Relaxation of excited spin, orbital, and valley qubit states in ideal silicon quantum dots

Abstract

We expand on previous work that treats relaxation physics of low-lying excited states in ideal, single electron, silicon quantum dots in the context of quantum computing. These states are of three types: orbital, valley, and spin. The relaxation times depend sensitively on system parameters such as the dot size and the external magnetic field. Generally, however, orbital relaxation times are short in strained silicon (10 −7 to 10 −12 s), spin relaxation times are long, (10 −6 to 1 s), while valley relaxation times are expected to lie in between. The focus is on relaxation due to emission or absorption of phonons, but for spin relaxation we also consider competing mechanisms such as charge noise. Where appropriate, comparison is made to reference systems such as quantum dots in III-V materials and silicon donor states. The phonon bottleneck effect is shown to be rather small in the silicon dots of interest. We compare the theoretical predictions to some recent spin relaxation experiments and comment on the possible effects of non-ideal dots.

Spin relaxation in a Si quantum dot due to spin-valley mixing

“…For comparison, we include in our discussion below spin relaxation due to intra-valley SO mixing (higher energy p-orbitals are involved), which has been studied extensively in the literature, especially for spin qubit in GaAs QD. 9,[33][34][35][39][40][41][42][43][44][45] For spin qubit in Si QD, this intra-valley SO mixing induced spin relaxation is also present, and is important in high B-field due to the stronger B-field dependence. 33,34 We use the existing results in the literature, and the corresponding spin relaxation rate is [33][34][35] …”

“…33 The spectrum of the phonon noise in the x-direction is therefore ( 2π 0 dϕ cos 2 ϕ = π) S E xx (ω) = Re 1 2π If the dipole approximation e i q · r ≈ 1 + i q · r is employed (for most spin qubit applications, the dipole approximation should be valid), so that f j (ω, θ) = 1, the relaxation rate would have taken the form given in Ref. 34. Furthermore, the temperature T of the lattice vibration is normally very low (T < 1 K), so that 2N ω + 1 = coth( ω/2k B T ) ≈ 1, in which case the spectrum of phonon noise shows a nice ω 5 dependence.…”

“…The relaxation rate through the intra-valley SO mixing is dominant in the high-magneticfield regime, which shows a B 7 0 dependence before the phonon bottleneck takes effect and the curves bend downward from the B 7 0 line. [33][34][35] The phonon bottleneck effect is due to the averaging of electron-phonon interaction matrix element for high-frequency phonons. This reduction in the effective coupling strength causes the spin relaxation rate to decrease from the B 7 0 curve in Fig.…”

“…Furthermore, a unified treatment is given here for both phonon and Johnson noise, and the phonon bottleneck effect is taken into account in a simplified manner. 34 To obtain the results for phonon noise, we need the correlation of the electric field E( r) = ∇U ph ( r)/e, which can be derived based on the electron-phonon interaction potential U ph ( r), 34,35 …”

“…On the other hand, it has been shown experimentally and theoretically that the presence of valleys in Si can significantly modify spin relaxation through spin-valley mixing, and a relaxation hot spot appears at the degeneracy point where the Zeeman splitting matches the valley splitting. 9,34 In this paper, we study spin relaxation of a single QDconfined electron in Si due to the presence of electrical noises (including Johnson noise, phonon noise, and the 1/f charge noise). One relaxation mechanism involves the mixing of spin and valley states, which should be particularly important when Zeeman energy E Z is comparable with valley splitting E VS .…”