Andreev Reflection. Fabrizio Dolcini Scuola Normale Superiore di Pisa, NEST (Italy) Dipartimento di Fisica del Politecnico di Torino (Italy)
|
|
- Michael Gallagher
- 5 years ago
- Views:
Transcription
1 Andreev Reflection Fabrizio Dolcini Scuola Normale Superiore di Pisa, NEST (Italy) Dipartimento di Fisica del Politecnico di Torino (Italy) Lecture Notes for XXIII Physics GradDays, Heidelberg, 5-9 October 9
2 Contents Andreev Reflection 3. The Blonder-Tinkham-Klapwijk model for N-S junction Solutions in N Solutions in S Supra-gap solutions (E > ): propagating waves Sub-gap solutions (E < ): evanescent waves Condition at the boundary Scattering Matrix Coefficient Solution in the Andreev Approximation Andreev Reflection The case of ideal interface (Z = ) Interface with arbitrary transparency Current-voltage Characteristics 6. Current and Conductance The limit of low transparency at arbitrary V The linear conductance at arbitrary transparency
3 Chapter Andreev Reflection. The Blonder-Tinkham-Klapwijk model for N-S junction z N y S x (x) =θ(x) e iϕ Figure.: Scheme of a N-S junction. Let us consider a normal(n) - superconductor(s) junction, as shown in Fig... We denote by x the longitudinal coordinate and by (y, z) the transversal coordinates. The junction is located at the coordinate x =. We model the system with the Bogolubov de Gennes Equation H e u u = E (.) v v H e 3
4 4 Solutions in N We describe the junction with a step-like order parameter (x) = θ(x) e iϕ (.) Assume that the system is separable, so that i) we can factorize the wavefunction into ψ(x) = longitudinal wavefunction Ψ(x, y, z) = ψ(x) Φ n (y, z) Φ n (y, z) = transversal wavefunction (.3) where n denotes the quantum number labeling the transversal mode ( ) ] [ m x + + V y (y, z) Φ n (y, z) = E n Φ n (y, z) (.4) with transversal energy E n, and V is the transversal confining potential. ii) The energy is the sum of the longitudinal and transversal energies E = E // + E n (.5) so that, for a given transversal mode n, the effective chemical potential for the longitudinal propagation reads ε F n = ε F E n (.6) where we assume that ε F already includes the self-consistent potential U. We include a Λδ(x) potential at the boundary in order to account for the contact resistance of the interface. In conclusion, the system is described by the following effective D BdG Hamiltonian m x ε F n + Λδ(x) (x) (x) m x + ε F n Λδ(x) u(x) v(x) = E u(x) v(x) (.7) and is called the Blonder-Tinkham-Klapwijk (BTK) Model, as described in Ref.[7]. The purpose is thus to find solutions with non negative energy E.
5 Chap.. Andreev Reflection 5 k e k h k h k e Figure.: Spectrum in the N region.. Solutions in N In the normal side N the equation (.7) reduces to ε m x F n u(x) u(x) = E (.8) + ε m x F n v(x) v(x) which exhibits two particle solutions Ψ e ±(x) = e ±ikex (.9) and two hole solutions where and Ψ h ±(x) = e ±ik hx (.) k e = k F n + E (.) ε F n k h = k F n E (.) ε F n k F n = mεf n (.3)
6 6 Solutions in S.3 Solutions in S q e q e q h q h Figure.3: Spectrum in the S region. In the superconductor side S the equation (.7) reduces to m x ε F n e iϕ We have to distinguish two cases e iϕ u(x) = E + ε x F n v(x) m u(x) v(x) (.4).3. Supra-gap solutions (E > ): propagating waves Here there are two particle-like solutions Ψ e ±(x) = u e iϕ/ v e iϕ/ e ±iqex (.5) and two hole-like solutions Ψ h ±(x) = v e iϕ/ u e iϕ/ e ±iq hx (.6) where q e = k E F n + ε F n q h = k E F n ε F n (.7) (.8)
7 Chap.. Andreev Reflection 7 and k F n = mεf n (.9) Here, the quantities u and v read u = v = + ( E ( E ) E e arccosh E (.) ) E e arccosh E (.) so that the wavefunctions (.5) and (.6) Ψ e ±(x) = E e arccosh E e iϕ/ e arccosh E e iϕ/ e ±iqex (.) and two hole-like solutions Ψ h ±(x) = E e arccosh E e iϕ/ e arccosh E e iϕ/ e ±iq hx (.3).3. Sub-gap solutions (E < ): evanescent waves In the subgap solutions q e/h acquire an imaginary part. A real part remains and is of the order of k F n. The solution is the analytic continuation of (.4) and (.5), and reads q e = k F n + i E ε F n q h = k F n i E ε F n Similarly the analytic continuation of (.) and (.) reads (.4) (.5) u = v = E e i arccos E (.6) i E e arccos E (.7)
8 8 Condition at the boundary Remark Notice that for the evanescent waves one has u + v. Instead one has u + v = ) (e i arccos E + e i arccos E = E = ) (arccos E cos E = (.8) Analytic continuation is important because we know from causality that S-matrix in the supra-gap regime admits an analytic continuation into the sub-gap regime. Such continuation is in general not unitary. Unitarity only holds for propagating modes, because evanescent waves carry no current and unitarity is related to the conservation of current..4 Condition at the boundary Integrating the equation u m x ε F nu(x) + Λδ(x)u(x) + (x)v(x) = Eu(x) around x =, one obtains the boundary conditions for the derivatives and x u( + ) x u( ) = mλ u() (.9) x v( + ) x v( ) = mλ v() (.3).5 Scattering Matrix Coefficient We now want to determine the coefficient of the Scattering Matrix. Let us start by considering the case of an incident electron, incoming from the N left electrode towards the interface Ψ in (x) = e +ikex (.3) π ve The wave reflected back into the N region is a left-moving electron or a left-moving hole, i.e. Ψ refl (x) = r ee e ikex r he + e +ik hx (.3) π ve π vh In contrast, the transmitted wave is a right-moving electron-like or a right-moving hole-like solution u t e iϕ/ v ee Ψ trans (x) = e +iqex t e iϕ/ he + e iq hx (.33) π we v e iϕ/ π wh u e iϕ/
9 Chap.. Andreev Reflection 9 Remarks We have denoted r ee = reflection coefficient e e (.34) r he = reflection coefficient e h (.35) t ee = transmission coefficient e e (.36) t he = transmission coefficient e h (.37) We have normalized the wavefunctions with their velocities, because they are different in general for particle and holes, and from normal to superconduting side. In this way, each wavefunction carries the same amount of flux of quasiparticle probability current[4], and therefore the above coefficients describe a unitary matrix. We recall that unitarity of the scattering matrix stems from the conservation of quasi-particle probability current. For the N side we have For the S side we have E = ke m ε F n v e = de dk e = k e m E = ε F n kh v h = de m dk h = k h m (.38) (.39) ( ) qe E = m ε F n + w e = de dq e = q e m ( ) E = εf n qh + w h = de m dq h = q h m (.4) (.4) The velocities are v e/h = k e/h m (.4) w e/h = E v e/h = (u v E )v e/h (.43) In the reflected wave the momenta of the electron and hole have opposite signs, just because we want to describe left-moving waves. Similarly for the right-moving transmitted waves (see Fig..4).
10 Solution in the Andreev Approximation left-moving electron right-moving hole left-moving hole right-moving electron Figure.4: Different signs of velocities in the electron-hole band. In order to find the solution we have to impose u( + ) = u( ) (continuity) (.44) v( + ) = v( ) (continuity) (.45) (.46) x u( + ) x u( ) = mλ u() (derivative) (.47) x v( + ) x v( ) = mλ v() (derivative) (.48) These conditions represent a set of four linear equations for the four unknowns r ee, r he, t ee, and t he. Exploiting the linearity of the above equations, one can find the other scattering matrix coefficients (such as r eh, t eh and so on) by setting e.g. an incoming hole from N or incoming electron/hole from S, and superimposing the various solutions..6 Solution in the Andreev Approximation The explicit solution of the linear set of equation is particularly simple in the so called Andreev approximation, which consists in envisaging low energies with respect to the Fermi level E, ε F n (.49) and thus retain the lowest order in E/ε F n and /ε F n. One can then approximate k e/h q e/h k F n (.5) and v e/h v F n (.5) w e/h E v F n = (u v E ) v F n (.5)
11 Chap.. Andreev Reflection with the Fermi velocity defined as v F n = k F n m (.53) Under the Andreev approximation one finds for the transmission and reflection amplitudes r he = u v u + Z (u v ) e iϕ (.54) r ee = (Z + iz)(v u ) u + Z (u v) (.55) t ee = ( iz)u u v u + Z (u v) e iϕ/ (.56) t he = izv u v u + Z (u v) e iϕ/ (.57) Here Z = Λm k F n = Λ v F n (.58) is the BTK dimensionless parameter of the interface transparency Z very transparent interface Z weakly transparent interface (tunnel limit) (.59) By the transparency we mean the transmission probability T N of the junction in the normal case, i.e. when the gap in the superconducting side is set to zero ( ) or the temperature is above the critical temperature T c. One can prove that the BTK parameter is related to T N through the relation T N = + Z (.6) The corresponding transmission and reflection coefficients read A. = A he LL B. = A ee LL C. = A ee RL D. = A he RL. = r he (.6). = r ee (.6). = t ee (.63). = t he (.64)
12 Solution in the Andreev Approximation where the first notation is the one of Ref.[7], and the second notation is in the style of Ref.[8]. Recalling the expression (.)-(.) and (.6)-(.7) for u and v we obtain: Supra-gap (E > ) A(E) = A he LL(E) = B(E) = A ee LL(E) = ( E + ( + Z ) ) (.65) E 4Z ( + Z )(E ) ( E + ( + Z ) ) (.66) E C(E) = A ee RL(E) = ( + Z ) E (E + E ) ( E + ( + Z ) ) (.67) E D(E) = A he RL(E) = Z E (E E ) ( E + ( + Z ) ) (.68) E One can easily verify that J=L/R β=e/h as required by unitarity of the S-matrix. Sub-gap (E < ) A βe JL = A + B + C + D = (.69) A(E) = A he LL = r he = B(E) = A ee LL = r ee = E + ( + Z ) ( E ) 4Z ( + Z )( E ) E + ( + Z ) ( E ) (.7) (.7) C(E) = A ee RL = t ee = (.7) D(E) = A he RL = t he = (.73) Notice that in the subgap case the transmission coefficients are vanishing C = D =. In fact E = is precisely the value at which the supra-gap coefficients (.67) and (.8) vanish. For E < one still has a non vanishing analytical continuations for the expressions (.67) and (.8). However, they cannot be interpreted as transmission coefficients, for in the superconductor there are no propagating modes for E >. One can again easily verify that A βe JL = A + B = (.74) J=L/R β=e/h
13 Chap.. Andreev Reflection 3 as required by unitarity of the S-matrix..7 Andreev Reflection.7. The case of ideal interface (Z = ) In order to discuss the physical consequences of the coefficients A, B, C and d found in the previous section, we start by considering the special case of ideal interface (Z = ). In this case the Andreev-reflection amplitude for the process e h coefficient reads r he = v e i arccos E E < e iϕ = e iϕ (.75) u e arccosh E E > Similarly, one can find for the process h e that r eh = v e i arccos E E < e iϕ = e iϕ (.76) u e arccosh E E > The coefficients in this case read:. A C.5 Z =. 3 4 E / Figure.5: The case of ideal interface Z = : The Coefficients A, and C are plotted as a function of energy. The coefficients B and D are vanishing.
14 4 Andreev Reflection Sub-gap regime (E < ) A(E) = (.77) B(E) = (.78) C(E) = (.79) D(E) = (.8) which shows that, for an ideal N-S interface, an injected electron can only be Andreev reflected as a hole, with % probability. This phenomenon is known as Andreev reflection[5, 6], and is depicted in Fig..6. An incoming electron is reflected as a hole. In contrast to an ordinary reflection, where momentum is not conserved and charge is conserved, in an Andreev reflection process momentum is almost conserved (in the sense that both the incoming electron and the reflected hole have momentum very close to the same k F, whereas charge is not conserved. Importantly, the velocities are reversed. reflected hole incoming electron N S Figure.6: hole. The phenomenon of Andreev reflection: the incoming electron is reflected as a Supra-gap regime (E > ) A(E) = ( E + ) (.8) E B(E) = (.8) C(E) = E E + E (.83) D(E) = (.84) Here we see that, for energies above the gap, the electron also has a finite probability to be transmitted as an electron, since single particle states are available in the superconductor above the gap. At high energies E, the effects of superconductivity
15 Chap.. Andreev Reflection 5 and normal transmission is in fact the most probable process, as shown by the curve C(E) in Fig Interface with arbitrary transparency Let us now consider the case of a non-ideal interface (Z > ). There is still a probability that electrons are Andreev-reflected as holes. However, in this situation, due to the presence of the barrier at the interface, electrons can also be ordinarily reflected as electrons. In the sub-gap regime the sum of probabilities of these two processes must equal (A + B = ), so that an increase of ordinary reflection leads to a decrease of Andreev reflection, as shown in Fig..7 for two different values of the interface parameter....5 A Z =..5 Z = B B A E / E / 3 4 Figure.7: The coefficients A and B are plotted as a function of energy for the case of Z =. (almost ideal interface with transmission coefficient T =.96) and Z = (interface with intermediate transmission T =.5).
16 Chapter Current-voltage Characteristics. Current and Conductance In the case of transport through a system connected to normal electrodes, the Landauer- Büttiker expression for the (single channel) current reads I = e de T (E) h }{{} = R(E) (f L (E) f R (E)) (.) where T (E) is the transmission coefficient of the sample, R(E) its reflection coefficient, the pre-factor stems from spin degeneracy, and f L/R (E) the Fermi functions of the Left and Right reservoirs f X (E) = X = L/R + e (E µ X)/k B T In the case of a (single channel) mesoscopic sample contacted to one normal and one superconducting electrode, the formula is modified as follows I = e de ( B(E) + A(E)) (f L (E) f R (E)) (.) h where B = r ee is the normal-reflection coefficient and decreases the current A = r he is the Andreev-reflection coefficient and increases the current In particular at zero temperature T =, we have where we have set I = e h ev de ( B(E) + A(E)) (.3) µ L = ε F + ev µ R = ε F V > 6
17 Chap.. Current-voltage Characteristics 7 The non-linear conductance at zero temperature then reads G NS (V ) =. di dv = e ( B(eV ) + A(eV )) (.4) h and explicitly G NS (V ) = e h (ev ) + ( + Z ) ( (ev ) ) ev ev + ( + Z ) (ev ) ev < ev > (.5) In particular we notice that In the subgap regime ev we have A + B = due to unitarity, so that we can also write G NS (V ) = 4e A(eV ) (.6) h In the limit of high voltage with respect to the gap (ev ), superconducting effects become negligible and we obtain the normal conductance (effectively this is equivalent to sending ) G NS (ev ) G NN = e h whence we read off the normal transmission coefficient T N = + Z (.7) + Z (.8) of the interface, as anticipated in Eq.(.6). Equivalently one often denotes by R N = G NN = the resistance of the normal junction. h e ( + Z ) (.9) The non-linear conductance is plotted at zero temperature in Fig.. for different values of the interface transparency. We observe that For high transparency the subpag regime is dominated by Andreev processes (A ) and therefore G NS is finite, whereas at low transparency Andreev reflection is strongly suppressed in favor of normal reflection, yielding a strong reduction of G NS (V ). G NS (V ) exhibits a cusp at ev =, corresponding to the singularity of the density of states of the superconductor at the gap.
18 8 Current and Conductance 4 3 Z =. Z =. R N G N S e V / Figure.: The current-voltage characteristics of an N-S junction within the BTK model is plotted at zero temperature and for different values of the barrier strength. The cusp at ev = corresponds to the singularity of the density of states of the superconductor at the gap... The limit of low transparency at arbitrary V We can now consider the particular case of a very strong barrier, i.e. a low-transparency interface Z T N (.) and consider the non-linear conductance to lowest order in O(/Z ) (i.e. lowest order in T N ) as a function of V. We obtain from (.5) ev < G NS (V ) = e h ev (.) Z ev > (ev ) Recalling that to lowest order e h Z e h T N = G NN (.) and using the definition of density of states for a supercondutor E N s (E) = N() θ(e ) (.3) E we can also rewrite that for a low transparency barrier G NS (V ) = G NN N s (ev ) N() Z (.4)
19 Chap.. Current-voltage Characteristics 9.. The linear conductance at arbitrary transparency We can now look at the limiting case of the linear conductance G NS () =. di dv = e ( B() + A()) (.5) V = h Recalling that in the subgap regime B(E) = A(E) because of unitarity, we can also write G NS () = 4e 4e A() = (.6) h h ( + Z ) The linear conductance is thus twice the quantum of conductance e /h multiplied by the Andreev reflection coefficient. One can also re-express the linear conductance in another form, exploiting the normal transmission coefficient derived above. T N = + Z Z = T N T N It is indeed straightforward to check that inserting the above expression into Eq.(.6) one obtains G NS () = 4e TN (.7) h ( T N ) We observe that Differently from the the normal conductance G NN () = e h T N the G NS conductance of an N-S junction is a non linear function of the normal transmission coefficient T N. Since T N one has the inequality At low transparency T N, we have that G NS () G NN () (.8) G NS () = O(T N) (.9) i.e. the linear conductance is vanishing to lowest order in the transmission. This is in agreement with the result of the tunneling approach, where one must compute conductance to higher orders in the tunneling amplitudes to obtain non-vanishing contributions.
20 Bibliography [] A. M. Zagoskin, Quantum Theory of Many-Body System, Springer Verlag New York (998). [] M. Tinkham, Introduction to Superconductivity, Dover Publications, New York (975). [3] Y. Blanter and M. Büttiker, Shot Noise in Mesoscopic Conductors, [ArXiv version cond-mat/9958] [4] C. W. J. Beenakker, in Transport Phenomena in Mesoscopic Systems (eds. H. Fukuyama, and T. Ando), Springer Series in Solid State Science 9, Springer Verlag Heidelberg (99). Articles [5] A. F. Andreev, Sov. Phys. JETP 9, 8 (964). [6] S. N. Artemenko, A. F. Volkov, and A. V. Zaitsev, JETP Lett. 8, 589 (978); Sov. Phys. JETP 49, 94 (979); Solid State Comm. 3, 77 (979); A. V. Zaitsev, Sov. Phys. JETP 5, (98). [7] G. E. Blonder, M Tinkham, T. M. Klapwijk, Phys. Rev. B 5, 455 (98). [8] C. J. Lambert and R. Raimondi, J. Phys. Cond. Matt., 9 (998); C. J. Lambert, J. Phys Cond. Matt. 3, 6579 (99); C. J. Lambert et al., J. Phys. Cond. Matt. 5, 487 (993). [9] C. W. J. Beenakker, in Quantum Transport in Semiconductor-Superconductor microjunctions, ArXiv cond-mat/94683
arxiv:cond-mat/ v1 [cond-mat.mes-hall] 27 Nov 2001
Published in: Single-Electron Tunneling and Mesoscopic Devices, edited by H. Koch and H. Lübbig (Springer, Berlin, 1992): pp. 175 179. arxiv:cond-mat/0111505v1 [cond-mat.mes-hall] 27 Nov 2001 Resonant
More informationarxiv:cond-mat/ v2 [cond-mat.mes-hall] 10 Sep 2005
Quantum transport in semiconductor superconductor microjunctions arxiv:cond-mat/9406083v2 [cond-mat.mes-hall] 10 Sep 2005 C. W. J. Beenakker Instituut-Lorentz, University of Leiden P.O. Box 9506, 2300
More informationSplitting of a Cooper pair by a pair of Majorana bound states
Chapter 7 Splitting of a Cooper pair by a pair of Majorana bound states 7.1 Introduction Majorana bound states are coherent superpositions of electron and hole excitations of zero energy, trapped in the
More informationIntroduction to the Scattering Matrix Formalism
Introduction to the Scattering Matrix Formalism Fabrizio Dolcini Scuola Normale Superiore di Pisa, NEST Italy Dipartimento di Fisica del Politecnico di Torino Italy Lecture Notes for XXIII Physics GradDays,
More informationLocal and nonlocal Andreev reflections in double barrier junctions
Journal of Physics: Conference Series Local and nonlocal Andreev reflections in double barrier junctions To cite this article: R J Wojciechowski and L Kowalewski J. Phys.: Conf. Ser. 3 8 View the article
More informationLecture 5. Potentials
Lecture 5 Potentials 51 52 LECTURE 5. POTENTIALS 5.1 Potentials In this lecture we will solve Schrödinger s equation for some simple one-dimensional potentials, and discuss the physical interpretation
More informationOne-dimensional potentials: potential step
One-dimensional potentials: potential step Figure I: Potential step of height V 0. The particle is incident from the left with energy E. We analyze a time independent situation where a current of particles
More informationarxiv: v2 [cond-mat.mes-hall] 18 Oct 2010
Tuning Excess Noise by Aharonov-Bohm Interferometry arxiv:13.511v [cond-mat.mes-hall] 18 Oct 1 Fabrizio Dolcini 1, and Hermann Grabert, 3 1 Dipartimento di Fisica del Politecnico di Torino, I-119 Torino,
More informationJosephson Effect in FS/I/N/I/FS Tunnel Junctions
Commun. Theor. Phys. Beijing, China 52 29 pp. 72 725 c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 4, October 5, 29 Josephson Effect in FS/I/N/I/FS Tunnel Junctions LI Xiao-Wei Department
More informationJosephson currents in two dimensional mesoscopic ballistic conductors Heida, Jan Peter
University of Groningen Josephson currents in two dimensional mesoscopic ballistic conductors Heida, Jan Peter IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you
More informationCONDUCTANCE OF SUPERCONDUCTOR-NORMAL METAL CONTACT JUNCTION BEYOND QUASICLASSICAL APPROXIMATION VLADIMIR LUKIC. BSc, University of Belgrade, 1997
CONDUCTANCE OF SUPERCONDUCTOR-NORMAL METAL CONTACT JUNCTION BEYOND QUASICLASSICAL APPROXIMATION BY VLADIMIR LUKIC BSc, University of Belgrade, 1997 DISSERTATION Submitted in partial fulfillment of the
More informationFerromagnetic superconductors
Department of Physics, Norwegian University of Science and Technology Pisa, July 13 2007 Outline 1 2 Analytical framework Results 3 Tunneling Hamiltonian Josephson current 4 Quadratic term Cubic term Quartic
More informationarxiv: v1 [cond-mat.mes-hall] 26 Jun 2009
S-Matrix Formulation of Mesoscopic Systems and Evanescent Modes Sheelan Sengupta Chowdhury 1, P. Singha Deo 1, A. M. Jayannavar 2 and M. Manninen 3 arxiv:0906.4921v1 [cond-mat.mes-hall] 26 Jun 2009 1 Unit
More informationSupplementary figures
Supplementary figures Supplementary Figure 1. A, Schematic of a Au/SRO113/SRO214 junction. A 15-nm thick SRO113 layer was etched along with 30-nm thick SRO214 substrate layer. To isolate the top Au electrodes
More information2 Metallic point contacts as a physical tool
2 Metallic point contacts as a physical tool Already more than 100 years ago Drude developed a theory for the electrical and thermal conduction of metals based on the classic kinetic theory of gases. Drude
More informationScattering in one dimension
Scattering in one dimension Oleg Tchernyshyov Department of Physics and Astronomy, Johns Hopkins University I INTRODUCTION This writeup accompanies a numerical simulation of particle scattering in one
More informationSuperconductivity at nanoscale
Superconductivity at nanoscale Superconductivity is the result of the formation of a quantum condensate of paired electrons (Cooper pairs). In small particles, the allowed energy levels are quantized and
More informationTheory of Lifetime Effects in Point-Contacts: Application to Cd 2 Re 2 O 7
Theory of Lifetime Effects in Point-Contacts: Application to Cd 2 Re 2 O 7 Božidar Mitrović Department of Physics Brock University St. Catharines, Ontario, Canada McMaster, May 24, 2013 Outline Tunneling
More informationSUPPLEMENTARY INFORMATION
Josephson supercurrent through a topological insulator surface state M. Veldhorst, M. Snelder, M. Hoek, T. Gang, V. K. Guduru, X. L.Wang, U. Zeitler, W. G. van der Wiel, A. A. Golubov, H. Hilgenkamp and
More informationSemiklassik von Andreev-Billards
Institut Physik I - Theoretische Physik Universität Regensburg Semiklassik von Andreev-Billards Diplomarbeit von Thomas Engl aus Roding unter Anleitung von Prof. Dr. Klaus Richter abgegeben am 12.05.2010
More information8.04: Quantum Mechanics Professor Allan Adams. Problem Set 7. Due Tuesday April 9 at 11.00AM
8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Thursday April 4 Problem Set 7 Due Tuesday April 9 at 11.00AM Assigned Reading: E&R 6 all Li. 7 1 9, 8 1 Ga. 4 all, 6
More informationDetecting and using Majorana fermions in superconductors
Detecting and using Majorana fermions in superconductors Anton Akhmerov with Carlo Beenakker, Jan Dahlhaus, Fabian Hassler, and Michael Wimmer New J. Phys. 13, 053016 (2011) and arxiv:1105.0315 Superconductor
More informationChapter 3 Properties of Nanostructures
Chapter 3 Properties of Nanostructures In Chapter 2, the reduction of the extent of a solid in one or more dimensions was shown to lead to a dramatic alteration of the overall behavior of the solids. Generally,
More informationarxiv:cond-mat/ v1 7 Jun 1994
Reflectionless tunneling through a double-barrier NS junction J. A. Melsen and C. W. J. Beenakker Instituut-Lorentz, University of Leiden P.O. Box 9506, 2300 RA Leiden, The Netherlands arxiv:cond-mat/9406034v1
More informationGraphene Field effect transistors
GDR Meso 2008 Aussois 8-11 December 2008 Graphene Field effect transistors Jérôme Cayssol CPMOH, UMR Université de Bordeaux-CNRS 1) Role of the contacts in graphene field effect transistors motivated by
More informationTransport through Andreev Bound States in a Superconductor-Quantum Dot-Graphene System
Transport through Andreev Bound States in a Superconductor-Quantum Dot-Graphene System Nadya Mason Travis Dirk, Yung-Fu Chen, Cesar Chialvo Taylor Hughes, Siddhartha Lal, Bruno Uchoa Paul Goldbart University
More informationPublication II. c (2007) The American Physical Society. Reprinted with permission.
Publication II Nikolai B. Kopnin and Juha Voutilainen, Nonequilibrium charge transport in quantum SINIS structures, Physical Review B, 75, 174509 (2007). c (2007) The American Physical Society. Reprinted
More information1.1 A Scattering Experiment
1 Transfer Matrix In this chapter we introduce and discuss a mathematical method for the analysis of the wave propagation in one-dimensional systems. The method uses the transfer matrix and is commonly
More informationarxiv: v1 [hep-th] 29 Sep 2017
Radiation enhancement and temperature in the collapse regime of gravitational scattering arxiv:1709.10375v1 [hep-th] 29 Sep 2017 (Dipartimento di Fisica, Università di Firenze and INFN Sezione di Firenze)
More informationNovel topologies in superconducting junctions
Novel topologies in superconducting junctions Yuli V. Nazarov Delft University of Technology The Capri Spring School on Transport in Nanostructures 2018, Anacapri IT, April 15-22 2018 Overview of 3 lectures
More informationShot-noise and conductance measurements of transparent superconductor/two-dimensional electron gas junctions
Shot-noise and conductance measurements of transparent superconductor/two-dimensional electron gas junctions B.-R. Choi, A. E. Hansen, T. Kontos, C. Hoffmann, S. Oberholzer, W. Belzig, and C. Schönenberger*
More informationLesson 1: Molecular electric conduction. I. The density of states in a 1D quantum wire
Lesson 1: Molecular electric conduction I. The density of states in a 1D quantum wire Consider a quantum system with discrete energy levels E n, n = 1,2, Take an energy (not necessarily an energy level)
More informationNew Quantum Transport Results in Type-II InAs/GaSb Quantum Wells
New Quantum Transport Results in Type-II InAs/GaSb Quantum Wells Wei Pan Sandia National Laboratories Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation,
More informationCharge carrier statistics/shot Noise
Charge carrier statistics/shot Noise Sebastian Waltz Department of Physics 16. Juni 2010 S.Waltz (Biomolecular Dynamics) Charge carrier statistics/shot Noise 16. Juni 2010 1 / 36 Outline 1 Charge carrier
More informationarxiv:cond-mat/ v1 23 Feb 1993
Subgap Conductivity of a Superconducting-Normal Tunnel Interface F. W. J. Hekking and Yu.V. Nazarov Institut für Theoretische Festkörperphysik, Universität Karlsruhe, Postfach 6980, 7500 Karlsruhe, FRG
More informationDensity of States in Superconductor -Normal. Metal-Superconductor Junctions arxiv:cond-mat/ v2 [cond-mat.mes-hall] 7 Nov 1998.
Density of States in Superconductor -Normal Metal-Superconductor Junctions arxiv:cond-mat/97756v [cond-mat.mes-hall] 7 Nov 1998 F. Zhou 1,, P. Charlat, B. Spivak 1, B.Pannetier 1 Physics Department, University
More informationAbsence of Andreev reflections and Andreev bound states above the critical temperature
PHYSICAL REVIEW B VOLUME 61, NUMBER 10 1 MARCH 2000-II Absence of Andreev reflections and Andreev bound states above the critical temperature Y. Dagan, A. Kohen, and G. Deutscher School of Physics and
More informationInterferometric and noise signatures of Majorana fermion edge states in transport experiments
Interferometric and noise signatures of ajorana fermion edge states in transport experiments Grégory Strübi, Wolfgang Belzig, ahn-soo Choi, and C. Bruder Department of Physics, University of Basel, CH-056
More informationLecture 12. Electron Transport in Molecular Wires Possible Mechanisms
Lecture 12. Electron Transport in Molecular Wires Possible Mechanisms In Lecture 11, we have discussed energy diagrams of one-dimensional molecular wires. Here we will focus on electron transport mechanisms
More informationModel Question Paper ENGINEERING PHYSICS (14PHY12/14PHY22) Note: Answer any FIVE full questions, choosing one full question from each module.
Model Question Paper ENGINEERING PHYSICS (14PHY1/14PHY) Time: 3 hrs. Max. Marks: 100 Note: Answer any FIVE full questions, choosing one full question from each module. MODULE 1 1) a. Explain in brief Compton
More informationGeneralization of the BTK Theory to the Case of Finite Quasiparticle Lifetimes
Generalization of the BTK Theory to the Case of Finite Quasiparticle Lifetimes by Yousef Rohanizadegan A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in
More informationarxiv:cond-mat/ v1 [cond-mat.mes-hall] 2 Feb 1998
Transport through an Interacting Quantum Dot Coupled to Two Superconducting Leads arxiv:cond-mat/9828v [cond-mat.mes-hall] 2 Feb 998 Kicheon Kang Department of Physics, Korea University, Seoul 36-7, Korea
More informationp 0 Transition in Superconductor-Ferromagnet-Superconductor Junctions 1
JETP Letters, Vol. 74, No. 6, 2001, pp. 323 327. From Pis ma v Zhurnal Éksperimental noœ i Teoreticheskoœ Fiziki, Vol. 74, No. 6, 2001, pp. 357 361. Original English Text Copyright 2001 by Chtchelkatchev,
More informationThe quantum mechanical character of electronic transport is manifest in mesoscopic
Mesoscopic transport in hybrid normal-superconductor nanostructures The quantum mechanical character of electronic transport is manifest in mesoscopic systems at low temperatures, typically below 1 K.
More information8.04 Quantum Physics Lecture XV One-dimensional potentials: potential step Figure I: Potential step of height V 0. The particle is incident from the left with energy E. We analyze a time independent situation
More informationSplitting Kramers degeneracy with superconducting phase difference
Splitting Kramers degeneracy with superconducting phase difference Bernard van Heck, Shuo Mi (Leiden), Anton Akhmerov (Delft) arxiv:1408.1563 ESI, Vienna, 11 September 2014 Plan Using phase difference
More informationCoulomb entangler and entanglement-testing network for waveguide qubits
PHYSICAL REVIEW A 72, 032330 2005 Coulomb entangler and entanglement-testing network for waveguide qubits Linda E. Reichl and Michael G. Snyder Center for Studies in Statistical Mechanics and Complex Systems,
More informationarxiv: v1 [cond-mat.supr-con] 1 Nov 2011
Signatures of d-wave Symmetry on Thermal Dirac Fermions in Graphene-Based F/I/d Junctions Morteza Salehi, a Mohammad Alidoust, b and Gholamreza Rashedi Department of Physics, Faculty of Sciences, University
More informationSupplementary Figures
Supplementary Figures Supplementary Figure 1 Point-contact spectra of a Pt-Ir tip/lto film junction. The main panel shows differential conductance at 2, 12, 13, 16 K (0 T), and 10 K (2 T) to demonstrate
More informationPhysics of Semiconductors
Physics of Semiconductors 13 th 2016.7.11 Shingo Katsumoto Department of Physics and Institute for Solid State Physics University of Tokyo Outline today Laughlin s justification Spintronics Two current
More informationProximity effect, quasiparticle transport, and local magnetic moment in ferromagnet d-wave-superconductor junctions
PHYSICAL REVIEW B VOLUME 61, NUMBER 2 1 JANUARY 2000-II Proximity effect, quasiparticle transport, and local magnetic moment in ferromagnet d-wave-superconductor junctions Jian-Xin Zhu Texas Center for
More informationCurrent and voltage fluctuations in microjunctions between normal metals and superconductors
Current and voltage fluctuations in microjunctions between normal metals and superconductors V.A. Khlus All-Soviet Institute for Research on Single-Crystal Materials (Khar 'kov) (Submitted 7 June 1987)
More informationJ10M.1 - Rod on a Rail (M93M.2)
Part I - Mechanics J10M.1 - Rod on a Rail (M93M.2) J10M.1 - Rod on a Rail (M93M.2) s α l θ g z x A uniform rod of length l and mass m moves in the x-z plane. One end of the rod is suspended from a straight
More informationarxiv:cond-mat/ v1 [cond-mat.mes-hall] 25 Jun 1999
CHARGE RELAXATION IN THE PRESENCE OF SHOT NOISE IN COULOMB COUPLED MESOSCOPIC SYSTEMS arxiv:cond-mat/9906386v1 [cond-mat.mes-hall] 25 Jun 1999 MARKUS BÜTTIKER Département de Physique Théorique, Université
More informationAndreev transport in 2D topological insulators
Andreev transport in 2D topological insulators Jérôme Cayssol Bordeaux University Visiting researcher UC Berkeley (2010-2012) Villard de Lans Workshop 09/07/2011 1 General idea - Topological insulators
More informationarxiv: v1 [cond-mat.mes-hall] 1 Aug 2013
International Journal of Modern Physics B c World Scientific Publishing Company arxiv:1308.0138v1 [cond-mat.mes-hall] 1 Aug 013 Electron-electron interaction effects on transport through mesoscopic superconducting
More informationUniversal conductance peak in the transport of electrons through a floating topological superconductor
Universal conductance peak in the transport of electrons through a floating topological superconductor von Jonas Stapmanns Bachelorarbeit in Physik vorgelegt der Fakultät für Mathematik, Informatik und
More informationarxiv: v1 [cond-mat.mes-hall] 23 Sep 2011
Normal reflection at superconductor - normal metal interfaces due to Fermi surface mismatch Elina Tuuli 1 3 and Kurt Gloos 1,3 arxiv:19.38v1 [cond-mat.mes-hall] 23 Sep 211 1 Wihuri Physical Laboratory,
More informationQuantum transport in nanoscale solids
Quantum transport in nanoscale solids The Landauer approach Dietmar Weinmann Institut de Physique et Chimie des Matériaux de Strasbourg Strasbourg, ESC 2012 p. 1 Quantum effects in electron transport R.
More informationSuperconductivity Induced Transparency
Superconductivity Induced Transparency Coskun Kocabas In this paper I will discuss the effect of the superconducting phase transition on the optical properties of the superconductors. Firstly I will give
More information+1-1 R
SISSA ISAS SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI - INTERNATIONAL SCHOOL FOR ADVANCED STUDIES I-34014 Trieste ITALY - Via Beirut 4 - Tel. [+]39-40-37871 - Telex:460269 SISSA I - Fax: [+]39-40-3787528.
More informationThe PWcond code: Complex bands, transmission, and ballistic conductance
The PWcond code: Complex bands, transmission, and ballistic conductance SISSA and IOM-DEMOCRITOS Trieste (Italy) Outline 1 Ballistic transport: a few concepts 2 Complex band structure 3 Current of a Bloch
More informationAndreev conductance of chaotic and integrable quantum dots
PHYSICAL REVIEW B VOLUME 6, NUMBER 15 15 OCTOBER 000-I Andreev conductance of chaotic and integrable quantum dots A. A. Clerk, P. W. Brouwer, and V. Ambegaokar Laboratory of Atomic and Solid State Physics,
More informationModeling Schottky barrier SINIS junctions
Modeling Schottky barrier SINIS junctions J. K. Freericks, B. Nikolić, and P. Miller * Department of Physics, Georgetown University, Washington, DC 20057 * Department of Physics, Brandeis University, Waltham,
More informationQUANTUM INTERFERENCE IN SEMICONDUCTOR RINGS
QUANTUM INTERFERENCE IN SEMICONDUCTOR RINGS PhD theses Orsolya Kálmán Supervisors: Dr. Mihály Benedict Dr. Péter Földi University of Szeged Faculty of Science and Informatics Doctoral School in Physics
More informationSummary of Last Time Barrier Potential/Tunneling Case I: E<V 0 Describes alpha-decay (Details are in the lecture note; go over it yourself!!) Case II:
Quantum Mechanics and Atomic Physics Lecture 8: Scattering & Operators and Expectation Values http://www.physics.rutgers.edu/ugrad/361 Prof. Sean Oh Summary of Last Time Barrier Potential/Tunneling Case
More informationThe Particle in a Box
Page 324 Lecture 17: Relation of Particle in a Box Eigenstates to Position and Momentum Eigenstates General Considerations on Bound States and Quantization Continuity Equation for Probability Date Given:
More informationSpin dynamics through homogeneous magnetic superlattices
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/243587981 Spin dynamics through homogeneous magnetic superlattices Article in physica status
More informationPhysics of Semiconductors (Problems for report)
Physics of Semiconductors (Problems for report) Shingo Katsumoto Institute for Solid State Physics, University of Tokyo July, 0 Choose two from the following eight problems and solve them. I. Fundamentals
More informationarxiv:hep-th/ v1 11 Mar 2005
Scattering of a Klein-Gordon particle by a Woods-Saxon potential Clara Rojas and Víctor M. Villalba Centro de Física IVIC Apdo 21827, Caracas 12A, Venezuela (Dated: February 1, 28) Abstract arxiv:hep-th/5318v1
More informationQuantum Size Effect of Two Couple Quantum Dots
EJTP 5, No. 19 2008) 33 42 Electronic Journal of Theoretical Physics Quantum Size Effect of Two Couple Quantum Dots Gihan H. Zaki 1), Adel H. Phillips 2) and Ayman S. Atallah 3) 1) Faculty of Science,
More informationQuasi-particle current in planar Majorana nanowires
Journal of Physics: Conference Series PAPER OPEN ACCESS Quasi-particle current in planar Majorana nanowires To cite this article: Javier Osca and Llorenç Serra 2015 J. Phys.: Conf. Ser. 647 012063 Related
More informationPlasmon Generation through Electron Tunneling in Graphene SUPPORTING INFORMATION
Plasmon Generation through Electron Tunneling in Graphene SUPPORTING INFORMATION Sandra de Vega 1 and F. Javier García de Abajo 1, 2 1 ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science
More informationSPIN-POLARIZED CURRENT IN A MAGNETIC TUNNEL JUNCTION: MESOSCOPIC DIODE BASED ON A QUANTUM DOT
66 Rev.Adv.Mater.Sci. 14(2007) 66-70 W. Rudziński SPIN-POLARIZED CURRENT IN A MAGNETIC TUNNEL JUNCTION: MESOSCOPIC DIODE BASED ON A QUANTUM DOT W. Rudziński Department of Physics, Adam Mickiewicz University,
More informationIS THERE ANY KLEIN PARADOX? LOOK AT GRAPHENE! D. Dragoman Univ. Bucharest, Physics Dept., P.O. Box MG-11, Bucharest,
1 IS THERE ANY KLEIN PARADOX? LOOK AT GRAPHENE! D. Dragoman Univ. Bucharest, Physics Dept., P.O. Box MG-11, 077125 Bucharest, Romania, e-mail: danieladragoman@yahoo.com Abstract It is demonstrated that
More informationLandau Bogolubov Energy Spectrum of Superconductors
Landau Bogolubov Energy Spectrum of Superconductors L.N. Tsintsadze 1 and N.L. Tsintsadze 1,2 1. Department of Plasma Physics, E. Andronikashvili Institute of Physics, Tbilisi 0128, Georgia 2. Faculty
More informationarxiv: v4 [cond-mat.quant-gas] 19 Jun 2012
Alternative Route to Strong Interaction: Narrow Feshbach Resonance arxiv:1105.467v4 [cond-mat.quant-gas] 19 Jun 01 Tin-Lun Ho, Xiaoling Cui, Weiran Li Department of Physics, The Ohio State University,
More informationSpin Superfluidity and Graphene in a Strong Magnetic Field
Spin Superfluidity and Graphene in a Strong Magnetic Field by B. I. Halperin Nano-QT 2016 Kyiv October 11, 2016 Based on work with So Takei (CUNY), Yaroslav Tserkovnyak (UCLA), and Amir Yacoby (Harvard)
More informationINTRODUCTION À LA PHYSIQUE MÉSOSCOPIQUE: ÉLECTRONS ET PHOTONS INTRODUCTION TO MESOSCOPIC PHYSICS: ELECTRONS AND PHOTONS
Chaire de Physique Mésoscopique Michel Devoret Année 2007, Cours des 7 et 14 juin INTRODUCTION À LA PHYSIQUE MÉSOSCOPIQUE: ÉLECTRONS ET PHOTONS INTRODUCTION TO MESOSCOPIC PHYSICS: ELECTRONS AND PHOTONS
More informationUniversity of Illinois at Chicago Department of Physics. Quantum Mechanics Qualifying Examination. January 7, 2013 (Tuesday) 9:00 am - 12:00 noon
University of Illinois at Chicago Department of Physics Quantum Mechanics Qualifying Examination January 7, 13 Tuesday 9: am - 1: noon Full credit can be achieved from completely correct answers to 4 questions
More informationWeek 5-6: Lectures The Charged Scalar Field
Notes for Phys. 610, 2011. These summaries are meant to be informal, and are subject to revision, elaboration and correction. They will be based on material covered in class, but may differ from it by
More informationTunneling Spectroscopy of Disordered Two-Dimensional Electron Gas in the Quantum Hall Regime
Tunneling Spectroscopy of Disordered Two-Dimensional Electron Gas in the Quantum Hall Regime The Harvard community has made this article openly available. Please share how this access benefits you. Your
More informationarxiv:cond-mat/ v1 [cond-mat.supr-con] 8 Jan 2006
JETP Letters 81, 10 (2005) Anomalous Behavior near T c and Synchronization of Andreev Reflection in Two-Dimensional Arrays of SNS Junctions arxiv:cond-mat/0601142v1 [cond-mat.supr-con] 8 Jan 2006 T.I.
More informationarxiv: v1 [cond-mat.mes-hall] 25 Feb 2008
Cross-correlations in transport through parallel quantum dots Sebastian Haupt, 1, 2 Jasmin Aghassi, 1, 2 Matthias H. Hettler, 1 and Gerd Schön 1, 2 1 Forschungszentrum Karlsruhe, Institut für Nanotechnologie,
More informationEffect of Spin-Orbit Interaction and In-Plane Magnetic Field on the Conductance of a Quasi-One-Dimensional System
ArXiv : cond-mat/0311143 6 November 003 Effect of Spin-Orbit Interaction and In-Plane Magnetic Field on the Conductance of a Quasi-One-Dimensional System Yuriy V. Pershin, James A. Nesteroff, and Vladimir
More informationPHYS 508 (2015-1) Final Exam January 27, Wednesday.
PHYS 508 (2015-1) Final Exam January 27, Wednesday. Q1. Scattering with identical particles The quantum statistics have some interesting consequences for the scattering of identical particles. This is
More informationQuasiclassical description of transport through superconducting contacts
PHYSICAL REVIEW B, VOLUME 64, 104502 Quasiclassical description of transport through superconducting contacts J. C. Cuevas Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-76128 Karlsruhe,
More informationarxiv:cond-mat/ v2 [cond-mat.mes-hall] 23 May 2006
Published in: Transport Phenomena in Mesoscopic Systems, edited by H. Fukuyama and T. Ando (Springer, Berlin, 1992). arxiv:cond-mat/0406127v2 [cond-mat.mes-hall] 23 May 2006 Three universal mesoscopic
More informationNonlocal transport properties due to Andreev scattering
Charles Univ. in Prague, 5 X 2015 Nonlocal transport properties due to Andreev scattering Tadeusz Domański Marie Curie-Skłodowska University, Lublin, Poland http://kft.umcs.lublin.pl/doman/lectures Outline
More informationLECTURE 3: Refrigeration
LECTURE 3: Refrigeration Refrigeration on-chip Thermoelectric refrigeration Peltier refrigerators, Peltier 1834 Thermionic refrigeration, Mahan, 1994 Korotkov and Likharev, 1999 Quantum-dot refrigerator,
More informationFractional charge in the fractional quantum hall system
Fractional charge in the fractional quantum hall system Ting-Pong Choy 1, 1 Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green St., Urbana, IL 61801-3080, USA (Dated: May
More informationSimple Explanation of Fermi Arcs in Cuprate Pseudogaps: A Motional Narrowing Phenomenon
Simple Explanation of Fermi Arcs in Cuprate Pseudogaps: A Motional Narrowing Phenomenon ABSTRACT: ARPES measurements on underdoped cuprates above the superconducting transition temperature exhibit the
More informationWeakly nonlinear ac response: Theory and application. Physical Review B (Condensed Matter and Materials Physics), 1999, v. 59 n. 11, p.
Title Weakly nonlinear ac response: Theory and application Author(s) Ma, ZS; Wang, J; Guo, H Citation Physical Review B (Condensed Matter and Materials Physics), 1999, v. 59 n. 11, p. 7575-7578 Issued
More informationarxiv:cond-mat/ v1 22 Aug 1994
Submitted to Phys. Rev. B Bound on the Group Velocity of an Electron in a One-Dimensional Periodic Potential arxiv:cond-mat/9408067v 22 Aug 994 Michael R. Geller and Giovanni Vignale Institute for Theoretical
More informationRe-Thinking Schroedinger Boundary Conditions: the Simple Rotator is Not Simple
Re-Thinking Schroedinger Boundary Conditions: the Simple Rotator is Not Simple Arthur Davidson ECE Dept., Carnegie Mellon University Pittsburgh, PA 1513 artdav@ece.cmu.edu ABSTRACT A contradiction arises
More informationChapter 8 Chapter 8 Quantum Theory: Techniques and Applications (Part II)
Chapter 8 Chapter 8 Quantum Theory: Techniques and Applications (Part II) The Particle in the Box and the Real World, Phys. Chem. nd Ed. T. Engel, P. Reid (Ch.16) Objectives Importance of the concept for
More informationManipulation of Majorana fermions via single charge control
Manipulation of Majorana fermions via single charge control Karsten Flensberg Niels Bohr Institute University of Copenhagen Superconducting hybrids: from conventional to exotic, Villard de Lans, France,
More informationKonstantin Y. Bliokh, Daria Smirnova, Franco Nori. Center for Emergent Matter Science, RIKEN, Japan. Science 348, 1448 (2015)
Konstantin Y. Bliokh, Daria Smirnova, Franco Nori Center for Emergent Matter Science, RIKEN, Japan Science 348, 1448 (2015) QSHE and topological insulators The quantum spin Hall effect means the presence
More information2004 American Physical Society. Reprinted with permission.
Pauli Virtanen and Tero T. Heikkilä.. Thermopower induced by a supercurrent in superconductor normal metal structures. Physical Review Letters, volume 9, number 7, 77. American Physical Society Reprinted
More informationMODELING THE FUNDAMENTAL LIMIT ON CONVERSION EFFICIENCY OF QD SOLAR CELLS
MODELING THE FUNDAMENTAL LIMIT ON CONVERSION EFFICIENCY OF QD SOLAR CELLS Ա.Մ.Կեչիյանց Ara Kechiantz Institute of Radiophysics and Electronics (IRPhE), National Academy of Sciences (Yerevan, Armenia) Marseille
More information