Electron mobility in the GaAs/InGaAs/GaAs quantum wells
The temperature dependence of the electron lateral mobility in quantum wells of the GaAs/InGaAs/GaAs heterostructures with delta-like doping has been studied. Two types of sample doping – in the quantum well and in the adjacent barrier at a small distance from the well – were used. In the case of...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2013
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irk-123456789-1176872017-05-27T03:03:29Z Electron mobility in the GaAs/InGaAs/GaAs quantum wells Vainberg, V.V. Pylypchuk, A.S. Baidus, N.V. Zvonkov, B.N. The temperature dependence of the electron lateral mobility in quantum wells of the GaAs/InGaAs/GaAs heterostructures with delta-like doping has been studied. Two types of sample doping – in the quantum well and in the adjacent barrier at a small distance from the well – were used. In the case of shallow wells, in such structures the experimental results may be well described by known electron scattering mechanisms taking into account the shape of real envelope wave functions and band bending due to non-uniform distribution of the positive and negative space charges along the growth direction of heterostructure layers. In the case of delta-like doping in the well, a good agreement between experiment and calculations is achieved, if one takes into account a contribution to electron transport of the states of the impurity band formed by the deltaimpurity beneath the bottom of the lowest quantum subband. 2013 Article Electron mobility in the GaAs/InGaAs/GaAs quantum wells / V.V. Vainberg, A.S. Pylypchuk, N.V. Baidus and B.N. Zvonkov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2013. — Т. 16, № 2. — С. 152-161. — Бібліогр.: 16 назв. — англ. 1560-8034 PACS 72.20.Fr, 72.80.Ey, 73.20.At, 73.21.Fg, 73.63.Hs, 81.07.St http://dspace.nbuv.gov.ua/handle/123456789/117687 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The temperature dependence of the electron lateral mobility in quantum wells
of the GaAs/InGaAs/GaAs heterostructures with delta-like doping has been studied. Two
types of sample doping – in the quantum well and in the adjacent barrier at a small
distance from the well – were used. In the case of shallow wells, in such structures the
experimental results may be well described by known electron scattering mechanisms
taking into account the shape of real envelope wave functions and band bending due to
non-uniform distribution of the positive and negative space charges along the growth
direction of heterostructure layers. In the case of delta-like doping in the well, a good
agreement between experiment and calculations is achieved, if one takes into account a
contribution to electron transport of the states of the impurity band formed by the deltaimpurity beneath the bottom of the lowest quantum subband. |
| format |
Article |
| author |
Vainberg, V.V. Pylypchuk, A.S. Baidus, N.V. Zvonkov, B.N. |
| spellingShingle |
Vainberg, V.V. Pylypchuk, A.S. Baidus, N.V. Zvonkov, B.N. Electron mobility in the GaAs/InGaAs/GaAs quantum wells Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Vainberg, V.V. Pylypchuk, A.S. Baidus, N.V. Zvonkov, B.N. |
| author_sort |
Vainberg, V.V. |
| title |
Electron mobility in the GaAs/InGaAs/GaAs quantum wells |
| title_short |
Electron mobility in the GaAs/InGaAs/GaAs quantum wells |
| title_full |
Electron mobility in the GaAs/InGaAs/GaAs quantum wells |
| title_fullStr |
Electron mobility in the GaAs/InGaAs/GaAs quantum wells |
| title_full_unstemmed |
Electron mobility in the GaAs/InGaAs/GaAs quantum wells |
| title_sort |
electron mobility in the gaas/ingaas/gaas quantum wells |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117687 |
| citation_txt |
Electron mobility in the GaAs/InGaAs/GaAs quantum wells / V.V. Vainberg, A.S. Pylypchuk, N.V. Baidus and B.N. Zvonkov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2013. — Т. 16, № 2. — С. 152-161. — Бібліогр.: 16 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT vainbergvv electronmobilityinthegaasingaasgaasquantumwells AT pylypchukas electronmobilityinthegaasingaasgaasquantumwells AT baidusnv electronmobilityinthegaasingaasgaasquantumwells AT zvonkovbn electronmobilityinthegaasingaasgaasquantumwells |
| first_indexed |
2025-07-08T12:37:42Z |
| last_indexed |
2025-07-08T12:37:42Z |
| _version_ |
1837082380200837120 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 152-161.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
152
PACS 72.20.Fr, 72.80.Ey, 73.20.At, 73.21.Fg, 73.63.Hs, 81.07.St
Electron mobility in the GaAs/InGaAs/GaAs quantum wells
V.V. Vainberg1, A.S. Pylypchuk1, N.V. Baidus2 and B.N. Zvonkov2
1Institute of Physics, National Academy of Sciences of Ukraine, 03680 Kiev, Ukraine
Phone: +38(044) 525-79-51, e-mail: vainberg@iop.kiev.ua
2Research Scientific Physical-Technical Institute of Lobachevskii State University, 603950 Nizhni Novgorod, Russia
Abstract. The temperature dependence of the electron lateral mobility in quantum wells
of the GaAs/InGaAs/GaAs heterostructures with delta-like doping has been studied. Two
types of sample doping – in the quantum well and in the adjacent barrier at a small
distance from the well – were used. In the case of shallow wells, in such structures the
experimental results may be well described by known electron scattering mechanisms
taking into account the shape of real envelope wave functions and band bending due to
non-uniform distribution of the positive and negative space charges along the growth
direction of heterostructure layers. In the case of delta-like doping in the well, a good
agreement between experiment and calculations is achieved, if one takes into account a
contribution to electron transport of the states of the impurity band formed by the delta-
impurity beneath the bottom of the lowest quantum subband.
Keywords: heterostructure, quantum well, electron mobility, lateral transport,
semiconductor.
Manuscript received 15.01.13; revised version received 27.02.13; accepted for
publication 19.03.13; published online 25.06.13.
1. Introduction
The GaAs/InGaAs/GaAs heterostructures with quantum
wells (QW) are successfully applied in modern
electronics. In particular, such structures enabled to
create low-noise UHF transistors HEMT [1] and
powerful laser diodes [2]. In the recent decade, the
possibility to generate or detect the middle and far
infrared radiation by using these structures was explored
[3]. The latter is based on the real-space transfer of
electrons under heating electric fields. In particular, the
delta-doped structures with two tunnel-coupled quantum
wells are used for this purpose. The current-voltage
characteristics and accompanying far IR radiation of
such structures in the regime of the lateral transport
under strong electric fields within the temperature range
4 to 100 K were studied in [4]. In that paper, the electron
mobility and real-space redistribution of electrons
between wells as functions of electron temperature were
calculated in the frame of a simple model of rectangular
quantum wells. The obtained dependences predicted
considerably higher redistribution and, consequently, a
stronger dependence of the mobility and IR radiation on
the electric field as compared to the experimental ones.
The successive measurements for other structures did
not eliminate this discrepancy.
For further studies of effects caused by the real-
space transfer of electrons in these structures, it is
important to investigate temperature and field
dependences of the electron mobility in the quantum
wells and analyze them with account of both the band
bending and real envelope wave functions [5].
The mobility of carriers in quantum wells is
sufficiently well studied in the case when the doping
impurity is introduced only into the barrier and is
separated from the well by a spacer. In these samples,
the highest mobility, especially at low temperatures, is
achieved. To generate IR radiation along with a high
mobility in one of the quantum wells, it is important to
have a low mobility in another one of the couple. It
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 152-161.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
153
Table 1. Parameters of samples under study.
Sample #
QW
width, Å
Barrier
width, Å
QW depth,
meV
Impurity
concentration,
211cm10
Impurity
position
Number of
periods
n4.2 K,
211cm10
5997 80 800 63 8.1 QW centre 10 7.85
5998 80 800 63 2 QW centre 10 1.7
6002 80 800 63 4.5 QW centre 10 4.05
6215 80 800 96 6.1
barrier, 100 Å
from QW
10 4.6
6291 100 800 50 2.1
barrier, 10 Å
from QW
20 1.6
could be achieved by inserting a delta layer of impurities
in the plane of the well. We do not know publications
concerning systematic investigations of mobility in such
samples, except of a few measurements for the systems
of AlGaAs/GaAs/AlGaAs [6], InAlAs/InGaAs/InAlAs
[7, 8] GaAs/InGaAs/GaAs [9]. Therefore, the purpose of
this work was to investigate temperature dependences of
the conductivity and Hall mobility of electrons within
the range 4 to 400 K in the GaAs/InGaAs/GaAs
quantum wells with different concentrations of delta-
doping impurity in the well plane. For the purpose of
comparison, the structures with quantum wells delta-
doped in adjacent barriers had been also investigated.
The results of studying the field dependence of the
electron mobility will be published later.
2. Samples, experimental details
and measurement results
The GaAs/InGaAs/GaAs heterostructures were grown
by the MOVPE method on semi-insulating GaAs (100)
substrates. The structures were delta-doped by Si. All
measurements were carried out with the samples listed in
Table 1 with parameters following from the
technological process data. The residual shallow
impurity concentration in these structures is estimated to
be within the limits of 1015 to 316 cm10 . To compensate
the influence of the surface states on the potential
profile, the structures were doped by an impurity delta-
layer near the surface. The samples for measurements of
resistivity and the Hall effect were cut from wafers in the
shape of the Hall bridge. The ohmic contacts were made
by deposition of the Pd/Ge/Au layers in vacuum and
further heating in the hydrogen ambient at 700 K.
Electric field was applied along 110.
Measurements of the resistivity and Hall effect in
the weak magnetic field (B = 0.2 T) were carried out
within the range 4 to 400 K in the DC regime
(I = 100 A). The applied electric field was less than
1 V/cm for all the studied samples. The error in
temperature measurements was no more than 0.1 K.
Measurements of the Hall effect were repeated after
several months, and results for the same sample
coincided with the accuracy within 1%. At the same
time, for different samples cut from one wafer, the Hall
concentration may differ considerably, while their
temperature dependences of mobility and Hall
coefficient coincide qualitatively quite well.
The temperature dependences of the Hall coefficient
RH for all 5 structures under study are presented in Fig. 1.
These curves show comparatively small variation of the
Hall coefficient over the whole temperature range. The
observed behaviour may be caused both by variation of
the charge carrier concentration and variation of the Hall
factor. Typical for all curves initial growth of RH with T at
low temperatures is related, from our point of view, with
the increase of the Hall factor, because a carrier
concentration decrease is unlikely in this temperature
range1. On the other hand, the further decrease of RH at
least at temperatures higher than 100 K is caused by
increasing the concentration of electrons with temperature
growth. Indeed, since in all the structures at temperatures
from 4 up to 10 K the electron gas is close to quantum
degeneracy, the Hall factor at these temperatures must be
in fact equal to unity2. Further, the high temperature Hall
coefficient values are less than those at 4 K. Hence, the
Hall factor should be considerably less than unity, if the
concentration is supposed to be constant with the growing
temperature. However, this is impossible. Listed in the
rightmost column of Table 1 are values of the impurity
concentration in the samples at 4 K calculated from data
presented in Fig. 1 when supposing that the Hall factor is
equal to unity.
The temperature dependences of the conductivity
for the samples with delta-doping in an adjacent barrier
and in QW are shown in Figs 2a and 2b, respectively.
The corresponding dependences of the Hall mobility
H = RH on temperature calculated for these structures
from the data shown in Figs 1, 2a and 2b are shown in
Figs 2c and 2d.
1 In principle, such a decrease may be related with capture of electrons
by very shallow traps.
2 Strictly, this statement is valid only in the case of common relaxation
time for all charge carriers. For example, in the case of filling several
subbands it may be violated.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 152-161.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
154
Fig. 1. Temperature dependences of the Hall coefficient for the
samples under study. Sample ##: 1 – 5998, 2 – 6291, 3 – 6215,
4 – 6002, 5 – 5997.
Comparing data in Fig. 2, one may conclude that,
as expected, the electron mobility and corresponding
conductivity at low temperatures are considerably larger
for samples doped in the barrier as compared to those
doped in QW. The temperature dependences both for
conductivity and mobility differ from each other
substantially in these two cases. In the case of doping in
the barrier, the electron mobility at first slightly
increases with growth of temperature from 4 K and then
strongly decreases. For the samples doped in the QW, it
increases by 4 to 5 times, exhibits a flat maximum and
slightly decreases with heating up to the room
temperature. Furthermore, one should notice the
following feature of the curves for the electron mobility
in the samples doped in QW (Fig. 2d). At low
temperatures (below 70-80 K) the higher is the impurity
concentration, the higher is the mobility. For the sample
with the lowest impurity concentration (approximately
211 cm102 , sample #5998) the mobility value at 4.2 K
differs from those in other samples by several times. At
the same time, at higher temperatures the higher is the
concentration, the lower is the mobility.
These results may be explained qualitatively as
follows. As it is well known, in delta-doped GaAs with
the concentration of the order of 21011 to 31011 cm–2
[10, 11] the states of impurities form an impurity band.
The conducting states below the conduction band bottom
appear. With an increase of the concentration, the
impurity band widens and at last overlaps with the 2D
conduction band. Furthermore, as mentioned in many
papers (for example, [11]), one should take into account
that, because of fluctuations of the distances between
impurity atoms in the delta-layer, the impurity band
widens even more, and the edge of the band (2D
subband) becomes smeared.
In the structures doped in a barrier independently of
the impurity concentration (up to 11012 cm–2), practically
all electrons tend to fall into the quantum well. Scattering
of carriers at low temperatures in these samples is mainly
caused by remote impurities, roughness of well
boundaries and alloy fluctuations in InGaAs. The mobility
in this case is comparatively high. At high temperatures, a
considerable portion of electrons is transferred into states
above the barrier. Then scattering of electrons occurs
mainly by lattice vibrations, and mobility decreases down
to the bulk value.
a
b
c
d
Fig. 2. Temperature dependences of conductivity and mobility of
electrons for the samples with the delta-layer of impurity in the
barrier (a and c) and QW centre (b and d). Sample ## in a and c:
1 – 6215, 2 – 6291, in b and d: 1 – 5998, 2 – 6002, 3 – 5997.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 152-161.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
155
In the structures doped in QW, the situation
changes. At low temperatures, if the impurity
concentration is lower than that corresponding to
overlapping of impurity and conduction bands, the
magnitude of mobility is very small. It increases with
temperature, because more and more electrons are
transferred into 2D subband states. Provided the
impurity and conduction bands overlap, then already at
low temperatures the conduction is determined by
electrons in the states of the 2D subband in QW.
However, mobility in this case has a lower value as
compared to doping in a barrier because of closeness of
the scattering impurity ions to electrons. At the same
time, it has a higher value as compared to the low doping
concentration when the states do not overlap. The
calculations performed in the next section take into
account these features of delta-doping in QW.
3. Numerical calculations
The aim of this section is, firstly, to construct the model
of the potential profile in the studied structures, which is
caused by a finite height of the barrier and distribution of
the space charge (concentration profile of the ionized
impurity and free electrons) along the direction of the
heterostructure layers growth, calculation of energy
levels and envelope functions of electrons in this
potential profile, and, secondly, calculations of mobility
in this model. Calculations were carried out for a set of
temperatures within the range from 4 to 400 K.
We have used an idealized model of an infinite
chain of independent quantum wells and neglected the
non-parabolicity of the conduction band. Its influence on
the energy levels and on the electron population is less
than a few percents even at the highest concentration
used in calculations. The positive charge of the
impurities in the delta-layer is assumed uniformly
distributed over the plane (the jelly model). This
assumption is justified in the case of doping in QW, at
least if the average distance between the ions is less than
a characteristic size of the electron wave function of the
impurity in the ground state (for GaAs it corresponds
approximately to NS > 211 cm105.2 ).
The system to be solved consists of the one-
dimensional Schroedinger and Poisson equations
supplemented with the electro-neutrality condition,
)()()(
)(
)(
1
2
2
zEzzU
dz
zd
zmdz
d
, (1)
)()(4)( 2 zNzne
dz
zdU
z
dx
d
, (2)
0)()(
dzzndzzN
L
L
L
L
D , (3)
where U(z) is the coordinate dependence of the energy
of the conduction band bottom, E – eigenvalue of the
energy, (z) – envelope wave function of an electron
along the growth direction of the layers, m(z), (z), n(z)
and N+(z) are the coordinate dependences of the electron
effective mass, dielectric permittivity, concentration of
free electrons and ionized impurities, respectively. Since
the quantum wells in the chain are identical, we consider
only one period of a heterostructure. The corresponding
boundary conditions for one period are written as
follows:
0)()( LL nn ; (4)
0)()( LULU ,
dz
LdU
dz
LdU )()(
. (5)
The quantum wells in adjacent periods are
separated by wide and non-transparent for electrons
barriers where the magnitude of the envelope wave
functions exponentially decreases down to zero. The
dependences of (z) and m(z) on the coordinate are
caused by presence of the quantum wells, where these
parameters have values corresponding to InGaAs with a
given composition of In.
The concentration of positively charged impurities
is determined as
kT
zЕE
g
zN
zN
imp )(
exp1
)(
)(
F
, (6)
where N(z) = Ndelta(z) + Nbg(z) is the coordinate
distribution of the impurity, g – degeneracy factor of the
impurity level. The delta-doped impurity concentration
is determined by the Gauss distribution function
2
2
0
2
exp
2
1
)(
zz
NzN deltadelta . (7)
Here, Nbg is the background impurity, EF – Fermi level,
Eimp(z) – energy of the impurity level in the point z
(different in QW and barriers). The concentration of free
electrons n(z) is determined in each point as the sum of
the concentrations in states above the barrier, which we
consider as 3D states, and in all subbands in quantum
wells (2D electron gas).
,exp1ln)(ψ
))(()()()(
F22
F2/1
3
23
kT
EE
Nz
zUEFNznznzn
CiD
C
i
i
c
D
C
i
DD
(8)
where D
CN 3 , D
CN 2 are the effective densities of states
in the conduction band for the 3D and 2D electron gas,
F1/2( zUE CF ) is the Fermi-Dirac integral for the
index ½ .
The parameters of GaAs and InAs used in our
calculations are listed in Table 2. The parameters of
AsGaIn x1x were determined by linear interpolation.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 152-161.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
156
Table 2. Parameters of GaAs and InAs.
Parameter InAs GaAs
Sound velocity, cm/s 4.28105 5.24105
Density, g/cm3 5.68 5.32
Lattice parameter, Å 6.0583 5.65325
Optical phonon energy, meV 30 35
Optical deformation potential, eV 41 41
Effective mass 0.023 0.063
Deformation potential, eV –5.04 –11
Static dielectric permittivity 15.15 12.9
HF dielectric permittivity 12.3 10.89
The system (1)-(3) was solved self-consistently
using numerical methods for various temperatures and
impurity concentrations. The equations (1) and (2) were
written in the finite-difference form for 1000 points
along the length of one period, and the solution was
found by a standard procedure for this approach.
Calculations were performed up to distances of
±500 Å from the centre of QW corresponding to the
period of our structures. The interval between adjacent
points, where the potential and wave functions were
calculated, was equal to 1 Å. The self-consistent
calculations were repeated until the difference between
two successive corrections to the potential becomes less
than a given value (less than 1%, which provided the
neutrality condition to be fulfilled with an accuracy no
worse than 4101 ).
The obtained data set for the energy spectrum,
energy of the Fermi level and envelope wave functions
were used in the next step for calculations of mobility. In
heterostructures based on III-V compounds, the main
mechanisms of electron scattering are scattering by
ionized impurities, polar optical phonons, interfaces
roughness, composition fluctuations (alloy scattering).
Though it is known that in these compounds the
scattering by acoustic and deformation potential optical
phonons do not play any substantial role, we included
them into consideration for completeness of description.
The corresponding expressions for the relaxation time,
Hall coefficient and mobility are given in the Appendix.
It should be noticed that for electron scattering by polar
optical phonons one cannot strictly introduce the
momentum relaxation time. Nevertheless, following [12]
we used for this scattering mechanism the expression for
the time of the average loss of the momentum
kdkkW
k
q
),(
cos1
, 'kkq ,
being the angle between q and k′. (9)
4. Results and discussion
Fig. 3 illustrates the profile of the conduction band
bottom for one period in the chain of the quantum wells
with parameters of the sample #5997 and delta-doped in
the centre of QW. In all our calculations, we also took
into account background impurities with the
concentration of 315 cm105 , which were assumed to
be uniformly distributed over a period. The energy of
quantum levels, the Fermi level and squared envelope
wave functions are also shown in Fig. 3.
Fig. 3. Energy spectrum in the conduction band for the samples
doped by a delta layer of impurity in the centre of QW. The
width and depth of QW and impurity concentration correspond
to parameters of the sample #5997. 1 – the conduction band
bottom; 2 – square of envelope wave function; 3 – Fermi level.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 152-161.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
157
Fig. 4. Energy spectrum in the conduction band for the samples
doped by a delta layer of impurity in the barrier. The width and
depth of QW and impurity concentration as well as its position
correspond to parameters of the sample #6215. 1 – the
conduction band bottom; 2 – square of envelope wave
function; 3 – Fermi level.
At low temperature (Fig. 3a), electrons from the
background impurities in the barrier are transferred into
the lowest energy states (states in QW) and the
remaining positive charge causes a rise of the electron
potential energy towards the well. Since the electrons
from the delta-doping remain in QW, the potential
profile weakly depends on this kind of impurity. With
temperature growth, electrons come back into the barrier
and partially neutralize the positive charge.
Approximately at 100 K (Fig. 3b) all donors in the
barrier are neutralized, and the potential profile
corresponds to that of rectangular QW. With further
temperature growth, electrons of donors located in the
central plane of QW begin to penetrate into the barrier.
The potential energy related with electric field of the
remaining positively charged donors increases with
growing the distance from this plane (Fig. 3c). The
higher is temperature, the wider and deeper becomes a
quantum well. At last, the width of the quantum well
becomes comparable with the period of the structure.
Fig. 4 illustrates similar dependences for the
structure doped in the barrier with parameters of the
sample #6215. Here, the impurity delta-layer forms the
second QW in the barrier with new quantum levels. The
profile of the potential energy becomes more
complicated, and it does not change qualitatively with
temperature growth.
Further it should be noticed that, for structures
delta-doped both in the centre of QW (Fig. 3) and in the
barrier (Fig. 4) the envelope wave function and,
consequently, the electron density do not vanish at the
well borders and penetrate quite deep into the barrier.
This effect was usually neglected in earlier calculations
of low temperature electron mobility in quantum wells
[7, 13, 14]; that may be justified in the case of a deep
QW or in the case of a large effective mass of charge
carriers (for example, in the case of holes in the quantum
well of the valence band [15]). Otherwise as shown, for
example, in [16] for the interface roughness scattering
mechanism, neglecting this factor may cause a
significant error in the calculated mobility.
The results obtained by the method described above
for a set of temperature values within the range from 4 to
400 K were used in calculations of the temperature
dependence of the mobility. Here, the following two
approximations were made, which substantially
influenced the obtained results. Firstly, according to the
qualitative analysis carried out above, for the structures
doped in QW, we assumed a spread of the impurity levels
with respect to energy in a shape of the Gauss distribution
and put additionally the states lying below the bottom of
the first conduction subband in the neutrality equation.
This leads to noticeable lowering the Fermi level and
considerable decreasing the calculated mobility. Secondly,
concerning the mobility in those 2D conduction subbands,
which bottoms lie below the states above the barrier by
value less than kT/4, it was assumed that it is close to the
electron mobility in the 3D states. Therefore, in
calculations of the mobility averaged over all bands the
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 152-161.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
158
concentration in these bands was determined with the 2D
density of states, while relaxation times were calculated
using the formulae for 3D states. Besides, changes in the
electron concentration with temperature were neglected in
the calculations, and concentration was taken equal to the
value measured at 4 K, because in those samples for
which the concentration increases considerably by
temperatures higher than 100 K the impurity scattering
does not play a noticeable role.
The parameters characterizing scattering processes
and used in our calculations are listed in Table 3.
Figs 5a and 5b show the temperature dependences
of the electron quasi-momentum relaxation time
averaged over the distribution function in the lowest
subband for all the scattering mechanisms taken into
account. Figs 5a and 5b correspond to delta-doping in
the centre of QW and the barrier, respectively.
Table 3. Parameters of the roughness and alloy scattering.
Roughness correlation length, Å 100
Average roughness height, Å 2.3
Scattering potential of alloy atoms, eV 0.33
a
b
Fig. 5. Temperature dependences of the averaged quasi-
momentum relaxation times of charge carriers in the lowest 2D
subband in the samples doped by a delta layer of impurity in
the centre of QW (a) and barrier (b). Scattering mechanisms:
1 – acoustic phonons; 2 – alloy; 3 – optical deformation
potential phonons; 4 – polar optical phonons; 5 – roughness;
6 – impurity.
Fig. 6. Fitting the measured temperature dependences of the
Hall mobility of electrons in the samples doped by a delta layer
of impurity in the barrier (1, 2) and in the centre of QW (3, 4).
1 and 3 – calculations; 2 and 4 – experiment.
As to be seen, in the case of delta-doping in the
centre of QW the prevailing scattering mechanisms are
scattering by charged impurities at low temperatures and
polar optical phonons at the high ones. For the structures
doped in the barrier, a considerable contribution
additionally results from scattering on the interface
roughness and composition fluctuations at low
temperatures and deformation potential optical phonons
at high temperatures. The calculated mobilities for
samples #5997 and 6215 are compared with experiment
in Fig. 6. As to be seen, all the qualitative features of
experimental curves considered above are correctly
described by the calculated data. Taking into account the
adopted approximations, we conclude that a satisfactory
quantitative agreement between measured and calculated
data is also achieved. Regarding the other investigated
samples a sufficient agreement between experimental
and calculated results is also obtained.
5. Conclusions
For successful applications of the GaAs/InGaAs/GaAs
heterostructures in devices based on the mechanism of
a real-space transfer of electrons between tunnel-
coupled quantum wells, one must provide a large
difference of the mobilities in coupled wells. As seen in
Fig. 6, the ratio of mobilities for wells with doping
either in the barrier or in one of coupled QW may
achieve 20 and even more at 4 K. And this value
quickly decreases with temperature growth. However,
one should take into account that the dependences of
mobility on lattice temperature in equilibrium and on
electron temperature under heating electric fields differ
in the region where scattering by phonons plays a
significant role. Although this ratio should decrease
less sharply with growth of electron temperature, the
demands to parameters of a structure destined to be
used in applications may remain sufficiently rigid, and
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 152-161.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
159
in order to obtain optimal structures, preliminary
calculations of their transport properties are needed. A
quite good agreement between calculated and
experimental dependences obtained in the present work
enables to perform sufficiently real calculations of
parameters for structures with different applied
destination.
Acknowledgments
The work was supported by State Target Scientific and
Technical Program (“Nanotechnologies and
Nanomaterials”, Project #1.1.7.18). The authors are also
grateful to Prof. O.G. Sarbey and Dr. Sc.V.N. Poroshin
for helpful discussion.
Appendix
Given below are expressions for the quasi-momentum
relaxation time used in calculations.
1. Acoustic phonons
ac
soundband
bandband
ac
FF
C
kTDmm
E 23
2
0
2)(
1
, (A.1)
dqJJFFac ; dzziqzJ z )exp()(
2 ,
dzziqzJ z )exp()(
2 , (A.2)
)(
)(1
2
zm
dzz
m
band
band
, (A.3)
mband and m0 are the effective and free electron masses, k
is the Boltzmann constant, Dband, ψband, Cband are the
deformation potential constant, density and sound
velocity.
2. Optical deformation potential phonons
ac
opt
FFeF
E
Pr
)(
1
, optE , (A.4)
ac
opt
opt
FF
kT
eF
E
exp1Pr
)(
1 , optE ,
(A.5)
1exp
1
4
Pr
2
0
2
kT
a
mmD
eF
optoptbandL
bandopt
, (A.6)
aL is the lattice parameter.
3. Polar optical phonons
,
)(
1
)(
1
)(
1exp
1
2
1
0
2
3
0
2
0
dz
zz
z
kT
mme
band
opt
optband
(A.7)
optE ,
,
42
)exp()()exp()(
2
2
0
2
2
02
22
1
dq
EE
mm
EE
mm
q
dziqzzdziqzz
FF
opt
band
opt
band
pol
(A.8)
q is the phonon wavevector participating in scattering.
1
01 )(
1
)(
1
)(
1
polFF
EEE
, (A.9)
optE ,
,
42
)exp()()exp()(
2
2
0
2
2
02
22
''
dq
EE
mm
EE
mm
q
dziqzzdziqzz
FF
opt
band
opt
band
pol
(A.10)
kT
FF
EEE
opt
pol
exp
)(
2
)(
1
)(
1 ''
01
. (A.11)
4. Alloy scattering
al
bandlatal
al
FF
mmxxaU
E 3
0
32 )1(
)(
1
, (A.12)
Ual is the scattering potential of alloy atoms.
L
al dzzFF
0
4
)( . (A.13)
5. Interface roughness scattering
,
)(1
)exp(
2
2
)(
1
1
0
22
422
0
2222
3
0
Sband
Crightleft
band
rough
xxx
dxxxk
dL
dE
mm
E
(A.14)
2
0
0
2
Emm
k band ,
0
2
0
2
0
k
emm
x band
S
, (A.15)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2013. V. 16, N 2. P. 152-161.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
160
, are the correlation length and magnitude of
roughness,
dL
dEC is the calculated subband bottom
fluctuation with the roughness-caused fluctuation of a
well width.
6. Impurity scattering
,)(),(
)(
1
)cos1(
4
)(
1
0
2
2
3
4
zNzqFdz
qq
d
me
E
(A.16)
02
sin2 kq
, (A.17)
'exp)'('),(
2
zzqzdzzqF , (A.18)
)(
2
)(
2
22
1)(
2
2
02
0
2
0 qF
me
qF
me
q
,
(A.19)
),()()(
2
zqFzdzqF . (A.20)
The total relaxation time for each subband
.
)(
1
)(
1
)(
1
)(
1
)(
1
)(
1
)(
1
EEE
EEEE
alloyimprough
poloptac
(A.21)
For each subband:
0
2
2
)(
4
1
)(
)(
band
CE Fnorm
kT
EE
ch
dEEE
IdEEf
dE
dE
df
EE
,
(A.22)
0
exp1band
CE F
norm
kT
EE
dE
I , (A.23)
bandmm
e
0
. (A.24)
The total drift mobility is determined as follows
bands
band
bands
bandband
drift
n
n
, (A.25)
bandn is the electron concentration in a given subband
.exp1ln
2
0
kT
EE
kTmm
n
CbandF
band
band
(A.26)
The total Hall mobility is determined as
bands
bandband
bands
bandband
n
nr 2
H
H , where
2
2
H
r . (A.27)
The quantum wells in the investigated structures
are not deep (less than 100 meV). Consequently, at high
temperatures from 200 to 400 K, there are a lot of
electrons in the 3D states that give a noticeable
contribution to the measured Hall concentration and
mobility. Participating in scattering of 3D electrons are
acoustic, optical deformation and polar phonons and
ionized impurity. The expressions for 3D electron
mobility at these scattering mechanisms are well known
and are not given here. They are included in the total
mobility as follows
D
bands
band
DD
bands
bandband
drift
nn
nn
3
33
, (A.28)
DD
bands
bandband
DD
D
bands
bandband
nn
nrnr
33
3
2
3
3
H
2
H
H
, (A.29)
where n3D is the integral of the above-barrier (3D)
electron concentration within one period.
References
1. Y. Ando, Field Effect Transistors, US Patent
5371387 (1994); C. Chang, GaAs-InGaAs high
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V.N. Poroshin, Lateral transport and far-infrared
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6. W. Ted. Masselink, High-differential mobility of
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// Phys. Rev. Lett. 66(11), p. 1513-1516 (1991).
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G. Galistu, Electron transport and optical properties
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properties of two-dimensional systems // Reviews
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11. Gold A. Gold, A. Ghazali, J. Serre, Electronic
properties of δ-doped GaAs // Semicond. Sci.
Technol. 7, p. 972-979 (1992).
12. B.K. Ridley, The electron-phonon interaction in
quasi-twodimensional semiconductor quantum-well
structures // J. Phys. C: Solid State Phys., 15,
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scattering limited mobility in a quantum well
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GaAs-GaAlAs multiple-quantum-well structures //
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PACS 72.20.Fr, 72.80.Ey, 73.20.At, 73.21.Fg, 73.63.Hs, 81.07.St
Electron mobility in the GaAs/InGaAs/GaAs quantum wells
V.V. Vainberg1, A.S. Pylypchuk1, N.V. Baidus2 and B.N. Zvonkov2
1Institute of Physics, National Academy of Sciences of Ukraine, 03680 Kiev, Ukraine
Phone: +38(044) 525-79-51, e-mail: vainberg@iop.kiev.ua
2Research Scientific Physical-Technical Institute of Lobachevskii State University, 603950 Nizhni Novgorod, Russia
Abstract. The temperature dependence of the electron lateral mobility in quantum wells of the GaAs/InGaAs/GaAs heterostructures with delta-like doping has been studied. Two types of sample doping – in the quantum well and in the adjacent barrier at a small distance from the well – were used. In the case of shallow wells, in such structures the experimental results may be well described by known electron scattering mechanisms taking into account the shape of real envelope wave functions and band bending due to non-uniform distribution of the positive and negative space charges along the growth direction of heterostructure layers. In the case of delta-like doping in the well, a good agreement between experiment and calculations is achieved, if one takes into account a contribution to electron transport of the states of the impurity band formed by the delta-impurity beneath the bottom of the lowest quantum subband.
Keywords: heterostructure, quantum well, electron mobility, lateral transport, semiconductor.
Manuscript received 15.01.13; revised version received 27.02.13; accepted for publication 19.03.13; published online 25.06.13.
1. Introduction
The GaAs/InGaAs/GaAs heterostructures with quantum wells (QW) are successfully applied in modern electronics. In particular, such structures enabled to create low-noise UHF transistors HEMT [1] and powerful laser diodes [2]. In the recent decade, the possibility to generate or detect the middle and far infrared radiation by using these structures was explored [3]. The latter is based on the real-space transfer of electrons under heating electric fields. In particular, the delta-doped structures with two tunnel-coupled quantum wells are used for this purpose. The current-voltage characteristics and accompanying far IR radiation of such structures in the regime of the lateral transport under strong electric fields within the temperature range 4 to 100 K were studied in [4]. In that paper, the electron mobility and real-space redistribution of electrons between wells as functions of electron temperature were calculated in the frame of a simple model of rectangular quantum wells. The obtained dependences predicted considerably higher redistribution and, consequently, a stronger dependence of the mobility and IR radiation on the electric field as compared to the experimental ones. The successive measurements for other structures did not eliminate this discrepancy.
For further studies of effects caused by the real-space transfer of electrons in these structures, it is important to investigate temperature and field dependences of the electron mobility in the quantum wells and analyze them with account of both the band bending and real envelope wave functions [5].
The mobility of carriers in quantum wells is sufficiently well studied in the case when the doping impurity is introduced only into the barrier and is separated from the well by a spacer. In these samples, the highest mobility, especially at low temperatures, is achieved. To generate IR radiation along with a high mobility in one of the quantum wells, it is important to have a low mobility in another one of the couple. It could be achieved by inserting a delta layer of impurities in the plane of the well. We do not know publications concerning systematic investigations of mobility in such samples, except of a few measurements for the systems of AlGaAs/GaAs/AlGaAs [6], InAlAs/InGaAs/InAlAs [7, 8] GaAs/InGaAs/GaAs [9]. Therefore, the purpose of this work was to investigate temperature dependences of the conductivity and Hall mobility of electrons within the range 4 to 400 K in the GaAs/InGaAs/GaAs quantum wells with different concentrations of delta-doping impurity in the well plane. For the purpose of comparison, the structures with quantum wells delta-doped in adjacent barriers had been also investigated. The results of studying the field dependence of the electron mobility will be published later.
2
11
cm
10
-
2. Samples, experimental details
and measurement results
The GaAs/InGaAs/GaAs heterostructures were grown by the MOVPE method on semi-insulating GaAs (100) substrates. The structures were delta-doped by Si. All measurements were carried out with the samples listed in Table 1 with parameters following from the technological process data. The residual shallow impurity concentration in these structures is estimated to be within the limits of 1015 to
3
16
cm
10
-
. To compensate the influence of the surface states on the potential profile, the structures were doped by an impurity delta-layer near the surface. The samples for measurements of resistivity and the Hall effect were cut from wafers in the shape of the Hall bridge. The ohmic contacts were made by deposition of the Pd/Ge/Au layers in vacuum and further heating in the hydrogen ambient at 700 K. Electric field was applied along (110(.
Measurements of the resistivity and Hall effect in the weak magnetic field (B = 0.2 T) were carried out within the range 4 to 400 K in the DC regime (I = 100 (A). The applied electric field was less than 1 V/cm for all the studied samples. The error in temperature measurements was no more than 0.1 K. Measurements of the Hall effect were repeated after several months, and results for the same sample coincided with the accuracy within 1%. At the same time, for different samples cut from one wafer, the Hall concentration may differ considerably, while their temperature dependences of mobility and Hall coefficient coincide qualitatively quite well.
The temperature dependences of the Hall coefficient RH for all 5 structures under study are presented in Fig. 1. These curves show comparatively small variation of the Hall coefficient over the whole temperature range. The observed behaviour may be caused both by variation of the charge carrier concentration and variation of the Hall factor. Typical for all curves initial growth of RH with T at low temperatures is related, from our point of view, with the increase of the Hall factor, because a carrier concentration decrease is unlikely in this temperature range
. On the other hand, the further decrease of RH at least at temperatures higher than 100 K is caused by increasing the concentration of electrons with temperature growth. Indeed, since in all the structures at temperatures from 4 up to 10 K the electron gas is close to quantum degeneracy, the Hall factor at these temperatures must be in fact equal to unity
. Further, the high temperature Hall coefficient values are less than those at 4 K. Hence, the Hall factor should be considerably less than unity, if the concentration is supposed to be constant with the growing temperature. However, this is impossible. Listed in the rightmost column of Table 1 are values of the impurity concentration in the samples at 4 K calculated from data presented in Fig. 1 when supposing that the Hall factor is equal to unity.
The temperature dependences of the conductivity for the samples with delta-doping in an adjacent barrier and in QW are shown in Figs 2a and 2b, respectively. The corresponding dependences of the Hall mobility (H = RH( on temperature calculated for these structures from the data shown in Figs 1, 2a and 2b are shown in Figs 2c and 2d.
Fig. 1. Temperature dependences of the Hall coefficient for the samples under study. Sample ##: 1 – 5998, 2 – 6291, 3 – 6215, 4 – 6002, 5 – 5997.
Comparing data in Fig. 2, one may conclude that, as expected, the electron mobility and corresponding conductivity at low temperatures are considerably larger for samples doped in the barrier as compared to those doped in QW. The temperature dependences both for conductivity and mobility differ from each other substantially in these two cases. In the case of doping in the barrier, the electron mobility at first slightly increases with growth of temperature from 4 K and then strongly decreases. For the samples doped in the QW, it increases by 4 to 5 times, exhibits a flat maximum and slightly decreases with heating up to the room temperature. Furthermore, one should notice the following feature of the curves for the electron mobility in the samples doped in QW (Fig. 2d). At low temperatures (below 70-80 K) the higher is the impurity concentration, the higher is the mobility. For the sample with the lowest impurity concentration (approximately
2
11
cm
10
2
-
×
, sample #5998) the mobility value at 4.2 K differs from those in other samples by several times. At the same time, at higher temperatures the higher is the concentration, the lower is the mobility.
These results may be explained qualitatively as follows. As it is well known, in delta-doped GaAs with the concentration of the order of 2(1011 to 3(1011 cm–2 [10, 11] the states of impurities form an impurity band. The conducting states below the conduction band bottom appear. With an increase of the concentration, the impurity band widens and at last overlaps with the 2D conduction band. Furthermore, as mentioned in many papers (for example, [11]), one should take into account that, because of fluctuations of the distances between impurity atoms in the delta-layer, the impurity band widens even more, and the edge of the band (2D subband) becomes smeared.
In the structures doped in a barrier independently of the impurity concentration (up to 1(1012 cm–2), practically all electrons tend to fall into the quantum well. Scattering of carriers at low temperatures in these samples is mainly caused by remote impurities, roughness of well boundaries and alloy fluctuations in InGaAs. The mobility in this case is comparatively high. At high temperatures, a considerable portion of electrons is transferred into states above the barrier. Then scattering of electrons occurs mainly by lattice vibrations, and mobility decreases down to the bulk value.
a
b
c
d
Fig. 2. Temperature dependences of conductivity and mobility of electrons for the samples with the delta-layer of impurity in the barrier (a and c) and QW centre (b and d). Sample ## in a and c: 1 – 6215, 2 – 6291, in b and d: 1 – 5998, 2 – 6002, 3 – 5997.
In the structures doped in QW, the situation changes. At low temperatures, if the impurity concentration is lower than that corresponding to overlapping of impurity and conduction bands, the magnitude of mobility is very small. It increases with temperature, because more and more electrons are transferred into 2D subband states. Provided the impurity and conduction bands overlap, then already at low temperatures the conduction is determined by electrons in the states of the 2D subband in QW. However, mobility in this case has a lower value as compared to doping in a barrier because of closeness of the scattering impurity ions to electrons. At the same time, it has a higher value as compared to the low doping concentration when the states do not overlap. The calculations performed in the next section take into account these features of delta-doping in QW.
3. Numerical calculations
The aim of this section is, firstly, to construct the model of the potential profile in the studied structures, which is caused by a finite height of the barrier and distribution of the space charge (concentration profile of the ionized impurity and free electrons) along the direction of the heterostructure layers growth, calculation of energy levels and envelope functions of electrons in this potential profile, and, secondly, calculations of mobility in this model. Calculations were carried out for a set of temperatures within the range from 4 to 400 K.
We have used an idealized model of an infinite chain of independent quantum wells and neglected the non-parabolicity of the conduction band. Its influence on the energy levels and on the electron population is less than a few percents even at the highest concentration used in calculations. The positive charge of the impurities in the delta-layer is assumed uniformly distributed over the plane (the jelly model). This assumption is justified in the case of doping in QW, at least if the average distance between the ions is less than a characteristic size of the electron wave function of the impurity in the ground state (for GaAs it corresponds approximately to NS >
2
11
cm
10
5
.
2
-
×
).
The system to be solved consists of the one-dimensional Schroedinger and Poisson equations supplemented with the electro-neutrality condition,
)
(
)
(
)
(
)
(
)
(
1
2
2
z
E
z
z
U
dz
z
d
z
m
dz
d
y
=
y
+
ú
û
ù
ê
ë
é
y
-
h
,
(1)
(
)
[
]
)
(
)
(
4
)
(
2
z
N
z
n
e
dz
z
dU
z
dx
d
+
-
p
=
ú
û
ù
ê
ë
é
e
,
(2)
0
)
(
)
(
=
-
ò
ò
-
-
+
dz
z
n
dz
z
N
L
L
L
L
D
,
(3)
where U(z) is the coordinate dependence of the energy of the conduction band bottom, E – eigenvalue of the energy, ((z) – envelope wave function of an electron along the growth direction of the layers, m(z), ((z), n(z) and N+(z) are the coordinate dependences of the electron effective mass, dielectric permittivity, concentration of free electrons and ionized impurities, respectively. Since the quantum wells in the chain are identical, we consider only one period of a heterostructure. The corresponding boundary conditions for one period are written as follows:
0
)
(
)
(
=
-
y
=
y
L
L
n
n
;
(4)
0
)
(
)
(
=
-
-
L
U
L
U
,
dz
L
dU
dz
L
dU
)
(
)
(
-
=
.
(5)
The quantum wells in adjacent periods are separated by wide and non-transparent for electrons barriers where the magnitude of the envelope wave functions exponentially decreases down to zero. The dependences of ((z) and m(z) on the coordinate are caused by presence of the quantum wells, where these parameters have values corresponding to InGaAs with a given composition of In.
The concentration of positively charged impurities is determined as
÷
÷
ø
ö
ç
ç
è
æ
-
+
=
+
kT
z
Е
E
g
z
N
z
N
imp
)
(
exp
1
)
(
)
(
F
,
(6)
where N(z) = Ndelta(z) + Nbg(z) is the coordinate distribution of the impurity, g – degeneracy factor of the impurity level. The delta-doped impurity concentration is determined by the Gauss distribution function
(
)
÷
÷
ø
ö
ç
ç
è
æ
s
-
-
s
p
=
2
2
0
2
exp
2
1
)
(
z
z
N
z
N
delta
delta
.
(7)
Here, Nbg is the background impurity, EF – Fermi level, Eimp(z) – energy of the impurity level in the point z (different in QW and barriers). The concentration of free electrons n(z) is determined in each point as the sum of the concentrations in states above the barrier, which we consider as 3D states, and in all subbands in quantum wells (2D electron gas).
,
exp
1
ln
)
(
ψ
))
(
(
)
(
)
(
)
(
F
2
2
F
2
/
1
3
2
3
÷
ø
ö
ç
è
æ
-
+
+
+
-
=
+
=
å
å
kT
E
E
N
z
z
U
E
F
N
z
n
z
n
z
n
Ci
D
C
i
i
c
D
C
i
D
D
(8)
where
D
C
N
3
,
D
C
N
2
are the effective densities of states in the conduction band for the 3D and 2D electron gas, F1/2(
(
)
z
U
E
C
-
F
) is the Fermi-Dirac integral for the index ½.
The parameters of GaAs and InAs used in our calculations are listed in Table 2. The parameters of
As
Ga
In
x
1
x
-
were determined by linear interpolation.
Table 2. Parameters of GaAs and InAs.
Parameter
InAs
GaAs
Sound velocity, cm/s
4.28(105
5.24(105
Density, g/cm3
5.68
5.32
Lattice parameter, Å
6.0583
5.65325
Optical phonon energy, meV
30
35
Optical deformation potential, eV
41
41
Effective mass
0.023
0.063
Deformation potential, eV
–5.04
–11
Static dielectric permittivity
15.15
12.9
HF dielectric permittivity
12.3
10.89
The system (1)-(3) was solved self-consistently using numerical methods for various temperatures and impurity concentrations. The equations (1) and (2) were written in the finite-difference form for 1000 points along the length of one period, and the solution was found by a standard procedure for this approach.
Calculations were performed up to distances of ±500 Å from the centre of QW corresponding to the period of our structures. The interval between adjacent points, where the potential and wave functions were calculated, was equal to 1 Å. The self-consistent calculations were repeated until the difference between two successive corrections to the potential becomes less than a given value (less than 1%, which provided the neutrality condition to be fulfilled with an accuracy no worse than
4
10
1
-
×
).
The obtained data set for the energy spectrum, energy of the Fermi level and envelope wave functions were used in the next step for calculations of mobility. In heterostructures based on III-V compounds, the main mechanisms of electron scattering are scattering by ionized impurities, polar optical phonons, interfaces roughness, composition fluctuations (alloy scattering). Though it is known that in these compounds the scattering by acoustic and deformation potential optical phonons do not play any substantial role, we included them into consideration for completeness of description. The corresponding expressions for the relaxation time, Hall coefficient and mobility are given in the Appendix. It should be noticed that for electron scattering by polar optical phonons one cannot strictly introduce the momentum relaxation time. Nevertheless, following [12] we used for this scattering mechanism the expression for the time of the average loss of the momentum
ò
¢
¢
Q
=
t
k
d
k
k
W
k
q
)
,
(
cos
1
,
'
k
k
q
-
=
,
( being the angle between q and k′.
(9)
4. Results and discussion
Fig. 3 illustrates the profile of the conduction band bottom for one period in the chain of the quantum wells with parameters of the sample #5997 and delta-doped in the centre of QW. In all our calculations, we also took into account background impurities with the concentration of
3
15
cm
10
5
-
×
, which were assumed to be uniformly distributed over a period. The energy of quantum levels, the Fermi level and squared envelope wave functions are also shown in Fig. 3.
Fig. 3. Energy spectrum in the conduction band for the samples doped by a delta layer of impurity in the centre of QW. The width and depth of QW and impurity concentration correspond to parameters of the sample #5997. 1 – the conduction band bottom; 2 – square of envelope wave function; 3 – Fermi level.
Fig. 4. Energy spectrum in the conduction band for the samples doped by a delta layer of impurity in the barrier. The width and depth of QW and impurity concentration as well as its position correspond to parameters of the sample #6215. 1 – the conduction band bottom; 2 – square of envelope wave function; 3 – Fermi level.
At low temperature (Fig. 3a), electrons from the background impurities in the barrier are transferred into the lowest energy states (states in QW) and the remaining positive charge causes a rise of the electron potential energy towards the well. Since the electrons from the delta-doping remain in QW, the potential profile weakly depends on this kind of impurity. With temperature growth, electrons come back into the barrier and partially neutralize the positive charge. Approximately at 100 K (Fig. 3b) all donors in the barrier are neutralized, and the potential profile corresponds to that of rectangular QW. With further temperature growth, electrons of donors located in the central plane of QW begin to penetrate into the barrier. The potential energy related with electric field of the remaining positively charged donors increases with growing the distance from this plane (Fig. 3c). The higher is temperature, the wider and deeper becomes a quantum well. At last, the width of the quantum well becomes comparable with the period of the structure.
Fig. 4 illustrates similar dependences for the structure doped in the barrier with parameters of the sample #6215. Here, the impurity delta-layer forms the second QW in the barrier with new quantum levels. The profile of the potential energy becomes more complicated, and it does not change qualitatively with temperature growth.
Further it should be noticed that, for structures delta-doped both in the centre of QW (Fig. 3) and in the barrier (Fig. 4) the envelope wave function and, consequently, the electron density do not vanish at the well borders and penetrate quite deep into the barrier. This effect was usually neglected in earlier calculations of low temperature electron mobility in quantum wells [7, 13, 14]; that may be justified in the case of a deep QW or in the case of a large effective mass of charge carriers (for example, in the case of holes in the quantum well of the valence band [15]). Otherwise as shown, for example, in [16] for the interface roughness scattering mechanism, neglecting this factor may cause a significant error in the calculated mobility.
The results obtained by the method described above for a set of temperature values within the range from 4 to 400 K were used in calculations of the temperature dependence of the mobility. Here, the following two approximations were made, which substantially influenced the obtained results. Firstly, according to the qualitative analysis carried out above, for the structures doped in QW, we assumed a spread of the impurity levels with respect to energy in a shape of the Gauss distribution and put additionally the states lying below the bottom of the first conduction subband in the neutrality equation. This leads to noticeable lowering the Fermi level and considerable decreasing the calculated mobility. Secondly, concerning the mobility in those 2D conduction subbands, which bottoms lie below the states above the barrier by value less than kT/4, it was assumed that it is close to the electron mobility in the 3D states. Therefore, in calculations of the mobility averaged over all bands the concentration in these bands was determined with the 2D density of states, while relaxation times were calculated using the formulae for 3D states. Besides, changes in the electron concentration with temperature were neglected in the calculations, and concentration was taken equal to the value measured at 4 K, because in those samples for which the concentration increases considerably by temperatures higher than 100 K the impurity scattering does not play a noticeable role.
The parameters characterizing scattering processes and used in our calculations are listed in Table 3.
Figs 5a and 5b show the temperature dependences of the electron quasi-momentum relaxation time averaged over the distribution function in the lowest subband for all the scattering mechanisms taken into account. Figs 5a and 5b correspond to delta-doping in the centre of QW and the barrier, respectively.
Table 3. Parameters of the roughness and alloy scattering.
Roughness correlation length, Å
100
Average roughness height, Å
2.3
Scattering potential of alloy atoms, eV
0.33
a
b
Fig. 5. Temperature dependences of the averaged quasi-momentum relaxation times of charge carriers in the lowest 2D subband in the samples doped by a delta layer of impurity in the centre of QW (a) and barrier (b). Scattering mechanisms: 1 – acoustic phonons; 2 – alloy; 3 – optical deformation potential phonons; 4 – polar optical phonons; 5 – roughness; 6 – impurity.
Fig. 6. Fitting the measured temperature dependences of the Hall mobility of electrons in the samples doped by a delta layer of impurity in the barrier (1, 2) and in the centre of QW (3, 4). 1 and 3 – calculations; 2 and 4 – experiment.
As to be seen, in the case of delta-doping in the centre of QW the prevailing scattering mechanisms are scattering by charged impurities at low temperatures and polar optical phonons at the high ones. For the structures doped in the barrier, a considerable contribution additionally results from scattering on the interface roughness and composition fluctuations at low temperatures and deformation potential optical phonons at high temperatures. The calculated mobilities for samples #5997 and 6215 are compared with experiment in Fig. 6. As to be seen, all the qualitative features of experimental curves considered above are correctly described by the calculated data. Taking into account the adopted approximations, we conclude that a satisfactory quantitative agreement between measured and calculated data is also achieved. Regarding the other investigated samples a sufficient agreement between experimental and calculated results is also obtained.
5. Conclusions
For successful applications of the GaAs/InGaAs/GaAs heterostructures in devices based on the mechanism of a real-space transfer of electrons between tunnel-coupled quantum wells, one must provide a large difference of the mobilities in coupled wells. As seen in Fig. 6, the ratio of mobilities for wells with doping either in the barrier or in one of coupled QW may achieve 20 and even more at 4 K. And this value quickly decreases with temperature growth. However, one should take into account that the dependences of mobility on lattice temperature in equilibrium and on electron temperature under heating electric fields differ in the region where scattering by phonons plays a significant role. Although this ratio should decrease less sharply with growth of electron temperature, the demands to parameters of a structure destined to be used in applications may remain sufficiently rigid, and in order to obtain optimal structures, preliminary calculations of their transport properties are needed. A quite good agreement between calculated and experimental dependences obtained in the present work enables to perform sufficiently real calculations of parameters for structures with different applied destination.
Acknowledgments
The work was supported by State Target Scientific and Technical Program (“Nanotechnologies and Nanomaterials”, Project #1.1.7.18). The authors are also grateful to Prof. O.G. Sarbey and Dr. Sc.V.N. Poroshin for helpful discussion.
Appendix
Given below are expressions for the quasi-momentum relaxation time used in calculations.
1. Acoustic phonons
ac
sound
band
band
band
ac
FF
C
kT
D
m
m
E
2
3
2
0
2
)
(
1
r
p
=
t
h
,
(A.1)
ò
¥
¥
-
-
+
=
dq
J
J
FF
ac
;
ò
y
=
+
dz
z
iq
z
J
z
)
exp(
)
(
2
,
ò
-
y
=
-
dz
z
iq
z
J
z
)
exp(
)
(
2
,
(A.2)
ò
y
=
)
(
)
(
1
2
z
m
dz
z
m
band
band
,
(A.3)
mband and m0 are the effective and free electron masses, k is the Boltzmann constant, Dband, ψband, Cband are the deformation potential constant, density and sound velocity.
2. Optical deformation potential phonons
ac
opt
FF
eF
E
×
=
t
Pr
)
(
1
,
opt
E
w
<
h
,
(A.4)
ac
opt
opt
FF
kT
eF
E
ú
ú
û
ù
ê
ê
ë
é
÷
÷
ø
ö
ç
ç
è
æ
w
+
=
t
h
exp
1
Pr
)
(
1
,
opt
E
w
³
h
,
(A.5)
1
exp
1
4
Pr
2
0
2
-
÷
÷
ø
ö
ç
ç
è
æ
w
w
r
p
=
kT
a
m
m
D
eF
opt
opt
band
L
band
opt
h
h
h
,
(A.6)
aL is the lattice parameter.
3. Polar optical phonons
,
)
(
1
)
(
1
)
(
1
exp
1
2
1
0
2
3
0
2
0
dz
z
z
z
kT
m
m
e
band
opt
opt
band
÷
÷
ø
ö
ç
ç
è
æ
e
-
e
y
´
´
-
÷
÷
ø
ö
ç
ç
è
æ
w
p
w
=
t
¥
ò
h
h
h
(A.7)
opt
E
w
<
h
,
(
)
(
)
,
4
2
)
exp(
)
(
)
exp(
)
(
2
2
0
2
2
0
2
2
2
1
ò
ò
ò
¥
¥
-
÷
÷
ø
ö
ç
ç
è
æ
w
+
-
÷
÷
ø
ö
ç
ç
è
æ
w
+
+
-
y
+
y
=
=
dq
E
E
m
m
E
E
m
m
q
dz
iqz
z
dz
iqz
z
FF
opt
band
opt
band
pol
h
h
h
h
(A.8)
q is the phonon wavevector participating in scattering.
1
0
1
)
(
1
)
(
1
)
(
1
pol
FF
E
E
E
×
t
=
t
=
t
,
(A.9)
opt
E
w
³
h
,
(
)
(
)
,
4
2
)
exp(
)
(
)
exp(
)
(
2
2
0
2
2
0
2
2
2
'
'
ò
ò
ò
¥
¥
-
÷
÷
ø
ö
ç
ç
è
æ
w
-
-
÷
÷
ø
ö
ç
ç
è
æ
w
-
+
-
y
+
y
=
=
dq
E
E
m
m
E
E
m
m
q
dz
iqz
z
dz
iqz
z
FF
opt
band
opt
band
pol
h
h
h
h
(A.10)
÷
÷
ø
ö
ç
ç
è
æ
w
×
t
p
+
t
=
t
kT
FF
E
E
E
opt
pol
h
exp
)
(
2
)
(
1
)
(
1
'
'
0
1
.
(A.11)
4. Alloy scattering
al
band
lat
al
al
FF
m
m
x
x
a
U
E
3
0
3
2
)
1
(
)
(
1
h
-
=
t
,
(A.12)
Ual is the scattering potential of alloy atoms.
ò
y
=
L
al
dz
z
FF
0
4
)
(
.
(A.13)
5. Interface roughness scattering
(
)
(
)
,
)
(
1
)
exp(
2
2
)
(
1
1
0
2
2
4
2
2
0
2
2
2
2
3
0
ò
+
-
l
-
ú
ú
û
ù
ê
ê
ë
é
÷
ø
ö
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
D
l
+
D
l
´
´
p
=
t
S
band
C
right
left
band
rough
x
x
x
dx
x
x
k
dL
dE
m
m
E
h
(A.14)
2
0
0
2
h
E
m
m
k
band
=
,
0
2
0
2
0
k
e
m
m
x
band
S
h
e
=
,
(A.15)
(, ( are the correlation length and magnitude of roughness,
dL
dE
C
is the calculated subband bottom fluctuation with the roughness-caused fluctuation of a well width.
6. Impurity scattering
[
]
,
)
(
)
,
(
)
(
1
)
cos
1
(
4
)
(
1
0
2
2
3
4
ò
ò
p
+
e
Q
-
Q
´
´
p
=
t
z
N
z
q
F
dz
q
q
d
me
E
h
(A.16)
0
2
sin
2
k
q
Q
=
,
(A.17)
(
)
ò
-
-
y
=
'
exp
)
'
(
'
)
,
(
2
z
z
q
z
dz
z
q
F
,
(A.18)
)
(
2
)
(
2
2
2
1
)
(
2
2
0
2
0
2
0
q
F
me
q
F
m
e
q
h
h
+
e
=
ú
ú
û
ù
ê
ê
ë
é
p
e
p
+
e
=
e
,
(A.19)
ò
y
=
)
,
(
)
(
)
(
2
z
q
F
z
dz
q
F
.
(A.20)
The total relaxation time for each subband
.
)
(
1
)
(
1
)
(
1
)
(
1
)
(
1
)
(
1
)
(
1
E
E
E
E
E
E
E
alloy
imp
rough
pol
opt
ac
t
+
t
+
t
+
+
t
+
t
+
t
=
t
(A.21)
For each subband:
ò
ò
ò
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ
-
t
=
t
=
t
0
2
2
)
(
4
1
)
(
)
(
band
C
E
F
norm
kT
E
E
ch
dE
E
E
I
dE
E
f
dE
dE
df
E
E
,
(A.22)
ò
÷
ø
ö
ç
è
æ
-
+
=
0
exp
1
band
C
E
F
norm
kT
E
E
dE
I
0
exp1
band
C
E
F
norm
kT
EE
dE
I
,
(A.23)
t
=
m
band
m
m
e
0
.
(A.24)
The total drift mobility is determined as follows
å
å
m
=
m
bands
band
bands
band
band
drift
n
n
,
(A.25)
band
n
is the electron concentration in a given subband
.
exp
1
ln
2
0
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ
-
+
´
´
p
=
kT
E
E
kT
m
m
n
Cband
F
band
band
h
(A.26)
The total Hall mobility is determined as
å
å
m
m
=
m
bands
band
band
bands
band
band
n
n
r
2
H
H
, where
2
2
H
t
t
=
r
.
(A.27)
The quantum wells in the investigated structures are not deep (less than 100 meV). Consequently, at high temperatures from 200 to 400 K, there are a lot of electrons in the 3D states that give a noticeable contribution to the measured Hall concentration and mobility. Participating in scattering of 3D electrons are acoustic, optical deformation and polar phonons and ionized impurity. The expressions for 3D electron mobility at these scattering mechanisms are well known and are not given here. They are included in the total mobility as follows
D
bands
band
D
D
bands
band
band
drift
n
n
n
n
3
3
3
+
m
+
m
=
m
å
å
,
(A.28)
D
D
bands
band
band
D
D
D
bands
band
band
n
n
n
r
n
r
3
3
3
2
3
3
H
2
H
H
m
+
m
m
+
m
=
m
å
å
,
(A.29)
where n3D is the integral of the above-barrier (3D) electron concentration within one period.
References
1.
Y. Ando, Field Effect Transistors, US Patent 5371387 (1994); C. Chang, GaAs-InGaAs high electron mobility transistor, US Patent 5653440 (1995).
2.
F. Bugge, G. Erbert, J. Fricke, S. Gramlich, R. Staske, H. Wenzel, U. Zeimer, and M. Weyers, 12 W continuous-wave diode lasers at 1120 nm with InGaAs quantum well // Appl. Phys. Lett. 79, p. 1965-1967 (2001).
3.
V.Ya. Aleshkin, A.A. Andronov, A.V. Antonov et al., Toward far- and mid-IR intraband lasers based on hot carrier intervalley/real-space transfer in multiple quantum well systems // SPIE Proc. 4318, p. 192-203 (2001).
4.
P.A. Belevski, V.V. Vainberg, M.N. Vinoslavskii, A.V. Kravchenko, V.N. Poroshin, and O.G. Sarbey, Real-space transfer and far-infrared emission of hot electrons in InGaAs/GaAs heterostructures with tunnel-coupled quantum wells // Ukr. J. Phys. 54(1-2), p. 117-122 (2009).
5.
N.V. Baidus, P.A. Belevskii, A.A. Biriukov, V.V. Vainberg, M.N. Vinoslavskii, A.V. Ikonnikov, B.N. Zvonkov, A.S. Pylypchuk and V.N. Poroshin, Lateral transport and far-infrared radiation of electrons in InxGa1–xAs/GaAs heterostructures with the double tunnel-coupled quantum wells in a high electric field // Semiconductors, 44, p. 1495-1498 (2010).
6.
W. Ted. Masselink, High-differential mobility of hot electrons in delta-doped quantum wells // Appl. Phys. Lett. 59, p. 694-696 (1991).
7.
W. Ted. Masselink, Ionized-impurity scattering of quasi-two-dimensional quantum-confined carriers // Phys. Rev. Lett. 66(11), p. 1513-1516 (1991).
8.
Il-Ho Ahn, G.Hugh Song, Young-Dahl Jho, Separating the contribution of mobility among different quantum well subbands // Jpn. J. Appl. Phys. 49, 014102-014105 (2010).
9.
V.A. Kulbachinskii, I.S. Vasil’evskii, R.A. Lunin, G. Galistu, Electron transport and optical properties of shallow GaAs/InGaAs/GaAs quantum wells with a thin central AlAs barrier // Semicond. Sci. Technol. 22, p. 222-228 (2007).
10.
T. Ando, A.B. Fowler, F. Stern, Electronic properties of two-dimensional systems // Reviews of Modern Physics, 54 (2), p. 437-672 (1982).
11.
Gold A. Gold, A. Ghazali, J. Serre, Electronic properties of δ-doped GaAs // Semicond. Sci. Technol. 7, p. 972-979 (1992).
12.
B.K. Ridley, The electron-phonon interaction in quasi-twodimensional semiconductor quantum-well structures // J. Phys. C: Solid State Phys., 15, p. 5899-5917 (1982).
13.
J. Lee, H.N. Spector, V.K. Arora, Impurity scattering limited mobility in a quantum well heterojunction // J. Appl. Phys. 54(12), p. 6995-7004 (1983).
14.
G. Fishman and D. Calecki, Electron concentration and buffer-width dependence of Hall mobility in GaAs-GaAlAs multiple-quantum-well structures // Phys. Rev. В, 29, p. 5778-5787 (1984).
15.
B. Laikhtman, R.A. Kiehl, Theoretical hole mobility in a narrow Si/SiGe quantum well // Phys.Rev. В, 47, p. 10515-10527 (1993).
16.
J.M. Li, J.J. Wu, X.X. Han, Y.W. Lu, X.L. Liu, Q.S. Zhu and Z.G. Wang, A model for scattering due to interface roughness in finite quantum wells // Semicond. Sci. Technol. 20, p. 1207-1212 (2005).
Table 1. Parameters of samples under study.
Sample #�
QW width, �
Barrier width, �
QW depth, meV�
Impurity concentration, � EMBED Equation.3 ����
Impurity position�
Number of periods�
n4.2 K,
� EMBED Equation.3 ����
�
5997�
80�
800�
63�
8.1�
QW centre�
10�
7.85�
�
5998�
80�
800�
63�
2�
QW centre�
10�
1.7�
�
6002�
80�
800�
63�
4.5�
QW centre�
10�
4.05�
�
6215�
80�
800�
96�
6.1�
barrier, 100 Å from QW �
10�
4.6�
�
6291�
100�
800�
50�
2.1�
barrier, 10 Å from QW �
20�
1.6�
�
� In principle, such a decrease may be related with capture of electrons by very shallow traps.
� Strictly, this statement is valid only in the case of common relaxation time for all charge carriers. For example, in the case of filling several subbands it may be violated.
© 2013, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
152
2
11
cm
10
-
_1431350310.unknown
_1431350759.unknown
_1431437176.unknown
_1434871072.unknown
_1437211380.unknown
_1437211388.unknown
_1437211954.unknown
_1434871109.unknown
_1434871205.unknown
_1434871264.unknown
_1434871168.unknown
_1434871097.unknown
_1434870350.unknown
_1434871056.unknown
_1431438513.unknown
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