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 inFig.. 2we sh
Trang 1Transport properties of a spin-polarized quasi-two-dimensional electron gas
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
Article history:
Received 29 March 2011
Received in revised form
18 May 2011
Accepted 25 May 2011
Available online 2 June 2011
a b s t r a c t
We investigate the mobility and resistivity of a quasi-two-dimensional electron gas in an InP/In1 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– ] 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
experimen-tal results with those of theoretical calculations [6– ] 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/In1 xGaxAs/InP quantum well at finite temperatures
There-fore, 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/In1 xGaxAs/InP quantum well
2 Theory
We consider a single InP/In1 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]
cðzÞ ¼
ffiffiffi
2
a
r
sin pz
a
and is zero for all other z
When the in-plane magnetic field is applied to the system, the carrier densities n7 for spin up/down are not equal At T¼0 we have[11]
n7¼n 17B
s
, BoBs
nþ¼n, n-¼0, B Z Bs
8
<
Here n¼nþþn is the total density and Bs is the so-called saturation field given by gmBBs¼2EFwhere g is the electron spin g-factor andmB is the Bohr magnetron For T40, n7 is deter-mined using the Fermi distribution function and given by[11,15]
nþ¼nt ln1e2b=tþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðe 2b=t 1 2 Þ þ4e ð2 þ 2bÞ=t p
2
n¼nnþ
8
<
where b¼ B/Bsand t ¼T/TFwith TFis the Fermi temperature The energy averaged transport relaxation time for the (7) compo-nents are given in the Boltzmann theory by[7,11]
/t7S ¼
R
detðeÞe½@f7ðeÞ=@e R
Heret(e) is the energy dependent relaxation time, and f7(e) is the Fermi distribution function
f7ðeÞ ¼ 1
whereb¼(kBT)1and
m7¼1
bln 1 þ expðbEF7Þ
ð6Þ
is the chemical potential for the up/down spin state (with the Fermi energy EF7) The energy dependent relaxation time t(e) depends on the scattering mechanism and given by[7– ] 1
tðkÞ¼
1
2p_e
Z 2k 0
/9UðqÞ92S
½ A ðqÞ2
q2dq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4k2q2
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Physica E
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n
Fax: þ848 8350096.
E-mail address: nqkhanh@phys.hcmuns.edu.vn
Trang 2where e¼_2k2=ð2mn
Þ, U(q) is the random potential for wave number q and[16–18]
AðqÞ ¼ 1 þ2pe2
AL
1
qFCðqÞ 1GðqÞ½ Pðq,TÞ 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 functionsc(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) ¼Pþ(q,T)þP(q,T) with P7(q,T) are the polarizabilities
of the polarized up/down spin states At finite temperature we
have[11,19]
P7ðq,TÞ ¼b
4
Z 1
0
dm0 P07ðq,m0Þ
where
P07ðq,EF 7Þ P07ðqÞ ¼ m
n
p_2 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2kF7 q
s
Yðq2kF 7Þ
2 4
3
5 ð10Þ with kF 7 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4pn7
p
is the 2D Fermi wave vector for the spin up/down carriers The Coulomb form factor is given by
FCðqÞ ¼
Z þ 1
1
dz9cðzÞ92
Z þ 1
1
dz09cðz0
and for our infinite quantum well model, we have[6]
FCðqÞ ¼ 1
4p2þa2q2 3aq þ8p2
aq
32p4
a2q2
1eaq
4p2þa2q2
ð12Þ
We will use the Hubbard approximation GHðqÞ ¼ 1
g s
q ffiffiffiffiffiffiffiffiffiffiffi
q 2 þk 2 F
p for the LFC [20] where gs is the spin degeneracy For the unpolarized
electron gas, we apply gs¼2 and for the fully polarized electron
gas, we use gs¼1 In this paper we will consider four scattering
mechanisms: surface-roughness (S), alloy disorder (A), remote
(R) and homogenous background (B) doping The random
poten-tials for these scattering mechanisms are given as follows[4,6]
/9USðqÞ92S ¼ 2 4p
a2
mn
mz
2
p
kFa
4
ðeFDLÞ2eq 2 L 2 =4 ð13Þ
/9UAðqÞ92S ¼ xð1xÞ A3
4a
ðdVÞ2 3 2
ð14Þ
/9URðqÞ92S ¼ ni
2pe2
AL
1 q
FRðq,ziÞ ¼8p2
aq
1
4p2þa2q2
1eqz ið1eaqÞ, zio0
11eqzi1eqðaziÞþa22pq22sin2 pzi
a , 0rzira
1eqðziaÞð1eaqÞ, zi4a
8
>
>
>
>
ð16Þ
/9UBðqÞ92S ¼ NBa 2pe2
AL
1 q
FBðqÞ ¼ 1
aq
4p2
4p2þa2q2
6
aqe
aq
þ 6
a2q2ðeaq1Þ
þ 2aq
p2 þ3a3q3
8p4 8ð1eaqÞ
4p2þa2q2
ð18Þ
In above expressions mz is the mass perpendicular to the
interface, D is the average height of the roughness in the
z-direction,Lis the correlation length parameter of the rough-ness in the xy direction, A3is 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 Numerical results
In this section, we present our numerical calculations for the mobility and resistivity of a Q2DEG in an InP/In1 xGaxAs/InP
QW using the following parameters [4]: NB¼1016cm 3
,
ni¼1011cm2, D¼1 ˚A, L¼50 ˚A, eL¼13.3, x ¼0.47, dV ¼0.6 eV, A¼5.9 ˚A and mn
¼mz¼0.041mo, where mo is the vacuum mass
of the electron
3.1 The mobility The mobility of the unpolarized and fully polarized 2DEG is given by m¼e/mn
ot4 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 inFig 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 no1012cm2 We note that our results are similar to those given earlier by Gold[4]
InFig 2we 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 a¼150 ˚A We observe that the alloy disorder scattering depends strongly on the magnetic field at low densities This dependence stems from the dependence of the screening function on the spin-polarization caused by the magnetic field At higher densities
n 1012cm 2the alloy disorder scattering shows a weak depen-dence 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 xAs/GaAs/Al
x-Ga1 xAs QW In the case of large aluminum concentration, the band edge discontinuity increases leading to increasing confine-ment of the electron wave function in the GaAs layer and correspondingly decreasing degree of wave function penetration
δV = 0.6 eV, x = 0.47
Λ = 50Å, Δ = 1Å
B R A S
2/Vs)
Fig 1 Mobilitymversus 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
Trang 3into the AlxGa1 xAs barrier layer Thus, our infinite confining
potential well model is reasonable and the alloy disorder
scatter-ing can be neglected[5] Furthermore, interfaces extremely flat
are obtainable by the state-of-art molecular-beam epitaxy
tech-nology 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 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 3indicate that the effect of the LFC is
remarkable at low densities
3.2 The resistivity
The resistivity of the polarized 2DEG is given byr¼1/swhere
s¼s þs is the total conductivity ands is the conductivity
of the (7) spin subband given by
s7¼n7e2/t7S
Results for the resistivity ratio r(Bs)/r(B¼0) versus electron density n at T¼0 for a ¼150 ˚A are shown inFig 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 ¼Bsis depicted inFig 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
InFig 6we plot the temperature dependence of the resistivity for a ¼150 ˚A As seen from the figure, the resistivity due to the 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 ¼B are plotted in Figs 7 and 8,
2/Vs)
δV = 0.6 eV, x = 0.47
B R A
Fig 2 Mobilitym 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.
2/Vs)
B R A
δV = 0.6 eV, x = 0.47
Fig 3 Mobilitym limited by alloy disorder, remote and background impurity
scattering versus electron density n at T¼0 and B¼ 0 for a¼150 ˚A The thin and
thick lines correspond to the cases of G(q) ¼ GH(q) and G(q) ¼0, respectively.
0 1 2 3 4 5
B R A
δV = 0.6 eV, x = 0.47
Fig 4 Resistivity ratio r(Bs)/r(B¼ 0) versus electron density n at T¼0 for a¼ 150 ˚A The thin and thick lines correspond to the cases of G(q) ¼0 and G(q) ¼ GH(q), respectively.
100 400 800 1200 1600 2000
B R A
NB = 1016cm-3
ni = 1011cm-2, zi = -a/2
a (Å)
120 140 160 180 200
n = 1011 cm-2
δV = 0.6 eV, x = 0.47
Fig 5 Resistivityr due to alloy disorder, remote and background impurity scattering versus the well width at T¼ 0 for B ¼0 and B¼ Bs (thick lines).
Trang 4respectively We observe that at high temperatures the resistivity
shows a weak temperature dependence
4 Discussion and conclusion
We now discuss the validity and limitations of our results
First, we note that our saturation field Bsis 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
authors in Ref.[24]have shown that, in the density range studied
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/Bsiis 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
Hubbard LFC used in this paper is not exact We believe, however, that our results are reasonable for carrier densities larger than
1011cm2 [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 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
no1012cm2 Our results and new measurements of transport properties can be used to obtain information about the scattering mechanisms in the InP/In1 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
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400
600
800
1000
1200
1400
1600
B R A
NB = 1016cm-3
ni = 1011cm-2, zi = -a/2
δV = 0.6 eV, x = 0.47
T/TF
n = 1011 cm-2
Fig 6 Resistivityr due to alloy disorder, remote and background impurity
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and B¼ Bs (thick lines).
4
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B R A
B = 0
NB = 1016 cm-3
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8 12 16 20 24 28 32 36 40
T (K)
Fig 7 Resistivityr 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 10 10 cm 2 and n¼ 10 11 cm 2 (thick lines).
4
102
103
104
105
T (K)
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B/Bc = 1
NB = 1016cm-3
ni = 1011cm-2, zi = -a/2
δV = 0.6 eV, x = 0.47
8 12 16 20 24 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 10 10
cm 2 and n¼ 10 11
cm 2 (thick lines).
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