Blocking-state influence on shot noise and conductance in quantum dots

Quantum dots (QDs) investigated through electron transport measurements often exhibit varying, state-dependent tunnel couplings to the leads. Under specific conditions, weakly coupled states can result in a strong suppression of the electrical current, and they are correspondingly called blocking states. Using the combination of conductance and shot noise measurements, we investigate blocking states in carbon nanotube (CNT) QDs. We report negative differential conductance and super-Poissonian noise. The enhanced noise is the signature of electron bunching, which originates from random switches between the strongly and weakly conducting states of the QD. Negative differential conductance appears here when the blocking state is an excited state. In this case, at the threshold voltage where the blocking state becomes populated, the current is reduced. Using a master equation approach, we provide numerical simulations reproducing both the conductance and the shot noise pattern observed in our measurements.

Figure 1

Measurement setup. (a) A SEM image of CNT device connected to a GHz impedance-matching circuit for efficient collection of noise signals. A bias tee enables simultaneous measurement of dc and rf properties. The source electrode is coupled to the on-chip matching circuit, while the drain is connected to the ground plane. Schematics of (b) a lumped $LC$ impedance transformer, used in device A and (c) a stub tuner based on coplanar transmission lines, used in device B. Both matching circuits are fabricated from Nb films and have a resonance frequency of $\approx 3$ GHz.

Figure 2

Conductance and noise measurements of device A (a)–(e) and device B (f)–(j). (a),(f) Differential conductance deduced from rf reflectance, as a function of bias voltage $V_{\mathrm{SD}}$ and side gate voltage $V_{\mathrm{G}}$. Dashed green lines serve as guides for the eye; the same guides are overlaid in the remaining panels. The purple arrow in (f) points to a segment of negative differential conductance. (b),(g) The Schottky noise, $|2e\langle I\rangle |$, provides a comparison reference for current noise. (c),(h) Measured current noise spectral density, $S_I$. (d),(i) The excess noise, $S_I^{\mathrm{EP}}=S_I-\left|2e\langle I\rangle \right|$. (e),(j) The Fano factor, obtained by dividing the current noise spectral density $S_I$ by the simultaneously measured Schottky value $|2e\langle I\rangle |$. Cyan areas inside the diamonds represent divergent domains, where uncertain noise values are divided by small currents.

Figure 3

(a) Energy levels in the considered model, with transitions marked as dashed arrows. (b) Electrochemical-potential levels, given by the arrows in (a) and a common offset. (c) QD transitions represented as electrochemical-potential levels, whose positions are determined in (b). (d) Schematic stability diagram with relevant transitions. Positive- (negative-) slope lines depict resonances of the source (drain) Fermi level with quantum state transitions depicted in (c). (e) Chemical-potential diagrams for the lead-resonant lines [circle and square points in (d)] and for a domain inside the cone of the full gray lines and the cone of the dashed gray lines [star points in (d)]. (f) Two paths for inelastic cotunneling [for region II in (d)].

Figure 4

Quantum state graph with associated electrochemical-potential diagrams, for device A, region II. The gray rectangles in tunnel barriers stand for weaker tunneling via the neighboring transition. $|N\rangle$ is a blocking state, from which the quantum dot can escape only by low-rate processes (inelastic cotunneling in the second diagram; excitation). Elastic cotunneling is not displayed. In the bottom-right chemical-potential diagram, tunneling into S is equally possible.

Figure 5

Quantum state graph with associated electrochemical-potential diagrams, for device B, region X. $|N^{*}\rangle$ is a blocking state if the main escape process, namely tunneling from S via transition $N^{*}\to N+1$, is hindered (marked with a gray rectangle in the tunnel barrier). Cotunneling transitions are not displayed.

Figure 6

Numerical calculations of conductance $G$ and Fano $F$ factor for (a)–(e) device A ($t_{N,N\pm 1}^{\mathrm{S}/\mathrm{D}}=0.33$) and (f)–(j) device B ($t_{N^{*},N+1}^{\mathrm{S}}=0.1,t_{N^{*},N+1}^{\mathrm{D}}=0.4$, the relaxation rate for the excited state of $N+1$ is $\mathrm{\Gamma }$). All other $t$ amplitudes are 1. The maximal tunneling rates are ${\mathrm{\Gamma }}^{\mathrm{S}/\mathrm{D}}=\mathrm{\Gamma }={10}^{-3}{U}_{\mathrm{c}}/\hslash$ and the relaxation rate is ${10}^{-3}\mathrm{\Gamma }$. In device A, $F$ is enhanced in regions I, II, III. In device B, $F$ is significantly high in region X, whose top corner is characterized by a bias voltage $-{\mathrm{\Delta }}_0/\left|e\right|$ and right corner by $-{\mathrm{\Delta }}_1/\left|e\right|$.

Figure 7

Graph of quantum states for a QD that exhibits telegraphic transport. While the QD oscillates in the left loop ($N+1\leftrightarrow N$), the current is switched on and an average number of $\langle n\rangle$ electrons tunnel sequentially from the source into the QD and farther into the drain. $|N^{*}\rangle$ is the blocking state. Cotunneling and internal excitation and relaxation are not taken into account.