Scattering approach to frequency-dependent current noise in Fabry-Pérot graphene devices

We study finite-frequency quantum noise and photon-assisted electron transport through a wide and ballistic graphene sheet between two metallic leads. The elementary excitations allow us to examine the differences between effects related to Fabry-Pérot-like interferences and signatures caused by correlations of coherently scattered particles in electron- and holelike parts of the Dirac spectrum. We identify different features in the current-current auto- and cross-correlation spectra and trace them back to the interference patterns of the product of transmission and reflection amplitudes, which define the integrands of the involved correlators. At positive frequencies, the correlator of the autoterminal noise spectrum with final and initial states associated to the measurement terminal is dominant. Phase jumps occur within the interference patterns of corresponding integrands, which also reveal the intrinsic energy scale of the two-terminal graphene setup. The excess noise spectra, as well as the cross-correlation ones, show large fluctuations between positive and negative values. Oscillatory signatures of the cross-correlation noise are due to an alternating behavior of the integrands.

Figure 1

Wide graphene strip ($W\gg L$) between two heavily doped, metallic graphene leads. The Fermi level of the sheet can be tuned by modifying the electrochemical potential $V_g$ in the graphene sheet, i.e., via a gate voltage. Electron and hole states are injected via the time-dependent bias-voltages in left and right leads ${\mu }_{L/R}\left(t\right)$.

Figure 2

Transmission probability $T_q\left(\epsilon \right)={\left|t_q\left(\epsilon \right)\right|}^2$ as a function of energy and transverse momentum q.

Figure 3

Conductivity $\sigma (\omega ,\alpha )=(L/W)G(\omega ,\alpha )$ as a function of dc voltage applied across the two-terminal setup. We show curves for various ac-driving strengths $\alpha$ applied to (a) both reservoirs ($a=0$) and (b) to the left reservoir only ($a=1$) with $\omega L/v_f=5$.

Figure 4

Schematic view of the different regions, which occur in the integrands of the correlators contributing to the finite-frequency shot noise spectrum.

Figure 5

Real parts of integrands of the four correlators [see Eq. (24)] contributing to the shot noise, namely, (a) $0.25|1-r_q^{*}\left(\epsilon \right)r_q{(\epsilon +\hslash \Omega )|}^2$, (b) $T_q\left(\epsilon \right)T_q(\epsilon +\hslash \Omega )$, (c) $R_q\left(\epsilon \right)T_q(\epsilon +\hslash \Omega )$, and (d) $R_q(\epsilon +\hslash \Omega )T_q\left(\epsilon \right)$. Here, the energy is fixed at $\epsilon =0$ corresponding to vanishing dc bias. The correlator in (a) cannot be written in terms of probabilities, except in the zero-frequency limit the integrand results in $T_q^2\left(\epsilon \right)$. Correlator (b) contains one transmission probability at zero energy that is only nonzero at small $q$. Since for small transversal momentum $R_q\left(\epsilon \right)$ decays as $q^{-2}$, the correlator (c) tends to zero in this regime and otherwise mimics the behavior of $T_q\left(\epsilon \right)$. Integrand (d) is also restricted to low transverse momentum because $T_q\left(\epsilon \right)=0$ otherwise.

Figure 6

Real parts of integrands of correlators [see Eq. (24), see also Fig. 5] contributing to the shot noise for fixed energy $\epsilon L/\hslash v_F=20$. At finite $\epsilon$, there is an additional interference pattern along $q$ if $\hslash v_F\left|q\right|<\left|\epsilon \right|$, leading to phase jumps in the integrand of correlator (a), the one where initial and final states belong to the measurement terminal $L$. When the integrand can be written as a product of probabilities, see (b)–(d), the phase jumps are absent but two independent interference patterns are found.

Figure 7

Real parts of integrands of the correlators [see Eq. (24), see also Fig. 5] contributing to the shot noise with fixed frequency $\Omega L/v_F=20$. Analogous to Fig. 6 but as a function of $(q,\epsilon )$. Phase jumps occur in the interval $-\hslash \Omega <\epsilon <0$ in integrand (a), region III${}_b$ of Fig. 4a. The interplay of the two interference patterns can also be observed at larger energies and transverse momenta for $\hslash v_F\left|q\right|<\left|\epsilon \right|$, $\hslash v_F\left|q\right|<|\epsilon +\hslash \Omega |$ in all integrands (a)–(d).

Figure 8

Real parts of integrands that appear in Eq. (27) contributing to the cross-correlation shot noise for fixed energy $\epsilon L/\hslash v_F=20$ (top) and fixed frequency $\Omega L/v_F=20$ (bottom), namely, (a) and (c) $\Re \left\{t_q^{*}(\epsilon +\hslash \Omega )t_q\left(\epsilon \right)[r_q^{*}(\epsilon +\hslash \Omega )r_q\left(\epsilon \right)-1]\right\}$ and (b) and (d) $4\Re \left[r_q^{*}\left(\epsilon \right)t_q(\epsilon +\hslash \Omega )r_q^{*}(\epsilon +\hslash \Omega )r_q\left(\epsilon \right)\right]$. Due to symmetry reasons, the integrands are identical when interchanging index labels $L,R$. As a function of frequency integrand (a) leads to strongly oscillating contributions to the noise spectrum. These oscillations are reduced due to the alternating behavior along $q$ in cross-terminal contributions (b). In (c) and (d), the integrands are plotted as a function of $(q,\epsilon )$ where they reveal a similar structural difference.

Figure 9

Real parts of integrands appearing in the correlators of Eq. (A1) contributing to the shot noise for driving frequency $\omega L/v_F=7.5$ and fixed frequency $\Omega L/v_F=20$. The integrands are (a) $0.25[1-r_q^{*}\left(\epsilon \right)r_q(\epsilon +\hslash \Omega )][1-r_q^{*}(\epsilon +\hslash \Omega +\hslash \omega )r_q(\epsilon +\hslash \omega )]$, (b) $t_q^{*}\left(\epsilon \right)t_q(\epsilon +\hslash \Omega )t_q^{*}(\epsilon +\hslash \Omega +\hslash \omega )t_q(\epsilon +\hslash \omega )$, (c) $r_q^{*}\left(\epsilon \right)t_q(\epsilon +\hslash \Omega )t_q^{*}(\epsilon +\hslash \Omega +\hslash \omega )r_q(\epsilon +\hslash \omega )$, and (d) $t_q^{*}\left(\epsilon \right)r_q(\epsilon +\hslash \Omega )r_q^{*}(\epsilon +\hslash \Omega +\hslash \omega )t_q(\epsilon +\hslash \omega )$. When two frequencies are present none of the correlators can be written in terms of probabilities and additional phase jumps come into play.

Figure 10

Real parts of integrands of the correlators contributing to the shot-noise with driving frequency $\omega L/v_F=7.5$ and fixed energy $\epsilon L/\hslash v_F=20$. Analogous to Fig. 9, but as a function of $(q,\Omega$). As a consequence of PAT, horizontal interference lines occur for transverse momenta $\hslash v_F\left|q\right|<|\epsilon +m\hslash \omega |$ as in (a) and (c).

Figure 11

Real parts of autocorrelation noise spectrum in units of $2\pi \hslash /e^2$. We compare a setup where dc-bias voltages are fixed symmetrically around the Dirac point (top,) with the case when $e{V}_{0}L/\hslash v_F=2eV/\hslash v_F=10$ (bottom). Thick lines are shot noise and correlators. Thin lines are derivatives with respect to frequency. Contributions from $C_{L\to L}\left(\Omega \right)$ are dominant at positive frequencies. (Top) Special features in the derivatives are seen for frequencies $\hslash \Omega <eV$ in the $R\to R$ contribution, when the lower bound of the energy-integration interval approaches the Dirac point (compare to Figs. 6 and 7). (Bottom) The distance to the Dirac point is increased by the offset voltage. Therefore oscillatory features appear in a larger frequency interval and in all four correlators, since integration boundaries in all contributions are crossing the Dirac point with increasing $\Omega$.

Figure 12

Real parts of autocorrelation excess-noise spectrum in units of $2\pi \hslash /e^2$ (thick lines, top) and derivatives (thin lines, bottom) with dc bias symmetrically applied around the Dirac point (top) and for finite $e{V}_0=2eV$ (bottom). By subtracting the noise at zero dc bias, the divergent background is removed. The structure and especially the oscillatory behavior are coined by autoterminal contributions of Eq.(27a) related to the measurement terminal $L$. The jump in the derivative of $C_{L\to R}\left(\Omega \right)$ is present because this correlator does not contribute for frequencies $\hslash \Omega <eV$. By applying an offset voltage $e{V}_{0}L/\hslash v_F=10$ a complicated structure emerges, best visible in the derivatives.

Figure 13

Real parts of autocorrelation current-current fluctuations in units of $2\pi \hslash /e^2$ as a function of dc bias for fixed frequency $\hslash \Omega$ (thick lines, top). We compare the symmetric setup without dc-bias offset (top) and when $e{V}_{0}L/\hslash v_F=10$ (bottom). Thin lines (lower panels) are used for the derivatives with respect to voltage. (Top) Due to symmetrically applied bias voltage, the noise and the autoterminal contributions are symmetric in the voltage dependence and ${\left.S_{\alpha \to \beta }\left(\Omega \right)\right|}_V={\left.S_{\beta \to \alpha }\left(\Omega \right)\right|}_{-V}$ if $\alpha \ne \beta$. (Bottom) By applying an offset voltage, we are breaking the setup's symmetry. Autoterminal terms are then symmetric with respect to $eV=\pm 2e{V}_0$, while the summed-up noise is asymmetric.

Figure 14

Real parts of cross-correlation spectrum in units of $2\pi \hslash /e^2$ when $e{V}_0=0$ (top) and $e{V}_{0}L/\hslash v_F=3eVL/\hslash v_F=18$ (bottom). Without an offset voltage ($e{V}_0=0$), autoterminal contributions are identical as well as cross-terminal ones at large frequencies $\hslash \Omega \gg eV$. At finite $e{V}_0$, the asymmetric bias voltage is reflected in the frequency dependence of the autoterminal correlators by their different heights and the shift of the oscillations maxima.

Figure 15

Real parts of current-current cross-correlations in units of $2\pi \hslash /e^2$ as a function of symmetrically applied dc-bias voltage and for a fixed frequency. Jumps in the derivatives at $\pm \Omega L/v_F$ are due to the onsets of the cross-terminal contributions.

Figure 16

Derivatives of current-current correlations real parts in units of $2\pi \hslash /e^2$ with respect to frequency as a function of frequency. We have chosen a symmetrically applied dc-bias voltage with additional harmonic ac driving ($\omega L/v_F=3.5$, $\alpha =0.5$ and $a=1$) in lead $L$. Dashed vertical lines mark the step positions, coloring specifies the correlator that shows the step at corresponding $\Omega L/v_F$.