Download A Raman scattering-based method to probe the carrier drift velocity

A Raman scattering-based method to probe the carrier drift velocity in
semiconductors: Application to gallium nitride
A. V. Andrade-Neto, A. R. Vasconcellos, R. Luzzi, and V. N. Freire
Citation: Applied Physics Letters 85, 4055 (2004); doi: 10.1063/1.1808231
View online: http://dx.doi.org/10.1063/1.1808231
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APPLIED PHYSICS LETTERS
VOLUME 85, NUMBER 18
1 NOVEMBER 2004
A Raman scattering-based method to probe the carrier drift velocity
in semiconductors: Application to gallium nitride
A. V. Andrade-Neto
Departamento de Física, Universidade Estadual de Feira de Santana, Campus Universitário, 40031-460
Feira de Santana, Bahia, Brazil
A. R. Vasconcellos and R. Luzzi
Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-970 Campinas,
São Paulo, Brazil
V. N. Freirea)
Departamento de Física, Universidade Federal do Ceará, Centro de Ciências, Caixa Postal 6030, Campus
do Pici, 60455-900 Fortaleza, Ceará, Brazil
(Received 21 May 2004; accepted 30 August 2004)
A single expression relating the carrier drift velocity in semiconductors under an electric field to
Raman scattering data is derived resorting to a full nonequilibrium picture for electrons and holes.
It allows one to probe with high optical precision both the ultrafast transient as well as the steady
state carriers’ drift velocity in semiconductor systems. This is achieved by simply modifying the
experimental geometry, thus changing the angle between the transferred wave vector Q and the
applied electric field E, and measuring the frequency shift promoted by the presence of the field to
be observed in the single-particle and plasmon scattering spectra. An application to zinc-blende
gallium nitride is presented to highlight the power of the method. © 2004 American Institute of
Physics. [DOI: 10.1063/1.1808231]
To overcome difficulties of classical transport measurement methods, optical-based techniques like time-resolved
absorption, transmission, reflection, luminescence, and Raman spectroscopy have been developed to measure the highest steady-state carriers’ drift velocity (negative differential
resistivity onset), as well as the drift velocity overshoot pattern in semiconductors. Nowadays, the most used are: (i) the
time-resolved electroabsorption technique,1,2 which follows
closely the seminal work of Shank et al.3 but with femtosecond resolution, which can give information on the electron
velocity–electric field characteristic and transient velocity
overshoot. With this technique, both the electron velocityfield characteristic and the transient electron velocity overshoot in GaN were measured,1,2 showing good agreement
with theoretical estimates; (ii) an electron spectrometerbased measure of the energy distribution of extracted
electrons,4,5 from which overshoot drift velocity characteristic curves of thin semiconductor films can be estimated; (iii)
a transient subpicosecond Raman spectroscopy-based
method, which allows for the measurement of both nonequilibrium carriers’ distribution and electron and hole drift velocities in the transient regime.6–9
The time-resolved electroabsorption technique requires
expensive equipment, and has the drawback of assuming that
the majority of the photoinduced bleaching is due to electron
transport, i.e., the hole drift velocity must be much smaller
than the electron drift velocity.1,2 On the other hand, electron
spectrometer-based measurements depend on the characteristic of the device structure,4,5 a thin AlN film with a semitransparent Au-deposited contact. Finally, the transient subpicosecond Raman spectroscopy-based method depends on
the assumption that the electron distribution function is nondegenerate, and that is possible to eliminate the lumines-
cence background of the semiconductor structure, which
makes it possible to calculate approximately the electron
drift velocity using the Raman signal from the single particle
spectrum.6–9
The purpose of this work is to describe how to probe
easily the carrier drift velocity in semiconductors through
Raman scattering measurements. Simulations are presented
for zinc-blende gallium nitride (GaN), which demonstrates
the feasibility and advantages of such method. Electrons and
holes from doping (or from photoinjection) are considered,
which are driven far from equilibrium by the applied electric
field, E. Their nonequilibrium thermodynamic state is characterized by the density n共t兲, which is constant in the case of
the doped material; the quasitemperature T*c 共t兲; and the drift
velocity, v␣共t兲, ␣ = e for electron and ␣ = h for heavy hole.
The evolution and steady state values of T*c 共t兲 and v␣共t兲 are
derived10 from a kinetic theory for far-from-equilibrium systems based on a nonequilibrium statistical ensemble
formalism.11 Within its framework, a single expression is
obtained which allows one to determine the carrier drift velocity from the Raman scattering data performing changes in
the angle between the transferred wave vector Q and the
applied electric field E, and measuring the frequency shift to
be observed in the plasmon and single-particle scattering
spectra.
Using the generalized fluctuation-dissipation theorem,11
there follows that the electric-field dependent differential Raman scattering cross section is given by an expected expression in which is present the wave vector and frequency dependent dielectric function ⑀共Q , ␻ 兩 E兲, also dependent on the
electric field strength, for an n-doped sample given by
a)
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4055
© 2004 American Institute of Physics
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Andrade-Neto et al.
Appl. Phys. Lett., Vol. 85, No. 18, 1 November 2004
d2␴共Q, ␻兩E兲
Im关− 1/⑀共Q, ␻兩E兲兴
,
⬃
d⍀d␻
兵1 − exp关− ␤共E兲ប共␻ − Q · ve共t兲兲兴其
共1兲
in arbitrary units. Im stands for the imaginary part, ␤共E兲
= 1 / kBT*c 共E兲 is the reciprocal of the field-dependent nonequilibrium temperature, and ប␻ 共Q兲 is the energy (momentum)
transfer in the scattering event. The dependence on the electric field is a consequence that the nonequilibrium thermodynamic variables are electric field dependent, i.e., T*c 共E兲 and
ve共E兲. In the limit E → 0, ve goes to zero and T*c becomes T0.
Resorting to the random phase approximation (RPA) one
obtains in the present case a Lindhart(or RPA)-type expression, however specialized to include the presence of the electric field and nonequilibrium conditions. Going over to the
continuum and using spherical coordinates 共k , ␪ , ␾兲, the real
and imaginary parts of the dielectric function are straightforwardly calculated to obtain that
⑀1共Q, ␻兩E兲 = 1 +
⑀2共Q, ␻兩E兲 =
冋 册
2me
␤共E兲
␲m1/2
e
1/2 2
kDH
3 兵D共y 1兲
បQ
2
kDH
3 兵exp共−
2␤共E兲 បQ
+ D共y 2兲其,
y 22兲 − exp共− y 21兲其,
共2兲
共3兲
2
= 4␲ne2 / 共⑀0kBT*c 兲 is the Debye–Huckel screening
where kDH
factor. For simplicity we have omitted to explicitly
indicate the dependence on E in the expressions
D共y兲 = exp共−y 2兲兰0y exp共x2兲dx (Dawson integral), and
y1 =
y2 =
冋
冋
␤共E兲ប2
2me
␤共E兲ប2
2me
册再
册再
1/2
1/2
冎
冎
Q me
+
共 ␻ − Q · v e兲 ,
2 បQ
共4兲
Q me
−
共 ␻ − Q · v e兲 ,
2 បQ
共5兲
Inspection of these results tells us that we do formally
have identical expressions to that in the absence of an electric field, except for the presence of a shift Q · ve in the frequency ␻, which is also present in the thermal factor in Eq.
(1). Therefore, we can conclude the relevant result that in the
presence of the electric field the Raman spectrum, let it be
resulting from single-particle scattering or from a plasmon
scattering, should present a shift in frequency given by
Q · ve共t兲, and, hence, depending on the angle ␪ between the
wave vector transfer and the electric field 共ve 储 E兲. A maximum shift follows from Q parallel to E, and a null shift for
Q perpendicular to E. Then for, say, angles ␪1 and ␪2, if we
call ⌬␻pl共␪1 , ␪2 兩 E兲 the separation of the peaks in the bands
due to scattering by plasmons in the two different experimental geometries, the value of the electron drift velocity
should follow from
ve共E兲 =
⌬␻pl共␪1, ␪2兩E兲
.
Q共cos ␪1 − cos ␪2兲
FIG. 1. Plasmon modes in doped zinc-blende GaN with n = 1017 cm3, and
field strength E = 50 kV/ cm. They were calculated using Q = 1.8
⫻ 105 cm−1, ve = 1.83⫻ 107 cm/ s, Tc* = 511 K, and angles ␪ = 0, ␪ = ␲ / 4, and
␪ = ␲ / 2 (from right to left); ␻pl ⬃ 1.2⫻ 1013 s−1.
共6兲
in Ref. 11. The results have very good agreement with those
obtained
in
computer-modeling
Monte
Carlo
simulations.12–14 The calculations have been restricted to
field intensities below 100 kV/ cm, when we can resort to
using the effective mass approximation (parabolic bands)
around the ⌫ point in this direct inverted-band semiconductors; for larger fields it would be necessary to include the
influence of side valleys.
Figure 1 presents the calculated spectra considering
zinc-blende GaN with n = 1017 cm3, and in the presence of a
field strength E = 50 kV/ cm, using Q = 1.8⫻ 105 cm−1, and
for angles ␪ = 0, ␪ = ␲ / 4, and ␪ = ␲ / 2, while we have that
12
ve = 1.83⫻ 107 cm/ s and T*c = 511 K. The plasma frequency
13 −1
is ␻pl ⬃ 1.2⫻ 10 s . In Table I are presented the differences in frequency for all three pairs of angles as obtained
from the position of the peaks in Fig. 1 共⌬␻pl共␪1 , ␪2 兩 E兲兲 and
those calculated using Q共cos ␪1 − cos ␪2兲. The agreement is
good, which indicates practically no influence of the bandwidth due to Landau damping. Finally, in Fig. 2 are shown
spectra corresponding to scattering by single-particle excitations (Raman–Doppler scattering), for E = 100 kV/ cm and
Q = 5 ⫻ 104 cm−1 (for smaller values of Q the band is
strongly damped). For ␪ = ␲ / 2 we can see the usual form
centered at ␻ = 0, but for ␪ ⫽ ␲ / 2 there occurs the expected
shifts as in the case of plasma bands. The maximum possible
shift is of the order of 1.2⫻ 1012 s−1, within the resolution of
typical Raman scattering detection apparatuses. The form of
the band reflects the thermal distribution of the carriers, having a Gaussian form ⬃exp兵−共␻ − Q · ve共t兲兲2 / Q2v2th其, with
mev2th / 2 = kBT*c , but presenting a depletion at very low frequencies due to screening effects.15 In the case of
InxGa1−xN / GaN, see Ref. 9.
All these results, obtained in the steady state, remain
valid in the transient regime, interpreting that Eqs. (1)–(6)
are valid at any time t (within the resolution time), and thus
We concentrate the attention on the steady state in this
case of n-doped polar semiconductor, and we consider zincblende GaN in the numerical calculations. The values of the
TABLE I. Frequency shift ⌬␻pl of Eq. (6).
average
internal
energy,
U共E兲 = 共3 / 2兲kBT*c 共E兲
␪1
␪2
⌬␻pl 共s−1兲
Qv共cos ␪1 − cos ␪2兲 共s−1兲
⌬␻pl / ␻pl
+ 共1 / 2兲mev2e 共E兲, and ve共E兲 versus the electric field strength
17
−3
were obtained for a concentration n = 10 cm and lattice
0
␲/4
9.7⫻ 1011
9.7⫻ 1011
0.08
temperature T0 = 300 K by solving numerically the coupled
12
0
␲/2
3.3⫻ 10
3.3⫻ 1012
0.28
set of evolution equations for the nonequilibrium thermody␲/4
␲/2
2.3⫻ 1012
2.3⫻ 1012
0.19
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namic state of the system, as shown elsewhere, and Chap. 6
143.106.1.143 On: Thu, 10 Jul 2014 17:18:52
Andrade-Neto et al.
Appl. Phys. Lett., Vol. 85, No. 18, 1 November 2004
4057
lium nitride highlights the usefulness of the proposed
method. We hope this work will stimulate its use by
experimentalist.
The authors acknowledge financial support provided by
the National Research Council (CNPq) through the grant
NanoSemiMat Project (No. 550.015/01-9) (V.N.F., A.R.V.,
R.L.) and the São Paulo State Research Foundation under
Grant No. 2002/06694-0 (A.R.V. and R.L.). A.R.V., V.N.F.,
and R.L. are CNPq Research Fellows, and A.V.A.-N. is a
predoctoral CAPES-PICD fellow.
1
FIG. 2. Single particle bands in doped zinc-blende GaN with n = 1017 cm3,
and field strength E = 100 kV/ cm. They were calculated using Q = 5.0
⫻ 104 cm−1, ve = 2.45⫻ 107 cm/ s, Tc* = 511 K, and angles ␪ = 0, ␪ = ␲ / 4, and
␪ = ␲ / 2 (from right to left); ␻pl ⬃ 1.2⫻ 1013 s−1.
resorting to ultrafast time-resolved laser spectroscopy, one
can determine the time evolution of the carrier’s drift velocity and nonequilibrium temperature. In the case of the photoinjected double plasma of electrons and holes (whereas the
system is on the metallic side of Mott transition), in Eq. (1)
there needs to be included the contribution to the dielectric
function of the fluid of holes, and so the ultrafast timeresolved Raman scattering can determine the evolution in
time of both ve共t兲 and vh共t兲.
Summarizing, we have presented a sounding derivation
of a very practical formula from which the carriers’ drift
velocity both in the steady state or in the ultrafast transient
regime can be obtained by simply modifying the experimental geometry of the Raman scattering in semiconductors under an electric field. The key step was to show the presence
of a frequency shift Q · ve in the Raman spectrum using a
fluctuation-dissipation theory for semiconductors in arbitrary
nonequilibrium conditions under the action of an applied
electric field. The application performed for zinc-blende gal-
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