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
Trang 1Transport properties of the two-dimensional electron gas in AlP
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 onμare remarkable for no1012cm 2or T 0.3TF
The effects of LFC and temperature onτt=τsare nearly canceled in the ratio for IRS
For CIS, the temperature effects onτt=τsare notable at T 0.3TFonly for GH.
At low n, the dependence of the resistance ratio on LFC decreases when T increases
With decreasing L, nMITincreases and becomes nearly independent of LFC and zi.
a r t i c l e i n f o
Article history:
Received 22 September 2013
Received in revised form
19 November 2013
Accepted 25 November 2013
Available online 4 December 2013
Keywords:
AlP quantum well
Magnetoresistance
Scattering time
Temperature effect
a b s t r a c t
We investigate the transport scattering time, the single-particle relaxation time and the magnetoresis-tance of a quasi-two-dimensional electron gas in a GaP/AlP/GaP quantum well at zero andfinite 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
1 Introduction
GaP/AlP/GaP quantum well (QW) structures, where the
elec-tron gas is located in the AlP, have been studied recently at low
temperatures via cyclotron resonance, quantum Hall effect,
Shubni-kov 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 gv¼1 for well width LoLc¼45 Å, and valley degeneracy
gv¼2 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
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 magneticfield 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 atfinite 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 andfinite 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 nMITfor 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
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Physica E
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n Corresponding author Fax: +848 38350096.
E-mail address: nqkhanh@phys.hcmuns.edu.vn (N.Q Khanh)
Trang 2plane with infinite confinement for zo0 and z4L For 0rzrL,
the electron gas in the lowest subband is described by the wave
functionψ(0rzrL)¼pffiffiffiffiffiffiffiffi2=Lsin(πz/L)[14]
When the in-plane magneticfield B is applied to the system,
the carrier densities n7for spin up/down are not equal[15–16] At
T ¼0 we have
n7¼n
2ð17B
sÞ; BoBs
Here, n ¼ nþþn is the total density and Bs is the so-called
saturationfield given by gμBBs¼ 2EF where g is the electron spin
g-factor,μBis the Bohr magneton and EFis the Fermi energy[15]
For T40, n7is determined using the Fermi distribution function
and given by[16]
nþ¼n
2t ln
1 e 2x=t þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðe 2x=t 1Þ 2 þ 4e ð2 þ 2xÞ=t p
2
n¼ nnþ
ð2Þ
where x ¼ B=Bsand t ¼ T=TFwith TFis the Fermi temperature The
energy averaged transport relaxation time for the (7)
compo-nents is given in the Boltzmann theory by[14–16]
τ7
¼RdετðεÞε½∂f7ðεÞ=∂ε
R
where
1
τðkÞ¼
1
2πℏε
Z 2k
0
〈jUðqÞj2〉
AðqÞ
q2dq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k2q2
A qð Þ ¼ 1þ2πe2
AL
1
Π7ðq; TÞ ¼β
4
Z 1
0
dμ= Π0
7q;μ=
Π0
7ðq; EF 7Þ Π0
7ðqÞ ¼gvmn
2πℏ2 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2kF7
q
s
Θðq2kF7Þ
2 4
3 5;
ð8Þ
FCðqÞ ¼ 1
4π2þa2q2
3aq þ8π2
aq 32π4
a2q2
1 e aq
4π2þa2q2
ð9Þ with f7ðεÞ ¼ 1=f1þexpðβ½εμ7ðTÞÞg, β¼ ðkBTÞ 1, μ7¼ ln ½1
þexpðβEF 7Þ=β, EF 7¼ ℏ2k2F 7=ð2mnÞ and ε¼ ℏ2k2=ð2 mnÞ Here,
mnis the effective mass in xy-plane, gvis the valley degeneracy, G
(q) is the local-field correction (LFC) describing the exchange–
correlation effects[17–20], AL is the background static dielectric
constant and〈 UðqÞ 2
〉 is the random potential which depends on the scattering mechanism[14] For IRS the random potential is
given by[14]
UIRSðqÞ
¼ 2 4π
a2
mn
mz
kFa
ðεFΔΛÞ2e q2Λ2
whereΔrepresents the average height of the roughness
perpen-dicular to the 2DEG,Λrepresents the correlation length parameter
of the roughness in the plane of the 2DEG and mzis the effective
mass perpendicular to the xy-plane
For CIS the random potential has the form
〈 U CISðqÞ 2
〉 ¼ ni
2πe2
A
1 q
where niis the 2D impurity density, ziis the distance between remote impurities and 2DEG, and FCISðq; ziÞis the form factor for the electron–impurity interaction given in Ref.[14]
The mobility of the nonpolarized and fully polarized 2DEG is given byμ ¼ e〈τ〉=mn The resistivity is defined byρ¼ 1=s where
s ¼ sþþsis the total conductivity ands7is the conductivity of the (7) spin subband given by s7¼ n7e2〈τ7〉=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]
4πn2
Z1 0
U qð Þ 2
ΠoðqÞ 2
qdq
A qð Þ
For n4nMIT, where Ao1, the Q2DEG is in a metallic phase and for
nonMIT, 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 LoLc using the following parameters [3–4]: AL¼9.8,
mn¼0.3mo and mz¼0.9mo, 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 approx-imation, only exchange effects are taken into account and the LFC has the form GHðqÞ ¼ q=½gvgs
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q2þk2 F
q
where gsis the spin degen-eracy We also use analytical expressions of the LFC (GGA) accord-ing 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
two different QW widths and temperatures Two approximations for the LFC, GHand GGA are used It is seen that exchange and correlation effects are very important for no1012cm 2and the mobility depends strongly on the approximation for LFC The LFC reduces the screening, increases the effective scattering potential,
1 0.1
0.1 1
10
T/TF=0 T/TF=0.3
GGA
GGA
L=40Å
GH
GH Δ=3Å; Λ=50Å; B/B s =0
4cm
2/Vs)
electron density n(1012cm-2)
5 L=30Å
Fig 1 Mobilityμversus electron density n for IRS for two different QW widths and
Trang 3and hence reduces the mobility The temperature effect is
remark-able for T 0.3TF ( 2.75 K for n ¼1011cm2) Note that the
mobility inFig 1 is shown on a log-plot On a linear scale the
changes due to afinite LFC and temperature are much larger
The ratio of the transport scattering time and the
single-particle relaxation timeτt=τsversus electron density is shown in
observe that the dependence ofτt=τson 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 ðkFΛÞ2=3 and the ratio allows us to determine kFΛ, and for a
given density, the parameterΛcan be determined
Results for the resistance ratioρðBsÞ=ρðB ¼ 0Þ versus electron
density for L ¼40 Å and two temperatures in different
approxima-tions for the LFC are shown inFig 3 We see that the resistance
ratio depends strongly on approximations for the LFC The
resis-tivity 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
spin-polarization At low density, the dependence of the resistance ratio
on the LFC decreases when temperature increases and the
tem-perature effect is remarkable for T 0.3TF
3.2 Results for charged impurity scattering
The mobility versus electron density for CIS, characterized by the
impurity density ni¼n and the distance ziof the impurity layer from
the QW edge at z¼0, is shown inFig 4 We observe that, the LFC is
very important at low density On the other hand, the dependence of
the mobility on ziis more pronounced at higher density Again, the
temperature effects on the mobility are remarkable at T 0.3TF
The ratioτt=τsversus electron density for CIS for two
tempera-tures and three values of ziin two G(q) models is displayed inFig 5
At low density, thefinite LFC decreasesτt=τsremarkably for zi¼L/2
or G(q)¼GGA[3] At high density, the ratioτt=τs for zi¼L/2 differs
strongly from that for zi¼ L/2 The temperature effects onτt=τsare
notable for T 0.3TFat low density only for the Hubbard LFC
The resistance ratio ρðBsÞ=ρðB ¼ 0Þ versus electron density for
impurities with density n¼n and z¼L/2 is plotted inFig 6 We see
0.6
τt
/τs
10 1
0.1
1
(kFΛ) 2
/3
2/3
G=0(T/TF=0)
GH(T/TF=0)
GGA(T/TF=0)
GGA(T/TF=0.3)
electron density n(1012 cm-2)
L = 40Å < LC
Δ = 3Å; Λ = 50Å; B/B s =0
2
4
6
Fig 2 Ratio of the transport scattering time and the single-particle relaxation time
τ t =τ s versus electron density for IRS.
1 0.1
1
10
T/TF=0 T/TF=0.3
GGA
GH
G = 0
electron density n(1012cm-2)
L = 40Å < LC
Δ = 3Å; Λ = 50Å
2
Fig 3 Resistance ratio ρðB s Þ=ρðB ¼ 0Þ versus electron density for IRS for L¼40 Å and two temperatures in different approximations for the LFC.
10 1
0.1 0.1 1 10
3cm
2/Vs) GH
GH
GH
GGA
GGA
GGA
electron density n(1012cm-2)
T/TF = 0 T/TF = 0.3
zi = -L/2
zi = 0
zi = L/2 L=30Å; n i = n; B/Bs =0
Fig 4 Mobility μ versus electron density n for CIS for two temperatures and three values of the position of impurities z i in two approximations for the LFC.
τt
/τs
10 1
0.1 1 10
GGA
GGA
GH
GH
zi = L/2
zi = -L/2
electron density n (1012cm-2)
L=30Å; ni = n; B/Bs =0 T/TF = 0 T/TF = 0.3
Fig 5 Ratio τ t =τ s versus electron density for CIS for two temperatures and z i in two approximations for the LFC.
Trang 4that the effects of LFC and temperature are notable at low density
only for Hubbard approximations
3.3 Results for the metal–insulator transition
The critical electron density versus QW width for two
approx-imations 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
effec-tive 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 (nMIT42 1012cm2) and becomes nearly
inde-pendent of the approximation used for the LFC (because at
n42 1012
cm 2 the exchange–correlation effect is small, see
nMITis determined mainly by IRS (CIS) for small (large) QW width Therefore, when L approaches Lc, the nMITfor zi¼L/2 is larger than that for zi¼ L/2 because the CIS is stronger for impurities inside the QW in comparison with the case of impurities located outside the QW
4 Conclusions
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 scatter-ing Wefind the remarkable effects of the LFC and temperature for
no1012cm 2and 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 ratioρðBsÞ=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 0.3TFonly for Hubbard approximation The LFC decreases the screening properties and hence increases the effec-tive 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 nMITis determined mainly by CIS and the nMITfor zi¼L/2 is larger than that for zi¼ 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
Acknowledgement This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 103.02-2011.25
References
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10 1
0.1
0
2
4
6
GGA
GH
electron density n (1012 cm-2)
G = 0 (T/TF = 0)
T/TF = 0
T/TF = 0.3
L = 30 Å
zi = L/2
n i = n
Fig 6 Resistance ratio ρðB s Þ=ρðB ¼ 0Þ versus electron density for CIS for L¼30 Å and
two temperatures in different approximations for the LFC.
40 30
20 10
0.1
1
10
nMIT
12cm
-2)
quantum well width L(Å)
insulating phase
conducting phase
B/Bs= 0; T/TF= 0
Δ = 3Å;Λ = 50Å
n = ni = 1x10 11 cm -2
IRS IRS+RIS (zi=L/2) IRS+RIS (zi=-L/2)
GH
GGA
Fig 7 Critical electron density versus QW width for two values of z i and two
approximations for the LFC.