- Email: [email protected]
Excitonic theory of Stark ladders in superlattices
Superlattices
and Microstructures,
375
Vol. 7, No. 4, 1990
EXCITONIC
THEORY
OF STARK
LADDERS
IN SUPERLATTICES
D.M.Whittaker Royal Signals and Radar Establishment,
St Andrews Road, Great Malvern,
(Received A theoretical describe
model for excitons
the excitonic
30 July
WR14
3PS. UK.
1990)
in superlattices
Stark ladder states
Worcestershire.
is developed
which occur in optical
and used to
spectra
when an
axial electric field is applied. The model is based on a superlattice scale envelope function type approximation, using a basis of states from a single pair of electron and hole minibands. to be obtained. simulated
This provides sufficient simplification to allow numerical solutions The results are presented both in the form of a fan diagram and as
absorption
spectra.
Comparisons
are made with published
experimental
results.
$1 Introduction The optical
properties
of superlattices
in an axial
To understand the spectra in detail, it is necessary to include the coulomb interaction in the theory. Yan et al.’ obtained variational results for the main exci-
electric field are currently attracting considerable experimentaland theoretical”‘-” attention. As a result of the field, Wannier-Stark localisation occurs, with the superlattice miniband breaking down into a series of localised states, separated in energy by approximately the potential drop across a single period. The effect can be detected optically by observing transitions between
However, their results give no indication of the behaviour in the interesting intermediate field regime, where the
electrons and holes peaked in different wells, giving a Stark ladder of roughly equally spaced peaks, labelled R = 0, ztl . . . according to the electron-hole separation in
lope function type approximation, based on a single pair of electron and hole minibands. A similar approach has been adopted by Dignam and Sipe12, yielding compara-
periods. The strength of the peaks falls off rapidly with increasing /nl, as the wavefunction overlap decreases with separation, and with field, as the localisation increases. Most theoretical treatments of the Stark ladder to date have considered only the single particle behaviour in a field, neglecting excitonic effects due to the Coulomb interaction between the electron and hole. This gives a reasonable qualitative, and for wide minibands (where the axial excitonic correlation is weak) fairly quantitative description of the behaviou?*. The zero field superlattice density of states is split into a series of steps, each corresponding to the 2d density of states associated with a given n. The magnitude of the steps oscillates as the field is raised through intermediate values, until at high fields, when the potential drop across a single period is large compared to the inter-well coupling, the only step which remains significant is n = 0, located at the centre of the miniband. Since strong excitonic effects are associated with rapid variations in the density of states, it is easily deduced that the field will shift the main excitonic peak upwards, at the same time increasing its binding energy and oscillator strength.
ton peak at zero and high field, showing t,he expected increase in strength and binding energy on localisation.
R # 0 peaks are important. More general calculations for all the quasi-bound states have been carried out by using a superlattice scale envethe present author”,”
ble results. In this paper, the calculations are extended to cover the whole spectrum, allowing detailed comparisons to be made with experimental results. $11 Theory The full exciton Hamiltonian for a superlattice is difficult to solve exactly since it involves three non-separable degrees of freedom: the axial coordinates of both electron and hole and their separation in the plane. However, in superlattices with periods sufficiently short to give an appreciable miniband width, the miniband separation is usually large. As a result, the coulomb interaction and electric field cause little mixing of the minibands, provided that the potential drop across a single period is small compared to the miniband separation. With this restriction, the superlattice exciton can be constructed out of a basis of states from a single pair of electron and hole minibands”. Only states with total wavevector K = 0 can couple to the photon, so optically active states are described by only the relative wavevectors, k in the plane and p in the axial direction. Thus the basis states take the form
376
Superlattices O(k,q; r,n, ic,i,,)
r is the in-plane
=
eik.‘eiqdf.(&)fh(ih)
separation.
The axial coordinates
(1) r,, z,,
have been split into two parts: an integer, n, giving the separation in periods (nd) and ‘unit cell’ coordinates i,, i,,. f.(&) and fa(&,) are the superlattice periodic envelope functions; their q and k dependence is neglected here, which is reasonable for structures with well confined minibands where the variation is too small to be of importance. The approximation made in neglecting the q dependence tions for fc(i,),
can be avoided by using Wannier funcfh(ih) r2, but the k dependence
easily be incorporated. The coulomb interaction matrix
cannot
mixes the basis states
with
J d’rV,(r)ei(k-k’).‘ei(q-qJ)~
(2)
elements Vkk,qq, =
where V,(r)
F
is an effective
potential
aging the Coulomb interaction cell of the superlattice:
obtained
by aver-
U(r, z) over a single unit
and Microstructures,
Coulomb interaction
when considering
Treating
the mixing by perturbation
theory to all ortrans-
the mixing of the
basis states. Incorporating this in equation (3) and using the fact that _fe, fh are normalised functions with definite parity, it can be seen that the corresponding term in the superlattice envelope Hamiltonian is simply eFnd. Equation (6) is the superlattice analogue of the envelope function theory for bulk semiconductors. However, owing to the similarity between the energy and length scales for miniband and exciton, some of the usual simplifying approximations can not be applied. The discrete form of the axial coordinate (n) must be retained in order for Wannier-Stark localisation to take place, and is necessary to understand the dimensional crossover which occurs when the exciton binding energy is comparable to the miniband width. Furthermore, the averaging of the Coulomb potential to V,(r) cannot be neglected as it reduces the calculated binding energies by a large amount, typically a factor - 2 for narrow minibands. These considerations are usually minor effects in the bulk case, where they correspond to non-parabolicity and ‘central cell’ corrections. $111 Bound
ders leads to an integral equation for the Fourier form of the envelope function +(n, r)
Vol. 7, No. 4, 1990
Though
a considerable
States
simplification
of the full su-
perlattice Schriidinger equation, (6) is still not easily solved, being an eigenvalue problem in two variables. However, for superlattices in which Stark ladders are studied the inter-well coupling is generally quite weak, so can be treated using perturbation theory. Without
E(q, k) is the superlattice
miniband
hakl
a(q,k)=-+Aa,+aA,cosqd+... 2Pt where pt is the reduced in-plane
the axial hopping term, Tf, equation (6) can be easily solved numerically: the eigenstates consist of 2d exci-
dispersion (5)
mass and As, Ai,
...
are the inter-well hopping matrix elements. As a result of the periodicity of the structure, the q dependence can always be expanded in this form. As the coulomb matrix element depends only on q q’, k -- k’, the potential energy term is a convolution of the Fourier transform of V”(r) with $(q, k). Thus the integral equation can be transformed into a Schriidinger equation
Izt T&(n,r)
1
energy operator,
(6)
which takes the
= A0$(n,r)+Aizlt(n+l,r)+A13(n-l,r)+...
An axial electric
is large only for pairs with the same nP. Thus a good description of the bound states can be obtained by diagonalising the Tc term within a basis consisting of the lowest few (- 4) bound states on each layer. In fact, there will always be weak coupling to continuum states further down the field but at the same energy, so there are no true bound states, only quasi-bound resonances
d +Z + K(T) ?l(n,r) = W(n, r)
Tl is the axial kinetic form
ton states, localised in individual layers and labelled by the in-plane radial quantum number n,, and layer number n. Only states with zero axial angular momentum need to be considered since only they can couple to the to photon. The matrix elements of T( are proportional the in-plane overlaps of the two states involved, which
field, F, can easily be included
(7) in
the derivation of equation (6). It is represented by a term eF(f, - ih + nd) which must be added to the
broadened by coulomb assisted tunnelling. This effect will be examined in more detail in $IV when the full spectra are discussed. The structure used for the calculations is a 65A period superlattice, with 3OA GaAs wells separated by 35A Als.ssGs+.ssAs barriers, discussed in detail in reference (1). The miniband width used in the calculation is 40meV, rather than the 28meV deduced in that paper, but this change is justified by the good agreement obtained with experiment.
Superlattices
and Microstructures,
377
Vol. 7, No. 4, 1990
../(+1 ,O)
65w
.:. .:’
Superlattice
I
1
0
20
...-./
I
ELECTRIC
I
1
40
,
I
60
80
FIELD (kV/cm)
Fig.1 Theoretical Fan Diagram for the 65A GaAsAlO,ssGasssAs superlattice of reference (1). The area of each plotted point is proportional to the oscillator strength for the transition. tions is discussed in $111.
Fig.1 gives the results of the calculation
as a fan dia-
The labelling
of the transi-
label the high field transitions
in the figure.
Since the
gram, with the energies of the quasi-bound states shown as a function of field. The size of the plotted circles is proportional to the oscillator strength for the transition. At high fields, when the peak separation is large com-
oscillator strength depends on j$(0,0)12, only the n = 0 contribution is strong, with the higher InI peaks falling off rapidly with lnl, and with field as the localisstion increases. It is apparent that in the high field regime the
pared to the inter-well coupling and exciton binding energy, the results are fairly easy to understand. The wells are nearly uncoupled, so the eigenstates are very similar to the uncoupled basis states described above. There is a series of 2d-like exciton spectra, displaced in energy by eFnd from the centre of the miniband. The 2d exciton quantum numbers (n,n,) are not exact, but give a good description of the states - hence they are used to
-_1n peak is always stronger than +Inl, an asymmetry which is primarily excitonic in origin, as can be understood by a simple perturbation argument”: Using the uncoupled states as a basis, the oscillator strength for the n # 0 peaks depends on the admixture of n, = 0 states in their wavefunctions. Since the n = 0 binding energy is larger than that of the finite n peaks, the - /nj state is always closer in energy to n = 0 than the +lnl
378
and Microstructures,
Superlattices
state, so it mixes more strongly oscillator strength.
Vol. 7, No, 4, 7990
and the peak has greater
At lower fields, the interpretation
of the spectra
be-
comes more complicated.
When eFd is comparable to or smaller than the inter-well coupling strength, A,, the states in adjacent wells mix strongly, and anticrossings occur between the Stark ladder peaks. The most prominent example takes place when, on decreasing the field, the strongest n = 0 transition is approached by n = -1 peaks. The states repel, as can be seen by their energy shifts, and the oscillator strength is transferred to the lower peak. As the field is further reduced, this process is repeated many times, and the strongest oscillator strength is transferred sequentially to increasingly negative n peaks, ending up in the superlattice exciton
24
position at zero field. As a result of the anticrossing, an appropriate description of the states is difficult to find. The strongest peak always corresponds to the state with wavefunction dominated by the (0,O) basis state, but as the field is reduced, it passes through all the (n < 0,O) levels.
-,/,p&_ __,_A---’
t.
-6(1
-40
-20
§IV Spectra In order to model the full Stark ladder spectra,
equa-
gives the full spectrum, which must be inhomogeneously broadened sufficiently to smooth out the structure which arises from the discretisation. The results of the calculation for the 65A GaAsAlo.ssGaosaAs superlattice are shown in fig.2 (solid lines) and compared with the experimental spectra (dashed) of reference (1). In order to reproduce the experimental line widths, the amount of inhomogeneous broadening has to be increased with field, from 5meV at zero field to 7meV at GOkV/cm. The increase in the experimental widths is believed to be due to the more important
role
which electric field non-uniformity plays at high fields. The experimental results are well reproduced, except at higher energies where the light hole contributions, not included in the theory, become important. At zero field, the spectrum shows the exciton peak below the band-edge,
and a broad
Field (kV/cm)
0 ENERGY
tion (6) must be solved for both quasi-bound and continuum states. The method adopted here is a straight forward extension of that used in $111. The continuum is discretised by enclosing the exciton in an axial cylinder with radius 30 times the Bohr radius, so that by employing a sufficiently large basis, of up to 30 states for each n, an adequate range of energies is covered. Diagonalisation of the coupling term within this basis
‘saddle point’ exci-
tonic enhancement’s’14 spread across most of the miniband width. The spectrum is similar to those calculated with comparable by Chu and Chang l5 for structures miniband widths. When a weak field is applied, the exciton peak aquires a pronounced tail on the high energy side and the peak position moves up slowly. This
6.
Fig.2
Comparison
20
40
60
80
(meV)
of theoretical
spectra
(solid lines)
and experimental results (dashed lines) for the 65A GaAsAlo.ssGao.ssAs superlattice. The discrepancies at high energies are due to the light hole Stark Ladder peaks which are not included in the theoretical results.
corresponds
to the unresolved
anticrossing
behaviour,
as can be seen by comparison with fig.1. At the same time, structure begins to appear within the miniband. As the field is raised further, the individual transitions within the broadened exciton peak become resolved and it splits up into the -_Inl Stark ladder states. The structure within the miniband evolves into the +lnl peaks. At high fields, the n = 0 contributions dominate as expected, and the spectra look like those of an isolated quantum well. In the high field regime, the spectra show the effects of Coulomb assisted tunnelling on the broadening of the R = 0 peak, though this is partially obscured by the inhomogeneous contributions. It occurs because the effective Coulomb interaction is different for each R, so the uncoupled basis states are not the same on each layer, as would be the case without the interaction. As a result, a bound state can mix with continuum states at the same energy from further down the field, leading to a broadening of the peak, -4meV in the field range considered. In fact, since the potentials are similar at large enough T, the overlap is largest for states nearest to the edge, falling off up through the continuum. Thus this contribution to the broadening will decrease at high fields.
Superlattices
and Microstructures,
379
Vol. 7, No. 4, 1990
$V Conclusion An envelope function type Hamiltonian
has been de-
6 KFujiwara, H.Schneider, R.Cingolani and K.Ploog, Solid State Communications 72, 935 (1989). 0 B.Jogai and K.L.Wang, Physical Review B 55,653
rived for excitons in superlattices. It has been used to model the development of the optical absorption spec-
(1987). 7 D.Emin and C.F.Hart,
trum from its zero field superlattice form to the high field Stark localisation regime. Good agreement with
(1987). 8 J.Bleuse, G.Bastard
experimental
Letters 00, 220 (1988). 0 R.H.Yan, F.Laruelle and L.A.Coldren, ics Letters 55, 2002 (1989).
results has been obtained.
@Controller
HMSC London 1990
10 D.M.Whittaker,
References 1 E.E.Mendez,
F.Agullb-Rueda
and J.M.Hong,
Phys-
ical Review Letters 60, 2426 (1988). 2 P.Voisin, J.Bleuse, C.Bouche, S.Gaillard, C.Alibert, and A.Regreny, Physical Review Letters 61,163s (1988). S J.Bleuse, P.Voisin, M.Allovon and M.Quillec, Applied Physics Letters 58, 2632 (1988). 4 F.Agullo-Rueda, E.E.Mendez and J.M.Hong, Physical Review B 40, 1357 (1989).
Physical Review B Se,7353
and P.Voisin, Physical Review Applied Phys-
Physical Review B 41,3238
(1990).
11 D.M.Whittaker, M.S.Skolnick, G.W.Smith and C.R.Whitehouse, to be published in Physical Review B 42. 12 M.M.Dignam and J.E.Sipe, Physical Review Letters 64, 1787 (1990). l!l B.Deveaud, A.Chomette, F.Clerot, A.Regreny, J.C.Maan, R.Romestain,G.Bastard,H.ChuandY.C.Chang, Physical Review B 40, 5802 (1989). 14 R.H.Yan, R.J.Simes, HRibot, L.A.Coldren and A.C.Gossard, Applied Physics Letters 54, 1549 (1989). 15 H.Chu and Y.C.Chang, 2946 (1987).
Physical
Review B SB,
and Microstructures,
375
Vol. 7, No. 4, 1990
EXCITONIC
THEORY
OF STARK
LADDERS
IN SUPERLATTICES
D.M.Whittaker Royal Signals and Radar Establishment,
St Andrews Road, Great Malvern,
(Received A theoretical describe
model for excitons
the excitonic
30 July
WR14
3PS. UK.
1990)
in superlattices
Stark ladder states
Worcestershire.
is developed
which occur in optical
and used to
spectra
when an
axial electric field is applied. The model is based on a superlattice scale envelope function type approximation, using a basis of states from a single pair of electron and hole minibands. to be obtained. simulated
This provides sufficient simplification to allow numerical solutions The results are presented both in the form of a fan diagram and as
absorption
spectra.
Comparisons
are made with published
experimental
results.
$1 Introduction The optical
properties
of superlattices
in an axial
To understand the spectra in detail, it is necessary to include the coulomb interaction in the theory. Yan et al.’ obtained variational results for the main exci-
electric field are currently attracting considerable experimentaland theoretical”‘-” attention. As a result of the field, Wannier-Stark localisation occurs, with the superlattice miniband breaking down into a series of localised states, separated in energy by approximately the potential drop across a single period. The effect can be detected optically by observing transitions between
However, their results give no indication of the behaviour in the interesting intermediate field regime, where the
electrons and holes peaked in different wells, giving a Stark ladder of roughly equally spaced peaks, labelled R = 0, ztl . . . according to the electron-hole separation in
lope function type approximation, based on a single pair of electron and hole minibands. A similar approach has been adopted by Dignam and Sipe12, yielding compara-
periods. The strength of the peaks falls off rapidly with increasing /nl, as the wavefunction overlap decreases with separation, and with field, as the localisation increases. Most theoretical treatments of the Stark ladder to date have considered only the single particle behaviour in a field, neglecting excitonic effects due to the Coulomb interaction between the electron and hole. This gives a reasonable qualitative, and for wide minibands (where the axial excitonic correlation is weak) fairly quantitative description of the behaviou?*. The zero field superlattice density of states is split into a series of steps, each corresponding to the 2d density of states associated with a given n. The magnitude of the steps oscillates as the field is raised through intermediate values, until at high fields, when the potential drop across a single period is large compared to the inter-well coupling, the only step which remains significant is n = 0, located at the centre of the miniband. Since strong excitonic effects are associated with rapid variations in the density of states, it is easily deduced that the field will shift the main excitonic peak upwards, at the same time increasing its binding energy and oscillator strength.
ton peak at zero and high field, showing t,he expected increase in strength and binding energy on localisation.
R # 0 peaks are important. More general calculations for all the quasi-bound states have been carried out by using a superlattice scale envethe present author”,”
ble results. In this paper, the calculations are extended to cover the whole spectrum, allowing detailed comparisons to be made with experimental results. $11 Theory The full exciton Hamiltonian for a superlattice is difficult to solve exactly since it involves three non-separable degrees of freedom: the axial coordinates of both electron and hole and their separation in the plane. However, in superlattices with periods sufficiently short to give an appreciable miniband width, the miniband separation is usually large. As a result, the coulomb interaction and electric field cause little mixing of the minibands, provided that the potential drop across a single period is small compared to the miniband separation. With this restriction, the superlattice exciton can be constructed out of a basis of states from a single pair of electron and hole minibands”. Only states with total wavevector K = 0 can couple to the photon, so optically active states are described by only the relative wavevectors, k in the plane and p in the axial direction. Thus the basis states take the form
376
Superlattices O(k,q; r,n, ic,i,,)
r is the in-plane
=
eik.‘eiqdf.(&)fh(ih)
separation.
The axial coordinates
(1) r,, z,,
have been split into two parts: an integer, n, giving the separation in periods (nd) and ‘unit cell’ coordinates i,, i,,. f.(&) and fa(&,) are the superlattice periodic envelope functions; their q and k dependence is neglected here, which is reasonable for structures with well confined minibands where the variation is too small to be of importance. The approximation made in neglecting the q dependence tions for fc(i,),
can be avoided by using Wannier funcfh(ih) r2, but the k dependence
easily be incorporated. The coulomb interaction matrix
cannot
mixes the basis states
with
J d’rV,(r)ei(k-k’).‘ei(q-qJ)~
(2)
elements Vkk,qq, =
where V,(r)
F
is an effective
potential
aging the Coulomb interaction cell of the superlattice:
obtained
by aver-
U(r, z) over a single unit
and Microstructures,
Coulomb interaction
when considering
Treating
the mixing by perturbation
theory to all ortrans-
the mixing of the
basis states. Incorporating this in equation (3) and using the fact that _fe, fh are normalised functions with definite parity, it can be seen that the corresponding term in the superlattice envelope Hamiltonian is simply eFnd. Equation (6) is the superlattice analogue of the envelope function theory for bulk semiconductors. However, owing to the similarity between the energy and length scales for miniband and exciton, some of the usual simplifying approximations can not be applied. The discrete form of the axial coordinate (n) must be retained in order for Wannier-Stark localisation to take place, and is necessary to understand the dimensional crossover which occurs when the exciton binding energy is comparable to the miniband width. Furthermore, the averaging of the Coulomb potential to V,(r) cannot be neglected as it reduces the calculated binding energies by a large amount, typically a factor - 2 for narrow minibands. These considerations are usually minor effects in the bulk case, where they correspond to non-parabolicity and ‘central cell’ corrections. $111 Bound
ders leads to an integral equation for the Fourier form of the envelope function +(n, r)
Vol. 7, No. 4, 1990
Though
a considerable
States
simplification
of the full su-
perlattice Schriidinger equation, (6) is still not easily solved, being an eigenvalue problem in two variables. However, for superlattices in which Stark ladders are studied the inter-well coupling is generally quite weak, so can be treated using perturbation theory. Without
E(q, k) is the superlattice
miniband
hakl
a(q,k)=-+Aa,+aA,cosqd+... 2Pt where pt is the reduced in-plane
the axial hopping term, Tf, equation (6) can be easily solved numerically: the eigenstates consist of 2d exci-
dispersion (5)
mass and As, Ai,
...
are the inter-well hopping matrix elements. As a result of the periodicity of the structure, the q dependence can always be expanded in this form. As the coulomb matrix element depends only on q q’, k -- k’, the potential energy term is a convolution of the Fourier transform of V”(r) with $(q, k). Thus the integral equation can be transformed into a Schriidinger equation
Izt T&(n,r)
1
energy operator,
(6)
which takes the
= A0$(n,r)+Aizlt(n+l,r)+A13(n-l,r)+...
An axial electric
is large only for pairs with the same nP. Thus a good description of the bound states can be obtained by diagonalising the Tc term within a basis consisting of the lowest few (- 4) bound states on each layer. In fact, there will always be weak coupling to continuum states further down the field but at the same energy, so there are no true bound states, only quasi-bound resonances
d +Z + K(T) ?l(n,r) = W(n, r)
Tl is the axial kinetic form
ton states, localised in individual layers and labelled by the in-plane radial quantum number n,, and layer number n. Only states with zero axial angular momentum need to be considered since only they can couple to the to photon. The matrix elements of T( are proportional the in-plane overlaps of the two states involved, which
field, F, can easily be included
(7) in
the derivation of equation (6). It is represented by a term eF(f, - ih + nd) which must be added to the
broadened by coulomb assisted tunnelling. This effect will be examined in more detail in $IV when the full spectra are discussed. The structure used for the calculations is a 65A period superlattice, with 3OA GaAs wells separated by 35A Als.ssGs+.ssAs barriers, discussed in detail in reference (1). The miniband width used in the calculation is 40meV, rather than the 28meV deduced in that paper, but this change is justified by the good agreement obtained with experiment.
Superlattices
and Microstructures,
377
Vol. 7, No. 4, 1990
../(+1 ,O)
65w
.:. .:’
Superlattice
I
1
0
20
...-./
I
ELECTRIC
I
1
40
,
I
60
80
FIELD (kV/cm)
Fig.1 Theoretical Fan Diagram for the 65A GaAsAlO,ssGasssAs superlattice of reference (1). The area of each plotted point is proportional to the oscillator strength for the transition. tions is discussed in $111.
Fig.1 gives the results of the calculation
as a fan dia-
The labelling
of the transi-
label the high field transitions
in the figure.
Since the
gram, with the energies of the quasi-bound states shown as a function of field. The size of the plotted circles is proportional to the oscillator strength for the transition. At high fields, when the peak separation is large com-
oscillator strength depends on j$(0,0)12, only the n = 0 contribution is strong, with the higher InI peaks falling off rapidly with lnl, and with field as the localisstion increases. It is apparent that in the high field regime the
pared to the inter-well coupling and exciton binding energy, the results are fairly easy to understand. The wells are nearly uncoupled, so the eigenstates are very similar to the uncoupled basis states described above. There is a series of 2d-like exciton spectra, displaced in energy by eFnd from the centre of the miniband. The 2d exciton quantum numbers (n,n,) are not exact, but give a good description of the states - hence they are used to
-_1n peak is always stronger than +Inl, an asymmetry which is primarily excitonic in origin, as can be understood by a simple perturbation argument”: Using the uncoupled states as a basis, the oscillator strength for the n # 0 peaks depends on the admixture of n, = 0 states in their wavefunctions. Since the n = 0 binding energy is larger than that of the finite n peaks, the - /nj state is always closer in energy to n = 0 than the +lnl
378
and Microstructures,
Superlattices
state, so it mixes more strongly oscillator strength.
Vol. 7, No, 4, 7990
and the peak has greater
At lower fields, the interpretation
of the spectra
be-
comes more complicated.
When eFd is comparable to or smaller than the inter-well coupling strength, A,, the states in adjacent wells mix strongly, and anticrossings occur between the Stark ladder peaks. The most prominent example takes place when, on decreasing the field, the strongest n = 0 transition is approached by n = -1 peaks. The states repel, as can be seen by their energy shifts, and the oscillator strength is transferred to the lower peak. As the field is further reduced, this process is repeated many times, and the strongest oscillator strength is transferred sequentially to increasingly negative n peaks, ending up in the superlattice exciton
24
position at zero field. As a result of the anticrossing, an appropriate description of the states is difficult to find. The strongest peak always corresponds to the state with wavefunction dominated by the (0,O) basis state, but as the field is reduced, it passes through all the (n < 0,O) levels.
-,/,p&_ __,_A---’
t.
-6(1
-40
-20
§IV Spectra In order to model the full Stark ladder spectra,
equa-
gives the full spectrum, which must be inhomogeneously broadened sufficiently to smooth out the structure which arises from the discretisation. The results of the calculation for the 65A GaAsAlo.ssGaosaAs superlattice are shown in fig.2 (solid lines) and compared with the experimental spectra (dashed) of reference (1). In order to reproduce the experimental line widths, the amount of inhomogeneous broadening has to be increased with field, from 5meV at zero field to 7meV at GOkV/cm. The increase in the experimental widths is believed to be due to the more important
role
which electric field non-uniformity plays at high fields. The experimental results are well reproduced, except at higher energies where the light hole contributions, not included in the theory, become important. At zero field, the spectrum shows the exciton peak below the band-edge,
and a broad
Field (kV/cm)
0 ENERGY
tion (6) must be solved for both quasi-bound and continuum states. The method adopted here is a straight forward extension of that used in $111. The continuum is discretised by enclosing the exciton in an axial cylinder with radius 30 times the Bohr radius, so that by employing a sufficiently large basis, of up to 30 states for each n, an adequate range of energies is covered. Diagonalisation of the coupling term within this basis
‘saddle point’ exci-
tonic enhancement’s’14 spread across most of the miniband width. The spectrum is similar to those calculated with comparable by Chu and Chang l5 for structures miniband widths. When a weak field is applied, the exciton peak aquires a pronounced tail on the high energy side and the peak position moves up slowly. This
6.
Fig.2
Comparison
20
40
60
80
(meV)
of theoretical
spectra
(solid lines)
and experimental results (dashed lines) for the 65A GaAsAlo.ssGao.ssAs superlattice. The discrepancies at high energies are due to the light hole Stark Ladder peaks which are not included in the theoretical results.
corresponds
to the unresolved
anticrossing
behaviour,
as can be seen by comparison with fig.1. At the same time, structure begins to appear within the miniband. As the field is raised further, the individual transitions within the broadened exciton peak become resolved and it splits up into the -_Inl Stark ladder states. The structure within the miniband evolves into the +lnl peaks. At high fields, the n = 0 contributions dominate as expected, and the spectra look like those of an isolated quantum well. In the high field regime, the spectra show the effects of Coulomb assisted tunnelling on the broadening of the R = 0 peak, though this is partially obscured by the inhomogeneous contributions. It occurs because the effective Coulomb interaction is different for each R, so the uncoupled basis states are not the same on each layer, as would be the case without the interaction. As a result, a bound state can mix with continuum states at the same energy from further down the field, leading to a broadening of the peak, -4meV in the field range considered. In fact, since the potentials are similar at large enough T, the overlap is largest for states nearest to the edge, falling off up through the continuum. Thus this contribution to the broadening will decrease at high fields.
Superlattices
and Microstructures,
379
Vol. 7, No. 4, 1990
$V Conclusion An envelope function type Hamiltonian
has been de-
6 KFujiwara, H.Schneider, R.Cingolani and K.Ploog, Solid State Communications 72, 935 (1989). 0 B.Jogai and K.L.Wang, Physical Review B 55,653
rived for excitons in superlattices. It has been used to model the development of the optical absorption spec-
(1987). 7 D.Emin and C.F.Hart,
trum from its zero field superlattice form to the high field Stark localisation regime. Good agreement with
(1987). 8 J.Bleuse, G.Bastard
experimental
Letters 00, 220 (1988). 0 R.H.Yan, F.Laruelle and L.A.Coldren, ics Letters 55, 2002 (1989).
results has been obtained.
@Controller
HMSC London 1990
10 D.M.Whittaker,
References 1 E.E.Mendez,
F.Agullb-Rueda
and J.M.Hong,
Phys-
ical Review Letters 60, 2426 (1988). 2 P.Voisin, J.Bleuse, C.Bouche, S.Gaillard, C.Alibert, and A.Regreny, Physical Review Letters 61,163s (1988). S J.Bleuse, P.Voisin, M.Allovon and M.Quillec, Applied Physics Letters 58, 2632 (1988). 4 F.Agullo-Rueda, E.E.Mendez and J.M.Hong, Physical Review B 40, 1357 (1989).
Physical Review B Se,7353
and P.Voisin, Physical Review Applied Phys-
Physical Review B 41,3238
(1990).
11 D.M.Whittaker, M.S.Skolnick, G.W.Smith and C.R.Whitehouse, to be published in Physical Review B 42. 12 M.M.Dignam and J.E.Sipe, Physical Review Letters 64, 1787 (1990). l!l B.Deveaud, A.Chomette, F.Clerot, A.Regreny, J.C.Maan, R.Romestain,G.Bastard,H.ChuandY.C.Chang, Physical Review B 40, 5802 (1989). 14 R.H.Yan, R.J.Simes, HRibot, L.A.Coldren and A.C.Gossard, Applied Physics Letters 54, 1549 (1989). 15 H.Chu and Y.C.Chang, 2946 (1987).
Physical
Review B SB,