Disentangling orbital and confinement contributions to $g$-factor in Ge/SiGe hole quantum dots

This study disentangles orbital and confinement contributions to the out-of-plane $g$-factor in Ge/SiGe hole quantum dots by comparing excitation and addition spectra, revealing gate-tunable $g$-factors that resolve previous discrepancies and highlight their potential for all-electric qubit control.

The Gist

The Big Picture: Tuning a Quantum Piano

Imagine you are trying to build a super-fast computer using tiny, invisible particles called holes (which are essentially the absence of an electron) trapped inside a microscopic box called a quantum dot. This box is made of Germanium and Silicon.

To make this computer work, you need to control the "spin" of these holes, which acts like a tiny magnet. How strong this magnet is depends on a number called the g-factor. Think of the g-factor as the "volume knob" for the magnet. If you know the exact volume, you can turn the knob precisely to play the right musical note (perform a calculation).

However, in this Germanium material, the "volume knob" is tricky. It's not just one knob; it's a complex machine where the magnet's strength is influenced by two things happening at once:

1. The Spin: The intrinsic magnetism of the hole itself.
2. The Orbit: How the hole is dancing or moving around inside the box.

Because of a phenomenon called Spin-Orbit Coupling, these two things are glued together. If the hole changes its dance move (orbit), the volume knob (g-factor) changes too. This makes it very hard to figure out what the "pure" magnetism is, which is a problem for building reliable quantum computers.

The Problem: Two Different Rulers

The researchers noticed a confusing problem. When they tried to measure the volume knob (g-factor) using two different methods, they got different answers:

• Method A (The "Addition" Method): They counted how many holes were in the box one by one. This is like weighing a backpack by adding one book at a time and seeing how much the scale jumps.
• Method B (The "Excitation" Method): They kept the number of holes the same but gave them a little energy "kick" to see how they jumped to a higher energy level. This is like tapping a guitar string to hear its specific note without changing the string itself.

The Conflict: Method A and Method B gave different numbers for the volume knob. The scientists needed to know: Is the knob actually changing, or are our measuring tools just seeing different things?

The Solution: Disentangling the Knot

The team realized that the "Addition" method was being tricked. When you add a new hole to the box, the whole system rearranges. The "dance floor" (the confinement potential) changes shape, and the holes push against each other. This extra movement (orbital contribution) was messing up the measurement, making the volume knob look different than it really was.

By comparing the two methods side-by-side, they managed to untangle the two effects:

1. The Pure Spin: The actual magnetic strength of the hole.
2. The Orbital Noise: The extra wiggle caused by the hole's movement and the changing shape of the box.

They found that the orbital movement was contributing about 10% to the total measurement. This explained why the two methods were giving different results. It wasn't that the physics was broken; it was just that one method was measuring the "pure spin" while the other was measuring "spin + dance moves."

The Surprise: The Volume Knob is Tunable!

Here is the most exciting part. Once they understood how the system worked, they discovered they could change the volume knob just by turning a voltage dial.

Imagine you have a piano, and instead of just pressing keys, you can slide the entire piano across the room. As you slide it, the pitch of the notes changes automatically.

In their experiment, by slightly adjusting the voltage on the gates (the walls of the box), they could shift the position of the hole. This shift changed the local environment (strain and electric fields), which in turn changed the g-factor by up to 15%.

Why Does This Matter?

1. Fixing the Confusion: Now, scientists know that if they measure a g-factor, they need to be careful about how they measured it. Are they seeing the pure spin, or is the hole's orbit messing it up? This helps everyone in the field compare their data correctly.
2. All-Electric Control: This is the "holy grail" for quantum computing. Usually, to control a quantum bit (qubit), you need big, clunky magnets or microwave pulses. But because this team found they could change the g-factor just by turning a voltage knob (electricity), they might be able to control these quantum bits using only electricity. This would make quantum computers smaller, faster, and easier to build.

Summary Analogy

Think of the quantum dot as a trampoline.

• The hole is a jumper.
• The g-factor is how high the jumper bounces.
• The Spin is the jumper's natural jumping ability.
• The Orbit is the tension of the trampoline springs.

Previously, scientists were confused because when they added more jumpers (holes), the springs stretched differently, changing the bounce height. They didn't know if the jumper got stronger or if the springs just changed.

This paper figured out exactly how much the springs (orbit) were affecting the bounce. Even better, they discovered that by moving the jumper to a different spot on the trampoline, they could change the bounce height without touching the jumper at all—just by shifting the trampoline itself. This means we can control the quantum computer using simple electrical switches instead of giant magnets.

Technical Summary

1. Problem Statement

Spin qubits in semiconductor quantum dots are typically operated in the lowest orbital state to minimize interference. However, in hole-based systems (specifically Germanium/SiGe), strong spin-orbit coupling (SOC) intrinsically links spin and orbital degrees of freedom. This coupling significantly influences the g-factor, a critical parameter for qubit control and manipulation.

Current challenges include:

• Discrepancies in Measurement: Different spectroscopic methods (e.g., Coulomb Blockade Addition Spectroscopy vs. Excitation Spectroscopy) often yield different g-factor values for the same system.
• Theoretical Gaps: Existing theories capture the general magnitude of g-factors but fail to reproduce precise values or angular dependencies, likely due to complex orbital structures, confinement potentials, and local strain.
• Lack of Understanding: It is unclear how much of the observed g-factor is due to the "bare" spin Zeeman effect versus orbital contributions or many-body interactions (screening, exchange).

2. Methodology

The authors investigated gate-defined quantum dots in a strained Ge/SiGe heterostructure (20 nm quantum well) using two nominally identical double quantum dot (DQD) devices. They employed an integrated charge sensor to probe the first few hole states.

The study utilized two complementary spectroscopic techniques to disentangle contributions:

1. Coulomb Blockade Addition Spectroscopy (CBAS):
   • Measures many-body energy differences between successive hole numbers ($N$ and $N+1$).
   • Extracts addition energies ($E_{add}$) by sweeping gate voltages and detecting charge transitions.
   • Separates charging energy ($E_C$) from single-particle level spacing ($\Delta \epsilon$) to approximate the single-particle spectrum.
2. Pulsed Excited-State Spectroscopy (PESS):
   • Measures excitation spectra at a fixed occupancy ($N$).
   • Uses high-frequency AC pulses on the plunger gate to shift energy levels relative to the reservoir Fermi level, allowing the detection of excited states (spin and orbital) without adding/removing holes.
   • Directly probes the "bare" spin splitting and orbital excitations.

Key Experimental Controls:

• Magnetic Field: Applied perpendicular to the quantum well ($B_\perp$) and at various tilt angles to probe anisotropy.
• Gate Tuning: Varied interdot barrier voltages ($V_{B12}$) and neighboring plunger gates ($V_{P2}$) to shift the quantum dot wavefunction spatially and modify the confinement potential without changing the hole number.

3. Key Contributions

• Methodological Separation: Successfully distinguished between pure Zeeman (spin) g-factors and orbital contributions by comparing CBAS (many-body) and PESS (fixed occupancy) data within the same device.
• Quantification of Orbital Effects: Demonstrated that orbital wavefunction changes contribute up to 10% to the apparent Zeeman splitting measured between states with distinct orbital characters.
• Gate-Tunability: Discovered and quantified the electrical tunability of the g-factor, showing it can be modulated by shifting the wavefunction relative to the confinement potential.
• Resolution of Discrepancies: Provided a physical explanation for why g-factors extracted via CBAS differ from those via PESS and why g-factors vary between different orbital states (e.g., $N=1$ vs. $N=3$).

4. Key Results

A. Disentangling Spin and Orbital Contributions

• CBAS vs. PESS Comparison:
  • CBAS (measuring $g_{\epsilon_2-\epsilon_1}$ and $g_{\epsilon_4-\epsilon_3}$) yielded values around 11.1 and 9.6, respectively. The reduction in the second pair was attributed to many-body effects and orbital mixing.
  • PESS (measuring pure spin transitions $g_{\uparrow o_1 - \downarrow o_1}$) yielded a value of ~9.86, consistent with the first CBAS pair but lower than the second.
  • Crucial Finding: When PESS measured transitions involving different orbitals (e.g., ground state to first excited orbital, $g_{\uparrow o_1 - \downarrow o_2}$), the g-factor increased significantly to ~13.7.
• Orbital Contribution: By comparing the linear Zeeman splitting of pure spin states against the splitting between different orbital states, the authors estimated that orbital wavefunction changes contribute ~10% to the total measured g-factor in the 0–1 T range. This explains the non-linearities and discrepancies between methods.

B. Gate-Tunability of the g-Factor

• Interdot Barrier Tuning ($V_{B12}$): Changing the barrier voltage shifted the confinement potential spatially. The CBAS g-factor ($g_{\epsilon_2-\epsilon_1}$) showed a tunability of up to 15%.
• Plunger Gate Tuning ($V_{P2}$): By sweeping the neighboring plunger gate, the hole wavefunction was delocalized across the barrier and relocalized in the second dot.
  • The pure spin PESS g-factor ($g_{\uparrow o_1 - \downarrow o_1}$) remained relatively constant.
  • However, the triplet-state g-factor ($g_{T_0-T_-}$) varied by approximately 20%.
• Mechanism: This sensitivity is attributed to changes in the local potential and strain as the hole's position shifts relative to the interdot barrier, confirming that the g-factor is highly sensitive to the local electrostatic environment.

C. Anisotropy and Orbital Structure

• The out-of-plane g-factor is large (10–12), while the in-plane g-factor is very small (0.01–1), confirming strong anisotropy.
• Discontinuities observed in the addition spectrum (CBAS) at level crossings were attributed to changes in the confinement potential and orbital energies as hole number increases, which are less pronounced in PESS (fixed occupancy).

5. Significance and Impact

• Qubit Control: The demonstration of 15–20% gate-tunability of the g-factor is a major breakthrough. It suggests that all-electric qubit manipulation (controlling spin states purely via gate voltages without magnetic field modulation) is feasible in Ge/SiGe systems.
• Characterization Standards: The paper establishes that comparing g-factors across different studies requires extreme caution. One must specify whether the value is derived from addition spectra (many-body) or excitation spectra (single-particle) and which orbital states are involved.
• Model Validation: The results highlight the limitations of simple single-particle models (like Fock-Darwin) and the necessity of including full Luttinger-Kohn Hamiltonians and many-body effects to accurately predict g-factors in hole systems.
• Scalability: Understanding how g-factors vary with hole number and orbital state is essential for scaling up quantum dot arrays, where uniformity and precise control are paramount.

In summary, this work provides a rigorous experimental framework for separating spin and orbital contributions to the g-factor in Ge hole quantum dots, revealing significant gate-tunability that opens new pathways for electrically controlled quantum computing.