# Understanding disorder in Silicon quantum computing platforms: Scattering
mechanisms in Si/SiGe quantum wells

Yi Huang  Sankar Das Sarma  Condensed Matter Theory Center and Joint Quantum
Institute, Department of Physics, University of Maryland, College Park,
Maryland 20742, USA

###### Abstract

Motivated by recent experiments on Si/SiGe quantum wells with a co-design of
high electron mobility and large valley splitting [B. Paquelet Wuetz, et al.,
Nature Communications 14, 1385 (2023); D. D. Esposti, et al.,
arXiv:2309.02832], suitable for a Si-based spin qubit quantum computing
platform, we examine the role of disorder by theoretically calculating
mobility and quantum mobility from various scattering mechanisms and their
dependence on the electron density. At low electron densities $n_{e}<4\times
10^{11}$ cm-2, we find that mobility is limited by remote Coulomb impurities
in the capping layer, whereas interface roughness becomes the significant
limiting factor at higher densities. We also find that alloy disorder
scattering is not a limiting mechanism in the reported high-mobility
structures. We estimate the critical density of the disorder-driven low-
density metal-insulator transition using the Anderson-Ioffe-Regel localization
criterion and qualitatively explain the breakdown of the Boltzmann-Born theory
at low densities. We also estimate the critical density by considering
inhomogeneous density fluctuations induced by long-range Coulomb disorder in
the system, and find a larger critical density compared to the one obtained
from the Anderson-Ioffe-Regel criterion. For quantum mobility, our calculation
suggests that remote and distant background impurities are likely the limiting
scattering sources across all density. Future measurements of quantum mobility
should provide more information on the distribution of background impurities
inside the SiGe barriers. Moreover, we extend our theoretical analysis to the
effect of quantum degeneracy on transport properties and predict the mobility
and the critical density for the metal-insulator transition in spin-polarized
high-mobility structures under an external parallel magnetic field.

##  I Introduction

For spin qubits in gate-defined silicon quantum dots, it is important to
minimize disorder potential fluctuations to better control charging energies
and tunneling between quantum dots, which requires high mobility of the hosted
two-dimensional electron gas (2DEG) Vandersypen  _et al._ (2017); Scappucci
_et al._ (2021); Burkard  _et al._ (2023). Moreover, background charge noise,
necessarily associated with unintentional random impurities in the
environment, is the primary mechanism causing charge noise induced decoherence
in semiconductor qubits, and therefore, a comprehensive understanding of the
background impurity disorder is essential for good qubit performance Hu and
Das Sarma (2006); Culcer  _et al._ (2009); Ahn  _et al._ (2021); Burkard  _et
al._ (2023). Additionally, a large valley splitting energy is important to
limit leakage from the computational Hilbert space in the lowest conduction
band, which is crucial for maintaining high-fidelity qubit operations Culcer
_et al._ (2010); Buterakos and Das Sarma (2021); Zwanenburg  _et al._ (2013);
Huang and Hu (2014); Tagliaferri  _et al._ (2018); Zwerver  _et al._ (2023);
Burkard  _et al._ (2023). Recent advancements in isotopically purified 28
Si/SiGe quantum wells (QWs) at a low temperature of 1.7 K have shown promising
improvements in channel static disorder, as indicated by an enhanced mobility
$3\times 10^{5}$ cm2/Vs and a smaller critical density of metal-insulator
transition $n_{c}=7\times 10^{10}$ cm-2, while maintaining a relatively high
mean valley splitting $0.2$ meV Paquelet Wuetz  _et al._ (2023); Esposti  _et
al._ (2023). (Isotopic purification suppresses nuclear spin noise for
semiconductor based spin qubits in the Si platform, providing long spin
coherence times and enabling high-fidelity gate operations Witzel  _et al._
(2010).) In comparison, previous experimental reports of large valley
splitting $>0.2$ meV in Si/SiGe QWs showed 5 times lower mobility $<6\times
10^{4}$ cm2/Vs Borselli  _et al._ (2011); Hollmann  _et al._ (2020); McJunkin
_et al._ (2022a). On the other hand, high mobility of $6.5\times 10^{5}$
cm2/Vs was reported in conventional Si/SiGe heterostructures but with low
valley splitting $35-70$ $\mu$eVMi  _et al._ (2017). Therefore, the co-design
of high electron mobility and large valley splitting calls for reinvestigating
the scattering sources inside the Si/SiGe QWs reported in Refs. Paquelet Wuetz
_et al._ (2023); Esposti  _et al._ (2023). This is the goal of our theoretical
work, particularly in the context of the importance of the Si/Ge quantum
computing platform.

In this study, we employ the Boltzmann-Born transport theory Ando  _et al._
(1982) to theoretically investigate both transport and quantum mobility ($\mu$
and $\mu_{q}$). Our focus is on quantifying the effects of various scattering
mechanisms, namely remote impurities at the semiconductor-oxide interface
(RI), background impurities (BI), interface roughness (IR), and alloy disorder
(AD). By analyzing experimental mobility data from 5 nm and 7 nm Si/SiGe
quantum wells reported in Refs. Paquelet Wuetz  _et al._ (2023); Esposti  _et
al._ (2023), we find that RI predominantly limits $\mu$ at electron densities
below $4\times 10^{11}$ cm-2, while IR becomes the limiting factor at higher
densities ranging from $4-6\times 10^{11}$ cm-2. Despite limited experimental
data on quantum mobility ($\mu_{q}=3.0\pm 0.5\times 10^{4}$ cm2/Vs at the
highest density $6\times 10^{11}$ cm-2 pri ), our theoretical models indicate
that $\mu_{q}$ is likely constrained by RI and distant BI scattering across
all electron densities. Since $\mu_{q}$ is much more sensitive to RI and
distant BI scattering compared to $\mu$ Das Sarma and Stern (1985), we need
more experimental data on $\mu_{q}$ across a wider range of densities to
better understand the distribution for BI in SiGe barriers. We also show that
AD is quantitatively irrelevant in all experimental density ranges for these
high-mobility structures. We address the limitations of the Boltzmann-Born
theory at low electron densities $n_{e}\lesssim 2\times 10^{11}$ cm-2 with
considerable disagreement between the theory and experimental data, although
our theory explains the higher density data quantitatively. A possible
explanation for this discrepancy at low densities is that the 2DEG is broken
into inhomogeneous electron puddles separated by long-range Coulomb disorder
potential barriers, so that both the perturbative Born approximation and
linear screening are no longer applicable. The system eventually undergoes a
metal-insulator transition (MIT) at a critical density $n_{e}=n_{c}$, where
transport in the low-density $n_{e}\geq n_{c}$ region should be described by
percolation through charged puddles Last and Thouless (1971); Kirkpatrick
(1973); Shklovskii and Efros (1975); Sarma and Hwang (2005); Tracy  _et al._
(2009); Manfra  _et al._ (2007); Das Sarma  _et al._ (2013, 2014). This MIT
is, in fact, a crossover (happening sharply over a narrow density range around
$n_{c}$ because of the rapid failure of screening of the long-range Coulomb
disorder) and not a quantum phase transition Sarma and Hwang (2005); Tracy
_et al._ (2009). We estimate $n_{c}$ in the experimental systems Paquelet
Wuetz  _et al._ (2023); Esposti  _et al._ (2023) by using the Anderson-Ioffe-
Regel (AIR) criterion Anderson (1958); Ioffe and Regel (1960), although the
failure of the Boltzmann-Born theory at low densities indicates that the
critical density estimated from the AIR criterion should be smaller than the
actual critical density (percolation threshold) observed in experiments and
should be used as a lower bound estimate of the percolation threshold. We also
estimate $n_{c}$ by considering inhomogeneous density fluctuations induced by
long-range Coulomb disorder in the system Efros (1988); Pikus and Efros
(1989), and find a larger $n_{c}$ compared to the one obtained from the AIR
criterion. Moreover, we extend our theoretical analysis to the effect of
quantum degeneracy on transport properties and predict the mobility and the
critical density for the metal-insulator transition in spin-polarized high-
mobility structures, which can be tested in future experiments by applying an
external parallel magnetic field.

The Si/SiGe heterostructures reported in Refs. Paquelet Wuetz  _et al._
(2023); Esposti  _et al._ (2023) consist of a strained Si QW of width $w$
surrounded by a Si0.69Ge0.31 upper barrier of thickness $d=30$ nm and a thick
strained relaxed Si0.69Ge0.31 bottom barrier of several $\mu$m. The upper
barrier is passivated with a 1 nm Si capping layer. A metallic top gate is
separated from the capping layer by a $d_{o}=10$ nm oxide dielectric spacer.
The electron density $n_{e}$ is tuned by the gate up to $6\times 10^{11}$
cm-2. The bottom barrier is placed on top of a step-graded Si1-xGex strain-
relax buffer layer, which has a decreasing Ge concentration $x$ until it
reaches the Si substrate with $x=0$. A schematic illustration of the Si/SiGe
heterostructure is shown in Fig. 1. In our calculations, we model the possible
disorder sources as follows. There are uniform background charged impurities
outside (inside) the QW with 3D concentration $N_{1}$ ($N_{2}$). The remote
charged impurities at the semiconductor-oxide interface have a 2D
concentration $n_{r}$. Because the top gate is relatively close to the
semiconductor-oxide interface, we take into account the gate screening when we
calculate the remote and background charged impurity scattering. The interface
roughness between the Si QW and the Si0.69Ge0.31 barrier is characterized by a
typical height $\Delta$ and a lateral size $\Lambda$. The conduction band
offset from the twofold valley degenerate conduction band $\Delta_{2}$ in the
strained Si to the sixfold valley degenerate conduction band $\Delta_{6}$ in
the relaxed Si0.69Ge0.31 is taken to be $V_{0}=180$ meV Stern and Laux (1992);
Burkard  _et al._ (2023). The 2DEG wave function penetrates the barriers and
scatters from the alloy disorder. Scattering mechanisms considered in our
theory are the only low-temperature scattering mechanisms operational in high
quality SiGe 2D structures, where the crystal quality is good enough to rule
out short-range scattering by atomic point defects and vacancies cry , but
random charged impurities are invariably present in the system along with
interface roughness and alloy disorder. Since our interest is in understanding
the low-temperature mobility, we neglect all phonon scattering and restrict
our calculations to $T=0$. We believe that our comprehensive transport theory
is quantitatively accurate at high carrier densities, and poorer at lower
densities although it should remain qualitatively (and semiquantitatively)
valid at lower densities above the critical density for the 2D MIT.

The rest of this paper is structured as follows. Section II discusses the
general formalism of the Boltzmann transport theory to calculate the mobility
and quantum mobility. We evaluate the BI and RI scattering rates using the
delta-layer 2DEG approximation in Section III. The infinite potential well
approximation together with the local field correction, applied to RI, BI, and
IR scattering, is discussed in Section IV, while Section V employs a finite
potential well approximation to calculate the AD scattering. Gate-screened
charged impurity scattering is explored in Section VI. In Section VII, we
estimate the critical densities of MIT using the Anderson-Ioffe-Regel (AIR)
criterion and qualitatively explain the breakdown of the Boltzmann-Born theory
at low densities. We mention that the AIR criterion is often referred to as
the Mott-Ioffe-Regel (MIR or IRM) or just the Ioffe-Regel criterion, but we
believe that AIR is a more appropriate terminology here since our criterion is
specific to disorder induced incoherent scattering associated with destructive
interference as envisioned by Anderson rather than the Mott criterion
involving the transport mean free path being equal to lattice spacing, which
plays no role in doped semiconductors of interest to us. We also estimate the
critical density by considering inhomogeneous density fluctuations induced by
long-range Coulomb disorder in the system and compare the result to the AIR
critical density. Section VIII examines the effect of spin/valley degeneracy
on the mobility and the AIR critical density. Finally, we conclude and
summarize in Section IX.

Figure 1: A schematic illustration of the high-mobility Si/SiGe
heterostructure reported in Refs. Paquelet Wuetz  _et al._ (2023); Esposti
_et al._ (2023), where $w$ is the thickness of the strained Si quantum well.
The oxide and the top Si0.69Ge0.31 barrier is separated by a Si capping layer
of thickness 1 nm.

##  II Mobility and quantum mobility

In this section, we apply the Boltzmann transport theory to calculate the
mobility and quantum mobility at $T=0$ Ando  _et al._ (1982); Das Sarma and
Hwang (1999); Hwang and Das Sarma (2005, 2007, 2013a); Das Sarma and Hwang
(2014a); Hu  _et al._ (2015). The transport mobility is given by
$\mu=e\tau/m^{\star}$, and the conductivity is $en_{e}\mu$, where $n_{e}$ is
the 2DEG concentration, $m^{\star}=0.19\,m_{0}$ is the transverse effective
mass of an electron in Si, $m_{0}$ is the free electron mass,

$\displaystyle\frac{1}{\tau}=\frac{4m^{\star}}{\pi\hbar^{3}}\int\limits_{0}^{2k_{F}}\frac{dq}{\sqrt{4k_{F}^{2}-q^{2}}}\quantity(\frac{q}{2k_{F}})^{2}\expectationvalue{\absolutevalue{U(q)}^{2}}\,,$
(1)

is the transport scattering rate, and $U(q)$ is the screened potential of a
given scattering source. $k_{F}=\sqrt{4\pi n_{e}/g}$ is the Fermi wave vector.
$g=g_{s}g_{v}$ is the total degeneracy of electrons, where $g_{s}$ is the spin
degeneracy and $g_{v}$ is the valley degeneracy. The quantum mobility is given
by $\mu_{q}=e\tau_{q}/m^{\star}$, where the quantum (single particle)
scattering rate reads

$\displaystyle\frac{1}{\tau_{q}}=\frac{2m^{\star}}{\pi\hbar^{3}}\int\limits_{0}^{2k_{F}}\frac{dq}{\sqrt{4k_{F}^{2}-q^{2}}}\,\expectationvalue{\absolutevalue{U(q)}^{2}}.$
(2)

The quantum scattering rate can be indirectly measured through the Dingle
temperature $T_{D}=\hbar/2\pi\tau_{q}$ which characterizes the damping of
Shubnikov-de Haas (SdH) oscillations. In general, $\tau_{q}<\tau$ since the
transport mobility ignores all forward scattering. (For strictly zero range
s-wave scattering the two scattering times are trivially equal, but for long-
range disorder scattering the two could vary by orders of magnitude.) For
charged impurity scattering, the potential correlator averaged over impurity
positions is given by

$\displaystyle\expectationvalue{\absolutevalue{U(q)}^{2}}=\int\limits_{-\infty}^{+\infty}dz\;N(z)U_{1}^{2}(q,z)\,,$
(3)

where $N(z)$ is the 3D concentration of impurities at a distance $z$ from the
center of the 2DEG, and $U_{1}(q,z)$ is the screened Coulomb potential for an
impurity located at $z$:

$\displaystyle U_{1}(q,z)=\frac{2\pi e^{2}}{\kappa
q\epsilon(q)}\int\limits_{-\infty}^{+\infty}dz^{\prime}\;\absolutevalue{\psi(z^{\prime})}^{2}e^{-q\absolutevalue{z-z^{\prime}}}\,,$
(4)

where $\psi(z)$ is the confinement wave function along $z$ direction and the
dielectric function is given by

$\displaystyle\epsilon(q)=1+(q_{\rm TF}/q)F_{c}(qw)[1-G(q)]\,.$ (5)

Here, $q_{TF}=g/a_{B}$ is the Thomas-Fermi screening wave vector,
$a_{B}=\kappa\hbar^{2}/m^{\star}e^{2}$ is the effective Bohr radius,
$\kappa=11.9$ is the dielectric constant of Si, and $w$ is the QW width.
$G(q)=q/(g\sqrt{q^{2}+k_{F}^{2}})$ is the local field correction using the
Hubbard approximation Jonson (1976); Gold (1994) which reflects the suppressed
screening at $q\gtrsim k_{F}$ (with $G=0$ being the RPA, which overestimates
short wavelength screening). The form factor $F_{c}(qw)$ is given by

$F_{c}(qw)=\iint\limits_{-\infty}^{+\infty}dzdz^{\prime}\absolutevalue{\psi(z)}^{2}\absolutevalue{\psi(z^{\prime})}^{2}\exp(-q\absolutevalue{z-z^{\prime}})\,.$
(6)

In Eq. (4) we ignore the screening from the top gate. To include the gate
screening, we should add the Coulomb potential of the image charges and
replace $e^{-q\absolutevalue{z-z^{\prime}}}$ by
$\quantity(e^{-q\absolutevalue{z-z^{\prime}}}-e^{-q\absolutevalue{2d_{g}-z-z^{\prime}}})$,
where $d_{g}=d+d_{o}+w/2$ is the distance from the gate to the center of the
2DEG.

##  III delta layer 2DEG approximation

For simplicity, assuming that the gate is far away $d_{g}\gg d$ and the
wavefunction perpendicular to the quantum well has a delta function profile
$\absolutevalue{\psi(z)}^{2}=\delta(z)$, the form factors reduce to identities
and $U_{1}(q,z)=\frac{2\pi e^{2}}{\kappa
q\epsilon(q)}e^{-q\absolutevalue{z}}$. In this section, we disregard the local
field correction, thus making the dielectric function the RPA form,
$\epsilon(q)=1+q_{TF}/q$. We will address the effect of form factors and local
field correction in Section IV, and the effect of gate screening in Section
VI. We use a 2-impurity model to calculate the scattering rate following Refs.
Hwang and Das Sarma (2013b); Das Sarma and Hwang (2014a). That is, we assume
that there are two delta layers of impurities. One layer is located at a
distance $d$ from the 2DEG, which represents the remote charged impurities in
the capping layer. The other layer is at $z=0$, which represents oxygen-
related background charges in the QW Mi  _et al._ (2015) or an effective
description of other short-range scattering sources such as the interface
roughness Huang  _et al._ (2012); Das Sarma and Hwang (2014b). The impurity
distribution is given by

$\displaystyle N(z)=n_{r}\delta(z-d)+n_{2}\delta(z).$ (7)

The corresponding transport scattering rate
$\tau^{-1}=\tau_{\mathrm{RI}}^{-1}+\tau_{\mathrm{S}}^{-1}$ reads Hwang and Das
Sarma (2013a, b); Das Sarma and Hwang (2014a)

$\displaystyle\frac{1}{\tau_{\mathrm{RI}}}=n_{r}\frac{2\pi\hbar}{m^{\star}}\quantity(\frac{2}{g})^{2}f\quantity(2k_{F}d,\frac{q_{TF}}{2k_{F}}),$
(8)
$\displaystyle\frac{1}{\tau_{\mathrm{S}}}=n_{2}\frac{2\pi\hbar}{m^{\star}}\quantity(\frac{2}{g})^{2}f\quantity(0,\frac{q_{TF}}{2k_{F}}).$
(9)

The quantum scattering rate
$\tau_{q}^{-1}=\tau_{q\mathrm{RI}}^{-1}+\tau_{q\mathrm{S}}^{-1}$ reads

$\displaystyle\frac{1}{\tau_{q\mathrm{RI}}}=n_{r}\frac{2\pi\hbar}{m^{\star}}\quantity(\frac{2}{g})^{2}f_{q}\quantity(2k_{F}d,\frac{q_{TF}}{2k_{F}}),$
(10)
$\displaystyle\frac{1}{\tau_{q\mathrm{S}}}=n_{2}\frac{2\pi\hbar}{m^{\star}}\quantity(\frac{2}{g})^{2}f_{q}\quantity(0,\frac{q_{TF}}{2k_{F}}),$
(11)

and the dimensionless functions $f(a,s)$ and $f_{q}(a,s)$ are defined as

$\displaystyle
f(a,s)=\int_{0}^{1}s^{2}\frac{2x^{2}e^{-2ax}\,dx}{\sqrt{1-x^{2}}(x+s)^{2}},$
(12) $\displaystyle
f_{q}(a,s)=\int_{0}^{1}s^{2}\frac{e^{-2ax}\,dx}{\sqrt{1-x^{2}}(x+s)^{2}}.$
(13)

In the density range

$\displaystyle\frac{g}{16\pi d^{2}}\ll n_{e}\ll\frac{g^{3}}{16\pi a_{B}^{2}},$
(14)

the limits $2k_{F}d\gg 1$ and $q_{TF}/2k_{F}\gg 1$ are satisfied and we have
$f(a,s)=(2a^{3})^{-1}$ and $f_{q}(a,s)=(2a)^{-1}$ RI_  which lead to the well-
known results for RI scattering Price (1981); Gold (1988); Monroe  _et al._
(1993); Dmitriev  _et al._ (2012)

$\displaystyle\frac{1}{\tau_{\mathrm{RI}}}=n_{r}\quantity(\frac{2}{g})^{2}\frac{\pi\hbar}{8m^{\star}(k_{F}d)^{3}},$
(15)
$\displaystyle\frac{1}{\tau_{q\mathrm{RI}}}=n_{r}\quantity(\frac{2}{g})^{2}\frac{\pi\hbar}{2m^{\star}k_{F}d}.$
(16)

and the corresponding ratio is given by

$\displaystyle\quantity(\frac{\tau}{\tau_{q}})_{\mathrm{RI}}=(2k_{F}d)^{2}\gg
1.$ (17)

The large difference between $\tau$ and $\tau_{q}$ reflects the diffusive
character of the electron dynamics on the Fermi surface scattered from remote
impurities Das Sarma and Stern (1985). The physical meaning of $\tau/\tau_{q}$
is the typical number of small-momentum forward scattering events needed to
change the direction of momentum by an angle of order $\pi$ Dmitriev  _et al._
(2012). Next, we calculate the short-range scattering rate
$\tau_{\mathrm{S}}^{-1}$ and $\tau_{q\mathrm{S}}^{-1}$ contributed from a
delta layer of impurities at $z=0$. The related integrals $f(0,s)$ and
$f_{q}(0,s)$ at $s>1$ can be calculated analytically:

$\displaystyle
f(0,s)=s^{2}\pi+\frac{2s^{3}}{1-s^{2}}+\frac{2s^{3}\left(2-s^{2}\right)\sec^{-1}(s)}{\left(s^{2}-1\right)^{3/2}},$
(18) $\displaystyle
f_{q}(0,s)=\frac{s}{1-s^{2}}+\frac{s^{3}\,\mathrm{sec}^{-1}(s)}{\left(s^{2}-1\right)^{3/2}}.$
(19)

In the density range $n_{e}\ll g^{3}/16\pi a_{B}^{2}$, the limit
$s=q_{TF}/2k_{F}\gg 1$ is satisfied and $f(0,s)=f_{q}(0,s)=\frac{\pi}{2}$, so
that $\tau_{\mathrm{S}}^{-1}$ and $\tau_{q\mathrm{S}}^{-1}$ are given by Hwang
and Das Sarma (2013b, a)

$\displaystyle\frac{1}{\tau_{\mathrm{S}}}=\frac{1}{\tau_{q\mathrm{S}}}=n_{2}\frac{\pi^{2}\hbar}{m^{\star}}\quantity(\frac{2}{g})^{2},$
(20)

which is independent of the electron concentration. The ratio
$\tau/\tau_{q}=1$ reflects that short-range impurities are efficient in
backscattering the electrons – this is the situation in regular 3D metals
where $\tau$ and $\tau_{q}$ are basically the same.

Figure 2: Mobility ($\mu$) versus electron density ($n_{e}$) for quantum wells
with widths of 5 nm (red) and 7 nm (blue). The experimental data are depicted
by red and blue curves, with corresponding error bars shown as shaded areas.
The solid black curves represent the theoretical transport mobility, while the
dashed black curves denote the quantum mobility, both calculated using the
delta-layer approximation for the 2DEG without local field correction.
Subfigures (a) and (b) incorporate remote and short-range impurity scattering
[cf. Eqs. (8), (9), and (10)], whereas (c) and (d) consider background and
short-range impurity scattering [cf. Eqs. (9), (22), and (23)]. Fitting
parameters: (a) $n_{r}=1.9\times 10^{12}$ cm-2 and $n_{2}=2.8\times 10^{9}$
cm-2; (b) $n_{r}=1.7\times 10^{12}$ cm-2 and $n_{2}=1.8\times 10^{9}$ cm-2;
(c) $n_{2}=2.1\times 10^{9}$ cm-2 and $N_{1}=11\times 10^{15}$ cm-3; (d)
$n_{2}=1.4\times 10^{9}$ cm-2 and $N_{1}=12\times 10^{15}$ cm-3.

In the other 2-impurity model, we replace the delta-layer remote impurities by
uniform background charged impurities outside the well with a 3D concentration
$N_{1}$. The impurity distribution is given by

$\displaystyle N(z)=N_{1}\Theta(\absolutevalue{z}-w/2)+n_{2}\delta(z),$ (21)

The corresponding scattering rates due to uniform background charged
impurities are given by

$\displaystyle\frac{1}{\tau_{\mathrm{BI}}}=N_{1}\frac{2\pi\hbar}{m^{\star}}\quantity(\frac{2}{g})^{2}\int\limits_{\absolutevalue{z}>w/2}f\quantity(2k_{F}z,\frac{q_{TF}}{2k_{F}})dz,$
(22)
$\displaystyle\frac{1}{\tau_{q\mathrm{BI}}}=N_{1}\frac{2\pi\hbar}{m^{\star}}\quantity(\frac{2}{g})^{2}\int\limits_{\absolutevalue{z}>w/2}f_{q}\quantity(2k_{F}z,\frac{q_{TF}}{2k_{F}})dz.$
(23)

In the electron density range

$\displaystyle\frac{g}{16\pi w^{2}}\ll n_{e}\ll\frac{g^{3}}{16\pi a_{B}^{2}},$
(24)

the limits $2k_{F}w\gg 1$ and $q_{TF}/2k_{F}\gg 1$ are satisfied and we get
the analytical expressions of the BI scattering rates

$\displaystyle\frac{1}{\tau_{\mathrm{BI}}}=N_{1}w\quantity(\frac{2}{g})^{2}\frac{\pi\hbar}{2m^{\star}(k_{F}w)^{3}},$
(25)
$\displaystyle\frac{1}{\tau_{q\mathrm{BI}}}=N_{1}w\quantity(\frac{2}{g})^{2}\frac{\pi\hbar\ln(2d_{s}/w)}{m^{\star}k_{F}w},$
(26)

where $d_{s}$ is the cutoff distance of the BI distribution so that impurities
at $\absolutevalue{z}>d_{s}$ can be ignored. Otherwise,
$\tau_{q\mathrm{BI}}^{-1}$ diverges logarithmically due to the contribution of
small-momentum scattering from the long distance. We choose $d_{s}=d_{g}/2$ as
half the distance from the gate to the center of the 2DEG, considering that
charged impurities close to the gate should be effectively screened Sammon
_et al._ (2018). We will discuss this choice of $d_{s}$ and the gate screening
effect in more detail in Section VI.

In the density range described by Eqs. (14) and (24), we see from Eqs. (15)
and (25) that $\tau_{\mathrm{RI}}^{-1}$ and $\tau_{\mathrm{BI}}^{-1}$ are
proportional to $k_{F}^{-3}$, which leads to a mobility proportional to
$n_{e}^{3/2}$. On the other hand, $\tau_{\mathrm{S}}^{-1}$ in Eq. (20) does
not depend on $k_{F}$, so it results in a constant mobility that does not
depend on $n_{e}$. As a result, the total mobility described by the 2-impurity
model $\tau^{-1}=\tau_{\mathrm{RI}}^{-1}+\tau_{\mathrm{S}}^{-1}$ or
$\tau^{-1}=\tau_{\mathrm{BI}}^{-1}+\tau_{\mathrm{S}}^{-1}$ should increase as
$n_{e}^{3/2}$ at low density and eventually saturate at high density. We can
compare the density range $n_{e}=1-6\times 10^{11}$ cm-2 in the high mobility
structures reported in Refs. Paquelet Wuetz  _et al._ (2023); Esposti  _et
al._ (2023) with the density ranges written in Eqs. (14) and (24). We use the
spin/valley degeneracy $g_{s}=2$ and $g_{v}=2$ for the high mobility
structures reported in Refs. Paquelet Wuetz  _et al._ (2023); Esposti  _et
al._ (2023). In the density range $n_{e}=1-6\times 10^{11}$ cm-2 the Fermi
energy calculated using $g=4$ is $E_{F}=0.6-3.8$ meV which is much higher than
the valley splitting $0.2$ meV reported in Refs. Paquelet Wuetz  _et al._
(2023); Esposti  _et al._ (2023), so our choice of $g=4$ is correct. In
Section VIII, we will discuss the effects of quantum degeneracy on the
transport properties in more detail. Using $g=4$, we find that the
experimental density range is well within the range described in Eq. (14)
where the upper bound is $g^{3}/16\pi a_{B}^{2}\approx 10^{13}$ cm-2 and the
lower bound is $g/16\pi d^{2}\approx 10^{10}$ cm-2. Therefore, the asymptotic
expressions for RI scattering Eqs. (15) and (16) should be a good
approximation. On the other hand, the lower bound in Eq. (24) is $g/16\pi
w^{2}\approx 2.4\times 10^{11}$ cm-2, which means that the asymptotic
expressions for BI scattering rates Eqs. (25) and (26) are good only when
$n_{e}\gg 2.4\times 10^{11}$ cm-2. At low densities $n_{e}\ll 2.4\times
10^{11}$ cm-2, we have

$\displaystyle\frac{1}{\tau_{\mathrm{BI}}}=\frac{1}{\tau_{q\mathrm{BI}}}=N_{1}(2d_{s}-w)\frac{\pi^{2}\hbar}{m^{\star}}\quantity(\frac{2}{g})^{2},$
(27)

and the background impurity scattering behaves similarly to short-range
impurity scattering with an effective 2D impurity concentration
$N_{1}(2d_{s}-w)$. Hence, as the density is reduced, $\tau_{\mathrm{BI}}^{-1}$
and $\tau_{q\mathrm{BI}}^{-1}$ transition from a long-range behavior, as
expressed in Eqs. (25-26), to a short-range behavior, as expressed in Eq.
(27). The crossover electron density is at $n_{e}=2.4\times 10^{11}$ cm-2.

Fig. 2 shows the results of the best-fit mobility curves (solid black) and the
corresponding prediction of quantum mobility (dashed black) by performing the
numerical integrals of Eqs. (8), (9) and (22). The experimental data from
Refs. Paquelet Wuetz  _et al._ (2023); Esposti  _et al._ (2023), represented
by blue and red curves with shaded areas indicating error bars, are also
included for comparison. The results of the combination of remote and short-
range impurity scattering are shown in Figs. 2 (a) and (b), while the results
of the combination of background and short-range impurity scattering are shown
in Figs. 2 (c) and (d). The fitting impurity densities are $n_{r}\approx
2\times 10^{12}$ cm-2, $n_{2}\approx 1.4-2.8\times 10^{9}$ cm-2 and
$N_{1}\approx 10\times 10^{15}$ cm-3, in reasonable agreement with the numbers
reported in Refs. Stern and Laux (1992); Tanner  _et al._ (2007); Gold (2011);
Mi  _et al._ (2015); Hwang and Das Sarma (2013a); Laroche  _et al._ (2015);
Ahn and Das Sarma (2022a). In both 2-impurity models, short-range impurity
scattering is the limiting factor for transport mobility at high densities
$n_{e}=4-6\times 10^{11}$ cm-2. The dominant scattering mechanism at lower
densities is RI or BI, depending on the choice of the impurity model. We see
that the first 2-impurity model
($\tau^{-1}=\tau_{\mathrm{RI}}^{-1}+\tau_{\mathrm{S}}^{-1}$) fits the mobility
data at intermediate to high electron density $n_{e}=2.5-6\times 10^{11}$ cm-2
within the experimental uncertainty. At the highest density $n_{e}=6\times
10^{11}$ cm-2 for both quantum wells, the predicted quantum mobility is around
$1.0\times 10^{4}$ cm2/Vs, while the measurements of SdH oscillations give a
larger value $(3.0\pm 0.5)\times 10^{4}$ cm2/Vs pri . On the other hand, the
other 2-impurity model
($\tau^{-1}=\tau_{\mathrm{BI}}^{-1}+\tau_{\mathrm{S}}^{-1}$) fits the
experimental mobility data in a narrower density range $n_{e}=3-6\times
10^{11}$ cm-2, with the predicted quantum mobility around $10\times 10^{4}$
cm2/Vs at the highest density $n_{e}=6\times 10^{11}$ cm-2, which is a factor
of 3 larger than the experimental value pri . Since the typical momentum
transfer of RI scattering is much smaller than that of BI scattering, the
ratio $(\mu/\mu_{q})_{\mathrm{RI}}\gg(\mu/\mu_{q})_{\mathrm{BI}}$ Das Sarma
and Stern (1985), which explains a factor of 10 difference in the predicted
quantum mobility shown in Fig. 2. In Section VI, we will explore how including
gate screening effects in RI scattering calculations can improve the agreement
with experimental quantum mobility data. However, it is important to note that
the experimental accuracy for quantum mobility data is generally less reliable
than for transport mobility, largely due to the indirect method of extracting
quantum mobility through the Dingle temperature Das Sarma and Stern (1985).
Given these considerations, a semi-quantitative agreement between our
theoretical predictions and experimental data should be regarded as
satisfactory. At low densities $n_{e}<2\times 10^{11}$ cm-2, the theoretical
mobility curves deviate and become higher than the experimental data. We
should emphasize that this deviation at low densities does not arise from any
new scattering mechanism ignored in the theory. The dominant scattering
mechanism at low densities $n_{e}<4\times 10^{11}$ cm-2 is RI if we use the
2-impurity model Eq. (7) [or BI if we use the 2-impurity model Eq. (21)].
Rather, this deviation indicates the systematic failure of the Boltzmann-Born
theory at low densities, a topic we will address in more detail in Section
VII, particularly in the context of the metal-insulator transition.

Figure 3: Mobility ($\mu$) versus electron density ($n_{e}$) for quantum wells
with widths of 5 nm (red) and 7 nm (blue). The experimental data are depicted
by red and blue curves, with corresponding error bars shown as shaded areas.
The solid black curves represent the theoretical transport mobility, while the
dashed black curves denote the quantum mobility, both calculated using the
infinite potential well approximation without the local field correction. (a)
and (b) assume only RI near the capping layer $n_{r}\neq 0$. (c) and (d)
assume only BI outside the quantum well $N_{1}\neq 0$ and no BI inside
$N_{2}=0$, while (e) and (f) assume $N_{1}=0$ and $N_{2}\neq 0$. Impurity
concentrations and roughness parameters for fitting: (a) $n_{r}=3.1\times
10^{12}$ cm-2; (b) $n_{r}=2.6\times 10^{12}$ cm-2; (c) $N_{1}=14\times
10^{15}$ cm-3 and $N_{2}=0$; (d) $N_{1}=12.6\times 10^{15}$ cm-3 and
$N_{2}=0$; (e) $N_{1}=0$ and $N_{2}=10.3\times 10^{15}$ cm-3; (f) $N_{1}=0$
and $N_{2}=5.5\times 10^{15}$ cm-3. Figure 4: Mobility ($\mu$) versus electron
density ($n_{e}$) for quantum wells with widths of 5 nm (red) and 7 nm (blue).
The experimental data are depicted by red and blue curves, with corresponding
error bars shown as shaded areas. The solid black curves represent the
theoretical transport mobility, while the dashed black curves denote the
quantum mobility, both calculated using the infinite potential well
approximation without the local field correction. Subfigures (a) and (b)
incorporate remote impurity and interface roughness scattering, whereas (c)
and (d) consider background impurity and interface roughness scattering.
Impurity concentrations and roughness parameters for fitting: (a)
$n_{r}=2.5\times 10^{12}$ cm-2, $\Delta=4.5$ Å, and $\Lambda=27$ Å; (b)
$n_{r}=2.5\times 10^{12}$ cm-2, $\Delta=8$ Å, and $\Lambda=28$ Å; (c)
$N_{1}=12\times 10^{15}$ cm-3, $N_{2}=0$, $\Delta=5$ Å, and $\Lambda=16$ Å;
(d) $N_{1}=12\times 10^{15}$ cm-3, $N_{2}=0$, $\Delta=7$ Å, and $\Lambda=20$
Å. Figure 5: Mobility ($\mu$) versus electron density ($n_{e}$) for quantum
wells with widths of 5 nm (red) and 7 nm (blue). The experimental data are
depicted by red and blue curves, with corresponding error bars shown as shaded
areas. The solid black curves represent the theoretical transport mobility,
while the dashed black curves denote the quantum mobility, both calculated
using the infinite potential well approximation with the local field
correction. (a) and (b) assume only RI near the capping layer $n_{r}\neq 0$.
(c) and (d) assume only BI outside the quantum well $N_{1}\neq 0$ and no BI
inside $N_{2}=0$, while (e) and (f) assume $N_{1}=0$ and $N_{2}\neq 0$. To fit
the data, the impurity concentrations and roughness parameters are chosen as
(a) $n_{r}=2.5\times 10^{12}$ cm-2; (b) $n_{r}=2.1\times 10^{12}$ cm-2; (c)
$N_{1}=9.7\times 10^{15}$ cm-3 and $N_{2}=0$; (d) $N_{1}=8.7\times 10^{15}$
cm-3 and $N_{2}=0$; (e) $N_{1}=0$ and $N_{2}=7.2\times 10^{15}$ cm-3; (f)
$N_{1}=0$ and $N_{2}=3.8\times 10^{15}$ cm-3. Figure 6: Mobility ($\mu$)
versus electron density ($n_{e}$) for quantum wells with widths of 5 nm (red)
and 7 nm (blue). The experimental data are depicted by red and blue curves,
with corresponding error bars shown as shaded areas. The solid black curves
represent the theoretical transport mobility, while the dashed black curves
denote the quantum mobility, both calculated using the infinite potential well
approximation with the local field correction. Subfigures (a) and (b)
incorporate remote impurity and interface roughness scattering, whereas (c)
and (d) consider background impurity and interface roughness scattering.
Fitting parameters: (a) $n_{r}=2.0\times 10^{12}$ cm-2, $\Delta=3.5$ Å, and
$\Lambda=30$ Å; (b) $n_{r}=2.0\times 10^{12}$ cm-2, $\Delta=6.5$ Å, and
$\Lambda=30$ Å; (c) $N_{1}=8.1\times 10^{15}$ cm-3, $N_{2}=0$, $\Delta=3$ Å,
and $\Lambda=22$ Å; (d) $N_{1}=8.1\times 10^{15}$ cm-3, $N_{2}=0$, $\Delta=6$
Å, and $\Lambda=22$ Å.

##  IV Infinite potential well

In this section, we use an infinite potential well approximation in order to
address the quantum well confinement better, where the 2DEG wave function is
given by $\psi(z)=\sqrt{2/w}\cos(\pi z/w)\Theta(w/2-\absolutevalue{z})$ Gold
(2010); Hwang and Das Sarma (2013b). The justification of the infinite
potential well approximation will be discussed at the end of this section.
Substituting this wavefunction into Eq. (4), we obtain the single-impurity
Coulomb potential

$\displaystyle{U}_{1}(q,z)=\frac{2\pi e^{2}}{\kappa
q\epsilon(q)}\begin{cases}F_{0}(qw)e^{-q\absolutevalue{z}},&\absolutevalue{z}>w/2\,,\\\
G_{0}(q,z),&\absolutevalue{z}<w/2\,,\end{cases}$ (28)

where

$\displaystyle F_{0}(x)=\frac{8\pi^{2}}{x(x^{2}+4\pi^{2})}\sinh(x/2)\,,$ (29)

and

$\displaystyle G_{0}(q,z)$
$\displaystyle=\frac{8\pi^{2}(1-e^{-qw/2}\cosh{qz})}{qw(4\pi^{2}+q^{2}w^{2})}$
$\displaystyle+\frac{2qw[1+\cos(2\pi z/w)]}{4\pi^{2}+q^{2}w^{2}}\,.$ (30)

Both $F_{0}(qw)$ and $G_{0}(q,z)$ monotonically decrease as a function of $q$
and weaken the scattering potential at large momentum $q$. The screening form
factor $F_{c}(qw)$ in Eq. (6) reads

$\displaystyle
F_{c}(x)=\frac{20\pi^{2}x^{3}+3x^{5}-32\pi^{4}(1-e^{-x}-x)}{x^{2}(4\pi^{2}+x^{2})^{2}}.$
(31)

This form factor enters the dielectric function $\epsilon(q)$ and weakens the
screening at large momentum $q$.

First, we calculate the charged impurity scattering rates separately and see
how well they can interpret the data individually. The 3D impurity
concentration is given by

$\displaystyle N(z)$ $\displaystyle=n_{r}\delta(z-d-w/2)+$
$\displaystyle+N_{1}\Theta(\absolutevalue{z}-w/2)+N_{2}\Theta(w/2-\absolutevalue{z}),$
(32)

where $n_{r}$ is the RI concentration in the capping layer, $N_{1}$ ($N_{2}$)
is the BI concentration outside (inside) the quantum well. The theoretical
results of mobility and quantum mobility are shown in Fig. 3, based on the
assumption that only one term in Eq. (32) is nonzero. Figs. 3 (a) and (b) show
the results of $n_{r}\neq 0$ and $N_{1}=N_{2}=0$. This RI-dominant mobility
can fit the data at intermediate densities $n_{e}=2-3\times 10^{11}$ cm-2
within the margin of experimental uncertainty. The predicted quantum mobility
is around $1.0\times 10^{4}$ cm2/Vs at the highest density $n_{e}=6\times
10^{11}$ cm-2. Figs. 3 (c) and (d) show the results of $N_{1}\neq 0$ and
$N_{2}=n_{r}=0$, which fit the data at intermediate densities
$n_{e}=2.5-5\times 10^{11}$ cm-2. The predicted quantum mobility is around
$10\times 10^{4}$ cm2/Vs at the highest density $n_{e}=6\times 10^{11}$ cm-2.
Figs. 3 (e) and (f) show the results of $N_{2}\neq 0$ and $N_{1}=n_{r}=0$,
which fit the data at high densities $n_{e}=4-6\times 10^{11}$ cm-2. The
corresponding ratio $\tau/\tau_{q}\approx 1$ is expected, as the background
impurities inside the QW should behave similarly to the short-range impurities
because of strong screening by the electrons. In general, switching the
dominant impurity distribution from RI ($n_{r}\neq 0$) to BI ($N_{1}\neq 0$)
and finally to a different BI distribution ($N_{2}\neq 0$) results in a
progressive flattening of the mobility $\mu(n_{e})$ and a decrease in the
ratio $\mu/\mu_{q}$ towards unity.

In order to better fit the mobility data, we use interface roughness as an
additional scattering mechanism that limits mobility at high density Gold
(1988); Feenstra  _et al._ (1995); Penner  _et al._ (1998); Das Sarma and
Hwang (2004); Tracy  _et al._ (2009); Gold (2010); Das Sarma and Hwang
(2014b). Using the correlator of local well width variations
$\expectationvalue{\Delta(\mathbf{r})\Delta(\mathbf{r}^{\prime})}=\Delta^{2}\exp(-\absolutevalue{\mathbf{r}-\mathbf{r}^{\prime}}^{2}/\Lambda^{2})$,
where $\Delta$ is the typical roughness height and $\Lambda$ is the roughness
lateral size, the scattering potential due to interface roughness is given by

$\displaystyle\expectationvalue{\absolutevalue{U_{\rm
IR}(q)}^{2}}=\frac{\pi}{\epsilon^{2}(q)}\Delta^{2}\Lambda^{2}\quantity(\partialderivative{E}{w})^{2}e^{-q^{2}\Lambda^{2}/4}\,,$
(33)

where $E$ is the ground state energy of the potential well along $z$
direction. For an infinite potential well, the ground state energy
$E=E_{0}=\hbar^{2}\pi^{2}/2m^{\star}_{z}w^{2}$, and $\partial E/\partial
w=2E_{0}/w$. Here $m_{z}^{\star}=0.92m_{0}$ is the longitudinal effective mass
of an electron in Si. Using Eqs. (33), (1), and (2), we obtain the transport
relaxation rate $\tau_{\mathrm{IR}}^{-1}$ and quantum rate
$\tau_{q,\mathrm{IR}}^{-1}$ due to interface roughness scattering. At low
electron densities $n_{e}\ll g/4\pi\Lambda^{2}$ where $k_{F}\Lambda\ll 1$ and
$q_{TF}/2k_{F}\gg 1$ are satisfied, we obtain

$\displaystyle\tau_{\mathrm{IR}}^{-1}=\frac{1}{\tau_{0}}\frac{3\pi}{8}\quantity(\frac{2k_{F}}{q_{TF}})^{2},$
(34)
$\displaystyle\tau_{q\mathrm{IR}}^{-1}=\frac{1}{\tau_{0}}\frac{\pi}{4}\quantity(\frac{2k_{F}}{q_{TF}})^{2},$
(35)

where $\tau_{0}^{-1}$ is given by

$\displaystyle\tau_{0}^{-1}=\frac{2\pi^{4}\hbar}{m^{\star}}\quantity(\frac{m^{\star}}{m^{\star}_{z}})^{2}\frac{\Delta^{2}\Lambda^{2}}{w^{6}}.$
(36)

We find that both $\mu_{\mathrm{IR}}$ and $\mu_{q\mathrm{IR}}$ are
proportional to $n_{e}^{-1}$ at low electron densities $n_{e}\ll
g/4\pi\Lambda^{2}$. In the high-density range $g/4\pi\Lambda^{2}\ll n_{e}\ll
g^{3}/16\pi a_{B}^{2}$, we obtain

$\displaystyle\tau_{\mathrm{IR}}^{-1}=\frac{1}{\tau_{0}}\frac{3\sqrt{\pi}}{4}\quantity(\frac{2k_{F}}{q_{TF}})^{2}(k_{F}\Lambda)^{-5},$
(37)
$\displaystyle\tau_{q\mathrm{IR}}^{-1}=\frac{1}{\tau_{0}}\frac{\sqrt{\pi}}{4}\quantity(\frac{2k_{F}}{q_{TF}})^{2}(k_{F}\Lambda)^{-3},$
(38)

where we find $\mu_{\mathrm{IR}}\propto n_{e}^{3/2}$ and
$\mu_{q\mathrm{IR}}\propto n_{e}^{1/2}$ IR_ , similar to the long-range remote
impurity scattering results [cf. Eqs. (15) and (16)]. The crossover of the
$\mu_{\mathrm{IR}}\propto n_{e}^{-1}$ to the $\mu_{\mathrm{IR}}\propto
n_{e}^{3/2}$ behavior is at $n_{e}=g/4\pi\Lambda^{2}$, where
$\mu_{\mathrm{IR}}$ reaches its minimum

$\displaystyle\mu_{\mathrm{IR},\min}\approx\frac{e\tau_{0}}{m^{\star}}\frac{q_{TF}^{2}\Lambda^{2}}{4}=\frac{e}{8\pi^{4}\hbar}\quantity(\frac{m^{\star}_{z}}{m^{\star}})^{2}\frac{q_{TF}^{2}w^{6}}{\Delta^{2}}.$
(39)

For the high mobility structures reported in Refs. Esposti  _et al._ (2023)
with $w=7$ nm, we equate the saturation mobility value $\approx 30\times
10^{4}$ cm2/Vs to $\mu_{\mathrm{IR},\min}$, and obtain a rough estimate of
$\Delta\approx 5$ Å. This value is in reasonable agreement with the roughness
height measured from scanning transmission electron microscopy (STEM) images
reported in Refs. Paquelet Wuetz  _et al._ (2023); Esposti  _et al._ (2023).

Figs. 4 (a) and (b) are calculated using the combination of RI and IR
scattering, while (c) and (d) are calculated using the combination of BI and
IR scattering. When we fit the experimental data, we restrict the RI (BI)
concentration to be the same for both the 5 nm and 7 nm quantum wells. This
restriction is based on the fact that the experimental uncertainty of mobility
is relatively small $\sim$10% for different devices with the same structure
design Paquelet Wuetz  _et al._ (2023); Esposti  _et al._ (2023), indicating
that the sources of disorder should be similar from device to device. As a
result, the best-fit charge impurity concentrations are $n_{r}=2.5\times
10^{12}$ cm-2 and $N_{1}=12\times 10^{15}$ cm-3. To explain the difference in
the peak mobility between the 5 nm and 7 nm quantum wells, the roughness
parameters are chosen differently within the range $\Delta=4.5-8.0$ Å and
$\Lambda=16-28$ Å. The roughness parameters that we choose are of the same
order of magnitude as those used in Refs. Das Sarma and Hwang (2004); Tracy
_et al._ (2009); Huang  _et al._ (2012); Das Sarma and Hwang (2014b). From
Fig. 4 we see that the 2-impurity model combining RI and IR scattering fits
the mobility data in a wider density range $n_{e}=1.8-6\times 10^{12}$ cm-2,
while the other 2-impurity model combining BI and IR fits the data in a
narrower density range $n_{e}=3-6\times 10^{12}$ cm-2. Therefore, RI
scattering is more likely the dominant scattering mechanism for transport
mobility at low densities, whereas at high densities the transport mobility is
limited by IR scattering. This behavior is similar to that in Si-SiO2 MOSFET
2DEG systems, where the IR scattering is much larger leading consequently to
lower maximum mobilities.

In the above calculations, we ignore the local field correction
$G(q)=q/(g\sqrt{q^{2}+k_{F}^{2}})$ in the dielectric function Eq. (5). Results
that include the local field correction are presented in Figs. 5 and 6.
Incorporating this correction leads to a reduction in the estimated impurity
concentrations and the ratio of $\tau/\tau_{q}$. This can be understood
because the local field correction weakens the screening at a large momentum
$q\gtrsim k_{F}$, so the disorder is more effective in scattering the
electrons. Given that $\tau$ is more sensitive to large-momentum scattering
compared to $\tau_{q}$, the local field correction consequently reduces $\tau$
to a greater extent than $\tau_{q}$ for the same disorder configuration,
leading to a smaller $\tau/\tau_{q}$. To analytically incorporate the local
field correction $G(q)$ into the scattering rate calculations, we evaluate the
$q$ integral in Eqs. (1) and (2) in an approximation where the square-root
singularity at $q=2k_{F}$ is integrated, but the form factors and $G(q)$ are
taken at $q=2k_{F}$. As a result, the expressions of charged impurity
scattering rates Eqs. (15), (16), (20), (25), (26), and (27) should be
multiplied by an additional factor
$F_{0}(2k_{F}w)^{2}[(2k_{F}/q_{TF})\epsilon(2k_{F})]^{-2}$, where the local
field correction $G(q)$ enters the dielectric function $\epsilon(q)$ defined
in Eq. (5). At low densities $k_{F}\Lambda\ll 1$, the expressions of IR
scattering rates Eqs. (34) and (35) should be multiplied by
$[(2k_{F}/q_{TF})\epsilon(2k_{F})]^{-2}$. While at high densities
$k_{F}\Lambda\gg 1$, the typical momentum transfer is $\Lambda^{-1}$ so Eqs.
(37) and (38) should be multiplied by
$[(2/q_{TF}\Lambda)\epsilon(2/\Lambda)]^{-2}$.

Now we address the justification of the infinite potential well approximation
employed in our calculations. This approximation is justified under two
conditions. The first condition is that the subband gap of the infinite
potential well, $E_{sg}=3\hbar^{2}\pi^{2}/2m_{z}^{\star}w^{2}$, should satisfy
the inequality $E_{F}\ll E_{sg}\ll V_{0}$, where $E_{F}$ is the Fermi energy
of the 2DEG and $V_{0}$ is the conduction band offset of the quantum well. In
the case of the 7 nm quantum well reported in Ref. Esposti  _et al._ (2023),
$E_{sg}\approx 25$ meV, $E_{F}<4$ meV, and $V_{0}\approx 180$ meV Stern and
Laux (1992); Burkard  _et al._ (2023), so the first condition $E_{F}\ll
E_{sg}\ll V_{0}$ is met. The second condition is that the asymmetric electric
field from the top gate should not significantly change the effective well
width. If the electric field is strong enough to tilt the bottom of the
conduction band toward a triangular potential well, then the effective well
width becomes smaller. Using the Fang-Howard wavefunction Fang and Howard
(1966), we can roughly estimate the effective well width at the highest
density where the effect of the electric field $E_{\mathrm{gate}}=4\pi
en_{e}/\kappa$ is the strongest. At the highest density $n_{e}=6\times
10^{11}$ cm-2, the estimated smallest effective well width is
$w_{FH}=6/b\approx 5.5$ nm, where $b=(48\pi
m_{z}^{\star}e^{2}n_{e}/\kappa\hbar^{2})^{1/3}$. Given that the estimated
smallest effective well width from the Fang-Howard wavefunction,
$w_{FH}\approx 5.5$ nm, does not significantly differ from the original well
width $w=5-7$ nm, it suggests that the bending of the conduction band bottom
due to the electric field from the top gate should not substantially alter the
results. This reasoning justifies the use of the infinite potential well
approximation in our calculations. One shortcoming of both the strict 2D and
the infinite square well potential is that they cannot incorporate the short-
range alloy disorder scattering arising from the SiGe alloy on the barrier
outside the well. We address this shortcoming in the next section by using a
finite well confinement model where the carrier wavefunction is allowed to
tunnel into the barrier regime.

Figure 7: Mobility ($\mu$) versus electron density ($n_{e}$) for quantum wells
with widths of 5 nm (red) and 7 nm (blue). The experimental data are depicted
by red and blue curves, with corresponding error bars shown as shaded areas.
The solid black curves represent the theoretical transport mobility, while the
dashed black curves denote the quantum mobility, both calculated using the
finite potential well approximation without the local field correction.
Subfigures (a) and (b) incorporate RI, IR, and AD scattering, whereas (c) and
(d) consider BI, IR, and AD scattering. Fitting parameters: (a)
$n_{r}=2.4\times 10^{12}$ cm-2, $\Delta=4.5$ Å, and $\Lambda=50$ Å; (b)
$n_{r}=2.4\times 10^{12}$ cm-2, $\Delta=9$ Å, and $\Lambda=37$ Å; (c)
$N_{1}=11\times 10^{15}$ cm-3, $N_{2}=0$, $\Delta=5$ Å, and $\Lambda=26$ Å;
(d) $N_{1}=11\times 10^{15}$ cm-3, $N_{2}=0$, $\Delta=7$ Å, and $\Lambda=30$
Å. Figure 8: Mobility ($\mu$) versus electron density ($n_{e}$) for quantum
wells with widths of 5 nm (red) and 7 nm (blue). The experimental data are
depicted by red and blue curves, with corresponding error bars shown as shaded
areas. The solid black curves represent the theoretical transport mobility,
while the dashed black curves denote the quantum mobility, both calculated
using the finite potential well approximation with the local field correction.
Subfigures (a) and (b) incorporate RI, IR, and AD scattering, whereas (c) and
(d) consider BI, IR, and AD scattering. Fitting parameters: (a)
$n_{r}=1.9\times 10^{12}$ cm-2, $\Delta=4.5$ Å, and $\Lambda=22$ Å; (b)
$n_{r}=1.9\times 10^{12}$ cm-2, $\Delta=8$ Å, and $\Lambda=35$ Å; (c)
$N_{1}=7.5\times 10^{15}$ cm-3, $N_{2}=0$, $\Delta=5$ Å, and $\Lambda=20$ Å;
(d) $N_{1}=7.5\times 10^{15}$ cm-3, $N_{2}=0$, $\Delta=7.5$ Å, and
$\Lambda=24$ Å. Figure 9: Mobility and quantum mobility due to alloy disorder
$\mu_{\mathrm{AD}}$ versus electron density $n_{e}$ for a $w=5$ nm (red) and
$w=7$ nm (blue) quantum well, calculated using Eq. (55) with the finite
potential well approximation.

##  V Finite potential well

In this section, to calculate the short-range alloy disorder scattering rate,
we employ the finite potential well approximation to determine the
wavefunction tail extending outside the quantum well. Furthermore, we
recalculate the RI, BI and IR scattering rates using the finite potential well
wavefunction, which enters the expressions of the form factors Eqs. (5, 6) and
the lowest subband energy in Eq. (33). The comprehensive results combining the
scattering mechanisms of RI (BI), IR and AD are shown in Figs. 7 and 8. The
finite potential well Schrödinger equation reads

$-\frac{\hbar^{2}}{2m(z)}\frac{d^{2}\psi(z)}{dz^{2}}+V(z)\psi(z)=E\psi(z)\,,$
(40)

where $m(z)=m^{\star}_{z}=0.92\,m_{0}$ is the longitudinal effective mass for
silicon at $\absolutevalue{z}<w/2$ and $m(z)=m_{B}\simeq m^{\star}_{z}$ at
$\absolutevalue{z}>w/2$ inside the Si1-xGex barrier, since at $x<0.85$,
Si1-xGex alloys are considered as “Si-like” material with almost the same
effective masses Schäffler (1997); F. (2001). Note that Eq. (40) is an
approximation for the situation with a variable effective mass, which applies
very well in our system since $m(z)$ varies little with $z$. The effective
finite potential well is described by

$V(z)=\begin{cases}0,\quad\,&\absolutevalue{z}<w/2\,,\\\
V_{0}-E_{F},\quad\,&\absolutevalue{z}>w/2\,,\end{cases}$ (41)

where $V_{0}=180$ meV Stern and Laux (1992); Burkard  _et al._ (2023) and
$E_{F}=\hbar^{2}k_{F}^{2}/2m^{\star}$ is the Fermi energy of the 2DEG. In the
following we use $V$ to represent $V_{0}-E_{F}$ for simplicity. The bound
state solutions ($E<V$) for the lowest subband are

$\psi(z)=\begin{cases}Ce^{-\eta\absolutevalue{z}},\,&\absolutevalue{z}>w/2\,,\\\
D\cos(kz),\,&\absolutevalue{z}<w/2\,,\\\ \end{cases}$ (42)

where $k=\sqrt{2m^{\star}_{z}E/\hbar^{2}}$ and
$\eta=\sqrt{2m_{B}(V-E)/\hbar^{2}}$. The boundary condition reads

$\displaystyle Ce^{-\eta w/2}=D\cos(kw/2)\,,$ (43)
$\displaystyle\frac{C}{m_{B}}\eta e^{-\eta
w/2}=\frac{D}{m^{\star}_{z}}k\sin(kw/2)\,.$ (44)

The normalization of $\psi(z)$ gives

$C^{2}\int_{w/2}^{\infty}e^{-2\eta
z}dz+D^{2}\int_{0}^{w/2}\cos^{2}(kz)dz=1/2.$ (45)

Combining with Eq. (43), one has

$C=\quantity[\frac{1}{\eta}e^{-\eta
w}+\frac{kw+\sin(kw)}{2k\cos^{2}(kw/2)}e^{-\eta w}]^{-1/2}.$ (46)

Dividing Eq. (44) by Eq. (43), one obtains

$\tan(kw/2)=\frac{m^{\star}_{z}}{m_{B}}\frac{\eta}{k}\,.$ (47)

Introducing dimensionless quantities $\tilde{E}=E/E_{0}$, $\tilde{V}=V/E_{0}$,
where $E_{0}=\hbar^{2}\pi^{2}/2m^{\star}_{z}w^{2}$ is the ground state energy
for infinite well, we arrive at 111See for example, Eq. (5c) in Ref. Gold,
1989 or Section 4.9 in Ref. Davies, 1997.

$\tilde{E}+\frac{m_{B}}{m^{\star}_{z}}\tilde{E}\tan^{2}{\quantity(\sqrt{\tilde{E}}\pi/2)}=\tilde{V}\,.$
(48)

The local energy fluctuation due to interface roughness reads

$\delta E(\mathbf{r})=\partialderivative{E}{w}\Delta(\mathbf{r})\,,$ (49)

where $\Delta(\mathbf{r})$ is the local variation of the well width at a
position $\mathbf{r}$. Taking the derivative with respect to $w$ for both
sides in Eq. (48), we obtain the expression for $\partialderivative*{E}{w}$

$\partialderivative{E}{w}=-\frac{2E}{w}\quantity{1+g\quantity(\frac{m_{B}}{m^{\star}_{z}},\frac{E}{V}){\tilde{V}}^{-1/2}}^{-1}\,,$
(50)

where

$g(x,y)=\frac{2}{\pi}\quantity[x^{-1/2}+y(x^{1/2}-x^{-1/2})]^{-1}(1-y)^{-1/2}\,.$
(51)

Substituting Eq. (50) into Eq. (33), we obtain the scattering potential for
the interface roughness for a finite potential well.

We now estimate the correction introduced by the finite potential well model
relative to the infinite potential well model. According to Eq. (48), we see
that if $\tilde{V}\to\infty$ and $m_{B}=m^{\star}_{z}$, then $\tilde{E}=1$.
For finite values of $\tilde{V}$, we have $0<\tilde{E}<1$, implying that the
effective well width is larger than $w$ due to wavefunction penetration into
the barriers. For example, at electron density $n_{e}=6\times 10^{11}$ cm-2,
we have $\tilde{V}\approx 10$ for a 5 nm QW and $\tilde{V}\approx 20$ for a 7
nm QW. In the limit $\tilde{V}\gg 1$ one can solve Eq. (48) perturbatively and
obtain

$\displaystyle\tilde{E}\approx
1-\frac{4}{\pi\tilde{V}^{1/2}}+\frac{12}{\pi^{2}\tilde{V}}+\mathcal{O}\quantity(\tilde{V}^{-\tfrac{3}{2}}),$
(52)

so that $\tilde{E}\approx 0.70$ for a 5 nm QW and $\tilde{E}\approx 0.77$ for
a 7 nm QW. The reduced $\tilde{E}$ enters directly into the interface
roughness scattering potential Eq. (33) and (50) and lowers the scattering
rate by a factor of $\tilde{E}^{2}$. The parentheses in Eq. (50) lead to a
further reduction in the scattering rate. As a result,
$\tau_{\mathrm{IR,inf}}/\tau_{\mathrm{IR,fin}}\approx 0.34$ for a 5 nm QW and
$\tau_{\mathrm{IR,inf}}/\tau_{\mathrm{IR,fin}}\approx 0.45$ for a 7nm QW. To
account for this reduction in the scattering rate due to the finite well
potential, the product of $\Delta\Lambda$ must be increased to fit the
mobility data, as evident from Eq. (33). Above, we see that the finite
potential well approximation significantly alters the interface roughness
scattering. The leading order correction to $\tau_{\mathrm{IR}}$ is
proportional to $\tilde{V}^{-1/2}$, which is substantial given that
$\tilde{V}$ is around 10 for a 5 nm QW and 20 for a 7 nm QW. Next, we consider
how the finite potential well model alters the background impurity scattering
rates. The model alters the form factors as per Eqs. (4) and (5). Since the
form factors are proportional to the electron density
$\absolutevalue{\psi(z)}^{2}$, the leading correction should be comparable to
the leakage probability outside the quantum well

$\displaystyle\mathrm{Prob}_{\absolutevalue{z}>w/2}=\int\limits_{\absolutevalue{z}>w/2}\absolutevalue{\psi(z)}^{2}dz=C^{2}\eta^{-1}e^{-\eta
w}.$ (53)

In the limit of $\tilde{V}\gg 1$, we have

$\displaystyle\mathrm{Prob}_{\absolutevalue{z}>w/2}\approx\frac{2}{\pi\tilde{V}^{3/2}}.$
(54)

Hence, $\mathrm{Prob}_{\absolutevalue{z}>w/2}\approx 0.02$ for a 5 nm QW and
$\mathrm{Prob}_{\absolutevalue{z}>w/2}\approx 0.007$ for a 7 nm QW. This
indicates that the finite potential well approximation only marginally
modifies the background impurity scattering rates, affecting them by at most a
few percent.

Alloy disorder within Si1-xGex barriers is another short-range scattering
mechanism that contributes equally to the transport and quantum scattering
rate Bastard (1983); Laikhtman and Kiehl (1993); Huang  _et al._ (2022)

$\displaystyle\frac{1}{\tau_{\mathrm{AD}}}=\frac{m^{\star}}{\hbar^{3}}(\Delta
E_{c})^{2}\Omega
x(1-x)\int\limits_{\absolutevalue{z}>w/2}\absolutevalue{\psi(z)}^{4}dz$ (55)

where $\Delta E_{c}\approx 0.8$ eV is the conduction band offset between Si
and Ge Venkataraman  _et al._ (1993), $\Omega=a^{3}/4$ is the scatter volume
with $a=5.4$ Å is the lattice constant. In the limit of $\tilde{V}\gg 1$, we
have $\int_{\absolutevalue{z}>w/2}\absolutevalue{\psi(z)}^{4}dz\approx
2\tilde{V}^{-5/2}/\pi w$, and the corresponding mobility is given by

$\displaystyle\mu_{\mathrm{AD}}\approx\frac{w^{6}e\quantity(2m^{\star}_{z}V)^{5/2}}{2\pi^{4}\hbar^{2}m^{\star
2}(\Delta E_{c})^{2}\Omega x(1-x)}$ (56)

For a $w=5$ nm quantum well, $\mu_{\mathrm{AD}}\approx 1.4\times 10^{6}$
cm2/Vs, while for a $w=7$ nm quantum well, $\mu_{\mathrm{AD}}\approx 1\times
10^{7}$ cm2/Vs, which are much larger than the peak mobility reported in Refs.
Paquelet Wuetz  _et al._ (2023); Esposti  _et al._ (2023). The complete
numerical results for alloy disorder scattering using the full finite
potential well approximation are shown in Fig. 9, and we see that the order of
magnitude agrees with the rough analytical estimation of Eq. (56). Therefore,
AD scattering is negligible in the high-mobility quantum wells reported in
Refs. Paquelet Wuetz  _et al._ (2023); Esposti  _et al._ (2023), where there
is no Ge present inside the well. (For other devices that incorporate Ge
concentration within the quantum well McJunkin  _et al._ (2022b), AD
scattering is much stronger, resulting in an electron mobility of $2-3\times
10^{4}$ cm2/Vs.) Regarding Eq. (55), we follow the approach of Refs. Bastard
(1983, 1988), omitting the dielectric function due to the short-range nature
of AD around a lattice constant. Though several other studies have included
the dielectric function in AD scattering calculations Ando (1982); Gold
(1988); Das Sarma and Hwang (2014b), the inclusion of dielectric screening
would only increase mobility, further diminishing the role of AD. Therefore,
our conclusion that AD scattering is not a limiting factor in the high-
mobility structures reported in Refs. Paquelet Wuetz  _et al._ (2023); Esposti
_et al._ (2023) remains valid.

The results of mobility and quantum mobility calculated using the finite
potential well approximation without the local field correction are shown in
Fig. 7. Here, Figs. 7 (a) and (b) are calculated using a combination of RI,
IR, and AD scattering. (c) and (d) are calculated using a combination of BI,
IR, and AD scattering. The corresponding results calculated with local field
correction are shown in Fig. 8.

Figure 10: Mobility ($\mu$) versus electron density ($n_{e}$) for quantum
wells with widths of 5 nm (red) and 7 nm (blue). The experimental data are
depicted by red and blue curves, with corresponding error bars shown as shaded
areas. The solid black curves represent the theoretical transport mobility,
while the dashed black curves denote the quantum mobility, both calculated
using the finite potential well approximation with gate screening and the
local field correction. Subfigures (a) and (b) incorporate RI, IR, and AD
scattering, whereas (c) and (d) consider BI, IR, and AD scattering. Fitting
parameters: (a) $n_{r}=5.2\times 10^{12}$ cm-2, $\Delta=4.5$ Å, and
$\Lambda=39$ Å; (b) $n_{r}=5.2\times 10^{12}$ cm-2, $\Delta=8$ Å, and
$\Lambda=36$ Å; (c) $N_{1}=7.5\times 10^{15}$ cm-3, $N_{2}=0$, $\Delta=5$ Å,
and $\Lambda=20$ Å; (d) $N_{1}=7.5\times 10^{15}$ cm-3, $N_{2}=0$,
$\Delta=7.5$ Å, and $\Lambda=24$ Å.

##  VI Gate screening

In our calculations above, the screening effect of the top gate is considered
primarily through the cutoff distance $d_{s}$ of the background impurity
distribution outside the quantum well. In Section III, we argued that
$d_{s}=d_{g}/2$ should be half the distance from the gate to the center of the
quantum well because the charged impurities close to the gate are effectively
screened by the image charges. In particular, we assume that the gate does not
screen the remote impurities. In this section, we refine our approach by
incorporating the Coulomb potential of the image charges induced by the gate
and compare the mobility results of the RI and BI scattering. The best-fit
mobility and the corresponding quantum mobility curves are depicted in Fig.
10, where the fitting concentrations of charge impurities are $n_{r}=5.2\times
10^{12}$ cm-2 and $N_{1}=7.5\times 10^{15}$ cm-3. To explain the difference in
the peak mobility between the 5 nm and 7 nm quantum wells, the roughness
parameters are chosen differently within the range $\Delta=4.5-8.0$ Å and
$\Lambda=20-39$ Å. Our analysis confirms that $d_{s}=d_{g}/2$ serves as a
reasonably accurate approximation for the cutoff distance of the background
impurity distribution within the upper SiGe barrier. Furthermore, we observe
that the gate-screened RI scattering results align more closely with the
quantum mobility data compared to the RI results that disregard gate
screening. Previous results shown in Fig. 8 indicate that at a high density of
$n_{e}=6\times 10^{11}$ cm-2, the quantum mobility for RI without gate
screening is $1.0\times 10^{4}$ cm2/Vs. This value is smaller than the
experimental data point $(3.0\pm 0.5)\times 10^{4}$ cm2/Vs pri . On the other
hand, as illustrated in Figs. 10, when gate screening is included, the quantum
mobility for RI increases to $3.0\times 10^{4}$ cm2/Vs at $n_{e}=6\times
10^{11}$ cm-2, aligning well with the experimental data. This enhancement in
quantum mobility due to the inclusion of gate screening can be attributed to
the suppression of small momentum forward scattering by image charges,
consequently reducing the ratio $\tau/\tau_{q}$. Therefore, for a given best-
fit mobility curve, the corresponding quantum mobility is expected to be
higher.

In the scattering rate calculations, the gate screening is taken care by
including the image charge potential into the remote impurity potential Eq.
(28) at $\absolutevalue{z}>w/2$:

$\displaystyle{U}_{1}(q,z)=\frac{2\pi e^{2}F_{0}(qw)}{\kappa
q\epsilon(q)}\quantity(e^{-q\absolutevalue{z}}-e^{-q\absolutevalue{2d_{g}-z}}),$
(57)

where $2d_{g}=2d+2d_{o}+w$. As a result, the RI scattering rates Eqs. (15) and
(16) are replaced by 222We assume the delta-layer approximation $w=0$ to get
the analytical expressions.

$\displaystyle\frac{1}{\tau_{\mathrm{RI}}}=\frac{\pi\hbar
n_{r}}{8m^{\star}k_{F}^{3}d^{3}}\quantity(1+\frac{d^{3}}{(2d_{g}-d)^{3}}-\frac{2d^{3}}{d_{g}^{3}})\quantity(\frac{2}{g})^{2},$
(58) $\displaystyle\frac{1}{\tau_{q\mathrm{RI}}}=\frac{\pi\hbar
n_{r}}{2m^{\star}k_{F}d}\quantity(1+\frac{d}{2d_{g}-d}-\frac{2d}{d_{g}})\quantity(\frac{2}{g})^{2}.$
(59)

The ratio of the scattering rates is given by

$\displaystyle\quantity(\frac{\tau}{\tau_{q}})_{\mathrm{RI}}=(2k_{F}d)^{2}I_{g}(d,d_{g}),$
(60)

where the dimensionless function $I_{g}(d,d_{s})$ reads

$\displaystyle
I_{g}(d,d_{g})=\frac{1+\frac{d}{2d_{g}-d}-\frac{2d}{d_{g}}}{1+\frac{d^{3}}{(2d_{g}-d)^{3}}-\frac{2d^{3}}{d_{g}^{3}}}.$
(61)

A comparison between Eqs. (17) and (60) reveals that the inclusion of gate
screening modifies the ratio by a factor of $I_{g}(d,d_{s})$, which is
independent of $k_{F}$. For the high mobility structures reported in Refs.
Paquelet Wuetz  _et al._ (2023); Esposti  _et al._ (2023), $d_{g}\approx 44$
nm and $d\approx 30$ nm, which gives $I_{g}(d,d_{g})\approx 0.3$. Thus,
incorporating gate screening leads to a decrease in the ratio
$\quantity(\tau/\tau_{q})_{\mathrm{RI}}$, and consequently, the corresponding
quantum mobility should increase by a factor of 3. This explains the
difference in the quantum mobility values between Figs. 8 and 10 (a) (b).

On the other hand, we see that the results in Figs. 8 and 10 (c) (d) are very
similar to each other, suggesting that our choice of $d_{s}=d_{g}/2$ describes
the gate-screened BI scattering within the upper SiGe barrier reasonably well.
We comment on the cutoff distance in the lower SiGe barrier in the context of
the high-mobility structures reported in Refs. Paquelet Wuetz  _et al._
(2023); Esposti  _et al._ (2023). When we discuss the gate screening effects
for BI in Section III, we assume that the cutoff distances are the same from
both sides of the SiGe barriers, although in experiments the heterostructure
is not symmetric. The bottom barrier (a few $\mu$m) is significantly thicker
than the upper spacer (30 nm). Given that the top gate can effectively screen
charge impurities only within the upper SiGe barrier, the cutoff distance for
the lower SiGe barrier remains an open question. However, as indicated by Eq.
(26), at high densities, the cutoff distance $d_{s}$ only logarithmically
changes the quantum mobility, without significantly affecting the transport
mobility. Therefore, the choice of the cutoff distance should not change the
fit to the experimental mobility data. By considering the cutoff distance as
an additional fitting parameter, one can always adjust the model to match a
specific quantum mobility data point. A more comprehensive future experimental
investigation of high-mobility Si quantum wells would be instrumental in
enhancing our understanding of the BI distribution within the SiGe barrieres.

In summary, the results of our model, which incorporates gate screening,
suggest that RI scattering is likely the predominant scattering mechanism for
transport mobility at low densities, whereas at high densities the transport
mobility is restricted by IR scattering. The quantum mobility predicted for
gate-screened RI scattering aligns closely with experimental data within the
margin of error at the highest density examined. This observation suggests
that RI scattering could be the limiting factor for quantum mobility across
all densities. On the other hand, since the cutoff distance of the BI
distribution inside the bottom SiGe barrier remains an open question, we
cannot rule out the possibility that some distant background impurities also
contribute to the quantum mobility. Future experiments focusing on the quantum
mobility over a broader range of densities could yield more quantitative
information on the concentration and the cutoff distance of background
impurities inside the bottom SiGe barrier.

Table 1: Critical density $n_{c}$ and $n_{cq}$ in units of $10^{11}$ cm-2 for different quantum degeneracy $g=4$, 2, and 1. The results are obtained from $k_{F}l=1$ and $k_{F}l_{q}=1$ using the same impurity parameters from Fig. 10 calculated in a finite potential well approximation with gate screening and local field correction. RI (BI) indicates the dominant scattering mechanism at low densities. 5 nm (7 nm) means the calculation for a 5 nm (7 nm) QW. | RI (5 nm) | RI (7 nm) | BI (5 nm) | BI (7 nm)
---|---|---|---|---
$n_{c}$ (g=4) | 0.52 | 0.49 | 0.15 | 0.14
$n_{cq}$ (g=4) | 0.65 | 0.62 | 0.12 | 0.11
$n_{c}$ (g=2) | 0.50 | 0.48 | 0.28 | 0.26
$n_{cq}$ (g=2) | 0.83 | 0.79 | 0.28 | 0.25
$n_{c}$ (g=1) | 0.49 | 0.47 | 0.42 | 0.38
$n_{cq}$ (g=1) | 1.1 | 1.0 | 0.48 | 0.43
Table 2: Critical density $n_{c}$ in units of $10^{11}$ cm-2 obtained by the power law fit $\sigma\propto(n_{e}-n_{c})^{p}$ to the conductivity data $\sigma=en_{e}\mu$ at low densities $n_{e}=1-2\times 10^{11}$ cm-2 reported in Refs. Paquelet Wuetz  _et al._ (2023); Esposti  _et al._ (2023). The parenthesis show the 95% confidence interval of the fitting parameters. The first column fixed $p=1.31$ assuming the 2D percolation. The second and third columns are results treating $p$ as a free parameter. |  $n_{c}$ ($p=1.31$) | $n_{c}$ | $p$
---|---|---|---
5 nm | (0.837, 0.855) | (0.436, 0.514) | (2.20, 2.41)
7 nm | (0.869, 0.889) | (0.447, 0.515) | (2.34, 2.52)

##  VII Metal insulator transition

In this section, we discuss the mobility data in the low-density regime, and
estimate the critical density of the metal-insulator transition (MIT) using
the Anderson-Ioffe-Regel (AIR) condition Anderson (1958); Ioffe and Regel
(1960). All our results in the previous sections are calculated using the
Boltzmann-Born transport theory, which is valid at high densities such that
$k_{F}l\gg 1$, where $l=v_{F}\tau$ is the transport mean free path. One might
expect that the Boltzmann-Born theory fails at a critical density
$n_{e}=n_{c}$ where $k_{F}l=1$. This is the well-known AIR criterion of MIT.
However, deviations in the metallic regime of our results, as shown in Fig.
10, suggest that the theoretical mobility curves, fitting well at high
densities, start overestimating mobility at low densities compared to the
experimental data. It is crucial to emphasize that these low-density
deviations are not due to any neglected scattering mechanism in the theory.
The dominant scattering mechanism at low densities is RI (or BI, depending on
the choice of the impurity model). For RI-dominant cases, as depicted in Figs.
10 (a) and (b), the deviation is observed at $n_{e}\lesssim 1.8\times 10^{11}$
cm-2, where $k_{F}l\lesssim 30$ or $k_{F}l_{q}\lesssim 6$ with
$l_{q}=v_{F}\tau_{q}$ Ahn and Das Sarma (2022b). On the other hand, for BI
dominant scenarios, shown in Figs. 10 (c) and (d), the deviation occurs at a
higher density $n_{e}\lesssim 2.8\times 10^{11}$ cm-2, around $k_{F}l\lesssim
100$ or $k_{F}l_{q}\lesssim 60$. The failure of the Boltzmann-Born theory at
low densities is due to the following. Long-range Coulomb disorder leads to
the fragmentation of the 2DEG into electron puddles, separated by disorder
potential barriers, creating an inhomogeneous 2DEG landscape. Although
$k_{F}l\gg 1$ is locally satisfied within each puddle, the uniform 2DEG
assumption, fundamental to linear screening theory, is no longer valid.
Instead, transport properties in the low-density region $n_{e}\geq n_{c}$ near
MIT should be described by percolation through charged puddles, following
$\sigma\propto(n_{e}-n_{c})^{p}$ with $p=1.31$ Last and Thouless (1971);
Kirkpatrick (1973); Shklovskii and Efros (1975); Sarma and Hwang (2005); Tracy
_et al._ (2009); Manfra  _et al._ (2007); Das Sarma  _et al._ (2013, 2014),
and cannot be captured by the Boltzmann-Born theory even including all high-
order scattering processes. The failure of the Boltzmann-Born theory at low
densities also indicates that the critical density estimated from the AIR
criterion should be smaller than the actual critical density (percolation
threshold) observed in experiments and should be used as a lower bound
estimation of the percolation threshold.

Nevertheless, we apply the AIR criterion, characterized by $k_{F}l=1$ and
$k_{F}l_{q}=1$, to estimate the critical densities $n_{c}$ and $n_{cq}$
Anderson (1958); Ioffe and Regel (1960). Utilizing the delta-layer 2DEG
approximation, as discussed in Sections III and VI, we can analytically
estimate the critical density of the metal-insulator transition (MIT). By
substituting the gate-screened remote impurity scattering rates from Eqs. (58)
and (59) into the AIR criterion, we derive:

$\displaystyle
n_{c}^{\mathrm{(RI)}}=\frac{g^{1/5}\quantity[1+\frac{d^{3}}{(2d_{g}-d)^{3}}-\frac{2d^{3}}{d_{g}^{3}}]^{2/5}}{2^{12/5}\pi^{3/5}}\quantity(\frac{n_{r}}{d^{3}})^{2/5},$
(62) $\displaystyle
n_{cq}^{\mathrm{(RI)}}=\frac{g^{-1/3}\quantity[1+\frac{d}{2d_{g}-d}-\frac{2d}{d_{g}}]^{2/3}}{2^{4/3}\pi^{1/3}}\quantity(\frac{n_{r}}{d})^{2/3},$
(63)

Eqs. (62) and (63) are applicable if $n_{c}^{\mathrm{(RI)}}$ and
$n_{cq}^{\mathrm{(RI)}}$ exceed $g/16\pi d^{2}$. This condition is met in
high-mobility quantum wells Paquelet Wuetz  _et al._ (2023); Esposti  _et al._
(2023) where $g/16\pi d^{2}\approx 10^{10}$ cm-2. For BI scattering,
substituting the rates from Eqs. (25) and (26) into the AIR criterion yields:

$\displaystyle
n_{c}^{\mathrm{(BI)}}=g^{1/5}2^{-8/5}\pi^{-3/5}\quantity(\frac{N_{1}}{w^{2}})^{2/5},$
(64) $\displaystyle
n_{cq}^{\mathrm{(BI)}}=g^{-1/3}2^{-2/3}\pi^{-1/3}\quantity[\ln(2d_{s}/w)]^{2/3}N_{1}^{2/3},$
(65)

which are valid if $n_{c}^{\mathrm{(BI)}}$ and $n_{cq}^{\mathrm{(BI)}}$ are
greater than $g/16\pi w^{2}$. Otherwise, we use Eq. (27) and obtain

$\displaystyle
n_{c}^{\mathrm{(BI)}}=n_{cq}^{\mathrm{(BI)}}=\frac{\pi}{g}N_{1}(2d_{s}-w),$
(66)

which is applicable if $n_{c}^{\mathrm{(BI)}}<g/16\pi w^{2}$. Given that the
crossover density $g/16\pi w^{2}\approx 2.4\times 10^{11}$ cm-2 is
significantly higher than the MIT critical density in high-mobility quantum
wells Paquelet Wuetz  _et al._ (2023); Esposti  _et al._ (2023), we should use
Eq. (66) for estimating $n_{c}$ in BI-dominated scenarios. Using $d=30$ nm,
$d_{s}=d_{g}/2=22$ nm, and the impurity concentration $n_{r}\approx 5.2\times
10^{12}$ cm-2 and $N_{1}\approx 7.5\times 10^{15}$ cm-3 from Fig. 10, we
obtain the critical densities $n_{c}^{\mathrm{(RI)}}\approx 5\times 10^{10}$
cm-2, $n_{cq}^{\mathrm{(RI)}}\approx 7\times 10^{10}$ cm-2,
$n_{c}^{\mathrm{(BI)}}=n_{cq}^{\mathrm{(BI)}}\approx 2\times 10^{10}$ cm-2.

Next, we estimate the critical density of MIT by solving the AIR conditions
$k_{F}l=(2/g)(hn_{e}\mu/e)=1$ and $k_{F}l_{q}=(2/g)(hn_{e}\mu_{q}/e)=1$ with
$\mu$ and $\mu_{q}$ numerically calculated from the best fits shown in Fig.
10. The complete numerical results for $n_{c}$ and $n_{cq}$ are summarized in
the first two rows of Table 1. These values closely align with the analytical
results from Eqs. (62), (63), and (66). Furthermore, Table 1 includes results
for various quantum degeneracies $g=4$, 2, and 1, while keeping the same
disorder parameters as used in Fig. 10. The changes in $n_{c}$ and $n_{cq}$ as
functions of $g$ are semi-quantitatively consistent with Eqs. (62), (63), and
(66). We should emphasize that our estimate of $n_{c}$ and $n_{cq}$ using the
AIR criterion shown in Table 1 should serve as a lower bound for the
experimental critical density of MIT.

Considering that the nature of 2D MIT in high-mobility Si quantum wells likely
resembles a density-inhomogeneity-driven percolation transition similar to Si
MOSFETs Tracy  _et al._ (2009), we performed a percolation fit of the
experimental data to estimate the experimental critical density. Refs.
Paquelet Wuetz  _et al._ (2023); Esposti  _et al._ (2023) reported the
percolation density $n_{c}=7\times 10^{10}$ cm-2 ($n_{c}=8\times 10^{10}$
cm-2) for their high-mobility 7 nm (5 nm) Si quantum wells. The percolation
density can also be estimated approximately by considering inhomogeneous
density fluctuations in the system Das Sarma and Hwang (2014c); Efros (1988);
Pikus and Efros (1989); Tracy  _et al._ (2009); Das Sarma  _et al._ (2013);
Ahn and Das Sarma (2023), leading to $n_{c}\approx 0.1\sqrt{n_{r}}/d\approx
7.5\times 10^{10}$ cm-2, where we used $d=30$ nm and $n_{r}\approx 5\times
10^{12}$ cm-2. This $n_{c}$ is higher than the one obtained from the AIR
criterion. Furthermore, we individually performed the percolation fit
$\sigma\propto(n_{e}-n_{c})^{p}$ using the conductivity data
$\sigma=en_{e}\mu$ at low densities $n_{e}=1-2\times 10^{11}$ cm-2 reported in
Refs. Paquelet Wuetz  _et al._ (2023); Esposti  _et al._ (2023). The results
of the best-fit $n_{c}$ and $p$ are presented in Table 2. The percolation fit
with fixed $p=1.31$ yields $n_{c}\approx 0.85\times 10^{11}$ cm-2. Allowing
$p$ as a fitting parameter results in $p\approx 2.4$ and $n_{c}\approx
0.5\times 10^{11}$ cm-2. Our percolation fit results are in reasonable
agreement with those reported in Refs. Paquelet Wuetz  _et al._ (2023);
Esposti  _et al._ (2023). Comparing Tables 1 and 2, we observe that the RI
results provide a more stringent lower bound of the critical density, as
$n_{c}^{\mathrm{(RI)}}$ and $n_{cq}^{\mathrm{(RI)}}$ are closer to the
percolation densities reported in Refs. Paquelet Wuetz  _et al._ (2023);
Esposti  _et al._ (2023). This suggests that RI scattering, rather than BI, is
likely the limiting mechanism at low densities in the studied high-mobility Si
quantum wells. Given that $n_{cq}^{\mathrm{(RI)}}>n_{c}^{\mathrm{(RI)}}$, we
find that the AIR criterion, which includes the single-particle scattering
time $\tau_{q}$ Ahn and Das Sarma (2022b), provides a more accurate estimate
of the percolation density. In the future, a more systematic experimental
study of the temperature and density dependence of the conductivity is needed
to gain a more comprehensive understanding of the 2D MIT in high-quality Si
quantum wells.

Figure 11: Mobility ($\mu$) versus electron density ($n_{e}$) for quantum
wells with widths of 5 nm (red) and 7 nm (blue). The dashed, solid, and dot-
dashed curves represent the theoretical mobility assuming the total quantum
degeneracy $g=4$, 2, and 1 respectively. The calculations are done using the
finite potential well approximation with gate screening and local field
correction. Subfigures (a) and (b) incorporate RI, IR, and AD scattering,
whereas (c) and (d) consider BI, IR, and AD scattering. The corresponding
disorder parameters are the same as in Fig. 10.

##  VIII Effects of Spin/Valley degeneracy

In earlier sections, our numerical calculations considered a total quantum
degeneracy $g=g_{s}g_{v}=4$. This assumption is valid under two conditions:
first, the Fermi energy must substantially surpass the valley splitting, and
second, the external magnetic field should be weak enough to avoid inducing
spin polarization. In this scenario, both the valley and spin degrees of
freedom maintain a twofold degeneracy, indicated by $g_{v}=g_{s}=2$. This
section theoretically addresses how the total quantum degeneracy $g$ affects
the electron transport properties in the high-mobility Si/SiGe structures for
a given disorder configuration. For example, a decrease in degeneracy from
$g=4$ to $g=2$ can be achieved by applying an in-plane magnetic field $B$
parallel to the 2DEG, which is assumed to only affect the spin degeneracy of
the 2DEG without orbital effect if the thickness of the 2DEG is much smaller
than the cyclotron radius $w\ll c\hbar k_{F}/eB$ Das Sarma and Hwang (2000,
2005a, 2005b); Hwang and Das Sarma (2013a). Such a reduced degeneracy is also
possible without any spin polarization if the valley splitting happens to be
larger than the Fermi energy so that $g_{v}=1$ in the system.

In the strong screening limit $q_{TF}/2k_{F}\gg 1$ (for the density range of
interest to us $n_{e}\ll g^{3}/16\pi a_{B}^{2}$), mobility increases
monotonically with increasing $g$ for a given impurity configuration. This
monotonic behavior of mobility versus $g$ has been shown in a comprehensive
analysis of valley-dependent 2D transport in (100), (110), and (111) Si
inversion layers with the same bare Coulomb disorder Hwang and Das Sarma
(2013a, 2007). A similar phenomenon was observed in an ambipolar gate-
controlled Si(111)-vacuum field effect transistor, where by tuning the
external gate voltage the 2D electron and hole transport can be studied within
the same device against the same background Coulomb disorder Hu  _et al._
(2015). For the ambipolar transistor discussed in Ref. Hu  _et al._ (2015),
the effective valley degeneracy is 6 for a 2D electron system,, whereas it is
1 for a 2D hole system, resulting in the peak electron mobility being
approximately 20 times greater than the peak hole mobility at low
temperatures. This can be qualitatively understood from the expression of the
Thomas-Fermi screening wave vector $q_{TF}=g/a_{B}$, implying that higher $g$
leads to improved screening and consequently, higher mobility. However, the
relationship between mobility and $g$ is generally nontrivial, as $g$
influences both $q_{TF}\propto g$ and $k_{F}\propto g^{-1/2}$. For remote
impurity scattering, it is evident from Eqs. (15), (16), (58), and (59) that
$\mu_{\mathrm{RI}}\propto g^{1/2}$ and $\mu_{q\mathrm{RI}}\propto g^{3/2}$. On
the other hand, for short-range impurity scattering,
$\mu_{\mathrm{S}}=\mu_{q\mathrm{S}}\propto g^{2}$, as indicated by Eq. (20).
As discussed in Section III, the $g$-dependence of background impurity
scattering crosses over from short-range to remote impurity scattering
behavior with increasing electron density, as detailed in Eqs. (25-27). The
crossover density is at $n_{e}=g/16\pi w^{2}$. For interface roughness
scattering, the mobilities $\mu_{\mathrm{IR}}$ and $\mu_{q\mathrm{IR}}$ are
proportional to $g^{3}$ at low densities $n_{e}<g/4\pi\Lambda^{2}$,
transitioning to a long-range behavior where $\mu_{\mathrm{IR}}\propto
g^{1/2}$ and $\mu_{q\mathrm{IR}}\propto g^{3/2}$ at higher densities, as shown
in Eqs. (34-38). However, IR scattering at low densities
$n_{e}<g/4\pi\Lambda^{2}$ is typically overshadowed by the dominant scattering
mechanism at these densities, which is usually remote Coulomb disorder. At
$n_{e}=g/4\pi\Lambda^{2}$, the minimum mobility due to IR scattering
$\mu_{\mathrm{IR},\min}\propto g^{2}$, as per Eq. (39). Since the unscreened
alloy disorder scattering rate, as given by Eq. (55), depends on $k_{F}$ only
through $V=V_{0}-E_{F}$, the mobility $\mu_{\mathrm{AD}}$ is not influenced by
$g$ provided that $V_{0}\gg E_{F}$ is fulfilled. The results of the above
analysis suggest that if a parallel magnetic field is used to lift the spin
degeneracy without any orbital effect, the metallic resistance can at most be
increased by a factor of 4 Okamoto  _et al._ (1999); Dolgopolov and Gold
(2000); Herbut (2001); Hwang and Das Sarma (2005); Das Sarma and Hwang (2005a,
b).

Fig. 11 presents the results of mobility $\mu(n_{e})$ for different quantum
degeneracies $g=4$, 2, and 1. For these calculations, the finite potential
well approximation with gate screening and local field correction is utilized,
employing the same disorder parameters as in Fig. 10. As expected, we see that
the mobility decreases monotonically with the reduction in $g$. The scattering
mechanism that limits the transport mobility is IR at high densities, while
the limiting factor of mobility at low densities is RI (or BI, depending on
the choice of the impurity model). The dashed curves in Fig. 11 replicate the
$g=4$ mobility results from Fig. 10. The solid curves represent our prediction
for the mobility in the presence of a parallel magnetic field without orbital
effect. In this scenario, the spin is fully polarized ($g_{s}=1$), resulting
in a total quantum degeneracy of $g=g_{v}=2$. The observed ratio of
$\mu_{g=4}/\mu_{g=2}$ falls within the range from $\sqrt{2}$ to $4$, which
aligns with the predicted $g^{1/2}$ to $g^{2}$ scaling of $\mu$ across various
scattering mechanisms. Our choice of the valley degeneracy $g_{v}=2$ is
appropriate for the high mobility structures reported in Refs. Paquelet Wuetz
_et al._ (2023); Esposti  _et al._ (2023), where the Fermi energy in the
density range $n_{e}=1-6\times 10^{11}$ cm-2 is $E_{F}\in(0.6,3.8)\times(4/g)$
meV. This Fermi energy is significantly greater than the valley splitting of
$0.2$ meV. Should the valleys split further such that only one valley is
occupied, combined with a strong parallel magnetic field, one could achieve a
quantum degeneracy of $g=1$. The dot-dashed curves in Fig. 11 are our
predictions in this scenario, indicating a further reduction in mobility.

The quantum degeneracy $g$ also has an effect on the critical density $n_{c}$
estimated from the AIR criterion. The specific relationship between $n_{c}$
and $g$ depends on the dominant scattering mechanism responsible for MIT and
the dimensionless parameters such as $q_{TF}/2k_{F}$, $k_{F}d$ for RI,
$k_{F}w$ for BI, and $k_{F}\Lambda$ for IR. These parameters determine the
effective screened potential that scatters electrons, thereby altering the MIT
critical density. In the strong screening limit $q_{TF}/2k_{F}\gg 1$ or
$n_{e}\ll g^{3}/16\pi a_{B}^{2}$, which is of interest to us for the high-
mobility structures reported in Refs. Paquelet Wuetz  _et al._ (2023); Esposti
_et al._ (2023), the critical densities estimated from the AIR criterion for
the RI and BI scatterings are summarized in Eqs. (62-66). We see that for
long-range Coulomb impurity scattering, $n_{c}\propto g^{1/5}$ while
$n_{cq}\propto g^{-1/3}$. For MIT induced by short-range impurity scattering,
the critical density is given by

$\displaystyle n_{c}^{\mathrm{(S)}}=n_{cq}^{\mathrm{(S)}}=\frac{\pi}{g}n_{2},$
(67)

which is the same as Eq. (66) with $N_{1}(2d_{s}-w)$ replaced by the effective
2D short-range impurity concentration $n_{2}$ nearby the 2DEG. For example,
Eq. (67) describes the case in Si MOSFETs where $n_{c}$ increases as $g$ (and
$\mu$) decreases Sarachik (2002); Das Sarma and Hwang (2014c); Ahn and Das
Sarma (2022b). Table 1 shows the value of $n_{c}$ and $n_{cq}$ for the same
disorder parameters taken from Fig. 10 but with different $g$. We should
emphasize that the disorder parameters in Fig. 10 are obtained by fitting the
experimental data with the theoretical mobility curves using $g=4$, which is
justified since $E_{F}$ is much larger than the valley splitting in the high-
mobility Si quantum wells. For completeness, Appendix A presents the best-fit
results of the mobility curves and the corresponding critical densities
assuming $g=2$. In this scenario, the fitting disorder parameters are lower
than those in Fig. 10 due to weaker screening.

##  IX Conclusion

In summary, as the main goal of our theoretical work in the context of the
importance of the Si/Ge quantum computing platform, we have systematically
examined the roles of different scattering sources on the transport and
quantum mobilities in the Si/Si0.69Ge0.31 quantum wells with a co-design of
high electron mobility and large valley splitting reported in Refs. Paquelet
Wuetz  _et al._ (2023); Esposti  _et al._ (2023). We identify remote charged
impurities and interface roughness as primary limitations for transport
mobility at low and high electron densities, respectively. Quantum mobility,
on the other hand, influenced by both remote and distant background
impurities, requires further experimental investigation to understand the
charged impurity distribution in SiGe barriers. Our analysis finds alloy
disorder scattering to be negligible in high-mobility structures. Other
scattering sources, such as inhomogeneous strain distributions and threading
dislocations cry  that originated from the bottom metamorphic substrate of
several $\mu$m, are also quantitatively irrelevant for the reported mobility
using similar estimations from Ref. Monroe  _et al._ (1993); Ismail  _et al._
(1994). We estimate the metal-insulator transition critical density using the
Anderson-Ioffe-Regel criterion and qualitatively explain the breakdown of the
Boltzmann-Born theory at low densities due to the long-range Coulomb disorder.
We also estimate the critical density by considering inhomogeneous density
fluctuations induced by long-range Coulomb disorder in the system and find a
larger critical density compared to the one obtained from the Anderson-Ioffe-
Regel criterion. To gain a more comprehensive understanding of the metal-
insulator transition, more extensive future experimental studies of the
temperature and density dependence of the conductivity should be conducted. We
also predict the electron mobility with reduced quantum degeneracy, where the
spin is fully polarized by an external parallel magnetic field, for future
experimental validation.

###### Acknowledgements.

The authors thank G. Scappucci and D.D. Esposti for discussions and
clarification of the experimental details. This work is supported by the
Laboratory for Physical Sciences.

##  Appendix A Best-fit results assuming $g=2$

In this appendix, we present the best-fit results of mobility curves and the
corresponding MIT critical densities assuming the total quantum degeneracy
$g=2$. Fig. 12 shows the mobility and quantum mobility as a function of the
electron density. By comparing Figs. 10 and 12, it is evident that the
disorder parameters for $g=4$ are generally higher than those for $g=2$, due
to the fact that screening increases monotonically as $g$ increases. Table 3
shows the corresponding results of MIT critical densities. We see that $n_{c}$
and $n_{cq}$ are in general smaller than the $g=2$ results shown in Table 1,
because the values of best-fit mobility and quantum mobility are higher.

Table 3: Critical density $n_{c}$ and $n_{cq}$ in units of $10^{11}$ cm-2. The results are obtained from $k_{F}l=1$ and $k_{F}l_{q}=1$ using the same impurity parameters as in Fig. 12 calculated in the finite potential well approximation with gate screening and local field correction. RI (BI) indicates the dominant scattering mechanism at low densities. 5 nm (7 nm) means the calculation for a 5 nm (7 nm) QW. | RI (5 nm) | RI (7 nm) | BI (5 nm) | BI (7 nm)
---|---|---|---|---
$n_{c}$ (g=2) | 0.44 | 0.41 | 0.16 | 0.15
$n_{cq}$ (g=2) | 0.68 | 0.63 | 0.15 | 0.14
Figure 12: Mobility ($\mu$) versus electron density ($n_{e}$) for quantum
wells with widths of 5 nm (red) and 7 nm (blue). The experimental data are
depicted by red and blue curves, with corresponding error bars shown as shaded
areas. The solid black curves represent the theoretical transport mobility,
while the dashed black curves denote the quantum mobility, both calculated
using the finite potential well approximation with gate screening and the
local field correction, assuming the total quantum degeneracy $g=2$.
Subfigures (a) and (b) incorporate RI, IR, and AD scattering, whereas (c) and
(d) consider BI, IR, and AD scattering. Fitting parameters: (a)
$n_{r}=3.6\times 10^{12}$ cm-2, $\Delta=3.5$ Å, and $\Lambda=20$ Å; (b)
$n_{r}=3.6\times 10^{12}$ cm-2, $\Delta=8$ Å, and $\Lambda=15$ Å; (c)
$N_{1}=3.3\times 10^{15}$ cm-3, $N_{2}=0$, $\Delta=4$ Å, and $\Lambda=15$ Å;
(d) $N_{1}=3.3\times 10^{15}$ cm-3, $N_{2}=0$, $\Delta=6.5$ Å, and
$\Lambda=18$ Å.

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