Asymmetric Double Quantum Wells with Smoothed Interfaces
ABSTRACT We have derived and analyzed the wavefunctions and energy states for an
asymmetric double quantum wells, broadened due to static interface disorder
effects, within well known discreet variable representation approach for
solving the onedimensional Schrodinger equation. The main advantage of this
approach is that it yields the energy eigenvalues and the eigenvectors in
semiconductor nanostructures of different shapes as well as the strengths of
the optical transitions between them. We have found that interface broadening
effects change and shift energy levels to higher energies, but the resonant
conditions near an energy coupling regions do not strongly distorted. A
quantummechanical calculations based on the convolution method (smoothing
procedure) of the influence of disorder on the motion of free particles in
nanostructures is presented.
 Citations (8)
 Cited In (0)

 [Show abstract] [Hide abstract]
ABSTRACT: A nonquasistatic (NQS) model suitable for simulating the small signalperformance of monolithic MESFET and HEMT devices, which is consistent with a largesignal model, is described. The hallmark of the model is that it incorporates NQS effects not only in the capacitors of the equivalent circuit, but also in the conductances, employing the same relaxation time in both types of elements. A parameter extraction technique which provides a correct identification of parasitic and intrinsic elements is also presented. The single relaxationtime NQS model has been successfully applied to predict the smallsignal performance of monolithic MESFET and HEMT devices in the 1 to 40 GHz band at a wide variety of bias conditions. The close agreement obtained between measured and simulated Sparameters at all bias points validates the proposed model and proves that it represents the main devices' behavior aspects.© (2004) COPYRIGHT SPIEThe International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.04/2004;  SourceAvailable from: hku.hk[Show abstract] [Hide abstract]
ABSTRACT: A FabryPerot reflectiontype modulator which uses interdiffused AlGaAsGaAs quantum wells as the active cavity material has been studied and optimized theoretically. An asymmetric Bragg reflector structure (modeled by transfer matrices), with a doped depletion layer in the heterostructure, has been considered. This is the first study to model such a material system in this type of modulator, and the results show improvement in modulation property over its asgrown rectangular quantumwell modulator. In particular, the change of reflectance in the diffused quantumwell modulator is almost 0.6 to 0.7, which is higher than that of the typically available values (~0.5 to 0.6), while the OFFstate onresonance reflectance is almost close to zero. The operation voltage is also reduced by more than half as the interdiffusion becomes extensive. The finesse of the more extensively diffused quantum well also increases. Both of these features contribute to an improvement of the change of reflectance in the modulator. The operation wavelengths can be adjusted over a range of 100 nm. However, the absorption coefficient change of the diffused quantum well increases only when there is a small amount of interdiffusionIEEE Journal of Quantum Electronics 04/1997; · 1.83 Impact Factor
Page 1
Cent. Eur. J. Phys. • 115
Author version
Central European Journal of Physics
Asymmetric Double Quantum Wells with Smoothed
Interfaces
Rapid Communication
Vladimir Gavryushin∗
Institute of Applied Research and Semiconductor Physics department, Vilnius University,
Sauletekio al. 9III, 10222 Vilnius, Lithuania
Abstract: We have derived and analyzed the wavefunctions and energy states for an asymmetric double quantum
well (ADQW), broadened due to interdiffusion or other static interface disorder effects, within a known
discreet variable representative approach for solving the onedimensional Schrodinger equation. The main
advantage of this approach is that it yields the energy eigenvalues, and the eigenvectors, in semiconductor
nanostructures of different shapes as well as the strengths of the optical transitions between them. The
behaviour of ADQW states for the different mutual widths of coupled wells, for the different degree of
broadening, and under increasing external electric field is investigated.
broadening effects change and shift energy levels, not monotonously, but the resonant conditions near an
energy of subband coupling regions do not strongly distort. Also , it is shown that an external electric
field may help to achieve resonant conditions for intersubband inverse population by intrawell emission
of LOphonons in diffuse ADQW.
We have found that interface
PACS (2008): 73.21.Ac, 73.63.Hs
Keywords: Multibarrier quantum well structures • Asymmetric double quantum wells; quasibound levels • Discreet
variable representation • Schrodinger equation
© Versita Warsaw and SpringerVerlag Berlin Heidelberg.
1. Introduction
When a thin (∼100 A layers of one semiconductor (e.g. GaAs) are sandwiched between layers of another semi
conductor with a larger band gap (e.g. AlGaAs), carriers are trapped and confined in two dimensions (2D),
due to the potential barriers. As a result of quantum confinement, discrete energy states (or ”subbands”) oc
cur, which change dramatically electronic and optical properties of such structures, known as the quantum wells
(QWs). When the quantum wells are coupled there exist probabilities for the electron tunnel, which can be in
either of the two wells. The novel optical properties of the 2D electron gas appear associated with the transitions
between quantized subbands, so called ”intersubband transitions”, which correspond to the range from mid
infrared to terahertz (THz) photon energies. They have narrow linewidths and extremely large dipole moments
of transitions.
∗Email: v.gavriusinas@cablenet.lt
1
Page 2
Asymmetric Double Quantum Wells with Smoothed Interfaces
In recent years, there has been considerable interest in asymmetrical multiplequantum well systems [1], because
many new optical devices based on intersubband transitions are being developed (”intersubband optoelectron
ics” [2]). This feature could fulfill the need for efficient sources of coherent infrared (IR) radiation for several
applications, such as communications, radar, and optoelectronics. Their most spectacular applications are quan
tum well IR photodetectors [3] and the quantum cascade (QC) lasers [4], that relies on the cascaded intersub
band transitions and resonant tunneling between adjacent QWs. These devices are made with epitaxially grown
GaAs/AlGaAs, InGaAs/AlInAs, GaN/AlGaN [1, 5], and Si/SiGe [6] systems. With the recent development on
semiconductor device growth technology, multibarrier quantum well structures are becoming the basic building
blocks of modern semiconductor devices, such as resonant tunneling diodes (RTD) [7], highspeed light modulators
[8], wavelength tunable lasers [9], farinfrared and THz lasers [10], quantum cascade (QC) lasers [4, 11], etc.
In an intersubband quantum cascade laser, the population inversion must be established by engineering the
lifetimes and oscillator strengths, i.e. by a suitable design of the active region. Calculations for quantum wells
are often performed in the approximation where the conduction and valence band offsets are taken as sharp step
functions. In practice, real QW structures tend to deviate from the ideal homogeneous heterostructures with
perfect abrupt interfaces due to the fluctuations of the band edges (Fig. 1), and interdiffusion which thus is
not very well defined. The reasons for this are the stochastic processes of the crystal growth leading to local
variations of chemical composition, well width, and lattice imperfections to name a few. Since a QW is generally
a heterostructure formed by a binary semiconductor (AB) and a ternary disordered alloy (AB1−xCx), as in
GaAs/InGaAs, there are two types of disorders responsible for the inhomogeneous broadening: compositional
disorder, caused by concentration fluctuations in a ternary component, and random diffusion across the interface.
Surface segregation during epitaxial growth [12] and thermal annealing [13] processes may result in the symmetric
interdiffusion at interfaces [13]. Both processes change the electronic behaviour of a system by narrowing the QWs
and degrading the barriers of QWs. In such interdiffusedor intermixed QW structures with smooth interface
profiles, significant changes in the subband spacing and carrier scattering rates in a Si/SiGe QW system [6], were
predicted.
The interdiffused QW structures such as GaAs/AlGaAs [14] and InP/InGaAs [15] have been actively investigated
for improved intersubband electroabsorptive light modulation [15], and widely used in several optical devices,
such as electroabsorptive [16] and lateral confinement [17] waveguides, light modulators [18], and wavelength
tunable lasers [9]. The improved quantum confinement and a higher tunneling rate achieved in the interdiffused
QW’s cannot be fulfilled by a rectangular QW structure simultaneously. The extensively interdiffused QW
reduces the required bias or increases the tunneling rate for EA modulation. Therefore, the diffused QW structure
is potentially attractive for developing highspeed modulators. The effect of interdiffusion has been simulated
and investigated below.
The genius of the QW heterostructure concept—and the quantum cascade in particular—lies in its innate en
gineerability. In order to design new devices or optimize the device performance, and thus properly predict
2
Page 3
Vladimir Gavryushin
their behavior, one needs to know the detailed information of quasibound levels in real disordered multibarrier
quantum well structures. Theoretical studies of the influence of the compositional and interface disorder on the
motion of free carriers in nanostructures have a long history [1921]. There have been calculations for phenomena
like exciton localization at lateral fluctuations of the well width and band tails [22] as shown in Fig. 1c due to
fluctuations in impurity concentration.
To understand the physical properties of the heterostructure devices, one needs to solve the eigenvalue problem
of carriers in QWs. It is well known that exact analytical solutions to such problems are only available for simple
structures such as a square or parabolic well [23], and even in these structures, in general, in the presence of
perturbations such as external fields, disorder effects [24], etc., the problem cannot be solved exactly. There have
been various numerical methods used to calculate the band profiles in QWs: the matrix approach (MA) [25], the
transfer matrix (TM) method [26, 27], the finite difference method (FDM) [28, 29], the ”shooting method” (SM)
[28, 30], the finite element (FE) technique [31, 32], discreet variable representation (DVR) approach [33], envelope
function (EF) method [34], Wentzel–Kramers–Brillouin (WKB) approximation [35], variational method (VM)
[36], and Monte Carlo (MC) simulations [37]. Among them, the WKB and EF methods adopt approximations,
thus giving unreliable results; the VM only works well at simple QWs and weak fields; the MC and FE methods
are highly computerorientated approaches; the MA usually require wave function to be well behaved; the SM’s
speed comes at the cost of stability. The DVR and TM methods overcome all the shortcomings listed above and
could be easily applied to any potential profiles of biased/unbiased multibarrier quantum well structures.
In this paper, we describe a numerical technique based on the DVR approach, as a gridpoint representation (grid
discretisation method) of a Hamiltonian matrix element [24, 33, 38] which is capable of solving the eigenvalue and
eigenfunction problems in an arbitrary QW under arbitrary perturbation. Calculation results for energy levels
and wave functions of an asymmetric double quantum well with sharp interfaces and interdiffused interfaces with
gradual variation of the potential, are presented and compared.
2. Calculation details
2.1. A model of the interface disorder effects
Randomly distributed charged dopants, or composition x in mixed crystals AxB1−x, lead to unavoidable fluctu
ations of the doping impurities concentration on a microscopic scale. Semiconductor heterostructures possess a
certain degree of disorder due to their alloy structure and/or imperfect interfaces. Two things influence interface
disorder effects: coordinate fluctuations of interface position (Fig 1 a), and gap energy Eg(x) fluctuations (Fig
1 b,c) due to the randomly distributed dopants. These fluctuations result in potential fluctuations giving rise
to band tails composed from localized states. This situation is schematically shown in Fig 1. The magnitude
of bandedge energy fluctuations (Fig 1 b,c) caused by the random distribution of charged donors and acceptors
was first calculated by Kane [39]. States with energy below the unperturbed conduction band edge or above
3
Page 4
Asymmetric Double Quantum Wells with Smoothed Interfaces
Figure 1. a) Characteristic length scales describing the interface roughness of a QW. (b) Spatially locally fluctuating band edges
caused by random distribution of impurities [40]. (c) Resulting densities of states in the conduction with tail states
extending into the forbidden gap. The dashed lines show the parabolic densities of states in undoped semiconductors.
the unperturbed valence band edge are called tail states, which significantly change the density of states in the
vicinity of the band edge.
Particularly simple 1D models of an electron moving in a random or diffused potential are possible. If the fluctua
tions are not too large, good approximation is obtained by calculating spectra for slightly different configurations
and adding them up using some broadening weight factor. Inhomogeneous broadening, due to site variation pro
duced by a random distribution of local crystal fields, results in a Gaussian type broadening [39]. Homogeneous
broadening, from dynamic perturbations on energy levels and equally on all ions, leads to a Lorentzian type
broadening. So, a QW’s barrier interface roughness may be approximated [24] by the convolution of Heaviside
step function Φ(xx0) with an area normalized, a moving Gaussian broadening envelope function of width σG [39]:
HG(z,z0,σ) =
1
√2πσG
∞
?
0
exp
?
−(z − x)2
2σ2
G
?
Φ(x − z0)dx, (1)
or with the Lorentzian broadening envelope function of width Γ:
HL(z,z0,Γ) =Γ
π
∞
?
0
1
(z − x)2+ Γ2Φ(x − z0)dx.(2)
A convolution (smoothing) procedure [41] is an integral that expresses the amount of overlap of envelope function
(i.e. Gaussian or Lorentzian) as it is shifted over another function Φ. Instead of the ∞ limit usually used in
integration, we take any value big enough for the resultant convergence.
The barrier steps may also be broadened in an extremely simple analytical way,  by applying of the phenomeno
logical atan(x) function against the Heaviside step function Φ(x), usual for an ideal heterostructure with perfect
interfaces:
Φ(z − zi) ⇒π
2+ arctanz − zi
Γi
(3)
4
Page 5
Vladimir Gavryushin
Figure 2. Comparison of convolution broadenings of the Heaviside stepfunction with different broadening functions: 1 – Gaussian
broadened step; 2 – Gaussian envelope; 3 – Lorentzian broadened step; 4  Lorentzian envelope; 5 – Analytical arctan(x)
approximation curve (3); Analytical error function profile (4) is fully overlapped with curve 1. Broadening parameter
Γ = 0.05 nm.
where Γi is the broadening parameter of the interfaces at zi. This function with zero mean zi, characterises
the deviation of the ithinterface from its average position. The interdiffusion of the QW composition profile is
described, usually, by an error function erf(x) [6, 13, 42]:
Φ(z − zi) ⇒1
2
?
1 + erf
?z − zi
√2Ld
??
(4)
The extent of the interdiffusion process is characterised by a diffusion length Ld. Examples of calculated functions
(1)(4) are presented in Figure 2. An analytical error function profile (4) is fully overlapped with the Gaussian
broadened step (curve 1). As we see in Figure 2, function (3) (red dots) may be a good approximation for
convolution of Heaviside function by Lorentzian envelope (2) (curve 3). To our knowledge, any attempts to use
the Lorentzian type broadening or interdiffusion modeling have been published, so annealed QW interfaces were
simulated below in such a kind.
2.2. Method of Discrete Variable Representation (DVR)
Analytical solutions for asymmetrical double and triple quantumwells are possible only for unbiased sharp rect
angular QWs [43]. We select discrete variable representation (DVR) as a numerical method [33, 44] for our
MathCAD calculations of a stationary 1D Schrodinger equation for confined eigenstates. Different types of DVR
methods have found wide applications in different fields of problems [45]. We also carried out DVR calculations
for QWs and unharmonic Morse potential [24, 38].
The DVR is a numerical gridpoint method in which the matrix elements of the local potential energy operator V(r)
is approximated as a diagonal matrix (mnemonic: DVR  diagonal V(r), or Discrete Variable Representation):
Vik = ?φiVφk? = δikV(xi) [46], and the kinetic energy matrix is full, but it has a simple analytic form, as
a sum of 1D matrices. DVR method is selected since it avoids having to evaluate integrals in order to obtain
5
Page 6
Asymmetric Double Quantum Wells with Smoothed Interfaces
the Hamiltonian matrix and since an energy truncation procedure allows the DVR grid points to be adapted
naturally to the shape of any given potential energy surface. The DVR method greatly simplifies the evaluation
of Hamiltonian matrix elements Hik = ?φiHφk? and obtains the eigenstates and eigenvalues by using standard
numerical diagonalization methods of MathCAD or Mathematica.
If we choose an equally spaced grid, xi = i∆x, (i = 0, ±1, ±2, ... ±N), then the DVR gives an extremely simple
gridpoint representation of the kinetic energy matrixˆTi,k= ¯ h2k2
i,k
?2m∗within the conditional formulation [33]:
?
Ti,k= g
π2?3,
(i−k)2 , i ?= k
i = k
2 (−1)i−k
. (5)
The only parameter involved being the grid spacing ∆x via an energetically weighted grid parameter g (“energy
quantum of the grid”):
g =
¯ h2
2m∗
?
1
∆x
?2
(6)
where m∗is the electron effective mass. So, if the grid points are uniformly spaced then numerical solutions of a
matrices elements of the full energy Hamiltonian operator
ˆ H =
?
T +ˆV = −¯ h2
2m∗
d2
dx2+ V (x)
is as [33]:
ˆ Hi,k=
?
Ti,k+ˆVi,k=
¯ h2
2m(∆x)2(−1)i−k
?π2
3δi,k+
2
(i − k)2(1 − δi,k)
?
+ V (xi)δi,k
(7)
when the δfunctions are placed on a grid that extends over the interval x = (∞,∞). First term in parenthesis
is a value of second term in the limit N → ∞ [33].
In our calculations the potentials of an ideal and of a broadened biased/unbiased double QW are used as [24]:
V1(xi) = U1[1 + Φ(xi− R1) − Φ(xi)] + Vbias(xi),
V2(xi) = V1(xi) + U2[−Φ(xi− Rb− R1) + Φ(xi− Rb− R1− R2)], (8)
and
V1(xi) = U1[1 +1
πatan
?xi− R1
?xi− Rb− R1
Γ
?
−1
?
πatan
?xi
Γ
?
] + Vbias(xi),
V2(xi) = V1(xi) + U2[−1
πatanΓ
−1
πatan
?xi− Rb− R1− R2
Γ
?
], (9)
correspondingly. Here Φ(x) is a Heaviside step function; U1(2)are the depths of potential wells (differences in the
offset band energies) for the 1stand 2ndQW of the widths R1(2); and Rb is the barrier width. The applied bias
potential here
6
Page 7
Vladimir Gavryushin
Vbias(xi) =
???????
Vcont, ifxi < Rcont
Ef · xi, otherwise
(10)
Vcont = U1[1 − Φ(R1− Rcont) + Φ(Rcont)] + Ef · Rcont,
is used when an electric field Ef perpendicular to the quantum well is applied in calculations.
3.Results and discussion
Interwell opticalphononassisted transitions are studied in an asymmetric doublequantumwell heterostructures
[46] comprising one narrow and one wide coupled quantum wells (QWs). It is shown that the depopulation rate
of the lower subband states in the narrow QW can be significantly enhanced thus facilitating the intersubband
inverse population, if the depopulated subband is aligned with the second subband of the wider QW, while
the energy separation from the first subband is tuned to the energy of optical phonon mode. Such mentioned
structure under applied bias is working as a ”resonantphonon type” active region of stimulated emission in the
farinfrared quantum cascade lasers (QCL) [47]. Laser operation [4] is based on stimulated radiative transitions
between inverse populated states (wavy arrows at Fig 7a). Depopulation of lower state is achieved by setting the
subband separation between 1stand 2ndlevels to the LOphonon resonance energy, which causes electrons to
quickly relax from level 2 via electronphonon scattering (bold arrow at Fig 7a). An overview of QCL physics,
including development and applications, are given in the review papers [48].
Seeking to reproduce mentioned effects, the eigenvalues (stationary energy states En) and the eigenvectors (wave
functions ψn) for a quantum number n, as the solutions of the Schrodinger equationˆ Hψn(r) = Enψn(r), were
calculated using standard numerical diagonalization methods (eigenvals(H) and eigenvec(H) commands in Math
CAD) for the DVR Hamiltonian (7).
Results of our DVR calculations of such a laser structure, as in [46], are depicted in Fig 3 as the eigenstates,
together with the corresponding wavefunctions, for the coupled asymmetric square quantum wells with the po
tentials of the form (8). The states over the dissociation energy U are unbound and delocalised. Dependencies
of the positions of the five lowest subbands Bi as a function of the narrow well width values R1 = 3, 5, 7, 10,
12, 15 nm for the fixed values of barrier width Rb = 2 nm and of the second QW width R2 = 15 nm are shown
in Fig 3. Broken lines are the same dependencies for the Ai states of a 1st, but single, quantum well with the
same width R1. Comparing the fans of dependencies for Ai and Bi states for single and double QW’s, we can
resolve the doublet nature of the states and especial coupling (anticrossing) regions of the levels in double QW.
The insets around in Fig 3 show the model band diagrams with energy levels and corresponding wavefunctions
of an ADQW heterostructures (AlGaAs/GaAs).
It is useful also to have some indication of how many grid points are necessary for this DVR procedure to provide
an accurate description of a quantum system. Convergence of the calculation can be checked by decreasing the
7
Page 8
Asymmetric Double Quantum Wells with Smoothed Interfaces
Figure 3. Unbroadened ideal ADQWs case. Dependencies of the positions of the five lowest subbands B0 – B4 (in eV) as a
function of the narrower well width R1 (= 3, 5, 7, 10, 12, 15 nm) for fixed values Rb = 2 nm and R2 = 15 nm.
Broken lines are the same dependencies of the A states for a single quantum well of R1 width. The insets around
show the model band diagrams with energy levels and corresponding wavefunctions of an asymmetric double quantum
well heterostructures. The wavy arrow indicates the stimulating lightemitting transition, assisted by the fast resonant
opticalphonon emission (arrow between states n=1 and n=0) in the heterostructure. Calculated region: 30 nm ÷ +30
nm with 500 grids.
number of grid used in the calculation. For full convergence of the calculation results, we found that a big enough
grid number is N ≥ 50, however we have used the number of calculus points N = 500 for the better shaping of
calculated wavefunctions.
All states in double QWs are split doublets, because the degeneracy is unmounted by different parity properties.
When the barrier thickness becomes smaller, quantum coupling due to the tunneling between the wells is dominant.
As a consequence, an energy splitting (anticrossing) occurs (coupling regions indicated in Fig 3 and Fig 4) and
the respective electron states, the socalled binding (symmetric) and antibinding (antisymmetric) states, are
delocalised over both wells. The minimum energy splitting or tunnel coupling ∆E (anticrossing gap) is determined
by the barrier thickness Rband height. The lowest coupled state is mainly localised in the wide well and the other
state is mainly localised in the narrow well. Due to the coupling of the two wells the two states have nonzero
probability density in both wells.
8
Page 9
Vladimir Gavryushin
Figure 4. Broadened ADQWs with interdiffused interfaces. Dependencies of the positions of the five lowest subbands B0– B4(in
eV) as a function of the narrower well width R1(= 3, 5, 7, 10, 12, 15 nm) for fixed values Rb= 2 nm and R2= 15 nm.
Broken lines are the same dependencies of the three Aistates for a single QW with the width R1. For comparison, thin
black lines presents the case of ideally rectangle shaped ADQWs as in Fig 3. The insets around shows the model band
diagrams with energy levels and corresponding wavefunctions. Interface roughness parameter Γ = 0.2 nm. Calculated
region: 30 nm ÷ +30 nm with 500 grids.
One embodiment of THz laser structure is depicted in Fig 3 and Fig 4 where the energy levels in coupling region
are separated by energy equal to the optical phonon energy. The wavy arrow corresponds to the stimulated
lightemitting transition, assisted by the fast resonant opticalphonon emission (arrow between states n=1 and
n=0). Phononassisted transitions between the coupled levels depend strongly on the phonon energy involved in
the transition.
Even with the sophisticated growth technologies used in current structures, the presence of interface and alloy
disorder induces noticeable effects. The results of DVR calculations of the same family of ADQWs as in Fig 3,
but with broadened interfaces with the potentials approximated by (9), using interface smoothing Γ = 0.2 nm, are
depicted in Fig 4. In the central part of Fig 4, for comparison, both energy fans for ideal rectangular interfaces
(as in Fig 3), and for broadened ones, are presented together. Here the thin black lines represent the case of sharp
rectangular shaped ADQWs. As we see, interface broadening changes shift energy levels to the higher energies,
9
Page 10
Asymmetric Double Quantum Wells with Smoothed Interfaces
but the resonant conditions near an energy coupling region are not strongly distorted. The blue shift is as a
sequence of an effective narrowing of the distorted QWs, and the changes seen in the coupling regions are caused
from the different interwell barrier profile.
Such modest onedimensional modeling results for QWs may correspond to the mean effects from the lateral
fluctuations in the plane of QWs. Interface roughness as 3D problem affects in general the dynamics of confined
exciton states [22] or scattering processes [49]. The ambiguity is due to the continuous spatial variation of interface
from alloy composition. It is usually assumed that the 3D interface roughness parameters are independent of inter
diffusion. In reality, however, annealinglike diffusion processes may reduce the interface roughness height.
Similar calculations, but for a single heterojunction with transition layer, were performed in [50], where also was
shown that interface smoothing gave small effects to the energy level differences,  they are relatively insensitive
to the interface profile. Intersubband carrier scattering in Si/SiGe quantum wells with diffuse interfaces was
investigated in [6].
3.1. Effect of the Interdiffusion degree to the states of ADQW
To estimate the effect of interdiffusion upon barrier degradation [6], same pairs of QWs separated by a thin
barrier were considered. Figure 5 shows the results for the coupled diffuse double quantum well, resolving the
effect of interdiffusion degree Γ on the subband spacing ∆Enn?. As the broadening factor Γ increases, the bottom
of the wells narrow and the top widens. Subbands, which are nominally near the bottom of the well, are therefore
pushed up in energy as interdiffusion increases, while those at the top drop relatively in energy. Conversely, the
effect is small in subbands near the middle of the QW depth.
The lower energy electrons are strongly confined in the wider well, and the higher energy electrons in both wells.
In the “weak coupling” regime, the subband spacing increases to a peak shift. At greater interdiffusion lengths,
the subband spacing decreases. Interdiffusion degrades the barrier between wells. Rightbottom inset of Figure 5
shows the “single well” regime, in which large interdiffusion merges the wells.
Three distinct regimes can be identified, as interdiffusion increases. For low interdiffusion the interfaces are
almost abrupt, and the barrier is well defined. This effectively uncouples the wells, resulting in very small overlap
between the lowest pair of subbands. As interdiffusion increases, the barrier degrades and the wells become
weakly coupled, leading to an increased overlap between subbands. The bottoms of the wells narrow, leading to
increased subband spacing. At very large interdiffusion lengths, the barrier potential is substantially reduced,
and the system resembles a single quantum well with the nominal “barrier” region acting as a perturbation.
The region of overlap between subbands now extends across the entire system, and the energy spacing between
subbands is determined approximately by the width of the wide, single well and is hence lower than the nominal
value. A blue shift in the interminiband emission frequency for GaAs/AlAs superlattices was observed as the
interdiffusion length increased in [51].
10
Page 11
Vladimir Gavryushin
Figure 5. Conduction band potential, confined electron states and wavefunctions shown schematically for varying degrees of
interdiffusion [Γ = 0.2, 0.5, 1, and 2 nm] for ADQW, same as in Figure 4. Left QWs width is R1 = 7 nm; right QWs
width is R2 = 15 nm; barrier width is 2 nm. Middle fan represents the subband spacing as a function of broadening
factor Γ.
3.2. Electric field effect to ADQW states
The resonant situation can be obtained for asymmetric coupled double quantum wells with applied bias [4, 46, 47].
We have performed the same calculations for the diffused ADQW as one of geometry from Figure 4 including the
influence of an external electric field Ef. Results were shown at Fig 6. When an external bias perpendicular to
the quantum well is applied, indicated in the Fig 6 by the changed constant slope, electrons are pushed to one
side of the well, thus the effective well width is reduced. At the chosen bias Ef potential tilts downwards from
right to left. The middle fan shows how it is possible to change the resonant conditions of longitudinal optical
(LO) phonon emission (breaked arrow) to achieve the inversion population at 3rdstate for laser effect with FIR
photon generation (wiggly arrow). Here, once more, we may see an anticrossing situation caused by bias for the
levels n=0 and n=1 around the 23 meV/nm region of electric field strength.
Investigating structure may work as an electroabsorptive light modulator [15] or as an active stimulated emission
region in the far/midinfrared quantum cascade (QC) lasers [47]. The schematic for this laser operation [4]
shown at Fig 7c is based on a stimulated and cascaded radiative transition from E3 to a lower subband E2.
Depopulation of lower subband is achieved by external bias, setting the subband separation between levels
2 and 1 to the LOphonon resonance energy, which causes electrons to quickly relax from level 2 via resonant
electronLOphonon scattering.
For QC lasers the lifetimes are ultimately limited by LOphonon emission. The computed [2] intersubband
lifetime due to optical phonon emission, is plotted as a function of transition energy for a square QW in Fig 7b
(zero initial kinetic energy in the upper subband and zero temperature were assumed). As shown in Fig 7b, a
strong reduction of the lifetime is predicted when the two states are spaced resonantly with the optical phonon
energy hωLO. This increase in scattering rate appears when ∆kscatt approaches zero at resonance (k21 = 0 in Fig
11
Page 12
Asymmetric Double Quantum Wells with Smoothed Interfaces
Figure 6. Confined electron states and wavefunctions dependence on an electric field [E = 0, 1.4, 2.7, 5.4 meV/nm] applied
perpendicular to the diffused (Γ = 0.2 nm) ADQW layer plane, same as in Figure 4. Left QW width R1 = 7 nm; right
QW width R2 = 15 nm; barrier width is 2 nm. Zero external field, or metal contact position Rcont is selected at 7
nm.
7c).
A strong reduction of an intersubband lifetime (up to ≈ 0.2 ps at Fig 7b) may be obtained by using resonant
conditions (electric field tunable) for LOphonon emission process, when the transition 2→1 approaches the LO
phonon energy value. Consider the structure shown in Fig 7c. In this threelevel system, the first two states are
separated by optical phonon energy [≈34(36) meV in InGaAs(GaAs)]. Resonant optical phonon emission between
these two states will reduce the lifetime of the state n=2 to about τ21
≈ 250 fs. LOphonon scattering is also
≈ 1.3 ps) due to the large the dominant nonradiative relaxation mechanism from level 3, but is reduced (τ32
inplane momentum k23
exchange necessary. In other words, 3→2 transitions will proceed with large wavevectors,
whereas 2→1 transitions proceed at nearly zero exchanged wavevectors over a whole subband (see Fig 7c).
The ”resonantphonondepopulation” design for QCLs was developed in [4, 52]. This design uses combination
of resonant tunneling and direct electronLOphonon scattering. Electrons are injected by tunneling into the n
= 3 excited state efficiently filling it. The population inversion (its requirement is simply that τ32
>τ21) is
achieved between the two excited states n = 3 and 2 (wavy arrow in Fig 7a,c  as 3→2 laser transition). Strong
relaxation by emission of optical phonons occurs between the strongly overlapped and closely spaced 2ndand 1st
subbands.
4.Conclusions
The discreet variable representation approach for solving the Schrodinger equation are performed to calculate the
electronic properties of asymmetric double quantum wells with interfaces broadened due to static interdiffusion
effects. Seeking to exploit the inherent flexibility in QW structures, we have presented our results, simulating
12
Page 13
Vladimir Gavryushin
Figure 7. a) Scheme of confined electron states, wavefunctions, and laser transitions at applied external bias 5.44 meV/nm
perpendicular to the diffused (Γ = 0.2 nm) ADQW layer plane. (b) The subband electron lifetime (inverted horizontal
axis) due to optical phonon scattering as a function of intersubband transition energy in a square QW. (c) Engineering
a population inversion using the longitudinal optical (LO) phonon resonance. In a ladder of three states, where the
two lower ones are spaced by LOphonon energy, E3state’s lifetime will naturally be longer than the E2because of the
smaller (k12≈ 0) exchanged wavevector for LOphonon emission between the 2ndand 1ststates. Small vertical arrows
show a resonant intersubband LOphonon scattering. (b,c  adopted from [2])
annealed interfaces with Lorentzian type broadened shapes, as an attempt to model real systems more accurately
than the abrupt rectangular QW approximation. The wave functions and energy states of ADQWs were derived
and analyzed in their changes: due to the different mutual widths of coupled wells, for different degree of the
broadening, and under increasing external electric field. Our calculations, however, are simulating the behavior
of diffused electroabsorptive light modulators [15] and the lightemitter regions of some types of quantum cascade
lasers [47].
Perhaps the main conclusion of this work is that the key features for sharp interfaces are preserved, but interface
broadening effects change and shift energy levels, but not monotonously. The resonant conditions near the energy
of subband coupling (anticrossing) regions are not strongly distorted, but anticrossing energy is slightly growing
when the interfaces are made smooth. We found, that electric field tunable intersubband energy separations
(controlling quantum cascade laser work) were not changed monotonously on the interdiffusion level, and therefore
are important for intersubband optoelectronics technology predictions [2]. It is shown how external bias may
help to achieve resonant conditions for inverse population by intrawell emission of LOphonons.
Acknowledgments
This work was partly supported by the Lithuanian State Science and Studies Foundation grand.
References
[1] K. Talele, E. P. Samuel, D. S. Patil, Optik – Int. J. for Light and Electron Optics, 122, 626 (2011)
13
Page 14
Asymmetric Double Quantum Wells with Smoothed Interfaces
[2] J. Faist, Intersubband Optoelectronics, ETH Zurich, 2009, www.phys.ethz.ch/∼mesoqc/lectures/QCL
lecturelong.pdf
[3] B. Levine, et al., Appl. Phys. Lett. 50, 1092 (1987)
[4] J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, A. Y. Cho, Science 264, 553 (1994)
[5] J. Faist, F. Capasso, C. Sirtori, D. Sivco, A. Hutchinson, A. Cho, Appl. Phys. Lett. 67, 3057 (1995)
[6] A. Valavanis, Z. Ikonic, R. W. Kelsall, Phys. Rev. B, 77, 075312 (2008)
[7] T. Sollner, W. Goodhue, P. Tannenwald, C. Parker, Appl. Phys. Lett. 43, 588 (1983)
[8] M. Ghisoni, G. Parry, M. Pate, G. Hill, J. Roberts, Jpn. J. Appl. Phys., 30, L1018 (1991); W. C. H. Choy
and E. H. Li, IEEE J. Quantum Elect., 33, 382 (1997)
[9] I. Gontijo, T. Krauss, R. M. De La Rue, J. S. Roberts, J. H. Marsh, Electron. Lett., 30, 145 (1994); J. J. He,
et al., Electron. Lett., 24, 2094 (1995)
[10] R. Kohler, et al., Nature 417, 156 (2002)
[11] J. S. Yu, A. Evans, J. David, L. Doris, S. Slivken, M. Razeghi, IEEE Photonics Technol. Lett., 16, 747 (2004)
[12] J. Zhang et al., Surf. Sci., 600, 2288 (2006)
[13] E. H. Li, B. L. Weiss, K.S. Chan, IEEE J. Quantum. Elect., 32, 1399 (1996)
[14] W. C. H Choy, E. H. Li, IEEE J. Quantum Elect., 34, 1162 (1998)
[15] E. H. Li, IEEE J. Quantum Electr., 34, 982 (1998); ibid. 34, 1155 (1998)
[16] J. D. Ralston, W. J. Schaff, D. P. Bour, L. F. Eastman, Appl. Phys. Lett., 54, 534 (1988)
[17] J. E. Zucker, et al., Electron. Lett., 28, 853 (1992)
[18] M. Ghisoni, G. Parry, M. Pate, G. Hill, J. Roberts, Jpn. J. Appl. Phys., 30, L1018 (1991); W. C. H. Choy,
E. H. Li, IEEE J. Quantum Elect., 33, 382 (1997)
[19] A. Efros, M. Raikh, in Optical Properties of Mixed Crystals, edited by R. Elliott, I. Ipatova (NorthHolland,
Amsterdam, 1988), p. 133
[20] M. Herman, D. Bimberg, J. Christen, J. Appl. Phys. 70, R1 (1991)
[21] E. Runge, in Solid State Physics, Vol. 57, edited by H. Ehrenreich, F. Spaepen (Academic, San Diego, 2002)
p. 149
[22] R. Zimmermann, F. Grobe, E. Runge, Pure and Applied Chemistry 69, 1179 (1997); E. Runge, Phys. status
solidi (a) 201, 389 (2004)
[23] H. Wang, H. Xu, Y. Zhang, Phys. Lett. A 340, 347 (2005)
[24] V. Gavryushin, SPIE Proc. 6596, 659619 (2007)
[25] A. K. Ghatak, K. Thyagarajan, M. R. Shenoy, Thin Solid Films, 163 (1988) 461; IEEE J. Quant. Elect. 24,
1524 (1988)
[26] E. Anemogiannis, E. N. Glysis, T. F. Gaylord, T.K. Gaylord, IEEE J. Quant. Elect. 29, 2731 (1993)
[27] E. P. Samuel, D. S. Patil, Progress In Electromagnetics Research Letters, 1, 119 (2008)
[28] P. Harrison, Quantum wells, wires and dots: Theoretical and computational physics, 2nd ed., Wiley, Chich
14
Page 15
Vladimir Gavryushin
ester, UK, 2005
[29] B. M. Stupovski, J. V. Crnjanski, D. M. Gvozdic Comput. Phys. Commun. 182, 289 (2011)
[30] S. F.P. Paul, H. Fouckhardt, Phys. Lett. A, 286, 199 (2001)
[31] K. Nakamura, A. Shimizu, M. Koshiba, K. Hayata, IEEE J. Quant. Elect. 25, 889 (1989)
[32] K. Q. Le, Microwave and Optical Technology Lett. 51, 1 (2009)
[33] D. Colbert, W. Miller, J. Chem. Phys. 96, 1982 (1992)
[34] E. J. Austin, M. Jaros, Phys. Rev. B 31, 5569 (1985)
[35] J. Heremans, D. L. Partin, P. D. Dresselhaus, Appl. Phys. Lett. 48, 644 (1986)
[36] D. Ahn, S. L. Chuang, Appl. Phys. Lett. 49, 1450 (1986)
[37] J. Singh, Appl. Phys. Lett. 48, 434 (1986)
[38] V. Gavryushin, arXiv:qbio/0510041v1 [qbio.BM]
[39] E. O. Kane, Phys. Rev. 131, 79 (1963)
[40] J. I. Pankove, Optical Processes in Semiconductors, Dover Publications Inc., New York, 1975
[41] E. W. Weisstein, Convolution. Wolfram Web Resource: http://mathworld.wolfram.com/Convolution.html
[42] T. E. Scholesinger and T. Kuech, Appl. Phys. Lett., 49, 519 (1986)
[43] R. BetancourtRiera, R. Rosas, I. MarinEnriquez, R. Riera, J. L. Marin, J. Phys. C, 17, 4451 (2005)
[44] G. W. Wei, J. Phys. B. 33, 343 (2000)
[45] H.S. Lee, J. Light, J. Chem. Phys. 120, 4626 (2004); J. Tennyson, et al., Comput. Phys. Commun. 163, 85
(2004)
[46] M. A. Stroscio, M. Kisin, G. Belenky, S. Luryi, Appl. Phys. Lett. 75, 3258 (1999)
[47] J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, A. Y. Cho, Quantum cascade lasers. In H. C. Liu, F. Capasso, eds,
Intersubband Transitions in Quantum Wells: Physics and Device Applications, Nr. 2, Ch. VIII. (Academic
Press, New York) 2000
[48] F. Capasso, et al., J. Quantum Elect., 38, 511 (2002); J. Faist, D. Hofstetter, M. Beck, T. Aellen, M. Rochat,
S. Blaser, J. Quantum Elect., 38, 533 (2002)
[49] A. Leuliet, A. Vasanelli, A. Wade, G. Fedorov, D. Smirnov, G. Bastard, C. Sirtori, Phys. Rev. B 73, 085311
(2006)
[50] F. Stern, S. Das Sarma, Phys. Rev. B 30, 840 (1984)
[51] T. Roch, C. Pugl, A. M. Andrews, W. Schrenk, G. Strasser, J. Phys. D, 38, A132 (2005)
[52] B. S. Williams, H. Callebaut, S. Kumar, Q. Hu, J. L. Reno, Appl. Phys. Lett., 82, 1015 (2003)
15
View other sources
Hide other sources
 Available from Vladimir Gavryushin · May 15, 2014
 Available from ArXiv