# Galvanomagnetic Phenomena

## Galvanomagnetic Phenomena

the aggregate of phenomena associated with the action of a magnetic field on the electrical (galvanic) properties of solid conductors (metals and semiconductors) through which a current is flowing. The most important are the phenomena in a magnetic field **H** that is perpendicular to the current (transverse galvanomagnetic phenomena). Among these is the Hall effect—the appearance of a potential difference (the Hall emf *V _{H}*) in the direction perpendicular to the field

**H**and the current

**j**(

**j**is the current density) and the change in the electrical resistance of a conductor in a transverse magnetic field. The difference Δ

_{ρ}between the resistance

*ρ*of the conductor in the magnetic field and without the field is often called the reluctance.

The Hall constant serves as a measure of the Hall effect:

Here *d* is the distance between the contacts used to measure the Hall electromotive force (emf). Within wide limits the Hall constant is independent of the magnetic field strength (and, for metals, the temperature as well). The linear dependence of *V _{H}* on the magnetic field

*H*is utilized to measure magnetic fields.

In electron conductors, in which the current is carried by “free” electrons (conduction electrons), the Hall constant is expressed according to the simplest concepts in terms of the number of conduction electrons *n* per cu cm: *R* = 1/*nec* (*e* is the electron charge, and *c* is the velocity of light). Therefore the measurement of *R* is one of the basic methods of evaluating the concentration of conduction electrons *n* in electron conductors. In this case *R* is negative. For semiconductors with hole conductivity, and for certain metals, the Hall constant is positive, corresponding to the positively charged current carriers, or holes. Since the Hall emf changes its sign when the direction of the magnetic field is reversed, the Hall effect is known as an odd galvanomagnetic phenomenon.

The relative change of resistance in a transverse field (Δ*ρ*/*ρ*)⊥ is very small under ordinary conditions (at room temperature): for good metals (Δ*ρ*/*ρ*)⊥ ~ 10^{-4} when *H* ~ 10^{4} oersteds. Bismuth (Bi), for which (Δ*ρ*/*ρ*)⊥ ≈ 2 when *H* = 3 × 10^{4} oersteds, is an important exception. It can therefore be utilized to measure magnetic fields. For semiconductors the change in resistance is somewhat greater than for metals: (Δ*ρ*/*ρ*)~ 10^{-2} to 10^{-1}, and it depends essentially on the concentration of impurities in the semiconductor, as well as on temperature. For example, for sufficiently pure germanium (Δ*ρ*/*ρ*)⊥ ≈ 3 when T = 90°K and *H* = 1.8 × l0^{4} oersteds.

A reduction in temperature and an increase in the magnetic field causes (Δ*ρ*/*ρ*)⊥ to increase. In 1929, P. L. Kapitsa, using magnetic fields of several hundred thousand oersteds and comparatively low temperatures (the temperature of liquid nitrogen), observed a substantial increase in the resistance of a large number of metals and showed that (Δ*ρ*/*ρ*)⊥ is linearly dependent on the magnetic field over a wide range of magnetic fields (Kapitsa’s law).

In weak magnetic fields, (Δ*ρ*/*ρ*)⊥ is proportional to *H*^{2}. The proportionality factor between (Δ*ρ*/*ρ*)⊥ and *H*^{2} is positive—that is, the resistance increases with the magnetic field. The change of resistance in a magnetic field is known as an even galvanomagnetic phenomenon, since (Δ*ρ*/*ρ*)⊥⊥ does not change its sign when the direction of the field *H* is reversed.

Since resistance is extremely sensitive to the quality of a specimen (the amount of impurities and flaws in the crystal lattice), as well as to temperature, each measurement results in a new dependence of *ρ* on *H*. The available experimental data for metals can be conveniently described by expressing (Δ*ρ*/*ρ*)⊥ in the form of a function *H*_{ef} = *Hρ*_{300}/ρ, where *ρ*_{300} is the resistance of a given metal at room temperature (*T* = 300° K) and *ρ* is the resistance at the experimental temperature. Thus, the various data relating to one metal fall on one curve (Kohler’s rule).

The main cause of galvanomagnetic phenomena is the warping of current-carrier trajectories (conduction electrons and holes) in a magnetic field. The trajectory of the carriers in a magnetic field can be considerably different from the trajectory of a free electron in a magnetic field, which is a circular spiral twisting around a line of magnetic force. The diversity of trajectories for current carriers in various conductors is the reason for the variety of galvanomagnetic phenomena, and the dependence of the trajectory on the direction of the magnetic field is the reason for the anisotropy of galvanomagnetic phenomena in single crystals. A measure of the effect of the magnetic field on an electron trajectory is the ratio of the free path length *l* of the electron to the radius of curvature of its trajectory in a field *H*: *r _{H}* =

*cp*/

*eH*(

*p*is the electron’s momentum). A magnetic field is considered to be weak with respect to galvanomagnetic phenomena if

*H*<<

*H*

_{0}=

*el*/

*cp*and strong if

*H*>>

*H*

_{0}.

At room temperatures, *H*_{0} ≈ 10^{5} to 10^{7} oersteds for various metals and good semiconductors; for poor semiconductors, *H*_{0} ≈ 10^{8} to 10^{9} oersteds. A reduction in temperature increases the path length *l* and therefore reduces the value of *H*_{0}. This makes it possible, through the use of low temperatures and ordinary magnetic fields (~ 10^{4} oersteds), to achieve conditions corresponding to a strong field *H*>>*H*_{0}.

The measurement of the resistance of single-crystal specimens of metals in strong magnetic fields is one of the important methods of studying metals. The dependence of resistance on the magnetic field strength and its direction with respect to the crystallographic axes is investigated. The theory of galvanomagnetic phenomena has shown that the dependence of the resistance on the field *H* is essentially associated with the energy spectrum of the electrons. The pronounced anisotropy of resistance in strong magnetic fields (for gold, silver, copper, tin, and other metals) indicates the essential anisotropy of the Fermi surface. On the other hand, the low anisotropy of the resistance in a magnetic field indicates virtual isotropy of the Fermi surface. In this connection, if *ρ* does not tend toward saturation as the magnetic field is increased for all directions (bismuth, arsenic, and others), then electrons and holes are present in the conductors in equal numbers. If the resistance tends to saturate, it means that either electrons or holes predominate (the type of carriers can be determined from the sign of the Hall constant).

A small change in the resistance of metals in a magnetic field parallel to the current *I* is observed along with transverse galvanomagnetic phenomena: (Δ*ρ*/*ρ*)║, which is called the longitudinal galvanomagnetic effect. Quantum effects, which are manifested in the nonmonotonic (oscillating) dependency of the Hall constant and the resistance on the field *H*, are observed in strong magnetic fields.

When galvanomagnetic phenomena are studied in thin films and fine wires, both (Δ*ρ*/*ρ*)⊥ and (Δ*ρ*/*ρ*)║ are found to depend on the specimens’ dimensions and shapes (size effects). As *H* increases, this dependency disappears under *r _{H}* << where

*d*is the smallest dimension of the specimen. In ferromagnetic metals and semiconductors (ferrites), galvanomagnetic phenomena have a number of specific characteristics resulting from spontaneous magnetization without a magnetic field. For example, the Hall emf in ferromagnetic materials depends not only on the average field

*H*in a specimen but also on the magnetization, and the resistance sometimes decreases in weak fields.

### REFERENCES

Lifshits, I. M., M. I. Kaganov, and M. Ia. Abzel’.*Elektronnaia teoriia metallov*. Moscow, 1971.

Ziman, J.

*Printsipy teorii tverdogo tela*. Moscow, 1966. (Translated from English.)

M. I. KAGANOV [6-218-6; updated]