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Resonant plasmonic terahertz detection in vertical graphene-base hot-electron
transistors
V. Ryzhii
1,2
, T. Otsuji
1
, M. Ryzhii
3
, V. Mitin
4
, and M. S. Shur
5
1
Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan
2
Center for Photonics and Infrared Engineering,
Bauman Moscow State Technical University and Institute of Ultra High
Frequency Semiconductor Electronics of RAS, Moscow 111005, Russia
3
Department of Computer Science and Engineering,
University of Aizu, Aizu-Wakamatsu 965-8580, Japan
4
Department of Electrical Engineering, University at Buffalo, SUNY, Buffalo, New York 1460-1920, USA
5
Department of Electrical, Computer, and System Engineering and Physics, Applied Physics,
and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
We analyze dynamic properties of vertical graphene-base hot-electron transistors (GB-HETs)
and consider their operation as detectors of terahertz (THz) radiation using the developed device
model. The GB-HET model accounts for the tunneling electron injection from the emitter, electron
propagation across the barrier layers with the partial capture into the GB, and the self-consistent
oscillations of the electric potential and the hole density in the GB (plasma oscillations), as well as
the quantum capacitance and the electron transit-time effects. Using the proposed device model,
we calculate the responsivity of GB-HETs operating as THz detectors as a function of the signal
frequency, applied bias voltages, and the structural parameters. The inclusion of the plasmonic
effect leads to the possibility of the HET-GBT operation at the frequencies significantly exceeding
those limited by the characteristic RC-time. It is found that the responsivity of GB-HETs with a
sufficiently perfect GB exhibits sharp resonant maxima in the THz range of frequencies associated
with the excitation of plasma oscillations. The positions of these maxima are controlled by the
applied bias voltages. The GB-HETs can compete with and even surpass other plasmonic THz
detectors.
I. INTRODUCTION
Recently, vertical hot-electron transistors (HETs) with
the graphene base (GB) and the bulk emitter and col-
lector separated from the base by the barrier layers -
the hot-electron graphene-base transistors (GB-HETs) -
made of SiO
2
and Al
2
O
3
were fabricated and studied [1–
4]. These HETs are fairly promising devices despite their
modest characteristics at the present. Similar devices can
be based of GL heterostructures with the hBN, WS
2
, and
other barrier layers [5–8]. The history of different ver-
sions of HETs, including those with the thin metal base
and the quantum-well (QW) base, in which the carri-
ers are generated from impurities or induced by the ap-
plied voltages, as well as HETs with resonant-tunneling
emitter, is rather long (see, for example, Refs. [9–16]).
Figure 1 shows schematically the GB-HET structure and
its band diagrams. Depending on the GB doping, the
base-emitter and collector-base voltages, V
B
and V
C
, and
the thicknesses of barrier layers separating the base from
the emitter and collector, W
E
and W
C
, respectively, the
GB can be filled either with electrons or holes [compare
Figs. 1(b) and 1(C)].
In GB-HETs (as in HETs) , a significant fraction of
the electrons injected from the top n-type emitter con-
tact due to tunneling through the barrier top crosses the
GB and reaches the collector, while the fraction of the
electrons captured into the GB can be rather small. The
tunneling electrons create the emitter-collector current
J
C
. The variations of the base-emitter voltage V
B
result
in the variation of J
C
.
In the case of the GB with a two-dimensional elec-
tron gas (2DEG), the GB-HET transistors can also be
referred to as the N-n-N heterostructure HETs with the
GB of n-type. In another case, when the GB comprises
a two-dimensional hole (2DHG), the GB-HET transis-
tors can also be called the N-p-N heterostructure bipolar
transistors (HBTs) with the GB of p-type.
The electrons captured into the GB support the
emitter-base current J
B
. The transistor gain in the
common-emitter configuration is equal to g
0
= J
C
/J
B
=
(1 − p)/p, where p is the average capture probability of
the capture of electrons into the GB during their transit.
One of the main potential advantages of GB-HETs is the
high-speed operation associated with the combination of
a short transit time (due to the vertical structure), a
high gain g
0
(because of a low probability, p, of the hot
electron capture into the GB), and a low GB resistance
(owing to a high mobility of the carriers in graphene).
As shown recently [17], p in the graphene heterostruc-
tures with proper inter-graphene barrier layers can be
very small. Very small values of p in GB-HETs can lead
to a fairly high gain. The use of the undoped base with
the induced carriers, leads to the exclusion of the scat-
tering of the hot electrons crossing the GB with the im-
purities and to an increase of the mobility of the carriers
localized in the GB.
By analogy with HETs with the QW-base made of the
standard materials [18, 19], GB-HETs can exhibit res-
onant response to the incoming signals associated with
the excitation of the plasma oscillations. The resonant
2
V
C
E m i t t e r
C o l l e c t o r
G B
( b )
( c )
e V
B
E m i t t e r
C o l l e c t o r
e V
B
e V
C
e V
C
E m i t t e r
C o l l e c t o r
V
B
( a )
B a r r i e r l a y e r s
G B
G B
( u n d o p e d )
d
V
w
d
V
w
V
B
FIG. 1: (a) Structure of a GB-HET with undoped GB and
GBT -HET band diagrams (potential profile in the direction
perpendicular to the GB plane) at different relations between
V
B
and V
C
: (b) with GB filled with holes and (c) with GB
filled with electrons. Arrows show propagation of electrons
across the barrier layers and capture of some portion of elec-
trons into GB.
plasma frequencies are determined by the characteristic
plasma wave velocity s (which increases with the carrier
density Σ
0
) and the lateral sizes of the GB 2L, while the
quality factor of the plasma resonances is mainly limited
by the carrier momentum relaxation time τ associated
with the scattering on impurities, various imperfections,
and phonons. Due to the specific features of the carrier
statistics and dynamics in the graphene layers [20], the
plasma velocity s > v
W
, where v
W
' 10
8
cm/s is the
characteristic velocity of the Dirac energy spectrum, and
s can markedly exceed that in the QW heterostructures.
This promotes the realization of with the plasmonic res-
onances in the terahertz (THz) range even in the GB-
HETs with fairly large lateral sizes. The possibility of
achieving the elevated carrier mobilities in the GB can
enable sharp plasmonic resonances in GB-HETs at room
temperatures or above.
The application of the ac voltages ±δV
ω
/2 at the sig-
nal frequency ω, associated with the incoming radiation
(received by an antenna) to the side contacts to the GB,
results in the variation of the local ac potential differ-
ence δϕ
ω
= δϕ
ω
(x) (the axis x is directed in the GB
plane) between the GB and the emitter contact. The
emitter, base, and collector current densities j
E
, j
B
, and
j
C
include the dc components j
E,0
, j
B,0
, and j
C,0
(de-
termined by the applied bias voltages V
B
and V
C
), the
ac components δj
E,ω
, δj
B,ω
, and δj
C,ω
(proportional to
δV
ω
, and (due to the nonlinear dependence of the tun-
neling injection current on the local potential difference
between the GB and the emitter) the rectified dc com-
ponents δj
E,ω
, δj
B,ω
, and δj
C,
(proportional to |δV
ω
|
2
,
i.e., to the intensity of the incoming radiation received by
an antenna. The net rectified current δJ
C,ω
can serve as
the output current in the GB-HETs radiation detectors.
The detector responsivity R
ω
∝ δJ
C,ω
/|δV
ω
|
2
. Due to
the possibility of the plasmonic resonances, the rectified
component (the detector output signal) can be resonantly
large, similar to that in the HET detector [21] and other
plasmonic THz detectors using different transistor struc-
tures, including those incorporating graphene [22–34].
In this paper, develop the GB-HET device model and
evaluate the GB-HET characteristics as a radiation de-
tector of radiation, in particular, in the THz range of
signal frequencies.
II. DEVICE MODEL AND RELATED
EQUATIONS
We consider a GB-HET with the highly conducting
tunneling emitter and collector of the n-type and the un-
doped GB (as shown in Fig. 1) with a 2DHG induced by
the applied bias voltages creating the N-p-N structure
with the band diagram shown in Fig. 1(b). The GB-
HET device model accounts for the tunneling injection of
hot-electrons from the emitter to the barrier layer(above
the barrier top), their propagation across the barrier lay-
ers, partial capture of hot-electrons into the GB, and the
excitation of the self-consistent oscillations of the elec-
tric potential and the hole density in the GB (plasma
oscillations). The quantum capacitance and the electron
transit-time effects are also taken into account.
We assume that the bias voltages V
B
and V
C
are ap-
plied between the GL-base and the emitter and between
the collector and the emitter contacts, respectively. In
the framework of the gradual channel approximation [35],
which is valid if W
E
, W
C
2L, the density of the two-
dimensional hole gas (2DHG)Σ = Σ(x, t) in the GL-base
and its local potential ϕ = ϕ(x, t) (counted from the po-
tential of the emitter) are related to each other as
Σ =
κ
4π e
ϕ − µ/e − V
bi
W
E
+
ϕ − µ/e − V
bi
− V
C
W
C
, (1)
where κ is the dielectric constant of these layers, e is
the hole charge, V
bi
is the built-in voltage between the
contact material and an undoped GL, and µ is the
2DHG Fermi energy in the GB. In a degenerate 2DHG,
µ = ~ v
W
√
πΣ, where ~ is the Planck constant. The de-
pendence of the right side of Eq. (1) on the hole Fermi
energy is interpreted as the effect of quantum capaci-
tance [36, 37].
The ac voltages ±δV
ω
/2 and ω are applied between the
side GB contacts connected with an antenna, so that the
ac potential of the GB δϕ
ω
= δϕ
ω
(x) obeys the following
(asymmetric) conditions:
δϕ
ω
|
x=±L
= ±δV
ω
/2. (2)
3
The side contacts can serve as the slot wave guide trans-
forming the incoming THz radiation signals being re-
ceived by an antenna into the ac voltage [see Fig.1(a)].
The ac component of the collector-emitter voltage δV
C,ω
can also arise due to the ac potential drop across the load
resistance. However, in the GB-HET with the wiring
under consideration, δV
C,ω
can be disregarded provid-
ing that the GB-HET structure is symmetrical (see be-
low). For GB-HETs with the degenerate 2DHG, the
relation between the variations of the hole density and
the potential, δΣ
ω
and δϕ
ω
(the ac components) are, as
follows from Eq. (1), can be expressed via the net ca-
pacitance per unit area C = C
g
C
quant
/(C
quant
+ C
g
),
which accounts for the geometrical capacitance C
g
=
(C
E
+ C
C
) = (κ/4π)(W
−1
E
+ W
−1
C
) (with C
E
∝ W
−1
E
and C
C
∝ W
−1
C
being the geometrical emitter and
collector capacitances, respectively) and the quantum
capacitance [36, 37] C
quant
= (2e
2
√
Σ
0
/
√
π~ v
W
) =
(2e
2
µ
0
/π~
2
v
2
W
), where Σ
0
and µ
0
are the pertinent dc
values of the hole density and the Fermi energy.
eδΣ
ω
= C δϕ
ω
. (3)
Considering the electron tunneling from the emitter
to the states above the top of the barrier (through the
triangular barrier), the emitter electron tunneling current
density j
E
can be presented as:
j
E
= j
t
E
exp
−
F
F
E
. (4)
Here F = (a
√
m∆
3/2
/e~), is the characteristic tunnel-
ing field, ∆ is the activation energy for electrons in the
contact, m is the effective electron mass in the bar-
rier material, a ∼ 1 is a numerical coefficient, F
E
=
(ϕ − V
bi
− µ/e)/W
E
is the electric field in the emitter
barrier, and j
t
E
is the maximum current density which
can be provided the emitter contact, As follows from
Eqs. (2) and (3), the ac component and the rectified com-
ponent of the emitter tunneling current δ j
E,ω
and δ j
ω
(for F F
0
) are, respectively, given by
δj
E,ω
= j
E,0
F
F
2
E,0
δF
E,ω
= σ
E
δF
E,ω
. (5)
δj
E,ω
'
j
E,0
2
1
2
F
F
E,0
2
−
F
F
E,0
δF
E,ω
F
E,0
2
'
σ
E
F
4
δF
E,ω
F
E,0
2
. (6)
Here j
E,0
= j
t
E
exp(−F/F
E,0
)), F
E,0
= (V
E
− V
bi
−
µ
0
/e)/W
E
, and σ
E
= j
0
(F/F
2
E,0
) are the emitter dc cur-
rent density (in the absence of the ac signals), the dc
electric field in the emitter barrier, and the emitter dif-
ferential conductance, respectively. The dc hole Fermi
energy in the GB µ
0
obeys the following equation:
µ
2
0
=
κ~
2
v
2
W
4e
V
E
− V
bi
−
µ
0
e
1
W
E
+
1
W
C
−
V
C
W
C
.
(7)
The ac electric field components perpendicular to the
GB plane in the emitter barrier, as well as in the collector
barrier, are respectively given by
δF
E,ω
=
C
quant
(C
quant
+ C
g
)
δϕ
ω
W
E
, (8)
δF
C,ω
= −
C
quant
(C
quant
+ C
g
)
δϕ
ω
W
C
. (9)
Using the Shockley-Ramo theorem, one can find the ac
component the electron current densities coming to the
base and the collector:
δj
EB,ω
= σ
EB,ω
δF
E,ω
, δj
EC,ω
= σ
EC,ω
δF
E,ω
. (10)
Here
σ
EB,ω
= σ
E
1
W
E
Z
W
E
0
dze
iω z/v
−
(1 − p)
W
C
Z
W
E
+W
C
W
E
dze
iω z/v
=
σ
E
iω
e
iωτ
E
− 1
τ
E
−
(1 − p)e
iωτ
E
(e
iωτ
C
− 1)
τ
C
, (11)
σ
EC,ω
= σ
E
(1 − p)
W
C
Z
W
E
+W
C
W
E
dze
iω z/v
=
σ
E
iω
(1 − p)e
iωτ
E
(e
iωτ
C
− 1)
τ
C
. (12)
where v is the drift velocity of the hot electrons cross-
ing the barriers above their tops (which is assumed to
be constant) and τ
E
= W
E
/v and τ
C
= W
C
/v are
the electron transit time across the emitter and collec-
tor barrier layers. The axis z is directed perpendicu-
lar to the GB plane. Since under the boundary condi-
tions (2), δϕ
ω
(x) = −δϕ
ω
(−x) (see below) and, hence,
δF
E,ω
(x) = −δF
E,ω
(x), the net ac currents
δJ
EB,ω
=
Z
L
−L
dxδj
EB,ω
= σ
EB,ω
Z
L
−L
dxδF
E,ω
= 0,
(13)
4
δJ
EC,ω
=
Z
L
−L
dxδj
EC,ω
= σ
EC,ω
Z
L
−L
dxδF
E,ω
= 0.
(14)
Simultaneously for the rectified components of the dc
current densities from the emitter to the GB and from
the emitter to the collector one obtains
δj
EB,ω
'
pσ
E
F
4
δF
E,ω
F
E,0
2
= pΓ |δϕ
ω
|
2
, (15)
δj
EC,ω
'
(1 − p)σ
E
F
4
δF
E,ω
F
E,0
2
= (1 − p)Γ |δϕ
ω
|
2
, (16)
where
Γ =
σ
E
F
4F
2
E,0
W
2
E
C
2
quant
(C
quant
+ C
g
)
2
. (17)
Consequently, the net rectified components of the dc
emittter-base and emitter-collector currents are given by
δJ
EB,ω
=
Z
L
−L
dxδj
EB,ω
' pΓ
Z
L
−L
dx|δϕ
ω
|
2
, (18)
δJ
EC,ω
=
Z
L
−L
dxδj
EC,ω
' (1−p)Γ
Z
L
−L
dx|δϕ
ω
|
2
, (19)
respectively.
The ac hole current along the GB is given by
δJ
BB,ω
= −σ
BB,ω
dδϕ
ω
dx
x=L
, (20)
where the lateral ac conductivity of the GB σ
BB,ω
is
given by (see, for example, Refs. [38–42]
σ
BB,ω
=
ie
2
µ
0
π~(ω + i/τ)
. (21)
III. PLASMA OSCILLATIONS AND RECTIFIED
CURRENT
To calculate the rectified current components using
Eqs. (15) and (16), one needs to find the spatial dis-
tributions of the ac potentials in the GB δϕ
ω
= δϕ
ω
(x).
For this purpose we use the hydrodynamic equations
for the hole transport along the GB [43, 44] (see also
Refs. [21, 22]) coupled with the Poisson equation solved
using the gradual channel approximation [i.e., using
Eq. (1)]. Linearizing the hydrodynamic equations and
Eq. (1) and taking into account that the ac component
of the hole Fermi energy is expressed via the variation of
their density and the potential, we arrive at the following
equation for the ac component of the GB potential δϕ
ω
(compare with Refs. [16, 21, 22] which should be solved
with boundary conditions given by Eq. (2):
d
2
δϕ
ω
d x
2
+
(ω + iν)(ω + iν)
s
2
δϕ
ω
= 0. (22)
Here s is the characteristic velocity of the plasma waves
in the gated graphene layers, which is given by s =
p
e
2
Σ
0
/mC ∝ Σ
1/4
0
[18], m = µ
0
/v
2
W
∝
√
Σ
0
being
the so-called ”fictituous” effective hole (electron) mass in
graphene layers [27], ν = (σ
B,ω
/W
E
C
g
) and ν = 1/τ + ˜ν,
where τ is the hole momentum relaxation time in the
2DHG and ˜ν = ˜ν
visc
+ ˜ν
rad
is associated with the con-
tribution of the 2DHG viscosity to the damping (see for
example, Ref. [21]) and with the radiation damping of
the plasma oscillations. The latter mechanism is associ-
ated with the recoil that the holes in the GB feel emitting
radiation (the pertinent term in the force acting on the
holes is referred to as the Abraham-Lorentz force or the
radiation reaction force [46, 47]). Taking into account
that the viscosity damping rate is proportional to the
second spatial derivative, for the gated plasmons with
the acoustic-like spectrum ˜ν
visc
= ζω
2
/s
2
(ζ is the 2DHG
viscosity),and considering that ˜ν
rad
∝ (2e
2
/3mc
3
)ω
2
[45–
47]), we put ν = 1/τ + ηω
2
), where η is the pertinent
damping parameter, and c is the speed of light. If the
viscosity damping surpasses the radiation damping, one
can set η ' ζ/s
2
.
The characteristic plasma-wave velocity s is deter-
mined by Σ
0
(as well as the thicknesses of the emitter
and barrier layers W
E
and W
C
) [20] and, hence, can be
changed by the variations of the bias voltages V
E
and
V
B
. Due to this the characteristic plasmonic frequency
Ω = π s/L can be effectively controlled by these voltages.
Equation (22) with Eq. (2) yield
δϕ
ω
=
sin(
p
(ω + iν)(ω + iν))x/s]
sin[
p
(ω + iν)(ω + iν)L/s]
δV
ω
2
. (23)
One can see from Eq. (23) that the spatial dependence of
δϕ
ω
is rather complex. In particular, when ω approaches
to nΩ, where n = 1, 2, 3, ..., this distribution can be os-
cillatory with fairly high amplitude of the spatial oscilla-
tions when Ω ν.
Substituting δϕ
ω
given by Eq. (23) to Eqs. (15) and
(16), we arrive at the following:
δj
EB,ω
' pΓ
×
sin[
p
(ω + iν)(ω + iν)x/s]
sin[
p
(ω + iν)(ω + iν)L/s]
2
|δV
ω
|
2
4
, (24)
5
δj
EC,ω
' (1 − p)Γ
×
sin[
p
(ω + iν)(ω + iν)x/s]
sin[
p
(ω + iν)(ω + iν)L/s]
2
|δV
ω
|
2
4
, (25)
δJ
EB,ω
' pΓ
×
Z
L
−L
dx
sin[
p
(ω + iν)(ω + iν)x/s]
sin[
p
(ω + iν)(ω + iν)L/s]
2
|δV
ω
|
2
4
, (26)
δJ
EC,ω
' (1 − p)Γ
×
Z
L
−L
dx
sin[
p
(ω + iν)(ω + iν)x/s]
sin[
p
(ω + iν)(ω + iν)L/s]
2
|δV
ω
|
2
4
. (27)
As follows from Eqs. (24) - (27), δj
EC,ω
= [(1 −
p)/p]δj
EB,ω
and δJ
EC,ω
= [(1 − p)/p]δJ
EB,ω
(as for the
pertinent dc current in the absence of the THz signals),
so that δJ
EC,ω
δJ
EB,ω
.
Using Eqs. (20) and (24), for the ac current between
the based contacts we obtain
δJ
BB,ω
= −
ie
2
µ
0
cot[
p
(ω + iν)(ω + iν)L/s]
π~ s
×
r
ω + iν
ω + iν
δV
ω
2
(28)
Due to the symmetry of the GB-HET structure and the
asymmetric spatial distribution of δϕ
ω
, there is no rec-
tified component of the lateral current in the GB-base,
i.e., δJ
BB,ω
= 0.
At very low signal frequencies ω ν, |ν| ' 4π pσ
E
/κ,
from Eqs. (23) and (22) we obtain
δϕ
ω
'
sinh(
√
ννx/s)
sinh(
√
ννL/s)
δV
ω
2
, (29)
δJ
EC,ω
' (1 − p)Γ L
sinh(2
√
ννL/s)
(2
√
ννL/s)
− 1
sinh
2
(
√
ννL/s)
|δV
ω
|
2
4
' (1 − p)Γ L
|δV
ω
|
2
6
. (30)
Since in reality |ν| ν, there is an intermediate
range of frequencies |ν| ω < ν. Assuming that
|ν| ν, Ω
2
/ω < ν, from Eq. (27) we arrive at
δJ
EC,ω
' (1 − p)Γ L
s
L
r
2
νω
|δV
ω
|
2
4
∝ (1 − p)
r
Ω
2
νω
|δV
ω
|
2
. (31)
FIG. 2: Spatial distributions of rectified component of the
emitter-collector current density δj
EC,ω
at different frequen-
cies ω (τ = 1 ps, s = 2.5 × 10
8
cm/s, and L = 1.5 µm).
If the characteristic frequency of the plasma oscilla-
tions Ω = π s/L ν, the ac GB potential amplitude
|δϕ
ω
| can markedly exceed the amplitude of the input
ac signal δV
ω
when the frequency is close to one of the
resonant plasma frequencies nΩ, where n = 1, 2, 3, ... is
the plasma resonance index. In this case, the rectified
emitter-collector current is pronouncedly stratified, i.e.,
its density δj
EC,ω
is a nearly periodic function of the
coordinate x.
Figure 2 shows the spatial distributions of the recti-
fied emitter-collector current density (corresponding to
the current stratification) calculated for different fre-
quencies using Eq. (25). It is assumed that τ = 1 ps,
s = 2.5×10
8
cm/s, and L = 1.5 µm. As seen in Fig. 2, the
spatial stratification of the current is rather pronounced
when ω is close to the plasma resonant frequencies (the
frequencies Ω = 5/6 ' 0.83 THz and 2Ω/2π = 5/3 '
1.66 THz): the current exhibits two streams centered
at |x|/L ' 0.5 when ω ' 0.83 THz and four streams
centered at |x|/L ' 0.25 when ω ' 1.66 THz. At the
frequencies far from the resonances, the spatial current
distribution becomes weakly nonuniform.
IV. GB-HET DETECTOR RESPONSIVITY
A. Current responsivity
The rectified current δJ
EC,ω
can be considered as the
output signal used for the detection of electromagnetic
radiation (in particular, THz radiation). The current
detector responsivity using this output signal is defined
as
6
R
ω
=
δJ
EC,ω
H
SI
ω
, (32)
where H is the GB-HET lateral size in the direction along
the contacts to the GB, I
ω
is intensity of the incident
radiation, and S is the antenna aperture. The latter
is given by S = λ
2
ω
G/4π [50], where G is the antenna
gain, λ
ω
= 2π c/ω is the radiation wavelength, and c is
the speed of light in vacuum. Taking into account that
I
ω
= cE
2
ω
/8π, where E is the radiation electric field in
vacuum, and estimating δV
ω
as δV
ω
= λE/π, one can
arrive at
|δV
ω
|
2
=
8λ
2
ω
I
ω
π c
=
32π c
ω
2
I
ω
. (33)
Considering Eqs. (27), (32), and (33), for the current
responsivity (in A/W units) we find
R
ω
=
R
L
Z
L
−L
dx
sin[
p
(ω + iν)(ω + iν)x/s]
sin[
p
(ω + iν)(ω + iν)L/s]
2
, (34)
where
R '
8(1 − p)Γ LH
cG
= ρ
LH
W
2
E
(35)
ρ =
8(1 − p)j
0
cG
F
F
2
E,0
2
. (36)
As follows from Eq. (34), at relatively long hole mo-
mentum relaxation times τ, the GB-HET responsivity
exhibits a series of very sharp and high peaks. It is in-
structive that in the GB-HETs under consideration, the
height of the resonant peaks does not decrease with an
increasing peak index (as in some other devices using
the plasmonic resonances). Although some decrease in
the peaks height with the increasing resonance index at-
tributed to the effect of viscosity and radiative damping
takes place, the pertinent effect is relatively weak (see
below).
In the limiting cases ω ν, |ν| and |ν| ν, Ω
2
/ω < ν,
corresponding to Eqs. (25) and (26), one obtains
R
ω
= R
0
'
2
3
R (37)
and
R
ω
' R
s
L
r
2
νω
= R
r
2Ω
2
π
2
νω
< R, (38)
respectively. The quantities R and R
ω
depend on the
geometrical and quantum capacitances and, therefore, on
the barrier layers thicknesses:
R
ω
∝ R ∝
1
W
2
E
C
2
quant
(C
quant
+ C
g
)
2
=
1
W
2
E
1 +
κ
4π C
quant
(W
−1
E
+ W
−1
C
)
−2
. (39)
For the doping level of the emitter contact N
D
=
10
18
cm
−3
, assuming that the thermal electron velocity
v
T
= 10
7
cm/s, we obtain j
t
E
= 1.6×10
6
A/cm
2
. Setting
∆ = 0.2 eV and the effective mass in the barrier layer
m ' 2.5 × 10
−28
g , we arrive at F ∼ 2 × 10
6
V/cm.
Setting also F
E,0
= 5 × 10
5
V/cm (to provide the hole
density in the GB about of Σ = 10
12
cm
−2
), we find
j
E,0
' 2.9 × 10
4
A/cm
2
and σ
E
' 0.23 Ohm
−1
cm
−1
.
At these parameters, one obtains also C
quant
' 2.6 ×
10
6
cm
−1
. Using these data and setting in addition
p 1, W
E
W
C
/(W
E
+ W
C
) = 5 nm, κ = 4, and G = 1.5,
from Eq. (36) we obtain ρ ' 2×10
−4
A/W. If L = 1.5 µm
and H = 10 µm, Eqs. (35) - (36) yield R ' 30 A/W and
R
0
' 20 A/W. These parameters are also used in the
estimates below.
In the case of high quality factor of the plasma res-
onances Ω/ν, the quantities
δJ
EB,ω
and δJ
EC,ω
and,
hence, the responsivity as functions of the signal fre-
quency ω described by Eqs. (27) and (34) exhibit sharp
peaks at ω ' nΩ, attributed to the resonant excitation
of the plasma oscillations (standing plasma waves). The
peak width is primarily determined by the frequency ν.
Indeed, for Ω/ν 1 Eq. (34) yields
maxR
ω
' R
Ω
'
3R
0
2
2Ω
πν
2
>> R
0
. (40)
Using the above estimate for R
0
, the peak values of the
responsivity at Ω = 5/3 THz and τ = 1 ps is approxi-
mately equal to maxR
ω
' R
Ω
' 1.33 × 10
3
A/W.
Figure 3 shows the dependence of the normalized GB-
HET current responsivity calculated using Eq. (34) for
several sets of parameters. As seen, the positions of the
responsivity peaks shift toward higher frequencies when
the plasma frequency increases, i.e., when s increases
and/or L decreases because Ω ∝ s/L (compare the curves
”1” and ”2”). A shortening of the momentum relaxation
time τ leads to a smearing of the peaks (compare the
curve ”3” corresponding to τ = 1 ps and the curve ”4”
corresponding to τ = 0.5 ps). Figure 4 shows the low-
ering and broadening of the resonance peaks of the GB-
HET current responsivity with the decreasing momen-
tum relaxation time τ described by Eq. (34). As seen, at
τ < 0.3 − 0.4 ps, the responsivity peaks vanish, while at
τ = 1 ps they are fairly sharp and high..
B. Voltage responsivity
The variation of the dc current component δJ
EC,ω
cause by the ac signals results in a change of the volt-
7
0
0.5
1
1.5
2
Frequency, ω/2π (THz)
0
10
20
30
40
50
60
70
Normalized responsivity, R
ω
/R
1
2
3
4
FIG. 3: Normalized respoinsivity versus signal frequency for
GB-HETs with different parameters:
1 - τ = 1 ps, s = 5 × 10
8
cm/s, L = 1.5 µm;
2 - τ = 1 ps, s = 5 × 10
8
cm/s, L = 2.0 µm;
3 - τ = 1 ps, s = 2.5 × 10
8
cm/s, L = 1.5 µm;
4 - τ = 0.5 ps, s = 5 × 10
8
cm/s, L = 1.5 µm.
These parameters correspond to Ω/2π = 5/3, 5/4, 5/6, and
5/3 THz, respectively.
FIG. 4: Normalized respoinsivity as a function of signal fre-
quency and momentum relaxation time (s = 2.5 × 10
8
cm/s
and L = 1.5 µm, i.e., Ω/2π = 5/6 THz).
age drop, δV
ω
= −δV
C,0
, across the load resistor in the
collector circuit (see Fig. 1). Considering that this leads
to an extra variation of the dc emitter and collector cur-
rents δJ
E,0
= [σ
E
C
g
/(C
quant
+C
g
)(W
E
+W
C
)]δV
C,0
and
δJ
EC,0
= [(1 − p)σ
E
C
g
/(C
quant
+ C
g
)]δV
C,0
because of
its dependence on the collector contact dc potential. The
latter dependence is essentially associated with the effect
of quantum capacitance. Taking this into account, for the
voltage responsivity of the GB-HET under consideration
R
V
ω
= δV
ω
H/SI
ω
= δV
C,0
H/SI
ω
we obtain
R
V
ω
=
R
V
L
Z
L
−L
dx
sin[
p
(ω + iν)(ω + iν)x/s]
sin[
p
(ω + iν)(ω + iν)L/s]
2
. (41)
Here
R
V
= R
r
C
1 +
C
g
(C
quant
+ C
g
)
2LH
(W
E
+ W
C
)
r
C
σ
E
, (42)
where r
C
is the load resistance. It is instructive that due
the the absence of the ac collector current (i.e., the ac
current through the load resistor), associated with the
GB-HET structure symmetry and the asymmetry of the
applied ac signal voltage and the ac potential spatial dis-
tribution along the GB) the RC-factor of the voltage re-
sponsivity is independent of the signal frequency.
For W
E
= W
C
= W , Eqs. (38) and (40) yield
R
V
ω
= R
ω
r
C
1 +
σ
E
r
C
(LH/W
2
)W
(1 + 2π C
quant
W/κ)
(43)
At r
C
(1 + 2πC
quant
W/κ)(W/LHσ
E
) = r
C
,
Eq. (43) yields the obvious formulas:
R
V
0
= R
0
r
C
(44)
at low frequencies, and
R
V
Ω
=
3R
0
2
2Ω
πν
2
r
C
(45)
at the plasma resonance ω = Ω.
At r
C
> (1 + 2πC
quant
W/κ)(W/LHσ
E
) = r
C
, from
Eq. (43) we obtain, respectively,
R
V
Ω
' R
0
(W
2
/LH)
σ
E
W
1 +
2π C
quant
W
κ
. (46)
and
R
V
Ω
'
3R
0
2
2Ω
πν
2
(W
2
/LH)
σ
E
W
1 +
2π C
quant
W
κ
. (47)
For τ = 1 ps, s = 5×10
8
cm/s, and L = 1.5 µm (as for the
curve ”1” in Fig. 3), so that Ω = 5/3 THz, as well as σ
E
=
0.23 A/V cm, C
quant
= 2.6 ×10
6
cm
−1
(see the estimate
in the previous subsection), and κ = 4 from Eq.(47) we
obtain the following estimate: R
V
Ω
' 2 × 10
5
V/W. For
the above parameters one obtains r
C
' 150 Ohm. Even
at smaller r
C
, the voltage responsivity can be fairly large.
Setting r
C
= 5−10 Ohm, we obtain R
V
0
' 100−200 V/W
and R
V
Ω
' (6.6 − 13.2) × 10
3
V/W, respectively.
8
Considering Eqs. (42) and (39), we find the following
dependences of the current and voltage responsivities R
ω
and R
V
ω
on the emitter and collector barriers thickness
W (at not too large r
C
):
R
V
ω
= R
ω
r
C
∝
4π
2
C
2
quant
κ
2
(1 + 2π C
quant
W/κ)
2
. (48)
According to Eq. (48), R
ω
and R
V
ω
markedly decrease in
the range W > κ/2πC
quant
' 10 nm.
V. DISCUSSION
As follows from Eqs. (34) and(40), an increase in ν
with increasing frequency ω due to the reinforcement
of the plasma oscillation damping associated with the
viscosity and the radiative damping, might lead to the
gradual lowering of the resonant peaks with their index
n. However, our estimates show that the contribution of
these two mechanisms to the net damping is small com-
pared to the damping associated with the hole momen-
tum relaxation (collisional damping). Indeed, disregard-
ing the radiative damping and assuming the 2DHG vis-
cosity to be ζ = 10 cm
2
/s (i.e., smaller than in the stan-
dard 2DEG and 2DHG in the GaAs based heterostruc-
tures [21, 49, 50]) and s = (2.5 − 5) × 10
8
cm/s, we
obtain η ' (4 − 16) × 10
−17
s. Hence, in the fre-
quency range ω/2π ≤ 2 THz (as in Fig. 2), we find
˜ν
visc
= ηω
2
≤ (6.3 − 25.3) × 10
9
s
−1
1/τ. Therefore,
the heights of the responsivity peaks in Fig. 2 in the curve
”3” at ω ' 0.8 and 1.6 THz are virtually equal. How-
ever, the peaks corresponding to higher resonances with
the frequencies in the range 5 - 10 THz can be markedly
lowered and smeared, because in this range ˜ν
visc
can be-
come comparable with 1/τ . For example, for the same
values of η, τ = 1 ps, and ω/2π = 5 - 10 THz, we obtain
˜ν
visc
τ ' 0.16 − 0.64.
Equation (42) describes the saturation of the voltage
responsivity R
V
Ω
peak value with increasing load resis-
tance r
C
. This is associated with the effect of the voltage
drop across the load on the potential drop between the
emitter and the base and, hence, the hole Fermi energy
in the GB, which determine the injection current. Such
an effect is due to the finite value of the GB quantum
capacitance (see, Refs. [12, 13]) - if C
quant
tends to infin-
ity, the emitter-base voltage becomes independent of r
C
,
and the saturation of the R
V
Ω
−r
C
dependence vanishes.
The THz detectors using a similar operation princi-
ple and InP double heterojunction bipolar transistors
(DHBTs) were recently fabricated and studied experi-
mentally [51–53]. The estimated responsivities of the
DHBTs in question for the non-resonant detection regime
are somewhat smaller but of the same order of magni-
tude than those given by Eq. (43) and the pertinent esti-
mates. However, the experimental values of the respon-
sivity are much smaller than the values predicted above
for the resonant detection [see Eqs. (44) and (46) and
the estimates based on these equations]. Apart from the
parasitic effects and the absence of any spatial coupling
antenna, this can be attributed to the doping of the base
in the InP-DHBTs, which inevitably leads to relatively a
shorter hole momentum relaxation time τ compared to
that in the GB (where the 2DHG is induced by the ap-
plied voltages). Possibly, a higher probability of the hot
electron capture into the InGaAs base (that had a rela-
tively large thickness of 28 nm) in the DHBTs in com-
parison with the GB-HETs can be an additional factor.
The GB-HET current and voltage responsivities are
determined by several characteristics: the characteris-
tics of the tunneling emitter, geometrical characteristics
of the GB-HET structure, materials of the emitter as
well as the emitter and collector barrier layers, and ap-
plied bias voltages. The diversity of these factors enables
the optimization of the GB-HETs operating as resonant
plasmonic detectors, in particular, an increase in the re-
sponsivity in comparison with the values obtained in the
above estimates. The resonant plasmonic THz detectors
can based on not only the GB-HET structure shown in
Fig. 1(a) (with an extra antenna connected to the GB
side contacts), but also based on lateral structures with
the GB contacts forming a periodic array.
The comparison of the GB-HETs [1–4] and InP-
DHBTs [51–53] with the GB-HETs under consideration
highlights the following advantages of the latter: (i) a
longer momentum relaxation time of holes τ in the GB;
(ii) a higher plasma-wave velocity s that enables higher
resonant plasma frequencies; (iii) a smaller capture prob-
ability of hot electrons into the GB and, consequently,
larger (or even much larger) fraction of the hot electrons
reaching the collector; (iv) coupling the incoming THz
signal to the GB resulting in the absence of the ac cur-
rent in the emitter-collector circuit and prevanting the
RC effects usually hindering the high-frequency opera-
tion.
VI. CONCLUSIONS
We developed an analytical model for vertical het-
erostructure HETs with the GB of the p-type sandwiched
between the wide-gap emitter and collector layers and the
N-type contacts. Using this model, we described the GB-
HET dynamic properties and studied the GB-HET oper-
ation as detectors of THz radiation. The main features
of the GB-HETs are high hole mobility in the GB, low
probability the capture of the hot electrons injected from
the emitter and crossing the GB, and the absence of the
collector ac current. These features enable pronounced
voltage-controlled plasmonic response of the GB-HETs
to the incoming THz radiation, high hot-electron injec-
tion efficiency, and the elimination of the RC-limitations
leading to elevated the GB-HET current and voltage re-
sponsivities in the THz range of frequencies, particularly
at the plasmonic resonances at room temperature. This
might provide the superiority of the GB-HET-based THz
9
detectors over other plasmonic THz detectors based on
the standard heterostructures . Thus, the THz detectors
based on the GB-HETs can be interesting for different
applications.
Acknowledgments
The authors are grateful to D. Coquillat and F. Teppe
for the information related to their experimental data
on InP HBTs operating as THz detectors. The work
was supported by the Japan Society for Promotion of
Science (Grant-in-Aid for Specially Promoted Research
23000008) and by the Russian Scientific Foundation
(Project 14-29-00277). The works at UB and RPI were
supported by the US Air Force award FA9550-10-1-391
and by the US Army Research Laboratory Cooperative
Research Agreement, respectively.
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