A
b
c
o
c
h
m
la
y
b
a
fl
o
to
H
g
b
a
c
o
h
a
b
a
a
p
o
p
H
g
s
u
u
n
si
m
p
h
(a
b
to
d
e
te
m
c
o
h
a
te
m
i
m
c
o
1
0
th
x
C
al
i
C
6
f
(
D
t
b
stract—Imple
o
ncept in Hg
C
h
allenging pro
b
aterials that
c
y
er in HgCd
T
a
nd offset at t
h
o
w of photoge
n
minimise th
g
CdTe alloy-
b
a
rrier structu
o
mbined with
t
a
s been emplo
y
a
nd effective
m
p
proach whic
p
timisation o
f
g
CdTe-based
u
perlattice bar
r
Index Ter
m
n
ipolar barrie
r
m
ulation, non-
The family
h
otodetectors
a
bsorber) stru
c
conventio
n
e
monstrated
i
m
peratures [
2
o
ncept utilizi
n
a
ve demonst
r
m
perature fr
o
m
plementatio
n
o
mpound se
m
0
]. This is du
e
e HgCdTe n
Cd
x
Te alloys
,
i
gnment that
r
S
p
h
o
N
The authors
C
omputer Engin
e
6009, WA. (ema
i
The authors
w
f
or financial sup
p
(grants DP12
D
P150104839),
t
t
he Office of Sci
e
mentation o
f
C
dTe-based co
m
b
lem, primaril
y
c
an be emplo
y
T
e nBn struct
u
h
e n-type/barri
e
n
erated minor
i
e valence ba
n
b
ased barrier
re. In this
p
t
he non-equili
b
y
ed to numer
i
m
ass approxi
m
h is valid
f
f
band align
nBn detector
r
ier.
m
s – mercur
y
r
, nBn detect
o
equilibrium
G
I. IN
T
of unipol
a
based on
t
c
ture represen
t
n
al photovo
i
mproved pe
2
, 3]. In parti
c
n
g III-V com
p
r
ated a sig
n
o
m the typica
l
n
of the nBn
m
iconductor s
y
e
to the fact t
h
n
Bn detector,
,
exhibit a
r
esults in a re
S
uper
l
o
todet
. D. Akhava
n
are with the
S
e
ering, The Uni
v
i
l: nima.dehdasht
i
w
ish to acknowle
d
p
ort of this wor
k
0104835, D
P
t
he Australian N
a
e
nce of the State
G
f
the unipol
a
m
pound semi
c
y
because pra
c
y
ed as the w
i
u
res present
a
e
r interface, t
h
i
ty carriers. H
n
d offset by
with a Cd
T
p
aper, an 8
×
b
rium Green’
s
i
cally demons
t
m
ation is an
f
or the mo
d
ment and c
a
s incorporati
n
y
cadmium
t
o
r, infrared ,
G
reen’s functio
n
T
RODUCTIO
N
a
r barrier
n
t
he n (cont
a
t
a new devic
e
ltaic techn
o
rformance a
t
c
ular, detecto
r
p
ound semic
o
n
ificant incr
e
l
77K to aro
u
detector con
c
y
stem is not
h
at the hetero
i
when forme
d
type-III h
e
latively large
l
attic
e
ector
s
n
, G. A. Um
a
S
chool of Elec
t
v
ersity of Weste
r
i
@uwa.edu.au)
d
ge Australian R
e
k
under the Disc
o
P
140103667,
D
a
tional Fabricatio
n
G
overnment of
W
a
r barrier d
e
c
onductor allo
y
c
tical lattice-
m
i
de bandgap
b
a
significant
v
h
us impeding t
h
H
owever, it is
p
replacing th
e
T
e-HgTe supe
r
×
8 k.p Hami
l
s
function for
m
t
rate that the
adequate nu
m
d
elling, desig
n
a
rrier transp
o
n
g a wide b
a
t
elluride (Hg
C
8×8 k.p, nu
m
n
(NEGF)
N
n
Bn infrared
a
ct) B (barr
i
e
concept co
m
o
logy, and
t
higher op
e
r
s based on t
h
o
nductor tech
n
ease in op
e
u
nd 150K; ho
w
c
ept in the H
g
straightforw
a
i
nterfaces pre
s
d
employing
e
terostructure
valence ban
d
e
barri
s
: vali
d
app
r
a
na-Membre
n
t
rical, Electroni
c
r
n Australia, Cr
a
e
search Council (
A
o
very Project pr
o
D
P140103667,
n
Facility (ANF
F
W
estern Australia.
1
e
tector
y
s is a
m
atched
b
arrier
v
alence
h
e free
p
ossible
e
bulk
r
lattice
l
tonian
m
alism,
single-
m
erical
n
, and
o
rt in
a
ndgap
C
dTe),
m
erical
(IR)
i
er) n
m
pared
have
e
rating
h
e nBn
n
ology
e
rating
w
ever,
g
CdTe
a
rd [4-
s
ent in
Hg
1-
band
d
offset
at
sig
n
car
r
illu
s
(Δ
E
nB
n
in
w
abs
o
Fig.
offs
e
a rel
in th
S
pro
p
the
r
des
i
het
e
acr
o
alte
r
str
u
sup
e
de
m
the
is
n
det
a
In t
h
b
ar
r
Pe
n
ord
e
Hg
T
lev
e
ma
s
em
p
the
er Hg
C
d
atio
n
r
oxi
m
n
o, R. Gu,
M
c
and
a
wley,
A
RC)
o
gram
and
F
) and
the
b
arrier
/
n
ificantly blo
c
r
iers from the
s
trated in Fig
E
V
) severely
d
n
detectors in
c
w
hich a neg
l
o
rber heteroi
n
1. (a) Ideal uni
e
t (ΔE
V
), and (b)
H
atively large ΔE
V
e absorber to the
S
everal ban
d
p
osed to mini
m
r
mal energy
o
i
gns propos
e
e
rointerface b
o
ss the inter
f
rnatively, pr
o
u
ctures. Of th
e
e
rlattice met
h
m
and the need
barrier layer,
n
oted, howeve
a
il as suitable
h
eir report, K
o
r
ier HgCdTe
n
ny model, a
n
e
r to calculat
T
e-CdTe qu
a
e
ls were then
s
s, bandgap
p
loyed in an
dark current
a
C
dTe
n
of th
e
m
ation
M
. Asadnia, J
/
n-type abs
o
c
king the fl
o
absorber lay
e
. 1, this relat
i
d
egrades the
p
c
omparison t
o
l
igible ΔE
V
i
n
terface.
polar nBn
p
hot
o
H
gCdTe-
b
ased n
B
V
, in which the f
l
contact layer is a
d
gap engine
e
m
ize ΔE
V
to
v
o
f minority c
a
e
lowering
y grading bo
t
f
ace formed
o
pose the us
e
e
se two band
g
h
od is the m
o
for highly co
n
which can be
r, that superl
a
barrier layer
s
o
pytko and c
o
nBn detecto
r
n
d the solutio
n
e the equilib
r
a
ntu
m
-well s
t
used to appr
o
and band
effective-mas
a
nd photogen
nBn
i
e
effe
c
. Antoszews
k
o
rber hetero
o
w of photog
e
r to the conta
c
i
vely large v
a
p
erformance
o
the ideal uni
p
s present at
o
detector with n
e
B
n detector with
b
l
ow of photogen
e
ffected by ΔE
V
.
e
ring approa
c
v
alues below
o
a
rriers [10, 1
ΔE
V
at t
h
t
h the dopin
g
employing
H
e
of superla
t
g
ap engineeri
n
o
st promising
n
trolled grade
problematic
i
a
ttices have y
e
s
for nBn pho
t
o
-workers mo
d
r
s employing
n
to the eige
n
r
ium energy
l
t
ructure. The
o
ximate an eq
u
alignment,
w
s commercia
l
erated curren
t
i
nfrar
e
c
tive
m
k
i and L. Fa
r
interface,
enerated mi
n
c
t layer [7-12
]
a
lence band
o
of HgCdTe-
b
p
olar nBn det
e
the barrier/n
-
e
gligible valence
barrier layer exh
i
e
rated minority c
a
c
hes have
o
r approachin
g
1, 13, 14].
T
h
e barrier/n
-
g
and compo
s
H
gCdTe alloy
s
t
tice-
b
ased b
a
n
g approache
s
since it doe
s
d p-type dopi
n
i
n HgCdTe [1
e
t to be studi
e
t
odetector des
d
elled superla
t
a simple Kr
o
n
value proble
m
l
evels in a C
obtained e
n
uivalent effe
c
w
hich were
l
solver to pr
t
of a superla
t
e
d
m
ass
r
aone
thus
n
ority
]
. As
o
ffset
b
ased
e
ctor
-
type
band
i
biting
a
rriers
been
g
the
T
hese
-
type
s
ition
s
or,
a
rrier
s
, the
s
not
n
g in
1]. It
e
d in
igns.
t
tice-
o
nig-
m
, in
C
dTe-
n
ergy
c
tive-
then
edict
t
tice-
2
barrier HgCdTe detector [11]. While this device modelling
method is convenient and relatively easy to implement, it does
not adequately capture the fundamental physical details
associated with the superlattice structure, such as
wavefunction overlap, density of states, nor the influence of
layer doping on the resulting band diagram. In this work, we
present results of a full quantum mechanical approach based
on the non-equilibrium Green’s function (NEGF) to predict
the carrier transmission and band diagram of the HgCdTe nBn
detector with superlattice barrier [15, 16]. Bulk 8×8 k.p
Hamiltonian parameters for HgTe and CdTe have been
employed to calculate the electronic structure. Although the
8×8 k.p calculations employed for modelling are more
computationally intensive than a simple one-band effective
mass approximation, our results indicate that the latter
approach yields accurate results which can also be used to
model the band structure and carrier transport in HgCdTe nBn
structures, and to predict the performance of nBn infrared
photodetectors [16-18].
II. NUMERICAL SIMULATION DETAILS
The electronic properties of a superlattice barrier nBn
HgCdTe detector structure are determined by the electronic
properties of the individual layers that form the superlattice
basis. From a semiconductor growth technology viewpoint,
the simplest superlattice basis structure in the HgCdTe alloy
system is to employ the binary compounds HgTe and CdTe.
However, it is noted that, in practice, the high Hg over-
pressure during molecular beam epitaxial growth of HgCdTe
is likely to result in a superlattice structure comprised of HgTe
and a high x-value Hg
0.05
Cd
0.95
Te alloy. Thus, the starting
point is the band structure calculation for a CdTe/HgTe/CdTe
basis structure of the superlattice, in which the band structure
parameters for the superlattice defined by the 8-band k.p
Hamiltonian at T=0 K are summarized in Table 1. It should be
noted that superlattice electronic properties have been
theoretically analysed employing several approaches,
including tight binding, pseudopotentials, and density
functional theory, in addition to 8×8 k.p. However, the k.p
envelope function approach has been shown to yield results
that are similar to other approaches [1, 19]. In addition, the k.p
method is particularly valid around the gamma point of the
band structure where, the relevant physics of an nBn detector
are determined. More importantly, and in contrast to the
parameters required for other theoretical approaches, the band
structure parameters for CdTe and HgTe materials are well
established in the k.p framework [1, 20]. Details of the 8×8 k.p
Hamiltonian and the numerical discretization is provided in
the Appendix.
Following the method in the Appendix, and setting up the
discretized Hamiltonian, the band structure of the bulk
material is calculated by solving:
(
)
iii
ik
i
ik
ii
EeDeDD
zz
ψψ
=++
−
−
+
+
+
1
(1)
where k
z
is the wavenumber in the transport direction, and E
i
,
ψ
i
are the eigenvalue and eigenfunction of layer i, respectively.
For the simplest case of a bulk material band structure, since
there is only one material type which does not vary along the
transport direction z, the matrix D takes the form:
−
+
−−
−
+
+
++
−
+−
==
==
==
11
11
11
iii
iii
iii
DDD
DDD
DDD
(2)
and hence, the band structure of bulk material can be
calculated by setting i=0, which gives,
(
)
000000
ψψ
EeDeDD
zz
ikik
=++
−
−
+
+
(3)
The band structure of bulk CdTe and HgTe thus calculated is
presented in Fig. 2, where it can be seen that CdTe exhibits a
normal direct band gap structure, with the conduction band
minimum (Γ
6
) located above the valence band maximum (Γ
8
).
In contrast, HgTe manifests an inverted band structure in
which hole states (Γ
8
) are located above the electron states
(Γ
6
). It is also possible to extract the effective mass of
electrons, heavy holes and light holes from the bulk band
structures. The effective mass at the Γ point is equal to the
inverse of the E(k) curvature at k=0, which is given by,
1
0
2
2
0
2
*
)(
−
=
=
k
dk
kEd
qm
m
(4)
where m
0
and ħ are the free electron rest mass and the reduced
Planck constant, respectively. The band structure of CdTe and
HgTe calculated using the 8×8 k.p Hamiltonian and the
equivalent parabolic band approximation using the effective
masses of Γ
6
and Γ
8
bands for electrons and heavy holes are
shown in Fig. 2. From the k.p calculations, the effective mass
of electrons in the Γ
6
band is equal to 0.031 and 0.090 for
HgTe and CdTe, respectively, whereas the effective mass of
heavy holes in the Γ
8
band is equal to 0.3226 and 0.4926 for
HgTe and CdTe, respectively. These values of effective mass
are in good agreement with reported values by other groups,
thus validating our approach [19, 21].
Having detailed the methodology to construct the
Hamiltonian for both bulk material and superlattice structures,
and verified that the parabolic effective mass approximation
yields band structure results consistent with the 8-band k.p in
Table 1. Band structure parameters of HgTe and CdTe. E
g
is the energy
gap, Δ is the spin-orbit splitting energy, Λ is the valence band offset
between the two materials, E
P
is the energy related to the Kane
momentum matrix element P, F is related to the normalized conduction
band effective mass m
c
/m
0
, and γ
i
’s are the valence band Luttinger
parameters [1].
Band parameter HgTe CdTe
E
g
(eV)
-0.303 1.606
Δ (eV)
1.08 0.91
Λ (eV)
0 0.350
E
P
(eV)
18.8 18.8
F
0 -0.09
γ
1
4.1 1.47
γ
2
0.5 -0.28
γ
3
1.3 0.03
bu
st
r
q
u
F
i
di
p
e
a
p
n
e
H
a
D
T
h
th
u
th
Q
W
H
w
i
D
D
u
lk HgCdTe
m
r
ucture of the
u
antum well (
i
g. 3. Thus, th
e
mensional su
p
e
riodic bou
n
p
proximation
e
ar the Γ poi
a
miltonian m
a
U
correspond
s
h
e D
QW
matri
x
us constructe
d
e complete
H
W
s in the sup
e
=
L
QW
D
H
i
th,
=
+
0
0
x
N
U
D
D
=
0
0
L
D
Fig. 2. Bulk ban
d
the 8×8 k.p cal
c
minimum and v
a
HgTe is opposit
e
Γ
6
electron state
s
p
arabolic energy
the effective m
a
valence band he
a
describes the ba
n
point at k=0.
−0.2 −0.1
−1.5
−1
−0.5
0
0.5
1
1.5
2
k
E (eV)
(a) CdTe
m
aterials, we
c
HgTe/CdTe
s
QW) "buildi
n
e
QW is take
n
p
erlattice in t
h
n
dary condit
i
to describe t
h
i
nt. In Fig. 3
a
trix of the Q
W
s
to the matr
i
x
is effectivel
y
d
exactly in
t
H
amiltonian
m
e
rlattice struc
t
L
QWL
UQW
D
DD
DD
0
0
−
0
0
0
N
D
d
structure (solid
c
ulations. The re
a
lence band maxi
m
e
to that in CdTe;
s
. Also shown b
y
dispersion band
s
a
ss approximati
o
a
vy holes. The e
f
n
d structure of e
l
0 0.1 0.2
0
k
(1/nm)
Γ
8
Γ
6
c
an proceed t
o
s
uperlattice b
a
n
g block" sho
w
n
as the unit c
e
h
e growth dir
e
i
ons, which
h
e physics o
f
, D
QW
repres
e
W
unit cell i
n
i
x connecting
y
equivalent
t
t
he same wa
y
m
atrix for p
e
t
ure is given
b
U
QW
U
DD
D
0
0
−
0
0
x
N
d
lines) of (a) Cd
T
lative position o
m
um at the mini
m
in HgTe, the Γ
8
h
y
the (×) symbol
s
s
tructure charact
e
o
n for conducti
o
f
fective mass ap
p
l
ectrons and hea
v
0
.3
−0.2
−
−1.5
−1
−0.5
0
0.5
1
1.5
2
E (eV)
(b) HgT
e
o
calculate th
e
a
rrier basis us
i
w
n schematic
e
ll in a period
i
e
ction z
b
y im
p
is a reas
o
f
quantum tr
a
e
nts the disc
r
n
the z directi
o
adjacent uni
t
t
o D
i
, in (2),
y
. The final f
o
e
riodically ar
r
b
y:
U
T
e and, (b) HgT
e
f the conductio
n
m
um bandgap at
k
h
ole states lie abo
s
are the corresp
o
e
ristics calculate
d
o
n band electro
n
p
roximation ade
q
v
y holes close to
−
0.1 0 0.1 0
.2
k (1/nm)
e
Γ
6
Γ
8
3
e
band
i
ng the
ally in
i
c one-
p
osing
o
nable
a
nsport
r
etized
o
n, and
t
cells.
and is
o
rm of
r
anged
(5)
(6)
(7)
wh
e
is t
h
or
and
cell
Q
W
(
D
Q
Th
e
str
u
cal
c
of
D
and
ho
w
all
o
sup
e
eff
e
T
qua
n
8×
8
Fig
.
dia
g
Cd
T
spe
c
ord
e
sig
n
an
n
qua
n
dia
g
T
fun
c
em
p
eff
e
in
F
lay
e
pre
d
dev
k.p
mi
x
tha
t
app
r
the
rea
d
Th
e
cal
c
det
e
to
h
e
from
n
band
k
=0 in
ve the
o
nding
d
using
n
s and
q
uately
the Γ
.2
0.3
F
i
"
b
a
n
e
re D
QW
, D
L
a
n
h
e number of
N
b
=1 for the
D
U
are matr
i
s in the perio
W
superlattice
i
eD
ik
U
QW
+
+
e
above equat
i
u
cture which,
c
ulation of ei
g
D
QW
, D
L
and
D
most releva
n
w
ever, setting
o
wing the cal
c
e
rlattice struc
e
ctive mass ap
III.
T
he energy l
e
n
tum well b
a
8
k.p theoreti
c
.
4 in relatio
n
g
ram as well
T
e and HgT
e
c
ific range (i.
e
e
r to achie
v
n
ificantly larg
e
n
Bn structure
n
tum well m
a
g
ram at k=0.
T
he energy l
e
c
tion of Hg
T
p
loying the 8
e
ctive mass a
p
F
ig. 5 clearly
e
r increases,
t
d
icted by th
e
iate from th
o
approach [2
2
x
ing arising f
r
t
is negle
c
roximation.
S
heavy hole
m
d
ily evident o
n
e
energy le
v
c
ulated witho
u
e
ctor structur
e
h
ave a finite n
u
i
g. 3. Schemati
c
b
uilding block" a
n
n
d D
U
are matrice
n
d D
U
are all
o
bands (N
b
=8
single-
b
and
e
i
ces that serv
e
dic represent
a
i
s then calcul
a
)
ψ
eD
zz
ik
L
+
−
i
ons allow cal
c
in the mo
s
g
envalues usi
n
D
U
over k
z
val
n
t case of k
z
k
z
=0 reduce
s
c
ulation of av
a
ture in the fr
a
proximation.
RESULTS
A
e
vels calculat
e
a
sis, obtained
c
al simulatio
n
n
to their en
as their ener
g
e
layer thic
k
e
. 1nm<HgT
e
v
e an equiv
a
e
r
t
han the b
a
[11]. It is ev
i
a
tch the ener
g
e
vels for the
T
e layer wi
d
-
b
and k.p H
a
p
proximation.
indicate tha
t
t
he conducti
o
e
one band
o
se obtained
f
2
]. This is a
c
r
om the influ
e
c
ted in th
e
S
ince the elec
t
m
ass in the v
a
n
the electron
v
els of the
u
t taking into
e
s, the HgTe/
C
u
mber of Q
W
c
representation
n
d its correspond
i
s which link two
o
f size (N
z
×N
b
for the 8-
b
an
d
e
ffective mass
e
to link two
a
tion. The ba
n
a
ted from:
ψ
ψ
E=
c
ulation of th
e
s
t general c
a
n
g the relativ
ues of interes
in the vicini
t
s
the comput
a
a
ilable energ
y
a
mework of
t
A
ND DISCU
S
e
d for a 2n
m
employing t
h
n
framework
,
n
ergy locatio
n
g
y-momentu
m
k
nesses were
e
<3nm and 5
n
a
lent barrier
a
ndgap of the
i
dent that the
g
y levels of
t
HgTe/CdTe
d
th were the
n
a
miltonian a
n
The calculate
d
t
as the thic
k
o
n band elec
t
effective m
a
f
rom the mor
e
c
onsequence
e
nce of remo
t
e
one-
b
and
t
ron mass is
a
lence band,
t
states in the c
HgTe/CdTe
account tha
t
C
dTe superla
t
W
building blo
c
of the HgTe/
C
i
ng periodic sup
e
adjacent cells.
b
)×(N
z
×N
b
), a
n
d
k.p Hamilto
n
Hamiltonian
)
adjacent QW
n
d structure o
e
superlattice
b
a
se, demands
ely large ma
t
t. For the si
m
t
y of the Γ
p
a
tional cost
w
y
levels withi
n
t
he parabolic
b
S
SION
m
/8nm HgTe/
C
h
e above det
,
are present
e
n
within the
b
m
dispersion.
chosen wit
h
n
m<CdTe<8n
m
bandgap th
a
absorber regi
o
eigenvalues
o
t
he E-k dispe
r
superlattice
n
calculated
n
d the single-
b
d
results pres
e
k
ness of the
H
t
ron energy l
e
a
ss approxim
a
e
rigorous 8-
b
of increased
b
t
e bands, an
e
effective
m
much lighter
t
his effect is
m
onduction ba
n
superlattice
w
t
, in practical
t
tice is constr
a
c
ks and is bou
n
C
dTe quantum
w
e
rlattice structure
.
n
d N
b
n
ian,
)
. D
L
unit
f the
(8)
b
and
the
t
rices
m
plest
p
oint,
w
hile
n
the
b
and
C
dTe
ailed
e
d in
b
and
The
h
in a
m
) in
a
t is
o
n in
o
f the
r
sion
as a
both
b
and
e
nted
H
gTe
e
vels
a
tion
b
and
b
and
e
ffect
m
ass
than
m
ore
n
d.
w
ere
nBn
a
ined
n
ded
w
ell
.
D
L
o
n
re
d
e
m
e
q
in
p
a
re
a
c
o
v
p
h
a
p
f
o
in
[
2
N
E
th
to
ba
e
n
tr
a
n
B
re
n
both sides
b
gions. For t
h
e
tector a mor
e
m
atrix theory,
q
uilibrium Gr
e
order to gai
n
a
rticipating in
gion [16-18,
2
c
count effects
v
erlap, charge
h
enomenon.
p
proach, whic
o
r the calculat
i
semiconduct
o
2
4].
E
GF modelli
n
In contrast t
o
e energy lev
e
the effectiv
e
ar
rier in nBn
n
ables theore
t
a
nsport proba
b
B
n device, as
gions are co
m
Fig. 4. (left) Ene
HgTe layer thic
k
and (right) band
s
−4 −
2
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
E (eV)
CdTe
410 mV
Fig. 5. Energy le
layer thickness
fo
8×8 k.p Ham
i
approximation a
t
1
−0.
1
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
Energy (eV)
b
y adjacent
n
h
e practical d
e
e
sophisticate
d
transfer mat
r
e
en’s functio
n
n
better insig
h
carrier trans
p
2
3, 24]. In p
a
associated wi
density, scatt
In what fo
l
h has been p
r
i
on of electro
n
o
r devices w
h
n
g approach
o
the periodic
e
ls of the sup
e
e
bandgap pr
photodetecto
r
t
ical calculati
b
ilities using
depicted in
F
m
posed of bul
k
rgy levels of Hg
T
k
ness of 2nm, cal
s
tructure of the q
u
2
0 2
4
X (nm)
Cd
Te
HgTe
H
H
E
1
vels of HgTe/Cd
T
f
or a fixed CdTe
i
ltonian and
e
t
the Γ point with
1
1.5 2
1
0
1
2
3
4
5
6
7
8
9
HgT
e
HH
1
C1
n
arrow gap a
b
e
sign of a sup
e
d
approach,
s
r
ix method (
T
n
(NEGF) fo
r
h
t into the av
a
p
ort across th
e
a
rticular, thes
e
th density of
s
ering mechan
i
l
lows, we e
r
oven to be
a
n
ic properties
a
h
ere quantum
e
arrangement
r
e
rlattice alon
e
esented by t
h
r
structures, t
h
on of energ
y
the actual la
y
F
ig. 6. The a
b
k
Hg
0.3
Cd
0.7
T
T
e/CdTe quantu
m
culated using th
e
u
antum well wit
h
4
Te
H
1
0 0.
02
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
E (eV)
410
mV
T
e quantum well
thickness of 8n
m
e
quivalent one-
b
k
||
(k
y
=0, k
x
=0) a
n
2.5 3
e
thickness (n
m
1
Γ
8
b
sorber and
c
e
rlattice-
b
arri
e
s
uch as the
d
T
MM) or th
e
r
malism, is r
e
a
ilable energ
y
e
superlattice
b
e
methods ta
k
s
tates, wavef
u
i
sms, and tun
n
mploy the
N
a
powerful ap
p
a
nd carrier tr
a
e
ffects are do
m
r
equired to ca
l
e
, which corr
e
h
e superlattic
e
he NEGF ap
p
y
states and
y
ered structur
e
b
sorber and
c
e material, a
n
m
well unit cell,
w
e
8×8 k.p Hamilt
o
h
k
||
(k
y
=0, k
x
=0).
02
0.04 0.06 0
.08
k (1/nm)
mV
as a function of
H
m
, calculated usi
n
b
and effective
n
d k
z
=0.
3.5 4
m
)
8−band
1−band
8
CdTe
4
c
ontact
e
r nBn
d
ensity
e
non-
e
quired
y
states
b
arrier
k
e into
u
nction
n
elling
N
EGF
p
roach
a
nsport
m
inant
l
culate
e
spond
e
as a
p
roach
carrier
e
of an
c
ontact
n
d thus
cor
r
mi
d
pre
v
rep
r
wh
e
lay
e
Th
e
exp
r
(G
wh
e
ma
t
nu
m
rig
h
the
cal
c
alg
o
eig
e
rec
u
sof
t
F
pra
c
car
r
cal
c
(
E
T
Γ
L
Γ
R
wh
e
b
ro
a
dia
g
Gr
e
dia
g
(L
D
pro
b
str
u
thi
c
pre
s
pro
b
loc
a
b
e
n
eig
e
ma
x
can
n
wel
b
ro
a
sup
e
w
ith a
o
nian,
.08 0.1
HH
E1
HgTe
n
g the
mass
Fi
g
b
a
r
r
esponds to
a
d
wave IR (
3
v
iously, the
d
r
esent the in
t
e
reas the off-
d
e
rs (denoted
D
e
NEGF eq
u
ressed as [25]
(
[
) += iEE
η
e
re G is the G
r
t
rices, E is t
h
m
ber. Since t
h
h
t extremities
o
self-energy
m
c
ulated using
r
o
rithm or dir
e
e
nvalue prob
l
u
rsive algorit
h
t
ware which i
s
F
ollowing ca
l
c
tical detecto
r
r
ier injection
c
ulated from [
2
(
)
G
trace
E
=
(
+
Σ−Σ=
L
L
i
(
+
Σ−Σ=
R
R
i
e
re T is the
a
dening matr
i
g
onal elemen
t
e
en’s function
g
onal elemen
t
D
OS). The ca
l
b
abilities, an
d
u
cture and,
w
c
knesses, a to
t
s
ented in Fig.
b
ability peaks
a
l density of
s
n
oted that wh
i
e
nvalue prob
l
x
imum LDO
S
n
ot predict th
e
ls and is thus
a
dening that
e
rlattice barri
e
g
. 6. Schematic r
e
r
rier treated as a l
a
a
n nBn struc
t
3
-5 µm wa
v
d
iagonal bloc
k
t
eraction wit
h
d
iagonal bloc
k
D
i
+
=D
i+1
), th
u
u
ation for q
u
:
)
Σ
−− DI
η
reen’s functi
o
h
e energy, an
d
h
e Hamiltoni
a
o
f a realistic
n
m
atrix concept
r
ecursive me
t
e
ct methods
w
l
em [26]. In
h
m impleme
n
s
based on the
l
culation of
r
structure, t
h
across the
b
2
5]:
)()(
G
EE
G
L
Γ
)
+
L
)
+
R
transmissio
n
i
ces, and “tr
a
t
s of the mat
r
matrix, with
t
t
s correspondi
n
l
culated local
d
eigenenergi
e
w
ith 1 nm
H
t
al superlattic
e
7. It is evid
e
at energy le
v
s
tates located
i
le the energi
e
l
em align wi
t
S
occur, a sol
u
e
overlap of t
h
unable to m
o
determines t
h
e
r.
e
presentation of
H
a
yered structure.
t
ure optimise
d
v
elength ba
n
k
s of the Ha
m
h
in each laye
r
k
s of matrix
D
u
s ensuring th
a
u
antum trans
p
()( Σ−
Σ
E
RL
o
n, Σ
L
and Σ
R
a
d
η is an in
f
a
n D is infin
i
n
Bn detector
s
to render it fi
n
t
hods such as
w
hich are bas
this study
n
tation, an i
n
previous wor
k
the Green’s
h
e transmissi
o
b
arrier at di
ff
()(
E
E
G
R
Γ
+
n
probability
,
a
ce” is the s
u
r
ix. The matr
i
t
he imaginar
y
ng to the loc
a
density of s
t
e
s for a Hg
C
H
gTe and 1
e
barrier thic
k
e
nt that the c
a
v
els correspon
d
at ~0.8 eV a
n
e
s obtained fr
o
t
h energy lo
c
u
tion of the e
i
h
e wavefunct
i
o
del the trans
m
h
e flow of
c
H
gCdTe nBn det
e
d
for detecti
o
n
d). As det
m
iltonian mat
r
r
of the det
e
D
couple adj
a
a
t D is Herm
i
p
ort can the
n
]
1
)
−
E
a
re the sel
f
-e
n
f
initesimally
s
i
te at the lef
t
s
tructure, we
a
n
ite, which c
a
the Sancho-
R
s
ed on solvin
g
we have us
e
n
-house devel
k
s [24,25].
function fo
r
o
n probabilit
y
ff
erent energi
e
)
)
E
,
Γ
L
and Γ
R
u
mmation ov
e
i
x G is a ret
a
y
component
o
a
l density of
s
t
ates, transmi
s
C
dTe nBn det
e
nm CdTe
l
k
ness of 5 n
m
a
rrier transmi
s
d
ing to the hi
g
n
d ~1eV. It s
h
o
m solution
o
c
ations wher
e
i
genvalue pro
b
i
ons from adj
a
m
ission proba
b
c
arriers acros
s
e
ctor with superla
o
n of
ailed
r
ix D
e
ctor,
a
cent
i
tian.
n
be
(9)
n
ergy
s
mall
t
and
a
dopt
a
n be
R
ubio
g
the
e
d a
oped
r
the
y
for
e
s is
(10)
(11)
(12)
R
are
e
r all
a
rded
o
f the
s
tates
s
sion
e
ctor
l
ayer
m
are
s
sion
g
hest
h
ould
o
f the
e
the
b
lem
a
cent
b
ility
s
the
ttice
C
d
s
c
th
h
e
th
th
v
a
K
o
c
a
s
o
a
b
F
s
L
F
s
e
e
(
F
e
e
1
Recently, K
o
d
Te/HgTe su
p
c
hematic diag
r
is diagram, t
h
e
avy-hole wel
l
e CdTe valen
c
e valence b
a
a
lues used fo
r
o
pytco et. al
.
a
lculated fro
m
o
lver (right).
O
b
ove the cond
u
F
ig. 7. From le
ft
s
uperlattice layer
L
DOS is maxim
u
F
ig. 8. Schemati
c
s
ystem used in [1
e
lectron and hea
v
e
lectrons (thick
s
(
thick dashed bla
c
F
ig. 9. Compa
r
e
igenvalue solve
r
e
ffective mass a
p
1
.5 nm, respectiv
e
0 10
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Po
Energy (eV)
E
o
pytco et. al.
d
p
erlattice as a
r
am of the qu
a
h
e bottom of
l
are located
a
c
e band edge
Γ
a
nd offset Λ
=
r each layer
. [11]. Fig.
9
m
an eigenva
l
O
f particular r
e
u
ction band e
d
ft
to right: LDO
S
s have a barrier
w
u
m, which repres
e
c
diagram of en
e
1]. Λ represents
t
v
y-hole states ar
e
s
olid black line)
c
k line).
r
ison of resona
n
r
, and (right) t
h
p
proximation. Th
e
e
ly.
20 30
Position (nm)
Eigenvalue
d
etermined th
function of
w
a
ntum well is
the electron
w
a
t the same en
e
Γ
8
, and has a
v
=
350 meV.
T
are the sam
e
9
compares t
h
l
ue solver (l
e
e
levance, are
d
ge of the bul
S
, transmission
p
w
idth of 1nm an
d
e
nt energy levels
a
Energy (eV)
e
rgy band align
m
t
he valence band
e
associated wit
h
and the quantu
m
n
ce states calc
u
h
e NEGF solver
e
CdTe-HgTe thi
0 10
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Energy (eV)
0 0.2
e energy leve
w
ell width [1
1
shown in Fi
g
w
ell and top
e
rgy level rel
a
v
alue determi
n
T
he effective
e
as those u
s
h
e resonance
e
ft) and the
N
the resonanc
e
k HgCdTe m
a
p
robability, and
e
d
a well width o
f
a
t which carrier t
r
0
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Energy (eV)
0
0.5
m
ent in the HgTe
-
offset. The super
l
h
the quantum w
e
m
well for heavy
u
lated from (le
f
using the singl
e
cknesses are 8 n
m
10 20 30
Position (nm)
0.4 0.6 0.8
Transmission
5
ls of a
1
]. The
g
. 8. In
of the
a
tive to
n
ed by
mass
s
ed by
levels
N
EGF
e
states
a
terial,
wh
i
sol
v
b
an
sol
v
car
r
res
o
stat
e
NE
G
b
el
o
the
of
H
the
co
m
the
plo
t
Hg
T
agr
e
and
stat
e
Ko
p
Ho
w
cal
c
wit
h
I
t
Ha
m
eff
e
Fig
.
qua
n
b
ot
t
edg
loc
a
tha
t
def
i
eig
e
8×
8
app
r
app
r
cal
c
ele
c
sha
r
e
igenvalues of
H
f
1n
m
. The peak
s
r
ansport across t
h
5
10
Position (nm)
0.5
1
Transmission
-
CdTe
l
attice
e
ll for
holes
f
t) an
e
-band
m
and
1
i
ch is at 0.5
v
er predicts t
w
d, whereas t
h
v
er indicates
t
r
ier transport.
o
nance states,
e
s; whereas t
h
G
F approach
o
w 0 eV can
c
calculated re
s
H
gTe thickne
s
resonance le
v
m
pared with o
u
two lowest e
n
t
ted and labe
l
T
e thickness
e
ement with
t
with the res
u
e
s, our eige
n
p
ytco et. al.,
w
ever, note
c
ulated from
o
h
energies ab
o
t
is now app
r
m
iltonian cal
e
ctive mass a
p
.
11 shows th
e
n
tum well a
t
om of the el
e
e of the Hg
T
a
ted at the Γ
8
t
this is diff
e
i
nition used
e
nvalues of a
n
8
k.p Hamilt
roximation.
roximation,
t
c
ulations that
c
trons; howev
e
r
ed between t
h
H
gCdTe nBn det
e
s
in the transmis
s
h
e barrier can tak
e
10 15
1.5 2
Energy (eV)
eV. It can b
e
w
o resonance
h
e transmissi
o
t
hat only one
A similar s
i
where the e
i
h
e transmissi
o
indicates th
a
c
ontribute to
c
s
onance states
s
s for a fixed
v
els predicte
d
u
r NEGF and
n
ergy levels i
n
l
led as eig-1
a
less than 2n
m
t
he transmissi
u
lts of Kopytc
o
n
value resul
t
which have
that the h
o
o
ur NEGF res
u
o
ve 0 eV do n
o
r
opriate to co
m
culations wi
t
p
proximation
e
schematic b
a
ssociated wi
t
e
ctron quantu
m
T
e, and the t
o
b
and edge o
e
rent from t
h
by Kopytco
n
nBn superla
t
onian and t
h
Compared
t
here are sev
e
arise from b
a
e
r, only the ei
g
h
e k.p and eff
e
e
ctor calculated
u
s
ion probability
c
e
place.
0
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Energy (eV)
e
observed t
h
states in the
b
o
n probabilit
y
of these stat
e
i
tuation exist
i
genvalue sol
v
o
n probability
a
t only thos
e
c
arrier transp
o
of the superl
a
CdTe thickn
e
d
by Kopytco
eigenvalue s
o
n
the conducti
o
a
nd eig-2. It
i
m
, the secon
d
on probabilit
y
o
et. al.. For
t
t
s match wi
t
not been p
l
o
le transmis
s
ults indicate
t
o
t contribute t
o
m
pare results
t
h those fro
for the nBn s
u
and diagram
o
t
h the super
l
m
well is loc
a
o
p of the hol
e
f the HgTe.
I
h
e electron/h
o
et. al.[11].
F
t
tice device c
a
h
e single-
b
a
n
to the
e
ral eigenval
u
a
nd mixing o
g
envalue loc
a
e
ctive mass c
a
using the 8×8 k
c
orrespond to en
e
5 1
0
Position (nm)
h
at the eigen
v
b
arrier condu
c
y
from the
N
e
s is availabl
e
for valence
b
v
er predicts
m
obtained fro
m
e
resonance
s
o
rt. Fig. 10 s
h
a
ttice as a fun
c
e
ss of 8 nm,
w
et. al. have
o
lver. In this f
i
o
n band have
i
s evident th
a
d
eigenvalue
y
from our
N
t
he hole reso
n
t
h the result
l
otted for cl
a
s
ion probabi
t
hat the hole
s
o
carrier tran
s
from the 8×
8
m a single-
b
u
perlattice de
o
f the CdTe/
H
l
attice, wher
e
a
ted at the Γ
6
b
e
quantum w
e
I
t should be
n
o
le quantum
F
ig.12 show
s
a
lculated usin
g
n
d effective
m
effective
m
u
es from th
e
f heavy-hole
s
a
ted at 0.75 m
e
a
lculations. Fi
g
.p Hamiltonian.
e
rgy levels wher
e
0
15
v
alue
c
tion
N
EGF
e
for
b
and
m
any
m
the
s
tates
h
ows
c
tion
w
here
been
i
gure
been
a
t for
is in
N
EGF
n
ance
t
s of
a
rity.
lities
s
tates
s
port.
8
k.p
b
and
vice.
H
gTe
e
the
b
and
e
ll is
n
oted
well
s
the
g
the
m
ass
m
ass
e
k.p
s
and
e
V is
g
. 13
The
e
the