Polarized photoluminescence clocks ultrafast pseudospin relaxation in graphene

TL;DR

This study uses polarization-resolved photoluminescence spectroscopy to measure ultrafast pseudospin relaxation in graphene, revealing a lifetime of approximately 12 femtoseconds, which advances understanding of pseudospin dynamics in 2D materials.

Contribution

The paper introduces a novel experimental method to measure ultrafast pseudospin relaxation in graphene, providing quantitative insights into pseudospin lifetime.

Findings

Pseudospin relaxation time in graphene is approximately 12 fs.

Polarization-resolved photoluminescence effectively probes pseudospin dynamics.

The approach enhances tools for graphene pseudospintronics research.

Abstract

Electronic states in 2D materials can exhibit pseudospin degrees of freedom, which allow for unique carrier-field interaction scenarios. Here, we investigate ultrafast sublattice pseudospin relaxation in graphene by means of polarization-resolved photoluminescence spectroscopy. Comparison with microscopic Boltzmann simulations allows to determine a lifetime of the optically aligned pseudospin distribution of $12\pm 2\,\text{fs}$. This experimental approach extends the toolbox of graphene pseudospintronics, providing novel means to investigate pseudospin dynamics in active devices or under external fields.

Equations

$$R(E)=\frac{I_{\parallel }(E)}{I_{\perp }(E)}$$

$$\frac{\text{d}{f}_{c,\parallel }}{\text{d}t}$$

$$\frac{\text{d}{f}_{c,\perp }}{\text{d}t}$$

$$R(E)=\frac{\left[{\left|⟨q|{\hat{\sigma }}_x|q⟩\right|}^2\right]}{\left[{\left|⟨q|{\hat{\sigma }}_y|q⟩\right|}^2\right]}.$$

Full text

Present-day electronic devices process, store, and transport information based on charge carriers. Recent developments in spintronics Wolf et al. (2001); Žutić et al. (2004); Han et al. (2014) have extended these capabilities by additionally making use of the electronic spin. Moreover, depending on the local environment, carriers can be equipped with additional pseudospin degrees of freedom, including sublattice, valley, and layer pseudospin, which may be exploited in future information technology. These angular momentum components exhibit rich physical phenomena, not unlike the ones observed for the intrinsic spin of electrons Mecklenburg and Regan (2011); Song et al. (2015); Xu et al. (2014); Schaibley et al. (2016).

For example, the valley pseudospin is exploited in valleytronics Schaibley et al. (2016) by manipulating the occupation of degenerate but inequivalent chiral electron states. In particular, in hexagonal 2D materials such as transition metal dichalcogenides (TMDCs), selection rules enable a direct optical manipulation of the carrier pseudospin Cao et al. (2012); Zeng et al. (2012); Xu et al. (2014); Mak et al. (2014); Schaibley et al. (2016). Manifestations of valley pseudospin polarization in TMDCs include the valley-spin analogue of the spin Hall effect Xiao et al. (2012); Mak et al. (2014), and the valley Zeeman effect Srivastava et al. (2015).

Even single-layer graphene as the most simple 2D material displays valley and sublattice pseudospin Geim and Novoselov (2007). While, similar to the case of TMDCs, the valley pseudospin distinguishes the occupation within the Dirac cones at the $K$ and $K^{\prime}$ points, the sublattice pseudospin controls the relative phase of the electron wave function on the two hexagonal sublattices Pesin and MacDonald (2012). In carrier momentum space, sublattice pseudospin is collinear with the carrier momentum relative to the Dirac point Geim and Novoselov (2007); Mecklenburg and Regan (2011). Sublattice pseudospin conservation enables Klein tunneling in graphene, i. e. the counter-intuitive carrier transmission through infinitely high potential barriers Klein (1929); Katsnelson et al. (2006); Young and Kim (2009).

In graphene, coupling between carriers and optical fields Nair et al. (2008) is governed by sublattice pseudospin selection rules Mecklenburg et al. (2010), so that the initial carrier populations created by linearly polarized interband excitation exhibit a pronounced angularly asymmetric population within the Dirac cones Malic et al. (2011). However, intrinsic carrier scattering mechanisms are expected to rapidly destroy the optically imprinted pseudospin alignment. Ultrafast pseudospin relaxation has been addressed in a series of polarization-resolved transient optical spectroscopy experiments Chen et al. (2014); Mittendorff et al. (2014); Yan et al. (2014); Trushin et al. (2015); Yao et al. (2015a, b); König-Otto et al. (2016); Yan et al. (2017), elucidating the role of carrier momentum isotropization in carrier thermalization and cooling processes Dawlaty et al. (2008); George et al. (2008); Sun et al. (2008); Newson et al. (2009); Breusing et al. (2011); Winnerl et al. (2011); Tani et al. (2012); Brida et al. (2013); Malard et al. (2013); Winnerl et al. (2013). Although it is challenging to observe the fastest relaxation dynamics, most of the previous investigations suggest that sublattice pseudospin relaxation is complete within $50\,\text{fs}$ to $150\,\text{fs}$ Mittendorff et al. (2014); Yan et al. (2014); Trushin et al. (2015). Theoretical models predict even faster dynamics Malic et al. (2012), and a precise relation between energetic and momentum relaxation timescales has not been established experimentally, yet.

Here, we present evidence for a transient sublattice pseudospin state in graphene, obtained by analyzing the polarization-resolved photoluminescence (PL) from optically induced non-equilibrium carrier populations Lui et al. (2010); Liu et al. (2010); Stöhr et al. (2010); Koyama et al. (2013). We demonstrate that the initial pseudospin alignment is lost within $12\pm 2\,\text{fs}$ by comparing the experimentally observed optical polarization degree of the PL with microscopic simulations of Boltzmann rate equations. The results highlight the importance of efficient pseudospin relaxation in graphene, proceeding equally fast as energetic carrier thermalization, and an order of magnitude faster than the excitation of strongly coupled optical phonons Kampfrath et al. (2005).

In our experiments, single-layer graphene on a sapphire substrate is optically excited using ultrashort laser pulses ($18\,\text{fs}$ pulse duration, $1.55\,\text{eV}$ photon energy, $80\,\text{MHz}$ repetition rate). The resulting blue-shifted PL is detected in a polarization-resolved manner by a grating spectrometer and a liquid nitrogen-cooled charge-coupled device (CCD). At the excitation conditions employed, no optically induced sample damage was observed. Special care was taken to ensure accurate calibration of the spectral and polarization responses of the detection system (see Supplemental Material 111See Supplemental Material at [url], which includes additional Refs. Li et al., 2009; Gulde et al., 2014; Mattevi et al., 2011; Wang et al., 2012; Ferrari et al., 2006; Malard et al., 2009; Ziman, 1960; Saito et al., 1998; Grüneis et al., 2003; Fermi, 1953; Mošková and Moško, 1994, for a detailed description of the experimental setup, of the sample preparation procedure, and of the theoretical fundamentals of the microscopic simulations.).

Figure 1a schematically depicts the experimental setup and the microscopic processes underlying PL emission. Ultrashort, linearly polarized optical pump pulses excite carriers from the valence into the conduction band of graphene (red arrow). Initially, the non-equilibrium carrier distribution is strongly anisotropic within each Dirac cone of the graphene band structure Malic et al. (2012). During relaxation, carrier-carrier and carrier-phonon scattering cause energetic broadening of the carrier distribution and a loss of momentum anisotropy (black arrows). The emerging high-energy tail of the carrier distribution gives rise to a blue-shifted component of the PL by electron-hole recombination (blue arrow), as previously observed Lui et al. (2010); Liu et al. (2010); Stöhr et al. (2010), with recombining carriers at an energy $\pm\varepsilon$ emitting a PL photon of energy $E=2|\varepsilon|$. The carrier and phonon systems jointly thermalize due to an efficient energy transfer between carriers and a set of high-energy, strongly coupled optical phonons (SCOPs) on a hundred femtosecond timescale Kampfrath et al. (2005); Breusing et al. (2011). The SCOPs in turn decay on a timescale on the order of a picosecond Bonini et al. (2007); Kang et al. (2010); Schäfer et al. (2011); Chatelain et al. (2014).

Energetic relaxation and azimuthal momentum randomization of the broadened carrier distribution can be characterized by two phenomenological relaxation times $\tau_{\text{rel}}$ and $\tau_{\text{iso}}$, respectively (see inset in Fig. 1a). Our experiments yield access to momentum and energy relaxation rates, as the degree of PL polarization is governed by the ratio of these time constants. Specifically, in the limiting case of $\tau_{\text{rel}}\gg\tau_{\text{iso}}$, completely unpolarized PL emission is expected, whereas $\tau_{\text{rel}}\lesssim\tau_{\text{iso}}$ would result in a PL preferentially polarized parallel to the pump.

Representative PL spectra obtained in our experiments are depicted in Fig. 1b. The PL intensity decreases monotonously over the observed photon energy range in a spectral shape that was previously described by black-body radiation Lui et al. (2010). Most importantly, the intensity of the PL polarized parallel to the pump (indicated by $\parallel$) is up to $20\,\text{\%}$ higher than the intensity of the perpendicular PL component (indicated by $\perp$), evidencing considerable azimuthal carrier anisotropy within the PL lifetime. The agreement of the spectra for both horizontally (blue curves) and vertically (red curves) polarized pump beams demonstrates a proper polarization-dependent calibration of the detection. In order to quantify the polarization properties of the PL, we employ a polarization ratio

[TABLE]

of the temporally integrated PL flux, $I_{\parallel}$ and $I_{\perp}$, polarized parallel and perpendicular to the pump, respectively, as a function of the photon energy $E$. Unpolarized PL emission corresponds to $R=1$, while fully polarized PL emission is represented by $R\rightarrow\infty$.

To gain a first phenomenological description of the expected PL polarization, we describe the azimuthal and energetic carrier relaxation in a simplified model by two coupled, exponentially decaying components $f_{c,\parallel}$ and $f_{c,\perp}$ of the relevant high-energy carrier population in the conduction band:

[TABLE]

In the low-energy optical regime and at a negligible doping level, the carrier occupation in the valence band is given by $f_{v}=1-f_{c}$ at any time. For both components, we adopt an azimuthal shape following the angular dependence of the off-axis carrier-field coupling matrix elements (for linear polarization) within the Dirac cones, i. e. the total carrier distribution is given by $f_{c}(\varphi,t)=f_{c,\parallel}(t)\,\sin^{2}\varphi+f_{c,\perp}(t)\,\cos^{2}\varphi$ (see inset in Fig. 1a for the definition of $\varphi$) Malic et al. (2011). Thereby, considering the limiting case of optical excitation pulses much shorter than the relevant relaxation times, the perpendicular component $f_{c,\perp}$ of the carrier distribution is initially zero, and is only populated subsequently due to the azimuthal momentum relaxation term.

Taking into account the azimuthal momentum distribution of both components, the temporally integrated PL flux and polarization ratio are deduced from the evolution of the carrier distribution Mecklenburg et al. (2010); Winzer et al. (2015) (see Supplemental Material Note (1)). In this model, the polarization ratio is only a function of $\tau_{\text{rel}}/\tau_{\text{iso}}$, and an upper limit of the polarization ratio is given by $R=5$ (as shown in Fig. 1c), as already the initial $f_{c,\parallel}$ population yields a small perpendicularly polarized PL flux. The experimentally observed maximum polarization ratio of $R\approx 1.25$ (cf. Fig. 1b) is obtained at $\tau_{\text{rel}}/\tau_{\text{iso}}\approx 8$.

In order to quantitatively analyze the experimental data, we combine the rate equations introduced above with a microscopic carrier scattering model. Specifically, in Eqs. 2 and 3, we consider energy-dependent carrier distribution functions $f_{c,\parallel}(\varepsilon,t)$ and $f_{c,\perp}(\varepsilon,t)$, and replace the phenomenological exponential decay (described by $\tau_{\text{rel}}$) by Boltzmann rate equations, which include optical pulse absorption Mecklenburg et al. (2010); Malic et al. (2011), Auger and non-Auger carrier-carrier scattering Li et al. (2010); Kim et al. (2011); Rana (2007), as well as carrier-phonon scattering with the relevant optical phonon branches Kampfrath et al. (2005); Butscher et al. (2007); Tse and Das Sarma (2009); Kang et al. (2010). Both azimuthal populations are assumed to possess half of the angularly integrated density of states $D_{\parallel}(\varepsilon)=D_{\perp}(\varepsilon)=D(\varepsilon)/2=|\varepsilon|/\pi\hbar^{2}v_{F}^{2}$, where $v_{F}$ is the Fermi velocity Castro Neto et al. (2009). Particularly, this approach not only provides a realistic description of carrier relaxation, but also gives direct access to the spectral shape of the PL emission. Different from conventional band gap emission in semiconductors, the change of the carrier distribution due to photon emission is negligible in the semimetal graphene and therefore not taken into account. (See Supplemental Material for a detailed description of the numerical approach Note (1).)

It should be noted that the incoherent PL due to radiative carrier-carrier recombination described in the microscopic modeling may be accompanied by a recently proposed broadband coherent contribution Winzer et al. (2015), which would be inherently polarized. However, we experimentally verified that the PL detected in our experiment is a fully incoherent emission originating from a non-equilibrium carrier population, and off-diagonal elements of the carrier density matrix can be neglected Gaida et al. .

Results of the microscopic simulations are depicted in Fig. 2a (solid lines) in comparison with experimental PL spectra (data points) for three exemplary absorbed fluences. In the simulation, the spectral shape is well reproduced by adopting absorbed fluences of $0.83\,\text{\textmu J}/\text{cm}^{2}$ to $1.20\,\text{\textmu J}/\text{cm}^{2}$ for the experimental incident fluence range of $82\,\text{\textmu J}/\text{cm}^{2}$ to $117\,\text{\textmu J}/\text{cm}^{2}$. Within the experimental uncertainties, the corresponding graphene absorbance agrees with reported values in the literature Nair et al. (2008); Merthe and Kresin (2016). The simulated PL spectral flux for an absorbed fluence of $1.20\,\text{\textmu J}/\text{cm}^{2}$ results in a quantum efficiency of $3\times 10^{-10}$, close to the experimental value of ${\sim}10^{-10}$, which was derived from the PL intensity by taking into account an estimated total detection efficiency of $1\,\text{\%}$ to $2\,\text{\%}$ in the considered photon energy range ($2.05\,\text{eV}$ to $2.85\,\text{eV}$).

Figure 2b shows a comparison of experimental data for the polarization ratio with simulated curves for a set of isotropization times, each averaged over the experimentally accessible photon energy interval. The time-integrated polarization ratio extracted from the numerical simulations increases with decreasing absorbed fluence and increasing isotropization time. Considering the experimentally observed polarization ratio, we obtain a fluence- and photon energy-averaged value of $\tau_{\text{iso}}=12\pm 2\,\text{fs}$. For comparison, in Ref. Malic et al., 2012, the authors predict the formation of an isotropic electron distribution within $50\,\text{fs}$ at comparable fluences, while first indications of momentum relaxation are observed after $10\,\text{fs}$. Notably, at optical excitation energies in the meV range, the lifetime of the anisotropic carrier distribution exceeds $1\,\text{ps}$ König-Otto et al. (2016).

For comparison with our phenomenological model, we also extract the effective PL lifetime $\tau_{\text{\tiny PL}}$ from the simulated time-resolved PL emission in a photon energy- and fluence-resolved manner using a mono-exponential model (see Fig. 2c). It should be noted that $\tau_{\text{\tiny PL}}$ is dominated by the energetic carrier relaxation timescale $\tau_{\text{rel}}$, and does not correspond to the intrinsic radiative lifetime of excited carriers in graphene (which is typically on the order of picoseconds Rana (2007); Limmer et al. (2013)). Specifically, as the carrier occupation enters quadratically into the PL flux, $\tau_{\text{rel}}$ is approximately twice as large as $\tau_{\text{\tiny PL}}$.

Values of $\tau_{\text{\tiny PL}}$ obtained from the fits in Fig. 2c are depicted in the inset, exhibiting slower PL decay at low photon energies and for increasing excitation levels, respectively. For the experimental parameters, the estimated $\tau_{\text{\tiny PL}}$ ranges from $21\,\text{fs}$ to $32\,\text{fs}$, corresponding to an energetic relaxation time constant $\tau_{\text{rel}}$ between $42\,\text{fs}$ and $64\,\text{fs}$. Thus, we obtain a ratio $\tau_{\text{rel}}/\tau_{\text{iso}}$ of 3.5 to 5.3, which is in reasonable agreement with the factor of 8 obtained from the simplified, phenomenological model (cf. Fig. 1c). Furthermore, the fluence dependence of $\tau_{\text{\tiny PL}}$ qualitatively accounts for the experimentally observed PL polarization decrease at high fluences, as, in this case, the PL emission contains larger contributions from the isotropic carrier distribution at later times. Besides, from the photon energy dependence of $\tau_{\text{\tiny PL}}$, a pronounced change of the polarization ratio across the PL spectrum is expected, which, however, is not observed experimentally.

To obtain further insights into the temporal evolution of graphene excitation underlying anisotropic PL emission, Fig. 3a displays the numerically obtained carrier density map $D_{\parallel}\,f_{c,\parallel}$ of the parallel component of the carrier population within the first picosecond after optical excitation. During pulse absorption, a pronounced occupation builds up at half the pump photon energy (indicated by arrow). Carrier-carrier scattering results in a fast fading of the initial occupation peak and a pronounced broadening of the carrier distribution within the first ten femtoseconds. The emerging high-energy tail is the source of the experimentally observed PL. Slower carrier-phonon-mediated carrier cooling on a $100\,\text{fs}$ timescale Kampfrath et al. (2005); Butscher et al. (2007) quenches PL emission. Figure 3b depicts the difference in carrier densities between the parallel and perpendicular components, illustrating the filling of $D_{\perp}\,f_{c,\perp}$ due to equienergetic azimuthal scattering. Notably, both energetic broadening and momentum isotropization take place on a timescale comparable to the duration of the pump pulses. Specifically, the transient carrier density difference promptly increases at early delay times, and the carrier anisotropy has almost vanished again after $25\,\text{fs}$.

Finally, we want to point out the connection between the experimentally observed PL polarization ratio $R$ and the quantum mechanical pseudospin degrees of freedom in graphene. Considering graphene’s optical matrix elements Mecklenburg et al. (2010), one arrives at the following connection between $R$ and the expectation values of the sublattice pseudospin in $q_{x}$ and $q_{y}$ directions:

[TABLE]

Here, $\hat{\sigma}_{x}$ and $\hat{\sigma}_{y}$ are the pseudospin operators and $\left|q\right>$ is the electron state with momentum $\bm{q}$ (relative to the $K$ and $K^{\prime}$ point). $[\cdot]$ denotes the temporal average over the occupied conduction electron states at energy $\varepsilon(\bm{q})=E/2$, i. e. $[\cdot]=\int\text{d}t\sum_{\bm{q}}\,f_{c}^{2}\,\delta(\varepsilon-E/2)$. Thereby, the degree of polarization in the PL spectrum in graphene gives a direct fingerprint of the non-equilibrium pseudospin alignment on ultrashort timescales.

In summary, we presented polarization-resolved PL spectroscopy as a robust tool to directly access anisotropic carrier distributions in graphene during the first $10\,\text{fs}$ to $20\,\text{fs}$ after ultrashort optical excitation. In combination with a microscopic Boltzmann simulation, we extract a characteristic isotropization timescale of $12\pm 2\,\text{fs}$, resulting in a rapid loss of initial pseudospin alignment imprinted by the optical field.