Plasma waves in graphene

As in conventional semiconductors, in graphene, an electron-hole gas can be considered as a plasma , and, accordingly, the question arises of which waves can be observed in a solid. Due to the difference between the dispersion law and the parabolic law, it is expected that the properties of the waves will be different. Plasma waves in 2DEG in graphene were theoretically considered in ^[1] .

Conclusion

The kinetic equation for electrons in graphene in the collisionless approximation can be written as

{\frac {\partial f}{\partial t}}+\mathbf {v} _{p}{\frac {\partial f}{\partial \mathbf {r} }}+e{\frac {\partial \phi }{\partial \mathbf {r} }}{\frac {\partial f}{\partial \mathbf {p} }}=0.\qquad (4.1)

{\ displaystyle {\ frac {\ partial f} {\ partial t}} + \ mathbf {v} _ {p} {\ frac {\ partial f} {\ partial \ mathbf {r}}} + e {\ frac {\ partial \ phi} {\ partial \ mathbf {r}}} {\ frac {\ partial f} {\ partial \ mathbf {p}}} = 0. \ qquad (4.1)}

Here is the electron distribution function $f=f(\mathbf {r} ,\mathbf {p} ,t)$ ${\ displaystyle f = f (\ mathbf {r}, \ mathbf {p}, t)}$ Depends on coordinates, impulses and time. $\phi =\phi (\mathbf {r} ,t)$ ${\ displaystyle \ phi = \ phi (\ mathbf {r}, t)}$ - potential created by DEG. Since graphene is a two-dimensional system, the momentum vector has only two coordinates $\mathbf {p} =(p_{x},p_{y})$ ${\ displaystyle \ mathbf {p} = (p_ {x}, p_ {y})}$ . The electron velocity is also given by the formula $\mathbf {v} _{\mathbf {p} }=v_{F}{\frac {\mathbf {p} }{p}}$ ${\ displaystyle \ mathbf {v} _ {\ mathbf {p}} = v_ {F} {\ frac {\ mathbf {p}} {p}}}$ where $p=|\mathbf {p} |$ ${\ displaystyle p = | \ mathbf {p} |}$ .

The Poisson equation , which relates the concentration and distribution of potential in graphene, can be reduced to the equation

{\frac {V_{g}-\phi }{W_{g}}}={\frac {4\pi e}{\varepsilon }}\Sigma ,\qquad (4.2)

{\ displaystyle {\ frac {V_ {g} - \ phi} {W_ {g}}} = {\ frac {4 \ pi e} {\ varepsilon}} \ Sigma, \ qquad (4.2)}

Where $V_{g}$ ${\ displaystyle V_ {g}}$ - the applied voltage at the gate, which can control the concentration, $W_{g}$ ${\ displaystyle W_ {g}}$ The thickness of the dielectric with permittivity $\varepsilon$ ${\ displaystyle \ varepsilon}$ , and the electron concentration $\Sigma$ ${\ displaystyle \ Sigma}$ is given by the formula

\Sigma ={\frac {g_{s}g_{v}}{(2\pi \hbar )^{2}}}\int {d^{2}\mathbf {p} f},\qquad (4.3)

{\ displaystyle \ Sigma = {\ frac {g_ {s} g_ {v}} {(2 \ pi \ hbar) ^ {2}}} \ int {d ^ {2} \ mathbf {p} f}, \ qquad (4.3)}

which is similar to expression (3.3).

The joint solution of equations (4.1) and (4.2) in the form of plane ones gives an answer to the question of plasma waves in graphene.

The solution to equation (4.1) is sought in the form

f(\mathbf {r} ,\mathbf {p} ,t)=f_{0}+\delta f(p)e^{i(kx-\omega t)},\qquad (4.4)

{\ displaystyle f (\ mathbf {r}, \ mathbf {p}, t) = f_ {0} + \ delta f (p) e ^ {i (kx- \ omega t)}, \ qquad (4.4)}

where a small correction in the form of a plane wave is added to the equilibrium distribution function ( Fermi - Dirac distribution ) $|\delta f|\ll f_{0}$ ${\ displaystyle | \ delta f | \ ll f_ {0}}$ ) The potential is also a small perturbation (compared to $V_{g}$ ${\ displaystyle V_ {g}}$ )

\phi (\mathbf {r} ,t)=\delta \phi e^{i(kx-\omega t)}.\qquad (4.5)

{\ displaystyle \ phi (\ mathbf {r}, t) = \ delta \ phi e ^ {i (kx- \ omega t)}. \ qquad (4.5)}

Substituting solutions of (4.4) and (4.5) into (4.1) and (4.2), we arrive at the equations on $\delta f(p)$ ${\ displaystyle \ delta f (p)}$ and $\delta \phi$ ${\ displaystyle \ delta \ phi}$ up to the first order of smallness

\left(kv_{F}{\frac {p_{x}}{p}}-\omega \right)\delta f=-ek{\frac {\partial f_{0}}{\partial p_{x}}}\delta \phi ,\qquad (4.6)

{\ displaystyle \ left (kv_ {F} {\ frac {p_ {x}} {p}} - \ omega \ right) \ delta f = -ek {\ frac {\ partial f_ {0}} {\ partial p_ {x}}} \ delta \ phi, \ qquad (4.6)}

\delta \phi =-{\frac {2eW_{g}}{\pi \varepsilon \hbar ^{2}}}\int {d^{2}\mathbf {p} f}.\qquad (4.7)

{\ displaystyle \ delta \ phi = - {\ frac {2eW_ {g}} {\ pi \ varepsilon \ hbar ^ {2}}} \ int {d ^ {2} \ mathbf {p} f}. \ qquad ( 4.7)}

These equations are easily solved if the electron gas is degenerate, i.e. $k_{B}T\ll E_{F}$ ${\ displaystyle k_ {B} T \ ll E_ {F}}$ . For $\omega >v_{F}k$ ${\ displaystyle \ omega> v_ {F} k}$ we obtain a linear dispersion relation for plasma waves in graphene

\omega ={\frac {kv_{F}}{\sqrt {1-\left({\frac {\alpha }{\alpha +1}}\right)^{2}}}}=ks,\qquad (4.7)

{\ displaystyle \ omega = {\ frac {kv_ {F}} {\ sqrt {1- \ left ({\ frac {\ alpha} {\ alpha +1}} \ right) ^ {2}}}} = ks , \ qquad (4.7)}

Where

\alpha ={\sqrt {\frac {4g_{s}g_{v}e^{3}W_{g}V_{g}}{\varepsilon \hbar ^{2}v_{F}^{2}}}}.\qquad (4.8)

{\ displaystyle \ alpha = {\ sqrt {\ frac {4g_ {s} g_ {v} e ^ {3} W_ {g} V_ {g}} {\ varepsilon \ hbar ^ {2} v_ {F} ^ { 2}}}}. \ Qquad (4.8)}

.

Phase and group velocities are equal

s={\frac {v_{F}}{\sqrt {1-\left({\frac {\alpha }{\alpha +1}}\right)^{2}}}}.\qquad (4.9)

{\ displaystyle s = {\ frac {v_ {F}} {\ sqrt {1- \ left ({\ frac {\ alpha} {\ alpha +1}} \ right) ^ {2}}}}. \ qquad (4.9)}

The inclusion of finite temperatures and, accordingly, thermally excited holes was considered in ^[2] .

Links

↑ Ryzhii V. "Terahertz plasma waves in gated graphene heterostructures" Jpn. J. Appl. Phys. 45 , L923 (2006) DOI : 10.1143 / JJAP.45.L923
↑ Ryzhii V. et al. "Plasma waves in two-dimensional electron-hole system in gated graphene heterostructures" J. Appl. Phys. 101 , 024509 (2007) DOI : 10.1063 / 1.2426904

[1] Ryzhii V. "Terahertz plasma waves in gated graphene heterostructures" Jpn. J. Appl. Phys. 45 , L923 (2006) DOI : 10.1143 / JJAP.45.L923

[2] Ryzhii V. et al. "Plasma waves in two-dimensional electron-hole system in gated graphene heterostructures" J. Appl. Phys. 101 , 024509 (2007) DOI : 10.1063 / 1.2426904

Plasma waves in graphene

Conclusion

See also

Links

More articles: