Low dimensional structures. Quantum transport. Low dimensional structures. Metallic and SmCQuantum transport in 1D. Conductance quantizationAharonov-Bohm effect2DEG in magnetic field. Classical and quantum descriptionQuantum Hall effect

2DEG: 2 Dimensional Electron Gas

Free electrons are confined in the triangular potential in z-

direction. Energy level quantization.

They can move freely only in the direction parallel to the

heterostructure, i.e. in x-y direction

One may also prepare a special heterostructure which looks like wire or even a dot.

Low dimensional structures. EM structures versus Hetero-structures

3D ω2

c2= k2

x + k2y + k2

z

E , H ∝ ei k ⋅ r

E( k ) =ℏ2

2m* (k2x + k2

y + k2z )

Around extreme point of the dispersion relation

2DE n = 0

kz ⋅ c = πn, n = 1,2,3,...ω2

c2= k2

x + k2y +

π2n2

c2

1Dω2

c2= k2

x +π2m2

b2+

π2n2

c2

EM field between two conducting planes

EM Waveguide (mainly > 40 GHz)

Running wave appears only at certain frequency range

0D Resonator

ω2

c2=

π2l2

a2+

π2m2

b2+

π2n2

c2

2DEG in AlGaAs-GaAs structures. Electrons moves inTriangle potential. Free along x, y, quantized along z

E(kx, ky, n) =ℏ2

2m* (k2x + k2

y ) +ℏ2π2

2m*z

n2

l2d

Quantum wire. Semiconductor or metallic implementation

E(kx, m, n) =ℏ2k2

x

2m*x+ E(m) + E(n)

Quantum dot or artificial atom

E(l, m, n) = E(l) + E(m) + E(n)

EM waves Electrons

Low dimensional structures. Implementation with semiconductors and metals

Quantum wire fabricated of silicon bridge

Quantum wire created from 2DEG by applying the proper voltage on the gate

Quantum well

Quantum transport in 1D. Important length scales

Electron

eikr+iφ0

Impurities or scattering centers: phonons, ions, spins

Δφ1

Δφ2

Δφ3

Electron wave-packet propagates through a conducting medium. The wave experiences a random (stochastic) or certain change of its phase.

Our initial guess: More impurities -> the faster will be the process of loosing the wave properties.

However, at small structures with the typical dimensions of the order of the wavelength of the electronic wave-packet, electronic transport can be non-trivial.

Quantumness of the electronic transport depend on relation between elastic or inelastic scattering.

Quantum transport in 1D. Important length scales

Electron

eikr+iφ0

Impurities or scattering centers: phonons, ions, spins

Δφ1

Δφ2

Δφ3

Elastic mean free path � . Elastic scattering: phase may shift by a fixed amount. Occurs at scattering of impurity

atoms, dislocations and surfaces. As far as I know, maximal � (usual numbers 4-6 orders below) was

demonstrated in ultra-pure Copper bulk rod at T ~ 1K. The sample was purified and annealed. The resistance was sensitive to the impact. The elastic mean free path is determined by the surface scattering. In 2DEG the usual

numbers �

Inelastic mean free path � . Inelastic scattering: interaction with phonons. Phase coherence is lost. Wave-packet

before and after the inelastic scattering do not interfere with each other.

Phase coherence length � . Spin-flip scattering. It may be elastic, but it breaks phase correlations. At low

temperatures � .

Fermi wavelength: ~1 nm (metals), ~10-100 nm (2DEG in SmC heterostructures)

Observation of quantum transport requires dimension of the conductor to be an order of Fermi-wavelength

lele ≃ 1 cm

l2DEGe ≃ 10 μm

lin

lφ ≤ lin

lφ ≫ le

Quantum transport in 1D. Transport types

lφ ≲ L, WElectron performs diffusive motion through a sample

If the system dimension � interference effects may

play an essential role. Electron wave can be trapped after many collisions - > Weak localization.

L, W ≃ lφ

lφ ≃ L, le

lφ, le ≫ L

Does this quantum wire possesses any resistance?

Electron propagates through a system without inelastic scattering, just like an electromagnetic wave though a waveguide

le ≪ L, W

Picture is from H. Van Houten et al. In „Physics and Technology of Submicron Structures“, Springer Berlin 1988

Quantum transport in 1D. Conductance quantizationConsider a ballistic transport through a quantum wire orthrough a conducting 1D channel

Fixed boundary conditions

Ψ(x, y) ∝ sin kxx sin kyy

Single conducting state (channel) condition if

π 2kF

< W <π 5

kF

Electrons remains a Bloch-wave for the motion in z-direction

E(m, n, kz) = Ex(n) + Ey(m) +ℏ2k2

z

2m*

Conductance quantization R0 =h

2e2= 12.906 kΩ

Quantum transport in 1D. Conductance quantization

K. Hansen et al. Phys Rev B 56, 2208 (1997)

Aharonov-Bohm experiment. Fundamental aspects

XVIII-XIX century. Newtonian dynamics. Forces acts on bodies and define their dynamics.

Lagrangian approach based on energies and potentials is not just an aid to the Newtonian dynamics based on forces.

In classical physics electromagnetic fields are physical, while the electromagnetic potentials are purely mathematical constructions.

Aharonov-Bohm: (1) Wether the potentials are „physical“ or just convenient tool for the calculating the fields and forces?(2) Whether action principles are fundamental?(3) The principle of locality

Aharonov-Bohm experiment

(1959) Y. Aharonov and D. Bohm proposed „double-slit“ experiment. Its kind of Mach-Zehnder interferometer demonstrating wave-particle duality of electrons.

The idea is to control the constructive and destructive interference of two electron paths. It proves that electron is a wave indeed. Since electrons moves in the region with zero magnetic field, the interference also proves the physical reality of the vector potential.

The conductance of the loop has been found to oscillate with the flux

period �

Feyman path idea: �

Φ0 =he

TAB = T1 + T2 + 2 T1T2 cos φ

φ = φ0 +2πΦΦ0

Phys Rev 115, 485 (1959) - ideaPhys Rev Lett 74, 4047 (1995) - 2DEG implementation

Phys Rev B 11, 085413 (2008) - Graphene ring implementation

le < 2πR < lφ

Persistent current in normal metal rings

Superconducting ring shows quantization of the trapped

flux � which manifests the macroscopic wave

function for the Cooper pair condensate

Consider a normal metal ring with the radius R and high purity. At low very temperature (<0.5 K) electron

coherence extends over the whole ring �

Wave-function can circumvent a ring and form a standing wave which would correspond to the current without a decay (persistent current as in SC)

Two cases: (a) ballistic single channel (b) diffusive single channel (metal ring with a presence

of weak disorder)

Measurement of the trapped field. Similar to AB experiment, but there magnetoresistance was measured

Φs =h2e

2πR ≲ lφ