Elementary charge-transfer processes in mesoscopic conductors

We determine charge-transfer statistics in a quantum conductor driven by a time-dependent voltage and identify the elementary transport processes. At zero temperature unidirectional and bidirectional single-charge transfers occur. The unidirectional processes involve electrons injected from the source terminal due to excess dc bias voltage. The bidirectional processes involve electron-hole pairs created by time-dependent voltage bias. This interpretation is further supported by the charge-transfer statistics in a multiterminal beam-splitter geometry in which injected electrons and holes can be partitioned into different outgoing terminals. The probabilities of elementary processes can be probed by noise measurements: the unidirectional processes set the dc noise level, while bidirectional ones give rise to the excess noise. For ac voltage drive, the noise oscillates with increasing the driving amplitude. The decomposition of the noise into the contributions of elementary processes reveals the origin of these oscillations: the number of electron-hole pairs generated per cycle increases with increasing the amplitude. The decomposition of the noise into elementary processes is studied for different time-dependent voltages. The method we use is also suitable for systematic calculation of higher-order current correlators at finite temperature. We obtain current noise power and the third cumulant in the presence of time-dependent voltage drive. The charge-transfer statistics at finite temperature can be interpreted in terms of multiple-charge transfers with probabilities which depend on energy and temperature.

Figure 1

The elementary processes shown in different regimes. (a) For a dc bias voltage $\overline{V}$ and at arbitrary temperature $T_e$, the electrons are transferred independently at different energies. The probability of transfer is given by occupation numbers $f_{1,2}\left(\mathcal{E}\right)$ of the leads and transmissions $\left\{T_p\right\}$ of the junction. The transport occurs when the incoming electronic state is occupied, the outgoing state is empty, and the electron is transferred across the junction. (b) On the other hand, an ac voltage drive $\Delta V\left(t\right)$ mixes the electron states of different energies. At zero temperature it creates the electron-hole pairs with probabilities $p_k$, in addition to electrons injected due to excess dc voltage. A created electron-hole pair contributes to transport only if one particle from a pair is transmitted and the other is reflected, which occurs with probability $T_p{R}_p$ ($R_p$ being reflection coefficient). (c) Finally, at finite temperature and in the presence of a time-dependent voltage drive, the statistics can be interpreted in terms of multiple $n$-charge transfers with probabilities $q_n$.

Figure 2

The schematic circuit-theory representation of a two-terminal junction. The left and right terminals are characterized by the quasiclassical Keldysh-Green’s functions ${\check{G}}_{1,2}\left(0\right)$. The counting field $\chi$ is related to the charges which enter the left terminal through the cross section indicated by dashed line. It can be incorporated into the corresponding Keldysh-Green’s function via gauge transformation given by Eq. (8).

Figure 3

The elementary charge-transfer processes in a voltage-driven junction. The unidirectional processes represent single electrons injected toward the contact due to excess dc offset voltage $\overline{V}$ applied. The bidirectional processes represent electron-hole pairs created by time-dependent voltage drive and injected toward the contact. The probabilities $p_k$ of pair creations depend on the ac voltage component $\Delta V\left(t\right)$.

Figure 4

The probabilities $p_k$ of electron-hole pair creations for harmonic drive $V\left(t\right)=V_0\text{cos}\left(\omega t\right)$ as a function of the amplitude $V_0$ [panel (a)]. With increasing $V_0$ more and more pairs are created per voltage cycle. The decomposition of the differential noise $\partial S_I/\partial \left(e{V}_0\right)$ (solid line) into contributions of elementary processes (dashed lines) is shown in panel (b).

Figure 5

The differential noise $\partial S_I/\partial \left(e{V}_0\right)$ as a function of the amplitude $V_0$ for different ac driving voltages $\left(\overline{V}=0\right)$ is shown in top panel: square (dashed-dotted), cosine (solid), triangle (dotted), sawtooth (dashed), and Lorentzian pulses (short-dotted line). The width of Lorentzian pulses is ${\tau }_L=0.1\tau$. The oscillations are due to elementary processes which are created as the voltage amplitude increases (cf. Fig. 4). The decomposition of the differential noise into contributions of elementary processes is shown in lower panels.

Figure 6

The excess noise $S_I-S_{\text{dc}}$ is shown in panel (a) for different time-dependent bias voltages: square (dashed), cosine (short-dotted), triangle (dashed-dotted), sawtooth (dotted), and Lorentzian pulses (solid line). The width of Lorentzian pulses is ${\tau }_L=0.1\tau$. The amplitude and the dc offset increase simultaneously, $V_0=\overline{V}$. The probabilities $p_{kL}$ (dotted lines) and $p_{kR}$ (dashed lines) of bidirectional elementary processes are shown in panel (b) for the cosine voltage drive $V\left(t\right)=\overline{V}+\overline{V}\text{cos}\left(\omega t\right)$. Only one elementary process of the $L$ type and one of the $R$ type can be created per period $\left(k=1\right)$. The decomposition of the excess noise $S_I-S_{\text{dc}}$ into contributions of elementary processes is shown in panel (c).

Figure 7

The differential noise $\partial S_I/\partial \left(e\overline{V}\right)$ as a function of dc offset $\overline{V}$ is shown in panel (a) for different time-dependent bias voltages: cosine (solid), square (dashed), and sawtooth (dashed-dotted line). The ac amplitude of the drive is kept constant, $e{V}_0/\omega =5$. The excess noise $S_I-S_{\text{dc}}$ as a function of $\overline{V}$ (solid line) and the decomposition into contributions of elementary processes (dotted lines) are shown for the cosine drive in panel (b) [cf. Eq. (18)]. The probabilities of elementary processes are shown in the inset.

Figure 8

Schematic representation of a multiterminal beam splitter. The source terminal is biased with a time-dependent voltage $V\left(t\right)$ with respect to the outgoing terminals. The total conductance $g_{\Sigma }={\sum }_i{g}_i$ of the outgoing leads is assumed to be much larger than the conductance $g=\left(e^2/\pi \right){\sum }_p{T}_p$ of the source contact.

Figure 9

The probabilities $q_{\pm n}$ of multiple-charge transfers for harmonic voltage drive $V\left(t\right)=V_0\text{cos}\left(\omega t\right)$ and transmission $T_p=0.5$. The temperature dependence of $q_n$ is shown in panel (a) for different driving amplitudes. The energy dependence of $q_n$ at low temperatures is indicated by plots for $\mathcal{E}/\omega =0$ (top line) and $\mathcal{E}/\omega =0.5$ (bottom line). The energy dependence of $q_n$ is shown in the thermal limit (b) $e{V}_0/\omega =0$ and for (c) $e{V}_0/\omega =1$. At zero temperature $q_{\pm 1}$ do not depend on energy and are given by $q_{\pm 1}=T\left(1-T\right)p_1\left(e{V}_0/\omega \right)$ (cf. Fig. 4). As the temperature increases, multiple-charge transfers start to enter the transport. The limit of thermal fluctuations is reached at temperatures larger than the driving amplitude. It can be recast again into single-charge transfers independent in the full energy range.