# Hole-Spin-Echo Envelope Modulations

## TL;DR

This paper presents a theoretical analysis of hole-spin-echo envelope modulation (HSEEM) to measure hyperfine interactions in semiconductor quantum dots, especially in strained silicon, revealing significant modulations useful for qubit characterization.

## Contribution

It provides a general theoretical framework for HSEEM and applies it to boron-acceptor hole spins in silicon, highlighting conditions for observable envelope modulations.

## Key findings

- In unstrained silicon, hyperfine and Zeeman interactions are isotropic, leading to negligible modulations.
- In strained silicon, hyperfine anisotropy causes detectable spin-echo envelope modulations.
- Maximum modulation depth can reach about 10% at moderate magnetic fields (~200 mT).

## Abstract

Hole spins in semiconductor quantum dots or bound to acceptor impurities show promise as potential qubits, partly because of their weak and anisotropic hyperfine couplings to proximal nuclear spins. Since the hyperfine coupling is weak, it can be difficult to measure. However, an anisotropic hyperfine coupling can give rise to a substantial spin-echo envelope modulation that can be Fourier-analyzed to accurately reveal the hyperfine tensor. Here, we give a general theoretical analysis for hole-spin-echo envelope modulation (HSEEM), and apply this analysis to the specific case of a boron-acceptor hole spin in silicon. For boron acceptor spins in unstrained silicon, both the hyperfine and Zeeman Hamiltonians are approximately isotropic leading to negligible envelope modulations. In contrast, in strained silicon, where light-hole spin qubits can be energetically isolated, we find the hyperfine Hamiltonian and $g$-tensor are sufficiently anisotropic to give spin-echo-envelope modulations. We show that there is an optimal magnetic-field orientation that maximizes the visibility of envelope modulations in this case. Based on microscopic estimates of the hyperfine coupling, we find that the maximum modulation depth can be substantial, reaching $\sim 10\%$, at a moderate laboratory magnetic field, $B\lesssim 200\,\mathrm{mT}$.