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 π 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 (2 ) 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 πσ16 1 R = + Γ()l / d Wexler formula d 3 d σd σ 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 = ; 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 betweenξ inelastic scatterings is the inelastic diffusive length: = Dτ in in ξin Ifboththemeanfreepathandtheinelastic diffusive length are d

much smaller than the contact diameter, l, ξin<

d >> l,ξin (thermal regime) ξin > d >> l l,ξin >> d (ballisticregime) V 2 V 2 2 2 2 2 V 2 σ 3πd Dissipated power: = V σd Dissipated power: = V σd Dissipated power: = V 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 π 2 2 2 V V=100mV 2 2 2 2 V 3 d 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 d Distribution 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) Spin-valve F N F F NF

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) ρ

Definitions of spin polarization For a ballistic contact:ρ ↑v↑ −ρ ↓v↓ ρF F F F ρ j ~ FevF ⇒ PC = ↑ ↓ ρ ↑ ↑ ↓ ↓ Density of states −ρ F vF + ρF vF P = F F τ polarization: ρ ↑ ↓ F + ρF For a diffusive contact:ρ ↑ ↓ 2 ↑ ↑ ρ ↓ ↓ Polarization of the current: I − I Fe ρF / meff − F / meff P = j ~ E ⇒ PC = (contact polarization) C ↑ ↓ ↑ ↑ ↓ ↓ I + I meff F / meff + ρF / meff ρ ε ε F ρ ↑ ↓ ε Simplified DOS for some ferromagnets: Magnetization: M ~ ∫ ( ) − ( )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 (typicallyε ρ Al/Al O /FM) ε 2 3 For normal metal-superconductorε tunnelingε the G(V) ρ ε curve shows the superconducting DOS: ρ ε ρ ε ε ρ ε I + ~ T ⋅ d ( −ε eV ) f ( − eV )⋅ ( )(1− f ( )) ∫ N N S S ρ ε ρ ε − ε ε () I ~ T ⋅ d ( ) f ( )⋅ ( − eV ) 1− f ( − eV ) ∫ S εS ρ N ε N ∆ eV ε () I = I + − I − ~ T ⋅ ( ) d ε( ) f ( − eV ) − fε( ) N F ∫ S N S ρ Source: dI ' ε ~ T ⋅ N ( F ) d S ( ) f N ( − eV ) = T N ( 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 2 ∆ ε −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. Kant []ε + (1+ 2Z ) Ph.D. thesis

E ε = E 2 − ∆2

-For Z=0 and E<∆ all the incoming electrons are Andreev reflected 0 -At E<∆ the probability for quasiparticle transmission is The Andreev reflection also causes zero, i.e. A+B=1. a phase shift, which is π/2 at E=0 (for arbitrary Z) Calculation of the current Let us calculate the current at the normal side: I = eS∫ v(E)ρ(E)[]1+ A(E) − B(E) [f (E − eV ) − f (E)]dE The area of the contact

The conductance, GNS=dI/dV: G = −e2Sv ρ []1+ A(E) − B(E) f '(E − eV )dE NS F F ∫ The normal state conductance (∆=0): e2Sv ρ G = F F NN 1+ Z2 G NS = −(1− Z 2 )∫[]1+ A(E) − B(E) f '(E − eV )dE GNN

Z=0 limit: G(eV<< ∆)=2GN, Z>>1 limit: conventional for each incoming electron NIS tunneling curve, a hole is reflected, and a G(eV< ∆)=0, sharp peak at ∆ Source: charge of 2e is transmitted C.H. Kant Ph.D. thesis Inclusion of spin polarization in the BTK theory

Spin polarization on the N side can be considered as a sum of fully polarized and unpolarized currents:

I =+=II↑↓2( I ↓ + II ↑↓ − ) {I 14243 unpol I pol

GV(,, T PC ,) Z=− (1 P C ) G unpol (,,) V T Z + PG C pol (,,) V T Z For the unpolarized current the original BTK result is used. In the polarized current~ the Andreev reflection is suppressed, A → A = 0

The probability for the normal reflection is rescaled to ~ preserve current conservation: B → B

Assumption: the ratio for the normal reflection and quasiparticle transmission is independent of spin-polarization: ~ R B B ~ B n = = ~ ⇒ B = Tn 1− A − B 1− B 1− A For more details see: G. J. Strijkers et al. Phys. Rev. B 63, 104510 (2001). I. I. Mazin et al. J. Appl. Phys. 89, 7576 (2001). Y. Ji etal. Phys. Rev. B 64, 224425 (2001). Source: GNS 2 ~ = −P (1− Z ) []1− B(E) f '(E − eV )dE − C.H. Kant C ∫ GNN Ph.D. thesis − (1− P )(1+ Z 2 ) []1+ A(E) − B(E) f '(E − eV )dE C ∫ First measurements: tip-sample approach R. J. Soulen Jr., J. M. Byers,* M. S. Osofsky, B. Nadgorny, T. Ambrose, S. F. Cheng, P. R. Broussard, C. T. Tanaka, J. Nowak, J. S. Moodera, A. Barry, J. M. D. Coey, Science 282, 85 (1998)

One of the first studies demonstrating the Andreev spectroscopy technique for various ferromagnetic metals.

The spin-polarization is determined by the simple formula:

GFS(V=0)=2(1-PC)GN

Probably rather large diffusive contacts were studied, as the BTK theory would not give good fit to the curves. First measurements: membrane with a nano-hole Shashi K. Upadhyay, Akilan Palanisami, Richard N. Louie, and R. A. Buhrman, PRL 81 3248 (1998)

The curves are fitted with the modified BTK theory. G (V ) − G (V ) g(V ) = FS FN GFN (0) SC is suppressed by magn. field

P=0.37±0.02 A the pattern of the nanohole is defined on a

Si3N4 membrane by e-beam litography. The hole is established by etching which is stopped right after the hole breaks through. With this method the size of the hole is much smaller P=0.32±0.02 than the resolution of the litography.

P=0.00±0.01

Phonon spectroscopy proves the good quality of the contact (ballistic) Some other spin polarization measurements

Material P (%) Ref. Co 37±2 ; 46±3 S.K. Upadhyay, et al, PRL 81, 3247 (1998). G.J. Strijkers et al, PRB 63, 104510 (2001). C.H. Kant, et al, PRB 66, 212403 (2002). Fe 45±3 G.J. Strijkers et al, PRB 63, 104510 (2001). C.H. Kant, et al,PRB66, 212403 (2002). Ni 32±2; 37±2 S.K. Upadhyay, et al, PRL 81, 3247 (1998). G.J. Strijkers et al, PRB 63, 104510 (2001).

NixFe1-x 45±3 B. Nadgorny, et al., PRB (R) 61, R3788 (2000). Gd 45±4 C.H. Kant, et al, PRB 66, 212403 (2002).

CrO2 96±2 Y. Ji, et al. PRL 86 5585 (2001). R.J. Soulen, et al, Science 282, 85 (1998).

La0.7Sr0.3MO3 75±15; B. Nadgorny, et al., PRB 63, 184433 (2001). Y. Ji, et al, PRB 66, 012410 78±2 (2002).

La0.6Sr0.4MO3 83±2 Y. Ji, et al, PRB 66, 012410 (2002). NiMnSb 58±3 R.J. Soulen, et al, Science 282, 85 (1998).

SrRuO3 50±8 B. Nadgorny, et al, APL 82, 427 (2003). P. Raychaudhuri, et al, PRB 67, 020411 (2003). InMnSb 52±3 R.P. Panguluri, et al, APL 84, 4947 (2004). Proximity effect (why shall we use ballistic contacts?)

Proximity effect: the Andreev reflection introduces super- conducting correlations at the normal side. The Andreev V reflected hole is travelling on the time-reversed path of the incoming electron, thus the electron and the hole form N S phase-conjugated pairs. Diffusive contact: Ballistic contact: In a diffusive contact an A electron and the Andreev reflected hole can bounce V back and forth on the same trajectory between N S different points of the contact, causing a coherent superosition.

An energy difference, ∆E destroys the phase coherence after a time: τ ~ h ∆E ξ D Thus the coherence length is: = Dτ = h N ∆E In a ballistic contact the reflected hole travels back to the reservoir, The phase coherence can be destroyed by magnetic field, where it thermalizes. The incoming temperature and applied voltage states at the NS interface all have + the inelastic the distribution of the left electrode, diffusiveξ length: and no superconducting correlations hD hD are present. ξ N = ξ N = = Dτ kT eV in in Proximity effects 1.: reentrance C.W.J. Beenakker, cond-mat/9909293, T.M. Klapwijk, Journal of Superconductivity 17, 593 (2004), C.W.J. Beenakker, Rev. Mod. Phys. 69, 731 (1997)

The diffusive region is Ina diffusivecontact, theincomingelectronreachestheinterface modelled by a single barrier through a lot of scatterings, however the Andreev reflected hole comes with transmission tD back on the time-reversed path, thus a fully phase coherent NS junction NS interface is expected to be completely transparent, GNS=2GN (Z=0)

AR The experiments, however show, that (π/2) the conductance increases below the Tc, AR but it drops at low enough temperature. (π/2) (H. Courtois et al., Superlattices and Microstructures 25, 721 (1999))

The incoming electron acquires a phase φ, whereas the Andeev reflected hole on the time-reversed AR path acqires a phase –φ, but the Andreev reflection π phase shift, causes a phase shift of π/2, thus the net phase (π/2) destructive between the two paths is π! interference! At low enough temperature the coherence length increases, and the destructive interference becomes important.

SN It can be shown, that at T=0 GNS=GN! Proximity effects 1.: reentrance C.W.J. Beenakker, cond-mat/9909293, T.M. Klapwijk, Journal of Superconductivity 17, 593 (2004), C.W.J. Beenakker, Rev. Mod. Phys. 69, 731 (1997)

The zero-bias conductance of an NS junction: 2e2 NS interface GNS = 2⋅ NRhe (Z=0) h tD In an AR 2 electron The probability that an incoming e- Number of channels charges are transmitted is Andreev reflected as a hole

2 ∗ 2 2 2 it t T T AR R ≈ t ∗ it + t∗ r ir ∗it +... = D D = D = D (π/2) he D D D D D D 1− iir ∗r 2 2 D D ()1+ RD ()2 −TD AR (π/2) 2e2 T 2 ! 2e2 G = 2⋅ n < 2G = 2⋅ T NS h ∑ 2 N h ∑ n n ()2 −Tn n

Averaging with random matrix theory:

AR π phase shift, 2 2 π 2e Tn ( /2) destructive G = 2⋅ = G NS h ∑ 2 N interference! n ()2 −Tn

Note: without the phase shift of „i” G =2G would come! SN NS N Proximity eff. 2.: reflectionless tunneling AR T.M. Klapwijk, Journal of Superconductivity 17, 593 (2004), C.W.J. Beenakker, Rev. Mod. Phys. 69, 731 (1997)

If the NS interface has small transmission,

tNS<<1, the amplitude of Andreev reflection is 2 even smaller, rA~ t NS (2 electron charges cross the AR barrier). However, the electron can be reflected back to the NS interface by the disordered region several times, thus it can repeatedly attempt the Andreev reflection. The zero-bias 2e2 conductance of GNS = 2⋅ NRhe AR an NS junction: h A. Kastalsky et al. Phys. Rev. 2 ∗ 2 2 2 Lett. 67, 3026 (1991) For a single process: Rhe = tDrAtD = RATD ≈ TNSTD The conductance is Summing up the multiple attempts: (the phase is same for all!) considerably larger! 2 ∗ 2 2 2 t r t T T R ≈ t∗ r t + t∗ r∗ r∗r r r t +... = D A D ≈ NS D he D A D D NS D A D NS D 1− r ∗ r r∗r 2 NS NS D D ()TNS +TD −TNSTD

General statement: (Beenakker, Rev. Mod. Phys. 69, 731 (1997))

t class h −1 −2 NS tD class RNS = 2 (g N + 2TNS ) 2e N RNS (B = 0,V = 0) ≈ RN T >> g h NS N class class −1 −1 RNS (B > 0,V = 0) ≈ RNS RN = (g N +TNS ) S N 2e2 N Proximity effects 3.: ferromagnetic electrode A.I. Buzdin, Rev. Mod. Phys. 77 935 (2005)

A Cooper pair in a superconductor consists of two electrons with opposite spins and momenta. In a ferromagnet the up-spin electron decreases its energy by µbHeff, while the downspin electron energy increases by the same value. To compensate this energy variation, the up-spin electron increases its kinetic energy, while the down-spin electron decreases its kinetic energy.

µBH eff kF ,− kF kF + hvF

2µB H eff

µBH eff − kF + hvF

2µBH eff πhvF δk = , λosc = Oscillation of the order parameter hvF µBH eff Why should we avoid diffusive contacts? 3. heating, induced magnetic field

For larger contacts negative dips are observed in the G(V) curves, which cannot be explained by BTK theory. A possible explanation: superconductivity is suppressed by Joule heating or the induced magnetic field.

The maximal temperature due to Joule heating in the thermal limit: 2 2 2 V 2 2 2 TPC = Tbath + ⇒ ()T [K] = (T [K]) +10.2⋅()V[mV] 4L PC bath

inserting the parameters: VC ≈ 2mV; Tbath =1.5K ⇒ TC ≈ 6.6K

the value is ~OK, but the assumprion for thermal regime (d>>ξin) is maybe not valid.

The magnetic field generated by the current: I V H = = inserting the parameters: Source: πd πdR C.H. Kant Ph.D. thesis VC ≈ 2mV; HC(bulk) ≈ 64kA/m; R = 21Ω ⇒ d ≈ 0.5nm The estimated contact diameter smaller by more than a factor of 10

Probably both effects give contribution.

Conclusion: for reliable measurement really small contacts (300-1000Ω) should be used! Measurements on magnetic semiconductors

III-V semiconductor + Mn The magnetic Mn2+ ions are randomly situated, but their

(Ga1-xMnx)As, (In1-xMnx)Sb… average distance is within the first, ferromagnetic region of the oscillating RKKY interaction x~0.01 (3N)2 π m Effect of Mn2+: J = eff r2J2 F()k r 2 ()2 pd F - acceptor ion h kFr - localized moments T. Dietl et al., Phys. Rev. B 63, 195205 (2001)

exchange with carriers InMnSb metallic hole-conduction even at low-T ferromagnetic ground state TCurie(InMnSb) < TSC(Nb)!

In spin polarization can be monitored Sb accross the magnetic transition Mn

In is substituted by Mn

T. Wojtowicz et al, Physica E 20 325 (2004) Magnetic semiconductors – magnetism can be switched on/off by electric field or pressure The magnetic coupling is mediated by holes

The hole-concentration can be tuned by a gate The coupling constant can be tuned by decreasing the lattice constant by high pressure (3N)2 π m J = eff r2J2 F()k r 2 ()2 pd F h kFr The paramagnetic material becomes ferromagnetic due to the high pressure

ρHall = R0B + RS M

M. Csontos et al., Nature Materials 4, 447 (2005) H. Ohno et al., Nature 408, 944(2000) Experimental technique

Positioning the tip: cone (1:20 attenuation) - thread: 5° -> 350nm resolution Nb tip - piezo: 3 nm/V -> max. 900nm thread liquid Helium dewar, +T is variable with a tempreture controller piezo actuator

magnetic semiconductor sample diff. ampl. Typical contact size: ~1-5nm filter d

I-V conv. Experimental examples, fitting

Au sample, Nb tip: (In,Mn)Sb sample, Nb tip:

BTK-fitting parameters: r - spin-polarization (P) We start from an initial parameter set, and a temperature: A,T -barrier strength (Z) r The „energy” is defined as: δ E = ∑ yi −Yi (A) - normal conductance (G ) 0 We change the r r r - contact temperature (T) parameters (random): A → A + A ⇒ E → E +δE

-SC gap(∆) If δE<0, we accept the new parameters + numerical integration If δE>0, we accept it with the probability: P = e-E/T T Monte Carlo fitting Meanwhile the temperature random walk in the param is changed simulating repeated space, annealing annealing processes: N Conclusions (Andreev spectroscopy)

- Andreev spectroscopy is a relatively simple and generally aplicable tool to study spin polarization on metallic surfaces

- The strength of BTK theory is that a lot of problems can be „packed” in a single parameter, Z (surface scattering, lattice mismatch, fermi momentum mismatch, dimensionality, moderately diffusive electrodes)

- For small, ballistic contacts BTK gives good fit and reliable results for spin polarization, for larger contacts cautions should be taken