Valley Phase and Voltage Control of Coherent Manipulation in Si Quantum Dots

Abstract

With any roughness at the interface of an indirect-bandgap semiconducting dot, the phase of the valley-orbit coupling can take on a random value. This random value, in double quantum dots, causes a large change in the exchange splitting. We demonstrate a simple analytical method to calculate the phase, and thus the exchange splitting and singlet-triplet qubit frequency, for an arbitrary interface. We then show that, with lateral control of the position of a quantum dot using a gate voltage, the valley-orbit phase can be controlled over a wide range, so that variations in the exchange splitting can be controlled for individual devices. Finally, we suggest experiments to measure the valley phase and the concomitant gate voltage control.The physics of valleys in Si is complicated, both theoretically and experimentally. On the theory side, calculations of the effects of valleys, especially with rough interfaces, often requires very large atomistic simulations; those simulations can sometimes make it more difficult to achieve a simple, intuitive understanding of the underlying physics. In this paper, we present a very simple and intuitively-appealing framework to understand and calculate the effect of the phase of the complex valley-orbit coupling in devices with rough interfaces. On the experimental side, the effects of the valleys can be bound up with spin and orbital effects, thus making effective classical and coherent control of the devices more challenging. We propose a simple experimental method (lateral gate voltages) to both analyze and to control these confounding experimental effects when they arise from the complex phase.Quantum coherent manipulation of electrons in Si has been a very active field of study recently [1][2]. The advantages of Si include low spin-orbit coupling and the ability to isotopically enrich 28 Si, both of which will tend to reduce the decoherence of spin qubits. In addition, the integration with CMOS classical circuitry holds the potential for monolithically integrating qubits with control circuits . The experimental work most relevant for our study is Shi et al [10], which showed experimentally a shift in the single-triplet splitting J for two electrons on a single dot, where J is dominated by the energy difference between single-particle ground and excited valley-orbit states in Si/SiGe. In particular, they measured J as a function of lateral shift using gate voltages; they showed about a 20 % shift, and interpreted it as coming from the change in single-particle valley-orbit energies arising from a rough interface (they solved this for the toy model of a single atomic step).In the vicinity of an interface, the lowest energy valleys are typically perpendicular to the interface, and are split by the valley-orbit coupling, which is defined aswhere|z , |z are the bare valley states centered at k z = ±k 0 = ±(0.85)2π/a 0 , V is the interfacial potential energy, F is the applied electric field, and a 0 is lattice constant; this leads to eigenstates which are symmetric superpositions of...

Quadrupolar Exchange-Only Spin Qubit

“…In the effective mass (EM) approximation the orbital ground state electron wavefunctions of a multi-valley system with valley ξ = z,z residing in quantum dot q = l, c, r is given by [69][70][71][72][73][74][75]…”

“…In realistic devices, however, miscuts, atomistic steps at the interface, or other effects couple the valley and orbital degrees of freedom. In general, the valley-orbit coupling can be described in the basis of each dot {χ zs q , χz s q , χ zp q , χz p q } by [75,77,81]…”

“…The second effect arises from local differences in the orientation of the z-valleys due to atomistic steps in the interface. The atomistic interface steps locally change the orientation of the valley pseudo-spin giving rise to a phase difference between valley pseudo-spin of electrons at different lateral position [75]. As a result, the valley pseudo-spins in the different quantum dots can have completely different phase factors φ v q with q = l, c, r. Setting φ v c = 0, atomistic steps give rise to |t l | ∼ |t l tan(φ v l /2)| and |t r | ∼ |t r tan(φ v r /2)| which yields identical tunnel couplings for φ v l = φ v r = π/2.…”

“…Tuning can be achieved by changing the position of the electron wavefunctions either by moving the quantum wells by the electrostatic gates or by adjusting the tunnel-barriers. [69,75]. Refs.…”

Valley Filtering in Spatial Maps of Coupling between Silicon Donors and Quantum Dots

“…Exchange coupling plays a key role in these proposals [1,2] and has been employed experimentally to couple spins over short distances [5,11], and to define multi-spin qubits [4,6,7,9,10,12] that can be coupled over larger distances via electric interactions [6], as also expected for spin-orbit qubits [13][14][15][16][17]. Because of the importance of exchange interactions, the impact of silicon's valley degrees freedom on electron tunneling and exchange has been the subject of many theoretical studies [18][19][20][21][22][23][24][25][26][27][28]. Notably, small changes in donor position [18][19][20][21][24][25][26] and QD surface roughness [27,28] are expected to produce large modulations of exchange coupling, affecting two-qubit gate fidelities, owing to the lattice-aperiodicity of the valley wavevector.…”