Download Contact and Space-Charge Effects in Quantum Well Infrared

Jpn. J. Appl. Phys. Vol. 38 (1999) pp. 5815–5822
Part 1, No. 10, October 1999
c
°1999
Publication Board, Japanese Journal of Applied Physics
Contact and Space-Charge Effects in Quantum Well Infrared Photodetectors
Victor RYZHII and H. C. L IU1
Department of Computer Hardware, University of Aizu, Aizu-Wakamatsu 965-8580, Japan
1 Institute for Microstructural Sciences, National Research Council, Ottawa, Ontario K1A 0R6, Canada
(Received April 19, 1999; accepted for publication July 5, 1999)
We present the results of a new self-consistent analytical model for the calculation of the dark current and the electric-field
and space-charge distributions in quantum well infrared photodetectors (QWIPs). This model takes into account thermionic
emission from the QWs, tunneling injection of electrons from the emitter contact, transport of the excited and injected electrons
across an active region of a QWIP and their capture into the QWs. It is shown that the electric-field and space-charge distributions in the QW structure are nonuniform in general. The character of their nonuniformity is determined by the relationships
between the structural parameters, parameters of elementary processes, and bias voltage.
KEYWORDS: quantum well infrared photodetector, electric-field distribution, space charge, dark current
1. Introduction
Quantum well infrared photodetectors (QWIPs) utilizing
intersubband electron or hole transitions have attracted much
attention over the recent years. The basic physics of QWIPs
has been well documented.1–3) QWIP operation is associated
with the photoexcitation of electrons (holes) from the bound
state in the QWs to the continuum states above the barriers, the electron injection from the emitter contact, and the
transport and capture of photoexcited and injected electrons.
Thermionic emission of electrons from QWs determines the
dark current. The collector contact enables the extraction of
electrons from an active region of a QWIP.
Different models have been proposed for steady-state and
transient processes in QWIPs.4–19) Early models of QWIPs
with multiple QWs are based on the assumption that the
electric–field distribution in the active region is uniform.4, 5)
Such simple QWIP models assume a perfectly injecting emitter contact,5) i.e., the contact injects as many electrons as
needed by the QW structure. Thus, the injected current density should be that which yields the required rate of electron capture into the QWs to compensate the escape of electrons due to their thermoemission and photoexcitation from
the QWs.
A real emitter contact provides such an injected current
density if the contact electric field has a proper value which
can differ from the average electric field in a QWIP. An appropriate value of the electric field at the emitter contact is
ensured by a space charge in the QW structure, which is created due to the difference in the electron and donor sheet concentrations in the QWs.8, 10–12) The space charge in the QW
structure can be either positive or negative depending on bias
voltage. Under certain bias voltages, determined by the parameters of the emitter contact, the space charge disappears,
hence, the electric field becomes constant and coincides with
the average field in the QW structure.12) The existence of a
space charge in the QW structure results, in general, in the
nonuniformity of the electric field. This effect has been considered previously for the cases of single20, 21) and multiple22)
QWIPs. It has been demonstrated, using numerical simulations,8, 10, 11, 15) that in QWIPs with a large number of QWs,
the region of nonuniform electric field can be relatively narrow and located near the injecting contact, while the electric
field in the other part of the QWIP active region is nearly constant. The nonuniformity in photoexcitation due to photon
flux attenuation adds complexity to the electric-field distribution.6, 9) It is instructive that in QWIPs in which the electric
field weakly affects the emission of electrons from the QWs
and their capture into the QWs, the electric field can be essentially nonuniform but rather smooth with a scale of nonuniformity comparable to the QW structure thickness.12) In such a
case, the electron concentrations in all QWs are almost the
same. The latter leads to a near-parabolic distribution of the
electric potential and, consequently, a linear distribution of
the electric field.12) This indicates that strongly nonuniform
distributions, like distributions with high electric field domain
near the emitter contact, are associated with relatively strong
dependences of the electron excitation and capture rates upon
the local value of the electric field. Properties of the emitter contact can also influence the space charge in the QW
structure and, hence, the distribution of the electric field in
it. Thus, the electric-field distribution in a QWIP and, consequently, its characteristics, are determined by both the emitter contact parameters and the field dependences of the rates
of electron excitation from and capture into the QWs. The
overall characteristics of QWIPs with multiple QWs can be
rather insensitive to the emitter contact parameters in a wide
range of applied voltages.12) Recently, a weak effect of the
shape of the emitter barrier in QWIPs with a large number of
QWs was confirmed experimentally.23) In contrast, the contact and space-charge effects in QWIPs with a moderate number of QWs can be essential in allowing some flexibility for
the optimization of such devices. Apart from this, properties
of the emitter contact can manifest themselves in nonlinear
effects in QWIPs, in particular, at a high power of incident infrared radiation12, 15) (see also experimental results presented
in refs. 24 and 25).
Phenomena in QWIPs can be well described by numerical simulations based on either drift-diffusion or Monte Carlo
models. However, QWIPs are characterized by numerous parameters. This hinders the use of numerical models for the
extraction of physical information on the details of QWIP operation. From this standpoint, analytical models leading to
explicit formulas for different characteristics of QWIPs can
be beneficial.
In this paper we present a study of contact and space-charge
effects using a self-consistent analytical model for QWIPs
with an arbitrary number of QWs and tunneling injection of
electrons. We derive explicit analytical expressions as functions of bias voltage and structural parameters, taking into ac-
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Jpn. J. Appl. Phys. Vol. 38 (1999) Pt. 1, No. 10
count both the nonideality of the emitter contact and the influence of the electric field on the processes of electron thermoemission and capture. This paper is a significant generalization of a previous paper by one of the authors.12)
2. Device Model and Main Equations
The QWIP under consideration consists of a multiple QW
structure, sandwiched between the emitter and collector barriers, followed by heavily doped contact layers. The QW
structure comprises thin doped narrow-gap QWs separated by
thick undoped wide-gap barriers. An example of the conduction band diagrams of a QWIP is shown in Fig. 1.
It is assumed that all the QWs, the barriers between the
QWs, and the barrier between the extreme QW and the collector layer are identical. The barrier separating the emitter contact layer and the first QW (the top-most barrier or the emitter
barrier) can differ from the interwell barriers in height. The
thickness of the QWs (L w ) is much smaller than the thickness
of the barriers (L b ). The thickness and the depth of the QWs
are chosen in such a way that the QW has a single bound state
while the first excited level lies near the top of the barriers.26)
It is assumed that the electron injection from the emitter contact into the QW structure is due to tunneling. Thermionic
emission is considered to be the main mechanism of electron
escape from the QWs if the QWIP is not illuminated.
The model of the QWIP is described by the Poisson equation and equations governing the electron balance in the
QWs:12)
N
d 2ϕ
4π e X
=
(6n − 6d ) · δ(x − nL),
dx 2
æ n=1
Gn =
j
pn .
e
(1)
(2)
Here ϕ = ϕ(x) is the self-consistent electric potential, 6n
is the electron sheet concentration in the nth QW (n =
1, 2, 3, . . . , N , where N is the number of QWs), 6d is the
g
sheet concentration of donors in each QW, G n = G(E n , 6n )
is the rate of the electron escape from the nth QW, pn =
p(E nc ) is the capture parameter (probability5) ) for an electron
g
traversing the nth QW, E n and E nc are the effective electric
fields which determine the excitation rate from the nth QW
and the capture rate into the latter (as seen below, these fields
can differ from the average electric field and actual electric
Emitter
QW structure
Collector
Bound state
L
Fig. 1. Schematic view of the QWIP conduction band diagram under bias
voltage.
fields at the left-hand and right-hand sides of the QW), j is
the current density, e is the electron charge, æ is the dielectric
constant, L = L w + L b ' L b is the QW structure period, δ(x)
is the Dirac δ function, and x is the coordinate in the direction
perpendicular to the QW plane (in the growth direction).
At temperatures near the temperature of liquid nitrogen,
electron escape from the QWs is associated with thermoemission.4) The rate of electron thermoemission from the QWs
increases exponentially with increasing local electric field,
primarily due to the lowering of the barrier height with respect to the Fermi energy of electrons in the QWs, which
leads to the decrease of the activation energy. The increase
of the electron sheet concentration resulting in the elevation
of the Fermi level also leads to the decrease of the activation
energy. Hence, when thermo-emission is the dominant mechanism,14, 27–29) the rate of electron escape from the QW can be
presented as
· w
g ¸
²act (6n ) − e E n l
g
.
(3)
G(E n , 6n ) = g · exp −
kB T
Here, g is the preexponential factor, kB is the Boltzmann conw
(6n ) = ²0w − ²Fw (6n ), ²0w and
stant, T is temperature, and ²act
w
²F are the energy of the bound state in the QW with respect
to the barrier top (the QW ionization energy) and the Fermi
energy of the two-dimensional electron gas in the QW, respectively, and l ' L w /2.
For the degenerate two-dimensional electron gas (K =
π h̄ 2 6d /mkT À 1) one has ²Fw = π h̄ 2 6n /m, where h̄ is
the Planck constant and m is the electron effective mass. Taking this into account, the rate of thermionic emission from the
QWs can be presented as
· g
¸
En
π h̄ 2
(6n − 6d ) ,
+
(4)
G(E ng , 6n ) = G · exp
Eg
mkB T
where G = g · exp(π h̄ 2 6d /nkB T ) is the thermoemission rate
when 6n = 6d and E nw = 0, and E g = kB T /el is the characteristic field of electron thermoemission from the QWs.
At liquid nitrogen temperatures the inequality K À 1 occurs for GaAs QWs with 6d ≥ 5 × 1011 cm−2 . At lower
temperatures, the two-dimensional electron gas is degenerate
even at smaller concentrations.
The term on the right-hand side of eq. (2) describes the
capture of electrons into the QWs under the assumption of
drift transport of electrons across the QW structure when the
electron diffusion can be neglected. In such a case, the concentration of electrons in the continuum states is proportional
to the current density. This is valid if the electric field in all
parts of the QW structure is sufficiently strong.
The QWIP model under consideration differs from that
used previously12) to derive analytical expressions via the inclusion of the dependences of the excitation and capture rates
on the electric field. To obtain the QWIP characteristics in
closed analytical form some model field-dependence for the
capture parameter p(E nc ) should be assumed.
The mechanisms of the electron capture into QWs have
been extensively studied (see, for example, ref. 30). However,
the nature of the field dependence of the capture probability
p(E n ) is not yet fully understood. In n-QWIPs made, for example, of A3 B5 heterostructures, the fraction of electrons in
the 0-valley having the energy lower than the optical phonon
energy can be small in comparison with the fractions of hot 0
Jpn. J. Appl. Phys. Vol. 38 (1999) Pt. 1, No. 10
V. RYZHII and H. C. L IU
electrons and electrons in the L- and X-valleys, and markedly
decreases with increasing electric field. Because the capture
processes are mainly associated with the emission of optical
phonons by low energy 0− electrons,31–33) the capture probability drops when the electric field increases due to the decrease of the number of such electrons. As shown by Monte
Carlo modeling,18, 34) the drop of the capture probability can
be notably strong. Taking this into account, we will use the
following field dependence of the capture probability:
µ
¶
E nc
c
,
(5)
p(E n ) = exp −
Ec
where E c is the characteristic electric field associated with
the capture of electrons. The “exponential” can fit real dependences and is in good agreement with recent Monte Carlo
simulations.34)
It is necessary to stress that the rates of thermoemission
g
and capture depend on the effective electric fields E n and E nc ,
which are given by the local electric field averaged over some
area in the vicinity of a particular QW. In the simplest case,
these effective fields are determined by the real electric fields
in the left (E n ) and right (E n+1 ) barriers. If the electric field is
uniform, the effective fields coincide with the average electric
field in the QWIP E = V /W , where V is the bias voltage
and W = (N + 1) L is the thickness of the QW structure.
g
Generally, the effective electric fields E n and E nc are different
from the average electric field and from each other. To take
into account the nonlocality effects in question we assume
that
g,c
E ng,c = E + αl (E n − E) + αrg,c (E n+1 − E).
g,c
αl
g,c
αr
(6)
and
are coefficients. These coefficients deHere,
termine the relative contributions of the real electric fields in
the left barrier (with the subscript “l”) and in the right one
g
(with the subscript “r”) to the effective electric fields E n and
c
E n . In the case of uniform electric field in the QWIP when
g,c
E n = E n+1 = E, one obtains E n = E, i.e., the effective fields are equal to the average field. It is obvious that
if the electric field in the left or right barriers becomes larger
than the average value, the rate of thermoemission increases,
g,c
g,c
while the capture rate drops. Thus, αl , αr ≥ 0. It is necessary to emphasize that the electron gas in the continuum
states is strongly heated under conditions typical of QWIP
operation (strong electric field and relatively low temperature). In such a situation, the characteristic length L ² of hot
electron relaxation can significantly exceed the QW structure
period L and be comparable to the QW structure thickness
W . It is conceivable that in AlGaAs/GaAs QWIPs at liquid nitrogen temperature, the energy relaxation length is of
L ² ' 102 –103 nm. Hence, for the usual period of the QW
structure of L = 30–50 nm, L ² /L ' 2–30. This indicates that
the association between the rates and the electric field can be
g,c
quite nonlocal. Strong nonlocality corresponds to small αl
g,c
and αr when the balance between thermoemission and capture is determined primarily by the average electric field. This
case has been studied analytically in ref. 12. The model used
g
in numerical simulations11) conforms, in fact, to αlc = αr = 1
g
c
and αl = αr = 0 (see also ref. 15).
It is reasonable to assume the boundary conditions in the
form
¯
¯
ϕ ¯¯
= 0,
and
x=0
¯
¯
ϕ ¯¯
= V.
5817
(7)
x=W
Because the current density j is the same throughout the
QW structure, it coincides with the density of the tunneling
current through the emitter barrier. Hence, taking into account
the triangular shape of the emitter barrier in the presence of
the electric field, one can write12, 16, 35)
µ
¶
Et
j = jm exp −
,
(8)
Ee
where E e = (dϕ/dx)|x=0 is the electric field at the emitter
contact (in the emitter barrier, hence, E e = E 1 ), E t is the
characteristic tunneling electric field, and jm is the maximum
current density from the emitter layer. The quantity jm depends mainly on the doping level of the emitter layer. The
characteristic tunneling field is determined by the activation
energy of electrons in the emitter contact ²act = ²0 −²F , where
²0 is the height of the emitter barrier for electrons in the emitter layer, and ²F is the Fermi energy√of electrons in this layer.
3/2
This field is expressed as36) E t = 4 2m²act /3eh̄. If the electron tunneling is assisted by thermoexcitation, E t can depend
on the temperature as well.
The model under consideration yields a self-consistent description of the QWIP under dark conditions. Using this
model, one can express the dark current in a QWIP as a
function of the bias voltage V (the average electric field
E = V /W ) and the characteristic fields related to the elementary mechanisms: E t (tunneling through the emitter barrier), E g and E c (thermoemission and capture, respectively).
Similar models have been previously used in numerical simulations.8, 10, 11, 15) In contrast to those works, below we obtain
the exact analytical solution of the equations of the model.
3. Dark Current
Equations (2) and (4) yield
6n − 6d =
¤
æ £
· γ E gc ln i dark + γl (E − E n ) + γr (E − E n+1 ) . (9)
4π e
Here, i dark = jdark /j0 (E) is the dark current reduction factor,
jdark is the dark current density, j0 (E) = e G(E, 6d )/ p(E) =
e G exp(E/E gc ),
4kB T
,
eaB E gc
µ g
¶
4kB T αl
αc
+ l ,
γl =
eaB
Eg
Ec
µ g
¶
4kB T αr
αc
+ r ,
γr =
eaB
Eg
Ec
γ =
where E gc = E g E c /(E g + E c ) and aB = æh̄ 2 /me2 is the
Bohr radius. The quantity j0 (E) is the dark current density
in a QWIP at the bias voltage V = (N + 1)E L À kT /e in
the case of a homogeneous electric field. The deviation of the
dimensionless dark current density i dark from unity indicates
the deviation of the dark current density derived using our
model from that obtained assuming a uniform electric field.
The QWIP characteristics and, in particular, the form of the
potential and space-charge distributions depend on the parameters γ , γl , γr and N .
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Jpn. J. Appl. Phys. Vol. 38 (1999) Pt. 1, No. 10
V. RYZHII and H. C. L IU
Substituting 6n − 6d from eq. (9) into eq. (1) and integrating over dx, we obtain the following equation for the selfconsistent electric field in the nth barrier:
·
¸
γ
E gc ln i dark ,
E n = a n−1 E e + (1 − a n−1 ) E +
(γl + γr )
(10)
where a = (1 − γl )/(1 + γr ).
Taking into account that the electric field in the nth barrier
E n does not depend on the coordinate, boundary conditions
(7) can be reformulated as
L
N +1
X
E n = V.
(11)
n=1
Substituting E n from eq. (10) into eq. (11), one can obtain
the equation which relates the emitter electric field E e , the
average electric field E, and the quantity i dark as:
Ee = E +
γ (1 − b N )
E gc ln i dark ,
(γl + γr )
(12)
where b N = (N + 1)(1 − a)/(1 − a N +1 ).
Substituting E e from eq. (12) into eq. (8), we obtain an
equation for the dark current reduction factor:
γ (1 − b N )
Et
E gc ln i dark ,
=E+
(13)
ln α − ln i dark
(γl + γr )
where ln α = ln[ jm /j0 (E)]. The latter equation can be rewritten as ln α = ln α0 − E/E gc , where α0 = jm /eG. Taking into
account that usually ln α À 1 (due to the very large value of
jm /eG), we obtain the following approximate solution for the
dark current reduction factor:
¶
µ
(γl + γr )
Et
.
(14)
E−
ln i dark =
γ (b N − 1) E gc
ln α
From eq. (14) for the dark current density as a function of
the average electric field (i.e., the bias voltage) and structural
parameters one obtains
¶¸
·
µ
(γl + γr )
Et
. (15)
E−
jdark = j0 (E) · exp
γ (b N − 1)E gc
ln α
If (γl + γr )N ¿ 1, eq. (15) yields the same formula for the
dark current as that obtained previously.12) This special case
corresponds to the rather weak dependences of the thermoemission and capture rates on the local value of the electric
field.
Equation (15) can be simplified if N = 1 or N À 1. For
QWIPs with N = 1, one has b1 (γ ) = 2/(1 + a), and from
eq. (15) we obtain
¶¸
·
µ
(2 + γr − γl )
Et
. (16)
E−
jdark ' j0 (E) · exp
γ E gc
ln α
When N À 1 and |a| < 1, eq. (15) leads to
¶¸
·
µ
(1 + γr )
Et
.
E−
jdark ' j0 (E) · exp
γ N E gc
ln α
(17)
4. Derivation of the Electric-Field and Space-Charge
Distributions
Using eqs. (12) and (14), one can obtain the value of the
self-consistent electric field in the nth barrier as:
En =
Et
+
ln α
µ
Et
−E
ln α
¶
b N (a n−1 − 1)
.
(b N − 1)
(18)
In particular, from eq. (18) one can obtain the following formula for the electric field E e = E 1 in the emitter barrier:
Et
.
(19)
Ee =
ln α
The electric fields in the collector barriers (E c = E N +1 ) in
QWIPs with N = 1 in the limiting case of (γl + γr )N À 1
are given, respectively, by
Et
,
E c = 2E −
ln α
µ
¶
(20)
Et
1 + γr
E−
' E.
Ec ' E +
(γl + γr )N
ln α
The obtained formulas result in the following equation for
the electron sheet concentration in the nth QW:
µ
¶
E t (a n−1 − a n ), b N
æ
E−
.
(21)
6n − 6d =
4π e
ln α
(b N − 1)
In the limiting case of (γl + γr )N ¿ 1, eq. (21) results in
µ
¶
Et
æ
E−
,
(22)
6n − 6d '
2π eN
ln α
i.e., the different QWs are charged equally. Only a few
QWs near the emitter contact are actually charged, forming a
charged domain, while the bulk of the QW structure is quasineutral.
The total charge accumulated in the QWs (per unit area) is
given by
µ
¶
Et
æc N
−E ,
(23)
Q=
4π ln α
where the factor c N = b N (1−a N )/(b N −1) varies from 2 to 1
with increasing (γl + γr )N . Hence, the magnitude of the total
space charge Q is halved when (γl + γr )N varies from a small
value to a large one. According to eq. (23), the total charge
Q > 0 if E < E t / ln α, and Q < 0 when E > E t / ln α. If
E ln α = E t , the total charge becomes zero, and the electricfield distribution becomes strictly uniform. The latter can be
clearly seen from eq. (18). In such a case, as seen from eqs.
(15)–(17), one obtains jdark = j0 (E). These conclusions were
reached previously12) using a simplified QWIP model.
5. Examples
The comparison of eqs. (15)–(17) reveals that in all limiting cases the dependences of the dark current upon the average electric field are very similar. The influence of the contact and space-charge effects on the dark current is described
by the dark current reduction factor i dark = jdark /j0 (E). The
latter is determined by the exponential factors in the righthand sides of eqs. (15)–(17). These factors describe the deviation of the current-voltage characteristic from that calculated under the assumption that the self-consistent electric
field is uniform, i.e., without the inclusion of the effects in
question. The dark current reduction factor also describes the
dependence of the dark current on the number of QWs because j0 is independent of N . If the number of QWs is large
enough, so that (γl + γr )N À 1 one obtains jdark ' j0 (E) ∝
exp(E/E gc ). In this limiting case, the dark current does not
depend on the number of QWs or any parameter related to
V. RYZHII and H. C. L IU
Jpn. J. Appl. Phys. Vol. 38 (1999) Pt. 1, No. 10
This is in agreement with experimental results.23)
In contrast to QWIPs with a large number of QWs, in
QWIPs with a moderate number of QWs, the contact and
space charge-effects markedly influence the dark current. If
N is small, the dark current jdark can be markedly lower than
j0 (see curves for N = 1 and 4 in Fig. 2). However, the contact and space-charge effects resulting in the change of the
dark current value essentially do not modify its dependence
on the average electric field. This is apparent from Fig. 2
where the ratio jdark /j0 (E) is almost independent of the electric field for QWIPs with the number of QWs varying in the
range of N = 1–32.
As shown in Fig. 3, the dark current is weakly dependent on
the emitter parameters (which determine the tunneling field
E t ) when N = 32, and is fairly sensitive to these parameters
for a moderate N (see curves in Fig. 3 for N = 8). Figures 4–6 show the spatial distributions of the self-consistent
electric field in the QWIPs (electric field versus normalized
distance x/L) with different N , E t for different average electric fields E. These figures reveal that, generally, the electric
field steeply decreases from high values in several barriers ad-
1.0
0.8
N=32
0.6
Reduction factor
the properties of the contacts. The current-voltage characteristic is determined only by the “combined” characteristic
field E gc = E g E c /(E g + E c ). The electric-field dependences
of the dark current obtained experimentally for the n-type
QWIPs made of AlGaAs/GaAs23, 37, 38) and InGaAlAs/InP39)
heterostructures at liquid nitrogen temperatures in the range
of the average electric fields E > 5 kV/cm correspond to
E gc ' 5 kV/cm. Assuming that the thermoemission characteristic electric field is given by27) E g ' kT /el ' 8kT /5e L w ,
for L w = 5 nm and T = 77 K, we obtain E g ' 20 kV/cm.
Thus, E gs is markedly smaller than E g . This indicates that the
electron capture processes affect the current-voltage characteristics more strongly than thermoemission does (E c < E g ).
Using the above data we obtain E c ' 6.7 kV/cm. This is in
excellent agreement with the results of Monte Carlo modeling
of AlGaAs/GaAs QWIPs.34)
Figures 2–9 show examples of the dark current reduction
factor as a function of the average electric field (i.e., as a
function of the bias voltage) and the spatial distributions of
the self-consistent electric field and charges in QWIPs with
different numbers of QWs, N , and different tunneling fields
E t at T = 80 K. The plots were calculated using the formulas obtained in the previous section. It was assumed that
E gc = 5 kV/cm, ln α0 = 20, γl = 0.5–1.5, and γr = 0.5.
The characteristic tunneling field E t was chosen to be in the
range of 300–2000 kV/cm. The span of the variation of E t
corresponds to the range of activation energies ²act for electrons in the emitter contact layer, ²act ' 70–240 meV. The
minimum value of E t corresponds to a QWIP with a relatively low emitter barrier for electrons in the contact layer.
The maximum value of E t corresponds to a QWIP with
a rather high emitter barrier which can differ from the interwell barriers. The parameter ln α0 can be estimated as
w
/kB T . For AlGaAs/GaAs QWIPs
ln α0 = ln( jm /eG) ' ²act
11
−2
with 6d = 5 × 10 cm designed for the wavelength range
λ = 8–12 mµ, ln α0 ' 15–20.
It is seen from Figs. 2 and 3 that in QWIPs with a large
number of QWs, the dark current reduction factor, i.e., the
ratio jdark /j0 (E), is on the order of unity in a wide range of
electric fields. In QWIPs with N = 32 and quite different
E t , the values of the dark current are very close to each other.
5819
Et=300kV/cm
1000kV/cm
2000kV/cm
0.4
1.0
N=8
0.8
0.6
0.4
10
20
30
Electric field (kV/cm)
40
50
Fig. 3. Dark current reduction factor as a function of electric field for
QWIPs with different characteristic tunneling fields.
1.0
80
70
E=30kV/cm
E=20kV/cm
E=10kV/cm
60
Electric field (kV/cm)
Reduction factor
0.8
0.6
0.4
Et=1000kV/cm
N=32
8
4
1
N=8
50
Et=1000kV/cm
40
30
20
0.2
10
0.0
10
20
30
Electric field (kV/cm)
40
50
Fig. 2. Dark current reduction factor as a function of electric field for
QWIPs with different numbers of QWs.
0
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
Normalized distance
1 2 3 4 5 6 7 8
Fig. 4. Electric-field spatial distributions in QWIP with eight QWs for different average electric fields (different biases).
5820
V. RYZHII and H. C. L IU
Jpn. J. Appl. Phys. Vol. 38 (1999) Pt. 1, No. 10
80
80
70
E=30kV/cm
E=20kV/cm
70
E=10kV/cm
γl=1.5
60
Electric field (kV/cm)
Electric field (kV/cm)
60
γl=1.0
γl=0.5
N=32
50
Et=1000kV/cm
40
30
30
20
10
10
2
8
14
20
26
32 2
8
14
20
26
32 2
8
14
20
26
0
32
Et=1000kV/cm
E=30kV/cm
40
20
0
N=8
50
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
Normalized distance
Normalized distance
Fig. 5.
The same as Fig. 4 but for QWIP with 32 QWs.
1 2 3 4 5 6 7 8
Fig. 7. Electric-field spatial distributions in QWIPs for different values of
parameter γl .
140
Et=2000kV/cm
Et=1000kV/cm
35
Et=300kV/cm
Et=2000kV/cm
1000kV/cm
300kV/cm
30
100
N=8
E=30kV/cm
Relative QW charge (%)
Electric field (kV/cm)
120
80
60
40
20
0
25
20
N=8
E=30kV/cm
15
10
5
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
Normalized distance
1 2 3 4 5 6 7 8
Fig. 6. Electric-field spatial distributions in QWIPs with different characteristic tunneling fields.
jacent to the emitter to relatively low values in the QWIP bulk
and near-collector region.
When E < E t / ln α and a > 0 (i.e., γl < 1), the electron sheet concentrations 6n in all the QWs are lower than
the donor sheet concentration 6d . This follows directly from
eq. (21). According to eqs. (21)–(23), such a situation corresponds to positive charges in QWs and in the QW structure
as a whole. The QWs adjacent to the emitter are depleted
more strongly than those in the QW structure bulk, hence, the
space charge is located near the emitter contact (see Fig. 8).
This is consistent with the results of previous numerical simulations.8, 10, 11, 15)
However, in QWIPs with a small E t and average electric
fields satisfying the inequality E > E t / ln α, the electricfield distributions can be quite different. As seen from Fig. 6,
the distribution for E t = 300 kV/cm at E = 30 kV/cm corresponds to the low electric-field domain localized near the
emitter and the relatively high electric field in the bulk. In
this case, the QWs are charged negatively due to their enrichment with electrons.
Figure 7 shows the spatial electric-field distributions for
different values of the parameter γl and the fixed parameter γr
(γr = 0.5). When γl = 1, the electric field in all the barriers,
except the emitter barrier, is constant, and it is significantly
smaller than that in the emitter barrier. It is interesting to note
0
5
1
2
3
4
5
QW index
6
7
8
Fig. 8. Space charge distributions for QWIPs with different characteristic
tunneling fields.
that if γl > 1, the electric-field distribution exhibits an oscillatory behavior. Such a distribution is shown in Fig. 7 (see
curve for γl = 1.5).
The relative charges of the QWs (6d − 6n )/6d for E t =
300–2000 kV/cm are shown in Fig. 8. It is seen that the QW
charges are positive for E = 30 kV/cm and E t = 2000
and 1000 kV/cm when E < E t / ln α. In contrast, when
E > E t / ln α, the QWs are charged negatively, as shown in
Fig. 8 (see curve for E t = 300 kV/cm). In the latter case, the
dark current reduction factor i dark exceeds unity.
The average electric field corresponding to the change of
the sign of the total charge, and consequently, to the transition from the high-field domain near the emitter to the lowfield domain, satisfies the equation E ln α = E t . Taking into
account the explicit dependence of α on E, the latter equation
can be presented as E ln α0 − E 2 /E gc = E t . This equation
has a solution if the characteristic field E gc is large enough,
namely, E gc ≥ 4E t / ln2 α0 .
Figure 9 shows the charges of the QWs for different values
of the parameter γl with γr = 0.5. These plots correspond
to the electric-field distributions shown in Fig. 7. As shown
in Fig. 9, when γl = 1, only the QW closest to the emitter
V. RYZHII and H. C. L IU
Jpn. J. Appl. Phys. Vol. 38 (1999) Pt. 1, No. 10
25
γl=0.5
1.0
1.5
Relative QW charge (%)
20
15
3)
N=8
Et=1000kV/cm
E=30kV/cm
10
4)
5
0
5)
5
10
Fig. 9.
1
2
3
4
5
QW index
6
7
8
The same as Fig. 8 but for different values of parameter γl .
6)
contact is charged. This is also evident from eq. (21). It is
interesting to note that if γl > 1 when a < 0, the sign of the
QW charge oscillates as a function of the QW index. As seen
from Fig. 9, the odd-numbered QWs are charged positively,
whereas the charges of the even-numbered QW are negative.
Such a charge distribution results in oscillatory electric-field
distribution damping in the direction from the emitter to the
collector. Large values of the parameter γl correspond to a
strong influence of the electric field in the preceding barrier
(left-hand-side barrier) on the electron balance in the QW.
The damping oscillations of the self-consistent electric field
and space charge can be explained as follows. A strong electric field in the emitter barrier leads to a small probability of
electron capture into the first QW. As a result, the electron
concentration in this QW significantly decreases and the QW
is charged positively. The positive charge of the first QW results in a substantially smaller electric field in the next barrier
compared with that in the first one. This, in turn, is the reason
for the rather low capture rate into the second QW leading to
the excess of electrons in the latter, i.e., to the negative QW
charge. The negative charge of the QW increases the electric
field in the next barrier and decreases the capture rate into the
successive QW.
6. Conclusions
We have proposed a self-consistent analytical model for
QWIPs with tunneling injection of electrons through the
extreme barrier. The QWIPs under dark conditions have
been investigated for the temperature range where thermionic
emission is the main mechanism of electron escape from
QWs. Explicit analytical expressions have been derived for
the dark current as well as for the electric-field and spacecharge distributions as functions of the bias voltage (average
electric field), and for the QWIP structural parameters, in particular, the number of QWs. The following has been shown
using the obtained analytical formulas.
1) The contact and space-charge effects weakly influence
the dark current in QWIPs with a large number of QWs,
whereas these effects can be essential in QWIPs with a
moderate number of QWs.
2) The dark current-voltage characteristics of QWIPs with
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a large number of QWs weakly depend on the number of
QWs and are primarily determined by the dependences
of the thermoemission and capture rates on the average
electric field in the QW structure.
The dark currents in QWIPs with moderate and large
numbers of QWs can be markedly different.
The electric-field distributions in the QW structure are,
in general, nonuniform regardless of the strength of the
dependences of the rates of the elementary processes
(thermoemission and capture) on the local electric field.
The electric-field distributions in QWIPs with relatively
strong dependence of the thermoemission and capture
rates on the local electric field are partitioned into two regions with markedly different values of the electric field
with the formation of a charged domain (positive or negative) adjacent to the emitter contact and quasi-neutral
region in the bulk.
The spatial distributions of the self-consistent electricfield and space charge can be oscillatory.
Acknowledgments
The authors thank M. Ryzhii and I. Khmyrova for essential
assistance and useful comments on the manuscript.
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