Electron transmission and phase time in semiconductor superlattices

0.28 MB·English

by  D. W. L. Sprung

Electron transmission and phase time in semiconductor superlattices
D. W. L. Sprunga, W. van Dijka,b and C. N. Veenstraa,b,c
aDepartment of Physics and Astronomy, McMaster University, Hamilton ON, L8S 4M1
bRedeemer University College, Ancaster ON L9K 1J4, Canada and
cDepartment of Physics & Astronomy, University of Britush Columbia, Vancouver BC, V6T 1Z1
J. Martorell
Departament d’Estructura i Constituents de la Materia,
Facultat F´ısica, University of Barcelona
Barcelona 08028, Spain
(Dated: February 2, 2008)
We discuss the time spent by an electron propagating through a finite periodic system such as
8
a semiconductor superlattice. The relation between dwell-time and phase-time is outlined. The
0
envelopes of phase-time at maximum and minimum transmission are derived, and it is shown that
0
thepeaksandvalleysofphase-timecanbewelldescribedbyparametersfittedattheextrema. Fora
2
many-periodsystem thiscoversmost of theallowed band. Comparison ismadetodirect numerical
n solutions of the time-dependent Schr¨odinger equation by Veenstra et al. {cond-mat/0411118} who
a
compared systems with and without addition of an anti-reflection coating (ARC). With an ARC,
J
the time delay is consistent with propagation at the Bloch velocity of the periodic system, which
3 significantly reducesthe time delay,in addition toincreasing the transmissivity.

] PACSnumbers: 03.65.Xp,73.63.-b,05.60.Cg
h
p
-
t I. INTRODUCTION dependent effective mass:
n
a d dψ(x) 2m∗(x)
u The question of “how long does it take for a non- m∗(x)dx m∗(x)dx + ¯h2 [E−V(x)]ψ(x)=0
q (cid:20) (cid:21)
[ relativistic particle to cross a barrier”, or in general 2m∗(x)
any potential, is a contentious one in quantum physics k2(x)= [E V(x)]
¯h2 −
1 [1, 2, 3]. There is for example the Hartmann effect
v [4], according to which a transmitted particle might be Ψ(x,t)=e−iEt/h¯ ψ(x) . (1)
6
found before it has reached the potential. Since 1990
The approximations leading to eq.1 are fully explained
7
the subject has taken on renewed interest in the con-
5 in Bastard’s monograph [17]. At a layer boundary, ψ(x)
0 text of electrons in semiconductor superlattices (SL) and ψ′ (h¯/m∗)dψ/dx are continuous, to conserve flux.
. [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. It is com- Itiscon≡venienttomatchψ(x)toplanewaves,normalized
1
monly assumedthat suchpropagationis ballistic, with a
0 to unit flux, at the boundaries of the unit cell a<x<b
mean free path larger than the device dimensions. But
8 [19]
0 because electrons can scatter from lattice vibrations, it
c d
: is of interest to know how long the electron is exposed ψ (x) = L eikL(x−a)+ L e−ikL(x−a) , x<a,
v to such interactions. Many theories have been put for- L √vL √vL
i
X ward as to how the time of passage should be defined c d
ψ (x) = R eikR(x−b)+ R e−ikR(x−b) , x>b(.2)
ar aanredthmeebaseustrerde;coagmnoiznegd.thVeemensdtwraelel-ttiaml.e[1a8n]dstpuhdaiesde-ttiimmee R √vR √vR
dependence of propagation by direct numerical solution Ateitherend,(B =L, R)kB isthewavenumberoutside
ofthetime-dependentSchr¨odingerequation(TDSE),us- the periodic system, and vB = h¯kB/m∗B is the velocity.
ing gaussian incident wave packets. Their results agreed We will consider systems without bias, so kL = kR = k,
well with phase-time. In this contribution, based on a etc.,butitwillbeconvenienttoretaintheindicesinorder
talk given at Theory-Canada3 in June 2007, we explain to know where various terms come from. The transfer
why phase-time should be applicable in the context of matrix relates the coefficients on either side:
semiconductor superlattices. c M M c
L = 11 12 R where
d M M d
L 21 22 R
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)
1/t r∗/t∗
M = (3)
r/t 1/t∗
II. TRANSFER MATRIX METHOD (cid:18) (cid:19)
in terms of the reflection and transmission amplitudes
r(k), t(k). It contains all the information about scatter-
Assuming ballistic transport, a conduction band elec-
ing from that cell. One can show [20] that
tron in a potential cell of arbitrary shape on a < x <
b satisfies the Schr¨odinger equation with a material- Mσ M† = σ c2 d2 =constant . (4)
z z
⇒| | −| |
2
which says that the probability flux is preserved by the composed of cells, each described by M. Mathemati-
action of M, and which implies detM =1. cally, M is a hyperbolic rotation of a two-dimensional
It is convenient to use Kard’s parameterization[21] of Dirac spinor, about an invariant axis whose polar angles
M. In an allowed band we write are (µ, χ) [20].
The usefulness of the transfer matrix for a periodic
M11 = cosφ−isinφ coshµ=1/t=M2∗2 system arises from the property
M = ieiχ sinφ sinhµ=r/t=M∗ . (5)
21 − 12 M(N) = MN(φ,µ,χ)=M(Nφ,µ,χ)
⇒
At a given energy, the Bloch phase is determined by 1
= cosNφ isinNφcoshµ . (8)
TrM = 2cosφ; the impedance parameter from the ratio t −
N
of M /ImM = tanhµ, and the asymmetry parame-
| 21| 11 Hence the transmission probability for N-cells
terχfromthephaseofM /M =exp(2iχ). (χ=0for
21 12
a reflection-symmetric potential cells. For the most part t 2 = [1+sin2Nφ sinh2µ]−1 1/cosh2µ (9)
N
| | ≥
thispaperisrestrictedtothesymmetriccase.) Forasim-
ple square barrier-cell,eµ is the ratio of averagevelocity shows narrow peaks determined by Nφ(E) = mπ, m =
1,2, N 1. Between these peaks the t 2 touches
outsidetoinside. Acrossanallowedband,φincreasesby N
the e·n·v·elo−pe of minima, 1/cosh2µ. At the| ce|nter of an
π, while µ is quite constant across most of the band, di-
allowed band, the envelope of minima crosses the curve
vergingatthebandedges. Anexampleofthis behaviour
t 2.
is showninFig. 1. takenfrom[18]. Panel(a)showcosφ 1
| |
In forbidden bands, the Bloch phase φ pπ + iθ
for a square barrier cell, and for a phase-shift equivalent
→
acquires an imaginary part; as a result the transmis-
gaussian barrier cell. Panel (b) shows the corresponding
sion goes rapidly to zero. Pacher et al. [22] used this
values for µ. The dotted lines in each panel are the pa-
property to design an electron band-pass filter. Further,
rametersofthesingle-layerARCcell. Theslopeofcosφ
A
by adding a quarter-wavecell (anti-reflectioncoating, or
is abouthalfthatofcosφ, while their Braggpoints coin-
ARC) at each end of the periodic array [23, 24], they
cide. The rule of thumb for ARC’s in optics wouldmake
wereabletoincreasetheaveragetransmissionwithinthe
µ =µ/2.
A
band from about 25% to about 75%. In a series of pa-
a) b) pers [20, 25, 26, 27], some of us showed how to design
anARC whichgivesoptimaltransmissionwithinagiven
2 5
miniband, by adding suitably configured potential cells
1.5
1 4 on each end of a periodic array. Without an ARC, elec-
cos o / -00..055 /u 23 tarsoinnsewqh.i9ch,saoreonteraenxspmeictttsedadsiognsioficvaiantntairmroewdreelasyo.naWncitehs
-1 1 an ARC, the incident plane wave is transformed into a
-1.5
Bloch wave of the periodic potential, and travels at the
-240 45 50 55 E (6 m0eV) 65 70 75 80 040 45 50 55 E (6m0eV) 65 70 75 80 Bloch velocity. This should reduce the transit time.
FIG. 1: Comparison of (a) Bloch phases of a square barrier
cell (dash line) and a gaussian cell (solid line); also shown III. TIME DELAY
is cosφA of a single-cell ARC (dotted line). (b) same for
impedance parameters µ, µA. A. Dwell time and phase time
In the Kardrepresentation,transfer matrix factorizes:
As already mentioned, the subject of time-delay in
scattering is controversial. Razavy’s book [3] is a useful
M =cosφ isinφ coshµ+sinhµ cosχσx+sinχσy σzintroduction. Nussenzweig [5, 6] has argued that dwell-
−
=e−i(χ/2)σz e(µ/2)σ(cid:2)x e−iφσz e−(µ/2(cid:0))σx ei(χ/2)σz (cid:1)i(6)time is the best founded, for description of wave packet
scattering. First we explain two of these theories.
Classically,
The matrix of eigenstates is easily seen to be
dx dx
U =e−i(χ/2)σze(µ/2)σx dt= = . (10)
v(x) ¯hk(x)/m
Substitution into eq. 6 gives
For a plane wave normalized to unit flux, as in eq. 2,
MU = U e−iφσz, m∗ 1
ψ (x,E)2 = = . (11)
e−iχ/2coshµ/2 e−iχ/2sinhµ/2 | 0 | ¯hk vcl
U = (7)
eiχ/2sinhµ/2 eiχ/2coshµ/2
Dwell-time delay is defined to be the difference
(cid:18) (cid:19)
Theeigenvaluesaree−iφ ande+iφ. Physically,theeigen- xR
τ = ψ(x,E)2 ψ (x,E)2 dx . (12)
states are the Bloch waves of an infinite periodic array D | | −| 0 |
ZxL
(cid:2) (cid:3)
3
Asymptotically, the scattering wave function, for waves problem (incident waves from left, or right). The S-
incident from the left, is usually written matrix is symmetric and unitary, and may be written
1 r t reiδ teiη
ψ(x,E) = √vL eikLx+re−ikLx , x<xL S =(cid:18)t r¯(cid:19)=(cid:18)||t||eiη −|r||e|i(2η−δ)(cid:19) (17)
1 (cid:2) (cid:3)
= teikRx , x>x , where r¯= r∗t/t∗ is the reflection amplitude for waves
√vR R incident fro−m the right. r2+ t2 =1.
| | | |
where t= teiη , r = reiδ (13) Smith [29] defined the time-delay matrix for a many-
| | | | channel system as
define the scattering phase shifts.
dS
Following Smith [29], the Schr¨odinger equation gives τ i¯hS† = i¯hS†
the identity ≡− dE − ×
(r′+irδ′)eiδ (t′+itη′)eiη
∂ψ (|t|′+i|t|η′)eiη (r′+|i|r(2η|′| δ′))ei(2η−δ)
ψ =(H E)
(cid:18) | | | | − | | | | − (cid:19)
− ∂E ⇒
(18)
¯h2 ∂ ∂ ∂ψ ∂ψ ∂ψ∗
ψ∗ψ = Re ψ∗ ,
−2m ∂x ∂x ∂E − ∂E ∂x whereprimesmeanderivativewithrespecttoenergy. Af-
(cid:26) (cid:20) (cid:18) (cid:19) (cid:21)(cid:27) ter some work we have
xR ¯h2 ∂ ∂ψ ∂ψ ∂ψ∗ xR
ψ∗ψdx= Re ψ∗ dη dδ
ZxL −2m (cid:20) ∂x(cid:18)∂E(cid:19)− ∂E ∂x (cid:21) (cid:12)(cid:12)xL τ11 = +h¯ |t|2dE +|r|2dE
((cid:12)14) (cid:18) (cid:19)
(cid:12) ∂ r dη dδ
τ = i¯hei(η−δ) tan−1(| |) +h¯r∗t
For a potential on a < x < b, xL < a and xR > b 12 − ∂E t dE − dE
(cid:18) | | (cid:19) (cid:18) (cid:19)
are positions at which a position measurement could be
dη dη dδ
carried out. Taking only the scattering wave function in τ22 = +h¯ + r2 . (19)
dE | | dE − dE
eq. 12, (cid:18) (cid:18) (cid:19)(cid:19)
In the case of a reflection symmetric potential, r¯= r,
¯h2 ∂ ∂ψ ∂ψ ∂ψ∗ x ∂η
Re ψ∗ = R +h¯ t2whichrequiresη =δ+π/2,sotheirderivativesareequal.
− 2m (cid:20) ∂x(cid:18)∂E(cid:19)− ∂E ∂x (cid:21)xR (cid:18)vR ∂E(cid:19)| |As a result, all elements τij are real, and the diagonal
¯h2 ∂ ∂ψ ∂ψ ∂ψ∗ elements are equal. For a reflection-symmetric potential
Re ψ∗ = we have the compact result
−2m ∂x ∂E − ∂E ∂x
(cid:20) (cid:18) (cid:19) (cid:21)xL
dη
xvL(1+|r|2)−¯h∂∂Eδ |r|2+ m¯hk|r2| sin(2kLxL−δ) . τ11 = h¯ dE =τ22
L L ∂
(15) τ12 = h¯ sin−1 r =h¯(r′/t) . (20)
∂E | | | | | |
Using eqs. 15 and 14 in eq. 12 gives the dwell-time as Thediagonalelements(cid:0)arecalled(cid:1)the“phase-timedelay”.
They agree with the first term in the top line of eq. 16.
∂η ∂δ ¯hr
τ = h¯ t2+ r2 | |sin(2k x δ) But due to coupling, the time associated with a process
D L L
∂E| | ∂E| | − 2E − depends on just which mixture of the two channels is
(cid:18) (cid:19)
x x 2x involved. The difference is that eq. 16 was derived as-
+ R L t2 L r2 . (16)
suming that one has specified the channel with incident
v − v | | − v | |
(cid:18) R L(cid:19) (cid:18) L (cid:19) waves from the left, ignoring the other channel. The os-
Sincethepotentialliesbetweenthelimits,wecanassume cillatoryterm of eq. 16 has been subject of muchdebate
that x is a negative distance, while x is positive. The also. Obviously it arises from interference between the
L R
terms on the second line have an obvious interpretation incident and reflected waves. Winful [31] intepreted this
as the “free passage time”, so the top line is the “dwell- term as the time taken to cross a distance of the or-
time delay”. But be careful: Nussenzweig [5] has a lucid der of the scattering length. On a more practical note,
discussion of the case of an incident wave-packet, and if x , and one averages k over a wave packet
L L
→ −∞
finds an additional effect which arises from the uncer- narrow in energy, this oscillatory term is exponentially
tainty principle: you cannot localize a quantum particle small. Therefore in a practical sense, either dwell-time
in less than a de Brogliewavelength. This effect cancels or phase-time are equivalent for SL scattering, at least
out between the free and interacting situations, so it af- for symmetric potentials.
fects dwell-time delay, but not dwell-time. We skip over Eq. 20 shows that there are two situations where the
this complication and rely on eq. 16. channelcouplingvanishes: atamaximumorminimumof
Phase time was introduced by Wigner and Eisenbud transmission. InaSL,thenumberofmaximaisN 1,so
−
[28],andsimilarlyrelatestimedelaytotherateofchange as N increases,these points of uncoupling become closer
ofthescatteringphase. Scatteringin1Disatwo-channel together.
4
B. A tale of two phases behind Nφ, then it has to catch up at the zeroes Nφ .
m
This is nicely illustrated in Fig. 5. The steeper slope
In the transfer matrix method, the wave functions are of /ηN near φm makes for a longer time delay at those
defined with a different phase than in usual scattering energies.
theory: weresetthephasetozerooneachside(x=a, b)
ofthe potentialarray,ratherthanatanarbitraryorigin.
B. Phase time near the maxima of transmission
That makes M translation invariant, so MN describes a
periodic system without having to include a shift in the
origin. In an infinite periodic array the electron would move
Adopting Nussenzweig’s notation [ψ ; ψ ] for the attheBlochvelocity. Letthetimetocrossacellofwidth
L R
asymptotic wave function to left and right [5], we com- d be τBl. Write φ/d=κ, the pseudo-momentum.
pare our ψ(x) with the usual conventionψ˜(x) as follows: ∂E 1∂E ¯h2κ ¯h ¯h
= = = v =
∂φ d ∂κ dm∗ d Bl τ
ψ(x) [eik(x−a)+re−ik(x−a); teik(x−b)] Bl
∼ ∂φ
ψ˜(x) [eikx+r˜e−ikx; t˜eikx ] ¯h = τBl . (25)
∼ ∂E
eikaψ(x) [eikx+re−ik(x−2a); teik(x−w)] ,(21) In the following, the prime will mean ∂/∂E. We can
∼
show using eqs. 23 and 24 that
where w = b a is the total width of the potential. It
follows that t−he phase η of our transmission amplitude ∂ηN [Nφ′ coshµ+sin2Nφ sinhµ µ′/2]
=
is related to the usual phase by η˜ = η kw. Then the ∂E 1+sinh2µ sin2Nφ
−
standard phase-time delay is [1+sin2Nφ tanhµ (µ′/(2Nφ′)]
τ = Nτ co(cid:2)shµ (cid:3)
dη˜ dη dk dη w ph Bl 1+sinh2µ sin2Nφ
τ =h¯ = h¯ w =h¯ .(22)
ph dE dE − dE dE − v (26)
cl (cid:2) (cid:3)
h i h i
Away from resonance, the denominator is a factor of
We conclude thatthe phaseη ofour transfermatrixam-
t 2 which cuts offthe phase time very sharply,causing
plitudes gives phase time, not time-delay, because the N
| |
it to mimic the shape of the transmission curve. Near a
free-passage time has to be subtracted from it.
resonance, φ mπ/N +ε, so that
∼
( )msinNφ sinNε N(E E )φ′ .
IV. PHASE TIME FOR SUPERLATTICE − ∼ ∼ − m m
TRANSMISSION This allows us to approximate tN 2 as a Breit-Wigner
resonance, with half-width Γ =| 2/|(Nsinhµ φ′ ):
m m m
A. Relation of scattering phases to Kard −1
E E
parameters t 2 = 1+( − m)2 . (27)
| N|m Γ /2
(cid:20) m (cid:21)
Similarly, in the same vicinity,
TheS-matrixandthetransfermatrixcontainthescat-
tering information in different forms. To see how phase- ∂η
N
τ = h¯ Nτ coshµ
time applies to a SL, we examine the relation between ph,m Bl,m m
∂E ∼ ×
the two descriptions. Equating (cid:12)m
(cid:12)
1 1 × 1+2bm(cid:12)(cid:12)(EΓ−/E2m) 1+(EΓ−/E2m)2
=cosNφ isinNφ coshµ e−iηN (23) (cid:20) m (cid:21).(cid:20) m (cid:21)
tN − ≡ |tN| with 2b = 1 2µ′ +cothµ φ′m′
m Nφ′ coshµ m mφ′
we find the following relations between the two sets of m m (cid:20) m(cid:21)
parameters: (28)
This is called a Fano resonance shape [30]. From eq.
cosη = t cosNφ ; sinη = t sinNφcoshµ
N | N| N | N| 26, we see that the locus of phase time at transmission
tanη tanη
N maxima is
tanη =tanNφcoshµ ; =coshµ= ,
N
tanNφ tanφ
τ =Nτ coshµ (29)
ph,max Bl
(24)
Similarly, at transmission minima, sin22Nφ = 1; the
whereη =η1isthephaseshiftforasinglecell. Inthefirst denominator of τph (eq. 26) becomes cosh2µ, giving a
allowed band, 0 < φ < π; tanη varies smoothly; coshµ downside locus of
divergesatthe bandedge,like 1/sinφ. Zeroesandpoles
τ =Nτ /coshµ (30)
oftanη coincidewiththoseoftanNφ. Thesepointsare ph,min Bl
N
the scatteringresonancesNφ =mπ andthe minima of for phase time at transmission minima. The Bloch time
m
transmission Nφ =(p+1/2)π. Near the poles, η lags for N cells is the geometric mean of the two loci.
p N
5
C. Phase time near the minima 2 cos φ
2φ/π
1.5 2η/π
Transmissionminima occur at φ =(p+1/2)π/N,for
p
wpr=itte1n, 2, ···n− 2. Close by, the denominator may be φos 1
c
∼t−N2co=shc2oµshp21µ(+1µ−′pttaannhh2µµp(cδoEsp2)N2φ)1−(Nφ′p tanhµpδEp)2 φπηπ/, /, 0.50
(cid:2) µ′ δE (cid:3)2 (cid:2) δE 2 (cid:3) -0.5
=cosh2µ 1+ p p 1 p (31)
p(cid:20) Nφ′p (cid:18)Γp/2(cid:19)(cid:21) " −(cid:18)Γp/2(cid:19) # -1
50 55 60 65 70 75
where δE =E E . The width E (meV)
p p
−
Γ =2/[Nφ′ tanhµ ] (32) FIG. 2: Play model: cosφ, and angles φ, and η, in units
p p p
of π/2. Remarkably, φ is reasonably linear over much of the
is large compared to the widths Γm of the resonances. band.
1
∂η
N
τ =h¯
ph,p
∂E p
(cid:12) 0.8
(cid:12) 1+C E−Ep
∼ cNosτhBµl,p (cid:12) h p(cid:16) Γp/2 (cid:17)i 2 on 0.6
p 1+2Dp EΓ−p/E2p +(Dp2−1) EΓ−p/E2p missi
(cid:20) (cid:16) (cid:17) (cid:16) (cid:17)(3(cid:21)3) Trans 0.4

φ′′ Γ µ′ 0.2 |t|2
whereC = p p and D = p . (34) |tN|2
p φ′ 2 p Nφ′ envelope
p p 0
50 55 60 65 70 75
Both C and D are of order 1/N. The prefactor is the E (meV)
p p
locus of time-delay at minima:
FIG. 3: Play model: Transmission probability for one cell,
Nτ N¯h dφ dcosφ ninecells, and envelopeof transmission minima.
Bl
= = N¯h /ImM (35)
11
coshµ coshµdE − dE
which involves only well-behaved single-cell quantities. betweenE =50and75meV.Inaddition, t2isspecified.
| |
Veenstraetal. [18]comparedtheircomputedtime de- All other parameters follow from these two. With such
lay to phase-time delay, and found good agreement. In a model it is easy to see how changing some parameter
particular they found that the locus of maxima and eqs. will affect the results.
27, 28 accounted for the overall picture. In an interest-
ingpaper,Pacher,BoxleitnerandGornik[14]pointedout cosφ = 0.08(62.5 E), 50 E 75 (meV),
− ≤ ≤
that all theories of time-delay agree at the transmission t−2 = 1+160/E . (36)
maxima, so they concentrated their attention on those | |
points and found the locus of maxima. The main differ- The phase η is determined by cosη = t cosφ, and the
encebetweentheirworkandthe presentoneis thatthey impedance parameter follows from | |
spoke of the mean velocity for an electron traversingthe
SL, rather than the time. Further they expressed their coshµ = sinη/(t sinφ) . (37)
| |
results in terms of the real and imaginary parts of the
transfer matrix elements M , e.g. eq. 35, rather than It diverges at each band edge, since t sinη is a smooth
ij
| |
the Kard parameters. It is our opinion that the Kard function of energy, while φ runs from pπ to (p+1)π.
parameters, given their simple behaviour, make the ex- In Fig. 2 we show the play model cosφ = λ(E
B
−
pressions more easily understandable. E), and the corresponding phases φ and η. η is quite
linear, crossing φ at the Bragg point. In this model,
φ′ =λ/sinφ, with λ=0.08 meV−1.
D. Play Model In Fig. 3 we show the play model transmission for
9 cells. The envelope of transmission minima is simply
Toillustratethe aboveresults,wetakeasimplemodel 1/cosh2µ, which givesdirectphysicalsignificanceto the
inwhichwespecify cosφtobelinearinenergyinaband impedance parameter.
6
2.5 1
τ
locus max τ
2.5 T
2 locus min τ on 0.8
ati
ps) xim
τme ( 1.5 appro 0.6
ase ti 1 B-W 0.4
Ph nd
a
0.5 N0.2
T T
resonance approNx
0 0
50 55 60 65 70 75 50 55 60 65 70 75
E ( meV ) E (meV)
FIG. 4: Play model: phase time, and loci of phase time at FIG.6: Transmissionprobabilitycomparedtoresonanceap-
maximaandminimaoftransmission. Transmission probabil- proximation (eq. 27).
ity is shown for orientation.
2.5
τ
4.45 ηN /π Nφ /π x. (ps) 2.0 Fanlooc suhsa apte m ata xmimaxa.
o
φπN / 3.35 no appr 1.5
π/ and N2.25 τ and Fa 1.0
ηPhases 1.15 ase time 0.5
h
P
0.5
0
50 55 60 65 70 75
0
50 52 54 56 58 60 62 E (meV)
E (meV)
FIG. 7: Play model: phase time, and its approximation by
resonance formula eq. 28. Locus of phase time at maximum
FIG. 5: Play model: ηN and Nφ, in units of π. Note that transmission is shown to demonstrate that phase-time max-
slope of ηN is largest at integer multiples of π. For clarity, ima are very little shifted from thelocus.
only half theallowed band is shown.
E. Realistic potential model
In Fig. 4 we show the play model phase-time, along
with the envelopes of maxima and minima. In the back-
Some results using a realistic semiconductor potential
groundfor orientationarethe transmissionpeaks,which
of Pacher and Gornik [23, 24] are shown in Figs. 8, 9.
line up well with the maxima of phase-time.
Here we took five potential cells, giving four resonances
InFig. 5we showthe danceofthe Blochphase forN- in the band, a number chosen for comparison with the
cells, and the corresponding transmission phase ηN, in calculationsofVeenstraetal. [18]. Fig. 8correspondsto
the lower half of the allowed band. The curve in the up- Fig. 4forthephase-time. Forthispotentialtheenvelope
per half of the band is a double reflection of this, ending of minima is lowermeaninga larger µ. Also shownis the
at 9π. Exceptat the band edges, the lines crossat every Bloch time, the geometric mean of the two envelopes.
half-integer multiple of π. Since η is catching up at in-
N Fig. 9 can be compared with Fig. 7. The Fano for-
teger multiples of π, the steeper slope leads to a longer
mula fitted at the maxima of transmission agrees with
phase-time at the transmission resonances.
the exact result over two standard deviations. In Fig. 9
In Fig. 6 we show (solid line) the transmission and we have included eq. 33, fitted at the minima of trans-
(dashed line) the Breit-Wigner resonance fitted at the mission. The fit to the minima is not so good, largely
peaks. This is truncated at two standarddeviations; the because the half-width at minima is so muchlargerthan
horizontal lines simply connect the B-W curves between at maxima. Still, we can say that between them the ap-
successive peaks. The agreement is excellent. proximations reproduce the phase time over about 90%
Fig. 7issimilartoFig. 6,butforthephase-time. The of the band. If there were more layers, the peaks and
Fano-shape formula fitted to the maxima also does an valleys would be narrower and the agreement would im-
excellent job of reproducing the exact calculation. prove, as in Fig. 7. For large N the phase time can be
7
4 1.6 1
right scale
left scale
3.5 1.4
3 1.2 0.75
τ and (ps) Nph12..255 Time Delay (ps) ..681 0.5Transmission
T 1 0.4 0.25
0.2
0.5
0 0
0 45 50 55 En6e0rgy (m6e5V) 70 75 80
50 55 60 65 70
E ( meV )
FIG. 10: Time-dependent numerical solution of wave equa-
FIG. 8: Pacher-Gornik 5-cell array: phase time (solid line),
tion for a Pacher-Gornik 5-cell array plus ARC: phase-time
alongwiththelociatmaximaandminima(longdashes),and
Bloch time (dash - dot), which is their geometric mean. The comparedtoBlochtime. Transmissionshowninbackground.
chain-line is four times the transmission probability.
V. CONCLUSION
2
Wehaveoutlinedthetransfermatrixmethodfortrans-
ps) mission of electrons in a one-dimensional AlGaAs/GaAs
xns. ( 1.5 scuompepralartetdicteh,eustiwnogmthoestKcaormdmpaornalmyeutseerdiztahtieoonr.iesWfeorththene
o
ppr 1 averagetimespentincrossingapotentialregion,namely
a
o phase time and dwell time. These were applied to SL
n
Fa transmission, first for a play model and then for the po-
d
n 0.5 tential that corresponds to experiments of Pacher and
a
τ ph Gornik. We noted that the phase-time is well defined

both at maxima and minima of transmission. In the
0
50 55 60 65 70 neighbourhood of these points, the transmission can be
E ( meV ) describedbyaBreit-Wignerresonanceformula,withthe
parameters extracted at the extrema. The same holds
FIG. 9: Pacher-Gornik 5-cell array: along with the Fano- for the phase-time, except it is a Fano shape resonance.
type approximation at each resonant peak and valley. The
Thefitsatmaximaaregoodovertwohalf-widths,andat
fit at the peaks is excellent over two half-widths, but not so
the minima overone. Taken together,this coversalmost
good at thephase-timeminima. Thelong dashanddash-dot
all the band width, for a system with more than a very
lines are the loci of phase-time at maximum and minimum
few periods.
transmission.
As the number of periods N of the SL is increased,
thewidthsofthepeaksandvalleysdecrease,andtheap-
proximate forms become more and more accurate. This
explainsthesuccessofphase-timefordescribingthetime
calculated reliably using only properties at the extrema,
spent in traversing a SL.
where the various theories of time-delay agree with each
other. The locus of phase-time at maxima was derived in
Veenstra et al. [18], and by Pacher et al. [14]. The
Finally, in Fig. 10 we show the results calculated by locus of phase-time at mimina is new. Their geometric
Veenstra et al. for the Pacher five cell array, plus a two average is the Bloch time for traversing a SL. An ARC
layer ARC. The dotted line shows the transmission is worksbyconvertingtheincidentplanewaveintoaBloch
closeto100%overmostofthebandwidth. Therearetwo wave of the periodic system. Veenstra et al. [18] studied
lines shown for the Bloch time. One takes into account the time delay for the SL plus ARC by direct numerical
only the periodic 5-cell system, and the other includes solution of the TDSE using gaussian wave packets. In
a correction for the ARC layers on each end. The two Fig. 10, taken from that reference, it can be seen that
estimates are close together. The time delay extracted indeed the time-delay with ARC agrees quite well with
from the TDSE calculations is indeed very close to the the Blochtime. The ARCnotonlyincreasesthe average
Bloch time, which is what we expect from the argument transmission, but it smooths out the dwell time, remov-
that the ARC converts the incident plane wave into a ingthepeaksandvalleysassociatedwiththeexponential
Bloch state. decay of the resonances.
8
Acknowledgments aSummerResearchAwardthroughRedeemerUniversity
College (CNV); and to DGES-Spain for continued sup-
We are grateful to NSERC-Canada for Discovery portthroughgrantFIS2006-10268-C03-01(JM).Wealso
Grants SAPIN-8672 (WvD), RGPIN-3198 (DWLS) and thank Gigi Wong for assistance in redrawing Fig. 10.
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