Anisotropic Keldysh interaction

TL;DR

This paper extends the classical models of Coulomb interaction in thin films to anisotropic materials, providing analytical expressions for weak anisotropy and highlighting unresolved challenges in strong anisotropy cases.

Contribution

It generalizes the Rytova-Keldysh model to anisotropic permittivity, deriving explicit formulas for weak anisotropy and identifying open problems for strong anisotropy.

Findings

Explicit asymptotic expressions for weak anisotropy

Analytical form of electrostatic potential energy

Unresolved inverse Fourier transform for strong anisotropy

Abstract

We generalize the classic calculations by Rytova and Keldysh of screened Coulomb interaction in semiconductor thin films to systems with anisotropic permittivity tensor. Explicit asymptotic expressions for electrostatic potential energy of interaction in the weakly anisotropic case are found in closed analytical form. The case of strong in-plane anisotropy, however, requires evaluation of the inverse Fourier transform of $1/(k+Ak_x^2+Bk_y^2)$, which, at present, remains unresolved.

Equations

$$V(\rho )=(\pi q{q}^{\prime }/ϵd)[H_0(\rho /{\rho }_0)-Y_0(\rho /{\rho }_0)],$$

$$V(\mathbf{\rho },z,z^{\prime })=\int \frac{d^2\mathbf{k}}{(2\pi {)}^2}{e}^{i\mathbf{k}\cdot \mathbf{\rho }}V(\mathbf{k},z,z^{\prime }),$$

$$V(\mathbf{k},z,z^{\prime })=\frac{4\pi q{q}^{\prime }}{{ϵ}_z}\frac{\cosh \left[\tilde{k}\left(\frac{d}{2}-z\right)+{\tilde{\eta }}_2\right]\cosh \left[\tilde{k}\left(\frac{d}{2}+z^{\prime }\right)+{\tilde{\eta }}_1\right]}{\tilde{k}\sinh [\tilde{k}d+{\tilde{\eta }}_1+{\tilde{\eta }}_2]},$$

$${\tilde{\eta }}_{1,2}=\frac{1}{2}\ln \left(\frac{{ϵ}_z\tilde{k}+{ϵ}_{1,2}k}{{ϵ}_z\tilde{k}-{ϵ}_{1,2}k}\right),\tilde{k}=\sqrt{\frac{{ϵ}_x{k}_x^2+{ϵ}_y{k}_y^2}{{ϵ}_z}},k=\sqrt{k_x^2+k_y^2},$$

$$V(\mathbf{\rho })$$

$${\rho }_{0x}\equiv \frac{{ϵ}_{x}d}{{ϵ}_1+{ϵ}_2},{\rho }_{0y}\equiv \frac{{ϵ}_{y}d}{{ϵ}_1+{ϵ}_2},$$

$$V(\mathbf{\rho })$$

$$ϵ(\mathbf{k})=1+\frac{1}{k}\left({\rho }_{0x}{k}_x^2+{\rho }_{0y}{k}_y^2\right),$$

$$F(x,y)=\int \frac{d{k}_{x}d{k}_y}{(2\pi {)}^2}\frac{e^{i(k_{x}x+k_{y}y)}}{k+A{k}_x^2+B{k}_y^2},$$

$$V(\mathbf{\rho })$$

$$=\frac{2q{q}^{\prime }}{({ϵ}_1+{ϵ}_2){\rho }_0}{\int }_0^{\infty }dt\frac{I(t,\alpha ,a,b)}{t+b},$$

$$I(t,\alpha ,a,b)=\frac{1}{2\pi }{\int }_0^{2\pi }d\theta \frac{e^{it\cos \theta }}{1-{\epsilon }^2(t,a,b){\cos }^2(\theta +\alpha )},$$

$${\epsilon }^2(t,a,b)\equiv \frac{at}{t+b},a\equiv 1-\frac{{\rho }_{0x}}{{\rho }_{0y}},b\equiv \frac{\rho }{{\rho }_0},{\rho }_0\equiv {\rho }_{0y},$$

$$\displaystyle\frac{1}{1-\varepsilon^{2}\cos^{2}\chi}=\frac{1}{\sqrt{1-\varepsilon^{2}}}\bigg{[}1+2\sum^{\infty}_{n=1}\bigg{(}\frac{\varepsilon}{1+\sqrt{1-\varepsilon^{2}}}\bigg{)}^{2n}\cos(2n\chi)\bigg{]},$$

$$I(t,\alpha ,a,b)$$

$$\displaystyle\quad+\frac{2}{\sqrt{1-\varepsilon^{2}(t)}}\sum^{\infty}_{n=1}\bigg{(}\frac{\varepsilon(t)}{1+\sqrt{1-\varepsilon^{2}(t)}}\bigg{)}^{2n}\bigg{(}\frac{1}{2{\pi}}\int_{0}^{2\pi}d\theta\,e^{it\cos\theta}\cos(2n\theta)\bigg{)}\cos(2n\alpha)$$

$$=\frac{J_0(t)}{\sqrt{1-{\epsilon }^2(t)}}$$

$$\displaystyle\quad+\frac{2}{\sqrt{1-\varepsilon^{2}(t)}}\bigg{(}\frac{\varepsilon(t)}{1+\sqrt{1-\varepsilon^{2}(t)}}\bigg{)}^{2}\,\frac{tJ_{0}(t)-2J_{1}(t)}{t}\,\cos(2\alpha)$$

$$\displaystyle\quad+\frac{2}{\sqrt{1-\varepsilon^{2}(t)}}\bigg{(}\frac{\varepsilon(t)}{1+\sqrt{1-\varepsilon^{2}(t)}}\bigg{)}^{4}\,\frac{t\left(t^{2}-24\right)J_{0}(t)-8\left(t^{2}-6\right)J_{1}(t)}{t^{3}}\,\cos(4\alpha)$$

$$\displaystyle\quad+\frac{2}{\sqrt{1-\varepsilon^{2}(t)}}\bigg{(}\frac{\varepsilon(t)}{1+\sqrt{1-\varepsilon^{2}(t)}}\bigg{)}^{6}\,\frac{t\left(t^{4}-144t^{2}+1920\right)J_{0}(t)-6\left(3t^{4}-128t^{2}+640\right)J_{1}(t)}{t^{5}}\,\cos(6\alpha)+\,\dots,$$

$$\displaystyle I(t)\approx J_{0}(t)+\frac{\varepsilon^{2}(t)}{2}J_{0}+\frac{\varepsilon^{2}(t)}{2}\bigg{(}J_{0}(t)-\frac{2J_{1}(t)}{t}\bigg{)}\,\cos(2\alpha),$$

$$V(b,\alpha )$$

$$V(b,\alpha )=V_0(b)+V_1(b,\alpha ),$$

$$V_0(b,\alpha )=\frac{\pi q{q}^{\prime }}{({ϵ}_1+{ϵ}_2){\rho }_0}[H_0(b)-Y_0(b)]$$

$$V_1(b,\alpha )$$

$$+\frac{a}{2}\frac{\pi q{q}^{\prime }}{({ϵ}_1+{ϵ}_2){\rho }_0}\left[\frac{2}{\pi }{b}^3+b\left[\left(b^2-2\right)Y_1(b)+b{Y}_0(b)\right]-b\left[\left(b^2-2\right)H_1(b)+b{H}_0(b)\right]-\frac{4}{\pi }\right]\frac{\cos (2\alpha )}{b^2}$$

$$V_0$$

$$V_1$$

$$\Delta E_0=\frac{2\pi aq{q}^{\prime }}{({ϵ}_1+{ϵ}_2){\rho }_0}{\int }_0^{\infty }\left(\ln \left(\frac{2}{b}\right)-\gamma -1\right)|{\psi }_0(b){|}^{2}bdb,$$

$$V(\rho )=\frac{2q{q}^{\prime }}{({ϵ}_1+{ϵ}_2){\rho }_0}\left(\frac{1}{b}-\frac{a\cos (2\alpha )}{b^2}\right)\to \frac{2q{q}^{\prime }}{({ϵ}_1+{ϵ}_2)\rho }.$$

$${\mathrm{Err}}_1\equiv \frac{V_{\mathrm{exact}}-V_{\mathrm{approx}}}{V_{\mathrm{exact}}}=\frac{V_{1\mathrm{exact}}-V_1}{V_{\mathrm{exact}}}$$

$${\mathrm{Err}}_2\equiv \frac{V_{\mathrm{exact}}-V_{\mathrm{approx}}}{V_{\mathrm{exact}}-V_0}=\frac{V_{1\mathrm{exact}}-V_1}{V_{1\mathrm{exact}}},$$

$$I$$

$$V(b,\alpha )$$

Full text

semiconductor films; dielectric screening; Keldysh interaction; anisotropy

I Introduction

The important role played by the dielectric screening in determining excitonic properties of various two-dimensional semiconductor heterostructures has been the subject of numerous investigations over the last several decades (for recent studies see, for example, cudazzo2010strong ; cudazzo2011dielectric ; chernikov2014exciton ; low2014plasmons ; wang2015highly ; chaves2015anisotropic ; latini2015excitons ; pedersen2016exciton ; trolle2017model ; hichri2017dielectric ; szyniszewski2017binding ; mostaani2017diffusion ; cavalcante2018electrostatics ). A particularly interesting direction of current experimental research involves perovskite chalcogenide films whose in-plane dielectric anisotropy gives rise to some rather unusual optical behavior niu2018giant ; niu2018mid . Past theoretical work on two-dimensional dielectric screening involved various ab initio calculations cudazzo2010strong ; cudazzo2011dielectric , the use of the nonlinear Thomas-Fermi model low2014plasmons , latini2015excitons , the modified Mott-Wannier approach pedersen2016exciton , the transfer matrix method cavalcante2018electrostatics , as well as various approaches based on effective mass approximation chaves2015anisotropic ; hichri2017dielectric , Here we pursue what is likely the simplest possibility — generalization to anisotropic films of classic calculations by Rytova Rytova1967 and Keldysh keldysh1979coulomb ; keldysh1997excitons . The motivation for this approach is rather obvious: we want to get a better sense of how the famous isotropic form of screened electrostatic interaction energy,

[TABLE]

is modified under the minimal number of microscopic assumptions. In what follows, we provide the general expression for the Fourier image of the anisotropic potential in momentum space, and analytically work out in real space the weakly anisotropic case only. Interested readers are invited to improve on that calculation by exploring the strongly anisotropic scenario.

II General considerations

The electrostatic potential energy of interaction between charges $q$ and $q^{\prime}$ located at $(\bm{\rho},z)$ and $({\bf 0},z^{\prime})$ ($z>z^{\prime}$, $\bm{\rho}=(x,y)$) inside an anisotropic semiconductor film of thickness $d$ surrounded by two isotropic media with dielectric constants $\epsilon_{1}$ and $\epsilon_{2}$ is given by (see Appendix for derivation and Fig. 6; compare with Rytova1967 ; keldysh1979coulomb )

[TABLE]

with

[TABLE]

where

[TABLE]

and the axes of the coordinate system $(x,y,z)$ coincide with the principal axes of the film’s permittivity tensor, $\epsilon={\rm diag}(\epsilon_{x},\epsilon_{y},\epsilon_{z})$. In the most interesting for practical applications scenario, $\epsilon_{1,2}\ll\epsilon_{x},\epsilon_{y},\epsilon_{z}$, and, for distances $\rho\gg|z-z^{\prime}|\sim d$, the main contribution to the integral in (2) comes from ${\bf k}$ satisfying $kd\ll k\rho\lesssim 1$. Under these conditions, ${\tilde{k}}d\ll 1$, ${\tilde{\eta}}_{1,2}\approx\epsilon_{1,2}k/(\epsilon_{z}{\tilde{k}})$, and, with the dependence on $z$ and $z^{\prime}$ disappearing, we get the two-dimensional form of the interaction,

[TABLE]

At this point it is convenient to introduce two “screening” lengths,

[TABLE]

characterizing polarizability of the film in the $x$ and $y$ directions, respectively, and write the interaction (5) in the form

[TABLE]

where $\epsilon({\bf k})$ is the dielectric function, formally defined by

[TABLE]

which generalizes the standard isotropic result cudazzo2011dielectric . In Ref. berkelbach2013theory , for the case of surrounding vacuum in the isotropic scenario, the authors have numerically verified that the screening length of a monolayer can be calculated with good accuracy on the basis of Eq. (6) provided the dielectric contrast is large and the relevant dielectric constant of the monolayer is the in-plane component of the permittivity tensor of the bulk material. We take that as an indication that the Keldysh model is a good approximation to realistic experimental situations and hypothesize that its anisotropic generalization proposed here should work reasonably well even for samples of monolayer thickness.

The problem thus reduces to the calculation of a two-dimensional Fourier integral,

[TABLE]

with $A,B>0$. To that end, working in polar coordinates, we write,

[TABLE]

where $t=k\rho$, $\alpha$ is the angle between the position vector ${\bm{\rho}}$ and the positive $x$-axis, as shown in Fig. 6,

[TABLE]

and

[TABLE]

with $a\in(-\infty,1]$ playing the role of the anisotropy parameter; the greater the $|a|$, the greater the anisotropy, with $a=0$ corresponding to the isotropic case.

Without loss of generality, we may assume that $0<\rho_{0x}\leq\rho_{0y}$, and thus $0\leq a<1$. Then, since $t\geq 0$, we have $0\leq\varepsilon^{2}(t)<1$. Taking into account the well-known Fourier series expansion mikhlin1964integral ,

[TABLE]

and using the fact that $\int_{0}^{2\pi}d\theta\,e^{it\cos\theta}\sin(2n\theta)=0$, we get for the $\theta$-integral in (11) the asymptotic multipole series,

[TABLE]

where $J_{0}$ and $J_{1}$ are the Bessel functions of the first kind.

III Weak anisotropy

In a rather straightforward manner (for a better approach see Sec. IV), assuming $0\leq a\ll 1$ and treating $\varepsilon^{2}(t,a,b)$ as a small parameter in (II), we get, in lowest order,

[TABLE]

and, thus,

[TABLE]

Performing the remaining $t$-integration we find,

[TABLE]

where

[TABLE]

is the standard Keldysh-Rytova result, and

[TABLE]

is the linear correction whose graph is shown in Fig. 1 (assuming $qq^{\prime}<0$). In the above, various $H_{i}$ and $Y_{i}$ denote the Struve and Neumann functions, respectively.

For $b\ll 1$, or ${d}\ll{\rho}\ll\rho_{0}$, we get

[TABLE]

where $\gamma\approx 0.577216$ is the Euler constant. Since $\int_{0}^{2\pi}\cos(2\alpha)d\alpha=0$, the excitonic ground state energy in this case experiences a first order shift,

[TABLE]

where $\psi_{0}(b)$ is the unperturbed axially symmetric ground state wave function. On the other hand, for $b\gg 1$, or ${\rho}\gg\rho_{0}$, Eqs. (18) and (III) reproduce the standard Coulomb asymptotics,

[TABLE]

To get a sense of the error involved in this linear approximation, we define two relative errors by

[TABLE]

and

[TABLE]

respectively, with $V_{1\,{\rm exact}}\equiv V_{\rm exact}-V_{0}$. Here, $V_{\rm approx}\equiv V_{0}+V_{1}$ is calculated on the basis of Eqs. (18) and (III), and $V_{\rm exact}$ is found by direct numerical integration of the double integral in (II). The corresponding results are summarized in Fig. 2. Notice that for all $\rho$ the error ${\rm Err}_{1}$ is greatest for points with $\alpha=\pi/2$. The error ${\rm Err}_{2}$ is particularly troublesome, as the blue curve clearly indicates.

IV Weak anisotropy: Renormalized Keldysh interaction

A better linear approximation can be achieved by “renormalizing” the zeroth order Keldysh contribution, $V_{0}$, as follows: we re-write the monopole term in (II) as shown below,

[TABLE]

and expand everything in square brackets to linear (leading!) order in $a$. The potential then becomes

[TABLE]

which should be compared with (16). We then find

[TABLE]

where

[TABLE]

is the renormalized isotropic Keldysh term, and

[TABLE]

is the corresponding linear correction consisting of a linear monopole and a linear dipole contributions, Fig. 3.

Now in the $b\ll 1$ limit we get

[TABLE]

and a perfectly reasonable first order correction

[TABLE]

which does not contain the logarithmic term. The excitonic ground state energy in this case undergoes a simple first order shift,

[TABLE]

Notice that our renormalization procedure eliminates logarithmic terms in all orders of the monopole perturbation, not just the first one. For example, keeping the second order monopole contribution in square brackets in Eq. (26) would add the term

[TABLE]

to the potential in (28), which in the $b\ll 1$ limit is just

[TABLE]

Returning to the linear approximation (28), we again define two relative errors,

[TABLE]

and

[TABLE]

with $V_{1\,{\rm ren}\,{\rm exact}}\equiv V_{\rm exact}-V_{0\,{\rm ren}}$ and $V_{\rm ren\,approx}\equiv V_{0\,{\rm ren}}+V_{1\,{\rm ren}}$. The corresponding numerical results presented in Fig. 4 show that our revised approximation scheme is indeed superior to the one used in Sec. III.

Finally, we also performed numerical simulations in the extreme anisotropic regime, as shown in Fig. 5. In this case, the “correction” $V_{1\,{\rm ren}\,{\rm exact}}$ becomes comparable to $V_{0\,{\rm ren}}$, and the linear approximation breaks down.

V Summary

The classic Keldysh-Rytova formula for screened Coulomb interaction in semiconductor thin films has been generalized by taking into account the anisotropy of the layer’s dielectric permittivity tensor. The Fourier image of the anisotropic potential in momentum space, as well as the linear correction to the isotropic potential in real space, have been worked out in closed analytical form. The case of strong in-plane anisotropy, however, remains unresolved due to the appearance of the function $I(t,\alpha,a,b)$ (see Eqs. (11) and (II)), whose explicit analytical expression is not known.

Acknowledgements.

The author thanks Robert Zaballa for useful discussions.

APPENDIX: Momentum space representation

Following Rytova1967 and keldysh1979coulomb , we consider a geometry in which the anisotropic semiconductor film occupies the region of space $-d/2\leq z\leq d/2$, as shown in Fig. 6. The half-space $z<-d/2$ (the substrate) is filled with an isotropic medium whose dielectric constant is $\epsilon_{1}$, while the half-space $z>d/2$ with an isotropic medium whose dielectric constant is $\epsilon_{2}$.

We are assuming that the axes of the coordinate system $(x,y,z)$ coincide with the principal axes of the film’s dielectric permittivity tensor. The electrostatic potential at point ${\bf r}\equiv(\bm{\rho},z)=(x,y,z)$ due to charge $q^{\prime}$ located at ${\bf r}^{\prime}=(0,0,z^{\prime})$ satisfies in regions 1, 2, and 3 (the film) the following system of equations:

[TABLE]

with the boundary conditions at the interfaces,

[TABLE]

and the boundary conditions at the two infinities,

[TABLE]

Fourier transforming,

[TABLE]

and substituting into (38), (39), and (40), we get the following equations for the corresponding Fourier components,

[TABLE]

where

[TABLE]

Conditions at infinity, (43), combined with Eqs. (45) and (46) give

[TABLE]

while Eq. (47) gives, for $z\neq z^{\prime}$,

[TABLE]

and, at $z=z^{\prime}$, the jump discontinuity in the $z$-derivative,

[TABLE]

Imposing the boundary conditions at the $z=+d/2$ interface,

[TABLE]

we get

[TABLE]

where

[TABLE]

Similarly, imposing the boundary conditions at the $z=-d/2$ interface, we get

[TABLE]

and, after defining

[TABLE]

find

[TABLE]

Now, for $z=z^{\prime}$, Eqs. (52), (55), and (60) give

[TABLE]

resulting in

[TABLE]

and

[TABLE]

Taking into account that

[TABLE]

we get

[TABLE]

For $z<z^{\prime}$, a similar calculation results in

[TABLE]