard LFC used in this paper is not exact. We believe, however,
that our results are reasonable for carrier densities larger than
1011 cm À 2 [27]. For lower densities, we have to use more exact
LFCs [28–30]. Third, we have excluded the phonon contribution
from our calculations. The phonon effects, however, are negligible
for the temperature range considered in this paper [21].
In conclusion, we have calculated the mobility and resistivity
of a Q2DEG in InP/In1 À xGaxAs/InP QW in an applied in-plane
magnetic field at arbitrary temperatures for three scattering
mechanisms: alloy disorder, remote and homogenous background
doping. We have investigated the dependence of the mobility and

resistivity on the carrier density, layer thickness and magnetic
field. We have shown that the contribution of surface-roughness
scattering to the mobility can be neglected for a $ 150 A˚ and
no1012 cm À 2. Our results and new measurements of transport
properties can be used to obtain information about the scattering
mechanisms in the InP/In1 À xGaxAs/InP QWs [3].

Acknowledgment
The author wishes to thank the Vietnam’s National Foundation
for Science and Technology Development (NAFOSTED) for the
financial support. He also thanks the referees for requiring him to
be more precise in preparing this manuscript.
References

4. Discussion and conclusion
We now discuss the validity and limitations of our results.
First, we note that our saturation field Bs is defined with respect to
a non-interacting system. In order to get better results, we have to
take into account the inter-particle interactions [23,24]. The
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in this paper, the saturation field for interacting systems Bsi is
somewhat less than that for non-interacting systems and the
dependence of the spin-polarization on the ratio B/Bsi is similar to
that of non-interacting systems. Therefore, except the saturation
field value, our results for the mobility and resistivity are still
acceptable at least for high densities and low temperatures. We
note that we can also include the temperature effects on the
saturation field using the classical-map hypernetted-chain
method [25,26]. Second, we admit that the zero-temperature

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