🔬 mesoscale physics

Complete measurement of tunnel- and valley-coupling parameters in a silicon double quantum dot

This paper presents a complete characterization of intravalley and intervalley tunnel couplings, including their complex valley phases, in a silicon double quantum dot, demonstrating how these phases govern measurable energy gaps and fill a critical gap in understanding sample-wide variations of key physical parameters like spin-orbit coupling and valley-orbit mixing.

Original authors: Daniel J. King, Minyoung Kim, J. Reily, Jonathan C. Marcks, Mark Friesen, Benjamin D. Woods, M. A. Eriksson

Published 2026-07-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniel J. King, Minyoung Kim, J. Reily, Jonathan C. Marcks, Mark Friesen, Benjamin D. Woods, M. A. Eriksson

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a silicon chip not as a flat, boring piece of sand, but as a tiny, bumpy landscape where electrons are like little hikers. In the world of quantum computing, we want these hikers to carry information. Usually, we think of them just hopping from one hill (a quantum dot) to another. But in silicon, there's a twist: the hills have hidden "valleys" inside them, like secret tunnels running deep underground.

This paper is about mapping out exactly how these hikers move between hills and through those secret tunnels. The researchers, working with a device from Intel, discovered that they can't just measure how fast the hikers jump; they have to measure the rhythm and direction of their jump, which they call "valley phases."

The Hidden Rhythm of the Jump
Think of the two quantum dots as two rooms in a house. Usually, we just care if a person can walk from Room A to Room B. But in this silicon house, the walls are made of a special material (a mix of Silicon and Germanium) that is slightly messy, like a floor covered in random pebbles. This messiness creates two types of tunnels:

The "Safe" Tunnel: Where the hiker stays on the same "floor" of the valley.
The "Cross-Valley" Tunnel: Where the hiker jumps from one valley to a completely different one while moving between rooms.

The big surprise here is that these tunnels aren't just simple doors; they have a complex "phase," which is like a specific timing or rotation in the hiker's step. If the timing is off, the hiker might get stuck or take a weird path. The authors measured these phases for the first time in a double-dot system, showing that they change depending on exactly where the hiker is standing in the house.

The "Ghost" Connection
For a long time, scientists thought that when a hiker moved between rooms, they only interacted with the "safe" tunnel. They assumed the "cross-valley" tunnel between the two rooms was zero or negligible. This paper explicitly argues against that idea. The researchers found that there is a significant "ghost" connection—a cross-valley tunnel between the two dots—that cannot be ignored.

They proved this by measuring four different "gaps" (energy levels where the hiker gets stuck) in the system. If you only measure three, you miss the ghost connection. By measuring all four, they showed that this cross-valley coupling is real and important, changing the behavior of the system in ways previous models missed.

The Moving Target
Here is where it gets playful: the researchers didn't just take one picture; they moved the hikers around. By changing the voltage on a "screening gate" (think of it as a dimmer switch that moves the rooms slightly), they shifted the position of the quantum dots by tens of nanometers.

As the dots moved, the "rhythm" (the valley phase) changed dramatically. Why? Because the dots were stepping over different patterns of the messy Germanium pebbles in the floor. The authors found that the phase depends entirely on the local atomic structure right under the dot. If you move the dot just a tiny bit, the rhythm changes. This explains why different devices on the same chip might behave differently—they are standing on different patches of the messy floor.

How Sure Are They?
The authors are very confident in their measurements. They didn't just guess; they used a technique called "delta-axis spectroscopy" (DAXS) to map the energy levels with extreme precision. They measured the gaps between energy states and fit them to a mathematical model (a four-level Hamiltonian).

To make sure their model wasn't just a lucky guess, they ran computer simulations. They simulated the messy atomic floor and the quantum dots, then "measured" the simulation just like they did in the real lab. The results matched up perfectly, confirming that their method works and that the "ghost" connection they found is real. They even calculated that the ratio of this ghost connection to the main tunnel is about 0.25, which is a big deal—it's not a tiny, ignore-able number.

What They Didn't Solve
It's important to note what this paper doesn't do. It doesn't say they have fixed the problem of device variability. In fact, they show that the variability is real and caused by the random atomic messiness. They also don't claim to have built a perfect quantum computer yet. Instead, they provided a "map" and a "compass" (the valley phases) that future engineers need to navigate these messy silicon landscapes.

The Takeaway
In short, this paper is like discovering that a video game character doesn't just walk in a straight line; they have a hidden dance move that changes depending on the texture of the floor they are standing on. The authors measured this dance move, proved it exists, and showed that if you want to build a reliable quantum computer, you can't ignore the dance. You have to know exactly where your dots are standing and what the rhythm of their jump is. Without this knowledge, the quantum information might get lost in the shuffle.

Problem
Silicon quantum dots are a leading platform for quantum information processing due to their scalability and high-fidelity operations. However, large-scale implementation faces challenges from device variability, particularly arising from random-alloy disorder in Si/SiGe heterostructures. A critical aspect of silicon qubit physics is tunneling, which connects not only qubit states but also the valley minima in the conduction band. These valley degrees of freedom introduce complex phase differences (valley phases) that govern tunnel couplings. While the magnitudes of valley splittings and tunnel couplings are routinely measured, a complete characterization including the complex phases of both intravalley and intervalley tunnel couplings has been lacking. Specifically, the role of interdot-intervalley coupling ( $\Delta_{RL}$ ) and its associated phase ( $\phi_{RL}$ ) has been largely overlooked in existing models. Without knowledge of these phases, understanding sample-wide variations in valley couplings and dependent physical parameters (such as spin-orbit coupling and Landé g-factors) remains incomplete.

Methodology
The authors experimentally characterize a silicon double quantum dot (DQD) fabricated in an Intel Tunnel Falls Si/SiGe device. The study employs a technique called delta-axis spectroscopy (DAXS), which maps the energy dispersion of the DQD as a function of detuning.

Experimental Setup: The DQD is formed using plunger gates (PL, PR) and controlled by barrier gates (BL, BC, BR) and a screening gate (SG). The device is tuned near the (0,0)-(1,0)-(0,1) charge-occupation triple point.
DAXS Measurement: Periodic square-wave voltage pulses are applied along the $\delta$ axis (shifting both dot energies together) to induce electron loading/unloading. The resulting charge sensor current reveals the energy spectrum.
Data Acquisition: The authors perform a two-step procedure: first, a global DAXS scan locates all four anticrossings between the ground and excited valley states of the two dots. Second, high-resolution one-dimensional DAXS scans are acquired near these anticrossings to determine energy levels with sufficient precision.
Theoretical Modeling: The data is fitted to a four-level Hamiltonian model. This model includes a single spatial orbital per dot with a valley degree of freedom ( $\pm z$ ). The Hamiltonian is transformed from the $\pm z$ -valley basis to a ground/excited-valley basis, yielding expressions for intravalley ( $\tilde{t}_{gg}, \tilde{t}_{ee}$ ) and intervalley ( $\tilde{t}_{ge}, \tilde{t}_{eg}$ ) tunnel couplings in terms of the valley-conserving tunnel coupling ( $t_c$ ), intradot-intervalley couplings ( $\Delta_L, \Delta_R$ ), interdot-intervalley coupling ( $\Delta_{RL}$ ), and their respective phases.
Parameter Extraction: By fitting the measured anticrossing gaps ( $E_{\mu\nu}$ ) to the model, the authors extract six parameters: $t_c$ , $|\Delta_L|$ , $|\Delta_R|$ , $|\Delta_{RL}|$ , the valley phase difference $\Delta\phi$ , and the interdot-intervalley phase difference $\Delta\phi_{RL}$ .
Validation: The authors use electrostatic simulations and effective-mass models with alloy disorder to verify the statistical consistency of the extracted parameters, particularly the ratio $|\Delta_{RL}|/t_c$ .

Key Contributions

Complete Characterization: The paper presents the first full experimental characterization of both the magnitudes and complex phases of intravalley and intervalley tunnel couplings in a Si DQD.
Inclusion of Interdot-Intervalley Coupling: The study explicitly includes and quantifies the interdot-intervalley coupling ( $\Delta_{RL}$ ), demonstrating that it is non-negligible and essential for accurately fitting experimental data.
Valley Phase Mapping: The authors demonstrate that valley phases are not static but evolve as a function of quantum dot gate voltages, which shift the dot positions relative to the underlying atomic disorder.
Gap Ratio Encoding: The work establishes that the ratios of the four anticrossing gaps encode the valley phases and coupling strengths, providing a direct link between measurable observables and the underlying microscopic physics.

Results

Hamiltonian Parameters: The authors successfully extracted all six Hamiltonian parameters. The valley phase difference $\Delta\phi$ was found to vary from approximately $5\pi/16$ to $\pi/2$ as the screening gate voltage ( $V_{SG}$ ) shifted the dot positions.
Role of $\Delta_{RL}$ : The interdot-intervalley coupling $|\Delta_{RL}|$ was found to be significant, with a ratio $|\Delta_{RL}|/t_c \approx 0.25$ across the tested barrier gate voltage range. This coupling breaks the symmetry of the anticrossing gaps (e.g., $E_{ee} \neq E_{gg}$ ), a feature previously unexplained in simpler models.
Dependence on Dot Position: Varying $V_{SG}$ (shifting dot positions) caused significant changes in valley splittings and valley phases, attributed to the sampling of different local atomic distributions of Ge in the quantum well.
Dependence on Barrier Voltage: Varying the barrier gate voltage ( $V_{BC}$ ) changed the tunnel couplings ( $t_c$ and $|\Delta_{RL}|$ ) but left the valley phase difference $\Delta\phi$ nearly constant, while $\Delta\phi_{RL}$ varied significantly. This suggests $\Delta\phi$ is determined by the dot wavefunction overlap with the local alloy disorder, whereas $\Delta\phi_{RL}$ is sensitive to the tunnel barrier region's atomic environment.
Theoretical Consistency: Statistical analysis of the ratio $|\Delta_{RL}|/|\Delta_i|$ showed that experimental values align with the mode of theoretically calculated probability distributions derived from effective-mass models with alloy disorder.

Significance
The paper claims that mapping valley phases is a crucial milestone for Si/SiGe quantum computing. Since interdot tunnel couplings and g-factors depend strongly on valley phases, which vary locally due to random-alloy disorder, understanding these variations is necessary for operating qubits at scale. The ability to measure these parameters using DAXS provides a method to create local maps of valley physics. This capability is a prerequisite for statistically significant sampling of valley parameters across wider portions of quantum wells, which is essential for confirming the existence of deterministically enhanced valley splitting and for improving the reliability of qubit gate operations. The work fills a key gap in understanding sample-wide variations of valley couplings and the physical parameters that depend on them.

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