atomic-sized contacts and wires. i - capri school · atomic-sized contacts and wires. i ... djukic,...

Capri, March 2007 1

Atomic-sized Contacts and Wires. I

Jan van RuitenbeekKamerlingh Onnes Laboratorium

N. Agrat, A. Levy Yeyati and JMvRPhysics Reports 377 (2003) 81-380.

Capri, March 2007 2

Electrical contacts to a single atom or molecule

A V

Capri, March 2007 3

In collaboration with

Leiden: Chris Muller, Martijn Krans, Niko van der Post, Helko van den Brom, Bas Ludoph, Alex Yanson, Yves Noat, Roel Smit, Carlos Untiedt, Darko Djukic, Ancuta Hulea, Annemarie Houkes, Robert Thijssen, Sander Otte, Oren Tal, Christian Martin, Tadashi Shiota, Manabu Kiguchi

Madrid: Nicolas Agrat, Gabino Rubio, Juan-Carlos Cuevas, Alvaro Martn-Rodero, Alfredo Levy Yeyati, Michael Haeffner

Saclay: Michel Devoret, Daniel Esteve, Cristian UrbinaKonstanz: Elke ScheerFreiburg: Daniel Urban, Hermann GrabertGteborg: Katja Bratus', Gran Johansson, Vitaly Shumeiko, Gran WendinTU Denmark: Mads Brandbyge, Kristian Thygesen, Sune Bahn, Karsten

JacobsenTucson, Arizona: Jerome Brki, Charles Stafford

Universiteit Leiden

Outline of Lectures

I. Atomic-sized contacts Experimental techniques: how to contact a single atom Theory of contact resistance: from classical to quantum limit The concept of eigenchannels Evolution of conductance and force in atomic contacts Conductance histograms Experimental investigation of conductance channels I:

Superconducting subgap structureII. The conductance channels for a single atom

Shot noise Conductance fluctuations Thermopower

III. Shell structure in metallic nanowires IV. Transport through atomic chains and molecules

Allow Nature to do the job

A single atom:physics becomes simple

A

Vtip

sample

traced path

sample

tip

piezoelement

Scanning Tunneling Microscope

Capri, March 2007 7

Contact formation by STM

Au at 4.2 KImportant problem: Clean surface preparation

Capri, March 2007 8

Combined STM and HRTEM

T. Kizuka et al. PRB 55 (1997) R7398

Capri, March 2007 9

Images of atomic contacts (room temp.)

V. Rodrigues et al. PRL 85 (2000) 4124

Capri, March 2007 10

Mechanically Controllable Break Junction


Mechanically Controllable Break Junction

Conductance for Au contacts at 4.2 K

0 50 100 150 200 250 300012345678

Gold, 4.2 K

Con

duct

ance

(2e2

/h)

Piezo-voltage (V)


Lithographically fabricated MCBJ

J.M. van Ruitenbeek et al. Rev. Sci. Instrum. 67 (1995) 108 2

6Lut

zI=

uL

t


Relays

K. Hansen et al. PRB 56 (1997) 2208

Touching wires: J.L. Costa-Krmer, et al., Surf. Science 342 (1995) L1144


Outline

Experimental techniques: how to contact a single atom Theory of contact resistance: from classical to quantum limit The concept of eigenchannels Evolution of conductance and force in atomic contacts Conductance histograms Experimental investigation of conductance channels I: shot noise


Classical limit: Maxwell resistance

Classical limit

hyperbolic constriction

Oblate spheroidal coordinates

sinsinhsincoscoshcoscoscosh

azayax

===

F,, llL >>charge neutral conductor

0)(2 = rV

solution

)arctan()( 02021

eVVV +=

total current obtained from Ohms law and by integrating:

)sin1(2 0M = aG

00 =For an orifice

aG 2M = (J.C. Maxwell)


Semiclassical approximation: Sharvin resistance

Semiclassical ballistic contact llL ,F


Corrections to the Sharvin resistance

Modes in a cavity (Weyl 1911)

For cylindrical wire with hard walls:

For small sizes integration over k introduces edge errors

( )[ ]L++= 61F212F212

S2 akakheG

Sharvinperimeter correction

topological correction

C. Hppler and W. Zwerger, PRL 80 (1998) 1792


Quantum conductance: Landauer formula

As a first step: assume 1D conductor, single channel

eV= RL

L R

( ) ( )

( )

=

==

)()(

)()()()(

RL

1

RLRL

kkk

kkkk

kkk

ffvdkdde

ffvdkeffvLeI

kvdkd

h=

he

VIG

22==and taking T = 0,


Quantum conductance: Landauer formula

Scattering problem

L RLN RNS

etc. matrix, a is ,

LR NNtrttr

S

=

==n

nThett

heG

2

2 2)Tr(2


2DEG experiments

B. Van Wees et al., PRL 60 (1988) 848D.A. Wharam et al., J. Phys. C 21 (1988) L209

Atomic contacts


Eigenchannel decomposition

lir

Incoming waves

ror

Outgoing waves

Matrix of transmission ampl.

lr itorr =

Landauer:

==n

nThett

heG

2

2 2)Tr(2

Mesoscopic PIN code


Eigenchannels

Element Type ofatomNumber of

modesConductance for

one atom

Au s 1 1 G0

Al s-p 3 ~0.8-1.2 G

Pb s-p 3 ~2.5-3 G

Nb s-d 5 ~2.5-3 G

J.C. Cuevas, A. Levy Yeyati & A.Martin-Rodero, PRL 80 (1998) 1066


Outline



Jump to contact and tunneling regime

J.M. Krans et al. PRB 48 (1993) 14721


Conductance for Au contacts at 4.2 K

0 50 100 150 200 250 300012345678

Gold, 4.2 K

Con

duct

ance

(2e2

/h)

Piezo-voltage (V)


force and conductance

Gold at room temperature

G. Rubio, N. Agrat and S. Vieira,PRL 76 (1996) 2302


sp-metal

75 100 125 150012345678

Aluminium, 4.2 K

Con

duct

ance

(2e2

/h)

Piezo-voltage (V)


d-metal

120 140 160 1800

2

4

6

8

10

12

Platinum, 4.2 K

Con

duct

ance

(2e2

/h)

Piezo-voltage (V)


Outline



Conductance histogram for Au at 4.2K

0 1 2 3 40

10

20

30

40

50

G [2e2/h]

# po

ints

(x 1

03)


Conductance histogram for aluminum

0 1 2 3 4 5 6 7

0,05

0,10

0,15

0,20

0,25

0,30

0,35Aluminum4.2 K

C

ount

Conductance (2e2/h)


Conductance histogram for niobium

0 1 2 3 4 50

500

1000

1500

2000

Cou

nts

G (2e2/h)

Nb13 K


Semimetal Bi, 4 K

J.G. Rodigo et al. Phys. Rev. Lett. 88 (2002) 246801


Semimetal Bi, 77K

J.G. Rodigo et al. Phys. Rev. Lett. 88 (2002) 246801


MCBJ for alkali metals


Conductance traces for sodium

10 15 20 25 30 35 40012345678

Sodium, 4.2 K

Con

duct

ance

(2e2

/h)

Piezo-voltage (V)


Conductance histogram for potassium

0 1 2 3 4 5 6 7 80.0

0.1

0.2

0.3

0.4

0.5Potassium, 4.2 K

Nor

mal

ized

nr.

coun

ts

Conductance [2e2/h]


Eigenchannels in a cylindrical contact:

0 1 2 3 4 5 6 7-0.6-0.4-0.20.00.20.40.60.81.01.2

Bessel functions

m=3m=2m=1

m=0

Ampl

itude

0 1 2 3 4 5 6 70123456789

Con

duct

ance

kFR

+

+ +

++

++

+

m, n

0, 1

1, 1

2, 1

0, 2


Local Density calculation for sodium contact

Nakamura, Brandbyge, Hansen & Jacobsen, Phys. Rev. Lett. 82, 1538 (1999).


Outline

Experimental techniques: how to contact a single atom Theory of contact resistance: from classical to quantum limit The concept of eigenchannels Evolution of conductance and force in atomic contacts Conductance histograms Experimental investigation of conductance channels

I. Superconducting subgap structureII. Shot noiseIII. Conductance fluctuations as a function of bias voltageIV. Thermopower


Superconducting contacts: Niobium

15 20 25 300

2

4

6tunnelling regimecontact regime

G (2

e2/h

)

VP (V)

0,01

0,1

1


Tunneling regime. Niobium

0 2 4 60

20

40

60C

urre

nt [n

A]

Voltage [/e]


Subgap structure

0 1 2 3 40.01

0.1

1

10

100

NbR=146kT= G/G0=0.0707

curr

ent

[nA

]

eV []

N. van der Post et al., Phys. Rev. Lett. 73, 2611 (1994)


Principle of Multiple Andreev Reflection

eV >2

Ef

eV >

eV > 23

T T2

a b c

T3

leftelectrode

rightelectrode


Subgap structure in tunneling regime

0 1 2 3 40.01

0.1

1

10

100

NbR=146kT= G/G0=0.0707

curr

ent

[nA

]

eV []0 1 2 3 4

0.01

0.1

1

10

100

NbR=146kT= G/G0=0.0707

curr

ent

[nA

]

eV []

N. van der Post et al., Phys. Rev. Lett.73, 2611 (1994)

Theory:E.N. Bratus, V.S. Shumeiko, and G. Wendin , Phys. Rev. Lett.74, 2110 (1995)


Full theory for single channel

Theory to all orders in transmission T

D. Averin and A. Bardas, Phys. Rev. Lett. 75, 1831 (1995).

J.C. Cuevas, A. Martin-Rodero and A. Levy Yeyati, Phys. Rev. B 54, 7366 (1996).

E.N. Bratus, V.S. Shumeiko, E.V. Bezuglyiand G. Wendin, Phys. Rev. B 55, 12666 (1997).

0 1 2 30

2

4

eI/G

ev/

T = 10.99

0.90.8 . .

... .

0.1


Contact regime: one atom for Al

0 1 2 3 4 50

2

4

Aluminium, T=100 mK

eI/G

eV/

1 20

2

4

E. Scheer et al., Phys. Rev. Lett. 78, 3535 (1997)

G = 1.747 T1 = 0.997, T2 = 0.46, T3 = 0.29

G = 0.85 T1 = 0.74, T2 = 0.11

G = 0.88 T1 = 0.46, T2 = 0.35, T3 = 0.07

G = 0.025 T1 = 0.025


d-metal Niobium

0 1 2 30

2

4

6

8

10 Nb

5432

I (2e/

h)

eV/

2 3 4 5 60.001

0.01

0.1

1

2 (a.u.)

# channels

B. Ludoph et al., Phys. Rev. B 61, 8561 (2000)


Monovalent metal: Au

E. Scheer et al., Nature 394, 154 (1998)


Monovalent metal: Au

E. Scheer et al., Phys. Rev. Lett. 86, 284 (2001)


Conclusions

Single-atom contacts can be produced by simple techniques for any metal

A single atom contact forms a nice, well-defined mesoscopic test sytem The number conductance modes through a single atom is determined by

the number of valence orbitals. Conductance is usually NOT quantized The number of channels can be obtained from SSGS

Next lecture: other techniques for analyzing the channels of an atomic contact

Atomic-sized Contacts and Wires. IElectrical contacts to a single atom or moleculeIn collaboration withOutline of LecturesAllow Nature to do the jobScanning Tunneling MicroscopeContact formation by STMCombined STM and HRTEMImages of atomic contacts (room temp.)Mechanically Controllable Break JunctionMechanically Controllable Break JunctionConductance for Au contacts at 4.2 KLithographically fabricated MCBJRelaysOutlineClassical limit: Maxwell resistanceSemiclassical approximation: Sharvin resistanceCorrections to the Sharvin resistanceQuantum conductance: Landauer formulaQuantum conductance: Landauer formula2DEG experimentsAtomic contactsEigenchannel decompositionEigenchannelsOutlineJump to contact and tunneling regimeConductance for Au contacts at 4.2 Kforce and conductancesp-metald-metalOutlineConductance histogram for Au at 4.2KConductance histogram for aluminumConductance histogram for niobiumSemimetal Bi, 4 KSemimetal Bi, 77KMCBJ for alkali metalsConductance traces for sodiumConductance histogram for potassiumEigenchannels in a cylindrical contact:Local Density calculation for sodium contactOutlineSuperconducting contacts: NiobiumTunneling regime. NiobiumSubgap structurePrinciple of Multiple Andreev ReflectionSubgap structure in tunneling regimeFull theory for single channelContact regime: one atom for Ald-metal NiobiumMonovalent metal: AuMonovalent metal: AuConclusions

atomic-sized contacts and wires. i - capri school · atomic-sized contacts and wires. i ... djukic,...

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