1. Theory

Definition: A Hall system passes a current through a sample in a perpendicular magnetic field and measures the transverse voltage, which gives the carrier type, sheet carrier density and, with the sheet resistance, the mobility (NIST).
Origin: The Lorentz force deflects moving carriers sideways until the charge that builds up at the edges produces a field that balances it (NIST).
Sign: Holes and electrons are deflected to the same side, so the sign of the Hall voltage gives the carrier type (NIST).
Scattering factor: The measured Hall mobility and density differ from the drift values by the Hall scattering factor, computed as 1.22 for electrons and 1.09 for holes in 4H-SiC (Deng et al.) and 1.1 to 1.7 in p-type ZnO (Makino et al.).

2. Working Principle

Van der Pauw geometry: Four small contacts at the edge of a thin, uniform, singly connected sample; current is forced across one diagonal and the Hall voltage read across the other (NIST).
Field and current reversal: Readings for both current directions and both field directions are combined so that misalignment and thermoelectric offsets cancel (NIST; Linseis).
Resistivity: The same contacts give the sheet resistance from the van der Pauw equation, needed for the mobility (NIST; Four-Point Probe page).
AC-field systems: An alternating field and a lock-in amplifier separate the small Hall signal from DC offsets for low-mobility materials (Linseis).

3. Applications

Material qualification: Carrier density and mobility of epitaxial layers, wafers and 2-D electron gases (NIST).
Temperature-dependent Hall: Two-carrier effects and Hall coefficient inversion as mobilities and carrier densities change with temperature (SES Hall manual; US 5,789,931).
Low-mobility materials: Organic, amorphous and oxide semiconductors with AC-field Hall (Linseis).

Hall Effect Relations

Name	Formula	Related Formulae	Variables/Constants
Lorentz force and Hall field	$F = - q (E + v \times B)$	$E_{y} = R_{H} J_{x} B_{z}, tan θ_{H} = μ_{H} B$	$F : f or ce o n an e l ec t r o n$ $q : e l e m e n t a r y c ha r g e$ $E : e l ec t r i c f i e l d$ $v : c a r r i er v e l oc i t y$ $B : ma g n e t i c f l ux d e n s i t y$ $E_{y} : t r an s v er se H a l l f i e l d$ $R_{H} : H a l l coe f f i c i e n t$ $J_{x} : c u r r e n t d e n s i t y$ $θ_{H} : H a l l an g l e$ $μ_{H} : H a l l m o bi l i t y; N I S T : t h e L or e n t z f or ce a s t h e ba s i s o f t h e H a l l e f f ec t; S E S H a l l man u a l : E_{h} = R_{H} J H; H a l l an g l e f r o m E_{y} / E_{x} = R_{H} σ B = μ_{H} B (co m p u t e d)$ $(co m p u t e d : μ = 1 c m^{2} / (V s) = 1 \times 1 0^{- 4} m^{2} / (V s) in 1 T g i v es tan θ_{H} = 1 \times 1 0^{- 4}, so t h e H a l l v o l t a g e i s o n l y 1 \times 1 0^{- 4} o f t h e l o n g i t u d ina l v o l t a g e)$
Hall voltage and sheet density	$∣ V_{H} ∣ = \frac{I B}{q n d}$	$n_{s} = n d = \frac{I B}{q ∣ V _{H} ∣}, n = \frac{n _{s}}{d}$	$V_{H} : H a l l v o l t a g e$ $I : c u r r e n t$ $B : f i e l d n or ma l t o t h e s am pl e$ $n : b u l k c a r r i er d e n s i t y$ $d : t hi c k n ess$ $n_{s} : s h ee t c a r r i er d e n s i t y; N I S T : t h ese e q u a t i o n s, V_{H} n e g a t i v e f or n - t y p e an d p os i t i v e f or p - t y p e in i t s s i g n co n v e n t i o n$ $(co m p u t e d : n_{s} = 1 \times 1 0^{12} c m^{- 2}, 10 μ A an d 0.5 T g i v e V_{H} = 3.1 mV; a 500 μ m w a f er a t 1 \times 1 0^{19} c m^{- 3} w i t h 10 mA an d 0.5 T g i v es 6.2 μ V)$
Hall coefficient and scattering factor	$R_{H} = \frac{V _{H} d}{I B} = \frac{r _{H}}{q p} or - \frac{r _{H}}{q n}$	$r_{H} = \frac{⟨ τ ^{2} ⟩}{⟨ τ ⟩ ^{2}}, μ_{H} = r_{H} μ$	$R_{H} : H a l l coe f f i c i e n t$ $r_{H} : H a l l sc a tt er in g f a c t or$ $τ : m o m e n t u m r e l a x a t i o n t im e$ $⟨ ⟩ : e n er g y a v er a g e$ $μ : d r i f t m o bi l i t y; D e n g e t a l . : r_{H} = μ_{H} / μ w i t h R_{H} = r_{H} / (n q), sc a tt er in g f a c t or a s t h e r a t i o o f e n er g y - a v er a g e d τ^{2} an d τ s q u a r e d, co m p u t e d 1.22 f or e l ec t r o n s an d 1.09 f or h o l es in 4 H - S i C a t 300 K; S E S H a l l man u a l : e x a c t c a l c u l a t i o n m o d i f i es t h e s im pl e r es u l t b y 3 π /8; M ak in o e t a l . : e f f ec t i v e H a l l f a c t or s 1.1 t o 1.7 in p - t y p e Z n O; N I S T d oes n o t d e f in e R_{H}$ $(co m p u t e d : t ak in g r_{H} = 1 w h e n i t i s 1.22 o v er es t ima t es t h e d e n s i t y b y 22 p er ce n t an d u n d er es t ima t es t h e d r i f t m o bi l i t y b y t h e s am e f a c t or)$

Mobility and Mixed Conduction

Name	Formula	Related Formulae	Variables/Constants
Hall mobility	$μ_{H} = \frac{∣ V _{H} ∣}{R _{s} I B} = \frac{1}{q n _{s} R _{s}}$	$ρ = R_{s} d, μ_{H} = \frac{∣ R _{H} ∣}{ρ}$	$μ_{H} : H a l l m o bi l i t y$ $R_{s} : s h ee t r es i s t an ce f r o m t h e v an d er P a u w m e a s u r e m e n t$ $ρ : r es i s t i v i t y; N I S T : μ = ∣ V_{H} ∣ / (R_{s} I B) = 1/ (q n_{s} R_{s}) an d ρ = R_{s} d; S E S H a l l man u a l : μ = R_{H} σ$ $(co m p u t e d : n_{s} = 1 \times 1 0^{12} c m^{- 2} w i t h R_{s} = 6.24 kΩ p er s q u a r e g i v es 1000 c m^{2} / (V s))$
Two-carrier Hall coefficient	$R_{H} = \frac{r _{H} ( p μ _{p}^{2} - n μ _{n}^{2} )}{q ( p μ _{p} + n μ _{n} ) ^{2}}$	$R_{H} = 0 when p μ_{p}^{2} = n μ_{n}^{2}$	$n, p : e l ec t r o n an d h o l e d e n s i t i es$ $μ_{n}, μ_{p} : m o bi l i t i es; S E S H a l l man u a l, E q . 5 (g a r b l e d in e x t r a c t i o n) : t hi s f or m w i t h o u t r_{H}; a co mm o n r_{H} f or b o t h c a r r i er s (m y r e a d in g)$ $(a ss u m e d : μ_{n} / μ_{p} = 3; co m p u t e d : R_{H} c han g es s i g n w h e n p = 9 n)$

Measurement Procedure

Name	Formula	Related Formulae	Variables/Constants
Van der Pauw Hall readings	$V_{C} = V_{24 P} - V_{24 N}, V_{D} = V_{42 P} - V_{42 N}, V_{E} = V_{13 P} - V_{13 N}, V_{F} = V_{31 P} - V_{31 N}$	$p_{s} = \frac{8 \times 1 0 ^{- 8} I B}{q ( V _{C} + V _{D} + V _{E} + V _{F} )} (B in gauss, p_{s} in c m^{- 2})$	$V_{24 P} : v o l t a g e b e tw ee n co n t a c t s 2 an d 4 w i t h c u r r e n t f r o m 1 t o 3 an d p os i t i v e f i e l d$ $N : n e g a t i v e f i e l d, o t h er s u b scr i pt s b y t h e s am e r u l e; N I S T : p os i t i v e s u m f or p - t y p e an d n e g a t i v e s u m f or n - t y p e (n_{s} i s t h e n t h e ma g ni t u d e), s i g n s o f m e a s u r e d v o l t a g es k e pt so o f f se t s c an ce l, d ia g o na l p ai r s s h o u l d a g r ee f or a u ni f or m s am pl e, f i e l d u ni f or m w i t hin 3 p er ce n t an d n or ma l w i t hin a f e w d e g r ees, 500 t o 5000 G f r o m a p er man e n t ma g n e t or e l ec t r o ma g n e t, 10 μ A t o 100 mA (d o w n t o 1 nA f or se mi - in s u l a t in g G a A s), v o l t m e t er co v er in g 1 μ V t o 1 V$ $(co m p u t e d : e a c h d i f f er e n ce i s 2 V_{H}, so t h e s u m i s 8 V_{H}; 1 G = 1 \times 1 0^{- 4} T an d 1 c m^{- 2} = 1 \times 1 0^{4} p er s q u a r e m e t r e co mbin e t o t h e 8 \times 1 0^{- 8} f a c t or)$
AC-field Hall measurement	$B = B_{0} cos ω t \Rightarrow V_{H} (t) = \frac{I B _{0}}{q n _{s}} cos ω t$	$V_{m e a s} = V_{H} (t) + V_{0} + V_{in d}, V_{in d} \propto \frac{d B}{d t} (9 0^{\circ} out of phase)$	$B_{0} : f i e l d am pl i t u d e$ $ω : an g u l a r f r e q u e n cy$ $V_{0} : f i e l d - in d e p e n d e n t o f f se t (mi s a l i g nm e n t an d t h er m oe l ec t r i c v o l t a g es)$ $V_{in d} : v o l t a g e in d u ce d in t h e s am pl e an d l e a d s; L in se i s : o f f se t s a pp e a r a s D C an d a r e se p a r a t e d b y l oc k - in d e t ec t i o n, t h e in d u ce d t er m i s c u r r e n t - in d e p e n d e n t an d in q u a d r a t u r e, D C - f i e l d m e t h o d l imi t e d b e l o w ab o u t 1 c m^{2} / (V s), A C m e t h o d d o w n t o ab o u t 1 \times 1 0^{- 3} c m^{2} / (V s); mi s a l i g nm e n t v o l t a g e p r o p or t i o na l t o r es i s t i v i t y an d c u r r e n t, t h er m oe l ec t r i c v o l t a g e f r o m t h e s am pl e - co n t a c t j u n c t i o n$ $(co m p u t e d : t h e A C m e t h o d r e a c h es m o bi l i t i es 1000 t im es l o w er, co n s i s t e n t w i t h L in se i s^{'} s f a c t or o f 1000)$

Important Questions

Ques: What does a Hall system measure, and how?

Carrier type, sheet carrier density and mobility. With current I through the sample and a field B normal to it, the Lorentz force pushes carriers sideways until the transverse Hall voltage balances it; NIST gives $∣ V_{H} ∣ = I B / (q n d)$ , so the sheet density is $n_{s} = I B / (q ∣ V_{H} ∣)$ , with no need to know the thickness. The sign of $V_{H}$ shows whether electrons or holes dominate. The same contacts give the sheet resistance from the van der Pauw method, and the mobility follows as $μ = 1/ (q n_{s} R_{s})$ . A layer with $n_{s} = 1$ e12 $c m^{- 2}$ and $R_{s} = 6.24$ $kΩ$ per square has a mobility of 1000 $c m^{2} / (V s)$ (computed). The bulk density and resistivity then need the thickness. Reference: NIST, ‘Hall Effect’ (basic equations, van der Pauw resistances, Hall procedure, sign convention, error sources) (fetch-tool summary).

Ques: Why are eight Hall readings taken with both field directions?

To cancel offsets that are often larger than the Hall voltage itself. The contacts are never exactly opposite each other, so a misalignment voltage, proportional to the resistivity and current, appears even at zero field, and junctions between contacts and sample add thermoelectric voltages (Linseis). NIST therefore measures the voltage across each diagonal for both current directions and both field directions, forming $V_{C}$ to $V_{F}$ as differences between positive- and negative-field readings, keeping their signs. The misalignment voltage does not change with the field and cancels in each difference, while each difference is twice the Hall voltage, so the sum is $8 V_{H}$ ; with B in gauss and density per square centimetre this gives the factor $8 \times 1 0^{- 8}$ (computed). Reference: NIST, ‘Hall Measurement’ procedure page, archived and no longer updated (eight Hall voltages, V_C to V_F, 8e-8 factor, checks, equipment ranges) (fetch-tool summary); Linseis, ‘DC and AC Hall Effect Measurements’ wiki page (offsets, AC-field method, mobility limits; equations as images) (fetch-tool summary).

Ques: How large is the Hall voltage in practice?

It ranges from millivolts to microvolts. Because $V_{H} = I B / (q n_{s})$ , it falls as the sheet density rises. A 2-D electron gas or thin epitaxial layer with $n_{s} = 1$ e12 $c m^{- 2}$ , measured at 10 $μ A$ in 0.5 $T$ , gives about 3.1 $mV$ (computed). A 500 $μ m$ wafer doped to $1 \times 1 0^{19}$ $c m^{- 3}$ has a sheet density of $5 \times 1 0^{17}$ $c m^{- 2}$ , and at 10 $mA$ in the same field the Hall voltage is only about 6.2 $μ V$ (computed), in line with NIST’s note that thick, heavily doped samples give voltages of the order of microvolts. NIST specifies fields of 500 to 5000 $G$ , which is 0.05 to 0.5 $T$ , and a voltmeter covering 1 $μ V$ to 1 $V$ . Current is limited by the 1 to 5 $mW$ heating guideline. Reference: NIST, ‘Hall Measurement’ procedure page, archived and no longer updated (eight Hall voltages, V_C to V_F, 8e-8 factor, checks, equipment ranges) (fetch-tool summary).

Ques: Why do DC Hall systems fail for low-mobility materials?

Because the Hall voltage shrinks relative to the longitudinal voltage as the mobility falls. The ratio is set by the Hall angle, $tan θ_{H} = μ_{H} B$ (computed from $E_{y} = R_{H} J_{x} B$ and $μ = R_{H} σ$ in the SES Hall manual); at 1 $c m^{2} / (V s)$ in 1 $T$ , it is only $1 \times 1 0^{- 4}$ (computed), so a misalignment voltage that is a small fraction of the longitudinal drop swamps the Hall signal. Linseis places the limit of the DC-field method at about 1 $c m^{2} / (V s)$ . AC-field systems drive the magnet with an alternating field, so the Hall voltage alternates while misalignment and thermoelectric offsets stay DC, and a lock-in amplifier extracts it. The voltage induced in the sample leads by the changing field is in quadrature and current-independent, so phase detection and current reversal remove it. This extends the range to about $1 \times 1 0^{- 3}$ $c m^{2} / (V s)$ . Reference: Scientific Equipment and Services, Roorkee, ‘User’s Manual: Hall Effect Setup’ (IISER Kolkata PH324 manual): E_h = R_H J H, R_H = 1/(nq), mu = R_H sigma, two-carrier R_H (Eq. 5, garbled in extraction), exact calculation modifies results by 3 pi/8, Hall coefficient inversion in p-type, Ettingshausen, Nernst and Righi-Leduc voltages, averaging of four readings (fetch-tool summary); Linseis, ‘DC and AC Hall Effect Measurements’ wiki page (offsets, AC-field method, mobility limits; equations as images) (fetch-tool summary).

Ques: What is the Hall scattering factor, and how does it bias the results?

It is the ratio between the Hall and drift mobilities, $r_{H} = μ_{H} / μ$ , so that $R_{H} = r_{H} / (q n)$ (Deng et al.); it is a ratio of energy averages of the relaxation time, $⟨ τ^{2} ⟩ / ⟨ τ ⟩^{2}$ , which exceeds one when carriers of different energies scatter at different rates. Assuming $r_{H} = 1$ therefore gives a density too high and a mobility too low by the factor $r_{H}$ (computed from these relations). The SES Hall manual notes that the exact calculation modifies the simple result by $3 π /8$ , which is 1.18 (computed). Deng et al. compute 1.22 for electrons and 1.09 for holes in 4H-SiC at 300 $K$ , and Makino et al. find effective values of 1.1 to 1.7 in p-type ZnO. Taking $r_{H} = 1$ when it is 1.22 overestimates the density by 22 per cent (computed). Reference: T. Deng, D. Yang and X. Pi, ‘Phonon-limited carrier mobilities and Hall factors in 4H-SiC from first principles’, arXiv:2210.02813, 2022: r_H = mu_H/mu, R = r_H/(ne), scattering factor <<tau^2>>/<>^2, computed 1.22 for electrons and 1.09 for holes at 300 K (fetch-tool summary); Scientific Equipment and Services, Roorkee, ‘User’s Manual: Hall Effect Setup’ (IISER Kolkata PH324 manual): E_h = R_H J H, R_H = 1/(nq), mu = R_H sigma, two-carrier R_H (Eq. 5, garbled in extraction), exact calculation modifies results by 3 pi/8, Hall coefficient inversion in p-type, Ettingshausen, Nernst and Righi-Leduc voltages, averaging of four readings (fetch-tool summary); T. Makino et al., ‘Hole Transport in p-Type ZnO’, arXiv:cond-mat/0605287: Hall factors 1.0 to 1.5 per band, effective Hall factors 1.1 to 1.7 (fetch-tool summary).

Ques: What happens when both electrons and holes conduct?

The Hall coefficient becomes a weighted difference and can vanish. With both carrier types, $R_{H} = (p μ_{p}^{2} - n μ_{n}^{2}) / [q (p μ_{p} + n μ_{n})^{2}]$ , weighting each by the square of its mobility (SES Hall manual, Eq. 5). If electron mobility is three times hole mobility (assumed), the coefficient changes sign when the hole density is nine times the electron density (computed), so a lightly p-type sample can look n-type at high temperature; the manual notes that such Hall coefficient inversion is characteristic of p-type semiconductors. A single-field measurement gives only averaged density and mobility, while multi-field data analysed as a mobility spectrum show each carrier type as a peak; US Patent 5,789,931 separates four species in p-type HgCdTe on CdTe, including holes at the substrate interface. Reference: Scientific Equipment and Services, Roorkee, ‘User’s Manual: Hall Effect Setup’ (IISER Kolkata PH324 manual): E_h = R_H J H, R_H = 1/(nq), mu = R_H sigma, two-carrier R_H (Eq. 5, garbled in extraction), exact calculation modifies results by 3 pi/8, Hall coefficient inversion in p-type, Ettingshausen, Nernst and Righi-Leduc voltages, averaging of four readings (fetch-tool summary); US Patent 5,789,931, ‘Quantitative mobility spectrum analysis for magnetic field dependent Hall and resistivity data’: single-field data give only averaged density and mobility, multi-field mobility spectrum with one peak per species, four species in p-HgCdTe on CdTe (fetch-tool summary).

Ques: Which errors do not cancel by reversing current and field?

Those that reverse in the same way as the Hall voltage, or that come from the sample itself. NIST lists four error sources: ohmic contact quality and size, sample uniformity and thickness, thermomagnetic effects in nonuniform temperature, and photoconductive and photovoltaic effects, reduced by measuring in the dark. The SES Hall manual adds that the Ettingshausen, Nernst and Righi-Leduc effects also put voltages on the Hall probes; averaging four readings removes the Nernst and Righi-Leduc terms, while the Ettingshausen term is negligible only when heat is carried mainly by the lattice or the thermoelectric power is small. Finite-size contacts short the Hall voltage, giving errors of order contact size over spacing (NIST). Field uniformity within 3 per cent and alignment within a few degrees are required, and NIST checks that both diagonals give the same Hall voltage. Reference: NIST, ‘Hall Effect’ (basic equations, van der Pauw resistances, Hall procedure, sign convention, error sources) (fetch-tool summary); NIST, ‘Hall Measurement’ procedure page, archived and no longer updated (eight Hall voltages, V_C to V_F, 8e-8 factor, checks, equipment ranges) (fetch-tool summary); Scientific Equipment and Services, Roorkee, ‘User’s Manual: Hall Effect Setup’ (IISER Kolkata PH324 manual): E_h = R_H J H, R_H = 1/(nq), mu = R_H sigma, two-carrier R_H (Eq. 5, garbled in extraction), exact calculation modifies results by 3 pi/8, Hall coefficient inversion in p-type, Ettingshausen, Nernst and Righi-Leduc voltages, averaging of four readings (fetch-tool summary).

References

NIST Nanoscale Device Characterization Division, “Hall Effect.” Available: https://www.nist.gov/pml/nanoscale-device-characterization-division/popular-links/hall-effect/hall-effect
NIST, “Hall Effect Measurements: Hall Measurement” (archived page, no longer updated). Available: https://www.nist.gov/node/430016
Linseis, “DC and AC Hall Effect Measurements,” Linseis wiki (author and date not stated). Available: https://www.linseis.com/en/wiki/dc-and-ac-hall-effect-measurements/
T. Deng, D. Yang and X. Pi, “Phonon-limited carrier mobilities and Hall factors in 4H-SiC from first principles,” arXiv:2210.02813, 2022. Available: https://arxiv.org/abs/2210.02813
Scientific Equipment and Services, Roorkee, User’s Manual: Hall Effect Setup, PH324 laboratory manual, IISER Kolkata. Available: https://www.iiserkol.ac.in/~ph324/ExptManuals/HallEffect.pdf
US Patent 5,789,931, “Quantitative mobility spectrum analysis for magnetic field dependent Hall and resistivity data.” Available: https://image-ppubs.uspto.gov/dirsearch-public/print/downloadPdf/5789931
T. Makino, A. Tsukazaki, A. Ohtomo, M. Kawasaki and H. Koinuma, “Hole transport in p-type ZnO,” arXiv:cond-mat/0605287. Available: https://arxiv.org/abs/cond-mat/0605287

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