Valentin Peschansky and puzzles of magnetotransport

Starting from the 1950s, the Kharkov school of theoretical physics was one of the world leaders in the theory of metals. In particular, the research by V.G. Peschansky for many years was focused on studying the relationship between magnetic field dependence of resistivity components and the electron...

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Опубліковано в: :	Физика низких температур
Дата:	2011
Автор:	Pudalov, V.M.
Формат:	Стаття
Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2011
Теми:	Органические проводники
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Цитувати:	Valentin Peschansky and puzzles of magnetotransport / V.M. Pudalov // Физика низких температур. — 2011. — Т. 37, № 9-10. — С. 970–976. — Бібліогр.: 34 назв. — англ.

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spelling	Pudalov, V.M. 2017-05-31T07:22:33Z 2017-05-31T07:22:33Z 2011 Valentin Peschansky and puzzles of magnetotransport / V.M. Pudalov // Физика низких температур. — 2011. — Т. 37, № 9-10. — С. 970–976. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 71.30.+h, 72.15.Rn, 73.40.Qv https://nasplib.isofts.kiev.ua/handle/123456789/118766 Starting from the 1950s, the Kharkov school of theoretical physics was one of the world leaders in the theory of metals. In particular, the research by V.G. Peschansky for many years was focused on studying the relationship between magnetic field dependence of resistivity components and the electron energy spectrum. V.G. Peschansky elaborated an elegant theory of magnetoresistance that took into account surface scattering of electrons. The physics of bulk 3D metals was almost exhausted by the end of 1970s and Peschansky extended his research to the low-dimensional electron systems. Through all his scientific life, V.G. Peschansky advocated the idea that magnetoresistance is a powerful tool that can be used to explore rich physics of electron systems. By now, numerous experimental and theoretical studies of magnetoresistance behavior in various systems, from simple to the most complex ones, confirm the fruitfulness of this idea. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Органические проводники Valentin Peschansky and puzzles of magnetotransport Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Valentin Peschansky and puzzles of magnetotransport
spellingShingle	Valentin Peschansky and puzzles of magnetotransport Pudalov, V.M. Органические проводники
title_short	Valentin Peschansky and puzzles of magnetotransport
title_full	Valentin Peschansky and puzzles of magnetotransport
title_fullStr	Valentin Peschansky and puzzles of magnetotransport
title_full_unstemmed	Valentin Peschansky and puzzles of magnetotransport
title_sort	valentin peschansky and puzzles of magnetotransport
author	Pudalov, V.M.
author_facet	Pudalov, V.M.
topic	Органические проводники
topic_facet	Органические проводники
publishDate	2011
language	English
container_title	Физика низких температур
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
format	Article
description	Starting from the 1950s, the Kharkov school of theoretical physics was one of the world leaders in the theory of metals. In particular, the research by V.G. Peschansky for many years was focused on studying the relationship between magnetic field dependence of resistivity components and the electron energy spectrum. V.G. Peschansky elaborated an elegant theory of magnetoresistance that took into account surface scattering of electrons. The physics of bulk 3D metals was almost exhausted by the end of 1970s and Peschansky extended his research to the low-dimensional electron systems. Through all his scientific life, V.G. Peschansky advocated the idea that magnetoresistance is a powerful tool that can be used to explore rich physics of electron systems. By now, numerous experimental and theoretical studies of magnetoresistance behavior in various systems, from simple to the most complex ones, confirm the fruitfulness of this idea.
issn	0132-6414
url	https://nasplib.isofts.kiev.ua/handle/123456789/118766
citation_txt	Valentin Peschansky and puzzles of magnetotransport / V.M. Pudalov // Физика низких температур. — 2011. — Т. 37, № 9-10. — С. 970–976. — Бібліогр.: 34 назв. — англ.
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fulltext	© V.M. Pudalov, 2011 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10, p. 970–976 Valentin Peschansky and puzzles of magnetotransport V.M. Pudalov P.N. Lebedev Physical Institute, 53 Leninskii Ave., Moscow 119991, Russia E-mail: pudalov@sci.lebedev.ru Received May 12, 2011 Starting from the 1950s, the Kharkov school of theoretical physics was one of the world leaders in the theory of metals. In particular, the research by V.G. Peschansky for many years was focused on studying the relation- ship between magnetic field dependence of resistivity components and the electron energy spectrum. V.G. Pe- schansky elaborated an elegant theory of magnetoresistance that took into account surface scattering of electrons. The physics of bulk 3D metals was almost exhausted by the end of 1970s and Peschansky extended his research to the low-dimensional electron systems. Through all his scientific life, V.G. Peschansky advocated the idea that magnetoresistance is a powerful tool that can be used to explore rich physics of electron systems. By now, nu- merous experimental and theoretical studies of magnetoresistance behavior in various systems, from simple to the most complex ones, confirm the fruitfulness of this idea. PACS: 71.30.+h Metal-insulator transitions and other electronic transitions; 72.15.Rn Localization effects (Anderson or weak localization); 73.40.Qv Metal-insulator-semiconductor structures (including semiconductor-to-insulator). Keywords: magnetoresistance, electronic properties of metals, low-dimensional carrier system, electron–electron interaction. 1. Introduction With age, I started feeling obligated to write occasional memoirs in order to reward distinguished scientists whom I met in my life. I appreciate them for sharing with students the overview of the architecture of science, for the object lessons in creativity and, quite important, for the lessons in regardful attitude towards the students. The latter ingre- dient makes a specific favorable friendly atmosphere in physical research. Particularly, I greatly appreciate Profes- sor Valentin Peschansky not only for his remarkable con- tribution to the theory of metals, but also for his perpetual benevolence. Physics is made by people, and without knowing personalities the physics would be colorless, like boring black-and-white textbooks. Young scientists must know the biographies of outstanding personalities, because major scientific achievements often appear to be linked with events in their life. I met Valentin Peschansky for the first time at the Ka- pitza Institute for Physical Problems (named after Kapitza after his death), in Moscow, where I was a graduate stu- dent of Professor M.S. Khaikin. The period from the 1950s to 1970s was triumphal for metal physics. Experimentalists succeeded in purifying materials and growing state-of-the- art single crystals of almost all metals. There was certainly a sort of competition among them in achieving record val- ues of the resistance ratio, 300 4.2/R R , or the electron mean free path. The high quality of the studied samples inspired respect to the published data. The research in experimental laboratories at the Kapitza Institute was in a full swing: Yu.V. Sharvin did beautiful experiments on electron focusing with point contacts. M.S. Khaikin with his young co-workers discovered numerous phenomena in microwave resonant electron transport in Bi, Sn, In, Pb, W, including the size effects in cyclotron mo- tion of electrons. N.E. Alexeevskii and collaborators stu- died magnetoresistance anisotropy in Be, V, Nb. N.V. Za- varitskii studied phonon drag and acousto-magnetic effects in Sn and Al. The next generation of experimentalists, V.F. Gantmakher, V.S. Edelman, Yu.P. Gaidukov and others, grew up in these parent laboratories and started their own research also from metal physics. The period from the 50s to the 70s was also a golden age for the Kharkov school of theoretical physics. One has to recall that Mark Ya. Azbel, Moissei I. Kaganov, Arnold M. Kosevich, Ilya M. Lifshitz, and Valentin G. Peschansky were highly creative, working at the same time and the same place. This short list of distinguished theorists could do an honor for any University all over the world. The Kharkov theorists were welcome at the Kapitza Institute; they often came to give talks at the famous Kapitza semi- Valentin Peschansky and puzzles of magnetotransport Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10 971 nars and visited experimental laboratories. Their activity in the 50s and the 60s was focused on studying the relation- ship between observable features in electron properties in magnetic field and electron energy spectrum [1,2]. The nearly-free-electron approach has just emerged and the Fermi surfaces for the majority of metals yet remained to be explored. Needless to say, there was a plethora of novel experi- mental data at the Kapitza Institute for theorists to explain; in their turn, theorists were eager to have their predictions tested experimentally. The mutual attraction of Moscow experimentalists and Kharkov theorists has made the Ka- pitza Institute the center of meetings and discussions, par- ticularly in the field of electronic properties of normal met- als. After Landau death, Kapitza wanted to strengthen the theory department of his Institute, and, soon after, Ilya Lif- shitz (in 1969) and Moissei Kaganov (in 1970) joined the Kapitza Institute. Even earlier, in 1966, Emmanuil Rashba, also an originally Ukranian theorist, joined the Landau In- stitute for Theoretical Physics. Peschansky, Azbel and Ka- ner remained working and lecturing in Kharkov, however I saw them very often in Moscow at IPP. Valentin Peschansky stood out among other Ukranian theorists by his friendly attitude and benevolence toward surrounding people. He spoke with a soft voice and never put the opponent in an awkward situation. My own gradu- ate research was also related to the electron energy spec- trum of metals, therefore, when the time came to defend the PhD thesis, my scientific supervisor M.S. Khaikin sug- gested Peschansky as an opponent. As usual, he kindly agreed. This was indeed a good choice that left the best reminiscence. As the result of experimental efforts in crystal growth, the quality of the metal samples reached such the state-of- the-art, where the electron mean free path at low tempera- tures became comparable with a sample size. It appeared unexpectedly that the electron scattering at the sample sur- face is an elastic and almost mirror-reflection process. As an example, M.S. Khaikin observed quantum transport of electrons skipping at the surface; Yu.V. Sharvin and V.S. Tsoi studied focusing of the electrons which experienced mirror reflection at the surface; V.S. Edel'man, V.F. Gant- makher, and Yu.P. Gaidukov also observed various size effects caused by mirror reflection of electrons at the sam- ple surfaces. Yet at the Institute for Physical Problems, Yu.P. Gaidu- kov studied magnetoresistance anisotropy and, using the theory by Peschansky et al., reconstructed the Fermi sur- face topology [3]. When Gaidukov moved to the Low-Tem- perature Department of the Moscow State University, he extended his research from bulk to whisker crystals of me- tals. These tiny specimens a typical size of 1 10 100× × μm behaved wonderfully. They could be easily charged elec- trostatically and bent into a spiral with no loss of quality and with no residual defects. Gaidukov and collaborators succeeded in growing whisker crystals of Bi, Sb, Zn, and Cd from vapor phase (he even found whisker crystals of tin which grew on his old tin-lined skates). Besides their extreme mechanical properties, whisker crystals were the ideal object for size effect studies owing to their small thickness and mirror-flat natural surfaces. Experimental studies of magnetoresistance in Gaidulkov's laboratory were very fruitful. To handle the tiny objects under microscope one had to have perfect vision and strong nerves, because moving the crystals, fixing them, and making tiny contacts took many hours of delicate work using eye-surgery tools. In Gaidukov's laboratory this re- search was performed by young female PhD and graduate students. One of the Gaidukov's PhD student, Elena Go- liamina, became my wife, and for this reason I also became indirectly involved in the “whisker business”, have read Peschansky's papers, and even have done some research on tiny whisker crystals grown by Gaidukov [4]. Semimetals such as Bi and Sb were good test objects for experimentalists because of the low melting point, high purity, and extremely small and almost cylindric electronic pockets at the Fermi surface. The latter lead to huge ampli- tude of quantum oscillatory effects, resonant effects, etc. On the theoretical side, Valentin Peschansky with col- laborators also intensively studied kinetic properties of semimetals [5,6]. The most elegant effect, elaborated theo- retically by Peschansky and collaborators was the “static skin effect” [5,7,8]. It appeared that the contribution of ordinary “bulk” electrons (which don't scatter by the sur- face) to conductivity could become much less than the con- tribution of electrons mirror-reflected by the sample sur- face. In other words, for a given sample thickness d and a given mean free path l , >d l , one can set such a strong magnetic field, where the current will be concentrated within a thin layer near the surface; the thickness of this layer is an order of the cyclotron radius, r . This concept radically changed the preceding viewpoint on conduction in thin samples and lead to such interesting observable effects as novel type of magnetoresistance, magnetoresis- tance anisotropy, and oscillatory dependent nonuniform conduction in the bulk. However, the “static skin effect” was expected to be very sensitive to the presence of a mi- nor diffusivity in the electron scattering at the surface, the drawback that made the experimental verification of the theoretical prediction rather difficult. Experiments at the Moscow State University were mo- tivated in part by the predictions of the “static skin effect” theory. Correspondingly, the paper by Peschansky et al. [5] was one of the “hand-books” at the Yurii Gaidukov's la- boratory. Within the period from 1973 to 1978, Gaidukov and collaborators studied the size effect in magnetoresis- tance of Cd, Sb, and Zn whisker crystals [9–11]. Some features of the observed magnetoresistance agreed qualita- tively with Peschansky theory. Particularly, the magnetore- sistance data scaled for samples with different thicknesses, V.M. Pudalov 972 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10 and in weak magnetic field the “parallel” magnetoresis- tance ( )B I exhibited a maximum. However, quantita- tively, the maximum didn't agree with the theory, for it was observed at fields ( )r B d∝ , while in the theory it should be located at ( )r B d∝ . The disagreement with the theory was sample dependent: it was minor for Sb, and more essential for Zn and Cd. The experimental results thus pointed to the insufficiently smooth surface and lack of ideal mirror reflection at the Zn and Cd surfaces. Despite lacking of quantitative agreement with theory, the experimental data have manifested unambiguously the striking difference in magnetoresistance for thin samples (d r∼ ) and that for thick 3D samples ( d r ). The most direct confirmation of the theory was achieved in experi- ments with Sb. Whereas the bulk Sb samples showed al- most isotropic magnetoresistance, the thin plate-like whiskers of Sb demonstrated a factor of 10 larger and ani- sotropic magnetoresistance. This fact clearly demonstrated the contribution of surface scattering to electron transport for thin 3D samples, which was the central point of the Peschansky theory. As often occurs in experiments, besides the sought for monotonic magnetoresistance, Gaidukov and Goliamina unexpectedly observed a novel oscillatory size effect, the Shubnikov–de Haas oscillations cut-off in such weak mag- netic fields [11,12], where the cyclotron orbit became larg- er than the whisker thickness. Remarkably, even earlier, Peschansky also wrote a theoretical paper [13] on Shubni- kov–de Haas effect in thin conductors, where he predicted the oscillation cut off, but didn't consider the mirror reflec- tion case. The experiments were made with Sb, the materi- al in which electron scattering at the surface was of the mirror type, and in which monotonic magnetoresistance demonstrated the best agreement with the “static skin ef- fect” theory. The experiments, however, revealed more rich physics, than the theory predicted: in magnetic fields lower than the cyclotron orbit cut-off, new series of oscil- lations emerged, due to quantization in magnetic fields of the truncated electron orbits with mirror reflected trajec- tories. Soon after, Elena Goliamina had her PhD thesis ready and Peschansky, as the major expert in the field, was in- vited to act as the opponent. After the official defence he was very surprised to learn that he played the same role for the second time for the same family. 2. Magnetoresistance in low dimensional systems The golden age of electronic properties of three- dimensional metals was over by the end of the 70s. In 1979, Peschansky published comprehensive reviews on kinetic size effects in metals [6,7]. Five years later, he and several other Russian “metal physicists”, including the author, wrote a book [8,14] to overview their preceding research in the physics of 3D metals. These and many oth- er reviews summarize the results of the 20-years long in- tensive investigations in physics of metals. The bulk metals appeared to be rather simple and their physics was quickly exhausted. Besides knowledge of the Fermi surfaces for the majority of metals [15], the net result of these studies was the development of a number of milestone concepts, including the approaches of nearly-free-electrons, strong and weak coupling, Fermi liquid paradigm, and Fermi sur- face. Another key result was the development of a number of powerful experimental and theoretical tools for studying electronic systems. Right at this time the physics of low-dimensional sys- tems started emerging worldwide and the seeds of the 3D metal physics fell on the good ground. With the develop- ment of semiconductor technology and the advent of high- quality low-dimensional systems many “metal physicists” in the 70s and 80s switched to the physics of “low-di- mensions”. Some of them started studying two-dimensio- nal (2D) electron systems at semiconductor interfaces, some other - to layered (1D and 2D) crystals. As mentioned above, the twenty-years-long studies of electronic properties of 3D metals resulted in a number of elaborated powerful concepts and tools. Their list includes, first of all, Shubnikov–de Haas and de Haas–van Alfven effects, cyclotron resonance, and magnetoresistance. In bulk metals, and even in thin conductors, the magnetoresis- tance was used as a probe to test electron orbital motion. The spin degree of electron freedom considered almost irrelevant. Much later, the researchers encountered the ef- fects of electron–electron (exchange) interactions, go- verned by physics of spin. Now, it is well recognized that the magnetoresistance can also be used to probe the elec- tron–electron interaction, i.e., the physics of spins. In this interesting swing, the physicists came back to the original Peschansky idea that the magnetoresistance can be a key tool for probing the unknown electron system and under- standing its microscopic architecture. 2.1. Magnetoresistance and cyclotron resonance in organic low-dimensional systems Searching for a new field of research, the author also started studying low-dimensional systems: in 1980 — the 2D electron systems in semiconductors, and in 1998 — quasi-one-dimensional organic conductors. The latter ob- jects, in particular, compounds of the (TMTSF)2X family, are very interesting: at low temperatures, they exhibit physics related with spin density wave state [16] in low or zero magnetic fields, whereas at elevated pressure the spin- density wave state is suppressed and (TMTSF)2PF6 at fi- nite temperatures behaves as a quasi-1D layered metal. The (TMTSF)2PF6 has a quasi-one-dimensional elec- tron system confined in a three dimensional host lattice. The electron system is therefore highly sensitive to exter- nal parameters and, depending on pressure, magnetic field, temperature etc., exhibits properties inherent of 1D, 2D, Valentin Peschansky and puzzles of magnetotransport Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10 973 and 3D systems [16]. The unique property of (TMTSF)2PF6 is that its – –P B T phase diagram contains numerous phases such as the spin density wave (SDW) state, field induced spin density wave (FISDW) state, quantum Hall effect, and superconducting state (at temperatures below 1≈ K). This is because the paramagnetic metallic state of the quasi-1D electronic system is unstable and due to electron–electron interaction, at lowering temperatures, undergoes a transi- tion to the SDW state, which is an antiferromagnetic (AF) spin-ordered state (an insulator). Increasing pressure de- stroys the SDW order and makes the paramagnetic metallic state more favorable [16]. Again, as in bulk 3D metals, the magnetoresistance appears to be the most powerful expe- rimental tool for studying the origin of various phases and phase transitions in this quasi-1D material. It is a coincedence that Valentin Peschnasky in his theore- tical studies also switched from 3D metals to quasi-two di- mensional conductors [17–19]. In particular, in Ref. 19, he studied theoretically cyclotron resonance in layered ma- terials. Almost at the same time, we sought for the cyclo- tron resonance, but in quasi-one-dimensional (TMTSF)2PF6. At first sight, the idea of cyclotron motion in 1-dimen- sional systems sounds odd. However, due to finite transfer integrals, the system under study is quasi-one-dimensional. In the metallic state it has an open Fermi surface; the mag- netic field (below the onset of the field-induced spin- density wave state, FISDW [16]) applied perpendicular to the conducting plane causes one-dimensionalization of the electron motion. Nevertheless, the finite transfer integrals (perpendicular to the most conducting direction) lead to the periodic motion of the electrons in magnetic field. This motion was detected in our experiments by observing the cyclotron resonance in the mm-wave range [20]. The mea- sured cyclotron mass was found larger than the theory ex- pected; the discrepancy motivated theorists to revise the existing “standard” models of the field-induced spin densi- ty wave state [16]. 2.2. Magnetoresistance in two-dimensional systems Historically, the physics of 2D systems emerged in 1966 when Allan Fowler, Frank Fang, et al. observed Shubnikov–de Haas oscillations in silicon MOSFET (met- al–oxide–semiconductor field effect transistor) [21]. By tilting the sample in magnetic field they found the period of quantum oscillations to be governed by perpendicular field component solely, and thus, had proven the two- dimensionality of the studied electron systems. Initially, the same theoretical ideas that were developed earlier for 3D metals were now applied for 2D systems and it ap- peared they worked, at least to the first approximation. For about twenty years the 2D physical community used con- ventional Lifshitz–Kosevich formulae for 3D systems with minor transparent modifications for the 2D density of states; V.G. Peschansky also contributed to this research and published in 2002 a paper on magnetotransport effects in organic layered conductors [17]. In the 1980s, the experimental studies in high perpendi- cular magnetic fields revealed huge effects of the electron– electron interactions, which were negligibly small in 3D metals and ignored therefore previously. To parametrize the electron–electron interaction, a dimensionless ratio sr of the potential Coulomb interaction energy to the kinetic Fermi energy is commonly used. In 2D systems, the sr values as high as 10∼ can be easily achieved by decreas- ing the electron density in high quality samples. The pres- ence of electron–electron interaction makes the 2D system much more complex, and its physics much more deep. During only 30 years of research, we evidenced three No- bel prizes for discovery of unexpecting phenomena in 2D electron systems: integer and fractional quantum Hall ef- fects, and physics of graphene. Surely, more discoveries are still waiting to be awarded. The most familiar interaction effect is the negative compressibility, κ , of the electron system, where κ chan- ges sign at 1.4sr ≈ . The effect was predicted theoretically by A.L. Efros [22] and observed in Ref. 23 in 2D system of Si-MOSFET, and later on, by J. Eisenstein on double-layer GaAs/AlGaAs heterostructure [24]. The physics of this ef- fect is straightforward: as one adds new electrons to the 2D interacting system, the gain in potential electron (exchange interaction) energy overpowers the growth in kinetic (Fer- mi) energy. The total system (2D electrons+lattice+gate) remains neutral due to the presence of the metallic gate, or due to the installed dopants near the interface. Therefore, the classical electrostatic energy maintains the stability of the system, in contrast to the black holes in cosmology which also have negative compressibility. It turns out that the interaction effects are strongly en- hanced in perpendicular field; this effect was initially de- scribed in terms of interaction between resolved Landau levels. The effects such as quantum oscillations of the Landau level splitting, enhanced and oscillatory spin- splitting, and Landau level broadening are the most known consequence of the interaction effects in perpendicular field. 3. Magnetoresistance in the in-plane field More delicate physics of inter-electron interaction can be revealed in magnetic fields parallel to the 2D plane. In such geometry, the field does not couple to the electrons motion and couples only to their spins. For a noninteract- ing ideal 2D system, the in-plane field doesn't cause a magnetoresistance. When the thickness of the 2D layer becomes comparable to the magnetic length ,Hl one has to take into account a diamagnetic shift of the energy le- vels. We ignore these strong field and finite thickness ef- fects and focus only on the electron–electron interaction induced magnetoresistance. V.M. Pudalov 974 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10 Application of the in-plane magnetic field to the strong- ly interacting and clean 2D electron system was found to cause a dramatic increase of the resistance, more than by two orders of magnitude, as seen in Fig. 1 [25,26]. At high fields, the resistance saturates. Such behavior was found in all high-mobility 2D samples with strongly interacting car- rier system. Thus, the magnetic field simply destroys the metallic state. At fields higher than a density dependent value ,B the magnetoresistance saturates; the saturation field approximately corresponds to full spin polarization of the 2D electron system, = 2B Fg B Eμ . This remarkable magnetoresistance effect was observed in 1997; it took about 5 years to understand the interaction- induced magnetoresistance in B field. In 2002, Zala, Na- rozhny, and Aleiner (ZNA) [27] developed a theory that took into account all interaction contributions to the con- ductivity, including the exchange ones. This theory offers a unified approach to both ballistic ( 1Tτ ) and diffusive ( 1Tτ ) interaction regimes by considering the quantum interference between electron waves scattered off a short- range random potential “dressed” by Friedel oscillations of the electron density. The theory was extended for the case of a long-range scattering potential by Gornyi and Mirlin (GM) [28]. The theories [27,28] naturally incorporate the Altshuler–Aronov (AA) results for the interaction correc- tions to the conductivity in the diffusive regime [29]. The theories [27,28] predict that the magnitude and sign of the the interaction correction to conductivity, ( , )T BΔσ is determined by the Fermi-liquid parameter 0Fσ , and there- fore, the 0Fσ value can be found from ( , )T BΔσ measure- ments [30]. Independently, this parameter was also found by measuring the Shubnikov–de Haas (18) oscillations in weak magnetic fields tilted to the plane of a 2D structure [31,32]. 3.1. Modern view on magnetoresistance in interacting 2D system The in-plane magnetic field, being coupled mostly to electron spins, provides a useful tool for exploring the inte- raction effects in the low-temperature conductivity of 2D system [25,26]. When the Zeeman energy =Z b BE g Bμ ( = 2bg is the bare g-factor, Bμ is the Bohr magneton) becomes much greater than T, the number of triplet terms that contribute to interaction correction ( )ee TΔσ is re- duced from 15 to 7. Similar reduction of triplet terms is expected for a valley splitting >V TΔ . These two effects have been accounted by the theory of interaction correc- tions [27,33]; in the presence of the magnetic field and/or valley splitting the interaction correction to the conductivi- ty can be expressed as follows [33]: 0( , , , , ) = ( ) 2 ( , )Z ee V ee ZT F B T E TσΔσ τ Δ Δσ + Δσ + 2 ( , ) ( , ) ( , ),Z Z Z V Z V Z VT E T E T+ Δσ Δ + Δσ + Δ + Δσ − Δ (1) where ( )ee TΔσ is given by Eq. (2): ( ) = ( ) 15 ( ).ee C TT T TΔσ δσ + δσ (2) Here Cδσ is the so-called “charge” contribution which combines Fock correction and the singlet part of Hartree correction, and Tδσ is the “triplet” contribution due to the triplet part of the Hartree term. The valley index can be considered as a pseudo-spin in multi-valley systems, and the valley degeneracy increases the number of triplet terms due to the spin exchange processes between electrons in different valleys. For the (100) Si-MOSFET system with two degenerate valleys, the total number of interaction channels is 4 4 = 16× , among them 1 singlet and 15 triplet terms (for comparison, there are 1 singlet and 3 triplet terms for a single-valley system). All the terms ( , )Z Z TΔσ have a form ( , ) ( , ) (0, ) = ( ) ( ) =Z b dZ T Z T T Z ZΔσ ≡ σ −σ δσ + δσ 0 0 0 0 21 ( ) , , , 2 21 b d F Z ZT K F K F T TF σ σ σ σ ⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪⎛ ⎞ ⎛ ⎞= τ +⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥π π⎝ ⎠ ⎝ ⎠+ ⎣ ⎦⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭ (3) if the relevant energies FZ E ( Z stands for ZE , VΔ , and combinations Z VE ± Δ ). The explicit expressions for Fig. 1. Resistivity vs parallel magnetic field, measured at = 0.29T K on Si-MOSFET sample. Different symbols corres- pond to the gate voltages from 1.55 to 2.6 V, or, equivalently, to the densities from 1.01 to 112.17·10 cm–2. Represented from Ref. 26. 0.01 1 10 100 0 2 4 6 8 10 12 1.55 1.60 1.65 1.70 1.80 2.0 2.2 2.6 B\|\| T = 0.29 K � , / h e 2 Magnetic field, T Valentin Peschansky and puzzles of magnetotransport Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, Nos. 9/10 975 the functions bK and dK are given in Ref. 27. In particu- lar, Eq. (3) describes the interaction-driven magnetocon- ductivity in the magnetic fields which are much weaker than the field of full spin polarization of a system. In Eq. (3), we neglected the crossover function which is nu- merically small and does not modify the value of ( , )Z TΔσ outside the ballistic-diffusive crossover region by more then one percent. 3.2. Magnetoconductivity. Comparison of the experiment with theory Since the theory calculates corrections to conductivity rather than resistivity, from now, we switch to the magne- toconductivity (MC). To test the theoretical predictions on the magnetoconductivity induced by in-plane magnetic fields, in Ref. 34 ( )Bσ dependences were measured at fixed temperatures. The MC for sample Si6-14 over the field range 4.5 T < < 4.5 TB− is shown for different den- sities and temperatures in Fig. 2. The theoretical ( )BΔσ dependences, see Eqs. (1), (3), plotted in Fig. 2 as solid curves, describe the observed MC very well in not-too-strong magnetic fields < 0.2 .b B Fg B Eμ The only adjustable parameter was the 0 ( )F nσ value ex- tracted for each density from fitting the MC at high tem- peratures ( 0.7≈ K) where the effects of valley splitting or intervalley scattering on ( , )ee T BΔσ can be neglected. The fitted 0 ( )F nσ values were found to agree with those inde- pendently measured earlier from Shubnikov–de Haas oscil- lation beats in tilted magnetic fields [32]. As B grows and/or density decreases, the data start deviating from the theoretical curves (see Fig. 2,d); this deviation can be attributed to the violation of the condition b B Fg B Eμ required for applicability of Eqs. (1), (3). 4. Conclusion The experiments show that the low-T behavior of the magnetoconductivity of interacting 2D electron system in Si MOSFETs is well described by the theory of interaction effects in systems with short-range disorder [27]. Over a wide range of intermediate temperatures ( < ),b B Fg B T Eμ the interaction effects are strongly enhanced in Si MOS- FETs due to the presence of two valleys in the electron spectrum. This factor, in combination with the interaction- driven renormalization of the Fermi-liquid parameter 0Fσ , leads to an increase of σ with decreasing T . The 0Fσ values obtained from fitting the experimental data with the theory [27] agree well with the 0Fσ data obtained from the analysis of oscillations in these samples. The considered above example demonstrates that the central idea by V.G. Peschansky that the magnetoresistance is a powerful tool to explore complex physics of the electron system, remains valid and fruitful until now. Fig. 2. (color online.) Magnetoconductance for Si-MOSFET sample at different electron densities and temperatures. Experi- mental data are shown by dots, the theoretical dependences calcu- lated according to Eqs. (1)–(3) — by solid curves. The 0Fσ value is the only fitting parameter in comparison with the theory [27], the corresponding values of 0Fσ are shown in Fig. 5 of Ref. 34. Arrows indicate the fields corresponding to the condition /2 = 0.1B Fg B Eμ . The values of n are shown in units of 1011 cm–2. 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Valentin Peschansky and puzzles of magnetotransport

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