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Ordinary Hall Effect

Spin and Charge Transport

The ordinary Hall effect occurs when a magnetic field is applied perpendicularly to the direction of a drift current. In this case both the electrons and the holes turn out from the flow direction and they are accumulated at same side of the wire. The charge accumulation at sides of the wire can be measured and the transport properties of the electron gas can be evaluated. Since there is a near equal amount of electrons and holes in a metal, they are of opposite charge and they both are accumulated at same side wire, the Hall voltage in a metal is not large.

Due to the ordinary Hall effect, the spin accumulation may be significantly enlarged at one edge of a ferromagnetic wire. It is because both the spin-polarized electrons and the spin-polarized holes are accumulated at one edge of the wire.


The same content can be found in this paper (http://arxiv.org/abs/1410.7511 or this site for a more upgraded version) .Chapter 12, pp. 35-36
Possible confusion!!: from 2014 to 2017 I have used names TIA and TIS for groups of spin-polarized and spin-unpolarized electrons, respectively. The reasons are explained here.


Main Result

Spin accumulation at a side of sample due to the Hall effect

The Hall effect in metal. The electron current (green balls) flows from “-” to “+” . In a magnetic field the electrons turn left in respect to direction of their movement. The holes current (blue balls) flows from “+” to “-” . In a magnetic field the holes turn right in respect to direction of their movement. Therefore, at the side of the sample both the spin-polarized holes and electrons are accumulated. it means that at this side of the sample there is no charge accumulation, but there is a large spin accumulation.

 

 

 


Relativistic origin of the Hall effect (the Lorentz force)

Relativistic origin of the Lorentz force (Hall effect). An object (red) is moving in a static magnetic field. In the coordinate system moving together with the object, the static magnetic field is relativistically transformed into the effective magnetic field H_eff and the effective electrical field E_eff. In case if the particle is charged, it experiences the force from the effective electrical field (the Lorentz force).

When an electron moves in a magnetic field, it experiences the Lorentz force. The Lorentz force has a relativistic origin. The Theory of Relativity states that a particle moving in a magnetic field experiences an effective electrical field, which is directed perpendicularly to the magnetic field and perpendicularly to the particle movement direction. It is important to emphasize that the direction and magnitude of the effective electrical field does not depend either on the particle charge or on the particle spin.

 

According to the Theory of The Relativity the electric and magnetic field mutually transformed into each other depending on the speed of an observer. For example, if in a coordinate system of static observer there is only a magnetic field, a movable observer will experience this field as both an electrical field and a magnetic field.

A particle moving in a static magnetic field experiences an effective electric field. The effective electrical field acts on the particle charge (the Lorentz force, Hall effect) and forces the particle to move along this field.

A particle moving in a static electrical field experiences an effective magnetic field. The effective magnetic field acts on the particle magnetic moment (spin-orbit interaction) and causes the precession of the magnetic moment around the direction of the effective magnetic field.

 

 

 

 

 

 

 

The Hall Effect and the Spin-Orbit interaction are close cousins

the Hall effect ==== results in ====> an effective electrical field

The Lorentz force- tween effect with the effect of the spin-orbit interaction

 

The Lorentz force (Hall effect) and the spin-orbit interaction, both are relativistic effects. The both effects are originated from fact that an electrical and magnetic field are relative to the velocities of observers. A particle, which moves in a static magnetic field, experiences an effective electrical field (The Lorentz force). A particle, which moves in a static electric field, experiences an effective magnetic field.

The spin-orbit interaction has a similar relativistic origin. In case if the particle has a magnetic moment (spin), there will be a spin precession around the effective magnetic field.

the Spin-Orbit interaction ===results in=====> an effective magnetic field

 

The effective electrical field due to the Hall effect does not depend either on the particle charge or the particle spin. it is only depend on the particle movement direction and the direction of the magnetic field !!!!

 

 

 

 


 

 

 

Results in short

Hall effect in n-type semiconductor. There is a spin accumulation at the right side, where there is an accumulation of the negative charge. There is no spin accumulation at the right side of the sample. Hall effect in p-type semiconductor. There is a spin accumulation at the right side, where there is an accumulation of the positive charge. There is no spin accumulation at the right side of the sample. Hall effect in metal. There is a spin accumulation at the right side. However. there is almost no charge accumulation!!

 

 

 

 

 

 

 

 

 

 

 

 

 


The Hall effect from the Boltzmann Equations

The Hall effect can be described by the force term in the Boltzmann transport equations

where

In an equilibrium in the electron gas the electrons move in all direction. Since the Lorentz force only changes the electron movement direction, the Lorentz force does not change the equilibrium electron distribution F_i,0

Eq. (1.3) can be verified as follows

Only if there is a drift current, the Lorentz force may change the electron distribution. In cases of the current of the running-wave electrons, substituting the solution for the current of the running-wave electrons Eq. (14.7) into Eq.(1.1) gives

Using the relaxation-time approximation Eqs. (9),(12), ignoring the term of order H^2 and substituting Eq. (1.5) into the Boltzmann equations Eqs. (18) gives

From Eq. (1.6) the Hall current can be calculated as (See Eq.(11.3) here)

or Hall current can be calculated as

 


 

In the case when a magnetic field is applied perpendicularly to the flow direction of a drift current, the charge carriers experience the Lorentz force in the direction perpendicular to both the magnetic field and the current. Because of this force, the carriers are accumulated at the edges of the sample and a voltage transverse to the drift current is built up. This voltage is called the Hall voltage. In n- and p-type semiconductors the Hall voltage is of opposite sign, because the holes and electrons are accumulated at the same side of sample, but they have opposite charge (Figs.14-15). Because of a nearly- equal number of holes and electrons in a metal, the Hall voltage in the metal is small and it is proportional to the gradient of the density of states at the Fermi level. When the electron gas in a metal is spin-polarized, both the electrons and holes are spin-polarized. Because of the Hall effect, the spin-polarized electrons and holes are accumulated at the same side of the sample. Therefore, at this side of sample a significant spin accumulation occurs. (Fig.16)

 

The Hall effect in n-type semiconductors.

Fig. 14. The Hall effect in a n-type semiconductor. The magnetic field is applied perpendicular to the film (blue arrows). There is only an electron current. Dark-green arrows show path of spin states, which transfer a negative charge and spin. The negative charge and spin are accumulated at the right side of the film. At left side of the film a positive charge accumulated. There is no spin accumulation at the left side.

 

 

 

 

 

 

In a n-type semiconductor there is only one type of carries. It is the electrons or "spin" states.

 

The electrons diffuse from a "-" source toward a "+" drain. As they diffuse in the magnetic field they turn to the left.

Therefore, the spin and the negative charge are accumulated at the right side of the sample.

 

 

 

 

 

 

 

 

 

 


 

 

 

 

The Hall effect in p-type semiconductors.

 

In a p-type semiconductor the holes are the carriers for the Spin and the Charge. In fact the hole current consists of two currents: the current of "spin" states and the current of "full" states.

 

The holes diffuse from a "+" source toward a "-" drain. As they diffuse in the magnetic field they turn to the right.

Therefore, the spin and the positive charge are accumulated at the right side of the sample. (see left figure)

 

Look closer!!!

The hole current consists of the currents of "spin" and "full" states

 

In a p-type semiconductor the "spin" states diffuses from a "+" source toward a "-" drain.

Therefore, the "spin" states are accumulated at the right side of the sample. (see right figure).

The "full" states moves in the opposite direction and they are accumulated at the right side of the sample.

Therefore, the spin and the positive charge are accumulated at the right side of the sample. (see right figure)

 

Remember!!!!

The Hall effect is the relativistic effect. It does not matter for the Hall effect the charge or the spin of the "spin" states. It is only matter their movement direction.

This is the reason why the "spin" states in the cases of the n- and p-type semiconductors turn in the opposite directions.(Compare Fig.14 and Fig.15(right))

 

note about the current of "full" states (click to expand)

The current of "full" states is the normal current.

This means that the negatively charged "full" states diffuses from a "-" source toward a "+" drain.

Due to the Hall effect they should turn to the left (similar to the electrons shown in Fig. 14).

However, the most of the electrons, which occupies the "full" states, are the standing-wave electrons. (See Fig.9 here)

The standing-wave electrons do not move, therefore they do not contribute to the Hall current.

In contrast, the electrons, which occupies the "spin" states, are the running-wave electrons.

For this reason, mostly the "spin" states contribute to the Hall effect in the case of the hole current.

 

 

Fig. 15. Hall effect in a p-type semiconductor. Magnetic field is applied perpendicular to the film (blue arrows). There is only an electron current Light-green arrows show path of spin states, which transfer a positive charge and spin. The positive charge and spin are accumulated at the right side of the film. At left side of the film a negative charge accumulated. There is no spin accumulation at the left side.

(Left side) Only path for flow of spin states are shown (Right side) Both flows of spin and full states are shown. The full states are accumulated at the left side and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

 

 

 

The Ordinary Hall effect in metals.

 

Fig. 16. The Hall effect in metal. The electron current (green balls) flows from “-” to “+” . In a magnetic field the electrons turn left in respect to direction of their movement. The holes current (blue balls) flows from “+” to “-” . In a magnetic field the holes turn right in respect to direction of their movement. Therefore, the charge does not accumulate. In contrast, the spin is accumulated at one edge of the sample.

The drift current in a metal consists of the electron and hole currents, which flow in the opposite direction. Since the charge of an electron and a hole is opposite, the direction of the charge transport is the same for the electron and hole current. This is the transport of the "-" charge from "-" to "+". Or what is the same, the transport of the"+" charge from "+" to "-".

Since the electrons and holes move in the opposite directions, in the magnetic field the electrons turn to the left and holes turn to the right.

As result, both the electrons and the holes are accumulated at the same side of the sample.

About the same amounts of the electrons and holes are accumulated

 

Since the charge of an electron and a hole is opposite, the charge is not accumulated. The Hall voltage in a metal is small.

 

In contrast, the spin direction of electrons and holes is the same, therefore there is a significant spin accumulation due to the Hall effect in a metal

 

 

 

 

 

 

Fig. 17. Hall effect in a metal. Magnetic field is applied perpendicular to the film (blue arrows). There Green arrows show path of spin states, which transfer a negative charge and spin. The negative charge and spin are accumulated at the right side of the film. AT left side of the film a positive charge accumulated. There is no spin accumulation at the left side.

 

 

Look closer!!!

Both the electron and hole currents are currents of "spin" states

The "spin" states of energies higher than the Fermi energy moves from "-" to "+" similar to positive particles
The "spin" states of energies lower than the Fermi energy moves from "+" to "-" similar to positive particles (See here)

Since the "spin" states of the different energies move in the opposite directions, they turn in the opposite directions due to the Hall effect.

The "spin" states of higher energies turn to the left and the "spin" states of lower energies turn to the right.

As result, the "spin" states of all energies are accumulated at the same side of the sample. There is a significant spin accumulation at this side of the sample.


Notes

1. The Hall voltage in a metal is small, but it is not zero.

It can be positive and negative. It depends on whether the electron or hole current is larger in a metal.

The Hall voltage in a metal is proportional to the energy derivative of the density of states at the Fermi energy. (See here)

 

2. From the Hall voltage in a metal the injection conductivity can be estimated.

The injection conductivity is proportional to the Hall voltage.

3. For a larger the spin torque or a better spin injection efficiency, a metal having largest the Hall voltage should be used.

The largest spin torque is required for efficient a MRAM cell

The spin torque is proportional to (See here).

The spin injection efficiency is proportional to (see here)

 


 

Q. How to choose a good metal for the spin injection or to achieve a largest spin-transfer torque???

 

A1. Use a metal, which shows a largest Hall voltage.

A2. If you want to inject the spin from one metal to another, use metals, which show the Hall voltage of the same sign.

 

Note 1.

The conductivity in the bulk of a metal and in the vicinity of the interface is different. The spin injection and the spin-transfer torque are the effects of an interface. The Hall effect is often measured in the bulk of a metal.



Hall coefficient and Hall angle in metals

from Colin M. Hurd "THE HALL EFFECT IN METALS AND ALLOYS" , PLENUM PRESS· 1972

Summary

note m3 /A/s =Ohm*m/T

Metals with hole-dominated transport

(spin is drifted from "+" to "-")

metal Cu Ag Au Al Pt Mg Pd Gd Tb  

Hall coefficient

(* 1E-11 Ohm*m/T)
-5.12 -8.81 -7.16 -3.44 -2.2 -8.3 -7.6 -37 -44  
Conductivity (*1E7 S/m) 5.96 6.21 4.1 3.5 0.96 2.27 0.95 0.076 0.087  
Hall angle/B (*mrad/T) -3.05 -5.471 -2.935 -1.204 -0.211 -1.884 -0.722 -0.281 -0.3828  
Hall angle/B (*mdeg/T) -174.75 313.47 -168.16 -68.984 -12.089 -107.95 -41.368 -16.1 -21.933  
crystal structure fcc fcc fcc fcc fcc hcp fcc cph hcp  

 

Metals with electron-dominated transport

(spin is drifted from "-" to "+")

metal Ta Ru Cr Rh W V Zn Mo Nb  

Hall coefficient

(* 1E-11 Ohm m/T)
10.1 22 37.9 5.05 10.8 8.2 20(2) 18 9  
Conductivity (*1E7 S/m) 0.76 1.40 0.8 0.0211 1.79 0.5        
Hall angle/B (mrad/T) 0.7676 3.08 3.032 0.01 1.933 0.41        
Hall angle/B (mdeg/T) 43.98 176.47 173.72 0.57296 110.75 23.491        
crystal structure bcc hcp bcc fcc bcc bcc hcp bcc bcc  

 

 

Iron Fe

The transport is electron-dominated.

Hall coefficient at 6T is 1E9 B* kG/Ohm/cm

non-ordinary Hall coefficient is Rs=30-40E-11 Ohm*m/T at RT (Hall angle=3-4E-3 1/T), it decreases almost to zero at lower T

ordinary Hall coefficient is 2E-11 Ohm*m/T at RT (Hall angle=0.2E-3 1/T), it weakly decreases at lower T.

Conductivity : 1.01E7(S/m)

bcc structure


Cobalt Co

Transport is ???-dominated. (maybe hole-dominated)

 

non-ordinary Hall coefficient is small positive Rs=2E-11 Ohm*m/T at RT(Hall angle=0.32E-3 1/T), it decreases almost to zero at lower T, it becomes becomes negative below 250 C

ordinary Hall coefficient is negative and large -8E-11 Ohm*m/T(Hall angle=-1.28E-3 1/T), it weakly increases to -12E-11at lower temparature;

Conductivity : 1.602E7(S/m)

hcp structure


Nickel Ni

Transport is hole-dominated

 

non-ordinary Hall coefficient is big negative Rs=-70E-11 Ohm*m/T (Hall angle=-10.1E-3 1/T), decrease almost to zero at lower T.

ordinary Hall coefficient is negative -4E-11 Ohm*m/T (Hall angle=-0.572E-3 1/T);

 

Conductivity : 1.43E7(S/m)


Iron-Cobalt Fe-Co (bcc)

non-ordinary Hall coefficient is positive and very large Rs=400E-11 Ohm*m/T, at Co=15%

ordinary Hall coefficient is negative and largest -25E-11 Ohm*m/T, at Co =35 %

Properties of FeCo

the spin polarization is negative for Co and positive for Fe ?? some calculated spin polarizations for FeCo , near 16 % of Co sp polarization changes its signs Fe=56 %; Co(25%) sp=-26%; Co(50%) sp=-67%; Co(75%) sp=-73% Co sp=-80% from book "Materials Design and Synthesis for Desirable Magnetic and Optical Properties By Hao Zhu"

magnetic moment of BCC Co maybe the same as BCC Fe

lattice constant =2.87 A x=0, it is near constant till 20 %, next it reduces to 2.83 A for BCC Co


Cromium

At 290 K the Hall coefficient is independent on magnetic field, but at 4.2 and 77 K, R decreased by a factor of three as the field was increased to 147 kG


Titanium

Ti has cph-structure Hall coefficient is of opposite sign for directions parrall and perpendicular to c-axis R_parallel=4.2-11 Ohm*m/T (electron-dominated) R_perpendicular= -7.7-11 Ohm*m/T

note: All metals of cph structures (Ti,Y,Gd) have significant difference of Hall resistance along and perpendiculary to the c-axis indicating very strong spin-orbit interaction.


Terbium

In the paramagnetic region the values obtained for Ro and Rs are, respectively, -44E-11Ohm*m/T and -420E-11 Ohm*m/T (very large)



Hall angle in semiconductors

 

  n-Ge p-Ge n-Si p-Si n-GaAs p-GaAs  
mobility (cm2/V/s) 3900 1900 1450 500 8000 400  
Hall angle/B (*1E-3 1/T) 390 190 145 50 800 40  

 

 

 

 

 

 

 

 

 

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