Quantum transport and its classical limit Piet Brouwer Laboratory of Atomic and Solid State Physics...
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Quantum transport and its classical limit Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University Lecture 1 apri spring school on Transport in Nanostructures, March 25-31, 2007
Quantum transport and its classical limit Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University Lecture 1 Capri spring school on
Quantum transport and its classical limit Piet Brouwer
Laboratory of Atomic and Solid State Physics Cornell University
Lecture 1 Capri spring school on Transport in Nanostructures, March
25-31, 2007
Slide 2
Quantum Transport About the manifestations of quantum mechanics
on the electrical transport properties of conductors These
lectures: signatures of quantum interference Quantum effects not
covered here: Interaction effects Shot noise Mesoscopic
superconductivity sample
Slide 3
Quantum Transport These lectures: signatures of quantum
interference What to expect? Magnetofingerprint Nonlocality R 1+2
=R 1 +R 2 B (mT) G 1+2 (e 2 /h) G 1 =G 2 =2e 2 /h B GG B (10 -4 T)
G (e 2 /h) Figures adapted from: Mailly and Sanquer (1991) Webb,
Washburn, Umbach, and Laibowitz (1985) Marcus (2005)
Slide 4
Landauer-Buettiker formalism sample x y W N: number of
propagating transverse modes or channels a n : electrons moving
towards sample b n : electrons moving away from sample Note: |a n |
2 and |b n | 2 determine flux in each channel, not density N
depends on energy , width W Ideal leads
Slide 5
Scattering Matrix: Definition sample |S mj;nk | 2 describes
what fraction of the flux of electrons entering in lead k, channel
n, leaves sample through lead j, channel m. Probability that an
electron entering in lead k, channel n, leaves sample through lead
j, channel m is |S mj;nk | 2 v nk /v mj. More than one lead: N j is
number of channels in lead j Use amplitudes a nj, b nj for
incoming, outgoing electrons, n = 1, , N j. Linear relationship
between a nj, b nj : S: scattering matrix
Slide 6
Scattering matrix: Properties sample Linear relationship
between a nj, b nj : S: scattering matrix Current conservation: S
is unitary Time-reversal symmetry: If is a solution of the
Schroedinger equation at magnetic field B, then * is a solution at
magnetic field B.
Slide 7
Landauer-Buettiker formalism Reservoirs sample Each lead j is
connected to an electron reservoir at temperature T and chemical
potential j. j, T Distribution function for electrons originating
from reservoir j is f( - j ).
Slide 8
Landauer-Buettiker formalism Current in leads sample j, T I
j,in I j,out In one dimension: = ( nk h) -1 Buettiker (1985)
Slide 9
Landauer-Buettiker formalism Linear response sample j, T I j,in
I j,out j = eV j Expand to first order in V j : Zero
temperature
Slide 10
Conductance coefficients sample j = -eV j IjIj Current
conservation and gauge invariance Time-reversal Note: only if B=0
or if there are only two leads. Otherwiseand in general.
Slide 11
Multiterminal measurements In four-terminal measurement, one
measures a combination of the 16 coefficients G jk. Different ways
to perform the measurement correspond to different combinations of
the G jk, so they give different results! I V V V I I Benoit,
Washburn, Umbach, Laibowitz, Webb (1986)
Slide 12
Landauer formula: spin Without spin-dependent scattering:
Factor two for spin degeneracy With spin-dependent scattering: Use
separate sets of channels for each spin direction. Dimension of
scattering matrix is doubled. Conductance measured in units of 2e 2
/h: Dimensionless conductance.
Slide 13
Two-terminal geometry r, r: reflection matrices t, t:
transmission matrices tr in out f()f() f()f() |t| 2 |r|2|r|2 t r
|t|2|t|2 |r| 2 eV (meV) f()f() Anthore, Pierre, Pothier, Devoret
(2003)
Slide 14
Quantum transport Landauer formula tr sample t r What is the
sample? Point contact Quantum dot Disordered metal wire Metal ring
Molecule Graphene sheet
Slide 15
Example: adiabatic point contact N(x)N(x) N min x g 10 6 2 4 8
0 V gate (V) -2.0-1.8-1.6-1.4-1.2 Van Wees et al. (1988)
Slide 16
Quantum interference In general: g small, random sign t nm, , t
nm, : amplitude for transmission along paths ,
Scattering matrix and Green function Recall: retarded Green
function is solution of In one dimension: k = and v = h -1 d k /dk
Green function in channel basis: r in lead j; r in lead k
Substitute 1d form of Green function If j = k:
Slide 19
Quantum transport and its classical limit Piet Brouwer
Laboratory of Atomic and Solid State Physics Cornell University
Lecture 2 Capri spring school on Transport in Nanostructures, March
25-31, 2007
Slide 20
Characteristic time scales h/Fh/F erg DD HH F l L Ballistic
quantum dot: ~ erg ~ L/v F, l ~ L Diffusive conductor: erg ~ L 2 /D
Inverse level spacing: relevant for closed samples Elastic mean
free time
Slide 21
Characteristic conductances Conductances of the contacts: g 1,
g 2 Conductance of sample without contacts: g sample if g >>
1 Bulk measurement: g 1,2 >> g sample Quantum dot: g 1,2
Assumptions and restrictions Always: F > 1. This implies
D
RMT: with time-reversal symmetry Quantum dot N 1 channels N 2
channels Weak localization correction is difference with classical
conductance For N 1, N 2 >> 1: Same as diagrammatic
perturbation theory Jalabert, Pichard, Beenakker (1994) Baranger
and Mello (1994)
Slide 37
RMT: conductance fluctuations Quantum dot N 1 channels N 2
channels Without time-reversal symmetry: With time-reversal
symmetry: Same as diagrammatic perturbation theory There exist
extensions of RMT to deal with contacts that contain tunnel
barriers, magnetic-field dependence, etc. Jalabert, Pichard,
Beenakker (1994) Baranger and Mello (1994)
Slide 38
Quantum transport and its classical limit Piet Brouwer
Laboratory of Atomic and Solid State Physics Cornell University
Lecture 3 Capri spring school on Transport in Nanostructures, March
25-31, 2007
Slide 39
Ballistic quantum dots Past lectures: Qualitative microscopic
picture of interference corrections in disordered conductors;
Quantitative calculations can be done using diagrammatic
perturbation theory Quantitative non-microscopic theory of
interference corrections in quantum dots (RMT). This lecture:
Microscopic theory of interference corrections in ballistic quantum
dots Assumptions and restrictions: F > 1 Method: semiclassics,
quantum properties are obtained from the classical dynamics
Slide 40
Semiclassical Green function Relation between transmission
matrix and Green function Semiclassical Green function (two
dimensions) : classical trajectory connection r and r S: classical
action of : Maslov index A : stability amplitude r r r
Slide 41
Comparison to exact Green function Semiclassical Green function
(two dimensions) Exact Green function (two dimensions) Asymptotic
behavior for k|r-r| >> 1 equals semiclassical Green
function
Slide 42
Semiclassical scattering matrix Insert semiclassical Green
function and Fourier transform to y, y. This replaces y, y by the
conjugate momenta p y, p y and fixes these to Result: Legendre
transformed action y Jalabert, Baranger, Stone (1990)
Slide 43
Semiclassical scattering matrix Legendre transformed action
Stability amplitude transverse momenta of fixed at y Transmission
matrix Reflection matrix
Slide 44
Diagonal approximation Reflection probability Dominant
contribution from terms = . probability to return to contact 1
Slide 45
Enhanced diagonal reflection Reflection probability If m=n:
also contribution if time- reversed of : Without magnetic field:
and have equal actions, hence Factor-two enhancement of diagonal
reflection Doron, Smilansky, Frenkel (1991) Lewenkopf,
Weidenmueller (1991)
Slide 46
diagonal approximation: limitations One expects a corresponding
reduction of the transmission. Where is it? Note: Time-reversed of
transmitting trajectories contribute to t, not t. No interference!
Compare to RMT: captured by diagonal approximation missed by
diagonal approximation We found The diagonal approximation
gives
Slide 47
Lesson from disordered metals =+ Hikami box + permutations Weak
localization correction to transmission: Need Hikami box. Weak
localization correction to reflection: Do not need Hikami box.
Slide 48
Ballistic Hikami box? In a quantum dot with smooth boundaries:
Wavepackets follow classical trajectories.
Slide 49
Ballistic Hikami box? Marcus group But quantum interference
corrections g and var g exist in ballistic quantum dots!
Slide 50
Ballistic Hikami box? Initial uncertainty is magnified by
chaotic boundary scattering. : Lyapunov exponent Aleiner and Larkin
(1996) Richter and Sieber (2002) Time until initial uncertainty ~ F
has reached dot size ~L: L= F exp( t) t = Ehrenfest time
Interference corrections in ballistic quantum dot same as in
disordered quantum dot if t E