Non-equilibrium transport reveals energy level degeneracy

Abstract

We demonstrate a method to determine energy level degeneracies using non-equilibrium electronic transport through voltage-biased quantum dots. We establish the general validity of this approach using single and double quantum dots in bilayer graphene and GaAs. Unlike established methods based on entropy measurements or time-resolved tunneling statistics, our approach achieves comparable precision without requiring calibrated electron heating or real-time charge detection. We resolve the predicted symmetric shell structure in bilayer graphene quantum dots, including a singlet ground state at half filling and the ground state degeneracies of the first 13 carriers. Extending the method to double quantum dots, we observe degeneracy doubling associated with bonding and antibonding orbitals for a single carrier and a fourfold degeneracy for two carriers, previously inaccessible with existing techniques. These results establish a conceptually general and experimentally straightforward approach for probing energy level degeneracies in complex quantum systems.

Figures (11)

• Figure 1: (a) Schematic energy diagram of an out-of-equilibrium quantum dot symmetrically coupled to two voltage-biased reservoirs with tunneling rate $\Gamma$. A resonant level within the bias window enables transitions from the empty-dot state $|0\rangle$ to a doubly degenerate spin-$1/2$-like level $|1\rangle$. Owing to this degeneracy, the effective tunneling-in rate from the left reservoir is doubled. (b) Two-level energy diagram for the $0$ and $1$ charge states with ground-state degeneracies $d_{0}=1$ and $d_{1}=2$. Blue arrows indicate allowed charge transitions, while the red arrow denotes a spin-flip process. $U$ denotes the single-particle quantization energy. (c) Rate-equation simulation of the average dot occupation as a function of plunger gate voltage and bias voltage. Within the bias window, the occupation approaches the universal value $2/3$, reflecting the degeneracy ratio between the final and initial charge states. (d) Horizontal line cuts from (c) taken at $V_{\mathrm{sd}} =\pm35µeV$, as indicated by the red and gray arrows. (e) Measured average charge occupation of a bilayer graphene quantum dot as a function of plunger gate voltage and bias voltage. The occupation inside the bias window approaches the same universal value of $2/3$, corresponding to the lowest Kramers doublet acting as an effective spin-$1/2$ state. (f) Horizontal line cuts from (e) taken at $V_{\mathrm{sd}} \approx\pm35µeV$, as indicated by the red and gray arrows.
• Figure 2: (a) Schematic of an out-of-equilibrium quantum dot coupled to two voltage-biased reservoirs with tunneling rates $\Gamma_{\mathrm{L}}$ and $\Gamma_{\mathrm{R}}$. A resonant level within the bias window enables transitions from the $|n\rangle$ ground state to the ground (solid) and excited (dashed) $|n\!+\!1\rangle$ charge states. The average dot occupation $\bar{n}$ is monitored by a nearby charge sensor, while the transport current $I$ is measured at the drain. (b) Two-level energy diagram for the $n$ and $n\!+\!1$ charge states with ground-state degeneracies $d_{\mathrm{n}}$ and $d_{\mathrm{n+1}}$. (c) Transitions between degenerate manifolds of one and two spin$-1/2$ particles energy states allowed by selection rules (blue arrows). The orange path illustrates an indirect transition between states that are not directly coupled. (d) Symmetric coupling, $\Gamma_{\mathrm{L}}=\Gamma_{\mathrm{R}}$. Average dot occupation as a function of plunger gate voltage $\alpha \Delta V_{\mathrm{g}}$ for different degeneracy ratios $d_{\mathrm{n+1}}/d_{\mathrm{n}}$, where $\alpha$ is the lever arm. The occupation plateau within the bias window directly reflects this ratio. (e) Strongly asymmetric coupling, $\Gamma_{\mathrm{L}}\gg\Gamma_{\mathrm{R}}$. Absolute transport current at positive ($|I_{+}|$) and negative ($|I_{-}|$) bias as a function of plunger gate voltage $\alpha eV_{\mathrm{g}}$ for different degeneracy ratios. In this regime, the ratio of current plateaus inside the bias window directly yields $d_{\mathrm{n+1}}/d_{\mathrm{n}}$.
• Figure 3: (a) Energy diagram of a three-level system for the $n$ and $n\!+\!1$ charge states of the quantum dot, with ground- and excited-state degeneracies $d^{\mathrm{g}}_{\mathrm{n}}$, $d^{\mathrm{g}}_{\mathrm{n+1}}$, and $d^{\mathrm{e}}_{\mathrm{n+1}}$. The excited state of the $n\!+\!1$ configuration is separated from the ground state by an energy $\Delta$. Relaxation between excited and ground states (red arrow) is slow compared with tunneling to and from the leads (blue arrows). (b) Single-carrier energy spectrum of a bilayer graphene quantum dot as a function of out-of-plane magnetic field. The two Kramers pairs are separated by the Kane--Mele spin--orbit gap $\Delta_{\mathrm{SO}} \approx 70µeV$. A magnetic field lifts the degeneracy of each pair via spin and valley Zeeman splittings with distinct effective $g$ factors, leading to a stepwise evolution of the effective degeneracy from $d_{1}=1$ to $d_{1}=4$ as more transitions enter the bias window. The empty-dot degeneracy is $d_{0}=1$. (c) Rate-equation simulation of the dot occupation as a function of source--drain bias $V_{\mathrm{sd}}$ and plunger gate voltage. The onset of the excited state appears as a step within the bias window as illustrated by the horizontal line cuts in (d). Line-cut positions are indicated by gray and red arrows. (e,f) Experimental charge-sensor data corresponding to the simulations in (c,d) for the first-electron transition. (g,h) Same as (e,f) for the first-hole transition.The hole spectrum is equivalent to electron up to inversion of the energy axis. (i,j) Rate-equation simulation (i) and experimental data (j) of the dot occupation as a function of out-of-plane magnetic field and bias voltage at fixed plunger gate voltage for the first-hole transition, indicated by the white arrow in (g). Solid lines indicate the single-carrier spectrum from (b).
• Figure 4: (a) Energy spectrum of the first electron shell in a bilayer graphene quantum dot as a function of perpendicular magnetic field, illustrating the lifting of degeneracies. Only the ground and first excited states are shown. Colored arrows indicate allowed transitions between the $n$ and $n\!+\!1$ charge states: black (ground--ground), blue (ground--excited), green (excited--ground), and yellow (excited--excited). A full shell of four electrons forms a new effective vacuum state. (b) Rate-equation simulation of the quantum-dot occupation for different charge transitions as a function of energy and bias voltage. Arrows indicate the corresponding transitions entering the bias window. (c--f) Measured dot occupation as a function of plunger gate voltage (converted to energy) and bias voltage for the first thirteen carriers, grouped into four-electron shells for comparison with the simulations in (b). Colored arrows correspond to the transitions predicted by the model spectrum in (a). (g) Extracted ratios of initial and final degeneracies, plotted on a logarithmic scale as a function of carrier number (red circles and dashed line), together with the theoretical values from the spectrum in (a) (blue circles and dashed line).
• Figure 5: (a) Theoretical dot occupation for the first-hole transition as a function of energy in the intermediate-coupling regime, shown for positive (purple) and negative (orange) bias. The dashed line indicates the symmetric-coupling case. The ratio of initial and final degeneracies is extracted from the plateau values $\delta\bar{n}_{+}$ and $\delta\bar{n}_{-}$ inside the bias window. (b) Hight of occupation plateaus at positive and negative bias plotted against each other for several degeneracy ratios $d_{\mathrm{n+1}}/d_{\mathrm{n}}$ and coupling asymmetries. Experimental data for the first hole (red circles) fall on the theoretical curve for $d_{\mathrm{n+1}}/d_{\mathrm{n}}=2$. The dashed line marks the symmetric-coupling case. (c) Extracted values of $\ln(d_{\mathrm{n+1}}/d_{\mathrm{n}})$ (entropy change) as a function of tunnel-coupling asymmetry. Insets show the measured first-hole transition as a function of bias voltage and plunger gate voltage for the corresponding asymmetries indicated by dashed arrows.