Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure

Superconductivity induced phase-controlled mesoscopic magnetic effects in a two-dimensional electron gas that bridges two superconducting reservoirs are investigated. Giant paramagnetic response of the junction, occuring at certain values of the phase difference of the order parameter, is predicte...

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Дата:	2008
Автори:	Romanovsky, I.A., Bogachek, E.N., Krive, I.V., Landman, U.
Формат:	Стаття
Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2008
Назва видання:	Физика низких температур
Теми:	Carbon nanotubes, quantum wires and Luttinger liquid
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Цитувати:	Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure / I.A. Romanovsky, E.N. Bogachek, I.V. Krive, U. Landman // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1098–1106. — Бібліогр.: 32 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1175532025-02-10T01:33:36Z Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure Romanovsky, I.A. Bogachek, E.N. Krive, I.V. Landman, U. Carbon nanotubes, quantum wires and Luttinger liquid Superconductivity induced phase-controlled mesoscopic magnetic effects in a two-dimensional electron gas that bridges two superconducting reservoirs are investigated. Giant paramagnetic response of the junction, occuring at certain values of the phase difference of the order parameter, is predicted. A geometrically similar system, consisting of a graphene ribbon stretched between two superconducting leads, is also considered. The magnetic effects in this system are found to be small and the difference between the magnetic properties of the two systems is discussed. The research of I.A.R., E.N.B., and U.L. was supported by the U.S. Department of Energy, grant No.FG-05-86ER 45234. I.V.K. acknowledges the financial support from the grant «Effects of electronic, magnetic and elastic properties in strongly inhomogeneous nanostructures» by the National Academy of Sciences of Ukraine and the financial support from the Royal Swedish Academy of Sciences 2008 Article Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure / I.A. Romanovsky, E.N. Bogachek, I.V. Krive, U. Landman // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1098–1106. — Бібліогр.: 32 назв. — англ. 0132-6414 PACS: 73.63.Nm;74.78.Na;74.45.+c;74.25.Bt;74.25.Ha https://nasplib.isofts.kiev.ua/handle/123456789/117553 en Физика низких температур application/pdf 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	Carbon nanotubes, quantum wires and Luttinger liquid Carbon nanotubes, quantum wires and Luttinger liquid
spellingShingle	Carbon nanotubes, quantum wires and Luttinger liquid Carbon nanotubes, quantum wires and Luttinger liquid Romanovsky, I.A. Bogachek, E.N. Krive, I.V. Landman, U. Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure Физика низких температур
description	Superconductivity induced phase-controlled mesoscopic magnetic effects in a two-dimensional electron gas that bridges two superconducting reservoirs are investigated. Giant paramagnetic response of the junction, occuring at certain values of the phase difference of the order parameter, is predicted. A geometrically similar system, consisting of a graphene ribbon stretched between two superconducting leads, is also considered. The magnetic effects in this system are found to be small and the difference between the magnetic properties of the two systems is discussed.
format	Article
author	Romanovsky, I.A. Bogachek, E.N. Krive, I.V. Landman, U.
author_facet	Romanovsky, I.A. Bogachek, E.N. Krive, I.V. Landman, U.
author_sort	Romanovsky, I.A.
title	Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure
title_short	Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure
title_full	Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure
title_fullStr	Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure
title_full_unstemmed	Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure
title_sort	giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2008
topic_facet	Carbon nanotubes, quantum wires and Luttinger liquid
url	https://nasplib.isofts.kiev.ua/handle/123456789/117553
citation_txt	Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure / I.A. Romanovsky, E.N. Bogachek, I.V. Krive, U. Landman // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1098–1106. — Бібліогр.: 32 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT romanovskyia giantmagnetizationofasuperconductortwodimensionalelectrongassuperconductorstructure AT bogacheken giantmagnetizationofasuperconductortwodimensionalelectrongassuperconductorstructure AT kriveiv giantmagnetizationofasuperconductortwodimensionalelectrongassuperconductorstructure AT landmanu giantmagnetizationofasuperconductortwodimensionalelectrongassuperconductorstructure
first_indexed	2025-12-02T12:18:25Z
last_indexed	2025-12-02T12:18:25Z
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fulltext	Fizika Nizkikh Temperatur, 2008, v. 34, No. 10, p. 1098–1106 Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure I.A. Romanovsky1, E.N. Bogachek1, I.V. Krive2,3, and Uzi Landman1 1 School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430 E-mail: igor.romanovsky@gmail.com 2 B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine 3 Department of Physics, Göteborg University, SE 41296 Göteborg , Sweden Received April 23, 2008 Superconductivity induced phase-controlled mesoscopic magnetic effects in a two-dimensional electron gas that bridges two superconducting reservoirs are investigated. Giant paramagnetic response of the junc- tion, occuring at certain values of the phase difference of the order parameter, is predicted. A geometrically similar system, consisting of a graphene ribbon stretched between two superconducting leads, is also consi- dered. The magnetic effects in this system are found to be small and the difference between the magnetic properties of the two systems is discussed. PACS: 73.63.Nm Quantum wires; 74.78.Na Mesoscopic and nanoscale systems; 74.45.+c Proximity effects; Andreev effect; SN and SNS junctions; 74.25.Bt Thermodynamic properties; 74.25.Ha Magnetic properties. Keywords: supercurrent, magnetization, Andreev levels, graphene. 1. Introduction Low-dimensional mesoscopic systems exhibit a num- ber of properties which are of fundamental scientific in- terest, as well as some that may offer opportunities for the use of such systems in future miniaturized electronic de- vices. The new behavior exhibited by systems in this size regime made them the subject of extensive theoretical and experimental research over the past two decades. Quantum wires (QW), are electric conductors with lat- eral (transverse) dimensions reduced such that the motion of the charge carriers in the wire becomes quantized. Quantum wires can be fabricated in many different ways. One of the common methods for the creation of QWs is the gate voltage method applied to semiconducting heterostructures with a two-dimensional electron gas (2DEG) at the interface [1,2]. Another often used method is the controlled break junction technique (CBJ). In this case a contact is pulled apart (for example, separating a contact formed between a tip and a surface) [3], or bent [4], in such a way that at the point where it breaks one gets a very narrow junction with a diameter comparable to the wavelength of the electrons at the Fermi level, resulting in conductance quantization [1,2,5,6] and/or force oscil- lations [3,4] that emerge upon continuous pulling of the contact. Quantum wires can also be made from quasi- two-dimensional conductors, such as graphene, by re- stricting the planar motion of the electrons in one of the two planar directions through the formation of sufficient- ly narrow ribbons [7]. Magnetic field effects in three-dimensional quantum wires connecting two superconductors have been studied previously in [8], where the phenomenon of giant magne- tization oscillations was predicted. Here we focus on me- soscopic magnetic effects of superconducting-normal-su- perconducting (SNS) constrictions, shown schematically in Fig. 1, where the narrow (normal) part that connects the two bulk superconductors is a quasi-two-dimensional conductor. We consider two cases: (i) a superconductor– two-dimensional electron gas–superconductor (S/2DEG/S) junction and (ii) a superconductor–graphene mono- layer–superconductor (S/GM/S) junction. Both types of � I.A. Romanovsky, E.N. Bogachek, I.V. Krive, and Uzi Landman, 2008 systems can be realized experimentally (see, e.g., [9–12]). In the following section we consider the S/2DEG/S system. We investigate the influence of the barriers and the carrier effective mass differences at the interfaces be- tween the semiconductor and superconductor on the Andreev levels and calculate the magnetization. In Sec. 3 we discuss the differences between the S/2DEG/S junc- tion and S/GM/S junction, and explain the smallness of the magnetic field effects in the S/GM/S junction. We summarize our results in Sec. 4. 2. Magnetic effects in S/2DEG/S junctions In this section we consider a S/N/S contact made from a semiconductor heterostructures, that bridges two bulk superconductors. The difference in the electron band structure of the InAs, In Ga Asx x1− and In Al Asx x1− semi- conductors leads to the formation of a narrow (∼ λ F ) po- tential well in the InAs layer [13]. As a result, some of the conduction electrons of the semiconductors are trapped in this well, forming a two-dimensional electron gas. The quantum wires are fabricated by restricting the lateral di- mensions of the two-dimensional electron gas in the semi- conductor heterostructure through the use of electrostatic voltage gates. The potential of the electric field of the gates forms a bottleneck whose size in the narrowest part can be reduced to reach the order of several Fermi wavelengths. In- side this narrow region motion of the electrons in the direc- tions that are transverse to the constriction axis is highly quantized and, consequently, at low temperatures only few transverse modes are populated and can conduct a current. Since the motion of the electrons in the transverse di- rection is quantized, the motion of the electrons in each transverse mode inside the 2DEG is effectively one-di- mensional. Assuming that reflection at the interfaces be- tween the semiconducting bridge and the superconduct- ing reservoirs is specular (i.e., electrons from different modes do not mix and, therefore, can be treated indepen- dently), we can describe each mode with a simplified one-dimensional model [15]. We assume that the left and right superconductors have uniform order parameters of the same magnitude Δ, but with different phases, φ1 and φ2. The order parameter inside the semiconducting bridge vanishes, i.e. Δ Δ Δ ( ) / , , / / , / . x x L L x L x L i i = < − − < < > + ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ φ φ e e 1 2 2 0 2 2 2 (1) In our model we neglect interelectron interactions inside the 2DEG. For such a quasi-1D constriction we could have used a Luttinger liquid model to describe the one-di- mensional transverse channels between the superconduc- tors (see, e.g., Refs. 16 and 17). It turns out, however, that for an adiabatic constriction both models produce the same final result. The influence of a magnetic field on the electrons in the 2DEG is twofold. First, the magnetic field acts on the magnetic moments of the electrons and holes, and second it affects the spatial motion of the charge carriers via the Lorentz force. However, if the magnetic field is applied parallel to the plane of the 2D electron gas, orbital effects can be neglected. In this article we consider only the ef- fects of Zeeman interactions. We assume that the properties of the 2DEG/S interface are very similar to the properties of a normal (N) metal/su- perconductor (S), N–S, interface, namely, the electrons and holes from the nonsuperconducting part of the con- striction, whose energies are lower than the energy gap Δ in the spectrum of the superconductors, cannot leave the normal part of our S/2DEG/S junction. Instead they are being Andreev reflected [18] at the 2DEG/S interfaces. The interference of the incident electron and the reflected hole produces a set of discrete Andreev–Kulik (AK) states [19], that are responsible for the occurence of a Josephson current through the system. The spectrum of the AK states can be found by solving the Bogolyubov– De Gennes equations [15] Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1099 Fig. 1. A schematic description of the superconductor–2DEG–su- perconductor junction. Two superconductors with different pha- ses of the order parameter are connected via a two-dimensional normal electron gas, or, alternatively, a narrow ribbon of gra- phene. A magnetic field is applied locally between the super- conductors, parallel to the plane of the 2D conductor and is taken to be negligibly weak near the SN interfaces. A similar system has been considered in Ref. 14, where the effects of Zeeman splitting and spin-orbit interaction on the Josephson current were studied. � ( ) � ( ) � * ( ) � ( ) ( ) ( ) ( H D D H u v u v x x x x x x x − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ = ( ) ε x ) ⎛ ⎝⎜ ⎞ ⎠⎟ , (2) where the two-component spinors u( ) ( ( ), ( ))x u x u x= ↑ ↓ and v( ) ( ( ), ( ))x v x v x= ↑ ↓ are, respectively, electron and hole probability amplitudes, and the ↑ and ↓ subscripts determine the quasiparticle spin projection. The Hamil- tonian �H is a 2 2× matrix � ( )H x he= ⎛ ⎝⎜ ⎞ ⎠⎟ + − ⎛ ⎝⎜ ⎞ ⎠⎟ 1 0 0 1 1 0 0 1 μ . (3) Here h p m x U xe F= + −2 2/ ( ) ( ) ε and μ μ= g B / 2. The first term in Eq. (3) describes the nonmagnetic part of the Hamiltonian with εF being the Fermi energy, and m x( ) the effective mass of the quasiparticles. The effective mass ms in the superconductors can be different from the effective mass mn in the semiconductor. The function U x( ) describes the external potential. We assume that the S/2DEG/S junction is clean and, therefore, U x( ) = 0 everywhere except at the interfaces. At the interfaces ( / )x L= ± 2 the difference in material composition of su- perconductor and semiconductor commonly leads to for- mation of potential barriers. We will model these poten- tial barriers by Dirac delta-functions of equal strength W , i.e. U x W x L x L( ) ( ( / ) ( / ))= + + −δ δ2 2 . (4) The second term in Eq. (3) is the Zeeman coupling of the magnetic momenta of the electrons to the external field B x( ). We assume that the magnetic field is weak ( ( )g B xBμ << Δ, where μB is the Bohr magneton and g is the Lande g-factor), and that it acts only in the nonsuper- conducting part of the junction B x B x L ( ) , \| \| / , , = <⎧ ⎨ ⎩ if otherwise . 2 0 These requirements are important in order to guarantee that the field does not create currents (or equivalently, phase gradients) along the S/2DEG interfaces. Finally, the off-diagonal potential �D( )x , which enters Eq. (2) is equal to � ( ) ( ) ,D x x= ⎛ ⎝⎜ ⎞ ⎠⎟ Δ 1 0 0 1 (5) where Δ( )x is the order parameter, Eq. (1). The use of the Bogolyubov–de Gennes equations to describe a S/2DEG/S structure is valid only when the nonsuperconducting seg- ment of the system is shorter than the electron–hole pair correlation decay length l v k TT F B= � / ( ) (see, e.g., dis- cussion in Ref. 20). The Andreev–Kulik levels are ob- tained by matching the solutions for the uniform regions at the S/2DEG interfaces. The result of the matching pro- cedure is quite cumbersome. To simplify it we will use the Andreev approximation [18,19]. That is, to the lowest nonvanishing order in max ( , ) /Δ E Fε we (approximate- ly) replace the wave vectors of the electrons and holes in- side the S and 2DEG parts of the junction with the Fermi wave vectors k k ke h F n≈ ≈ ( ) and q q ke h F s≈ ≈ ( ) , where k k m m F s F n s n ( ) ( ) / /= ; and for the differences between the wave vectors we write k k m ke h n F n− ≈ ε / ( ) ( ) � 2 and q q m ke h s F s− ≈ Δ / ( ) ( ) � 2 . In a long junction (L vF>> =ξ 0 � / Δ) the bound state energies close to the Fermi level, ε << Δ, are ε πω π σ μσn L Bn g B, ( ) ( ) ± φ = ± +⎛ ⎝⎜ ⎞ ⎠⎟ +Θ 2 2 , (6) where the new phase Θ is defined as Θ ≈ − φ + +⎛ ⎝ ⎜ ⎞ ⎠ ⎟π arccos R R Lk R Lk D F F Z 1 2 32 2cos ( ) cos ( ) sin ( ) , [ ( ) ] , , [( ) D m m m m Z R m m R m m m Z s n s n n s n s s = + + = = − − 4 42 2 2 1 3 2 2 2 + − + = − + 8 3 16 8 4 2 2 4 4 3 m m m m Z m Z R m m Z m m m m n n s s n n s n s s ( ) )] , [( ) nZ2 2] . (7) Here Z Wm k F= / ( )� is the parameter which describes the barrier strength and ωL Fv L= � / is the energy level spac- ing. If the effective masses of the charge carriers in the superconducting leads and in the 2DEG are equal, the re- sults obtained in Refs. 21–23 are recovered. Equation (6) describes two sets (±) of discrete levels, labeled by inte- ger indices n = ± ± …0 1 2, , , and an additional index σ = ±1 which characterizes the splitting of the energy levels in a magnetic field. Since the scattering properties of the interfaces are assumed to be spin-independent, spin is conserved in both the Andreev and normal reflections. The magnetization of each populated AK level is, accord- ing Eq. (6), g Bμ / 2. The mesoscopic harmonic factors, sin ( )2Lk F and cos ( )2Lk F in Eq. (7), are associated with interference of the incident and normally reflected quasi- particle waves. They appear due to a large change in the quasimomentum of the quasiparticle at the interface in normal reflection. In a transparent SNS junction (Z = 0) with equal effective masses, these oscillations are absent since in pure Andreev reflection processes momentum is 1100 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 I.A. Romanovsky, E.N. Bogachek, I.V. Krive, and Uzi Landman approximately conserved. As a result, Eq. (6) reduces to the well known Kulik spectrum [19]. For a transparent SNS junction in zero magnetic field, B, the AK levels, corresponding to two different sets ( ± φ), intersect at φ = +r rπ π2 (r = ± ± …0 1 2, , , ). At these special points the levels are four-fold degenerate for a single channel junction. The situation is changed if the junction is not transparent. Even if the barriers are very small, levels do not intersect and they oscillate periodi- cally with phase, approaching each other at the points φ = = ± ±πk k( , , )0 1 2 . Every AK level is now two-fold degenerate. An external magnetic field removes the re- maining degeneracy. The magnetization generated by the AK state nσ( )± of the transverse mode l is given by M n l l n , ( ) , , ( ) ( ) exp ( ( )) σ σ μσ βε ± ± = − + − φ1 . (8) The subscript l which enters this formula implies that the transverse modes are different. The total magnetization of a single transverse mode is given by a sum over all the AK states. It is useful to define the magnetizations produced by pairs of levels with opposite directions of the mag- netic moment M n M n M n l l l ( ) , ( ) ,( ) ( ) ( ) ( )± + ± −= + ± 1 1 and then sum up the pair contributions M n l ( ) ( ) ± over the index n using the Poisson summation formula M n M d M l n l l k ( ) ( ) ( ) ( ) ( ) Re ± = −∞ +∞ ± −∞ ∞ ± −∞ ∞ = +∞ ∑ ∫ ∫∑ = = +ν ν 2 1 ( ) .ν νπ νe 2 ik d ⎧ ⎨ ⎪ ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ (9) Such ordering guarantees convergence of the first and second terms in Eq. (9). The total magnetization of the single transverse mode is [8] M g B Tg k k kT l B L B L Lk = + = 2 2 1 4 2 μ πω μ ω χ π ω cos ( ) sin ( ) ( / ) Θ sinh ∞ ∑ , (10) where χ μ ω= g BB L/ . The first term in Eq. (10) does not depend on the phase difference and the temperature. This term describes magnetization of the junction in the absence of Andreev reflection; that is, for nonsuperconducting reservoirs (Pauli magnetization). It is the second term which is re- sponsible for superconductivity-induced oscillations of the magnetization. At high temperatures (T L>> ω ), when the Fermi distribution is smeared over many AK levels the amplitude of these oscillations is small in comparison with the first term in Eq. (10). The sum in Eq. (10) can be truncated at k = 1 so that the magnetization of each trans- verse mode oscillates as A cos Θ with an amplitude A g BT TL L= −( / ) exp ( / )4 22 2 2μ ω π ω . In the opposite limiting case, at low temperatures ( )T g BB< μ , only a small number of states near the Fermi energy contributes to the magnetization. The AK levels are shifted in energy because of the change of the phase difference φ between the superconductors. At certain val- ues of the phase difference some AK states approach very close to the Fermi energy. At these phase differences the magnetization of the junction will increase. Taking into account only a single pair of states which is the closest (at the given φ = φr) to the Fermi energy and neglecting the contribution of the other states, we can ap- proximate the previous formula by a simple expression M g T g T B g T g l B L B B L (osc) ≈ −⎛ ⎝⎜ ⎞ ⎠⎟ + − − + μ ω μ μ ω μ exp exp 2 2 1 2 Θ Θ B T B 2 1 ⎛ ⎝⎜ ⎞ ⎠⎟ + , (11) where the angle Θ is defined by Eq. (7). The same result can be achieved if we use the Euler–Maclaurin summa- tion formula to approximate Eq. (10) in the vicinity of the points φ = +r rπ( )1 2 . Formally Eq. (10) describes only the part of the mag- netization that is due to the discrete spectrum of AK states. The continuous spectrum (\| \|ε > Δ) also contributes to the magnetization [24]. At temperatures T << Δ the ef- fect of continuous spectrum is to compensate the nonsmooth contributions of the discrete spectrum arising when summing up to the finite number of bound states in- side the energy interval \| \|ε < Δ. In our calculation method (see Eqs. (9) and (10)) the superconducting gap was for- mally put equal to infinity, and as a result only the discrete spectrum survives. This method reproduces the correct results for thermodynamic properties of a long SNS junc- tion at temperatures T << Δ (see the corresponding discussion in a recent review [25]). In order to evaluate the total magnetization of the junc- tion we have to sum up the contributions of all the open transverse modes (channels). A channel is open if the en- ergy of its lowest longitudinal mode is smaller than the Fermi energy of the superconducting leads that it is con- nected to. Since the barriers at the S/2DEG interfaces are assumed small the motion in the longitudinal direction is almost unperturbed. This situation corresponds to the case of junctions fabricated in InGaAs heterostructures with Nb electrodes [9]. Confinement in the lateral direc- tion is usually created by an electrostatic potential gene- rated by gate electrodes etched on the surface of the semiconductor. Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1101 We obtained results for two simple models of the con- fining potential: i) a hard wall potential, and ii) a para- bolic potential. For more accurate description one needs to solve the Schr�dinger and Poisson equations self-con- sistently as described in Refs. 26 and 27. For hard wall boundary conditions, we can express the velocity of the electrons and the holes, with energies near the Fermi energy in the lth channel, in the form v m m l dF l n F n ( ) = − ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 2 1 2 2 ε π� . (12) The velocities of quasiparticles in a multichannel junc- tion with parabolic confinement U y m y /n( ) = ⊥Ω 2 2 2 are v m l F l n F ( ) = − +⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟⊥ 2 1 2 ε �Ω , (13) where the lateral frequency Ω⊥ = 8 2εF nm d/ ( ) was chosen such that U y d F( / )= ± =2 ε . From this formula we see that the velocities and, con- sequently, the separation between the AK levels ω ω= =l F l v L� ( ) / are different for different transverse modes. If the 2DEG junction is wide the number of modes is large and we can replace the summation over them by an integration and write the magnetization of the S/2DEG/S junction as M N g D dB≈ ⊥ ∫μ ξ α β λ ξ ξ( , , ) ( ) 0 1 . (14) The function D f f( , , ) ( ) ( )ξ α β αξ β αξ β= − − + is the dif- ference between the occupation of the two states with op- posite momenta, f ( ) / ( )η η= +1 1e is the Fermi distribu- tion, α = �v LTFΘ / ( )2 and β μ= g B TB / ( )2 . The weight function λ ξ( ) is equal to 2ξ for the «soft walls» potential and to ξ ξ/ 1 2− for the «hard walls» potential. The integral on the right hand side of Eq. (14) cannot be calculated analytically, but we can find the asymptotic behavior of the magnetization in the vicinity of several values of the of phase difference φ. In the vicinity of the resonance points φr where Θ << LT vF/ ( )� , the magneti- zation is the highest, and it is approximately equal to M N g g B T C v LT g B B F B ≈ × × − ⎛ ⎝⎜ ⎞ ⎠⎟ ⊥ μ μ μ tanh sech ( / ) ( 4 1 12 1 2 2Θ� B T/ ) .4 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ (15) The constant C1 is equal to 1 2/ in the «hard wall» po- tential and to 2 3/ in the «soft wall» potential model. On the other hand, if the phase is far away from resonance ( / ( ))Θ >> LT vF� the magnetization approaches slowly the value M C N g BT L v B F ≈ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟⊥2 2 2 2 2 2μ � Θ ln . (16) The constant C 2 is equal to 1 for the «hard wall» model and it takes the value 2 for the «soft wall» model. While Eq. (15) works well for transparent S/2DEG/2 junction, it may be inaccurate if the barriers at the interfaces are not small enough. In this case the condition Θ << 1 cannot be satisfied at any phase φ. In Fig. 2 we show the result of numerical calculations for the magnetization as a function of the phase differ- ence, and compare this numerical result with the asymp- totic behavior in Eqs. (15) and (16). Figure 3 shows the behavior of the magnetization at several different temper- atures. Higher temperatures lead to smearing of the Fermi distribution and the resonance peaks become smaller and broader. Since for transparent clean junction the resonance con- dition φ = φr is the same for all transverse modes, magne- tization at the resonances is proportional to the number of transverse modes. This effect is analogous to the giant oscillations of the conductance considered in Ref. 28. Therefore, the wider the junction, the higher would be the magnetization at the resonances. This behavior is illus- 1102 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 I.A. Romanovsky, E.N. Bogachek, I.V. Krive, and Uzi Landman 0.9 1.0 1.1 0 4 8 12 φ π/ M /μ B M MA MB Fig. 2. Magnetization of a transparent (Z = 0) S/2DEG/S junc- tion with a harmonic (parabolic) lateral confining potential in a magnetic field B = 10 Oe, plotted as a function of the phase difference. Also shown are its asymptotes MA (Eq. (15)) and MB (Eq. (16)). For this demonstration we choose materials with the same effective masses. We assume that the superconduc- tors are made from niobium, with a Fermi energy εF ( ) .Nb erg= ⋅ −8 52 10 12 , and effective masses m ms n= . The length of the 2DEG part of the junction is L = −10 4 cm, and it's width is d = ⋅ −6 10 5 cm. Results are shown for a temperature T = 0 1. K, which is much lower than the critical temperature of niobium Tc( ) .Nb K= 9 2 . Note that the resonance peaks are ex- tremely narrow, their width δ μ ωφ ∼ g BB L/ is approximately equal to the phase change necessary to shift the spectrum of AK levels by an amount equal to Zeeman splitting. trated in Fig. 4. On the other hand, varying the length of the junction will affect only the width of the peaks, whereas their height will remain unchanged (see Fig. 5). In the case of nontransparent junctions, the electrons and holes incident on the surface may be reflected nor- mally from the S/2DEG interface. This process modifies the spectrum of the AK levels. The AK levels do not inter- sect and do not cross the Fermi energy at any value of the phase φ. As a consequence, the amplitude of the magneti- zation peaks decreases. Differences between the effective masses of the 2DEG and the superconductors have a simi- lar effect on the magnetization as the presence of barriers at the interfaces. Namely, because of the mismatch be- tween the Fermi velocities some of the electrons are being normally reflected at the 2DEG/S surface. For larger dif- ference between the effective masses, larger fraction of all of the incident electrons will be reflected from the interfaces normally. In clean S/2DEG/S or S/N/S junctions every Andreev ref lected electron picks up an addi t ional phase δφ = ± φ +1 2, ( / )arccos E Δ at the interface, whereas nor- mal electrons do not pick up such a phase. The wave func- tion of the reflected electron-hole pair is a mixture of Andreev and normally reflected electrons and holes. The resonance conditions for this mixture are different from the resonance conditions for the wave function in the clean S/2DEG/S junction, where only Andreev reflection can occur. The resonance conditions in the junction with barriers, or with different effective masses, depend not only on the phase difference, but also on the length of the junction. Although it is always possible to achieve reso- nance for one channel, it is difficult to satisfy the reso- nance conditions for many channels simultaneously, since different transverse channels have different longitu- dinal velocities. The resonant peaks for magnetization of each single channel are very narrow, with the majority of the channels being off resonance even for small barriers or slightly different materials. As a result, the total mag- netization of the junction for both barrier models (hard and soft walls) and for different masses, is strongly sup- pressed. Therefore, for observation of strong resonance behavior the use of materials with similar effective masses is recommended. The magnetization, as well as the superconducting current in a S/2DEG/S junction, result from the difference in the population of the different AK levels. Josephson current through the junction is possible only when the en- Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1103 0.95 1.00 1.05 0 5 10 15 20 T = 0.05 K T = 0.10 K T = 0.15 K M /μ B φ π/ Fig. 3. Magnetization of the transparent S/2DEG/S junction in a magnetic field B = 10 Oe at several temperatures plotted versus the phase difference. The width of the junction is d = ⋅ −5 10 5 cm, and all the other parameters of the junction are as in Fig. 2. 14 12 10 8 6 4 2 0 M /μ B φ π/ 0.95 1.00 1.05 d = 2·10 cm –5 d = 6·10 cm –5 d = 4·10 cm –5 Fig. 4. A magnetization as a function of the width of the 2DEG part of a transparent S/2DEG junction. The length of the junc- tion is L = −10 4 cm and the temperature T = 0 1. K. The strength of the magnetic field and all other parameters are the same as in Fig. 2. 0.9 1.0 1.1 0 2 4 6 8 10 12 M /μ B φ π/ L = 2·10 cm 4– L = 3·10 cm 4– L = 1·10 cm –4 Fig. 5. The magnetization of a transparent S/2DEG/S junction versus the phase difference φ, for different lengths of the 2DEG part. The width of the junction is d = ⋅ −5 10 5 cm, temper- ature T = 0 1. K and the magnetic field B = 10 Oe. ergy levels of the two sets (dE dn / φ > 0 and dE dn / φ < 0) are unequally populated. Similarly, magnetization of the S/2DEG/S junction is a result of different population of states with opposite directions of the magnetic moments. It is most interesting to note that the magnetization is more sensitive to the barriers and mass differences than the Josephson current. Indeed, since we are constrained to use weak external magnetic fields in order to avoid de- struction of superconductivity, Zeeman splitting of the AK levels is small (μ ωB Z LB = <<Δ ). In this situation, impurities, barriers, or effective mass difference that modify the spectrum of the AK levels can move them away from the region (∼ T ) where the gradient of the Fermi distribution is high. Even a small shift Δ b of the en- ergy levels from the Fermi level may result in the situa- tion T Z b<< <<Δ Δ when both levels with opposite direc- tion of the magnetic moment are almost equally populated, combining to give a small magnetization M l B<< μ . This situation does not occur for levels which belong to different sets. These levels are well separated ∼ Δ L and small change Δ Δb L<< in their positions cannot affect their population significantly. 3. Differences between properties of S/2DEG/S and S/GM/S junctions We turn now to analysis of a junction with the geome- try similar to that of the S/2DEG/S junction dicussed above, but where the two superconducting leads are con- nected to each other through a monolayer of graphene. Graphene attracted much recent attention. In this material the dispersion relation for the low-energy electrons and holes is similar to the dispersion relation of relativistic massless fermions. This quasi-relativistic behavior of the electrons in graphene has many interesting consequences. One of these is the existence of two types of Andreev re- flections at the graphene/superconductor boundary [29]. Another consequence is the effect of Klein tunneling through the a potential barrier [30]. One might expect that Klein tunneling and specular Andreev reflection would enhance magnetic effects by reducing the strong destruc- tive interference between normally and Andreev reflected electrons and holes in the nonsuperconducting (graphene) part of the junction. In addition to specular Andreev re- flection and Klein tunneling one should also take into ac- count the fact that reflection of the electrons from the edges of the graphene ribbon depends on the orientation of the crystallographic axes with respect to the line of the edge at the sides of the graphene ribbon — this affects the quantization of the transverse motion within the ribbon [31]; in our calculations we assumed infinite-mass boundary conditions at the edges, and as a result our transverse momentum is quantized as q l Wl = +( / ) /1 2 π . Similar to the S/2DEG/S junction, by matching the so- lutions of the Schr�dinger equation for each part of the contact (see Ref. 32) one can find the spectrum of the AK levels in the S/GM/S contact and use this spectrum to cal- culate the total magnetization of the S/GM/S junction un- der the same conditions as for the S/2DEG/S one. It turns out, however, that the magnetization of the S/GM/S junc- tion is very small even in the absence of potential barriers at the interfaces between the superconductor leads and 1104 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 I.A. Romanovsky, E.N. Bogachek, I.V. Krive, and Uzi Landman 10 8 6 4 2 0 10 8 6 4 2 0 ε, ar b . u n it s ε, ar b . u n it s –1 –10 01 12 23 3 φ π/ φ π/ Fig. 6. The spectrum of AK levels in a S/2DEG/S junction (left) and in a S/GM/S junction (right). Both figures are schematic, that is, these figures are plotted for junctions with different geometrical parameters with the main purpose to illustrate the basic charac- teristic features of the energy levels in these systems. Note, that unlike the S/2DEG/S junction where the lowest AK levels of all transverse modes approach the Fermi level and become degenerate at the resonant phases (φ = +r rπ π2 , r = ± ± …0 1 2, , , ), the lowest AK levels of the S/GM/S junction, belonging to different modes, never become degenerate at the resonant phases, and all (except one level of the lowest transverse mode) stay away from the Fermi level. the graphene. In fact the magnetization is much smaller than that found for the S/2DEG/S junction (see above). There are several reasons for such a small effect. First is the fact that the density of states of graphene near the Fermi level (for undoped graphene) is very small and therefore the number of AK levels in the range of energies near the Fermi level for the S/GM/S junction should be much smaller than for the S/2DEG/S junction. The other reason for the smallness of magnetization is that the AK levels in different transverse modes of the S/GM/S junc- tion behave differently (see Fig. 6) and they do not con- verge to a single degenerate state at the Fermi level for the resonant phase differences, as happens for S/2DEG/S junctions (see above). Since the energy levels of the trans- verse modes in the S/GM/S junction never approach the Fermi level and always stay away from the region with maximal gradients of the Fermi distribution, their contri- bution to the total magnetization is very small. We conclude that the different behavior of the AK le- vels in the semiconductor 2DEG and in graphene origi- nate from the different type of dispersion relations of the electrons in the two cases: a quasi-relativistic linear dis- persion relation for the electrons in graphene versus the regular quadratic dispersion relation in the two-dimen- sional electron gas. 4. Summary In summary, we considered magnetic effects in a two-dimensional electron gas bridging two superconduct- ing reservoirs. We demonstrated that this system can ex- hibit interesting superconductivity-induced magnetic res- onance effects. Namely, we predict sharp increases in the magnetic susceptibility of the junction at special values φ = +r rπ π2 of the phase difference φ = φ −φ1 2 of the or- der parameter between the two superconductors. This effect results from a change of the population of the AK levels near the Fermi energy. In general, magneti- zation of a single transverse mode due to the Andreev lev- els is very small but since for transparent junctions the resonance conditions are identical for all transverse modes, the magnetic response at the resonances (at low temperatures) is proportional to the number of transverse modes. Consequently, when the number of transverse mo- des is large, the total magnetization of the junction may become large enough to allow experimental detection. We also considered a junction made from a graphene ribbon bridging the two superconductors. We found that the giant magnetization oscillations that we predict for the S/2DEG/S junction are absent in the Josephson con- tact made with the monolayer graphene ribbon. Reasons for the differences in magnetization response between the S/2DEG/S and S/GM/S systems were discussed. The research of I.A.R., E.N.B., and U.L. was sup- ported by the U.S. Department of Energy, grant No. FG-05-86ER 45234. 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Giant magnetization of a superconductor–two-dimensional electron gas–superconductor structure

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