Tunneling dynamics in cryocrystals: localization and delocalization

The phenomenon of quantum diffusion of muonium in cryocrystals with rotational degrees of freedom is discussed. The quantum tunneling dynamics and electron transport are considered taking into account the effects of disorder.

Збережено в:

Бібліографічні деталі
Дата:	2003
Автор:	Storchak, V.G.
Формат:	Стаття
Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2003
Назва видання:	Физика низких температур
Теми:	Electronically Induced Phenomena
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/128921
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Tunneling dynamics in cryocrystals: localization and delocalization / V.G. Storchak // Физика низких температур. — 2003. — Т. 29, № 9-10. — С. 992-1000. — Бібліогр.: 45 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-128921
record_format	dspace
spelling	irk-123456789-1289212018-01-15T03:03:40Z Tunneling dynamics in cryocrystals: localization and delocalization Storchak, V.G. Electronically Induced Phenomena The phenomenon of quantum diffusion of muonium in cryocrystals with rotational degrees of freedom is discussed. The quantum tunneling dynamics and electron transport are considered taking into account the effects of disorder. 2003 Article Tunneling dynamics in cryocrystals: localization and delocalization / V.G. Storchak // Физика низких температур. — 2003. — Т. 29, № 9-10. — С. 992-1000. — Бібліогр.: 45 назв. — англ. 0132-6414 PACS: 67.40.Yv, 73.20.Jc http://dspace.nbuv.gov.ua/handle/123456789/128921 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Electronically Induced Phenomena Electronically Induced Phenomena
spellingShingle	Electronically Induced Phenomena Electronically Induced Phenomena Storchak, V.G. Tunneling dynamics in cryocrystals: localization and delocalization Физика низких температур
description	The phenomenon of quantum diffusion of muonium in cryocrystals with rotational degrees of freedom is discussed. The quantum tunneling dynamics and electron transport are considered taking into account the effects of disorder.
format	Article
author	Storchak, V.G.
author_facet	Storchak, V.G.
author_sort	Storchak, V.G.
title	Tunneling dynamics in cryocrystals: localization and delocalization
title_short	Tunneling dynamics in cryocrystals: localization and delocalization
title_full	Tunneling dynamics in cryocrystals: localization and delocalization
title_fullStr	Tunneling dynamics in cryocrystals: localization and delocalization
title_full_unstemmed	Tunneling dynamics in cryocrystals: localization and delocalization
title_sort	tunneling dynamics in cryocrystals: localization and delocalization
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2003
topic_facet	Electronically Induced Phenomena
url	http://dspace.nbuv.gov.ua/handle/123456789/128921
citation_txt	Tunneling dynamics in cryocrystals: localization and delocalization / V.G. Storchak // Физика низких температур. — 2003. — Т. 29, № 9-10. — С. 992-1000. — Бібліогр.: 45 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT storchakvg tunnelingdynamicsincryocrystalslocalizationanddelocalization
first_indexed	2025-07-09T10:14:06Z
last_indexed	2025-07-09T10:14:06Z
_version_	1837163923937165312
fulltext	Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10, p. 992–1000 Tunneling dynamics in cryocrystals: localization and delocalization Vyacheslav G. Storchak Russian Research Centre «Kurchatov Institute», 46 Kurchatov Sq., Moscow 123182, Russia E-mail: storchak@dnus.polyn.kiae.su The phenomenon of quantum diffusion of muonium in cryocrystals with rotational degrees of freedom is discussed. The quantum tunneling dynamics and electron transport are considered tak- ing into account the effects of disorder. PACS: 67.40.Yv, 73.20.Jc 1. Introduction A vast amount of kinetic processes in chemistry and biology, nuclear and solid state physics, disordered systems and liquids, etc. deals with the mass and charge transport, i.e. dynamics of neutral (typically atomic) particles and charged particles (typically electrons or electronic complexes), limited by poten- tial barriers. At low temperatures there is no other way to overcome a potential barrier than by quantum tunneling of a particle through it. This phenomenon is called quantum diffusion (QD). The concept of quan- tum diffusion is introduced for diffusing particles which are heavy compared with electron. On the other hand, a quantum-mechanical evaluation suggests that the tunneling probability is crucially enhanced for light particles. Therefore, in the context of QD the role of the positive muon (� �) is of particular interest because of its intermediate mass (about 200 times more than that of the electron, but about an order of magnitude less than that of the proton). Being a com- plete chemical analogue of the proton, � � captures an electron and forms the light hydrogen isotope known as muonium (Mu = � � �� e ). This happens in insula- tors and semiconductors, while in metals we deal with the «bare» muon. Because of the unique mass of the muon one can hardly mention any other example where quantum diffusion was observed in such a wide temperature range as for � � and Mu [1]. The other reason for the success of the quantum diffusion study using muons is the sensitivity of the muon spin relax- ation (�SR) techniques (see e.g. [2]) to � � and Mu dynamics. The basic issue in nonclassical transport is whether a wave-like or particle-like description is appropriate, i.e., whether the tunneling is coherent or incoherent. This depends on whether the interaction with the en- vironment is such as to lead to spatial localization of the wave function or to bandlike (Bloch wave) mo- tion. One of the possible channels for localization of a particle is through its interaction with lattice exci- tations (phonons, librons, magnons etc.). In a dissi- pative environment [3] the lattice excitations can be represented as a bath of harmonic oscillators; interac- tion with this environment causes a crossover from co- herent quantum tunneling to incoherent hopping dy- namics when the particle «dressed» with the lattice excitations can be effectively thought of as a polaron. At low temperatures, the environmental excitations are frozen out. In this case, conventional understanding suggests that the only possible channel for particle lo- calization is the introduction of crystal disorder, which thus may dramatically change the transport properties of a solid. A well-known example is the spatial localiza- tion of electron states near the Fermi level in a disor- dered metallic system, which leads to a transition into a dielectric state (the Anderson transition) [4]. The con- cept of Anderson localization suggests that the wave function of a particle in a random potential may change qualitatively if the randomness becomes large enough. Coherent tunneling of a particle is possible only be- tween levels with the same energy (e.g. between equi- valent sites); in the case of strong randomness, states with the same energy may be too distant (spatially separated) for tunneling to be effective. Although the concept of localization by disorder has been introduced primarily in order to describe the electronic transport properties of condensed matter, it may also be applied to the quantum dynamics of heavier particles, whether charged or neutral [5,1]. Recent experimental results for positive muons as well © Vyacheslav G. Storchak, 2003 as for muonium atoms clearly indicated that inte- raction with crystal excitations and crystal disorder dramatically changes the nature of tunneling dynam- ics for particles ~ 200 times heavier than the electron. In this lecture we discuss recent studies on both quantum tunneling dynamics and electron transport in cryocrystals using � �SR techniques paying particular attention to processes of particle localization and delocalization. 2. Quantum diffusion via � �SR Under-the-barrier tunneling dynamics of particles in crystalline lattice is pure quantum mechanical pheno- menon which has no analogue in classical physics. Typically, tunneling occurs between two or more po- tential wells which would be degenerate in a pure sy- stem. In this case the quantum-mechanical coherence between the particle’s states in different wells mani- fests itself (the well-known example is the Bloch wave propagation of electrons in crystalline solids). The ba- sic concept introduced to describe this phenomenon is that of a band motion (coherent tunneling) of a par- ticle with a bandwidth � determined by the amplitude of the particle’s resonance transitions between the po- tential minima [6,7]. Particle dynamics in perfect crys- tal at T = 0 presents the simplest case of a band motion. The standard expression for the tunneling amplitude between the two nearest wells is given in the semiclassical approximation (see, e.g., [8], � � 1) � � �2 0 0Z e S� . (1) Here we assume that for the particle with the mass m me�� , where me is the electron mass, zero-point vibrations ZPV around local minima of the crystal potential are small as compared with the interwell separation a (or lattice constant, if there is only one minimum in the unit cell). This condition implies that tunneling splitting of the lowest levels in each well is much less then ZVP frequency � �0 02� , and the lowest states are well separated from the rest of the particle spectrum. The tunneling action S p x0 1 2 � d r r is given by the integral along the optimal path connecting turning points r1 and r2 on different sides of the barrier; Z being the coordination number. Typically, in solids S0 1�� which in fact is already satisfied when the barrier height UB is only few times larger than the ZPV energy [1]. Therefore even for particles with the intermediate mass like muons or muonium atoms the bandwidth turns out to be exponen- tially small. Nevertheless, at T = 0 in a perfect crystal any particle is completely delocalized. At T � 0, however, tunneling occurs on the back- ground of the coupling with the excitations of the me- dium. Since � is so small the interaction of the particle with environmental excitations may easily destroy the coherence and lead to particle localization. The basic characteristic of the particle interactions with the me- dium excitations is frequency � of phase correlations damping at neighboring equivalent positions of the particle. Even at low temperatures � could be as large as �; the temperature raise results in an exponential decrease of the coherent tunneling transition [5,1]. Here one have to distinguish different frequency re- gimes: those modes which have frequencies signifi- cantly larger than � will follow the motion of the par- ticle adiabatically and can at best renormalize �; while those of frequencies of order of � or less can ex- tract energy from the system during the tunneling pro- cess. The latter effect is known as dissipation in quan- tum tunneling [3,9] which causes strong particle localization. Destraction of bandlike propagation and eventual localization of muonium atom in molecular crystals of solid methanes [10] and solid nitrogen [11] due to coupling to molecular rotations at low tempera- tures are typical examples of this kind of effect: inter- action with low-frequency rotational modes causes the crossover from coherent quantum tunneling to inco- herent hopping dynamics at low temperatures. Since � is small with respect to all other energy pa- rameters in a solid, quantum dynamics is extremely sensitive to crystal imperfections. Therefore, localiza- tion of the particle often takes place at a relatively low defect concentration. Until very recently studies of Mu diffusion have fo- cussed on nearly perfect crystals, in which bandlike mo- tion of Mu persists at low temperatures. Crystalline de- fects have been treated mainly as local traps [12] with trapping radii on the order of the lattice constant a. The justification for such an approach was that the charac- teristic energy of the crystalline distortion, U a( ), is usually much less than the characteristic energy of lat- tice vibrations, . Unfortunately, since it does not take the particle bandwidth � into consideration, this com- parison turns out to be irrelevant to the problem of par- ticle dynamics, for which the crucial consideration is that � is usually several orders of magnitude less than U a( ). For example, a typical Mu bandwidth in insula- tors is on the order of � ~ 0.01–0.1 K [1], whereas U(a) could be as large as 10 K. In metals the mismatch is even more drastic: typical values [U a( ) ~ 103 K vs. � ~ 10–4 K] differ by about seven orders of magnitude. Under these circumstances, the influence of crystalline defects extends over distances much larger than a. If the «dis- turbed» regions around defects overlap sufficiently, complete particle localization can result. Tunneling dynamics in cryocrystals: localization and delocalization Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 993 In this lecture we concentrate our attention on Mu quantum diffusion phenomena in cryocrystals with ro- tational degrees of freedom. A more general review of muon and muonium diffusion in a variety of materials may be found elsewhere [1]. 2.1. Destruction of bandlike propagation in orientationally ordered crystals: the two-phonon quantum diffusion regime Studies of the diffusion of hydrogen atoms [15] and � � in metals, as well as Mu diffusion in insulators and semiconductors [1] have convincingly shown the quan- tum mechanical character of the phenomenon, most clearly seen at low temperatures where the particle hop rate �c �1 increases with decreasing temperature T according to the power law � � c T� ��1 , thus manifest- ing the onset of the coherent process. In metals, cou- pling to conduction electrons is the dominant scatte- ring mechanism [16] and causes � � 1. In insulators, where phonon scattering processes prevail, a is pre- dicted [5,6] to be 7 or 9 at low temperatures where the absorption of single phonons shifts the energy of the diffusing particle too much for tunneling to occur and so two-phonon diagrams (which can leave the energy almost unchanged) are expected to dominate. Surpris- ingly, the experimental results on Mu diffusion in ionic insulators [17] indicate that � is generally close to 3; this «universal» power-law behaviour with � � 3 prompted the authors of Ref. l8 to conclude that muonium diffusion is governed by one-phonon scatter- ing. On the other hand, in Ref. 19 it was shown that � � 3 can also be obtained from two-phonon scattering processes if the actual phonon spectrum of the ionic crystal is taken into account; unfortunately that proce- dure requires introduction of adjustable parameters. This basic problem on the validity of the former or the latter remained open until recent results on Mu quan- tum diffusion in solid nitrogen [11], methanes [10], and carbon dioxide [20] presented direct experimental evidence of the dominance of two-phonon scattering mechanism in insulators at low temperatures. In the harmonic approximation, the transport proper- ties of a neutral particle in a simple crystalline insulator (e.g. a monatomic or ionic crystal) depend only on the phonon modes of the lattice. For crystals composed of molecules, two additional contributions enter from the internal vibrational and rotational degrees of freedom of the molecules. Internal vibrations of molecules scarcely change the particle dynamics because of their extremely high frequencies. Molecular rotation, however, is a differ- ent matter. Two extremes are possible: the molecules may rotate almost freely in the crystal or the rotational motion may be severely restricted and hence transformed into torsional excitations (librons). Since typical rotational frequencies are still much higher than the particle band- width, in the first extreme the energy levels in different unit cells are degenerate and therefore particle dynamics remain unperturbed. In the second extreme the anisotropic interaction between molecules (which causes orientational ordering in the first place) changes the crystalline poten- tial so that this degeneracy is lifted. As far as the particle dynamics are concerned, this splitting of the energy levels of adjacent sites acts as an effective disorder, creating the bias �. To demonstrate this, a suitable molecular lattice should be found where (a) this disorder is essentially weak and short range and (b) both extremes can be reached in the accessible temperature range. The simplest molecular solids are the cryocrystals formed by the small lightweight molecules, namely solid H2, D2, CH4, CD4, N2, N2O, CO2 etc. In solid N2O and CO2 the anisotropic part of intermolecular interaction is so strong that the lat- tice keeps its orientational order in the entire solid phase. In solid para-H2 and ortho-D2, by contrast, this interac- tion is so weak that orientational order cannot be reached even at the lowest temperatures. Here we discuss our study of muonium dynamics in solid nitrogen and methanes (CH4 and CD4) which undergo orientational ordering in the solid phase. In solid N2 this transition takes place at T = 35.6 K, in CH4 – at T = 20.4 K while in solid CD4 partial orientational ordering occurs at T = 27 K with a further transition to complete molecular ordering at T = 22.1 K. These crystals show similar nonmonotonic temperature dependences of muonium re- laxation rate T2 1� . Figure 1 presents the temperature de- pendence of the muonium hop rate �c �1 in solid nitrogen. For temperatures T �� (the Debye temperature) quan- tum diffusion is believed [5] to be governed by two-phonon processes, for which �c �1 is given by �c �1 ~ ~ ( ) ( ) , � � � 0 2 2 2 T T � � (2) 994 Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 Vyacheslav G. Storchak Fig.1. Temperature dependence of the muonium hop rate in ultra high purity solid N2. Stars correspond to the com- bined longitudinal field measurements; circles, triangles, diamonds and inverted triangles correspond to transverse field measurements in different samples. where ~�0 is the renormalized bandwidth for Mu dif- fusion and � is the typical difference between energy levels of the particle at adjacent tunneling sites due to static disorder. The-main feature of Eq. (2) is the minimum of �c T( ) at � ~ �( )T . Note that �c T � �1 0 2~ ( ) � � for � � �, (3) whereas � � c T� �1 0 2 2 ~ ( )� � for � �� (4) giving the opposite temperature dependence so that Mu atoms are localized as T � 0. In the T � 0 limit only acoustic phonons are important and �( ) ( )T T� �7 2 (5) The two additional powers of T appear only in the case of muonium tunneling between absolutely equiv- alent sites. In the temperature range 30 K < T < 50 K, the mea- sured Mu hop rate in solid nitrogen exhibits an empirical temperature dependence � � c T� ��1 with � = 7.3(2); since, from Eq. (3), �c T� ��1 1� ( ), we have �( )T T� 7 as expected [Eq. (5)]. This is the first experimental con- firmation of the T �7 dependence of �c �1 predicted by the two-phonon theory of quantum diffusion [5]. Below about 30 K the Mu, hop rate levels off, due to band motion with an estimated [11] renormalized bandwidth of ~�0 ~ 10–2 K [11]. Similar experiments in solid methanes give the following values for muonium bandwidth: about 3 10 2� � K in CH4 and about 10–3 K in CD4. This values for the Mu bandwidth in solid ni- trogen and methanes should be compared with the bandwidth � ~ 10–4 K obtained for the quantum diffusion of 3He atoms in 4He crystals [21]: the qualita- tive similarity of these results suggests a common dy- namical behaviour for light particles in insulators, as opposed to metals, where different scattering mecha- nisms lead to quite different impurity dynamics. Muonium motion in solid nitrogen slows down again below about 20 K, probably due to the orientational ordering of N2 molecules. For T < 18 K the data in Fig. l obey � � � c T� �� 1 0 1 , (6) with = 83 K, �0 1 13 13 6 8 10� �� . ( ) s and � � 6 7 1. ( ). The change in the temperature dependence of the Mu hop rate from a T7 to a T �7 law reflects a cros- sover from Eq. (3) to Eq. (4). Muonium diffusion in solid methane isotopes and solid carbon dioxide exhi- bits similar temperature dependences of �c �1 [10,20]. In all four crystals, at low temperatures gradual Mu localization takes place which reflects a suppression of band motion by static disorder introduced by orien- tational ordering. 2.2. Coherent quantum diffusion of muonium atom in highly disordered material: orientational glass To date most of our knowledge on tunneling dy- namics of particles in solids comes from the extensive studies of crystalline or nearly crystalline materials. However, in reality, the crystalline state is the excep- tion rather then the rule. Disorder exists in varying degree, ranging from a few impurities in an otherwise perfect crystalline host to the strongly disordered limit of allows or glassy structures. All the studies on muon and muonium localization so far have been fo- cused on crystals with weak disorder [1]. In this sec- tion we present experimental studies of muonium tun- neling dynamics under conditions of strong disorder in orientational glass [22,23]. The term «orientational glasses» usually refers to randomly diluted (or randomly mixed) molecular crystals. Molecular crystals without such randomness in their chemical constitution undergo an order-disor- der phase transition from the «plastic crystal» phase at high temperatures, where the multipole moments associated with the molecules can rotate more or less freely, into a phase with a long-range orientational or- der at lower temperature (e.g. N2, ortho-H2, CH4, CD4, etc.). This order gets severely disturbed by dilu- tion of the material with atomic species which have no multipole moment (e.g. Ar in N2, Kr in CH4, para-H2 in ortho-H2, etc.); strong enough dilution leads to a new type of phase, where the multipole moments are frozen into random directions. These glass phases are believed to result from the combined effect of the frustration of the highly anisotropic interactions between the molecules (e.g., electrostatic multi- pole-multipole) and the disorder introduced by the random substitution of molecular multipoles by non-interacting shperical atoms or molecules [22]. The frustration in these systems arises from the geo- metrical impossibility of realizing the minimum possi- ble energy configuration for all pairs of neighboring molecular quadrupoles in close 3D lattices, and disor- der simply comes from the replacement of multipole bearing molecules by non-interacting diluents such as Kr in the CH4–Kr system. Although orientational glasses have many common features with structural glasses (like amorphous SiO2) and spin glasses (like CuMn) there is un important dif- ference even in qualitative description of these glass systems. Unlike the canonical spin glasses such as Tunneling dynamics in cryocrystals: localization and delocalization Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 995 CuMn alloys for which the frustration and the disorder go hand-in-hand, the orientational glasses belong to a new class of systems characterized by independent ef- fects of both frustration and disorder. The interaction between two molecules responsible for the orientational ordering (the short range highly anisotropic electric quadrupole-quadrupole or octopole-octopole interac- tion) is explicitly, known. This fact allows one to ex- tract the influence of disorder which can be easily var- ied (or even switched on and off) by changing the temperature and/or composition, allowing a detailed investigation of the effects of strong disorder on quan- tum tunneling of muonium atoms. Since the early heat-capacity measurements [24], it has been known that the specific-heat anomalies in CH4 at the orientational transition vanish if a suffi- cient amount of Kr is added to CH4. It has been estab- lished by heat-capacity, NMR [25] and dielectric tech- niques [26] that as the Kr concentration increases the temperature of orientational transition gradually de- creases. Above the critical concentration (about 25%) ordered phase never forms. Instead, as the tempera- ture goes down the dynamical orientational disorder eventually freezes into a static pattern of randomly oriented octupoles, the orientational glass. Figure 2 shows the temperature dependencies of the muonium hop rate in pure CH4 and CH4 + 25% Kr, ex- tracted in the regime of dynamical averaging using the values of � obtained from the low temperature values of T2 1� [10]. The plateaus in �c T�1( ) (around 45–55 K in pure CH4 and 50–60 K in the mixture) manifest the onset of muonium band motion [1,5]. The bandwidth ~� ~ 10–2 K determined in CH4 + 25% Kr mixture turns out to be remarkably high: it is only slightly less than ~� ~ 3 10 2� � K in pure CH4, about the same as that in pure solid nitrogen [11] and an order of magni- tude higher than ~� ~ 10–3 K in pure solid CD4 [10]. Addition of 16% of Kr to CH4 does not change the bandwidth for Mu tunneling dynamics. This fact sug- gests that substitution of 2 nearest neighbours out of 12 does not destroy the coherence in Mu band regime. This is a remarkable feature never observed in quan- tum diffusion studies so far: presence of impurities in a crystal even on the level of 10–3 is typically enough to destroy the coherent tunneling regime [1,5]. Although addition of 25% of Kr to CH4 does change the muonium bandwidth, in the temperature range be- tween 50 and 60 K Mu atom still exhibits coherent tunneling which means that substitution of 3 nearest neighbours out of 12 still does not destroy the coher- ence. The question why does such a high concentration of foreign atoms fail to destroy coherence in muonium dynamics still remains open. 3. Electron transport via m �SR lonization of matter by high energy charged parti- cle radiation inevitably produces excess electrons and thus may cause electrical breakdown even in wide-gap insulating materials subjected to high electric field. These materials are used in a large number of applica- tions ranging from power generation equipment to mi- croelectronic devices. It is therefore important to understand the transport mechanisms of radiolysis electrons in insulators. In condensed matter, the transport of a charged particle depends upon the adiabaticity of its interac- tion with excitations of the environment. For particles slow enough that electronic excitations are prohib- ited, the general picture depends critically on the in- terplay of two characteristic times. The first repre- sents the typical time that a charged particle spends interacting with a given atom or molecule: � i a/v� , where a is the lattice constant and v is the velocity of the charged particle. The other characteristic time is ��1, where �� is the characteristic phonon energy. Fast particles (�� i �� 1) move through the medium retaining their «bare» identity, whereas charge carri- ers moving so slowly that �� i � 1 are followed «in- stantaneously» by phonon modes and are best thought of as a polaron [27] whose mobility is drastically de- creased. This crossover from the fast (�� i � 1) to the slow (�� i � 1) regime thus leads to a dramatic change in charge transport properties. In different insulators, electron transport is deter- mined by qualitatively different interactions of elec- trons with the medium. Measurements on Ar, Kr and Xe crystals [28,29] show clearly that electron mobili- ties in these solids are comparable to those found in wide-band semiconductors (be ~ 103 cm2 s–1V–1), which encouraged different authors to apply Shockley’s well-known theory [30]. An approxima- 996 Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 Vyacheslav G. Storchak Fig. 2. Temperature dependencies of muonium hop rate in pure solid methane (circles) and solid mixture of CH4 + + 25% Kr (stars). tion in which the free charge carriers are completely delocalized and the electron-phonon interaction is treated as a perturbation gave an adequate description of the observed electron transport. The rather low electron mobilities found in the di- atomic solids of N2, CO, and O2 [31] (be ~ 10–2 – 10–3 cm2s–1V–1 ) suggest that a fundamentally differ- ent mechanism of electron transport occurs in these materials. The localization of excess electrons with the formation of a small polaron [27] due to strong inter- actions with excitations of the medium was proposed to explain such low values of drift mobility. Measurements of electron mobility by time-of-flight (TOF) technique represent a very direct approach to the study of charge transport properties in solids. It should be noted, however, that in such experiments the path length between electrodes is macroscopic (~ 10–2–10–1 cm), making the results highly susceptible to spurious TOF changes if electrons interact with crystalline defects such as impurities, strains, and microcracks. The muon spin rotation technique avoids these difficulties inherent to the traditional TOF technique because the distances involved are much shorter (~ 10–6 – 10–4 cm). In �SR experiments each incoming several-MeV � � leaves behind an ionization track of liberated electrons and ions. Although this circumstance has been disre- garded in a great majority of experimental and theoret- ical studies of condensed matter by � �SR techniques, the liberation of electrons by muon radiolysis is far from a negligible effect — in fact, in some insulators and semiconductors it may determine much of the sub- sequent behaviour of the system. Recent � �SR, experi- ments in liquid helium [32], solid nitrogen [33–36], liquid [37] and solid neon and argon [29,38] have shown that the spatial distribution of the ionization track products is highly anisotropic with respect to the final position of the muon: the � � thermalizes well «downstream» from the end of its track. Some of the excess electrons generated in this track turn out to be mobile enough to reach the thermalized muon and form the hydrogen-like muonium (Mu � � �� e ) atom. The phenomenon of delayed muonium formation (DMF) described above is crucially dependent on electron interaction with the environment through its influence on electron mobility. Thus DMF forms the basis of a new technique [34] for measurements of the electron mobility be in insulators [36–38] and semi- conductors [39–41] on a microscopic scale: be can be estimated whenever one can measure both the charac- teristic time for Mu atom formation and the character- istic distance between the stopped muon and its «last» radiolysis electron. In this section we consider several examples where � �SR techniques allow one to deter- mine whether excess electron in cryocrystals becomes a polaron or occupies the conduction band (in other words, whether electron is localized or delocalized). 3.1. Electron delocalization in solid �-nitrogen Both muonium (Mu) and diamagnetic (D) signals are evident in solid N2 at all temperatures [33]. We found strong correlation between the muonium ampli- tude and the electron mobility in solid nitrogen: both have similar temperature dependences. The straightfor- ward implication is that Mu formation in s-N2 is at least partially due to convergence of the � � and a radiolysis electron. In s-N2 positive charges have been found to be immobile [31], so the e� must move to the � � . Rather strong evidence in support of this picture comes from the electric field dependences of the dia- magnetic and Mu amplitudes (Fig. 3). A positive sign for E signifies that the electric field is applied in the same direction as the initial muon momentum. The re- sults show that, on average, muons thermalize down- stream from the last radiolysis electrons of the muon’s ionization track; in this case a positive E will pull the � � and e� apart, giving rise to an increased D ampli- tude, whereas a negative E will push the � � and e� to- gether. The characteristic muon-electron distance R in solid �-N2 was estimated from these measurements to be about 5 10 6� � [34,35]. Analogous measurements in �-N2 at T = 59 K reveled a much weaker electric field dependence, giving an estimate of the characteristic � �—e� distance of about half that in �-N2 [36]. The characteristic time � for e� transport to the � � can be determined by measurement of the magnetic field de- pendence of the Mu amplitude. Assuming that the muonium formation process is governed by a first-or- der kinetic equation dn t dn t n tMu( ) ( ) ( )� � �� , where � �� 1/ is the characteristic formation rate, the muonium amplitude has been shown to be Tunneling dynamics in cryocrystals: localization and delocalization Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 997 Fig. 3. Electric field dependences of muonium (Mu, circles) and diamagnetic (D, stars) amplitudes in �-N2 at T = 20 K. AMu Mu � � � � �2 2 . (7) For a weak local electric field E the electron mobil- ity be is independent of E and the charge drift velocity v is defined by v b Ee� . (8) In the absence of an applied field, the electric field at a distance r from the muon is E e/ r� º 2, which can be integrated to give an expression for the Mu forma- tion time � � R ebe 3 3 º . (9) Very near the muon, E is large and be is no longer constant; however, Eq. (9) turns out to be a good ap- proximation anyway because � is determined mainly by slow motion at large distances in low electric fields. Expressions (7) and (9) allow one to estimate the electron mobility be . Figure 4 shows the magnetic field dependence of AMu in �-N2 (circles) and in �-N2 (stars). In �-N2 the estimate of the electron mobility from � �SR measure- ments using Eqs. (7) and (9) gives a value of the same order of magnitude as that extracted by time-of-flight technique. The dashed curve shows numerical calcula- tions according to Eqs. (7) and (9) with the values of electron mobility determined from TOF measurements [31] in �-N2. The experiment, however, show that AMu is field independent (i.e. � �� Mu), which means that the Mu formation time is much shorter than expected from TOF measurements. Using Eq. (7) one can estimate a lower limit for the electron mo- bility in �-N2: be � 100 cm2s–1V–1— a value more than 105 times higher than the electron mobility in �-N2. The discrepancy between TOF and � �SR results in �-N2 is probably connected to crystal cracking at the � �— transition of s-N2. The TOF technique [31], which relies on electron drift over the macroscopic dis- tances between electrodes, is inevitably sensitive to crystal imperfections. We claim that the � �SR tech- nique, which involves microscopic characteristic dis- tances, avoids these difficulties. Such a high electron mobility suggests that the electron transport mecha- nism in �-N2 is fundamentally different from that in �-N2. Probably the localization of electrons does not occur in �-N2 and Shockley’s delocalized approxima- tion [30] can be applied. A possible mechanism for electron localization in �-N2 may be interactions with the rotational modes of N2 molecules — a scattering mechanism that is absent in �-N2 due to the orientational ordering of the molecules. 3.2. Electron localization in orientational glass Most of our understanding of electron transport in solids is modelled on nearly perfect crystalline materi- als, but even in this limit disorder plays a crucial role [42]. The most familiar phenomenon governing elec- tron transport in disordered metals is «Anderson local- ization» [4]: introduction of sufficiently strong disor- der into a metallic system causes spatial localization of electron states near the Fermi level and thus drives a transition to an insulating state. In metals, however, electron-electron interactions dramatically modify the density of states at the Fermi level, leading to forma- tion of the Coulomb pseudogap [43]. To observe the effects of disorder on electron transport without the complications of electron-electron interactions, one must therefore study electron dynamics in a disor- dered insulating host [44]. Orientational glasses formed by random mixtures of molecular and atomic species [22] offers a unique opportunity for such studies. One of the best studied orientational glass systems is the N2–Ar mixture [45]. Pure N2 has two low-pressure crystalline forms, the hex- agonal close-packed (hcp) high temperature phase and the cubic Pa3 (fcc) low temperature phase. Despite in- trinsic geometrical frustration, pure N2 undergoes a first-order phase transition to a long-range periodic orientationally ordered �-phase below T�� 35 6. K; the high temperature �-phase is orientationally disordered. Solid (N2)1-xArx is obtained by simply cooling li- quid mixtures, as nitrogen and argon are completely miscible. As the Ar concentration x is increased, the hcp-to-fcc transition temperature decreases. Above the critical Ar concentration xc � 0 23. , the hcp lattice ap- pears to be stable down to T = 0. The dynamical orientational disorder of the high-T phase eventually 998 Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 Vyacheslav G. Storchak Fig. 4. Magnetic field dependences of muonium ampli- tudes in �-N2 at T = 20 K (circles) and in �-N2 at T = 59 K (stars). Smooth curves represent numerical calculations in the framework of a localized electron model (see text). freezes into a static pattern of randomly oriented N2 molecules, the orientational glass [45]. Being a mixture of insulators, the N2–Ar system has a very large energy gap (~ 10 eV), so that even at high temperature the ambient density of free elec- tronic states is exponentially low. Experimental study of electron transport in this system therefore requires that the empty conduction band be «injected» with free carriers, ideally in low enough concentrations that electron-electron interactions can be safely ig- nored. The ionization of molecules and/or atoms by high energy charged particles (e.g. positive muons) offers just such a source of free carriers. Figure 5 depicts the temperature dependences of the asymmetries (amplitudes) of the various signals in solid (N2)1-xArx for x = 0, 0.09, 0.16, and 0.25. At high temperature (above about 40 K), all the mixtures have roughly the same Mu and � D asymmetries as pure N2. At low temperatures, however, adding argon causes dramatic changes. In pure N2 below about 30 K there is a large Mu signal and a small � D signal, indi- cating efficient DMF; as Ar is added there is a pro- gressively larger � D signal, indicating reduced DMF, until at x = 0.25 there is only a small Mu signal. In solid N2 muonium formation has been shown [34, 36] to proceed via two channels: the thermal DMF process outlined above and the epithermal prompt process which takes place prior to the � � thermalization and is therefore independent of temperature, electron mo- bility, etc. The small, temperature independent Mu amplitude in the x = 0.25 sample (see Fig. 5) is the same as the prompt Mu amplitude in pure solid nitro- gen [36], suggesting a complete absence of DMF in the orientational glass. The hypothesis that Mu formation in the x = 0.25 mixture is essentially all via the prompt channel at 20 K is further supported by the observation that AMu and AD do not depend on an externally applied elec- tric field for that, sample, as shown in Fig. 6. Both amplitudes show significant electric field dependence in pure N2 at 20 K, from which the characteristic muon-electron distance R is estimated to be about 50 nm [34,36]; about the same value of R is found in solid Ar, which exhibits almost 100% DMF [38]. The absence of DMF at this length scale at low tempera- ture in the x = 0.25 mixture suggests that electrons are localized in orientational glass [44]. 4. Conclusions Recent studies on both quantum diffusion of muonium atom and electron transport in condensed matter have demonstrated considerable power of muon spin relaxation techniques in determination of the quantum state of these particles. The dynamics of neu- tral and charged particles is basically governed by dif- ferent mechanisms of localization and delocalization. This work was supported by the INTAS Founda- tion, the Royal Society of London, NSF, and NATO. Tunneling dynamics in cryocrystals: localization and delocalization Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 999 Fig. 5. Temperature dependences of muonium (top, H � 5 G) and diamagnetic (bottom, H � 100 G) signal ampli- tudes in pure solid nitrogen (squares) and solid (N2)1-xArx (circles: x = 0.25; triangles: x = 0.16; stars: x = 0.09). Fig. 6. Electric field dependences of 2AMu and AD in pure solid nitrogen (crosses and stars, respectively) and in solid 75% N2 + 25% Ar (circles and triangles, respectively) in a transverse magnetic field H = 36 G at T = 20 K. The muoniuin amplitudes are doubled to compensate for the 50% depolarization of Mu by hyperfine oscillations [2]. 1. V.G. Storchak and N.V. Prokof’ev, Rev. Mod. Phys. 70, 929 (1998). 2. A. Schenck, Muon Spin Rotation: Principles and Applications in Solid State Physics, Adam Hilger, Bristol (1986); S.F.J. Cox, J. Phys. C20, 3187 (1987); J.H. Brewer, Muon Spin Rotation/Relaxation/Resonance, in: Encyclopedia of Applied Physics, G.L. Trigg (ed.), VCH, New York (1995). 3. A.O. Caldeira and A.J. Leggett, Phys. Rev. Lett. 46, 211 (1981); Ann. Phys. (N.Y.) 149, 374 (1983). 4. P.W. Anderson, Phys. Rev. 109, 1492 (1958). 5. Yu.M. Kagan and N.V. Prokof’ev, Quantum Tunneling Diffusion in Solids, in: Quantum Tunneling in Condensed Media, A.J. Leggett and Yu.M. Kagan (eds.) North-Holland (1992), p.37. 6. A.F. Andreev and I.M. Lifshitz, Zh. Eksp. Teor. Fiz. 56, 2057 (1969) [Sov. Phys. JETP 29, 1107 (1969)]. 7. R.A. Guyer and L.I. Zane, Phys. Rev. 188, 445 (1969). 8. L.D. Landau and E.M. Lifshitz, Quantum Mechanics. Non Relativistic Theory, Nauka, Moscow (1974). 9. A.J. Leggett et. al., Rev. Mod. Phys. 59, 1 (1987). 10. V.G. Storchak et. al., Phys. Rev. Lett. 82, 2729 (1999). 11. V.G. Storchak, J.H. Brewer, W.N. Hardy, S.R. Kreitman, and G.D. Morris, Phys. Rev. Lett. 72, 3056 (1994). 12. R. Kadono in: Perspectives in Meson Science, T. Yama- zaki, K. Nakai, and K. Nagamine (eds.) North-Holland, Amsterdam (1992), p. 113. 13. S.F.J. Cox and D.S. Sivia, Hyperfine Int. 87, 971 (1994). 14. C.P. Slichter, Principles of Magnetic Resonance, Springer-Verlag, Berlin (1980). 15. D. Steinbinder et.al., Europhys. Lett. 6, 535 (1988). 16. J. Kondo, Physica B125, 279 (1984); B126, 377 (1984); K. Yamada, Prog. Theor. Phys. 72, 195 (1984). 17. R. Kadono, Hyperfine Int. 64, 615 (1990). 18. P.C.E. Stamp and C. Zhang, Phys. Rev. Lett. 66, 1902 (1991). 19. Yu.M. Kagan and N.V. Prokof’ev, Phys. Lett. A150, 320 (1990). 20. D.G. Eshchenko, V.G. Storchak, J.H. Brewer, S.P. Cottrell, S. F.Y. Cox, E. Karlsson, and Waeppling, Fiz. Nizk. Temp. 27, 1153 (2001) [Low Temp. Phys. 27, 854 (2001)]. 21. M.G. Richards et. al., J. Low Temp. Phys. 24, 1 (1976); V.A. Mikheev, B.N. Esel’son, V.N. Grigor’ev and N.P. Mikihn, Fiz. Nizk. Temp. 3, 387 (1977) [Sov. J. Low. Temp. Phys. 3 (1977)]. 22. U.T. Hochli, K. Knorr, and A. Loidl, Adv. Phys. 39, 405 (1990). 23. K. Binder and J.D. Reger, Adv. Phys. 41, 547 (1992). 24. A. Eucken and H. Veith, Z. Phys. Chem. B34, 275 (1936). 25. P. Calvani et. al., Phys. Lett. A86, 490 (1981). 26. R. Bohmer and A. Loidl, Z. Phys. B80, 139 (1990). 27. T. Holstein, Ann. Phys. 8, 343 (1959). 28. L.S. Miller, S. Howe, and W.E. Spear, Phys. Rev. 166, 871 (1968). 29. D.G. Eshchenko, V.G. Storchak, J.H. Brewer et. al., Phys. Rev. B66, 035105 (2002). 30. W. Shockley, Bell System Tech. J. 30, 900 (1951). 31. R.J. Loveland, P.L. Le Comber, and W.E. Spear, Phys. Rev. B6, 3121 (1972). 32. E. Krasnoperov, E. Meilikhov, R. Abela, D. Herlach, E. Morenzoni, F. Gydax, A. Schenck, and D.G. Eshchenko, Phys. Rev. Lett. 69, 1560 (1992). 33. V. Storchak, J.H. Brewer, and G.D. Morris, Phys. Lett. A193, 199 (1994). 34. V. Storchak, J.H. Brewer, and G.D. Morris, Phys. Rev. Lett. 75, 2384 (1995). 35. V. Storchak, J.H. Brewer, and G.D. Morris, Phil. Mag. 72, 241 (1995). 36. V. Storchak, J.H. Brewer, G.D. Morris, D.J. Arseneau, and M. Senba, Phys. Rev. B59, 10559 (1999). 37. V. Storchak, J.H. Brewer, and G.D. Morris, Phys. Rev. Lett. 76, 2969 (1996). 38. V. Storchak, J.H. Brewer, and D.G. Eshchenko, Appl. Mag. Reson. 13, 15 (1997). 39. V.G. Storchak et. al., Phys. Rev. Lett. 78, 2835 (1997). 40. D.G. Eshchenko, V.G. Storchak, and G.D. Morris, Phys. Lett. A264, 226 (1999). 41. D.G. Eshchenko, V.G. Storchak, J.H. Brewer, and R.L. Lichti, Phys. Rev. Lett., in press. 42. N.F. Mott and E.A. Davis, Electron Processes in Non-Crystalline Materials, Clarendon Press, Oxford (1979). 43. N.F. Mott, Metal-Insulator Transitions, Taylor and Francis, London (1974). 44. V.G. Storchak, D.G. Eshchenko, J.H. Brewer, G.D. Morris, S.P. Cottrell, and S.F.J. Cox, Phys. Rev. Lett. 85, 166 (2000). 45. J.A. Hamida, E.B. Genio, and N.S. Sullivan, J. Low Temp. Phys. 103, 49 (1996). 1000 Fizika Nizkikh Temperatur, 2003, v. 29, Nos. 9/10 Vyacheslav G. Storchak

Tunneling dynamics in cryocrystals: localization and delocalization

Репозитарії

Схожі ресурси