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theory

Last edited by Philipp Rüssmann

Perform Impurity calculations using kkrflex

Theory

Motivation

STS (Scanning tunnelling spectroscopy) is an experimental technique which provides information about the electrons charge density resolved on energy and position. This allows to extract a compound surface properties such its surface states and densities. One main features STS allows to access are the standing waves patterns which correspond to charge oscillations generated by an impurity and which decays with a power law from it. Due to the presence of many impurities at different positions and the superposition of different standing waves the topographic data is very difficult to interpret. For this reason a Fourier transform (FT) is performed in order to access main scattering processes.

Stationary phase approximation (SPA)

The spin, energy and position resolved density in presence of an impurity is:

\Delta n (\vec{r}\sigma;E)=-\frac{1}{\pi}\mathrm{Im} \left\langle \vec{r}\sigma \left| G_{imp}(E)-G_{host}(E) \right| \vec{r} \sigma \right\rangle
. In a non-degenerate case this Green's function difference can be written under the form:
\Delta G(\vec{r} + \vec{R}_n,\vec{r} + \vec{R}_n;E) = - \frac{1}{\Omega_{rec}^2} \int\limits_{0}^{+\infty} dt \ dt\' \int d^2\vec{k} \d^2\vec{k}\' \Psi_{\vec{k}}(\vec{r}) \mathcal{T}_{\vec{k}\vec{k}\'} \Psi^{\*}_{\vec{k}\'}(\vec{r}) e^{i\Phi}
where
\Phi=(E-\epsilon_{\vec{k}}+\eta)t + (E-\epsilon_{\vec{k}\'}+\eta\')t\' + (\vec{k}-\vec{k}\')\cdot\vec{R}_n
.

The main idea behind the SPA is to suppress the quick

\vec{k}
and
t
oscillations in the above integral. The waves having the same phase will sum constructively while the ones with different phases will sum incoherently cancelling their contribution.This can be done looking for points
\{ \vec{k}_0,\vec{k}_0\',t_0,t_0\' \}
which makes the phase
\Phi
stationary. This gives the following set of equations:
\left\{\begin{array}{ll}\vec{R}_n &= - v_{\vec{k}_0\'}t_0\' \vec{R}_n &= v_{\vec{k}_0}t_0 E &= \epsilon_{\vec{k}_0} E &= \epsilon_{\vec{k}_0\'}\end{array}\right.

This means that the velocities at

\vec{k}_0
and
\vec{k}_0\'
must be parallel but with an opposite direction. Furthermore the states defined by the previous wave vectors must lie in the same energy shell. Within these conditions one can obtain the difference in the density performing the integrals and trace calculation on spins, which results in :

\Delta n (\vec{R}_n+\vec{r};E)=|A_{\vec{k}_0,\vec{k}_0\'}(\vec{r})| \frac{\sin((\vec{k}_0-\vec{k}_0\')\cdot \vec{R}_n+\phi_{\vec{k}_0,\vec{k}_0\'})}{|\vec{R}_n|}

where

|A_{\vec{k}_0,\vec{k}_0\'}(\vec{r})| \propto |\mathcal{T}_{\vec{k}_0,\vec{k}_0\'}|
. The above relation shows that the density oscillates with respect to the position of the site n. It is also observed that this oscillation on the density has an
1/|\vec{R}_n|
envelope. More generally the envelope has a power law shape which depends on the type of impurity (point defect, edge defect, magnetic defect ... ) and of the dimensionality of the crystal:
\frac{1}{R}
for a surface and
\frac{1}{R^2}
for the bulk. This envelope also rely on the symmetries connecting the scattered states, in particular TRS. On Q. Liu et al.((Liu et al. Phys. Rev. B 85 2012)) established a chart depicting the shape of the envelop with respect to these last considerations.

Accessing scattering processes beyond JDOS : the extended joint density of states (exJDOS)

STS experiments allow to observe standing waves in the density however mainly because of the presence of multiple impurities and because of the superposition of distinctive standing waves with different scattering vectors it's difficult to extract information from the raw density map. For this reason a FT is performed on the density map highlighting the main scattering processes.

There is a general method exhibiting directly the FT depicted above and is commonly named joint density of states (JDOS). The formula describing this method gives the contribution of all scattering processes, having a scattering vector

\vec{q}=\vec{k}_f-\vec{k}_i
, lying in the same energy shell :

\mbox{JDOS}(\vec{q})=\int d^2 \vec{k} \ A_{\vec{k}}(E) A_{\vec{k}+\vec{q}}(E) % \quad \mbox{with} \quad % A_{\vec{k}}(E)=\int\limits_{surface} d^2 \vec{r} \ |\Psi_{\vec{k}}(\vec{r})|^2 \delta(E-\epsilon_{\vec{k}})

However according to this formula all scattering processes have the same weight which is in general not the case. Furthermore this formula leads to homogeneous quasiparticle interferences which is erroneous. In order to highlight the real scattered processes the extended joint density of states (exJDOS) method is used. Within this method the scattering processes are weighted as following:

 * by the transition rate between states thanks to "Fermi's golden rule" and the 

\mathcal{T}_{\vec{k},\vec{k}'}
 components: 
\mbox{P}_{\vec{k},\vec{k}'}=2\pi |\mathcal{T}_{\vec{k},\vec{k}'}|^2\delta(\epsilon_{\vec{k}}-\epsilon_{\vec{k}'})
 * by the velocity-dependent scattering probability.States having velocities with opposite directions are most likely to scatter: 
1-\cos(\vec{v}_{\vec{k}},\vec{v}_{\vec{k}'})

The first point gives the probability of transition from one state to another in presence of a perturbation (here an impurity in the host system). The second point stems directly from the SPA. With the above considerations the convenient quantity exhibiting the FT of the quasiparticle interference map is :

\mbox{exJDOS}(\vec{q})=\int d^2 \vec{k} \ A_{\vec{k}}(E) M_{\vec{k},\vec{k}+\vec{q}} A_{\vec{k}+\vec{q}}(E)

with

M_{\vec{k},\vec{k}+\vec{q}}=\mbox{P}_{\vec{k},\vec{k}+\vec{q}}
\left( 1-\cos(\vec{v}_{\vec{k}},\vec{v}_{+\vec{q}}) \right)

The method called JDOS-

\gamma
STM only takes into account the group velocity contribution of exJDOS. In the case where the transition rate is not known there exists a phenomenological result, obtained by Rushan, and valid in topological insulators surfaces which asserts that states having spins with same directions are most likely to scatter :
\mathrm{P}_{\vec{k},\vec{k}\'} \approx \left( 1-\cos(\vec{s}_{\vec{k}},\vec{s}_{\vec{k}+\vec{q}}) \right)
. This approach is called spin-dependent scattering probability (SSP).

A key object : the
\mathcal{T}
-matrix

In the calculation of electron scattering properties induced by defects within the KKR method, the transition matrix

T_{kk\'}
(where
k=(\mathbf{k},\nu)
comprises the crystal momentum
\mathbf{k}
and another index
\nu
for additional degeneracies) takes the central role. It is defined as

T_{k'k} =\int{ \mathrm{d} \mathbf{x} \, \mathrm{d} \mathbf{x}\,' ~ \boldsymbol{\psi}^{\dagger}_{k'}(\mathbf{x}) \, \Delta \mathbf{V}(\mathbf{x},\mathbf{x}\,') \, \boldsymbol{\psi}^{\mathrm{imp}}_{k}(\mathbf{x}\,') }~,`

and, using the expansions that are made in the KKR formalism, takes the form

T_{k'k} = \sum_{i,i'} \sum_{\Lambda,\Lambda'} {c^i}^*_{k',\Lambda} T^{ii'}_{\Lambda \Lambda'} ~ c^{i'}_{k,\Lambda'}~.`

with

T^{ii'}_{\Lambda \Lambda'} = \sum_{\Lambda`*`}`` ``\Delta^{i}_{\Lambda`` ``\Lambda`*`} \, \left( \delta_{ii'} \, \delta_{\Lambda'' \Lambda'} + \sum_{\Lambda`**`}`` ``G^{\mathrm{imp},ii'}_{\Lambda''`` ``\Lambda`**`} ~ \Delta t^{\mathrm{imp},i'}_{\Lambda''' \Lambda'} \right)~.`

The matrix

T^{ii\'}_{\Lambda \Lambda\'}
is independent on the state vector
k
, but it depends only on the energy (often equal to the Fermi energy). All
k
-dependencies enter via the .

In order to calculate scattering properties off defects, several steps have to be performed.

 - converge a potential of an ideal host system with the Jülich-München code  - calculate the Fermi surface and wave-functions on the Fermi surface   - converge a potential of the defect (=impurity cluster) with the KKRimp code

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