Nonadiabatic charge pumping by oscillating potentials in one dimension: Results for infinite system and finite ring

We study charge pumping when a combination of static potentials and potentials oscillating with a time period $T$ is applied in a one-dimensional system of noninteracting electrons. We consider both an infinite system using the Dirac equation in the continuum approximation and a periodic ring with a finite number of sites using the tight-binding model. The infinite system is taken to be coupled to reservoirs on the two sides which are at the same chemical potential and temperature. We consider a model in which oscillating potentials help the electrons to access a transmission resonance produced by the static potentials and show that nonadiabatic pumping violates the simple $\text{sin}\varphi$ rule which is obeyed by adiabatic two-site pumping. For the ring, we do not introduce any reservoirs, and we present a method for calculating the current averaged over an infinite time using the time evolution operator $U\left(T\right)$ assuming a purely Hamiltonian evolution. We analytically show that the averaged current is zero if the Hamiltonian is real and time-reversal invariant. Numerical studies indicate another interesting result, namely, that the integrated current is zero for any time dependence of the potential if it is applied to only one site. Finally we study the effects of pumping at two sites on a ring at resonant and nonresonant frequencies, and show that the pumped current has different dependences on the pumping amplitude in the two cases.

Figure 1

(Color online) Current pumped through a symmetric double barrier by an oscillating potential applied at one site as a function of $E_F$. The barrier strengths $u_i$ are both equal to 4, and their positions are at 0 and $\pi$. The resonant energy is $E_r=0.917$, and no bias is applied. The oscillating potential is applied at the position $-\pi$ with an amplitude 0.2 and frequency 0.035. We set $v_F=1$. The locations of the two maxima and the minimum in the middle are discussed in the text.

Figure 2

(Color online) Current pumped through a symmetric double barrier by oscillating potentials applied at two sites. The barrier strengths are both equal to 4, and their positions are at 0 and $\pi$. The resonant energy is $E_r=0.917$, and no bias is applied. The oscillating potentials are applied at the positions $-\pi$ and $-2\pi$ with the same amplitude 0.2 and frequency 0.035 but with a phase difference of $\pi /2$. We set $v_F=1$. The locations of the maxima and the minimum are discussed in the text.

Figure 3

(Color online) Current pumped through a symmetric double barrier by oscillating potentials applied at two sites as a function of the phase difference ${\varphi }_2-{\varphi }_1$, for four different Fermi energies given by 0.893 (magenta solid line), 0.910 (blue dashed line), 0.920 (green dotted-dashed line), and 0.934 (red dotted line). The barrier strengths are both equal to 4, and their positions are at 0 and $\pi$. The resonant energy is $E_r=0.917$ and no bias is applied. The oscillating potentials are applied at the positions $-\pi$ and $-2\pi$ with the same amplitude 0.2 and frequency 0.035. We set $v_F=1$.

Figure 4

(Color online) Instantaneous current at various times over one time period $T=10\pi$ (corresponding to $\omega =0.2$) for an oscillating potential applied at one site, for the third eigenstate of $U\left(T\right)$ for a six-site system. Two cases are considered: the oscillating potential is (i) time-reversal invariant (blue dashed line) and (ii) not time-reversal invariant (red solid line). A static potential of strength 1 is applied at site 2, while the oscillating potential is applied at site 3. For case (i) with an oscillating potential $\text{cos}\left(\omega t\right)+0.2\text{cos}\left(2\omega t\right)$, we see an exact cancellation between the values at times $t$ and $T-t$, while case (ii) with an oscillating potential $\text{cos}\left(\omega t\right)+0.5\text{sin}\left(2\omega t\right)$ does not show such a pairwise cancellation. In both cases, the integrated current contributed by the third eigenstate is zero.

Figure 5

(Color online) Total pumped current versus the overall phase for oscillating potentials applied at two sites, for a six-site ring with three electrons. A static potential of strength 1 is applied at site 2, while the oscillating potentials are applied at sites 3 and 4 with an amplitude $b=0.1$, a frequency $\omega =0.2$, and a phase difference of $\pi /2$.

Figure 6

(Color online) Pumped currents for all the eigenstates of $U\left(T\right)$ for oscillating potentials applied at two sites versus the oscillation amplitude $b$, for a six-site ring. A static potential of strength 1 is applied at site 2 while the oscillating potentials are applied at sites 3 and 4 with a phase difference of $\pi /2$ and a nonresonant frequency $\omega =0.2$. In all cases, the current is proportional to $b^2$ for small $b$.

Figure 7

(Color online) Total pumped current for oscillating potentials applied at two sites versus the oscillation amplitude $b$, for a six-site ring with three electrons. All the parameters are the same as in Fig. 6.

Figure 8

(Color online) Pumped currents for all the eigenstates of $U\left(T\right)$ for oscillating potentials applied at two sites versus the oscillation amplitude $b$, for a six-site ring. A static potential of strength 1 is applied at site 2 while the oscillating potentials are applied at sites 3 and 4 with a phase difference of $\pi /2$ and a resonant frequency $\omega =1.7046$. In four cases, the current is proportional to $b^2$, but in two cases, the current is proportional to $b$ for small $b$ (see text for details).

Figure 9

(Color online) Total pumped current for oscillating potentials applied at two sites versus the oscillation amplitude $b$, for a six-site ring with three electrons. All the parameters are the same as in Fig. 8.