Anisotropic plasmons, Friedel oscillations and dynamical screening in $8-Pmmn$ borophene

TL;DR

This paper analytically investigates the anisotropic plasmon modes, Friedel oscillations, and screening properties of $8-Pmmn$ borophene, revealing undamped plasmon modes and anisotropic decay patterns in oscillations.

Contribution

It provides the first analytical calculation of the density-density response function and explores the anisotropic plasmon dispersion and Friedel oscillations in $8-Pmmn$ borophene.

Findings

Anisotropic plasmon mode remains undamped at higher energies along mirror symmetry.

Friedel oscillations decay as $r^{-3}$ with anisotropic amplitude and periodicity.

Screening properties are direction-dependent due to anisotropic Dirac cone.

Abstract

$8-Pmmn$ borophene is a polymorph of borophene which hosts an anisotropic tilted Dirac cone. Using a low energy effective Hamiltonian for borophene, we analytically calculate the density-density response function of borophene and study its anisotropic plasmon dispersion and screening properties. We find that the anisotropic plasmon mode in $8-Pmmn$ borophene, remains undamped for higher energies along the mirror symmetry direction. The friedel oscillation in borophene is found to decay as $r^{-3}$ in the large $r$ limit, with both the amplitude as well as the periodicity of the Friedel oscillations being anisotropic.

Equations

$$\mathcal{H}=\hslash \sum _{\mathbf{k}}{\psi }_{\mathbf{k}}^{†}\left[v_x{k}_x{\sigma }_x+v_y{k}_y{\sigma }_y+v_t{k}_y\text{openone}\right]{\psi }_{\mathbf{k}}.$$

$${\Pi }^{(0)}$$

$$E_{kx}=\hslash v_x{k}_x,E_{ky}=\hslash v_y{k}_y,\mathrm{and}{E}_k=\sqrt{E_{kx}^2+E_{ky}^2}.$$

$$f_{s{s}^{\prime }}(\mathbf{k},\mathbf{q})=\frac{1}{2}\left(1+s{s}^{\prime }\frac{E_k+E_q\cos {\varphi }_E}{\sqrt{E_k^2+E_q^2+2E_k{E}_q\cos {\varphi }_E}}\right),$$

$$\frac{\Pi^{(0)}({\bf q},\omega)}{N_{0}}=-1+\frac{1}{8\mu}\frac{E_{q}^{2}}{\sqrt{E_{q}^{2}-\hbar^{2}\omega^{\prime 2}}}\big{[}\Gamma^{+}(\nu^{+})+\Gamma^{-}(\nu^{-})\big{]}~{}.$$

$${\Gamma }^{\pm }(z)\equiv z\sqrt{1-z^2}\pm i{\cosh }^{-1}(z),$$

$${\nu }^{\pm }(\mathbf{q},\omega )\equiv \frac{2\mu \pm \hslash (\omega -v_t{q}_y)}{E_q}.$$

$${\Pi }^{\mathrm{RPA}}(\mathbf{q},\omega )=\frac{{\Pi }^{(0)}(\mathbf{q},\omega )}{1-v_q{\Pi }^{(0)}(\mathbf{q},\omega )},$$

$$ϵ(\mathbf{q},\omega )\equiv 1-v_q{\Pi }^{(0)}(\mathbf{q},\omega )=0.$$

$${\omega }_{+}^{\mathrm{intra}}(\mathbf{q})=v_{x}q\left[f({\theta }_q)+v_t\sin {\theta }_q/v_x\right],$$

$${\omega }_{-}^{\mathrm{intra}}(\mathbf{q})=v_{x}q\left[f({\theta }_q)+v_t\sin {\theta }_q/v_x\right]-2\mu /\hslash ,$$

$$f^2({\theta }_q)={\cos }^2{\theta }_q+\frac{v_y}{v_x}{\sin }^2{\theta }_q.$$

$${\omega }^{\mathrm{inter}}=2\mu /\hslash -v_{x}q[f({\theta }_q)-v_t\sin {\theta }_q/v_x].$$

$$\Pi^{(0)}({\bf q},\omega)\approx-N_{0}\bigg{[}1-\frac{\omega^{\prime}}{\sqrt{\omega^{\prime 2}-v_{x}^{2}q^{2}f^{2}(\theta_{q})}}\bigg{]}~{}.$$

$${\omega }_{\mathrm{pl}}(q\to 0)=v_{x}qf({\theta }_q)\frac{1+v_q{N}_0}{\sqrt{1+2v_q{N}_0}}+v_{t}q\sin {\theta }_q.$$

$$\frac{{\Pi }^{\mathrm{st}}(\mathbf{q})}{N_0}$$

$$q_0({\theta }_q)\equiv \frac{2k_F}{f({\theta }_q)-v_t\sin ({\theta }_q)/v_x},$$

$${\varphi }_{\mathrm{sc}}(\mathbf{r})=\int d\mathbf{q}{e}^{i\mathbf{q}.\mathbf{r}}\frac{{\varphi }_{\mathrm{ext}}(\mathbf{q})}{ϵ(\mathbf{q},0)}{e}^{-qs}.$$

$$ϵ(q<q_0,0)=1+q_{\mathrm{TF}}/q,$$

$${\int }_0^{2\pi }{e}^{iqr\cos {\theta }_q}d{\theta }_q=2\pi J_0(qr)\simeq \frac{\sqrt{8\pi }}{\sqrt{qr}}\cos (qr-\pi /4).$$

$${\varphi }_{\mathrm{sc}}(\mathbf{r})=\sqrt{\frac{8\pi }{r}}{\int }_0^{\infty }dq{q}^{1/2}\frac{{\varphi }_{\mathrm{ext}}(q)}{{ϵ}^{\prime }(q,{\theta }_r)}\cos (qr-\pi /4).$$

$${\varphi }_{\mathrm{sc}}(\mathbf{r})=A[q_0({\theta }_r)]\int \delta {\Pi }^{\mathrm{st}}(q,{\theta }_r)S(q,r,{\theta }_r)e^{-qs}dq,$$

$$S(q,r,{\theta }_r)=\frac{\sqrt{8\pi }}{\sqrt{q_{0}r}}\cos [(q_0+q)r-\pi /4],$$

$$A[q_0({\theta }_r)]=\frac{q_0{\varphi }_{\mathrm{ext}}(q_0)v_{q=q_0}}{[1-v_{q=q_0}{\Pi }^{\mathrm{st}}(q_0,{\theta }_r){]}^2}.$$

$${\varphi }_{\mathrm{sc}}(\mathbf{r})=F({\theta }_r)\frac{\cos (q_{0}r)}{r^3}.$$

$$F({\theta }_r)=\frac{gA(q_0)}{\sqrt{8\pi }{q}_0}\frac{f({\theta }_r)}{\sqrt{1-\frac{v_t^2{\sin }^2{\theta }_r}{v_x^2{f}^2({\theta }_r)}}}{\left(1-\frac{v_t\sin {\theta }_r}{v_{x}f({\theta }_r)}\right)}^{3/2}.$$

$${\Pi }_{\mathrm{imp}}^{(0)}(\mathbf{q},\omega )=\frac{(\omega +i{\tau }^{-1}){\Pi }^{(0)}(\mathbf{q},\omega +i/\tau )}{\omega +i{\tau }^{-1}{\Pi }^{(0)}(\mathbf{q},\omega +i/\tau )/{\Pi }^{(0)}(\mathbf{q},0)},$$

$$\Pi_{\rm imp}^{(0)}({\bf q},\omega)\approx-N_{0}\bigg{[}1-\frac{\omega^{\prime}}{\sqrt{(\omega^{\prime}+i/\tau)^{2}-q^{2}v_{x}^{2}f^{2}(\theta)}-i/\tau}\bigg{]}~{}.$$

$${\omega }_{\mathrm{pl}}^{\mathrm{imp}}(\mathbf{q})$$

$$q_c({\theta }_q)=-q_{\mathrm{TF}}+\sqrt{q_{\mathrm{TF}}^2+1/(v_x^2{f}^2({\theta }_q){\tau }^2)}.$$

Full text

I Introduction

Collective density oscillations (plasmons) of an interacting electron gas, and their screening properties, are of fundamental interest from both experimental and theoretical perspective Polini_review ; Maier ; Stauber ; GV . From an application point of view, plasmons in graphene and related two dimensional (2D) materials are very promising on account of their longer lifetime, gate tunability and optoelectronic applications – facilitating the localization and guiding of light into electrical signalsappl . Plasmons and the dynamical dielectric function have been actively explored in a variety of 2D materials including spin-orbit coupled 2D electron gas fabian_2010 ; SOC_plasmon ; Persistent , 2D hole gas scholz_2012 , graphene stauber_2006 ; sarma_2007 ; polini_2011 ; amit_2015 ; Polini_review ; Peres , MoS2, scholz_2013 , topological insulators TI_plasmons , massive Dirac materials pyat_2009 ; thakur_2015 ; thakur_2017 , and phosphorene TLow1 ; Rodin_plasmon ; Barun_plasmon . In this paper, we study the plasmon dispersion, dynamical dielectric function and static screening properties in one of the 2D polymorphs of borophene Poly1 ; Poly2 ; Feng ; Mannix , which has a symmetry protected tilted and anisotropic Dirac cone zabo_2016 .

The polymorph of borophene with a tilted and anisotropic Dirac cone (called $8-Pmmn$ borophene) was predicted in Ref. [Zhou1, ], and has recently been experimentally confirmed Feng2 . The ab-initio properties of $8-Pmmn$ borophene monolayer have been studied in detaillopez_2016 , and based on symmetry considerations an effective low energy Hamiltonian, for energies and wave-vectors in vicinity of the Dirac point (laying on the $\Gamma-Y$ axis), was proposed in Ref. [zabo_2016, ]. Using the low energy effective Hamiltonian of $8-Pmmn$ borophene, we analytically calculate the density-density response function (also called the polarization/Lindhard function) and the dynamical dielectric function within the random phase approximation (RPA). We show that $8-Pmmn$ hosts anisotropic plasmons, and has highly anisotropic static screening properties in the long wavelength screening as well as the Friedel oscillations, which decays as $r^{-3}$ far away from the impurity.

This article is organized as follows: In Sec. II we use the anisotropic 2D Dirac Hamiltonian of borophene to analytically calculate the density density response function. In Sec. III, we use the analytical polarization function to study the anisotropic plasmon dispersion. This is followed by a detailed discussion of the long wavelength screening and anisotropic Friedel oscillations in Borophene, in Sec. IV. In Sec. V we explore the impact of small disorder on the plasmon dispersion of borophene. Finally we summarize our findings in Sec. VI.

II Non-interacting Polarization Function for $8-Pmmn$ Borophene

Our starting point is the effective low energy Hamiltonian of $8-Pmmn$ borophene, which describes a tilted anisotropic Dirac cone in 2D using three parameters zabo_2016 :

[TABLE]

Here $\psi_{\bf k}=(a_{\bf k},b_{\bf k})^{T}$, where $a_{\bf k}$ and $b_{\bf k}$ are the the destruction operators of the Bloch states of the two triangular sub-lattices in $8-Pmmn$ borophene. The three velocities are given by zabo_2016 $\{v_{x},v_{y},v_{t}\}=\{0.86,0.69,0.32\}\times 10^{6}$ m/s. The corresponding energy dispersion in given by $\varepsilon_{s{\bf k}}=\hbar v_{t}k_{y}+s\hbar\sqrt{v_{x}^{2}k_{x}^{2}+v_{y}^{2}k_{y}^{2}}$, and $s=+1$ ($-1$) denotes the conduction (valance) band. Similar to the crystal structure of $8-Pmmn$ borophene (see Ref. [lopez_2016, ; zabo_2016, ]), its energy dispersion also has mirror symmetry about the ${\hat{y}}$ axis (${\hat{y}}-{\hat{z}}$ mirror plane at $x=0$). This mirror symmetry will also reflect in the polarization function, plasmon dispersion, and the static screening properties discussed below.

The non-interacting dynamical polarization function of an electron gas with two bands, for a given (${\bf q},\omega$) is given by

[TABLE]

where $g=2$ denotes the spin ($=2$) and valley ($=1$) degeneracy, $\omega^{+}=\omega+i\eta$ with $\eta\to 0$, and $f_{ss^{\prime}}({\bf k},{\bf q})$ is the orbital overlap function. The Fermi function is given by $n_{s{\bf k}}=(e^{\beta(\varepsilon_{s{\bf k}}-\mu)}+1)^{-1}$, where $\beta=1/(k_{B}T)$ and $\mu$ denotes the chemical potential. At zero temperature the Fermi function acts as a step function and it sets the integration limits. In what follows we will work in the $\omega>0$ regime, since we have $\Pi^{(0)}({\bf q},-\omega)=\Pi^{(0)}({\bf q},\omega)^{*}$.

To simplify the calculation of the anisotropic polarization function, we define the following modified variables:

[TABLE]

In terms of the new variables the energy eigenvalue is $\varepsilon_{s{\bf k}}=\hbar v_{t}k_{y}+sE_{k}$. The band overlap function, in terms of the redefined variables, is given by

[TABLE]

where $\phi_{E}=\tan^{-1}({E_{ky}}/{E_{kx}})-\tan^{-1}({E_{qy}}/{E_{qx}})$.

The non-interacting polarization function in Eq. (2), can be expressed as a sum of the polarization function of the intrinsic part $\Pi^{(0)}_{0}$ (which is finite for $\mu=0$), and an extrinsic part $\Pi^{(0)}_{1}$ (which vanishes for $\mu\to 0$): $\Pi^{(0)}=\Pi^{(0)}_{0}+\Pi^{(0)}_{1}$. Both the intrinsic and the extrinsic contributions are evaluated explicitly in appendix A, in a manner similar to that in graphenepyat_2009 . The total polarization function can be expressed in a compact form as

[TABLE]

Here $N_{0}\equiv{g\mu}/({2\pi\hbar^{2}v_{x}v_{y}})$ is the density of states in 2D anisotropic massless Dirac systems, and we have defined $\omega^{\prime}\equiv\omega-v_{t}q\sin\theta$. In Eq. (5) we have defined the complex function

[TABLE]

and the dimensionless variables,

[TABLE]

In Fig. 1 (a)-(b), we display the real and the imaginary part of the polarization function in the $\omega-q_{y}$ plane. Since the anisotropic term in the $8-Pmmn$ borophene Hamiltonian [see Eq. (1)] involves $q_{y}$ only, thus along the $q_{x}$ direction (for $q_{y}=0$) the polarization function resembles that of graphene. Panels (c) and (d) of Fig. 1 show the real and imaginary part of the polarization function respectively as a function of $\omega$ for different $q_{y}$ values, while panels (e) and (f) explore the same as a function of the orientation angle of $\bf q$ in the $q_{x}-q_{y}$ plane for a fixed value of $\omega$ and $q$. Note the mirror symmetry of the polarization function about the ${\hat{y}}$ axis, i.e., about the $\theta_{q}=\pi/2$ and $\theta_{q}=3\pi/2$ line in Fig. 1(e)-(f). No such symmetry exists along the $\hat{x}$ axis or alternately around $\theta_{q}=0$ and $\theta_{q}=\pi$ lines in Fig. 1(e)-(f). In the next section, we use the obtained polarization function to calculate the dispersion of the dynamical collective density excitations, i.e., plasmons.

III Plasmon Dispersion

The dynamical interacting density-density response function, within the random phase approximation (RPA), is given by

[TABLE]

where $v_{q}=2\pi e^{2}/(\kappa q)$ is the Coulomb interaction in the momentum space in 2D with $\kappa$ being the static dielectric constant. Collective density excitations or plasmons (within RPA) are now given by the poles of $\Pi^{\rm RPA}({\bf q},\omega)$, or alternately by the zeros of the dynamical dielectric function $\epsilon({\bf q},\omega)$, i.e.,

[TABLE]

From an experimental perspective, plasmons appear as resonance peaks in the momentum resolved electron energy loss spectrum, which directly measures the loss function Barun_plasmon : ${\cal{E}}_{\rm loss}({\bf q},\omega)=-\mathfrak{Im}[1/\epsilon({\bf q},\omega)]$. The anisotropic loss function for $8-Pmmn$ borophene is shown in Fig. 2 for $\theta_{q}=\pi/2$, $\pi/4$, $-\pi/2$ and $-\pi/4$ in panels a), b) c) and d) respectively. In all the panels, there is a dominant plasmon peak in the region without any (single) particle-hole excitations, and it broadens (gets damped) significantly on entering the inter-band particle hole continuum – whose boundary is also $\theta_{q}$ dependent.

The particle hole boundary for the intra-band transitions is given by lines specified by

[TABLE]

and

[TABLE]

where we have defined the function

[TABLE]

In terms $f(\theta_{q})$ we have $E_{q}=\hbar v_{x}q{f(\theta_{q})}$. Similarly the particle hole boundary of the inter-band single particle excitations is given by

[TABLE]

For $\theta_{q}=\pi/2$ ($\theta_{q}=-\pi/2$), both the boundaries of the intra-band and the inter-band particle hole continuum are pushed to higher (lower) energies for a given $q$. Now if the plasmon mode decays by entering into the inter-band particle hole continuum, then it remains undamped for relatively higher (lower) energies for $\theta_{q}=\pi/2$ (for $\theta_{q}=-\pi/2$) – as shown in panel (a) (panel (c)) of Fig. 2.

To obtain the exact plasmon dispersion, Eq. (9) has to be solved numerically. However analytical expressions can be obtained in the long wavelength limit. In the dynamical long wavelength limit with ${\bf q}\to 0$ and $\omega>v_{x}q$, the polarization function can be approximated asamit_2015 ,

[TABLE]

Using Eq. (14) in Eq. (9), the long wavelength plasmon dispersion can be obtained to be

[TABLE]

The second term in the r.h.s of Eq. (15) is linear in $q$ and the co-efficient becomes negative for $\theta_{q}\in(0,-\pi)$. Thus it seems that the long wavelength plasmon frequency first increases and then decreases with increasing $q$ as the linear term becomes dominant. However, this turns out to be a peculiarity of the long wavelength limit, and the actual (exact) plasmon dispersion remains monotonically increasing and eventually decays by entering the inter-band particle hole continuum. This is highlighted in Fig. 2, which displays the exact plasmon dispersion (solid black line) along with the corresponding long wavelength limit (green-dashed line), on the backdrop of the momentum resolved loss function for different values of $\theta_{q}$.

IV Static Response and Charged impurity screening

We now turn our attention to the static scenario ($\omega\to 0$) which is useful for studying the long wavelength screening of coulomb interactions in an electron gas, and the corresponding Friedel oscillations of a charged impurity. The static non-interacting response function is given by $\Pi^{\rm st}({\bf q})\equiv\Pi^{(0)}({\bf q},0)$ and

[TABLE]

where $k_{F}={\mu}/{\hbar v_{x}}$ and we have defined a real valued functionstauber_2006 $G_{<}(x)\equiv x\sqrt{1-x^{2}}-\cos^{-1}(x)$. In the long wavelength limit, or more generally for $q<q_{0}$ where

[TABLE]

we have $\Pi^{(0)}(q<q_{0},0)=-N_{0}$, a constant. However for finite $q>q_{0}$ the static dielectric function is anisotropic. The anisotropic static response function is shown in Fig. 3(a), for $q$ along three different directions. Figure  3(b) shows the static response function in the $q_{y}-q_{x}$ plane, where the mirror symmetry about the ${\hat{y}}$ axis ($q_{x}=0$) is evident.

The effective screened potential of a charged impurity at origin, of potential $\phi_{\rm ext}(\bf q)$ in vacuum, is given by

[TABLE]

Here $s\rightarrow 0$ is supposedly the distance of the charged impurity from the 2D plane of the electron gas. We now discuss the long wavelength Thomas-Fermi screening, and the Friedel oscillations of a charged impurity in the next two subsections.

IV.1 Long wavelength Thomas Fermi screening

For large length scales or in the long wavelength ($q\to 0$) limit, the static screening of a charged impurity in an electron liquid is dictated by the long wavelength static dielectric constant. Using Eq. (IV), we have

[TABLE]

where we have defined a density (or $\mu$ dependent) Thomas Fermi wavevector, $q_{\rm TF}\equiv 2\pi e^{2}N_{0}/\kappa=ge^{2}\mu/(\hbar^{2}\kappa v_{x}v_{y})$ in all directions. The $q_{\rm TF}$ for $8-Pmmn$ borophene is almost identical to that of graphene with $v_{\rm F}^{2}\to v_{x}v_{y}$, barring the extra valley degeneracy factor in graphene. The Thomas Fermi screening wavevector is arising specifically from the extrinsic contribution of the static polarization function which vanishes for the intrinsic case ($\mu\to 0$). The long wavelength screening in 8mmn borophene reduces the impact of a Coulomb impurity (with $\phi_{\rm ext}({\bf r})=e^{2}/r$), and its effective behaviour decreases as $r^{-3}$ for large distances ($r>1/q_{\rm TF}$), similar to that in graphenestauber_2006 .

IV.2 Anisotropic Friedel oscillations

We now discuss the impact of screening of a charged impurity by the electron gas in $8-mmn$ borophene arising from the dielectric contribution for finite wavevectors. This typically leads to Friedel oscillations in the system with a power law decay thakur_2017 .

In the asymptotic $r\to\infty$ limit, the integral in Eq. (18) has a very rapidly varying phase: $qr\cos(\theta_{q}-\theta_{r})$, where $\theta_{r}$ is the the real space angle denoting the direction of ${\bf r}$. Consequently the dominant contribution to it arises when the phase is stationary. Now following Ref. [fabian_2010, ] we use the method of stationary phase which allows us to simplify Eq. (18). The $\theta_{q}$ dependence of $\epsilon({\bf q},0)$ in Eq. (18) is replaced by $\theta_{q}\to\theta_{r}$. With the $\theta_{q}$ dependence of $\epsilon({\bf q},0)$ taken care of, the angular integration simion_2005 in Eq. (18) can be easily performed to yield,

[TABLE]

Substituting Eq. (20) in Eq. (18) leads to a one dimensional $q$ integral given by

[TABLE]

where $\epsilon^{\prime}(q,\theta_{r})\equiv\epsilon({\bf q},0)$ with $\theta_{q}\to\theta_{r}$.

The one dimensional integral of Eq. (21) is now highly oscillatory around zero in the asymptotic limit of $r\rightarrow\infty$, for finite $q$. Thus using the Riemann-Lebesgue lemma, the asymptotic behaviour of the integral in Eq. (21) is small and it is dominated by the contribution from the the non-analytic points of the integrand (the static dielectric function in our case). The anisotropic static dielectric function of borophene appearing in Eq. (21) has a discontinuity in its first derivative at $q=q_{0}(\theta_{q}\to\theta_{r})$; see Eq. (17) and Fig. 3(a). Now following Ref. [thakur_2017, ], Eq. (21) can be approximated as

[TABLE]

where $\delta\Pi^{\rm st}(q,\theta_{r})\equiv\Pi^{\rm st}[q_{0}(\theta_{r})+q]-\Pi^{\rm st}[q_{0}(\theta_{r})]$ is the increment of the polarization function in vicinity of the non-analytic point $q_{0}(\theta_{r})$. In Eq. (22), $S(x,r,\theta_{r})$ originates from the angular integration of Eq. (20) and it is given by,

[TABLE]

and the function $A$ is explicitly given by

[TABLE]

On evaluating Eq. (22), the asymptotic anisotropic behaviour of the screened potential is found to be similar to that in graphene:

[TABLE]

Where the amplitude of Friedel oscillation $F(\theta_{r})$ is given by,

[TABLE]

However unlike graphene, here both $q_{0}$ (the period of the Friedel oscillations) and the proportionality constant (amplitude of Friedel oscillations) are direction ($\theta_{r}$) dependent. The directional dependence of the Friedel oscialltion amplitude and oscillation period is highlighted in Fig. 4(a). Figure 4(b) shows the anisotropic Friedel oscillations in the $x-y$ plane. From Eq. (26) it can be inferred that the oscillation amplitude is maximum along the $-{\hat{y}}$ direction. The period of oscillation ($2\pi/q_{0}$) is also maximum along the $-{\hat{y}}$ direction.

V Effect of small static disorder on the plasmon dispersion

In this section, we study the effect of the small static disorder on the dynamical polarization function and the plasmon mode. For dilute concentration of the impurities, the polarization function gets modified by a disorder averaged polarization function. Within the relaxation time approximation, the disorder averaged effective polarization is given by the Mermin formulamermin_1970 ; GV ; amit_2015 ,

[TABLE]

with $\tau$ being an effective relaxation time arising from the presence of the disorder. The dynamical long wavelength limit of the disorder averaged response function can be obtained by substituting Eq. (14) in Eq. (27), and it is given by

[TABLE]

Now we can easily calculate the disorder averaged plasmon dispersion in the long wavelength limit by substituting Eq. (28) in Eq. (9) to obtain,

[TABLE]

Note that $N_{0}v_{q}=q_{\rm TF}/q$ for the Coulomb potential. The imaginary part of the plasmon dispersion is given by $\tau^{-1}\times(1+v_{q}N_{0})(1+2v_{q}N_{0})$ which has a finite value of $\tau^{-1}/2$ even in the $q\to 0$ limit. In the limit of vanishing disorder i.e., $1/\tau\to 0$ Eq. (V) reduces to Eq. (15). As expected from the case of 2DEG and graphene quinn_1984 ; amit_2015 , here also the collective density excitations can only exist beyond a certain critical wavevector $q_{c}(\theta_{q})$ given by

[TABLE]

The existence of the critical wavevector, is a consequence of the fact that the collective excitations cannot exist for length scales larger than the disorder length scale (effective mean free path).

Figure 5 shows the effective energy loss function (calculated using Eq. (5) in (27)) for $\hbar\tau^{-1}=0.1\mu$ in the $\omega-q$ plane for different directions, along with the long wavelength plasmon dispersion of Eq. (V) in black line. The existence of the anisotropic critical wavevector, and the corresponding anisotropy in the plasmon dispersion for collective density excitations is evident.

VI Conclusion

To summarize, we have calculated the density-density response function of $8-Pmmn$ borophene, which has a tilted and anisotropic Dirac cone. The polarization function is in turn used to calculate the anisotropic plasmon dispersion. The analytical form of the polarization function calculated in the dynamical long wavelength limit, matches reasonably well with the exact numerical results. We find that for wave-vectors along the mirror symmetry axis of borophene ($\hat{y}$ axis), both the inter- and intra-band particle-hole continuum are pushed up in energy. Consequently the plasmon dispersion is undamped for relatively higher energies for $q$ along the $\hat{y}$ axis. In case of small finite disorder, the plasmon mode ceases to exist below a critical anisotropic wave-vector, and beyond which it develops a finite damping even for small $q$. The static screening properties of $8-Pmmn$ borophene is found to be similar to that in graphene, with the coulomb potential having a asymptotic decay of $r^{-3}$. The Friedel oscillations in borophene also decay as $r^{-3}$ as in graphene, though they are highly anisotropic, with both the amplitude and periodicity of the Friedel oscillations being direction dependent.

Appendix A Calculation of the polarization function

To evaluate Eq. (2), we proceed in a manner similar to that in graphene, and define the retarded function stauber_2006

[TABLE]

In Eq. (A), $f^{\gamma}({\bf k},{\bf q})$ is the band overlap function given by,

[TABLE]

and $E_{k+q}=\sqrt{E_{k}^{2}+E_{q}^{2}+2E_{k}E_{q}\cos\phi_{E}}$. Here $\gamma=+1$ (-1) corresponds to intra (inter) band transitions and $D$ sets the general upper limit of the integration. For $\mu=0$ the intrinsic polarization is given by

[TABLE]

where $\Lambda$ is the ultra violet band cut-off. In the limiting case of $\Lambda\rightarrow\infty$, Eqs. (A)-(33) can be evaluated to yield

[TABLE]

For the doped case there is finite chemical potential which gives an extra contribution to the polarization function, and it is given by

[TABLE]

Evaluation of Eq. (35) lead to

[TABLE]

In Eq. (36), $\Theta(x)$ is the unit step function, $\omega^{\prime}=\omega-v_{t}q_{y}$, and we have defined the function

[TABLE]

Equations (34)-(37) are very similar to that of graphenestauber_2006 ; thakur_2017 are reproduce the results for graphene in the limiting case of $v_{t}\to 0$ and $v_{x}=v_{y}=v_{\rm F}$.