Dynamics of electronic transport in a semiconductor superlattice with a shunting side layer

0.82 MB·English

by  Huidong Xu

Dynamics of electronic transport in a semiconductor superlattice with a shunting side
layer
Huidong Xu,1 Andreas Amann,2 Eckehard Scho¨ll,3 and Stephen W. Teitsworth1
1Duke University, Department of Physics, Box 90305, Durham, NC 27708-0305, USA
2Tyndall National Institute, Lee Maltings, Cork, Ireland
3Institut fu¨r Theoretische Physik, Technische Universita¨t Berlin, Hardenbergstraße 36, 10623, Berlin, Germany
(Dated: January 12, 2009)
Westudyamodeldescribingelectronictransportinaweakly-coupledsemiconductorsuperlattice
9
with a shunting side layer. Key parameters include the lateral size of the superlattice, the con-
0
nectivity between the quantum wells of the superlattice and the shunt layer, and the conduction
0
properties of the shunt layer. For a superlattice with small lateral extent and high quality shunt,
2
static electric field domains are suppressed and a spatially-uniform field configuration is predicted

n tobestable,aresultthatmaybeusefulforproposeddevicessuchasasuperlattice-basedTeraHertz
a (THz) oscillators. As the lateral size of the superlattice increases, the uniform field configuration
J
loses its stability to either static or dynamic field domains, regardless of shunt properties. A lower

2 qualityshuntgenerallyleadstoregularandchaoticcurrentoscillationsandcomplexspatio-temporal
1 dynamicsinthefieldprofile. Bifurcationsseparatingstaticanddynamicbehaviorsarecharacterized
and found to be dependenton theshuntproperties.
]
i
c PACSnumbers:
s
-
rl I. INTRODUCTION x=0 -x
t
m (s)
j
km−1→m
. Theoretical work by Esaki and Tsu1 in 1970 was the
t
a first to propose a Bloch oscillator based on a superlat-
m tice (SL) structure. In that paper, they derived current- j(s)
⊥m
- voltage (I-V) characteristics of a SL which showed neg- l n(ms)
d ative differential conductivity (NDC) associated with
n Bloch oscillations2,3 of the miniband electrons under a
o
DC bias. However, direct observation of Bloch oscil-
c
(s)
[ lations is difficult due to decoherence caused by elec- j
km→m+1
tron scattering. In other important early work, Kti- z δ L -
1 torov,SiminandSindalovskii4predictedanegativehigh- ? x x E
v
frequencydifferentialconductivity andassociatedampli-
2 FIG. 1: Schematic of the shunted SL. The growth direction
7 fication of high frequency signals thereby suggesting an isalongthez directionandthequantumwells areparallel to
6 alternative means of THz oscillation. This dynamic con- the x direction. The SL is located at x>0 and the shunt is
1 ductivityremainsnegativeuptotheBlochfrequencyωB at x< 0. The thick line on the right is the potential energy
1. and reachesa resonanceminimum at a frequency closely of an electron in theconduction band of theSL.
0 below ωB, suggestingthat the SL may serveas anactive
9 medium for THz radiation.
0 However, no such devices have been realized to date tum cascade laser devices have been shown to oper-
:
v morethanthreedecadesaftertheirproposalbecausethe ate in the THz range for temperatures up to 164 K.11
i NDCcausesspace-chargeinstability. AlthoughBlochos- On the other hand, if superlattice-based Bloch oscilla-
X
cillations have been observedexperimentally in undoped tors could be successfully realized they might be ex-
ar SLs5 by studying optical dephasing of Wannier-Stark pected to have certain advantages relative to the quan-
ladder6excitationsusingdegeneratefour-wavemixing,7,8 tum cascade structures.12 Recently, rapid progress in
the gain achievable is too small to build electrically ac- THz technology13 including biomedical sensing, three-
tive Bloch oscillators. For high current densities, the dimensional imaging and chemical agent detection has
space-chargeinstability causes moving charge accumula- attracted renewed attention to Bloch oscillators. Some
tion layers (CALs) and charge depletion layers (CDLs) structureshavebeenproposedtostabilizethefieldinthe
andthus the SLexhibits oscillationssimilartothe Gunn SL against NDC-related instabilities. One scheme theo-
effect.9 Whiledevicesbasedontheseoscillationsmayop- retically proposed by Hyart et al.14 is the dc-ac-driven
erate in the microwave range, they do not extend to the SL which requires the presence of an initial THz pump.
THz region.10 The SL is biased in the NDC regionunder a DC electric
The lack of suitable THz radiation sources and detec- field,initially superposedwithanACpump electricfield
tors hampers the technological exploitation of the fre- which stabilizes the field distribution.15 Then the initial
quencyregimespanningfrom300GHzto10THz. Quan- pump field can be gradually turned off when THz oscil-
2
3 0.02 ns 0.02ns,max(Jx)=5.14×109A/m2 (a) 5 F1 3
(cid:0)(cid:9)
n()ND 12 610 V/m) 34 (cid:27)(cid:27)Fn01 122..55 nsity n (N)D
0 0.2 ns 0.2ns,max(Jx)=4.12×108A/m2 Field ( 2 1 arge de
1 6 0.5 ch
n2
0 0
00 1100 2200 3300 4400
SL period
(b)20
20 6 ns 6ns,max(Jx)=1.79×108A/m2 (F0,Jk0→1) Jk1→2(F,n1,n2)
15 @R Jk2→3(F,n2,n3)
m) 2m) Jk(F,ND,ND)
x(µ 40 A/10 Jk0→1(F)=σF
1
0 J (
20 40 5 (F2,Jk2→3)
SLperiod A (cid:0)(cid:9)(F1,Jk1→2)
AU
FIG.2: Chargedensityplots(leftcolumn)andcurrentvector 00 2 4 6 8 10
plots of j⊥m(x) (right column) for a SL with Lx = 20 µm, F(106 V/m)
U =2.1Vandσ=0.04(Ωm)−1 at0.02, 0.2and6ns. Initial
conditionisaCALatthecenteroftheSL.Theshuntisatthe
FIG. 3: The steady state for a SL with Lx = 20 µm, U =
bottom. The color bar on the left of the first contour plot is 2.1 V and σ = 0.04 (Ωm)−1 at x = Lx: (a) field profile
thescaleencodinginunitsofND usedthroughoutthepaper. (solid line) and charge density (dashed line), (b) The solid
dotsindicatetheactualcurrentoperation pointson thelocal
vertical current field characteristics jkm→m+1(F,nm,nm+1).
lation has been already established in the SL. Another
suggestion is to stack a few short SLs, where domains
are not able to form.16 These short SLs are separated inSLstructuresboththeoreticallyandexperimentally.20
by heavily doped material, and an increase in terahertz In similar work by Feil et al.,21 a side layer is grown
transmission at dc bias has been observed. onthecleavededgeofalightlydopedGaAs/AlGaAsSL,
Yet anotherschemeis to openashunting channelpar- such that a 2D electron gas is formed at the interface
alleltotheSL,similartoamethodthathasbeenusedto betweenthe SL andthe side layer. The lightly dopedSL
stabilize tunnel diode circuits.17,18 Daniel et al.19 used a servestwopurposes: (i)toprovideamodulatedpotential
distributed nonlinear circuit model to simulate the elec- forthe2Delectrongasattheinterfacesothatunderthis
tric field domain suppression in a SL. They have shown periodicpotential,theelectrongasbecomesasurface SL
that the shunt is able to suppress the voltage inhomo- withonelateraldimension;(ii)toprovideauniformfield
geneity above a critical bias voltage which depends on tothissurfaceSLsincealightlydopedSLcanmaintaina
the shunt width, the SL width, and the shunt resistiv- uniform field under externalbias. While the suppression
ity. However, the circuit model does not include aspects of field instabilities has been reportedin this type of SL,
of the electronic tunneling transportthat appear to play it is still not clear whether this lateral structure will be
an important role in SL behavior. The model possesses useful as a THz oscillator.
only a global coupling since the elements are connected In this paper, we study an extension of a well-
in series and the I V characteristic of each element is established model of electronic transport in weakly-
−
fixed. Ontheotherhand,theSLmodelhasamorecom- coupled superlattices by adding a shunting side layer.
plexstructurethathasbothaglobalcouplingduetothe Our treatment includes the effect of lateral electronic
applied voltage constraint as well as a nearest neighbor (i.e., horizontal) transport within each of the quantum
coupling arising from the varying charge densities that well layers. Here, the vertical electron dynamics is
dynamically change the local current density vs. field associated with sequential resonant tunneling between
(J F)characteristics. As aresult,the nonlinearcircuit weakly-coupled quantum well layers, rather than mini-
−
model of Daniel et al. is not able to exhibit connected band transport or Wannier-Stark hopping as occurs for
fielddomainsorcurrentself-oscillationsthatareobserved strongly-coupledSLs. AlthoughtheBlochoscillatorgen-
3
(a)160 (d)640 1.4
µm) µm) 2m)1.3
x ( x ( A/
4
0
1
0 20 40 0 20 40 nt (1.2
SL period SL period e
r
r
u
(b)160 (e)640 L c1.1
S
m) m)
µx ( µx ( 1 1 2 3 4 5 6
time (µs)
0 0 1755ns 1969ns 2211ns 2313ns
0 20 40 0 20 40 1.28
SL period SL period
m)
(c) (f) m
6 3 3 (
6Field (10 V/m)24 12charge density n (N)D 6Field (10 V/m)246 12charge density n (N)D xFIG0. 6: Same as Fig. 5, butwith Lx =1.28 mm.
000 1100 2200 3300 44000 000 1100 2200 3300 44000 tricfieldprofileinashuntedSL.InsectionIII,wediscuss
SL period SL period
theextremelydifferenttimescalesinvolvedinthismodel,
whicharechallengesto numericallysolvingit. Insection
FIG. 4: Steady states: (a), (d) Charge density plots, (b),(e)
current vector plots and (c), (f) field profile (solid line) and IV, we numerically explore the effect of a high quality
chargedensity(dashed line) at x=Lx forLx =160 µm (left shuntonthedynamicsofSLsasthelateralsizeoftheSL
column) and Lx =640 µm (right column), respectively, with is varied, and show that the uniform field configuration
U =2.1 V and σ=0.04 (Ωm)−1. is stable, provided that the shunt and shunt connection
have high enough quality and the SL lateral extent is
1.2 nottoo great. In sectionV, we choosea laterallynarrow
SL andstudy the dependence of the SL dynamics onthe
2)1.18 shunt properties. The transition from a stable uniform
m
field configuration to static field domains is found to be
A/
40 1.16 complex and the bifurcations involved in this transition
1 are discussed. The Appendix presents details of the nu-
nt ( merical methods employed.
e1.14
r
r
u
c
SL 1.12 II. LATERALLY EXTENDED MODEL OF THE
SUPERLATTICE WITH SHUNT LAYER
1.1
0.5 1 1.5 2
time (µs)
Weakly-coupledsemiconductorsuperlatticeshavebeen
1016ns 1044ns 1081ns 1126ns successfully described by the sequentialresonanttunnel-
0.8
ing model overthe past severalyears.20,22,23,24 However,
m) previousworksusually consideronlythe dynamicsalong
m
x( 0 the growth (vertical) direction of the SL and ignore the
dynamics in the in-plane (lateral) direction, i.e., treat
FIG. 5: SL current density JSL/Lx and snapshots of charge each period as an infinitely large plane with uniform
density distribution for Lx = 0.8 mm, U = 2.1 V and σ = chargedensity. More recently, Amann et al.25 developed
0.04(Ωm)−1. Thetimesofthesnapshotsaremarkedassolid atheoreticalframeworkwhichdescribesbothlateraland
circles in theupperpanel.
verticalelectronicdynamics. Here,weextendthisframe-
work to include the effects of a shunting side layer.
The structure of the shunted SL is shown in Fig. 1.
erallyrequiresastrongly-coupledSL,theweakly-coupled Each quantum well forms a slab that is parallel to the
SL has similar NDC features in I-V characteristics and x y plane, with cross sectional dimensions L and L .
x y
similar current self-oscillations occur due to recycling of Th−ere are N such quantum wells stacked on top of each
electronic fronts.12,22 other in the z direction, sandwiched between an emitter
In the next section, we establish a two-dimensional layer and a collector layer. The shunt layer is located
modelfordescribingthecurrentflowanddynamicalelec- between δ x 0, with thickness δ . The SL period
x x
− ≤ ≤
4
isl =w+d,wherewanddarethewidthofthequantum with the two-dimensional density of states ρ =
0
well and width of the barrier, respectively. The external m∗/(π¯h2), where m∗ is the electron effective mass. Here
voltage is applied in the z direction, across the emitter we assume that µ and D are fixed.
0
and the collector. Both the lateral and vertical currents depend on the
Inside the SL, the electrons are localized within one electricalfieldswhichinturndependonthescalarpoten-
quantum well due to the relatively thick quantum bar- tialφ (x,y). ThepotentialcanbesolvedbythePoisson
m
riers. Furthermore, the electrons are assumed to be at equation
local equilibrium and the local two-dimensional charge
e
density at time t is denoted by n (x,y,t), where m is ∆φ (x,y)=(∆ +∆ )φ (x,y)= (n N ),
m m ⊥ k m −lǫ ǫ m− D
the well index, x, y are the in-plane coordinates. The r 0
(7)
charge continuity equation in the SL can be written as:
with
en˙ (x,y,t)=j j j , (1)
m km−1→m− km→m+1−∇⊥· ⊥m ∂2
∆ φ (x) = φ (x), (8)
where ⊥ m ∂x2 m
φ (x) 2φ (x)+φ (x)
∂ ∂ ∆ φ (x) = m−1 − m m+1 , (9)
⊥ =ex +ey , (2) k m l2
∇ ∂x ∂y
whereǫ andǫ aretherelativeandabsolutepermittivity,
andj denotesthethreedimensionalverticalcur- r 0
km−1→m respectively. Then the field can be calculated as
rentin z directiontunneling througheachbarrier(units:
[A/m2]) and j is the lateral two-dimensional current
⊥m φm+1(x) φm(x)
density(units: [A/m]). Theelectronchargeise<0. The Fkm(x,y) = l− ,
y-dependence is ignoredandEq.(1) canbe rewrittenas:
∂φ (x)
m
F (x) = . (10)
∂j⊥m(x) ⊥m − ∂x
en˙ (x,t)=j j . (3)
m km−1→m − km→m+1 − ∂x
Here we solve the Poissonequation using an approxima-
The localverticaltunneling currentj through tion method assuming that the typical structures in the
km→m+1
each barrier is described by the sequential resonant tun- lateraldirectionvaryona length scalemuchlongerthan
neling model which has been derived using different the mean free path of the degenerate electrons.25
methods;20,22,24 in this paper, we have used the same The drift-diffusion dynamics of the shunting layer is
form as in Refs. 20,24. This tunneling current depends similar to that of the lateral dynamics within each SL
on the electric field F (x) across the barrier through quantum well. First, we neglect x-dependence in the
km
which the tunneling occurs and the electron charge den- shunt, that is, the shunt is collapsed into a single layer
sities n (x) and n (x) in the neighboring quantum alongthez-direction. NotealsothatunliketheSL,which
m−1 m
wells ofthis barrier. Thus, the tunneling currenthas the possesses an intrinsic discreteness along z direction, the
functional form: shunt is a continuous layer. Therefore, we make a fur-
ther approximationthat the shunt is divided into blocks
j (x)= j [F (x),n (x),n (x)].
km−1→m km−1→m km m−1 m aligned with the periods of the SL and that the charge
(4)
density is locally uniform within each block. This as-
The tunneling current densities through the emitter
sumption not only provides the discretization required
and collector layers are modeled by Ohmic bound-
by numerical simulation, but also matches the dynamics
ary conditions,25 that is, j (x) = σF (x), and
k0→1 k0 oftheshuntwiththatoftheSL.Withthesetwoassump-
j (x) = σF (x)n /N , with contact conduc-
kN→N+1 kN N D tions, we can write down the continuity equation in the
tivity σ and two-dimensionaldoping density N in each
D m-th shunt block as follows:
well.
Thelateraldynamicsiscausedbythein-planecurrent en˜˙(s)(t)δ lL = j(s) δ L j(s) δ L ˜j(s) lL ,
j which consists of a drift part and a diffusion part. m · x y km−1→m· x y− km→m+1· x y− ⊥m· y
⊥m (11)
When the y-dependence is ignored, this becomes
where the superscript (s) denotes the quantities in the
∂n shuntandthetildedenotesthatthequantitiesarethree-
m
j (x)= eµn F eD (5)
⊥m m ⊥m 0 dimensional, i.e.,
− − ∂x
where F⊥m(x) is the in-plane component of the electric n(s) =n˜(s) l; j(s) =˜j(s) l. (12)
field at x in well m, µ is the mobility and D is the dif- m m · ⊥m ⊥m·
0
fusioncoefficient. The generalizedEinsteinrelation26 es- Here, the quantity j(s) denotes the lateral current that
tablishes the connection between µ and D for arbitrary ⊥m
0 flows between the shunt and the SL through their inter-
two-dimensional electron densities including the degen-
face. Then we can write Eq. (11) in the form:
erate regime:
D0(nm)= −eρ0(1−exp[n−mnm/(ρ0kBT)])µ (6) en˙(ms)(t)= jk(sm)−1→m − jk(sm)→m+1 − j⊥δ(sxm) , (13)
5
Note that the vertical current in the shunt has a very SLcancausebandbending effectsatthe interface. Even
different form than the tunneling current in the SL. It iftheshuntisdopedtohavethesameFermilevelasthat
follows a similar dynamics as the in-plane current in the in the SL so that little band bending might be expected,
SLquantumwellsandisrelatedtothethree-dimensional thereareotherissuesthatimpactthe connectionquality
charge density in the shunt: betweenthe shuntandthe SL,for example,the presence
oftrapstatesorathinoxidelayer. Toquantifythequal-
∂n˜(s) ity of the connection between the SL and the shunt, we
j(s) = eµn˜(s)F(s) eD m . (14)
km−1→m − m km − 0 ∂z introduce a parameter 0 a 1 such that a = 1 corre-
≤ ≤
sponds to a perfect connectionand a=0 correspondsto
Here we assume the mobility µ and the diffusion coeffi- no connection. We modify Eq. (15) to be
cient D have the same values as in the SL.
0
Next,weexaminethelateralcurrentthatconnectsthe
shunt and the quantum well layer within the SL: j⊥(sm) =a·(cid:18)−eµnm(x=0)F⊥m−D0∇⊥nm(cid:12) (cid:19).
(cid:12)x=0+
(cid:12) (19)
j⊥(sm) =−eµnm(x=0)F⊥m−D0∇⊥nm(cid:12) . (15) The relationship between specific values of pa(cid:12)rameter a
(cid:12)x=0+ and microscopic models of conduction across the shunt-
(cid:12)
(cid:12) SL interface are discussed elsewhere.27
In this equation, the boundary should be defined at
x = 0+ for calculation of both the current and the po- Similarly, we introduce a separate parameter b > 0
tential in the shunt. Since the shunt is assumed to be thatallowsustomodeltheeffectofhavingdifferentdop-
uniform in x direction, defining the above equation at ingdensityand/ormobilityintheshuntvs. SLquantum
x = 0− implies that F and n(s) are zero which wells. Also recognize that the field in the shunt is al-
wouldleadtozerobound⊥amrycurre∇n⊥t. Amnotheradvantage most uniform and n(ms) ≈ ND(s) when the conductance in
ofchoosingthe boundary atx=0+ is that the potential the shunt is high, where N(s) is the doping density in
D
in the shunt should be equal to the potential in the SL the shunt. This leads to the following modification of
close to its boundary, i.e., φ(s)(x < 0) = φ (x = 0+), Eq. (14),
m m
since the potential is continuous everywhere. This rela-
tion allows us to equate the potential in the shunt with ∂n˜(s)
that at the inner boundary of the SL. So the potential jk(sm)−1→m =−eµn˜(ms)Fk(ms)−eD0 ∂zm ≈−ebµ(s)N˜DFk(ms),
at the boundary of the solution of Eq.(7) is just the po- (20)
tential in the shunt. The fields required to calculate the where bµN˜ = µ(s)N˜(s). Note that b > 1 when the
current in Eq. (15) can be obtained by D D
doping density in the shunt is greater than that in the
quantum wells and b is much less than one when the
φ(s) (x) φ(s)(x)
F(s)(x) = m+1 − m , shunt is weakly conducting so that only a small fraction
km l of the total vertical current flows through it.
F (0+) = φ (x) . (16) It is also useful to point out that the total current,
⊥m −∇⊥ m (cid:12)
(cid:12)x=0+
(cid:12) Lx
Thechargedensityanditsnormalgra(cid:12)dientatthebound- J = ǫ ǫ F˙(s)+j(s) δ + ǫ ǫ F˙ +j dx,
ary are (cid:16) r 0 km km→m+1(cid:17)· x Z0 (cid:16) r 0 km km→m+1(cid:17)
(21)
is the same for each period. To show this, note that the
n (0+)+n(s)(0−)
n (x=0) = m m , (17) Poissonequation can be written as
m
2
n (∆x) n(s) e
∇⊥nm(cid:12)(cid:12)x=0+ = ∆xli→m0+ m ∆x− m . (18) ∇·(F⊥+Fk)= lǫrǫ0(nm−ND), (22)
(cid:12)
(cid:12) or
Here we also note the possible effects of energy band
structureoftheshuntedSLandthedopingdensityinthe
F F ∂F e
shunt. In the above discussion, the situation has been km− km−1 + ⊥ = (n N ). (23)
m D
simplified because no band bending is included. How- l ∂x lǫrǫ0 −
ever, variations in doping densities in the shunt and the
Substituting the above equation into Eq. (3) yields
d F F ∂F ∂j (x)
lǫ ǫ km− km−1 + ⊥ =j j ⊥m . (24)
r 0dt(cid:18) l ∂x (cid:19) km−1→m − km→m+1 − ∂x
6
Then, one integrates both sides of the preceeding equation with respect to x from δ to L . Due to the vanishing
x x
−
boundary conditions F ( δ )=F (L )=0 and j ( δ )=j (L )=0, the lateral terms in the above equation
⊥ x ⊥ x ⊥m x ⊥m x
− −
integrate to zero. This yields
d Lx Lx d Lx Lx
ǫ ǫ F dx+ j dx=ǫ ǫ F dx+ j dx. (25)
r 0dtZ km Z km→m+1 r 0dtZ km−1 Z km−1→m
−δx −δx −δx −δx
which shows that the total current is independent of the 1.7
well index m. Note that the current through the shunt
will be the dominating contribution to the total current 2m)1.6
of a SL if the shunt is thick and well-conducting. Even A/
a completely disconnected shunt (i.e. a=0) contributes 40 1.5
1
a constant current of J0(s) = δxeµNDU/(Nl+d) to the nt (
totalcurrentJ ofa homogeneousSL. Since we areinter- re1.4
r
u
ested in effects arising from the interaction between the c
SL and the shunt, we will in the following discuss the SL 1.3
current dynamics on the basis of the SL current defined
by JSL(t)=J(t)−J0(s). 1 2 tim3e (µs)4 5 6
2258ns 2449ns 2643ns 3000ns
III. PARAMETERS AND TIME SCALES 2.56
m)
m
Theparametersthatweuseinthesimulationarelisted (
x 0
in Table I. We found that there are very different time
3083ns 3295ns 3394ns 3582ns
2.56
TABLE I: Parameters used for theshuntedSL. m)
m
(
N ND w d µ D0 T ǫr x 0
- (m−2) (nm) (nm) (m2/Vs) (m2/s) (K) -
40 1.5×1015 9 4 10 0.015 5 13.18 FIG. 7: Same as Fig. 5, butwith Lx =2.56 mm.
scalesinthiscomplexstructurewhichrequiresanimplicit
method of numerical iteration. The first time scale τ is
b
the dielectric relaxation time in the bulk material both works, we also know that the behavior of the electrons
in the shunt and in each quantum well in the SL. It is in the vertical direction is not simply dielectric relax-
determined by the doping density. We know that the ation. More complex phenomena, such as current self-
conductivity g is proportional to the charge density oscillation,orinjecteddipolerelocationduetoswitching,
havemuchlongertimescalesranginguptomicroseconds.
g eµN /l 1.6 10−19 10 1023(Ωm)−1 105(Ωm)−1
≈ D ∼ × × × ∼ The time scale τt sets a lower limit of the time scales for
(26)
these nonlinear processes.
So the dielectric relaxation time in the shunt layer and
within each quantum well is approximated as
Another important time scale τ is the time that it
i
ǫ ǫ 0.1 10−9 takes to carry away or supply the electrons in the SL
τb = rg0 ∼ ×105 (s)∼10−15(s) (27) through the shunt. Because the vertical processes are
relatively slow, if the shunt has good connection and
whichisrelativelyfastduetothehighconductivity. This high conductance, the electrons will move laterally, pass
is the time it takes for a fluctuation in the charge den- through the intersection between the quantum well and
sity to be neutralized within either the shunt layer or the shunt, and drift away through the shunt. This time
quantum wells. scale τ is considerably larger than τ since the electrons
i b
The second time scale τ is the one in the vertical dy- havetomoveintotheshuntfirst. Laterwewillseethatit
t
namics. According to the sequential resonant tunneling takes1 ns to deplete a full CAL in a smallSL. The pres-
model,theverticalcurrentistotheorderof10−4(A/m2) ence of extremely different time scales means that the
and the positive differential conductivity g is of order numericalintegrationis astiff problemandthis suggests
t
0.1(Ω m)−1. Thus,τ =ǫ ǫ /g 10−9 s,a muchlarger the use of an implicit method. The numerical procedure
t r 0 t
∼
time scale than τ . Moreover, from numerous previous is described in the Appendix.
b
7
2.1 (a)10−2
2m) 2 10−3 A
A/
4nt (10 1.9 a10−4 oscillations
e1.8
r
r
cu 10−5
SL 1.7 B
10−6
1.6 0 1 2 3 4
4 6 8 10 Voltage (V)
time (µs) (b)
5.12 5215ns 5478ns 5696ns 6332ns 20
m)
m) mA/
x(m 0 ation (15
IV. FDIGEO.P8NE:NSTaDHmEEeNaLsCAFETigOE. 5RF,AbSuLHtUSwINiZthTELINOxFG=5TD.1HY2ENmAmM. ICS Amplitude of current oscill105 Frequency (MHz)051.46 1.48 1.5 Al Au
0
1.46 1.47 1.48 1.49 1.5 1.51
SUPERLATTICE
Voltage U (V)
(c) 106
Inthis section,we discussthe effects ofthe lateralsize
L of the SL with a high quality shunting layer, i.e.,
x
a = b = 1. The shunting layer has a width δ such that
x z)
varying δx does not affect the dynamics in the shunt. y (H
This is numerically confirmed even for the chaotic case nc
e
u
thatwe willdiscussbelow,where a80nm shuntinglayer q
e
has the same effect as a 8 mm one. This is because τ is Fr
b
muchsmallerthanτ andtheelectronsenteringtheshunt
i
are carried away so fast that a change in the shunt con-
105
ductancedoesnotchangeτi. We willstudy theSLswith 10−5 10−4 10−3 10−2
a relatively high contact conductivity σ = 0.04 (Ωm)−1. U−U (V)
critB
At this value of σ, without a shunt, the SL has a static
high field domain near the emitter and a static low field FIG. 9: (a) Bifurcation diagram for σ = 0.04 (Ωm)−1,
domainnearthecollectorseparatedbyastaticchargeac- Lx = 20 µm, b = 1.00. Dashed curve shows the approxi-
cumulation layer (CAL). Due to the high quality shunt mateboundaryoftheoscillatory region andlocation ofstud-
ied bifurcation points A and B. (b) Bifurcation scenario at
the totalcurrentis dominatedby the contributionofthe
A for a = 1.00×10−3: amplitude vs. voltage (main figure)
current through the shunt. As discussed at the end of
and frequency vs. voltage (inset). Points Au and Al denote
SectionII, we will thereforeconsiderthe SL currentJ .
SL the endpoints of the upper and the lower branches, respec-
Also,sincewearevaryingLx,wescalecurrenttocurrent tively. (c) Bifurcation scenario at B for a = 1.00×10−5:
density.
scaling of frequency vs. voltage (double logarithmic plot);
UcritB =2.30441 V.
A. High quality shunting layer with small Lx
of the current) into the shunt. We can see that when
Figure 2 shows charge and current density plots for a the systemreachessteady state,the net chargeis almost
relatively narrow SL with lateral extent Lx = 20 µm. neutral, i.e., n = ND, everywhere in the SL and the
Theinitialstateispreparedasachargeconfigurationfor shunt. There are still some small lateralcurrent flows at
the SL without shunt at total applied voltageU =2.1 V the first and the last period.
andshowsastaticchargeaccumulationlayeratthe 20th If we take a close look at the steady state, we find
period. After an interval of about 1 ns, the space charge that there is a small CAL at the first period and a CDL
configuration is almost uniform. The in-plane current is nearby(Fig.3(a)). The situationis almostinvertednear
plotted as a vector field and shows the electrons in the the collector. To better understandthis, we focus onthe
CALmoveinthelateraldirection(theoppositedirection operation points near the emitter shown in Fig. 3(b) at
8
(a) 8 (a) 250
249.5 248
m) 249 247.5
A/
µtime (s)7.5 current (m24284.85 24726 tim2e6 .(5µs) 27
L 247.5
S
247
246.5
20 40 60 80 100 120
10 20 30 40 time (µs)
SL period
(b) (c) (b)1 (c) 100
6Field (10 V/m)12345 123charge density n (N)DSL current (mA/m)222225667750505 µtime (s)0.5 10 20 30 40µtime (s)99.5 10 20 30 40
000 SL 22p00eriod 44000 2507 time7. 5(µs) 8 SL period SL period
270
FIG. 10: (a) Charge density distribution evolving in time, (d)
270
(b) a snapshot of field profile (solid line) and charge density 265
profile (dashed line) at t = 8 ns and (c) SL current JSL on m) 260
the upper branch of Fig. 9b. Parameters: a = 1.00×10−3, A/
U =1.46 V,σ=0.04 (Ωm)−1. nt (m260 250
e
urr255 240 30 30.5 31 31.5
x = 20 µm. In this case, the field is almost uniform in L c time (µs)
S
theSLandeachperiodisbiasedintheNDCregion. The 250
field across the first barrier between the emitter and the
first well will also have this same value in the absence 245
5 10 15 20 25 30 35
of charge accumulation in the first well. This causes a time (µs)
verticalcurrentfromthe emitter tothe firstperiod(thin 0.2
(e)
solid line in Fig. 3(b)) which is much larger than the
0.15
vertical current in the corresponding NDC region of the
SL. Close to the shunt this extra currentwill give rise to 0.1
a lateral current which will quickly reach the shunt and
1) 0.05
is carried away by the shunt. A little further away from −
s
µ
the shunt where the lateral current is not sufficient to λ ( 0
completely neutralize this extra current, a small CAL is
−0.05
formedinthefirstwellwhichlowerstheelectricfieldand
thereforethecurrentacrossthefirstbarrier. Atthesame
−0.1
time, the electric field in the second barrier is pushed
above the uniform field, causing a very small CDL next −0.115.48 1.482 1.484 1.486 1.488
to the CAL. Similar arguments can be applied to the Voltage U (V)
collector to explain the appearance of a small CDL in
the lastquantumwell. The overalleffect is thata nearly FIG. 11: (a) SL current J vs. time for U = 1.486 V (the
SL
uniform vertical electric field configuration is stabilized insetshowsanenlargement),(b),(c)chargedensitydistribu-
for these conditions. tionsevolvingfortwodifferenttimeintervalsforvoltagenear
bifurcation point Al. (d) SL current for U = 1.481 V. (e)
The rate λ of exponential decay λ<0 (or increase λ>0) of
oscillation amplitude versus applied voltage U. Parameters:
B. High quality shunting layer with large Lx a=1.00×10−3, σ=0.04 (Ωm)−1.
As the lateral size L of the SL becomes larger, the
x
CAL andCDL nearthe emitter become more prominent
(cf. Fig. 4(a)-(c), L = 160 µm) since with increasing
x
distance to the shunt the lateral current becomes less
9
(a)270 (a) 35 (b)450
270
mA/m)226605 oscillations 225600 19.6 19.8 20 µtime (s)32.5 SL current (mA/m)334050000
ent ( 10 SL p20eriod 30 40 250 31 3ti2me (µ3s3) 34 35
urr255 (c) 15 (d)450
c
SL 224550 µtime (s)12.5 SL current (mA/m)334050000
10 20 30 40 50
time (µs) 10 20 30 40 250 11 12 13 14 15
SL period time (µs)
(b) 30
FIG. 13: Bifurcation scenario at point B of Fig.9: (a), (c)
chargedensitydistributionsvs. time,(b),(d)SLcurrentJ
SL
for U =2.3 V (upperpanel) and U =2.304 V (lower panel),
respectively, with a=1.00×10−5, σ=0.04 (Ωm)−1.
s)
µ
e (27.5 200 200
m (a) (b)
ti
µme (s)199.5 µme (s)199.5
ti ti
10 20 30 40 10 20 30 40
10 20 30 40 SL period SL period
SL period (c) 200 (d)460
(ower spectrumc)1100−−150 UU == 11..449999785 ( d)T (s)1100−−43 µtime (s)199.5 10 SL p20eriod 30 40 SL current (mA/m)334446802400000 192 1t9im4e (µ1s9)6 198 200
P10−150 2 4 6 101−50−7 10−6 10−5 10−4 10−3 FIG. 14: Charge density distribution vs. time for a=1.00×
Frequency (107 Hz) U − UA u (V) 10−3, σ =0.04 (Ωm)−1 at (a) U = 1.2 V, (b) U = 1 V, and
(c) U =0.5 V.(d) SL current at U =0.5 V.
FIG. 12: (a) SL current J (the inset shows an enlarge-
SL
ment), and (b) charge density distribution evolving in time
for U = 1.49985 V with a = 1.00×10−3, σ = 0.04 (Ωm)−1.
(c)PowerspectrumdataforoscillationsatU =1.4997Vand is plotted in Fig. 4(f). Field domains are forming as
U =1.49985V.(d)ThetimeT forwhichthesystemexhibits the field is low to the left of the CAL and high to the
transient oscillations versus the applied voltage U − UAu. right of the depletion region. In this case, the upstream
UAu =1.499791 V. CAL (closer to the emitter, at the left bottom corner
of Fig. 4(d)) and the downstream CAL (closer to the
collector, the wider one in Fig. 4(d)) are still connected
efficient at carrying away the excess current from the and this is a time-independent steady state.
emitter to the shunt. In the above case, the lateral size of the SL is just
For wider SL (cf. Fig. 4(d)-(f), L = 640 µm), the below a characteristic value for which the steady state
x
field closer to the shunt is more uniform and the CAL loses stability to oscillatory behavior. Figure 5 (Lx =
is still attached to the emitter. However, away from the 800µm)showsthesimulationsofaslightlywiderSLthan
shunt, the CAL detaches from the emitter and locates consideredabove. The largedownstreamCAL stillstays
itself in the first few periods and the nonuniform field in that position. However, due to the large size of the
regionbecomeslarger. Thisbehaviorisduetothelateral SL, the lateralcurrentis not able to sustain a connected
currentbeinginsufficienttocarryawaytheextracurrent stableCAL.ThesmallupstreamCALtouchesandbreaks
fromtheemitter. Thus,theCALgrowsbiggerandtends off from the downstream CAL periodically. There is a
to move toward the collector. With the center of the small amplitude oscillation in the total current which is
CALlocatedindifferentwellsatdifferentxpositions,the shown in the top panel.
lateralgradients can be increasedand a sufficient lateral For an even wider SL (Fig. 6 with L = 1.28 mm),
x
currentcanbesustained. Thefieldprofileatx=640µm the upstream and downstream CALs are mostly discon-
10
10−2 weak on the opposite side of the SL. Thus, the down-
(a)
streamCALislocatedveryclosetothe20thperiodwhere
itwouldbeintheabsenceofashuntinglayer. Themerg-
10−3
ingofthe CALsdescribedinlastparagraphalsoappears
here except that the merging events are now difficult to
C
a10−4 predict and manifestly not periodic. Figure 8 shows the
behaviorofaSL withL =5.12mm. Itshouldbe noted
x
oscillations thatrealSLsamplesrarelyhavesuchalargesize. Inthis
10−5
case, the unstable dynamics only occurs in the portion
of the SL closest to the shunt. In the portion of the SL
10−6 away from the shunt, a CAL is located at the 20th well,
0 1 2 3 4
Voltage (V) where the shunt has no apparent influence. Over time,
(b) 2 (c) 2 the lateral extension of this CAL changes. When a large
CDL collides with itat 5.696ms, the static CAL shrinks
µme (s)1.5 µme (s)1.5 tTohae spmreaslelnsciezeo,fcasuucshingchaarlagregteridpiopleincothnefigcuurrarteinotnst2r8acoef.
ti ti one CDL and two CALs has already been shown to be
associated with chaotic behavior in one-dimensional SL
10 SL 2p0eriod 30 40 10 SL 2p0eriod 30 40 models without lateral dynamics.29
(d) 2 (e) 2 To summarize,we are able to identify three character-
istic length scales in the x direction. The shortest one
µme (s)1.5 µme (s)1.5 cishatrhgeeddeecnasyityleningtthheL¯xfir(sotfqouradnetrum10wµemll)inactrewasheicshfrtohme
ti ti
N at the SL-shunt interface to its maximum value (cf.
D
Fig. 4(a)-(c)). The next length scale (of order 200 µm)
10 20 30 40 10 20 30 40
SL period SL period is the range above which the vertical field configuration
loses uniformity and static field domains start to form
FIG. 15: (a) Bifurcation diagram for σ = 0.016 (Ωm)−1.
(cf. Fig. 4(d)-(f)). The longest length scale (of order
Charge density distribution vs. time near bifurcation point
700 µm) is the width of the SL above which the steady
C, a = 1.00×10−4 at (b) U = 1.6 V, (c) U = 2.1 V, (d)
state loses stability to oscillatory behavior. This implies
U =2.7 V, and (e) U =3.0 V.
that lateral uniformity in the electric field distribution
can be expected when L is smaller than the intermedi-
x
atecharacteristiclengthscale. Theshortestdecaylength
nected. TheupstreamCALextendslaterallyintotheSL
L¯ can be estimated by noting that the extra current
and moves toward the downstream CAL, (at time 1.969 x
coming from the emitter must be directed to the shunt
ms). For certain times during the dynamical evolution
by the negative gradient of the lateral current J , i.e.,
(notshowninFig.6),theupstreamCALbreaksofffrom ⊥
∂J⊥(x) = J (x) J (x) < 0. Then there is ap-
theemitterandreachesandmergeswiththedownstream ∂x k0→1 − k1→2
CAL.Mostly,thereisadepletionregionformingbetween proximately a decay length L¯x, at which the quantities
the upstream and downstream CALs (2.211 ms). This suchas J⊥(x), n(x) andFx(x) approachasymptotic val-
depletion region grow and dies away as it merges with ues exponentially. Calculation shows that L¯x is of order
the upper CAL (1.969 ms). For certain times, it grows 10µmfortheparametersusedinTableI,27 inagreement
into a full CDL extending throughoutthe structure and, with our numerical results.
inthiscase,theupstreamCALalsogrowsintoafullCAL
(1.755ms). Thenallthreefrontsmovedownstream. The
olddownstreamCALandtheCDLquicklydisappearand V. DEPENDENCE OF DYNAMICAL
the new CAL formed by the upstream CAL replaces the BEHAVIOR ON THE SHUNT PROPERTIES
old CAL and stays at its position. Although these be-
haviorsarequite complicated,they arestill periodic and In the previous section, we have seen that the width
duringeachperiod,theupstreamanddownstreamCALs of the SL determines the lateral dynamics of electronic
merge several times. transport and that the shunt can stabilize a nearly uni-
However, for an extremely wide SL (Fig. 7, L = form field configuration in sufficiently narrow SLs. Now
x
2.56mm), the behavior is apparently chaotic. The effect we investigate the effects of the shunt properties on a
of the shunt is to cause a CAL attached to the emitter small SL with width of 20 µm where the lateral field
near the shunt. For large values of x, the shunt has less and electron density profiles are almost uniform. Since
effect and this CAL detaches from the emitter, tends to thechargedensityisalmostuniformlaterally,wemodify
move downstream to the collector and thus extends to- the model such that the SL is collapsed to one point in
ward the downstream CAL. Due to the large lateral size x direction. This modification significantly reduces the
of the SL, the impact of the shunt layer becomes very complexity of the simulation. We first study the effects