Polar optical modes in semiconductor nanostructures

Polar optical modes in semiconductor nanostructures

V.R. Velasco Instituto de Ciencia de Materiales, CSIC, Cantoblanco, 28049 Madrid, Spain

and F. Garcia-Moliner Cfitedra de Ciencia Contempor~nea, Universitat Jaume I, Campus de Borriol, 12071 Castell6n, Spain

ELSEVIER

Amsterdam-Lausanne-New York-Oxford-Shannon-Tokyo

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Contents

1. Introduction 2. Microscopic calculations 3. Macroscopic models 3.1. Dielectric continuum model 3.2. Hydrodynamic model 3.3, Continuum models with ad hoc constants 3.4, Complete phenomenological model for long-wavelength polar optical modes in semiconductor nanostructures 3.5. Polar optical phonons at surfaces and interfaces 3.6. Long-wavelength polar optical modes in GaAs based quantum wells 3.7. Long-wavelength polar optical modes in GaAs-A1As superlattices 3.8. Long-wavelength polar optical modes in quantum wires and quantum dots 4. Electron-phonon interaction 5. Envelope-function theory for polar optical phonons in semiconductor heterostructures 6. Further developments of the complete phenomenological model for more complicated heterostructures 7. Conclusions

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Acknowledgements References

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surface science reports Surface Science Reports 28 (1997) 123-176

ELSEVIER

Polar optical modes in semiconductor nanostructures V.R. V e l a s c o a , . , F. G a r c i a - M o l i n e r b a lnstituto de Ciencia de Materiales, CSIC, Cantoblanco, 28049 Madrid, Spain b C&edra de Ciencia Contemporgtnea, Universitat Jaume I, Campus de Borriol, 12071 Castell6n. Spain

Manuscript received in final form 21 February 1997

Abstract Polar optical modes play an important role in electron-phonon processes such as scattering rates, polaron effects and resonant Raman scattering in quantum wells and superlattices. Because of this there has been in recent years a strong interest in the development of a long-wave theory for optical modes in semiconductor nanostructures. This theory would be the equivalent of the effective mass theory for electrons. Besides microscopic calculations it should provide a satisfactory theoretical model to study the long-wave limit, to which most experimental evidence is circumscribed. Important elements in this type of theory are the inclusion of the bulk spatial dispersion of the optical modes together with the fact that, at an interface between two media, mechanical and electromagnetic boundary conditions must be satisfied. In some cases, like InAs/GaSb and related superlattices, the details of the interface structure are also important. We discuss here the different approaches employed to study the long-wave limit in these systems, including other approaches in which the envelope function model is derived directly from microscopic lattice dynamics. Keywords: Phonons; Heterostructures; Quantum wells; Superlattices

1. Introduction In recent years c o n s i d e r a b l e a t t e n t i o n has been paid to the theoretical [1 42] and e x p e r i m e n t a l [ 4 3 - 6 7 ] s t u d y o f p o l a r optical p h o n o n s in l a y e r e d s e m i c o n d u c t o r h e t e r o s t r u c t u r e s , m a i n l y superlattices (SLs) a n d q u a n t u m wells (QWs). O n the theoretical side, besides m i c r o s c o p i c calculations [ 1 - 1 9 ] there have been n u m e r o u s efforts to d e v e l o p l o n g - w a v e p h e n o m e n o l o g i c a l models, simple to use a n d c a p a b l e o f p r o v i d i n g sufficiently reliable results which can be c o n t r a s t e d with the e x p e r i m e n t a l d a t a a n d the m o r e accurate, b u t m o r e costly, m i c r o s c o p i c calculations. In fact all the

* Corresponding author. Tel.: + 34 1 3349045; fax: + 34 1 372 06 23; e-mail: [email protected]. 0167-5729/97/$32.00 © 1997 Elsevier Science B.V. All rights reserved P I I S0167- 5729(97)00006-X

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basic qualitative features can be obtained with these models, and it is then clear that they provide, in principle, a very useful tool to study the polar phonon modes in semiconductor heterostructures. To this end several theoretical models have been proposed and used [-20-42]. They represent substantially different and even diverging viewpoints, and have met with varying degrees of success in the interpretation of observed experimental facts. These long wavelength models would be the equivalent of the envelope function approximation for electrons [68-72], which in practice has proved very useful. On the experimental side, different techniques have been used to study the vibrational properties of semiconductor nanostructures. Raman-scattering data are available for a wide range of systems, especially of the GaAs/A1As type [43-46], and also for InAs/GaSb and related heterostructures [47-53]. The experimental data show the existence of different types of polar optical modes. Some of these modes have amplitudes mainly concentrated in one of the constituent slabs, and they are usually termed confined modes [-4,43,44], while those with the amplitudes spreading to both constituents, but tending to concentrate close to the interfaces are called interface modes [-24,46]. The Raman experiments do not provide the spatial dependence of the displacement amplitudes, but this can be surmised from the fact that the observed frequencies are forbidden in one of the constituent materials (guided or confined modes) or in both (interface modes). The latter have been attributed to interface disorder [-44]. Moreover the geometry of the Raman-scattering experiment can be chosen to detect modes with different symmetries [73]. It is then possible to establish experimentally whether the amplitude (or the potential) of the observed modes is even or odd with respect to the center of the slabs [44]. Micro-Raman spectroscopy, which takes advantage of the focusing of light to a micrometer sized spot allows one to make backscattering measurements on surfaces with different orientations. In this way it gives access to in-plane wave vectors [57,58]. This has revealed the zone-center anisotropy of optical phonons and shown that the sequence of confined modes is modified when the propagating wave vector is changed from perpendicular to parallel to the interfaces. Besides Raman spectroscopy some time-resolved optical techniques have been developed [5967] which allow for the study of electron and hole dynamics on a picosecond and subpicosecond timescale. This temporal regime is important for the study of relaxation of photoexcited carriers as well as for the incoherent tunneling across thin barriers, and is dominated by the carrier interaction with the optical phonon modes. It has been shown that the coupling to acoustic phonons leads to much slower processes [17]. All these facts established by experimental evidence, together with the key features of the results obtained in microscopic calculations should be taken as the criterion to decide whether a phenomenological model is reasonable or not, while some lack of quantitative accuracy can, to some extent, be expected and accepted. We shall discuss here the theoretical aspects, focusing mainly on the long-wavelength regime. In Section 2 we shall discuss the microscopic calculations and the results obtained from them. In Section 3 we shall examine the different long-wavelength models used. We shall analyze their foundation together with their differences and shortcomings when compared with the microscopic calculations and experimental data. A brief analysis of the electron-phonon interaction is presented in Section 4, while Section 5 deals with the envelope-function theory for polar optical phonons. Further developments of the complete phenomenological model are presented in Section 6 and finally conclusions are given in Section 7.

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2. Microscopic calculations Early Raman work on semiconductor heterostructures was mainly carried out in the backscattering geometry [-72]. This does not probe into modes with momentum parallel to the interfaces, which intervene in the interaction processes with confined electrons. The bulk of this work was devoted, as indicated above, to the study of GaAs/A1As and GaAs/AIGaAs structures grown along the [0 0 1] direction [-43-46]. Likewise, the main effort on the theoretical side initially focused on the modes which were accessible to Raman backscattering for GaAs/A1GaAs and GaAs/A1As systems with (0 0 1) interfaces. Along the [-0 0 1] direction in the zincblende structure, both in bulk and in heterostructures, each atomic plane contains only one type of atoms (anions or cations) and in these modes vibrates as a whole. Longitudinal and transverse vibrations are then decoupled by symmetry and the full three-dimensional (3D) problem can be mapped exactly onto three one-dimensional ( 1D) problems. These are those of a linear chain of atomic planes connected by interplanar force constants. All these facts allowed for the intensive use of 1D microscopic models [1-8]. These models covered a wide range of sophistication, going from simple linear chains with nearest-neighbor interactions to 1D models which make use of interplanar force constants derived from ab initio local density calculations of the phonon spectra of bulk GaAs [74]. Because GaAs and AlAs have well-separated optical branches, due to the fact that the mass of Al is much lighter than that of Ga, the resulting heterostructure phonon spectrum is easy to understand in terms of matching of bulk waves. In the optical range spanning the frequencies of one of the constituent materials, the vibration is forbidden in the other one, and vice versa. The resulting phonon modes are oscillatory solutions in one material, exponentially decaying into the other one with an attenuation rate given by the imaginary part of the wave vector. These are called confined modes, AlAs-like or GaAs-like, depending on the vibrating material. The decay length into the adjacent material is typically of the order of one monolayer [3] leading to the uncoupling of the successive slabs of the constituent materials and to dispersionless phonon branches along the [0 0 1] direction in the case of the SL. It is possible to obtain some additional information by considering a SL made ofn ~monolayers of GaAs and n 2 monolayers of AlAs. A Kronig-Penney linear chain model with nearest-neighbor interactions only [2], yields a simple dispersion relation for the longitudinal modes propagating along the growth direction, in closed form c o s [ q a ( n 1 + n2) ] -----c o s ( n a k ~ a ) c o s ( n z k z a ) - F s i n ( n l k l a ) s i n ( n 2 k 2 a )

~l)

F = 1 - cos(kla) cos(k2a) sin(kla) sin(k2a) '

(2)

with

where ki is the wave vector (real or complex) at frequency ~o in constituent medium i (i = 1,2), q the SL wave vector along the growth direction, and a is the common monolayer thickness in GaAs and AlAs. Let us assume k 2 be infinite and pure imaginary, i.e. total confinement in the GaAs slab. Then (1) becomes sin((nl + 1)kla)= 0,

(3)

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and the frequencies of the phonon modes confined in the GaAs slab are those of the bulk GaAs ones at wave vector kx

m/¢

m

(n l + l ) a '

m=+l,+2, m

m

°

,

°

,

+ n 1. m

(4)

This expression can be easily understood by taking into account that the interracial As atom is still vibrating and that the first pinned atom is the first AI atom in the AlAs slab. A similar expression can be obtained for the phonon modes confined in the AlAs slab. In this way one obtains a correspondence between the SL confined modes and the bulk dispersion relation of the constituent materials. It is clear that an exact correspondence between the SL mode frequencies and the bulk frequencies, at specified wave vectors, has a twofold interest: in the first place, it would be possible to have an exact determination of the layer thicknesses, or if this is known by other techniques, it could yield an estimate.of the departure from the predicted behavior, due to disorder, impurities, etc. Alternatively, it could be used to obtain the bulk phonon dispersion relation from measurements on SLs when the bulk phonon dispersion relation is not known, because of the difficulty in growing stable bulk samples, like in the AlAs case, or for other reasons. In fact this idea was employed to obtain the dispersion relation of AlAs from measurements on several (GaAs),-(A1As), SLs (n = 4, 6, 8) [54]. It is clear that (4) can be generalized by substituting (n + 1) by an n¢rf [8], which is the number of monolayers over which the mode actually extends. Different determinations for this value have been given by several authors [11,12,23]. It has been found that GaAs-like modes coincide to a very good approximation with the known bulk dispersion relation if neff-- n 4- 1 is employed to evaluate the appropriate bulk wave vector k,, [54]. For modes of high index rn a departure from the predicted behavior has been found in the different experimental measurements [47,54-56]. A comparison of the calculated envelope of the displacement pattern associated with each SL confined mode with cos(k,,z) and sin(kmz ) was made in [8] for a (GaAs)s-(A1As)8 SL. In this case a 1D model with interplanar force constants derived from ab initio local density calculations was employed. Because of the similar bonding of all III-V semiconductors [74], both constituent materials and the interfaces in the SL were considered to have the same force constants, the differences being in the different atomic masses and effective charges, only. It was found that all the GaAs-like phonon modes had displacements which were well represented by sin or cos functions. The same was true for the first four AlAs-like phonon modes. However, for the low-lying AlAs-like phonon modes, a deviation of the sinusoidal form was found, because the amplitude of the oscillations in the AlAs slab was not constant. This can be understood, in principle, because of the fact that the As plane at the interface is shared between the two constituent slabs, and it has a bonding which is different from those of the As planes inside the constituent slabs. In the case of SL modes with high m values, which have a large vibration amplitude close to the interface, the specific bonding at the interface may lead to a modification of the SL confined mode frequencies as compared with those of a clamped slab, even in the case of ideal interfaces. In order to interpret micro-Raman experiments [57,58] it was necessary to put forward 3D microscopic models [9-18] giving access to in-plane wave vectors. From these models it was possible to obtain the phonon dispersion relations and displacements at any wave vector. They also showed the anisotropic behavior of optical phonons at the SL zone-center, which is due to the anisotropic Coulomb interaction in layered systems. This can be easily understood by following the arguments of Ren et al. [11], who extended the analysis for simple crystals [76-79]. The Coulomb

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interaction between the Ith and the/'th ionic layers when k ~ 0 can be written as

k . Dis(l,l'), c CC(k,l,l') - 4nQt -~pQr ki~-~+

(5)

where i a n d j denote the three spatial directions x, y and z, and Qt and Qt, represent atomic-transfer charges of atoms in the/th and the/'th ionic layers, respectively, with l, l' = 1, 2,..., 2Np, where Np is the total number of monolayers in a SL period. The first term on the right-hand side of (5) takes different values when k approaches zero from different directions. The second term on the right-hand side is direction independent. The short-range interaction matrix between sublattices I and l' at k = 0 can be similarly written as DSR(l, l') and it can be shown that D c and D sR have the same symmetry property. For a SL the direction independent term D sR q-- D c in the dynamic matrix at k = 0 has the following form:

a

.

(6)

0 When the direction dependent terms k i k f k 2 are added on, the phonon frequencies for the SL at k = 0 become direction (angular) dependent. Obviously these highly sophisticated 3D models would be applied to the study and interpretation of the time-resolved optical techniques [59-67], probing the interaction of the carriers with the optical phonon modes. In order to study the electron-phonon interaction, which plays an important role in the determination of the fast optical and electrical response of quasi-2D systems to external disturbances, Molinari et al. [15] performed a study based on a fully microscopic ab initio calculation of the phonon spectra in quasi-2D semiconductor systems [18]. They obtained the dynamical matrix ofa SL in terms of real-space interatomic force constants derived from an ab initio local density linear-response calculation performed with nonlocal norm-conserving pseudopotenrials and a large plane wave basis set. Due to the great similarity between the Ga and Al cations, they showed that the difference in the force constants of GaAs and AlAs was very small, and also that the force constants of the virtual crystal (a periodic crystal with average Ga and A1 pseudopotential as cationic potential) reproduced the phonon spectra of bulk GaAs and AlAs and of GaAs/A1As SLs rather accurately [14,16,18]. They also obtained the dynamical matrix by using such virtual-crystal force constants with the appropriate masses. Phonon frequencies and displacements were then obtained by direct diagonalization. The method has basically the accuracy and predictive capacity associated with ab initio calculations, while remaining tractable for systems with a large number of atoms in the unit cell. Because it is much easier to perform lattice-dynamical calculations for systems having 3D periodicity, the case of the GaAs QW was simulated within a supercell geometry. The QW frequencies and displacements were extracted from the results of a SL calculation with a thick AlAs barrier. In this way they studied a 20-monolayer GaAs QW from the results of a (GaAs)2o-(A1As)2o SL [ 15] and obtained the angular dependence of the phonon mode frequencies when k ~ 0. In their results two separate GaAs and AlAs-like frequency ranges can be observed. For k parallel to the growth direction ([0 0 1]), the longitudinal and transverse phonons are decoupled. For arbitrary

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directions this is not true. The z-component of the vibration amplitude u z for the highest GaAs-like longitudinal optic (LO) modes (LO1,LO2 and LO3) is strictly confined in the GaAs slab and vanishes in the AlAs side very close to the interface. When the direction of the vanishing wave vector varies from I-0 0 1] to [1 0 0] the LO 1 mode shifts to a lower frequency and ~LO2becomes the highest mode. When k x increases all the confined modes are deformed and show a sharper decay at the interfaces. The mode coming from LO1 exhibits increasing localization at the interfaces with increasing parallel wave vector, and is then identified as an interface mode (IF1). A second interface mode in the GaAs-like range (IF2) comes from the lowest-order confined transverse optic (TO) mode. The same qualitative behavior is true for the optical phonons in the AlAs-like range. In this range the interface modes fall in the gap between the remaining phonon modes of the LO and TO branches. Therefore, they have a minor hybridization with the confined modes of the same parity. All these modes were calculated for a (GaAs)2o-(A1As)6 o SL, in order to avoid the coupling of neighboring layers in a sufficiently large range of parallel wave vectors. For the smallest values of the parallel wave vector a thicker AlAs barrier would be needed. In the case of polar crystals the dynamical charges associated with the ions produce long range electric fields which can couple to the charged carriers 1,76]. The electric potential at the r point created by the vibrating ions is given by e~ o(r) =

1

-

v i, " _ R R , rl

(7)

oo

e~ being the effective charge of the nth ion located at r, in cell R and eo~is the high-frequency dielectric constant of the appropriate constituent material. The effective charges were obtained from the ab initio calculation I-16,18]. In the case of a single QW the envelope of the electric potential associated to the mode up is given by [15] e - X~z-z,i ,

(8)

where A is the area of the 2D unit cell, N Othe number of lattice points in the normalization volume and I¢ is the parallel wave vector. The question of the field normalization will be discussed in Section 4. From here it is possible to obtain the scattering rates for the electron-phonon interaction by using Fermi's "golden rule"

F ( i , f ) = 2--~I ( f l H o

phi/)12b(Ef

- E,),

(9)

where i a n d f represent the initial and final states of the crystal (electrons and phonons), respectively. The interaction Hamiltonian, H0 ph is given b y - etp, where e is the electron charge and ~p is the electrostatic potential associated to the lattice vibrations and discussed above. For a single QW grown along the [0 0 1] direction, the total scattering rate of an electron with parallel wave vector a" from subband i to subband j is given by [15] e'ij(K) = ~ F/~(I¢, K', v) a", v

2~ze2 -

h

..[_l (h2K2 h2K'2 I¢',v

*

2m

) '

(10)

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with t¢ = K" + g, where N v is the Bose-Einstein function for the phonon occupation, m* the electron effective mass, and the coupling factor Sij is given by S(j(g, v) -- ~bi(z) ~bj(z) ~o(~, g, z) d z.

(11 )

4~i(z)and ~bj(z)are the envelope functions for an electron in subbands i and j, respectively, and ~0is the electrostatic potential associated with the quantized phonon mode #. For intersubband transitions (from i to j) (12)

heo* = hcov(g ) +_ (E i - E,)

is the sum (difference) of the energy of the emitted (absorbed) optical phonon and the energy difference between the initial and final electronic subband ground states. For intrasubband transitions (i =j), it follows from (11) that Si~= 0, except for phonons with potentials of the same parity. Rficker et al. [15] applied this scheme to a GaAs single QW, having a thickness of 56Aand a barrier of leV. For the electron states they used the effective-mass approximation [80], and they considered the two lowest electronic subbands, only. They obtained the following results: The first confined mode contribution amounts approximately to 28% of the total intrasubband rate. The third mode contribution is much weaker. The contribution of the second confined mode is the dominant one for the intrasubband transition. The interface GaAs-like modes have a significant contribution, totalling the 14% of the interaction. The AlAs-like interface modes provide a noticeable contribution to the scattering rate, due to the fact that the electrostatic potential associated to them penetrates far into the GaAs slab for the wave vectors of interest. The contribution of the remaining AlAs-like modes is negligible. The times for intrasubband and intersubband emission at room temperature, resulting from these calculations, were 0.09 and 1.0 ps, respectively, thus confirming that the slow cooling rate detected experimentally could not be associated with phonon confinement and had to be related to hot-phonon effects [17,67]. It must be noted that the relative contribution of the interface modes to the total scattering depends very strongly on the well thickness. It is clear that the relative contribution of GaAs-like and AlAs-like interface modes also changes. This also produces changes in the total scattering rates, because the coupling of these modes is much stronger in AlAs than in GaAs. The microscopic models and subsequent calculations provide a series of trends and quantitative results for the polar optical phonons in semiconductor heterostructures which, besides their value by themselves, when added to the experimental data constitute a validation ground for the simpler macroscopic models and their predictions. o

3. Macroscopic models As mentioned above, most of the available experimental information on the modes here considered pertains to the long-wave range. Also, Fr61ich-type electron-phonon scattering is strongly inhibited for high momentum transfers and long-wave modes predominate again. Consequently, a substantial amount of literature on the subject is devoted to efforts to develop reasonable long-wave phenomenological models [20-42], with substantially smaller computational demands.

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These can also deal with heterostructures involving larger widths without the corresponding increase in computer time and memory required by microscopic calculations. Several proposals have been made which have met with varying, degrees of success in the interpretation of observed experimental facts and when compared with the results of microscopic calculations. The essence of a long-wavelength phenomenological model is that one uses differential calculus and it is necessary to find a way to match the solutions at the interfaces. Of course it could be argued [21,81], that in principle the differential equations are not valid in the immediate vicinity of the interfaces, when finite changes take place over microscopic distances. Any phenomenological model of the matching problem must necessarily be an approximation, just as it is in all the existing matching calculations for long-wavelength acoustic or piezoelectric modes. Differential calculus is used to match at abrupt interfaces and the model works quite well. The same can be said of envelope function matching calculations for electronic states [69-72,82]. It is then clear that the standard type of phenomenological model cannot be rigorously justified on formal grounds. Nevertheless, it is possible to start from a simple phenomenological model and obtain a solution which has all the general properties one may require and reproduces to a good approximation experimental results and microscopic calculations. The key issue is related to the matching boundary conditions, although another important aspect [31] will be discussed in Section 3.4. One option has been, for instance, to approach the problem with a dielectric model, like in the first study for the modes of a film [85], and impose electrostatic continuity. One then achieves electrostatic continuity but at the expense of mechanical discontinuity. Alternatively, one can start from a mechanical equation of motion for the vibration amplitude [20] and then impose mechanical continuity, but this produces a discontinuous electrostatic potential. The incompatibility between the separate use of these two approaches was soon noted and stressed [4,24]. The physical inconsistencies in these models led to partial failures in their predictions, when compared to microscopic calculations and experimental data. It is then clear that we must require the theoretical internal consistency of the model proposed, besides satisfactory agreement with the key features of experimental evidence and with the results of accurate microscopic calculations. Different proposals have been made to this effect. One is to start from a phenomenological model for the vibrational wave and then introduce ad hoc boundary conditions which force the electrostatic continuity in the analysis [21,84]. The idea is to save the simplicity of the phenomenological model with parameters determined from a fit to bulk phonon-dispersion relations, for use in problems of practical interest like the calculation of Raman-scattering efficiencies and the derivation of an electron-phonon interaction Hamiltonian. Another approach was considered [22] in a study of these problems with a model in which the vibrations are totally confined by infinitely rigid barriers. The mechanical amplitudes then vanish at the boundaries but not so the electrostatic potential which is continuous. This model describes well the confined modes, which are totally confined due to the rigid barrier condition, and gives correctly the observed symmetries, but it is severely constrained by the approximation inherent to it and cannot describe interface modes. The simultaneous account of the mechanical and electrostatic field and the corresponding, and compatible, boundary conditions are the significant differences of this approach as compared to others. This approach was later used, but relaxing the restriction to rigid barriers [28,31,32,34-36,38-41]. Another possible way is to derive the envelope function model directly from microscopic lattice dynamics [25] or to develop an envelope-function formalism for the treatment of lattice dynamics

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problems [42] and from there to derive a set of long-wavelength equations which have the same form as the phenomenological equations. We shall now discuss the various approaches in order to have a perspective on the strengths and limitations of these different models. 3.1. Dielectric continuum model This is the oldest model and was first invoked in [43] to explain the observations in right-angle Raman scattering from GaAs/A1GaAs heterostructures. It is based on the Kliewer-Fuchs analysis for the optical phonons of a slab [85]. It is well known that in the zincblende structure the k = 0 optical phonons are both Raman and infrared active. The infrared activity is due to the existence of an electric dipole m o m e n t associated with the relative atomic displacement, vibrational amplitude, u. Because of this the Raman phonons are split into a LO singlet and a TO doublet, except for forward scattering, where polariton effects are observed [86]. The electrostatic fields associated with these phonons are subject also to boundary conditions. The theoretical frame of the model is standard electrostatics. For the LO phonons, neglecting retardation effects, the electrostatic field E derives from an electrostatic potential ~o E = -V~o

(13)

while the long-wave, frequency dependent dielectric function of the ionic crystal is [76]: ~((0) =

(D 2 - - ( 0 2 0

2 ,

(14)

gcc,(02 __ (0TO

where (0To and (0to are the transverse (TO) and longitudinal (LO) optic p h o n o n frequencies, respectively. In the absence of free charges, and neglecting retardation effects, the electric field E a n d the electric displacement must satisfy Maxwell's equations curiE = 0,

divD = 0,

D = e((0)E,

(15)

whence the Laplace equation e((0)vZ(p

~--- 0.

(16)

If we consider heterostructures formed by two different constituent materials, Eq. (16) must be solved in the two media. Two different solutions, corresponding to the different dielectric functions of the two media, will be obtained. These solutions must be matched at the interfaces by imposing the boundary conditions of continuity of ~0 and D z. In the case of a SL the electrostatic potential must also satisfy the Bloch theorem. In order to show the main features of this model let us briefly discuss a 1/2 SL. e~((0) is the dielectric function of medium # = 1, 2. If the SL has been grown along the z axis the solutions will be labeled by a (x, q) wave vector in the SL Brillouin zone. A solution to Eq. (16) is given by ~o = e i~Pf(z),

(17)

where p = (x, y). The functionf(z) satisfies the Bloch condition f ( z + d) = eiqdf(z),

(18)

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where d = N l d ~ + N 2 d 2 is the SL period. We shall first study solutions with el(co)=0, which correspond to the bulk L O - m o d e in m e d i u m 1. With this condition, (16) is automatically satisfied in the slabs of m e d i u m 1. For the slabs of m e d i u m 2, (16) becomes d2f(z) xEf(z). dS- -

(19)

It is reasonable then to consider the following set of solutions, having placed the origin z = 0 at the center of the m e d i u m 1 slab:

f(z) = A 1sin(qlz ) + B lcos(klz )

(20)

in m e d i u m 1, and f(z) = A 2 sinh (~cz) + B 2 cosh (~cz) in m e d i u m 2. By applying the b o u n d a r y conditions at the interfaces we obtain

A 1 =0,

mn ql - N l d l ,

B 1 = 0,

ql - N l d l ,

(21) A 2 -- B E = 0,

and

m = 1,3,5,...

(22)

m = 2, 4, 6,...

(23)

or mT~

These modes are confined to m e d i u m 1. Similar solutions confined to m e d i u m 2 are obtained for e2(~o) = 0. The p h o n o n amplitude u corresponding to L O - m o d e s is proportional to the polarization, which is also proportional to the electric potential u ocP = - 4roE = 4roVe0.

(24)

F o r m odd, with the normalization convention chosen by the authors for the eigenvectors u, these are [87]: ULo(X, Z)

x/NTd 1 (q~ + 1x2) 1/2e i~x0x • cos(qlz)ex, 0, ql sm(qlz)ez),

(25)

ex and e z being the unit vectors in the x and z directions, respectively. We can see from Eq. (25) that: (i) the modes do not depend on q, reflecting the total confinement within this model; (ii) all confined modes have the same bulk L O frequency determined from el(~o) = 0. This had to be expected, because the bulk model embodied in (14) does not include dispersion; (iii) for q << x << ql, the p h o n o n s have essentially T O character but L O frequencies. This fact was invoked in 1-89] to explain the observation of modes with LO frequency but with R a m a n selection rules characteristic of T O phonons. A distinctive feature emerging from Eq. (25) is that while the electrostatic potential q9 vanishes at the 1/2 interfaces (z = _ N~d~/2), the vibrational amplitudes, which are proportional to V~o, have m a x i m a at these points and fall abruptly to zero. This discontinuity is in contradiction with microscopic models, where displacement patterns are continuous, and is clearly unphysical.

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If we compare the mth confined mode of the dielectric continuum model with those obtained from microscopic calculations, the situation is even worse, because they have opposite symmetry. The z component of Eq. (25) for the m = 1 mode is proportional to sin(q~z), while the m = 1 mode coming out of microscopic calculations has a u z amplitude proportional to cos(qaz) [8,15]. This has important consequences for the electron-phonon interaction in heterostructures, which is also important in transport properties, as q~(z) has correspondingly well-defined symmetries characteristic of the model. In the case of QWs the electronic wave functions have alternating even-odd symmetry under inversions with respect to the z = 0 plane, the ground state being even and the first excited state is odd. Thus the symmetry of the potential is decisive in determining which electronp h o n o n matrix elements vanish and a model with the wrong symmetries makes in this respect just the wrong prediction. The strongest resonant contribution to the Raman cross-section usually is given by intrasubband terms of the Fr61ich-type interaction. Then the electrostatic potential must be even to couple an electronic subband with itself, and this implies that u~ is odd. According to the microscopic calculations this is the situation for phonons having m = 2, 4, 6 .... , and consequently one expects that these modes should result in strongly enhanced near resonances. This was the result obtained in [44], which proved that even-number phonons are the dominating ones near resonance. Besides the confined modes in medium 1, there are those confined in medium 2, which are obtained when imposing e2 (m) = 0. But besides these modes there are additional solutions, obtained when ~1(~) :~ 0 and ~2 ((2)) -7(: 0. The normal mode dispersion relation is obtained [87] from the usual equation of the Kronig-Penney type cos(qd) - e~ + e~ sinh(•Nld0 sinh(KN2d2 ) + cosh(KNld0 2~ag 2

cosh(KN2d2).

(26)

This is a fourth-order equation, as ~ (co) and ~2(~) are quadratic in ~, and it gives four solutions for each wave vector, two being AlAs-like and two GaAs-like phonons. For finite ~c, the amplitude of these modes decays exponentially away from the interfaces, and these are the interface modes. R~cker et al. [15] calculated the scattering rates for the electron-optical p h o n o n interaction in a QW. In order to obtain comparable results from the microscopic and macroscopic calculations, they used in the macroscopic models the p h o n o n frequencies and effective charges given by the microscopic calculations. The results obtained with the dielectric continuum model turned out to agree rather well with those of the microscopic calculations. They explained this fact by noting that the relevant wave vectors • for the electron-phonon scattering in QWs have the largest component parallel to the interfaces. It can be seen that for K= 0 and small values of q the LO1 mode has a nodeless displacement amplitude, nearly vanishing at the interfaces, while the successive modes exhibit an increasing number of nodes and an analogous behavior at the interfaces. For q = 0 and small values of ~ the LO1 mode goes to a lower frequency, while the LO2 mode becomes the highest frequency mode. For increasing values of ~ the LO1 mode is increasingly more localized in the interface region, and was therefore identified as an interface mode. Thus, they concluded that although the dielectric continuum model does not reproduce accurately the amplitude pattern at all K, as indicated above, their deviations are more important in a small region near the interface, for the most relevant values of K. The electronic wave functions have a small amplitude near the interface, and it is then clear that the discrepancies of the displacement pattern near the interface will have a small effect. We shall discuss this subject further later.

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In spite of the reasonable agreement found for the scattering rates, the displacement discontinuities found in the dielectric c o n t i n u u m model are unphysical, and some other models were proposed to circumvent this limitation.

3.2. Hydrodynamic model A different model was p r o p o s e d by Babiker [20], by generalizing the Born and H u a n g [76] c o n t i n u u m approach to the optical p h o n o n s in polar materials. This generalization included the spatial dispersion, which was absent in the dielectric c o n t i n u u m model. The relative ionic displacement u(r, t) in a given m e d i u m is coupled to the electric field E and the polarization field P by the equations ii = - 092u + [(eo -- e~)/4rC]'/Z09T E -- flzav(V'u) -- fl~Vzu, e = [(eo - eo~)/4rc]1/Z09TU + [(e~ -- 1)/4rolE,

(27)

where 09+ stands for 09TOand eo is the static dielectric constant. The terms including fla and/3 b are a generalization of the Born and H u a n g approach, and describe the effects of spatial dispersion, fla and fib have the nature of velocity parameters. Eq. (27) must be supplemented with Maxwell's equations d i v e = - 47tdive

(28)

c u r l H = 1(/~ + 4rclO),

(29)

and

coupling E and H to the polarization charge and current densities pp and Jp pp = - d i v e ,

Jp = P.

(30)

The set of equations is completed by the source-free Maxwell's equations d i v H = 0,

curl E = - II:L c

(31)

F o r normal m o d e s with time dependence e -i°~t the electric field E(r, 09) and the polarization field P(r, 09) were expressed in terms of the amplitude u(r, 09) as e = [4rc/(e o - e~)09231/2 [092 _ 092 + fl~Vdiv + flzVZ]u, P = [1/41r092(eo - ~oo)] 1/2 {(e 0 - - 1)09 2 - - ( E ~ -

1)09 2 + ( e ~ - -

(32) l)fl]Vdiv + f12V2]} u.

The electric displacement D is given by D = E + 47rP.

(33)

F r o m (32) we obtain D = [4rc/(e o -/3~)(.z)2]1/2 {eo(D 2 - - ec~09 2 -~- e~ [flEVdiv + f12V23} u

=- Ou

(34)

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137

As D must satisfy (15), this gives divOu = 0,

(35)

which opens two possibilities div u = 0,

Ou ~ 0

(36)

divu ¢: 0.

(37)

or

Ou = 0,

The solutions satisfying (36) are transverse and have the bulk dispersion relation k2c 2 (% - eo )o) O)-z- - e~ + O)2 _ O)2 _ fl~k 2.

(38)

In the absence of the dispersive term including f12, (38) is identical to the usual polariton dispersion relation. The fields obeying (37) are longitudinal with displacement D, which in terms of the relative ionic displacement u, can be written as [(EO/e~)O)2 __ O))2 .31_(fla2 .31_fl2)V 2]//(0), r) = 0.

(39)

By defining the limiting LO frequency o) L by the usual relation O)~ = %o)2,

(40)

and a new velocity parameter fl by f12 = flJ + fib2,

(41)

Eq. (39) can be written as (O)L-- O)2 + fl2V Z)u(r ' O)) = 0.

(42)

In the bulk this yields the dispersion relation (43)

09 2 = 092 -- f12k2.

In the absence of dispersion (fl = 0) we recover the usual Born and Huang description of the LO modes [76]. In order to obtain an estimate of the velocity parameter fl, Babiker [20] employed the analogy with the LO phonons ofa 1D diatomic chain. The dispersion relation for these modes in the long-wavelength limit, ka << 1, is given by 4cZa2k 2

(DE ~ O)2 -- M+ M_ O)2,

(44)

c being the force constant and a the lattice parameter. The LO frequency in this case is given by 2c(M+ + M_) O)~ =

M + M_

(45) '

M+ and M_ being the ionic masses. Babiker [20] estimated that for polar semiconductors fl is of the order of 105 cm s- x, typical of sound velocities.

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Associated to this longitudinal displacement u(r, 09) are the electric field E and polarization field P, which can now be written as P(r, co) = - ~--~E(r, co) = ~u(r, co),

(46)

with 2 0~= [(e 0 --/300) ~ ]

1/2

(47)

Babiker considered as the most important feature of this approach the introduction of effective hydrodynamic pressure terms as directly responsible for the dispersion of the LO modes. One can then write for the pressure V ~ ~ pfl2 Vdiv u,

(48)

where p is the reduced mass density, as u is related to a relative atomic displacement. The proportionality sign appears in (48) due to the fact that the vibrational amplitude u is given in terms of the true relative ionic displacement field w by the Born and Huang convention [76]: U = pl/2W.

(49)

The reduced mass density was defined as [20]: p = Ytlvo,

(50)

Va being the volume of the lattice cell and ]~ = [M+ M _ / ( M + + M _ ) ] the reduced mass in the unit cell. Then from (48) we can obtain = fl2pVZdiv u.

(51)

As usual, in order to apply this theory to describe the LO phonons in heterostructures it is necessary to specify the matching boundary conditions at the interfaces between the different materials. Babiker employed the hydrodynamic boundary conditions and imposed the continuity of the normal component of the velocity ~ and of the pressure ~ . In terms ofu this requires the continuity of p- v2fiz,

(52)

while the pressure boundary condition is equivalent to the continuity of flZ pl/Z div u.

(53)

This model was applied to the study of the phonons in a GaAs (medium 1)/GaA1As (medium 2) QW, yielding the following dispersion relation for the longitudinal optical phonons [20]: ''=

(54)

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139

with q,=~--I

(i=1,2)

(55)

and /q6 °2 _ x2

qi=x/W

(i = 1,2).

(56)

The solutions of (54) with ql real and q2 pure imaginary, decay exponentially in medium 2 and were called guided modes [20], which are equivalent to the confined modes discussed above. The solutions with both ql and q2 pure imaginary decay exponentially in medium 2 and are given by a combination of exponentially increasing and exponentially decreasing functions of z in the slab of medium 1. They were called double-interface modes [20] by analogy with the polariton case. It was found that in the GaAs QW studied there were at most two modes of the double-interface type and only a finite number of guided-type modes were allowed [20]. These double-interface modes are quite different from the interface modes of the dielectric model due to the fact that the present model leads to an electric discontinuity, which is unphysical and the limiting frequency of both modes at x = 0 is ~ LO. This subsequently influences the nature of the interface modes. 3.3. Continuum models with ad hoc constants

As we have seen, both the dielectric continuum model and the hydrodynamic model are only partially correct and in different aspects. Some modifications of these models have been proposed in order to retain their key features while making them basically correct. Huang and Zhu [21] in order to study the limitations of the dielectric continuum model introduced a simple microscopic model. This took into account the long-range Coulomb interaction and simulated the optical vibrations between the oppositely charged lattice atoms by a simple-cubic lattice of charged oscillators. To model a SL formed by two different materials A and B they assumed that the only difference between them was a change in their restoring force constant. They showed that they could treat this difference ~s a "perturbation", using as basic vectors to express the dynamical matrix of the SL those of the material A. When comparing the results for the LO-confined bulk-like modes at • 4:0 of their microscopic model and of the dielectric continuum model they found that on a first analysis they showed an apparent agreement. But after a more detailed analysis some important differences were present. So, the potential curves calculated from the microscopic model approach the interfaces at z = _+ d/2 with zero slope whereas the potential curves calculated from the dielectric continuum model have maximum slopes at the interfaces. That is to say, the electric field has nodes at the interfaces according to the microscopic model, whereas the dielectric continuum model leads to maximum electric field at the interfaces. It was also observed that the sequence of confined modes m = 1,2, 3,... from the dielectric continuum model contained, respectively, 1/2, 1, 3/2 .... wavelengths, whereas the corresponding sequence of modes coming out of the microscopic model contained 1,3/2,... wavelengths. The conclusion was that, according to the usual sequence of standing waves limited to a confined length, the microscopic model appeared to have lost its half-wavelength mode. This is due to the fact that when the wave vector k changes from strictly the z-direction to other

140

V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (I 997) 123-176

directions, the half-wavelength mode changes its confined character and becomes an interface mode. This indicates that in the case analyzed in [21] the half-wavelength mode derived from the continuum model actually should not be there, and that the ordering of the modes coming out of the dielectric continuum model is faulty. This can be understood because, due to the neglect of the phonon dispersion in the dielectric continuum model, all the LO-confined modes are completely degenerate and any of their linear combinations can be taken as modes of the system. They concluded 1-21] that the dielectric continuum model by itself could not provide a useful criterion for determining the realistic bulk-like modes. However, they found that by following some of the features coming out from the microscopic calculations, it was possible to obtain simple analytical expressions for the bulk-like modes exhibiting a good agreement with the results obtained from their microscopic model. Clearly the single sinusoidal solutions used in the dielectric continuum model [21] cannot satisfy the requirements of both vanishing potential and field at the interfaces. They found by studying the electrostatic potential curves calculated from the microscopic model, that the requirement is met by some simple composite functions and obtained even modes composed of a cosine function plus a constant, and odd modes consisting of a sine function plus a linear term. By considering that the potential, and its derivative, of the phonon modes as calculated from the microscopic model vanish in the neighborhood of z = _+ (m + 1)a/2,

(57)

in the case of a SL, they wrote for the electrostatic potential of these even phonons ~o,(z) = cos[nTzz/(m + 1)a] - ( -

1)"/z, n = 2,4,...,

(58)

and tp.(z) = sin[p, rcz/(m + 1)a] + C,z/(m + 1)a

(59)

for the odd modes. #, and C, were constants to be determined by the conditions that q~, and its derivative must vanish at the positions given by (57), i.e., sin(#,~r/2) = - C./2,

cos(#, re/2) = - C,/#, rt,

(60)

which is equivalent to tan(#, zt/2) = #, re/2.

(61)

This equation gives the following solutions for #,: #3 = 2.8606, #5 =4.918, #7 =6.95, #9 8.9548 .... , which tend to be increasingly closer, with increasing n, to the corresponding odd integers, which characterize these phonons in the dielectric continuum model. The corresponding C, values are: C3 = 1.9523, C5 = - 1.983, Cv = 1.992, C 9 = - 1.995 .... The corresponding amplitudes have a vanishing z component as well as vanishing components at the interfaces. Another approach, also employed, imposed phonon confinement by requiring vanishing amplitudes at the interfaces, thus implying the vanishing of the electric field in the barrier region, as follows from (24), which is common to all models assuming from the start the existence of purely longitudinal solutions for matched systems. The phonon potential associated with the totally confined modes was written as [84]: =

q~.(z)=Csin(qz),

n=1,3,5,..,

q~,(z)=Ccos(qz),

n = 2 , 4 , 6 ....

(62)

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V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123.-176

by following the hydrodynamic model [20]. This model rules out the existence of interface modes and has a discontinuity in the electrostatic potential of the interfaces. The predictions of this model for the scattering rates were found to be inconsistent with those of microscopic calculations [15]. 3.4. Complete phenomenological model for long-wavelength polar optical modes in semiconductor nanostructures The various phenomenological models discussed so far share the starting assumption that a matched system with interfaces admits purely longitudinal solutions, which is a formally invalid hypothesis when vector fields are involved [-90 -92]. A consequence of this is that the field equations for q) and u are decoupled and, at this stage, two alternative options are followed: Some models focus on the Poisson equation for (p and others on Newton's law for u. Irrespective of details, this kind of bifurcation in the analysis is essentially the key issue [-31]. Ad hoc modifications achieve some useful improvements, but the formal difficulty is still at the root and partial successes are never rid of some flaws in the theory. In order to solve these difficulties a complete phenomenological model has been developed which is not subject to the said assumption and satisfies simultaneously mechanical and electrical boundary conditions [-28,31,34,39]. Besides a harmonic oscillator term containing the resonant frequency, the full equation of motion for u denoting now the actual relative ionic displacement u+ - u contains terms in the nature of mechanical stresses introducing spatial dispersion and a term which introduces the coupling to the electric field. In this extension of the B o r n - H u a n g scheme [76], the equation of motion for u is p((o2 - OJTO)// 2 -~- V" r - - o~Vq) = O,

2 ~2 = e~ToP(e o -- e~)/4n,

(63)

it is r that has the nature of a mechanical stress tensor. For an isotropic medium the form of this tensor is given by ziJ = _ p ( f i ~ - 2f12)6,:V.u - pfl.r(Viuj 2 + V~ui),

[64)

where flL and fly are empirical parameters which take into account the spatial dispersion, that is the k dependence of the bulk frequencies, up to terms of order k 2. The sign shown in (64) is the opposite to that holding for acoustic waves, on account of the negative dispersion of the optical modes. The essence of the phenomenological model is that (63) is accepted as it stands, even for inhomogeneous systems where the material parameters depend on the position coordinate z normal to the interface, and this includes abrupt interfaces, in line with the general comments made above. This equation is to be solved simultaneously with Poisson's equation for q~in the quasistatic limit. Since there is no external free charge, V. D = 0.

(65)

In the simplest isotropic model the general constitutive relation [-39] takes the form [-76] P=~u+

~ 4zt -

1E

"

(66)

Then, by noting that the source of the electrostatic potential is the polarization charge, we have V2q) = 4nV-P

(67)

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142

and by using the above expressions together with the Lyddane -Sachs -Teller relation, one obtains -eoe~ ---

4rtp

V2~0 =

OgLoV:U

(68)

This and (63) cast as p(o92 - ~O2o)U+ V. r - aVtp = 0

(69)

constitute the system of coupled differential equations for u and ~p. The problem then is to solve this coupled system with simultaneous mechanical and electrical matching conditions, the number of unknowns and conditions being compatible. It proves useful to introduce the longitudinal and transverse projectors in Fourier transform 1

L = ~ kk,

(70)

T = I- L

and then cast (69) in the form 2 + f12k2] T'u = 0, /)(0)2--0)20 + flL2 k 2 ) L . u _ ~L'V~o + p[o~ 2 --O~TO

(71)

so the transverse and longitudinal parts are now evident. Now, if and only if, one assumes u to be longitudinal, so that u = L. u and T. u = 0, then one is left in (71) with an equation for a purely longitudinal field, which as stressed above is not a valid assumption for systems with interfaces. Then u and E are directly proportional, i.e. 0~

u

p[a~2_a~T2O+f12k2 ] E

(72)

and depending on which one of these fields one eliminates, one is led to either of the two alternative models just discussed. It has been stressed [31] that the difference between these does not concern only the matching boundary conditions, but also the field equations on which the analysis is based. Indeed, the inverse of the differential equation operator of the hydrodynamic model in Fourier transform is 1

(Q~ _ kzZ),

(73)

where 2

tn2 __ f12k2 fiE~

,

(74)

while the corresponding one in the dielectric model is 2 + f12(r2 + k2)] 1 [(D2_(DTO (x 2 + k 2) e~(k~ Q2)

(75)

These expressions indicate a very important difference. The poles of (73) and (75) in the complex-k z plane control the z-dependence of the corresponding field amplitudes. It is easy to see that both have poles at ___QL, which originate terms going like exp( + iQLZ). Besides this, (75) has also poles at +_ix, and these originate terms going like e x p ( - tc]z ]), which cannot be obtained from (73) or from any

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143

other model based only on a mechanical field equation. It is easy to see that these poles yield the dominant contributions in the case of the interface modes [31,40]. The full formulation of the complete phenomenological model is generally valid for any nanostructure with arbitrary geometry [34] and, in particular, for planar heterostructures. The essence of the question is that the electrostatic potential is created by the polarization charge / P ~o - e~ \1/2 Ppol = - - 1 . . . . ] (DLodlVu ,

(76)

/

so E is only related to the longitudinal part of u but not to the full u vector and the two differential equations (63) and (68) cannot be decoupled. For planar heterostructures with an interface at z = 0 the analysis simply requires the standard continuity conditions ~zj(z = + O) = zz;(z = --0)

(77)

(j = x , y , z )

and Dz(z = + 0 ) = D=(z = - 0 ) ,

(78)

amounting to four scalar conditions for the four independent amplitudes, so there is no difficulty in satisfying all formal requirements. A simple example demonstrates the basic soundness of the model. Consider a GaAs well with AlAs barriers and study GaAs-like phonons. The frequency difference in this case is very large and it is an excellent approximation to assume that the interfaces and the AlAs barriers are mechanically clamped, that is u(z = +_-d/2) = 0, and besides u - 0 for Izl > d/2, d being the well width. By studying the simplest case of (0 0 1) interfaces and vibrations along the [0 0 1] direction with x = 0, we only need to study the longitudinal part of u, because in this case the transverse part ux - 0 for all z ~ ( - d/2, d/2). From the vanishing of curl u, the mechanical field system of equations reduces to a single second-order differential equation for Uz with the amplitude vanishing for [zl > d/2. Eq. (68) then reduces to

l [ - 4g°-g°~71/2 gP

d2q~ dz2 - L

%%

J

du= ~LO dz '

(79)

and ~p must be continuous at the interface. Condition (78) is automatically satisfied in the present case. The full solutions of (63) and (79) are [22]: u=(z)= A , sin

z + d/2

,

(80)

n= l,2,...

and z < - d/2,

(El - ( - 1)", /

dnrc 7 { [ ( - 1)" - 1],

[1 + ( - 1)"],

Izl <

d/2,

n = 1,2 ....

(81)

z > d/2,

with •

C. = A.

E4z~p(e~ 1

-

So

1)]1/20)£O

(82)

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and A, being a normalization constant. These results agree quite well with those of the microscopic calculations 1,5,15,21]. The electrostatic potential is continuous and the symmetry pattern agrees with experimental evidence. Raman-scattering data [24,44,46] give LO modes with q~ odd, corresponding to n odd in (81), when the geometry of the experiment is ~(y,x)z and modes with q~ even, corresponding to n even in (81) when the geometry of the experiment is £(x,x)z, and this pattern is obtained from (81) as well as from the microscopic calculations. It can also be seen that this is quite similar to the results obtained by Huang and Zhu [21] and compacted in Eqs. (58) and (59). This simple example demonstrates the basic soundness of the model, in which the matching boundary conditions are satisfied without ad hoc manipulations. We now discuss the uses of this model for any situation not restricted to the case of a mechanically rigid wall.

3.5. Polar optical phonons at surfaces and interfaces The formulation of the matching problem for the complete phenomenological model is in essence similar to that for piezoelectric surface or interface waves [93-96], as in both cases there are two coupled fields and both must comply simultaneously with matching conditions. The case of a single surface or interface has been studied [40] by employing the surface Green function matching (SGFM) method as a technique for studying matched systems [92]. By studying the r-resolved local density of states at z = 0, it was possible to obtain the eigenvalues co(x) as the frequencies at which the peaks in the density of states appear. The dispersion relations were obtained by varying x. For the GaAs free surface, besides the transverse (~o ,-~ 268.4 cm -1) and longitudinal (~o ~ 292.3 c m - 1) thresholds two additional features were present for wave vectors beyond x = 105 cm - 1 The first one was a peak beyond the maximum frequency of the bulk crystal although quite close to it, which was assigned to a localized surface mode. The second feature appeared at frequencies ~o ,~ 280 c m - 1 ( ~ 34.7 meV), in the continuum of bulk scattering states, which thus corresponds to a resonant mode. This resonance was identified as the Fuchs and Kliewer [85] surface phonon, which has been detected by using EELS by several authors in the range 36 -38 meV [97 100]. The localized surface mode merged with the bulk LO frequency at around K = 105 c m - 1. These features are presented in Fig. 1 which corresponds to the spectral strength for two values of x ((a) ~: = 4 × 106 c m - 1, and (b) ~: = 2 × 106 c m - 1). No experimental evidence of the localized surface mode appears to exist, but there are two possible reasons. One is that the mode is very close to the longitudinal threshold and thus makes very difficult its distinct experimental detection. The other one can be related to the simplicity of the theoretical model, in which an isotropic approximation was made. It is quite plausible that some further improvement of the model, like the inclusion of anisotropy, might cause the disappearance of the local mode which is so close to the longitudinal bulk threshold. In order to see the influence of the boundary conditions and in analogy with the case of acoustic surface waves in piezoelectric systems 1-93 -96] the case of a metallized surface was considered [40]. This consists of a surface which is coated with an infinitesimally thin metallic layer serving to ground the surface, thus keeping it at a constant potential while not affecting appreciably the mechanical vibrations 1-93 -96]. In this case it was found 1-40] that the surface mode peak was present and that, on varying tc it had also practically the same dispersion relation as in the free surface case. On the other hand, when the electrical potential at the surface was clamped then the surface resonance or

V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123-176 LDOS(arb.unlts)

145

(a)

17500 15G60 12500 10000 7500 5O00 2500 j

J 265

270

•

280

2~

~o

2~'

'

29O

2~

z~o

2~

W (cm "1 )

LDOS{arb.unlts) 17500 150OO IlJO0 10000 7500 5OO0 250O j

J 2~

'

' 2~

' '

275

Fig. 1. Local d e n s i t y o f states ( L D O S ) in a r b i t r a r y (b) K = 2 x 1 0 6 c m -x"

W ( ¢ m -1 )

units at the free G a A s

surface, for (a) • = 4 x 1 0 6 c m - 1;

Fuchs and Kliewer mode disappeared, which shows the essentially electrical nature of the surface resonant mode. The electrostatic spectral strength was found to be concentrated in the surface resonance, the rest of the spectrum being almost unaffected. No clear experimental evidence of the disappearance of the Kliewer and Fuchs mode with a metallized surface was found, although this mode is not detected when depositing thin Ag layers on GaAs (1 0 0) [ 101 ], and the layers are smooth films. Fig. 2 gives the spatial variation of (ur, uz,~0) for the localized mode of the free surface, for = 4 × 106 c m - 1 in the y-direction. The surface character of the mode is quite evident, as manifested in the damping of the amplitudes when penetrating into the crystal. Next the GaAs/Ga0.1A10.gAs interface was also studied in [40]. The ternary c o m p o u n d AlxGa 1 xAs was described in the two-mode model due to the reasons given for this on the basis of experimental evidence [103]. The idea of the two-mode model arises from the observation that at a given value of 0 < x < 1, the ternary c o m p o u n d exhibits two LO and two TO modes. When x tends to zero the two modes of highest frequency tend to the LO and TO modes of the bulk AlAs in the F point, and they are called AlAs-like modes, while the two low frequency modes coalesce into the single frequency corresponding to an A1 impurity in GaAs. When x tends to 1, the two lowest frequency modes tend to the GaAs bulk LO and TO modes in the F point, and they are called GaAs-like modes, while the two high frequency modes coalesce into the single frequency corresponding to a Ga impurity in AlAs. The idea is that the G a - A s and M - A s bonds vibrate following their own frequencies, which are very similar to those of the corresponding bulk crystals and are only slightly modified by their environment in the ternary compound.

146

V.R. Velasco, F. Garcia-M oliner/Surface Science Reports 28 (1997) 123-176 (a)

1

.d o -!

-2. I0

.6

-6 -l.f 10

-6 -I. 10

.5.10

7.(cm)

-7

(b) 3

'2

-6 - I J IO

-6 -2. I0

-6 -L 10

-J. 10-7

Z(cm)

(c) O.OOt~ 0.00~1

; e

o -O.O000l

-0.00002 i

.2. 104

,

-I.~ I0

-6 -l. I0

-6

Z(©m)

4 . I0

-7

Fig. 2. Spatial variation of the mechanical uy (a), u~ (b) and electrical tp (c) amplitudes of the surface localized mode, for ~---4 x 106cm -1.

In the case of the GaAs/Ga0.1Al0.9As interface [40] it was found that no localized mode existed, while the resonant mode survived. This mode decreased its frequency quite slowly with decreasing x and had a frequency slightly higher than in the surface case, o 9 ~ 2 8 2 c m -1 for 102 < x < 4 × 106 c m - 1. For the metallized interface it was found that the resonant mode moved

V.R. l/elasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123-176

147

towards the transverse threshold and it was very close to it. As in the case of the surface, the interface-resonant mode was drastically affected by a change in the electrical boundary conditions at the interface, thus putting in evidence the predominant electrical character of this mode. Fig. 3 gives the spatial variation of (u r, u z, ¢p) for the resonant mode of the interface, when x = 4 x 10 6 c m - 1 , (a)

-3

'

'

' -8,1o

' .7

" 6 1 0 "7

.4.10 °7

z(cm)

.2.10 "7

2 . 1 0 -7

(b)

"i § -!

-8.10 .7

-6.10 `7

"4. I 0 .7

-2.10 -7

2 . 1 0 -7

Z(cm)

(c) 0.~02

0.00001

" ~ " -0.00001

-0.00002

-8,10 "7

-6. I0 "7

.4.10 .7

-2.10 -7

2 . 1 0 `7

Fig. 3. Spatial variation of the mechanical uy (a), uz (b) and electrical q~ (c) amplitudes of the interface resonant mode, for x = 4 × 106cm -1.

148

V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (I 997) 123-176

showing the tendency to a stationary bulk oscillation on moving away from the interface and into the GaAs bulk, since we study GaAs modes. These results explain why the dielectric continuum model successfully accounts for the interface modes found in QW systems. In the case of a single interface there is only one such mode. When the symmetric QW is formed then two of these appear and have predominant electrical character. 3.6. Long-wavelength polar optical modes in GaAs based quantum wells

A GaAs quantum well of width d = 20A with A10.gGa0.1As barriers which has been studied experimentally [44] was analyzed in detail in [34,35] with the complete phenomenological model and full matching in every respect. Except for the numerical values particular to the specific constituent materials of the system under consideration all the relevant physical features would be present in other GaAs-based QWs. The dispersion relation was obtained by studying the LDOS at the left (l) or right (r) interface, covering the range from ~OLO(GaAs) down to below ogTo(GaAs) for different values of x, as shown in Fig. 4. The different eigenvalues appear as peaks in the LDOS, as can be seen in Fig. 5. The peaks would be in principle strict 6-functions but have a small width due to a small imaginary part added to o9 and needed in practice to perform the numerical calculations. The modes were labeled by a discrete label m = 1, 2,..., and the value m = 1 was ascribed to the highest mode, so m grows for decreasing frequencies at x = 0. It was found that the eigenvalues corresponding to the first few values of m, in the vicinity ofogLO(GaAs), do not differ in a significant way from those coming out of the rigid-barrier model 1-22-1and given by the expression 2

ogre ~

(83)

2 _fl~m2n21d 2.

ogLO

300

290

I E

280

0 V 270

f

3 260

250 1 ,

,

i

• (106cm

,

,

-1 )

Fig. 4. Dispersion relations for the eigenmodesof a GaAs quantum well of width d --- 20 ~.

V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123 176

149

e-

© ..a

260

270

280

o3(cm-')

290

30o

Fig. 5. LDOS, in arbitrary units, at the interfaces of the q u a n t u m well of width d = 20 ~.

An insight into the physical nature of the normal modes was obtained from the nonvanishing mechanical (u) and electrical (q~) amplitudes as a function of z for different values of ~c. Fig. 6 gives these amplitudes for ~c= 0, for the first three modes of Fig. 1. In the isotropic version of the model it is possible to choose x = (0, ~:) in the y-direction without loss of generality. With that choice the amplitude u x is factorized as a purely mechanical transverse horizontal vibration and u r vanishes identically. Thus only u= and q0 need to be studied for the modes of interest. With this geometry the only nonvanishing c o m p o n e n t of curl u is (V Au)~=ixu=

dur dz '

(84)

while div u is du z

V-u = ixu r + d--z"

(85)

Ifu r = 0, as is the case for ~: = 0, then there is no surviving term in (84), while V'u ~ 0 always and the modes are longitudinal but this strictly holds for x = 0. The symmetry pattern, for instance the parity of the electrostatic potential, follows the sequence odd (m = H/even (m = 2)/odd(m = 3)/... and the values of q~ outside the well vanish for the even modes and are nonvanishing constants for the odd ones. These facts agree with experimental evidence [44], as well as with the results of microscopic calculations 1,5,15,16-]. It can be seen in Fig. 6 that u= does not vanish at the interfaces, where its magnitude is actually quite significant, and this is also in agreement with microscopic calculations [5,15,16-]. Fig. 6 shows a very fast decrease of u= beyond the interfaces, typically over a distance of a few A, and then vanishes. This behavior is associated to the nature o f the model and puts in evidence that it is not only the matching b o u n d a r y conditions that matter, but also the mathematical structure o f the differential operator describing the model as discussed above. By keeping both fields

150

V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123-176 oz (I)

(i)

(21

12)

-,

!

\

J

/ \

(3)

/

(3)

J

-i

I l

@

Fig. 6. Spatial dependence of the mechanical (u,) and electrical (~0)amplitudes for x = 0, modes m = 1, 2, 3, in arbitrary units. The mode number is indicated in brackets and the abscissa z is in/~.

(u and q~) in the 4 x 4 differential system, the differential o p e r a t o r (for given co, x and as a function of k=) has two types of poles, n a m e l y (i) those of an electrical nature at kz = + i x , and (ii) those of a mechanical nature, at k z = + QL and k z = +--QT, where =

+ [o 2

2

2 1/2

- ~OrO]/fiv } •

(86)

The poles of type (i) give spatially d e p e n d e n t a m p l i t u d e s going like exp( - x lz I), and c o n s e q u e n t l y do not d e c a y for x = 0, while the poles of type (ii) give a spatial d e c a y outside the well going as exp( - QL,TI Z I), if Z is m e a s u r e d from the interface one studies, This is the origin of the fast decay seen in Fig. 6 and since the m o d e s are purely longitudinal the d e c a y length is of the order 1/QL. The dispersion relations seen in Fig. 4 have s o m e significant features which the m o d e l explains in physical terms. Consider, for instance, m o d e s labeled m = 1 and 2 at x = 0, which crossover at x c. F o r x < x¢ m o d e m = 1 is the m o r e dispersive, while the o p p o s i t e holds for x > x~. The electrical and mechanical spectral strengths of these m o d e s are s h o w n in Fig. 7 for ~c in the n e i g h b o r h o o d of x¢, s h o w i n g that for x < x~ m o d e m = 1 is of p r e d o m i n a n t electrical character and m o d e m = 2 is mainly mechanical, while the o p p o s i t e holds for x > x~. This change in the character of the m o d e s is further seen in Fig. 8 where u z and q~ are s h o w n as functions o f z for the m o d e s m = 1, 2, 3, w h e n x > x¢. It was

V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123-176

,ll

A

290

•~

151

(b)

29().5

A

[ 290

290'.S

r"

-a

--i

.6 L_

.6

t¢

t¢

(d)

L_

290

CO

290.5

290

(cm -1)

290.5

w (cm -1)

Fig. 7. Electrical (a,c) and mechanical (b,d) spectral strengths for the upper and lower modes, in the immediate neighborhood of the crossover between modes m = 1 and rn = 2, x = 1.1 × 106 c m - 1 for (a, b) and x = 1.4 x 106 c m - 1 for (c, d). Spectral strengths are in arbitrary units and e~ in c m - 1.

co 1

W2

,-

-~

"I

K

~

tO3 ~~

z (A)

z (A)

Fig. 8. Spatial dependence of the mechanical (uz) and electrical (~0) amplitudes for x = 2 x 106 c m - 1, modes m = 1, 2, 3, with z extending up to + 60/~. The mode number is indicated in brackets.

152

V.R. Velasco, F. Garcla-Moliner/Surface Science Reports 28 (1997) 123-176

seen by explicit evaluation that IV A u] << IV'uJ, so these modes are still in practice quasilongitudinal for this value of •. Figs. 6-8 put in evidence how the predominance of electrical or mechanical character and the parity of ~o are interchanged at ~ = Kc. If we denote by O / E (odd/even) z-dependence, then the parity sequence for ~01 to tp 5 is O, E, O, E, O when ~c< G and it changes to E, O, O, E, O when ~c> K¢. This description of the mixed character of the normal modes provides a useful basis for the study of the e l e c t r o n - p h o n o n coupling when not so small values of ~c are involved, as the electronic wave functions in a symmetric Q W also have definite parities. The m o d e m = 6 (starting from ~ = 0) originates the second crossover with m o d e m = 5 for low •. The system of eigenmodes presents two modes (for every K) having predominantly electrical character. These are m = 1 and m = 6 for K < G, while they are m = 2 and m = 6 for K > G. The electrical and mechanical contributions to the L D O S of the m = 5 and m = 6 modes are transferred in a similar way to that depicted in Fig. 7. It was verified by explicit evaluation of the space dependence of ~o that for every K there are two modes which (i) are the two most dispersive ones, (ii) have opposite parities for tp(z), (iii) have predominant electrical character and (iv) have amplitudes ~o(z) accumulating near the interfaces. These are the two interface modes, one even and one odd, resulting from the long range electrostatic coupling of the interface modes at the barriers of the QW. As a consequence of the symmetry of the system each amplitude q~(z)by itself has a definite parity and this determines which e l e c t r o n - p h o n o n matrix elements vanish. If the modes were all purely longitudinal, the first one would simply cross the second one, which has opposite parity, without any mixing, while it would mix strongly with the third one which has again the same parity. The repulsion between the corresponding dispersion branches is evident in Fig. 4. For ~c¢ 0 the longitudinal and transverse parts are in principle coupled and some mode mixing takes place between modes 1 and 2 on one hand and 5 and 6 on the other hand. In the case of ~c~ K~ a strong mixing of the longitudinal and transverse polarizations for m = 5 and m = 6 takes place, where the terms ]V A u] and FV-u] are comparable. In the case of GaAs/AIAs QWs and SLs there are physical considerations which can introduce acceptable simplifications allowing for the obtainment of analytical solutions for the optical modes of these heterostructures [37,38,41]. The LO and T O bulk frequencies of the AlAs are ,-~ 100 c m - 1, both higher than the ~OLO/tOWOin GaAs. It would then be necessary for an optical mode to vibrate in the AlAs slab with a frequency in the range of the GaAs slab and that the wave vector in the AlAs slab were very large in order to reduce its frequency from that at the F point value. As the in-plane wave vector components have the same value in both materials the k z components of the LO and T O modes in the AlAs should be very large as compared to the values of these components in the GaAs. The displacement in the AlAs layer would then be rapidly varying waves with an extremely small amplitude, so the assumption that u -- 0 in the AlAs slab, for modes with frequencies in the range e)xo < e) < COLOof the GaAs material is a very good approximation. In practice the interface with AlAs behaves as a completely rigid wall for the GaAs vibration modes. The situation can be quite different for AlxGa I xAs/GaAs heterostructures, when the GaAs-like modes in the AlxGal_xAs slabs can have their frequencies at the F point very close to those of the GaAs material. Then one must observe full matching b o u n d a r y conditions with no simplifications [28,34,35]. Putting u = 0 in the AlAs slab, allows for an important simplification in the solution of the equations, thus providing the possibility to obtain all amplitudes of QWs [38] and SLs [37,41] in closed analytic form. Some of these results are summarized in Figs. 9-11. Fig. 9 gives the dispersion relation ofa GaAs/A1As Q W having a width of 20/~. The branches crossing each other have different

300

153

GaAs -like d=2nm

LO1 LO 2

L03\ ,..-

280

~vOE TO1\ LO 4 3

TO2 -

LOs / TO4- -

0

I

I

I

I

2

4

6

8

~.(106cm -1) Fig. 9. Dispersion relation for the optical modes ofa GaAs/A1As QW having a width of 20/~. The dashed lines indicate the shear horizontal modes uncoupled to the rest of the spectrum.

GaAs -like d = 5.6 nm

LO3.J~']~~

29oI 3

L°'L

288

0

0.5

1

1.5

2

~. ( 107cm -1) Fig. 10. Dispersion relation for the first four optical modes of a GaAs/AIAs QW with a width of 56,~.

154

(a)

~.=0

, <

.......... /....

l~

t-

~ ~

to2 I

~

OINDQO000i

O000QO000

N

oo

I

I

,.........,-........,......... -28

0 z

28

I

I

(A)

(b)

A-I

•~ = 0 . 1 5

I ~, , ~ ~ / ~ ,m

t-

00 2 d~

000000000

.....

00001100110

N

-28

0 z

28

(A)

Fig. 11. u z components of the first optical modes (solid lines) of the GaAs Q W with d = 56/~, as a function of z, (a) for x = 0 and (b) x = 1.5 x 107 c m - 1. The G a and As displacements are shown separately, represented by full triangles and circles, respectively.

V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123 176

155

parities while branches with the same parities repel each other and originate gaps whenever they come close [38]. All branches are almost flat except in the ~cregions where crossings and gaps appear. The ranges of t¢ exhibiting the stronger dispersion are those in which the states with stronger electric character are present. Therefore, the "interface" character is stronger for these modes. Fig. 10 presents the dispersion relation of the first four optical modes of a similar QW but having a width of 56 ~. The crossings and gaps are clearly visible there. Fig. 11 gives the amplitudes u z obtained from the calculation with the complete phenomenological model in the u - - 0 approximation [38], compared with the results of ab initio microscopic calculations [16]. The agreement between both calculations is excellent, not only for the ~ = 0 case, but for ~- = 0.15 ~, which is a relatively large value of the in-plane wave vector.

3.7. Long-wavelength polar optical modes in GaAs-AlAs superlattices A key feature for the SLs is that the long range electrostatic fields couple different slabs even if the mechanical vibrations characteristic of each constituent material would not by themselves penetrate the adjoining slabs sufficiently to couple to the next slab of the same species. The case of the GaAs AlAs SLs has the advantage that the infinitely rigid walls approximation can be used for the mechanical vibrations, as discussed above, thus allowing for an analytical study [37,41]. Both studies are similar but have some differences. The calculation in [37] starts from a trial form of the solution written as a linear combination of LO, TO and interface modes, and then proceeds with the standard type of solution for the coupled differential equations, while in [41] the problem is solved directly in terms of a basis. The eigensolutions u,(r) and q~,(r) are then exactly obtained in the basis formed by independent subspaces of transverse horizontal, odd potential and even potential modes. This provides a very convenient way of obtaining the field amplitudes u(r, t) and q~(r, t) and hence the electron-phonon interaction Hamiltonian. A distinct feature of the SL modes is the dependence on q, the wave vector associated with the propagation in the growth direction. For arbitrary q the resulting ~0(z) has no definite parity, but for q = O, n/d it has a definite parity. Fig. 12 displays the angular dispersion of the first four long wave modes for a 12-12 SL (d6aAs = dA~As= 34.32]k). These are modes near the center of the Brillouin zone, corresponding to 2 = 3500/~ in a micro-Raman experiment. With a refractive index of 3.5 this corresponds to a wave vector K = 8.298 x 105 cm-1 . The • and q components of the intervening normal modes are tc -- K sin 0 and q = K cos 0. The figure shows the eigenfrequencies of the normal modes as a function of 0 for the above value of K. In [37] several (1 1 1) and (1 00) GaAs/A1As SLs were studied, and the results of the phenomenological model compared with those of a lattice dynamics shell-model calculation [ 104,105]. The agreement with the lattice dynamics calculations was found to be particularly good for the AlAs modes and GaAs modes of(1 1 1) SLs. For the GaAs modes of the (1 0 0) SLs some disagreement was found due to the use of rigid mechanical barriers which decreases the number of confined modes seen in the range ~Ovo<< ~o<< ~LO" The origin of this is that in this approximation the effective confined wave vector is equal to mn/(Na), where m is the confined mode number, N the number ofmonolayers and a is the monolayer thickness. It has been seen from the microscopic calculations and the fitting of the confined mode frequencies obtained by Raman scattering that the wave vector is more correctly given by Eq. (4). It was found in [37] that if the GaAs nominal layer width was increased while

V.R. Velasco, F. Garcla-Moliner/Surface Science Reports 28 (1997) 123-176

156 282

291

290

!

E t.)

289

..,,.,.

3

288

287

286

0.8'

2%

Fig. 12. Angular dispersion of the first four polar optical modes near the zone center of a 12-12 GaAs-A1As superlattice. For a fixed wavelength 2 = 35000/~ the first four eigenvalues are given as a function of the angle 0 of the propagation direction relative to the growth direction of the superlattice.

simultaneously reducing the AlAs layer width by one monolayer, then it was possible to simulate, within the rigid barrier model, the fact that the vibrations really go to zero at the first A1 layers on either side of the interfaces. It was then found that the results obtained with this approximation gave a much better agreement for (1 0 0) SLs, even those having small periods. The modes of the (1 1 1) SLs were found to exhibit greater mechanical confinement, and the agreement was excellent even without the modification which increases the effective width of the GaAs slabs. When considering the angular dispersion of phonons with wave vectors of the order of 108 c m - 1 and larger it was found [37] that the upper interface mode also anticrosses with the even confined modes, due to the fact that there exists a Bloch wave vector along the direction of growth allowing interface phonons with Bloch k z and in-plane k x components which are no longer purely symmetric or antisymmetric with respect to the well layer. On the other hand, the modes of QWs are either symmetric or antisymmetric with respect to the center of the well so the antisymmetric interface mode does not anticross with the even confined modes 1-27,34].

3.8. Long-wavelength polar optical modes in quantum wires and quantum dots Recent techniques make possible to grow heterostructures with quasi-lD (quantum wires) and quasi-zero-dimensional (quantum dots) confinement. It must be noted that the sample quality and

KR. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123-176

157

the size of the different samples which are actually produced are such that no clear indication of the reduced dimensionality effects can be found in the phonon spectra obtained by means of light scattering experiments 1-106]. However, several spectroscopic techniques, including Raman scattering, are being actively used to study confined LO-phonons [107,108], surface optical phonons [109,110] and confined acoustic phonons [111,112], so it is interesting to give a brief summary of the theoretical work performed on these systems, in view of possible future developments. The same continuum models discussed above have been used to study the long-wave polar optical modes in quantum wires and quantum dots. The polar optical modes of a QW wire have been studied by using the dielectric continuum model [113-116] and the hydrodynamic model [117-119]. The case of a rectangular GaAs wire embedded in AlAs has also been studied by using a microscopic approach [120]. The phonon frequencies in microcrystals exhibiting different geometries have also been studied by using macroscopic models which do not take into account the effect of the mechanical boundary conditions [85,111,121]. The electron-phonon interaction of spherical semiconductor microcrystallites has been studied for the case of confined phonons by using the dielectric continuum [122] and hydrodynamic [123] models. The complete phenomenological model has also been applied quite recently to the study of the polar optical modes in semiconductor quantum wires and quantum dots [124-126]. The starting equations are again (63) and (68) but the geometry is now substantially different. This system of four coupled differential equations can be solved by defining the auxiliary functions tp= V'u,

A = V/x u,

(87)

which satisfy the equations (V2 + Q2)A = 0

(88)

(V2 + q2) iF= 0

(89)

and

with Q2

2 aJ'ro -

092

,

2

__ 092

q2 __ ~OLO

(90)

It is found [125] that the fields of interest are then given by 4~ q~= ~oH e~q2 ~,

(91)

where (PH is a solution of the Laplace equation and by u=-V

[ ~ q e~ n +

p&Q

~22] + 1 V A A .

(92)

The functions A, ~ and (PH are the solutions of their respective differential equations satisfying the boundary conditions appropriate to the system they describe. The results thus obtained for u and ¢pare valid, irrespective of the type of nanostructure (quantum wires or quantum dots) and cross-section geometry, provided it is formed by isotropic and homogeneous materials separated by interfaces. In order to solve for a particular case one must

158

V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123-176

obtain the general solutions of the Helmholtz equations for ~u and A together with that of the Laplace equation (q~u) inside each constituent m e d i u m and then use the matching boundary conditions at the interfaces. The solutions are strongly dependent on the geometrical conditions of the problem [126]. We can consider a q u a n t u m wire with circular cross-section r o and the z-axis along the center of the cylinder. The most general solution of this problem is quite complicated, but sufficient information can be gleaned from the study of the important case, qz = 0 [124,126]. In the case of a free standin9 wire the mechanical vibrations are free on the b o u n d a r y surface S and the condition r - N = 0 for r e S must be satisfied (N is the outward normal to the surface), while u only exists in the active medium. The p h o n o n dispersion relations are obtained by solving the following secular equation:

2n(n -- 1)( flL']272 tn( q, Q) Jo+ 2(Qro) + 2n(Oro) 2 J,+ 2(Qro)[qroJ',( qro) - J,( qro) ]

\t TJ

+ 9,(q)[2QroJ',(Qro) + (Q2r2 - 2nZ)Jo(Qro)]

= 0,

(93)

1(qro),

(94)

with

~2 / ro'~ 2. 2

2

=~K) ~('/)LO--O)TO),

9.(q)

(flL'~2

= \flTJ (qr°)2 J"(qr°)

t.(q, Q) = L\

[(Qro) 2 J.(qro)

\YT/

-- 2(n

+ 1)qr o J.+

J. +2(Qro) + ~-~ 9.( q) J.( Qro) ] J.+ (Qro)

,

and J.(x) is the cylindrical Bessel function of order n. Analytical expressions for u and ~p can be found in [124,126]. If the Q W wire is formed by GaAs for r < r 0 and AlAs for r > ro, then it is possible to use the approximation u = 0 at reS, as in the case of planar interfaces, and one has completely confined polar optical modes [123]. The dispersion relation for this case is obtained by solving the secular equation 1 ( 2, o~+ e2~)[ J. -1 (qro) J.

+1(Qro) + J.-I (Qro) J. +1(qro)] ~L~' 2 n ~2~ _ +e,~(flT--~o)) [J,_,(qro)+-~oJ.(qro)(~ 1)]J,+l(Qro)=O.

(95)

Analytical expressions for u and ~p can be found in [124,126]. It can be shown, both for the free standing wire and completely confined modes, that for n -- 0 the L and T solutions can be decoupled and therefore pure L and T modes are obtained. The pure LO-modes are associated with the electrostatic potential ~0, while the pure TO-modes consist only of mechanical vibrations with amplitude u. Fig. 13 shows the q u a n t u m wire frequencies as a function of the radius r o for n -- 2. When n # 0 a strong mixing between the L and T parts is present. It is also

V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123 176

40

)

38

159

In= 2

>.~~36 34 32 30 28 40 38 P I,o,L ~...~36

9

J

34 3 32 +C: 30 28 10 20

30

40

50

60

Fig. 13. P h o n o n energies o f a GaAs quantum wire having circular cross section as a function of the radius r o for n = 2. (a) Confined q u a n t u m wire and (b) free standing wire. The broken lines give the bulk L O and T O phonon energies cov denotes the Fr61ich frequency.

evident from the figure that when a LO-curve approaches a TO-curve an anticrossing effect occurs. These anticrossings are more evident for small values of r o . The coupling between the electrostatic potential q~ and the displacement u due to the matching b o u n d a r y conditions give origin to the so-called surface modes. For large values of r o the influence of the short range forces on the matching b o u n d a r y conditions becomes negligible and consequently the p h o n o n spectrum is essentially determined by the long range electrostatic forces. In this case a new m o d e is obtained with a frequency (the Fr61ich frequency ~OV)between (DLOand ~T o. In the case of the cylindrical geometry ~oF is given by

o9F = _/sx°+

e2°~COTo.

~ l c c + g2oo

(96)

V.R. Velasco,F. Garcia-Moliner/SurfaceScienceReports 28 (I 997) 123-176

160

The case of a single q u a n t u m dot having spherical symmetry and radius R has also been studied [125,126]. F o r a GaAs q u a n t u m dot in an AlAs matrix the approximation u(R) = 0 can again be used, and it leads to completely decoupled TO-modes with a dispersion relation given by: =COTO e T \ R J

'

(97)

#,, being determined by the conditionjz(#,. ) = 0,jz being the spherical Bessel function. These modes correspond to oscillations along Xz,. and have no associated electrostatic potential with them. Xt,. is defined by i Xl

m

=

{-0,~

rAVYlm(O'(a)'lv~O

(98)

l=0, Yu,(0, ~b) being the spherical harmonics with m = - l.... , I and 1 = 0, 1,... F o r these T O modes

D being a constant to be determined [126]. All the remaining b o u n d a r y conditions give coupled phonon oscillations. The secular equation for these modes is given by

qRjt(qR ) F~(QR) = l(l + 1)jz(qR ) G,(QR)

(100)

with

Fl(x ) _ ~OLOfl~ -- ~TO

G,(x

f12

l(xfi(x)-ljt(x))+ \xJ

l+

(l+l)

(ljt(x) -- xh(x)) + l +

., • (xj,(x)+ji(x)), (101)

(l + 1) jl(x).

In the case of large R, Eq. (100) leads to the solution (.02

2 elol + ~2c¢ (1 + 1) = ~OT°e 1~----~+ e2~ (l + 1)'

(102) which is the frequency corresponding to the interface m o d e of the dielectric model [111]. If we use free surface b o u n d a r y conditions the frequencies of the TO-decoupled modes are obtained from the equation

jt(qR) = qRj; (qR),

(103)

while the secular equation for the coupled modes is given by •

.!

.!

.

2

.t

.v

ctljz(qR)jt(QR) + fllQRjI(qR)jt (QR) + tltqRJt (qR)jt(QR) = ~zR qQJt (qR)jI(QR),

(104)

V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123-176

161

with

52[

QR % = l(l 2 - 1)(l + 2) Is(70/+ l + 1) + l] + ~

l ~(70 + 1)2Q 2R2 + e(l + 1)(70 + 1)

]

I/Q2R2 ) × L----~ - + 1 - 2el(l + 1)(7ol + 1 + 1) - / ( 2 / 2 + 2 1 - 1 + 27o/2 + 3 7 o / + 72l - 70) ,

fl,=7o 2+ 1 Q2R2El(y° +

1) + (1 + 1)e] - e7o/(/2 - 1)(I + 2)],

rh = Q 2R2 [I(~o -4- I) 4- (l + i) e] + (I - I)(l + 2) [/(7oI - I) - (l + I) e],

(lo5)

~, = (I - 1)(I + 2) [I(~o + i) + e(l + 1)-I, 1 7 o - Q2

2

2

O)LO - - (/)TO

,

~2~ /31~c

In Fig. 14 the p h o n o n energies of the completely confined case are shown as a function of the radius R. The anticrossings evident in the figure correspond to situations in which a curve resembling a LO

32

~

h~ T

31 30

28 27 26 25 ,

10

,

20

30

40

50

60

R Fig. 14. Phonon energies of a spherical quantum dot, for the completely confined phonons case, as a function of the radius R.

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phonon dispersion curve (with the vector displacement along the spherical coordinate er) approaches the corresponding "TO-phonon curve" (with the displacement along e, A X~,,), and the behavior changes from LO to TO phonon dispersion and vice versa.

4. E l e c t r o n - p h o n o n i n t e r a c t i o n

Studies of the electron-phonon interaction based on microscopic calculations [15] were referred to in Section 2. A general study of the problem has been carried out in terms of the complete phenomenological model for arbitrary systems, irrespective of the geometry of the interfaces [39]. The analysis starts from a Lagrangian density of the form

= ~pu'u + 1

•

•

1

1

~ V u : ) t : V u + ~u" ?'-u - u. ~. Vcp,

1

+ ~ V ~o-(I + 4rq#) • V ~o,

(106)

where ~, #, 2t are second rank tensors and ,,1,is a fourth rank tensor playing the role of an "elastic stiffness" tensor. This must be considered only as an analogy because u is not the actual atomic displacement but, like the other tensors, ~, has the symmetries stemming from general invariance arguments. Thus # and ~, are symmetric and 2,jr. = ;tj,t,. = 2,jmt = 2mtij =...

(107)

The problem is then formulated for arbitrary structure and constituent anisotropic crystalline media, always in the long-wave limit. Additional symmetries can be obtained from the symmetry of the crystal. The form of the Lagrangian follows from general considerations: it is defined in the nonrelativistic limit for the generalized coordinates (u, tp), from gauge invariance arguments it depends on V q~ but not on ~por ~b, and in order to yield linear differential equations for u and q~it depends on these amplitudes in a quadratic or bilinear form. Electrostatic effects are not included. The form of this Lagrangian density is well known [29,127], but it must be stressed that the longitudinal and transverse parts of u are coupled and the analysis is then quite general. From the Hamiltonian density derived from (106) the electric displacement vector is ~Yg =4rc[ot.u_l(l+4rcp).Vcp] D = - 4rc--~--g

(108)

whence the constitutive relation P = ~.u -

#. Vcp.

(109)

This has the same form as in the original Born-Huang model [76] for isotropic media, but holds for arbitrary (anisotropic) media. The differential system formed by the coupled Euler-Lagrange equations is then written as the following eigenvalue problem: L" F~= ~ .

F~,

(110)

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where v is the set of labels denoting the eigenmodes, and

(111)

0 The 4 x 4 linear differential matrix L is partitioned in the form (112) with

LMM'U= - - 7 " u + V ' g : V u ,

LMECP=ot'Vq~,

L M' u = - v .

=

u],

1

v- [(I +

v

(113)

where M refers to mechanical and E to electrical parts. Then Eqs. (110)-(113) are compacted in the form LMM'Uv + LME(Pv= pOj2Uv, LEM'Uv + LEEqgv:O ,

(114)

which are Newton and Poisson equations. The asymptotic boundary conditions, which together with the differential equations define the eigenvalue problem, are

[q~l<~

lul<~,

(115)

Vr,

because u and ~p must be bounded everywhere. It is proved [39] that the linear differential operator is Hermitean, which is an important analytical property, whence the orthogonality relation

~v d3rp(r)u*(r)'u~(r) = 0

(Ia # v)

(1 16)

with the weight factor p, noting that (i) p is the reduced mass density and (ii) this holds generally for any position-dependent p, in particular for matched systems where p(r) is piecewise constant if the constituent media are homogeneous but different. It is furthermore shown that a first integral of the differential system yields at the interfaces the continuity conditions [,~2 :Vuu(2) - A1:Vuu(1)]" N = 0,

[D~(2) - D~(1)]. N = 0,

(117)

where N is the outer normal of medium 1 pointing into medium 2. This provides a theoretical frame which encompasses general boundary conditions as well as particular cases, such as free surfaces or mechanically rigid walls. The analysis of [39] discusses in detail the question of normalization, a point which is often glossed over. The distinction is stressed between the normalization of the eigenvectors u u, which is arbitrary and the determination of the amplitude of the electrostatic potential field, written in second quantized form as • q~(r, t) = ~ A~ [b~0~ (r) e - io~t _.{._ b tVq~* (r) e"~a ], V

(118)

164

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which is subject to a canonical rule. If one chooses to normalize the u v as ~v d3r p(r) u~(r)'uv(r) = 6zv,

(119)

then it follows from a formal argument based on canonical commutation rules that At = ~ .

(120)

If the normalization of (119) were chosen to be different, which would be a perfectly valid option, then A t would consequently be different This form of the At coefficients was well known for acoustic modes with planar geometry [128-130]. It must be stressed that this derivation for the long-wavelength polar optical modes has substantial differences with the acoustic case, because (120) is valid independently of the geometry and structure of the system under study, and besides there is the coupling of u and go and therefore a different analysis is required. The crucial point to be stressed is that the orthogonality of the eigenvectors concerns only the u u, while full account is taken of the u and go coupling. It then follows that the correctly normalized electron-phonon Hamiltonian, when the normalization convention (119) is adopted, is quite generally [39]

He _ ph=~ue tZc% - )

[b~go~(r)e-i,ov,+b~go~(r)e * • i,~v,].

(121)

w

Detailed expressions for quantum wires and quantum dots can be found in [125,126].

5. Envelope-function theory for polar optical phonons in semiconductor heterostructures The envelope-function theory in this case can be considered as a bridge between the microscopic calculations and the phenomenological models, discussed in the previous sections. The envelopefunction approximation was derived directly from microscopic lattice dynamics models by Akera and Ando [25]. They described the properties of the interface by means of a transfer matrix relating the envelope functions and their derivatives on either side of the interface. Very recently Foreman [42] has proposed another approach for the envelope-function approximation for the optical phonons in heterostructures. Starting from an exact envelope-function formalism Foreman derived a set of long-wavelength equations of motion having the same form as those provided by the complete phenomenological model, and reproducing also all the essential features of the discrete theory (lattice dynamics) and the Akera-Ando envelope-function approximation. Specific interface features, previously ignored in phenomenological studies, were incorporated through the addition of 6-function terms in the lowest order material parameters, such as the reduced mass and the zone-center force constant. In fact the inclusion of interface properties in this way is a well-known method in the study of acoustic properties in solids and fluids [131-133]. For simplicity Foreman restricted his study to a 1D model. The problem which must be addressed by this theory is, given a discrete function defined on a uniform lattice, how to choose, from among the infinity of possible continuum interpolations, that interpolation which describes best the properties of the original discrete function. The deciding criterion must be the "smoothness" of the interpolation, due to the fact that fast variations within less than the lattice parameter are not unique and cannot represent any real properties of the discrete

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function. In order to do this one must restrict the interpolatory functions to wave vectors inside the first Brillouin zone. For a lattice with N points, there are exactly N Fourier components in the first Brillouin zone and this choice gives an envelope-function which is exactly and uniquely related to the original discrete function. The restricted space of continuum functions obtained in this way was named the "quasicontinuum" by Foreman [42]. An application of this technique to lattice dynamics problems was made by Krumhansl [134 136]. A similar approach was developed independently by Kunin [137-140], who coined the name "quasicontinuum" to emphasize the differences between this theory and the "continuum" approaches in which no restriction is imposed on the wavelength. Foreman [42] seeking the best description of the long-wavelength phenomena transforms the equations to a set of variables that describe best the zone-center modes of vibration, for the crystal with a diatomic basis. A long-wavelength acoustic vibration is a motion of the center of mass of the unit cell, while an optical mode generates a relative motion of the ions within the unit cell. In a discrete theory all discrete physical quantities are sampled at the ionic coordinates x,j = x, + xj, where as usual the index n labels the unit cells, whilej = 1, 2 labels the ions (cation or anion) within the unit cell. Due to the fact that acoustic and optical variables must be independent of the ionic index j, it is convenient to rewrite the discrete theory in a way in which any explicit dependence on the ionic coordinates is eliminated [42]. This can be accomplished by sampling all quasicontinuum functions at the same physical location, for example the unit cell location x,. Foreman 1-42] distinguishes this representation by the use of capital letters (x N) and calls the resulting formalism the N representation. The change of basis between u~(n) - uj(x.~) and uj(N) =- u~(xN) is given by uj( U) = a ~ uj(n') 6B(X N -- X,,j),

uj(n) = a ~ uj( U') bB(X,j -- xN, ),

n"

(122)

N'

where a is the lattice parameter and bB(x) is the first Brillouin zone part of the Dirac b-function: ba(x)=lf

B(k)e'k~'dk-sin(rcx/a)rtx '

(123)

]kl < re~a, Ikl >re/a,

(124)

with B(k)=

1, 0,

fiB(X) is a finite and continuous function everywhere [140] and at integer multiples of the lattice constants, it takes the values (125)

bn(na) = a - 1 b,o.

In the case of a general nonlocal operator @, one has ~i(N,N')

= a 2 ~ bn(X u -- x,,7) ~s(n", n " ) b B ( x u, -- x,,,,j),

(126)

n', n'"

and for the mass operator mj( N, N') = a 2 ~ ms(n") b , ( x N - x,,,j) bB(X N, -- x,,,j) n"

= a3 2

m j~iN"" ) OB(X ~ " u -- X,,,~) ba(X N, -- X,,,,~) bB(XN,, -- X,,,,i).

(127)

166

V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123 176

It must be noted that mj is local in the n representation, but nonlocal in the N representation, unless xj is an integral multiple of a. The original equations are then transformed to center-of-mass (acoustic) and relative (optical) displacement variables U and u, respectively, given by M U = m l u I + m2u2,

u = u I - u 2.

(128)

The operator M represents the total unit cell mass, associated to the acoustic motion, while the operator associated to the optical vibrations is the reduced mass #. They are defined as M = rn 1 + m2,

St = m l M -

lm2 = m 2 M - l m j = ( m l 1 + m ; 1)- 1,

(129)

where for the inverse operator one must use the relations [42] M - l(n, n") M ( n " , n') = 6,,,,, a f M - 1(k, k . .) .M. ( k ,

a

m

- l

k') dk" = -1 B(k) 6(k - k'), a

. . . . . . .

(x,x)M(x

,x) dx

1

(130)

= -a 6(x - x'),

In the case of bulk media the operators M and # are local, and they reduce to the usual definitions of the total and reduced mass. After elimination of optical variables in the acoustic regime and of acoustic variables in the optical regime Foreman introduces the "boundary conditions" [42]. The meaning of this term is some statement of the interface behavior valid on a macroscopic scale but not involving any of the microscopic details of the interface. This is performed by means of a local transformation and after some involved steps, one can write the equation of motion for the optical modes as an infinite-order differential equation

(Co_

+L

l

.... 0,

(131)

where M o and #o are the total and reduced mass densities (mass per unit length in one dimension), respectively. The remaining kinetic energy terms such as m2, #2, M4, #4, and so on, represent spatial dispersion effects in the kinetic energy which arise near an interface. These terms vanish in the homogeneous bulk, because the mass operators are nonlocal in inhomogeneous media only. c o is a zone-center optical force constant, while c2, c4, etc. account for the second and higher order optical spatial dispersion. All these functions depend explicitly on the frequency. Obviously the infinite-order differential equation (130) as such cannot be used in practical calculations. It would be necessary to neglect the higher-order derivatives. It is clear that this can be easily done for slowly varying envelopes in homogeneous bulk media, but the situation is more complicated near an interface. One can integrate (130) across the interface region. To do this Foreman introduced a parameter e representing the approximate length scale of the interface. The parameter must be small on the macroscopic scale, but sufficiently large that the interval [xl < e covers all of the rapidly varying interface region. For Ix [ > e the material properties and the envelope functions will be slowly varying

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167

bulk functions. Then one can write

(Co-CO21~o)udx+ (c2-co #2)~-S|

+~S

(c,-o92/~4)~--~2|

+...=0.

(132)

The spatial dispersion terms enter (131) only in terms of their values at the borders of the bulk regions x = + e. If the bulk envelopes are slowly varying the infinite series (131) will converge very rapidly, and only the second order terms will be important in the long-wavelength limit. One may discard all terms higher than second order in (130) because the truncated differential equation has the same boundary conditions to this order. Then the only material coefficients having an important interface behavior in the case of the optical modes are c o and #o. If c o and/~o do not deviate significantly in the interface region Ix] < e from their bulk values, then we can see from (130) that Co - (~2~o is of order (a/2) 2. The interface term in (131) is therefore of order (a/2) 3, and the optical boundary conditions are exactly the same as the acoustic boundary conditions [42]. On the other hand, if the interface values of c o and #o are quite different from their bulk values, the function Co - ~O2#omust be considered as of order (a/2) °, and the interface term in (131) is of order (a/2). Then, this term will be the dominant one in the boundary conditions, and it may produce a qualitatively different behavior in such a way that mechanical interface modes exist at frequencies outside the bulk optical spectrum. This effect may be incorporated into a long-wavelength theory by using Dirac 6-functions [72,131-133]. They represent the limiting behavior ofc 0 and/~o as e tends to zero, but with the condition that the integral in (131) does not change. The theory was applied to study the p h o n o n spectra of InAs/GaSb SLs, and good agreement was found when compared with microscopic calculations [6,7]. When applying the theory to the GaAs/AIAs interfaces [42] the molecular perspective was used, and the only bonds in the structure are GaAs and AlAs, with no distinct interface layer. The mechanisms for generating interface modes near the zone center are very weak, and they would come from differences in the second-neighbor force constants in the bulk and the modification of the nearest-neighbor force constants at the interface. As a consequence no mechanical interface mode near the zone center of GaAs/AIAs heterostructures would be expected. This is an example of a case where the interface terms are negligible and the boundary conditions are equivalent to those of elasticity theory. On the other hand, if one considers InAs/GaSb heterostructures the situation is quite different. The reduced masses of InAs and GaSb are nearly equal and consequently there is considerable overlap of their bulk optical bands. Besides this fact, if one adopts a molecular viewpoint, the interface will be formed by one layer of either InAs or GaAs, depending on the growth sequence of the heterostructure. This layer will exhibit vibrational properties quite different from those of the bulk constituents, because InSb is much heavier than both InAs and GaSb, while GaAs is much lighter. It is then quite reasonable to expect the appearance of localized polar optical modes at frequencies above (GaAs) and below (InSb) the bulk p h o n o n spectra of InAs and GaSb. A detailed study of this case can be found in [42]. It is then clear that the primary interface effects coming out of this theory are those contained in the reduced mass, since this parameter is the determining factor in the existence or nonexistence of mechanical interface modes. When these modes exist it is also important to include the effects of the force-constant term describing the difference between bond-bending and bond-stretching forces [42].

168

V.R. Velasco, F. Garcla-Moliner/Surface Science Reports 28 (I 997) 123 176

6. Further developments of the complete phenomenological model for more complicated heterostructures We have seen that this model gives a very good description of the GaAs-A1As heterostructures. Nevertheless the model has been put forward in the simplest form, and the equivalent of the stiffness tensor (64) is written in an isotropic form. Crystalline anistropy, for the case of cubic crystals has been included in the model 1-141] by using for ~ the anisotropic form of a stress tensor and treating the anisotropy as a perturbative correction to the isotropic case. Analytic expressions for the u and ~0of the polar optical modes were thus obtained for QWs with propagation direction along the [0 1 0] and [1 10] directions. It was found in [141] that for the GaAs modes the corrections are always small (< 8 cm-1), but they could be observed experimentally. The decoupled transverse modes have a simple parabolic correction induced by the change in the transverse dispersion from the isotropic case. The modes having longitudinal and transverse components exhibit anticrossings in their dispersion for the isotropic case. These anticrossings are also present in the frequency corrections due to the anisotropy. It has been mentioned when discussing the envelope-function approach that the interface effects may easily be incorporated by adding 6-function terms in the lowest order material parameters. Quite recently [142] the complete phenomenological model has been extended in such a way that one can study the case of interfaces with no common ion, by means of the introduction of extra terms in the boundary conditions, in a similar way to that considered in the study of acoustic properties of solids and fluids [131-133]. The case studied in 1-142] is that of the localized interface phonon modes of a homogeneous material with one atomic plane of extraneous atoms (a planar defect). Two kinds of solutions were found, in the form of transverse uncoupled modes with zero electrostatic potential and coupled modes with a mixed character. It was also found in [142] that the eigenfrequency for the coupled solution is solely a function of the parameters of the extraneous atomic planes. There are also more complicated symmetric nanostructures involving a larger number of nonequivalent interfaces. We can consider, among these, the double barrier structures grown for the study of resonant tunneling, where phonon replicas are observed in the tunneling current [ 143]. This indicates that there is a significant interaction of the tunneling electrons with the phonons of the double barrier resonant tunneling (henceforth DBRT) structure and therefore it is interesting to study the optical modes of such a system. Now, the mechanical amplitudes of the modes centered at the GaAs well decay very fast on penetrating the A1GaAs barriers, but the electrostatic potential does not, as seen in explicit calculations of the spatial dependence of amplitudes and spectral strengths for QWs 1-35] and single interfaces [40]. Since mechanical and electrical amplitudes are coupled, the normal modes of the GaAs well are ultimately coupled to the continuum of the bulk GaAs modes outside the barriers and this requires studying the modes of the entire DBRT structure, which involves matching at four interfaces. The surface Green function matching method [92] has been recently extended to an arbitrary number of interfaces [144] so that matching is simultaneously effected in compact form at all the interfaces. This can be directly applied to the four interfaces involved in the DBRT structure but, since this is symmetric, it was considered interesting and useful to set up a method for separating solutions of opposite parity [145] for the following reasons: It is known that the frequencies of the optical modes in a quantum well tend to concentrate rather densely in narrow frequency intervals and, as the number increases substantially with the well

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169

thickness dw, it is easy to find consecutive eigenvalues so close to each other that it can be numerically very awkward to resolve them. It is then clearly helpful to be able to separate odd and even solutions from the start. The effect of this separate "combing" of the spectrum is that one always has a comfortable numerical resolution of consecutive eigenvalues. Moreover, it was shown that this can be done while having the number of interfaces at which it is necessary to match, which is intuitively obvious from the symmetry of the structure and is also advantageous from a practical point of view. In this case one needs only match at two interfaces instead of four. The method was applied to the study of the DBRT structures considered in 1-143]. We shall quote here, in order to demonstrate the effects of electrostatic coupling through the barriers, the more relevant results for the structure having the widest barriers (db = 111 A) and largest composition difference (x = 0.4). This is the case when one would expect the weakest coupling effects. The well width is d b = 58 A and the other input parameters are as follows: reduced mass densities in g c m - 3. 1.34 for GaAs and 1.10 for A1GaAs (x = 0.4); resonant frequencies in c m - ~: see Table 1. We shall consider the GaAs-like modes which are the ones of interest for the phonon-assisted tunneling [144] as the fraction of time spent by the tunnelling electrons in the A1GaAs barriers is comparatively negligible. After making full use of the formal symmetry analysis [145] the states were classified as even/odd (E/O) by reference to the parity of the electrostatic potential in the literal sense that q~(z) = _ qg~(- z). In this - literal - sense this is the same as the parity ofu r but opposite to that o f Uz.

In order to see the effects of long range electrostatic coupling between the central layer, the well region, and the external media across the barriers the actual double barrier structure A - B - A - B - A was compared with the corresponding well B - A - B in which the barriers extend to + ~ . The focus being mainly on the long-wave limit it sufficed to study the electrical part of the density of states, which is the part of the total spectral strength which matters for the electron-phonon interaction. For a full study of the problem one would need the spatial dependence of the electrostatic potential ~0~(z) properly normalized [39]. This can be obtained if desired, but in order to demonstrate the effects of electrostatic coupling it suffices to note that for very small K the predominantly electrical modes are the interface modes [35,40], and it is possible to concentrate only on the electrical spectral Table 1 Spectral strength (in brackets) of the main electrical peaks, in common arbitrary units ~o 273.53 273.58 278.57 a 292.22 292.35 292.37

~OLO

QW

DB

DB

Eow(1.44 x 103)

EDB(2.38 x 10-1)

O~B(1.34x 10 -2)

E~w(1.38 x 10 -a) Oow(1,00 x 10 -2)

OOB(2.02× 10 -z)

A1GaAs

E~B(8.10 × 103) GaAs

a The calculation has been made for ~ = 106 c m - a. For the rest x = 10 c m - 1. First column: frequencies in units of cm-1. Second colum: bulk values, of ~Oeo. A1GaAs is the ternary compound with x = 0.4. Eow/OQw denote the two interface (predominantly electrical) modes (even/odd) of the quantum well. For the double barrier (DB) structure they appear at the same frequencies and two new interface modes ObB and EbB appear very close to these.

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170

strength projected at the matching interfaces, which is sufficiently meaningful to demonstrate the changes in the spectrum of the well modes due to coupling across finite barriers. The simple QW was considered first as a testing ground for the method. With two matching interfaces there are two interface modes, one odd (OQw) and one even (EQw), in order of decreasing frequencies. These appear in the electrical spectral strengths as peaks at the corresponding eigenfrequencies, given in Table 1 together with their peak intensities. The calculation was made for a very small ~ = 10 cm-1. It is striking that the strength of EQw is five orders of magnitude higher than that of OQw, a fact which can be understood by noting that EQw, for x = 10 c m - 1, is extremely close to the bulk resonant frequency COLofor the barrier material (A1GaAs) outside the well. Fig. 15

200

(a)

~150

~'100 0 ci _.1

50

JL_ ,

.

.

.

.

.

i

250

,

,

,

,

260

i

,

270

J

,

,

.

280

.

.

.

.

.

.

.

.

,

~ (cm-l) 290

300

0.00012

(b)

0.0001

"~0.00008 ~'0.00006 0 Q

..a0.00004 0.00002

r

250

,

,

,

,

i

260

,

,

,

,

I

i

270

280

.

,

,

,

(cm')

i

290

,

,

,

,

r

300

Fig. 15. Electrical spectral strength for even potential states in the simple q u a n t u m well described in the text. (a) Calculated for x = 10 c m - 1. (b) F o r ~c= 10 6 c m - 1. Spectral strength in arbitrary units, the same everywhere in this and the following figures. Peak intensities are given in Table 1 in the same units.

V.R. Velasco, F. Garcia-M oliner/Surface Science Reports 28 (I 997) 123-176

171

shows the electrical spectra of the even modes for x = 10 c m - 1 (a) and for x = 106 c m - 1 (b), which is still in the long-wavelength range, two orders of magnitude smaller than a "radius" of the Brillouin zone. The corresponding peak (E~w) has moved upwards in frequency, away from ~OLO(AIGaAs) and its strength (Table 1) has gone down by five orders of magnitude, now being comparable to the strength of Oow. Fig. 16 shows the odd spectra for QW and the DBRT structure. The eigenvalue of OQw, again for x = 10cm -1, is quite insensitive to the change on going over from the simple QW to the DB structure, and the same holds for EQw,but other features are rather more strongly affected. Thus, the strength of EbB, fifth column in Table 1, goes down by four orders of magnitude with respect of that of Eow. The mode still resonates with O)LO(A1GaAs) but outside the well there are onlyfinite barriers of

0.0001

(a)

0.00008

==o.oooo6 0';I

00.00004 ...I

0.00002 _

,,,

jJ~ i

250

260

270

L

280

\ i

i

290

30O

to (cm")

0.0~25

(b)

0.0002

0.00015

J

v (O

Oo 0.0001 _.J

0.00005

0

i

i

250

260

,

,...j

i

270

280

i

J

290

30O

~0(cm-~) Fig. 16. As in Fig. 1, electrical spectrum of odd potential states for K = 10cm 1: (a) for the simple QW, (b) for the DBRT strucuture.

172

V.R. Velasco, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123-176

A1GaAs and the effect is much weaker. However, the strength of EDR is still one order of magnitude higher than that of ODB, which remains essentially unchanged, although it increases by a factor of 2 because the spectra for the DB structure contain the contributions from the projection at four matching interfaces, instead of the two of the QW. Besides, two new modes (ODB and EoB) appear in the DB structure. The first one appears, accidentally, at the same frequency as EDB and the second one very close to ODB-The formal method for factorizing separately the even and odd spectra is here very useful. Without this the problems of numerical resolution appearing in this case would be uncomfortable in practice. The mode EDB is now extremely close to the resonant frequency COLOof GaAs material of the well, where these modes mainly reside, and its strength goes up, again, to the

6 1.10

(a)

8O00OO

5 6ooooo

~400000 200000

250

260

270

280 co (cm ~) 290

300

6 1.5 10

6

(b)

1.25 10 '~"

6

~

1.10

~750000

g "J

500000

2soooo

d i

250

,

,

,

,

i

260

,

,

,

,

i

270

,

,

,

,

i

280

j ,

,

,

,

(cm~)

i

290

,

,

,

,

i

300

Fig. 17. Mechanicalspectrumof evenpotentialstates for x = 10cm- :(a)for the simpleQW; (b) for the DBRT structure. Commonarbitrary units for (a) and (b), unrelatedto those of Figs. 2 and 3.

V.R. Velaseo, F. Garcia-Moliner/Surface Science Reports 28 (1997) 123-176

173

order of magnitude of EQw. It is significant that the peak ODB, although also at the resonant frequency COLO(A1GaAs), is one order of magnitude weaker than EDa and of a strength similar to that of ODB. A glimpse at a possible interpretation can be seen in Fig. 17, which is also interesting to consider for the sake of completeness. This figure compares the mechanical spectral strength of the QW (a) with that of the DB structure (b), again for x = 10 c m - 1 and in arbitrary common units. The numerical values of the mechanical spectral strength must not be related to those of the electrical spectrum, as they have different physical dimensions. What matters is the comparison between the two mechanical spectra of Fig. 17. The point is that the polar modes under study have a mixed character and the electrical and mechanical vibrations are mutually coupled. With x = 0.4 the mechanical barriers are already in practice totally rigid and all GaAs-like modes would be separately confined in the three disconnected GaAs regions of the structure with no coupling across the barriers, but the electrical vibrations are coupled and this drives an indirect coupling of the mechanical vibrations, as seen in Fig. 17(b). The mechanical spectrum undergoes substantial changes on going from the QW to the DB structure which means that some spectral strength is transferred from electrical to mechanical. In this respect it can be seen that there is a distinctly confined mode of essentially mechanical character at a frequency very close to 273.53 cm-1. The question of why this should affect the odd electrical mode ODB rather than the even one EDB could only be elucidated by a more detailed study than the one presented in [145], but it is clear that irrespective of all the details the spectrum of the DB structure may exhibit quite substantial changes with respect to that of the simple QW.

7. Conclusions The theoretical description of long-wave polar optical modes in nanostructures in terms of simple phenomenological models has reached a reasonably satisfactory situation. The basic issues have been clarified and, provided all couplings between the electrical and mechanical fields are taken into account and all matching boundary conditions are consistently satisfied, the results obtained in this way appear to satisfy all quantitative and even to a large extent qualitative reliability tests. This also provides a sound scheme to study the electron-phonon interaction and to include further improvements such as crystal anisotropy and interface effects.

Acknowledgements We are grateful to Profs. R. P6rez-Alvarez and C. TraUero-Giner for useful discussions and for providing us with some figures and results prior to publication. This work was partially supported by the Spanish DGICYT through Grant No. PB93-1251.

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