Extracting inter-dot tunnel couplings between few donor quantum dots in silicon

The long term scaling prospects for solid-state quantum computing architectures relies heavily on the ability to simply and reliably measure and control the coherent electron interaction strength, known as the tunnel coupling, $t_c$. Here, we describe a method to extract the $t_c$ between two quantum dots (QDs) utilising their different tunnel rates to a reservoir. We demonstrate the technique on a few donor triple QD tunnel coupled to a nearby single-electron transistor (SET) in silicon. The device was patterned using scanning tunneling microscopy-hydrogen lithography allowing for a direct measurement of the tunnel coupling for a given inter-dot distance. We extract ${t}_{{\rm{c}}}=5.5\pm 1.8\;{\rm{GHz}}$ and ${t}_{{\rm{c}}}=2.2\pm 1.3\;{\rm{GHz}}$ between each of the nearest-neighbour QDs which are separated by 14.5 nm and 14.0 nm, respectively. The technique allows for an accurate measurement of $t_c$ for nanoscale devices even when it is smaller than the electron temperature and is an ideal characterisation tool for multi-dot systems with a charge sensor.

The entanglement of multiple quantum particles is becoming an established practice for enhanced measurement protocols in quantum metrology [1], secure communications in quantum key distribution [2,3] and is the central tenant of quantum computation [4]. Entanglement is created by a coherent coupling between quantum particles. In solid-state architectures, the spin-spin interaction between single electrons isolated to quantum dots (QDs) enables multi-qubit operations needed for universal quantum computation [5]. The strength of this interaction is governed by the coherent tunnel coupling term, t c between two electron charge states of neighbouring QDs [6].
Unlike in conventional QD architectures where electrons are confined using metallic surface gates, donor based systems rely on the attractive Coulomb potential of the donor atoms themselves [7][8][9][10][11]. Nanoelectronic devices based on phosphorus doped silicon (Si:P) have recently demonstrated electron transport at the few electron level where spin-spin interactions can be observed [12][13][14]. Following this, the singlet-triplet states of a strongly coupled donor pair have been readout [15] and electrons confined to double QDs formed from donor clusters have been investigated using charge-sensing [16]. Importantly, for the scalability of multi-donor systems in a quantum computing architecture, the ability to simply and reliably measure t c between donors is vital.
Unlike gate defined QDs [17][18][19], the value of t c between isolated phosphorus donors or clusters in Si:P qubit architectures is fixed by the physical distance between the donors [20][21][22], and is difficult to tune using external gates since the donors are only separated by tens of nanometres [7,13,23]. Therefore, knowledge of t c as a function of donor separation is extremely important for the design, fabrication, and operation of donor based qubits [20][21][22]24]. Several methods to investigate and determine t c based on electron transport [25][26][27][28][29][30], spin funnel experiments [31,32] and the response of a quantum point contact across an inter-dot charge transition [17] have already been established. However, these techniques either require multiple-electron spin readout at low magnetic fields [31] or a large capacitive difference between the QDs and charge sensor [17]. The second condition requires that both QDs are at vastly different distances to the charge sensor; which is not ideal for Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. single electron spin readout since complex shuttling protocols must be developed to determine the individual spin states [33].
In this letter we demonstrate a new method to determine t c between QDs based on a simple time-resolved charge sensing technique. We use our method to determine t c between adjacent QDs in a triple QD device that uses a single-electron transistor (SET) as a charge sensor [34]. SETs have been used extensively for charge sensing in Si:P where they can perform high-fidelity single-electron spin readout [11,[35][36][37]. We show how our timeresolved SET charge sensing technique can be used to determine t c and at the same time allows for the extraction of electron temperature, T e . The conversion factor from gate voltage to energy known as the lever arms, α, can also be measured without the possibility of artificial broadening of the SET Coulomb blockade peaks [38] present in previously reported methods [35].
The device, shown in figure 1, was fabricated using a scanning tunneling microscope (STM) to selectively remove hydrogen from a passivated Si(100) 2×1 reconstructed surface. The lithographic mask is subsequently dosed with PH 3 and annealed to incorporate P atoms into the silicon substrate [39]. The lithographic outline of the device is shown in figures 1(a), (b). The QDs, L, M, and R (left, middle, and right) are formed with~5 P donors in each QD, determined by examining the size of the lithographic patches [11,13]. Three gates, G L , G M and G R control the electron numbers on the QDs. The electrons are able to tunnel to a SET island used as the charge sensor. The SET is defined as a much larger QD in between source and drain leads with a control gate G SET . It is operated with a source-drain bias of 0.3 mV and has a charging energy of~5 meV.
In our experiment, G L and G R are used to detune QDs L and R with respect to the SET, while G M is used as global gate to shift the potential of the QDs. Figure 1(c) shows the SET transport current as a function of G L and G R . Enhanced current lines running at~ 45 due to Coulomb blockade of the SET can be seen with breaks corresponding to charge transitions of the three QDs. Due to the different capacitive coupling of the gates to the QDs, three lines of SET breaks with distinct slopes are visible. In addition a characteristic pentagon structure associated with the quadruple point of a triple QD [14,40,41] can be seen, confirming the presence of three separate QDs. We note that the absolute electron number has not been determined for this device; however, for the purpose of this work we assign the charge states shown in figure 1(c) where (n n n , , L M R ) represents the relative electron numbers on QDs L, M and R, respectively. Next, we describe the method of determining t c using the L-M transition as an example, see figure 2. The protocol involves measuring the tunnel rate from the QDs to the SET across an inter-dot transition, figure 2(a). From the detuning, D dependency of the measured tunnel rates, the t c can be extracted. We define D = 0 as the charge degeneracy point between the L and M QD ((ii) in figures 2(a) and (c)). Using a two level pulse scheme, the system is first initialised in the equivalent single electron state, (1, 0, 0) after which the second pulse moves into the (0, 0, 0) to unload this electron. This pulse duration, t p =0.1 ms, is adiabatic with respect to t c but faster than either of the independent tunnel rates from the QDs to the SET. An exponential decay is fitted to the average current trace (200 cycles) and a tunneling rate ( ) G D extracted accordingly, see figure 2 where P i is the probability of the electron occupying QD = i L, M and G i is the effective tunnel rate which takes into account assisted tunneling via a neighbouring QD to the SET. Importantly both the occupation probabilities and effective tunnel rates will depend on the parameters, Δ, t c and temperature of the system, T [17]. We have also performed numerical calculations based on a Linblad master equation approach and achieve the same form for the tunnel rate, ( ) G D (see appendix). To find the general expression of ( ) G D we must compute the probabilities P i of the electrons occupying the QDs in equation (1). Assuming that the QDs are in thermal contact with the SET with a temperature, T, since  g k T i B [42], the probabilities of finding the electron in either of the two QDs is given by [17]    The width of the plateau is directly related to the strength of t c . We can eliminate the possibility that the plateau is caused by an external charge fluctuation by looking at the stability map, figure 1(c). There is no charge offset in the SET current that can be attributed to external charge movement. Equation (4) diverges from the theoretically predicted results using the Lindblad formalism for  t k T c B where the sequential tunneling approach cannot be used. It is worth noting that if g g = L M , then ( ) G D does not vary as a function of D and our method cannot be used to gain any information about t c . As a result, dg ¹ 0 LM is required which is most likely the case for any system, in particular for donor based QDs where the coupling decays exponentially with distance [20,21] meaning differences in donor position even on the atomic scale will change the tunnel rate significantly.
By fitting equation (4) to the tunnel rate across the transition we can determine t c and the temperature, T. By varying the temperature we are able to determine both the minimum electron temperature, T e as well as the lever arm for the inter-dot transition, α [35]. Figure 4(a), shows the form of ( ) G D at two different temperatures = T 100 and 550 mK where we see the impact of thermal broadening. The lever arm is related to the temperature, T by since the width of the plateau only depends on t c . Therefore, t c can also be measured as a function of temperature. The t c should not have a temperature dependence since it corresponds to an energy separation. This is confirmed experimentally in figure 4(c) where t c remains constant as the temperature is varied. The coherent coupling term between the QDs is much greater than the inelastic tunnel rate to the SET despite having a similar separation (14 nm compared to ∼18 nm). In general this would be expected since inelastic tunnel rates are much smaller than coherent electron transfer processes [13]. In addition, the observed tunnel rate is extremely sensitive to the shape of the electron wavefunctions in the SET and QDs and the crystallographic orientation, which could cause a large difference in the tunnel rates. We also note that it is difficult to make a direct comparison to previously measured or theoretical results for single donors [20,21,24] since both the absolute electron and donor numbers are not known for our device. However, the t c values obtained from this work are consistent with previously reported values with slightly different donor numbers and inter-dot distances [13,16].
In summary, we have demonstrated a relatively simple method to determine the tunnel coupling between adjacent QDs or donors that are weakly tunnel coupled to a reservoir. The method can be applied to any system with a charge sensor where tunnel times can be measured and can also be used to obtain the minimum electron temperature and lever arms. The simplicity of the technique makes it ideal as a characterisation tool for future experiments examining multi-donor exchange interactions. Combining this method with atomic-precision lithography using a STM we can investigate the relationship between the inter-dot distance and the exchange coupling predicted from theoretical calculations [20,21,24].