András Halbritter: Point contact Andreev spectroscopy Budapest University of Technology and Economics Department of Physics Low temperature solid state physics laboratory Coworkers: correlated electron systems, magnetic semiconductors (Pressure cell (up to 30kBar), MOKE setup Prof. György Mihály TEP measurements,heat conductivity Dr. Szabolcs Csonka measurements) Attila Geresdi MCBJ, Andreev spectroscopy Péter Makk Outline 1. Theory of point contacts -ballistic contacts -diffusive contacts -”thermal” regime -point contact spectroscopy 2. Andreev spectroscopy -What is spin polarization? N S -BTK theory -diffusive SN contacts -study of magnetic semiconductors In Sb Mn History of point contacts MCBJ technique Touching wires Spear-anvil geometry First application of point-contact spectroscopy: Study of electron-phonon interaction (Yanson, 1974) Later: study of two-level systems, magnetic impurities, etc. Taking advantage of the Possibility to create extreme stability single- heterocontacts atom and single molecule contacts can be investigated Study of unconventional superconductors with NS junctions (heavy fermion superconductors, MgB2, etc.) Spin polarization measurements with FS junctions (Soulen; Upadhyay 1998) V Ballistic point contacts d << l In a ballistic contact the contact diameter is much smaller than the mean free path of the electrons, thus the electrons are only scattered on the walls d 2e d3k j(r) = v f (r) = 2e v f (r) ∑ k k ∫ 3 k k Vsample k (2π ) eV eV ε= ky dk = I = dA jx (x = 0) = A⋅ jx (x = 0) d k ∫ Distribution h vk y A function at the middle of k the contact: x kx Integration over the half sphere In a ballistic contact all the electrons at the contact surface with k >0 come x 2e eV from the left electrode, and all with j (x = 0) = ⋅ ⋅ v cos(ϑ) k <0 come from the right electrode x 3 ∫ k x (2π ) h vk S { 14243 dk 2 kF vk π Sharvin formula in 3D: The electrical potential: (~<µ>) 2 2e2 k d V Ω(r) I = ⋅ F ⋅V Φ(r) = − 1− sgn(z) h 4 2 2π 1424 434 The voltage drops within a GSharvin distance of ~d from the contact In a diffusive contact the Diffusive point contacts contact diameter is much larger that the mean free V path, thus an electron d >> l coming e.g. from the left side of the contact can stem from both electrodes The resistance can be determined by solving the Maxwellequations in a hyperbolic coordinate system: j(r) = σE(r), ∆Φ(r) = 0, d Φ(∞) − Φ(−∞) = eV The Maxwell 1 resistance R = σd Intermediate regime In the intermediate region between the diffusive and ballistic regime an interpolating formula can be set up by solving the Boltzmann equation for d ~ l arbitrary ratio of the contact diameter and mean free path l 16 1 R = + Γ()l / d Wexler formula d 3πσd σd Γ(l/d) is a monotonous dρ / dT The first term is the Sharvin function with d ≈ resistance by inserting the 2 3 dR / dT ne l kF Γ(0) =1; Γ(∞) = 0.694 Drude conductivity σ = ; n = 2 (independent of l!) hkF 3π V Self heating of the contact In a diffusive contact the electron looses its momentum frequently due to elastic scattering. However, to relax its energy inelastic scattering is needed. The average distance between inelastic scatterings is the inelastic diffusive length: ξ = Dτ in in ξin Ifboththemeanfreepathandtheinelastic diffusive length are d much smaller than the contact diameter, l, ξin<<d, then both the electrical and heat conduction can be treated by classical equations: j = −σ∇Φ; q = −κ∇T ∇ σ∇Φ = 0 boundary conditions: ( ) V ∇j = 0; ∇q = −j∇Φ + T(±∞) = T , Φ(±∞) = ± L 2 2 bath ∇ σ∇T +σ ()∇Φ = 0 2 + Wiedemann-Franz law: () 2 2 The contact temp. is κ π 2 k = L ⋅σ , L = B for general contact geom. independent of the diameter! T 3 e 2 2 2 2 2 V Φ (r) 2 2 V T (r) = Tbath + − , TPC = Tbath + Lorentz number 4L L 4L d >> l,ξin (thermal regime) ξin > d >> l l,ξin >> d (ballisticregime) 2 2 2 2 V 2 V 2 V 2 3πd Dissipated power: = V σd Dissipated power: = V σd Dissipated power: = V σ R R R 16l Length-scale of dissipation: d Length-scale of dissipation: ξ in Length-scale of dissipation: ξin 2 V 2 d 2 2 2 2 V V=100mV 2 2 2 2 V 3πd T = T + TPC ≈ Tbath + T ≈ T + PC bath ->T ~300K 4 PC bath 4L PC L ξin 4L 16lξin A.G.M. Jansen et al. J. Phys. C 13, 6073. (1980) Point contact spectroscopy A. Halbritter, L. Borda, A. Zawadowski Advances in Physics 53, 939 (2004) The inelastic scattering can cause backscattering through the contact, d << l V which is reflected by a nonlinearity in the I-V curve: g(E) excitation spectrum d2I/dV2 shows the spectrum of the inelastic excitations in the close hω E vicinity of the contact I ω eV h eV ε= ky dI/dV Distribution d function at the middle of kx the contact: hω eV d2I/dV2 At T=0 an electron at the contact going hω to the right cannot be scattered to the eV occupied right-going states, but it can be scattered to the unoccupied left-going states Andreev spectroscopy nm-sized ballistic contacts with good stability screw-thread Nb tip BUTE, 2005 SC tip sample piezo actuator ferromagnetic sample For a normal metal with P=0 an incoming electron is In a half-metal (P=1) Andreev reflection is - Andreev reflected, thus for each incoming e a charge prohibited, GFS=0 of 2e is transmitted, GNS=2GN The fit of the I-V curves tells the spin-polarization! As a first approx.: GFS(V=0)=2(1-PC)GN The research on spin polarization is not only essential for better The inportance of fundamental understanding, but the wide application range of spin polarization magnetoresistive devices makes it technologically relevant as well. GMR (Giant MagnetoResistance, 1988) F N F F NF Spin-valve Low High Low Resistance High Resistance The magneto-resistance is ~10% at room-T! Magnetic Tunnel Junction Magneto-resistance up to 70%! Application: MRAM FFI FFI (Magnetic Random Acces Memory) 2004, IBM -> 16Mb Much faster than flash memory! Low Resistance High Resistance Spin –Transistor concept (Datta, Das, 1990) V Not yet realized. g A FET with spin-polarized source and drain electrodes. In the 2DEG the spin is Spintronics review: precessed by the gate due to the Rashba 2DEG I. Zutic et al. Rev. Mod. Phys. effect. 76 323 (2004) For a ballistic contact: Definitions of spin polarization ↑ ↑ ↓ ↓ ρF vF − ρF vF j ~ ρFevF ⇒ PC = ↑ ↓ ↑ ↑ ↓ ↓ Density of states ρ − ρ ρF vF + ρF vF P = F F polarization: ↑ ↓ ρF + ρF For a diffusive contact: ↑ ↓ 2 ↑ ↑ ↓ ↓ Polarization of the current: I − I ρFe τ ρF / meff − ρF / meff P = j ~ E ⇒ PC = (contact polarization) C ↑ ↓ ↑ ↑ ↓ ↓ I + I meff ρF / meff + ρF / meff ε F ↑ ↓ Simplified DOS for some ferromagnets: Magnetization: M ~ ∫ ρ (ε ) − ρ (ε )dε −∞ M>0, P>0, PC>0 M>0, P<0, PC≈0 Spin-pol. is a Fermi surface property, while the magnetization counts for all the electrons! d-character band, split due to s-character band, mobile electrons The so-called half-metals are fully spin-polarized exchange interaction, immobile with large vF and small meff electrons with small vF and large meff Other methods to measure spin-polarization 1. Spin-polarized photoemission spectroscopy -The spin orientation of the e photoelectrons is detected photon e -Disadvantage: low energy resolution (100-200 meV) P = 0 2. Tunneling spectroscopy Insulator Ferromagnet Superconductor •Hard to fabricate •High energy resolution (<1 meV) H •Tunnel junction (typically Al/Al2O3/FM) For normal metal-superconductor tunneling the G(V) curve shows the superconducting DOS: I + ~ T ⋅ dε ρ (ε − eV ) f (ε − eV )⋅ ρ (ε) 1− f (ε) ∫ N N S ( S ) I − ~ T ⋅ dε ρ (ε) f (ε)⋅ ρ (ε − eV ) 1− f (ε − eV ) ∫ S S N ()N ∆ eV I = I + − I − ~ T ⋅ ρ (ε ) dε ρ (ε) f (ε − eV ) − f (ε) N F ∫ S ()N S Source: dI ' ~ T ⋅ ρN (ε F ) dε ρS (ε) f N (ε − eV ) = TρN (ε F )ρS (eV ) C.H. Kant ∫ T =0 dV Ph.D. thesis To detect spin-polarization a Zeeman splitting is applied by an external field R. Meservey, P.M. Tedrow, Phys. Rep. 238, 173 (1994) The Bogoliubov-de Gennes equation: „Electron-like” state BTK theory H ∆ f (x) (conductance of a ballistic NS junction) Ψ = EΨ, where Ψ(x) = ∗ ∗ ∆ − H g(x) G.E. Blonder, M.Tinkham, measured from EF N S T.M.Klapwijk, PRB 25, 4515 (1982) 2 d2 „Hole-like” state H = − h − E +V (x) 2m dx2 F coupling 2E dimensionless V (x) = Z F δ (x) „barrier strength” 2 2 2 k kF 2 h 2 E = − EF + ∆ 2m electron-like band hole-like band –thegroup velocity has opposite sign to the wave number ∆ N = 0 ∆S = ∆ 1 0 1 u i(k +k )x v i(−k +k )x i(kF +kN )x i(kF −kN )x −i(kF +kN )x Ψ (x) = c e F S + d e F N ΨN (x) =1 e + a e + b e S 0 1 0 v u Andreev reflection normal reflection quasiparticle transmission Matchingthewavefunctions: 2EF 2m ΨN (0) = ΨS (0) ≡ Ψ(0), Ψ'N (0) − Ψ'S (0) = Z 2 Ψ(0) kF h Reflection probabilities: 2 The probability for Andreev reflection: A = a 2 The probability for normal reflection: B = b E < ∆ E > ∆ ∆2 ε 2 −1 A = 2 2 2 2 2 A = 2 E + (∆ − E )(1+ 2Z ) []ε + (1+ 2Z 2 ) 4Z 2 (1+ Z 2 ) B =1− A B = Source: 2 2 C.H.