# Role of optical rectification in photon-assisted tunneling current

P. Février, J. Basset, J. Estève, M. Aprili, J. Gabelli  Laboratoire de
Physique des Solides, CNRS, Université Paris-Sud, Université Paris-Saclay,
Orsay, France

###### Abstract

We study the optical rectification in a metallic tunnel junction. We consider
a planar junction in a Kretschmann configuration and measure the photon-
assisted tunneling under infrared illumination at
$\lambda=1.5\,\mu\mathrm{m}$. To address the microscopic mechanism at the
origin of the optical rectification, we compare the photon assisted current
and the current-voltage characteristics of the junction measured on a voltage
range much greater than $V_{0}=\frac{hc}{e\lambda}=0.825\,\mathrm{V}$. The
experimental results do not agree with the Tucker theory based on the exchange
of energy quanta between electrons and photons and describing the dc current
induced by photon-assisted processes in terms of a linear combination of the
shifted characteristics $I(V)$ and $I(V\pm V_{0})$. We show instead that the
illumination power mainly goes into heating and that the rectification results
mainly from the non-linearity of the tunnel junction at optical frequency.

##  I Introduction

Recent years have seen tremendous progress in nanophotonics with the
realization of optical antennas which can receive Muhlschlegel2005 ;
Bharadwaj2009 and emit radiation at optical frequencies Qian_2018 ; Zhang_2019
; Qian_2021 . In terms of detectors, nanoantennas allow an efficient transfer
of radiation into surface plasmons and they close the gap between the
characteristic lengths of optics (micrometer scale) and electronic devices
(nanometer scale). Thus, if coupled with a non-linear element, they can be
used for optical rectification. One thus refers to rectenna: a nanoantenna
converts the optical radiation into an ac voltage which is in turn converted
to a dc current through a rectifier (diode). However, optical rectification in
conductors raises central questions. (i) Is it possible to define a voltage at
optical frequencies as used in the microwave domain? A microwave excitation of
frequency $\omega$ at low temperature ($k_{\mathrm{B}}T\ll\hbar\omega$)
actually induces discontinuities in the derivative of distribution function at
energies $\epsilon=n\hbar\omega$ with $n$ integer Kozhevnikov_2000 . (ii) Is
it possible to observe this kind of step like distribution function at optical
energies? While they are linear elements at low voltage bias (i.e
$V_{\mathrm{dc}}\leq 10\,\mathrm{mV}$ ), Metal-Insulator-Metal (MIM) junction
becomes strongly non-linear at optical frequencies ($\hbar\omega\sim
1\,\mathrm{eV}$) and are promising rectifiers. The goal of this article is to
focus on MIM junctions as an optical rectifier and to address the underlying
microscopic mechanism.

The photon-assisted tunnelling (PAT) has been investigated in experiments
using microwave radiation, where the illumination is simply described by an ac
bias voltage, $V_{\mathrm{ac}}\cos\omega t$, across the conductor Tucker_1985
; Kouwenhoven_1994 ; Gabelli_2008 . In this regime, the PAT is well described
in the quantum limit, $\hbar\omega\gg k_{\mathrm{B}}T$, by the approach of
Tien and Gordon Tien_1963 where this time dependent potential, creates a non-
equilibrium distribution function in the reservoir absorbing the radiation and
corresponding to energy quanta exchange between electrons and microwave
photons. It is tempting to describe the optical rectification within the same
theory by describing the illumination by an optical voltage $V_{\mathrm{ac}}$.
In the following, we consider the excess current due to illumination: $\delta
I(V_{\mathrm{dc}},V_{\mathrm{ac}})=I(V_{\mathrm{dc}},V_{\mathrm{ac}})-I_{0}(V_{\mathrm{dc}})$
where $I_{0}$ is the current in the absence of incident light and
$I(V_{\mathrm{dc}},V_{\mathrm{ac}})$ the current under illumination. As
demonstrated below , the time-averaged photon-assisted tunneling current
calculated from the distribution functions predicted by Tien and Gordon, is
then given at the first non-vanishing order in $V_{\mathrm{ac}}$, by the
Tucker formula (TF):

$\begin{split}\delta
I(V_{\mathrm{dc}},V_{\mathrm{ac}})=&\left(\frac{eV_{ac}}{2\hbar\omega}\right)^{2}\left[I_{0}\left(V_{\mathrm{dc}}+V_{0}\right)\right.\\\
&\left.+I_{0}\left(V_{\mathrm{dc}}-V_{0}\right)-2I_{0}\left(V_{\mathrm{dc}}\right)\right],\end{split}$
(1)

where $V_{0}=\hbar\omega/e$, $\hbar$ is the Planck constant and $e$ the
elementary charge. Below we will address the full quantum regime $V_{0}\approx
V_{\mathrm{dc}}$. This is technically demanding as it requires to be able to
polarize a MIM junction with a voltage bias $V_{\mathrm{dc}}>V_{0}\approx 1V$
without deteriorating the device. We are not aware of previous work in this
regime. To do so, we have chosen to perform the experiment on a robust
Al/AlOx/Al tunnel junction at low temperature to prevent junction aging
effects as experimentally verified in our previous works Fevrier_2018 . We use
infrared radiation $\lambda=1550\,\mathrm{nm}$ corresponding to
$V_{0}=0.825\,\mathrm{V}$. We found that PAT in MIM junctions can not be
explained by the theory of quantum detection as expected by eq. (1). Knowing
that the Fermi-liquid theory Pine_Nozieres_1966 gives lifetime of an excited
electron of the order of
$\tau\sim\tau_{0}(\epsilon_{\mathrm{F}}/\delta\epsilon)^{2}$, the lifetime of
quasiparticles excited by optical photons is more than eight order of
magnitude shorter than the ones excited by microwaves. For electrons with
energy $\delta\epsilon=eV_{0}=0.825\,\mathrm{eV}$ above the Fermi energy, we
estimate $\tau\sim 100\,\mathrm{fs}$. This lifetime can only be accessed by
time-resolved photoemission spectroscopy Fann_1992 ; Campillo_1999 . As a
consequence, the tunnel spectroscopy under continuous wave illumination can
hardly reveal non-thermal distributions Sivan2019 . As a result, our data are
well described by thermal distribution functions characterized by a
temperature in each electrode and highlight a tunneling Seebeck effect.

Figure 1: a Experimental setup used to measure the photon assisted tunneling
induced by a laser illumination at infrared frequencies. b We consider a
Kretschmann configuration where a right angle prism allowed to enhance the
coupling between the tunneling current and the light via the excitation of
surface plasmons. c The tunnel barrier is modeled by a trapezoidal barrier.

The article is organized as follows. The sample and the experimental setup are
described in section II. To increase the amount of radiation absorbed by the
MIM, we convert the far field radiation into a local electromagnetic mode by
means of a surface plasmon polariton (SPP) resonance. We probe the SPP
resonance by PAT in section III. We then measure the right and left hand sides
of eq. (1) independently. We discuss the validity of eq. (1) in the optical
regime by using the Simmons tunnel barrier model in section IV Simmons_1964 .
In order to probe in more detail the interplay between illumination and the
electron distribution in metallic thin films, we compare our measurement with
the two possible senari: the hot carrier distribution functions characterized
by a step like function at optical energies and the equilibrium distribution
function at different temperatures.

##  II Sample fabrication and experimental setup

The sample is a $100\times 100\,\mu\mathrm{m}^{2}$ metallic cross-junction
deposited on a standard microscope cover glass. The aluminum electrodes
thickness is $a=8\,\mathrm{nm}$ and the tunneling barrier is obtained by Al
oxidation performed in a glow discharge. The $I(V)$ characteristics fit
Simmons_1964 enables to determine the barrier characteristics depicted in Fig.
1 c: thickness $d=1.6\,\mathrm{nm}$, barrier height $U_{0}=2.9\,\mathrm{eV}$
and barrier asymmetry $\Delta U_{0}=-0.78\,\mathrm{eV}$. A right angle prism
is glued with a UV-curing glue on the rear of the glass to allow the SPP
excitation by the incoming optical radiation in a Kretschmann configuration
Kretschmann_1972 (see Fig. 1 b). The experimental setup is shown in Fig. 1a.
The junction is mounted on a rotating stage in a cryogen-free refrigerator at
$T_{0}=2.8\,\mathrm{K}$ enabling the excitation of the SPP resonance (SPR).
The characteristics of the junction is measured using a standard four-point
probe method simultaneously with the photon assisted current. The current
generated due to the radiation is given by a lock-in amplifier referenced to
the optical chopper at frequency $f_{\mathrm{chopper}}$ ranging from
$80\,\mathrm{Hz}$ to $900\,\mathrm{Hz}$. The SPP mode is transverse magnetic.
It is excited by a p-polarized light at the resonance angle
$\arcsin(1/n)\simeq 41.1^{\circ}$ with $n=1.52$ the refractive index of the
cover glass (see Appendix A). The p-polarization is set by adjusting a linear
polarizer in the fridge at room temperature and small bias voltage before
cooling. Figure 2 clearly shows the effect of the field enhancement due to
surface plasmon resonance on the PAT Berthold_1985 at $\alpha_{res}=\pm
39.7^{\circ}$. In the following, we will consider PAT current induced by this
resonance. It corresponds to an enhancement of the electromagnetic field in
the tunnel junction by a factor 3.

Figure 2: a. Theoretical value of the ratio between the absorbed power
$P_{\mathrm{abs.}}$ and the laser power $P_{\mathrm{laser}}$ as a function of
the angle $\alpha$. b Detection of the SPP resonance by measuring the PAT at
room temperature.

##  III Photon-assisted tunneling at infrared frequencies

The responsivity $\mathcal{R}$ of a rectifier is given by the ratio between
the photon-assisted current $\delta I$ and the optical absorbed power $P$.
Let’s assume that optical excitation induces an ac signal of amplitude
$V_{\mathrm{ac}}$ at frequency $\omega$. The related ac current is given by
$I_{\mathrm{ac}}=G(\omega)V_{\mathrm{ac}}$ where $G(\omega)$ stands for the
conductance at frequency $\omega$. In the classical limit of small ac
excitation ($\hbar\omega\ll eV_{\mathbf{dc}}$ and $V_{\mathrm{ac}}\ll
V_{\mathrm{dc}}$), the conductance is associated to the adiabatic response of
the $I(V)$ characteristics $G(\omega)=I^{\prime}(V)=\frac{dI}{dV}$ and the
time-averaged current is proportional to its second derivative $\delta
I=I^{\prime\prime}(V_{\mathrm{dc}})\times V_{\mathrm{ac}}^{2}/2$. Surprisingly
this classical limit seems to account for the experimental results of Ward et
al Ward_2010 out of its regime of validity $V_{0}\gg V_{\mathrm{dc}}$. With
the hypothesis that the optical absorbed power is equal to the electrical one
$P=G(\omega)V_{\mathrm{ac}}^{2}$, the responsivity can be rewritten by
$\mathcal{R}=\frac{I^{\prime\prime}(V_{\mathrm{dc}})}{2I^{\prime}(V_{\mathrm{dc}})}$
as for a classical diode. Following the same assumption and using the Tucker’s
description of the electronic transport in a metallic tunnel junction at
finite frequencies, the conductance is given by
$G(\omega)=\frac{I(V_{\mathrm{dc}}+V_{0})-I(V_{dc}-V_{0})}{2V_{0}}$ and the
responsivity in the quantum regime reads Grover2013 :

$\mathcal{R}(V_{\mathrm{dc}})=\frac{1}{V_{0}}\frac{I_{0}(V_{\mathrm{dc}}+V_{0})+I_{0}(V_{dc}-V_{0})-2I_{0}(V_{\mathrm{dc}})}{I_{0}(V_{\mathrm{dc}}+V_{0})-I_{0}(V_{\mathrm{dc}}-V_{0})}.$
(2)

Figure 3 shows the experimental responsivity
$\mathcal{R}^{(\mathrm{exp.})}(V_{dc})=\delta
I^{\mathrm{(PAT)}}/P_{\mathrm{abs.}}$ measured as a function of the bias
voltage for different laser powers at SPR. The absorbed power at SPR is
proportional to the laser power: $P_{\mathrm{abs.}}\simeq
0.07P_{\mathrm{laser}}$ (see Fig. 2a and Appendix A). As shown in eq. (2), the
responsivity does not depend on the illumination power. The conversion
efficiency of the metallic tunnel junction is independent of the incident
optical power and acts as linear photon detector for the investigated power
range ($P_{\mathrm{laser}}<100\,\mathrm{mW}$). We estimate the maximal
experimental responsivity at SPR:
$\mathcal{R}^{(\mathrm{exp.})}(V_{\mathrm{dc}}=1.7\,\mathrm{V})\sim
400\mathrm{nA}.\mathrm{W}^{-1}$. By using the measured $I(V)$ characteristics
between $-1.7\,\mathrm{V}$ and $1.8\,\mathrm{V}$ (see inset Fig. 3), we
estimate the theoretical responsivity $\mathcal{R}(V_{\mathrm{dc}})$ between
$-0.88\,\mathrm{V}$ and $0.97\,\mathrm{V}$. We see in Fig. 3 that
$\mathcal{R}^{(\mathrm{exp.})}$ does not have the same voltage dependence
(black dashed line) and is more than seven orders of magnitude smaller than
the theoretical value given by eq. (2). This requires some more investigation
which we do here below.

Figure 3: Normalized PAT current $\delta I^{\mathrm{(PAT)}}/P_{\mathrm{abs}}$
for laser power ranging between $11\,\mathrm{mW}$ and $53\,\mathrm{mW}$. The
width of the laser spot on the sample is
$w_{\mathrm{laser}}=650\,\mu\mathrm{m}$. The measurement is performed at
$T_{0}=2.8\,\mathrm{K}$ at the SPP resonance
$\theta_{\mathrm{res}}=-39^{\circ}$. Inset bottom right: $I(V)$ and $I(V\pm
V_{0})$ characteristics of the tunnel junction measured at low temperature
$T_{0}=2.8\,\mathrm{K}$. The fit of $I(V)$ with the Simmons theory gives
$d=1.6\,\mathrm{nm}$, $U_{0}=2.9\,\mathrm{eV}$, $\Delta
U_{0}=-0.78\,\mathrm{eV}$.

##  IV Validity of the Tucker formula

Equation (2) is convenient because it directly gives the photon responsivity
of the junction from its electrical characteristics $I(V)$. However, this
description is based on two strong hypothesis: (i) the photon-assisted
processes can be described by an optical voltage biasing of the junction and
(ii) the absorbed power is given by the electric resistance of the junction
evaluated at optical frequency. Let us briefly show how to establish eq. (1).
The effect of voltage bias
$V(t)=V_{\mathrm{dc}}+V_{\mathrm{ac}}\cos\left(\frac{eV_{0}}{\hbar}t\right)$
due to the dc bias and the laser illumination is to shift adiabatically the
Fermi sea of the upper reservoir sustaining the surface plasmon while the
lower reservoir is grounded. The resulting time averaged out-of-equilibrium
distribution function $\tilde{f}$ in the upper reservoir is then given by the
Tien-Gordon formula Tien_1963 ; Blanter_2000 :

$\tilde{f}(\epsilon,T_{0})=\sum_{n=-\infty}^{+\infty}\left|J_{n}\left(\frac{V_{\mathrm{ac}}}{V_{0}}\right)\right|^{2}f(\epsilon+neV_{0},T_{0}),$
(3)

where $J_{n}$ is the $n^{\mathrm{th}}$ Bessel’s function of the first kind and
$f(\epsilon,T_{0})=1/\left(1+\exp((\epsilon-\epsilon_{F})/k_{B}T_{0})\right)$
is the Fermi–Dirac distribution with $\epsilon_{F}$ the Fermi energy and
$T_{0}$ the electron temperature. Using a Landauer-Büttiker scattering
approach, we express the time-average PAT current as:

$\begin{split}\delta I(V_{\mathrm{dc}},V_{\mathrm{ac}})&=\frac{e}{h}\int
d\epsilon\mathcal{T}(\epsilon,eV_{\mathrm{dc}})\left[\tilde{f}(\epsilon-
eV_{\mathrm{dc}},T_{0})\right.\\\ &\left.-f(\epsilon-
eV_{\mathrm{dc}},T_{0})\right].\end{split}$ (4)

Using $\sum_{n}\left|J_{n}\right|^{2}=1$ and assuming that transmission is
voltage independent $\mathcal{T}(\epsilon,eV_{\mathrm{dc}})=\mathrm{cte}$ ,
one deduces:

$\delta
I=\sum_{n=-\infty}^{+\infty}\left|J_{n}\left(\frac{V_{\mathrm{ac}}}{V_{0}}\right)\right|^{2}I\left(V_{\mathrm{dc}}+nV_{0}\right)-I\left(V_{\mathrm{dc}}\right),$
(5)

If one only considers the one photon apsorption/emission processes at energies
$e\left(V_{dc}\pm V_{0}\right)$ ($V_{\mathrm{ac}}\ll V_{0}$), eq. (5)
corresponds to eq. (1) by retaining the lowest order terms in
$V_{\mathrm{ac}}/V_{0}$ . Note that eq. (4) is explicitly not gauge invariant
Aguado2003 . This hypothesis is valid in the microwave regime where the
alternative electric field is well described by a voltage drop across the
junction (one electrode is effectively grounded). However, it remains a strong
assumption in optics where the lower electrode is not grounded at optical
frequencies and the associated vector potential does play a role. In what
follows, we focus on the experimental verification of eq. (1). To do so, we
probe $\delta I$ in a range of $\pm V_{\mathrm{exp}}$ and the $I(V)$
characteristics in a range of $\pm\left(V_{\mathrm{exp}}+V_{0}\right)\simeq\pm
1.8\,\mathrm{V}$. Figure 4 shows the PAT current measured by optically
chopping of the laser beam at $\sim 120\,\mathrm{Hz}$. The data are compared
to the TF photon assisted tunneling current $\delta I_{\mathrm{Tucker}}$ given
by eq. (1). Note that this quantity is directly measured. The only free
parameter is the optical voltage $V_{\mathrm{ac}}$ which is adjusted with the
slope of $\delta I^{\mathrm{(PAT)}}$ at zero dc bias. We then deduce an
optical voltage $V_{\mathrm{ac}}^{(\mathrm{Tucker})}=1.52\,\mathrm{mV}$. This
value is overestimated by more than four orders of magnitude in view of the
laser power (according to Appendix A $P_{\mathrm{laser}}=53\,\mathrm{mW}$
corresponds to $V_{ac}\simeq 70\,\mathrm{nV}$). More importantly equation (1)
does not account for the experimental data neither qualitatively or
quantitatively. As mentioned above, this equation assumes that tunnel
transmission is voltage independent:
$\mathcal{T}(\epsilon,eV_{\mathrm{dc}})=\mathcal{T}_{0}(\epsilon)$. This is
not the case at bias voltage of the order of the tunnel barrier height.
Instead, the transmission $\mathcal{T}(\epsilon,eV_{\mathrm{dc}})$ is
calculated by modelling the junction by a trapezoidal barrier Simmons_1964
(see inset Fig. 3 and Appendix B). A numerical evaluation of eq. (4) gives a
smaller but still unrealistic optical voltage $V_{\mathrm{ac}}^{(\mathrm{Tien-
Gordon})}=0.48\,\mathrm{mV}$. In conclusion, the fitting parameters using TF
are not physical. Our experiment confirms that the description of the photon-
assisted tunneling as an optical rectification cannot explain the photon-
assisted processes in a tunnel junction. A more realistic description is to
consider the Boltzmann equation (BE) describing electron dynamics in metallic
systems Voisin_2000 .

Figure 4: PAT current as a function of dc voltage bias under an illumination
at $\lambda=1550\,\mu\mathrm{m}$ and $T_{0}=2.8\,\mathrm{K}$. The light
intensity is $16\,\mathrm{W}.\mathrm{cm}^{-2}$. The black dashed line
corresponds to the measured right hand side of eq. (1). The only fitting
parameter is the value of $V_{\mathrm{ac}}=V_{\mathrm{op}}^{\mathrm{Tucker}}$
which is adjusted with the slope at zero bias. The black solid line
corresponds to the numerical simulation of eq. (4) using the barrier
parameters $U\simeq 2.9\,\mathrm{eV}$, $\Delta U\simeq-0.78\,\mathrm{eV}$ and
$d\simeq 1.6\,\mathrm{nm}$.

##  V Hot-carrier vs thermal distribution

The PAT current of eq. (4) can be re-written in terms of the deviation of the
electron distribution from the equilibrium:

$\begin{split}\delta I(V_{\mathrm{dc}},\delta f_{\mathrm{up}},\delta
f_{\mathrm{dn}})=&\frac{e}{h}\int
d\epsilon\;\mathcal{T}(\epsilon,eV_{\mathrm{dc}})\times\\\ &\left[\delta
f_{\mathrm{up}}(\epsilon+eV_{\mathrm{dc}})-\delta
f_{\mathrm{dn}}(\epsilon)\right],\end{split}$ (6)

where $\delta f_{\mathrm{up/dn}}=f(\epsilon)-f(\epsilon,T_{0})$ states for the
deviation of the electron distribution from the equilibrium distribution at
base temperature $T_{0}$. This steady-state non-equilibrium electron
distribution function under continuous wave illumination is given by the
master equation:

$\left(\frac{\partial
f}{\partial\epsilon}\right)_{\mathrm{e-ph}}+\left(\frac{\partial
f}{\partial\epsilon}\right)_{\mathrm{e-e}}+\left(\frac{\partial
f}{\partial\epsilon}\right)_{\mathrm{exc}}=0,$ (7)

where $f$ is the electron distribution function, the first term stands for
electron-phonon energy transfer between electrons and lattice, the second
stands for electron-electron thermalization (characterized by an electron
collision time of the order of few 100 fs) while the third term describes the
excitation due to photon absorption in the contacts. The first two terms will
mainly lead to an equilibrium distribution characterized by an effective
temperature. The excitation term could however lead to the existence of a
small fraction of hot electron. we evaluate below what would be the effect of
a hot electron distribution on the PAT. It can be evaluated by considering a
detailed balanced equation and the Fermi golden rule (Appendix C):

$\begin{split}\delta f_{\mathrm{exc}}=\left(\frac{\partial
f}{\partial\epsilon}\right)_{\mathrm{exc}}=&A_{\mathrm{exc}}\left[\sqrt{\epsilon-
eV_{0}}f(\epsilon-eV_{0})\left(1-f(\epsilon)\right)\right.\\\
&\left.-\sqrt{\epsilon+eV_{0}}f(\epsilon)\left(1-f(\epsilon+eV_{0})\right)\right],\end{split}$
(8)

where $A_{\mathrm{exc}}$ is a constant proportional to the energy absorbed by
electrons. In the relaxation time approximation for the electron-electron
thermalization , we deduce the time-average PAT current of eq. (4) due to hot-
carrier contribution by considering the deviation of the electron distribution
$\delta f_{\mathrm{up/dn}}=\delta
f(\epsilon,A_{\mathrm{up/dn}})\propto\left(\frac{\partial
f}{\partial\epsilon}\right)_{\mathrm{up/dn}}$. $A_{\mathrm{up}/\mathrm{dn}}$
is a constant proportional to the energy absorbed in the upper/lower
electrode. It is convenient to characterize this amount of energy by
expressing it in terms of equivalent temperature of excitation $\delta
T_{\mathrm{exc}}$. To do so, we consider a Fermi-Dirac distribution at
$T_{0}+\delta T_{\mathrm{exc}}$ containing the same energy as the non-
equilibrium distribution function:

$\int\epsilon\rho(\epsilon)\left[f(\epsilon,T_{0})+\delta
f_{\mathrm{exc}}\right]\,d\epsilon=\int\epsilon\rho(\epsilon)f(\epsilon,T_{0}+\delta
T_{\mathrm{exc}})\,d\epsilon,$ (9)

where $\rho(\epsilon)\propto\sqrt{\epsilon}$ is the density of states of the
free electron gas. Figure 5 shows the photon-assisted current $\delta
I^{(ph)}$ as a function of dc current flowing through the junction biased
between $-1.78\,\mathrm{V}$ and $+1.84\,\mathrm{V}$. It is mainly proportional
to the current. Dashed blue line in figure 5 corresponds to the theoretical
prediction given by eqs. (8,6). We clearly see that the proportionality cannot
be explained by a hot-carrier distribution function. We fit $\delta
I(I_{\mathrm{dc}})$ by adjusting the absorption coefficients
$A_{\mathrm{up}}\sim 1.42\times 10^{-8}$ and $A_{\mathrm{dn}}\sim 0.69\times
10^{-8}$ with an optimization software. Note that
$A_{\mathrm{up}}>A_{\mathrm{dn}}$ is in agreement with the fact that upper
electrode experiences the field enhancement of the surface plasmon. They
corresponds to equivalent temperatures of excitation of $\delta
T_{\mathrm{exc,up}}=0.30\,\mathrm{K}$ and $\delta
T_{\mathrm{exc,dn}}=0.15\,\mathrm{K}$ respectively. Even if this term leads to
a Tien-Gordon-like distribution given by eq. (3), the hot carrier distribution
remains extremely small because of electron heating and heat leakage to the
lattice as studied by Dubi and Sivan Sivan2019 ; Bouhelier_2021 . Moreover,
the equivalent temperature of excitation is smaller than $0.3\,\mathrm{K}$ and
we see in the following that this temperature remains smaller than the
temperature induced by the power absorption in the lead. It follows that the
photon absorption mainly induces heating rather than generating hot carriers.
In the following, we analyze the PAT current as resulting from two different
thermal distributions in the contacts and can be refereed as a the tunneling
Seebeck effect Stovneng_1990 . However, unlike the Seebeck effect which
depends on the metallic density of states of the electrode materials, the
tunneling Seebeck effect depends on the energy dependence of the tunneling
transmission and so that on the non linearity of the $I(V)$ characteristic
Smith_1980 . The PAT current is then described by eq. (6) with the deviation
of the electron distribution $\delta f_{\mathrm{up/dn}}=\delta
f(\epsilon,T_{\mathrm{up/dn}})=f(\epsilon,T_{0}+\delta
T_{\mathrm{up/dn}})-f(\epsilon,T_{0})$. $T_{\mathrm{up}/\mathrm{dn}}$ is the
temperature of upper/lower electrode.

Figure 5: PAT current as a function of dc current with the same experimental
settings as Fig. 4. Symbols are experimental data. Blue dashed line is the
numerical simulation from eqs. (8) and (6) dealing with hot carrier
distribution in the electrodes whereas the red dashed line corresponds to two
different thermal distributions in the contacts described. Inset: photon-
induced current measured in the upper (orange circles) and lower contacts
(purple circles).

Figure 5 shows the proportionality between the photon assisted current and the
current. This proportionality is explained by a differential heating between
the upper and lower electrodes. Our data are in good agreement with the
heating description (red dashed line in Fig. 5). We have checked that the PAT
measured across the junction is one order of magnitude greater than the photon
assisted current in each electrode. The cross geometry of the tunnel junction
allows us to measure the photon-assisted current in the upper/lower electrode
by using a current polarization of the upper/lower electrode. The power supply
is connected on the $V_{B}$ contact (resp. $V_{A}$) to measure the upper
(resp. lower) electrode (see Fig. 2) while the current amplifier is connected
to the opposite side of the electrode. Inset of figure 5 shows the photo-
assisted current measured in the two electrodes independently. This photon
assisted current is attributed to a variation of the aluminum conductivity.
The Drude conductivity is indeed proportional to the relaxation time which
depend on the temperature so that a laser induced temperature modulation will
induce a conductance modulation. The energy and voltage dependent transmission
$\mathcal{T}(\epsilon,eV_{\mathrm{dc}})$ of the barrier is deduced from the
fit of the $I(V)$ curve. The fit of the curve $\delta I^{(ph)}(I)$ gives
$\delta T_{\mathrm{up}}\simeq 3.22\,\mathrm{K}$ and $\delta
T_{\mathrm{dn}}\simeq 2.54\,\mathrm{K}$. The temperature rise can be
associated with the optical power absorbed by the electrodes. The junction in
the Kretschmann configuration forms a well defined plasmonic multilayer stack
and the absorbed power at the resonance angle $\alpha_{res}=-39.7^{\circ}$ are
evaluated to $P_{\mathrm{up}}\simeq 265\,\mathrm{\mu W}$ and
$P_{\mathrm{dn}}\simeq 84\,\mathrm{\mu W}$ respectively (Appendix A). We
neglected the power leakage due to thermal conduction of the contacts and
identify the absorbed power and the power dissipated to the lattice of volume
$\Omega=a\times w\times w_{\mathrm{laser}}\simeq 5.2\times
10^{-16}\,\mathrm{m}^{-3}$. The temperature rise in the electrodes is thus
estimated to $\delta
T_{\mathrm{up/dn}}=\left(\frac{P_{\mathrm{up/dn}}}{\Sigma\Omega}\right)^{1/5}$
where the electron-phonon coupling constant in aluminum Meschke_2004 is
$\Sigma\simeq 0.3\,\mathrm{nW}.\mu\mathrm{m}^{-3}.\mathrm{K}^{-5}$. It gives
$\delta T_{\mathrm{up}}\sim 1.7\,\mathrm{K}$ and $\delta T_{\mathrm{dn}}\sim
1.0\,\mathrm{K}$. Note that the estimated temperature rise is slightly
underestimated. It can be attributed to the value of the dielectric constant
that we have considered Rakic_1995 knowing we consider here an unusual ultra-
thin aluminum film. The photon-assisted tunneling is strictly speaking a
thermo-assisted tunneling. Our measurement could also be sensitive to the
variation of the electrode conductance due to their temperature modulation.
Thanks to the cross-geometry of the junction, we have checked that this effect
is negligible compare to the PAT measured on the junction. The inset of Fig. 5
shows that the photon-induced current in the electrodes is lower by more than
one order of magnitude compared to the PAT current. The modulation of the
light power by the chopper (laser on/off) induces a modulation of the
resistance of the electrodes:
$R_{\mathrm{up/dn}}^{\mathrm{on}}=R_{\mathrm{up/dn}}^{\mathrm{off}}+\delta
R_{\mathrm{up/dn}}$. When the electrode is voltage bias, this translates in a
photon-induced current given by $\delta
I=I^{\mathrm{on}}-I^{\mathrm{off}}=-\frac{\delta
R_{\mathrm{up/dn}}}{R_{\mathrm{up/dn}}}I$. The inset of Fig. 5 shows that the
photon induced current in the electrodes corresponds to a relative increase of
their resistance ($\delta R>0$) Herzog_2014 : $\eta_{\mathrm{up}}=\frac{\delta
R_{\mathrm{up}}}{R_{\mathrm{up}}}\simeq(7.3\pm 0.2)\times 10^{-7}$ and
$\eta_{\mathrm{dn}}=\frac{\delta
R_{\mathrm{dn}}}{R_{\mathrm{dn}}}\simeq(3.6\pm 0.2)\times 10^{-7}$ . Using the
Sommerfeld expansion in the electrical conductivity and assuming a linear
dependence of the relaxation time, we expect a quadratic temperature
dependence of the conductivity so that
$\frac{\eta_{\mathrm{up}}}{\eta_{\mathrm{dn}}}=\left(\frac{\delta
T_{\mathrm{up}}}{\delta T_{\mathrm{dn}}}\right)^{2}\simeq 2.0\pm 0.3$ (see
Appendix D). We then deduce a ratio
$\sqrt{\frac{\eta_{\mathrm{up}}}{\eta_{\mathrm{dn}}}}\simeq 1.4\pm 0.2$ which
is consistent with the measured ratio $\frac{\delta T_{\mathrm{up}}}{\delta
T_{\mathrm{dn}}}\simeq 1.27$.

##  VI Conclusion

Optical rectification in metallic tunnel junctions produces dc photocurrent
assuming that illumination acts as an ac voltage at optical frequencies across
the nonlinear tunnelling conduction. The photon-assisted transport theory in
tunnel junctions states that the dc photocurrent is a well known function of
the $I(V)$ characteristics on a voltage range of the photon voltage
$\hbar\omega/e$. This theoretical prediction is based on the creation of a
non-thermal hot carrier distribution function in the metals under optical
illumination. We have shown experimentally that this is incorrect in a planar
tunnel junction. By comparing the photocurrent generated in a planar junction
in the Kretschmann configuration and the measured $I(V)$ characteristics, we
have demonstrated that the photocurrent can be described by heating and
thermal distribution functions instead. While we have only considered a planar
tunnel junction without nanoantenna, the question of the generation of hot
electrons in metallic nanostructures for optical rectification or optical
detection still raises theoretical questions. As pointed out by Iñarrea et al.
in the case of THz detectors in double barrier systems Inarrea_1994 , it is
crucial to calculate the transmission coefficient by considering the coupling
term $\mathbf{A}.\mathbf{p}$ where $\mathbf{A}$ is the vector potential of the
electromagnetic field and $\mathbf{p}$ the electronic momentum operator. Note
that the coupling between electrons and the electromagnetic field no longer
takes place in a single region of the structure but interacts with the
barrier. In conclusion, the planar tunnel junction is not a good candidate to
highlight the hot carrier distribution in illuminated metallic thin films. The
metallic density of states is indeed to smooth and it does not allow the
realization of the necessary energy filter as it is in a molecular junction
Dubi_2022 .

## Appendix A: Plasmonic multilayer stack

We consider an $\mathrm{Al/Al_{2}O_{3}/Al}$ MIM stack on top of a righ angle
prism (see Fig. 6). The thickness of the metallic films is $a=8\,\mathrm{nm}$
and that of dielectric barrier is $d\simeq 1.6\,\mathrm{nm}$. The dielectric
constants of metal and barrier are respectivelly
$\epsilon_{m}=-242+49.4\mathrm{i}$ and $\epsilon_{m}=3.051+0.0629\mathrm{i}$
at $\lambda=1550\mu\mathrm{m}$ Rakic_1995 ; Dodge_1986 .

Figure 6: Schematics of the MIM stack. $\alpha$ is the incident angle in the
vacuum whereas $\theta$ is the angle in the prism. The Snell–Descartes laws
give
$\sin\left(\frac{\pi}{4}-\alpha\right)=n\sin\left(\frac{\pi}{4}-\theta\right)$.

We consider the Fresnel coefficients for a TM-polarised light (p polarized).
For a single interface from (1) to (2) between two dielectric materials
characterized by the dielectric constant $\epsilon_{1}$ and $\epsilon_{2}$
respectively, the reflexion and transmission Fresnel coefficients read:

$r_{1\hookleftarrow
2}=\frac{\epsilon_{2}k_{z1}-\epsilon_{1}k_{z2}}{\epsilon_{2}k_{z1}-\epsilon_{1}k_{z2}},\;t_{1\rightarrow
2}=\frac{2\epsilon_{2}k_{z1}}{\epsilon_{2}k_{z1}-\epsilon_{1}k_{z2}}.$ (10)

By using the Fresnel’s coefficients on the interfaces at $z=0,a,a+d,2a+d$, the
amplitude of the incoming $H_{i}^{(-)}$ and outgoing $H_{i}^{(+)}$ magnetic
field are connected by a transfer matrix:

$T_{i,i+1}=\left(\begin{array}[]{cc}t_{i\rightarrow
i+1}-\frac{r_{i\hookleftarrow i+1}r_{i+1\hookleftarrow i}}{t_{i+1\rightarrow
i}}&\frac{r_{i+1\rightarrow i}}{t_{i+1\rightarrow i}}\\\
-\frac{r_{i\rightarrow i+1}}{t_{i+1\rightarrow i}}&\frac{1}{t_{i+1\rightarrow
i}}\\\ \end{array}\right).$ (11)

Figure 7: Magnitude of the magnetic field inside the junction for different
incident angle $\alpha$ at SPR.

The p-polarized magnetic fields
$\mathbf{H}=\left(H^{(+)}_{i}e^{ik_{zi}z}+H^{(-)}_{i}e^{-ik_{zi}z}\right)\mathbf{e_{\mathrm{y}}}$
in the different regions are defined by:

$\displaystyle\left(\begin{array}[]{c}H_{2}^{(+)}\\\ H_{2}^{(-)}\\\
\end{array}\right)$
$\displaystyle=T_{1,2}\times\left(\begin{array}[]{c}H_{1}\\\ H_{1}^{(-)}\\\
\end{array}\right),$
$\displaystyle\left(\begin{array}[]{c}e^{ik_{z3}a}H_{3}^{(+)}\\\
e^{-ik_{z3}a}H_{3}^{(-)}\\\ \end{array}\right)$
$\displaystyle=T_{2,3}\times\left(\begin{array}[]{c}e^{ik_{z2}a}H_{2}^{(+)}\\\
e^{-ik_{z2}a}H_{2}^{(-)}\\\ \end{array}\right),$
$\displaystyle\left(\begin{array}[]{c}e^{ik_{z4}(a+d)}H_{4}^{(+)}\\\
e^{-ik_{z4}(a+d)}H_{4}^{(-)}\\\ \end{array}\right)$
$\displaystyle=T_{3,4}\times\left(\begin{array}[]{c}e^{ik_{z3}(a+d)}H_{3}^{(+)}\\\
e^{-ik_{z3}(a+d)}H_{3}^{(-)}\\\ \end{array}\right),$
$\displaystyle\left(\begin{array}[]{c}e^{ik_{z5}(2a+d)}H_{5}^{(+)}\\\ 0\\\
\end{array}\right)$
$\displaystyle=T_{4,5}\times\left(\begin{array}[]{c}e^{ik_{z4}(2a+d)}H_{4}^{(+)}\\\
e^{-ik_{z4}(2a+d)}H_{4}^{(-)}\\\ \end{array}\right),$

with,

$\displaystyle k_{\mathrm{x}}$
$\displaystyle=\sqrt{\epsilon_{0}}\left(\frac{\omega}{c}\right)\sin\theta,$
$\displaystyle k_{zi}$
$\displaystyle=\sqrt{\epsilon_{i}\left(\frac{\omega}{c}\right)^{2}-k_{\mathrm{x}}^{2}}\hskip
8.5359pt\forall i\in\left\\{1,...,5\right\\}.$

The color plot Fig. 7 shows the surface plasmon resonance (SPR) at the angle
$\alpha_{0}=\pm 39.7^{\circ}$. This angle corresponds approximatively to the
critical angle
$\alpha_{\mathrm{c}}=\frac{\pi}{4}-\arcsin\left[n\sin\left(\frac{\pi}{4}-\theta_{\mathrm{c}}\right)\right]\simeq
39.1^{\circ}$ with $n=1.52$ the refractive index of the cover glass.
$\theta_{c}=\arcsin\left(\frac{1}{n}\right)$ is the incident angle enabling
the coupling between the light in the cover glass and the surface plasmon
propagating on the interface upper electrode/vacuum. Figure 7 shows the
amplitude of the normalized magnetic field $|H/H_{0}|$ inside the junction.
The measured SPR (see Fig. 2) is broader than the simulated one probably due
to the roughness of our ultra-thin film deposition. At angle $\alpha_{0}$, the
field enhancement occurs on the upper electrode. The electric field is deduced
from the magnetic field:

$\mathbf{E_{i}}=-\frac{H_{i}}{\omega\epsilon_{0}\epsilon_{i}}\left(k_{z}\mathbf{\mathbf{e_{\mathrm{x}}}}+k_{x}\mathbf{\mathbf{e_{\mathrm{z}}}}\right).$
(14)

Figure 8: Magnetic field as a function of $z$ at normal incidence
$\alpha=0^{\circ}$ and at SPR $\alpha\simeq 39.7^{\circ}$

The optical voltage between the two upper and lower electrodes is then given
by:

$V_{\mathrm{op}}(\alpha)=\left|\int_{a}^{a+d}\mathbf{E_{i}}.\mathbf{\mathbf{e_{\mathrm{z}}}}\,dz\right|=\left|\frac{k_{x}}{\epsilon_{0}\epsilon_{i}\omega}\right|.\left|\int_{a}^{a+d}H_{i}(z,\alpha)\,dz\right|.$
(15)

and it is maximum around the SPR: $V_{\mathrm{op}}(\alpha_{0})\simeq
25\,\mu\mathrm{V}$ (see Fig. 9 a). The absorbed power in the different regions
of the junction is given by:

$P_{\mathrm{\mathrm{abs.},i}}(\alpha)=\frac{\omega\mathrm{Im}\left(\epsilon_{i}\right)}{2\epsilon_{0}\left|\epsilon_{i}\right|}\int_{{i}}\left|H_{i}(z,\alpha)\right|^{2}\,dz.$
(16)

Figure 9b shows that the upper electrode absorbs more power than the lower one
at SPR: $P_{\mathrm{up}}\simeq 79\,\mathrm{\mu W}$ whereas
$P_{\mathrm{dn}}\simeq 24\,\mathrm{\mu W}$. Note that the power dissipated in
the barrier is negligible and that it is reasonable to assume that the barrier
transmission is not affected by any thermal effects.

Figure 9: a. Optical voltage between the two electrode as a function of angle
$\alpha$ for $P_{\mathrm{laser}}=53\,\mathrm{mW}$. b. Absorbed power in the
tunnel function as a function of angle $\alpha$ for
$P_{\mathrm{laser}}=53\,\mathrm{mW}$.

## Appendix B: Tunnel transmission

Figure 10: Transmission $\mathcal{T}(\epsilon,eV_{\mathrm{dc}})$ for a
trapezoidal barrier characterized by $U_{0}=2.87\,\mathrm{eV}$, $\Delta
U=-0.78\,\mathrm{eV}$ and $d=1.6\,\mathrm{nm}$.

In order to calculate the different quantities related to the photon-assisted
tunneling current in the far from equilibrium regime, we model the tunnel
barrier by a trapezoidal barrier $U(z)$ characterized by a thickness $d$, a
mean height $U_{0}$ and an asymmetry $\Delta U$ (see Fig. 11). This allowed us
to obtain an analytical expression of the tunnel transmission
$\mathcal{T}(\epsilon,eV_{\mathrm{dc}})$. This analytical expression is
essential to perform numerical optimization on the $I(V)$ characteristics (see
Fig. 3) and the photon assisted current (see Fig. 5 and eq. (6)).

Figure 11: Trapezoidal barrier defined by de thickness $d$, a mean high
$U_{0}$ and an assymetry $\Delta U$ .

We are considering the Simmons’ model where an effective potential includes
the potential barrier $U(z)$ and the biasing energy
$U_{\mathrm{bias}}(z,eV)=-eV(1-z/d)$,
$U_{\mathrm{eff}}(z)=U_{\mathrm{bias}}(z,eV)=U(z)+U_{\mathrm{bias}}(z,eV)$.
The 3D transmission can be expressed as a function of the
Wentzel–Kramers–Brillouin (WKB) transmission through the 1D effective
potential Simmons_1964 ; Fevrier_2018 :

$\mathcal{T}=\exp\left\\{-\frac{\sqrt{8m}}{\hbar}\int_{0,z_{0}}^{z_{0},d}\,dz\sqrt{U(z)-eV\left(1-\frac{z}{d}\right)-\epsilon}\,\right\\}.$
(17)

We can distinguish three cases: (i) the energy $\epsilon$ is smaller that the
barrier and the electrons see a trapezoidale barrier
($\epsilon<\min_{0<z<d}{U(z)}$) (ii) the electrons see a triangular barrier
($\min_{0<z<d}{U(z)}<\epsilon<\max_{0<z<d}{U(z)}$ and (iii) the electrons go
over the barrier ($\epsilon>\max_{0<z<d}{U(z)}$). These cases depend on the
values of $\epsilon$ and $eV$:

  * •

if $\epsilon<\mathrm{min}\left(U_{\mathrm{L}}-eV,U_{\mathrm{R}}\right)$ then

$\log\mathcal{T}=-\frac{4d\sqrt{2me}}{3}\frac{\left(U_{\mathrm{R}}-\epsilon\right)^{3/2}-\left(U_{\mathrm{L}}-eV-\epsilon\right)^{3/2}}{U_{\mathrm{R}}-U_{\mathrm{L}}+eV},$

  * •

if $U_{\mathrm{L}}-eV<\epsilon<U_{\mathrm{R}}$ then

$\log\mathcal{T}=-\frac{4d\sqrt{2me}}{3}\frac{\left(U_{\mathrm{R}}-\epsilon\right)^{3/2}}{U_{\mathrm{R}}-U_{\mathrm{L}}+eV},$

  * •

if $U_{\mathrm{R}}<\epsilon<U_{\mathrm{L}}-eV$ then

$\log\mathcal{T}=-\frac{4d\sqrt{2me}}{3}\frac{\left(U_{\mathrm{L}}-eV-\epsilon\right)^{3/2}}{U_{\mathrm{L}}-U_{\mathrm{R}}-eV},$

  * •

if $\epsilon>\mathrm{max}\left(U_{\mathrm{L}}-eV,U_{\mathrm{R}}\right)$ then

$\mathcal{T}=1.$

The Landauer-Büttiker formula gives the $I(V)$ characteristics:

$I(V)=\frac{2e}{h}\int
d\epsilon\,\mathcal{T}(\epsilon,eV)\left[f(\epsilon)-f(\epsilon+eV)\right].$
(18)

A numerical optimization fitting of the $I(V)$ characteristics with eq. (18)
and the analytical transmission gives $U_{0}=2.87\,\mathrm{eV}$, $\Delta
U=-0.78\,\mathrm{eV}$ and $d=1.6\,\mathrm{nm}$. Fig. 10 shows the color plot
of $\mathcal{T}\left(\epsilon,V_{\mathrm{dc}}\right)$ for the considered
barrier.

## Appendix C: Non-equilibrium distribution function

The interplay between photons and electrons in the metallic contact may lead
to ”hot” electron distribution junction or heating. For a moderate increase of
temperature, the heating affects a large number of quasiparticles around the
Fermi sea whereas the photon-assisted processes brings a small amount of
quasiparticles at high energy corresponding to the photon energy $\hbar\omega$
(see Fig. 12). The excitation of the conduction electrons is done by the
absorption of photons of energy $\hbar\omega$. Assuming that the processes of
excitation involve only intraband transitions, electrons of energy $\epsilon$
reach energy $\epsilon+\hbar\omega$. The change in occupancy number $\delta
f_{\mathrm{exc}}$ is given by the difference between the number
$dN_{+}(\epsilon)$ of electrons that access the energy $\epsilon$ and the
number $dN_{-}(\epsilon)$ of electrons that leave it:

$\displaystyle dN_{+}(\epsilon)$
$\displaystyle=A\rho(\epsilon-\hbar\omega)f_{0}(\epsilon-\hbar\omega)\rho(\epsilon)\left(1-f_{0}(\epsilon)\right),$
$\displaystyle dN_{-}(\epsilon)$
$\displaystyle=A\rho(\epsilon)f_{0}(\epsilon)\rho(\epsilon+\hbar\omega)\left(1-f_{0}(\epsilon+\hbar\omega)\right),$

Figure 12: Non-equilibrium distribution function due to photon-assisted
processes/heating. The settings parameters correspond to the measured one in
section V: $\left(T_{0}=2.8\,\mathrm{K},\delta T=3,\ \mathrm{K}\right)$ for
heating and $A_{\mathrm{exc}}=10^{-8}$ for photon illumination.

where $\rho(\epsilon)\propto\sqrt{\epsilon}$ is the density of states of the
free electron gas, $A$ is a constant proportional to the energy transfered to
the electron gas and $f_{0}$ the equilibrium Fermi-Dirac distribution. The
change in the occupancy number is given by $\delta
f_{\mathrm{exc}}=\left(dN_{+}(\epsilon)-dN_{-}(\epsilon)\right)/\rho(\epsilon)$.
The excitation term in eq. (8) is then directly deduced.

Figure 13: Normalized difference $\eta(A,\delta T)$ for the equivalent
temperature of excitation in the upper electrode characterized by a power
coefficient $A_{\mathrm{up}}=1.42\times 10^{-8}$ (orange line) and the lower
electrode characterized by a power coefficient $A_{\mathrm{dn}}=0.69\times
10^{-8}$ (purple line).

The normalized difference between the energy contained in the non-equilibrium
distribution function characterized by absorbed power coefficient $A$ and the
equilibrium function at temperature $T_{0}+\delta T$ is given by:

$\kappa(A,\delta
T)=\frac{\int\epsilon\rho(\epsilon)\left[f(\epsilon,T_{0})-f(\epsilon,T_{0}+\delta
T)+\delta
f_{\mathrm{exc}}\right]\,d\epsilon}{\int\epsilon\rho(\epsilon)f(\epsilon,T_{0})\,d\epsilon}.$

The equivalent temperature of excitation $\delta T_{\mathrm{exc}}$ corresponds
to $\kappa(A_{\mathrm{exc}},\delta T_{\mathrm{exc}})=0$. Figure 13 gives the
values of $\delta T_{\mathrm{exc}}$ for the upper electrode
($A_{\mathrm{up}}=1.42\times 10^{-8}$) and the lower one
($A_{\mathrm{dn}}=0.69\times 10^{-8}$).

## Appendix D: Temperature dependence of the conductivity

Using the free electron gas model to describe the electron transport in the
metallic electrodes, the Boltzmann equation and the relaxation time
approximation give the electrical conductivity:

$\sigma=\frac{ne^{2}\langle\tau\rangle}{m^{\star}},$ (20)

where $n$ is the density of electrons, $m^{\star}$ the effective masse of
electrons, $e$ the elementary charge and $\langle\tau\rangle$ the mean value
of the relaxation time $\tau(\epsilon)$. By considering the density of states
$\rho\propto\epsilon^{3/2}$ in 3D, the mean value is given by:

$\langle\tau\rangle=\frac{\int_{0}^{+\infty}\tau(\epsilon)\epsilon^{3/2}\left(-\frac{\partial
f_{0}}{\partial\epsilon}\right)d\epsilon}{\int_{0}^{+\infty}\epsilon^{3/2}\left(-\frac{\partial
f_{0}}{\partial\epsilon}\right)d\epsilon},$ (21)

In the first order approximation near the Fermi energy, the relaxation time
can be written as
$\tau(\epsilon)\simeq\tau(\epsilon_{\mathrm{F}})+\tau^{\prime}(\epsilon_{\mathrm{F}})\left(\epsilon-\epsilon_{\mathrm{F}}\right)$.
The Sommerfeld expansion of $\langle\tau\rangle$ gives:

$\langle\tau\rangle=\tau(\epsilon_{\mathrm{F}})+\frac{\pi^{2}}{2}\frac{\tau^{\prime}(\epsilon_{\mathrm{F}})}{\epsilon_{\mathrm{F}}}\left(k_{\mathrm{B}}T\right)^{2},$
(22)

and the relative increase of the resistance is thus:

$\eta_{\mathrm{up/dn}}=\frac{\delta
R_{\mathrm{up/dn}}}{R_{\mathrm{up/dn}}}=-\frac{\pi^{2}\tau^{\prime}(\epsilon_{\mathrm{F}})}{2\epsilon_{\mathrm{F}}\tau(\epsilon_{\mathrm{F}})}\left(k_{\mathrm{B}}\delta
T_{\mathrm{up/dn}}\right)^{2}.$ (23)

Assuming that the energy dependence of the relaxation time is the same in both
aluminum electrodes, the ratio $\eta_{\mathrm{up}}/\eta_{\mathrm{dn}}$ gives
access to the ratio $\delta T_{\mathrm{up}}/\delta T_{\mathrm{dn}}$.

_Acknowledgements._ We acknowledge fruitful discussions with Marc Bescond. We
also thank Sylvie Gautier for the help in sample fabrication and Gilles
Guillier for the cryognetic rotation realization. This work has been supported
by Region Ile-de-France in the framework of DIM SIRTEQ.

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