quantum coherent transport

8/11/2019 Quantum Coherent Transport

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Quantum coherent transport in

Meso- and Nanoscopic Systems

Philippe [email protected]

U of Arizona


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General Literature:Electronic Transport in Mesoscopic Systems

S. Datta, Cambridge Unversity Press, 1995(good, simple introduction)

Introduction to Mesoscopic PhysicsY. Imry, Oxford University Press, 1997

Mesoscopic Physics of Electrons and Photons

E. Akkermans and G. Montambaux, Cambridge UniversityPress, 2007

(a must if you want to enter the field and plan to stay)


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Mesoscopic vs. nanoscopic physics

Nanometer = length scale = 10-9mNanophysics = physics at the sub-micron length scale

Mesoscopic physics = physics intermediate betweenmacroand micro

That the humaneye can see

Atomic level

In what sense ?-> Phase-coherent nature of the electron matters!


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Nanoscopic is entirely defined by length scales

Mesoscopic is a regime (in a sense to be defined)

Nanometer = length scale = 10-9mNanophysics = physics at the sub-micronic length scale

Mesoscopic physics = physics intermediate betweenmacroand micro

Corollary:

The smallest, cleanest mesoscopic systems fall intothe class of nanoscopic systems

Mesoscopic vs. nanoscopic systems


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What is

Mesoscopic Physics ?


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system is INCOHERENT (no phase) use CLASSICAL MECHANICS

V

Conductivity : !=n e "/mConductance : G=!W/L

2

Solid-State Physics 101:Drude-Boltzmann Theory of Transport

Electrons = point particles ; (charge,mass)=(-e,m)

Metal is disordered uncorrelated collisions at impurities DIFFUSION

~Ohmic conductor with intensive conductivity


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system is INCOHERENT (no phase) use CLASSICAL MECHANICS

V

Conductivity : !=n e "/mConductance : G=!W/L

2

Is this all ? Are there quantum corrections ?

What happens once # is taken into account ?F

Solid-State Physics 101:Drude-Boltzmann Theory of Transport

Electrons = point particles ; (charge,mass)=(-e,m)

Metal is disordered uncorrelated collisions at impurities DIFFUSION


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Mesoscopic Physics - Length Scales

# l L L$0L >> L %system is coherent !T-dependent!

%use Quantum Mechanics !

L >># %system is semiclassical

%use small parameter # /L

$

F

F

F

FermiWavelength

Elastic MeanFree Path System Size

CoherenceLength

Between microscopic and macroscopic ; N. van Kampen 81

Diffusive systems ~ lL


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Some Mesoscopic Systems

Webb 84 Heiblum 97

Kouwenhoven 10

van Wees 08

Marcus 03


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What is new in

Mesoscopic Transport ?


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Mesoscopic Physics : Novel Anomalous Properties I

Nonlocality:

Details of sample out of direct current path matter!!

See: Washburn and Webb,

Phys. Today 88


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Sample-Specificity:

Nonlocality + Sample Specificity:

%Breakdown of Ohms Law

G = !W/L%Look at conductance, not conductivity

Samesamples, i.e. with same material

& same fabrication, exhibit discrepancies

in transport properties

Mesoscopic Physics : Novel Anomalous Properties II


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Dependence onsample geometry

From: Ford et al., PRL 89

Anomalous Hall Effect

due to underlying

classical dynamics

Mesoscopic Physics : Novel Anomalous Properties III


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Violationof macroscopic/classical symmetries

E.g. Onsager 31:

G(H)=G(-H)

Dependence on measurement set-up

From: Benoit et al., PRL 86

Mesoscopic Physics : Novel Anomalous Properties IV

Kobayashi et al., J. Phys. Soc. Jpn !02

II I

IVV V

V=


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Main Novel Fundamental Effects

inQuantum Coherent Transport


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Quantum Corrections to Transport I:Weak localization


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Coherent Mesoscopic Corrections to Transport I:

Weak Localization

Th. : Gorkov, Larkin, Khmelnitskii, JETP Lett. 79

Abrahams, Anderson, Ramakrishnan, PRL 79

Rev.: Bergmann, Phys. Rep. 83; Lee and Ramakrishnan, RMP 85

QM destructive interferences induce negative corrections

to Drude conductivity

Expansion in # /lIn 2D :

F

Fig. From : van den Dries et al., PRL 81


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Coherent Mesoscopic Corrections to Transport I:

Weak Localization - Quasiclassical Picture

A B

C

!Break TRS - magnetic field

!Probability to be at C:

P(C) ~ |P+P| =2 |P| (1+cos[2&$/$0])22

QM interferences disappear = positive magnetoconductance

$


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Weak Localization - Quasiclassical Picture

A B

C

!Break TRS - magnetic field

!Probability to be at C:

P(C) ~ |P+P| =2 |P| (1+cos[2&$/$0])

!Averaging over loop length/flux distribution

22

$


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(Chang et al., PRL 94) (Chan et al., PRL 95)

Trademark of Weak Localization - Magnetoresistance


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Quantum Corrections to Transport II:Universal Conductance Fluctuations


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Searching for Aharonov-Bohmoscillations in ring with L


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Quantum Corrections to Transport III:Aharonov-Bohm effects


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Sharvin and Sharvin 81

~Aharonov-Bohm oscillations


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Magnetoresistance

~Aharonov-Bohm oscillations

Measurement sample

diam ~1m, width ~0.04m

Amplitude of oscillations decreases

with increasing temperature

~decoherence/dephasing


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Quantum Corrections to Transport III:Conductance Quantization


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[001][100]

[010]

two-dimensional

electron gas

(2DEG)

quantum point contact

(QPC)

Quantization plateaux at

multiples of G0=2e2/h

Gate voltage ~ tunes width of QPvan Wees et al./Wharam et al. 88


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Quantum Corrections to Transport IV:Coulomb Blockade


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Electrons are verticallyConfined to 2DEG

Electrons are laterallyConfined by gates

Average conductance controlledby tuning opening of contacts

[001][100]

[010]

two-dimensional

electron gas

(2DEG)

What is a quantum dot ?


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(Marcus lab.,

Harvard)

(Alhassid, RMP 01)


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The two-slit experiment (textbook version)

|'|2=|'1+'2|2

1

2


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The two-slit experiment (non-textbook version)

?


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Regular cavityChaotic cavity


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"IS THIS DECOHERENCE ?

"DO CHAOTIC / COMPLEX SYSTEMS DECOHERE

WHEREAS REGULAR / INTEGRABLE SYSTEMS

DO NOT ?


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NO!It is multiple random scattering


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Regular cavityChaotic cavity

Despite multiple chaotic scattering

the Gaussian envelope still exhibits

(small) modulations !

A.k.a. weak localization

quantum coherent transport

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