Condensed Matter Physics 1 − 응집물질물리학 1
Hongki Min
[email protected]
Department of Physics and Astronomy, Seoul National University, Korea
Orientation, March 2, 2017
Syllabus
Syllabus
References [1] Fundamentals of Condensed Matter Physics Marvin L. Cohen and Steven G. Louie
[2] Condensed Matter Physics Michael P. Marder
[3] Solid State Physics Neil W. Ashcroft and N. David Mermin
References for many-body physics [1] Introduction to Many Body Physics Piers Coleman [2] Quantum theory of many-particle systems Alexander L. Fetter and John Dirk Walecka [3] Quantum theory of the electron liquid Gabriele Giuliani and Giovanni Vignale [4] Many-particle physics (3rd Ed.) Gerald D. Mahan
Topics for CMP1 Part I. Basic Concepts: Electrons and Phonons 1. Concept of a solid: qualitative introduction and overview 2. Electrons in crystals 3. Electronic energy bands 4. Lattice vibrations and phonons Part II. Electron Interactions, Dynamics and Responses 5. Electron dynamics in crystals 6. Many-electron interactions: the interacting electron gas and beyond 7. Density functional theory (DFT) 8. The dielectric function for solids
Chapter 1
Concept of a solid: qualitative introduction and overview
Condensed matter theory Subjects • Properties of condensed phases of matter ― Solids, liquids, superconductors, magnets, … ― Electronic structure, optical & transport properties, interaction effects, quasi-particle properties, … Tools • Modeling ― Tight-binding model, effective theory, … • Analytic approach ― Many-body theory, quantum field theory, … • Numerical approach ― Density functional theory, Monte Carlo,…
Many-body problem System of interacting electrons and ions
H H el H el ion H ion •
•
•
Electron Hamiltonian 2 2 1 e2 H el i 2 i j ri r j i 2m Electron-ion interaction Z I e2 H el ion i , I ri R I Ion Hamiltonian 2 Z Z e 2 1 H ion 2I I J 2 I J R I R J I 2M I
Statistics, symmetries, effective low-energy theories
[1] Quasiparticle self-energy Corrections to the particle’s energy due to interactions
𝜀 𝑘 = 𝜀0 𝑘
See Mahan Ch.5.8 non-interacting
1 𝐴 𝒌, 𝜀 ~Im 𝜀 − 𝜀(𝒌)
𝜀0(𝒌)
𝜀
[1] Quasiparticle self-energy Corrections to the particle’s energy due to interactions
𝜀 𝑘 = 𝜀0 𝑘 +Σ(𝑘)
See Mahan Ch.5.8 non-interacting
Σ 𝑘 = Σ𝑒−𝑒 𝑘 + Σ𝑒−𝑝ℎ 𝑘 + ⋯
ReΣ(𝑘)~𝛿𝜀(𝑘) ℏ ImΣ(𝑘)~ 2𝜏𝑘
1 𝐴 𝒌, 𝜀 ~Im 𝜀 − 𝜀(𝒌)
ReΣ~𝛿𝜀 ImΣ~ℏ/2𝜏 interacting
𝜀0(𝒌)
Interactions renormalize the energy dispersion and gives rise to lifetime.
𝜀
[2] Response function Linking measurements to correlations See Giuliani and Vignale, Ch.3~5
n ~ nVext
n ~ n n
J ~ Eext
~ J J
M ~ M H ext
M ~ M M
Response to the experimental probes can be expressed in terms of correlation functions, which contain information of the unperturbed system.
Chapter 2
Electrons in crystals
Hamiltonian Time independent Schrödinger equation
H (k ) n (k ) En (k ) n (k ) k En
n
wave vector energy band band index
Lattice translational symmetry Periodic system •
Hamiltonian is invariant under the lattice translation
TR f ( x) f ( x R)
[TR , H ] 0 •
H and TR can have simultaneous eigenstates
TR C R
H E TRTR TR R •
CR e
Bloch’s theorem
( x R) e ( x) ikR
ikR
k is a quantum number
representing an eigenstate in a periodic system.
Chapter 3
Electronic energy bands
Tight-binding model Bloch wavefunction and atomic orbitals
(k )
= Linear combination of atomic orbitals
E (k ) Variational method
(k ) H (k ) (k ) (k )
H (k )c E(k )S (k )c k
wave vector
c
coefficients vector
S
overlap matrix
Tight-binding model Example: 1D chain composed of two kinds of atoms 2
a εβ
t
α
t
β
α H * f (k )
f (k ) t e
ik ( a / 2 )
α
β
β f (k ) e
ik ( a / 2 )
E/|t|
εα
0
-2
-π
0
ka
π
0.2 | t |
ka 2t cos 2
Monolayer graphene
6C=1s22s22p2
Tight-binding model , *: sp2
β R1 α
R2
α 0 H * f (k )
K M
t
β
20
R3
Energy (eV)
β
30
β f (k ) 0
, * : pz σ , σ*
π, π*
K
10 0 -10
-20
K
f (k ) t eikR1 eikR 2 eikR3 v0 (k x ik y ) Expand around K
M
K
Chapter 4
Lattice vibrations and phonons
Phonons and electron-phonon interaction •
Harmonic oscillators and phonons
𝑝𝑖2 1 𝐻0 = + 𝐾 𝑥𝑖 − 𝑥𝑖−1 2𝑀 2 𝑖
ℏ𝜔𝑘 † † → 𝑎𝒌 𝑎𝒌 + 𝑎−𝒌 𝑎−𝒌 2
2 See Coleman, Ch.2.4; Mahan Ch.1.1
𝑘
•
Electron-phonon interactions 𝑍𝑒 −𝑒 𝑑 𝑑 ′ ′) 𝐻int = ∫ 𝑑 𝑥𝑑 𝑥 𝑛 𝑥 𝛿𝑛 (𝑥 b 𝑥 − 𝑥 ′ el → 𝑀𝑞 𝜌(−𝑞) 𝑎𝒒 + 𝑞
† 𝑎−𝒒
See Coleman, Ch.8.7; Fetter and Walecha, Ch.12
Chapter 5
Electron dynamics in crystals
Berry phase Quantum phase by adiabatic evolution of a quantum state
𝑒 𝑖𝛾12 = 𝑅1 𝑅2
𝑅1
Not well defined
𝑅3
𝑒 𝑖𝛾123 = 𝑅1 𝑅2 𝑅2 𝑅3 𝑅3 𝑅1 Well defined for a closed path
𝑒
𝑖𝛾(𝐶)
= lim 𝑅1 𝑅2 … 𝑅𝑁 𝑅1
𝑅2
𝑅1
𝑅2 1 2 3 𝑁
𝑁→∞
𝐶
Berry connection and Berry curvature Geometric phase for a closed path
𝑒 𝑖Δ𝛾 = 𝑹 𝑹 − Δ𝑹 ≈ 1 − Δ𝑹 𝑹 𝛻𝑹 Δ𝛾 = 𝑖Δ𝑹 ⋅ 𝑹 𝛻𝑹
𝛾(𝐶) = ∫𝐶 𝑑𝑹 ⋅ 𝑨(𝑹)=∫𝑆 𝑑𝑺 ⋅ 𝛀(𝑹) 𝑨(𝑹) = 𝑖 𝑹 𝛻𝑹 𝛀 𝑹 = 𝛻 × 𝑨 𝑹 = 𝑖 𝛻𝑹 × 𝛻𝑹
𝑆
𝐶
Analogies with electromagnetism Similar structures to electromagnetism Berry connection
Vector potential
𝑨(𝑹) = 𝑖 𝑹 𝛻𝑹
𝑨(𝒓)
Berry curvature
Magnetic field
𝜴 𝑹 =𝛻×𝑨 𝑹
𝑩 𝒓 = 𝛻 × 𝑨(𝒓)
Berry phase
Magnetic flux
𝛾 = ර 𝑑𝑹 ⋅ 𝑨 = න 𝑑𝑺 ⋅ 𝜴
Φ𝐵 = ර 𝑑𝒓 ⋅ 𝑨 = න 𝑑𝑺 ⋅ 𝑩
Chern number 1 𝒅𝑺 ⋅ 𝛀 𝑹 𝐶1 = 2𝜋
Magnetic monopole 1 𝒅𝑺 ⋅ 𝑩 𝒓 𝑒𝑀 = 4𝜋
Anomalous velocity Semiclassical equations of motion
𝜕𝜀𝑛 𝒓ሶ = −𝒌ሶ × 𝛀𝑛 𝛀𝑛 𝒌 = 𝑖 𝛻𝒌 𝑢𝑛 (𝒌) × 𝛻𝒌 𝑢𝑛 (𝒌) ℏ𝜕𝒌 𝒓ሶ ℏ𝒌ሶ = (−𝑒) 𝑬 + × 𝑩 𝑐 𝜕𝜀 𝑒 2 − 𝑬 × ∫ 𝑑𝒌𝑓 𝒌 𝛀𝑛 (𝒌) 𝑱 = (−𝑒)∫ 𝑑𝒌𝛿𝑓(𝒌) ℏ𝜕𝒌 ℏ Intrinsic contribution Ashcroft and Mermin, "Solid state physics" Xiao, Chang and Niu Rev. Mod. Phys. 82, 1959 (2010)
Chapter 6
Many-electron interactions: the homogeneous interacting electron gas and beyond
Many-body problem System of interacting electrons and ions
H H el H el ion H ion •
•
•
Electron Hamiltonian 2 2 1 e2 H el i 2 i j ri r j i 2m Electron-ion interaction Z I e2 H el ion i , I ri R I Ion Hamiltonian 2 Z Z e 2 1 H ion 2I I J 2 I J R I R J I 2M I
Electron-electron interaction Interaction-induced ordered states
•
Magnetism
•
Superconductivity
•
Exciton-condensation ...
–,↑
–,↓
+
+
+
–
–
–
Mean-field theory Weiss molecular-field approximation 1 H J ij S i S j 2 i , j S i S i S i
S i S j S i S i S j S j
S i S j S i S j S i S j S i S j Si S j S j Si Si S j H MF Β iMF S i i
Β iMF J ij S j j
Mean-field theory Weiss molecular-field approximation T T c Β iMF 0
Si 0
T T c Β iMF 0
Si 0
Mean-field theory Hartree-Fock approximation –
–
–
–
–
–
–
–
–
Interacting electrons
Non-interacting electrons in an effective potential
Ground state energy Exchange and correlation energies 𝐻 = 𝐻0 + 𝑉𝑒𝑒
𝐸𝐻𝐹 = 0 𝐻 0 = 𝐸0 + 𝐸𝑒𝑥
ȁ0ۧ: ground state for 𝐻0
𝐸 = Ω 𝐻 Ω = 𝐸𝐻𝐹 + 𝐸𝑐𝑜𝑟𝑟
ȁΩۧ: ground state for 𝐻
Hartree-Fock
𝐸0 = 0 𝐻0 0 , 𝐸𝑒𝑥 = 0 𝑉𝑒𝑒 0 , 𝐸𝑐𝑜𝑟𝑟 = 𝐸 − 𝐸𝐻𝐹 Exchange energy 𝐸𝑒𝑥
1 𝑑𝑑 𝑞 ℏ ∞ 𝑑𝜔 = න 𝑣𝑞 − න 𝜒0 𝑞, 𝑖𝜔 − 1 𝑑 2 2𝜋 𝑛0 0 𝜋
RPA correlation energy 𝑅𝑃𝐴 𝐸𝑐𝑜𝑟𝑟
RPA
𝜒𝜆 →
𝜒0 1−𝜆𝑣𝑞 𝜒0
ℏ 𝑑𝑑 𝑞 ∞ 𝑑𝜔 = න න 𝑣𝑞 𝜒0 (𝑞, 𝑖𝜔) + ln 1 − 𝑣𝑞 𝜒0 (𝑞, 𝑖𝜔) 𝑑 2𝑛0 2𝜋 𝜋 0
Chapter 7
Density functional theory (DFT)
First-principle calculations System of interacting electrons and nuclei
H H el H ion H el ion •
•
•
Electron-electron interaction ― Density functional theory
1 e2 H el el 2 i j ri r j
Electron-ion interaction H el ion i,I ― Pseudopotential approximation
Z I e2 ri R I
2 Z Z e 2 1 H ion 2I I J 2 I J R I R J I 2M I
Ion relaxation ― Adiabatic (Born-Oppenheimer) approximation ― Electron dynamics in frozen ionic configurations
Density functional theory P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)
Hohenberg-Kohn theorem •
The ground state properties of a many-electron system are uniquely determined by an electron density
•
Energy functional
External potentials from ions
E[n] T [n] Vel el [n] Vext [n]
The ground state energy and many-body wavefunctions are obtained by minimizing E[n]
Vel el [n] VH [n] Vxc [n] 1 e 2 n(x)n(x) : Hartree functional VH [n] dxdx 2 x x
Vxc [n] : Exchange-correlation functional
Density functional theory Kohn and Sham, Phys.Rev.140, A1133 (1965)
Kohn-Sham equations •
Parameterize the particle density in terms of a set of auxiliary non-interacting single electron orbitals
2 2 H (x) xc (x) ext (x) i i i 2m eff (x) 2 e n(x) Vxc [n] H [ n ] d x xc (x) x x n(x)
Mapping of the interacting many-electron system •
onto a system of non-interacting electrons moving in an effective potential due to all the other electrons 2 n(x) i (x) Self-consistent method
n n (1)
(1) eff
( 2)
( 2) eff
iocc
Density functional theory Exchange-correlation •
All many-body effects of exchange and correlation are grouped in Vxc[n]
•
In principle, the Kohn-Sham equations are exact, but we do not know the exact form of Vxc[n].
•
Local density approximation (LDA) ― The exchange-correlation potential is assumed locally that of a homogeneous system
•
Extensions of LDA ― Local spin density approximation (LSDA) ― Generalized gradient approximations (GGA)
Pseudopotential approximation Replacement of a strong potential to a weaker one •
Near the atomic region, the kinetic energy of the electrons is large resulting in rapid oscillations.
•
Strong ionic potential is replaced by a weaker pseudopotential which acts on the pseudowavefunctions Adapted from Wikipedia
•
The pseudopotential is much weaker than the true potential making the solution much simpler.
Chapter 8
The dielectric function for solids
Linear response theory Linear response to an external perturbation
𝐻 𝑡 = 𝐻0 + 𝐻ext (𝑡) 𝐻ext 𝑡 = 𝑛 𝑡 𝑉ext (𝑡) 𝑛 𝑡
ext
𝑖 𝑡 − න 𝑑𝑡′ 𝑛 𝑡 , 𝐻ext 𝑡′ ℏ −∞
≈ 𝑛 𝑡
up to first-order of 𝐻ext
+∞
𝛿𝑛 𝑡
≈න −∞
𝑑𝑡 ′ 𝜒𝑛 𝑡 − 𝑡 ′ 𝑉ext (𝑡 ′ )
𝜒𝑛 𝑡 − 𝑡 ′ = Θ(𝑡 − 𝑡 ′ ) 𝑛 𝑡 , 𝑛 𝑡′ Response function : Retarded correlation function
Linear response theory Linking measurements to correlations Experimental probes can be regarded as small perturbations
𝛿𝑛 ∼ 𝜒𝑛 𝑉ext
𝜒𝑛 ∼ 𝑛 𝑛
𝛿𝐽 ∼ 𝜎𝐸ext
𝜎∼ 𝐽𝐽
𝛿𝑀 ∼ 𝜒𝑀 𝐻ext
𝜒𝑀 ∼ 𝑀 𝑀
Response to the experimental probes can be expressed in terms of linear response functions, which contain information of the unperturbed system.
Linear response theory Response function in frequency space
i AB (t t ) (t t ) Aˆ (t ), Bˆ (t )
AB ( ) dteit AB (t ) i m t i n t i mn t ˆ ˆ m A(t ) n e m An e Amn e
Pm Pn Amn Bnm AB ( ) m , n m n i 0 Pn: Occupation probability for state n
Non-interacting response function Density response of non-interacting electrons k+q
k
𝑛𝒌,𝑠 − 𝑛𝒌+𝒒,𝑠′ 𝑑𝑑 𝑘 𝜒0 𝑞, 𝜔 = 𝑔 න 2𝜋 𝑑 ℏ𝜔 + 𝜀𝒌,𝑠 − 𝜀𝒌+𝒒,𝑠′ + 𝑖0+ 𝑠,𝑠′
𝑛𝒌 𝑔 𝑠
Fermi distribution function for k spin/valley degeneracy factor band index
0 (q 0, 0) N 0 N0: DOS at EF
Linear response theory : Interacting system Density response of interacting electrons n n 0 Vext Veff Veff Vext Vint
Veff: Effective self-consistent field acting on each particle independently Vint: Internal field generated by the density deviations
n Veff Vint n 0 0 1 int 0 1 Veff Vext n Vext 1 int 0 int Vint int C xc Random phase approximation n Response of interacting system to an external field can be obtained from the response of non-interacting system to an effective self-consistent field
Linear response theory : Interacting system Polarization and dielectric function •
Transport properties ― Screening
–
–
–
+
–
–
•
•
Collective modes ― Ex: Plasmon Collective density oscillations
Correlation ― Electron motion ↔ Surrounding electrons
–
–
–
–
–
– –
– – – –
– –
Example: Plasmons Semi-classical approach Density oscillations due to the restoring force by a selfconsistent electric field generated by local excessive charges 𝑥 + + + +
𝐸
− − − −
See Ashcroft & Mermin, Ch.1
Displaced electron gas with the positive background of ions
𝑁𝑚𝑥ሷ = 𝑁(−𝑒)𝐸 𝐸 = 4𝜋𝜎 = 4𝜋𝑛𝑒𝑥 2 𝑥ሷ + 𝜔𝑝𝑙 𝑥=0
𝜔𝑝𝑙 =
4𝜋𝑛𝑒 2 𝑚
Example: Plasmons Quantum mechanical approach The frequencies of collective modes in a many-body system are determined by the poles of the response function.
𝛿𝑛 𝜒𝑛 = ~ 𝑛𝑛 𝛿𝑉ext
Response ~ Correlation
𝜒𝑛 → ∞ 3𝐷 𝜔𝑝𝑙 =
~√𝑞(𝑐1 + 𝑐2 𝑞) ~𝑐1 + 𝑐2 𝑞2
Collective modes
4𝜋𝑛𝑒 2 1 + 𝒪(𝑞 2 ) 𝑚
Topics for CMP2 (Next semester) Part III. Optical and Transport Phenomena 9. Electronic transitions and optical properties of solids 10. Electron-phonon interactions 11. Dynamics of crystal electrons in a magnetic field 12. Fundamentals of transport phenomena in solids Part IV. Superconductivity, Magnetism, and Lower Dimensional Systems 13. Using many-body techniques 14. Superconductivity 15. Magnetism 16. Reduced-dimensional systems and nanostructures
Chapter 9
Electronic transitions and optical properties of solids
Optical conductivity Kubo formula 2
ie xx ( )
2
n, k v x n, k d k f n ,k f n ,k d n , n ( 2 ) n ,k n,k n ,k n,k i d
η: broadening by impurity 2 ie 2 intra ( ) D ( ) M n F nn ( k F ) ( i ) n 2 e inter real ( 0)
D
n n
nn
( ) M (k ) 2 nn
T→0 limit, n : states crossing εF
T, η→0 limit 2EF n : occupied states n': unoccupied states
g s g v k g s g v k Dn ( ) (2D DOS) Dnn ( ) (Joint DOS) 2 nk nk 2 nk 2 d 2 (Angular integral of velocity M nn (k ) n, k vx n, k operator for given k ) 2
Optical conductivity Monolayer graphene at n=5x1012 cm-2
σ(E)/σuni
2EF
2EF
uni
E (eV)
e2 2 h
Exciton effects Coulomb interaction between the excited electron and hole
© Cambridge University Press 2016
Chapter 10
Electron-phonon interactions
Phonons and electron-phonon interaction •
Harmonic oscillators and phonons
𝑝𝑖2 1 𝐻0 = + 𝐾 𝑥𝑖 − 𝑥𝑖−1 2𝑀 2 𝑖
ℏ𝜔𝑘 † † → 𝑎𝒌 𝑎𝒌 + 𝑎−𝒌 𝑎−𝒌 2
2 See Coleman, Ch.2.4; Mahan Ch.1.1
𝑘
•
Electron-phonon interactions 𝑍𝑒 −𝑒 𝑑 𝑑 ′ ′) 𝐻int = ∫ 𝑑 𝑥𝑑 𝑥 𝑛 𝑥 𝛿𝑛 (𝑥 b 𝑥 − 𝑥 ′ el → 𝑀𝑞 𝜌(−𝑞) 𝑎𝒒 + 𝑞
† 𝑎−𝒒
See Coleman, Ch.8.7; Fetter and Walecha, Ch.12
Phonon renormalization by electron gas See Ashcroft and Mermin, Ch.26
Because of the huge difference in masses between electrons and ions, electrons follow the ion motion dressing the bare ions with the cloud of screening electrons. 𝜖ion
Ω20 =1− 2 𝜔
Ω2 =Ω20 /𝜖el
dress 𝜖ion
→
, 𝜖el ≈ 1 +
Ω2 =1− 2 𝜔
2 𝑞TF 𝑞2
dress 𝜖ion = 0 → 𝜔 = 𝑣s 𝑞 (long wavelength oscillations)
The long-range Coulomb interaction between ions are screened by electrons and the phonon-like collective oscillations occur in metals.
Effective electron-electron interaction See Ashcroft and Mermin, Ch.26
The Coulomb interaction between two electrons is screened not only by other electrons but also by ions. 𝑉scr
𝑉C = 𝜖el
→ 𝑉eff =
𝑉C dress 𝜖el 𝜖ion
𝑉C 𝜔2 = 𝜖el 𝜔 2 − Ω2
The frequency dependence reflects that the screening action of the ions is not instantaneous but limited by the phonon velocity thus the interaction is retarded.
Note that at low frequencies the effective action becomes attractive rather than repulsive, leading to the instability of the Fermi surface and the phase transition from a normal to superconducting state.
Electron self-energy for el-phonon interaction Consider the self-energy correction due to el-ph interaction
Example: Einstein phonons
† 𝐻int = 𝑀𝜆 𝒒 𝜌(−𝒒)(𝑎𝒒𝜆 + 𝑎𝒒𝜆 )
Re𝛴
𝒒,𝜆 (0)
𝐷𝜇 (𝒒, 𝑖𝜈𝑚 )
Σ(𝒌, 𝑖𝜔𝑛 ) = 𝑀𝜆
𝒒
𝑔
0
Im𝛴 𝑀𝜆 𝒒
(𝒌 + 𝒒, 𝑖𝜔𝑛 + 𝑖𝜈𝑚 )
𝑀𝜆 𝒒 𝑀𝜆 𝒒 = 𝜖(q)
𝜔 (Ω0 ) See Coleman Ch.8.7; Mahan Ch.7.3~4
Chapter 11
Dynamics of crystal electrons in a magnetic field
Hall measurements Hall effect •
Voltage difference transverse to an electric current under a perpendicular magnetic field
–
+
–
+
– –
+ +
v (e) B c Adapted from Wikipedia
Electrons under magnetic fields Classical Drude model •
Lorentz force B
v dv v m (e) E B c dt
–
Steady state τ: relaxation time
E x 1 j x R H Bj y E y 1 j y R H Bj x
ne m 2
1 RH n ( e)c
RH depends only on the carrier density
Electrons under magnetic fields Classical Hall effect •
Diagonal resistivity is finite and independent of B field
xx •
1
m 2 ne
Hall resistivity is proportional to B field while does not depend on the relaxation time.
B xy R H B nec •
Hall measurement can determine the sign of carrier density.
Integer quantum Hall effect (IQHE) Vanishing diagonal resistivity and quantized Hall resistivity
Paalanen, Tsui, and Gossard Phys. Rev. B 25, 5566 (1982)
Integer quantum Hall effect (IQHE) Experimental observation •
Vanishing diagonal resistivity and dissipation
xx 0 •
Quantization of Hall resistivity/conductivity
h xy 2 e •
ν: integer
h/e2=25812.807572 Ω
The quantization is universal and independent of all microscopic details such as the material type, sample purity, and value of magnetic field.
Landau levels Quantized Hall conductivity
1 n c n 2
eB c mc
(n=0,1,2,3,∙∙∙)
Landau levels Formation of discrete degenerate energy levels •
The energy eigenvalues are discretized into 1 n c n like a harmonic oscillator. 2 Landau levels
•
The Landau levels are degenerate with respect to the freedom in the positions of cyclotron orbits. Nd
•
Lx L y 2l B2
BLx Ly 0
Total number of magnetic flux measured in units of flux quanta Φ0=hc/e
The ratio for the number of electrons to degenerate states is called the filling factor. N n 0 Nd B
N n (charge density) Lx L y
Edge states Conduction through 1D channel Electrons localized in orbits
Electrons moving along edges
Adapted from Nagaosa, Science 318, 758 (2007)
•
Landauer formula
dk k e e2 I ( e) V 2 k h h
e2 I VH h ν separate edge channels
Chapter 12
Fundamentals of transport phenomena in solids
Boltzmann transport theory •
Non-equilibrium distribution function 𝑓𝒌 (𝑡) 𝑑𝒌 𝑛=න 𝑓𝒌 𝑡 𝑑 2𝜋 𝑑𝒌 𝑱 = −𝑒 ∫ current density 𝑑 𝑓𝒌 𝑡 𝒗(𝒌) 2𝜋
•
Relaxation time approximation 𝑑𝑓𝒌 𝜕𝑓𝒌 𝑑𝒌 = ⋅ 𝑑𝑡 𝜕𝒌 𝑑𝑡 (0) 𝑓𝒌
𝑓𝒌 − ≈− 𝜏𝒌
𝑑𝒌 ℏ = −𝑒 𝑬 𝑑𝑡 (0)
𝑓𝒌
equilibrium
= distribution function 𝜏𝒌 = relaxation time
Boltzmann transport theory •
•
For example, see Relaxation time 𝒌′ Ashcroft & Mermin, Ch.13 𝜃𝒌𝒌′ 1 𝑑𝒌′ 𝒌 ′ ′ =න 𝑊 (1 − cos 𝜃 ) 𝒌𝒌 𝜏𝒌 2𝜋 𝑑 𝒌𝒌 2𝜋 Fermi golden rule 𝑊𝒌𝒌′ = 𝑛imp 𝑉𝒌𝒌′ 2 𝛿(𝜀𝒌 − 𝜀𝒌′ ) ℏ
Electrical conductivity 2 𝑣 F 2 𝜎 = 𝑒 𝐷(𝜀F )𝕯 ∝ 𝑛imp 𝑉 2 DOS
𝑱 = 𝜎𝑬 1 2 1 𝜕𝜀F 𝕯 = 𝑣F 𝜏F 𝑣F = 𝑑 ℏ 𝜕𝑘 Diffusion constant in d-dimensions
Fermi velocity
Chapter 13
Using many-body techniques
Occupation number representation Identical particles and indistinguishability
[𝑃12 , 𝐻] = 0,
2 𝑃12
=1
𝑃12 ȁ𝜆ۧ1 ห𝜆′ ൿ2 = ห𝜆′ ൿ1 ȁ𝜆ۧ2
⇒ 𝑃12 = ±1 Boson/Fermion Example: Two identical particles 1 ห𝜆1 𝜆2 ൿ± = ห𝜆1 ൿ1 ห𝜆2 ൿ2 ± ห𝜆2 ൿ1 ห𝜆1 ൿ2 2 ห𝑛𝜆1 = 1, 𝑛𝜆2 = 1ൿ In general, ห𝑛𝜆1 , 𝑛𝜆2 , 𝑛𝜆3 , …ൿ
Occupation number representation
Many-body operators Operators in the occupation number representation See Giuliani and Vignale, App.2
One-body operators: Density, current, … † መ 𝐹 = 𝑓𝑖 → 𝑎ො𝜆′ 𝑓𝜆′𝜆 𝑎ො𝜆
𝑓𝜆′𝜆 = 𝜆′ 𝑓መ 𝜆
𝜆,𝜆′
𝑖
Two-body operators: Coulomb interaction, … 1 1 † † 𝑉 = 𝑣ො𝑖𝑗 → 𝑎ො𝜆′ 𝑎ො𝜆′ 𝑣𝜆′1𝜆′2𝜆1𝜆2 𝑎ො𝜆2 𝑎ො𝜆1 2 2 ′ 1 2 𝑖,𝑗
𝜆,𝜆
𝑣𝜆′1𝜆′2𝜆1𝜆2 = 𝜆1′ 𝜆′2 𝑣ො 𝜆1 𝜆2
Chapter 14
Superconductivity
Superconductivity: Experimental background
Zero resistance
Critical field
Meissner effect (expulsion of magnetic flux) © Cambridge University Press 2016
Origin of the attractive interaction
See Tinkham, Ch.3.2
The bare or screened Coulomb interaction between electrons are repulsive. How is it possible to have attractive effective interactions between electrons? ⇒ Attractive interactions are possible when the motion of ions are taken into account.
After the first electron has polarizes the medium, a second electron feels the distorted ionic potential, and the net effect is an attractive interaction between the two electrons. ⇒ Phonon-mediated attractive interaction inducing Cooper pairs with opposite spin and momentum
–,↑
–,↓
Broken symmetry and order parameter See Coleman Ch.11
Spherical symmetry
Six-fold symmetry
Landau-Ginzburg theory Express the free energy as a function of the order parameter near the critical point taking into account the inhomogeneity in space and the symmetry of the system.
1 𝑓 𝜙 = 𝛻𝜙 2 𝛿𝑓 𝛿𝜙
=0→
2
𝑟 2 𝑢 4 + 𝜙 + 𝜙 − ℎ𝜙 2 4
free energy minimum
𝑟 and 𝑢 are unknown constants depending on physics at the atomic scale while the generic power-law dependence follows from the structure of the free energy independent of the microscopic details showing a universal behavior.
BCS theory
See Tinkham, Ch.3.5
−𝑘, ↓
Assume that the most important 𝑘, ↑ interactions are those involving ′ 𝑘 ,↑ Cooper pairs. 𝐻=
† 𝜉𝑘 𝑐𝑘𝜎 𝑐𝑘𝜎 𝑘,𝜎
−
† 𝑈𝑘𝑘 ′ 𝑐𝑘↑ 𝑘,𝑘 ′
𝑈𝑘𝑘 ′
† 𝑐−𝑘↓ 𝑐−𝑘 ′ ↓ 𝑐𝑘 ′ ↑
In a mean-field theory, † † † 𝐻 = 𝜉𝑘 𝑐𝑘𝜎 𝑐𝑘𝜎 − Δ𝑘𝑘 ′ 𝑐𝑘↑ 𝑐−𝑘↓ + ℎ. 𝑐. 𝑘′
𝑘,𝜎
Δ𝑘𝑘 ′ = 𝑈𝑘𝑘 ′ 𝑐−𝑘 ′ ↓ 𝑐𝑘 ′↑ 𝑘′
⇒1=
ℏ𝜔𝑐 𝑁𝑈 ∫0
𝑑𝜉 𝜉 2 +Δ2
tanh
𝜉 2 +Δ2 2𝑘B 𝑇
Gap equation ⇒ 𝑇𝑐 , Δ(𝑇)
−𝑘 ′ , ↓
Chapter 15
Magnetism
Paramagnetism
Pauli paramagnetic spin susceptibility
𝜒 = 𝜇𝐵2 𝑁(𝐸𝐹 ) © Cambridge University Press 2016
Ferromagnetism and antiferromagnetism Heisenberg Hamiltonian
𝐻 = − 𝐽𝑖𝑗 𝑺𝑖 ⋅ 𝑺𝑗 𝑖≠𝑗
𝐽𝑖𝑗 > 0: Parallel spins, 𝐽𝑖𝑗 < 0: Antiparallel spins
Spin waves
© Cambridge University Press 2016
𝐸 𝑘 ∼ 𝑘 2 for a ferromagnetic system; 𝐸 𝑘 ∼ 𝑘 for a antiferromagnetic system
Chapter 16
Reduced-dimensional systems and nanostructures
Quantum confinement of electrons Density of states
Changes in the DOS due to reduced dimensionality © Cambridge University Press 2016
Ballistic transport Quantization of conductance
Plateaus at multiples of 2𝑒 2 /ℎ © Cambridge University Press 2016
Coulomb blockade Quantum dot in contact with metallic leads
Interplay between the charging energies and the excited-state energy levels of the quantum dot © Cambridge University Press 2016
Nanostructures Graphene and graphene nanostructures
© Cambridge University Press 2016
