# Quasiballistic electron transport in cryogenic SiGe HBTs studied using an
exact, semi-analytic solution to the Boltzmann equation

Nachiket R. Naik ID Division of Engineering and Applied Science, California
Institute of Technology, Pasadena, CA 91125, USA  Austin J. Minnich ID
Division of Engineering and Applied Science, California Institute of
Technology, Pasadena, CA 91125, USA<EMAIL_ADDRESS>
###### Abstract

Silicon-germanium heterojunction bipolar transistors (HBTs) are of interest as
low-noise microwave amplifiers due to their competitive noise performance and
low cost relative to III-V devices. The fundamental noise performance limits
of HBTs are thus of interest, and several studies report that quasiballistic
electron transport across the base is a mechanism leading to cryogenic non-
ideal IV characteristics that affects these limits. However, this conclusion
has not been rigorously tested against theoretical predictions because prior
studies modeled electron transport with empirical approaches or approximate
solutions of the Boltzmann equation. Here, we study non-diffusive transport in
narrow-base SiGe HBTs using an exact, semi-analytic solution of the Boltzmann
equation based on an asymptotic expansion approach. We find that the computed
transport characteristics are inconsistent with experiment, implying that
quasiballistic electron transport is unlikely to be the origin of cryogenic
non-ideal IV characteristics. Our work helps to identify the mechanisms
governing the lower limits of the microwave noise figure of cryogenic HBT
amplifiers.

##  I Introduction

Silicon-germanium heterojunction bipolar transistors (HBTs) are widely used in
microwave applications such as radar and communication systems [1, 2] and show
potential in space science [3] and imaging applications,[4, 5, 6] owing to
their competitive microwave performance combined with ease of integration with
the CMOS process, high yield, and low cost relative to III-V technologies [7,
8]. SiGe HBTs are now approaching the THz domain due to fabrication process
advancements that enable the continued scaling of key parameters such as base
width and resistance [2, 9]. Moreover, at cryogenic temperatures they have
demonstrated DC gains exceeding 45 dB, noise temperatures $\lesssim 5$ K at 3
GHz with bandwidths of 2-3 GHz [10, 11], lower $1/f$ noise relative to III-V
devices [12], and power consumption in the hundreds of microwatts [13]. These
competitive figures of merit make SiGe HBTs of interest for low-noise
amplifiers and cryo-CMOS circuits [14, 15] in radio astronomy and quantum
computing applications [16, 13] and in oscillators with low phase noise [12,
17, 18].

Following the first reports of SiGe HBTs grown by molecular beam epitaxy [19,
20], studies of the cryogenic performance of HBTs were performed in the early
1990s [19, 21]. Subsequent work focused on understanding and optimizing base
doping and Ge grading profiles for cryogenic performance [22]. While these
works demonstrated the enhanced collector current and transconductance
expected at cryogenic temperatures, later studies reported marked deviations
from the DC current-voltage characteristics predicted from drift-diffusion
theory [23, 24]. Specifically, it was observed that below $\sim 77$ K the
collector $(I_{C})$ and base current $(I_{B})$ exceeded the predicted values
at a given temperature and bias and were independent of temperature, with the
transconductance $g_{m}$ saturating instead of increasing as $T^{-1}$. These
non-ideal $IV$ characteristics limit the gain, cutoff frequency, and
ultimately the microwave noise figure.

The origin of this behavior has been attributed to several mechanisms,
including trap-assisted tunneling at base-emitter voltage ($V_{BE}$) well
below the built-in potential; [23] and non-equilibrium carrier transport, in
which electrons quasiballistically traverse the base, at biases comparable to
the built-in potential [24]. The latter effect has been phenomenologically
described using an elevated electron temperature that is taken as the
effective temperature for thermionic emission at the base-emitter junction
[11, 24, 10]. As base widths were scaled further down to $\sim 20$ nm in
recent years, direct tunneling of electrons was reported to contribute to the
collector current at biases approaching the built-in potential [25, 26, 27].

There are several discrepancies with these explanations, however. First, both
base and collector currents exhibit non-idealities at cryogenic temperatures
[11, 25], and holes are unlikely to transport ballistically or tunnel across
the emitter region. Further, at base doping levels above $\sim 3\times
10^{18}$ cm-3 common in modern devices, [27, 25] ionized impurity scattering
is expected to dominate. Therefore, quasiballistic transport is not expected
to be markedly more pronounced at cryogenic temperatures relative to room
temperature as evidenced by the weak temperature dependence of the minority
carrier mobility [28, 29, 30, 11]. Finally, while electrons may be heated as
they traverse the base by the built-in field, the collector current and
transconductance depend on the electron temperature at the base-emitter
junction prior to transport across the base and thus cannot be affected by the
built-in field.

A quantitative, non-empirical description of the quasiballistic transport
process would allow a more thorough test of whether quasiballistic transport
is a possible origin of the cryogenic DC non-idealities. Steady particle
transport across a slab of thickness comparable to the MFPs of the particles
is described by the Boltzmann equation [31, 32]. The slab problem has been
extensively investigated for radiative [32, 33, 34], neutron [35], phonon [32,
36], and electron transport [37, 38, 39, 40] as well as for rarefied gases
[41, 42, 43]. Although analytical solutions under the ’one-speed’ or ’gray’
approximations are possible,[37, 34] solutions considering energy-dependent
relaxation times are less tractable analytically. Alternative numerical
approaches are computationally expensive owing to the need to discretize both
the spatial and reciprocal space coordinates [37, 38, 44]. Several works have
reported asymptotic series expansion methods, based on the original theory by
Grad [45], for rarefied gas dynamics [46] and phonon transport [47]. These
methods could enable the efficient solution of the present problem of
electronic transport in a narrow base, but they have yet to be adapted for
electronic transport.

Here, we report a study of quasiballistic electron transport across a narrow
base using an exact, semi-analytic asymptotic expansion approach to solve the
Boltzmann equation. We show that quasiballistic transport leads to decreased
collector current for a given base-emitter voltage and does not alter the
transconductance. Both of these findings contradict experimental observations,
indicating that quasiballistic transport is unlikely to be the origin of
cryogenic non-idealities in SiGe HBTs. Instead, we hypothesize that the origin
of these non-idealities could involve defect-mediated leakage currents which
are readily observed in heavily doped p-n junctions. Our work advances the
understanding of cryogenic transport processes in HBTs that is required to
improve their microwave noise performance.

##  II Theory

We begin by describing the semi-analytical asymptotic approach used to solve
the one-dimensional Boltzmann equation describing electron transport across
the base. Focusing on the DC characteristics, we assume steady transport
across a base of width $L$ with prescribed chemical potential boundary
conditions. Then, the Boltzmann equation for the absolute electronic
distribution function $f_{\lambda}(x)$ is given by:

$\displaystyle\frac{\partial f_{\lambda}}{\partial\overline{x}}$
$\displaystyle=$
$\displaystyle\frac{L}{v_{x\lambda}}\sum_{\lambda^{\prime}}C_{\lambda\lambda^{\prime}}f_{\lambda^{\prime}}=\sum_{\lambda^{\prime}}M_{\lambda\lambda^{\prime}}f_{\lambda^{\prime}}$
(1)

where $C_{\lambda\lambda^{\prime}}$ is the collision matrix, $v_{x\lambda}$ is
the group velocity for electronic state $\lambda$,
$M_{\lambda\lambda^{\prime}}$ is a dimensionless matrix defined as above, $x$
is the spatial coordinate, $\overline{x}\equiv x/L$, and $f_{\lambda}$ is the
desired distribution function normalized so that
$V^{-1}\sum_{\lambda}f_{\lambda}(x)=n(x)$, where $V$ is the sample volume and
$n(x)$ is the carrier density. Observables such as carrier density and
electric current are computed by the appropriate Brillouin zone sum [32].

In this work, we will solve this equation using an exact, semi-analytic
asymptotic expansion method originally reported for rarefied gases [46] and
recently adapted for phonons [47]. The theory for phonons is given in detail
in Ref. [47] for an isotropic crystal under the relaxation time approximation.
Here, we generalize this theory to incorporate the full electron collision
matrix and arbitrary crystals and extend its applicability beyond linear
deviations from a reference distribution.

In the asymptotic expansion approach, $f_{\lambda}$ and associated observables
are written as series expansions with the average Knudsen number
$\epsilon\equiv\braket{\Lambda}/L$ as the expansion parameter, where
$\braket{\Lambda}\equiv\sum_{\lambda_{k}}v_{x\lambda}\tau_{\lambda}f_{\lambda}^{eq}/n_{eq}$.
Here $f_{\lambda}^{eq}$ and $n_{eq}$ are the reference equilibrium
distribution function and electron density at zero bias voltage, and
$\tau_{\lambda}$ is the wave vector dependent relaxation time. As an example,
$f_{\lambda}$ is written as:

$\displaystyle f_{\lambda}$ $\displaystyle=$
$\displaystyle\sum_{l}\epsilon^{l}f_{\lambda,l}$ (2)

where $f_{\lambda,l}$ is the distribution function for order $l$. Substituting
this expansion into Eq. 1, we get

$\sum_{l}\epsilon^{l}\frac{\partial f_{\lambda,l}}{\partial
x}=\sum_{l}\epsilon^{l}\sum_{\lambda^{\prime}}M_{\lambda\lambda^{\prime}}f_{\lambda^{\prime},l}$
(3)

Since $M_{\lambda\lambda^{\prime}}$ scales with $\epsilon^{-1}$, the left and
right side of Eq. 3 are offset by one order of $\epsilon$. Equating terms of
the lowest order $l=0$ we find that the zeroth-order solution satisfies

$\sum_{\lambda^{\prime}}M_{\lambda\lambda^{\prime}}f_{\lambda^{\prime},0}(x)=0=g_{0}(x)\sum_{\lambda^{\prime}}M_{\lambda\lambda^{\prime}}e_{0,\lambda^{\prime}}$
(4)

where we have split the solution into a pure $x-$dependent term $g_{0}(x)$ and
wave vector dependent term $e_{0,\lambda}$, i.e.
$f_{\lambda,0}(x)=e_{0,\lambda}g_{0}(x)$. We note here that the null
eigenvector $e_{0,\lambda}$ can be any Boltzmann distribution function, but
for this work we choose $e_{0,\lambda}=f_{\lambda}^{eq}$ corresponding to the
Boltzmann distribution for the case where no bias is applied.

We now consider the first-order terms. From Eq. 3, the solution can be written
using the zeroth-order solution:

$\displaystyle
f_{\lambda,1}(x)=\sum_{\lambda^{\prime}}\epsilon^{-1}M_{\lambda\lambda^{\prime}}^{-1}\left(\frac{\partial
f_{\lambda^{\prime},0}}{\partial\overline{x}}\right)+g_{1}(x)e_{0,\lambda}$
(5)

This solution is also determined only up to an $x$-dependent multiple of the
null eigenvector, denoted $g_{1}(x)$, which will be obtained after deriving
the governing equation for $x$-dependence. The higher-order solutions proceed
similarly.

The $x$-dependence of the zeroth-order solution can be derived by enforcing
current continuity for the electric current density $J_{x}$ in the
$x$-direction for a 1D steady slab:

$\frac{\partial
J_{x}}{\partial\overline{x}}=0=V^{-1}\frac{\partial}{\partial\overline{x}}\sum_{\lambda}qv_{x\lambda}\sum_{l}\epsilon^{l}f_{\lambda,l}$
(6)

which must be satisfied at each order of the expansion. Substituting from Eq.
5 for the first-order term, we find that

$\sum_{\lambda^{\prime}}v_{x\lambda}\epsilon\frac{\partial}{\partial\overline{x}}\left(\epsilon^{-1}M_{\lambda\lambda^{\prime}}^{-1}\frac{\partial
f_{\lambda^{\prime},0}}{\partial\overline{x}}+g_{1}(x)e_{0,\lambda}\right)=0$
(7)

The second term in the parentheses vanishes since $v_{x\lambda}$ is odd in
$\lambda$ whereas $e_{0,\lambda}$ is even. Therefore, we find

$\frac{\partial^{2}f_{\lambda,0}(x)}{\partial\overline{x}^{2}}=0$ (8)

Applying $V^{-1}\sum_{\lambda}f_{\lambda,0}(x)=n_{0}(x)$ to the above
equation, we see that the zeroth-order electron density $n_{0}(x)$ satisfies
the diffusion equation. Similarly, it can be shown that higher-order terms of
$f_{\lambda,l}(x)$ also satisfy the diffusion equation (see Appendix A of Ref.
[47]).

To solve these equations, the boundary conditions at each order must be
specified. The boundary conditions at zeroth order are simply the prescribed
electron density at the edges of the slab, $n_{L}$ and $n_{R}$. Thus, the
solution at zeroth order is just the diffusion equation solution. Note that
the boundary conditions of the original problem are completely satisfied at
zeroth order.

We now discuss the boundary conditions at higher orders. Because the zeroth
order solution completely satisfies the boundary conditions of the original
problem, to enforce the boundary conditions at higher orders we must introduce
a boundary correction term $\Phi_{\lambda}(x)$ that satisfies the Boltzmann
equation in the boundary region. This boundary correction must exactly cancel
the contribution from $f_{\lambda}(x)$ at the boundaries and vanish in the
bulk.

The Boltzmann equation for the boundary solution at first-order is:

$\frac{\partial\Phi_{\lambda,1}(\eta)}{\partial\eta}=\sum_{\lambda^{\prime}}\epsilon
M_{\lambda\lambda^{\prime}}\Phi_{\lambda^{\prime},1}(\eta)$ (9)

where we introduce the stretched boundary-region coordinate $\eta\equiv
x/\epsilon$. Focusing on the left boundary at $\overline{x}=0$, the condition
enforced on this first-order boundary term is that it must cancel the first-
order bulk term: $\Phi_{\lambda,1}\rvert_{0}=-f_{\lambda,1}\rvert_{0}$. Using
Eq. 5, the condition for the left boundary is:

$\Phi_{\lambda,1}\rvert_{0}=-c_{1}e_{0,\lambda}\left.\frac{\partial
g_{0}}{\partial
x}\right|_{0}-\sum_{\lambda}\epsilon^{-1}M_{\lambda\lambda^{\prime}}^{-1}\left.\frac{\partial
g_{0}}{\partial x}\right|_{0}e_{0,\lambda^{\prime}}$ (10)

where, as in Ref. [47], we anticipate the boundary term to scale as the
gradient of the previous order solution:

$g_{1}\rvert_{x=0}=-c_{1}\left.\frac{\partial
g_{0}}{\partial\overline{x}}\right\rvert_{x=0}$ (11)

An analogous boundary condition applies to the right boundary. The unknown
constant $c_{1}$ captures the jump-type boundary condition for non-diffusive
corrections at first order. Solving for $c_{1}$ involves obtaining the
eigenvalues and eigenvectors of $M$ and is described in detail in Ref. [47].
In brief, the general solution to Eq. 9 can be written as a linear combination
of the eigenvectors corresponding to the negative eigenvalues of $M$:

$\Phi_{\lambda,1}=\sum_{i}A_{i}h_{\lambda,i}e^{\rho_{i}\eta}$ (12)

where $A_{i}$ are unknown coefficients, and $\rho_{i}$ and $h_{\lambda,i}$ are
the negative eigenvalues and the corresponding eigenvectors, respectively.
Only the negative eigenvalues are used so that the boundary solution tends to
zero in the bulk. Then, Eq. 10 gives a linear system of equations for $A_{i}$
and $c_{1}$. We note that in the formulation in Ref. [47], $c_{1}$ represents
a deviation in a linearized quantity, such as temperature, relative to an
equilibrium value. However, in the present formulation, $c_{1}$ multiplies the
absolute distribution function rather than a deviation and is not restricted
to linear deviations from that distribution.

After calculating $c_{1}$, the first-order solution is completely specified
using the diffusion equation for $g_{1}(x)$ and the boundary condition from
Eq. 11. The analysis progresses similarly for higher-order solutions. For the
specific slab problem here, it can be shown that the jump coefficients
associated with second-order derivatives vanish as in Appendix C of Ref. [47].
Therefore, $c_{1}$ is the only required coefficient, allowing the asymptotic
expansion to be summed over all higher-order terms. After summing
$f_{\lambda}$ over the Brillouin zone to obtain the carrier density, we
obtain:

$n(x)=n_{0}+(n_{R}-n_{L})(1-2\overline{x})\sum_{n=1}^{\infty}\epsilon^{n}(-2)^{n-1}c_{1}^{n}=n_{0}+\frac{\epsilon
c_{1}}{1+2\epsilon c_{1}}(n_{R}-n_{L})(1-2\overline{x})$ (13)

where $n_{0}(x)$ is the zeroth-order carrier density, and the higher-order
terms correct for non-diffusive transport. Because the macroscopic
constitutive relation between electric current density and number density
gradient applies in the bulk, [47] the current density including the effect of
quasiballistic transport is given by:

$J_{e,x}=-qD_{e,Si}\frac{\partial n(x)}{\partial x}$ (14)

Here, $D_{e,Si}$ is the bulk diffusivity. These results have been extensively
validated by comparison to Monte Carlo simulations in Ref. [47]. We also
observe from Eq. 13 that the dependence of carrier density and thus electric
current on base-emitter voltage $V_{BE}$ is only through the left boundary
condition $n_{L}\sim e^{qV_{BE}/kT}$, allowing the current-voltage
characteristics for arbitrary $V_{BE}$ to be obtained once $c_{1}$ is
computed.

We briefly discuss the features of the present asymptotic approach compared to
those reported previously for electronic transport. An early study of the slab
electron transport problem employed a direct scattering matrix solution to the
Boltzmann equation [38] that required discretization in wave vector and
spatial coordinates and was thus numerically expensive. Approximate methods
such as the McKelvey flux method [39, 48] and the Landauer approach [49, 44]
were next applied for electron and phonon transport. However, while the
McKelvey flux method is numerically less expensive than the full Boltzmann
equation because the wave vector coordinates are averaged into forward and
backward fluxes, information is lost in this process. The Landauer approach
further simplifies the McKelvey flux method by only requiring the forward- and
reverse-going boundary conditions, but as consequence cannot resolve carrier
density across the slab.

The advantage of the present approach is that it provides the exact solution
to the Boltzmann equation with the full collision matrix while requiring only
discretization in wave vector space rather than in both real space and wave
vector space. The resulting calculation is thus orders of magnitude faster
than the purely numerical solution of the Boltzmann equation. Further,
compared to Ref. [47] this work does not require linearization around an
equilibrium distribution and can accommodate arbitrary crystals, rather than
isotropic ones; and the full collision matrix, rather than the relaxation time
approximation. Finally, with a standard numerical treatment, the current
density would need to be recomputed when any variable is changed. With the
present method, the expensive calculation for $c_{1}$ only needs to be
performed once at each temperature because the average Knudsen number and
$V_{BE}$ are independent of $c_{1}$ and included directly in Eq. 13.

##  III Results

We apply this approach to study 1D steady-state electron transport in a narrow
base HBT ($L\sim 40$ nm) consisting of non-degenerately doped silicon
($N_{A}=3\times 10^{-18}$ cm-3). The carrier density at the left and right
boundaries is given as $n_{L}=n_{L,eq}\exp{(qV_{BE}/kT)}$ and $n_{R}=0$,
respectively, where $n_{L,eq}$ is the equilibrium minority carrier
concentration. Owing to lack of precise knowledge of the electronic structure
and scattering mechanisms in the strained SiGe:C films used in devices, in
this work we assume parabolic bands and employ the relaxation time
approximation, with energy-dependent impurity and acoustic phonon scattering
rates based on the forms given in Chapter 3 of Ref. [31]. Several works report
that the temperature dependence of minority carrier mobility in p-doped SiGe
does not obey the $\sim T^{1.5}$ scaling typical of impurity scattering at
cryogenic temperatures [28, 29, 30]. Therefore, we extract coefficients and
exponents for relaxation time $\tau=\mu_{0}T^{p}E^{s}$ by fitting the computed
mobility to minority carrier mobility data reported in Fig. 3 of Ref. [30] and
combining the relaxation times for acoustic and impurity scattering using
Matthiessen’s rule. The fitting coefficients and exponents for acoustic
(impurity) scattering rates are $\mu_{0}=350~{}(700)$ cm2V-1s-1,
$p=-1.5~{}(0.25)$ and $s=-0.5~{}(-0.5)$, respectively. Our conclusions are
robust to these assumptions. We solve for $c_{1}$ at the temperatures
corresponding to Ref. [25], assuming an isotropic crystal using a $2200\times
2200$ grid in energy and angular coordinate space. The collision matrix used
in this work corresponds to Eq. B.5 of Ref. [47]. For example, at 300 K we
find $c_{1}=0.7341$.

Figure 1: (a) Electron density $n(x)$ versus normalized distance
$\overline{x}$ at 300 K and $V_{BE}=0.8$ V at zeroth order (blue dash-dotted
line), first order (orange dashed line) and infinite order (green solid line).
(b) Normalized electric current density versus Knudsen number $\epsilon$ at
300 K (green dash-dotted line) and 36 K (blue solid line) derived from carrier
concentration gradient. The current density is normalized to the diffusion
current density obtained at zeroth order (dashed line). Quasiballistic
transport predicts a comparable decrease in current relative to drift-
diffusion at both room and cryogenic temperatures as Knudsen number increases.

Figure 1 shows the carrier density versus normalized distance at 300 K for
$L=40$ nm and $V_{BE}=0.8$ V calculated at different orders of the expansion.
The zeroth-order solution is by definition the drift-diffusion solution and
serves as a reference against which to compare higher-order solutions. The
higher-order non-diffusive corrections predict a reduced carrier concentration
gradient across the base relative to the zeroth-order case. This feature
occurs because at $\epsilon=0.38$, transport is quasiballistic, with electron
mean free paths being on the order of the base width. Under these conditions,
the ballistic propagation of some electrons across the base lowers the carrier
density at the left boundary and increases it at the right boundary, resulting
in a shallower carrier density across the base. The solution at the first-
order over-corrects for non-diffusive effects, but the infinite series
accounts for higher-order terms and lies in between the zeroth and first-order
solutions. This non-monotonicity occurs because the closed form solution
switches signs at each order in Eq. 13.

To study how the electric current density is impacted by quasiballistic
transport, we calculate the current density versus Knudsen number in Fig. 1
for two temperatures values reported in Ref. [25]. The electric current
density from Eq. 14 is proportional to the carrier concentration gradient in
Fig. 1 for a given voltage, temperature and Knudsen number. At 300 K, the
current density is observed to decrease from the diffusive limit as
$\epsilon\rightarrow 1$ (quasiballistic limit). As seen from Fig. 1,
quasiballistic corrections lower the concentration gradient, and the
contribution of the correction terms is multiplied by Knudsen number,
resulting in greater deviation with increasing Knudsen number. Therefore, a
shallower concentration gradient and consequently a decreased current density
is expected with increasing Knudsen number. At 36 K, the calculation for
$c_{1}$ is repeated to give a value of $0.733$. At this temperature, we see
that the deviation from the diffusive limit is comparable to that at 300 K.
The difference between the two curves is due to changes in bulk diffusivity at
36 K that are independent of $c_{1}$ and predict a slightly smaller current
density than at 300 K. This calculation shows that even at cryogenic
temperatures, quasiballistic effects predict a lower collector current density
compared to drift-diffusion, opposite to that observed experimentally.

Figure 2: (a) Collector current density $J_{c}$ versus base-emitter voltage
$V_{BE}$ from drift-diffusion (dashed line) and quasiballistic calculations
(solid line) compared to measured data (triangles) at various temperatures.
(b) Transconductance per unit area $g_{m}$ versus temperature for an ideal
diode (dashed line), from calculation (circles), and experiment (triangles,
Ref. [25]) for a fixed $J_{C}$ derived from the $J-V$ characteristics in (a).
The calculations do not predict the increased current density or decreased
transconductance observed in measurements at cryogenic temperatures.

We now compare the predictions of our calculation with experimentally reported
DC IV characteristics [25]. Figure 2 plots calculated and measured collector
current density versus base-emitter voltage $V_{BE}$. We observe that at all
temperatures, quasiballistic transport predicts a weakly reduced current
density compared to the drift-diffusion prediction, as expected from the
discussion of Fig. 1. The calculation nearly coincide in Fig. 2 because the
deviation of quasiballistic current density from the diffusive current density
is small on the logarithmic scale in Fig. 2. At 300 K, 200 K, and 77 K, both
predictions agree well with measured data. However, at 36 K, quasiballistic
and drift-diffusion current density predictions are similar in magnitude but
are both orders of magnitude lower than the measured data for a given
$V_{BE}$. Further, the predicted slope of the $J-V$ curve is unchanged from
the drift-diffusion prediction and does not saturate to a value similar to
that at 77 K as observed in the measured data.

To further examine this result, we present transconductance versus temperature
at a fixed value of collector current density in Fig. 2. The transconductance
(per unit area) $g_{m}$ is calculated as the slope of the current-voltage
curve at a fixed bias, and for an ideal SiGe HBT it is given by
$g_{m}=qJ_{C}/kT$. The measured transconductance exhibits a saturation to a
lower value than the ideal value as temperature decreases. However, the
transconductance values calculated from the slope of the calculated results in
Fig. 2 are identical to the ideal values. Therefore, quasiballistic transport
does not change the temperature dependence of the transconductance. The reason
can be seen in Eq. 13 in which $V_{BE}$ affects the electron density only
through the left boundary condition $n_{L}$; therefore, a change in voltage
affects current density in a way that is independent of the mechanism of
carrier transport and thus preserves the diffusive temperature dependence
regardless of the Knudsen number.

##  IV Discussion

Our analysis has showed that the predicted influence of quasiballistic
transport on the IV characteristics of HBTs is inconsistent with the
experimental observations. We now address several mechanisms that we have not
included in our analysis. First, we have not incorporated the built-in field
in the base region due to Ge grading. However, as in Ref. [50], the only
effect of a field term is to modify $M$ and thus change the value of $c_{1}$,
and our conclusions are robust to such changes. Second, modern devices may
have base doping levels exceeding $10^{19}$ $cm^{-3}$ [51, 25, 27] to minimize
base resistance and prevent carrier freeze-out. This high level of doping
leads to degenerate carrier statistics, while we have assumed non-degenerate
statistics. However, non-idealities in the IV characteristics are observed at
temperatures up to $\sim 80$ K for which the electrons are non-degenerate, and
the present analysis thus applies under relevant conditions. Lastly, the
asymptotic solution is restricted to values of average Knudsen number such
that $2\epsilon c_{1}<1$, which limits our analysis to a minimum base width of
40 nm for the chosen scattering rates. However, DC non-idealities were
reported in devices with base widths on this order,[26, 25, 27] and again our
analysis applies. Given these considerations, we conclude that quasiballistic
transport is not responsible for non-ideal cryogenic DC characteristics of
SiGe HBTs.

Finally, we discuss alternate explanations for the cryogenic non-idealities.
Prior works have suggested that direct tunneling from the emitter to the
collector is a possible origin of non-idealities in devices with narrow base
widths $\sim 10$ $nm$ [25, 26]. However, non-ideal cryogenic transconductance
has been observed in first-generation SiGe HBTs with base widths on the order
of $100$ nm [24, 27] for which the direct tunneling current is expected to be
negligible. Further, non-idealities observed in the hole current cannot be
explained by direct tunneling as the hole potential barrier terminates at the
poly-Si emitter contact, which is hundreds of nanometers away from the base-
emitter junction. Lastly, abrupt changes in the slope of I-V characteristics
have been attributed to a switch from trap-assisted transport to a tunneling
or quasiballistic mechanism [27, 26]. However, similar trends are observed in
p-n diodes [52, 53, 54] for which direct tunneling or quasiballistic transport
is not relevant, and the possibility of a similar mechanism occurring in SiGe
HBTs has not been excluded.

We instead offer a hypothesis for the origin of cryogenic non-idealities as
leakage currents related to inhomogeneities in the base-emitter potential
barrier. The earliest works on heavily doped p-n junctions reported
temperature independent slopes of cryogenic forward-bias I-V characteristics
for voltages where band-to-band tunneling is unlikely to occur [55, 56, 57].
Similar non-ideal behavior, observed as an increase in ideality factor with
decreasing temperature, has been reported in Schottky diodes [58, 59] and
III-V diodes [60, 61]. In fact, the dependence of ideality factor versus
physical temperature for Schottky diodes (e.g. Fig. 11 of Ref. [59], Fig. 1 of
Ref. [58]) exhibits the same qualitative trend observed in SiGe HBTs (see Fig.
8 of Ref. [10]). In Schottky diodes, these excess currents and increased
ideality factors have been attibuted to local Schottky barrier inhomogeneities
and are predicted to be more pronounced with increased doping [59]. Recent
studies have demonstrated a correlation of the reduced barrier heights with
deep-level defect concentration [62]. With the high levels of doping use in
SiGe:C HBTs to optimize high-frequency performance [51] and the presence of
electrically active carbon defects [63], such leakage currents due to barrier
inhomogeneities may play a role in non-ideal cryogenic base transport. These
phenomena will be the focus of our future studies.

##  V Conclusion

We have reported a study of quasiballistic transport in SiGe HBTs using an
exact, semi-analytic solution to the Boltzmann equation based on an asymptotic
expansion method. We find that the predicted IV characteristics including
quasiballistic transport are inconsistent with experiment. Specifically, our
calculations including quasiballistic transport predict collector currents
that are orders of magnitude smaller than measurements for a given base-
emitter voltage and an unaltered temperature dependence of transconductance
relative to the ideal value. We hypothesize that the origin of non-ideal
cryogenic I-V characteristics is linked to potential barrier inhomogeneities
that have been studied in Schottky diodes. Our work advances the understanding
of electronic transport in cryogenic SiGe HBTs that is required to further
improve their microwave noise performance.

###### Acknowledgements.

The authors thank Mark Lundstrom, J.P. Peraud, and Nicolas Hadjiconstantinou
for useful discussions. This work was supported by NSF Award Number 1911926.

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