ELECTRON TRANSPORT PHENOMENA IN LOW-DIMENSIONAL SYSTEMS

1. ELECTRON TRANSPORT PHENOMENA IN LOW-DIMENSIONAL SYSTEMS Baku State University Baku, Azerbaijan Sophia R. Figarova 2012

2. CONTENTS •Introduction: a little of history. • Different low-dimensional systems. Density of states . • Basic transport phenomena in low-dimensional systems. •Electron transport phenomena in superlattices in a nonquantized magnetic field. •Electron transport phenomena in superlattices in a quantized magnetic field taking into account spin splitting. •Conclusions

3. In solids, sizes of which are comparable with • mean free path, • de Broglie wavelength, • coherence length, • localization length, there appear new physical properties, called size effects, caused mainly by quantum effects. Among these effects are • oscillations of conductivity, • quantum Hall effect, • resonant tunneling, • negative differential conductivity, • giant resistance, • spin Hall effects,

4. which can show itself in specially fabricated nano-systems (from 1 to 100nm), for example in • quantum films, • quantum wires, • quantum dots, • heterojunctions, • superlattices. • Not so many years passed since Leo Esaki was awarded to the Nobel Prize for discovery of new effects in superlattices. In fact, the work in Nanophysics has begun since the Eighties of the XX century. The outstanding achievements in this field were awarded to the Nobel prizes in Physics

5. A LITTLE OF HISTORY Nobel prizes in nano - physics • 1973 - The Nobel Prize in Physics 1973 was divided, one half jointly to Leo Esaki and Ivar Giaever "for their experimental discoveries regarding tunneling phenomena in effects in quantum wells and superlatticessemiconductors". • 1985 - The Nobel Prize in Physics 1985 was awarded to Klaus von Klitzing "for the discovery of the quantized Hall effect". • 1986 - The Nobel Prize in Physics 1986 was divided, one half awarded to Ernst Ruska "for his fundamental work in electron optics, and for the design of the first electron microscope", the other half jointly to Gerd Binnig and Heinrich Rohrer "for their design of the scanning tunneling microscope".

6. 1998 - The Nobel Prize in Physics 1998 was awarded jointly to Robert B. Laughlin, Horst L. Störmer and Daniel C. Tsui "for their discovery of a new form of quantum fluid with fractionally charged excitations". • 2000 - The Nobel Prize in Physics 2000 was awarded "for basic work on information and communication technology" with one half jointly to Zhores I. Alferov and Herbert Kroemer "for developing semiconductor heterostructures used in high-speed- and opto-electronics" and the other half to Jack S. Kilby "for his part in the invention of the integrated circuit". • 2007 - The Nobel Prize in Physics 2007 was awarded jointly to Albert Fert and Peter Grünberg "for the discovery of Giant Magnetoresistance“. • 2010 – The Nobel Prize in Physics 2010 was awarded jointly to Andre Geim and Konstantin Novoselov "for groundbreaking experiments regarding the two-dimensional material graphene“.

7. Experimental observation of size effects is possible thanks to development of new technological methods such as molecular beam epitaxy. • However, in far 1959 at annual meeting of the American Physical Society theorist Richard Feynman has given well-known, legendary lecture entitled “ There is Plenty of Room at the Bottom: Invitation to enter a new field of physics”, where he predicted Nano-Physics. • In March of 1959, Richard Feynman challenged his listeners to build “Computers with wires no wider than 100 atoms, a microscope that could view individual atoms, machines that could manipulate atoms 1 by 1.”This assumption has been realized in the creation of scanning electron microscopes, which allows one to study microscopic objects and purposefully manipulate these objects.

8. Over the past two decades in Physics of low-dimensional systems a number of great discoveries was made. Let us name major of them: weak and strong localization of guantum states; quantization of conductivity in ballistic transport; Coulomb blockade of tunneling in nanostructures; spin Hall effect. The behavior of charge carriers in low-dimensional systems is determined by the following basic phenomena: • Quantum confinement • Ballistic transport of charge carriers • Tunneling of charge carriers • Spin effects

9. THREE DIMENSIONAL (3D) SYSTEMS The energy: The density of states therewith is proportional to the square root of energy: Bulk samples, their energy diagrams and density of states

10. or QUANTUM CONFINEMENT Quantum confinement arises if the free motion of electrons in one of the directions becomes confined by potential barriers.-Transport of electrons can realize parallel and perpendicular to the potential barriers. In the case of the motion of carriers along the barriers ballistic transport is dominant effects. The passage of carriers through the barriers takes place via tunneling. Size quantum effect in infinitely deep potential well. The wave fun-ction is standing ones and equal to: The energy spectrum is quantized and has the form: Potential well and electron wave functions in it The confinement of the motion of charge carriers, leading to nonzero minimum energy and discreteness of energy spect-rum, is called the quantum confinement.

11. TWO-DIMENSIONAL (2D) SYSTEMS Quantum films– d ~ are two-dimensional (2D) structures, in which quantum confinement acts only in one direction-the direction perpendicular to the film (the z-direction). The film thickness is of the order of de Broglie wave length. Charge carriers can freely move in the xy - plane. Their energy equals: In the k-space, an energy diagram of a quantum film represents a family of parabolic bands, which, overlapping, form subbands. The dependence of the electron density of states on energy in a quantum film has a step-like character:

12. Quantum films, their energy diagrams and density of states Density of state as a function of film thickness

13. ONE DIMENSIONAL (1D) SYSTEMS Quantum wiresare one-dimensional (1D) structures. Charge carriers can freely move only in one direction-along the wire axis. The energy spectrum in this case has the form: The density of states in a quantum wire is inversely proportional to the square root of energy -1/2: Quantum wires, their energy diagrams and density of states

14. Quantum films in a quantized magnetic field – If a quantized magnetic field is directed along the z-axis of a size-quantized film, the energy spectrum becomes completely discrete: ZERO DIMENSIONAL (0D) SYSTEMS Quantum dotsare (0D) structures, in which the motion of charge carriers is confined in all three directions. In each of these directions electron energy turns out to be quantized in accordance with the formula: The density of states represents a set of sharp peaks: Quantum dots, their energy diagrams and density of states

15. SUPERLATTICES Superlattices are solid periodic structures in which apart with the usual potential of the lattice, there is additional potential. The semiconductor superlattice is made of two layers of semiconductors with different band gap , which thickness is a few nm. In this case the superlattice can be considered as a periodic system of quantum wells, separated by narrow barrier. Additional periodicity leads to the fact that the energy spectrum component, connected with the motion of an electron along this axis, represents a system of narrow strips-minibands. Such systems have the very strong anisotropy of the energy spectrum, at which the motion of electrons in the layer plane is free and is described in the effective mass approximation. However in the direction perpendicular to the layer plane, the motion of electrons is strongly hindered and is described by strong coupling approximation. An electron gas in this direction can be described by a cosinusoidal dispersion law. In a whole, the energy spectrum of charge carriers can be written as: where is ithe miniband number, iis its width. In this case the effective mass in direction perpendicular to the layer plane can change sign for electrons with the wave vector kz=  / 2d :

16. Superlattice energy diagrams with minibands  = a,b The density of states in a superlattice has the following form: (- z) is a Heaviside function. It is seen that the density of state has a step-like character and if the energy of electron is large than mini-band width the density of state does not depend on energy, the caracteristic for two-dimensional systems. Density of states in a superlattice vs. energy

17. Dichalcogenides of transition metals of the NbSe2, TaS2 type, the III-V layered semiconductor compounds, intercalated compounds (synthetic metals) can be considered as natural superlattices and the electron gas in these systems are described by a cosinusoidal dispersion law. For narrow minibands wave functions of electrons along axis of superlattices are overlapped and electron spectrum consists of discrete levels. The electron gas behaves itself as a two-dimebsional gas. Semiconductor heterostructureAlGaAs Conduction band around a heterostructure between n-AlGaAs and undoped GaAs. Electrons are separated from their donors to form a two-dimensional electron gas The peculiarity of transport phenomena in superlattices is the fact that there takes place longitudinal and transverse transport of charge carriers. In additional depending on the degree of filling of the mini-cand the electron gas can be either two-dimensional or quasi-two-dimensional.

18. Basic type of ideal nanostructures DOS in Low-Dimensional Electron Systems

19. We in detail consider the density of states in low dimensional structures since its behavior essentially influences many physical characteristics. Moreover, there are physical quantities which are directly proportional to the density of states and their behavior is completely dictated by it, e.g. Entropy, Heat capacity, Thermopower and Magnetization. As an example, consider a superlattice in a quantized magnetic field. Entropy and electron heat capacity. In the case of a degenerate electron gas, these quantities depend on the density of states at the Fermi surface and have the form:

20. 2.5 30 2.0 S2D ∕ S3D 20 S2D∕ S3D 1.5 1.0 10 0.5 0.0 0 2 4 6 8 10 12 0 1 2 3 4 B (T) Z0 Fig.1. Ratio of entropy of a two-dimensional degenerate electron gas to entropy of a three-dimensional gas vs. a magnetic field. Fig.2. Ratio of entropy of a two-dimensional degenerate electron gas to entropy of a three-dimensional gas vs. the degree of band filling. The parametersa = 10nm, m = 0.067m0, 0= 0.1meV for the GaxAl1-xAssuper-lattices are used. From the Figures it is seen that depending on the magnetic field, entropy oscillates and in the quantum limit, the entropy of a two-dimensional electron gas S becomes larger than the entropy of a three-dimensional electron gas: This fact, apparently, is connected by the fact that the radius of a cyclotron orbit in a two-dimensional case is larger (since m>m ). Therefore cyclotron orbits are crossed and the confusion in a two dimensional gas becomes larger [Askerov B.M., S.R.Figarova S.R., Mahmudov M.M., Figarov V.R., Jap. J. Appl. Phys., 50, 05FE10, 2011].

21. 50 25 M (Z0),(А/m) 0 25 50 1 1.5 2 2.5 3 3.5 Z0 Thermo-electrical power is determined by entropy from the formula  = - S/en, the behaviour of the thermo electrical power is completely determined by the density of states . Thermo-electrical power of quantum films and superlattices oscillates in a strong magnetic field [B.M.Askerov, S.R.Figarova, V.R.Figarov, Nanotechnology, 18, 424024, 2007]. Such a behaviour of thermo-electrical power was experimentally confirmed in superlattices of the GaAlAs type at low temperatures [W. Zawadzki, Physica B+C, 127, 388, 1984]. Magnetization in the case of a degenerate electron gas is directly proportional to the density of states, too. Fig.1.7.Diamagnetic magnetization of a degenerate quasi-two- dimensional electron gas versus the magnetic field at the following parameters:0 = 1meV, a = 10nm, n = 1023m-3. Fig. 1.6.Diamagnetic magnetization of a degenerate quasi-two- dimensional electron gas versus the degree of band filling at the following parameters:0 = 1meV, a = 10nm, n = 1023m-3.

22. From the Figures it is seen that diamagnetic magnetization of a quasi-two-dimensional electron gas depending on the degree of band filling changes the sign and in the two-dimensional case becomes positive. In a magnetic field magnetization oscillates. Such a behaviour of magnetization in superlattices is explained by existence of a negative-effective-mass region in the mini-band. Therewith a conduction electron moves in the direction opposite to the free electron motion. In a magnetic field the conduction electron rotates in the opposite direction; this fact leads to the positive magnetization [B.M.Askerov, S.R.Figarova, M.M.Mahmudov, V.R.Figarov, Proc. Royal Soc. A, 464, 3213, 2008]. The sign change of magnetization in superlattices was observed in the experimental work [S.D.Prado, M.A.de Aguiar, Phys. Rev., E, 54, 1369, 1996]. The sign change essentially influences optical phenomena.

23. . BALLISTIC TRANSPORT OF CHARGE CARRIERS In low-dimensional structures the size of which is less than the mean free path, transport of charge carriers occurs without the carrier scattering. Such a transport is called the ballistic transport. Main effects, related to ballistic transport, are determined by the ratio between structure sizes and mean free path at the elastic and the inelastic scattering of carriers, the phase coherence length and Fermi wavelength.

24. If the structure size is comparable with these lengths, quantization of energy becomes essential. The ideal ballistic transport of charge carriers in nanostructures is characterized by the universal ballistic conductance, which is independent of the material type and is determined only by fundamental constants. The conductivity is quantized in terms of 2e2/h. This fact is observed at quantum point contacts. In the conductivity curve there appear steps (see Fig). As the electron motion becomes coherent, its wave function conserves its phase. Therewith various interference effects arise. Conductance at ballistic transport: a - scheme of quantum point contacts, b - conductance.

25. TUNNELING OF CHARGE CARRIERSIN LOW-DIMENSIONAL SYSTEMS Tunneling means transport of particles through the region, confined by a potential barrier, the height of which is larger than the total energy of the given particles. Such an effect is impossible from the point of view of classical mechanics, however it takes place for quantum particles. The interaction of quantum particles with various potential barriers was illustrated in the Figure.

26. Existence of the wave having passed through the barrier, corresponding to a quantum particle with energy that less than the barrier height is called the tunneling effect. In low-dimensional structures tunneling has specific features, the fact which distinguishes it from effects in bulk systems. One of these features is connected with the discrete nature of the charge carriers and is called “single-electron tunneling” [Tinkham Am.J.Phys., 1996, N 64, p.343]. Current–voltage characteristic (CVC) at Coulomb blockade

27. Another feature, determined by discreteness of energy states of charge carriers in semiconductor nano-structures is called resonant tunneling. Resonant tunneling occurs if the following conditions are satisfied: 1. de Broglie wavelength should be comparable with the width of the quantum well, 2. the free mean path should be large and the electron is scattered specularly at the edges of the quantum well, 3. electron energy should coincide with the energy of quantum levels in the well. In the current-voltage characteristics (CVC) there appears the region with a negative differential resistance, which promotes to the generation of energy and leads to light amplification. An increase in the tunneling current occurs if the Fermi level coincides with the discrete level of the quantum well. The CVC for the resonant tunneling is shown in the figure below. [Chang L.L., Esaki L., Tsu R., Appl. Phys. Lett., 1974, v.24, p.593]

30. SPIN EFFECTS Taking into account spin of charge carrier in low-dimensional structures leads to new features of transport. The spin effects in low-dimensional systems manifest itself through Hall effect and magnetoresistance. In nonmagnetic materials spin effects represent spin splitting of energy levels in a magnetic field and Rashba spin-orbital splitting. In these cases splitting of energy levels schematically has the shape:

31. heregis the spin splitting factor,  = e/2m is the Bohr magneton, • = 1/2 is the electron spin quantum number, В is the magnetic field induction. . Each Landau level is split into two sublevels. The magnitude ofg-factor depends on the band gap widthg, spin-orbital interaction,number of the Landau level and the magnetic field magnitude. In the Hamiltonian additional terms appear: here=3x10-11eVm is theRashba constant, p is the impulse operatorin the confinement direction. In the energy spectrum, there appears a term proportional to the wave vector. For a two-dimensional electron gas, placed in a perpendicular magnetic field, the energy spectrum with regard to these two mechanisms has the form: here

32. Spin effects in magnetic materials arise if there is the spin misbalance of population of the Fermi level. Such a misbalance is presented in ferromagnetic materials, where the densities of vacant states for electrons with different spins are identical, however these states are distinguished by energy, as it is schematically shown in Fig. • Two main transport effects, namely giant magnetoresistance and tunneling magnetoresistance are connected with electron spin in low-dimensional systems. • Besides in low-dimensional systems owing to spin-orbital interaction of the Rashba type, Hall spin effect occurs.

33. The spin-Hall effect • It turns out that the RashbaHamiltonian gives rise to a pure transverse spin current in response to a charge current • 2DEG • The associated spin-Hall conductivity has a universal value in the 2D plane, and is of much interest to spintronics. Transistors

34. SOME DEVICES BASED ON CONSIDERED EFFECTSIN LOW-DIMENSIONAL SYSTEMS • MOS (metal-oxide-semiconductor) field transistors are based on quantum confinement. • Quantum interference transistors are based on interference of electron waves and ballistic transport of current carriers. • Resonant-tunnel diodes are based on resonant tunneling

35. ELECTRON TRANSPORT PHENOMENA IN SUPERLATTICES IN A NONQUANTIZED MAGNETIC FIELD • In superlattices, anisotropy of the structure, the energy spectrum of conduction electrons and the scattering mechanisms lead to fundamentally new phenomena. For example: • resistance oscillations depending of the orientation of the magnetic field • negative magnetoresistance, if a magnetic field is situated in the layer plane [A.A.Bykov, G.M.Gusev, J.R.Leite, A.K.Bakarov, A.V.Goran, V.M.Kudryashev, A.I.Toropov. Phys. Rev. B, 65, 035302, 2001] and perpendicular to it [N.M.Sotomayor G.M.Gusev, J.R.Leite, A.A.Bykov, A.K.Kalagin, V.M.Kudryashev, A.I.Toropov, Phys. Rev. B, 70, 235326, 2004]. Such a behaviour of magnetoresistance was usually associated with with strong scattering of electrons by inhomogeneities. However, as we demonstrate, main causes of these special effects in superlattices are form of the energy spectrum and quantum confinement. • To construct a theory of electron transport phenomena first we should determine the relaxation time in superlattices.

36. Relaxation time at scattering of current carriers by phonons and impurity ions For a cosinusoidal energy spectrum: we calculated the relaxation times at scattering by different types of phonons and impurity ions. At scattering by different types of phonons, the relaxation time can be generalized and written as [B.M.Askerov, B.I.Kuliev, S.R.Figarova, I.R.Gadirova, J. Phys.: Cond. Matt., 7, 843, 1995]: where =(, II) are transverse and longitudinal components of the relaxation time, k are transverse and longitudinal components of the wave vector. At low temperatures the current carrier scattering by impurity ions is one of dominant scattering mechanisms. At the weak screening the Coulomb potential of impurity ions for components of the inverse relaxation time tensor we have [B.M.Askerov, G.I.Guseynov, V.R.Figarov, S.R.Figarova, Physics of the Solid State, 50, 780, 2008]: In the case of the strong screening we have:

37. Galvanomagnetic effects in superlattices Due to the strong anisotropy of the energy spectrum in quasi-two-dimensional systems, the character of the motion of charge carriers parallel and perpendicular to the layers is essentially distinct. An external magnetic field binds the current carrier motion in the layer plane and in the direction perpendicular to it. In connection with this fact, transport phenomena can be divided into two classes: longitudinal and transverse ones. The magnetoresistance and the Hall coefficient strongly depend on orientation of the magnetic field. We consider a two direction of the magnetic field: a)the magnetic field is perpendicular to the layer plane, b) the magnetic field is in the layer plane. Geometry of the problem

38. For the first geometry of the problem, Hall coefficientRand specific resistance in the layer planeare expressed through the components of the galvanomagnetic tensor as follows: For the second geometry of the problem, Hall coefficientRII(Ez=RjxB) and specific resistancein a magnetic field situated along the layer plane are determined by the formulas: where components of the electric conductivity tensorikare given by the following expressions:

39. where where and the angle brackets denote and the angle brackets denote if BIIoz ifBoz

40. where the plus sign corresponds to the case when the magnetic field is perpendicular to the layer plane of a superlattice, and the minus sign does when it is parallel to the layer plane. The concentration, entering in the expression of Hall coefficient is not the full one, but it is the effective concentration of current carriers: i.е. Hall coefficient in the two-dimensional case depends only on parameters of the superlattice. Separately consider cases of a two-dimensional gas, if the Fermi surface is open-the corrugated cylinder and a quasi-two-dimensional gas with the closed Fermi surface of a shape like a rugby ball Hall coefficient Hall coefficient of a two-dimensional electron gas (the open Fermi surface) is determined only by the effective concentration of charge carriers with the formula:

41. Depending on the geometry of the problem Hall coefficient changes its sign. When a magnetic field is situated in the layer plane, Hall coefficient of a two-dimensional electron gas is positive. From the dependence of Hall coefficient of a quasi-two-dimensional electron gas (the closed Fermi surface) on the degree of band filling (Fig.) it is seen that Hall coefficient of a quasi-two-dimensional electron gas can be both positive and negative . Such a behaviour of Hall coefficient is connected with the sign of the effective mass [Figarova S.R., Figarov V.R. , Phil. Mag. Lett., 2007, v.87, p.373-378].The positive sign of Hall coefficient in quasi-two-dimensional electron systems with the cosinusoidal dispersion law is due to the existence in the electron miniband a region with the negative effective mass. To the existence of regions of the negative effective mass in the superlattice miniband, it was pointed out in the work [Yu.A.Romanov, Physics of the Solid State 45, pp. 559–565 (2003)]. Singularity in the behavior of Hall coefficient takes place at z= /2.

42. TMR of a quasi-two-dimensional electron gas vs. the degree of band filling Z0 in a strong magnetic field perpendicular to the layer plane at phonon scattering (B)/(0) vs. the ratio between the miniband width and Fermi level k = 20/ in a strong magnetic field at impurity ion scattering. The two-dimensional electron gas Transverse magnetoresistance (TMR) in a perpendicular magnetic field,B IIoz Magnetoresistance is change in resistance in a magnetic field. Positive magne-toresistance corresponds to an increase in resistance, and negative one does to its decrease. When the direction of an external magnetic field is perpendicular to the current the magnetoresistance is called transverse magnetoresistance. From the Figures it follows that for a magnetic field directed perpendicular to the layer plane in a strong magnetic field TMR of a two-dimensional electron gas is negative for scattering by phonons and impurity ions. Whereas for the quasi-two-dimensional case depending on the scattering mechanism and the degree of band filling, TMR can become positive and negative. In a weak magnetic field TMR is positive.

43. TMR of a quasi-two-dimensional electron gas vs. the degree of band filling Z0 in a weak magnetic field perpendicular to the layer plane at impurity ions (r0 /a=5). Quasi-two-dimensional electron gas. TMR of a quasi-two-dimensional electron gas vs. the degree of band filling Z0 in a strong magnetic field perpendicular to the layer plane at impurity ions (r0 /a=5). Quasi-two-dimensional electron gas. Transverse magnetoresistance in a magnetic fieldsituated in the layer plane, B II oy With such a geometry of the problem, in contrast to preceding, in a strong magnetic field, TMR of a two-dimensional electron gas is positive, and in a weak one, TMR is negative. [S.R.Figarova, V.R.Figarov, Euro Phys. Lett., 89, 37004, 2010].

44. TMR vs. the magnetic field parallel to the layer plane0 for a quasi-two-dimensional electron gas. TMR vs. the degree of band filling Z0 in the strong magnetic field parallel to the layer plane(0 =6) for quasi-two-dimensional electron gas. TMR vs. the degree of band filling Z0 in the weak magnetic field parallel to the layer plane(0 =0.1) for quasi-two-dimensional electron gas. From the Figures it is seen that transverse magnetoresistance of a quasi-two-dimensional electron gas in the magnetic field parallel to the layer plane is negative in the weak field and is positive in the strong field. Therefore changing the direction and magnitude of the magnetic field, one can change the TMR sign. The fact that TMR is positive in a strong magnetic field parallel to the layer plane is explained by the fact that conduction electrons moving in the z - direction become localized, and resistance grows. Negative TMR in superlattices was experimentally revealed in the work [D.N.Bose, S.Pal, Phys. Rev. B 63, 235321, (2001)] where negative magnetoresistance in the magnetic field in the layer planewas observed at 10K for fields of 0.4T inGaTelayered semiconductors.

45. Motion of an electron in an electrical field: in the k - space (a), in the coordinate space (b). Electron effective mass and dynamics in the superlattice Peculiarities in the behaviour of galvanomagnetic phenomena in superlattices are connected with the effective mass in the direction perpendicular to the layer plane (which can takes negatives values): and the electron dynamics in the superlattice (see Fig.). We have the Bloch oscillations, whose frequency lies in the terahertz range. When the electric field is applied, an electron begins to accelerate in the field direction. If the crystal is ideal (there are no defects), then action of the force, the quasi-impulse begins to grow until it does not reach kx=/a. Note that the electron effective mass becomes negative as it approaches the value of kx=/a . This means that in the coordinate space an electron, going from the О point, at first accelerates, then slows down, when it approaches the А point and finally again begin accelerating, but only in the opposite direction (moving to the point В), although the direction and magnitude of the external force conserve invariable. At kx=0the electron again turns to be at rest. Therefore, under influence of an external field, the electron executes a jump-like motion along the kx -axis and vibrate in the confined section of the Х-axis in the coordinate space with the amplitude A=/2eE and frequency =eEa/h( is the energy band width, a is the lattice period).

46. ELECTRON TRANSPORT PHENOMENA IN SUPERLATTICES IN A QUANTIZED MAGNETIC FIELD TAKING INTO ACCOUNT SPIN SPLITTING • A strong magnetic field, perpendicular to the two dimensional layer, quantizes the motion of current carriers in the layer plane and leads to the following experimentally observed effects: • semimetalic-to-semiconductor transition in a superlattice in a quantized magnetic field [N.J.Kawai, L.L.Chang, G.A.Sai-Halasz, C.A.Chang, L.Esaki, Appl. Phys. Lett., 36, 369, 1980]. • conductivity oscillations. The oscillation period is determined by the magnetic length and superlattice constant perpendicular to the layer plane, [B.Laikhtman, D.Menashe,Phys. Rev. B, 52, 8974, 1995]. • unusually sharp growth of resistance with increasing the magnetic field in alternating layers of GaAs and AlGaAs [V.Renard, Z.D.Kvon, G.M.Gusev, J.C.Portal,, Phys. Rev. B, 70, 033303, 2004] and in the semiconductor layer of InSb [S.A.Solin, D.R.Hines, A.C.H.Rowe, J.S.Tsai, Yu.A.Pashkin, S.J.Chung, N.Goel, M.B.Santos, Appl. Phys. Lett., 80, 4012, 2002]. • existence of a region of negative differential conductivity in the superlattice at room temperature (see, e.g. [Estibals O., Kvon Z.D., Gusev G.M., Arnaud G., Portal J.C. Physica E, 2004, v.22, 446, 2004-449]). Negative differential conductivity in a solid is caused by the negative effective mass and Bloch oscillations.

47. existenceof the vertical magnetoresistance in a magnetic field, directed perpendicular to the layer [Yu.A.Pusep, G.M.Gusev, A.J.Chiquito, S.S.Sokolov, A.K.Bakarov, A.A.Toropov, J.R.Leite, Phys. Rev. B, 63, 165307, 2001] • linear growth of magnetoresistance in a superlattice with the magnetic field, so-called the Kapitsa effect [P.V. Gorskii, Semiconductors, 38, 830, 2004]. • in a strong magnetic field of the order of 30 Т there are observed maxima and minima in magnetoresistance in GaAs/AlGaAs structure [M.V. Vakunin, G.A. Al`shanskii, Yu.G. Arapov, V.N. Neverov, G.L. Kharus, N.G. Shelushinina, B.N. Zvonkov, E.A. Uskova, A.deVisser, L.Ponomarenko, Semiconductors, 39, 107, 2005]. This fact is explained by change in the density of states at the Fermi level owing to spin splitting of energy. • In low-dimensional systems, placed in a magnetic field, spin effects appear. Their main characteristic is magnetoresistance. Because of this fact, we theoretically studied resistance in ideal superlattices with the cosinusoidal dispersion law in a strong magnetic field taking into account spin splitting. • As known, transport phenomena are closely connected with the density of states. Therefore at first consider the density of states of a quasi-two-dimensional electron gas taking into account spin splitting in a quantized magnetic field.

48. Density of statesin a quantized magnetic field A strong magnetic field, parallel to the z-axis, quantizes the electron motion in the layer plane and removes the spin degeneracy; the energy spectrum has the form: where g* is the factor of spin splitting of the electron energy. In the energy spectrum, an additional term, connected with spin splitting, appears. Each Landau level is split into two spin sublevels. The density of states has the form: where is the magnetic length.

49. From the Formula it is seen that the density of states has a singularity. The density of states significantly depends on the ratio between the Fermi level and miniband width. In a two-dimensional electron gas (>20) there are oscillations of the density of states, characteristic for two-dimensional electron systems in a strong magnetic field [T.Ando, A.B. Fowler, F.Stern, Review of Modern Physics, 54, 437 (1982)], which vanish in the quasi-two-dimensional case (<20). Besides, from Figure1 it is seen that spin splitting significantly influences the density of states and at large values of the g*- factor, the density of states linearly depends on the magnetic field. Density of states as a function of the magnetic field. The solid line corresponds to no spin splitting, the dashed line does to g*=5,the dotted does tog*=2. a -  >20, b -  <20.

50. From this expression it is seen that magnetoresistance takes an infinitely large value, if the condition is satisfied. Vertical longitudinal magnetoresistance in a quantized magnetic field taking into account spin splitting If the direction of an external magnetic field Band current j coincide and they are directed along the z-axis, i.e., BII jII oz magnetoresistance is called vertical longitudinal magnetoresistance. The electron motion quantization in the magnetic field, leads to longitudinal magnetoresistance, due to the fact that in a quantized magnetic field the probability of the current carrier scattering and the Fermi level substantially depends on the magnetic field. In a quantized magnetic field, the relaxation time at the scattering by acoustic phonons is inversely proportional to the density of states of electrons in a magnetic field: Taking this fact into account for electrical conductivityzz= II , we have: