blackbody source at temperatureTsin a nar-
row bandwidth,Dn, isS=dP/dA, whereS¼
M 0 ðn;TsÞDnwithM 0 ðn;TsÞbeing the spectral
exitanceofablackbodyinfreespace(eq.15
in the supplementary materials). The incident
free space electric field amplitude is obtained
from the Poynting flux,S¼jE 0 ðwÞj^2 = 2 Z 0 ,
whereZ 0 is the permittivity of free space.
The resulting instantaneous voltage at any
point along the metal width isVðx;tÞ¼Vdc Vmeiwtcosp
w
xþw
2
ð 1 Þthe complex voltage profile across the de-
vice (Fig. 2B). A small dc bias (Vdc) is added
to the harmonic time-varying term to ac-count for self-bias of the device. Here,Vm¼
dgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 Z 0 M 0 ðn;TsÞDnp
is the ENZ enhanced
blackbody induced voltage, anddis the tun-
nel oxide thickness. Estimates for the incident
voltage amplitude in a fixed bandwidth from
the blackbody source are shown in Fig. 2C.
Thenodeofthevoltageinthemodeloscillates
around the center of the metal. The voltage
node,xc(t), is a function of the time, but we
can use the two end points for the limits of
integration for the spatially averaged current
shown in Fig. 2B. Explicitly, we find thatxTc¼T
w
2þ
w
pcos ^1
Vdc
Vm
ð 2 Þwhich splits the current into two regions that,
in general, vary with time. We can define two
instantaneous current amplitudes asInðtÞ¼ ∫
xcðtÞ w= 2dx
wVðx;tÞ
ZnðtÞð 3 ÞIpðtÞ¼∫
w= 2xcðtÞdx
wVðx;tÞ
ZpðtÞð 4 ÞwhereZnandZpare the time-dependent diode
impedances for each section, respectively. The
half-cycle time average of the current ampli-
tudes is of interest. Because the nodal position
of the voltage and the impedance of the n and
p sections vary with time, we need to make
some assumptions to simplify the integration.
To do so, we split the integration into two quar-
ter cycles; cycle 1, where 0≤t<T/4, and cycle 2,
whereT/4≤t<T/2.Forcycle1,thenodalposi-
tion is assumed to bexþcfor the whole cycle, and
impedance is constant and given byZn=Rn
andZp=Rpin this time interval (RnandRp
are the forward bias tunnel resistances). Both
tunnel diodes are forward-biased, and a large
tunnel current flows from the metal to the n+
region,Ifn, and from p+ to the metal,Ifp. Like-
wise, for cycle 2, the nodal position is assumed
to bexc , and impedance is constant and given
byZn=rnandZp=rpin this time interval (rn
andrpare the reverse bias resistances for the
n and p tunnel diodes, respectively; see Fig. 2,
D and E, for the resistances). The two tunnel
diodes are reverse-biased, and small backflow
currentIrnandIrpis observed. The magnitude
and direction of the tunnel currents are indi-
cated by arrows in Fig. 2B. The real parts of
the half-cycle averaged currents areIn¼Vdc
41
Rnþ1
rn
þxc
wVdc
21
Rn1
rn
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Vm^2 Vdc^2pp^21
Rn1
rn
ð 5 Þ20 MARCH 2020•VOL 367 ISSUE 6484 1343°Fig. 2. Bipolar grating-coupled tunnel diode model. (A) Equilibrium band diagram of the bipolar device
under a metal gate showing electron and hole particle currents. Inset shows unit cell geometry. The period
of the grating (P) is 3 mm, the metal width (w) is 1.8 mm, and the tunnel oxide thickness (d) is 3 to
4.5 nm. (B) The instantaneous voltage profile in the device at t = 0 and at t = T/2. The spatially varying
currents occur in both n and p+ regions, and the voltage node shifts to a negative x position. The half-period
instantaneous voltage profile and currents across the device. The voltage node shifts to a positive x position.
(C) Integrated blackbody source power per unit area (red curve) for bandwidth between c/8.0 mm and
c/7.0 mm with field enhancement g = 20 and d = 4 nm. The blue curve is the associated ac voltage
amplitude Vm.(D) Measured tunnel diode characteristic for typical n+ MOS tunnel diode with resonant
PAT single-photon voltage marked. (E) Extracted resistance from n+ MOS tunnel diode. Rn ≃ 200 ohms
and rn ≃ 50,000 ohms at the indicated photovoltages (Fig. 1E).
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