# On the Evaluation of the Spin Galvanic Effect in Lattice Models with Rashba Spin-Orbit Coupling

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## Abstract

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## 1. Introduction

## 2. Response Functions

## 3. Analytic Evaluation of the SGE in the Clean and Dirty Limit for a Free Electron Gas with RSOC

#### 3.1. Clean Limit

#### 3.2. Disorder Limit

## 4. Evaluation of the Spin Galvanic Effect for a Disordered Lattice Model with RSOC

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Drude coefficient of the spin galvanic response ${D}^{SGE}$ for the single-band Rashba lattice model as a function of chemical potential (solid lines). The symbols correspond to ${D}_{SGE}=-e\alpha /\left(2B\right)$ obtained from Equation (16) with the replacement $m=2/B$ and the bandwidth parameter $B=4t$.

**Figure 2.**Optical conductivity (full dots) evaluated for disorder strength ${V}_{0}/t=0.3$, chemical potential $\mu =-3t$, and RSOC $\alpha /t=0.3$. The blue solid line is a fit to the Drude model Equation (26) with ${\sigma}_{0}=11.5{e}^{2}t$ and $\tau =43.81/t$. Inset: The SGE response Equation (7) as a function of the lifetime parameter $\eta $ for the same parameters. The red solid line is a polynomial fit to the data points with $\eta >0.005$ corresponding to the average level spacing of eigenvalues (vertical dotted line). It extrapolates to a value of ${\sigma}^{SGE}\approx -2e$ in agreement with the value expected from the SCBA analysis (horizontal dashed line).

**Figure 3.**(

**a**) Comparison of the SGE response evaluated with the Kubo formula Equation (7) and performing the $\eta $-extrapolation (squares) with the continuum result obtained from Equation (22). The SGE Drude coefficient ${D}_{SGE}$ (diamonds, error corresponds to symbol size) computed from Equation (6) is also shown. (

**b**) The Drude formula parameters as a function of $\mu $ as extracted from fits to the optical conductivity. Calculations have been performed on $30\times 30$ lattices for disorder potential ${V}_{0}/t=0.3$ and RSOC $\alpha /t=0.3$.

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**MDPI and ACS Style**

Seibold, G.; Caprara, S.; Grilli, M.; Raimondi, R.
On the Evaluation of the Spin Galvanic Effect in Lattice Models with Rashba Spin-Orbit Coupling. *Condens. Matter* **2018**, *3*, 22.
https://doi.org/10.3390/condmat3030022

**AMA Style**

Seibold G, Caprara S, Grilli M, Raimondi R.
On the Evaluation of the Spin Galvanic Effect in Lattice Models with Rashba Spin-Orbit Coupling. *Condensed Matter*. 2018; 3(3):22.
https://doi.org/10.3390/condmat3030022

**Chicago/Turabian Style**

Seibold, Götz, Sergio Caprara, Marco Grilli, and Roberto Raimondi.
2018. "On the Evaluation of the Spin Galvanic Effect in Lattice Models with Rashba Spin-Orbit Coupling" *Condensed Matter* 3, no. 3: 22.
https://doi.org/10.3390/condmat3030022