Budapest University of Technology and EconomicsDepartment of Physics

Low temperature solid state physics laboratory

Coworkers:

Prof. György MihályDr. Szabolcs Csonka Attila GeresdiPéter Makk

András Halbritter:Point contact Andreev spectroscopy

correlated electron systems, magnetic semiconductors(Pressure cell (up to 30kBar), MOKE setupTEP measurements,heat conductivitymeasurements)

MCBJ, Andreev spectroscopy

Outline1. Theory of point contacts

-ballistic contacts-diffusive contacts-”thermal” regime-point contact spectroscopy

2. Andreev spectroscopy-What is spin polarization? -BTK theory-diffusive SN contacts-study of magnetic semiconductors

N S

InSbMn

History of point contacts

MCBJ technique

First application of point-contact spectroscopy:Study of electron-phonon interaction (Yanson, 1974)

Later: study of two-level systems, magnetic impurities, etc.

Possibility to createheterocontacts

Taking advantage of theextreme stability single-atom and single moleculecontacts can be investigated

Touching wires Spear-anvil geometry

Study of unconventional superconductors withNS junctions

(heavy fermion superconductors, MgB2, etc.)

Spin polarization measurements with FS junctions(Soulen; Upadhyay 1998)

Distributionfunction atthe middle ofthe contact: kx

kydε=e

Vd

V

∑ ∫==k

kkkk rvkrvrj )()2(

d2)(2)( 3

3

sample

fefVe

π

kx

kykvheVk =d )0()0(d =⋅=== ∫ xjAxjAI xx

A

43421h

π

ϑπ

kv

kk

vv

2F

)cos()2(

2)0(

d

3

k

S

k

xeVexj ∫⋅⋅==

VdkheI

G

⋅

⋅=

4434421Sharvin

2F

2

42

Sharvin formula in 3D:

Ballistic point contacts

Integration over the half sphereIn a ballistic contact all the electronsat the contact surface with kx>0 comefrom the left electrode, and all withkx<0 come from the right electrode

In a ballistic contact the contact diameter is muchsmaller than the mean free path of the electrons, thus the electrons are only scattered on the walls

ld <<

)sgn(2

)(12

zV

Ω

−−=)(Φπrr

The electrical potential: (~<µ>)

The voltage drops within a distance of ~d from the contact

V

deV=−∞Φ−∞Φ

=∆Φ=)()(

,0)(),()( rrErj σ

dR

σ1

=

Diffusive point contacts

Intermediate regime

( )d

dldd

lRσπσ1/

316

Γ+=

ld >>

In a diffusive contact thecontact diameter is muchlarger that the mean freepath, thus an electroncoming e.g. from the leftside of the contact canstem from both electrodes

The resistance can be determined bysolving the Maxwellequations in a hyperbolic coordinate system:

The Maxwellresistance

In the intermediate region between the diffusive and ballistic regime aninterpolating formula can be set up by solving the Boltzmann equation forarbitrary ratio of the contact diameter and mean free path

ld ~

Wexler formula

The first term is the Sharvinresistance by inserting theDrude conductivity(independent of l!)

Γ(l/d) is a monotonousfunction with TR

Tdd/dd/dρ

≈

2

3F

F

2

3;

πσ kn

klne

==h

694.0)(;1)0( =∞Γ=Γ

Self heating of the contact

T∇−=Φ∇−= κσ qj ;

2B

3

=

2

ekπL

Dissipated power:

,σκ⋅= L

T

+ Wiedemann-Franz law:

Lorentz number

If both the mean free path and the inelastic diffusive length aremuch smaller than the contact diameter, l, ξin<<d, then both theelectrical and heat conduction can be treated by classical equations:

Φ∇−=∇=∇ jqj ;0( )

( ) ( ) 02

022 =Φ∇+∇∇

=Φ∇∇

σσ

σ

TL

,)(4

)(22

2bath

2

LLrr Φ

−+=VTT

2)( V

±=±∞Φ,)( bathTT =±∞

L4

22

bath2

PCVTT +=

V

dξin

In a diffusive contact the electron looses its momentum frequentlydue to elastic scattering. However, to relax its energy inelasticscattering is needed. The average distance between inelasticscatterings is the inelastic diffusive length:

inin τξ D=

boundary conditions:

+

)regimethermal(, inξld >>

Length-scale of dissipation: d

L4

22

bath2

PCVTT +=

dVRV σ2

2

= Dissipated power:

ld >>>inξ

Length-scale of dissipation: ξin

in

22

bath2

PC 4 ξdVTT

L+≈

dVRV σ2

2

= Dissipated power:

)regimeballistic(, in dl >>ξ

Length-scale of dissipation: ξin

in

222

bath2

PC 163

4 ξπldVTT

L+≈

lV

RV

16d3 2

22 πσ=

The contact temp. is independent of the diameter!

V=100mV->TPC~300K

for general contact geom.

Distributionfunction atthe middle ofthe contact:

V

kx

kydε=e

V

Point contact spectroscopy

At T=0 an electron at the contact goingto the right cannot be scattered to theoccupied right-going states, but it can be scattered to the unoccupied left-goingstates

The inelastic scattering can cause backscattering through the contact,which is reflected by a nonlinearity in the I-V curve:ld <<

eV

eV

eV

dI/dV

d2I/dV2

IE

g(E)excitation spectrum

d2I/dV2 shows the spectrum of theinelastic excitations in the closevicinity of the contactωh

ωh

ωh

ωh

A.G.M. Jansen et al. J. Phys. C 13, 6073. (1980)A. Halbritter, L. Borda, A. Zawadowski

Advances in Physics 53, 939 (2004)

piezo actuator

screw-thread

sample

Nb tip

For a normal metal with P=0 an incoming electron isAndreev reflected, thus for each incoming e- a chargeof 2e is transmitted, GNS=2GN

The fit of the I-Vcurves tells thespin-polarization!

As a first approx.:GFS(V=0)=2(1-PC)GN

nm-sized ballistic contacts with good stabilityAndreev spectroscopy

In a half-metal (P=1) Andreev reflection is prohibited, GFS=0

BUTE, 2005SC tip

ferromagnetic sample

2 DEG

V gSpin –Transistor concept (Datta, Das, 1990)

Low Resistance

F N F

High

F N F

Low

High Resistance

F FI F FI

High ResistanceLow Resistance

Magnetic Tunnel Junction

The research on spin polarization is not only essential for better fundamental understanding, but the wide application range of magnetoresistive devices makes it technologically relevant as well.

The inportance ofspin polarization

Spin-valve

A FET with spin-polarized source anddrain electrodes. In the 2DEG the spin isprecessed by the gate due to the Rashbaeffect.

The magneto-resistance is ~10% at room-T!

GMR (Giant MagnetoResistance, 1988)

Application: MRAM (Magnetic Random Acces Memory)2004, IBM -> 16MbMuch faster than flash memory!

Magneto-resistance up to 70%!

Not yet realized.

Spintronics review:I. Zutic et al. Rev. Mod. Phys.76 323 (2004)

Definitions of spin polarization

↓↑

↓↑

+−

=FF

FF

ρρρρP

Density of statespolarization:

↓↑

↓↑

+

−=

IIII

PCPolarization of the current:(contact polarization)

For a ballistic contact:

↓↓↑↑

↓↓↑↑

+−

=⇒FFFF

FFFFCFF~

vvvvPevj

ρρρρρ

For a diffusive contact:

↓↓↑↑

↓↓↑↑

+−

=⇒effFeffF

effFeffFC

eff

2F

////~mmmmPE

mej

ρρρρτρ

Simplified DOS for some ferromagnets:

s-character band, mobile electronswith large vF and small meff

d-character band, split due toexchange interaction, immobileelectrons with small vF and large meff

The so-called half-metalsare fully spin-polarized

Spin-pol. is a Fermisurface property, while the magnetizationcounts for all theelectrons!

M>0, P>0, PC>0 M>0, P<0, PC≈0

Magnetization: εερερε

d~F

∫∞−

↓↑ )(−)(M

2. Tunneling spectroscopy

1. Spin-polarized photoemission spectroscopy

•Hard to fabricate

•High energy resolution (<1 meV)

•Tunnel junction (typically Al/Al2O3/FM)

eephoton

SuperconductorFerromagnet

H

Insulator

Other methods to measure spin-polarization

-Disadvantage: low energy resolution (100-200 meV)

-The spin orientation of the photoelectrons is detected

0=P

( )( )

( )

)()()()(d)(~dd

)()()(d)(~

)(1)()()(d~

)(1)()()(d~

SFN0

'NSFN

SNSFN

NNSS

SSNN

eVTeVfTVI

feVfTIII

eVfeVfTI

feVfeVTI

Tρερεερεερ

εεερεερ

εερεερε

εερεερε

=

−+

−

+

=−⋅

−−⋅−=

−−−⋅⋅

−⋅−−⋅

∫

∫∫∫

eV∆

For normal metal-superconductor tunneling the G(V)curve shows the superconducting DOS:

To detect spin-polarization a Zeeman splitting is applied by an external fieldR. Meservey, P.M. Tedrow, Phys. Rep. 238, 173 (1994)

Source:C.H. Kant Ph.D. thesis

BTK theory(conductance of a ballistic NS junction)

where,Ψ=Ψ

−∆

∆∗∗ E

HH

=Ψ

)()(

)(xgxf

x

)(dd

2 2

22

xVExm

H F +−−=h

22

F

222

2∆+

−= E

mkE h

xkkixkkixkki bax )()()(N

NFNFNF e01

e10

e01

1)( +−−+

+

+

=Ψ xkkixkki

uv

dvu

cx )()(S

NFSF ee)( +−+

+

=Ψ

)(2)(F

F xkEZxV δ=

),0()0()0( SN Ψ≡Ψ=Ψ )0(22)0(')0(' 2F

FSN Ψ=Ψ−Ψ

h

mkEZ

„Electron-like”state

„Hole-like” state

The Bogoliubov-de Gennes equation:

measured from EF

dimensionless„barrier strength”

0N =∆ ∆=∆S

Andreev reflection normal reflection quasiparticle transmission

coupling

electron-like band

hole-like band – the groupvelocity has opposite signto the wave number

Matching the wave functions:

N SG.E. Blonder, M.Tinkham,

T.M.Klapwijk, PRB 25, 4515 (1982)

Reflection probabilities:

The probability for Andreev reflection:

The Andreev reflection also causesa phase shift, which is π/2 at E=0

(for arbitrary Z)

2aA =2bB =

22222

2

)21)(( ZEEA

+−∆+∆

=

[ ]22

22

)21()1(4

ZZZB

++

+=

ε

[ ]22

2

)21(1Z

A++

−=

εε

AB −=1

∆<E ∆>E

22 ∆−=

EEε

The probability for normal reflection:

-For Z=0 and E<∆ all theincoming electrons areAndreev reflected

-At E<∆ the probability forquasiparticle transmission iszero, i.e. A+B=1.

Source:C.H. Kant

Ph.D. thesis

0

Calculation of the current

[ ][ ] EEfeVEfEBEAEEveSI d)()()()(1)()( −−−+= ∫ ρ

Let us calculate the current at the normal side:

The conductance, GNS=dI/dV:

The area of the contact

[ ] EeVEfEBEASveG d)(')()(1FF2

NS −−+−= ∫ρ

The normal state conductance (∆=0):2

FF2

NN Z1+=

ρSveG

[ ] EeVEfEBEAZGG d)(')()(1)1( 2

NN

NS −−+−−= ∫

Z>>1 limit: conventionalNIS tunneling curve, G(eV< ∆)=0, sharp peak at ∆

Z=0 limit: G(eV<< ∆)=2GN, for each incoming electrona hole is reflected, and a charge of 2e is transmitted

Source:C.H. Kant

Ph.D. thesis

Inclusion of spin polarization in the BTK theorySpin polarization on the N side can be considered as a sum offully polarized and unpolarized currents:

BB ~→

2 ( )unpol polI I

I I I I I I↑ ↓ ↓ ↑ ↓= + = + −14243

( , , , ) (1 ) ( , , ) ( , , )C C unpol C polG V T P Z P G V T Z P G V T Z= − +For the unpolarized current the original BTK result is used. In the polarized current the Andreev reflection is suppressed,

The probability for the normal reflection is rescaled topreserve current conservation:

Assumption: the ratio for the normal reflectionand quasiparticle transmission is independent of spin-polarization:

ABB

BB

BAB

TR

n

n

−=⇒

−=

−−=

1~

~1

~

1

0~=→ AA

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).

[ ][ ] EeVEfEBEAZP

EeVEfEBZPGG

d)(')()(1)1)(1(

d)(')(~1)1(

2C

2C

NN

NS

−−++−−

−−−−−=

∫

∫Source:

C.H. Kant Ph.D. thesis

First measurements: tip-sample approachR. 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 Andreevspectroscopy technique for various ferromagneticmetals.

The spin-polarization is determined by the simple formula:

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

Probably rather large diffusive contacts werestudied, as the BTK theory would not give goodfit to the curves.

First measurements: membrane with a nano-holeShashi K. Upadhyay, Akilan Palanisami, Richard N. Louie, and R. A. Buhrman, PRL 81 3248 (1998)

A the pattern of the nanohole is defined on a Si3N4 membrane by e-beam litography. The holeis established by etching which is stopped rightafter the hole breaks through. With thismethod the size of the hole is much smallerthan the resolution of the litography.

Phonon spectroscopy proves thegood quality of the contact (ballistic)

The curves are fitted with themodified BTK theory.

P=0.37±0.02

P=0.32±0.02

P=0.00±0.01

)0()()()(

FN

FNFS

GVGVGVg −

=

SC is suppressed by magn. field

Some other spin polarization measurements

B. Nadgorny, et al, APL 82, 427 (2003). P. Raychaudhuri, et al, PRB 67, 020411 (2003).

50±8SrRuO3

R.J. Soulen, et al, Science 282, 85 (1998).58±3NiMnSb

R.P. Panguluri, et al, APL 84, 4947 (2004).52±3InMnSb

Y. Ji, et al, PRB 66, 012410 (2002).83±2La0.6Sr0.4MO3

B. Nadgorny, et al., PRB 63, 184433 (2001). Y. Ji, et al, PRB 66, 012410 (2002).

75±15; 78±2

La0.7Sr0.3MO3

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

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

S.K. Upadhyay, et al, PRL 81, 3247 (1998). G.J. Strijkers et al, PRB 63, 104510 (2001).

32±2; 37±2Ni

G.J. Strijkers et al, PRB 63, 104510 (2001). C.H. Kant, et al, PRB 66, 212403 (2002).

45±3Fe

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).

37±2 ; 46±3Co

Ref.P (%)Material

V

N S

V

N S

A

Proximity effect: the Andreev reflection introduces super-conducting correlations at the normal side. The Andreevreflected hole is travelling on the time-reversed path of theincoming electron, thus the electron and the hole formphase-conjugated pairs.

Proximity effect (why shall we use ballistic contacts?)

In a ballistic contact the reflectedhole travels back to the reservoir, where it thermalizes. The incomingstates at the NS interface all havethe distribution of the left electrode,and no superconducting correlationsare present.

Ballistic contact:Diffusive contact:

In a diffusive contact anelectron and the Andreevreflected hole can bounceback and forth on thesame trajectory betweendifferent points of thecontact, causing a coherent superosition.

An energy difference, ∆E destroys the phase coherence aftera time:

E∆h~τ

Thus the coherence length is:EDDN ∆

==hτξ

The phase coherence can be destroyed by magnetic field, temperature and applied voltage

kTD

Nh

=ξeVD

Nh

=ξ

+ the inelasticdiffusive length:

inin τξ D=

Proximity effects 1.: reentranceC.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)

In a diffusive contact, the incoming electron reaches the interfacethrough a lot of scatterings, however the Andreev reflected hole comesback on the time-reversed path, thus a fully phase coherent NS junctionis expected to be completely transparent, GNS=2GN

The experiments, however show, thatthe conductance increases below the Tc,but it drops at low enough temperature.(H. Courtois et al., Superlattices andMicrostructures 25, 721 (1999))

The incoming electron acquires a phase φ, whereasthe Andeev reflected hole on the time-reversedpath acqires a phase –φ, but the Andreev reflectioncauses a phase shift of π/2, thus the net phasebetween the two paths is π!

At low enough temperature the coherence length increases,and the destructive interference becomes important. It can be shown, that at T=0 GNS=GN!

AR(π/2)

S N

NS interface(Z=0)

The diffusive region ismodelled by a single barrierwith transmission tD

π phase shift,destructiveinterference!

AR(π/2)

AR(π/2)

Proximity effects 1.: reentranceC.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)

AR(π/2)

S N

NS interface(Z=0) tD

π phase shift,destructiveinterference!

AR(π/2)

AR(π/2)

The zero-bias conductance of an NS junction:

heNS NRheG

222 ⋅=

In an AR 2 electroncharges are transmitted Number of channels The probability that an incoming e-

is Andreev reflected as a hole

( ) ( )2

2

2

222

211...

D

D

D

D

DD

DDDDDDDDhe

TT

RT

riirtititirrtittR

−=

+=

−=++≈ ∗

∗∗∗∗

( ) ∑∑ ⋅=<−

⋅=n

nNn n

nNS T

heG

TT

heG

2!

2

22 2222

22

Averaging with random matrix theory:

( ) Nn n

nNS G

TT

heG =

−⋅= ∑ 2

22

222

Note: without the phase shift of „i” GNS=2GN would come!

Proximity eff. 2.: reflectionless tunnelingT.M. Klapwijk, Journal of Superconductivity 17, 593 (2004), C.W.J. Beenakker, Rev.

Mod. Phys. 69, 731 (1997)AR

AR

AR

S N

If the NS interface has small transmission, tNS<<1, the amplitude of Andreev reflection is even smaller, rA~ t2

NS (2 electron charges cross thebarrier). However, the electron can be reflectedback to the NS interface by the disorderedregion several times, thus it can repeatedlyattempt the Andreev reflection.

The zero-biasconductance ofan NS junction:

heNS NRheG

222 ⋅=

For a single process: 2222

DNSDADADhe TTTRtrtR ≈== ∗

Summing up the multiple attempts: (the phase is same for all!)

A. Kastalsky et al. Phys. Rev.Lett. 67, 3026 (1991)

NSt Dt

( )2

2222

1...

DNSDNS

DNS

DDNSNS

DADDNSDADNSDDADhe

TTTTTT

rrrrtrttrrrrrttrtR

−+≈

−=++≈ ∗∗

∗∗∗∗∗

The conductance is considerably larger!

( )212

class 22

−− += NSNNS TgNehR

( )112

class

2−− += NSNN Tg

NehR

class

class

)0,0(

)0,0(

NSNS

NNS

RVBR

RVBR

≈=>

≈==NNS gT >>

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

Proximity effects 3.: ferromagnetic electrodeA.I. Buzdin, Rev. Mod. Phys. 77 935 (2005)

eff2 HBµ

F

effB

vHkFh

µ+FF kk −,

F

effB

vHkFh

µ+−

effB

F

F

effB ,2Hv

vHk osc µ

πλµδ h

h==

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 downspinelectron energy increases by the same value. Tocompensate this energy variation, the up-spin electronincreases its kinetic energy, while the down-spin electrondecreases its kinetic energy.

Oscillation of the order parameter

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 inducedmagnetic field.

L4

22

bath2

PCVTT += ( ) ( ) ( )22

bath2

PC ]mV[2.10]K[]K[ VTT ⋅+=⇒

K6.6K 5.1mV;2 CbathC ≈⇒=≈ TTV

dRV

dIH

ππ==

nm5.0 21 kA/m;64mV;2 C(bulk)C ≈⇒Ω=≈≈ dRHV

The maximal temperature due to Joule heating in the thermal limit:

inserting the parameters:

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

The magnetic field generated by the current:

inserting the parameters:

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!

Source:C.H. Kant

Ph.D. thesis

Measurements on magnetic semiconductors

Effect of Mn2+:- acceptor ion- localized moments

exchange with carriers

ferromagnetic ground state TCurie(InMnSb) < TSC(Nb)!

spin polarization can be monitoredaccross the magnetic transition

InSbMn

T. Wojtowicz et al,

Physica E 20 325 (2004)

( )( )

( )rkFJrrkmN3J F

2pd

22

F2

eff2

h

π=

T. Dietl et al., Phys. Rev. B 63, 195205 (2001)

The magnetic Mn2+ ions are randomly situated, but theiraverage distance is within the first, ferromagneticregion of the oscillating RKKY interaction

(Ga1-xMnx)As, (In1-xMnx)Sb…

x~0.01

IIIIII--V V semiconductorsemiconductor + + MnMn

In is substituted by Mn

InMnSbmetallic hole-conduction even at low-T

H. Ohno et al., Nature 408, 944(2000)

The magnetic coupling is mediated by holes

( )( )

( )rkFJrrkmN3J F

2pd

22

F2

eff2

h

π=

The paramagnetic material becomesferromagnetic due to the high pressure

MRBR SHall += 0ρ

M. Csontos et al., Nature Materials 4, 447 (2005)

Magnetic semiconductors – magnetism can be switched on/off byelectric field or pressure

The hole-concentration can be tuned by a gate The coupling constant can be tuned bydecreasing the lattice constant by high pressure

Nb tip

Typical contact

size: ~1-5nm

d<l reachable

Experimental technique

thread

piezo actuator

Positioning the tip:

- thread: 5° -> 350nm resolution

- piezo: 3 nm/V -> max. 900nm

cone (1:20 attenuation)

magnetic semiconductorsample

filter

diff.

ampl.

DAQ card1:100

I-V

conv.

liquid Helium dewar,

+T is variable with atempreture controller

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

Experimental examples, fitting

BTK-fitting parameters:

- spin-polarization (P)

-barrier strength (Z)

- normal conductance (G0)

- contact temperature (T)

- SC gap (∆)

+ numerical integration

Monte Carlo fitting

random walk in the paramspace, annealing

EEEAAA δδ +→⇒+→rrr

We start from an initial parameter set, and a temperature:

The „energy” is defined as: ∑ −= )(AYyE ii

r

We change theparameters (random):

If δE<0, we accept the new parametersIf δE>0, we accept it with the probability: -E/Te=P

TA,r

T

N

Meanwhile the temperatureis changed simulating repeatedannealing processes:

Conclusions (Andreev spectroscopy)

- Andreev spectroscopy is a relatively simple and generally aplicable toolto 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