Follow presentation live and in HD:
pandascience.github.io/PhDdefense
2021-02-26 - TU Bergakademie Freiberg - GEL-0001 / BBB Virtual Room - 2 p.m.
Universal Electromagnetic Response Relations
applied to the free homogeneous electron gas
— PhD Thesis —
René Wirnata
Outline
- Motivation
- Functional Approach to ED in Media
Source Splitting & Field Identifications
Fundamental Response Tensor & URR
- Quantum Field Theory
Kubo-Greenwood Formalism
Full Current Operator
- Impact on Materials Models
London Conductivity as Toy Model
Optical & Magnetic Properties of FEG
Lindhard Integral Theorem
- Summary & Conclusion
coil: Killian Eon via Pexels / CC0 quantum field: Ahmed Neutron via via Wikimedia Commons CC BY-SA 4.0 magnet: Peter nussbaumer via Wikimedia Commons CC BY-SA 3.0
Motivation
Goal: Description of optical & magnetic response
properties based on microscopic field theories
| ☐ | Why microscopic theories?
|
| ☐ | Possible applications?
|
Functional Approach to
Electrodynamics in Media
img: Skitterphoto via Pexels / CC0
Source Splitting & Field Identifications
Fundamental Response Tensor (1)
Basis of (linear) Response Theory:
Induced fields can be regarded as functionals of external ones.
| Q: | Which functional is the "correct" one? \E_\ind [\E_\ext, \B_\ext], \quad \vcj_\ind[\E_\ext], \quad \B_\ind[\rho_\ext, \vcj_\ext], \; \ldots \; ? |
| A: | Postulate of Functional Approach: \;\; j^\mu_\ind = j^\mu_\ind[A^\nu_\ext] |
Lorentz 4-vector notation:
% \quad x^\mu = \pmat{ct \\ \vcx} \,, \quad j^\mu = \pmat{c \rho \\ \vcj} \,, \quad A^\mu = \pmat{\varphi / c \\ \A}
Fields & Potentials:
\begin{align} \E &= - \nabla \varphi - \del_t \A \\[1ex] \B &= \nabla \times \A \end{align}
Linear response tensor:
\quad\quad\gbox{ \quad \chimn(x, x') = \dfrac{\delta j^\mu_\ind(x)}{\delta A^\nu_\ext(x')} \quad}
2nd order tensor \chimn contains complete information on EM response.
Fundamental Response Tensor (2)
Continuity equation and gauge invariance imply the constraints:
\del_\mu \blue{\chimn}(x, x') = \del^{'\nu} \blue{\chimn}(x, x') = 0
⇒ at most 9 independent linear response functions!
\begin{align}
\quad x^\mu &= \pmat{ct \\ \vcx} \\[1.5ex]
\quad \del^\mu &= \pmat{ -\del_t \\ \nabla} \\[1.5ex]
\quad k^\mu &= \pmat{ \omega/c \\ \vck}
\end{align}
For homogeneous systems: \chi(x, x') = \chi(x - x')
Equivalently in Fourier space: \chi(k, k') = \chi(k) \; \delta(k-k')
this implies in | \quad \blue{\chimn}(\vck,\omega) = \begin{pmatrix} -\dfrac{c^2}{\omega^2} \, \vck^\T \, \green{\boldsymbol{\tsr\chi}} \, \vck & \dfrac{c}{\omega} \, \vck^\T \, \green{\boldsymbol{\tsr\chi}} \, \\[1.5ex] -\dfrac{c}{\omega} \, \, \green{\boldsymbol{\tsr\chi}} \, \vck & \, \green{\boldsymbol{\tsr\chi}} \, \end{pmatrix} \quad with \quad \gbox{\; \tsr\chi = \rlap{\phantom{\Huge\langle_{a_a}}} \dfrac{\delta\vcj_\ind}{\delta\A_\ext} \;} |
Remaining components form the 3x3 Current Response Tensor \green{\tsr\chi}(\vck,\omega).
Universal Response Relations
Central Claim:
Current Response Tensor \tsr\chi
determines all linear EM
materials properties.
\gbox{
\begin{align}
\tsr\chi_\EE = \dfrac{\de \E_\ind}{\de \E_\ext} \,,\quad
\tsr\chi_\EB = \dfrac{\de \E_\ind}{\de \B_\ext} \\[2ex]
\tsr\chi_\BE = \dfrac{\de \B_\ind}{\de \E_\ext} \,,\quad
\tsr\chi_\BB = \dfrac{\de \B_\ind}{\de \B_\ext}
\end{align}}
⇒ can all be expressed in terms of \orange{\tsr\chi} or
alternatively in terms of conductivity \tsr\sigma via
basic relation: \tsr\chi(\vck, \omega) = \i\omega \h \tsr\sigma(\vck,\omega)
Combined with the power of
Total Functional Derivatives yields
Universal Response Relations (URR)
model and material-independent relations
between EM response functionsalready include all effects of anisotropy,
rel. retardation and ME cross-couplingall standard relations known in ab initio
theory recovered in suitable limiting cases
e.g. optical limit → \tsr\eps_\tn{\!r}(\omega) = 1 -\dfrac{\ttsr\sigma(\omega)}{\i\omega \h\eps_0}conductivity routinely calculated ab initio
↝ URR as post-processing
↝ see also Elk Optics Analyzer
Quantum Field Theory
img: Giulio Schober / with permission
Kubo-Greenwood Formalism (1)
| Q: | How to calculate response functions from first principles? |
| A: | Employ Kubo Formalism: |
\chi_\AB(t-t') \eqdef \bbox{\frac{\delta A(t)}{\delta \green{P(t')}} \bigg|_{P=0} }
= -\frac{\i}{\hbar} \, \Theta(t-t') \,
\orange{\bra{\Psi_0}} [ \purple{\hat{A}}_\mathrm{I}(t), \purple{\hat{B}}_\mathrm{I}(t') ] \orange{\ket{\Psi_0}}
Let P(t) be a (weak) time-dependent perturbation
which couples to an operator \hat{B} in the Hamiltonian:
\Ham(t) = \Ham_0 + \green{P(t)} \h \purple{\hat{B}} | \quad \Rightarrow \quad
\i\hbar\,\del_t \h \ket{\Psi(t)} = \Ham(t) \h \ket{\Psi(t)} \quad \Rightarrow \quad \ket{\Psi(t)} = \green{\big(\,} \ket{\Psi(t)}\green{\big) \bold{[} P \bold{]} } |
Expectation value of some other observable \hat{A} is then given by
\bbox{A(t)\green{ [ P ] } }= \braket{\Psi(t) |\, \hat{A} \,}{\Psi(t)}
note: \ket{\Psi} is a many-body state!
Kubo formula allows to express the response \chi of the observable w.r.t.
the perturbation solely in terms of unperturbed quantities !
Kubo-Greenwood Formalism (2)
For non-interacting systems, MB-WF can be chosen as Slater determinant
\ket{\Psi^N_0} = \ket{\tn{SL}(\blue{\varphi_1}, \ldots, \blue{\varphi_N})}
Many-body problem decomposes into effective single-particle system
\Ham {}^N = \sum_N \Ham {}^1 \,, \quad \Ham {}^1 \ket{\blue{\varphi_i}} = \epsilon_i \ket{\blue{\varphi_i}}
In Grand Canonical Ensemble, Kubo formula reverts
to its Spectral Representation in Fourier space
\chi_\AB^\tns{R}(\omega) = \sum\limits_{i,j=0}^\infty \dfrac{ \Big( \green{f_{\beta,\mu}}(\epsilon_i) - \green{f_{\beta,\mu}}(\epsilon_j) \Big) \h A_{ij} \h B_{ji} }{\hbar(\omega + \i\eta) - (\epsilon_j - \epsilon_i)} \qquad
A_{ij} = \braket{\blue{\varphi_i} | \hat{A}}{\blue{\varphi_j}} \,, \qquad \green{f_{\beta,\mu}}(\epsilon_i) = \dfrac{1}{1+ \e^{\beta(\epsilon_i -\mu)}}
In plane waves basis …
\quad \blue{\varphi_\vck}(\vcx) = \braket{\vcx}{\vck} = \dfrac{1}{(2\pi)^{3/2}} \h\, \e^{\i \vck \cdot \vcx} \quad
… and for density response
\quad \begin{align} \hat{A} \mapsto \hat\rho(\vcx) &= (-e) \h \hat\psi {}^\dagger(\vcx) \h \hat\psi(\vcx) \\ \hat{B} \mapsto \hat\rho(\vcx') &= (-e) \h \hat\psi {}^\dagger(\vcx') \h \hat\psi(\vcx') \end{align}
\hat{\psi}(\vcx) \h \ket{\varphi} \\= \braket{\vcx}{\varphi} \ket{0}
Kubo formula yields in Thermodynamic Limit
\epsilon_\vck = \hbar \omega_\vck
\chi_{\rho\rho}(\vcq,\omega) = 2 \h e^2 \idkpi
\dfrac{f(\epsilon_\vck) - f(\epsilon_{\vck+\vcq})}{\hbar(\omega + \i\eta) +
\epsilon_\vck - \epsilon_{\vck+\vcq}}
⇒ famous Lindhard density response
Full current operator (1)
For current response tensor \tsr\chi \equiv \tsr\chi_{\vcj\vcj} we need \vcjh instead of \hat\rho .
Start with free Hamiltonian
\i\hbar \h\del_t \h \psi(\vcx,t) = \Ham_0 \h \psi(\vcx,t) \,,\quad \Ham_0 = \dfrac{|\hat\vcp|^2}{2m} \quad \Rightarrow \quad \boxed{ \vcj = \underbrace{ \dfrac{(-e)\hbar}{2m\i} \Big(\psi^* (\nabla \psi) - (\nabla \psi)^* \psi \Big) }_{\mathsf{orbital}} }
Extend by U(1) gauge theory → minimal-coupling of EM fields
\begin{align} \del_t &\mapsto \del_t - \tfrac{\i}{\hbar} \, \green{e \varphi} \\ \nabla &\mapsto \nabla + \tfrac{\i}{\hbar} \, \green{e \A} \\ \end{align} \;\; \Bigg\} \quad \Rightarrow \quad \Ham_\tn{min} = \dfrac{\left| \hat{\vcp} + \green{e\A} \right|^2}{2m} - \green{e \varphi} \quad \Rightarrow \quad \boxed{ \vcj = \vcj_\orb + \underbrace{\dfrac{e}{m} \, \rho \A}_{\mathsf{diamagnetic}} }
| Q: | But what about spin-induced magnetism? |
| A: | Commonly not considered for \vcj , but: Yes , we can! — By invoking the Pauli Equation … |
Full Current Operator (2)
The Pauli Equation is of Schrödinger-type and incorporates spin.
\begin{align}
&\text{Pauli Matrices} \\[1.5ex]
\sigma_1 &= \pmat{0 & 1 \\ 1 & 0} \\[1ex]
\sigma_2 &= \pmat{0 & -\i \\ \i & 0} \\[1ex]
\sigma_3 &= \pmat{1 & 0 \\ 0 & -1}
\end{align}
Leads to a total current | \boxed{\quad \vcjh_\tot = \vcjh_\dia + \hspace{-3em} \underbrace{ \vcjh_\orb + \green{\vcjh_\spin} }_{ \quad\qquad\mathsf{"paramagnetic"} \rightarrow \vcjh_\tn{p} = \vcjh_\tot\big|_{A=0} }} |
with the additional spinorial contribution
\vcjh_\tn{dia} = \dfrac{e}{m} \, \hat\rho \A \,,\qquad \vcjh_\tn{orb} = \dfrac{(-e)\hbar}{2m\i} \Big(\psih {}^\dagger (\nabla \psih) - (\nabla \psih)^\dagger \psih \Big)
\gbox{\vcjh_\tn{spin} = \dfrac{(-e)\hbar}{2m} \, \nabla \times \left( \sum\limits_{s,s'=\uparrow, \downarrow} \psih{}^\dagger_s \, \vc{\sigma}_{ss'} \, \psih_{s'} \right) }
This is the most general yet
non-relativistic current.→ can also be derived from Dirac
equation in non-relativistic limit
Full Current Response Tensor
Use Generalized Kubo Formula for the current response tensor
\tsr\chi(\vcx,t; \vcx',t') =
\boxed{
\frac{\phantom{i}\!\!e}{m} \, \rho(\vcx) \h \delta^3(\vcx-\vcx') \h \delta(t-t') \h \identity
} + \boxed{
\frac{\i}{\hbar} \, \Theta(t-t') \, \Big\langle \hll{[} \vcjh_\tn{p}(\vcx,t),
\vcjh_\tn{p}(\vcx', t') \hll{]} \Big\rangle
}
\bold{\green{[}} \hat{A}, \hat{B} \bold{\green{]}} := \hat{A}\hat{B} - \hat{B}\hat{A} \\ \vcjh_\tn{p} = \vcjh_\orb + \vcjh_\spin
Because of bilinearity, commutator yields three contributions
\tsr\chi = \tsr\chi_\tn{dia} + {\large\purple{(}} \tsr\chi_\corb +
\tsr\chi_\cspin + \tsr\chi_\ccross {\large\purple{)}}
\begin{align}
\tsr\chi_\dia &\;=
\;\mathsf{local \; contribution \; (left \, box \; above)} \\[1ex]
\hline
%
\tsr\chi_\corb &\ppt \Big\langle [ \vcjh_\corb, \vcjh_\corb' ]\Big \rangle \\[1ex]
%
\tsr\chi_\cspin &\ppt \Big\langle [ \vcjh_\cspin, \vcjh_\cspin' ] \Big \rangle \\[1ex]
%
\tsr\chi_\ccross &\ppt \Big\langle [ \vcjh_\corb, \vcjh_\cspin' ] +
[ \vcjh_\cspin', \vcjh_\corb ] \Big\rangle
\end{align}
known: diamagnetic + orbital
novel: spin + cross-correlation
but: cross-correlation vanishes for
non spin-polarized systems like FEG
Impact on Materials Models
magnet: Peter nussbaumer via Wikimedia Commons CC BY-SA 3.0
From Covariant Wave Equation to London model
Inserting \E= -\nabla \varphi -\del_t \A and \B = \nabla \times \A into the Maxwell equations leads to the fundamental Lorentz-covariant Wave Equation
(\eta^\mu{}_\nu\Box + \del^\mu \del_\nu) \h A^\nu_\tot = \mu_0 \h (\purple{\vcj_\ind} + \blue{\vcj_\ext)}
Eliminating the induced current in the spirit of response theory and setting \vcj_\ext \equiv 0 produces
(\eta^\mu{}_\nu\Box + \del^\mu \del_\nu - \mu_0 \h \green{\chimnt}) \h A^\nu_\tot = 0
\chimnt = \dfrac{\delta j^\mu_\ind}{\delta A^\nu_\tot}
\qquad\qquad\qquad\qquad\green{\chimnt} \equiv 0 → vacuum case
| Q: | What happens for the most simple assumption of a constant \green\chimnt? |
| A: | We obtain the London model with purely diamagnetic current \vcj = \vcj_\dia. |
The London Model of Superconductivity is the simplest possible materials model from a response theoretical point of view.
\boxed{\text{London:} \quad \vcj = \green{-\frac{ne^2}{m}} \, \A } with \tsr\chi = \tsr\chi_\cdia
completely isotropic, local and instantaneous (proper) response
Meißner Effect, zero electrical "DC resistance" and plasma frequency via URR
highly related to Drude Model and with some cheating also Lorentz Oscillator Model
→ Spin-correction has no considerable effect.
London Model vs. Free Electron Gas
Based on QM, the Free Electron Gas describes free charge carriers in a solid and is surprisingly successful in reproducing many exp. phenomena.
| Q: | What changes when we replace the London Model with the Free Electron Gas and explicitly include spinorial current contributions ? |
Optical Properties:
→ FEG adds \green{q^2} term
\Im\sigma_\T^\tn{ns} = \green{\dfrac{ne^2}{\omega m}} \left(\green{1} + \frac{1}{5} \, \frac{\green{\vc{q^2}} v_\fermi^2}{\omega^2} + \frac{3}{35} \, \frac{q^4 v_\fermi^4}{\omega^4} + \small \mathcal{O}\left(q^6\right)\right)
→ spin-correction adds order \green{q^4} contributions
\Im \sigma^\tn{spin}_\T = \green{\dfrac{ne^2}{\omega m}} \left( \frac{1}{4} \, \frac{v_\fermi^2}{k_\fermi^2} \, \frac{\green{\vc{q^4}}}{\omega^2} + \small \mathcal{O}(q^6) \right)
⇒ no significant impact on dispersion relation!
Magnetic Properties:
→ (dia + orb) parts lead to Landau Diamagnetism
\chi_\tn{m}^\tn{ns} \overset{\omega\to 0}{=} \green{ \mu_0 \mu_\tns{B}^2 \h g(E_\fermi) } \Big( \green{-\frac{1}{3} } + \frac{1}{60}\h\frac{q^2}{k_\fermi^2} + \small \mathcal{O}(q^4) \Big)→ spin-correction adds Pauli Paramagnetism
\chi_\tn{m}^\tn{spin} \overset{\omega\to 0}{=} \green{\mu_0 \mu_\tns{B}^2 \h g(E_\fermi) } \left( \green{1} - \frac{1}{12}\h\frac{q^2}{k_\fermi^2} + \small \mathcal{O}(q^4)\right)
⇒ reproduces Landau and Pauli mag. via URR
Lindhard Integral Theorem
Central Result: Current response of FEG can be reduced to 3 dimensionless parameter integrals. Complete response is then determined by these integrals, Lindhard density response and constant charge density.
\orange{(\tsr\chi)_{ij}}(\vcq,\omega) = -\left( \frac{e^2 \green{n}}{m} +
\frac{\hbar^2
\abs{\vcq}^2}{4m^2} \, \green{\rchi}(\vcq,\omega) \right) \delta_{ij} +
\green{\alpha_{ij}}(\vcq,\omega) + q_i \h \green{\beta_j}(\vcq,\omega) +
\green{\beta_i}(\vcq,\omega) \h q_j
\left.
\begin{align}
\green{\alpha_{ij}}(\vcq,\omega) &= -\frac{\hbar^2}{4m^2} \left( 2e^2
\vint{\vck} \green{4 k_i k_j} \, \frac{f_\vck - f_{\vck+\vcq}}{
\hbar \omega\plus + \eps_\vck - \eps_{\vck + \vcq}} \right) \\[1ex]
%
\label{eq:beta_i}
\green{\beta_i}(\vcq,\omega) &= -\frac{\hbar^2}{4m^2} \left( 2e^2
\vint{\vck} \green{2 k_i} \, \frac{f_\vck - f_{\vck+\vcq}}{
\hbar \omega\plus + \eps_\vck - \eps_{\vck + \vcq}} \right)
\end{align} \quad \right\} | ⇒ can be reduced from 12 to 3 ⇒ can be solved analytically for ⇒ can be expressed by Lindhard Int. |
\green{n} = 2 \vint{\vck} f_\vck = \int \de\omega \, g(\omega) f(\omega)
\,, \qquad
\green{\rchi}(\vcq,\omega) = 2e^2 \vint{\vck} \green{\;1\;} \frac{f_\vck - f_{\vck + \vcq}}{
\hbar \omega\plus + \eps_\vck - \eps_{\vck + \vcq}}
Summary & Conclusion
Based on exclusively microscopic field theory it is shown that via URR
that all relevant linear optical and magnetic materials properties
of the FEG follow from the full current response tensor. (Central Claim)
In particular:
By Ampères Law, every magnetization is generated by a microscopic current
As a matter of principle, this also includes
spin-induced magnetism
Implication
All contributions to the current should be
treated equally in Linear Response Theory
Derived in this Thesis
Full current operator based on the Pauli Equation
General form of \tsr\chi for the FEG
Analytic expressions in Zero Temperatur Case
Postulation & proof of Lindhard Integral Theorem
Further Work
Algorithm for n_\tn{e} and n_\tn{o} from given \sigma
2nd-order non-linear analogon of \chimn
Thanks for the attention!
— Acknowledgements —
Prof. Jens Kortus
(supervisor)
Prof. Caterina Cocchi
(2nd referee)
Haushalt
(funding)
Institute for
Theoretical Physics
(hospitality)
Any Questions?
— Special Thanks —
Dr. Ronald Starke
(person in charge)
Dr. Giulio Schober
(2nd person in charge)
My Wife
(not in charge but calls
the tune anyways)
Friends & Family
(always helpful…
…most of the time)
Lindhard Integral Theorem (2)
For zero temperature case: \begin{align}
\green{\alpha_{xx}} &=
{\small E_\fermi \h g(E_\fermi)} \,
\tfrac{e^2}{m} \, \tfrac{\red{(-2)}}{4\hat q} \,
\big(\purple{I_{\alpha{xx}}(\nu_-)} - \purple{I_{\alpha{xx}}(\nu_+)}\big)
\\[1ex]
\green{\alpha_{zz}} &=
{\small E_\fermi \h g(E_\fermi)} \,
\tfrac{e^2}{m} \, \tfrac{\red{(-4)}}{4\hat q} \,
\big(\purple{I_{\alpha{zz}}(\nu_-)} - \purple{I_{\alpha{zz}}(\nu_+)}\big)
\\[1ex]
\green{\beta_z} &=
\tfrac{E_\fermi \h g(E_\fermi)}{k_\fermi} \,
\tfrac{e^2}{m} \, \tfrac{\red{(-2)}}{4\hat q} \,
\big(\purple{I_{\beta z}(\nu_-)} + \purple{I_{\beta z}(\nu_+)}\big)
\\[1ex]
\tfrac{\hbar^2}{4 m^2} \, \blue\rchi &=
\tfrac{E_\fermi \h g(E_\fermi)}{k_\fermi^2} \,
\tfrac{e^2}{m} \, \tfrac{\red{(+1)}}{4\hat q} \,
\big(\blue{I_\chi(\nu_-)} - \blue{I_\chi(\nu_+)}\big)
\end{align} | Master Formula \hat\gamma(\hat q, \hat\omega) = \frac{\gamma_0}{4 \hat{q}} \,
\big( \purple{I_\gamma(\nu_-)} + \purple{I_\gamma(-\nu_+)}\big) \begin{array}{c|c c c c}
\gamma & \blue\rchi & \green{\alpha_{xx}} & \green{\alpha_{zz}} & \green{\beta_z} \\
\gamma_0 & \red{+1} & \red{-2} & \red{-4} & \red{-2}
\end{array} |
\begin{align}
\purple{I_{\alpha{xx}}(z)} &\eqdef \int_0^1 \de x \, x^4 \int_0^\pi \de\theta \;
\frac{\sin^3\theta}{z - x\cos\theta}
= \frac{1}{3} \, z + \frac{1-z^2}{2} \, \blue{I_\chi(z)}
\\[1ex]
\purple{I_{\alpha{zz}}(z)} &\eqdef \int_0^1 \de x \, x^4 \int_0^\pi \de\theta \;
\frac{\sin\theta \h \cos^2\theta}{z - x\cos\theta}
= -\frac{2}{3} \, z + z^2 \blue{I_\chi(z)}
\\[1ex]
\purple{I_{\beta z}(z)} &\eqdef \int_0^1 \de x \, x^3\int_0^\pi \de\theta \;
\frac{\sin\theta \h \cos\theta}{z - x\cos\theta}
= -\frac{2}{3} + z \h \blue{I_\chi(z)}
\phantom{------------------------------------------------}
\end{align}
\qquad \nu_\pm = \dfrac{\omega}{q\h v_\fermi} \pm \dfrac{q}{2k_\fermi} \\[2ex] \blue{I_\chi(z)} = z + \dfrac{1-z^2}{2} \, \Ln\left(\dfrac{z+1}{z-1}\right)
All characteristic integrals can be expressed
in terms of the Lindhard integral!
Universal Electromagnetic Response Relations applied to the free homogeneous electron gas — PhD Thesis — Follow presentation live and in HD:
pandascience.github.io/PhDdefense
2021-02-26 - TU Bergakademie Freiberg - GEL-0001 / BBB Virtual Room - 2 p.m.René Wirnata