# A minimal double quantum dot

## Abstract

Double quantum dots (DQDs) are a versatile platform for solid-state physics, quantum computation and nanotechnology. The micro-fabrication techniques commonly used to fabricate DQDs are difficult to extend to the atomic scale. Using an alternative approach, which relies on scanning tunneling microscopy and spectroscopy, we prepared a minimal DQD in a wide band-gap semiconductor matrix. It is comprised of a pair of strongly coupled donor atoms that can each be doubly charged. The donor excitation diagram of this system mimicks the charge stability diagram observed in transport measurements of DQDs. We furthermore illustrate how the charge and spin degrees of freedom of the minimal DQD may be used to obtain a single quantum bit and to prepare a Bell state. The results open an intriguing perspective for quantum electronics with atomic-scale structures.

## Introduction

Quantum dots (QDs) are artificial nanometer scale structures in which quantum confinement causes the formation of discrete energy levels from continuous electronic bands of a solid. A single QD may be considered as an artificial atom, while a double QD (DQD) can be viewed as a molecule1. Depending on the strength of the coupling between the QDs, DQDs may be categorized as weakly-coupled or strongly-coupled1. Strongly-coupled DQDs have attracted much research attention, due to their fundamental properties and their significant applications. The Coulomb staircase, the spin blockade effect, the use as a single spin quantum bit, the realization of a micromaser are just a few of many examples1,2,3,4. A typical DQD involves source and drain leads, which are coupled to the DQD with tunneling contacts, and two gate electrodes, which individually control the local potential of each QD (Fig. 1a). By varying both gate voltages and measuring the source-drain conductance, the charge states of the DQD may be determined and displayed in the so-called stability diagram (Fig. 1c). The stability diagram may be used to extract the coupling strength and also reveals additional key parameters, such as the on-site binding energy and Coulomb interaction.

To further shrink the dimensions of DQDs to the atomic scale scanning tunneling microscopy and spectroscopy (STM/S) is an obvious choice. Indeed, STM has successfully been used to investigate single impurities in semiconductors like Si or GaAs5. Examples of recent achievements are the controlled switching of the charge state of a single impurity, the manipulation of individual donor binding energies, the magnetization of individual dopant, and the observation of a valley interference effect in single dopants6,7,8,9,10,11. However, the lack of a gate electrode is a significant drawback of STM in transport measurements, which are essential for studies of QDs. Here, we have present a data acquisition and analysis method that enables measurements of the stability diagram of atomic-scale DQDs. The DQDs are comprised of dopant dimers.

Monomers and dimers of donors that can carry a single charge have been characterized by STM6,7,8,9, 12,13,14. However, only donors that can be multiply charged are suitable for implementing a DQD. Therefore, we used the third generation semiconductor material ZnO15. In ZnO double-donors, which can be charged with up to two electrons, are available16, 17. From measured donor excitation diagrams (Fig. 1d) we found that a dimer of double-donors can be viewed as a DQD. In order to obtain such data, a set of differential conductance spectra (dI/dV) measured along a line crossing two neighboring double-donor atoms is required. The spectra are then represented as a two-dimensional map that displays the conductance vs. the bias voltage and the lateral tip position. The ionization of the donors leads to a peak in dI/dV that evolves with the position of the tip and separates the map into several regions. The distinct areas of such a map correspond to different occupation numbers (n 1, n 2) of the two donors, where n 1 (n 2) is the number of electrons on the left (right) dot.

The donor excitation diagrams from our STM experiment closely resemble the stability diagrams obtained in transport measurements on DQDs. In other words, pairs of double-donors in ZnO represent minimal DQDs, with each QD involving a single donor and the surrounding ZnO lattice.

The clean ZnO(0001) surface displays terraces with triangular islands of adatoms and vacancies18, 19 as shown in Fig. 1e. Despite the rough surface morphology, the dI/dV map exhibits sharp rings, similar to what has been observed on other materials, such as GaAs, Bi2Se3 and MoSe2. These rings are due to the ionization of subsurface dopant atoms6, 20, 21. Importantly, ZnO features concentric double and multiple rings, which result from double or multiple charging of a donor or donor dimer at their centre.

In Fig. 1f, two neighboring elliptical double-rings (yellow lines) are observed. Their centers are ≈10 nm apart and their overlapping contours apparently do not affect each other. These properties are expected from weakly coupled QDs. In addition, the spectroscopic map displays a system exhibiting three concentric rings (red lines). Cross-sectional profiles of the multiple-ring system (Fig. 2a and c), reveal a donor excitation diagram that is typical of a strongly coupled DQD.

Previous works have shown that the binding energy (ionization threshold) of a subsurface donor is largely determined by its depth underneath the surface22, 23. In other words, donor atoms at the same depth exhibit identical donor excitation diagram and ionization rings in real-space dI/dV maps. Therefore, a pair of donors at identical depths is expected to generate a symmetric DQD while different depths introduce a degree of asymmetry. The capability of producing both symmetric and asymmetric DQDs is essential for application as a quantum light source24, 25. In the STM/S data of Figs 2 and 3, we indeed observed both types of DQDs. Using the evolution from a weakly-coupled DQD to the strongly-coupled case (Figs 2e and 3e), we can identify charge occupation numbers (n 1, n 2). As shown in Figs 3d and 4d, the coherent superposition of particular DQD charge states, e.g. at the boundary between two areas with different (n 1, n 2), exhibits potential application to quantum electronics.

To describe the dominant features of the donor excitation diagram and to reveal the potential application of our minimal DQD we build an analytic model. The total Hamiltonian is given by

$$\begin{array}{rcl}H & = & {H}_{1}+{H}_{2}+{H}_{C}={H}_{1}+{H}_{2}+{H}_{T}+{H}_{m}+{H}_{s}\\ {H}_{1} & = & {n}_{1}({\epsilon }_{d}-{\mu }_{t1})+U{n}_{1}({n}_{1}-1);{H}_{2}={n}_{2}({\varepsilon }_{d}-{\mu }_{t2})+U{n}_{2}({n}_{2}-1)\\ {H}_{T} & = & -{t}_{12}{c}_{1}^{\dagger }{c}_{2}+h.c\mathrm{.}\\ {H}_{m} & = & {U}_{m}{n}_{1}{n}_{2}.\\ {H}_{s} & = & -{n}_{1}{\gamma }_{12}e(V-{V}_{F}).-{n}_{2}{\gamma }_{21}e(V-{V}_{F})\end{array}$$
(1)

H 1 (H 2) is the Hubbard Hamiltonian for the left (right) donor atom. H C , the inter-donor coupling, is comprised of three terms. H T describes coherent tunneling, H m is the Coulomb interaction between two donors, and H s represents the indirect chemical potential shift. In these equations, $${\epsilon }_{d}$$ is the first electron’s on-site binding energy; μ t describes the effective local chemical potential due to tip-induced band-bending (TIBB); U stands for the on-site Coulomb repulsion between two electrons at same donor; t 12 is the coherent coupling strength; U m is an inter-atom Coulomb repulsion; V F denotes the flat band voltage; V is the bias of the tip; γ ij is the TIBB of site i induced by TIBB of site j via the coupling between the donor atoms.

In our experiment, the two sites of the DQD are spatially separated (about 6.5 nm), direct coherent tunneling is suppressed by the barrier, so H T is negligible. The physics of the DQD can be understood in a semi-classical picture. In this regime, n 1 and n 2 are both good quantum numbers, and the Hamiltonian can be directly diagonalized. By taking into account the position-dependent TIBB (see details in Supplementary Note 1 and Supplementary Fig. 1), we are able to simulate the symmetric DQD donor excitation diagram and deduce all key parameters as listed in Table 1. Specifically, we obtained $${\epsilon }_{d}$$ = −200 meV and U 90 meV for individual donors and U m  30 meV for the inter-donor coupling.

By shining microwave radiation with a suitable wavelength into the STM junction26, 27, coherent tunneling (microwave assisted tunneling) between the two donor atoms may be induced. In such a scenario, it appears possible to build a quantum bit and a Bell state in our minimal DQD by utilizing its charge and spin degrees of freedom. To obtain a quantum bit, the DQD system is first initialized to an occupation (0,1), where the donors carry 0 and 1 electron, respectively. Next, the microwave energy is tuned close to resonance with the (0,1)–(1,0) transition and a coherent coupling between these two states is built up. This process turns the minimal DQD into a minimal single solid-state charge qubit, similar to proposals based on a micro-fabricated QD with larger size28. To prepare a Bell state, the system is initialized to the (0,2) state. The double donors in our ZnO samples most likely are interstitial Zn atoms16, 17. When the donor carries two electrons, they occupy the Zn 4s orbital, thus forming a spin singlet state. Moving one electron to the other donor atom thus produces a fourfold degenerate (1,1) state. Since the coherent tunneling conserves the total spin S 2 and its z-component S z , only the spin singlet (1,1) state, i.e. the Bell state $$|{{\rm{\Psi }}}_{B}^{-}\rangle =\frac{1}{\sqrt{2}}(|\uparrow ,\downarrow \rangle -|\downarrow ,\uparrow \rangle )$$, couples to the singlet (0,2) state. By applying the stimulated Raman adiabatic passage protocol (see details in Supplementary Note 2) to the (0,2)–(1,1) transition, the DQD is adiabatically driven from spin-singlet-(0,2) to a spin-singlet-(1,1) state29, 30. The destination of our DQD is thus in the Bell state $$|{{\rm{\Psi }}}_{B}^{-}\rangle$$.

In summary, we have experimentally realized a strongly-coupled DQD which is comprised of a dimer of donor atoms. The electronic levels of the individual donors depend on the depth below the surface, which can break the symmetry between the donors of a pair. As recently demonstrated the depth of an individual donor in ZnO can be manipulated23 and consequently the electron transport properties of the proposed DQD may be controlled. An analytical model is capable of reproducing the electronic states of the system. We propose that microwave radiation may be used to prepare interesting entangled quantum states of this minimal DQD.

For the time being, STM-based techniques enable probing these electronic properties. Defect densities in ZnO crystals are high and the experiments require a time-consuming search for a suitable sample area. With improved crystal quality, however, it may become possible to more routinely obtain these DQD presented above.

## Methods

### Experiment

All experiments were performed in an ultra-high vacuum system equipped with a home-built STM operated at 5 Kelvin. Single-crystalline ZnO(0001) was prepared by cycles of Ar+ bombardment and high temperature annealing. Au tips were cut from a polycrystalline wire and in situ annealed prior to transfer to the STM. STM imaging was performed in a constant-current mode with the bias voltage being applied to the sample. A sinusoidal voltage modulation of 10 mVrms and a lock-in amplifier were employed to measure dI/dV spectra.

### Theory

We build a total Hamiltonian to describe the strongly-coupled donor dimer. In the semi-classical regime, the Hamiltonian can be directly diagonalized enabling a simulation of the predominant features in the experimental data. In the quantum regime, we used two effective total Hamiltonians to describe the charge quantum bit and spin Bell state. Details of the theory are provided in the Supplementary Figs 1 and 2 and Supplementary Notes 1 and 2.

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## Acknowledgements

We thank A. Weismann, S. Zhang and Y. Liu for discussions. H.Z. thanks the supports from National Natural Science Foundation of China (No. 11674226), the National Key Research and Development Program of China (2016YFA0300403) and the Young 1000-Talent Program. Financial support by the Deutsche Forschungsgemeinschaft through SFB 855 is acknowledged.

## Author information

H.Z. performed the experiments. J.Z. did the modelling. All authors discussed the data and prepared the manuscript.

Correspondence to Hao Zheng.

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