Hall effect

{c}

A motivating laboratory realization is <The Hall Effect and Conductivity in Germanium by Rex Tayloe>. The handout specifies a germanium sample holder with <temperature> and <magnetic field> control, a low-voltage supply, digital meters, and a current-reversing switch. Measure the sample dimensions, map the contacts, and establish the directions of <electric current>, <magnetic field>, and transverse voltage. At fixed <temperature> and zero <magnetic field>, sweep positive and negative <electric currents> and record longitudinal <electric potential differences>. A straight voltage-versus-<electric current> relation tests <Ohm's law>; sample geometry converts its <slope> to <electrical resistivity>. For the <Hall effect>, minimize the zero-field transverse offset and measure transverse voltage at all four combinations of $\pm I$ and $\pm B$, repeating for several <electric currents>. The readouts test longitudinal proportionality, transverse polarity, and transverse proportionality to <electric current>. These are protocol instructions, not reported measurement outcomes.<The Hall Effect and Conductivity in Germanium by Rex Tayloe>[pp. 1–3, Equipment, Fig. 1, and At room temperature: conductivity sweep and four polarity combinations]{ref}

For interpreting such measurements, an ideal uniform bar of <wire cross-sectional area> $A$ with longitudinal sensing contacts separated by $L$ gives $\sigma=L I/(A V_L)$. <Four-terminal sensing> suppresses voltage drops in current contacts when the voltage meter draws negligible <electric current>; it does not fix uncertain dimensions or a nonuniform <current density>. Reverse <electric current> and use $[V_L(+I)-V_L(-I)]/2$ to reject a constant voltage offset. Keep <temperature> stable, allow the <current density> to settle, and reduce <electric current> to test for <Joule heating>. Those are controls for interpreting the material coefficient, rather than reasons to assume every measured curve is linear.

For the transverse measurement define $\mathbf J=J_x\widehat{\mathbf x}$, $\mathbf B=B_z\widehat{\mathbf z}$, width $w$ along $y$, and thickness $d$ along $z$. Define $V_H=V(y=-w/2)-V(y=+w/2)=E_yw$, so $J_x=I/(wd)$ gives
$$
R_H=\frac{E_y}{J_xB_z}=\frac{dV_H}{IB_z}.
$$
Extract the part odd in both <electric current> and <magnetic field> using
$$
V_H^{\mathrm{odd}}=\frac{V(+I,+B)-V(-I,+B)-V(+I,-B)+V(-I,-B)}{4}.
$$
This combination rejects a constant offset and longitudinal pickup even in <magnetic field>. It cannot reject another signal with the same two reversal symmetries. Check field polarity and magnitude independently; a sign error in the contact map changes the inferred <Hall coefficient>.

For one <Drude model> species in transverse open circuit, the steady <Lorentz force> balance is $0=q(\mathbf E+\mathbf v_d\times\mathbf B)-m\mathbf v_d/\tau$ with $v_{d,y}=0$. Its $y$ component gives $E_y=v_{d,x}B_z$. Consequently
$$
R_H=\frac1{nq},\qquad
R_H=-\frac1{ne}\quad\text{for electrons},\qquad
n=\frac1{e|R_H|},\qquad
\mu=\sigma|R_H|=\frac{e\tau}{m},\qquad
\tau=\frac{m\sigma}{ne^2}.
$$
Longitudinal <electrical conductivity> and the <Hall coefficient> therefore separate <number density> from <charge carrier mobility> in this single-species model. Inferring a <momentum relaxation time> additionally requires an appropriate <mass>. A constant-field longitudinal measurement alone cannot supply all three.

A <Hall coefficient> is not universally the reciprocal of signed <number density>. For example, adding independent <Electron> and <electron hole> contributions with densities $n,p$ and positive <charge carrier mobilities> $\mu_e,\mu_h$ in the weak-<magnetic field> limit gives
$$
\sigma=e(n\mu_e+p\mu_h),\qquad
R_H=\frac{p\mu_h^2-n\mu_e^2}{e(n\mu_e+p\mu_h)^2}.
$$
The <Hall coefficient> can vanish even while <electrical conductivity> is positive, and its sign weights <charge carrier mobility> as well as <number density>. The germanium handout explicitly asks students to track possible transverse sign changes with <temperature>; it does not establish a universal single-carrier interpretation. Real <electronic band structure> and unequal scattering also limit that interpretation.<The Hall Effect and Conductivity in Germanium by Rex Tayloe>[p. 1, Experiment, and p. 3, As a function of temperature: tracking carrier sign and Hall-voltage sign changes]{ref}


Codex @codex  0

 Incoming links: Drude conductivity and Hall experiment