1. Theory

Definition: The four-point probe passes a current through two outer probes and measures the voltage across two inner probes, so contact and spreading resistances at the current probes do not enter the measurement (Ossila; AggieFab).
Thin layers: For an infinite thin sheet with equally spaced collinear probes, $R_{s} = (π / ln 2) V / I = 4.532 V / I$ , independent of probe spacing (AggieFab; Ossila).
Thick samples: For a semi-infinite body, $ρ = 2 π s V / I$ , which depends on spacing; intermediate thicknesses need a correction factor (AggieFab; Doolittle).
Finite samples: Edges and finite diameter or width require geometric correction factors, tabulated by Smits and in NBS Technical Note 199 (AggieFab; NBS TN 199; Ossila).
Van der Pauw: Four contacts on the periphery of an arbitrarily shaped, uniform, thin sample give its sheet resistance from two measured resistances (NIST).

2. Working Principle

Separate current and voltage paths: The high-impedance voltmeter draws no current, so the probe contact resistance does not drop any voltage (Doolittle; Ossila).
Current spreading: Current spreads radially from each outer probe; in a thin sheet the potential falls logarithmically, giving the $ln 2$ in the formula (Doolittle).
Polarity reversal: Measuring with both current directions and averaging removes thermoelectric offsets (NIST).
Power limits: The current is chosen so that heating stays below a few milliwatts, avoiding resistivity drift (NIST).

3. Applications

Implant and diffusion monitoring: Sheet resistance maps after implant and anneal track dose and activation (Doolittle; From sheet resistance to doping row).
Thin-film metrology: Metal and silicide film resistivity and thickness control (Ossila; Doolittle).
Wafer acceptance: Bulk resistivity of incoming wafers with thickness and edge corrections (NBS TN 199).
Material research: Van der Pauw and Hall structures for new films and 2-D materials (NIST).

Basic Four-Point Probe Relations

Name	Formula	Related Formulae	Variables/Constants
Sheet resistance of a thin, wide layer	$R_{s} = \frac{π}{l n 2} \frac{V}{I} = 4.532 \frac{V}{I}$	$ρ = R_{s} t = 4.532 t \frac{V}{I}$	$R_{s} : s h ee t r es i s t an ce (o hm p er s q u a r e)$ $V : v o l t a g e b e tw ee n t h e inn er p r o b es$ $I : c u r r e n t t h r o ug h t h e o u t er p r o b es$ $ρ : r es i s t i v i t y$ $t : l a y er t hi c k n ess; O ss i l a : R_{s} f r o m Δ V / I an d ρ = R_{s} t, v a l i d f or l a y er s n o t hi c k er t han 40 p er ce n t o f t h e p r o b e s p a c in g o n s am pl es t y p i c a l l y 40 t im es t h e s p a c in g; A g g i e F ab (a f t er S mi t s 1958) : ρ = 4.5324 t (V / I) f or t hin, w i d e s am pl es w i t h 4.5324 \approx π / ln 2$ $(co m p u t e d : π / ln 2 = 4.5324; 2.2 mV a t 1 mA g i v es 9.97 Ω/ sq, an d f or a 0.5 μ m f i l m 4.99 \times 1 0^{- 4} Ω cm)$
Semi-infinite (thick) sample	$ρ = 2 π s \frac{V}{I}$	$f_{1} \to 2 ln 2 \frac{s}{t} (t ≫ s), 4.532 t \cdot 2 ln 2 \frac{s}{t} = 2 π s$	$s : p r o b e s p a c in g$ $f_{1} : t hi c k n ess cor r ec t i o n f a c t or; D oo l i ttl e : se mi - in f ini t e f or m ρ = 2 π s V / I; A g g i e F ab : f_{11} \approx 1 f or t ≪ s an d f_{11} = f_{12} \approx 2 ln 2 (s / t) f or t ≫ s, in s u l a t in g (f_{11}) an d co n d u c t in g (f_{12}) b o tt o m b o u n d a r i es$ $(co m p u t e d : 4.5324 x 2 l n 2 = 6.2832 = 2 π, so t h e t hin - l a y er f or m u l a w i t h t h e t hi c k - s am pl e f a c t or r e d u ces e x a c tl y t o t h e se mi - in f ini t e o n e)$

Geometric Corrections

Name	Formula	Related Formulae	Variables/Constants
Thickness correction	$ρ = 4.532 t \frac{V}{I} f_{1} (\frac{t}{s}) f_{2} (\frac{d}{s})$	$f_{1} = 0.9995, 0.9974, 0.9214, 0.8490, 0.6336 (t / s = 0.4, 0.5, 1.0, 1.25, 2.0)$	$f_{1} : t hi c k n ess cor r ec t i o n$ $f_{2} : l a t er a l - s i z e cor r ec t i o n$ $d : s am pl e d iam e t er or w i d t h; O ss i l a : t h ese t hi c k - s am pl e f a c t or s (in s u l a t in g b o tt o m); N B S T N 199 : t hi c k n ess cor r ec t i o n s b e l o w 0.3 p er ce n t f or t b e l o w ha l f t h e s p a c in g an d b e l o w 0.05 p er ce n t b e l o w 0.4 t im es t h e s p a c in g, w i t h S mi t s^{'} f a c t or n ee d e d f or t hi c k er s am pl es (see t h e Q / A); A g g i e F ab : t h e co mbin e d f or m a f t er S mi t s$ $(co m p u t e d : a t t / s = 2 t h e t ab u l a t e d f a c t or 0.634 l i es b e l o w t h e a sy m pt o t i c t hi c k - s am pl e v a l u e 2 ln 2/ (t / s) = 0.693, w hi c h t h e f a c t or a pp r o a c h es a s t / s g r o w s)$
Lateral-size and edge corrections	$C (\frac{w}{s}, \frac{l}{w}) = 0.2204, 0.4292, 0.7887, 0.9345, 1 (w / s = 1, 2, 5, 10, \infty)$	$ρ = \frac{V}{I} t F (\frac{s}{r}, \frac{A}{r}) (circular wafers, NBS)$	$C : r ec t an g u l a r - s am pl e f a c t or w i t h p r o b es ce n t r e d an d p a r a l l e l t o t h e l o n g e d g e l$ $w : w i d t h$ $F : c i r c u l a r - s am pl e f a c t or$ $r : s am pl e r a d i u s$ $A : p r o b e - a r r a y o f f se t f r o m t h e ce n t r e; O ss i l a : t h e r ec t an g u l a r t ab l e (l / w = 2 t o 4, a l m os t in d e p e n d e n t o f l / w ab o v e w / s = 2), c u bi c - s pl in e in t er p o l a t i o n, an d a w or k e d e x am pl e l = 20 mm, w = 10 mm, s = 2 mm g i v in g 0.7887; N B S T N 199 : t ab l es F_{1} (p r o b es a l o n g a d iam e t er), F_{2} (p er p e n d i c u l a r) an d F_{3} (s q u a r e a r r a y) f or s / r o f 0.01 t o 0.70 an d A / r o f 0 t o 0.99, f or e x am pl e F_{1} = 4.49925, F_{2} = 4.49961 an d F_{3} = 9.02097 a t s / r = 0.02, A / r = 0.81; A g g i e F ab : f_{2} \approx 1 f or d / s ≫ 1$ $(co m p u t e d : F_{1} = 4.49925 i s 0.73 p er ce n t b e l o w π / ln 2 = 4.5324, t h e er r or f r o m an e d g e a t 81 p er ce n t o f t h e r a d i u s)$

Van der Pauw and Doping

Name	Formula	Related Formulae	Variables/Constants
Van der Pauw sheet resistance	$exp (- \frac{π R _{A}}{R _{s}}) + exp (- \frac{π R _{B}}{R _{s}}) = 1$	$R_{A} = \frac{R _{21, 34} + R _{12, 43} + R _{43, 12} + R _{34, 21}}{4}, R_{s} = \frac{π R}{l n 2} (R_{A} = R_{B} = R)$	$R_{A}, R_{B} : c ha r a c t er i s t i c r es i s t an ces o f t h e tw o or i e n t a t i o n s$ $R_{ij, k l} : v o l t a g e b e tw ee n co n t a c t s k an d l d i v i d e d b y c u r r e n t f r o m i t o j; N I S T : e i g h t p os i t i v e r es i s t an ces m e a s u r e d w i t h b o t h c u r r e n t p o l a r i t i es, R_{A} an d R_{B} a s a v er a g es, ρ = R_{s} d, co n s i s t e n cy c h ec k s R_{21, 34} = R_{12, 43} an d r ec i p r oc i t y c h ec k s w i t hin 5 p er ce n t (p r e f er ab l y 3), co n t a c t s an d t hi c k n ess m u c h s ma l l er t han co n t a c t s p a c in g w i t h er r or s o f or d er D / L, d i ss i p a t i o n b e l o w 5 mW (p r e f er ab l y 1) v ia I < (200 R)^{- 0.5}, c u r r e n t s f r o m 10 μ A t o 100 mA (d o w n t o 1 nA f or se mi - in s u l a t in g G a A s n e a r 1 \times 1 0^{7} Ω cm) an d 1 μ V v o l t m e t er r eso l u t i o n (K e i t h l ey n o t e 3180 : v an d er P a u w e q u a t i o n)$ $(co m p u t e d : R_{A} = 100 an d R_{B} = 150 Ω g i v e R_{s} = 558.6 Ω/ sq, 1.4 p er ce n t b e l o w π (R_{A} + R_{B}) / (2 ln 2) = 566.5; I < (200 R)^{- 0.5} f or R = 100 Ω i s 7.07 mA, e x a c tl y 5 mW)$
From sheet resistance to doping	$ρ = \frac{1}{q ( n μ _{n} + p μ _{p} )} \approx \frac{1}{q N μ ( N )}$	$R_{s} = \frac{1}{q \int _{0}^{x_{j}} N ( x ) μ ( N ( x ) ) d x} (diffused layer)$	$q : e l e m e n t a r y c ha r g e$ $n, p : c a r r i er d e n s i t i es$ $μ : m o bi l i t y$ $N : d o p in g$ $x_{j} : j u n c t i o n d e pt h; D oo l i ttl e : R_{s} = 1/ (q n μ t) f or a u ni f or m n - t y p e l a y er an d R_{s} = 1/ (q \int n (x) μ d x) f or n o n u ni f or m l a y er s, t h e h o l e t er m a dd e d h er e b y sy mm e t r y (m y r e a d in g); m o bi l i t y a g ain s t d o p in g f r o m t h e T C A D M o bi l i t y M o d e l s p a g e$ $(co m p u t e d : n = 1 \times 1 0^{19} c m^{- 3} w i t h μ = 108 c m^{2} / (V s) g i v es 5.78 \times 1 0^{- 3} Ω cm, an d a u ni f or m 0.2 μ m l a y er ha s R_{s} = 289 Ω/ sq)$

Important Questions

Ques: Why does a four-point probe use four probes instead of two?

To keep contact resistance out of the measurement. In a two-probe measurement the current flows through the probe contacts and the spreading region beneath them, and their resistance, often far larger than the sheet resistance, adds to the reading. The four-point probe drives current through the outer probes and measures voltage with the inner pair through a high-impedance voltmeter, so almost no current flows in the voltage contacts and their resistance drops no voltage (Doolittle). For a thin, wide layer the result is $R_{s} = (π / ln 2) V / I$ , with $π / ln 2 = 4.5324$ (Ossila; AggieFab, computed). A reading of 2.2 $mV$ at 1 $mA$ is 9.97 $Ω/ sq$ , and for a 0.5 $μ m$ film a resistivity of $4.99 \times 1 0^{- 4}$ $Ω cm$ (computed). Reference: W. A. Doolittle, ECE 4813 lecture ‘Resistivity and Sheet Resistance’, Georgia Tech: rho = 2 pi s V/I, thin-layer pi/ln2 form, current flowing radially from the tips, Kelvin separation of current and voltage pairs, Rsh = 1/(q n mu t) and Rsh = 1/(q integral n(x) mu dx) for nonuniform layers, implant and metal-film sheet-resistance mapping, thermal variations from drive current (fetch-tool summary); Ossila, ‘Calculate Sheet Resistance Using the Four-Probe Method’, Ossila application page (author and date not stated; equations as images) (fetch-tool summary); AggieFab (Texas A&M University), ‘Four-point probe correction factor’ handout, after F. M. Smits, Bell System Technical Journal 37, 711-718 (1958) (formulas partly garbled; tables only as charts) (fetch-tool summary).

Ques: Is a film thin enough below 0.4 or 0.5 of the probe spacing?

Both limits appear, and they are consistent. Ossila says the simple formula is valid for layers no thicker than 40 per cent of the probe spacing, and its correction table gives 0.9995 at a thickness-to-spacing ratio of 0.4 and 0.9974 at 0.5. NBS Technical Note 199 says its thin-sample tables apply below about half the probe spacing, where the thickness correction is under 0.3 per cent, and under 0.05 per cent below 0.4. The two sources therefore describe the same curve with different tolerances: 0.4 for 0.05 per cent accuracy, 0.5 for 0.3 per cent (computed agreement with the table: 0.26 and 0.05 per cent). For a typical 1 $mm$ probe spacing, both conditions hold for films and diffused layers, but a 525 $μ m$ wafer, at 0.53, needs the 0.3 per cent correction (computed). Reference: Ossila, ‘Calculate Sheet Resistance Using the Four-Probe Method’, Ossila application page (author and date not stated; equations as images) (fetch-tool summary); L. J. Swartzendruber, ‘Correction Factor Tables for Four-Point Probe Resistivity Measurements on Thin, Circular Semiconductor Samples’, NBS Technical Note 199, National Bureau of Standards (15 April 1964) (correction-factor formulas garbled in the OCR) (fetch-tool summary).

Ques: How does the thin-layer formula connect to the thick-sample formula?

Through the thickness correction factor. For a thick sample, Smits’ factor tends to $2 ln 2 (s / t)$ (AggieFab), and multiplying the thin-layer result $4.532 t V / I$ by it gives $4.532 \times 2 ln 2 s V / I = 2 π s V / I$ (computed: $4.5324 \times 2 ln 2 = 6.2832$ ), the semi-infinite result given by Doolittle. Ossila’s tabulated factor falls from 0.9995 at a ratio of 0.4 to 0.6336 at 2.0; the asymptotic expression $2 ln 2/ (t / s)$ gives 0.693 at 2.0 (computed), so the table is still approaching the thick limit from below at that thickness. In the thin limit the spacing cancels and only sheet resistance is measured, while in the thick limit thickness cancels and the spacing matters. Probe spacing must therefore be known accurately only for thick samples. Reference: AggieFab (Texas A&M University), ‘Four-point probe correction factor’ handout, after F. M. Smits, Bell System Technical Journal 37, 711-718 (1958) (formulas partly garbled; tables only as charts) (fetch-tool summary); W. A. Doolittle, ECE 4813 lecture ‘Resistivity and Sheet Resistance’, Georgia Tech: rho = 2 pi s V/I, thin-layer pi/ln2 form, current flowing radially from the tips, Kelvin separation of current and voltage pairs, Rsh = 1/(q n mu t) and Rsh = 1/(q integral n(x) mu dx) for nonuniform layers, implant and metal-film sheet-resistance mapping, thermal variations from drive current (fetch-tool summary); Ossila, ‘Calculate Sheet Resistance Using the Four-Probe Method’, Ossila application page (author and date not stated; equations as images) (fetch-tool summary).

Ques: How large are edge and finite-size corrections?

Large near edges, negligible on big samples. Ossila’s table for rectangles, probes centred and parallel to the long edge, gives correction factors of 0.2204 at a width of one probe spacing, 0.4292 at two, 0.7887 at five and 0.9345 at ten, almost independent of the length-to-width ratio; its worked example, a 20 by 10 $mm$ sample with 2 $mm$ spacing, gives 0.7887. Ossila recommends samples about 40 times the probe spacing. For circular wafers, NBS Technical Note 199 tabulates factors for probes along a diameter, perpendicular to it and in a square array; at a probe spacing of 0.02 radius and an offset of 0.81 radius, the collinear factor is 4.49925, 0.73 per cent below 4.5324 (computed). So probing near a wafer edge biases the reading low by up to several per cent unless corrected. Reference: Ossila, ‘Calculate Sheet Resistance Using the Four-Probe Method’, Ossila application page (author and date not stated; equations as images) (fetch-tool summary); L. J. Swartzendruber, ‘Correction Factor Tables for Four-Point Probe Resistivity Measurements on Thin, Circular Semiconductor Samples’, NBS Technical Note 199, National Bureau of Standards (15 April 1964) (correction-factor formulas garbled in the OCR) (fetch-tool summary).

Ques: How does the van der Pauw method work, and how is it checked?

By measuring two four-terminal resistances around the edge of a thin sample. NIST takes eight resistances with both current polarities, averages them into $R_{A}$ and $R_{B}$ , and obtains $R_{s}$ from the van der Pauw equation $exp (- π R_{A} / R_{s}) + exp (- π R_{B} / R_{s}) = 1$ (as in Keithley note 3180, with $R_{s} = ρ / d$ ). For $R_{A} = 100$ and $R_{B} = 150$ $Ω$ , $R_{s} = 558.6$ $Ω/ sq$ , 1.4 per cent below the simple average formula (computed). NIST checks consistency, such as $R_{21, 34} = R_{12, 43}$ , and reciprocity sums, which must agree within 5 per cent and preferably 3 per cent, and requires contacts and thickness much smaller than the contact spacing, with errors of order contact size over spacing. Keithley and Penn State require a thin, homogeneous sample of uniform thickness, free of holes, with small contacts on the perimeter; its shape may be irregular. Reference: NIST Engineering Physics Division, ‘Hall Effect Measurements: Resistivity’, NIST web page (van der Pauw equation not shown in the text) (fetch-tool summary); Keithley Instruments application note 3180, ‘Performing van der Pauw Sheet Resistance Measurements Using the Keithley S530’, 2012: exp(-pi R_AB,CD d/rho) + exp(-pi R_BC,AD d/rho) = 1, thin uniform homogeneous sample, small contacts on the perimeter (fetch-tool summary, equation garbled); Penn State Materials Research Institute, ‘Van Der Pauw Method for Resistivity and Hall Mobility’: samples isotropic, homogeneous, flat, of uniform thickness, free of island hole defects, irregular shapes allowed, contacts along the perimeter (fetch-tool summary).

Ques: How much current should be used?

Enough for a clean voltage, little enough to avoid heating. NIST recommends keeping dissipation below 5 $mW$ , preferably 1 $mW$ , by choosing $I < (200 R)^{- 0.5}$ from the resistance R between opposing contacts; for 100 $Ω$ this gives 7.07 $mA$ , exactly 5 $mW$ (computed), so for the preferred 1 $mW$ the current should be about 3.2 $mA$ (computed). The current source should span 10 $μ A$ to 100 $mA$ , and down to 1 $nA$ for semi-insulating GaAs near $1 \times 1 0^{7}$ $Ω cm$ , with a voltmeter resolving 1 $μ V$ . Heating matters, since drive current causes thermal variations (Doolittle), and high fields in lightly doped samples can inject carriers. Reversing the current removes thermoelectric offsets, which NIST’s eight-measurement sequence does. Reference: NIST Engineering Physics Division, ‘Hall Effect Measurements: Resistivity’, NIST web page (van der Pauw equation not shown in the text) (fetch-tool summary).

Ques: How is sheet resistance converted to doping?

Through the mobility. For a uniformly doped layer, $ρ = 1/ (q N μ (N))$ , so with the doping-dependent mobility from the TCAD Mobility Models page, $1 \times 1 0^{19}$ $c m^{- 3}$ of electrons with 108 $c m^{2} / (V s)$ gives $5.78 \times 1 0^{- 3}$ $Ω cm$ , and a 0.2 $μ m$ layer has $R_{s} = 289$ $Ω/ sq$ (computed). For diffused or implanted layers, $R_{s} = 1/ [q \int N μ d x]$ over the junction depth; Doolittle notes that it depends on the total number of implanted or diffused impurities and on the layer thickness, so the integral is dominated by the heavily doped part of the profile (my reading). Because mobility depends on doping, the conversion must be iterated, and it assumes full activation and no compensation. A four-point probe alone cannot separate carrier density from mobility; that requires a Hall measurement, which uses the same van der Pauw contacts (NIST). Reference: TCAD Mobility Models page of this vault; W. A. Doolittle, ECE 4813 lecture ‘Resistivity and Sheet Resistance’, Georgia Tech: rho = 2 pi s V/I, thin-layer pi/ln2 form, current flowing radially from the tips, Kelvin separation of current and voltage pairs, Rsh = 1/(q n mu t) and Rsh = 1/(q integral n(x) mu dx) for nonuniform layers, implant and metal-film sheet-resistance mapping, thermal variations from drive current (fetch-tool summary); NIST Engineering Physics Division, ‘Hall Effect Measurements: Resistivity’, NIST web page (van der Pauw equation not shown in the text) (fetch-tool summary).

References

Ossila, “Calculate Sheet Resistance Using the Four-Probe Method,” Ossila application note (author and date not stated). Available: https://ossila.com/pages/sheet-resistance-theory
AggieFab Nanofabrication Facility, Texas A&M University, “Four-point probe correction factor” (after F. M. Smits, Bell System Technical Journal, vol. 37, pp. 711-718, 1958). Available: https://aggiefab.tamu.edu/wp-content/uploads/2024/07/4-point-correction-factor.pdf
L. J. Swartzendruber, “Correction Factor Tables for Four-Point Probe Resistivity Measurements on Thin, Circular Semiconductor Samples,” NBS Technical Note 199, National Bureau of Standards, Apr. 1964. Available: https://nvlpubs.nist.gov/nistpubs/Legacy/TN/nbstechnicalnote199.pdf
NIST Engineering Physics Division, “Hall Effect Measurements: Resistivity.” Available: https://www.nist.gov/engineering-physics-division/hall-effect-measurements-resistivity
W. A. Doolittle, “Resistivity and Sheet Resistance,” ECE 4813 Semiconductor Device and Material Characterization, lecture 2, Georgia Institute of Technology. Available: https://alan.ece.gatech.edu/ECE4813/Lectures/Lecture2ResistivityAndSheetResistance.pdf
Keithley Instruments, “Performing van der Pauw Sheet Resistance Measurements Using the Keithley S530 Parametric Tester,” application note 3180, 2012. Available: https://docs.ampnuts.ru/eevblog.docs/Keithley/SCS/S530/S530_VanDerPauwSheetRstnce.pdf
Penn State Materials Research Institute, “Van Der Pauw Method for Resistivity and Hall Mobility.” Available: https://www.mri.psu.edu/materials-characterization-lab/characterization-techniques/electrical-characterization-lab/van-der

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