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Contribution of effective mass variation to electro-acoustic phonon interaction in semiconductor nanostructures
ELSEVIER
Microelectronic Engineering 47 (1999) 373-375
Contribution of effective mass variation to electro-acoustic teraction in semiconductor nanostructures
phonon in-
V. V. Mitin”, V. I. Pipaa16,and M. A. Stroscio” ODepartment of Electrical and Computer Engineering Wayne State University, Detroit, MI 48202, USA %istitute of Semiconductor Physics, Kiev, 252650, Ukraine “U.S. Army Research Office, P.O. Box 12221, Research Triangle Park, NC 27709
The finite-size effects of the electron-acoustic phonon of the deformation-potential interaction in semiconductor heterostructures are studied. Modification of the interaction originates from the phonon induced changes of the interface spacing and electron effective mass. Calculations of the electron mobility for a quantum well (QW) show that for narrow GaAs-based QWs, the contribution of the additional mechanisms to the intrasubband scattering depends on the sign of the deformation potential constant and can exceed that from the usual deformation potential.
1. PROBLEM
FORMULATION
The accepted procedure for calculating the electron-acoustic phonon interaction via the deformation potential (DP) in semiconductor heterostructures is to use the bulk expression for the energy of interaction. In a cubic crystal with a conduction band minimum at the P-point, the interaction Hamiltonian is given by HDP=DdivU, int
(1)
where D is the deformation potential constant, and u is the acoustic displacement. There exists extensive literature on electron-acoustic phonon scattering in semiconductor heterostructures of different types where bulk expressions, similar to Eq. (l), are used (see, e.g., Ref.[l] and references therein). Also, there is uncertainty in the literature concerning the value of the DP constant, D, when the bulk interaction (1) is applied to a low-dimensional electron gas. For GaAs/AZGaAs heterostructures, the D value ranges mainly from 6 eV to 11 eV (when screening is neglected). 0167-9317/$@/$ - seefront matter 0 1999 Elsevier Science B.V. All rights memd. PII: SO167-9317(99)00237-3
374
WI Mitin et al. / Microelectronic
Engineering 47 (1999) 373-375
It has been shown in [2-41 in systems with interfaces additional
mechanisms
tion potential interface
of electron-acoustic
(MDP)
[2] (or ripple mechanism
spacing to change.
HMDP = int
-(WV)
This interaction
coupling.
A macroscopic
[3]) arises when acoustic is described
waves cause the
by
(2) potential
anism considered
originates
in Ref.[4]
mass, m*, in the direction
case of bulk semiconductors
for electrons
in the undeformed
from the phonon perpendicular
induced
crystal.
change
to the interfaces.
where this contribution
m* can yield a perturbation
comparable
the tensor for the inverse effective t&j,
as rn,il = m*( 1 +xuij)&j
>
of
and the change of
is expanded
in terms of the strain tensor,
For the case of a position-dependent
to the coefficient
mass, using
gives
)
(3)
to the lowest order in the displacement. x is related
the minimum
by Eq. (1). In a cubic crystal,
of the form ( -ti2/2)V(m-‘(r)V),
. v. , &&vi c8 (m
y
with that from
with that described
mass, rn;‘,
parameter.
energy operator
=-HT” ant
quantization,
to the
where m+ is the effective mass in the absence of deformation
and x is a phenomenological the kinetic
of the electron
is small compared
energy is reduced to the finite energy of spatial
The mech-
In contrast
the DP for energies near the band edge, for the case of heterostructures electron
deforma-
)
V(r)
where V(r) is a confinement effective
phonon
that one has to take into account
dE,/dP
For narrow-gap
semiconductors,
of the pressure bandgap
the parameter
energy dependence
as [4],
w h ere K is the modulus of the hydrostatic compression. Using x =(3K&)(d&IdP), data [5] for GaAa, we obtain x II 17. The condition x >> 1 implies that interaction of Eq. (3) dominates
over interaction
2. RESULTS
AND
In order to illustrate we have calculated scattering (2),
the mobility,
than
We neglect
V(z),
defined by Eqs. (2) and (3),
two-dimensional
electron
by the sum of interactions
outside
of a rectangular
QW (0 5 z 5 d).
the QW to be infinite.
of infinitely
high barriers,
itself, become invalid, and more sophisticated changes of the effective
in addition
phonons
to the bulk interaction
and (ii) for carrier scattering
gas for
from Eqs. (l), Note, that
as well as the consideration
mass, i.e. only u,, # 0 in
rate, T, is derived from the general form of Ref.[6].
the following contributions with transverse
determined
the lowest subband
also in-plane
The relaxation
mechanisms
of a degenerate
30 w the approximation
mass approximation
is necessary.
interact
er/m’,
we take the potential,
for QWs thiner
(3).
the roles of additional
El ec t rons occupy
For simplicity,
Eq.
DISCUSSION
due to the total interaction
and (3).
effective
of Eq. (2).
of Eq. (1):
We obtain
(i) the electrons
with longitudinal
phonons
375
V.V Mitin et al. I Microelectronic Engineering 47 (1999) 373-37-f
there is an interference
between the DP interaction
(2). The latter ph enomenon t, D. Calculations
of 2 - 10" cmm2. The calculated
of QW width for several temperatures
6
of Eqs. (1) and
of mobility on the sign of the constan-
have been carried out using the material parameters
electron concentration functions
leads to a dependence
and the mechanisms
of
GaAs, for an
relative changes in the mobility as
are given in Fig. 1. 6
(1) 1K (2) 5K
3 . :: ZL
4
3. .
4
2 2
2
40
60
80
100
60
width, A
interaction
interaction
100
width, A
Figure 1. Ratio of the electron mobility for scattering via the modified phonon
I
80
to that for the scattering
for three different temperatures:
electron-acoustic
due to only the bulk deformation
potential
(1) 1 K; (2) 5 K; (3) 20 K. (a) for D = 8 eV,
(b) for D = - 8 eV. We see that for narrow QWs the contribution by Eqs. moreover,
(2) and (3) prevails over that from the commonly their contribution
consideration the comparison of
of the additional
increases as the temperature
depends strictly on the interference
mechanisms
defined
used bulk DP interaction;
decreases, and the effect under
between the mechanisms
of (a) and (b) plots which have all parameters
as seen from
the same except the sign
D.
Acknowledgement:
This work was supported
by the US Army Research Office.
REFERENCES 1. B. K. Ridley, Electrons
and Phonons
in Semiconductor
Multilayers,
Cambridge
Uni-
versity Press, New York, 1997. 2. F. T. Vasko and V. V. Mitin, Phys. Rev. B, 52 (1994) 1500. 3. P. A. Knipp and T. L. Reinecke, Phys. Rev. B, 52 (1994) 5923. 4. V. I. Pipa, V. V. Mitin, and M. A. Stroscio, J. Appl. 5. G. Ji. D. Huang, U. K. Reddy, T.S. Henderson, Phys., 62 (1987) 3367. 6. P. J. Price, Solid State Commun.,
51 (1984) 607.
Phys. (to be published).
R. Houdrk, and H. Morcos,
J. Appl.
Microelectronic Engineering 47 (1999) 373-375
Contribution of effective mass variation to electro-acoustic teraction in semiconductor nanostructures
phonon in-
V. V. Mitin”, V. I. Pipaa16,and M. A. Stroscio” ODepartment of Electrical and Computer Engineering Wayne State University, Detroit, MI 48202, USA %istitute of Semiconductor Physics, Kiev, 252650, Ukraine “U.S. Army Research Office, P.O. Box 12221, Research Triangle Park, NC 27709
The finite-size effects of the electron-acoustic phonon of the deformation-potential interaction in semiconductor heterostructures are studied. Modification of the interaction originates from the phonon induced changes of the interface spacing and electron effective mass. Calculations of the electron mobility for a quantum well (QW) show that for narrow GaAs-based QWs, the contribution of the additional mechanisms to the intrasubband scattering depends on the sign of the deformation potential constant and can exceed that from the usual deformation potential.
1. PROBLEM
FORMULATION
The accepted procedure for calculating the electron-acoustic phonon interaction via the deformation potential (DP) in semiconductor heterostructures is to use the bulk expression for the energy of interaction. In a cubic crystal with a conduction band minimum at the P-point, the interaction Hamiltonian is given by HDP=DdivU, int
(1)
where D is the deformation potential constant, and u is the acoustic displacement. There exists extensive literature on electron-acoustic phonon scattering in semiconductor heterostructures of different types where bulk expressions, similar to Eq. (l), are used (see, e.g., Ref.[l] and references therein). Also, there is uncertainty in the literature concerning the value of the DP constant, D, when the bulk interaction (1) is applied to a low-dimensional electron gas. For GaAs/AZGaAs heterostructures, the D value ranges mainly from 6 eV to 11 eV (when screening is neglected). 0167-9317/$@/$ - seefront matter 0 1999 Elsevier Science B.V. All rights memd. PII: SO167-9317(99)00237-3
374
WI Mitin et al. / Microelectronic
Engineering 47 (1999) 373-375
It has been shown in [2-41 in systems with interfaces additional
mechanisms
tion potential interface
of electron-acoustic
(MDP)
[2] (or ripple mechanism
spacing to change.
HMDP = int
-(WV)
This interaction
coupling.
A macroscopic
[3]) arises when acoustic is described
waves cause the
by
(2) potential
anism considered
originates
in Ref.[4]
mass, m*, in the direction
case of bulk semiconductors
for electrons
in the undeformed
from the phonon perpendicular
induced
crystal.
change
to the interfaces.
where this contribution
m* can yield a perturbation
comparable
the tensor for the inverse effective t&j,
as rn,il = m*( 1 +xuij)&j
>
of
and the change of
is expanded
in terms of the strain tensor,
For the case of a position-dependent
to the coefficient
mass, using
gives
)
(3)
to the lowest order in the displacement. x is related
the minimum
by Eq. (1). In a cubic crystal,
of the form ( -ti2/2)V(m-‘(r)V),
. v. , &&vi c8 (m
y
with that from
with that described
mass, rn;‘,
parameter.
energy operator
=-HT” ant
quantization,
to the
where m+ is the effective mass in the absence of deformation
and x is a phenomenological the kinetic
of the electron
is small compared
energy is reduced to the finite energy of spatial
The mech-
In contrast
the DP for energies near the band edge, for the case of heterostructures electron
deforma-
)
V(r)
where V(r) is a confinement effective
phonon
that one has to take into account
dE,/dP
For narrow-gap
semiconductors,
of the pressure bandgap
the parameter
energy dependence
as [4],
w h ere K is the modulus of the hydrostatic compression. Using x =(3K&)(d&IdP), data [5] for GaAa, we obtain x II 17. The condition x >> 1 implies that interaction of Eq. (3) dominates
over interaction
2. RESULTS
AND
In order to illustrate we have calculated scattering (2),
the mobility,
than
We neglect
V(z),
defined by Eqs. (2) and (3),
two-dimensional
electron
by the sum of interactions
outside
of a rectangular
QW (0 5 z 5 d).
the QW to be infinite.
of infinitely
high barriers,
itself, become invalid, and more sophisticated changes of the effective
in addition
phonons
to the bulk interaction
and (ii) for carrier scattering
gas for
from Eqs. (l), Note, that
as well as the consideration
mass, i.e. only u,, # 0 in
rate, T, is derived from the general form of Ref.[6].
the following contributions with transverse
determined
the lowest subband
also in-plane
The relaxation
mechanisms
of a degenerate
30 w the approximation
mass approximation
is necessary.
interact
er/m’,
we take the potential,
for QWs thiner
(3).
the roles of additional
El ec t rons occupy
For simplicity,
Eq.
DISCUSSION
due to the total interaction
and (3).
effective
of Eq. (2).
of Eq. (1):
We obtain
(i) the electrons
with longitudinal
phonons
375
V.V Mitin et al. I Microelectronic Engineering 47 (1999) 373-37-f
there is an interference
between the DP interaction
(2). The latter ph enomenon t, D. Calculations
of 2 - 10" cmm2. The calculated
of QW width for several temperatures
6
of Eqs. (1) and
of mobility on the sign of the constan-
have been carried out using the material parameters
electron concentration functions
leads to a dependence
and the mechanisms
of
GaAs, for an
relative changes in the mobility as
are given in Fig. 1. 6
(1) 1K (2) 5K
3 . :: ZL
4
3. .
4
2 2
2
40
60
80
100
60
width, A
interaction
interaction
100
width, A
Figure 1. Ratio of the electron mobility for scattering via the modified phonon
I
80
to that for the scattering
for three different temperatures:
electron-acoustic
due to only the bulk deformation
potential
(1) 1 K; (2) 5 K; (3) 20 K. (a) for D = 8 eV,
(b) for D = - 8 eV. We see that for narrow QWs the contribution by Eqs. moreover,
(2) and (3) prevails over that from the commonly their contribution
consideration the comparison of
of the additional
increases as the temperature
depends strictly on the interference
mechanisms
defined
used bulk DP interaction;
decreases, and the effect under
between the mechanisms
of (a) and (b) plots which have all parameters
as seen from
the same except the sign
D.
Acknowledgement:
This work was supported
by the US Army Research Office.
REFERENCES 1. B. K. Ridley, Electrons
and Phonons
in Semiconductor
Multilayers,
Cambridge
Uni-
versity Press, New York, 1997. 2. F. T. Vasko and V. V. Mitin, Phys. Rev. B, 52 (1994) 1500. 3. P. A. Knipp and T. L. Reinecke, Phys. Rev. B, 52 (1994) 5923. 4. V. I. Pipa, V. V. Mitin, and M. A. Stroscio, J. Appl. 5. G. Ji. D. Huang, U. K. Reddy, T.S. Henderson, Phys., 62 (1987) 3367. 6. P. J. Price, Solid State Commun.,
51 (1984) 607.
Phys. (to be published).
R. Houdrk, and H. Morcos,
J. Appl.