Recent Progress on Nonlocal Graphene/Surface Plasmons

Abstract

We review recent experimental and theoretical studies of the non-local plasmon dispersion relations of both single and double layers of graphene which are Coulomb-coupled to a thick conducting medium. High-resolution electron energy loss spectroscopy (HREELS) was employed in the investigations. A mean-field theory (R.P.A.) formulation was used to simulate and explain the experimental results, with the undamped plasmon excitation spectrum calculated for arbitrary wave number. Our numerical calculations show that when the separation a between a graphene layer and the surface is less than a critical value a_c = 0.4k ⁻¹_F , the lower acoustic plasmon is overdamped. This result seems to explain the experimentally observed behavior for the plasmon mode intensity as a function of wave vector. The damping, as well as the critical distance, changes in the presence of an energy bandgap for graphene. We also report similar damping features of the plasmon modes for a pair of graphene layers. However, the main difference arising in the case when there are two layers is that if the separation between the layer nearest the surface and the surface is less than a_c, then both the symmetric and antisymmetric modes become damped, in different ranges of wave vector.

Appendices

Appendix 1: Dynamic Nonlocal Polarization Function for Free-Standing Graphene with no Bandgap; Brief Summary of the Results Derived in [19, 20]

The 2D RPA density perturbation response function, $\delta \rho /\delta V = R_{2D}^{(0)} ({\mathbf{p}},\omega ) \equiv D_{0} \widetilde{R}(x,v)$, for free-standing gapless Graphene in the T = 0 degenerate limit is given in terms of dimensionless frequency and wavenumber variables defined by ν = ω/E_F = ω/μ and x = p/p_F, respectively, as (note that $D_{0} \equiv \gamma^{ - 1} \sqrt {g_{s} g_{v} \rho_{2D} /\pi }$; g_s and g_v are spin and valley degeneracies, g_s = g_v = 2; ℏ → 1):

$$\widetilde{R}(x,\nu ) = \widetilde{R}^{ + } (x,\nu ) + \widetilde{R}^{ - } (x,\nu ),$$

(9.47)

with (θ(z) ≡ η₊(z) =  Heaviside unit step function)

$$\widetilde{R}^{ + } (x,\nu ) = \widetilde{R}_{1}^{ + } (x,\nu )\theta (\nu - x) + \widetilde{R}_{2}^{ + } (x,\nu )\theta (x - \nu )$$

(9.48)

where (define $\widetilde{\varPi } \equiv - \widetilde{R}$)

$$\begin{aligned} - {\text{Re}}\widetilde{R}_{1}^{ + } \left( {x,\nu } \right) & = {\text{Re}}\widetilde{\varPi }_{1}^{ + } \left( {x,\nu } \right) = 1 - \frac{1}{{8\sqrt {\nu^{2} - x^{2} } }}\left\{ f_{1} \left( {x,\nu } \right)\theta \left( {\left| {2 + \nu } \right| - x}\right) \right.\\ & \left. \quad +\, {\text{sgn}}\left( {\nu - 2 + x} \right)f_{1} \left( {x, - \nu } \right)\theta \left( {\left| {2 - \nu } \right| - x} \right) \right. \\ & \left. \quad + f_{2} \left( {x,\nu } \right)\left[ {\theta \left( {x + 2 - \nu } \right) + \theta \left( {2 - x - \nu } \right)} \right]\right\} , \\ \end{aligned}$$

(9.49)

$$\begin{aligned} - {\text{Re}}\widetilde{R}_{2}^{ + } \left( {x,\nu } \right) & = {\text{Re}}\widetilde{\varPi }_{2}^{ + } \left( {x,\nu } \right) = 1 - \frac{1}{{8\sqrt {x^{2} - \nu^{2} } }}\left\{ f_{3} \left( {x,\nu } \right)\theta \left( {x - \left| {\nu + 2} \right|} \right) \right.\\ & \left.\quad + f_{3} \left( {x, - \nu } \right)\theta \left( {x - \left| {\nu - 2} \right|} \right) \right.\\ & \left. \quad + \,\frac{{\pi x^{2} }}{2}\left[ {\theta \left( {\left| {\nu + 2} \right| - x} \right) + \theta \left( {\left| {\nu - 2} \right| - x} \right)} \right]\right\} , \\ \end{aligned}$$

(9.50)

$$\begin{aligned} - {\text{Im}}\widetilde{R}_{1}^{ + } \left( {x,\nu } \right) & = {\text{Im}}\widetilde{\varPi }_{1}^{ + } \left( {x,\nu } \right) = \frac{ - 1}{{8\sqrt {\nu^{2} - x^{2} } }}\left\{ f_{3} \left( {x, - \nu } \right)\theta \left( {x - \left| {\nu - 2} \right|} \right) \right.\\ & \left.\quad + \frac{{\pi x^{2} }}{2}\left[ {\theta \left( {x + 2 - \nu } \right) + \theta \left( {2 - x - \nu } \right)} \right]\right\} , \\ \end{aligned}$$

(9.51)

$$- {\text{Im}}\widetilde{R}_{2}^{ + } \left( {x,\nu } \right) = {\text{Im}}\widetilde{\varPi }_{2}^{ + } \left( {x,\nu } \right) = \frac{{\theta \left( {\nu - x + 2} \right)}}{{8\sqrt {x^{2} - \nu^{2} } }}\left[ {f_{4} \left( {x,\nu } \right) - f_{4} \left( {x, - \nu } \right)\theta \left( {2 - x - \nu } \right)} \right]$$

(9.52)

and

$$- \widetilde{R}^{ - } \left( {x,\nu } \right) = \frac{{\pi x^{2} \theta \left( {x - \nu } \right)}}{{8\sqrt {x^{2} - \nu^{2} } }} + i\frac{{\pi x^{2} \theta \left( {\nu - x} \right)}}{{8\sqrt {\nu^{2} - x^{2} } }}.$$

(9.53)

The quantities f₁(x, ν), f₂(x, ν), f₃(x, ν), f₄(x, ν) are defined as

$$f_{1} \left( {x,\nu } \right) = \left( {2 + \nu } \right)\sqrt {\left( {2 + \nu } \right)^{2} - x^{2} } - x^{2} \ln \frac{{\sqrt {\left( {2 + \nu } \right)^{2} - x^{2} } + \left( {2 + \nu } \right)}}{{\left| {\sqrt {\nu^{2} - x^{2} } + \nu } \right|}},$$

(9.54)

$$f_{2} \left( {x,\nu } \right) = x^{2} \ln \frac{{\nu - \sqrt {\nu^{2} - x^{2} } }}{x},$$

(9.55)

$$f_{3} \left( {x,\nu } \right) = \left( {2 + \nu } \right)\sqrt {x^{2} - \left( {2 + \nu } \right)^{2} } + x^{2} \mathop {\sin }\nolimits^{ - 1} \frac{2 + \nu }{x},$$

(9.56)

$$f_{4} \left( {x,\nu } \right) = \left( {2 + \nu } \right)\sqrt {\left( {2 + \nu } \right)^{2} - x^{2} } - x^{2} \ln \frac{{\sqrt {\left( {2 + \nu } \right)^{2} - x^{2} } + \left( {2 + \nu } \right)}}{x}.$$

(9.57)

Appendix 2: Dynamic Nonlocal Polarization Function for Graphene with a Finite Energy Bandgap; Brief Summary of the Results Derived in [18]

The 2D RPA ring diagram polarization function for graphene with a gap Δ may be expressed as

$$\begin{aligned} \varPi_{2D}^{(0)} (q,{\omega }) & = \frac{g}{{4\pi^{2} }}\int d^{2} k\sum\limits_{{s,s^{\prime} = \pm }} \left( {1 + ss^{\prime}\frac{{{\mathbf{k}} \cdot ({\mathbf{k}} + {\mathbf{q}}) +\Delta ^{2} }}{{\varepsilon_{k} \varepsilon_{{\left| {{\mathbf{k}} + {\mathbf{q}}} \right|}} }}} \right) \\ & \quad \times \frac{{f(s{\kern 1pt} \varepsilon_{k} ) - f(s^{\prime}\varepsilon_{{{\mathbf{k}} + q}} )}}{{s\varepsilon_{{\mathbf{k}}} - s^{\prime}\varepsilon_{{{\mathbf{k}} + {\mathbf{q}}}} - \hbar {\omega } - i\hbar {\delta }}} \\ \end{aligned}$$

(9.58)

Since we limit our considerations to zero temperature, T = 0, the Fermi-Dirac distribution function is reduced to the Heaviside step function f(ɛ, μ; T → 0) = η₊(μ − ɛ), so (9.58) is simplified to

$$\varPi_{2D}^{(0)} (q,\omega ) = - \chi_{ - }^{\infty } (q,\omega ) + \chi_{ + }^{\mu } (q,\omega ) + \chi_{ - }^{\mu } (q,\omega ){\kern 1pt} ,$$

(9.59)

where

$$\begin{aligned} \chi_{ \pm }^{\alpha } & = \frac{g}{{4\pi^{2} }}\int d^{2} {\mathbf{k}}{\kern 1pt} \eta_{ + } (\alpha^{2} - \varDelta^{2} - q^{2} )\sum\limits_{{s,s^{\prime} = \pm }} \left( {1 \pm ss^{\prime}\frac{{{\mathbf{k}} \cdot ({\mathbf{k}} + {\mathbf{q}}) + \varDelta^{2} }}{{\varepsilon_{k} {\kern 1pt} {\kern 1pt} \varepsilon_{{|{\mathbf{k}} + {\mathbf{q}}|}} }}} \right) \\ & \quad \times \left( {\frac{1}{{\hbar \omega + \varepsilon_{k} \mp \varepsilon_{{|{\mathbf{k}} + {\mathbf{q}}|}} + i\hbar \delta }} - \frac{1}{{\hbar \omega - \varepsilon_{k} \pm \varepsilon_{{|{\mathbf{k}} + {\mathbf{q}}|}} + i\hbar \delta }}} \right) \\ \end{aligned}$$

(9.60)

and

$$\varPi_{2D}^{(0)} (q,\omega ) = \varPi_{(0)} (q,\omega ) + \eta_{ + } (\mu -\Delta )\varPi_{(1)} (q,\omega ),$$

(9.61)

where

$$\begin{aligned} \varPi_{(0)} (q,\omega ) & = - \chi_{ - }^{\infty } (q,\omega ) \\ \varPi_{(1)} (q,\omega ) & = \chi_{ + }^{\mu } (q,\omega ) + \chi_{ - }^{\mu } (q,\omega ). \\ \end{aligned}$$

(9.62)

The following notations are employed to specify the expressions involved in the polarization function:

$${\mathcal{F}}(q,\omega ) = \frac{{g{\kern 1pt} \hbar v_{F}^{2} }}{16\pi }\frac{{q^{2} }}{{\sqrt {|\omega^{2} - (v_{F} q)^{2} |} }}$$

(9.63)

and

$${\mathcal{X}}_{0} = \sqrt {1 - \frac{{4\Delta ^{2} }}{{\hbar^{2} \left( {\omega^{2} - (v_{F} q)^{2} } \right)}}} ,$$

(9.64)

along with definitions of the following functions:

$${\mathcal{G}}_{ < } (x) = x\sqrt {{\mathcal{X}}_{0}^{2} - x^{2} } - (2 - {\mathcal{X}}_{0} ){\kern 1pt} \arccos (x/{\mathcal{X}}_{0} ),$$

(9.65)

$${\mathcal{G}}_{ > } (x) = x\sqrt {x^{2} - {\mathcal{X}}_{0}^{2} } - (2 - {\mathcal{X}}_{0} ){\kern 1pt} {\text{arccosh}}(x/{\mathcal{X}}_{0} ),$$

(9.66)

$${\mathcal{G}}_{ > } (x) = x\sqrt {x^{2} - {\mathcal{X}}_{0}^{2} } - (2 - {\mathcal{X}}_{0} ){\text{arcsinh}}(x/\sqrt { - {\mathcal{X}}_{0}^{2} } ).$$

(9.67)

Finally, the polarization function is given in the following form:

Real part

$$\begin{aligned} {\text{Re}}\varPi^{(0)} (q,\omega ) & = - \frac{g\mu }{{\hbar^{2} v_{F}^{2} }} + {\mathcal{F}}(q,\omega ) \\ & \quad \times \,\left\{ {\begin{array}{*{20}l} 0 \hfill & \Rightarrow \hfill & {Q_{1} } \hfill \\ {{\mathcal{G}}_{ < } \left( {\frac{{\mathcal{M}}}{{\hbar v_{F} q}}} \right)} \hfill & \Rightarrow \hfill & {Q_{2} } \hfill \\ {{\mathcal{G}}_{ < } \left( {\frac{{\mathcal{P}}}{{\hbar v_{F} q}}} \right) + {\mathcal{G}}_{ < } \left( {\frac{{\mathcal{M}}}{{\hbar v_{F} q}}} \right)} \hfill & \Rightarrow \hfill & {Q_{3} } \hfill \\ {{\mathcal{G}}_{ > } \left( {\frac{{\mathcal{P}}}{{\hbar v_{F} q}}} \right) - {\mathcal{G}}_{ < } \left( {\frac{{\mathcal{P}}}{{\hbar v_{F} q}}} \right)} \hfill & \Rightarrow \hfill & {Q_{4} } \hfill \\ {{\mathcal{G}}_{ > } \left( {\frac{{\mathcal{P}}}{{\hbar v_{F} q}}} \right) - {\mathcal{G}}_{ < } \left( {\frac{{\mathcal{M}}}{{\hbar v_{F} q}}} \right)} \hfill & \Rightarrow \hfill & {\varOmega_{1} } \hfill \\ {{\mathcal{G}}_{ > } \left( {\frac{{\mathcal{P}}}{{\hbar v_{F} q}}} \right)} \hfill & \Rightarrow \hfill & {\varOmega_{2} } \hfill \\ {{\mathcal{G}}_{ > } \left( {\frac{{\mathcal{P}}}{{\hbar v_{F} q}}} \right) - {\mathcal{G}}_{ > } \left( {\frac{{\mathcal{M}}}{{\hbar v_{F} q}}} \right)} \hfill & \Rightarrow \hfill & {\varOmega_{3} } \hfill \\ {{\mathcal{G}}_{ > } \left( {\frac{{\mathcal{M}}}{{\hbar v_{F} q}}} \right) + {\mathcal{G}}_{ > } \left( {\frac{{\mathcal{P}}}{{\hbar v_{F} q}}} \right)} \hfill & \Rightarrow \hfill & {\varOmega_{4} } \hfill \\ {{\mathcal{G}}_{0} \left( {\frac{{\mathcal{P}}}{{\hbar v_{F} q}}} \right) - {\mathcal{G}}_{0} \left( {\frac{{\mathcal{M}}}{{\hbar v_{F} q}}} \right)} \hfill & \Rightarrow \hfill & {\varOmega_{1} } \hfill \\ \end{array} } \right. \\ \end{aligned}$$

(9.68)

Imaginary part

$$\begin{aligned} {\text{Im}}\varPi^{(0)} (q,\omega ) & = - {\mathcal{F}}(q,\omega ) \\ & \quad \times \,\left\{ {\begin{array}{*{20}l} {{\mathcal{G}}_{ > } \left( {\frac{{\mathcal{P}}}{{\hbar v_{F} q}}} \right) - {\mathcal{G}}_{ > } \left( {\frac{{\mathcal{M}}}{{\hbar v_{F} q}}} \right)} \hfill & \Rightarrow \hfill & {Q_{1} } \hfill \\ {{\mathcal{G}}_{ > } \left( {\frac{{\mathcal{P}}}{{\hbar v_{F} q}}} \right)} \hfill & \Rightarrow \hfill & {Q_{2} } \hfill \\ 0 \hfill & \Rightarrow \hfill & {Q_{3} } \hfill \\ 0 \hfill & \Rightarrow \hfill & {Q_{4} } \hfill \\ 0 \hfill & \Rightarrow \hfill & {\varOmega_{1} } \hfill \\ { - {\mathcal{G}}_{ < } \left( {\frac{{\mathcal{M}}}{{\hbar v_{F} q}}} \right)} \hfill & \Rightarrow \hfill & {\varOmega_{2} } \hfill \\ {\pi \left[ {2 - {\mathcal{X}}_{0}^{2} } \right]} \hfill & \Rightarrow \hfill & {\varOmega_{3} } \hfill \\ {\pi \left[ {2 - {\mathcal{X}}_{0}^{2} } \right]} \hfill & \Rightarrow \hfill & {\varOmega_{4} } \hfill \\ 0 \hfill & \Rightarrow \hfill & {\varOmega_{5} } \hfill \\ \end{array} } \right. \\ \end{aligned}$$

(9.69)

where

$${\mathcal{P}} = 2\upmu + \hbar \omega \;{\text{and}}\;{\mathcal{M}} = 2\upmu - \hbar \omega .$$

(9.70)

The analytic expressions provided in the left columns above for the real and imaginary parts of Π⁰ pertain to the frequency wavenumber regions marked by the Q’s and Ω’s in the corresponding right column as indicated in Fig. 9.4. Specifically, these ω-q regions are defined as follows:

$$\begin{array}{*{20}l} {Q_{1} } \hfill & \Rightarrow \hfill & {\hbar \omega < \mu - \sqrt {\left( {\hbar v_{F} } \right)^{2} (q - k_{F}^{\mu } )^{2} + \varDelta^{2} } } \hfill \\ {Q_{2} } \hfill & \Rightarrow \hfill & { \pm \mu \mp \sqrt {\left( {\hbar v_{F} } \right)^{2} (q - k_{F}^{\mu } )^{2} + \varDelta^{2} } < \hbar \omega < } \hfill \\ {} \hfill & {} \hfill & { - \mu + \sqrt {\left( {\hbar v_{F} } \right)^{2} (q + k_{F}^{\mu } )^{2} + \varDelta^{2} } } \hfill \\ {Q_{3} } \hfill & \Rightarrow \hfill & {\hbar \omega < - \mu + \sqrt {\left( {\hbar v_{F} } \right)^{2} (q_{{}} + k_{F}^{\mu } )^{2} + \varDelta^{2} } } \hfill \\ {Q_{4} } \hfill & \Rightarrow \hfill & { - \mu + \sqrt {\left( {\hbar v_{F} } \right)^{2} (q + k_{F}^{\mu } )^{2} + \varDelta^{2} } < \hbar \omega < \hbar v_{F} q,} \hfill \\ \end{array}$$

(9.71)

and

$$\begin{array}{*{20}l} {\varOmega_{1} } \hfill & \Rightarrow \hfill & {q < 2k_{F} \quad \& \quad \sqrt {\left( {\hbar v_{F} } \right)^{2} q^{2} + 4\varDelta^{2} } < \hbar \omega < } \hfill \\ {} \hfill & {} \hfill & { < - \mu + \sqrt {\left( {\hbar v_{F} } \right)^{2} (q - k_{F}^{\mu } )^{2} + \varDelta^{2} } } \hfill \\ {\varOmega_{2} } \hfill & \Rightarrow \hfill & {\mu + \sqrt {\left( {\hbar v_{F} } \right)^{2} (q - k_{F}^{\mu } )^{2} + \varDelta^{2} } < } \hfill \\ {} \hfill & {} \hfill & {\mu + \sqrt {\left( {\hbar v_{F} } \right)^{2} (q - k_{F}^{\mu } )^{2} + \varDelta^{2} } } \hfill \\ {\varOmega_{3} } \hfill & \Rightarrow \hfill & {\hbar \omega > \mu + \sqrt {\left( {\hbar v_{F} } \right)^{2} (q + k_{F}^{\mu } )^{2} + \varDelta^{2} } } \hfill \\ {\varOmega_{4} } \hfill & \Rightarrow \hfill & {q > 2k_{F} \quad \& \quad \sqrt {\left( {\hbar v_{F} } \right)^{2} q^{2} + 4\varDelta^{2} } < \hbar \omega < } \hfill \\ {} \hfill & {} \hfill & { < - \mu + \sqrt {\left( {\hbar v_{F} } \right)^{2} (q - k_{F}^{\mu } )^{2} + \varDelta^{2} } } \hfill \\ {\varOmega_{5} } \hfill & \Rightarrow \hfill & {\hbar v_{F} q < \hbar \omega < \sqrt {\left( {\hbar v_{F} } \right)^{2} q^{2} + 4\varDelta^{2} } .} \hfill \\ \end{array}$$

(9.72)