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Magneto transport. Family of Hall effects and AMR effects. Spin and Charge TransportAbstract:The family of the Hall effects and the magneto resistance effect are described. The effect are distinguished by their symmetry with respect to a reversal of a substantially large magnetic field and direction in which magnetically created current flows. When it flows parallel to the main current, the resistivity of the wire is changed and the effect called the Anisotropic magneto resistance (AMR). When the magnetically created current flows perpendicularly to the wire, a charge is accumulated at sides of the wire and the voltage is created across wire, which is called the Hall voltage, and the effect is called the Hall effect. Additionally, magneto transport effects are distinguished by whether their due to spin features of either localized d electrons or conduction electrons or both.Content
1.Odd and even magneto transport effects2. 1st order odd magneto transport effects2.1. Ordinary Hall Effect (OHE)2.2 Anomalous hall Effect (AHE)2.3 Inverse Spin Hall effect (ISHE)2.4 Spin Hall effect3. 2nd order even magneto transport effects3.1 AMR and Planar Hall effect: two features of one single effect3.2 AMR3.3 Planar Hall effect3.4 Kondo type AMR/ Planar Hall effect3.5 Spin dependent conductivity3.5a Inplane GMR effect3.6 AHE type AMR/ Planar Hall effect3.7 ISHE type AMR/ Planar Hall effect3.8 OHE type AMR/ Planar Hall effect3.9 MH type AMR/Planar Hall effect4. DC measurement of the Hall effect5. RF measurement of magneto transport effects6 Influence of a Hall probe on a Hall measurement7. Magneto capacitance induced by the Hall effect(10). Questions & Answers(q1) Planar Hall/ AMR vs. AHE(q2) How to distinguish Planar Hall effect from Anomalous Hall effect experimentally?
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Distinguish features of a magneto transport effect: (Distinguish feature 1) a magnetically generated current is linearly proportional to current j_{bias}(Distinguish feature 2) a magnetically generated current is reversed its polarity when to current j_{bias} is reversed
(Distinguish feature 3) a magnetically generated current is proportional either an external magnetic field H or/and total spin of localized electrons S_{local} or/and total spin of conduction electrons S_{cond} .(Distinguish feature 4) an electrical current should change distribution of at least one magnetic property of conduction electrons. It should change the distribution of spins or orbital momentsIn the case of an ISHE like effect, the electron current changes the distribution of the spins. In the case of an AHE like effect, the electron current changes the distribution of the orbital moments.Odd and even magneto transport effectsAll magneto transport effects can be distinguished by their symmetry against reversal of substantially large magnetic field, which fully reverses spins of localized and conduction electrons.
(Odd magneto transport effects) (), which polarity is not reversed when the magnetic field and the spins are reversedPolarity of effect is Reversed, when when the magnetic field + spin are reversed Ordinary Hall effect (OHE), Anomalous Hall effect, Inverse Spin Hall effect
(Even magnetotransport effects) , which polarity is not reversed when the magnetic field and the spins are reversedPolarity of effect is NOT Reversed, when when the magnetic field + spin are reversed Anisotropic Magneto resistance (AMR), Planar Hall effect, , spin detection effect
1st order odd magneto transport effectsThis is effects which are linearly proportional either to an external magnetic field H or the total spin S_{local} of localized d electrons or the total spin S_{cond} of conduction electronsThree types to linear odd Hall effects(Linear odd Hall effect 1) Ordinary Hall effect (OHE)
More details see here(origin) 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. (interact with) Electron Charge of conduction electrons OHE is linearly proportional to an external magnetic field H (formula): a_{OH} is the rotation angle of the ordinary Hall effect (in mdeg/kG). H is external magnetic field. a_{OH} is positive for the hole dominated conductivity. a_{OH} is negative for the electron dominated conductivity. j_{V} 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.
(Linear odd Hall effect 2) Anomalous Hall effect (AHE)
More details see here(origin) Dependence of scattering of conduction electrons on the spin of localized electrons (interact with) Rotational (Orbital) Moment of conduction electrons AHE is linearly proportional to the total spin of localized electrons. (formula): a_{AH} is the rotation angle of the Anomalous Hall effect (in mdeg). S_{local }is the total spin of localized electrons. Since in the most of realistic cases, only the direction , but not magnitude of S_{local} changes, Eq. can be simplified as where M is an unit vector in direction of magnetization.
(note) The AHE depends on the total spin of localized d electrons, but they are independent of the total spin of conduction electrons.
(Linear odd Hall effect 3) Inverse Spin Hall effect (ISHE)
More details see here(origin) Dependence of scattering of conduction electrons on the spin of localized electrons (interact with) Rotational (Orbital) Moment of conduction electrons ISHE is proportional to the total spin of conduction electrons (formula): a_{AH} is the rotation angle of the Anomalous Hall effect (in mdeg). S_{local} is the total spin of localized electrons. Since the the total spin of the spinpolarized electrons is linearly proportional to the number of spin polarized electrons, the Eq. can be simplified as where P_{s} is spin polarization. m is unity vector along spin direction of spinpolarized conduction electrons
(origin of the Inverse Spin Hall effect) Spin dependent scatterings(explanation in short) The spin dependent scatterings means that the scattering probability of spin up electron is higher to the left and the scattering probability of spin down electron is higher to the right . For example, if the spin direction of the spin polarized conduction electrons is up, there are more electrons scattered to the left and as a result there is a charge current flowing to the left
(Linear odd Hall effect 3a) Spin Hall effect (ISHE)
More details see here(origin) Dependence of scattering of conduction electrons on the spin of localized electrons (interact with) Rotational (Orbital) Moment of conduction electrons SHE is proportional to the total spin of conduction electrons (formula):
a_{AH} is the rotation angle of the Anomalous Hall effect (in mdeg). S_{local} is the total spin of localized electrons. Since the the total spin of the spinpolarized electrons is linearly proportional to the number of spin polarized electrons, the Eq. can be simplified as
where P_{s} is spin polarization. m is unity vector along spin direction of spinpolarized conduction electrons
(note) Spin Hall effect (SHE) and Inverse Spin Hall effect (ISHE) are fully complementary effect. They have identical origins and in a material they have the same magnitude (note) Both the SHE and ISHE depends on the total spin of conduction electrons, but they are independent of the total spin of localized electrons. (note) The ISHE is linearly proportional to the number of spin polarized conduction electrons. The SHE is linearly proportional to the number of spin unpolarized conduction electrons. 2nd order even magneto transport effectsThree contributions to linear even Hall effect in a ferromagnetic nanomagnet(contribution 1) Anomalous
Two parts of the same effect: the Hall effects and Anisotropic Magneto Resistance (AMR) effect.
The Hall effect and AMR effect are in the same family of effects. The have similar properties, similar origins and similar symmetry. (a good example) the classic AMR and the Planar Hall effect describes one single effect, which is the magnetic generation of a current parallel to the magnetization direction. The component of this current, which is parallel to the bias current, describes the AMR effect. The component of this current, which is perpendicular to the bias current, describes the Planar Hall effect. Two twins effects: Hall effect and Anisotropic magneto resistanceThe Hall effect is defined
(generated current flows perpendicularly to bias current) Hall effect(current
Inplane Giant Magneto resistance (GMR) effect
The inplane GMR effect describes the fact that resistance of a metallic wire, which consists of two ferromagnetic layers separated by a nonmagnetic layer, depends on mutual magnetization directions of two ferromagnetic layers. It is the smallest, when magnetization directions are parallel and it is the largest, when the magnetization directions is opposite.For experimental discovery of the inplane GMR effect, Prof. Fert and Prof. Grünberg were awarded the Nobel Price in Physics in 2007.
(origin 1 of inplane GMR effect) Spin proximity effect The spin proximity effect (See details here) describes the fact that the spin polarized conduction electrons diffuses from the first ferromagnetic layer to the second ferromagnetic layer, change the spin polarization in the second layer and as a results the resistivity of the second ferromagnetic layer increases due to the effect of the spin dependent conductivity. (origin 2 of inplane GMR effect) spin dependent conductivity. The conductivity of a ferromagnetic metal depends on mutual directions of spins of localized electrons and spins of conduction electrons. When spinpolarized conduction electrons diffuses from one ferromagnetic metal to the second ferromagnetic layer of a different magnetization directions, they make different the in the spin directions of localized and conduction electrons in the second layer and as a result the resistivity of the second layer becomes larger.
(Origin of inplane GMR effect):When the magnetization directions in ferromagnetic layers are parallel, the spin directions of the spinpolarized conduction electrons are also the same and parallel to the magnetization (the spins of localized electrons). In this cases, the resistance of each layer is smallest. When the magnetization directions are opposite, the spin directions of conduction electrons are also opposite. In the case when in the first ferromagnetic layer the number of the spin polarized electrons is substantially larger than in the second ferromagnetic layer, a significant amount of the spin polarized electrons from first layer diffuses into the second ferromagnetic layer and the spin direction in there become the same as in the first layer and opposite to the magnetization of localized electrons. As a result, the resistivity of the second layer becomes larger. The resistivity of a material is largest when the spin direction conduction electrons is opposite to the spin direction of the localized electrons due to the effect of the spin dependent conductivity.
(note)When the total thickness of wire becomes smaller than the electron meanfree path, the electron gas becomes common through both ferromagnetic layers and the spin polarization is always the same in both layers. When magnetizations directions are parallel, the common spin polarization is the largest and parallel to each magnetization and therefore the resistance of each layer is smallest. When magnetizations directions are opposite, the common spin polarization is small (close to zero).As a result, the resistance becomes larger in each layer.
Influence of a Hall probe on a Hall measurement
The Hall voltage V_{Hall} is linearly proportional to the width w of nanowire ( See HallAMRbasic.pdf) where V is the voltage applied to the nanowire, α_{Hall} is the Hall angle, which is a material parameter, w is the width and L is the length of the nanowire. Fig.10a shows a conventional structure for a Hall measurement in a metallic nanowire. Two metallic contact (probes) contact the opposite sides of the nanowire to measure Hall voltage. However, the effective width at the measurement point is wider than the nanowire width w. The effective width w_{p} also includes the length of the probe. It may cause a systematic error in the measurement of the Hall angle α_{Hall}
Fig.10b shows an optimized design with a Hall probes, which are narrowing near contact. In this case, the charge is accumulated at the contacts and a possible systematic error is minimized. To test all metal transistor I have fabricated a hall probe as narrow as 30 nm.Usually, the width of Hall probe about 12 mm is still OK but critical. The maximum allowed width strongly depends on the sample structure. (calculate it) For a reliable Hall measurement it is always better to simulate numerically the structure using e.g. Comsol.
There are several designs of Hall bars, which nearly fully exclude the undesired influence of the Hall probe.
RF measurements of magnetotransport effects (Hall effects).
(main idea): The sample is illuminated by microwave at frequency of the Ferromagnetic resonance (FMR). The microwave excites the spin precession and additionally the microwave excites the electrical current. Since the Hall voltage proportional to both the spin direction and the current, which are both modulated by microwave, there are frequency beating of these two contribution. As a result, there is a DC component of the Hall voltage which is measured.
(merit of the method): It is possible to separate a studied Hall contribution from other contribution and measure a really weak Hall effect. For example, it is possible to measure a very weak ISHE effect in a paramagnetic metal.
(Method 1). RF measurement of 1st order magnetotransport effects: AHE and ISHE(what is modulated by RF):spin direction; electrical current
(Method 2). RF measurement of 2nd order magnetotransport effects: Planar Hall effect/ AMR and spin dependent conductivity(what is modulated by RF):spin direction of localized d electrons; spin direction of conduction electrons
Questions & AnswersIs 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 viceversa. 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 magnetotransport effect. The AMR/PHE is a second order magnetotransport effect. In a nanomagnet there are 3 independent variables, which timeinverse symmetry is broken: (1) externallyapplied magnetic field H; (2) the total spin Sd of localized d electrons (or the magnetization M) and (3) the total spin Scond of the spinpolarized conduction electrons (or the spin polarization). A linear magnetotransport 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 magnetotransport effect is proportional to a product of a pair from H, Sd and Scond. Additionally to the AMR/PHE, the inplane GMR is also a 2nd order magnetotransport 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 spindependent 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 spindependent 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 magnetotransport effect (Anomalous Hall effect, Inverse Spin Hall effect & Ordinary Hall effect) can be easily distinguished experimentally from a 2nd order magentotransport effect (AMR/PHE, inplane GMR etc.). Since the 1st order magnetotransport 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 magentotransport 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 magnetotransport 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 magnetotransport effects and the symmerical contribution is associated with the 2st order magnetotransport effects. In this way any unwanted contribution of 1st order magnetotransport effects to a measurement of a 2st order magnetotransport effect can be avoided and vice versa. As an example see my AMR/PHE measurement for nanomagnets here.
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