DSpace at VNU: Transport properties of a quasi-two-dimensional electron gas in a SiGe Si SiGe quantum well including temperature and magnetic field effects

Superlattices and Microstructures 64 (2013) 245–250

Contents lists available at ScienceDirect

Superlattices and Microstructures
journal homepage: www.elsevier.com/locate/superlattices

Transport properties of a quasi-two-dimensional
electron gas in a SiGe/Si/SiGe quantum well
including temperature and magnetic field effects
Nguyen Quoc Khanh ⇑, Nguyen Minh Quan
Department of Theoretical Physics, National University in Ho Chi Minh City, 227-Nguyen Van Cu Street, 5th District,
Ho Chi Minh City, Viet Nam

a r t i c l e

i n f o

Article history:
Received 14 July 2013
Accepted 27 September 2013
Available online 5 October 2013
Keywords:
Scattering mechanisms
Magnetoresistance
Quantum well

a b s t r a c t
We investigate the mobility and resistivity of a quasi-twodimensional electron gas in a SiGe/Si/SiGe quantum well at
arbitrary temperatures for two cases: with and without in-plane
magnetic field. We consider two scattering mechanisms: remote

charged-impurity and interface-roughness scattering. We study
the dependence of transport properties on the carrier density, layer
thickness, magnetic field and temperature. Our results can be used
to obtain information about the scattering mechanisms in the
SiGe/Si/SiGe quantum well.
Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction
During the last few decades, much attention has been devoted to the transport properties of
modulation-doped Si/SiGe heterostructures because of their high mobility and perfectives for applications [1–7]. Recently, Gold has calculated the zero temperature mobility of the nonpolarized
quasi-two-dimensional electron gas (Q2DEG) in a SiGe/Si/SiGe quantum well (QW), taking into
account many-body effects, beyond the random-phase approximation via a local-field correction
(LFC) [8], and obtained good agreement with recent experimental results [4,5]. The scattering mechanism, which is responsible for limiting the mobility, can be determined by comparing experimental
results with those of theoretical calculations [1–9]. Recent measurements and calculations of transport properties of a 2DEG in a magnetic field give additional tool for determining the main scattering
mechanism [10–16]. To the author’s knowledge, there is no calculation of transport properties of the

⇑ Corresponding author. Fax: +84 8 38350096.
E-mail address: (N.Q. Khanh).
0749-6036/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved.
/>

246

N.Q. Khanh, N.M. Quan / Superlattices and Microstructures 64 (2013) 245–250

spin-polarized Q2DEG in a SiGe/Si/SiGe QW at finite temperatures. Therefore, we calculate, in this paper, the finite temperature mobility of Q2DEG in a SiGe/Si/SiGe QW for charged-impurity scattering
and study the effects of the LFC, magnetic field and layer thickness on transport properties. We also
discuss the importance of interface-roughness scattering (IRS).
2. Theory
We consider a 2DEG with parabolic dispersion determined by the effective mass mÃ. We assume

that the electron gas is in the xy plane with infinite confinement for z < 0 and z > L. For 0 6 z 6 L,
the electron pffiffiffiffiffiffiffiffi
gas in the lowest subband is described by the wave function
wð0 6 z 6 LÞ ¼ 2=L sin ðpz=LÞ [1]. When the in-plane magnetic field B is applied to the system, the
carrier densities n± for spin up/down are not equal [11,17]. At T = 0 we have n± = n(1 ± B/Bs)/2 for
B < Bs with n+ = n and nÀ = 0 for B P Bs. Here n = n+ + nÀ is the total density and Bs is the so-called saturation field given by glBBs = 2EF where g is the electron spin g-factor, lB is the Bohr magneton and EF
is the Fermi energy. For T > 0, n± is determined using the Fermi distribution function in the standard
manner [11,17]. The energy averaged transport relaxation time for the (±) components are given in the
Boltzmann theory by:

h
i
Æ
desðeÞe À @f @eðeÞ
h
i
h sÆ i ¼ R
Æ
de e À @f @ eðeÞ
R

ð1Þ

where [1,11]

1
1
¼
sðkÞ 2phe

2 ðqÞ ¼ 1 þ

Z

2k

hjUðqÞj2 i
2

½2 ðqފ

0

q2 dq
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
2
4k À q2

ð2Þ

2pe2 1
F C ðqÞ½1 À GðqފPðq; TÞ;
2L q

ð3Þ

Pðq; TÞ ¼ Pþ ðq; TÞ þ PÀ ðq; TÞ;
PÆ ðq; TÞ ¼

0

Æ ðq; EFÆ Þ

P

F C ðqÞ ¼

b
4

Z

P0Æ ðq; l0 Þ

1

dl0

0

0
Æ ðqÞ

P

2b
ð
2

lÆ À l0 Þ

cosh

¼

2
g v mà 4
2ph


2

ð4Þ
ð5Þ

;

3
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2
2kF Æ
1À 1À
Hðq À 2kF Æ Þ5;
q

ð6Þ



1

8p2 32p4 1 À eÀaq
3aq
þ
À
4p2 þ a2 q2
aq
a2 q2 4p2 þ a2 q2
À1

ð7Þ
2 2

with f Æ ðeÞ ¼ 1=f1 þ expðb½e À lÆ ðTފÞg; b ¼ ðkB TÞ , lÆ ¼ ln½À1 þ expðbEFÆ ÞŠ=b, EFÆ ¼ 
h kFÆ =ð2mÃ Þ and
2 2

Ã

e ¼ h k =ð2m Þ. Here, m is the effective mass in xy-plane, gv is the valley degeneracy, G(q) is the LFC
Ã

describing the exchange–correlation effects [8,18–21] and hjUðqÞj2 i is the random potential which depends on the scattering mechanism [1]. For charged-impurities of density Ni located on the plane with
z = zi we have:

hjU R ðqÞj2 i ¼ Ni


2
2pe2 1
F R ðq; zi Þ2

2L q

ð8Þ

with the form factor FR(q, zi) for the electron–impurity interaction as given in Ref. [1]. Here eL is the
background static dielectric constant. For the interface-roughness scattering the random potential is
given by [1]:


N.Q. Khanh, N.M. Quan / Superlattices and Microstructures 64 (2013) 245–250

hjU S ðqÞj2 i ¼ 2

  Ã 2  4
4p m
p
2 2
ðeF DKÞ2 eÀq K =4
a2
mz
kF a

247

ð9Þ

where D represents the average height of the roughness perpendicular to the 2DEG and K 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.
The mobility of the nonpolarized and fully polarized 2DEG is given by l0 ¼ e < s > =mà . The resistivity is defined by q = 1/r where r = r+ + rÀ is the total conductivity and r± is the conductivity of the

(±) spin subband given by [11]:

rÆ ¼

nÆ e2 hsÆ i
mÃ

ð10Þ

The authors of Ref. [5] have found the strong decrease of the mobility at low electron densities. This
behavior is likely to be a precursor of localization which may lead to a metal–insulator transition
(MIT). It was shown that multiple-scattering effects (MSE) can account for this MIT at low electron
density where the mobility is determined by impurity scattering [22–23]. We use the symbol l for
the mobility when MSE are taken into account. For n > nMIT the mobility can be written as l = lo(1 À A)
with A 6 1. The parameter A describes the MSE and depends on the random potential, the screening
function and compressibility of the electron gas, and is given by [8,23]:

A¼

1
2pn2

Z

1

2

hjUðqÞj2 i½Po ðqފ qdq

0

2

½2 ðqފ

:

ð11Þ

For n < nMIT, where A > 1, the mobility vanishes: l = 0.
3. Numerical results
In this section, we present our numerical calculations for the mobility and resistivity of a Q2DEG in
a SiGe/Si/SiGe QW using the following parameters [8]: eL = 12.5, gv = 2, mà = 0.19mo and mz = 0.916mo,
where mo is the free electron mass. We study the dependence of the mobility and resistivity on the
LFC, impurity position, layer-thickness, magnetic field and temperature.
3.1. The many-body effects
To treat the many-body 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 has the form
[5,24]:

GH ðqÞ ¼

1
q
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gv gs
2
q2 þ kF

ð12Þ

where gs is the spin degeneracy. We also use numerical results for the LFC G(q) as reported in Ref. [21]
where both exchange and correlation effects are taken into account. The may-body effects on the
mobility lo of the nonpolarized Q2DEG in silicon QW of width a = 100 Å for remote impurity scattering (RIS) are displayed in Fig. 1 of Ref. [8]. Similar results for the mobility lo of the fully polarized
Q2DEG and the resistance ratio q(Bs)/q(B = 0) for RIS with Ni = 1 Â 1012 cmÀ2 and zi = À100 Å are plotted in Fig. 1. We see that the use of a LFC is very important and the Hubbard approximation is not sufficient at very low densities. We note that Dolgopolov and co-workers have analyzed the mobility as a
function of electron density and concluded that the LFCs are approximately double the Hubbard form
[5].
3.2. The impurity position and layer-thickness effects
We have calculated the critical density nMIT as a function of the impurity density Ni and the well
width a for a nonpolarized 2DEG at zero temperature. The results shown in Fig. 2 indicate that nMIT


248

N.Q. Khanh, N.M. Quan / Superlattices and Microstructures 64 (2013) 245–250
10
105

9

G
GH

G=0

8

G=0

4

7

ρ(Βs)/ρ(0)

mobility μ0(cm2/Vs)

10

G
GH

103

6
5
4
3

102

2
1

10

1

0.1

1

0
0.1

10
11

1

10
11

-2

electron density n(10 cm )

-2

electron density n(10 cm )

Fig. 1. The mobility lo of the fully polarized Q2DEG (left) and the resistance ratio qðBs Þ=qðB ¼ 0Þ (right) for RIS in different
approximations for G(q).

-2
11

electron density nMIT(10 cm )

-2
11

electron density nMIT(10 cm )

10

1

a = 100 Å
z i = +a/2
zi = 0
z i = -a/2

1

11

z i = +a/2
zi = 0
z i = -a/2
z i = -a

z i = -a
0.1
0.1

-2

N i = 10 cm

0.1
1

10
11

-2

impurity density N i (10 cm )

40

80

120

160

200

well width ( Å)

Fig. 2. The electron density nMIT as a function of the impurity density (left) and the well width (right) for charged-impurity
scattering.

decreases with increase in the well width and the distance of the impurities from the 2DEG. The
dependence of the critical density nMIT on the well width for zi = Àa, is much stronger compared to
the case with zi = a/2.
3.3. The magnetic field and temperature effects

In order to describe the temperature and magnetic field effects we display in Fig. 3 the mobility lo
as a function of electron density for several temperatures in two cases B = 0 and B = 2Bs. It is seen from
the figure that the temperature effect is considerable for T $ 0.5TF. Besides, we have found that the
mobility decreases with increasing temperature and differs remarkably from its zero-temperature value for T > 0.15TF (%11 K for n = 5 Â 1011 cmÀ2). The figure also indicates that the mobility of nonpolarized 2DEG limited by remote impurities is higher compared to that of the fully polarized case.
This effect is due to spin-splitting in the parallel magnetic field leading to reduced screening in a
spin-polarized electron gas. We note that for B = 2Bs the 2DEG is almost fully polarized at low
temperature.


N.Q. Khanh, N.M. Quan / Superlattices and Microstructures 64 (2013) 245–250

L=200 Å
z i=-125 Å

L=200 Å
z i=-125 Å
11

-2

N i=9.2x10 cm

106

11

B=2BS

mobility μ 0 (cm /Vs)

2

2

-2

N i=9.2x10 cm

106

B=0

mobility μ0 (cm /Vs)

249

T=0K
T=0.5TF

T=0K
T=0.5TF
105

T=TF

T=TF
5

6

7

8

9
11

105

10
-2

5

6

7

8

9
11

electron density n (10 cm )

10
-2

electron density n (10 cm )

Fig. 3. The mobility lo versus electron density at B = 0 (left) and B = 2Bs (right) versus electron density for charged-impurity
scattering.

G
GH

G=0

G=0

10

ρ(Bs )/ρ(0)

Δ=6 Å , Λ=30 Å
a=100 Å

2

mobility μ0 (cm /Vs)

20

G
GH

6

0.1

1

10
11

-2

electron density n (10 cm )

Δ=6 Å , Λ=30 Å

a=100 Å
10

0
0.1

1

10
11

-2

electron density n (10 cm )

Fig. 4. The mobility lo (left) and the resistance ratio qðBs Þ=qðB ¼ 0Þ (right) versus electron density for IRS with = 6 Å and = 30 Å
in different approximations for G(q).

3.4. The interface-roughness scattering

Up to now, we have considered only RIS. To evaluate the importance of other scattering mechanisms we have calculated the mobility lo and the resistance ratio q(Bs)/q(B = 0) for IRS. The results
for the case of a = 100 Å, D = 6 Å and K = 30 Å [1] with different approximations for G(q) are plotted
in Fig. 4. We observe that the LFC is very important at low densities and the mobility limited by IRS
is much higher than that of RIS and can be neglected. The figure again shows that the mobility of a
nonpolarized 2DEG limited by IRS is higher compared to that of the fully polarized case.

4. Conclusions
In this paper, we have investigated the effects of the LFC, impurity position, layer-thickness and
spin-polarization on the mobility and resistivity of the Q2DEG in a SiGe/Si/SiGe QW at arbitrary temperature. At zero temperature, our results reduced to those given in Ref. [8]. We have shown that the


250

N.Q. Khanh, N.M. Quan / Superlattices and Microstructures 64 (2013) 245–250

LFC is very important at low densities, and the critical electron density nMIT decreases with increase in
the well width and the distance of the impurity layer from the Si/SiGe interface. We have found that
the temperature effect is remarkable for T > 0.15TF and the mobility of the fully polarized 2DEG is lower than that of the nonpolarized 2DEG. We have also shown that the mobility limited by IRS is much
higher than that limited by RIS.
Acknowledgement
This research is funded by Vietnam National Foundation for Science and Technology Development
(NAFOSTED) under Grant Number 103.02-2011.25.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]

[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]

A. Gold, Phys. Rev. B 35 (1987) 723.
F. Schäffler et al, Semicond. Sci. Technol. 7 (1992) 260.
J. Tutor et al, Phys. Rev. B 47 (1993) 3690.
Z. Wilamowski et al, Phys. Rev. Lett. 87 (2001) 026401.
V.T. Dolgopolov et al, Superlattices Microstruct. 33 (2003) 271.
T.M. Lu et al, Appl. Phys. Lett. 90 (2007) 182114.
A. Gold, Semicond. Sci. Technol. 26 (2011) 045017.
A. Gold, J. Appl. Phys. 108 (2010) 063710.
T. Ando, A.B. Fowler, F. Stern, Rev. Mod. Phys. 54 (1982) 437.
A. Gold, V.T. Dolgopolov, Physica E 17 (2003) 280.
S. Das Sarma, E.H. Hwang, Phys. Rev. B 72 (2005) 205303.
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).
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, R. Marty, Phys. Rev. B. 76 (2007) 165309.