Biperiodic superlattices and the transparent state

0.49 MB·English

by  D. W. L. Sprung

Biperiodic superlattices and the transparent state
D. W. L. Sprunga, L. W. A. Vanderspekb, W. van Dijka,b, J. Martorellc and C. Pacherd
aDepartment of Physics and Astronomy, McMaster University
Hamilton, Ontario L8S 4M1 Canada
bDepartment of Physics, Redeemer University College
Ancaster, Ontario L9K 1J4 Canada
cDepartament d’Estructura i Constituents de la Materia, Facultat F´ısica,
University of Barcelona, Barcelona 08028, Spain and
dAustrian Research Centers GmbH - ARC, Smart Systems Division, Donau-City-Str. 1, 1220 Vienna, Austria
(Dated: February 1, 2008)
We study biperiodic semiconductor superlattices, which consist of alternating cell types, one
with wide wells and the other narrow wells, separated by equal strength barriers. If the wells were
8 identical,itwouldbeasimplyperiodicsystemofN =2nhalf-cells. Whenasymmetryisintroduced,
0
an allowed band splits at the Bragg point into two disjoint allowed bands. The Bragg resonance
0
turns into a transparent state located close to the band edge of the lower(upper) band when the
2
first(second) well is thewider. Analysisof this system gives insight intohow bandsplitting occurs.

n Further we consider semi-periodic systems having N = 2n+1 half-cells. Surprisingly these have
a very different transmission properties, with an envelope of transmission maxima that crosses the
J envelopeof minima at thetransparent point.

3
PACSnumbers: 73.21.Cd,73.61.Ey03.65.Nk

]
h
p I. INTRODUCTION for them to be distinguished. When a = c, the sim-
-
t ply periodic system of N =2n symmetric cells is known
n
[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
a Coquelin et al. [1, 2] carried out experiments
to exhibitN 1transmissionresonancesineachallowed
u on electron transmission through a finite biperiodic −
q band, according to
GaAs/AlGaAs superlattice consisting of alternating
[ types of unit cells. Biperiodic systems occur naturally t 2 = [1+sinh2µ sin2Nφ ]−1 cosh−2µ , (1)
3 in crystals and polymers [3], but in layered semiconduc- | N| h ≥
v tor heterostructures one has control over the properties where φh is the Bloch phase of the half-cell, and µ is
5 ofthecells. Asillustratedbythered(solid)lineinFig. 1, its impedance parameter in the Kard parameterization
5 Coquelin’ssystemhadidenticalbarriersofwidthb=3.8 Eq. (10)ofthetransfermatrix[21,22]. Thefundamental
7
nm,whiletherearetwoalternatingwellwidths,2a=4.3 band structure depends on those properties of the half-
1
nm (wide) and 2c = 3.8 nm (narrow), which changes it cell,whilethewidthandspacingofindividualresonances
.
8 from a simply periodic to a biperiodic system. depends on the number N of half-cells, via Nφh = mπ,
0
with m=1, 2, ...N 1.
7 We will consider the basic unit, called a half-cell, to −
0 comprise three layers: two well segments of GaAs hav-
: ing widths a, c separatedby anAlGaAs barrierof width Schematic of a biperiodic potential
v
b. A full or “double-cell” consists of a half-cell plus an-
i
X otherwhichisitsmirrorimage. Thedouble-cellofwidth
r 2d=2(a+b+c)centeredontheorigin,ismarkedinFig.
a
1 at the left of the three-cell array, in red. It has layers
of widths c, b, 2a, b, c and is overall reflection symmet-
ric about its mid-point. Due to the two barriers enclos- x)
V(
ing a well of width 2a, there will be quasi-bound states
which show up as resonances in electron scattering from
w n w n w (n)
the double cell. When N such cells are juxtaposed, the
c b 2a b c
wellswillbealternatelywideandnarrow;hencethename
← d →← d →
biperiodicarray. Eachdoublecellisasymmetriccell,and
thecompletearrayhasreflectionsymmetry. Byexchang-
ing the values of a and c, the character of the array will
FIG. 1: (Colour online) Solid (red) line: Biperiodic array [1,
change from say w,n,w, to n,w,n, , where w and
··· ··· 2]ofthreedouble-cells,eachofwidth2d;short-dashed(blue)
n stand for wide and narrow wells respectively.
line an additional half-cell could be added at right, creating
The experimental device [1, 2] had n=3 double cells, an additional narrow well as discussed in Sec. IV.
sothere were2n 1=5 wellsenclosedbetweenbarriers.
−
This numberwaschosenbecauseitprovidesenoughsep- When a = c, the half-cell is asymmetric under reflec-
6
aration between the measured transmission resonances tion, though the double cell remains symmetric. Each
2
allowed band develops a band gap near the Bragg point matrix is
φh = π/2. An allowed band of the n = 3 double-cell g u u′ u
system should show two resonances. As is seen in Fig. W =W−d,d = WRWL = g′ u′ g′ g
2, one of the allowed bands contains a third resonance (cid:18) (cid:19) (cid:18) (cid:19)
′ ′
which we will identify as a transparent state at which gu +g u 2ug
= ′ ′ ′ ′ . (4)
the impedance parameter µ 0, causing the envelope 2ug gu +g u
→ (cid:18) (cid:19)
ofminimum transmissionprobabilityto be pushedup to
To transform to the ingoing/outgoing waves representa-
unity. Such a state occurs at a fixed energy, indepen-
∗
tionweproceedasinAppendixAwithν =h¯k/(mm )as
dent of the number of double cells included, as can be
the velocity outside the potential. Unlike W, the matrix
seen fromeq. 1. This transparentstate lies veryclose to
a band edge, in the lower (upper) split band when the M−d,d operates from right to left. For the full cell it is
given by eq. A8.
wide (narrow) well is first in line for incident electrons.
The purpose of this paper is to explain why and how
the transparent state arises when the half-cell becomes wnwnw biperiodic 3 cell array
asymmetric, and why it is located very close to the split
1
band edge. In Sec. IV we will consider the surprising
effectofincludinganadditionalhalfcell,assuggestedby
0.8
the dashed blue line at the right in Fig. 1.
2| N0.6
II. TRANSFER MATRIX ANALYSIS |t
0.4
A. General 0.2
We will follow the line of argument of Shockley [23], 0
90 95 100 105 110 115 120
who studied surface states of a finite static periodic po- a) E ( meV )
tential whose unit cell is symmetric about its mid-point.
nwnwn 3 cell array
We extend his method to allow for the position and
energy-dependent effective mass, required in a semicon- 1
ductor superlattice. Since the full cell has reflection
symmetry, it is sufficient to solve the Schr¨odinger equa- 0.8
tion for the half-cell 0 < x < d. The even (g(x)) and
og(d0d) (=u(1x,)g)′(p0a)r=ity0saonludtiuon(0s)t=ake0,tuh′(e0)bo=un1daatrythvealoureis- 2| N0.6
gin. [Note: Following the development of Appendix A, |t
0.4
the prime means take the derivative, and then divide by
∗
thevariablefactormm /¯htoallowfortheeffectivemass,
∗ 0.2
m 0.07, which is dimensionless.] The transfer matrix
∼
for the half-cell is
0
90 95 100 105 110 115 120
g u b) E ( meV )
WR = W0,d = g′ u′ , and
(cid:18) (cid:19)
′ FIG.2: Transmission(solidline)forthe3-cellbiperiodicarray
WL = W0−,−1d = ug′ ug . (2) of Fig. 1: (a) wide well first; and (b) narrow well first. The
(cid:18) (cid:19) dashed line is the envelope of transmission minima in the
allowed zones and an upperbound in the band gap.
where the wave functions without argument are always
evaluated at the point x = d. Placing the half-cells in
Shockley’sfirststepwastoderivetheBlochphaseφof
′
reverseorder,simply interchangesthe elementsg andu. the double cell in terms of the solutions of the half cell,
Since the Wronskian of two solutions is a constant,
by appealing to Floquet’s theorem. He obtained
′ ′
detW = gu g u = 1. In dealing with a constant po-
tential and co−nstant m∗, for example, one has tan2φ/2= γ = g′u , (5)
−λ −gu′
∗
g(x)=cosqx u(x)=(mm /¯hq) sinqx (3)
whereγ,λarethelog-derivativesofthehalf-cellsolutions
g,u respectively. One finds easily that
whicharechosensothatatq =0(wellbottom),wehave
solutions g(x)=1 and u(x)=mm∗x/¯h. 1 tan2φ/2 gu′+ug′
cosφ= − = , (6)
Forthesymmetricdoublecell d<x<d,thetransfer 1+tan2φ/2 gu′ ug′
− −
3
wherethe denominatorisdetW =1. We write W inthe We have written s = a c; s/d is the asymmetry. The
−
parameterized form double-cell has two barriers and a well of width 2a be-
tween. Reversing the sign of s interchanges a and c,
cosφ (1/Z)sinφ
W = with which is equivalent to putting the two half-cells in the
Zsinφ cosφ
opposite order. When s > 0 a wide well occurs on both
(cid:18)− (cid:19)
Z2 = γλ . (7) ends of the biperiodic superlattice.
−
Thebandstructuredependsonlyonthelocationofthe
The log-derivatives γ, λ determine the effective velocity
zeroes and poles of γ, λ. These locations do not change
Z andtheBlochphaseφateachenergy,forasymmetric ′
if the ν in front of the off-diagonal elements g and u is
double cell. Again, in case of a constant potential across
multipliedbyaconstantfactor. Toreducethenumberof
the cell, they would be parametersinplay,wereplacethatν kd,equivalentto
∗ →
∗ ∗ sayingthath¯/mm =ν/k =d. Thenthetransfermatrix
γ = (h¯q/mm )tanqd; λ=(h¯q/mm )cotqd
− of the delta-barrier model is a function of dimensionless
Z = (h¯q/mm∗) ; tan2φ/2=tan2qd . (8)
variableskd,s/d=ks/kdandΩd. Thefiguresaredrawn
as functions of kd; the lowest allowed band ends when
It is evident that in an allowed band γ and λ must have
kd π.
opposite signs, to make both φ and Z real. Conversely ≈
in a forbidden band γ and λ have the same sign.
Inserting eq. 7 into eq. A8, gives M =M−d,d = γ
4 λ
cosφ isinφ(ν/Z+Z/ν) isinφ(ν/Z Z/ν) asym 10-5 0.10
− 2 2 −
isinφ(ν/Z Z/ν) cosφ+isinφ(ν/Z +Z/ν) 2
(cid:18) − 2 − 2 (cid:19)
(9)
tDheefinpiontgenetµia=l rνe/gZioans,twheeroabttiaoinoftvheeloKcitairedsopuatrsaimdee/tienrsiizdae- γ, λ 0
tion in our standard form [21, 22]:
-2
cosφ isinφ coshµ i sinφ sinhµ
i−sinφ sinhµ cosφ+i sinφ coshµ (10) -4
(cid:18) − (cid:19)
Wecallµtheimpedanceparameter,sinceaslowerveloc- 0 0.5 1 1.5 2
x = kd/π
ity inside the cell corresponds to a greater impedance.
FIG. 3: (Colour online) Log-derivatives γ, λ of the solutions
B. Asymmetric delta-barrier cells
g(kd) and u(kd) for the delta-barrier system, for very small
10−5 andmoderate0.10asymmetry;γ isshownassolid(red)
Togaininsightintohowbandsplittingoccurs,wecon- line and short-dashed (blue) line; λ as long-dashed (green)
sider a simple model which replaces the square barrier and dotted (mauve) lines, respectively.
cells illustrated in Fig. 1 by delta-function barriers of
strength Ωd = 1.403π, (a value chosen to give results
similartothoseoftheCoquelinpotential[1]). Inthewell
5
∗
sections, we have ν =h¯k/mm (see Eqs. (A3, A5)).The cos φh
transfer matrix for a half-cell of width d=a+c is 4 cos φ (a)
cos φ (b)
coskc (sinkc)/ν 3
WR = WcWδWa =(cid:18)−νsinkc coskc (cid:19) × φcos 2
1 0 coska (sinka)/ν , h
(cid:18)2Ων/k 1(cid:19) (cid:18)−νsinka coska (cid:19) φcos 1 A D
g u 0
= ′ ′ with
g u
(cid:18) (cid:19) -1 B C
Ωd
g = coskd+ (sinkd sinks) -2
kd − 0 0.5 1 1.5 2
x = kd/π
Ωd
′
g = ν sinkd (coskd+cosks)
− − kd
(cid:20) (cid:21)
1 Ωd FIG.4: (Colouronline)cosφhofthehalf-cell(solid/redline),
u = sinkd (coskd cosks)
ν − kd − and cosφ of the double-cell, for the delta-barrier system; (a)
(cid:20) (cid:21) forverysmall(10−5)asymmetry,(long-dash/greenline),and
Ωd
′
u = coskd+ (sinkd+sinks) . (11) (b) for moderate (0.10) asymmetry (short-dash/blueline).
kd
4
Figure 3 shows the evolution of the log-derivatives γ Thisaccountsforthethirdtransmissionresonanceinthe
and λ versus kd, for two values of the asymmetry pa- lower allowed band of Fig. 2(a). Conversely, if we take
rameter, s/d= 10−5, and 0.10. For a fixed cell width d, the opposite asymmetry, (so the narrow well is first in
the plot shows the energy dependence (via kd). Larger line), then it is the alternate function Z˜2 which applies.
s/d moves the poles to the left, seen by the redand blue This pole has the opposite sign residue, so it is the di-
linesforγ. Forλ(greenandmauvelines)thenodeshifts vergenceofZ˜2 nearpointlabelledC atthe loweredgeof
to the right. Allowed bands occur when γ and λ have the upper allowedband, whichproduces the transparent
opposite signs, which covers much of the interval for kd state.
between 0.7π and π. For the s = 10−5 asymmetry, the
In Fig. 6(a), the two poles are very close together
poleofγ atapproximately0.83π almostcoincideswitha
at BC. In the limit of exact reflection symmetry, they
nodeofλ,creatinganinfinitesimalforbiddenbandthere.
would coincide, and their sum would be zero. The band
For a reflection symmetric half-cell, they would exactly
gap would disappear, and cosφ would touch 1 without
coincide, and cancel, giving a vanishing gap. For the −
crossing below that line.
larger asymmetry, the separation between the pole and
node increases, widening the gap. A magnified view of
this region is shown in Fig. 6. Z2
In Fig. 4 we see the corresponding Bloch phases. The 4 − γ / λ
∼
red (solid) lines are the cosφ of the half cell, while the Z2
h
blue(dotted)andgreen(dashed)linesarethecosφofthe 2
db2φoeluho,bwsleocw1eh.lle.nHFoφowhre=vlienπre,/2f(oa,r)thmtehodedaeasrnhagetdleelaiφnsyeimsscmaalmrectoerylsyt(delieqnsueca(ebln)dt)os, ∼ φ22an /2, Z 0
theund−ershootisevident,andasizeablebandgapopens 2 Z, t-2
upbetweenkd=0.77(B)and0.89π(C).Theouterband
edges, markedby A and D, shift outwards a little at the
-4
same time.
In Fig. 5, the green (long dashed) line is γ/λ = 0 0.5 1 1.5 2
tan2φ/2; positive values are necessary for an−allowed a) x = kd/π
band to exist. The solid line is Z2 defined in eq. 7. Ζ2
The short dashed line is the alternative value Z˜2 = 4 − γ / λ
g′u′/(gu), which results when we interchange the val- ∼Ζ 2
−
ues of a and c, choosing the opposite asymmetry. Since ∼ 2Ζ 2
tianhngidsttchhheearvneagfloeusreetshφoefsguignancnhodafnusg′,,ediwt.hiIsincehqpulaeniavevalele(snat)thZte˜o2tirlnaiteceseracohlfmaWnogsRt- φ2n /2, 0
h a
oonnlytoapsomfaZll2,glbitecchau(sdeuethteoafisnyimtemsetetrpysiisnpdrraacwtiicnagl)lymzaerrkos; 2Z, t-2

the location on the curve near 0.83π. In panel (b) the
poles of Z2 and Z˜2 separate cleanly. When a > c, Z2 is -4
largebelowthebandgap,andsmallabovethebandgap,
0 0.5 1 1.5 2
which runs from kd = 0.77 to 0.89π. When a < c, Z˜2 b) x = kd/π
applies and those properties reverse.
The two panels of Fig. 6 provide a magnified view of
the split band region. The straight mauve (dotted) and FIG. 5: (Colour online) tan2φ/2: long-dashed (green) line;
′
black (double-dash) lines are multiples of g and u, the Z2: solid (red) line; and Z˜2: short-dash (blue) line, for the
−
negativesignimposedsothattheircrossingpointcanbe delta-barriersystem;(a)caseofverysmall(10−5)asymmetry;
easily identified. This is the point at which TrWR = 0, (b) moderate (0.10) asymmetry. In (a) the two poles almost
whichmakes φh =π/2. For a symmetric half-cell, this is coincide and cancel, leaving only a small glitch.
the energywhere the pole and node coincide, and cancel
eachother. ThepointlabelledBisanodeofg andapole
′
ofγ =g /g,attheloweredgeofthebandgap. Totheleft
of the pole, γ (red, solid) diverges, and so does Z .
→∞
SinceZ risesthoughallpositivevaluesbetweenthreshold
III. GENERAL CASE
at A and the pole at B, at some point it must equal the
externalvelocityν, whichmakesthe impedance parame-
terµ=logν/Z 0. Thatdefinesthetransparentpoint, To discuss the general case of barrier-type cells ar-
at which t −2→= 1+sinh2µsin2Nφ = 1 independent rangedfromlefttorightinorderLRLR L/R,wewrite
N
| | ···
of the number of cells or the value of the Bloch phase. the transfer matrices (in an allowed band) in terms of
5
three real parameters as follows: system has reflection symmetry and φ = 2nβ, but eα
N
appears on the off-diagonal elements making Z = zeα.
WR = gg′ uu′ ≡ e−zαscionsββ (1e/αzc)ossinββ eFlolerdododn othrdeehrsalNf-c=ell,2n(h+er1e,Wthe),trwanhsicfehrims aretrpiexNatisedmoonde-
(cid:18) (cid:19) (cid:18)− (cid:19) L
u′ u eαcosβ (1/z)sinβ extra time. Then
WL = g′ g ≡ zsinβ e−αcosβ
(cid:18) (cid:19) (cid:18)− (cid:19) cosφ = coshα cos(2n+1)β
cos2β (e−α/z)sin2β N
W = WRWL = zeαsin2β cos2β (12) ZN = z . (14)
(cid:18)− (cid:19)
Superlatticeswithanoddnumberofhalf-cellsarebiperi-
ComparingtoW ofeq. 7,weseethatβ =φ/2,halfthe
Blochphase ofthe symmetricdouble cell, while Z =zeα odic but not reflection symmetric. They exist in two
forms depending on the sign of α, (or what is the same
is the corresponding velocity parameter. Interchanging
thing, whether the firstwellon the left is of wide ornar-
the half-cells is equivalent to reversing the sign of α,
rowtype). Wewilldiscusstheirsurprisingpropertiesbe-
whichmeasuresthe degreeofasymmetryofthe half-cell,
low;fornowweconcentrateonthecaseN =2nwhichare
but leaving β and z unchanged. Incrementing the num-
symmetric biperiodic systems involving n double-cells.
ber of half-cells leads to the following rule:
eα cos3β (1/z)sin3β
W(3) = WLW = zsin3β e−αcos3β IV. TRANSMISSION IN SYMMETRIC
(cid:18)− (cid:19)
cos4β (e−α/z)sin4β BIPERIODIC SYSTEMS
W(4) = W2 = (13)
zeαsin4β cos4β ···
(cid:18)− (cid:19) The transmission probabilities shown in Fig. 2 were
calculated for the system described in the first paper of
2.5 Coquelin et al. [1]. We took into account the variable
∼Z 2
Z2 effectivemassandothermaterialpropertiesasin[24,25,
2 -30 g(x)
g’(x) 26]. Specifically, the barrier height is 288.09 meV, and
1.5 30 uu’((xx)) the effective masses m∗ are approximately 0.074 (well)
1 and 0.080 (barrier).
∼22 Z, Z 0 .05 A B C D tThheReettwsouotlatdlsewsvihidcotewhsnowfiintthhFe‘iwaglnslo.wwn2ewd(’baa)annaddns‘dnisw(tbnhw)encs’oarmwreeesllpfoaorrnrbdaoyttsho.
cases because the Bloch phase of the double cell is the
-0.5
same for both orderings[24]. What is different between
-1
the two cases is the impedance parameter µ = µ ;
wn nw
6
-1.5 this accounts for the different results obtained. In Fig.
0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05
a) x = kd/π 2(a) the lower allowed band runs from 92 to 97.5 meV.
The long dashed line is the envelope of minimum trans-
2.5
∼ZZ 22 mission, the curve 1/cosh2µ: see eq. 1. The presence of
2 -g(x) the transparent state with µ = 0 just inside the allowed
g’(x)
1.5 u’(x) u ( x ) band,pushestheenvelopeofminimauptounity. Thisis
the causeofthe thirdresonancesitting closetothe band
1
∼22Z, Z 0.5 A B C D eednvgeel;oiptsew(didatshheids dlirnive)enisbcyotnhteineunevdelaocpreososftmheinfiomrba.idTdhene
0 band, (where it has other significance,) but it is seen to
decay slowly across the forbidden zone, and is very low
-0.5
in the second allowed band. In panel (b), the order of
-1 the half-cells is reversed, and the extra state occurs in
the upper band.
-1.5
0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 Itisimpressivethatevenasmalldeparturefromstrict
b) x = kd/π
periodicity has such a largeeffect onthe band structure,
opening a sizeable gap from 98. to 112. meV, while nar-
FIG. 6: (Colour online) Pole region of Z2 and Z˜2, and wave rowing the allowed band width as compared to the case
functions g(x),g′(x) and u(x),u′(x); (a) case of 10−5 asym- of all wide or all narrow wells. The difference between
metry; (b) moderate (0.10) asymmetry. Lines identified at the3.8and4.3nmwidthsisonly13%. Sincecosφandµ
upperright. are single-cell properties, increasing the number of cells
has no effect on the band structure; it simply squeezes
The number (3)canbe replacedbyany oddinteger,and more resonances into the bands. In a forbidden zone,
(4) by any even integer. When the index is even the φ pπ +iθ. Since cosh3θ increases rapidly above 98
→
6
5
1
4
0.8
3
2 |t| N0.6 µ 2
0.4
1
0.2 -∞ -∞
0
0 90 95 100 105 110 115 120
90 95 100 105 110 115 120 E ( meV )
a) E ( meV )
1 FIG. 8: (Colour online) Impedance parameter µ of the3-cell
arrays of Fig. 7: solid (red) line, zero asymmetry; short-
0.8 dashed(blue)line,smallasymmetry;long-dashed(green)line,
opposite asymmetry. The arrows imply divergenceto −∞.
2| N0.6
|t the outer band edges are little shifted.
0.4
The transparentstate clearlydevelops fromthe Bragg
state of the symmetric double-cell, at E = 104.7 meV.
B
0.2 The double cell has strong barriers, so the energies of
its quasibound states 1/L2 are determined primarily
090 95 100 105 110 115 120 by the well width [27].∼When asymmetry is introduced,
b) E ( meV ) 2L=d s, in the notation of eq. 11. The energy differ-
±
ence of the two cases, wide or narrow well, is therefore
1 of order (s/d)EB 0.125EB = 13 meV, in Fig. 2. This
∼
agreeswellwiththelocationsofthetransparentstatesat
97.4and110.5meV,when2d=8.1nmand2s=0.5nm.
0.8
In Fig. 7, 2s = 0.1 nm and the splitting is 2.6 meV. In
2| N0.6 astattigehotf-boinnedionfgthmeosdpelli,ttbhaendtrsa,ninspwahreicnht ostnalytethiseawnideedgoer
|t only the narrow wells are occupied at resonance, which
0.4
is consistent with the quasibound state picture.
In Fig. 8, the impedance parameters µ of the three
0.2
situations are drawn. The red (solid) line exhibits typ-
ical behaviour [22] with a divergence of µ at the band
0
90 95 100 105 110 115 120 edges. In blue (dotted) the band splitting has caused µ
c) E ( meV ) todescendsteeplythroughzeroatthe transparentstate,
beforedivergingto atthebandgapedge,102.5meV.
−∞
In the forbidden band, µ ξ +iπ/2; ξ rises from
FIG. 7: Transmission of AlGaAs-barrier 3-cell biperiodic → −∞
arrays: (a) zero asymmetry; (b) 2(a − c) = 0.1 nm; (c) and diverges to +∞ at the upper edge of the band gap.
2(a−c)=−0.1 nm. Lines as in Fig. 2. The behaviour in the upper allowed band reverts to the
typical one. In green (dashed), the negative divergence
of µ is transferred to the lower edge of the upper band.
The subsequent rise in the allowed band produces the
meV,the transmissiondoes cutoffsharplyinthe forbid- transparent state and the consequent large values of the
den zone, even for just three double-cells. envelopeofminimumtransmission,justabovethatband
Further insight is gained by looking at a sequence of edge.
models very close to the symmetric (a = c) limit. In Inref. [22]weinterpretedthetransfermatrixasamap-
Fig. 7(a) we start with a simply periodic system using ping of the system point in the complex plane, or as a
wells of the average width 4.05 nm. Then we made a hyperbolic rotationof a Dirac spinor arounda fixed axis
small excursion into asymmetry by using 2a = 4.1 and whose orientation is determined by µ. The axis passes
2c = 4.0 nm for the wide and narrow wells respectively. through a fixed point at distance tanhµ/2 from the ori-
In Fig. 7(b) it is the wnwnw configuration and in panel gin. In the first allowed band of a symmetric system, as
(c)thenwnwn. Thegapinducedisabout2.6meV,while a function of energy, the fixed points start and end at
7
z =+1atthebandedges,movingontherealaxis. Then we get a lower bound on the inverse transmission prob-
in the forbiddenband, two fixed points movearoundthe ability, and from the third line an upper bound. That
unit circle in complex conjugate positions, from +1 to is,
1. In the ensuing allowed band the motion starts and
− 1 1
ends at z = 1. The motion which arises here is quite t 2 . (18)
different, in t−hat (in Fig. 8, blue line) the interior fixed cosh2α ≥ | N| ≥ cosh2η
point begins at +1 at the lower edge of the first allowed
When η < α, the upper and lower bounds are ex-
band,butdoesnotreturnto+1atthebandedge;rather | | | |
changed. Atapointwhereη =α, (whichimplies µ=0),
it remains on the realaxis and moves through the origin
the bounds cross,and t 2 is caughtbetweenthem. For
to reach 1 at the lower edge of the gap. Its passage | N|
− a symmetric half-cell, α = 0, so the upper bound on
through the origin produces the transparent point. In
transmissionrevertstounity,andη playsthe roleofµin
thebandgap,thefixedpointsdomoveontheunitcircle,
producingthe envelopeofminimum transmissionfor the
initially very quickly to reach i and then more slowly
± double cell.
reaching+1attheloweredgeofthesecondallowedband.
When η > α, transmissionreachesthe upper bound
The fixed points then move on the real axis in the nor- | | | |
when Nβ = pπ, and the lower when Nβ = (m+0.5)π.
malmanner,startingandendingat+1intheuppersplit
(p and m integers.) In the opposite case, η < α, the
band. Thefixedpointsmustreach 1atabandedge,so | | | |
± bounds are reversed, and the upper bound is reached
this is perhaps the only way that a new forbidden band
at Nβ = (m+0.5)π. At the “transparent” point where
can be inserted, without disturbing higher bands.
η = α,thetransmissionispinchedbetweenthebounds
|an|d ta|k|es the value cosh−2α.
V. TRANSMISSION FOR ODD N SYSTEMS
1
Now we return to the the case of an odd number
N =2n+1ofhalf-cells. Sincethehalf-cellalreadyencap-
0.8
sulates the band structure of the symmetric double-cell,
ictonmsiigdhetrisnegemthethaadtdtihtieorne iosfnaontheixntgrafufratchteorrtWoLleaarsninfroemq. 2| N 0.6
13. Surprisingly, it changes everything. When we form |t
0.4
the M-matrix from W , we obtain the elements
L
0.2
M = coshα cosβ icoshη sinβ
11
−
M = sinhα cosβ isinhη sinβ
21 0
∗ − ∗ 95 100 105 110 115
M22 = M11; M12 =M21; where a) E ( meV )
′
1 g 0.14
ImM = νu+
21
− 2 ν ⇒
(cid:18) (cid:19) 0.12
1 ν z
sinhη . (15)
≡ 2 z − ν 0.1
(cid:16) (cid:17)
1)βFolreaavniyngodthdenruemstbuerncohfacnegllesd,.simItpislyeraespillyaccehβec→ked(2tnha+t 2| N 0.08
|t 0.06
det M = 1, and putting the extra half-cell on the left,
rather than the right, only reverses the sign of α. 0.04
In comparison, for the double-cell one has
0.02
1 ν Z 1 ν zeα
sinhµ = . (16) 0
≡ 2 Z − ν 2 eαz − ν 96.8 97 97.2 97.4 97.6 97.8
(cid:18) (cid:19) (cid:18) (cid:19) b) E ( meV )
It follows that µ=η α.
−
Using 1/t =M of eq. 15 we have
N 11 FIG. 9: (Colour online) Transmission of 7 half-cell array as
solid (red) line: (a) showing both split bands; (b) detail of
1
= cosh2α cos2Nβ+cosh2η sin2Nβ region around the transparent state. Dash-dot (turquoise)
tN 2 lineisenvelopeofmaxima; dotted(mauve)lineisenvelopeof
| |
= cosh2α+[cosh2η cosh2α] sin2Nβ minima up to thetransparent point.
−
= cosh2η [cosh2η cosh2α] cos2Nβ .(17)
− − Some results are presented in Fig. 9, for the same po-
Suppose that η > α, making the factor in square tentialcellusedinFig. 2,appropriateforGaAs/AlGaAs
| | | |
brackets positive. Then from the second line in eq. 17 superlattices. To start we take N = 7 half-cells. Panel
8
(a) shows the transmission in both parts of the split asymmetrycauseseachallowedbandtosplitattheBragg
bands, with three resonances in each. One sees that energywherecosφ =0. ExtendingShockley’slineofar-
h
the envelope of maxima, 1/cosh2α crosses the envelope gument for a generic single-barrier cell, we have proved
of minima, 1/cosh2η, at the transparent point µ = 0. that this induces a transparentstate which lies in one of
From there to the band edge, they exchange their roles the split bands, and very close to the band edge. The
as upper/lower bounds. The upper bound reduces the rule is that the transparent state lies in the lower band
maximumtransmissionbyalargeamountincomparison when the wide well is first in line for incident electrons.
to the situation in Fig. 2, which is for six half-cells. In The transparent state is a resonance which occurs at a
the forbidden band, it can be shown that what was the fixed energy, independent of the number of cells in the
envelope of mimina becomes an upper bound on trans- array;otherwise the transmissionfollows the well known
mission, while the lower bound is zero. The cross-over rule expressed in eq. 1.
at the transparent point ensures that these bounds are For an odd number of half cells, the picture is com-
continuous. pletely different. These systems are asymmetric, so if
Fig. 9(b) shows detail near the transparent point. there is a wide well first on the left, there will be a
Where the bounds cross, tN 2 is pinched between them. narrow well first from the right. The asymmetry due
| |
To the left of the transparent point (which it no longer to the additional half-cell causes an envelope of maxi-
is!), the maxima occur when Nφ = pπ, with integer p. mum transmissionto appear, which crossesthe envelope
Fromthe thirdline of eq. 17,any maxima thatoccur af- of minimum transmission at the “transparent” point, in
terthetransparentpointsatisfyNφm =(m+0.5)π,with both split bands. The transmission probability is given
integer m < n. Such a situation must occur for a large by eq. 17 and is bounded on both sides as in eq. 18. At
enough N, because the energy of the transparent point the transparent point α = η, the bounds cross, and the
isfixed,whilehavingmorecellssqueezesadditionalreso- transmissionprobabilityispinchedbetweenthem. Reso-
nancesintoeachallowedband. InFig. 10weshowdetail nances occurring between the transparent state and the
of the region near the lower edge of the upper band, for band edge satisfy a different rule, Nφ = (m+1/2)π,
m
N =35 half-cells. Here one of the resonancesclearly lies not the Nφ =pπ which would follow from eq. 1.
p
between the band edge and the transparent state. In a separate article we will discuss these systems in
Innumerable papers have been written on symmetric a tight binding model, showing what can and cannot be
periodic systems, for which the relation eq. 1 applies, reproducedinthatapproximation. Forexample,intight
andtheirresonancesalwaysinvolveperfecttransmission. binding, the transparent state occurs exactly at a band
Therefore it is a surprise to see how adding a half-cell edge, rather than just inside.
introduces a non-trivial upper bound on transmission,
greatly reducing it in the neighbourhood of the “trans-
parent” state.
Acknowledgments
0.12
We are grateful to NSERC-Canada for support un-
0.1 der Discovery Grants RGPIN-3198 (DWLS), SAPIN-
8672(WvD)andaSummerResearchAwardthroughRe-
0.08 deemer UniversityCollege (LWAV); alsoto DGES-Spain
forcontinuedsupportthroughgrantsFIS2004-03156and
2| N 0.06 FIS2006-10268-C03(JM).
|t
0.04
APPENDIX A: TRANSFER MATRICES
0.02
0 For completeness, we write down our conventions
110.4 110.6 110.8 111 111.2
E ( meV ) for transfer matrices in the presence of an energy and
position-dependent effective mass. The Schr¨odinger
equation is
FIG. 10: (Colour online) Transmission of a 35 half-cell array
showingdetailinregionofthetransparentstate,intheupper ∗
d 1 dψ 2mm
split band. Lines as in Fig. 9. m∗ = [E V(x)]ψ(x)
− dx m∗ dx ¯h2 −
(cid:20) (cid:21)
q2(x)ψ(x) . (A1)
≡
VI. CONCLUSION Thedimensionlesseffectivemass(inunitsofthefreeelec-
∗
tron mass m) is m ; q is the wavenumber inside the po-
Wehavestudiedtransmissionthroughbiperiodicsemi- tentialregion,and q(x) k in the exteriorregionwhere
∗ →
conductorsuperlattices. Foranevennumberofhalf-cells, both V(x) and m become constant. (In this paper, the
9
substrate and cap of the heterostructure are GaAs, the One can show as in [4] that M is related to W by
same as the wells of the superlattice.)
If g(x) and u(x) are two independent solutions, then M = L−L1W−1LR, where
1/√ν 1/√ν
¯h du dg L = i√ν i√ν
[g u ]=constant (A2)
mm∗ dx − dx (cid:18) − (cid:19)
L−1 = 1 √ν −i/√ν . (A7)
where the constant may be set equal to one by choice of 2 √ν +i/√ν
(cid:18) (cid:19)
the boundary condition at some initial point, say x=0.
′ For a double cell as in eq. 4, this leads to
For example, take g(0) = 1, g (0) = 0, u(0) = 0 and
∗
[du/dx](0)=mm /¯h.
The transfer matrix W relates values of a “spinor” c˜ M−d,d =L−1W−−d1,dL=
whose components are ψ(x) and (h¯/mm∗)(dψ/dx) be- cosφ i(νug u′g′/ν) i(νug+u′g′/ν)
tween two points. i−(νug+u−′g′/ν) cosφ+i(νug u′g′/ν)
(cid:18) − − (cid:19)
(A8)
ψ(d)
c˜(d)= =W c˜(0)=
(cid:18) mh¯m∗ddψx (d)(cid:19) 0,d and 2cosφ=(ug′+gu′). With bias, the ν in the imagi-
g(d(cid:2)) (cid:3) u(d) ψ(0) nary parts is replaced by √νLνR, and the real parts be-
(A. 3)
(cid:18) mh¯m∗ddxg (d) mh¯m∗ddux (d)(cid:19) (cid:18) mh¯m∗ddψx (0)(cid:19) come( νL/νR± νR/νL)/2timescosφ,forthediagonal
and off-diagonal elements respectively.
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) p p
The Wronskian relation gives detW = 1, and in an al-
lowed band, TrW =2cosφ, where φ is the Bloch phase.
It is easily shown that APPENDIX B: KARD REPRESENTATION IN A
FORBIDDEN BAND
W2 = 2cosφW I,
− ⇒
sin(Nφ) sin(N 1)φ The parameterizationgivenin eqs. 12 or 13 is valid in
WN = W − I . (A4) allallowedbands,ifwepermitallrealvaluesforβ andz.
sinφ − sinφ
Inforbiddenbandsmattersarealittlemorecomplicated.
The band structure associated with the potential is all Wediscussherethebehaviourofthehalf-cellmatrixWL,
contained in the W-matrix for a single cell. If we agree in relation to the bands of the multi-cell system.
that the prime symbol on the wave function means to We have in mind a half-cell of type well-barrier-well,
take the derivative, and then multiply by h¯/(mm∗), all and the exterior energy is measured from the well-
the above equations reduce to the standard form for the bottom. At zeroenergy one is in a forbidden zone which
casewithconstanteffectivemass,inwhichm∗ 1. (i.e. we will label (FZ0). All four elements of WL are posi-
the effective mass can be absorbed into m, or→one may tive, as are the ratios g′/g and u′/u. The allowed zone
′
use an averagevalue 0.071 and leave it explicit.) (AZ0) begins when g becomes negative. The parame-
∼ ters α, β, z are all positive in AZ , with β in the first
In discussing transmission properties it is more con- 0
quadrant. If we change β iβ¯ and z iz¯ in FZ , the
venient to represent the wave function in terms of plane 0
→ →
correct signs and magnitudes are obtained: see line one
wavestates,normalizedtounitflux. Totheleft(x<x )
L
of eq. B1.
and right (x>x ) of the potential we write:
R
The band AZ ends when g also becomes negative, so
0
ΨL(x) = aL eikL(x−xL)+ bL e−ikL(x−xL) itnhatturthneelnodgsdwerhievnatuiv′ebseacroembeosthnengeagtaivteivea,nidntFhZe1s.ecTohnids
√νL √νL ′
allowed band AZ begins. Eventually g again becomes
1
a b
ΨR(x) = R eikR(x−xR)+ R e−ikR(x−xR)(A5) positive, and FZ2 occurs, now with both log-derivatives
√νR √νR negative. The signs are summarized in Table I, for ener-
gies up to 305 meV for a GaAs/AlGaAs SL. In AZ , the
∗ 1
wheretheexternalvelocityisνL,R =h¯kL,R/(mm ),with diagonal elements are negative, which is accommodated
kL = 2mm∗[E V(xL)]/¯h, and similarly for kR. By by placing β in the secondquadrant. Sinceu is still pos-
−
definition the transfer matrix M relates the plane wave itive, z remains positive. Similarly in AZ , placing β in
p 3
coefficients across the system the third quadrant replicates the signs.
Inthe forbiddenzoneFZ ,allfourelementsofW are
0 L
aL =M aR (A6) positive. In FZ1, both g and g′ are negative. Because
bL bR β =π/2 at the band edge, the diagonal elements (cosβ)
(cid:18) (cid:19) (cid:18) (cid:19)
aresmallerinmagnitudethantheoff-diagonal(sinβ),so
Different wave numbers at left and right allow for bias cosβ sinhβ¯ for continuity. The cosh and sinh must
across the potential [28], but here we will not consider chang→e places. This is accomplished by β π/2+iβ¯,
→
that situation further, so that ν =ν =ν. α α+iπ/2, and z unchanged. The phase β increases
L R
→
10
monotonicallythroughallowedzones,andits realpartis In FZ ,
0
stationaryin forbiddenzones. When α acquiresa phase,
z doesnot. ThefirstfewlinesofTableIsummarizethese M = coshα coshβ¯ isinhη¯sinhβ¯
11
adjustments. −
Similarly,inFZ3,theprescriptionβ 3π/2+iβ¯,α M21 =−sinhα coshβ¯−icoshη¯sinhβ¯ where
→ → 1 ν z¯
α+iπ/2givesthe correctsigns. There is ample scope to
sinhη¯= ,
manoeuvre things to fit whatever pattern of signs arises, 2 z¯ − ν
but we doubt that there is any general prescription that 1 = cosh2α+ (cid:16)cosh2α(cid:17)+sinh2η¯ sinh2Nβ¯ . (B3)
will fit every possible potential. t 2
| N| (cid:20) (cid:21)
In FZ the upper limit on transmission is given by
TABLE I: Signs of elements of the transfer matrix WL, and 0
required adjustments of the parameters. (A blank means no 1/cosh2α, as in an allowed band. Also, for large N,
adjustment.) For real β, thequadrant is shown. tN 2 approaches zero, as for periodic systems. In the
| |
next forbidden zone FZ ,
1
′ ′ α
Band u u g g e z β M = sinhα sinhβ¯ icoshη¯coshβ¯
11
FZ0 + + + + iz¯ iβ¯ M = −coshα sinhβ¯− isinhη¯coshβ¯
21
AZ0 + + − + I 1 − −
FZ1 + + − − α+iπ/2 z π/2+iβ¯ t 2 = cosh2η¯+
N
AZ1 − + − − II | |
FZ2 − + + − −iz¯ π+iβ¯ sinh2α+cosh2η¯ sinh2Nβ¯ . (B4)
AZ2 − − + − III (cid:20) (cid:21)
FZ3 + − + − α+iπ/2 −z 3π/2+iβ¯ The last line of B4 shows that an upper limit on the
transmissionprobabilityis givenby 1/cosh2η¯, butthere
Explicit forms for the first four forbidden bands are is no lower limit. In Figs. 9 and 10, it can be see that
the upper limit in the forbidden zone is the continuation
eαcoshβ¯ (1/z¯)sinhβ¯ u′ u of cosh−2η from the adjacent allowed zone.
W (FZ ) = =
L 0 z¯sinhβ¯ e−αcoshβ¯! g′ g! The roles of α and η¯ are exchanged between eqs. B3
and B4. This pattern applies to all FZ with even and
eαsinhβ¯ (1/z¯)coshβ¯
W (FZ ) = odd indexed subscripts.
L 1 z¯coshβ¯ e−αsinhβ¯!
− − In FZ2, the expressionsfor M11 and M21 are the com-
eαcoshβ¯ (1/z¯)sinhβ¯ plex conjugates of those in eq. B3, and in FZ3 they are
WL(FZ2) = −z¯sinhβ¯ e−αcoshβ¯! complex conjugates of those in eq. B4. This pattern
− appears to persist in going to higher bands, but may de-
eαsinhβ¯ ( 1/z¯)coshβ¯ pendonthedetailedformofthepotential. Theforbidden
WL(FZ3) = z¯coshβ¯ −−e−αsinhβ¯ ! . (B1) zdouneetsowthitehneeveedntoanedxchodandgleatbheelsrohlaevseofdciffoesrheβn¯tanpdatstienrhnsβ¯,
in the latter case.
Transforming from W to M (for the half-cell),
L L
The main virtue of the Kardparameterizationis that,
1 g+u′ i(νu g′/ν) g u′+i(νu+g′/ν) inallowedbands, it makesthe relationsbetween the sin-
ML = 2 g−u′−−i(νu−+g′/ν) g−+u′+i(νu−g′/ν)! gcolenvaenndiemntuilntipfolerbciedldlesnysbtaemndss,vebruytowbivthiousus.ffiKcieanrdtciasrleesist
(B2) offers the same advantages.
[1] M. Coquelin, C. Pacher, M. Kast, G. Strasser and E. Albuquerque and M.G. Cottam, Elsevier, (Amsterdam,
Gornik, “Transport studies on doubly periodic superlat- 2004), ISBN = 044516271.
tices by hot electron spectroscopy”. Contribution P-154 [4] D.W.L. Sprung, Hua Wu and J. Martorell, “Scattering
toInt.Conf.Phys.ofSemiconductors,Edinburgh(2002). by a finite periodic potential”, Am. J. Phys. 61 (1993)
PublishedonCDinIOPConferenceSeries,Vol.171,eds. 1118-24.
A R Long and J H Davies, (2003) ISBN:-7503-0924-5 [5] M. Cvetiˇc and L. Piˇcman, “Scattering states for a finite
[2] M. Coquelin, C. Pacher, M. Kast, G. Strasser and E. chain in one dimension”, J. Phys. A: Mathematical and
Gornik, “Wannier-Stark level anti-crossing in biperiodic General, 14 (1981) 379-382.
superlattices”, Physica Stat. Sol. B 243 (2006) 3692-5. [6] M. Pacheco and F. Claro, “Simple results for one-
[3] Polaritons in periodic and quasiperiodic structures, E.L. dimensional potential models”, Phys. Stat. Sol. B, 114