Magnetotransport Part 2. Hall Effects

Origin of OHE is the Lorentz force. An electron experience an relativistic electrical field due to electron movement perpendicularly to the magnetic field. The The relativistic electrical field interacts with the electron charge (not spin) forcing the electron to move in its direction.

FM 물질을 특정하는 또 다른 방법은 magnetotransport measurments를 통해서이다. 

(interact with) 

 Electron Charge of conduction electrons

 OHE is linearly proportional to an external magnetic field H

(formula):

 

aOH is the rotation angle of the ordinary Hall effect (in mdeg/kG). H is external magnetic field. aOH is positive for the hole- dominated conductivity. aOH is negative for the electron- dominated conductivity. jV is the bias current along metallic wire (from electrical source to electrical drain). The hole- dominated conductivity in a material, in which density of states decreases at the Fermi level. The electron- dominated conductivity in a material, in which density of states decreases at the Fermi level

 

(note) 

The OHE is independent of the spins of localized and conduction electrons. It depends on the charge of carrier and its transport properties.

(definition)  The OHE describes the fact that charge is accumulated at sides of metallic wire, when an external magnetic field H is applied perpendicularly to the wire. The origin of the OHE is the Lorentz force

The movement of the conduction electrons (green balls) turns from a straight path due the Lorentz force induced by the magnetic field H. As a result, the electrons are accumulated at the side of the wire.

The Lorentz force turns the electron (green bowl) from a straight movement. The electron, which moves in a magnetic field H, experience an electrical field perpendicularly to its movement (relativistic effect). This electric field interacts with the electron charge and turns the electron movement direction. It creates an electron current (Hall current) perpendicularly to the bias current.

Hall angle vs an applied magnetic field H measured in ruthenium Ru (a non-magnetic metal). The Ru thickness is 25 nm. The dependence is nearly perfectly linear.

 

 

 

Under an applied voltage the electrical current (drift current) flows in the metal wire. When the metal is ferromagnetic, the drift current is spin-polarized. Therefore, there are more electrons with spin directed up. It causes more electrons be scattered into the left than into into the right. This is the reason for the charge accumulation at the right side of the wire.

Under an applied voltage the drift current flows in the non-magnetic metal wire. The drift current is spin-unpolarized and the electrons have spin in any direction with an equal probability. Since the probability to be scattered to the left is higher for electrons with spin directed up and the probability to be scattered to the right is higher for

There is a region of spin accumulation at backside of the wire. The diffusive spin current flows from the region of a higher spin accumulation to the region of a lower spin accumulation. This means that spin polarized electrons flow from back to front of the wire. In the opposite direction the spin-unpolarized electrons flow. The scattering probability of spin-up electrons into the right is higher. This is the reason for the charge accumulation at the left side of the wire.

(Linear odd Hall effect 2)

 Anomalous Hall effect (AHE)

 

Dependence of scattering of conduction electrons on the spin of localized electrons

Rotational (Orbital) Moment of conduction electrons

AHE is linearly proportional to the total spin of localized electrons.

 

Is there any clear or evident relation between AMR/PHE and AHE? In my experience materials with strong AMR/PHE not necessarily exhibit strong AHE and vice-versa. But since they are both "magnetization angle dependent phenomena" I was just wondering if from a fundamental point of view they could be related, or if some reasonable theory relates them.

There is no relation between AMR/PHE and AHE.

From an experiment, this fact is known for a while. See, for example, T.R. Mcguire and R.I. Potter, IEEE Trans. Magn. (1975).

From theory point of view, the AMR/PHE and AHE have different symmetries and different physical origins. Therefore, they are two very different effects.

Difference 1: difference in the symmetry

The AHE is a linear magneto-transport effect. The AMR/PHE is a second- order magneto-transport effect. In a nanomagnet there are 3 independent variables, which time-inverse symmetry is broken: (1) externally-applied magnetic field H; (2) the total spin Sd of localized d- electrons (or the magnetization M) and (3) the total spin Scond of the spin-polarized conduction electrons (or the spin polarization). A linear magneto-transport effect is linearly proportional either to H or Sd or Scond. The ordinary Hall effect is proportional to H. The AHE is linearly proportional to Sd. The inverse spin Hall effect (ISHE) is linearly proportional to Scond. A 2nd order magneto-transport effect is proportional to a product of a pair from H, Sd and Scond. Additionally to the AMR/PHE, the in-plane GMR is also a 2nd order magneto-transport effect.

Difference 2: difference in the physical origin.

The AHE is dependent only on the magnetization (Sd) and is independent of spin polarization of the conduction electrons (Scond). In contrast, the AMR/PHE depends on both the magnetization and the spin polarization. The origin of the AHE is the spin-dependent scatterings of conduction electrons, which depend on the spin of a d- electron, but is irrelevant to the spin of a conduction electrons. The origin of the AMR/PHE is also spin-dependent scatterings of conduction electrons, but of different type, which depend on the angle between spin of a d- electron and the spin of a conduction electron.

 

 How to distinguish Planar Hall effect from Anomalous Hall effect experimentally??

A 1st order magneto-transport effect (Anomalous Hall effect, Inverse Spin Hall effect & Ordinary Hall effect) can be easily distinguished experimentally from a 2nd order magento-transport effect (AMR/PHE, in-plane GMR etc.). Since the 1st order magneto-transport effect is linearly proportional to magnetization M + external magnetic field H, it reverses its polarity when H+M are reversed. In contrast, the 2nd order magento-transport effect is proportional to a square/product of magnetization M + external magnetic field H, it does not reverse its polarity when H+M are reversed.

It is a common rule for any magneto-transport measurement that two measurements are always done with the magnetization in the forward and reversed direction. Next, the symmetric and antisymmetric contributions of measurements are calculated. The anisymmerical contribution is associated with the 1st order magneto-transport effects and the symmerical contribution is associated with the 2st order magneto-transport effects. In this way any unwanted contribution of 1st order magneto-transport effects to a measurement of a 2st order magneto-transport effect can be avoided and vice versa.

As an example see my AMR/PHE measurement for nanomagnets here.

 

 

2 . 4 . 1 . An isotrop ic magnetores istance and p lanar Ha l l e f fect Another way to characterize ferromagnetic materials is via magnetotransport measurements. In 1856, William Thomson observed an increase in the longitudinal resistance of ferromagnets for an in-plane magnetic field applied parallel to the electric current, while the resistance decreased for magnetic field applied perpendicular to the current [52, 54]. This effect, called anisotropic magnetoresistance (AMR), can be mathematically formulated em terms of resistivity as [3]:

Planar Hall effect

대부분의 3d ferromagnet에서의 Planar Hall effect의 유력적인 원인으로는 강자성체 내의 s-band electron과 d-band electron 속에서 SOC에 의해 섞이는 SOC의 영향을 받는것으로 알려져있다. 단순히 말해, SO항이 $\vec{L}\cdot \vec{S}$에 비례하기 때문에, SOC를 통해 spin dipole moment 는 외부자기장에 정렬한다. 이는 즉 $L=0$인 경우에는 isotropiccharge distribution을 가지지만, $L\ne0$인 경우에는 the non-spherical charge distribution leads to different scattering cross-sections for the conduction electrons (electric current) as the relative orientation of the current and the magnetic field changes [52]. Although AMR (or PHE) is common in ferromagnets and it will be used in combination with ferromagnetic resonance to quantify the amplitude of the spin-orbit torques studied in this thesis (see Chap. 3), Chap. 4 will discuss the observation of this effect in non-magnetic 3D topological insulators and its possible origins.

Emerges due to the anisotropic scattering rates of the conduction carriers when the magnetic field is applied parallel or perpendicular to the electrical current. 4.2.

Planar Hall effect는 in-plane magnetic field가 존재할 때 transversal resistance $R_{xy}$의 angular dependence를 나타낸다.

PHE studies explore how the sample magnetoresistance changes in the presence of an in-plane magnetic field, which requires a careful alignment between the sample and the in-plane magnetic field planes.

If the sample and the magnetic field planes are misaligned, an out-of-plane component of the magnetic field generates an orbital magnetoresistance that can hinder the magnetoresistance response one is interested to investigate.

AMR arising from an in-plane magnetic field and the orbital magnetoresistance

 

 

. In paramagnetic materials, Goldberg and Davis have suggested that PHE should arise if the Fermi contours are non-spherical in k-space [5]. Schliemann and Loss have suggested that the interplay of Rashba and Dresselhaus SO couplings in 2DEGs cause the Fermi contours to be anisotropic which, in turn, leads to anisotropies in the conductivity