PHYSICALREVIEW
B
VOLUME
42,
NUMBER
8
15
SEPTEMBER
1990I
Properties
of
the
Landauer
resistance
of
finite
repeated
structures
M.
Cahay
Nanoelectronics
Laboratoryand
Department
of
Electrical
and
ComputerEngineering,
University
of
Cincinnati,Cincinnati,
Ohio
45221
S.
Bandyopadhyay
Department
of
Electrical
and
ComputerEngineering,
University
of
Notre
Dame,
Notre
Dame,
Indiana
46556
(Received
25
January
1990;
revised
manuscriptreceived
12
April
1990)
Several
properties
of
the
Landauerresistance
of
finite
repeatedstructures
are
derived.Atheorem
relatingtheenergies
of
unity
transmission
through
a
finite
repeated
structure
to
the
band
structure
of
aninfinite
superlattice
formed
by
periodicrepetition
of
the
finite
structure
[Vezzetti
and
Cahay,
J.
Phys.
D
19,
L53
(1986)]
is
generalized
to
the
case
of
structures
with
spatially
varying
effective
mass.
Wealso
establish
a
sum
rule
for
the
Landauerresistances
of
periodic
structures
formed
by
periodicallyrepeatinga
basic
subunit.
Finally,
we
derive
an
analyticalexpression
for
the
boundary
resistance
of
a
structure,
as
introduced
by
Azbel
and
Rubinstein
in
connection
with
pseudolocalization,
and
prove
several
properties
of
thisquantity.
I.
INTRODUCTION
The
Landauerformula'
for
calculatingtheresistance
of
a
dissipationless
mesoscopicstructure
hasbeenusedquite
widely
in
the
study
of
quantum
transport
phenome
na.The
formula
relates
in
a
simple
way
theresistance
of
a
structure
(in
the
linearresponse
regime)
to
theprobabil
ity
of
transmission
of
an
electron
throughthe
structure.
The
usefulness
of
the
formula
lies
in
the
fact
that
it
reduces
the
problem
of
quantum
mechanically
calculat
ing
resistance
—
rather
difficult
problem
—
o
a
muchsimplerproblem
of
calculating
just
the
transmission
probability.
In
this
paper,
we
prove
several
interesting
properties
of
theLandauerresistance
(i.
e.
,
theresistance
in
the
linearresponse
regime)
of
a
finite
repeated
struc
ture
such
asa
semiconductorsuperlattice.Theseproper
ties
are
all
derived
fromthe
properties
of
thetransmis
sion
coefficient
of
an
electron
through
aperiodic
potential
of
finite
spatial
extent.
In
Sec.
II
of
this
paper,
we
firstemploy
atransfermatrix
technique
to
derive
a
general
expression
for
the
transmissionprobability
of
an
electron
through
an
arbi
trary
potentialprofile.
We
then
extend
this
result
in
Sec.
III
to
calculatethe
transmissionprobability
~
Ttt
~
of
an
electron
through
N
subunits
of
a
finite
repeated
structure.
Using
this
expression,
we
extend
an
earlier
result
relat
ing
the
energies
of
unity
transmissionthrough
a
finite
repeated
structure
to
the
energy
—
wavevectorrelation
for
an
infinite
structure
formed
by
periodically
repeating
the
basic
subunit
of
the
finite
structure.
In
Sec.
IV,
we
prove
a
set
of
theorems
that
establishinteresting
and
useful
re
lationships
between
the
transmission
probabilities
(and
hencethe
Landauer
resistances)
associated
with
the
sub
units
of
a
finite
repeated
structure.
These
theorems
are
all
illustrated
with
numericalexamples
dealing
with
com
positional
and
effectivemass
superlattices.In
Sec.
V,
we
establish
a
sum
rule
for
theLandauerresistances
of
periodicstructures
formed
by
successively
repeating
abasic
subunit,and
in
Sec.
VI,
we
derive
an
exact
analyti
cal
expression
for
the
boundary
resistance
of
a
struc
ture
as
introduced
by
Azbel
and
Rubinstein
in
connection
with
pseudolocalization.
Finally,
in
Sec.
VII,
we
summa
rize
ourconclusions.
II.
TRANSMISSION
OF
AN
ELECTRON
THROUGH
AN
ARBITRARYPOTENTIAL
g=(b(z)e
(2)
In
this
section,
we
first
derive
an
expression
for
the
transmissioncoefficient
of
an
electron
through
an
arbitrary
onedimensiona/
potential
of
finite
spatial
extent.
For
the
sake
of
generality,
we
allow
for
spatialvariation
of
the
electron's
eff'ective
mass
but
assume
it
varies
only
in
one
direction.The
timeindependent
Schrodinger
equationdescribing
thesteadystate
(ballistic)
motion
of
an
electron
throughsuch
apotential
is
fi
B
g
fi
B
g
A'
B
1
B1b
2m
*(z)
Bx
2m
*(z)
By
2
Bz
m
*(z)
+E,
(z)
=Ef,
(1)
where
E,
(z)
is
the
onedi.
mensional
potential
that
varies
in
the
z
direction
and
m*(z)
is
the
spatially
varying
effective
mass.
In
a
semiconductor
heterostructure,
E,
(z)
is
theconductionband
edgeprofilewhich
incorporates
any
bandbending
due
to
spacecharges,
variations
due
to
compositional
inhomogeneity,
and
also
variations
due
to
any
external
electric
field.
Because
theHamiltonian
in
Eq.
(1)
is
invariant
in
the
x
and
y
directions,
the
transverse
wave
vector
k,
is
a
good
quantum
number.
Furthermore,
sincethe
z
component
of
the
electron's
motion
is
decoupled
from
the
transversemotion
in
thexy
plane,
the
wave
function
P
can
bewrit
tenas
42
5100
1990
The
AmericanPhysical
Society
42
PROPERTIES
OF
THE
LANDAUER
RESISTANCE
OF
FINITE...
5101where
k
t
=
(
k,
ky
)
and
p
=
(
x,
y
).
The
z
component
of
the
wave
function
P(z)
now
satisfies
the
Schrodinger
equation
2
pl
dz
y(z)
dz
+
[E
+E,
[1
—
y(z)
']
(.
„)
y(z„)
dz
(t
(z„)
~(n)
pr(n)
'
11
12
pr(
n)
~(
n)
21
22
1
d(tt
+
)
y(z„+,
)
dz
P(z„+,
)
(4)
where
8'
'
are
the
elements
of
the
transfermatrix,
and
z„+,
and
z„stand
for
z„,
+e
and
z„—
,
respectively,
with
e
being
a
vanishinglysmall
positive
quantity.
Expli
cit
expressions
for
the
elements
of
the
transfermatrixare
givenin
theAppendix.
Assuming
continuity
of
P(z)
and
[I/y(z)]/(dtttldz)
everywhere
in
the
structure,
the
overall
transfer
matrix
W '
describing
theentire
region
[O,
L]
(see
Fig.
1)
can
be
foundby
simply
cascading
(multiplying)
the
individual
I
I
IIII
IIIIII
III
I
II
I
I
Ie
I
I
I
II
contact
I
I
I
0
z,
z2
n
n+1
''
contact
I
E
FIG.
1.
An
arbitrarypotential
profile
approximated
by
a
series
of
potential
steps.
Within
each
interval,the
potential
and
e6'ectivemass
are
assumed
to
be
spatially
invariant.
E,
—
(z)
I
tI)(z)
=0,
(3)
where
m,*
is
the
effectivemass
of
the
electrons
in
the
contacts
sandwiching
the
region
of
interest
(m,
'
is
spa
tially
invariant
within
the
contacts
and
isotropic),
y(z)=m'(z)/m,
*,
E,
=iri
k,
/2m,
*,
and
E
is
the
kinetic
energy
associated
with
the
zcomponent
of
themotion
in
the
contacts
(E
=Pi
k,
/2m,
*).
The
above
equation
cannot
be
solved
exactly
for
an
ar
bitrary
potential
E,
(z)
How.
ever,
an
approximate
solu
tion
can
be
found
by
approximating
the
potential
profile
by
a
series
of
potentialsteps
(see
Fig.
1)
or
byusing
a
piecewiselinear
approximation
for
the
potential.
In
the
former
scheme,the
region
over
which
thepotential
variesis
broken
down
into
a
finite
number
of
intervals.Within
each
interval
thepotential
and
the
effective
mass
are
as
sumed
to
be
constant.
In
that
case,
the
wave
function
and
its
first
derivative
at
the
left
and
right
edges
of
an
interval
are
related
through
asocalled
transfer
matrix,
characteristic
of
that
interval,
whoseelements
donot
dependon
the
z
coordinate
and
can
bedetermined
analytically.
The
transfer
matrix
for
the
nthinterval
[z„„z„]
sdefined
according
to
transfer
matrices
for
the
individual
intervals:
gytot
~(
X)
~(
1
)
(5)
where
8 '
is
thetransfer
matrix
for
the
nth
interval
as
definedin
Eq.
(4).
The
overall
transfer
matrix
8
relatesthe
wave
functions
and
their
first
derivatives
at
the
left
and
right
con
tacts:
dtt
(L+)
1
d((t
y(L+)
dz
y(0
—
)
dz
ttt(L+)
(0
)
In
Eq.
(6),
ttt(0
)
and
P(L+)
are
the
electronic
states
inside
the
leftand
right
contacts.
Theyare
given
by'
Ekoz
—
ikoz
e
'
+Re
,
z(0
iko(z
—
L)
(7)Te
',
z)L
tt(z)=
'
where
ko
[=(2m,
E
/trt)'
]
is
the
z
component
of
the
electron's
wave
vector
in
the
contact
and
R
and
T
are
the
overallreflection
and
transmission
coeScients
throughthe
region
[O,
L].
Using
these
scatteringstatesfor
the
wave
functions
at
z
=0
andz
=L+
and
noting
that,
bydefinition,
y(L+)=y(0
)=1,
we
obtainfrom
Eq.
(6)iko
T
~tot
1
iko(1
—
R)
1+R
Equation
(8)
finally
gives
us
the
twoequations
for
the
two
unknowns
T
and
R.
From
these
twoequations
T
and
R
can
be
found
by
straightforward
algebra.
Eliminating
R
gives
0
1122
12
21
'I
(
WtotWtot
Wtot
Wtot
}
ik
(
W' +
W ')+(
W' king
—
W'
)
(9)
where
8',
'
are
the
elements
of
the
matrix
W'
that
are
found
from
Eq.
(5).
Since
8
is
aunimodular
matrix,
theterm
within
parentheses
in
thenumerator
ofEq.
(9)
is
unity.
In
addition
(see
theAppendix),
W~
is
alwayspurely
real.Therefore
Eq.
(9)
gives
[
T/'=
4ko
k
2(
Wtot
+
Wtot
)2+
(
Wtotk
2
Wtot
)2
0
ll
2221
0
12
(10)
III.
TRANSMISSION
OF
AN
ELECTRON
THROUGH
A
FINITE
REPEATED
STRUCTURE
Havingfound
a
general
expression
for
~
T~,
we
now
proceed
to
evaluate
the
transmissionprobability
(and
hencethe
Landauer
resistance)
associated
with
a
finite
repeated
structure
formed
by
the
periodicrepetition
of
a
structure
with
arbitrarily
varying
potential.
Considerthe
potential
profile
in
Fig.
2
formed
by
theThe
aboveequation
givesus
a
general
expression
for
the
transmission
probability
of
an
electron
through
an
ar
bitrarypotential.
The
transmissionprobability
~
T~
is,
of
course,
related
to
the
reflection
probability
~R~
accord
ing
to
therelation
~T~
+
~R~
=1
as
required
by
currentconservation.
5102
M.
CAHAYAND
S.
BANDYOPADHYAY
42
Ec(z)
I
+sUBUNIT
—
N1
~~~
X
h
1
2e'
~T~
'
h
[sin
(NO)]
2e
kpW2,
—
W,
2kpsinO
—
+1
FIG.
2.
The
potential
profile
for
a
finite
repeated
structure
formed
by
periodic
repetition
of
a
region
with
arbitrarily
varying
potential.
R4
h
x
2e'
/T,
,
'
=R
h
L
where
kp
is
the
wave
vector
of
the
incident
electron.
periodic
repetition
of
an
arbitrarypotential.
Every
period
in
this
structure
has
the
same
transfer
matrix
(say
W)
characterizingthat
period
and
the
grandoverall
transfer
matrix
8
describing
theentire
structure
is,
as
before,
obtained
by
cascading
the
transfermatrices
for
the
individual
periods.
It
is
easy
to
see
that
for
a
structure
with
N
periods
with
each
periodidentical,
Wtot
—
(
W)N
As
shown
in
Ref.
10,
the
elements
of
the
matrix
8
can
be
expressed
in
terms
of
the
elements
of
thematrix
W
IV.
TRANSMISSION
THEOREMS
FOR
A
FINITE
REPEATED
STRUCTURE
We
now
prove
a
set
of
theoremsrelated
to
transmission
through
finite
repeated
structures.
First,
we
prove
a
theorem
that
relatesthe
energies
of
unit
transmission
(i.
e.
,
the
values
of
theincident
energy
for
which
the
transmissioncoefficientis
exactly
unity)
through
a
finite
repeated
onedimensional
structure,
to
the
band
structure
of
the
associated
infinite
lattice
formed
by
periodic
repetition
of
the
onedimensional
structure.
This
theorem
was
stated
for
the
first
time
in
Ref.
5.
A
more
detailed
proof
isgiven
here
with
generalization
to
the
case
of
a
structure
with
a
variable
effective
mass.
W«,
W
sin(NO)
sin[(N
—
1)O]
(12)
where
I
is
a
2
X
2
identity
matrix
and
8
depends
on
theei
genvalues
of
the
matrix
W
andis
givenby
exp(iO)=A,
=A&
'=
—
+
r(
W)
Tr(
W)
2
'2
1/2
(13)
~Tz~
=
[sin
(NO)]
2
'2
k
p
W2]
W]2
2kpsinO
—
+1
(14)
whichis
our
main
result.
The
two
and
fourprobe
(2p
and
4p)
Landauer
resis
tances
for
astrictly
onedimensional
repeated
structurecan
now
befound
easily
by
substituting
Eq.
(14)
for
the
transmissionprobability
~
Tz~
in
the
singlechannel
Lan
dauerformula:
where
A,
]
2
are
the
eigenvalues
of
the
2
X
2
matrix
W
and
thesecond
equalityfollows
from
the
fact
that
the
matrix
W
is
unimodular.
Wecan
now
find
the
overalltransmissionprobability
~TN~
through
a
periodicstructure
with
Nperiods.
For
this,
weuse
Eq.
(10)
with
the
elements
of
W '
now
givenby
Eq.(12).
This
gives
Theorem
I.
The
transmissioncoefficient
of
a
particle
through
a
periodic
structure,
formed
by
N
repetitions
of
a
basic
subunit,
reaches
unity
atthe
following
energies:
(a)
energies
at
which
thetransmissionthrough
the
basic
sub
unitis
unity,and
(b)
N
—
energies
in
each
energy
band
of
the
lattice
formed
by
infinite
periodic
repetition
of
the
basic
subunit,where
these
N
—
energies
are
given
by
E
=E,
(k
=+nrrlNL)(n
=1,
2,
3,
...
,
N
—
1,and
L
is
the
length
of
a
period).
Here
E,
(k)
is
the
energy
—
wave
vector
relation
(or
the
dispersion
relation)
for
the
ith
band
of
the
infinite
lattice.
l=~T,
~
=
.
[sin
(O)]
22
k
P
W2]
W]2
2kpsin0
—
+1
(17)
Part
(a)
of
thetheorem
is
actually
fairly
obvious.
All
it
states
is
that
by
connectingidenticalstructures
of
transmission
unity,
one
always
obtains
unit
transmission
throughthecomposite
structure.
Although
this
is
intui
tive,
we
proveitnevertheless
for
the
sake
of
cornplete
ness.
For
this,
we
first
note
from
Eq.
(14)that
the
transmission
~
T~
~
through
N
periods
reaches
unity
when
the
term
within
the
largesquare
brackets
vanishes.
The
term
within
the
largesquare
brackets
vanishes
when
2
2
k
o
W2]
—
W]2
(16)
2kpsinO
We
now
show
that
this
corresponds
to
the
conditionthat
~
T,
~
(i.
e.
,
the
transmissionthrough
one
period,
or
the
basic
subunit)
is
unity.
Substituting
N
=1
in
Eq.
(14),
we
get
that
the
condition
for
unit
transmission
throughone
subunit
isgiven
by
42
PROPERTIES
OF
THE
LANDAUER
RESISTANCEOF
FINITE.
.
.
5103
1.
C.
,
+
2m
+
3m
(N
—
1)m
W'e
now
have
to
prove
that
the
above
values
of8
also
cor
respond
to
the
wave
vectors
k
=+n~/NL
where
L
is
the
period.
For
this,
we
firstapply
the
Bloch
theorem
to
the
infinite
structure.The
Bloch
theorem
gives
P(z
+L)
=P(z}exp(ikL),
where
k
satisfies
the
relation
(20)
det[
WJ
5„exp—
ikL)]
=0
.
(21)
In
the
above
equation,
8',

is
the
ijth
element
of
the
transfer
matrix
W
describing
oneperiod
and
5,
is
a
Kronecker
delta.
From
Eq.
(21)
we
immediatelysee
that
exp(ikL)
is
the
eigenvalue
of
the
2X2
unimodular
matrix
8'and
hence
exp(ikL
)
=
I,
,
=
A,
2
'
=
+
r(W)
2
Tr(
W)
2
1/2
(22)
The
righthand
sides
of
Eqs.(13)
and(22)
are
identical
so
that
their
lefthand
sides
must
also
be
identical.
Therefore
exp(ikL)
=exp(i8),
(23)
or
which,
after
simplification,
reduces
exactly
to
Eq.(16).
This
proves
the
first
part
of
the
theorem,
viz.
,
that
the
en
ergies
of
unit
transmission
throughoneperiod
are
alsothe
energies
of
unit
transmission
through
all
thepcl
iods.
To
prove
the
second
part
of
the
theorem,
we
note
from
Eq.
(14)
thatthe
transmission
~
T~
~
also
reaches
unity
for
those
values
of0
that
satisfy
the
conditions
sin(N8)
=0,
sin(8}WO;
The
locations
of
the
band
edges
can
be
found
directlyfromthe
following
property,
which
we
prove:
The
states
characterized
by
wave
vectors
k
for
which
~Tr(
W}~
)
2
are
theevanescent
states
corresponding
to
the
stop
band
of
a
finite
repeated
structure.The
states
charac
terized
by
wave
vectors
k
for
which
~Tr(
W)~
&2
are
thepropagating
states
corresponding
to
the
passband
of
the
finite
repeated
structure.
To
provethe
property,
we
invoke
Eq.
(22).
If
~Tr(
W)
~
&
2,
then
the
righthand
side
ofEq.
(22}
is
purely
real
and
greater
than
unity.
In
that
case,
the
wave
vector
k
must
be
purely
imaginary
which
means
that
the
state
is
an
evanescent
state
corresponding
to
the
stop
band
of
the
finite
repeated
structure.
On
the
other
hand,
if
~Tr(W)~
&2,
the
righthandside
of
Eq.
(22)
becomes
complex
which
permits
k
to
be
real.In
the
latter
case,
the
state
isa
propagating
state
corresponding
to
the
passband
of
the
structure.The
values
of
wave
vector
k
for
which
~Tr[W]~=2
evidently
correspond
to
the
edgesbetween
the
pass
bands
and
the
stopbands.
Theorem
II.
At
the
energies
of
unity
transmissionthrough
a
finite
repeated
structure
with
N
periods,the
followingequality
holds:
~
Ttv
~
=
~
Ttt
~
whenever
1
2
N,
+Nz=N.
Here
~Ttv
~
and
~Ttt
(
are
thetransmis
sion
probabilitiesthrough
two
subsections
with
N&
and
N2
periodsrespectively.
As
stated
in
the
proof
of
theorem
I,
the
transmissionthrough
N
periods
reaches
unity
under
two
conditions:
(a)when
thetransmission
through
each
of
the
N
periods
is
unity,
and
(b)
when
8=+
(n
=1,
2,
3,...
,
N
—
1)
.
~
(25)
In
case
(a),
the
proof
of
theorem
II
is
trivial.
If
the
transmission
through
each
period
isunity,
then,
of
course,
the
transmission
through
any
arbitrary
number
of
periods
is
also
unity.
In
that
case,
obviously,
(26)
kL
=8(mod2vr)
.
(24}Consequently
whenever
k
=+n
m/NL,
the
quantity
8
=+n
m
/N.
Thusthe
energies
corresponding
to
k
=+trlNL,
+2m
INL,
+3m
INL,
...
,
+[(N
—
1)7r)/NL
are
the
energies
corresponding
to
8=+m
/N,
+2m/N,
+3ttlN,
...
,
+[(N
—
)m.
]IN,
which,
in
turn,
are
the
energies
corresponding
to
unity
transmissionthrough
the
finite
repeated
structure
with
N
periods
as
previously
noted.
Stated
in
other
words,
thismeans
that
the
energies
associated
with
unity
transmissionthrough
an
Nperiod
structure
are
the
bandenergies
E(k„)
corre
sponding
to
the
wave
vectors
k,
=+n
~/NL
in
an
infinite
repeated
structure.
This
gives
us
the
E(k„)versusk„re
lation
and
proves
the
theorem.The
usefulness
of
theorem
I
lies
in
the
fact
that
by
evaluating
theenergies
of
unit
transmission
througha
finitestructure
[which
we
can
do
from
Eq.
(14)],
we
cancalculate
the
band
structure
of
an
infinite
superlatticeformed
by
theperiodicrepetition
of
the
finite
structure.
regardless
of
what
N,
and
N2
might
be.
This
provesthetheorem
forcase
(a).
The
proof
for
case
(b)
proceeds
as
follows.
We
first
note
that
sinN,
8
=
sin(N
Nz
)8=
sin(+
n
n
—
N28
)—
=(
—
1)
+'sinN28,
(27)
where
we
used
Eq.
(25)
to
obtain
the
second
equality.
Us
ing
the
above
equality
in
Eq.
(14),
we
immediately
see
that
which
proves
case(b).
Theorem
III.
At
the
energies
of
unity
transmission
(~T&~
=1)
through
a
finite
repeated
structure
with
N
periods,
the
following
equality
holds:
~
Tz+M
~2
for
all
M
such
that
1
&M
&
N.
5104
M.
CAHAY
AND
S.
BANDYOPADHYAY
42
The
proof
of
this
theorem
is
very
similar
to
that
of
theorem
II
andis
thereforenotpresented.
A.
Numericalexamples
To
illustratetheorem
I,
we
show
in
Fig.
3
the
construc
tion
of
the
energyband
diagram
of
an
infinitely
repeated
structure
whose
basic
subunitis
shown
in
the
inset.
The
points
Q,
Q'
are
the
twolowestenergies
at
which
the
transmission
through
two
subunits
is
unity,whereas
thepoints
P,
P'
and
R,
R
'
are
thetwo
lowestenergies
for
which
the
transmission
through
three
subunitsis
unity.
These
points
are
on
the
two
lowestenergy
bands.
Other
points
on
the
energyband
diagram
can
be
found
similarly
by
steadily
increasing
the
number
of
periods
and
search
ing
for
the
energies
of
unit
transmission.Finally,thepoints
PI,
P~
and
Q„Q2
correspond
to
theband
edgesand
are
found
from
thecondition
~Tr(
W)
=2.
To
illustrate
theorems
II
and
III,
we
provide
thefol
lowing
numerical
examples.
E
xample
1.
We
have
calculatedthe
transmission
~
T~
~
[using
Eq.
(14)]
through
a
compositional
superlattice
con
P2
C
0.
3ev
l
soA
0.
3
)
~
0.
2—
CC
LLI
z
k
0.
1—
IIIIII
'k
IIIIII
I
=
m/4L
r
I
I
IIIII
II
IIII
III
I
=
7tj2L
I
IIII
,
'
k
I
IIII
t
I
I
I
III
370)'4L
',
Q,
p
,
Q
',
R
I
I
II
I
0.
05
0.
0079
0.
0157
0.
0236
0.0314
k
=
70'L
BLOCH
WAVE
NUMBER
(A
'
}
FIG.
3.
Ener
gyband
diagram
of
an
infinitely
repeated
struc
ture
whose
basic
subunit
is
shown
in
the
inset.
The
conduction
band
is
constructed
by
numericallyevaluating
theenergies
at
w
ich
the
transmission
throughincreasing
number
of
periods
go
to
unity.
The
points
Q,
Q'
correspondto
the
two
lowest
ener
gies
at
which
transmissionthrough
two
bo
su
units
is
unity,
whereas
thepoints
P,
P'
and
R,
R'
correspond
to
the
lowesten
ergies
for
which
transmissionthrough
three
subunits
is
unity.
The
pomts
P,
,
P2
and
Q,
,Q2
correspondto
theband
edgesand
are
found
fromthe
condition
Tr[
W]
=2.
Theorem
IV.
If
the
Fermi
energy
of
a
finite
repeated
onedimensional
structure
lies
at
the
boundary
between
a
passband
anda
stop
band,
then
thefourprobe
Lan
dauerresistance
of
N
periods
of
the
structure
is
equal
to
N
times
the
fourprobe
Landauer
resistance
of
one
period.This
means
that
the
fourprobeLandauer
resis
tance
increases
with
the
structure's
length
as
L
instead
0.
15
LU
0
0.
10—
U
LU
0
O
Z',
0
(0
CO
5
005
(0
Z
K
I—
I
I
II
I
I
II
II
III
I
I
II
III
I
I
I
IT3)~
III
I
I
I
II
I
0
3eV
I
50A
II
I
I
I
II
0
105
0.
00
0.
075
0.
081
0.
087
0.093
0.
099
ENERGY
(ev)
FIG.
4.
Transmissioncoefficientsthrough
a
periodicstruc
tureformed
by
repeatingthe
subunitshown
in
theinset.
The
subunit
consists
of
a
GaAs
well
andan
Al,
Ga
As
barrier
both
50
A
thick.
The
barrier
heightis
0.
3
eV
and
the
effectivemassisassumed
to
be
0.
067mo
everywhere.
Notethat
when
IT31'=1,
T,
'=IT,
/'.
Also
whenever
[T
/'=1,
/T
f'=/IT2/I'
These
illustrate
theorems
II
and
III,
respectively.
s&sting
of
rectangular
wells
and
barriers
in
which
the
bar
rier
and
well
thicknesses
are
50
A.
The
effective
mass
was
assumed
to
be
0.
067mo
everywhereand
the
barrier
height
was
taken
to
be
0.
3
eV.
Figure
4
shows
the
transmissioncoefficient
through
one,
two,
and
three
bar
riers
in
the
vicinity
of
the
lowest
resonant
energy
through
two
barriers.
(Resonant
transmissionthrough
two
bar
riers
has
been
studied
extensively
in
connection
with
thedoublebarrierresonant
tunneling
diode.
'
)
Figure
4
is
a
clear
illustration
of
theorem
III.
It
shows
that
when
the
transmissionthrough
two
barriers
is
unity,
the
transmissionthrough
threebarriers
is
equal
to
the
transmissionthrough
one
barrier,
i.
e.
T,
I2
~
with
N
=
2and
M
=
1.
Figure
4
also
shows
that
whenever
~T3~
=1,
~TI
~
=~T2
illustrating
theorem
II
for
the
case
N
=
1,
N
=2.
~
2
Exam
le
2.
.
In
Fig.
5
we
show
the
transmissionthrough
an
effectivemass
superlattice'
in
which
theconductionband
edges
in
the
differentlayers
are
assumed
to
be
aligned
but
the
effective
masses
are
different.
We
assumeeffectivemasses
of
0.
039mo
and
0.
073mo,
respec
tively,
in
two
alternating
layers.(These
correspond
to
thetransmissionsthrough
one,two,
and
three
layers
were
calculated
from
Eq.
(14)
at
theresonant
energy
through
threelayers.Clearly,
when
~
T3
~
=
1,
~
TI
~
=
~
T2
This
illustrates
theorem
II.
Also
when
~T
~
=1,
T
'
=(T3(
~
1
3
as
stated
in
theorem
III.