Directly visualizing the energy level structure of quantum dot molecules

Abstract

The orbital, spin and valley degrees of freedom in silicon quantum dots support many modes of spin qubit operation. However, it is generally challenging to obtain information about the energy level spectrum over large ranges of parameter space. We demonstrate a form of spectroscopy that is capable of mapping the energy level structure of a double quantum dot as a function of level detuning, interdot tunnel coupling, and magnetic field. In the one electron regime, we directly observe the transition from the atom like energy levels of isolated quantum dots to molecular like bonding and anti bonding states with increasing interdot tunnel coupling. We also resolve the Zeeman splitting of ground and excited valley states in a magnetic field. In the two electron regime, we gain access to the detuning dependent singlet triplet splitting. Our work may be extended to a broader class of systems, such as strong spin-orbit materials or proximitized quantum dots, allowing the direct extraction of various energy gaps.

Figures (9)

• Figure 1: Molecular spectroscopy.a,b, Level diagrams illustrating the spectroscopy of a DQD in both the atomic (a) and molecular (b) regimes. Both QDs are initially empty. Lowering their chemical potentials allows an electron to tunnel into the ground state, localized in one of the QDs at $|\varepsilon| \gg t_{c}$Oosterkamp1998. At small detuning $|\varepsilon| \ll t_{c}$, the degenerate states hybridize into bonding and anti-bonding states separated by $2t_{c}$Oosterkamp1998Petta2004. Further lowering of the chemical potentials permits tunneling into either the ground or the excited states. c, False-color scanning electron microscope image of the device. Square waves are applied to the plunger gates, P1 and P2. The tunneling of electrons on and off the DQD is detected by measuring the charge sensor conductance $g_{s}$. Bottom panel: Time-averaged $g_{s}$, illustrating the characteristic $RC$ response. When the excited state becomes accessible, the loading rate increases. The inset shows the applied square wave with amplitude $\Delta_{p}$ and period $2\tau$. d, DQD stability diagram measured as a function of $\varepsilon$ and $\Delta$ with no pulses applied. White axes indicate the range of $\varepsilon$ and $\Delta_{p}$ used for the spectroscopy measurement in e. $\Delta_{0}$ represents the reference level at which electron loading into the lower triple point occurs. e, Single-electron spectrum showing the detuning dependence of the energy levels. The vertical axis $\Delta_{p}$ is offset by $\Delta_{0}$. At $\epsilon$ = 0, the ground states of the left and right QDs are degenerate.
• Figure 2: Transition from artificial atom to molecule.a,b, One-electron spectra of the DQD at (a) weak and (b) strong $t_{c}$. $E_{O1,L}$ and $E_{O1,R}$ denote the first orbital excited state energies in the left and right QDs, respectively. $E_{O2,R}$ is the energy of the second orbital excited state in the right QD. At small $t_{c}$, the orbitals of the left and right QDs are localized. As $t_{c}$ increases, the localized orbitals hybridize to form bonding and anti-bonding states Oosterkamp1998, as indicated by the avoided crossings with a splitting of $2t_c$ in panel b.
• Figure 3: Zeeman splitting.a, One-electron spectrum taken at $B$ = 1 T, showing Zeeman splitting $E_{Z}$ of the lowest orbital. The orange dashed line indicates the value of detuning used to acquire the data in panel b. b, The level spectrum measured as a function of $B$ with $\varepsilon$ = 1 meV. The orange dashed line corresponds to $B$ = 1 T, the field at which the panel a data are acquired. From the level splitting of the lower- and upper-valley states, we extract $g$ = 1.98 $\pm0.12$.
• Figure 4: Two-electron state spectroscopy. a, Two-electron spectrum acquired at $B$ = 0 T, showing the ground and excited states of the (2,0), (1,1) and (0,2) charge configurations. The detuning axis $\varepsilon$ is measured relative to the lower (2,0) $\leftrightarrow$ (1,1) triple point. The vertical axis $\Delta_p$ is offset by the reference level $\Delta_0$, corresponding to the energy at the same triple point (see the red star in the inset). The stability diagram in the inset shows the spectroscopy measurement window. b--d, Cartoons illustrating tunneling processes in the two-electron regime. At small positive $\epsilon$ (b), a single electron initially resides in the left QD. A $\Delta$ pulse injects an extra electron into the right QD. The spectrum, in this case, reveals the valley and orbital excited states of the right QD. At a larger $\epsilon$ (c), the DQD is in the (0,1) configuration with an electron occupying the right QD. The excited state spectrum is acquired by adding a second electron into the left QD. At even larger $\epsilon$ (d), the DQD remains in the (0,1) configuration, but the second electron can tunnel into the (0,2) singlet or triplet state. e, High resolution spectrum acquired near the (1,1) $\leftrightarrow$ (0,2) transition. In the (1,1) charge configuration, the ground state consists of two electrons occupying the lower-valley states of the left and right QDs. In the excited state, one electron occupies the upper-valley state Philips2022. As $\epsilon$ increases, these (1,1) states moves up in energy and cross the singlet and triplet states in the (0,2) charge configuration. $E_{ST}$ denotes the energy gap between the singlet and triplet states.
• Figure Fig. 1: Raw charge sensor conductance and spectral broadening.a, Raw $g_{s}$ data for the one-electron spectrum presented in Fig. 1e. The raw data clearly exhibit the valley states of the left and right QDs. The charge sensing contrast is smaller for the left QD, as it is located further away from the charge sensor. b, Broadening of the (0,0) $\leftrightarrow$ (0,1) charge transition. The black circles represent a vertical cut of the $g_{s}$ data at $\varepsilon = \text{0.55 meV}$ (see the red line in a). The data are fit to a Fermi function with $T = \text{94 mK}$ (red line). c, The derivative $dg_s/d\Delta_{p}$ data (black circles) fitted to the derivative of a Fermi function (red line). The full width at half maximum corresponds to $\sim 3.5 k_{B}T$.