Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Roland
Jago
,
Raül
Perea-Causin
,
Samuel
Brem
and
Ermin
Malic
*

Chalmers University of Technology, Department of Physics, SE-412 96 Gothenburg, Sweden. E-mail: ermin.malic@chalmers.se

Received
25th February 2019
, Accepted 21st April 2019

First published on 13th May 2019

Temporally and spectrally resolved dynamics of optically excited carriers in graphene has been intensively studied theoretically and experimentally, whereas carrier diffusion in space has attracted much less attention. Understanding the spatio-temporal carrier dynamics is of key importance for optoelectronic applications, where carrier transport phenomena play an important role. In this work, we provide a microscopic access to the time-, momentum-, and space-resolved dynamics of carriers in graphene. We determine the diffusion coefficient to be D ≈ 360 cm^{2} s^{−1} and reveal the impact of carrier–phonon and carrier–carrier scattering on the diffusion process. In particular, we show that phonon-induced scattering across the Dirac cone gives rise to back-diffusion counteracting the spatial broadening of the carrier distribution.

The time- and momentum-resolved carrier dynamics in graphene is meanwhile well understood,

However, a full microscopic view on the spatio-temporal dynamics revealing the interplay between diffusion and momentum- and time-dependent scattering processes is still missing.

Exploiting the density matrix formalism^{19,20} and the Wigner representation,^{21} we provide microscopic insights into the temporally, spectrally, and spatially resolved dynamics of optically excited carriers in graphene including carrier diffusion, carrier–light, carrier–phonon, and carrier–carrier scattering processes on the same microscopic footing, cf. Fig. 1. In particular, we determine the diffusion coefficient and show that the diffusion process can be tuned with experimentally accessible knobs, such as pump fluence, substrate and temperature. Furthermore, we reveal how carrier–phonon scattering counteracts the diffusion through efficient scattering across the Dirac cone resulting in an efficient back-diffusion, cf. Fig. 1(b).

To introduce spatial effects, we transform the occupation probability into the Wigner formalism.^{22,23} Here, we consider fluctuations of the occupation probability and perform the Fourier transformation with respect to the momentum difference q resulting in the Wigner function

(1) |

The carrier dynamics is determined by a many-particle Hamilton operator H, where we take into account the free carrier and phonon contribution H_{0}, the carrier–carrier H_{c–c} and the carrier–phonon H_{c–ph} interaction accounting for Coulomb- (orange arrows) and phonon-induced scattering (green arrows), and the carrier–light coupling H_{c–l} (red arrows) that is treated on a semi-classical level. Details on the contributions of the many-particle Hamilton operator including the calculation of the matrix elements can be found in ref. 1 and 5.

Exploiting the Heisenberg equation of motion, we derive the equation of motion for the carrier fluctuation ρ^{λ}_{k,q}. Taking into account the free-particle Hamilton operator H_{0} leads to with the electronic dispersion ε^{λ}_{k}. To determine an equation for the Wigner function we perform a Fourier transformation resulting in . To simplify this integro-differential equation we expand the Wigner function to the first order . By using r'e^{iqr‘} = −i∇_{q}e^{iqr‘} and shifting the q-derivative to the electron dispersion via partial integration, the r′-integral depends only on the exponential function resulting in δ_{k,q}, whereby the zeroth order of the expansion of the Wigner function vanishes. Finally, the equation of motion for the Wigner function for the free Hamilton operator reads

(2) |

To derive the equations of motion for the Wigner function, the polarization and the phonon number with the full Hamilton operator we make the following assumptions: (i) we consider diffusion processes in the polarization to be small, since the latter quickly decays in momentum space and vanishes directly after the optical excitation.^{5} In contrast, the relaxation of carriers occurs on a picosecond timescale which is comparable to diffusion processes, and therefore the diffusion term can not be neglected in the equation for the Wigner function. (ii) We also neglect the phonon diffusion, since it is expected to be much slower than the electronic diffusion due to the flat phonon dispersion. (iii) We expect scattering processes between different spatial positions to be small compared to the diffusion. Now, using the Heisenberg equation of motion, we derive the full spatio-temporal graphene Bloch equations in second-order Born-Markov approximation

(3) |

(4) |

(5) |

The derived set of equations resemble the semiconductor Bloch equations for spatial homogeneous systems (cf.ref. 1 and 5) up to the additional term , which describes the diffusion of carriers in the direction ∇_{k}ε^{λ}_{k} ∝ e_{k} = k/|k|. As a result, carriers with different sign in momentum move in opposite directions generating locally asymmetric carrier distributions in momentum space and resulting in a local current with the Fermi velocity v_{F}. The sum contains both electrons in the conduction band and holes in the valence band and in a spatially homogeneous system, the mean current vanishes.

To quantify the diffusion and to estimate the diffusion coefficient for graphene, we fit the carrier density with a Gaussian exp(−x^{2}/w^{2}(t)) for every time step. The temporal evolution of the width w(t) is depicted in Fig. 2(d). It is connected to an effective diffusion coefficient D via^{27}w^{2}(t) = w_{0}^{2}+ 4Dt resulting in D ≈ 360 cm^{2} s^{−1} for the investigated graphene sample on a SiC substrate. Our results fit well to the experimentally obtained values^{17} for the diffusion coefficient of D = 250 ± 140 cm^{2} s^{−1}. In Fig. 2(e) we show the influence of pump fluence, substrate and temperature on the diffusion coefficient. We find that the temperature has the largest impact. The underlying processes will be discussed later.

So far we have investigated the diffusion behaviour by taking into account the full carrier dynamics. Now, to understand the fundamental interplay of many-particle scattering and diffusion processes we study separately the impact of different scattering mechanisms on the diffusion process, cf. Fig. 3. We start with the case without any scattering channels just considering the electron–light interaction. After the optical excitation, carriers with positive/negative momenta diffuse in opposite spatial directions according to the diffusion term in eqn (3). After approximately 100 fs the carrier separation becomes visible, as the intial carrier density distribution splits into two pronounced peaks of the same width but with half of the amplitude, cf. Fig. 3(a). Including the carrier–phonon scattering, we observe a strongly reduced spatial broadening of the carrier density and no splitting appears (Fig. 3(a)). Phonon-induced relaxation processes counteract the diffusion via back-scattering across the Dirac cone and the following back-diffusion (cf.Fig. 1). The impact of carrier–phonon scattering will be further microscopically resolved in the next section. Including only the carrier–carrier scattering, the density diffuses with the same speed as in the case without any scattering channels (cf.Fig. 3(c)). This is a consequence of the symmetry of Coulomb matrix elements, which favor parallel scattering.^{28,29} Scattering across the Dirac cone is relatively inefficient and back-scattering is even forbidden. In contrast to the case without scattering, the spatial region between the two peaks contains a non-zero density. This reflects the weak but not vanishing Coulomb scattering processes bringing carriers from one to the other side of the Dirac cone.

To investigate the impact of diffusion in more detail we performed the same calculation twice, but in the second computation we excluded diffusion processes. Illustrating the difference of both calculations, i.e. f_{k}(x) − f_{k}(x)^{nodiff}, we can directly observe the impact of diffusion on carrier–phonon scattering (Fig. 4(c) and (d)). As already discussed in the theory section carriers with positive/negative momentum diffuse in opposite spatial direction. This behaviour is illustrated in Fig. 4(c), where carriers with positive momentum diffuse from x < 0 positions (orange spots) to x > 0 positions (red spots). After 1 ps the carriers have already relaxed to energies close to the Dirac cone and below the optical phonon energy. Consequently, the scattering with acoustic phonons becomes dominant. Due to the flat dispersion of acoustic phonons with respect to the Dirac cones back-scattering across the Dirac cone is preferred, such that carriers with positive momenta are scattered to negative momenta and vice versa (Fig. 1). The inversion of momenta results in a back-diffusion, such that the overall carrier distribution stays bunched in space, cf. Fig. 3(b). The back-diffusion is shown in Fig. 4(d) by the the multiple sign change in the colored regions (red to orange to red).

Fig. 5 Impact of carrier–carrier scattering. The same as Fig. 4, but now only including the carrier–carrier instead of carrier–phonon scattering. Coulomb interaction is most efficient for parallel scattering along the Dirac cone in the momentum space. As a result, carrier–carrier scattering does not efficiently counteract the diffusion of carriers in opposite direction as clearly observed in (b). |

In summary, we provide a microscopic view on the spatio-temporal carrier dynamics in graphene based on the density matrix formalism in Wigner representation. We investigate the interplay of diffusion and many-particle scattering processes after a local optical excitation. In particular, we determine a diffusion coefficient of D ≈ 360 cm^{2} s^{−1} that agrees well with recent experimental values. Furthermore, we reveal that carrier–phonon scattering across the Dirac cone and the resulting back-diffusion are crucial ingredients to understand the spatial broadening of the carrier distribution. The gained insights are important e.g. for graphene-based photodetectors,^{35–39} that are governed by the thermoelectric effect, which relies on spatial temperature gradients.

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