Joule heating and electronic Gurzhi effect in hydrodynamic differential transport in an electron liquid

This paper investigates hydrodynamic electron transport in GaAs/AlGaAs quantum wells, demonstrating that a dc-current-induced resistance valley arises from Joule heating and confirming that the resulting viscosity resistivity follows a $T^{-2}$ dependence consistent with the electronic Gurzhi effect.

The Gist

Imagine a crowded dance floor where thousands of people (electrons) are moving around. Usually, in a standard electrical wire, these people bump into walls, furniture, and each other constantly, moving in a chaotic, slow shuffle. This is what we call "Ohmic" or normal resistance.

But in this paper, the scientists created a special, super-smooth dance floor (a high-quality semiconductor) where the people are so skilled and the floor so slippery that they rarely bump into the walls. Instead, they mostly bump into each other. When this happens, they stop acting like individuals and start moving together like a thick, sticky fluid—like honey or water flowing through a pipe. This is called Hydrodynamic Electron Flow.

Here is the simple breakdown of what the scientists discovered in this "electron fluid":

1. The "Traffic Jam" That Disappears

When the scientists sent a small, steady stream of people through this fluid, they noticed a strange pattern. If they applied a magnetic field (like a giant invisible hand trying to push the dancers sideways), the flow usually gets easier in a specific way, creating a "dip" or a valley in the resistance.

However, when they turned up the volume and pushed a stronger current (more people running faster), something weird happened:

• The smooth "valley" in the flow started to flatten out.
• Eventually, the fluid stopped acting like a thick liquid and started acting like a normal, chaotic gas again.

The Analogy: Imagine a river flowing smoothly (hydrodynamic). If you suddenly pour a massive amount of water in very quickly, the river gets turbulent and chaotic. The "smoothness" (viscosity) disappears because the water is moving too fast to stay organized.

2. The "Overheating" Secret (Joule Heating)

The big question was: Why did the smooth flow break down when they pushed harder?

The scientists realized it wasn't just about speed; it was about heat.

• When you push electrons hard, they get "friction" from each other, which heats them up.
• Think of it like rubbing your hands together. The faster you rub, the hotter your hands get.
• In this experiment, the electrons got so hot that they forgot how to dance in a coordinated fluid. They started bumping into the walls and the "floor" again, turning back into a normal, messy electrical current.

The paper calls this the Joule Heating Effect. The strong current heated the "electron liquid," changing its temperature and breaking the special fluid behavior.

3. The "Gurzhi Effect" (The Viscosity Rule)

There is a famous rule in physics called the Gurzhi Effect. It basically says: "The hotter the electron fluid gets, the less 'sticky' (viscous) it becomes."

The scientists proved this rule works even when you heat the fluid by pushing a strong current through it. They showed that the "stickiness" (viscosity) of the electron liquid drops dramatically as the current increases, exactly as if the liquid had been heated by a fire.

4. Why Does This Matter?

This research is like finding a new way to measure temperature without a thermometer.

• The Problem: It's very hard to measure the temperature of electrons inside a tiny chip because they are so small and fast.
• The Solution: The scientists found that by watching how the "smoothness" of the electron flow changes when they push a current, they can calculate exactly how hot the electrons are getting.

Summary

Think of the electrons as a school of fish.

• Normal wires: The fish are scared, bumping into rocks and each other, swimming in a panic.
• This experiment: The fish are in a calm, clear pool, swimming in a perfect, synchronized school (hydrodynamic flow).
• The discovery: When the scientists told the fish to swim really fast (high current), the school broke apart. The fish got too hot and agitated (Joule heating) and started bumping into everything again.

The paper shows us that by watching how the "school of fish" breaks up, we can learn how to control electricity better in future super-fast computers and understand the hidden heat inside electronic devices.

Technical Summary

1. Problem Statement

The study addresses the behavior of high-mobility two-dimensional electron gases (2DEGs) in the hydrodynamic regime, where electron-electron (e-e) collisions dominate over scattering from impurities or phonons. While the hydrodynamic transport of electrons (resembling viscous fluid flow) is well-understood under low-bias conditions, the effects of strong external DC electric fields (bias) on these electron liquids remain an open question.

Specifically, the authors investigate:

• How a strong DC current bias alters the equilibrium state of the electron liquid.
• The interplay between Joule heating (raising electron temperature, $T_e$) and the viscous transport properties (specifically the second momentum relaxation time, $\tau_{2,ee}$).
• The mechanism behind the observed nonlinear differential resistance features, particularly the transition from hydrodynamic to Ohmic regimes under high current.

2. Methodology

Sample Preparation:

• Experiments were conducted on high-quality GaAs/AlGaAs quantum wells grown via Molecular Beam Epitaxy (MBE).
• Three wafers (A, B, C) were used with high electron mobilities ($\mu \sim 3-5 \times 10^6 \, \text{cm}^2/\text{V}\cdot\text{s}$) and carrier densities $n_e \sim (2-3) \times 10^{11} \, \text{cm}^{-2}$.
• Hallbar devices were fabricated with a channel width $w = 4 \, \mu\text{m}$, designed such that the mean free path $l > w$ (ballistic/Ohmic condition) but the e-e correlation length $l_{ee} < w$ (hydrodynamic condition).

Experimental Setup:

• Environment: Measurements were performed in a cryogen-free VTI cryostat at a base temperature of $1.5 \, \text{K}$ with a vector magnet (up to $\pm 8 \, \text{T}$).
• Measurement Configurations:
  1. U-shape (U-turn) configuration: Current flows in a U-turn loop; voltage probes are on the opposite side. This setup is sensitive to non-local viscous effects.
  2. Ordinary (4-terminal) configuration: Standard source-drain current flow.
• Stimuli: A combination of low-frequency AC current (for lock-in detection) and a tunable DC current bias ($I_{dc}$ from $0.1 \, \mu\text{A}$ to $15 \, \mu\text{A}$) was applied to probe differential resistance ($R^*_{xx}$).

Theoretical Framework:

• The analysis utilizes hydrodynamic equations (Navier-Stokes for charge-compensated fluids) and Boltzmann transport theory.
• The authors model the system using a "heated electron" approximation, where the DC bias creates a non-equilibrium state characterized by an electron temperature $T_e$ distinct from the lattice temperature $T_l$.

3. Key Results

A. Differential Resistance and Lorentzian Profiles

• At zero magnetic field ($B \approx 0$) and low currents, the differential resistance exhibits a Lorentzian profile (a peak in resistance), characteristic of viscous electron flow.
• As the DC current ($I_{dc}$) increases:
  1. The Lorentzian peak broadens and flattens, eventually disappearing.
  2. A distinct dip (valley) appears at $B=0$, which deepens with increasing current before vanishing at very high currents ($>6 \, \mu\text{A}$).
  3. In the U-turn configuration, the system transitions from a hydrodynamic regime (viscous dominated) to an Ohmic regime as the current increases.

B. Joule Heating and Electron Temperature ($T_e$)

• The observed transport valley is attributed to Joule heating. The DC current heats the electron gas, raising $T_e$ significantly above the lattice temperature $T_l$.
• The second momentum relaxation rate ($1/\tau_{2,ee}$), which governs viscosity, shows a quadratic dependence on the current density ($j_{dc}^2$).
• The authors derive a relationship where the relaxation rate follows:
  $$\frac{1}{\tau_{2,ee}(j_{dc})} = \frac{1}{\tau_2} + A_{ee}^j j_{dc}^2$$
  This quadratic dependence mirrors the temperature dependence of e-e scattering in the Gurzhi effect ($1/\tau_{2,ee} \propto T_e^2$).

C. The Electronic Gurzhi Effect

• The study confirms that the viscosity resistivity ($\Delta \rho$) is proportional to $T_e^{-2}$.
• By bridging the current-dependent relaxation rate (Eq. 5) and the temperature-dependent relaxation rate (Eq. 8), the authors establish a quantitative link between the applied DC current and the effective electron temperature.
• The data shows that at high currents, the electron liquid effectively "cools" in terms of viscosity (viscosity drops), causing a transition from hydrodynamic flow to Ohmic flow.

D. Boundary Condition Dependence

• The transport features are highly sensitive to measurement geometry.
  • U-turn configuration: Shows a clear Lorentzian peak that evolves into a valley.
  • Ordinary configuration: Shows a "M-shaped" curve (peak with a central dip) at low currents, evolving into a "W-shaped" curve at high currents.
• This confirms the non-local nature of hydrodynamic transport and the influence of boundary slip lengths ($l_s$).

4. Key Contributions

1. Quantification of Joule Heating in Hydrodynamics: The paper provides a rigorous quantitative demonstration that the nonlinear transport features in the hydrodynamic regime are driven by Joule heating, which modifies the electron temperature and consequently the e-e scattering rate.
2. Observation of the Gurzhi Effect via DC Bias: The authors demonstrate that the electronic Gurzhi effect (where resistivity scales as $T^{-2}$ due to e-e collisions) can be induced and controlled simply by varying the DC current bias, without changing the lattice temperature.
3. New Method for Probing $T_e$: The study establishes a novel methodology to monitor the electron temperature ($T_e$) in ultra-clean electron systems by analyzing differential transport traces under DC bias. This is particularly useful for systems where direct thermometry is difficult.
4. Phase Transition Characterization: The work details the transition from the hydrodynamic regime to the Ohmic regime driven purely by drift velocity (current), providing a clear experimental signature of this phase transition.

5. Significance

• Fundamental Physics: The results deepen the understanding of non-equilibrium states in quantum fluids. It clarifies how external fields manipulate the collective behavior of electrons, validating hydrodynamic theories under strong non-linear conditions.
• Device Engineering: Understanding the boundary conditions and the transition between hydrodynamic and Ohmic regimes is crucial for designing next-generation low-power electronic devices and high-speed interconnects where viscous effects can be exploited (e.g., superballistic flow).
• Thermometry: The proposed method offers a robust, transport-based technique to estimate electron temperatures in 2DEGs, which is vital for characterizing quantum materials and topological systems where electron-phonon coupling is weak.
• Validation of Theory: The agreement between the experimental data and the hydrodynamic equations (including the specific $T^{-2}$ dependence of viscosity) strongly supports the theoretical models of viscous electron fluids in semiconductors.

In summary, this paper successfully decouples the effects of Joule heating from intrinsic transport mechanisms, revealing that the "electronic Gurzhi effect" is the dominant factor governing the nonlinear differential resistance in hydrodynamic electron liquids under DC bias.