atomic-sized contacts and wires. i - capri school · atomic-sized contacts and wires. i ... djukic,...
TRANSCRIPT
-
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
-
Capri, March 2007 11
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)
-
Capri, March 2007 13
Lithographically fabricated MCBJ
J.M. van Ruitenbeek et al. Rev. Sci. Instrum. 67 (1995) 108 2
6Lut
zI=
uL
t
-
Capri, March 2007 14
Relays
K. Hansen et al. PRB 56 (1997) 2208
Touching wires: J.L. Costa-Krmer, et al., Surf. Science 342 (1995) L1144
-
Capri, March 2007 15
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
-
Capri, March 2007 16
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)
-
Capri, March 2007 17
Semiclassical approximation: Sharvin resistance
Semiclassical ballistic contact llL ,F
-
Capri, March 2007 18
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
-
Capri, March 2007 19
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,
-
Capri, March 2007 20
Quantum conductance: Landauer formula
Scattering problem
L RLN RNS
etc. matrix, a is ,
LR NNtrttr
S
=
==n
nThett
heG
2
2 2)Tr(2
-
Capri, March 2007 21
2DEG experiments
B. Van Wees et al., PRL 60 (1988) 848D.A. Wharam et al., J. Phys. C 21 (1988) L209
-
Atomic contacts
-
Capri, March 2007 23
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
-
Capri, March 2007 24
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
-
Capri, March 2007 25
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
-
Capri, March 2007 26
Jump to contact and tunneling regime
J.M. Krans et al. PRB 48 (1993) 14721
-
Capri, March 2007 27
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)
-
Capri, March 2007 28
force and conductance
Gold at room temperature
G. Rubio, N. Agrat and S. Vieira,PRL 76 (1996) 2302
-
Capri, March 2007 29
sp-metal
75 100 125 150012345678
Aluminium, 4.2 K
Con
duct
ance
(2e2
/h)
Piezo-voltage (V)
-
Capri, March 2007 30
d-metal
120 140 160 1800
2
4
6
8
10
12
Platinum, 4.2 K
Con
duct
ance
(2e2
/h)
Piezo-voltage (V)
-
Capri, March 2007 31
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
-
Capri, March 2007 32
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)
-
Capri, March 2007 33
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)
-
Capri, March 2007 34
Conductance histogram for niobium
0 1 2 3 4 50
500
1000
1500
2000
Cou
nts
G (2e2/h)
Nb13 K
-
Capri, March 2007 35
Semimetal Bi, 4 K
J.G. Rodigo et al. Phys. Rev. Lett. 88 (2002) 246801
-
Capri, March 2007 36
Semimetal Bi, 77K
J.G. Rodigo et al. Phys. Rev. Lett. 88 (2002) 246801
-
Capri, March 2007 37
MCBJ for alkali metals
-
Capri, March 2007 38
Conductance traces for sodium
10 15 20 25 30 35 40012345678
Sodium, 4.2 K
Con
duct
ance
(2e2
/h)
Piezo-voltage (V)
-
Capri, March 2007 39
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]
-
Capri, March 2007 40
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
-
Capri, March 2007 41
Local Density calculation for sodium contact
Nakamura, Brandbyge, Hansen & Jacobsen, Phys. Rev. Lett. 82, 1538 (1999).
-
Capri, March 2007 42
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
-
Capri, March 2007 43
Superconducting contacts: Niobium
15 20 25 300
2
4
6tunnelling regimecontact regime
G (2
e2/h
)
VP (V)
0,01
0,1
1
-
Capri, March 2007 44
Tunneling regime. Niobium
0 2 4 60
20
40
60C
urre
nt [n
A]
Voltage [/e]
-
Capri, March 2007 45
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)
-
Capri, March 2007 46
Principle of Multiple Andreev Reflection
eV >2
Ef
eV >
eV > 23
T T2
a b c
T3
leftelectrode
rightelectrode
-
Capri, March 2007 47
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)
-
Capri, March 2007 48
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
-
Capri, March 2007 49
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
-
Capri, March 2007 50
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)
-
Capri, March 2007 51
Monovalent metal: Au
E. Scheer et al., Nature 394, 154 (1998)
-
Capri, March 2007 52
Monovalent metal: Au
E. Scheer et al., Phys. Rev. Lett. 86, 284 (2001)
-
Capri, March 2007 53
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