Energy relaxation of hot carriers in graphene via plasmon interactions

Abstract

Energy relaxation of hot carriers in graphene is studied theoretically and experimentally at low temperatures, where the loss rate may differ significantly from that predicted for electron–phonon interactions. We show here that plasmons, important in the relaxation of energetic carriers in bulk semiconductors, can also provide a pathway for energy relaxation in transport experiments in graphene. Reflecting the linear nature of graphene’s bands, we obtain a total loss rate to plasmons that is independent of carrier density. This results in energy relaxation times whose dependence on temperature and density closely matches that reported experimentally.

Appendices

Appendix 1

We note that the emission and absorption of plasmons by hot carriers is not the normal Fermi liquid behavior. If the Fermi sea is completely full at zero Kelvin, then there can be no emission at the Fermi level since the lower energy states are full. We need a spreading of the distribution function at an elevated temperature arising from the hot electron effect, induced by the electric field as shown in Fig. 1. The phase space interaction, illustrating conservation of both energy and momentum can be sketched as in Fig. 5, for both emission and absorption. Here, we show two circles which are the constant energy rings for the states $E(\mathbf{k})$ and $E~(\mathbf{k}\pm \mathbf{q})$. These rings are separated by the plasmon energy, which is q dependent. The red ring is the Fermi circle for the initial state, and the blue ring is the Fermi circle for the final state. The initial and final momenta are, of course, connected vectorially by the momentum of the plasmon, so that momentum conservation, as well as energy conservation, is assured in the interaction. Panel (a) represents the situation for plasmon emission, while (b) represents the situation for plasmon absorption.

Fig. 5

Diagrams for the conservation of momentum and energy in the plasmon interaction for a emission and b absorption

We begin by writing the energy conservation for the emission process in terms of the magnitudes of the various vectors as

$$\begin{aligned} \hbar v_F \left( {\left| \mathbf{k} \right| -\left| {\mathbf{k}-\mathbf{q}} \right| } \right) =\hbar \omega _p =\hbar \sqrt{{\chi _{2}} q} \end{aligned}$$

(13)

where the plasma frequency has been given in the main text by (3). Here, we have introduced the proper dynamic mass for graphene in the last term. We can now expand the terms in the first equation to get

$$\begin{aligned} k-\sqrt{k^{2}+q^{2}+2kq\cos \vartheta }=\frac{\sqrt{{\chi _{2}} q}}{v_F }. \end{aligned}$$

(14)

It is important to note here that the vector properties of Fig. 5a mean that $\cos \vartheta <0$, or that the angle is in the second or third quadrants. We can now rearrange the terms and square the result to get

$$\begin{aligned} k^{2}+\frac{{\chi _{2}} q}{v_F^2 }-2k\frac{\sqrt{{\chi _{2}} q}}{v_F }=k^{2}+q^{2}+2kq\cos \vartheta , \end{aligned}$$

(15)

or

$$\begin{aligned} \cos \vartheta= & {} -\frac{q^{2}-\frac{{\chi _{2}} q}{v_F^2 }+2k\frac{\sqrt{{\chi _{2}} q}}{v_F }}{2kq}\nonumber \\= & {} -\left[ {\frac{q}{2k}-\frac{{\chi _{2}} }{2kv_F^2 }+\frac{1}{v_F }\sqrt{\frac{{\chi _{2}} }{q}}} \right] . \end{aligned}$$

(16)

Thus, we require the term in square brackets to be positive. The solutions are shown in Fig. 6.

Fig. 6

The lines for which the delta function is satisfied for the emission of plasmons at two different values of energy of the carriers, as given by (16)

Now, we turn to the absorption process and write the energy conservation in terms of the magnitudes of the various vectors as

$$\begin{aligned} \hbar v_F \left( {\left| {\mathbf{k}+\mathbf{q}} \right| -\left| \mathbf{k} \right| } \right) =\hbar \omega _p =\hbar \sqrt{{\chi _{2}} q}. \end{aligned}$$

(17)

We can now expand the terms in the first equation to get

$$\begin{aligned} \sqrt{k^{2}+q^{2}+2kq\cos \vartheta }-k=\frac{\sqrt{\chi _2 q}}{v_F }. \end{aligned}$$

(18)

It is important to note here that the vector properties of Fig. 5b mean that $\cos \vartheta >0$, or that the angle is in the first or fourth quadrants. We can now rearrange the terms and square the result to get

$$\begin{aligned} k^{2}+\frac{{\chi _{2}} q}{v_F^2 }+2k\frac{\sqrt{{\chi _{2}} q}}{v_F }=k^{2}+q^{2}+2kq\cos \vartheta , \end{aligned}$$

(19)

or

$$\begin{aligned} \cos \vartheta= & {} \frac{-q^{2}+\frac{{\chi _{2}} q}{v_F^2 }+2k\frac{\sqrt{{\chi _{2}} q}}{v_F }}{2kq}\nonumber \\= & {} \left[ {-\frac{q}{2k}+\frac{{\chi _{2}} }{2kv_F^2 }+\frac{1}{v_F }\sqrt{\frac{{\chi _{2}} }{q}}} \right] . \end{aligned}$$

(20)

Thus, we require the term in square brackets to be positive (Fig. 7).

Fig. 7

The lines for which the delta function is satisfied for the absorption of plasmons at two different values of energy of the carriers, as given by (20)

It is clear that there are solutions to the above equations which allow for both emission and absorption of plasmons. In the emission case, the scattering lies in a very small range of angles whose “cone” is 6$^\circ$–10$^\circ$ for the curves shown. This is around the back-scattered direction and has a momentum almost equal to that of the initial state. Hence, this is almost back scattering, but not quite (we note that pure back-scattering in graphene is forbidden in the equilibrium case by chirality). For absorption, there is a very large range of angles for the forward process, but again it occurs for rather large values of the momentum. In both emission and absorption, the scattering involves rather large values of the plasmon energy. There is almost no density dependence in the range of $10^{11}-10^{12}\,\hbox {cm}^{-2}$, but one would expect the interaction to get stronger nearer to the Dirac point, although the plasmon may be damped strongly in this case by the interband single particle interactions discussed above.

Appendix 2

The integral in (10) may be written as

$$\begin{aligned} I=\int \limits _0^{u_{\max } } {\frac{u^{2}du}{e^{u}-1}} \end{aligned}$$

(21)

where

$$\begin{aligned} u=\frac{\hbar v_F q}{k_B T}, \quad u_{\max } =\frac{2\hbar v_F k_F }{k_B T}. \end{aligned}$$

(22)

The first term arises from the energy conserving delta function that appears in the integration over the frequency. The second term arises from the fact that the maximum value of q that can occur is approximately the span of the Fermi circle in two dimensions, which translates into a maximum value for u. Generally, this maximum is twice the ratio of the Fermi energy to the thermal energy, and can be quite large. In Fig. 8, we plot both the integrand and the value of the integral as a function of u or $u_{\mathrm{max}}$, as appropriate. It may be seen that the peak of the argument occurs at a relatively small value of u, due to the dominance of the exponential in the denominator. The integral approaches its limiting value fairly rapidly. The smallest value of the limit occurs for low density and high temperature. In the experiments [16], the lowest value of density is about $2 \times 10^{11}\,\hbox {cm}^{-2}$ while the highest carrier temperature is about 40 K. These numbers lead to a minimum value of $u_{max}$ of about 2.4 where the integral has already reached more than half its final value. So, except for low densities at the highest input power per electron, the integral can reasonably be assumed to be the limiting value of approximately 2.4.

Fig. 8

The argument and value of the integral (21) as a function of the argument or limit

Appendix 3

The mobility that is measured in the samples as a function of temperature for various densities is shown in Fig. 9. To get to the needed scattering time, one must understand how the various scattering rates are averaged over the scattering angle (See, e.g., [56]). With several scattering mechanisms active in graphene, this is actually rather difficult to unfold. We have considered impurity scattering to be dominant. The actual scattering rate differs from the momentum relaxation time determined from the measured mobility. The actual scattering rate from ionized impurity scattering in graphene can be written as [57]

$$\begin{aligned} \frac{1}{\tau _{imp} }=\frac{N_{imp} }{4\pi \hbar E_k }\left( {\frac{Ze^{2}}{\varepsilon _s }} \right) ^{2}\int \limits _0^\pi {\frac{\left[ {1+\cos (2\vartheta )} \right] e^{-4k\sin (\vartheta )d_s }d\vartheta }{\left[ {\sin (\vartheta )+\frac{q_s }{2k}} \right] ^{2}}}\nonumber \\ \end{aligned}$$

(23)

where $e^{2}k_F /2\pi \varepsilon _s$ is the screening wave vector and $2\theta$ is the angle between the incoming and scattered wave vectors—the elastic scattering occurs with $q = 2k\hbox {cos}( \theta /2)$, and the angle has been doubled in the equation. In the momentum relaxation time, there is an additional factor of $[1-\hbox {cos}(2\theta )]$ in the numerator (See, e.g., [58]). So, the ratio of the momentum relaxation time to the actual scattering time is the ratio of the integrals with, and without, this additional term. This ratio is shown in Fig. 10, with the scattering rates all evaluated at the Fermi surface, appropriate for low temperatures. In graphene, the mobility in suspended graphene is quite high, and can be of the order of several times $10^{5}\,\hbox {cm}^{2}/\hbox {s}$ [59]. The mobility in the experiments discussed in Ref. [16], and used in the manuscript, are much lower and so are assumed to dominated by impurity scattering. Hence, the ratio in Fig. 6 was used to convert the momentum relaxation time into the actual scattering time used in the computation of the energy loss rate per electron.

Fig. 9

The momentum relaxation time as a function of temperature for several values of the carrier density

Fig. 10

The ratio of the momentum relaxation time to the actual scattering time for ionized impurity scattering in graphene