of 7
PHYSICAL REVIEW B
109
, 235201 (2024)
Hot electron diffusion, microwave noise, and piezoresistivity in Si from first principles
Benjamin Hatanpää
and Austin J. Minnich
*
Division of Engineering and Applied Science,
California Institute of Technology
, Pasadena, California 91125, USA
(Received 4 October 2023; revised 16 April 2024; accepted 13 May 2024; published 3 June 2024)
Abinitio
calculations of electron-phonon interactions in materials without adjustable parameters have provided
microscopic insights into their charge-transport properties. Other transport properties such as the diffusion
coefficient provide additional microscopic information and are readily accessible experimentally, but few
ab
initio
calculations of these properties have been performed. Here, we report first-principles calculations of the
hot electron diffusion coefficient in Si and its dependence on electric field over temperatures from 77–300 K.
While qualitative agreement in trends such as anisotropy at high electric fields is obtained, the quantitative
agreement that is routinely achieved for low-field mobility is lacking. We examine whether the discrepancy can
be attributed to an inaccurate description of
f
-type intervalley scattering by computing the microwave-frequency
noise spectrum and piezoresistivity. These calculations indicate that any error in the strength of
f
-type scattering
is insufficient to explain the diffusion coefficient discrepancies. Our findings suggest that the measured diffusion
coefficient is influenced by factors such as space-charge effects, which are not included in
ab initio
calcu-
lations, impacting the interpretation of experimental measurements in terms of microscopic charge-transport
processes.
DOI:
10.1103/PhysRevB.109.235201
I. INTRODUCTION
Ab initio
calculations of linear transport coefficients such
as the electrical mobility of materials without adjustable pa-
rameters are now routine [
1
5
]. The approach is based on
density-functional theory (DFT) and density-functional per-
turbation theory (DFPT) to compute the electronic structure,
phonons, and electron-phonon matrix elements, followed by
Wannier interpolation to the fine grids needed for trans-
port calculations [
6
,
7
]. Transport properties are obtained by
solving the Boltzmann equation with the collision matrix
computed from the
ab initio
inputs. The accuracy of these
calculations for low-field mobility has been established for
many materials, including Si [
1
,
2
,
8
], GaAs [
4
,
5
,
8
], and others
[
9
].
High-field transport properties, as well as noise proper-
ties such as the power spectral density (PSD) of current
fluctuations, are also experimentally accessible and provide
additional information about microscopic transport phenom-
ena [
10
,
11
]. For instance, intervalley noise, in which carrier
number fluctuations between valleys are observable as current
fluctuations, can only be observed in a multivalley semicon-
ductor such as
n
-Si [
10
,
12
]. The PSD of hot electrons in Si
has been experimentally investigated at a range of frequencies,
temperature, and electric-field strengths. Measurements of the
electron-diffusion coefficient (proportional to low-frequency
PSD) at room temperature along the [111] direction indi-
cated a pronounced decrease with increasing electric field
[
13
]. A subsequent study at lower temperatures showed an
initial increase of the diffusion coefficient with increasing
*
Corresponding author: aminnich@caltech.edu
field, followed by the decrease seen at higher temperatures
[
14
]. A clear anisotropy in the diffusion coefficient was ob-
served between the [100] and [111] directions for high electric
fields (

2kVcm
1
) across 77–300 K [
14
], despite the cubic
symmetry of Si, and was attributed to intervalley noise. The
frequency dependence of the PSD at low temperatures
80 K
was also examined, showing how thermal, convective, and
intervalley noise contribute at different frequencies for these
two directions [
15
].
Despite extensive experimental investigation, transport
and noise properties beyond the low-field regime have his-
torically been evaluated using semiempirical Monte Carlo
methods [
14
,
16
18
]. These models utilized various approx-
imations, such as dispersionless optical phonons, Debye
acoustic phonons, and model band structures. More recently,
the Monte Carlo method in n-Si has been extended to device
simulation [
19
] and full-band studies [
20
23
]. Computations
of high-field transport properties using
ab initio
methods have
only recently been reported [
24
28
]. In GaAs, the drift veloc-
ity characteristics up to several kV cm
1
have been computed
and have provided evidence for the role of two-phonon scat-
tering [
26
,
27
]. The warm electron tensor of
n
-Si, including
two-phonon scattering, has also been computed and directly
compared to experiment [
29
]. An
ab initio
formalism to calcu-
late fluctuational properties like PSD has also been developed
recently [
24
]. Previous studies have used the formalism to
compute the electric-field dependence of the PSD in
p
-Si [
28
]
and
n
-GaAs, in the latter case including two-phonon scattering
[
26
,
27
]. However, whether
ab initio
methods can accurately
account for hot electron transport and noise properties in
n
-Si
has not yet been determined.
Here, we report first-principles calculations of the hot
electron diffusion coefficient and frequency-dependent PSD
2469-9950/2024/109(23)/235201(7)
235201-1
©2024 American Physical Society
BENJAMIN HATANPÄÄ AND AUSTIN J. MINNICH
PHYSICAL REVIEW B
109
, 235201 (2024)
in
n
-Si, including two-phonon scattering. We find that al-
though some qualitative features of the diffusion coefficient
are correctly predicted, such as an anisotropy at high electric
fields, quantitative agreement is in general poor. To identify
the origin of the discrepancies, we computed the microwave-
frequency PSD and piezoresistivity. The computed properties
are in reasonable qualitative agreement, constraining the mag-
nitude of inaccuracy in the computed intervalley scattering
rate. Together, these observations indicate that the diffu-
sion coefficient discrepancies may be attributed to factors
which are not included in the
ab initio
formulation of charge
transport, for instance real-space gradients and space-charge
effects. This finding has relevance to the interpretation of
diffusion coefficient measurements in terms of microscopic
charge-transport processes.
II. THEORY AND NUMERICAL METHODS
Our approach to solve for the high-field transport and noise
properties of charge carriers has been described previously
[
24
,
26
,
28
,
29
]. In brief, for a spatially homogeneous, nonde-
generate electron gas subject to an applied electric field, the
Boltzmann transport equation (BTE) is given by
q
E
̄
h
·∇
k
f
k
=−
k


kk


f
k

.
(1)
Here,
q
is the carrier charge,
E
is the electric field vector,
f
k
is the distribution function describing the occupancy of
the electronic state indexed by wavevector
k
,

f
k

is the
perturbation to the equilibrium electron distribution function
f
0
k
, and

kk

is the linearized collision matrix given by Eq. (
3
)
of Ref. [
24
]. We assume that only one band contributes to
charge transport and thus neglect the band index. This for-
mulation of the BTE is applicable to arbitrarily high fields for
nondegenerate electrons as shown in Ref. [
26
].
For sufficiently large electric fields, the reciprocal space
derivative of the total distribution function must be evalu-
ated numerically. In the present formulation, the derivative
is computed using a finite-difference approximation given in
Refs. [
30
,
31
]. The BTE then takes the form of a linear system
of equations
(
Eq. (
5
)inRef.[
24
]
)
that can be solved by
numerical linear algebra:
k


kk


f
k

=
γ
qE
γ
k
B
T
v
k
f
0
k
.
(2)
Here,
E
γ
and
v
k
are the electric field and electron drift ve-
locity along the
γ
-Cartesian axis, and the relaxation operator

kk

is defined as

kk

=

kk

+
γ
qE
γ
̄
h
D
kk

,
(3)
where
D
kk

is the momentum-space derivative represented in
the finite-difference matrix representation given by Eq. (24) in
Ref. [
30
]. The BTE is solved using numerical linear algebra
to obtain the steady-state electron distribution function, from
which transport properties can be obtained using an appropri-
ate Brillouin zone sum. For instance, the drift velocity in the
β
direction is given by
V
β
=
1
N
k
v
k
f
k
,
(4)
where
N
=
k
f
k
is the number of electrons in the Brillouin
zone. Similarly, the mobility is given by
μ
αβ
(
E
)
=
2
e
2
k
B
T
V
0
k
v
k
k


1
kk

(
v
k

f
0
k

)
,
(5)
where
V
0
is the supercell volume,
α
is the direction along
which the current is measured, and
β
is the direction of the
applied electric field [
1
]. Once mobility is found at different
stresses, the inverse (normalized to zero stress) yields the
piezoresistivity.
The fluctuations in the occupancy of electronic states mani-
fest in experiment as current noise, which can be characterized
by the power spectral density (PSD). As derived in Ref. [
24
],
the current PSD
S
j
α
j
β
can be calculated at a given angular
frequency
ω
as
S
j
α
j
β
(
ω
)
=
2
(
2
e
V
0
)
2

[
k
v
k
k

(
i
ω
I
+

)
1
kk

×
(
f
s
k

(
v
k

V
β
)
)
]
,
(6)
where
j
α
and
j
β
are the currents along axes
α
and
β
, and
I
is the identity matrix. In the limit
ωτ

1, where
τ
is a char-
acteristic relaxation time,
S
j
α
j
β
is proportional to the diffusion
coefficient, a relation known as the fluctuation-diffusion re-
lation [
32
]. Therefore, the diffusion coefficient may also be
computed from Eq. (
6
).
In this work, we also computed the piezoresistivity of
electrons in Si. For the calculations with compressive stress
in the [001] direction, a small uniaxial compressive strain was
applied in the [001] direction, and the other two lattice vec-
tors were then relaxed. For the calculations with compressive
stress in the [011] direction, the lattice vectors were changed
manually until the desired stress state was reached.
The numerical details are as follows. For all calculations,
the electronic structure and electron-phonon matrix elements
are computed on a coarse 14
×
14
×
14 grid using DFT
and DFPT with Q
UANTUM
E
SPRESSO
[
33
]. This finer coarse
grid was used, in comparison to the 8
×
8
×
8[
29
] that was
found to be sufficient to converge the unstressed mobility.
A wave-function energy cutoff of 40 Ryd was used for all
calculations, and a relaxed lattice parameter of 5.431 Å was
used for the unstrained properties. The electronic structure
and electron-phonon matrix elements were interpolated onto
the fine grid using P
ERTURBO
[
34
]. For every combination
of applied stress value and direction, the electronic structure
and electron-phonon matrix elements were recomputed. The
piezoresistivity was then computed by calculating the trans-
verse mobility using Eq. (
5
) from which the resistivity may be
obtained, and then normalizing by the calculated unstrained
resistivity value.
For temperatures of 160–300 K, a grid density of 100
×
100
×
100 for the electron states was used, while a grid den-
sity of 50
×
50
×
50 was used for the phonons. We report
235201-2
HOT ELECTRON DIFFUSION, MICROWAVE NOISE, AND ...
PHYSICAL REVIEW B
109
, 235201 (2024)
convergence tests on the PSD values, as they are more sen-
sitive to the details of the band structure and electron-phonon
interaction than the mobility. Using a phonon grid with the
same density as the electron grid resulted in a PSD change of
17% at 10 kV cm
1
, and using a grid density of 120
×
120
×
120 for the electron states and 60
×
60
×
60 for the phonon
states resulted in a PSD change of 20% at 10 kV cm
1
.The
quantity of relevance to our findings is the anisotropy between
[111] and [100], defined as (PSD
[111]
PSD
[100]
)
/
PSD
[111]
.
This quantity only changed by 0.5% when changing the grid
density, indicating that the PSD anisotropy was well con-
verged. From 160–300 K we used an energy window of
284 meV above the conduction-band minimum with a Gaus-
sian smearing parameter of 5 meV, and increasing this energy
window to 342 eV resulted in a PSD change at 10 kV cm
1
of 1%.
For the grid density at 77 K, a grid density of 140
×
140
×
140 for the electron states and 70
×
70
×
70 for the
phonon states was used, with an energy window of 145 meV
and a Gaussian smearing parameter of 2.5 meV. For 77 K,
increasing the energy window to 284 meV resulted in a PSD
change of 7% at 1 kV cm
1
. Using a phonon grid with the
same density as the electron grid resulted in a change of 12%
in the PSD at 1 kV cm
1
, the largest field used. Using a
grid density of 160
×
160
×
160 for the electron states and
80
×
80
×
80 for the phonon states resulted in a PSD change
of 18% at 1 kV cm
1
. However, as before, our findings are
not affected by these changes, as the qualitative trend of the
PSD compared to experiment is unchanged, and the sign of
the anisotropy is unchanged. Therefore, the calculations are
converged adequately at 77 K for the PSD.
For the piezoresistivity calculations (all performed in the
low-field limit), a grid density of 400
×
400
×
400 (200
×
200
×
200) for electrons (phonons) was required. Increasing
the electron grid density to 500
×
500
×
500 and the phonon
grid to 250
×
250
×
250 resulted in mobility changes of 15%.
The relevant quantity for our conclusions is the piezoresistiv-
ity, which is normalized by the zero-stress resistivity value,
and the normalized values changed only on the order of 5%.
In this case, an energy window of 20 meV was employed
for computational tractability. Increasing the energy window
from 20 meV to 25 meV resulted in mobility changes of 4%.
The final linear system of equations used to obtain the
mobility and the PSD was then solved by a Python implemen-
tation of the GMRES method [
35
]. For all calculations and
temperatures, the Fermi level was adjusted to yield a carrier
density of 4
×
10
13
cm
3
. Spin-orbit coupling was neglected,
as it has a weak effect on electron transport properties in
Si [
3
,
36
]. Similarly, quadrupole electron-phonon interactions
were neglected [
37
]. For all calculations of the diffusion co-
efficient, a frequency of 1 GHz was used, selected to ensure
that
ωτ
1

1, where
τ
is a characteristic relaxation time,
while avoiding too low frequencies which result in numerical
instabilities.
In our past work [
29
], it has been shown that two-phonon
scattering (2ph) is non-negligible in
n
-Si. Thus, for all PSD
calculations, two-phonon scattering was included. For the
piezoresistivity calculations, 2ph scattering could not be in-
cluded due to the computational cost. However, we do not
expect the absence of 2ph scattering for piezoresistivity to
affect our conclusions, as it was shown in Ref. [
29
] that the
energy dependence of 2ph scattering rates exhibited the same
qualitative trends as those of one-phonon rates, and further
that most of the effect of 2ph scattering can be accounted for
by scaling the 1ph scattering rates. As this scaling would be
present at all applied stresses, we therefore do not expect that
neglecting 2ph would affect the piezoresistivity values and our
conclusions.
III. RESULTS
A. Electric-field dependence of hot electron diffusion coefficient
We begin by examining the dependence of the diffusion
coefficient on electric field at various temperatures. We first
compare the experimental low-field values of the diffusion
coefficient to the computed ones. We considered four temper-
atures (300, 200, 160, and 77 K), corresponding to those for
which experimental data is available. At these temperatures,
the computed (experimental) diffusion coefficients were 29.7
(37) cm
2
s
1
, 59.3 (62) cm
2
s
1
, 58.8 (71) cm
2
s
1
, and 1120
(141) cm
2
s
1
. For all temperatures besides 77 K, the compu-
tation underestimates the experimental data. The magnitude
of the underestimate for
T
>
77 K is consistent with a prior
calculation of the electron mobility of Si when two-phonon
scattering is included [
29
]. However, at 77 K, the computed
value is
8
×
larger than experiment. This overestimate is
possibly attributable to ionized impurity scattering which is
neglected in the present calculations.
To facilitate the comparison of trends with electric field in
the subsequent plots, the computed data has been normalized
to the calculated low-field diffusion coefficient, while the ex-
perimental data has been normalized to the value at the lowest
electric field reported. We note that due to the requirement
that the transit time in the time-of-flight experiment be less
than the dielectric relaxation time, no data was reported below
a minimum field at each temperature [
14
]. The electric-field
dependence of the diffusion coefficient at 300 K is shown
in Fig.
1(a)
. In experiment, it is observed that at low fields,
the diffusion coefficient along the [100] and [111] directions
are equal. Starting at less than 2 kV cm
1
, the diffusion co-
efficient along the [111] direction is less than in the [100],
an anisotropy that has been attributed to intervalley diffusion
[
14
]. The magnitude of this anisotropy continues to increase
with field, reaches a maximum, and then decreases with field.
The same qualitative trend with field is seen at 200 K in
Fig.
1(b)
, with the main difference being the anisotropy man-
ifesting at lower fields than at higher temperatures. At 160 K,
shown in Fig.
1(c)
, there is a slight peak of the diffusion coeffi-
cient in the [100] direction at low fields and then a monotonic
decrease for higher fields. At 77 K, shown in Fig.
1(d)
, initial
increases of the diffusion coefficient with field are seen for
both directions.
The calculated results generally predict these trends qual-
itatively. At 300 K and 200 K, the correct trend of the
anisotropy is reproduced, as the [111] diffusion coefficient is
less than the [100] value once field values exceed 5 kV cm
1
and 2 kV cm
1
, respectively. While the initial increase seen in
experiment at 160 K with field applied in the [100] direction
is not captured by computation, the qualitative anisotropy
at high fields is reproduced. Similarly, at 77 K, the [111]
235201-3
BENJAMIN HATANPÄÄ AND AUSTIN J. MINNICH
PHYSICAL REVIEW B
109
, 235201 (2024)
FIG. 1. Normalized diffusion coefficient versus electric field at
(a) 300 K, (b) 200 K, (c) 160 K, and (d) 77 K, with field applied along
the [100] direction (red solid line) and [111] direction (purple dashed
line). Experimental data along the [100] direction (red circles) and
[111] direction (purple squares) from Figs. 3 and 4, Ref. [
14
]. In (d),
noise conductivity (NC) measurements (purple triangles) included
for comparison at low electric fields.
diffusion coefficient is less than in the [100] once the electric
field exceeds 0.2 kV cm
1
.
However, a number of quantitative discrepancies can be
seen. At 160, 200, and 300 K, the computed anisotropy starts
to manifest at higher fields than in experiment. In experiment,
at 300 K the anisotropy is observed once the electric field
exceeds 2 kV cm
1
, while at 200 K and 160 K the anisotropy
manifests even below 1 kV cm
1
. Similarly, the magnitude of
the anisotropy is underestimated, particularly for 160 K and
200 K, where the agreement with the [111] data is excellent,
but the [100] data lies much above the computed values.
At 77 K, the qualitative behavior of the diffusion coeffi-
cient with field changes greatly. We note that the electric-field
range used in this calculation is smaller than in the other cases
due to lack of convergence at high fields. In Fig.
1(d)
, in both
directions measured an initial increase in the experimental
PSD is seen. This increase is observed in computation, but
at lower fields than in experiment. Given the relative impor-
tance of ionized impurity scattering at 77 K compared to
higher temperatures, we examined whether the omission of
this scattering mechanism in the calculation could play a role
in the discrepancy. We implemented a simple model of ionized
impurity scattering [
38
] with a density of 10
14
cm
3
. The non-
monotonic features were observed to shift to higher electric
fields, suggesting that ionized impurity scattering could be
partly responsible for this discrepancy.
The anisotropy in the diffusion coefficient seen in experi-
ment has been attributed to a mechanism known as intervalley
diffusion. [
14
] To understand this mechanism, consider the
general expression for the intervalley diffusion coefficient
D
int
, given by
D
int
=
n
1
n
2
(
v
1
v
2
)
2
τ
int
.[
12
,
14
] Here,
n
1
and
n
2
are the fractions of electrons in valleys of type 1 and 2,
v
1
and
v
2
are the drift velocities in valleys of type 1 and 2, and
τ
i
is the characteristic intervalley relaxation time. When the field
is applied in the [111], the average velocities in each valley
type are equal, and this extra contribution vanishes. While
many transport properties such as mobility are insensitive to
the balance between
g
- (between equivalent valleys) and
f
-
type (between inequivalent valleys) scattering,
τ
int
is inversely
proportional to the square of the
f
-type coupling constant
[
14
]. A possible origin of the underpredicted anisotropy in the
diffusion coefficient is therefore computed
f
-type scattering
rates which are too large compared to experiment. To test
this hypothesis, we compute other transport and noise prop-
erties which are sensitive to the distinct types of intervalley
scattering.
B. Microwave-frequency PSD
We
first
compute
the
microwave
frequency
(
0
.
1–100 GHz) PSD at 77 K and 200 V cm
1
,for
which experimental data is available for comparison [
15
].
Here, the frequency ranges computed are much higher
than those in which sources of noise such as 1
/
f
noise
or generation-recombination noise would be relevant.
However, if the frequency is comparable to an inverse
time constant
τ
such as the momentum or energy relaxation
time, nonmonotonic features or rolloffs in the PSD with
increasing frequency around frequencies satisfying
ωτ
1
will be observed [
10
]. Comparing the frequencies at which
these features occur therefore provides an independent test of
the accuracy of the
ab initio
diffusion coefficient calculations.
Figure
2
shows the calculated spectral density of cur-
rent fluctuations versus frequency, at 77 K and 200 V cm
1
and with electric fields applied along the [111] and [100]
directions, along with experimental data. At frequencies
below 3 GHz, the [100] PSD is greater than the [111], due
to the presence of intervalley diffusion. As intervalley scat-
tering is characterized by a significantly smaller relaxation
rate than either the energy or momentum relaxation rates, a
rolloff in the [100] direction is observed around the relatively
low frequency of 1 GHz. The presence of the “convective”
mechanism away from equilibrium rolls off at a frequency
corresponding to the energy relaxation rate. Here, the convec-
tive peak occurs around 25 GHz. For semiconductors with a
sublinear current-voltage characteristic, this convective con-
tribution is negative [
10
]. This mechanism is present in both
the [100] and [111] cases, but is more obviously present in the
[111] due to the lack of intervalley noise. Finally, as the fre-
quency exceeds the momentum relaxation rate, the PSD rolls
off to zero as the electronic system is not able to redistribute
in response to the oscillating external field.
Over the entire calculated frequency range, the computed
results qualitatively capture the trends seen in experiment. The
anisotropy seen at low frequencies due to intervalley noise,
the rolloff in the [100] direction starting around 1 GHz due to
frequency exceeding the characteristic intervalley scattering
rate, and the convective noise peaks are all reproduced. At
frequencies above 100 GHz, the PSD is higher in the [111]
direction, simply due to the greater mobility in this direction
235201-4
HOT ELECTRON DIFFUSION, MICROWAVE NOISE, AND ...
PHYSICAL REVIEW B
109
, 235201 (2024)
FIG. 2. Microwave PSD versus frequency at 77 K and
200 V cm
1
applied electric field, with field applied along the [100]
direction (red solid line) and [111] direction (purple solid line).
Experimental data along the [100] direction (red circles) and [111]
direction (purple squares) from Fig.
1
,Ref.[
15
]. In both cases, the
data is normalized to the value of the PSD at the lowest-frequency
data point (computation, 0.19 GHz; experiment, 0.1 GHz) in the
[111] direction.
at 200 V cm
1
. Relaxation times for the various noise sources
(thermal, convective, and intervalley) can be obtained by fit-
ting the computed curves to Lorentzians parameterized by the
various relaxation times, as given in Eq. (9.5) in Ref. [
10
].
For the [111] direction, an energy relaxation time of 15 ps
was calculated using Monte Carlo simulation, as well as a
momentum relaxation time of 2 ps, while our computation
yields an energy relaxation time of 9 ps, and a momentum
relaxation time of 4 ps [
10
]. For the [100] direction, Monte
Carlo simulation reported an energy relaxation time of 5 ps
[
10
], and an intervalley relaxation time of 50 ps [
15
], while
our computation yields an energy relaxation time of 10 ps,
and an intervalley relaxation time of 79 ps. The magnitudes
of the relaxation times and relative difference between the
momentum and energy relaxation times are thus in qualitative
agreement with prior works. However, data only exists up to
intermediate frequencies (around 10 GHz), so it is difficult to
draw quantitative conclusions, especially for the momentum
relaxation time.
As the difference in the PSD at low frequency is due
to intervalley noise, and the magnitude of this difference is
captured accurately by our computation, the results of Fig.
2
suggest that the
f
-type scattering rates in computation are
compatible with their actual values. In addition, the frequency
of the intervalley rolloff and convective mechanism being well
captured imply that both the intervalley and energy relaxation
rates are qualitatively consistent with experimental values.
C. Piezoresistivity
We next compute the piezoresistivity at 300 K and 77 K,
for which experimental data is available [
39
]. Piezoresistivity
FIG. 3. Computed normalized transverse resistivity versus stress
at (a) 300 K and (b) 77 K, with stress applied along the [001]
direction (red triangles) and [011] direction (purple crosses). Experi-
mental data along the [001] direction (red circles) and [011] direction
(purple squares) from Figs. 3 and 4, Ref. [
39
].
provides information about
f
-type intervalley scattering due
to the following considerations. When compressive stress is
applied along a crystallographic direction, valleys parallel to
the stress axis decrease in energy compared to the other val-
leys. Therefore, if the stress is applied in the [001] direction, in
the limit of high stress all electrons will be in the [001] valleys.
Similarly, if stress is applied in the [011] direction, electrons
will be in the [010] and [001] valleys. In the first case ([001]
stress), all
f
-type scattering will be eliminated. However, in
the second case ([011] stress),
f
-type intervalley scattering
between [010] and [001] valleys remains. Therefore, if
f
-type
scattering is negligible [such as at 77 K as seen in experiment
in Fig.
3(b)
], the transverse resistivity (for instance resistivity
measured along [100]) at high stress in both cases is expected
to be identical [
40
]. If
f
-type scattering is present [as at 300 K
as seen in experiment in Fig.
3(a)
], the case with the stress
oriented along the [001] will have a lower resistivity due to
the lack of
f
-type scattering.
In Fig.
3(a)
, the computed transverse resistivity versus
stress in the [001] and [011] directions at 300 K is presented.
The computed anisotropy exhibits qualitative agreement with
experiment, as in both cases the resistivity is less when the
stress is applied along the [001] compared to the [011] case.
Due to the non-negligible contribution of
f
-type scattering at
300 K, applying pressure along the [001] eliminates
f
-type
scattering in the high-stress limit and thereby decreases the
resistivity by a greater amount than in the [011] case. How-
ever, the computation underpredicts the transverse resistivity
at all pressures for both applied stress directions.
In Fig.
3(b)
, the computed transverse resistivity versus
stress at 77 K is shown along with experimental data. Here, it
is observed in experiment that at high stresses, the resistivity
along both directions saturates to closer to the same value than
at 300 K. The computed resistivity saturates with pressure to
a slightly lower value than in experiment, but the difference
between the two directions is considerably smaller than at
300 K (69% at 300 K versus 8% at 77 K). The relatively small
difference in the high-pressure 77 K resistivity between the
two directions indicates that
f
-type scattering is negligible
at this temperature, while at 300 K the computed difference
between the two directions is comparable with experimental
results. The agreement at both temperatures indicates that the
235201-5
BENJAMIN HATANPÄÄ AND AUSTIN J. MINNICH
PHYSICAL REVIEW B
109
, 235201 (2024)
magnitude of
f
-type scattering at these temperatures is being
qualitatively captured.
IV. DISCUSSION
Figure
1
indicates that the anisotropy of the diffusion co-
efficient in
n
-Si is qualitatively captured in the calculation,
with the diffusion coefficient in the [111] direction being less
than in the [100] in the high-field limit for all temperatures
measured. The primary discrepancies between experiment
and computation between 160 and 300 K are the anisotropy
in the computed results being smaller and not manifest-
ing until higher fields compared to experiment. The smaller
anisotropy in the computed results suggests that the com-
putation underestimates the amount of intervalley noise, and
thus overestimates the amount of
f
-type scattering. However,
Fig.
2
indicates that the computed intervalley scattering rates
and intervalley noise magnitude are qualitatively compatible
with experiment, and Fig.
3
shows that the variation of
f
-type
scattering with temperature is qualitatively captured as well.
The amount of error in the computed
f
-type scattering rate is
therefore constrained to values that are insufficient to explain
the discrepancies in the diffusion coefficient.
Given these observations, and that we have used the highest
level of
ab initio
theory presently available which includes
two-phonon scattering, our findings suggest an external mech-
anism not contained in the computation is responsible for
the discrepancy. We suggest that this mechanism could be
the neglect of spatial inhomogeneities present in experiment.
The
ab initio
method used here does not include real-space
effects such as concentration gradients or space-charge ef-
fects. Although the time-of-flight experiment was carefully
implemented to avoid dielectric relaxation in the sample, it
is conceivable that fluctuations in drift velocity associated
with intervalley scattering within the generated electron pulse
could lead to space-charge effects which would spatially
broaden the pulse and hence increase the measured diffusion
coefficient. This effect would be present only in the [100]
direction due to the absence of intervalley scattering in the
[111] direction. Further, these effects would not appear in the
microwave PSD as these frequencies are much higher than
those associated with any dielectric relaxation phenomena.
Additional study will be required to determine the origin of
the diffusion coefficient discrepancies.
V. SUMMARY
We have computed the hot electron diffusion coefficient,
microwave PSD, and piezoresistivity in Si from first principles
from 77–300 K. We find that while qualitative features of the
diffusion coefficient such as the anisotropy at high electric
fields are generally predicted, several trends of the calculated
values differ from experiment. We computed the piezoresistiv-
ity and microwave PSD to investigate whether an inaccurate
description of
f
-type intervalley scattering could explain the
discrepancies. However, the good qualitative agreement of
these properties with experiment excluded this possibility,
leading to the hypothesis that the measured diffusion co-
efficient is influenced by factors not included in
ab initio
calculations such as real-space gradients and space-charge
effects. This finding indicates that care must be taken when
interpreting diffusion coefficient measurements in terms of
microscopic charge transport processes.
ACKNOWLEDGMENTS
B.H. was supported by a NASA Space Technology Gradu-
ate Research Opportunity under Grant No. 80NSSC21K1280.
A.J.M. was supported by AFOSR under Grant No. FA9550-
19-1-0321. The authors thank J. Sun, S. Sun, D. Catherall, and
T. Esho for helpful discussions.
[1] W. Li, Electrical transport limited by electron-phonon coupling
from Boltzmann transport equation: An
ab initio
study of Si,
Al, and MoS
2
,
Phys. Rev. B
92
, 075405 (2015)
.
[2] M. Fiorentini and N. Bonini, Thermoelectric coefficients of
n
-doped silicon from first principles via the solution of the
Boltzmann transport equation,
Phys.Rev.B
94
, 085204 (2016)
.
[3] S. Poncé, E. R. Margine, and F. Giustino, Towards predictive
many-body calculations of phonon-limited carrier mobilities in
semiconductors,
Phys. Rev. B
97
, 121201(R) (2018)
.
[4] J.-J. Zhou and M. Bernardi,
Ab initio
electron mobility and
polar phonon scattering in GaAs,
Phys. Rev. B
94
, 201201(R)
(2016)
.
[5] T.-H. Liu, J. Zhou, B. Liao, D. J. Singh, and G. Chen,
First-principles mode-by-mode analysis for electron-phonon
scattering channels and mean free path spectra in GaAs,
Phys.
Rev. B
95
, 075206 (2017)
.
[6] M. Bernardi, First-principles dynamics of electrons and
phonons,
Eur.Phys.J.B
89
, 239 (2016)
.
[7] F. Giustino, Electron-phonon interactions from first principles,
Rev. Mod. Phys.
89
, 015003 (2017)
.
[8] S. Poncé, F. Macheda, E. R. Margine, N. Marzari, N. Bonini,
and F. Giustino, First-principles predictions of Hall and drift
mobilities in semiconductors,
Phys. Rev. Res.
3
, 043022
(2021)
.
[9] S. Poncé, W. Li, S. Reichardt, and F. Giustino, First-principles
calculations of charge carrier mobility and conductivity in
bulk semiconductors and two-dimensional materials,
Rep. Prog.
Phys.
83
, 036501 (2020)
.
[10] H. L. Hartnagel, R. Katilius, and A. Matulionis,
Microwave
Noise in Semiconductor Devices
(John Wiley & Sons, New
York, 2001), Chap. 8.
[11] E. Conwell,
High Field Transport in Semiconductors
(Academic
Press, New York, 1967).
[12] P. J. Price, Intervalley noise,
J. Appl. Phys.
31
, 949 (1960)
.
[13] C. Canali, C. Jacoboni, G. Ottaviani, and A. Alberigi-Quaranta,
High-field diffusion of electrons in silicon,
Appl. Phys. Lett.
27
,
278 (1975)
.
[14] R. Brunetti, C. Jacoboni, F. Nava, L. Reggiani, G. Bosman, and
R. J. J. Zijlstra, Diffusion coefficient of electrons in silicon,
J.
Appl. Phys.
52
, 6713 (1981)
.
235201-6
HOT ELECTRON DIFFUSION, MICROWAVE NOISE, AND ...
PHYSICAL REVIEW B
109
, 235201 (2024)
[15] V. Bare
̆
ıkis, V. Viktoravichyus, and A. Gal’dikas, Frequency
dependence of noise in
n
-type Si in high electric fields, Fiz.
Tekh. Poluprovodn.
16
, 1868 (1982) [Sov. Phys. Semicond.
16
,
1202 (1982)].
[16] C. Jacoboni, R. Minder, and G. Majni, Effects of band non-
parabolicity on electron drift velocity in silicon above room
temperature,
J. Phys. Chem. Solids
36
, 1129 (1975)
.
[17] C. Jacoboni and L. Reggiani, The Monte Carlo method for the
solution of charge transport in semiconductors with applications
to covalent materials,
Rev. Mod. Phys.
55
, 645 (1983)
.
[18] E. Pop, R. W. Dutton, and K. E. Goodson, Analytic band
Monte Carlo model for electron transport in Si including acous-
tic and optical phonon dispersion,
J. Appl. Phys.
96
, 4998
(2004)
.
[19] Z. Aksamija and U. Ravaioli, Joule heating and phonon
transport in silicon mosfets,
J. Comput. Electron.
5
, 431
(2006)
.
[20] M. V. Fischetti, Monte Carlo simulation of transport in tech-
nologically significant semiconductors of the diamond and
zinc-blende structures. I. homogeneous transport,
IEEE Trans.
Electron Devices
38
, 634 (1991)
.
[21] B. Fischer and K. R. Hofmann, A full-band Monte Carlo model
for the temperature dependence of electron and hole transport
in silicon,
Appl. Phys. Lett.
76
, 583 (2000)
.
[22] P. H. Nguyen, K. R. Hofmann, and G. Paasch, Compara-
tive full-band Monte Carlo study of Si and Ge with screened
pseudopotential-based phonon scattering rates,
J. Appl. Phys.
94
, 375 (2003)
.
[23] M. V. Fischetti, P. D. Yoder, M. M. Khatami, G. Gaddemane,
and M. L. Van de Put, “Hot electrons in Si lose energy mostly to
optical phonons”: Truth or myth?
Appl. Phys. Lett.
114
, 222104
(2019)
.
[24] A. Y. Choi, P. S. Cheng, B. Hatanpää, and A. J. Minnich,
Electronic noise of warm electrons in semiconductors from first
principles,
Phys.Rev.Mater.
5
, 044603 (2021)
.
[25] I. Maliyov, J. Park, and M. Bernardi, Ab initio electron dynam-
ics in high electric fields: Accurate prediction of velocity-field
curves,
Phys.Rev.B
104
, L100303 (2021)
.
[26] P. S. Cheng, J. Sun, S.-N. Sun, A. Y. Choi, and A. J. Minnich,
High-field transport and hot-electron noise in GaAs from first-
principles calculations: Role of two-phonon scattering,
Phys.
Rev. B
106
, 245201 (2022)
.
[27] J. Sun and A. J. Minnich, Transport and noise of hot electrons in
GaAs using a semianalytical model of two-phonon polar optical
phonon scattering,
Phys.Rev.B
107
, 205201 (2023)
.
[28] D. S. Catherall and A. J. Minnich, High-field charge transport
and noise in
p
-Si from first principles,
Phys. Rev. B
107
, 035201
(2023)
.
[29] B. Hatanpää, A. Y. Choi, P. S. Cheng, and A. J. Minnich,
Two-phonon scattering in nonpolar semiconductors: A
first-principles study of warm electron transport in Si,
Phys.
Rev. B
107
, L041110 (2023)
.
[30] A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt,
and N. Marzari, wannier90: A tool for obtaining maximally-
localised Wannier functions,
Comput. Phys. Commun.
178
, 685
(2008)
.
[31] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D.
Vanderbilt, Maximally localized Wannier functions: Theory and
applications,
Rev. Mod. Phys.
84
, 1419 (2012)
.
[32] S. V. Gantsevich, V. L. Gurevich, and R. Katilius, Theory
of fluctuations in nonequilibrium electron gas,
Riv. Nuovo
Cimento
2
, 1 (1979)
.
[33] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C.
Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I.
Dabo, A. Dal Corso, S. De Gironcoli, S. Fabris, G. Fratesi, R.
Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri,
L. Martin-Samos
et al.
,
QUANTUM ESPRESSO
: A modular and
open-source software project for quantum simulations of mate-
rials,
J. Phys.: Condens. Matter
21
, 395502 (2009)
.
[34] J.-J. Zhou, J. Park, I. Lu, I. Maliyov, X. Tong, and M. Bernardi,
Perturbo: A software package for
ab initio
electron-phonon
interactions, charge transport and ultrafast dynamics,
Comput.
Phys. Commun.
264
, 107970 (2021)
.
[35] V. Frayssé, L. Giraud, S. Gratton, and J. Langou, Algorithm
842: A set of GMRES routines for real and complex arithmetics
on high performance computers,
ACM Trans. Math. Softw.
31
,
228 (2005)
.
[36] J. Ma, A. S. Nissimagoudar, and W. Li, First-principles study of
electron and hole mobilities of Si and GaAs,
Phys. Rev. B
97
,
045201 (2018)
.
[37] J. Park, J.-J. Zhou, V. A. Jhalani, C. E. Dreyer, and M. Bernardi,
Long-range quadrupole electron-phonon interaction from first
principles,
Phys.Rev.B
102
, 125203 (2020)
.
[38] D. Long and J. Myers, Ionized-impurity scattering mobility of
electrons in silicon,
Phys. Rev.
115
, 1107 (1959)
.
[39] K. V. Hansen, Some investigations of the intervalley scattering
in N-type silicon, Ph.D. thesis, Technical University of Den-
mark, 1974.
[40] M. H. Jørgensen, Electron-phonon scattering and high-field
transport in
n
-type Si,
Phys.Rev.B
18
, 5657 (1978)
.
235201-7