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

We study the dependence of transport properties on the carrier density, layer thickness, magnetic field and temperature.. Recently, Gold has calculated the zero temperature mobility of th

Transport properties of a quasi-two-dimensional

electron gas in a SiGe/Si/SiGe quantum well

including temperature and magnetic field effects

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-two-dimensional 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 applica-tions [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 mech-anism, 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 trans-port 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

0749-6036/$ - see front matter Ó 2013 Elsevier Ltd All rights reserved.

⇑Corresponding author Fax: +84 8 38350096.

E-mail address: nqkhanh@phys.hcmuns.edu.vn (N.Q Khanh).

Contents lists available atScienceDirect

Superlattices and Microstructures

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / s u p e r l a t t i c e s

spin-polarized Q2DEG in a SiGe/Si/SiGe QW at finite temperatures Therefore, we calculate, in this pa-per, 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 m 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,

wð0 6 z 6 LÞ ¼ ffiffiffiffiffiffiffiffi

2=L

p

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 < Bswith n+= n and n= 0 for B P Bs Here n = n++ nis the total density and Bsis the so-called sat-uration field given by glBBs= 2EFwhere g is the electron spin g-factor,lBis 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:

hsi ¼

R

desðeÞe @fðeÞ

@e

R

de e @fðeÞ

@e

where[1,11]

1

sðkÞ¼

1

2phe

Z 2k 0

hjUðqÞj2i

½2 ðqÞ2

q2dq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k2 q2

2 ðqÞ ¼ 1 þ2pe2

2L

1

Pðq; TÞ ¼b

4

Z 1 0

dl0 P0

ðq;l0Þ

P0ðq; EFÞ P0ðqÞ ¼gvm



2ph2 1 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1  2kF 

q

s

Hðq  2kF Þ

2 4

3

FCðqÞ ¼ 1

4p2þ a2q2 3aq þ8p2

aq 

32p4

a2q2

1  eaq

4p2þ a2q2

ð7Þ

with fðeÞ ¼ 1=f1 þ expðb½elðTÞÞg; b ¼ ðkBTÞ1,l¼ ln½1 þ expðbEFÞ=b, EF¼ h2k2F=ð2mÞ and

e¼ h2k2=ð2mÞ Here, mis the effective mass in xy-plane, gvis the valley degeneracy, G(q) is the LFC describing the exchange–correlation effects[8,18–21]and hjUðqÞj2i is the random potential which de-pends on the scattering mechanism[1] For charged-impurities of density Nilocated on the plane with

z = ziwe have:

hjURðqÞj2i ¼ Ni

2pe2

2L

1 q

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

hjUSðqÞj2i ¼ 2 4p

a2

mz

p

kFa

eFD

whereDrepresents the average height of the roughness perpendicular to the 2DEG andKrepresents the correlation length parameter of the roughness in the plane of the 2DEG and mzis the effective mass perpendicular to the xy-plane

The mobility of the nonpolarized and fully polarized 2DEG is given byl0¼ e <s> =m The resis-tivity is defined byq= 1/rwherer=r++ris the total conductivity andr±is the conductivity of the (±) spin subband given by[11]:

r¼ne

2hsi

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 symbollfor the mobility when MSE are taken into account For n > nMITthe mobility can be written asl=lo(1  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]:

2pn2

Z 1

0

hjUðqÞj2i½PoðqÞ2qdq

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, m= 0.19moand mz= 0.916mo, where mois 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]:

GHðqÞ ¼ 1

gvgs

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q2þ k2F

where gsis 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 mobilityloof the nonpolarized Q2DEG in silicon QW of width a = 100 Å for remote impurity scatter-ing (RIS) are displayed in Fig 1 of Ref.[8] Similar results for the mobilityloof the fully polarized Q2DEG and the resistance ratioq(Bs)/q(B = 0) for RIS with Ni= 1  1012cm2and zi= 100 Å are plot-ted inFig 1 We see that the use of a LFC is very important and the Hubbard approximation is not suf-ficient 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 nMITas a function of the impurity density Niand the well width a for a nonpolarized 2DEG at zero temperature The results shown inFig 2indicate that nMIT

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

decreases with increase in the well width and the distance of the impurities from the 2DEG The dependence of the critical density nMITon the well width for zi= a, 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 inFig 3the mobilitylo

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 va-lue for T > 0.15TF( 11 K for n = 5  1011cm2) The figure also indicates that the mobility of nonpo-larized 2DEG limited by remote impurities is higher compared to that of the fully pononpo-larized 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

10 1

10 2

10 3

10 4

10 5

0 1 2 3 4 5 6 7 8 9 10

μ 0

2/Vs)

electron density n(1011cm-2)

G

G H G=0

electron density n(1011cm-2)

G

G H G=0

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

0.1 1 10

0.1

1

a = 100 Å

impurity density Ni(1011cm-2)

nMIT

11 cm

-2 )

zi = +a/2

zi = 0

zi = -a/2

zi = -a

Ni = 1011cm -2

nMIT

11 cm

-2 )

zi = +a/2

zi = 0

zi = -a/2

zi = -a

well width ( Å)

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

3.4 The interface-roughness scattering

Up to now, we have considered only RIS To evaluate the importance of other scattering mecha-nisms we have calculated the mobilityloand the resistance ratioq(Bs)/q(B = 0) for IRS The results for the case of a = 100 Å,D= 6 Å andK= 30 Å[1]with different approximations for G(q) are plotted

inFig 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 tem-perature At zero temperature, our results reduced to those given in Ref.[8] We have shown that the

10 5

106

10 5

106

L=200 Å

zi=-125 Å

Ni=9.2x1011cm-2 B=0

μ0

2 /Vs)

electron density n (1011 cm-2)

T=0K T=0.5T F

T=T F

L=200 Å

zi=-125 Å

Ni=9.2x10 11

cm -2

B=2B S

μ0

2 /Vs)

electron density n (1011 cm-2)

T=0K T=0.5T F

T=T F

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

106

0 10 20

μ0

a=100 Å

G

GH G=0

electron density n (1011 cm-2) electron density n (1011 cm-2)

(Bs

G

GH G=0

a=100 Å

Fig 4 The mobilitylo (left) and the resistance ratioqðBs Þ=qðB ¼ 0Þ (right) versus electron density for IRS with = 6 Å and = 30 Å

in different approximations for G(q).

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 nMITdecreases 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 > 0.15TFand the mobility of the fully polarized 2DEG is

low-er 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

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