Spin-dependent transport of interacting electrons in mesoscopic systems [Elektronische Ressource] / vorgelegt von Andreas Laßl

Andreas Lassl

Sujets

• Physik

Informations

• Publié par: universitat_regensburg
• Publié le: 01 janvier 2007
• Nombre de lectures: 17
• Poids de l'ouvrage: 7 Mo

Extrait

Spin-dependent Transport of Interacting
Electrons in Mesoscopic Systems
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften (Dr. rer. nat.)
der Naturwissenschaftlichen Fakult¨at II – Physik
der Universit¨at Regensburg
vorgelegt von
Andreas Laßl
aus Regenstauf
Oktober 2007Promotionsgesuch eingereicht am 23. Oktober 2007
Promotionskolloquium am 16. November 2007
Die Arbeit wurde angeleitet von Prof. Dr. Klaus Richter
Pru¨fungsausschuß:
Vorsitzender: Prof. Dr. Christoph Strunk
1. Gutachter: Prof. Dr. Klaus Richter
2. Gutachter: Prof. Dr. Milena Grifoni
Weiterer Pru¨fer: Prof. Dr. Andreas Scha¨ferContents
List of symbols iii
1 Introduction 1
1.1 Towards nanoscale devices . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Approaches to transport phenomena . . . . . . . . . . . . . . . . . . . 3
1.3 Purpose of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Green function formalism for electronic transport 9
2.1 Basic definitions and properties . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Perturbation expansion of the Green function . . . . . . . . . . . . . . 14
2.3 Nonequilibrium Green functions . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Discretization of the system . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 Dimensionless quantities . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.3 Lattice Green functions . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.1 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.2 Particle density . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.3 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.4 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Short-range interactions and the 0.7 anomaly 39
3.1 Phenomenology of the 0.7 anomaly . . . . . . . . . . . . . . . . . . . . 40
3.2 The model of interacting electrons . . . . . . . . . . . . . . . . . . . . . 43
3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.1 Influence of the interaction strength . . . . . . . . . . . . . . . . 48
3.3.2 Level splitting and polarization . . . . . . . . . . . . . . . . . . 50
3.3.3 Magnetic field dependence and the 0.7 analog . . . . . . . . . . 53
3.3.4 Zero field case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56ii Contents
3.3.5 Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.6 Temperature dependence . . . . . . . . . . . . . . . . . . . . . . 62
3.4 Connection with Coulomb interaction . . . . . . . . . . . . . . . . . . . 64
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.5.1 General features of the model . . . . . . . . . . . . . . . . . . . 66
3.5.2 Limitations of the model . . . . . . . . . . . . . . . . . . . . . . 68
3.5.3 Transport of cold fermionic atoms . . . . . . . . . . . . . . . . . 69
4 The self-consistent potential drop 73
4.1 A biased nanosystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 The electrostatic potential . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 The self-consistent calculation scheme . . . . . . . . . . . . . . . . . . . 79
4.4 Example: potential drop in a quantum wire . . . . . . . . . . . . . . . 82
5 Quantum ratchet systems 89
5.1 Directed transport in asymmetric potentials . . . . . . . . . . . . . . . 90
5.2 Coherent ratchet devices and adiabatic driving . . . . . . . . . . . . . . 92
5.3 A quantum dot charge ratchet . . . . . . . . . . . . . . . . . . . . . . . 94
5.4 A resonant tunneling spin ratchet . . . . . . . . . . . . . . . . . . . . . 97
5.5 Recapitulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6 Summary and perspectives 109
A Interaction self-energy in Hartree-Fock approximation 113
B Green function of a semi-infinite lead 119
C Recursive Green function algorithm 121
References 125
Acknowledgments 137List of Symbols iii
List of symbols
Symbol Description
a lattice constant of the numerical grid
0A(~x,~x;E) spectral function
~A vector potential of the magnetic field
~ ~ ~ ~B magnetic field, B =∇×A
β inverse temperature, β = 1/(k T)B
d(E) density of states
δ(x) Dirac delta distribution
δ Kronecker symbol, δ = 1 for i =j, otherwise δ = 0i,j i,j i,j
−19e charge quantum e = 1.60×10 As; the electron charge is−e
ε relative dielectric constant
−12ε vacuum dielectric constant, ε = 8.85×10 As/(Vm)0 0
E energy
E Zeeman energy, E =gμ BZ Z B
β(E−μ) −1f(E,μ) Fermi-Dirac distribution function, f(E,μ)= [e +1]
g gyromagnetic ratio, for free electrons g≈ 2
G conductance
2 −5G conductance quantum, G = 2e /h= 7.75×10 A/V0 0
G time-ordered Green function
r(a)G retarded (advanced) Green function
<(>)G lesser (greater) Green function
(r,a,>,<)G Green function of the noninteracting system0
γ interaction strength for the delta-type interaction
−34h Planck’s constant, h = 6.63×10 Js
−34~ scaled Planck’s constant, ~ =h/(2π)= 1.05×10 Js
H Hamilton operator of noninteracting particles0
H full Hamiltonian including interactions
I current
−23k Boltzmann constant, k = 1.38×10 J/KB B
2 2κ energy unit in the lattice representation, κ= ~ /(2ma )
m effective electron mass, for GaAs m = 0.07m0
−31m free electron mass, m = 9.11×10 kg0 0
μ chemical potential
−24μ Bohr magneton, μ =e~/(2m ) = 9.27×10 J/TB B 0
−7μ magnetic permeability, μ = 4π×10 Vs/(Am)0 0
n(~x) particle density ¡ ¢
∂ ∂ ∂~ ~∇ differential operator,∇ = , ,
∂x ∂y ∂z
−15φ magnetic flux quantum, φ =h/e = 4.14×10 Vs0 0iv List of symbols
Symbol Description
(†)ˆΨ (~x,t) field operator that destroys (creates) a particle at point ~x at time t
~r two-dimensional vector in the plane of the electron gas, ~r = (x,z)
σ spin index or spin quantum number,↑= +1/2,↓=−1/2
~σ vector of Pauli matrices
Σ self-energy
0 0 † 0ˆ ˆS(t,t) S-matrix describing the time evolution, S(t,t) =U(t)U (t)
t time
T temperature
T transmission
T rocking period
Θ(x) Heaviside step-function, Θ(x) = 1 for x > 0, Θ(x)= 0 otherwise
U(~x) external (confinement) potential
ˆU(t) time evolution operator
V source-drain voltage
V (~x) electrostatic potentiales
0 0w(~x,~x) interaction potential between particles at ~x and ~x
W width of the quantum wire
W interaction Hamiltonian
~x three-dimensional vector, ~x = (x,y,z)
brackets:
{A,B} anti-commutator of the operators A and B,{A,B} =AB+BA
[A,B] commutator of the operators A and B, [A,B] =AB−BA
hAi thermal expectation value of the operator A
hQ(~r)i spatial average of the quantity Q(~r)
hQ(t)i temporal average of the quantity Q(t)
sub- and superscripts:
a advanced function
L refers to the left lead or contact
r retarded function
R refers to the right lead or contacteX the tilde marks a dimensionless quantity
Q(x) the bar refers to quantities Q(x,z), averaged over the z-coordinate
abbreviations:
2DEG two-dimensional electron gas
a.u. arbitrary units
AC alternating current
DFT density functional theoryThe scientist does not study nature because it is
useful; he studies it because he delights in it, and
he delights in it because it is beautiful. If nature
were notbeautiful, itwouldnotbeworthknowing,
and if nature were not worth knowing, life would
not be worth living.
Jules Henri Poincar´e (1854-1912)CHAPTER1
Introduction
1.1 Towards nanoscale devices
The development of modern electronic equipment such as notebooks, mobile phones
or multimedia devices experiences a rapid evolution towards always smaller and more
powerful units. This trend continues thanks to the technological progress in designing
smaller and smaller electronic components, which simultaneously increases the num-
ber of transistors per unit area on a chip. Whereas the first transistor built in the
early 1950s had the extensions of several millimeters, nowadays commercial transistors
have a channel length down to 45nm. Already in the early days of electronics Moore
realized that the packing density doubles every year, which leads to an exponential
increase of the number of transistors per chip [1]. Although the doubling time was
later corrected to about two years, the exponential shrinking of the size of electronic
components known as Moore’s law is still valid to date. The reason why the contin-
uous miniaturization process could take place over the last decades is that even the
smallest commercial transistors still work in the classical diffusive transport regime.
The working principle and the electronic properties did not change essentially during
the downscaling. However, there is a natural limit for the validity of Moore’s law as
soon as the extensions of the devices reach the atomic scale or the scale of the mean
free path of the electrons.
Inthelab,nanostructureslikequantumdotsorquantumpointcontactscanbereal-
izedandyetcurrentsthroughmoleculesorsingleatomscouldbemeasured[2]. Whereas
the former systems rely on the semi