Carrier interaction in quantum nanosystems тема автореферата и диссертации по физике, 01.04.10 ВАК РФ

YEREVAN STATE UNIVERSITY

MINISTRY OF EDUCATION AND SCIENCE. REPUBLIC OF ARMENIA

Badalian Samvel Michael

Carrier Interaction in Quantum Nanosystems

Speciality: A.04.10 Semiconductor and dielectric Physics

A dissertation submitted to the 049 Specialized Council of Yerevan State University for the scientific degree of

Doctor of Physical and Mathematical Sciences

Yerevan - 1997

I lie wölk ih (lullC al Yerevan Slale I ! II i Vi'l'hi I V .

Official opponents Dr. pliys. math. sc. A A Kirakosyan l)r. pliys. math. sc. I] R Minasyan Dr. pliys. math. sc. A Ya Sliik

Leading institution Institute for Iiadiophysics and Electronics.

the National Academy of Sciences of Armenia

The public defense will take place on December 27. 1997 at 13:00 at the session of YSU Specialized Council 0-19 (1 A Manoogian St.. Yerevan. Armenia. 375049).

The dissertation is available at YSU library.

The abstract of the dissertation is sent out on October 27. 1997.

Secretary of the Specialized C<

caiul. pliys. math. sc.

V P Qalantaryan

1 GENERAL

I. I IIII roil uc I ion

l'or decades. I lie microelecl ionics is I Ik- key technology whose device si rial mes are widely adopted for industry production being in great market demand. Giant efforts and resources are directed to the growth of the microelectronics in such industry developed countries as Japan. Germany and the USA. Alone in Germany. expenditures in the lield of the microelecl mnics account lor iiuhv than a third of the annual turnover (circa 600 milliard DM) with 3 million workers involved in litis sphere '. Nowadays, development of submicron electronic structures is being pursued in many countries because these structures give promise of new material systems with enhanced optical, transport and thermalization properties that could make an impact in a variety of technologies. including semiconductor lasers and modulators, detectors, and optical filters.

liecent rapid advances ill semiconductor technology of ultraline liihog-raphy and modern etching technique have enable the fabrication of a large diversity of ultra-narrow synthetic structures with perfect atomic interfaces separating different materials including silicon. Ill-V and 1I-IV semiconductors and others- "3. In these material systems, both composition and doping can lie controlled on a scale of the order of the de Broglie wavelength of carriers. Such a class of artificial semiconductor nanostructures (contacts of metal-insulator-semiconductor types, heterojunctions. superlattices. tic.) with dimensions in the range of i — 100 inn has opened up a new dimension in solid slate and semiconductor physics in the regime when the quantum size effect appears This new freedom in engineering the electronic states and their properties offer a feasibility lo study plivsics laws in real low dimensional systems ami ;m exciting potential lor elect ionic applical ions as well for invest igal loii1-ol new tantalizing phenomena principal for fundamental physics.

During the last decade, the semiconductor nanoscale systems with canier confinement in one (quantum wells [QW]), two (quantum wires [QYVr]). and all three dimensions (quantum dots [QD]) have been studied intensively in theory and.experiment 4 5 6 to characterize, understand conceptually, and exploit quantum effects in technologically important semiconductor device structures. Already, a set of novel results such as the weak and Anderson localization '. the effect of very high electron mobility in modulation doped heterojunctions'4 has been discovered in 2D nanosystems. Quantum effects in 2D nanosystems have been exploited to provide enhanced semiconductor lasers, a new class of infrared detectors, and resonant tunneling diodes. The high electron mobility I ransisl ors on (!aAs hel erosl ruct ures an' used in a source-drain channel lo

construct high frequency amplifiers wit li the limiting frequency above than 100 c;iiz '.

However, investigations of naiiosyslems willi a I wo dimensional electron g; is (Vl)KC) is frequently connected with I lie use of high magnetic fields. l'or example. Iiiv.li molality samples willi the 2l)l'!(i of iianoscale dimensions ex |u>sed In the normal high magnetic lield exhibit quantized resistance (the integer quantum Hall died [QUE]'" ") and exotic charge correlation (the fractional QIIK 12 l:l). Understanding the integer and the fractional QHE has intrigued and challenged researchers for more than 10 years now. both for the wide range of new phenomena discovered 14 1516 and for the potential for high precision metrology based on the QII10 1'■

Another key feature of the '2DE(J in the QUE geometry is the existence of quantum edge states localized in the vicinity of the sample boundary. The quantization of the Hall resistance into the li/c- portions can be explained easily under assumption that the edge states are not influenced by scattering and the transport is adiahalic 1 !l

(¿Wis and QlJs give promise to provide even greater enhancement in op-lical and relaxation properties of semiconductor device structures because the u a \ <• In i u'l ions are furl her com pressed by la I era 1 coulinemenl and I he quant inn ellects can be used lo concentrate the optical oscillator strengths at the active transitions-1 -'-'-3. In this limit these structures behave as manmadeartificial i/ui/jialoms""1. Furthermore, these structures under different new environments exhiliil dis<r«li- charging effects and give promise of devices operating in I he IiiiiiI of single e|ed ion Iraiisporl.

Carrier scattering is a fundamental process that always exist in crystals. Characterizing and understanding carrier scattering processes in the quantum naiiosyslems arc critical both for unraveling the basic phenomena of quantum effects in nanopliysics and for controlling carrier dynamics in the nanoscale device structures, such as relaxation rates for carrier thermalization. equilibration and Auger scattering rates for carrier distribution, and diffusion rates for carrier transport. Carrier scattering is strategic for identifying the lateral confinement effects and for developing enhanced, useful device struc-lures. For example, blueshifts in QWr phololuminescence are typically attributed to lateral confinement. Recently, blueshifts observed in QWr magneto-l<li'/l(>liiiiiiin s< < ncc liavi' lni'ii at I rib ill <d lo suppression of carrier I ransporl and incomplete relaxation in QWrs rather than to lateral confinement induced le\el >hifts. It has been suggested that this poor luminescence has an intrinsic origin, resulting from a suppression of plionon scattering in small QDs that leads to a "plionon bottleneck'' for electron relaxation2627. If this intrinsic

mechanism for suppressing luminescence dominates ex I rinsic eifeel s due lo pro-

cessing, then the possibilities for using the effectively OD nanosystems (QDs. QYVrs and 2D naiiostructtires in normal quantizing magnetic fields) in any application that relies on carrier relaxation, such as injection lasers, would be severely limited. Thus it is critical that the character of carrier scattering, relaxation and transport in effectively OD nanost met tires be established thoroughly to separate transport from confinement effects and better control and relieve I lie suppression of t ransport. M nit iphoiion processes provide additional evidence for this 'phonon bottleneck'' in these OD systems. It has been suggested that multiphonon processes-'8 and Auger scattering29 can break this ' bottleneck''. Thus a careful analysis of all of these scattering mechanisms for low dimensional nanostrncl ures is imperative.

In this work 1 present theoretical investigations of carrier scattering in quantum nanosystems carried out during I he lasl ten years by coworkers and me. Carrier interact ion wilh phonon.s. photons, impurities, and electrons in semiconductor nanoscale systems with carrier confinement in one and two dimensions in zero and quantizing magnetic fields has been addressed. Most importantly, our calculations allow to better understand phonon signature in optical, Ihermalizalion. and transport experiments that can be used to identify and characterize the basic phenomena of quantum confinement in these quantum nanostructures.

2 Scientific novelty and practical value of the dissertation

We have investigated peculiarities of the polaron (elementary excitations in I lie elect roii + plionoii sysl en I) spectrum near the longil udinal opl iral phonon emission threshold. In spite of weak electron-phonon coupling, we have obtained that in the 2DEG in the QHE geometry, new complex quasiparticles. electron-phonon bound states, appear in magneto-polaron spectrum. They const it lite an infinite set which is coagulated lo the threshold both above and below it'1'. In contrast to the virtual phunons taking part in the formal ion of the usual Frölich polaron. phonons in the bound slates are almost real The characteristic scale of the binding energies is essentially greater than i hi-correspondiiig scale in massive samples.

The existence of the electron-phonon bound states results to the fine structure of the cyclotron-phonon resonance33. According to the perturbation theory. photon absorption was to be expected at the phonon emission I liretlin|<l In reality, the pert urbat ion theory becomes inapplicable in the im mediate \ iciii-ity of the threshold. The true spectrum is obtained from the solution of an integral equation for the electron-phonon scattering amplitude. Absorption entirely governed by the bound states with the total angular momentum ±1.

I lie ali>ui |il inn spectrum consist* of two groups of peaks wliitli are approximately of I lie same ampliliide and are located approximately asymmetrically relative to the ' threshold''.

Our calculations34 35 36 37 38 39 40 41 explain the dynamics of carrier relaxation in the ID and "2D quantum nanosystems both in zero and quantizing magnetic fields. Understanding and conl rolling this dynamics is critical because rapid carrier relaxation is crucial for many oil lie I echnulogical applications proposed for semiconductor nanoscale quantum devices. We have investigated t lioiuiiglily I In- main relaxation characteristics of I D and 21) electron systems due to acoustic phonon (deformation DA and piezoelectric PA) and polar optical I'O phonon scat lerilig.

Special attention has been paid to the presence of various interfaces separating different materials in 2D nanostructures. These interfaces influence the acoustic phonon normal modes and can affect essentially electron-phonon interaction. We have proposed a new method for calculating the probability of electron scattering from the deformation potential of acoustic phonons34. Such a probability summed over all phonon modes of the layered elastic medium can be expressed in terms of the elasticity theory Green function which contains all information about structure geometry.

Exploiting this method, the energy and momentum relaxation times of a test electron as well as the relaxation rate of electron temperature for the whole Fermi '2DEG located in the vicinity of an interface between elastic semi-spaces have been calculated 3t>. Analysis of limiting cases for an interface between solid and liquid semi-spaces, for a free and rigid surfaces has shown that there are situations when the phonon reflection from various interfaces alters the energy (or electron temperature) dependence of the relaxation times and leads In a strung reduction of the relaxation rales.

To illustrate the interface effect in quantizing magnetic fields, electron relaxation between discrete Landau levels in 2DEG located near free crystal surface has been studied 37. The interface effect is obtained to be highest in the magnetic field since the 2DEC internets willi almost mouochromatic i yclui run phonons in this case. The electron transition probability has an oscillating behavior of the magnetic field and of the distance from the 2DEG to the interface. Scattering from the deformation potential of acoustic phonons has been only considered since fn quantizing magnetic fields, scattering from piezoelectric potential is strongly suppressed 40.

In the 2DIX! subjected to strong magnetic fields with rather thin election layers and subjected to rather strong magnetic fields, a large separation between Landau levels cannot be covered by an acoustical LA phonon 37 so the mulliphonon 2LA or an optical phonon assisted 41. processes become

more efficient. Longitudinal optical phonon assisted inter Landau level transitions via one-phonon emission mechanism requires a precise resonance. Away from the resonance, efficiency of this process falls steeply. Non-resonant optical phonon emission in the effectively OD systems should be accompanied by-acoustic phonon emission via the two-phonon emission mechanism. VVe have calculated polar optical PO phonon assisted electron relaxation as a function of the inter Landau level spacing in the 2DEG in the QHE geometry41. The interface optical SO phonon relaxation has been found to be at least by an order weaker than relaxation via polar optical PO phonon emission. To obtain a finite relaxation rate associated with one-phonon emission, the allowance for the Landau level broadening and for the PO phonon dispersion has been made. Immediately below the phonon energy, htjpo. the PO phonon dispersion contribution gives rise to a sharp peak with the peak value approximately 0.17 fs-1. The Landau level broadening contribution has a rather broad peak with relatively lower peak value. Below h^Jpo within an energy range of the order of h s/uJb It• the one-phonon relaxation rate exceeds 1 ps-1 (r is the relaxation time deduced from the mobility. In GaAs/AlGaAs with mobility // = 25 V-1 s-1 ni- this range makes up 0.7 ineV).

Tvvo-phonon emission is a controlling relaxation mechanism above fi^po 41. For energies A/Ziu-c immediately above lujpo- PO+DA phonon relaxation has a sharp onset. The relaxation rate increases as a fifth power in the magnetic field achieving to the peak value exceeding 1 ps at energy separations of the order of hs/aa (s is the sound velocity, as is the magnetic length. In GaAs at B — 7 T we have hs/aa s; 0.-1 ineV). At higher magnetic fields in the energy range sa^1 < A/u-73 — wpo (in GaAs with d = 3 inn we haw

hs/d zz 1.2 meV). the two-phonon peak decreases linearly in the magnetic field. Above L'po within the wide energy range (in GaAs this range makes up approximately 5 meV). the magnetic field dependence of the relaxation rate is rather weak and the subnanosecond relaxation between Landau levels can be achieved via the two-phonon emission mechanism. Our analysis has demonstrated that in some experimental situations the PO+DA-phonon emission mechanism is more efficient than relaxation in two consecutive emission acts: PO phonon emission (even under the sharp resonance) with subsequent emission of either LA- or 2LA-phonons.

As distinct from the conventional relaxation experiments where the relaxation rates are measured directly, in the ballistic phonon emission experiment> the intensity and the angular distribution of the phonon signal are detected 011 the sample reverse face which provide a worthy source of information on electronic properties. Such experiments are especially valuable in systems of reduced dimensionality since carriers are confined to small nanoscale regions

while other quasiparticles sucli as phonons can be not localized and more reachable for study.

In the dissertation I present calculations of ballistic acoustic phonon emission at electron transitions between fully discrete Landau levels in a 2DEG with account of the phonon reflection from a GaAs/AlGaAs type interface 43 44. In accordance with the experimental results, we have obtained that the angular distribution of emitted phonons has a distinctly expressed peak for small angles. Account for the interface effect affects essentially the intensity and the composition of the detected phonon field. Upon their reflection and conversion at the crystal surface, longitudinal acoustic LA phonons propagate backwards in the forms of LA and transverse acoustic TA phonons. The reflected LA phonons interfere with the initial LA phonons emitted by the 2DEG in the same direction. Therefore, we have obtained that under the deformation electron-phonon interaction, the detector records on the sample reverse face both the interference field of the LA phonons and. which is most intriguing for experiment, the conversion field of the TA phonons43. Our calculations of emission spectrum for surface acoustic phonons show that an exponential suppression of the emission of surface acoustic phonons occurs in a wide range of the magnetic field variation44. So. the cooling of the heated 2DEG is only at the expense of bulk LA and TA phonons.

Investigation of edge state scattering 38 39 45 is one of key stages of the dissertation. The single particle energy spectrum in a 2DEG exposed to a homogeneous magnetic field normal to the electron plane is separated into the edge states and the bulk Landau states The edge states correspond to the classical skipping orbits and are confined near sample edges. Edge states exist also in QWrs. If the wire width L is much greater than the magnetic length mi. L then the edge states both in 2DEGs and QWrs can be treated

in the same way as a ID electron system. The edge states play an important role both in conventional transport measurement experiments and in ballistic phonon emission and absorption experiments. In the latter case, equally with the bulk Landau states, the quantum edge states also give a contribution to emission and absorption of ballistic phonons. We have calculated ballistic acoustic (both for deformation DA and piezoelectric PA interactions) 45 and polar optical phonon emission by the quantum edge states. An analytic expression for the ballistic acoustic energy flux emitted by the quantum edge states has been derived. Detailed analysis of the phonon emission intensity distribution has been made in low and high temperature regimes and for different positions of the Fermi level. At the same time as phonon emission by the bulk Landau states is concentrated within a narrow cone around the magnetic field, at the inter edge state transitions and at low temperatures, the emitted

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phonon Held is |>rodoi11iii111ly concentrilled within a narrow cone around the direction of edge state propagation, while at high temperatures - around the magnetic field normal to the electron plane. At low temperatures the emission intensity decreases exponentially with decreasing filling of Fermi level. In contrast to the case of bulk Landau states where piezoelectric interaction is always suppressed with respect to the deformation interaction, in the edge state case, the relative contributions of piezoelectric and deformation interactions depend on the magnetic field and temperature.

We have studied an optical phonon assisted edge state relaxation in QWrs with a rectangular cross section exposed to the normal magnetic field. The ¡111 Iasiibbaiid scattering rates as a function of I lie initial electron energy ('•! different values of the magnetic field has been calculated. By considering dil-ferent limiting cases of the ratio of the cyclotron frequency to the strength of the lateral confinement, results for edge state relaxation both in 2DEG& and QWrs as well as for the magnetic field free case can be obtained.

We have calculated the inter edge state scattering length for an arbitrary confining potential3839. Phonon (deformation acoustic DA and piezoeiectiic PA interactions) and impurity scatterings are discussed and analytical expressions for scattering lengths are derived. As follows from energy and momentum conservation, only phonons with frequencies above some threshold can participate in the transitions between edge states. As a result, phonon scattering is exponentially suppressed at low lemperal ures. According lo our exaliia lions. I lie observed tempera! lire dependence ol I lie scat I erilig lengl h can hut I >c attributed to phonon scattering.

In recent magneto-luminescence experiments by Potemski ei at. 4>J on one-side modulation doped GaAs/AlGaAs quantum wells, an up-conversiori has been observed and interpreted as being due to an Auger process. The luminescence spectrum under interband excitation at low temperatures and for low excitation powers shows two peaks: besides the luminescence due to recombination of an electron from lowest I.andau level with a hole in a valence band, a second peak is observed above the exciting laser energy and is related lo recombination of an Auger up-converted electron with a hole. We have developed a theory of Auger up-conversion in quantum wells in quantizing magnetic fields to explain these experimental results'10. We have calculated the characteristic times of electron-electron scattering processes between Landau levels of the lowest electric subband and of electron-acoustic phonon scattering between Landau levels of the two lowest electric subbands as well as the lifetime of a test hole both with respect lo the Auger process and phonon emission By analyzing rate equations for these processes as well as for the pumping by inlerband excitation and llie recombination of electrons with photo-induced

holes. wo have found llic Anger process lime. As well the magnetic field and llie excitation power dependencies of the two luminescence peaks have been obtained which are consistent with the experimental findings. Thus, an understanding of the Auger up-conversion observed in the magneto-luminescence in quantum wells is provided.

Recently a research group from the Toshiba Cambridge Research Center

I.id. and Cavendish Laboratory has proposed a new technique to produce non-homogeneous magnetic lields 1' IM A remotely doped (.¡aAs/Al^iai_xAs heterojunction is grown over wafer previously patterned with series of facet. The use of in situ cleaning technique enables to regrow uniform high quality 2DEC.s which are no longer planar but follow I he contour of the original wafer. Application of a homogeneous magnetic lield lo this structure results a spatially varying field component normal lo the 2I)E(j. Thus, this technique offers possibilities lo investigate the effect of varying the topography of an electron gas in addition lo varying the ilniii nsioiiaht//.

We have investigated theoretically the magneto-transport of the non-pla-

II.II' •/1)1.(1'". A> an example I he elerl I'ic lield (I isl I'i I >111 io|| of III«' '¿DEC with a magnetic field barrier is calculated. The system satisfies the I'oisson equation

ii i wln'li hm .ii <liai;y . <l<t"|,,|, ;i i lie i n /n< in ii i.ij'.nel ic lieM inlei face.

'1 he magneto-resistance across the facet as well as in the planar regions of the '2DEG have been calculated which explain the main features of the magnetic field dependencies observed experimentally

2.1 Main goals and tasks of the dissertation

1. Investigation of threshold peculiarities of the polaron spectrum in the 2DEC! in a quantizing magnetic lield normal to the electron sheef. Demonstration of the existence of the spectrum new branches which describe the hound stales of an electron and an optical phonon.

2. Study of photon absorption on these electron-phonon bound states. EsI. > [ > 11.11111 • III '/I lip line -.1 rill | llie III llie lyilolliill phnlloll lesol I a 11 Co ill

quantum wells and hcteroslructtires. Consideration both bulk and surface optical phonons m formation of the bound stales.

.'i. Development of a new met hod for calculating llie probability of electron scattering from the deformation potential of acoustic phonons. Use of this method to treat the phonon reflection from a crystal surface and interfaces separating dill'erent materials.

I. Exploiting the proposed method lo calculate llie electron energy and 1110-iiieiiluni relaxation times for a test electron as well as the relaxation rale

of electron temperature for the whole Fermi 2DEG taking into account t lie phonon reflection from the interface between semi-infinite elast ic media.

r>. ( 'a Ici i I; it ic >11 of I lie t rnnsil ion pro lia bility h el ween fully quant ized Land: m levels and taking into accounl the crystal Iree surface ellect on interaction ■of the 2DEG with acoustic phonons.

6. Study of ballistic acoustic phonon emission in quantizing magnetic fields normal to the 2DEG plane when reflection of these phonons from a GaAs/AlGaAs type interface is taken into account.

7. Investigation of longitudinal polar I'O optical phonon assisted electron relaxation in the 2DEG in the QUE geometry. Consideration of inter Landau level relaxation via one phonon (for bulk PO and interface SO phonons) and two PO+DA phonon (for deformation acoustic DA and -PC) phonons) emission processes.

8. Calculation of the scattering length for inter edge states transitions in quantizing magnetic fields due to acoustic phonons (DA and PA interactions) and short- and long-range impurities, assuming that the shape of the confining potential is arbitrary.

!). Study of emission of ballistic acoustic phonons (due to déformai ion and piezoelectric interactions) by quant um edge states in quant izing magnet ic fields. Detailed analysis of the emission intensity and the angular distribution in low and high temperature regimes. Consideration of different positions of the Fermi level.

10. Calculation of the optical phonon assisted edge state relaxation rates in quantum wires exposed to quantizing magnetic fields. Derivation of the PO phonon emission rate dependence cn the electron initial energy and magnetic field.

11. Construction of a theory of the Auger up-conversion observed in recent magneto-luminescence experiment. Calculation of the relaxation rales of the carrier-carrier (between discrete Landau levels of the lowrsl electric subband) and the carrier-phonon (between discrete Landau levels of the two lowest electric subbands) scattering processes in quantum wells exposed to the normal quantizing magnetic field.

12. Investigation of the magneto-lransporl of a non-planar 2DEG. Calculation of I lie electric field distribution in the presence of a magnetic tunnel

barrier oi'/jin width. Derivation of the magnetic lield dependence of the magneto-resistance both across the facet and in the planar regions of the 215IX;

tl.'J Basic theses of the dissertation submitted to the defense

J. Despite to the weak electron-phonon coupling, an infinite set of bound states of an electron with an optical phonon exists both above and below the threshold of optical phonon emission in the magneto-polaron spectrum in the 2DEG.

2. The spectrum of the cyclotron-phonon resonance has a fine structure which is governed entirely by the electron-phonon bound states. There-lore. electromagnetic absorption is concentrated not at the threshold but below and above il ;il I lie .separations of I lie binding energies. The absorption spectrum consists of two groups of peaks which constitute an ' asymmetric doublet''..

,'J. It is easy to account for the phonon reflection from various interfaces separating different materials using the proposed method for calculating tlie probability of electroii scattering IViiiii Hie deformation potential of' acoustic plionotis.

4. 1 here are situations when the presence of an interface, near which tlie Fermi 2DEG is located, leads to a strong reduction of the energy and the momentum relaxation and alters the dependence of the relaxation rates on the election energy (or on electron temperature).

5. The interface effect in the 2DEG is strongest in quantizing magnetic fields normal to the plane of electrons and for the free crystal surface. Surface effect becomes weaker with a distance between the 2DEG and the crystal surface for two reasons: because of the spread of momenta and I lie spread of frequencies of eiilil led phonoiis. Even ill high-quality lielerusliucluivs. the Landuu level broadening is essential.

6. The transition probability between two Landau levels via deformation acoustic phonons is an oscillating function both of the magnetic field B and of the distance ¿o- The magnetic field oscillations are on top of a smooth background of the transition probability which has a subnanosec-ond peak Cor intermediate values of Ii (in CJaAs. for fields of the order of 1 T) and decreases as a fourth power of B for large B. For vanishing fields, selection rules force the transition probability to fall to zero.

7. In quantizing magnetic fields, inter Landau level transitions in the 2l)K(i via piezoelectric acoustic phonon interaction are suppressed with respect to the transitions via the deformation interaction mechanism.

8. Acoustic energy flux of ballistic cyclotron phonons emitted from the "2DEG in a normal quantizing magnetic field is concentrated in a narrow cone around the magnetic field. The interface affects essentially both tin-intensity and composition of the emitted phonon field so on the sample reverse face, the detector records an interference field of longitudinal LA phonons and a conversion field of transverse TA phonons.

9. Exponential suppression of emission of cyclotron surface acoustic SA phonons occurs in a wide range of the magnetic field variation, h^n '2mcji (jjb is the cyclotron frequency. m,: is the electron mass and c/< is the velocity of surface waves). So cooling of the heated 2DECJ is only at the expense of emission of bulk LA and TA phonons.

10. An allowance for the polar optical PO phonon dispersion and the Landau level broadening yield a finite relaxation rate associated with one-phonon emission. Immediately below the PO phonon energy. the PO phonon dispersion contribution give rise to a very sharp peak with peak value approximately 0.17 fs-1. The Landau level broadening cont ribut ion has a rather broad peak with relatively lower peak value. Within an energy range of the order of h(r is a relaxation time deduced from the mobility.), the one-phonon relaxation rate exceeds 1 ps_1. Relaxation via surface SO phonon emission mechanism at least by an order is weaker than via bulk phonons.

11. The two-phonon emission is a controlling relaxation mechanism abo\e hjjpo. For energies Alhu,u immediately above huipo. PO-f DA phonon relaxation rate increases as a fifth power in the magnetic field B. At energy separations of the order of ¿ajj1 (a is the sound velocity, ah is the magnetic length). PO+DA phonon emission provides a mechanism of subpicosecond relaxation. At higher B. the two-phonon relaxation peak decreases linearly in B and within a wide energy range of the order of /iu;/}. subnanosecond relaxation can be achieved.

12. Inter edge state relaxation due to acoustic phonon scattering is -trough suppressed at low temperatures in comparison with short- and longrange impurity scattering since only phonons with frequencies above some threshold can cause transitions.

I.'i. At low inuperalures emission of ballistic acoustic phonons (due to deformation and piezoelectric interactions) at inter edge stale transitions is predominantly concentrated within a narrow cone around the direction of edge state propagation while at high temperatures - around the magnetic field normal to the electron plane. The emission intensity decreases willi decreasing filling of I lie I'ei iiii level. This diliiimil ion is exponent ial .t I I' >w I > m| m i .11 ii 11 .

II. In contrast to the bulk Landau state, relative contributions of piezoelectric and deformation interactions depend on the magnetic field, electron temperature, and on the shape of the confining potential. At low temperatures atul for the non-smooth confining potential. DA interaction suppressed with respect to PA while for the smooth potential as well as for both cases at high temperatures. DA and PA interactions give roughly the same contribution to the emitted phonon field.

1"» Polar opl ieal phonon emission provides a picosecond edge si ate relaxation I'll .1 I' ■ 11 I t 11 >11 III < III a III Hill Wile:,. '| lie :,c a I lei lllg i a I e a.i III nil ioll.'i 1 >|'

the electron initial energy has a peak for magnetic fields not far from the resonance field while for lower fields, the scattering rate shows a monotonous increase in energy.

Itj. Our theory of the Auger up-conversion in a 2DEG in a normal quantizing magnetic field provides an understanding of the up-conversion observed in magneto-luminescence of one-side modulation doped GaAs/AlGaAs quantum wells'"', in particular, ils dependence on the excitation power and the magnetic field.

17 A < ■< ■< ii'liuj1. loom I lieiji'y of I lie inagnel o I ran spur I in a noli planar 20I'X •. the system satisfies the Poisson equation in which a line charges develop at the magnetic/non-magnetic field interface. The magneto-resistance calculated across the facet and in planar regions of the 2DEG explains the main features of the magnetic field dependencies observed in the experiment

.'.I :\[>iiivIhiIii>i) (>f llit work

The results of the investigation presented in the dissertation have been reported and discussed at the following conferences, schools, and seminars:

199(5 Seminar. Institute for Theoretical Physics. University of Re-genshurg. (¡ermauy

199(5 121 h Internal ional Conference "on I lie A|>plicat ion of High Magnet ic fields' . Win/burg. (ierniany

199(5 9th International Conference on "Superlat tices. Microst ruc-tures. and Microdevices. July. M-19 1996. Liege. Belgium.

1996 Research Workshop on Condensed Matter Physics. ICTP. Italy

1996 Seminar. Department of Condensed Matter Theory. University of Antwerp. Belgium

1995 Research Workshop on Condensed Mailer Physics. ICTP. It; 11 >

1993 Winter school. University of Wroclaw. Poland

1992 NATO ASI; Ultrashort Processes in, Lucca. Italy

1992 Seminar. Institute for Theoretical Physics, University of Re-genshurg. (¡ermany

1993 13th General Conference of Condensed Matter Division. European Physical Society. Regensburg. Germany

1989 14th Ail-Union Conference on ' Theory of Semiconductors". Donetsk. The Ukraine

1988 Seminar. Institute of Physical Investigations. Ashtarak. Armenia

1988 Seminar. Landau Institute for Theoretical Physics. Institute of Solid State Physics, and Institute of Microelectronics Technology and High Purity Materials. Chcrnogolovka. Moscow ili> tricl. Russia

1987 All-Union Conference on "Non-classical crystals". Sevan. Armenia

2-4 Structure and volume of the dissertation

The dissertation is presented on 227 pages including 41 figures and 5 tables. It consists of 7 Chapters, the Summary. 3 Appendixes, and the Bibliography of 327 names.

:t CONTKNT OF TIIK WOÜ.K

Chii/ilii I: (it in ml

This is ail introductory part of the dissertation and contains general information concerning to I lie work.

Chapter 2: Electron and optical phonon bound states

Sill)ji f'l of lliis chapter is the hound stall's of an electron and an longililili n.il optical I.O piioiion in the 2D naiiost luct Ules exposed to (lie quuul izing magnetic field normal to the electron sheet. In section 2.1 (as well as in the first sections of all other chapters) a review of works on relevant aspects of the carrier interaction in quantum nanosystems is given.

In section 2.2 the threshold approximation, applicable to the 2DEG. is formulated and the spectrum of bound slates is found. The bound states are sought as poles of I lie electron piioiion s( altering amplitude in a total energy parameter of the electron and the phonon. Calculation of the scattering amplitude by the usual perturbation theory cannot be considered as satisfactory because in the near-threshold energy range £ ~ £„ (f = hu!¡y + u-o is the threshold of the optical phonon emission wifli respect lo I lie Landau level n. J.U and ^,/1 UN' |||||||||>I| and cyclul 11 ill I rei |i|e|icies). (lie series of the perturba lion ilieoiy diverges. 1 lie diagrams responsible for this divergence are those which have dangerous sections along one electron and one phonon line. i.e. in which the process of phonon emission is almost real. The direct calculation of contributions of some first diagrams of perturbation theory shows that the perturbation theory becomes inapplicable in the energy range |e — en \ <ahujQ. In order to find the true spectrum in this region it is necessary to sum the complete series of the perturbation theory or. which is equivalent, to solve an integral equation for the scattering amplitude. When the spacing between levels of the spatial quantization AE hu¡B: huo- this integral equation for the components R'„ of the Fourier transformation of the dimensionless scattering amplitude can be represented in the standard form of the Fredholm equation for I lie resol Vi "111 willi I lie parameter A(f)

Rl„(r.t.t') = K'„{t.t') + \(e) r dtKln[t.i)R'„(e: t, t'), X(e)

Ju

u

£ - £„ + lO

ahuQ

(1)

where / = U.±l.±2____is the total angular momentum and the kernel A'/_ is

given by

(t — .s + ii

bi..

n .1 •

(2)

Hero Ji is I Ik- liessel function. /." is I In- Lagurrre polynomial. I lie parameter rr = u-u/u:[). I stands for the dimensionless inplane momentum of phonons. The form factors $ and the coupling constants a depend on the electron-phonon interaction mechanism and on the nature of the phonon localization. The spectrum of bound states is obtained from the equation A„(£:) = X'lt , determining the poles of the scattering amplitude H'n. The subscript r numbers III«- differenl eigenvalues A(, ,. of the kernel /\ '. Therefore, the energies of Unbound slates are

uum of two-particle electron+phonon stales in the 2DEG so that bound »tales appear both below the threshold (W > 0) and above it (Ur < 0).

In section 2.3 the spectrum of the cyclotron-phonon resonance is studied when I lie role of I lie final slates at I lie electron transitions are played by I lie bound stales. Absorption of the electromagnetic lield with frequency u near the v,| = tojJb is considered. On absorbing a photon of this energy an electron is transferred from the Landau level s — 0 to the s = n level and a phonon which is bound to the electron is created. According to the simplest ideas based on perturbation theory, a delta-like absorption peak should be observed at the threshold //„. However, there is no such peak in absorption. The absorption is gc iverned by I lie density of si ales of a sysl em of I wo part ides: an el eel roll on a level n and a phonon. Since phonon dispersion is ignored, bulb particles h.i\e an inlinite mass so that this system does not have a continuous spectrum. All the electron and phonon states are bound. This is the fundamental difference between the cyclotron-phonon resonance in the 2DEG and in the 3D electron

The fraction of electromagnetic radiation, incident normal to the plain- of tin-electrons (along the magnetic field) and absorbed by the 2DEG is represented

(3)

gas.

by

w± = 2itN

( D

Неге Л' is the number of electrons per 1 cm" at the n = 0 level, s/ф) is the refractive index at the frequency v. The ' ±v signs represent radiation with the right- and left-hand polarizations. The oscillator strengths /± are expressed in terms of t lie eigenfuiicl ions of t lie kernel /\" *1. From (4) it is clear t hat the absorption is governed by the bound stales with angular momentum I = ±1 and concentrated not at the threshold but below and above it at separations

The act и al calculations of t he binding energy and of the oscillator strengths inside a quantum well and in a heterojuuction both for the bulk LO and surface SO phonons are carried oil! ill section 2.-1. The investigation of the kernel ('2) is \<-r\ complicated problem and we do not know its solution in the general case. We overcome this dilliculty by considering separately the cases of strung and (i luk magnetic fields.

From the results of the calculations it follows that upon increasing the quantum number r. causing the binding energy |IV,.| to fall, there is a simultaneous fall also in the oscillator strength /,-• In strong fields the binding energies and oscillator strengths of the states above the threshold are of the same order of magnitude as loi the stalls below the threshold. Therefore, the absorption spectrum should consist of two groups of peaks, which have approximately the same amplitude and are located approximately asymmetrically relative to the threshold. The separation between these two groups of peaks is of the same onler of magnitude for the right- and left-hand polarizations but absorption for the right-hand polarization is (¿и/u;u)-' times greater. In weak lields three bound states with maximum binding energies and oscillator si lengths (two with / = +1 and oik4 with I = —1) lie above the threshold (sec lable I). However, ill colli last to llie case of strong fields, the absorption is of

Iable I: Binding energies iiiul oM-illalor strengths in weak magnetic fields for tlie SO phuiiun . I •.'.(• 11 <1 I........I -.1 ,(l ■ 11' t'[ ' is cli.-ii ;irlf| isl if limiting «ЧИТКУ sc;ilr.

11 binding energies Oscillator strengths

11=0 " V - n~ "-л/

II 1 It'1 Jiti -M tii/IT

the same order of magnitude irrespective of the polarization.

Comparing the binding energies for the LO and SO phonons one can see that

II /.o/IV'sr; — {Ii^h/AE)1/'* <C 1. This relationship applies in strong and weak

fields. A similar relationship describes the oscillator strengths. Therefore, tie-states bound to the SO phonons should be easier to observe.

In section 2.5 the results of this chapter are discussed. According to the numerical estimates, in samples GaAs/AIGaAs of good (but not exceptional) quality at the magnetic fields 10 T and the localization width 15 mil we have W = 1 meV and / = 0.01 which correspond to the absorption w = 5. 1 ■ lu-'1. This means that a ten layer superlattice can give rise to perceptible absorption amounting to a few percent.

ChapUrS: Eltciron-plionoii relaxation. hiltrfact tjjtti

In this chapter the energy and the moment mil relaxation rates of a test elect ton and for the whole Fermi 2DEG caused by acoustic phonons are calculated. The influence of various interfaces separating different elastic materials on the electron-phonon scattering is studied. The vector nature of the phonon field makes it practically impossible to calculate the complex normal modes of a layered medium using traditional methods. In section '¿.'2. a new method for calculating of the scattering probability summed over all phonon mode* is developed. This probability, evaluated in the Born approximation, can In expressed in terms of a phonon field correlation function and the correlation function itself is given in terms of the Green function of elasticity theory which contains all information about geometry of a layered structure.

In section 3.3. employing this method, a general formula is derived I'm the scattering probability in the 2DEG located near the interface between tun elastic media. It is assumed that the scattering takes place in a non-degenerate isotropic band with an energy minimum at the center of the Brillouin zone It is shown that the electron transition probability caused by the deformation potential of acoustic phonons can be represented as an average of a kernel />,' for the contact of two elastic semi-spaces weighted by the wave functions ol electron transverse-motion. The kernel A' is expressed in terms of the Green function of elasticity theory. An analytic expression for K is obtained. Limiting cases of an interface between solid and liquid semi-spaces, of a free and a rigid boundary are analyzed. In the case of the solid-liquid interface, the electrons interact not only with bulk and surface Stoneley waves but also with leaky waves. The ' leaky'' waves are damped Rayleigh waves modified somewhat by reaction of the liquid medium. These waves continuously transfer the energy of the 2DEG to the liquid forming an inhomogeiieous wave moving away from the interface. This is due to the fact that, strictly speaking, the ' leaky' waves are not of the surface type and a pole on a nonphysical sheet of the Green function of elasticity theory corresponds to the "leaky'' wave. When the differences

between elastic properties of the contacting media decrease, the "leaky" wave becomes even more strongly damped so that one cannot regard it as a surface wave. In the contrasting case of a free surface, the leaky'" wave becomes a surface Rayleigh wave.

In the section 3.4 an explicit calculation is carried out for the Fenni '2DEG located at the distance z0 from the interface separating elastic semi-spaces. The scattering probability is obtained in three ranges of a test electron energy £ — €f in which the scattering is qualitatively different 51. It is shown that scattering is strongly suppressed in the middle energy range B in which large-angle inelastic scattering is dominant. While in the lower energy range A and upper energy range C, in which small-angle inelastic scattering and large-angle elastic scattering are dominant, the interface effect is weak. The strongest effect is given by the free surface.

One may easily understand the free surface effect if the electron in the 2DEG is considered as an acoustic waves emitter. The waves emitted into the crystal bulk interfere with the waves reflected from the crystal surface. As a result the emission efficiency depends oii the distance zo. In the energy range B and C the 2DEG interacts with phonons predominantly propagated within a narrow cone near the normal to the 2DEG plane. The frequencies of these phonons are hu; ~ £ — £f. The frequency dispersion is of the order of the frequency itself, i.e. Alj ~ ui. Such a wave packet with momentum dispersion Aq ~ £ — £f/s is attenuated at a distance s/£ — £f- If the distance 20 is larger z0 s/e — £f (the range C). the emitted phonon field will not reach the surface and the latter has no effect on the electron-phonon interaction. In the energy range A due to strong mixing of longitudinal (LA) and transverse (TA) acoustic modes upon reflection of the LA phonons as well as in the case of the interfaces due to reaction of the second medium, the deformation potential node is eliminated. This ieads to a decrease of the interface effect.

In section 2.5 the energy and the momentum relaxation times of a test electron are calculated. Dependence of these quantities on the distance 20 and 011 the parameter A = ps/p's' is studied (ps is the normal impedance of one of

QK)

Figure 1: The energy-loss power Q of the Fermi 2DEG located near the free crystal surface. The clotted line corresponds to the case of the infinitely distant, ~♦ oc, boundary.

I he contacting media and tlie A determines the reaction of the second medium ill I hi' case of tlie interfaces).

The contributions of different phonon modes to electron relaxation are distinguished. The contributions of surface and bulk acoustic phonons to the momentum relaxation are comparable for low energies (in the range A and a lower part of the interval //) while for the higher energies, the contribution of the b.ulk phonons is dominant. Energy relaxation in the ranges B and C is determined only by bulk acoustic phonons while in the energy range A. contributions of I lie surface and the bulk phonons become of the same order of magnitude.

In the following.sect ion the relaxation of electron temperature 'l\ is calculated. It is found that in the middle of the interval B. when Te ss 10 K. the presence of a free surface reduces the relaxation rate of electron temperature by a factor of the order of 10 (see Fig. 1). The characteristic distance which is required for a surface to have an effect on electrons with such energies is up to 10 nm.

Our model of a free surface applies directly to inversion layers which are formed pn natural semiconductor surfaces (for example Ge or InAs) and to structures in which the 2DEG is located at distances of the order of 10 nm from the surface. Our model is most applicable to the systems where the insulator in I lie inel aMiisiilalor-seiiiicuiidiiclor (MIS) Nlriicliiie is uol deposited on I Insurface, but simply clamped (Mylar foil). The acoustic contact is then poor and phonons in the semiconductor are reflected from the semiconductor-insulator interface as if it was a free surface. We also wish to mention MIS structures with a thin oxide layer and a thin (optically transparent) gate. The reflection coefficient for the semiconductor-insulator and insulator-metal interfaces are relatively small. Neglecting, in the first approximation, the differences between acoustic properties of various media, we can describe such a system using our model.,

C'linptrr Phonon emission by f.andnv stales

The situation is drastically changed if tin- elerlrons are subjected lo the quantizing magnetic field normal to the 2DEG plane. In Chapter 4 theoretical study of acoustic and optical phonon scattering in the 2DEG in the QUE geometry is presenled. The calculations ¡»corporate one l)A or PO phonon and two DA+PO phonon emission processes for electron relaxation between bulk Landau states.

In section 4.2 (he inter Landau level electron transitions are studied in the 2DEG caused by acoustic phonon emission via a deformation potential.

'1 lie frequencies of I lie* emit led phonons in this regime equal approximately to t lie cyclotron frequency % lJu while the width of the frequency band of the (-milted phonons is of the order of I lie cyclotron resonance line width'5'. i.(. Aa,' & I /t. where t is I In- moment inn relaxat ion t ime. 't his corresponds lo I lie plioi/on wave pacl;cl alIeniialion length sr. In high mobility helerostruclures the length st is larger than s/Tf even at the liquid helium temperatures. Therefore in quantizing magnetic lields. electron-phonon interaction can be modified by a more remote surface. In good quality structures, the free surface has an effect on distances of the order of 100 nm from the 2DEG. The inter Landau level transition probability has an oscillatory dependence

oil the 111; IJ11 * • I i»" I i e | (| and distance •(,. 'I'||e oscillation period is given by I lie condit ion I ha I mi is a mull I pie of hall -wavelength of t he ell ill led phonons A/2 — x.i/*;//. In real structures typical values of Cu usually lie between 50 and 150 nm. Therefore, taking r(i = 50 um we lind the period of oscillation in the magnetic field as A13 = 0.1 T and the period of oscillation in the distance as A;(l = (i nm al H — I T (see Fig. 2). It can be seen from Fig. 2 that the

M«8n«l'c tiaid |TJ

Figure 2: Transitiun proljability Wi_o between Landau levels /j = 1 and l? = 0 versus magnetic field B for tlie distance zo = 10U nm from the 2DEG to the free surface (left) and versus t he distance "o fur I tie magnetic field B = 0.5 T (right). The transition probabilities air shown also wlit.-n no phonon icllect ion is tiiken into account, i.e. ¿0 —r oc.

transition probability strongly decreases in the magnetic field for large fields. i.c. in effectively zero dimensional (0D) systems with rather thin electron layers and subjected to rather strong magnetic fields, a large separation between Landau levels cannot be covered by an acoustical LA phonon37. In this case the

multiphonon 2LA phonon"1*' or a longitudinal optical plionon agisted LA + LO phonon11 emission processes can be efficient.

One optical phonon emission requires a precise resonance ALß = uio- A/ = I.2.3.-- in OD systems (uju and uii.o are the cyclotron and the 1,0 phonon frequencies). As far as it moves oil from the resonance, the elliciency of this process is steeply falls so that out of resonance. LO phonon emission becomes

I ii issiblr 111 a III II n I i.'ini 1111-111 I if I. A |>ln HI I 111 i-I Mission \ l.l I lie I Wi i |dn >non el 11 i-.moii mechanism.

In section 4.4 and 4J> optical phonon assisted electron relaxation is investigated via one. LO. and two. LO+LA. phonon mechanisms, respectively. 'Hie phonon emission rates versus inter-Landau-level separations are calculated Electron interaction with bulk PO and interface .SO phonons arc- considered. In quantizing magnetic fields, emission of LA phonons via piezoelectric interaction is suppressed in comparison with deformation interaction40. Therefore the calculations of LO+LA phonon emission are carried out for the deformation potential of DA phonons and for the Frölich coupling of PO phonons. To obtain a finite relaxation rate associated with one-phonon emission, the allowance for the Landau level broadening or for the LO plionon dispersion is made. Below the LO phonon energy, hui^o- within an energy range ol the order of h\Jtoßjt. the one-phonon relaxation rate exceeds lps-1, r is the relaxation time deduced from the mobility. In GaAs/AlGaAs heterostructure with the mobility /y = 25 V-1 s-1 m2. this range makes up 0.7 meV. Relaxation via surface SO phonon emission mechanism at least by an order is weaker than relaxation via bulk phonons.

Two-plionon emission has a significant contribution to relaxation above At energy separations of the order of sa^1 (s is the sound velocity, ao is the magnetic length). LO+LA phonon emission provides a mechanism of subpi-cosecond relaxation while in a wide energy range of the order of Ziu,//. sub-nanosecond relaxation can be achieved. Particular attention is given to the

comparison of elect run relaxation on in two dillennl \va\N first rela\

alion in two consecutive emission acts (emission of a PO phonon with subsequent emission of a LA or 2LA phonons) and. second, PO+DA phonon relaxation via the multiphonon emission mechanism between the same Landau levels. Numerical results are illustrated for PO phonon assisted relaxation between Landau levels / = 3 and / = 0 (see Fig. 3).

Study of emission and absorption of ballistic plionon pulses in systems with the 2DEG gives the most detailed information about the character and specific features of electron-phonon interaction. In this case one can easily trace contributions of separate phonon modes to the interaction. Previous works 52-53-54-55. analyzing these effects both in the presence and absence of

lignic I. I In- I..1111I.111 level Iiiu.idriiing (I' ll) ;iii*I llic optical |iIiiiiiiiii dispersion (iniiiilli ) colli riljutions to tlie IjO phonon emission rule us well as the lJ(_)+L)A phonon emission rale (light) versus on the inter Landau level spacing A/ikjj for the electron transitions between I = 'i and I' = 0 levels in the vicinity of the PO phonon energy uipo-

the quantizing magnetic field, however, did not take into account the ballistic phonon reflection fnii n lite interlaces mar which I lie 2DIXJ is placed.

In section 4.3 calculations of the emission spectrum of ballistic acoustic phouons in quantizing magnetic fields normal to the plane of the 2DEG are presented when the reflection of these phonons from a GaAs/AlGaAs type interface is taken into account 43 It is shown that the interface affects essentially the intensity and the composition of the emitted phonon field. Re-ceni works •'''. where I lie acoustic phonon omission from the healed 2DEG in ii ie| al-oxide-seinicondiiclor iianosl rue 11 ires has been si in lied I lieorel ically by taking into accoiinl of tin1 phonon rolled ion from llie interface, also indicate an important role the interface plays in electron-phonon interaction.

In spite of we considered only deformation electron-phonon interaction and 1 lie parabolic energy band (i.r. I lie electrons interact directly only willi I.A phonons;. we obtain Ilia! the detected phonon lield on the reverse face of the »ample consists of both bulk LA and TA phonons. This situation can be explained as follows. The electrons of the heated 2DEG emit only LA phonons in all directions. Therefore the detector at the point i-o will record all t lie LA phonons which are emitted in the direction of negative 2-axes at the angle of 0. However, the detector will also respond lo all those phonons which have been primary emitted in the direction of positive ;-axes at the angles of' 0 and 0/. lo the free surface. Such LA phonons. upon their rcJlrclwti and (onnrsion at I lie free surface, propagate backwards in the form of LA and TA phonons. respectively, in the negative ¿-direction both at the same 6 angle. The reflected LA phonons interfere with the initial LA phonons. Thus the detector will respond to the iiitrrfcrrnrf field of the LA phonons and to

(ho conversion field of the TA phonons. Now. since the phase difference of the initial and reflected LA phonons is t In- fund ion of t he angle 0 and t lie distance' ru. the intensity of the detected interference of the LA phonons should also depend oil 0 and in.

Emission angle [degree] Emission angle [degree]

Figure I: Angular ihstribnlion of t lie phonon emission intensity at Zq = I OS inn (left ) .nul o - III!» tun (riglil) when ihr l.amlau level I >i oatlrniiig is m»l taken inlo a< < mini ln-.il sliuws dependence of the latio vv' /vv": on I lie emission angle

In Fig. 4 the angular distribution of phonon emission from the heated 21)K(i in a ("¡aAs/AKJaAs heterojunct ion is shown. One can see that the pliouoii emission is practically absent lor angles larger than .'!()" The I A phonon field is concentrated in a narrower cone around the magnetic lield than the LA phonon field. The angular pattern of the LA phonon emission is very much different for ru " 105 urn and zu = 11)9 nm. Namely, for ;(J = 105 nni. LA phonon emission has a peak at 0 11° while for ;() = 109 nm. emission is strongly suppressed at I he same angle so 1 hat I here appear I wo other peaks. This is due to the interference oscillations discussed above. These oscillations are more distinctly demonstrated on insets of Figs. 4. The oscillations almost preserve their form with varying zq. they are simply shifted along the tf-axis. so that emission can be strongly suppressed by choosing a proper However, for large values 2o>A2 (even for high-quality heterostructures with mobility // = 100 m2 V-1 s-1 we have for the attenuation length A? = .it 100 nm).

the Landau level broadening smoothes these oscillations, as it is shown uu inset of Figs. 5. So the emission character will not be impressively different for different values of ;lp. However, the most intriguing for experiment remains the

Emission angle [degree] Emission angle [degree]

("i^ini' ■>: Angular'distribution of the phonon emission intensity at zo = 105 nin and 20 = 109 11m when I lie Landau level broadening is taken into account.

possibility to detect the '['A phonons on the reverse face of the sample under the defornialional electron-phonon interaction. Notice that the intensity of the TA phonons is nearly 3 — 6 times smaller than that of the LA phonons but still remains a quite measurable quantity in experiment.

In the dissertation, study of the emission spectrum of surface acoustic

11 III III' III.11 I . |I| ■ Ill' d Our i nl' illal 11 his sin iw I li.'il .'ill <-X| >< >ll«-|il ial silppl'i'ssii ill ol emission of ill«' siirlace »cousin' phoiniiis occurs m a wide range of tin-magnetic field variation. Hub '¿mcjh (mc is the electron mass and cr is the velocity of surface waves). So the cooling of the heated 2DEG is only at the expense of bulk LA and TA phonons.

('liiijili i fj: l.ili/i slnli stalh rnifi

. One of the aims of the present chapter is to calculate the inter-edge-state scattering length due to phonons and impurities under more general assumptions than used previously, for realistic models of the electron-phonon and electron-impurity interaction and for a realistic confining potential3839.

To calculate the impurity scattering we use the standard model of the heterostructures 58 which takes into account long-range potential fluctuations due to the layer of ionized donors as well as short-range fluctuations due to the uniformly distributed acceptors. Analytical expressions for the scatlenng length are derived for an arbitrary confining potential.

Phonon scattering is discussed in the next subsection. We consider de-formalion acoustic (DA) and piezoelectric (PA) interactions and again derive analytical expressions for an arbitrary confining potential. Since ¿> <C ''/. t /-. I lie scattering is quasielastic. i.e.. the energy of the emitted or absorbed phonons huiq <C huiB- As follows from energy and momentum conservation, only plioiious with frequencies above some minimal energy An> — /i.-Mv/. can j >. 1 rt n ipate ill the electron transitions I — /' between edge states, l'he change ol the momentum k is bku*. We consider low temperatures T A//». In this case, due to the phonon Bose factor and the Pauli exclusion principle, the phonon energy huv is close to the threshold A;/,. As a result the calculations of the scattering length is greatly simplified. In the Born approximation we get for DA scattering

= (l/'2)[ln(l +exp(i -7/) + expU)ln(l +exp(-^/)] ((¡)

The velocity (■/' and function \t< of the final state correspond to the energy of Ibis stale •• i:<|ii;it ioti (5) is valid il'exp( - ;/• A//'/'/') < exp( A.v/7")

or. in other words, if — £>1 < A//< and if £ — £p — A/y <C A;/>. In the iirst case hu; — A//' ~ T. while in the second case huj — Aw ~ c — £p ~ Aw ■ 1' has been also assumed that T ms~ and Aw <C hs/d. where d is the scale of the electron wave function <J(:). For GaAs/AlGaAs heterost met lire ;/ = 3 inn. a = 5 x 10-i in s~ 1. nm2 = 0.1 K and ht>/d = 13 K. Taking bku< = uj^. we have for B = 2 T: a/j = 18 nm. Aw = 2 K and Iiu^b = 39 K. The inverse scattering length (5) is to be averaged near Fermi energy. Since the function F grows exponentially with £ — £p for £ — £p > 0, the average value is rather due to hot electrons (e — £p ~ A;;< T) than to thermal ones (£ — £p ~ T). Until Aa' huB- on<- can put e — £p. With the above mentioned assumptions foi DA scattering we get

) = -¿—¿iAbkwaB?-— expf-^V (7)

\L(_n' / DA Lda \ 1 J

For piezoelectric PA scattering calculations are similar:

1 = -ir— Al,{6kwaB)— exp (-^f-) . (Hj

PA LPA ViH> \ I J

•n

lli'iv \vr ilcliiu'il nominal «-¡tllcring lengths

(t/i)"

Coa ATTlifiti'^ii'jj CPA Avhp^ms

I or (¡aAs at II "2 '{' wo lia\e L'i>.\ ~ 'I.S //in anil Ci>,\ = 0.3H //in (E"' anil A1 taken from Hef. f'1']). The results presented above are valid provided that the chemical potential dillerenie |A//| between edge levels is small compared to temperature. II' this condition is not satislied. then Kqs. (7) and (N) should be multiplied by the additional factor <,xp(|//|/'/')It can be seen (hat phonon scattering is exponentially suppressed at low temperatures. This suppression of the scattering rale is because of the deficit of the superthermal phonons for absorption and deficit of the free liual stales below t~y lor emission. According lo our evaluations, the observed temporal lire dependence of the scattering length cannot be allributed lo phonon scattering.

The .second goal of present chapter is lo calculate the ballistic acoustic (both deformation and piezoelectric interaction) and polar optical phonon emission by quantum edge slates. In contrast to the above discussed conventional transport measurement expor iments. we study I he inter-edge stale scattering also in the phonon emission experimental technique. The basic dilforence between these two technique is thai in the latter case. I he phonon signal is measured in the certain phonon emission angle ami al (he lixed excitation energy. At tutor-l.andau-slale transitions, the phonon emission energy is fixed by the cyclotron energy, therefore the angular dependence of the emitted phonon field remains as the sole important characteristics of emission. Tor edge states scattering. the energy distribution of emission is also important. Recently the frequency spectrum and angular distribution of the total energy-loss rate due to l.A phonon emission have boon obtained in a QWr*'1. Equally witli the bulk Landau slates, the quantum edge states also give contribution lo <mission and absorption of ballistic phonons by the 2DEG. Absorption of ballistic acoustic

Figure (i: Kniission intensity distribution (pie/oelecl l ie niti iaclion) in plioiioii momenta al low Irinpeiiiliiier. I, h^ih = ■I'lth = sbkio "iiil fur tin' uiin-.sinoolIt eunlining potential.

phonons by edge states can play an important role in the phonon-drag effect in the QHE regime02.

In section 5.3 an analytic expression for the ballistic acoustic energy flux emitted by quantum edge states is derived. Detailed analysis of the phonon emission intensity distribution is made in the low and high temperature regimes as well as different positions of the Fermi level ep are considered. It is shown that at low temperatures, phonon emission at the inter edge state transitions is predominantly concentrated within a narrow cone around the direction of edge state propagation while at high temperatures it is around the magnetic field normal to the electron plane. At low temperatures the emission intensity decreases exponentially with decreasing filling of the Fermi level. Phonon emission due to the piezoelectric interaction is discussed. In contrast to the case of bulk Landau states where piezoelectric interaction is always suppressed in comparison with deformation interaction, in the case of edge state scattering, the relative contribution of piezoelectric and deformation interactions depends on the magnetic field, on the shape of the confining potential, and on electron temperature. To illustrate our calculations we estimate the acoustic energy flux power emitted by the edge states at the peak of emission for the case of the non-smooth confining potential. In the low temperature regime emission goes mainly via piezoelectric coupling. At the peak of emission. qr = bkw and qy = qz = 0 and the emission intensity is given by

vPA _ 1 ms VB |^"'YCX1 _ [t3)-pn ( 0 (2tt)- fpA \ PB J ' \ Tt J ' TpA 2-xhps2

and ps — a^1 = invu is the magnetic momentum. For GaAs we have ms =

3.1 • 10-28 J s m-1 and at B = 2 T. and the nominal time of the piezoelectric interaction fpA = 36.4 ps. Taking ¿iiio = vb and 6k\Q — ps we have s6k\o =

2.2 K for B = 2 T so that at Te = 0.5 K we obtain VqA = 3.1 • 10-21 W s m-1 for the electron transitions between I = 1 and / = 0 edge states. For the emission cone angle we find 9 — tan-1 qr/yjq'2 + <?2 = 25° (see Fig. 6).

At high temperatures and are given by

_ _J_ms_VB_ ( T\_\4 1 1 _ _--p1.

- -

T>UA — a c ~ - ' _ ~ rif /it.

(2ttfDA 6v„. \sPB J Il + ^-r-]3'^ 2vhps* 1 ' PA = 1 ms vB ( Tt \a x2e~r

(2t y-fPA 6vw \spB J [1

Here x = anc^ J1 — ~hff2- F01" the GaAs/AlGaAs heterojunction d. = 3 nm. s = 5 • 103 m s-1 and so hs/d = 13 K. Taking T = 10 K we obtain that

the emission peaks for DA and PA interaction are determined, respectively, from x = 1.21 and x = 0.85. This means that at the emission peak, frequencies of phonons emitted due to the deformation coupling are approximately 1.4 times larger than frequencies of phonons emitted due to the piezoelectric coupling. At B = 2 T we have tda = 382 ps so that taking again ¿tjo = vb we obtain VgA = 1.03 • 10"18 W s m"1 and V%A = 0.53 • 10"18 W s m"1 for the electron transitions between edge states / = 1 and 1 = 0. For the emission cone angle we find 0 = tan-1 yj<il + <ly = 10° for DA p1 onons and 0 = 14° for PA phonons. It should be observed that in the smooth confining potential, phonon emission is suppressed exponentially. At low fejnperatures suppression takes place for two reasons. First, because of the threshold nature of emission, electrons are forced to emit phonons with frequencies larger than ¿th = ¿¿hi' apu- Second, because of the exponential smallness of the overlap integral. While at high temperatures suppression takes place only for the last reason.

In the next section polar optical PO phonon assisted edge state relaxation is investigated in QWrs with a rectangular cross section exposed to the normal magnetic field. Scattering rates as a function of the initial electron energy is calculated. The results of numerical calculations in the limit of the smooth

I'igure 7: The PO phonon emission rale versus the electron initial energy for different values

of the magnetic field.

confinement (we take ¿q = 1.754 meV) for inter edge state transitions between hybrid subbands / = 1 — /' = 0 (left) and / = 2 — /' = 0 (right) are shown in Figs. 7. The diagrams represent the dependence of the PO phonon emission rate on the electron initial energy for several values of the magnetic field. One can see that for the transition / = 1 —+ /' = 0 at low magnetic

fields B"= 15. 17 T corresponding to the detuning Aj100 = 9.89. 6.38 meV. the emission rate increases slowly but monotonously thought it is remaining sufficiently small in the whole range of the electron initial energy variation. It is exceeding the value of 1 ps-1 only near the upper edge of the energy' variation. Such behavior is conditioned by the sullicieutly large value of tin-transferred momentum qT and by its monotonous decrease in the same energy range f Inequality Ai/j-m^I. which remains from the momentum conservation

after quantization, is not satisfied. The frequency lu = +"-o determines

the hybrid subbands. a = \Jh/mcu; is the characteristic length of the hybrid quantization. A =-u;b/u;). The curves ol the emission rale for the magnetic fields B = 19. 20 i' corresponding to the detuning Aj100 = 2.87. 1.12 meV exhibit peaks at the energies for which the inequality Ai/.-ri < 1 starts to lake place. The peak values exceeding 40 ps-1 and 10ps_1. respectively. At higher energies, the inequality A '¡¿a <C 1 takes place in the strong sense, therefore the overlap integral becomes weakly depending on the energy and the energy dependence of the emissio*n rate is mainly determined by the behavior of the density of states (DOS), i.e.. it slowly decreases with an energy increase. On the low-energy side we have Aijra I causing an exponential increase of the emission rate with energy increase. For very sharp detuning (Aj10u < 0.5 meV) we have Xqxa <C 1 in the whole energy range and features of the emission rate are mainly determined by an energy dependence of the DOS. Particularly, a divergence of the emission rate at low energies can arise connected with (he divergence of the DOS' at the bottom of (he electronic subband. At the electron transitions between subbands it = 0. / = 2 and u' = 0./' = 0 we obtain an analogous behavior for the scattering rate.

Chapter 6: Theory of Anger itp-comersion

In receill magneto-luminescence experiment by Polemski ft al. '1r> on one side modulation doped quantum well», an np-coiiverstoii has been obseiwd and interpreted as being due to an Auger process. These authors studied photoluminescence in an asymmetric GaAs/AIGaAs single quantum well of width d = 25 urn with an electron density of A', = 7.6 ■ 10" cm--' with a magnetic field B applied in growth direction. The characteristic energy level scheme for 7.9 '1 < B < 12.9 T is depicted as inset in Fig. H: the lowest Landau level of the second electric subband L'0 lies between the second L\ and third I.-j Landau levels of t he lowest electric subband while due to the doping concentration the Fermi energy is pinned to the level L\. Changing the magnetic field in this interval allows to tune L'0 between Li and ¿2- The luminescence spectrum under interband excitation into L\ at low temperatures

(/' = 1.8 K) aiul for low excitation power (/'<x.- < 10 W cm-") shows two peaks: besides the luminescence due lo recombination of an electron from Lu with a hole in a valence band, a second peak is observed above the exciting

lo [.'... ¡.I.

& CTJ

s

Magnetic lietd IT]

I- iv,'H S: ,M;i^iii'l if IHIII <l«-|M-iHlence uf tin-probabilities for tile transitions between Landau levels l'u — Li({/}) and L2 — l'0({ii}) caused by acoustic phonon emission.

is the high intensity of the [.[, luiniiiesceiice which can he of the same order as

the /.,i luminescence.

laser energy and is rel recombination of an up-converted electron with a hole. In order Io explain I his second peak, the following processes have been supposed: al

ter an interband excitation of electrons into the partially filled level L\ (i). a recombination takes place between electrons from ¿0 and photo-induced holes (ii). then in an Auger process two electrons in L1 are scat-lered to l.i and /,(j (¡ii). and a relaxation process brings the electron from I.-2 to /-,) (iv). from where it recombines with a photo-induced hole (v) I o give I lie Up convert ed llllllilies cence or. emitting a phonon. relaxes into the level L\ (vi). The most surprising result obtained in experiment

In this chapter, we calculate the characteristic times of processes (iii) and

(vi)

41)

Auger scattering between Landau levels of the lowest electric

subband and electron-acoustic phonon scattering between Landau levels of the two lowest electric subbands (L-j —■ /,[, and ¿{, — L\). as well as the lifetime

of a lesl hole iii level /.,, wilh respect Id liotli I lie Auger process and the |.||'.II'||| I linv.ioii lly .Ilialy/iii)', I'.ile <■• 111:11 h his fur tile processes (i) (vi) we lind an I'siiinate for the time of the Auger process as well as magnetic lield and excitation power dependencies of the two luminescence peaks which are consistent with the experimental findings.

Phenomenological calculations of the electron-electron scattering rate for free electrons of the 2DLG exposed to the normal quantizing field are presented in section 6.'2. (Quite recently these calculations have been generalized by I.eviusoiifor a smooth random potential depending on only one coordinate.) The scattering time for a single electron due to the Auger process, in which two electrons are scattered from single particle states 1.2 into states l';2'. is

4

represented III 1 111' lor III

= WAultrMl -/l')(l -fr)

aug tr

where the probability of the Auger process is

(13)

l,+l2=l,>+l7

WAuytr — ^

*u aB V -0 ,

(II)

The overlap integral <I> ^ depends on magnetic field via I lie dilueusioiiless parameter ob/zq where 2o is a characterizing length parameter of the lowest electric subband. We have calculated the overlap integral o) for the Auger process involving Landau levels /j = /2 = 1 and l[ — 0. /\ — 2 (Auger process from the level L\ into the levels Lq and L2) (see the lower part of Fig. 9). The value of the parameter r0 = 3/6 = 10.5 11m has been chosen to reproduce the separation of the two lowest subbands of the actual quantum well in by a triangular potential model. The magnetic field de-

pendence of the probability of Aug<

<u

8 &

8 I"

a 090' 0 045

wA

- *<» 10 pa

« -10 9 nm

/

/ / / -

6 • 10 12 14 18

Magnetic llald (T)

Figure 9: Probability of the Auger process WAuger from the level L\ into the levels Lq awl 1-2. Dashed line shows the overlap integral "J>(<ii,/;0).

process M'au,,,,- is plotted in Fig. 9 One can see I hat ill t he range of magnetic fields between 7.9 T and 12.9 T. I lie fttticl ion <I> is slowly' varying func-I ion in t hi" magnetic lield so | hat 1 lie probability WAuger is approximately a linearly increasing function in B in the same range. When the magnetic field is varied from 1 to 20 T. the probability WAuger increases approximately from 1 to 10 fs. Notice that, recently, such a fast electron-electron thermalization (faster than 10 fs) has been observed in modulation-doped GaAs quantum wells64.

However, the occupation factors, which have to be included in order lo obtain the time of the Auger process

TAugtr■ drastically increase this value. For the case under consideration in the experiment of46 (with N, = 7.6 • 10'5 cin_2: T = 1.8 K and B -9.5 T ) but without pumping, we find by including the occupation factors 7Auger — ^^^Auger - ,£- ^e Auger process is not possible at all because the

lower Landau level /.,, is almost completely filled. It becomes possible only by optical pumping into the level /.| and subsequent recombination from Lu. thus creating the empty stales required for the Auger process. Without pumping. I lie Auger process is possible only al much lower magnetic fields (for lower carrier density) 111 11 ■ 11 <~ 11 higher temperatures.

Applying the same considerations to rjj r we obtain an expression as (13) but with the factor instead of the factor 1 — /{. This gives for half-filled L\ level t\juy(,r = 2 fs. In contrast to rlaugtt we see that rjjuyer does not

depend on available free places iu the level /.n ami shows I he efficiency of the Auger process in comparison with oilier processes which add (by emission of plioiions) or remove (by inlerband recombination) electrons in /.(j.

In seel ion 0.3 ncoust ic phoiion relaxai ion between Landau levels of different electric subbands has been studied. We obtain the relaxation times —

(1 - /n.i'oJWjl/^ ~ [f„i are occupation numbers of the level nI)

where the probabilities for the transitions {/}.{//} are given by

,|l) _ 1 up 4(s/;)r' Tb Z

= <16>

' TB Z (¿IMB-UE)

where 1Jis the magnetic lield when um = u(/iu,£ is the spacing between electric subbands). For CJaAs at B = 9.5 T and z = 6.7 nm (this corresponds to = 22.4 meV taken from46), we find = (32.3 //s)"1 and li;1/1 = (l.5//s)_l. Notice. that the relaxation probability calcillaled for ili-trasnbband scattering in lower magnetic fields It ~ I'/ '3' is iiiucli greater. Such a suppression ol electron pliouou 1111 er a c I i oi i at u. ami (u-'/j/.s); I follows from the conservation laws. The elect roll slates in this regime constitute a wave packet so that the states with l.n ~ 1 have momenta of the order of cijj1 in the (j\ i/)-plane and of the order of ¿-1 in ¿-direction. Therefore, only for a small number of electron states in this packet, the momentum conservation law is fulfilled al the interaction with acoustic phonons with momenta

/ ,-t .-I s-y a u . - •

The dependence of the probabilities (15) and (16) on the magnetic fields is plotted in Fig. 8. At low fields B fs 8T. the transition —* L'0 is predominant. As B increases, the probability W^1^ rapidly falls while W^P slowly increases so that already at B — 8.6 T these two transitions are equally probable. At high fields transition L-> —• L'0 is suppressed with respect to L'0 —* L\ and 1!'^'' rapidly increases with increasing B so that at fields near /?/.; achieves to

value« corresponding to times less than 1 ns.

We obtain I lie lifetime of a test hole which is larger than the relaxation

times Tpp ~ 1 /js and thus much larger than ~ Ifa). Hence, the

Auger process is much more ellicient to fill holes in the level /.u than the pltonon emission. This fact makes possible the observation of the Auger lip-conversion by iillerband optical piimpiiiK'11'

In section O.'l the pholuluiiuiicscence intensities Jo and l'u for recombinations from Lq and L'0. respectively have been determined from the rate equations using the characteristic times of the processes (i)-(vi). Because rjj r is much smaller than all other characteristic times of this system, the electron-electron Auger up-conversion mechanism immediately fills 'all arising holes in Lq due to the recombination of electrons from Lq and creates electrons ill the level f.j. Thus the number of I lie up t'ol I Ver I ed elect lolls Is .'ll

ways equal to the number of recombining electrons from Iq- The intensity of the up-

converted luminescence V0 is determined by Figure 10: Magnetic field dependence

"competition'' of the processes (v) and (iv). of <l>e luminescence intensities I0 and

. ... //1 . Ii for recombination from the Landau

i.e. by the ratio of r0 and rp\ . In order to levels Lq and respec(ivel>,

obtain the dependencies of Iq and on the

magnetic field and the excitation power we consider two different cases. In a case when process (v) is dominant over the process (vi). i.e. t^JJ r^ (r,, is the recombination time of electrons from level L'0 with photo-induced holes), we find

/„. /,',-x /',,.. (l (I7>

Here Ptxc is the excitation power. B\ is the magnetic field corresponding to the full occupation of two Landau levels Lq and L\. With the help of formulas of the calculated characteristic times, it is seen that the inequality » rf' corresponds to the situation when the magnetic field is close to the lower bound of the considered interval (B\,B/;) and at the same time the excitation power is high. In opposite case when r'^' <C Tq. which corresponds to magnetic fields close to the upper bound and low powers, we find

'¡«C (18)

■ 0 U tIJ Mtytie MM [T|

while In is given by Eq. (17) as before. Now it is clear that at B near B\ both intensities J0 and 1increase linearly will) B. As B increases further, ¡a continues to increase in accordance with (17) but not so sharply as near B\. However. I'0 shows another behavior and decreases with B as (Be — B)5 when B is near Be (Fig- 10). At low powers I'0 depends quadratically on Pexc while at high powers for I'0 and in the whole range of the power variation for /o this dependence is linear. In conclusion, our calculations provide an understanding of the main features of I he experiment reported in46.

Clio¡ilt v 7: Magneto-transport in a non-planar 2DHG

In this chapter we study a more complex situation of the 2DEG in a nonuniform magnetic field which has attracted considerable interest in the last several years 1,r'. Depending on the strength of the local magnetic field, the electron motion in the plane of the 2DEG can be tuned from regular to chaotic b0-6'. The motion of ballistic electrons in a periodic magnetic field is also believed to be closely related to the motion of composite fermions in a density modulated 2DEG in the fractional QHE regime68 69. Such magnetic supersystems offer the possibility of producing magnetic confinement and a wave-vector selective filter for electrons ,u.

Uecently ;i research group from I he Toshiba Cambridge Research ('enter Ltd. and Cavendish Laboratory have reported an alternative approach to produce spatially varying magnetic fields47. They have proposed a more flexible and potentially fruitful solution of this problem. A remotely doped GaAs/Alj-Gai-j-As heterojunction is grown over wafer previously patterned with series of facet. The electron gas is confined to a sheet at the heteroface which is no longer planar but follows the contour of the original wafer. The use of in sttu cleaning technique enables to regrow uniform high quality 2DEGs over etched substrates4'J. Application of a homogeneous magnetic field to this structure results a spatially varying field component normal to the 2DEG. Rotating the plane of the sample allows to find different non-homogeneous magnetic structures: magnetic barriers, magnetic wells and completely novel situations where I lie normal component of the field changes its sign on the facet (see Fig. li).

In this chapter we investigate theoretically the magneto-transport of the non-planar 2DEG r,u. As an example, the electric field distribution has been calculated in the presence of a magnetic tunnel barrier of/vm width. The system satisfies the Poisson equation in which line charges develop at the magnetic/non-magnetic-field interface. We have found that most of the electrons are injected at the edges of the magnetic barrier. The magneto-resistance

across I he facet as well as in 1 lie planar regions of the 2DECi have been 1 culated which provide an understanding of the main features of th-- ma; ^ Held dependencies observed experimentally by M I, beadbeater 11 til4>i. 1 hoy have constructed a noil-planar 2DEG which has been fabricated by growth of a

GaAs/AlGaAs heterojunction on with facets at 20° to the substrate. With the field in the plane of the substrate an effective ijiagnetic barrier has been created located at the facet (see the 9 = 90° case in Fig. 11). The resistance across such an etched facet has shown oscillations which are periodic in 1/5 and which are on top of a positive magneto-resistance background which increases quadrat ically with the magnetic field for small fields B anil linear in B for large B. In experiment the dimensions of the facet have been 40 /tm wide and 3 pm long, the voltage probes close to the facet are situated 10 /mi apart across the facet. The magneto-resistance has been measured using also the voltage probes on the planar regions of the 2DEG. There is no perpendicular component of the magnetic field in these regions, however, for probes directly adjacent to the facet, a strong magneto-rosislauco has been observed. Pairs of probes on opposite the sides of the mesa

a wafer pre-patterned

T_r

Figure 11: Applying a uniform magnetic field produces a spatially non-lioniogeiitous lield component numial to the 2DEG. Different magnetic su-perstruct urescan be obtained, depending on the 0 angle between the mag netic field and the substrate normal, magnetic barriers (6 = 90°), magnetic wells (0 = 0°). and novel situations when the normal cuinponenl uf tin-field changes its sign on the magnetic interface.

and on the opposite sides of the facet have shown the same symmetry. While there has been a pronounced asymmetry in each of traces with reversal of the magnetic field direction.

To explain, qualitatively, the main features of the experimental measurement« in lief, f8], namely the smooth background of the magnetic field dependence of the resistance across the facet and the symmetries of the resistance traces measured between various probes in the planar regions, we rely on a classical model since the width of the magnetic field barrier is much larger, than the magnetic length. When the electric field represented as the gradient of a potential function it satisfies the 2D Laplace equation in the separate planar and non-planar regions of the 2DF.G. However, this potential does nut satisfy the Laplace equation in the whole 2DEG region. It satisfies the Poisson equation with a non-trivial right side part which is determined by the field itself.

w

Thus at the magnetic interfaces there is an accumulation of linear charges. We luive hoIvihI directly l.nplihi' h equation Fur whole region of the 2DEG with tin1 same boundary conditions. Then, we have included the line charges developed at magnetic interfaces which implies that the electric field exhibits jumps at these points nud snlislies | he corresponding I'oisson equation. To construct the solution of the 21) l.aplace equal ion. the conforinal mapping method '1 has been exploited and the electric field has been obtained-which satisfies to the 2D Laplace equation . . . . ^

F,, . . = f [«"(*• y) + 8h(l.a)][8n(x, y) - sn(l: /?)]

where s//(j.//) _ *ii (l\ (ui)r //\''(///);/. hi) is the Jacobi elliptic function, in is determined by the sample aspect ratio. I\ and K' are the complete elliptic integrals of first kind. The field (19) has singularities in the corners of the magnetic interfaces. The same behavior exhibits the electric field in the Hall effect regime, in both cases an analytic function changes its phase from 0 to ( in one point which results a power-law singularity of this function at that point. The above field E' does not conserve current and the normal component of J jumps at the magnetic interfaces. We can remedy this as follows: in the planar legions I lie real electric field E (j'.ij) is given by Kq. (19) while ill the non-planar region, due to the jump of the «/-component of the field, we have

U) = —vrlu1 L"(x. y) - tan C 1m E'(x, y), E,(x: y) = Im E'(x: y). (20) cos- (,

The above equation together with the auxiliary Eq. (19) gives the solution of our problem.

The spatial distributions of the argument ArgE(x.y) and the absolute value AbsE(x.y) of the electric field E(x. y) in the neighborhood of the facet are shown in Fig. 12. One can see that both components of the electric field are small outside the facet region. The field exhibits power law singularity in the diagonally opposite corners of the facet while in the two other corners E(x. y) — 0. Near the sides of the planar and non-planar regions. ArgE(x. y) is near to zero and (,'. respectively. Arg/i'(j\//) lias local maximum along the y-axis near the edges of the magnetic barrier and sharply drops at the magnetic interface remaining near to zero in the whole non-planar region. The current llous between the ends of the 2DF(.i crossing the magnetic interfaces mainly in the singular points, i.i. it passes along the points for which AigE(x. y) is close io zero. Fieri rons entering or exiting the small regions of the facet corners will have large velocities proportional to the electric field at these locations to

Figure 12: Spatial distribution of the argument (in units of it) and the absolute value of /i for the facet situated between the points t!IS..r> and .r,0l.r> /1111. /. = 1000/ли. 2n = Id //in.

и = i 1 .

account for current conservation with a large number of electrons drifting with slow and uniform velocities in the middle of the facet were the electric field is smaller and uniform.

The magneto-resistance is determined by the electric field edge profile. The magnetic field dependence of the magneto-resistance across the facet is shown in Fig. 13. The classical origin of the positive background has been confirmed experimentally where it has been found that it persists at temperatures higher than 100 K. Note that experimental configuration is effectively a two terminal measurement where the measured resistance is determined both by the Hall resistance and magneto-resistance. For small magnetic fields the Hall resistance is small and thus the resistance is determined by the magneto-resistance and consequently the resistance increases as llJ. For larger magnetic fields a quasi linear behavior of the resistance as a function of В is found which is due to the fact that now it is Hall resistance which mainly limits the current. However, the resistance increases from 13.2 to 1020.8 Q when В : 0 —- 10 Т. which is approximately four times less than observed experimentally. The reason for such a diirerence could be the low ratio of the facet (3 //"/) to the mesa ( 1000 //771) length. The right figure in Fig. 13 shows the magneto-resistance calculated in the planar regions. The top two curves are the resistance for probes above the facet on the left (12-13) and right (3-4) of the mesa and the center two curves show tin- pairs (10-11) and (5-6) below the facet, (see

1J-1J

c

J-i J-4

u-:o 2-i

H«i!t*:c fit.a'îti;«

Figure 13: Magneto-resistance across the facet (left) and in the planar regions (right) for l(3 - 4) - 255 vm, l{5 - 6) = 220 nm, and /,(2 - 3) = 200 ^m.

Hef. Ps]). In agreement with experiment, pairs of probes on opposite sides of lhe mesa and on opposite sides of the facet show the same symmetry while there is a strong asymmetry in each of the traces with reversal of the direction of the magnetic field. These results agree qualitatively with experiment but there are problems with the quantitative values of the resistance. The main variation of the resistance takes place for small R and the magnitude of the variation strongly decrease-, with distant«- from (In- laci l. TlieriTure. in tin-scale of this figure one cannot see the resistance variation for probes (2-3) situated 300 //in away from the facet.

SUMMARY

In Summary of the dissertation, the basic results obtained in the thesis are restated.

ACKNOWLEDGMENTS

I I hank Prof. Y IJ Levitison. U Itossler. and F. 1'eelens for their common work on the problems presented in the dissertation.

I am grateful to Prof. Y M Haroutliounian for acceptance me in his division al YSl and for support during my work.

1 am thankful to all my colleagues from Radiophysics and Solid state physics depart tin-ills of VST. from the theoretical group of YrPhl. from Institute of

Microelectronics in Chernogolovka. from Regensburg and Antwerp Universities. 1 acknowledge especially to Prof. J Keller. К F Renk. A A Kirakosyan. I. M Kazaryan. Ё 1 Rashba. A V Chaplik. D L Maslov. L J Cballis. A Ya Shik. S G Petrosyan. G W Bryant. J T Devreese. A II Melikyan. and A G Sedrakyan for helpful discussions, suggestions, and comments.

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50. S. M. Badalian, I. S. Ibrahim, and F. M. Peeters, in High Magnetic Fields in the Physics of Semiconductors II. edited by G. Landwehr and W. Ossau (World Scientific, Singapore. 1997). pp. 327-330.

51. V. Karpus. Sov. Phys. Semicond. 20. 12 (1986).

52. G. A. Toombs. F. W. Sheard. D. Neilson. and L. J. Challis. Solid Slate Commun. G4. 577 (1987).

53. M. Rothenfusser. L. Köster, and W. Dietsche. Phys. Rev. B 34. 5518 (1986).

54. J. C. Ilensel, B. J. Ilalperin, and R. C. Dynes. Phys. Rev. Lett. 51, 2302 (1983).

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57. H. Kitagawa and S. Tamura, Phys. Rev. B 46, 4277 (1992). i

58. T. Ando. J. Phys. Soc. Japan 51, 3900 (1982). | I |

59. V. F. Gantmakher arid Y. B. Levinson, Carrier Scattering in metals and Semiconductors (Nauka, Moscow. 1984).

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List. of publication»« underlying the dissertation

1. S ¡VI Badalyan and A II M< likyan. Threshold anomalies in disintegration of an optic phonon into two acoustic phonons in one dimensional anharmonic crystal. Sov. Phys Low Temp.. 13:670. 1987.

'J S M 11; I • 1: 11V ■ 11 .Hid V И Levinson Hound stales of electron and optical l>liniiou ш a quatii urn well Sot I'ln/s. .//.77'. 67:611 645.1988.

'.',. S M Hadalyan and Y H Levinson. Cyclotron-phonon resonance in a two dimensional electron gas. Sov. Phys. Se.micond.. 22:1278-1281. 1988.

■1. S M Badalyan and Y В Levinson. Effect of an interface on the scattering of a two dimensional electron gas from acoustic phonons. Sov. Phys. Solid Stalt. 30:1592 1597: 1988.

5. S M Badalian and Y В Levinson. Interface effect on the interaction of a two-dimensional electron gas with acoustic phonons in a quantizing magnetic held. Proceedings of Mill All-Union Conference on Theory of Semiconductors. Donetsk. 1989.

() S M Hadalyan. F.hctvon-Phonon Interaction in Quasi Two Dimensional kin trim Systt ins. PhD thesis. Yerevan Slate University. Yerevan. 1989.

7. S M Badalyan. Scattering of electrons in a two dimensional Fermi gas by acoustic phonons near au interface between elastic half-spaces. Sov. I'ln/s. Si hi ii о и it . :М()87 10!) I. 1989.

8 S M Badalian and Y В Levinson. Free surface effect on the interaction of a two dimensional electron gas with acoustic phonons in a quantizing magnet ic field. I'ln/s. /.<//. A. 140:62 ()(i. 1989.

9 S M Badalian and Y В Levinson. Ballistic acoustic phonon emission by a two dimensional electron gas in a quantizing magnetic field with account of the phonon relied ion from a (¡aAs/AIGaAs interlace. Pltys. Lett. A. 155:200-206. 1991.

10. S M Badalian. Y В Levinson. and D L Maslov. Scattering of electron '•I;1/ lab- in .i i n a11* I ie I i • -1 • I by iiii|>uril ie:. and phonons Sue I'ln/s II. 77' 1.1:1 T . Г,:{.Г)!)Г> 599. 1991.

11. D L Maslov. Y В Levinson. and S M Badalian. Interedge relaxation in a magnetic field. Phys. Rev. B.46:7002-7010. 1992.

12. S M Badalian and Y B Levinson. Suppression of the emission of surface acoustic plionons from a two dimensional electron gas a quantizing magnetic field. Phys. I.(t1. A. 170:229 -231. 1902.

13. S M Hadaliau. I1 Mossier, and M I'olemski. Theory of Auger up-conver sion in quantum wells, page M33. Hegensburg. Germany. 1993. 13th General Conference of the Condensed Matter Division. European Physical Society.

14. S M Badalian. U Rossler. and M Potemski. Theory of Auger up-conver-sion in quantum wells in a quantizing magnetic field. J. Physics Cond. Mat., 5:6719-6728: 1993.

15. S M Badalian. Ballistic acoustic phonon emission by quantum edge states. Abstract 1022. Madrid. 1994. 14th General Conference of the Condensed Matter Division. European Physical Society.

16. S M Badalian. Emission of ballistic acoustic phonons by quantum edge slates. J. Physics Cond. Mat.. 7:3929 3936. 1995.

17. S M Badalian. Electron relaxation in the quantum Hall effect geometry: One-and two-phonon processes. Phys. Rev. B. 52:14 781 14 788. 1995.

18. S M Badalian. I S Ibrahim, and F M Peeters. Theory of the magnetotransport of electrons in a non-planar two dimensional electron gas. Abstract TuP 75. Wiir/.burg. Germany. 1996. 12th International Conference on the Application of High Magnetic Fields ill Semiconductor Physics.

19. S M Badalian. I S Ibrahim, and F M Peeters. Magneto-transport of electrons in a non-homogeneous magnetic field. Abstract TuP-52. Liege. Belgium. 1996. 9th International Conference on the ' Superlattices. Microstructures and Microdevices. July 14-19. 1996. (ICSMM-9).

20. I S Ibrahim. S M Badalian. and F M Peeters. Magneto-transport of electrons through a magnetic barrier. Eindhoven. Holland, 1996. Semiconductor days Meeting in Eindhoven.

21. I S Ibrahim. V A Schweigert. S M Badalian. and F M Peeters. Magnetotransport of electrons in a non-homogeneous magnetic field. Superlatttccs and Mtcroslruclures, 22:203-207. 1997.

22. S M Badalian. I S Ibrahim, and F M Peeters. Theory of the magnetotransport of electrons in a non-planar two dimensional electron gas. In G Landwehr and W Ossau. editors. High Magnetic Fields tn the Physics of Semiconductors II. pages 327 330. World Scientific. Singapore. 1997.

Biographical Sketch

Present. employ incut

I I'OI II 1995

Research in other institutions

I u'.ii. <ji-y:i

19% ■

1990. 95 1987-1991

Doit male

19x9

Subject of exaniinal ion: Sclent ific advisor:

Official opponents:

Leading orgam/al ion Post^raduati • st. i lilies

I9,v, 19«»

Graduate studies 1977-1982

Awards

199-1

199-1 1997

Leading researcher, supervisor of a scientific theme. Ye van Stale University. eiuaihbadalyan"1 Ix2.yerplii.am

Institute for Theoretical Physics. University of liege burg. Germany

Department of Condensed Matter Theory. University Antwerp. Belgium

International Center for Theoretical physics. Italy

Institute of Microelectronics Technology and High Pur Materials. Ciicriiogolovka. Russia

Candidate of Physics and Mathematics (Ph.D)

Eleclron-phonon interaction in 2D-electron systems

Prof. Y. B. Lev'inson. Institute of Microeleclroni ( 'heniogolovka. Russia

Prof. A. V. Chaplik. Institute of Semiconductor Physi Novosibirsk. Russia. Prof. A. A. Kirakosyan. Yere\ Slate University. Armenia

Institute of Solid Slate Physics. Chernogolovka. Russ I'h.D student. Institute for Physical Researches. Arme

Department of Physics (Theoretical physics). Ye^ State University (Honour degree)

' Young scientists 93': Honour of "Armenia" foundat for academic research.

International science foundation, grant No. RYU000.

Civilian research and development foundation, princi co-invest igator. grant No. AP1-375.

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