An acoustic wave amplifier for radiofrequency signal processing applications
must have significant levels of terminal gain, meaning that the acoustic gain
must be large enough to overcome all acoustic losses and then provide a
significant level of additional acoustic gain. This is accomplished by
increasing the applied drift field, which leads to larger dissipated power and
concomitant heating. Ultimately, the ability to operate continuously and to
reach peak acoustoelectric gain is limited by thermal dissipation under these
conditions. However, with respect to the acoustoelectric-Brillouin devices
described in this work, improving the performance and observing new dynamical
regimes only require that the applied drift field provides enough acoustic
gain to overcome the losses inherent to the acoustoelectric effect and the
intrinsic phonon losses. This requires substantially less gain—and thus
substantially less heat—than achieving net terminal gain in radiofrequency
acoustic amplifier systems.

To illustrate the feasibility of continuous operation, we analyze the heat
generated and temperature increase for an example device, using the lithium
niobate-on-silicon and In0.712Ga0.288As0.625P0.375 platform introduced in this
work. We assume an interaction length of 1 cm, a semiconductor width, $w$, of
0.5-1.5 µm, a semiconductor thickness, $t$, of 100-300 nm, an
electromechanical coupling coefficient, $k^{2}$, of 1-9%, and an acoustic
wavelength of approximately 350 nm (corresponding to a phonon frequency of
$\Omega_{m}$/2$\pi$ = 7-11 GHz). A phononic quality factor of 250 (assumed for
the prior calculations) corresponds to a phonon propagation loss of 0.11
dB/$\Lambda$, where $\Lambda$ is the acoustic wavelength. Under these
conditions, it is necessary to achieve an acoustic gain of 0.11 dB/$\Lambda$
or higher, which corresponds to 3100 dB/cm. The bulk semiconductor mobility
for In0.712Ga0.288As0.625P0.375 is approximately 5000 cm2/V$\cdot$s. Based on
our previous work [41], we expect the as-processed thin film mobility in these
type of acoustoelectric-Brillouin devices to be approximately 2000
cm2/V$\cdot$s and the expected carrier concentration is $1\times 10^{16}$ cm-3
(see Supplementary Section A for more details). This carrier concentration and
mobility lead to a conductivity-thickness product, $\sigma t$, that is 32-96
µS, which is one of the key factors that determine the strength of the
external electric field required to reach a given level of gain (see, for
example, Supplementary Information section A.6) as well as the amount of Joule
heating that electric field produces to achieve that gain.

Figure 15: Contour plots of the dissipated DC power as a function of (a)
$\sigma t$ and $w$ and (b) $k^{2}$ and $\Omega$/2$\pi$. (c) $\Delta
T_{\text{max}}$ as a function of $w$ for a 100 nm thick and 1 cm long
In0.712Ga0.288As0.625P0.375 semiconductor layer on a bulk lithium niobate
substrate or thin film lithium niobate on silicon substrate. The semiconductor
acts as the heat source for the finite element method model according to the
dissipated DC power.

From these values, we can estimate the dissipated power. Figure 15a shows the
dissipated power required to achieve an acoustic gain of 3100 dB/cm as a
function of $\sigma t$ and $w$ for $\Omega$/2$\pi$ = 8.78 GHz and $k^{2}$ =
5.87%. Figure 15b shows the dissipated power as a function of $k^{2}$ and
$\Omega_{m}$/2$\pi$ for a semiconductor width of 1.1 µm and $\sigma t$ = 32
µS. The dissipated power varies depending on the device configuration, but can
be 1 mW or less, which is significantly less than previous work where large
terminal gain was achieved [40, 41]. Low dissipated DC powers are achieved due
to two factors. One factor is the device geometry and semiconductor material
parameters have been optimized to make the device highly resistive. The second
factor is that $\Omega_{m}$ and $k^{2}$ are both larger, resulting in larger
acoustic gain being produced with smaller bias fields.

Another aspect that should contribute to the feasibility of continuous
operation in acoustoelectric Brillouin devices is that the interactions take
place in a waveguide with small cross-sectional width of approximately 0.5-1.5
µm. This makes the device more resistive and enables more efficient lateral
heat transport by increasing the surface to volume ratio compared to wider
devices.

Beyond reducing the power dissipated, the temperature increase is also
decreased by using a piezoelectric compound substrate that consists of a
lithium niobate film on bulk silicon as opposed to a bulk lithium niobate
substrate, both of which were demonstrated in recent work [41]. The thermal
conductivity of lithium niobate is 4.6 W/m$\cdot$K [112] while the thermal
conductivity of silicon is 150 W/m$\cdot$K [113]. Therefore, the 30$\times$
increase of thermal conductivity of silicon over lithium niobate leads to a
significantly smaller temperature rise for a given dissipated power. Although
terminal gain was not achieved, continuous operation of an acoustic wave
amplifier has been experimentally demonstrated for both an acoustic waveguide
[114] and a thin film lithium niobate on silicon substrate [40]. Continuous
operation and terminal gain has been achieved in a system using a similar
approach, in particular, leveraging a highly thermally conductive sapphire
substrate [33]. We also reiterate that only net phonon gain, as opposed to
terminal gain (which must overcome all acoustic, electronic, and transducer
losses), is necessary to enhance Brillouin optomechanical interactions via the
acoustoelectric effect—a much less stringent requirement.

A finite element method model was used to model conductive and convective heat
transfer for this exemplary device operating in air. We model the 100 nm
thick, 1.1 µm wide, and 1 cm long In0.712Ga0.288As0.625P0.375 semiconductor
film on a thin film lithium niobate (5 µm thick) on silicon substrate or a
bulk lithium niobate substrate. The In0.712Ga0.288As0.625P0.375 density is
5204 kg/cm3, the heat capacity at constant pressure is 335 J/kg$\cdot$K, the
thermal conductivity is 4.35 W/m$\cdot$K, and the coefficient of thermal
expansion is 5.26$\times$10-6 1/K [93]. Figure 15c shows the maximum
temperature difference $\Delta T_{\text{max}}$, defined as the difference
between the maximum steady-state temperature and the ambient temperature, as a
function of $w$ for the case of a bulk lithium niobate substrate and a lithium
niobate film on a silion substrate. For a semiconductor width of 1.1 µm on the
thin film lithium niobate on silicon substrate, $\Delta T_{\text{max}}$ = 0.08
K, which is approximately 200X smaller than the temperature rise associated
with achieving terminal gain in our recent acoustic wave amplifiers [41],
where the maximum gain applied was limited by needing to avoid significant
temperature increases. In conventional acoustoelectric devices, predicting
device behavior is complicated by Joule heating. For example, heating leads to
a temperature-dependent semiconductor conductivity which modifies the acoustic
gain. The results shown here suggest that these thermal effects do not need to
be considered for acoustoelectric-Brillouin devices, at least in the parameter
space explored in this work, as the dissipated DC power is relatively small
and thermal dissipation is effective.

##  Supplementary Information C Acoustoelectric dynamics

In this section, we formulate a model of acoustoelectric dynamics within
photonic and phononic waveguide structures of arbitrary, but translationally
invariant, cross-sectional geometry. This model assumes; (1) the validity of
the quasistatic limit, enabling the electric field to be expressed in terms of
a scalar potential, (2) describes the free carriers using a hydrodynamic
description, (3) neglects the dispersion of the material lattice, and (4)
captures perturbations of the charge density and velocity to first order.
Assumptions (1)-(4) are well satisfied for the candidate devices considered
here.

We begin with Lagrangian given by

$\displaystyle L=\int_{V_{sc}}d^{3}x\bigg{[}-m(\dot{n}+{\bf v}_{\rm
d}\cdot\nabla
n)\dot{\psi}-\frac{1}{2}mn_{0}(\nabla\dot{\psi})^{2}-en\varphi+\frac{1}{2}\epsilon(\nabla\varphi)^{2}\bigg{]}+\int_{V_{out}}d^{3}x\frac{1}{2}\epsilon(\nabla\varphi)^{2},$
(97)

where the displacement of the free carriers from equilibrium is expressed as
the gradient of the scalar potential $\psi$ (discussed in more detail below),
$n$ is the perturbation of the free-carrier density from equilibrium $n_{0}$,
$m$ and $e$ are the free carrier mass and charge, $\epsilon$ is the spatially-
dependent permittivity of the waveguide structure, and $\varphi$ is the
electric potential. The term containing ${\bf v}_{\rm d}$ accounts for a
constant background drift velocity where the free-carrier drift velocity ${\bf
v}_{\rm d}$ is directed parallel the waveguide surfaces. For later
convenience, the volume integration of the Lagrangian has been explicitly
divided into regions that do ($V_{sc}$, $sc$ standing for ‘semiconductor’) and
do not contain free carriers ($V_{out}$).

The use of the displacement potential $\psi$ to represent the motion of the
free carriers is justified in the quasistatic limit where the electric field
is well-described by $-\nabla\varphi$. In this limit the Lorentz force acting
on the charged fluid is the gradient of a scalar, and therefore is curl free.
Defining the displacement of the electrons from equilibrium by ${\bf\xi}$, the
linearized (hydrodynamic) equations of motion for the electron fluid about a
constant drift velocity ${\bf v}_{\rm d}$ are given by $m\ddot{\bf\xi}+m{\bf
v}_{\rm d}\cdot\dot{\bf\xi}=e\nabla\varphi$. Taking the curl of both sides of
this equation shows $\nabla\times{\bf\xi}=0$, justitfying the representation
of the electron displacement by ${\bf\xi}=-\nabla\psi$.

To describe the coupling between the potential and the elastic field, we use
the interaction Lagrangian given by

$L_{int}=\int d^{3}x\ \varphi\partial_{k}d_{ijk}\partial_{i}u_{j}=-H_{int}$
(98)

which reproduces the classical equations of motion, $d_{ijk}$ is the
piezoelectric tensor (or more generically, any coupling electro-mechanical
coupling), $u_{j}$ is the jth component of the elastic displacement, and
$\partial_{k}$ represents the $k$th component of the gradient.

Neglecting the coupling to the elastic field, the least action principle
yields the following equations

$\displaystyle\dot{n}+{\bf v}_{\rm d}\cdot\nabla n=n_{0}\nabla^{2}\dot{\psi}$
(99) $\displaystyle\ddot{\psi}+{\bf v}_{\rm
d}\cdot\nabla\dot{\psi}=\frac{e}{m}\varphi$ (100)
$\displaystyle-\nabla\cdot\epsilon\nabla\varphi=en.$ (101)

Noting that the perturbation to the free-carrier velocity is given by ${\bf
v}=-\nabla\dot{\psi}$, Eqs. (99) and (100) reproduce the linearized continuity
and hydrodynamic equations in the presence of drift, and Eq. (101) gives
Gauss’ Law. Eliminating the charge density, we find the effective equation of
motion for the electric and free-carrier displacement potential given by

$(\nabla\cdot\epsilon\nabla\partial_{\tau}^{2}+\epsilon_{in}\omega_{e}^{2}\nabla^{2})\begin{bmatrix}\varphi\\\
\psi\end{bmatrix}=0$ (102)

where $\partial_{\tau}=\partial_{t}+{\bf v}_{\rm d}\cdot\nabla$ (and
$\partial_{\tau}^{2}=(\partial_{t}+{\bf v}_{\rm d}\cdot\nabla)^{2}$),
$\epsilon_{in}$ is the permittivity within the region containing free
carriers, and $\omega_{e}=\sqrt{e^{2}n_{0}/(m\epsilon_{in})}$ is the
permittivity normalized plasma frequency. Within the region containing free
carriers (i.e. $\epsilon$ = $\epsilon_{in}$ = constant), Eq. (102) admits two
solution classes: (1) bulk modes that satisfy
$(\partial_{\tau}^{2}+\omega_{e}^{2})\varphi=0$, and (2) surface modes that
satisfy $\nabla^{2}\varphi=0$ everywhere [115, 116]. With the appropriate
boundary conditions, these modes can be fully characterized. Namely, the
electric potential must be continuous across all interfaces and must also
satisfy the Fourier domain boundary condition

$\epsilon_{out}\hat{n}\cdot\nabla\varphi_{out}=\epsilon_{in}\bigg{(}1-\frac{\omega_{e}^{2}}{\beta^{2}}\bigg{)}\hat{n}\cdot\nabla\varphi_{in}$
(103)

where $\beta=\omega-v_{\rm d}q$, $\hat{n}$ is a unit vector oriented normal to
the surface containing the free carriers, and the subscript ‘in’ and ‘out’
respectively denote quantities evaluated on the respective inner and outer
sides of the interface between the region with free carriers. For bulk modes,
the right hand side of Eq. (103) vanishes, requiring the potential outside
$V_{sc}$ to vanish [116]. Consequently, bulk modes do not couple to charges
outside of $V_{sc}$ for this model, and therefore for the candidate system
proposed in this paper—with a piezoelectric domain outside the semiconducting
region—bulk modes do not produce acoustoelectric gain [115, 116]. However,
with diffusion, bulk modes are not confined within the semiconducting region
[115]. As an example, for the surface modes on a planar interface without
drift ($v_{\rm d}=0$) Eq. (103) yields the well-known surface plasmon
frequency
$\omega_{S}^{2}={\epsilon_{in}/(\epsilon_{in}+\epsilon_{out})\omega_{e}^{2}}$.

###  C.1 Acoustoelectric Hamiltonian

To derive the Hamiltonian for the acoustoelectric dynamics, we select $n$ as a
generalized coordinate, find the conjugate momentum ($\delta
L/\delta\dot{n}\equiv P=-m\dot{\psi}$), and perform a Legendre transform of
$L$ to give

$\displaystyle H_{0}=\int_{V_{sc}}d^{3}x\bigg{[}\frac{n_{0}}{2m}(\nabla
P)^{2}+P{\bf v}_{\rm d}\cdot\nabla
n+en\varphi-\frac{1}{2}\epsilon(\nabla\varphi)^{2}\bigg{]}-\int_{V_{out}}d^{3}x\frac{1}{2}\epsilon(\nabla\varphi)^{2}.$
(104)

The equations of motion for this system can be derived from the Poisson
bracket $\\{n({\bf x}),P({\bf x}^{\prime})\\}=\delta^{3}({\bf x}-{\bf
x}^{\prime})$, reproducing Eqs. (99)-(101). We quantize this system by
requiring the equal-time commutation relation (ETCR) $[n({\bf x}),P({\bf
x}^{\prime})]=i\hbar\delta^{3}({\bf x}-{\bf x}^{\prime})$ (i.e., we promote
the classical Poisson bracket to a commutator multiplied by $i\hbar$).

Free carrier dissipation is critical to the acousto-electric effect. To model
these effects we use an open systems treatment tailored to reproduce the
permittivity of the Drude-Lorentz model for free carriers in the frame of the
drift current, captured by the “bath” Hamiltonian given by

$\displaystyle H_{bath}=\int_{V_{sc}}d^{3}x$
$\displaystyle\int_{0}^{\infty}d\nu\bigg{[}\frac{1}{2m}P_{\nu}^{2}+P_{\nu}{\bf
v}_{\rm d}\cdot\nabla
X_{\nu}+\frac{1}{2}m\nu^{2}X_{\nu}^{2}-m\hat{c}_{\nu}X_{\nu}n\bigg{]}.\quad\quad$

Here $P_{\nu}$ and $X_{\nu}$ denote momenta and position coordinates for the
$\nu$th bath mode and $\hat{c}_{\nu}$ is a mode-specific system bath coupling,
i.e., $\hat{c}_{\nu}$ takes on a specific form when the fields are expanded in
normal modes, depending on the normal mode eigenvalues. The classical
equations of motion for the bath variables can be derived from the Poisson
bracket $\\{X_{\nu}({\bf x}),P_{\nu^{\prime}}({\bf
x}^{\prime})\\}=\delta(\nu-\nu^{\prime})\delta^{3}({\bf x}-{\bf x}^{\prime})$
and the quantum dynamics can be recovered by replacing the Poisson bracket a
commutator divide by $i\hbar$.

After simplification, Hamilton’s equations for this coupled system give

$\displaystyle(\nabla\cdot\epsilon\nabla\partial_{\tau}^{2}+\epsilon_{in}\omega_{e}^{2}\nabla^{2})\varphi=-en_{0}\int_{0}^{\infty}d\nu\
\hat{c}_{\nu}\nabla^{2}X_{\nu}\quad$ (105)
$\displaystyle({\partial_{\tau}}^{2}+{\nu}^{2})X_{\nu}=\frac{\hat{c}_{\nu}}{e}\nabla\cdot\epsilon\nabla\varphi\quad$
(106)

which can be analyzed in terms of bulk modes and surface modes. Owing to the
linearity of this coupled system, Eqs. (105) and (106) also give the quantum
dynamics of the system when these fields (i.e., $\varphi$ and $X_{\nu}$) are
promoted to operators.

####  C.1.1 Bulk modes

For bulk modes, orthonormal eigenfunctions of the Helmholtz equation,
satisfying $\nabla^{2}(\psi_{\ell}\exp\\{iqz\\}/\sqrt{2\pi})=-Q_{\sigma
q}^{2}(\psi_{\ell}\exp\\{iqz\\}/\sqrt{2\pi})$ and $\int_{V_{sc}}d^{3}x\
\psi^{*}_{\ell}\psi_{\ell^{\prime}}exp\\{i(q-q^{\prime})z\\}/(2\pi)=\delta_{\ell\ell^{\prime}}\delta(q-q^{\prime})$
subject to Dirichlet boundary conditions on the boundary of $V_{sc}$, which
denote from hereon as $\partial V_{sc}$, yield the spatial dependence of the
potential and charge density. Using these eigenfunctions as a basis

$\displaystyle\varphi=\sum_{\ell}\int
dq\bigg{(}\psi_{\ell}\frac{e^{iqz}}{\sqrt{2\pi}}\varphi_{\ell q}+c.c.\bigg{)}$
(107) $\displaystyle X_{\nu}=\sum_{\ell}\int
dq\bigg{(}\psi_{\ell}\frac{e^{iqz}}{\sqrt{2\pi}}X_{\nu\ell q}+c.c.\bigg{)},$
(108)

we find

$\displaystyle(\partial_{\tau}^{2}+\omega_{e}^{2})\varphi_{\ell
q}=-\frac{en_{0}}{\epsilon_{in}}\int_{0}^{\infty}d\nu\ c_{\nu\ell q}X_{\nu\ell
q}\quad$ (109) $\displaystyle({\partial_{\tau}}^{2}+{\nu}^{2})X_{\nu\ell
q}=-\frac{c_{\nu\ell q}\epsilon_{in}}{e}Q^{2}_{\ell q}\varphi_{\ell q}.\quad$
(110)

Here, we assume that the waveguide is translationally invariant along the
z-direction so that $\psi_{\ell q}\propto\exp\\{iqz\\}/\sqrt{2\pi}$ where $q$
is the wavevector of a plane and the system-bath coupling is taken as
$\hat{c}_{\nu}\to c_{\nu\ell q}$ when expressed in this mode basis. Explicitly
solving for $X_{\nu\ell q}$, including both homogeneous ($X^{0}_{\nu\ell q}$)
and particular solutions, inserting the solution into the equation for
$\varphi_{\ell q}$ and assuming that $\sqrt{n_{0}}c_{\nu\ell q}Q_{\ell
q}=(2\gamma_{\rm e}\nu^{2}/\pi)^{1/2}\equiv G_{\nu}$ (which produces Ohmic
coupling to the bath), we obtain driven damped motion for the potential,
reproducing the physics of the Drude-Lorentz model in the presence of drift
current [117], given by

$\displaystyle(\partial_{\tau}^{2}+\gamma_{\rm
e}\partial_{\tau}+\omega_{e}^{2})\varphi_{\ell
q}=-\frac{en_{0}}{\epsilon_{in}}\int_{0}^{\infty}d\nu\ c_{\nu\ell
q}X^{0}_{\nu\ell q}.\quad$ (111)

Using the solution for $\varphi_{\ell q}$, $X_{\nu\ell q}$ can be obtained
from Eq. (106) [116]. One can show that the following mode expansions for the
potential and the bath satisfy Eqs. (105) & (106) as well as the ETCR

$\displaystyle\varphi=\omega_{e}\sum_{\ell}\int
dq\int_{0}^{\infty}d\omega\sqrt{\frac{\hbar}{4\pi\omega\epsilon_{in}}}\frac{1}{Q_{\ell
q}}\bigg{(}\chi(\omega)\psi_{\ell}e^{iqz}a_{\omega\ell q}+H.c.\bigg{)}\quad$
(112) $\displaystyle n=\sum_{\ell}\int
dq\int_{0}^{\infty}d\omega\sqrt{\frac{\hbar n_{0}}{4\pi m\omega}}Q_{\ell
q}\bigg{(}\chi(\omega)\psi_{\ell}e^{iqz}a_{\omega\ell q}+H.c.\bigg{)}\quad$
(113) $\displaystyle P=-i\sum_{\ell}\int
dq\int_{0}^{\infty}d\omega\sqrt{\frac{\hbar m\omega}{4\pi
n_{0}}}\frac{1}{Q_{\ell q}}\bigg{(}\chi(\omega)\psi_{\ell}e^{iqz}a_{\omega\ell
q}-H.c.\bigg{)}\quad$ (114) $\displaystyle X_{\nu}=\sum_{\ell}\int
dq\int_{0}^{\infty}d\omega\sqrt{\frac{\hbar}{4\pi
m\omega}}\bigg{(}\chi_{\nu}(\omega)\psi_{\ell}e^{iqz}a_{\omega\ell
q}+H.c.\bigg{)}\quad$ (115) $\displaystyle P_{\nu}=-i\sum_{\ell}\int
dq\int_{0}^{\infty}d\omega\sqrt{\frac{\hbar
m\omega}{4\pi}}\bigg{(}\chi_{\nu}(\omega)\psi_{\ell}e^{iqz}a_{\omega\ell
q}-H.c.\bigg{)}\quad$ (116)

when

$\displaystyle[a_{\omega\ell
q},a^{\dagger}_{\omega^{\prime}\ell^{\prime}q^{\prime}}]=\delta_{\ell\ell^{\prime}}\delta(\omega-\omega^{\prime})\delta(q-q^{\prime})$
(117) $\displaystyle\left[a^{\dagger}_{\omega\ell
q},a^{\dagger}_{\omega^{\prime}\ell^{\prime}q^{\prime}}\right]=[a_{\omega\ell
q},a_{\omega^{\prime}\ell^{\prime}q^{\prime}}]=0$ (118)
$\displaystyle\chi(\omega)=\frac{G_{\omega}}{-\omega(\omega+i\gamma_{\rm
e})+\omega_{e}^{2}}$ (119)
$\displaystyle\chi_{\nu}(\omega)=\delta(\omega-\nu)-\frac{G_{\nu}}{-\omega^{2}+\nu^{2}}\frac{G_{\omega}}{-\omega(\omega+i\gamma_{\rm
e})+\omega_{e}^{2}}$ (120)

and the functions $\chi(\omega)$ and $\chi_{\nu}(\omega)$ satisfy the
Lippmann-Schwinger orthogonality conditions [116]

$\displaystyle\int_{0}^{\infty}d\omega\chi(\omega)\chi^{*}(\omega)=1$ (121)
$\displaystyle\chi(\omega)\chi^{*}(\omega^{\prime})+\int_{0}^{\infty}d\nu\chi_{\nu}(\omega)\chi^{*}_{\nu}(\omega^{\prime})=\delta(\omega-\omega^{\prime})\quad$
(122)
$\displaystyle\int_{0}^{\infty}d\omega\chi_{\nu}(\omega)\chi^{*}_{\nu^{\prime}}(\omega)=\delta(\nu-\nu^{\prime})\quad$
(123)
$\displaystyle\int_{0}^{\infty}d\omega\chi_{\nu}(\omega)\chi^{*}(\omega)=0.\quad$
(124)

Inserting Eqs. (112)-(116) in the total Hamiltonian (i.e., $H=H_{0}+H_{bath}$)
and using the orthonormality conditions Eqs. (121)-(124) gives the
diagonalized Hamiltonian for the coupled modes of potential and charge

$\displaystyle H=\sum_{\ell}\int dq\int_{0}^{\infty}d\omega\
\hbar(\omega+v_{\rm d}q)a^{\dagger}_{\omega\ell q}a_{\omega\ell q}.$ (125)

Note that the background drift current has the effect of Doppler shifting the
frequency of these charge-potential modes.

###  C.2 Envelope formulation of acoustoelectric dynamics

To describe slowly varying changes to the elastic amplitude, this section
develops an envelope formulation of the acoustoelectric effect. The envelope
${\Phi}_{\omega\ell}(z)$ for charge density, velocity potential and electric
potential is related to the normal mode amplitudes $a_{\omega\ell q}$ by

$\displaystyle a_{\omega\ell q}=\int\frac{dz}{\sqrt{2\pi}}\ e^{i(q_{\rm
m}-q)z}\Phi_{\omega\ell}(z)$ (126)

where $q_{\rm m}$ is the carrier wavevector describing the spatial changes
along the waveguide. Equations (126) and (117) show that the envelope
operators satisfy the commutation relations

$\displaystyle\big{[}\Phi_{\omega\ell}(z),\Phi^{\dagger}_{\omega^{\prime}\ell^{\prime}}(z)\big{]}=\delta_{\ell\ell^{\prime}}\delta(\omega-\omega^{\prime})\delta(z-z^{\prime})$
(127)
$\displaystyle\big{[}\Phi_{\omega\ell}(z),\Phi_{\omega^{\prime}\ell^{\prime}}(z)\big{]}=0$
(128)
$\displaystyle\big{[}\Phi^{\dagger}_{\omega\ell}(z),\Phi^{\dagger}_{\omega^{\prime}\ell^{\prime}}(z)\big{]}=0$
(129)

Using the envelope description for the potential and elastic field, we find

$\displaystyle\varphi\approx\omega_{e}\sum_{\ell}\int_{0}^{\infty}d\omega\sqrt{\frac{\hbar}{2\omega\epsilon_{in}}}\frac{1}{Q_{\ell
q_{\rm m}}}\bigg{(}\chi(\omega)\psi_{\ell}e^{iq_{\rm
m}z}\Phi_{\omega\ell}(z)+H.c.\bigg{)}\quad$ $\displaystyle{\bf
u}\approx\sum_{\lambda}\sqrt{\frac{\hbar}{2\Omega_{\lambda q_{\rm
m}}}}\bigg{(}\vec{\mathcal{U}}_{\lambda}e^{iq_{\rm
m}z}B_{\lambda}(z)+H.c.\bigg{)}.$ (130)

where $\Omega_{\lambda q_{\rm m}}$ is the eigenfrequency for the $\lambda$th
mode with wavevector $q_{\rm m}$ and $\mathcal{U}_{\lambda,j}$ are orthonormal
eigenfunctions of the elastic equation satisfying the eigenvalue equation for
the medium’s mechanical motion
$\partial_{j}C_{ijkl}\partial_{k}[\mathcal{U}_{\lambda,l}\exp\\{iqz\\}]=-\rho\Omega^{2}_{\lambda
q}[\mathcal{U}_{\lambda q,i}\exp\\{iqz\\}]$ and the orthonormality condition
$\int d^{3}x\ \rho\
\vec{\mathcal{U}}_{\lambda}\cdot\vec{\mathcal{U}}^{*}_{\lambda^{\prime}}\exp\\{i(q-q^{\prime})z\\}/(2\pi)=\delta_{\lambda\lambda^{\prime}}\delta(q-q^{\prime})$.
Here, $\rho$ is the spatially-dependent mass density. When restricted to a
single mode of the elastic field (i.e. we suppress the sum over $\lambda$ from
hereon), the interaction Hamiltonian Eq. (98) can be expressed in the envelope
picture as

$\displaystyle H_{int}=-\sum_{\ell}\int_{0}^{\infty}d\omega\int dz\
\hbar(\kappa_{\omega\ell}\Phi_{\omega\ell}(z)B^{\dagger}(z)+H.c.)\quad\quad$
(131)

where the coupling rate (with units of $1/\sqrt{s}$) is given by

$\displaystyle\kappa_{\omega\ell}=\sqrt{\frac{1}{4\epsilon_{in}\omega\Omega_{\rm
m}}}\frac{\omega_{e}\chi(\omega)}{Q_{\ell q_{\rm m}}}\int d^{2}x\
\psi_{\ell}\tilde{\partial_{k}}d_{ijk}\tilde{\partial_{i}}\mathcal{U}^{*}_{j}\quad\quad\
\ $ (132)

with the integral taken over the waveguide cross section and
$\tilde{\partial_{i}}=\\{\partial_{x},\partial_{y},-iq_{\rm m}\\}$ for $i$
equal $x$, $y$ and $z$ respectively. Inserting Eq. (126) into Eq. (125) and
adding $H_{int}$ yields Eq. (1) listed in the main text.

###  C.3 Acoustoelectric gain and dispersion for bulk modes

The acoustoelectric gain and dispersion can be obtained from the coupled
envelope equations. Neglecting Brillouin coupling, the Heisenberg equations
give

$\displaystyle\dot{B}(z)+i\Omega_{\rm m}B(z)+v_{\rm
g,b}\partial_{z}B(z)=i\sum_{\ell}\int_{0}^{\infty}d\omega\
\kappa_{\omega\ell}\Phi_{\omega\ell}(z)\quad$ (133)
$\displaystyle\dot{\Phi}_{\omega\ell}(z)+i(\omega+v_{\rm d}q_{\rm
m})\Phi_{\omega\ell}(z)+v_{\rm
d}\partial_{z}\Phi_{\omega\ell}(z)=i\kappa^{*}_{\omega\ell}B(z).\quad$ (134)

To find steady-state time harmonic solutions, we assume that
$\Phi_{\omega\ell}$ and $B$ oscillate at the same frequency $\Omega$, giving
the following solution for $\Phi_{\omega\ell}$

$\displaystyle\Phi_{\omega\ell}(z)=i\frac{\kappa^{*}_{\omega\ell}}{v_{\rm
d}}\int_{-\infty}^{z}dz^{\prime}e^{-i(\Omega-\omega-v_{\rm d}q_{\rm
m})(z-z^{\prime})/v_{\rm d}}B(z^{\prime}).$ (135)

Inserting the solution for $\Phi_{\omega\ell}$ into the equation for $B$ we
find

$\displaystyle-i(\Omega-\Omega_{\rm m})B+v_{\rm
g,b}\partial_{z}B=-\sum_{\ell}\int_{-\infty}^{z}dz^{\prime}\int_{0}^{\infty}d\omega\frac{|\kappa_{\omega\ell}|^{2}}{v_{\rm
d}}e^{-i(\Omega-\omega-v_{\rm d}q_{\rm m})(z-z^{\prime})/v_{\rm
d}}B(z^{\prime}).$ (136)

The $\omega$-integral,
$\int_{0}^{\infty}d\omega\frac{|\kappa_{\omega\ell}|^{2}}{v_{\rm
d}}e^{-i(\Omega-\omega-v_{\rm d}q_{\rm m})(z-z^{\prime})/v_{\rm d}}$,
exponentially decays as $\sim\exp\\{-\gamma_{\rm e}(z-z^{\prime})/(2v_{\rm
d})\\}$, which far exceeds typical spatial decay rates for phonons. Under
these conditions the phonon envelope $B(z^{\prime})$ can be replaced with
$B(z)$ and brought outside the integral so that

$\displaystyle-\sum_{\ell}\int_{-\infty}^{z}dz^{\prime}\int_{0}^{\infty}d\omega\frac{|\kappa_{\omega\ell}|^{2}}{v_{\rm
d}}e^{-i(\Omega-\omega-v_{\rm d}q_{\rm m})(z-z^{\prime})}B(z^{\prime})\approx-
iP.V.\sum_{\ell}\int_{0}^{\infty}d\omega\frac{|\kappa_{\omega\ell}|^{2}}{\Omega-\omega-
v_{\rm d}q_{\rm m}}B(z)-\pi\sum_{\ell}|\kappa_{\Omega-v_{\rm d}q_{\rm
m},\ell}|^{2}B(z)$

where we have used $\int_{0}^{\infty}dz\ \exp{(ikz)}=iP.V.\ 1/k+\pi\delta(k)$
and P.V. denotes the Cauchy principal value. These assumptions lead to the
effective acoustoelectric dynamics including gain and dispersion given by

$\displaystyle v_{\rm g,b}\partial_{z}B-i(\Omega-\Omega_{\rm
m}-\Delta\Omega_{0})B-\frac{1}{2}G_{AE}B=0.\quad\quad$ (137)

where

$\displaystyle\Delta\Omega_{\rm
AE}=\sum_{\ell}\int_{0}^{\infty}d\omega|\kappa_{\omega\ell}|^{2}P.V.\frac{1}{\Omega-\omega-
v_{\rm d}q_{\rm m}}$ (138) $\displaystyle
G_{AE}=-2\pi\sum_{\ell}|\kappa_{\Omega-v_{\rm d}q_{\rm m},\ell}|^{2}.$ (139)

While this analysis was completed for bulk modes, the same expressions apply
for the case of the surface modes (described below).

###  C.4 Gain and dispersion for bulk plane-waves

In the uniform plane-wave limit the coupling rate for compressional waves is
given by

$\displaystyle\kappa_{\omega\ell}\approx\frac{\omega_{e}q_{\rm
m}d_{zzz}\chi(\omega)}{\sqrt{4\rho\epsilon\omega\Omega_{\rm
m}}}\delta_{\ell,0}$ (140)

yielding gain and dispersion given by

$\displaystyle G_{\rm AE}=-\frac{\pi\omega_{e}^{2}d^{2}_{zzz}q_{\rm
m}^{2}}{2\rho\epsilon_{in}\Omega_{\rm m}}\frac{|\chi(\Omega-v_{\rm d}q_{\rm
m})|^{2}}{\Omega-v_{\rm d}q_{\rm m}}$ (141) $\displaystyle\Delta\Omega_{\rm
AE}=\frac{\omega_{e}^{2}d^{2}_{zzz}q_{\rm
m}^{2}}{4\rho\epsilon_{in}\Omega_{\rm
m}}P.V.\int_{0}^{\infty}\frac{d\omega}{\omega}|\chi(\omega)|^{2}\frac{1}{\Omega-\omega-
v_{\rm d}q_{\rm m}}.\quad\quad$ (142)

###  C.5 Surface modes

The acoustoelectric dynamics for the surface modes follows by close analogy
with the bulk mode analysis above. In contrast with the bulk modes, the
surface modes have distinct spatial dependence being strongly localized to
surfaces, nontrivial boundary conditions at interfaces, and include the
effects of a singular surface charge density.

####  C.5.1 Surface mode Hamiltonian

The Hamiltonian can be derived from Eq. (97) by: (1) taking
$n\to\sigma\delta(x_{\perp}\in\partial V_{sc})$, where $\sigma$ is a surface
charge density, $\delta(x_{\perp}\in\partial V_{sc})$ is a delta function with
vanishing argument on the boundary of $V_{sc}$, and $x_{\perp}$ is the
coordinate locally normal the boundary of $V_{sc}$, (2) performing an
integration by parts and dropping terms proportional to $\nabla^{2}\psi$ and
$\nabla^{2}\varphi$ that vanish for surface modes, (3) finding the conjugate
momentum $p=-m\dot{\psi}$ (with $\dot{\psi}$ restricted to $\partial V_{sc}$)
to the surface charge density $\sigma$, and (4) performing a Legendre
transform. These steps lead to the Hamiltonian for the surface modes $H_{S}$,
including the impact of a bath (momenta $p_{\nu}$ and position $x_{\nu}$),
given by

$\displaystyle H_{S}=\oint_{\partial
V_{sc}}da\bigg{[}\frac{n_{0}}{2m}p\frac{\partial p}{\partial x_{\perp}}-p{\bf
v}_{\rm
d}\cdot\nabla\sigma+e\sigma\varphi_{S}-\frac{1}{2}\varphi_{S}\bigg{(}\epsilon_{in}\frac{\partial\varphi_{in}}{\partial
x_{\perp}}-\epsilon_{out}\frac{\partial\varphi_{out}}{\partial
x_{\perp}}\bigg{)}+\int_{0}^{\infty}d\nu\bigg{(}\frac{p_{\nu}^{2}}{2m}+\frac{1}{2}m\nu^{2}x_{\nu}^{2}+\hat{\mu}_{\nu}x_{\nu}\sigma\bigg{)}\bigg{]}.\quad\quad$
(143)

Here, $\varphi_{S}$ in the potential restricted to the boundary $\partial
V_{sc}$, $\hat{\mu}_{\nu}$ is the mode-specific (i.e., depending on mode
eigenvalues) system-bath coupling, and it is assumed that the drift velocity
is parallel to the waveguide. The dynamics for this coupled system follow from
Hamilton’s equations, yielding

$\displaystyle\partial_{\tau}\sigma=\frac{n_{0}}{m}\frac{\partial p}{\partial
x_{\perp}}$ (144)
$\displaystyle\partial_{\tau}p=-e\varphi_{S}-\int_{0}^{\infty}d\nu\
\hat{\mu}_{\nu}x_{\nu}$ (145) $\displaystyle
e\sigma=\varphi_{S}\bigg{(}\epsilon_{in}\frac{\partial\varphi_{in}}{\partial
x_{\perp}}-\epsilon_{out}\frac{\partial\varphi_{out}}{\partial
x_{\perp}}\bigg{)}$ (146) $\displaystyle\partial_{\tau}x_{\nu}=p_{\nu}/m$
(147) $\displaystyle\partial_{\tau}p_{\nu}=-\hat{\mu}_{\nu}\sigma.$ (148)

###  C.6 Quantization of surface modes

By generalization of the classical Poisson bracket, i.e., $\\{\sigma({\bf
x}),p({\bf x}^{\prime})\\}=\delta^{2}({\bf x}-{\bf x}^{\prime})$ with ${\bf
x}$ and ${\bf x}^{\prime}$ on the boundary $\partial V_{sc}$ to the ETCR
$[\sigma({\bf x}),p({\bf x}^{\prime})]=i\hbar\delta^{2}({\bf x}-{\bf
x}^{\prime})$ the charge-potential system can be quantized. The bath variables
satisfy an analogous ETCR $[x_{\nu}({\bf x}),p_{\nu^{\prime}}({\bf
x}^{\prime})]=i\hbar\delta(\nu-\nu^{\prime})\delta^{2}({\bf x}-{\bf
x}^{\prime})$. By expressing the charge, potential and bath variables in terms
of the eigenfunctions of the equations of motion Eqs. (144)-(148), the
Hamiltonian can be expressed in terms of creation and annihilation operators
for surface mode quanta.

Noting that both the electric potential and $p$ satisfy the Laplace equation,
we express $\varphi\propto\phi_{\ell}(x_{\|})\exp\\{iqz\\}f_{\ell
q}(x_{\perp})/\sqrt{2\pi}$ where
$\nabla^{2}(\phi_{\ell}(x_{\|})\exp\\{iqz\\}f_{\ell
q}(x_{\perp})/\sqrt{2\pi})=0$ and
$\phi_{\ell}(x_{\|})\exp\\{iqz\\}/\sqrt{2\pi}$ form a complete set of
orthogonal eigenfunctions on the surface of $V_{sc}$. Here, we denote
coordinates of the surface as $\\{x_{\|},z\\}$ where $z$ is directed along the
waveguide. Consequently, the eigenfunctions for the potential satisfy the
following relations

$\displaystyle(\nabla_{\|}^{2}+\partial_{z}^{2})(\phi_{\ell}\exp\\{iqz\\})=-K^{2}_{\ell
q}\phi_{\ell}\exp\\{iqz\\},$ (149) $\displaystyle\oint_{\partial V_{sc}}da\
\phi_{\ell}\phi^{*}_{\ell^{\prime}}\exp{iq(z-z^{\prime})}/(2\pi)=\delta_{\ell\ell^{\prime}}\delta(q-q^{\prime})\quad{\rm
and}$ (150) $\displaystyle\frac{\partial^{2}}{\partial x_{\perp}^{2}}f_{\ell
q}-K^{2}_{\ell q}f_{\ell q}=0.$ (151)

Likewise, $p$ (the velocity potential) can be expressed as
$p\propto\phi_{\ell}(x_{\|})\exp\\{iqz\\}h_{\ell q}(x_{\perp})/\sqrt{2\pi}$
where $\frac{\partial^{2}}{\partial x_{\perp}^{2}}h_{\ell q}-K^{2}_{\ell
q}h_{\ell q}=0$. Critical to the surface mode dynamics are the boundary
conditions for the functions $f_{\ell q}$ and $h_{\ell q}$ on the surface of
$V_{sc}$. These boundary conditions, following directly from Eqs. (144),
(146), and the ETCRs up to normalization, require

$\displaystyle\bigg{[}\frac{\partial h_{\ell q}(x_{\perp})}{\partial
x_{\perp}}-\epsilon_{in}\frac{\partial f_{\ell q}(x_{\perp})}{\partial
x_{\perp}}\bigg{]}_{x_{\perp}\in\partial
V_{sc}-0+}=-\epsilon_{out}\frac{\partial f_{\ell q}(x_{\perp})}{\partial
x_{\perp}}\bigg{|}_{x_{\perp}\in\partial V_{sc}+0+},$ (152) $\displaystyle
h_{\ell q}(x_{\perp})\frac{\partial h_{\ell q}(x_{\perp})}{\partial
x_{\perp}}\bigg{|}_{x_{\perp}\in\partial V_{sc}-0+}=K_{\ell q},$ (153)

and $f_{\ell q}$ to be continuous. Additionally, the appropriate boundary
conditions on the bounding surface of the total system must be satisfied.
Within this eigenfunction basis, and setting
$\hat{\mu}_{\nu}\sqrt{n_{0}K_{\ell q}}\to G_{\nu}\equiv\sqrt{2\gamma_{\rm
e}\nu^{2}/\pi}$, one can show that the following mode representations satisfy
the equations of motion and ETCRs.

$\displaystyle\varphi=\omega_{e}\sum_{\ell}\int
dq\int_{0}^{\infty}d\omega\sqrt{\frac{\hbar\epsilon_{in}}{4\pi\omega K_{\ell
q}}}\bigg{(}\Delta_{\ell q}(\omega)\phi_{\ell}(x_{\|})e^{iqz}f_{\ell
q}(x_{\perp})a_{\omega\ell q}+H.c.\bigg{)}\quad$ (154)
$\displaystyle\sigma=\sum_{\ell}\int
dq\int_{0}^{\infty}d\omega\sqrt{\frac{\hbar n_{0}}{4\pi m\omega K_{\ell
q}}}\bigg{(}\Delta_{\ell q}(\omega)\phi_{\ell}(x_{\|})e^{iqz}H_{\ell
q}a_{\omega\ell q}+H.c.\bigg{)}\quad$ (155) $\displaystyle p=-i\sum_{\ell}\int
dq\int_{0}^{\infty}d\omega\sqrt{\frac{\hbar m\omega}{4\pi n_{0}K_{\ell
q}}}\bigg{(}\Delta_{\ell q}(\omega)\phi_{\ell}(x_{\|})e^{iqz}h_{\ell
q}(x_{\perp})a_{\omega\ell q}-H.c.\bigg{)}\quad$ (156) $\displaystyle
x_{\nu}=\sum_{\ell}\int dq\int_{0}^{\infty}d\omega\sqrt{\frac{\hbar}{4\pi
m\omega}}\bigg{(}\Delta_{\nu\ell
q}(\omega)\phi_{\ell}(x_{\|})e^{iqz}a_{\omega\ell q}+H.c.\bigg{)}\quad$ (157)
$\displaystyle p_{\nu}=-i\sum_{\ell}\int
dq\int_{0}^{\infty}d\omega\sqrt{\frac{\hbar
m\omega}{4\pi}}\bigg{(}\Delta_{\nu\ell
q}(\omega)\phi_{\ell}(x_{\|})e^{iqz}a_{\omega\ell q}-H.c.\bigg{)}\quad$ (158)

when $a_{\omega\ell q}$ and
$a^{\dagger}_{\omega^{\prime}\ell^{\prime}q^{\prime}}$ satisfy Eqs.
(117)-(118) and

$H_{\ell q}=\frac{\partial h_{\ell q}(x_{\perp})}{\partial
x_{\perp}}\bigg{|}_{x_{\perp}\in\partial V_{sc}}.$ (159)

Much like the susceptibility $\chi$ for the bulk modes, the surface modes
satisfy an analogous set of Lippmann-Schwinger equations yielding

$\displaystyle\Delta_{\ell
q}(\omega)=\frac{G_{\omega}}{-\omega(\omega+i\gamma_{\rm e})+\omega_{\ell
q}^{2}}$ (160) $\displaystyle\Delta_{\nu\ell
q}(\omega)=\delta(\omega-\nu)-\frac{G_{\nu}}{-\omega^{2}+\nu^{2}}\Delta_{\ell
q}(\omega)$ (161)

where the surface mode resonance frequency $\omega_{\ell q}$ is determined by
the functions $f_{\ell q}$ and $h_{\ell q}$

$\displaystyle\omega_{\ell q}^{2}=\frac{e^{2}n_{0}}{m}\frac{f_{\ell
q}}{h_{\ell q}}\bigg{|}_{x_{\perp}\in\partial V_{sc}-0+}.$ (162)

The functions $\Delta_{\ell q}$ and $\Delta_{\nu\ell q}$ satisfy the
orthonormality relations Eqs. (121)-(124) with substitution of
$\chi\to\Delta_{\ell q}$ and $\chi_{\nu}\to\Delta_{\nu\ell q}$.

Substitution of Eqs. (154)-(158) into Eq. (143) leads to the Hamiltonian for
the surface modes represented in terms of creation and annihilation operators.
This Hamiltonian, and its formulation in terms of mode envelopes, takes the
exact same form as Eqs. (125) and (1) with the understanding that $\ell$
labels and counts surface modes.

####  C.6.1 Acousto-electric coupling with surface modes

Using the interaction Hamiltonian defined in Eq. (98) and the formal
expression for the electric potential for surface modes Eq. (112), the
coupling rate defined in the envelope picture is

$\displaystyle\kappa_{\omega\ell}=\omega_{e}\sqrt{\frac{1}{4\omega\Omega_{\rm
m}K_{\ell q_{\rm m}}}}\Delta_{\ell q_{\rm m}}(\omega)\int d^{2}x\
\phi_{\ell}(x_{\|})f_{\ell q_{\rm
m}}(x_{\perp})\tilde{\partial}_{k}d_{ijk}\tilde{\partial}_{i}\mathcal{U}^{*}_{q_{\rm
m},j}\quad\quad\ \ $ (163)

where the integral is taken over the waveguide cross-section.

###  C.7 Connection between normal mode picture and $k^{2}$

In this section, we show how the acoustoelectric coupling
$\kappa_{\omega\ell}$ relates to the standard expression for the coupling
$k^{2}$. Generally, the coupling $k^{2}$ is defined as the fractional change
in the square of the speed of sound for a free system and grounded system, or
equivalently in terms of the ration of stored interaction energy to injected
energy [118]. Given the direct connection between the resonance frequency and
the sound speed we estimate $k^{2}$ by the shift in the resonance frequency
$\Omega_{\rm m}$. Assuming that $\Omega_{\rm m}\gg|\Delta\Omega_{\rm AE}|$ we
find

$\displaystyle k^{2}=\frac{\Omega_{\rm m}^{2}-(\Omega_{\rm
m}+\Delta\Omega_{\rm AE})^{2}}{\Omega_{\rm
m}^{2}}\approx-2\frac{\Delta\Omega_{\rm AE}}{\Omega_{\rm m}}.$ (164)

Solving for the frequency shift then allows $k^{2}$ to be calculated. In the
limit that $\omega_{\ell q_{\rm m}}\sim\omega_{e}\gg\Omega_{\rm m}$ and
$\Omega_{\rm m}-v_{\rm d}q_{\rm m}$, the frequency integrals in the expression
for $\Delta\Omega_{\rm AE}$ can be approximated by

$\displaystyle P.V.\int_{0}^{\infty}d\omega\
\frac{|\chi(\omega)|^{2}}{\omega}\frac{1}{\Omega-v_{\rm d}q_{\rm
m}-\omega}=P.V.\int_{0}^{\infty}d\omega\ \frac{2\gamma_{\rm
e}\omega/\pi}{(\omega^{2}-\omega_{e}^{2})^{2}+\gamma_{\rm
e}^{2}\omega^{2}}\frac{1}{\Omega-v_{\rm d}q_{\rm
m}-\omega}\approx-\frac{1}{\omega_{e}^{2}}$ (165) $\displaystyle
P.V.\int_{0}^{\infty}d\omega\ \frac{|\Delta_{\ell q_{\rm
m}}(\omega)|^{2}}{\omega}\frac{1}{\Omega-v_{\rm d}q_{\rm
m}-\omega}=P.V.\int_{0}^{\infty}d\omega\ \frac{2\gamma_{\rm
e}\omega/\pi}{(\omega^{2}-\omega_{\ell q_{\rm m}}^{2})^{2}+\gamma_{\rm
e}^{2}\omega^{2}}\frac{1}{\Omega-v_{\rm d}q_{\rm
m}-\omega}\approx-\frac{1}{\omega_{\ell q_{\rm m}}^{2}}.$ (166)

We find $k^{2}$ for bulk and surface modes given by

$\displaystyle k^{2}=\sum_{\ell}\frac{1}{2\epsilon_{in}\Omega_{\rm
m}^{2}Q_{\ell q_{\rm m}}^{2}}\bigg{|}\int d^{2}x\
\psi_{\ell}\tilde{\partial_{k}}d_{ijk}\tilde{\partial_{i}}\mathcal{U}^{*}_{j}\bigg{|}^{2}\quad({\rm
bulk})$ (167) $\displaystyle
k^{2}=\sum_{\ell}k_{\ell}^{2}=\sum_{\ell}\frac{1}{2\Omega_{\rm m}^{2}K_{\ell
q_{\rm m}}}\frac{\omega^{2}_{e}}{\omega_{\ell q_{\rm m}}^{2}}\bigg{|}\int
d^{2}x\ \phi_{\ell}(x_{\|})f_{\ell q_{\rm
m}}(x_{\perp})\tilde{\partial}_{k}d_{ijk}\tilde{\partial}_{i}\mathcal{U}^{*}_{q_{\rm
m},j}\bigg{|}^{2}\quad({\rm surface}).$ (168)

Using
$\epsilon(\Omega)=\epsilon_{in}[1-\omega_{e}^{2}(\Omega(\Omega+i\gamma_{\rm
e}))^{-1}]$ and $\epsilon_{\ell q_{\rm
m}}(\Omega)=\epsilon_{in}[1-\omega_{\ell q_{\rm
m}}^{2}(\Omega(\Omega+i\gamma_{\rm e}))^{-1}]$ we find

$\displaystyle\frac{|\chi(\omega)|^{2}}{\omega}=-\frac{2\epsilon_{in}}{\pi\omega_{e}^{2}}{\rm
Im}\left[\frac{1}{\epsilon(\omega)}\right]$ (169)
$\displaystyle\frac{|\Delta_{\ell q_{\rm
m}}(\omega)|^{2}}{\omega}=-\frac{2\epsilon_{in}}{\pi\omega_{\ell q_{\rm
m}}^{2}}{\rm Im}\left[\frac{1}{\epsilon_{\ell q_{\rm m}}(\omega)}\right]$
(170)

yielding the compact expressions for the gain given by

$\displaystyle G_{AE}=2k^{2}\Omega_{\rm m}\epsilon_{in}{\rm
Im}\left[\frac{1}{\epsilon(\Omega-v_{\rm d}q_{\rm m})}\right]\quad({\rm
bulk})$ (171) $\displaystyle G_{AE}=\sum_{\ell}2k_{\ell}^{2}\Omega_{\rm
m}\epsilon_{in}{\rm Im}\left[\frac{1}{\epsilon_{\ell q_{\rm m}}(\Omega-v_{\rm
d}q_{\rm m})}\right]\quad({\rm surface}).$ (172)

##  Supplementary Information D Noise dynamics

In this section, we examine the noise dynamics of acoustoelectric enhanced
Brillouin interactions. We begin with the envelope equations of motion as
detailed in Section III, which are given by

$\displaystyle\frac{\partial\bar{B}}{\partial t}$
$\displaystyle=-i(\Omega_{\rm m}-\Omega)\bar{B}-\frac{\Gamma}{2}\bar{B}\pm
v_{\rm g,b}\frac{\partial\bar{B}}{\partial z}-ig_{0}^{*}\bar{A}_{\rm
s}^{\dagger}\bar{A}_{\rm p}+\eta$ (173)
$\displaystyle\frac{\partial\bar{A}_{\rm p}}{\partial t}$
$\displaystyle=-\frac{\alpha v_{\rm g,p}}{2}\bar{A}_{\rm p}\pm v_{\rm
g,p}\frac{\partial\bar{A}_{\rm p}}{\partial z}-ig_{0}\bar{A}_{\rm
s}\bar{B}+\xi_{\rm p}$ $\displaystyle\frac{\partial\bar{A}_{\rm s}}{\partial
t}$ $\displaystyle=-\frac{\alpha v_{\rm g,s}}{2}\bar{A}_{\rm s}\pm v_{\rm
g,s}\frac{\partial\bar{A}_{\rm s}}{\partial z}-ig^{*}_{0}\bar{A}_{\rm
p}\bar{B}^{\dagger}+\xi_{\rm s}$

We again note that Eq. 173 includes the effects of dissipation in an open
system, and as such, we include thermal and vacuum noise terms $\eta$,
$\xi_{\rm p}$, and $\xi_{\rm s}$ according to the fluctuation-dissipation
theorem for the phonon, probe, and pump fields, respectively. These Langevin
terms have auto correlation functions given by
$\langle\eta(z^{\prime},t^{\prime})\eta^{\dagger}(z^{\prime},t^{\prime})\rangle=(n_{\rm
th}+1)\Gamma\delta(z^{\prime}-z)\delta(t^{\prime}-t)$, $\langle\xi_{\rm
s}(z^{\prime},t^{\prime})\xi_{\rm
s}^{\dagger}(z^{\prime},t^{\prime})\rangle=\gamma_{\rm
s}\delta(z^{\prime}-z)\delta(t^{\prime}-t)$, and $\langle\xi_{\rm
p}(z^{\prime},t^{\prime})\xi_{\rm
p}^{\dagger}(z^{\prime},t^{\prime})\rangle=\gamma_{\rm
p}\delta(z^{\prime}-z)\delta(t^{\prime}-t)$, respectively, where $\alpha$ is
the optical spatial loss rate and $n_{\rm th}$ is the thermal phonon
occupation given by the Bose-Einstein distribution [25].

For simplicity, we neglect noise processes intrinsic to acoustoelectric
interactions, such as those from trapping effects [111], which for the
proposed system are expected to be smaller than contributions from
thermomechanical phonon noise. From the normal mode theory (and given the
parameters considered in this work), we estimate sources of intrinsic
acoustoelectric noise to be approximately 40% of the background thermal noise
[111] (see Section A and Fig. 14).

We once again treat the pump wave as undepleted and transform these equations
into the Fourier domain as

$\displaystyle-i\omega\bar{A}_{\rm s}[z,\omega]+\frac{\alpha v_{\rm
g,s}}{2}\bar{A}_{\rm s}[z,\omega]+v_{\rm g,s}\frac{\partial\bar{A}_{\rm
s}[z,\omega]}{\partial z}$ $\displaystyle=-ig_{0}^{*}\bar{A}_{\rm
p}\bar{B}^{\dagger}[z,\omega]+\xi_{\rm s}[z,\omega]$ (174) $\displaystyle
i(\Omega-\Omega_{\rm m}-\Delta\Omega_{\rm
AE}-\omega)\bar{B}[z,\omega]+\frac{\Gamma-G_{\rm
AE}}{2}\bar{B}[z,\omega]+v_{\rm g,b}\frac{\partial\bar{B}[z,\omega]}{\partial
z}$ $\displaystyle=-ig_{0}^{*}\bar{A}_{\rm p}\bar{A}^{\dagger}_{\rm
s}[z,\omega]+\eta[z,\omega]$

We solve these equations in the acoustoelectric enhanced Brillouin (AEB)
limit, in which, despite significant phonon amplification, the coherence
length of the phonon field is still much smaller than that of the optical
fields. Given this hierarchy, we eliminate the spatial dynamics of the phonon
field, such that

$\displaystyle\bar{B}[z,\omega]=\chi^{\rm AE}_{\rm
B}[\omega]\left(-ig_{0}^{*}\bar{A}_{\rm p}\bar{A}_{\rm
s}^{\dagger}[z,\omega]+\eta[z,\omega]\right),$ (175)

and the decoupled Stokes dynamics are given by

$\displaystyle\frac{\partial\bar{A}_{\rm s}[z,\omega]}{\partial
z}=-M\bar{A}_{\rm s}[z,\omega]+\frac{N_{\rm s}[z,\omega]}{v_{\rm g,s}}$ (176)

where $M=(-i\omega+\alpha v_{\rm g,s}/2-\chi_{\rm s}^{\rm AEB*})/v_{\rm g,s}$
and the dressed Langevin term $N_{\rm s}[z,\omega]=\xi_{\rm
s}[z,\omega]-ig_{0}^{*}\bar{A}_{\rm p}\chi_{\rm B}^{\rm
AE^{*}}\eta^{\dagger}[z,\omega]$. As defined in the main text, $\chi^{\rm
AE}_{\rm B}[\omega]=(i(\Omega-\Omega_{\rm m}-\Delta\Omega_{\rm
AE}-\omega)+(\Gamma-G_{\rm AE})/2)^{-1}$ and $\chi_{\rm s}^{\rm
AEB}[\omega]=|g_{0}|^{2}|\bar{A}_{\rm p}|^{2}\chi_{\rm B}^{\rm AE}[\omega]$.

The first order ordinary differential equation has the integral solution

$\displaystyle\bar{A}_{\rm s}[z,\omega]$ $\displaystyle=\frac{1}{v_{\rm
s}}\int_{0}^{z}N_{\rm s}[z^{\prime},\omega]e^{-M(z-z^{\prime})}$ (177)
$\displaystyle+\Big{(}\bar{A}^{\rm c}_{\rm s}[0,\omega]+\bar{A}^{\rm N}_{\rm
s}[0,\omega]\Big{)}e^{-Mz}$

where $\bar{A}^{\rm c}_{\rm s}[0,\omega]$ and $\bar{A}^{\rm N}_{\rm
s}[0,\omega]$ represent the coherent and noise input of the Stokes wave.

The number spectral density per unit length (defined by $\mathscr{A}_{\rm
s}[z,\omega]=\lim_{T\rightarrow\infty}(1/T)|\bar{A}_{\rm s}[z,\omega]|^{2}$)
is

$\displaystyle\mathscr{A}_{\rm s}[z,\omega]$ $\displaystyle=(\mathscr{A}_{\rm
s}^{\rm c}[0,\omega]+\mathscr{A}_{\rm s}^{\rm
N}[0,\omega])e^{-2\Re[M]z}+\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{z}dz_{1}\int_{0}^{z}dz_{2}\frac{\langle
N_{\rm s}[z_{1},\omega]N^{\dagger}_{\rm s}[z_{2},\omega]\rangle}{v_{\rm
g,s}^{2}}e^{-M^{*}(z-z_{1})-M(z-z_{2})}$ (178)
$\displaystyle=\underbrace{\mathscr{A}_{\rm s}^{\rm
c}[0,\omega]e^{-2\Re[M]z}}_{\rm Amplified\>signal}+\underbrace{\frac{1}{v_{\rm
g,s}}e^{-2\Re[M]z}+\frac{\alpha v_{\rm g,s}+|g_{0}|^{2}|A_{\rm
p}|^{2}|\chi_{\rm B}^{\rm AE}[\omega]|^{2}\Gamma n_{\rm th}}{2v_{\rm
g,s}^{2}\Re[M]}\Big{[}1-e^{-2\Re[M]z}\Big{]}}_{\rm
Amplified\>vacuum\>and\>thermal\>fluctuations}$

where $\Re[M]$ represents the real part of $M$ as defined above.

The noise factor ($F$) is defined by the ratio of the input
($\textup{SNR}_{1}$) to output ($\textup{SNR}_{2}$) signal to noise ratios,
such that

$\displaystyle F\equiv\frac{\textup{SNR}_{1}}{\textup{SNR}_{2}}.$ (179)

For simplicity, we assume an input signal defined by $\mathscr{A}_{\rm s}^{\rm
c}[0,\omega]=|A^{\rm in}_{\rm s}|^{2}\delta(\omega)$. The relevant input noise
for an optical amplifier is the vacuum noise [119], such that the input signal
to noise ratio is given by

$\displaystyle\textup{SNR}_{1}$
$\displaystyle=\frac{\int_{-\Delta\omega}^{\Delta\omega}d\omega^{\prime}|A_{\rm
s}^{\rm
in}|^{2}\delta(\omega^{\prime})}{\int_{-\Delta\omega}^{\Delta\omega}d\omega^{\prime}\frac{1}{v_{\rm
g,s}}}$ (180) $\displaystyle=\frac{|A_{\rm s}^{\rm in}|^{2}v_{\rm
g,s}}{2\Delta\omega}.$

The output SNR can be obtained from the amplified signal and noise in Eq. 178,
which yields

$\displaystyle\textup{SNR}_{2}=\frac{|A_{\rm s}^{\rm
in}|^{2}e^{-2\Re[M]z}}{2\Delta\omega\Bigg{[}\frac{1}{v_{\rm
g,s}}e^{-2\Re[M]z}+\frac{|g_{0}|^{2}|A_{\rm p}|^{2}|\chi_{\rm B}^{\rm
AE}[\omega]|^{2}\Gamma n_{\rm th}}{2v_{\rm
g,s}^{2}\Re[M]}\bigg{[}1-e^{-2\Re[M]z}\bigg{]}\Bigg{]}}.$ (181)

where $\Delta\omega$ is a bandwidth much smaller than the acoustoelectric
modified Brillouin gain bandwidth (i.e., $\Delta\omega\ll\Gamma-G_{\rm AE}$).

The resulting noise factor is given by

$\displaystyle F$
$\displaystyle\equiv\frac{\textup{SNR}_{1}}{\textup{SNR}_{2}}$ (182)
$\displaystyle=1+\frac{\alpha v_{\rm g,s}+|g_{0}|^{2}|\bar{A}_{\rm
p}|^{2}|\chi_{\rm B}^{\rm AE}[\omega]|^{2}\Gamma n_{\rm th}}{2v_{\rm
g,s}\Re[M]}\bigg{[}e^{2\Re[M]z}-1\bigg{]}.$

In the limit of large Brillouin amplification and low optical loss, the noise
factor simplifies to

$\displaystyle F\approx 1+n_{\rm th}\left(\frac{\Gamma}{\Gamma-G_{\rm
AE}}\right),$ (183)

which is the central result of this section. This derivation suggests that for
systems in which the thermomechanical noise is dominant, AEB amplifiers may
achieve near quantum-limited performance in the limit of low temperatures
(i.e., $k_{\rm B}T\ll\hbar\Omega_{\rm m}$).

##  Supplementary Information E Index of Notation

Acoustoelectric frequency shift |  $\Delta\Omega_{\rm AE}$ |  |  Phonon group velocity |  $v_{\text{g,b}}$
---|---|---|---|---
Acoustoelectric gain |  $\alpha_{\text{AE}}$ |  |  Phonon lifetime |  $\tau_{\text{m}}$
Acoustoelectric gain rate |  $G_{\rm AE}$ |  |  Phonon phase velocity |  $v_{\text{m}}$
Acoustoelectric phase delay |  $q_{\text{AE}}$ |  |  Phonon scaling factor |  $C_{\text{m}}$
Acoustoelectric time constant |  $\tau$ |  |  Phonon wave vector |  $q_{\text{m}}$
Boltzmann constant |  $k_{\text{B}}$ |  |  Phonon wavelength |  $\lambda_{\text{m}}$
Compliance matrix constant |  $S_{ij}$ |  |  Photoelastic coupling coefficient |  $g_{\text{pe}}$
Composition parameters |  $x\,\,\,\,y$ |  |  Photoelastic matrix constant |  $p_{ij}$
Debye length |  $\lambda_{\text{d}}$ |  |  Photon dissipation rate |  $\gamma$
Density |  $\rho$ |  |  Photon energy |  $E$
Dielectric gap height |  $h$ |  |  Piezoelectric coupling coefficient |  $k^{2}$
Dielectric gap permittivity |  $\epsilon_{\text{g}}$ |  |  Piezoelectric permittivity |  $\epsilon_{\text{p}}$
Dielectric relaxation frequency |  $\omega_{\text{c}}$ |  |  Plasma frequency |  $\omega_{\text{e}}$
Diffusion frequency |  $\omega_{\text{D}}$ |  |  Propagation angle |  $\beta$
Diffusion term |  $D$ |  |  Pump photon dissipation rate |  $\gamma_{\text{p}}$
Effective index |  $n_{\text{eff}}$ |  |  Pump photon envelope operator |  $A_{\text{p}}$
Effective mass |  $m^{*}$ |  |  Pump photon frequency |  $\omega_{\text{p}}$
Elastic mode loss coefficient |  $\alpha_{\text{m}}$ |  |  Pump photon group velocity |  $v_{\text{g,p}}$
Elastic mode quality factor |  $Q_{\text{m}}$ |  |  Pump photon scaling factor |  $C_{\text{p}}$
Elastic strain |  $\varepsilon_{ij}$ |  |  Pump photon wave vector |  $k_{\text{p}}$
Elastic wave displacement |  $\mathrm{u}_{i}$ |  |  Pump photon wavelength |  $\lambda_{\text{p}}$
Elasticity matrix constant |  $C_{ij}$ |  |  Radiation pressure coupling coefficient |  $g_{\text{rp}}$
Electric displacement field |  $\mathrm{D}_{i}$ |  |  Refractive index |  $n_{\text{r}}$
Electric field |  $\mathrm{E}_{i}$ |  |  Scattering efficiency |  $\eta_{\text{ef}}$
Electron charge |  $e$ |  |  Semiconductor permittivity |  $\epsilon_{\text{s}}$
Electron mass |  $m_{\text{e}}$ |  |  Space charge potential factor |  $M$
Free carrier concentration |  $N$ |  |  Space charge reduction factor |  $R$
Free carrier drift velocity |  $v_{\text{d}}$ |  |  Stokes photon dissipation rate |  $\gamma_{\text{s}}$
Free carrier mobility |  $\mu$ |  |  Stokes photon envelope operator |  $A_{\text{s}}$
Free electron scattering rate |  $\gamma_{\text{e}}$ |  |  Stokes photon frequency |  $\omega_{\text{s}}$
Interaction Impedance |  $Z_{\text{m}}$ |  |  Stokes photon group velocity |  $v_{\text{g,s}}$
Inverse Debye length |  $\gamma_{\text{d}}$ |  |  Stokes photon scaling factor |  $C_{\text{s}}$
Linear piezobirefringence coefficient |  $\alpha_{\text{pe}}$ |  |  Stokes photon wave vector |  $k_{\text{s}}$
Non-dimensionalized velocity |  $\gamma_{v}$ |  |  Stokes photon wavelength |  $\lambda_{\text{s}}$
Optomechanical coupling coefficient |  $g_{0}$ |  |  Temperature |  $T$
Permittivity of free space |  $\epsilon_{0}$ |  |  Top dielectric permittivity |  $\epsilon_{\text{d}}$
Phonon dissipation rate |  $\Gamma_{\text{m}}$ |  |  Waveguide thickness |  $t$
Phonon envelope operator |  $B$ |  |  Waveguide width |  $w$
Phonon frequency |  $\Omega_{\text{m}}$ |  |  |

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