Out-of-plane angle resolved second harmonic Hall analysis in perpendicular magnetic anisotropy systems

This paper presents an experimental approach using out-of-plane angle-resolved Second Harmonic Hall measurements and anomalous Hall effect-based spin-torque ferromagnetic resonance to quantify damping-like and field-like spin-orbit torque efficiencies in perpendicular magnetic anisotropy systems, revealing a unique magnetization-dependent anomalous field-like torque in Ta/CoFeB bilayers.

The Gist

Imagine you have a tiny, invisible compass needle (the magnetization) sitting inside a sandwich made of different metal layers. Scientists want to figure out exactly how hard they need to push this needle to make it flip direction, which is the key to building faster, smaller computer memory.

This paper is about a new, clever way to measure those "pushes" (called Spin Orbit Torques) without breaking the sandwich.

Here is the breakdown using simple analogies:

1. The Problem: The Invisible Push

In modern electronics, we use electricity to spin electrons. When these spinning electrons hit a magnetic layer, they give it a little nudge.

• The Nudge: There are two types of nudges.
  • The "Damping" Nudge: This is like pushing a swing to keep it moving in the right rhythm.
  • The "Field-Like" Nudge: This is like a sudden gust of wind that tries to tilt the swing sideways.
• The Challenge: To build better computers, engineers need to know exactly how strong these nudges are. But measuring them is tricky because the magnetic layers are incredibly thin (thinner than a human hair), and the signals are very weak.

2. The Old Way vs. The New Way

• The Old Way (In-Plane Sweep): Imagine trying to figure out how a windmill works by only looking at it from the side and rotating it left and right. You get some data, but you miss what happens when the wind blows from above or below. Previous methods mostly looked at the magnetic needle from the side.
• The New Way (Out-of-Plane Sweep): The authors in this paper decided to look at the needle from every possible angle, including tilting it up and down (like a globe spinning on its axis). They call this an "Out-of-Plane Angle Resolved" measurement.

The Analogy:
Think of the magnetic needle as a weather vane on a roof.

• Old Method: You only walk around the house looking at the vane from the ground level. You can see which way it points left or right, but you can't tell if a strong updraft is pushing it up.
• New Method: You take a drone and fly around the vane in a full 360-degree circle, going up, down, and all around. This gives you a complete 3D picture of how the wind (the electric current) is actually pushing the vane.

3. The Experiment: The "Harmonic" Dance

To measure these tiny nudges, the scientists didn't just push the needle once; they made it vibrate.

• They sent a tiny, fast-pulsing electric current through the metal sandwich.
• This current made the magnetic needle wobble back and forth very quickly (like a guitar string vibrating).
• Because the needle is wobbling, the electrical resistance changes slightly in a rhythmic pattern.
• The scientists used a special detector (a "lock-in amplifier") to listen for the second beat of this rhythm (the "Second Harmonic").
  • Why the second beat? The first beat is just the main vibration. The second beat contains the secret information about the strength and type of the push (the torque). It's like listening to a drumbeat to hear the specific rhythm of the drummer's hand, not just the sound of the drum.

4. The Big Discovery: The "Chameleon" Effect

The team tested two different types of metal sandwiches:

1. Platinum/Cobalt (Pt/Co): This behaved exactly as expected. The "wind gust" (Field-Like torque) was steady and predictable, regardless of which way the needle was pointing.
2. Tantalum/CoFeB (Ta/CoFeB): This one was a surprise! They found that the "wind gust" changed its strength depending on which way the needle was pointing.
   • The Metaphor: Imagine a sailboat. Usually, the wind pushes the sail with the same force no matter the angle. But in this Ta/CoFeB system, the wind acts like a chameleon. When the sail is at one angle, the wind is gentle; when it tilts slightly, the wind suddenly becomes a hurricane.
   • This "anomalous" behavior was only visible because they used their new 360-degree drone method. If they had only looked from the side (the old method), they would have missed this weird, changing behavior.

5. Why This Matters

• Better Memory: By understanding exactly how these nudges work, engineers can design computer chips that switch data faster and use less energy.
• New Physics: The discovery that the "wind" changes based on direction in the Tantalum system suggests there are hidden rules of physics at play that we don't fully understand yet. This opens the door for new discoveries.
• Double Check: The authors also showed that their new vibrating method gives the same results as a different, more complex method (called STFMR), proving their new technique is reliable and accurate.

Summary

In short, this paper is about looking at a magnetic needle from every angle to understand how electricity pushes it. They found that while one type of metal sandwich behaves predictably, another has a "chameleon" nature where the push changes strength depending on the angle. This new, 360-degree view helps us build better technology and understand the hidden rules of the magnetic world.

Technical Summary

1. Problem Statement

Spin-orbit torques (SOT) are critical for manipulating magnetization in heavy metal/ferromagnet (HM/FM) bilayers, serving as the mechanism for switching in next-generation spintronic memory and logic devices. Accurate estimation of SOT efficiency—specifically the damping-like ($\xi_{DL}$) and field-like ($\xi_{FL}$) components—is essential for device optimization.

While established techniques like Spin-Torque Ferromagnetic Resonance (STFMR) and Second Harmonic Hall (SHH) measurements exist, they face limitations in Perpendicular Magnetic Anisotropy (PMA) systems:

• STFMR: Standard Anisotropic Magnetoresistance (AMR)-based STFMR is unsuitable for PMA systems because PMA layers are extremely thin, yielding negligible AMR signals. While Spin Hall Magnetoresistance (SMR)-based STFMR exists, it is complex.
• SHH: Conventional SHH methods typically rely on low-range field sweeps or in-plane (azimuthal) angle sweeps. These approaches often fail to fully capture the dependence of SOT on the magnetization direction in PMA systems and may miss unique angular dependencies of the torque terms.

The authors address the need for a robust, comprehensive characterization method that can extract SOT efficiencies and reveal the angular dependence of torque terms in PMA systems.

2. Methodology

The authors propose and demonstrate an Out-of-Plane (OOP) Angle Resolved Second Harmonic Hall (SHH) measurement technique.

• Experimental Setup:

  • Samples: Two PMA systems were studied:
    1. Pt/Co: Ta(1)/Pt(5)/Co(1)/MgO(1.5)/Ta(1.5) (thickness in nm).
    2. Ta/CoFeB: Ta(5)/CoFeB(1.3)/MgO(1.5)/Ta(1.5).
  • Device Geometry: Hall bar devices ($10 \times 20 \, \mu m^2$) patterned via optical lithography.
  • Measurement Protocol: A low-frequency AC current ($1277 \, \text{Hz}$) is passed through the heavy metal layer. An external DC magnetic field is swept through a full 360° polar angle ($\theta_H$) in both longitudinal ($xz$-plane) and transverse ($yz$-plane) configurations.
  • Data Acquisition: First ($V_{1\omega}$) and second ($V_{2\omega}$) harmonic Hall voltages are recorded using a lock-in amplifier. DC voltage ($V_{dc}$) measurements were also performed to verify equivalence.

• Theoretical Framework:

  • The analysis utilizes the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation.
  • The authors solve for the magnetization dynamics in the low-frequency limit of magnetic susceptibility.
  • By expanding the transverse resistance (combining Anomalous Hall Effect and Planar Hall Effect) and the oscillating magnetization, they derive analytical expressions linking $V_{2\omega}$ and $V_{dc}$ to $\xi_{DL}$ and $\xi_{FL}$.
  • The method allows for the extraction of anisotropy fields ($H_{ani}$) and SOT efficiencies by fitting the angular dependence of the harmonic signals.

3. Key Contributions

1. Novel Measurement Technique: Introduction of a full 360° OOP angle sweep SHH method specifically tailored for PMA systems, overcoming the limitations of in-plane sweeps.
2. Discovery of Anomalous Torque Behavior: Identification of a unique, magnetization-direction-dependent field-like torque in the Ta/CoFeB system, which is not observed in Pt/Co.
3. Theoretical Validation: Demonstration of the equivalence between the second harmonic Hall voltage ($V_{2\omega}$) and the DC voltage ($V_{dc}$) in the low-frequency limit, validating the theoretical derivation.
4. Cross-Validation: Successful comparison of SHH results with an AHE-based STFMR technique, confirming the reliability of the extracted SOT efficiencies.

4. Key Results

Pt/Co System

• Parameters: Anisotropy field ($H_{ani}$) $\approx 9400 \, \text{Oe}$.
• Efficiencies:
  • Damping-like ($\xi_{DL}$): 0.12
  • Field-like ($\xi_{FL}$): -0.05
• Behavior: The transverse $V_{2\omega}$ signal fits the standard model over the full 360° range without modification. The field-like term is constant and independent of the magnetization direction ($m_z$).

Ta/CoFeB System

• Parameters: Anisotropy field ($H_{ani}$) $\approx 6000 \, \text{Oe}$.
• Efficiencies:
  • Damping-like ($\xi_{DL}$): -0.16
  • Field-like ($\xi_{FL}$): Variable (Standard fit yields 0.2, but fails to fit the full range).
• Anomalous Finding: The standard model failed to fit the transverse $V_{2\omega}$ data across the full 360° range. Specifically, the slope around $\theta = 90^\circ$ required a $\xi_{FL} \approx 0.6$, while the slope around $180^\circ$ required $\xi_{FL} \approx 0.2$.
• Resolution: The authors found that the effective field-like torque ($h_{FL}$) depends on the magnetization direction, specifically proportional to $m_z^2$. The modified field term is $h_{FL} \times (1 + a \cos^2\theta)$ with $a = -0.66$. This indicates an anomalous field-like torque unique to the Ta/CoFeB interface.

Equivalence of AC and DC Signals

• The study experimentally demonstrated that $V_{2\omega}$ and $-V_{dc}$ signals overlap perfectly in the low-frequency limit, confirming that DC measurements can serve as a valid proxy for SHH analysis in these systems.

STFMR Validation

• AHE-based STFMR was performed on the Pt/Co sample.
• Results: $\xi_{DL} = 0.12$ and $\xi_{FL} = -0.04$.
• These values align closely with the SHH results, validating the SHH methodology.

5. Significance

• Advanced Characterization: The OOP angle-resolved SHH method provides a more complete picture of SOT behavior in PMA systems than previous in-plane or limited field-sweep methods.
• Material Insights: The discovery of the $m_z^2$-dependent field-like torque in Ta/CoFeB suggests that interfacial symmetry and spin scattering mechanisms in this system are more complex than in Pt/Co. This has profound implications for the design of SOT-MRAM devices using Ta/CoFeB, as the switching efficiency may vary with the magnetization state.
• Methodological Robustness: By cross-validating with STFMR and demonstrating the AC/DC equivalence, the paper establishes a reliable, versatile protocol for quantifying SOT efficiencies in thin-film PMA heterostructures, which are the backbone of modern spintronic memory.

In conclusion, this work not only refines the measurement technique for SOT efficiency but also uncovers fundamental physical phenomena regarding the angular dependence of spin-orbit torques in technologically relevant materials.